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+{"text":"\\chapter{Introduction} \n\n\n\n\nLet $(M, \\omega)$ be a closed symplectic manifold. The group $\\Symp(M, \\omega)$ of symplectomorphisms, equipped with the standard $C^\\infty$-topology, is an infinite dimensional Fr\u00e9chet Lie group. A fundamental theorem of Banyaga~\\cite{Banyaga-Isomorphism} states that the isomorphism type of $\\Symp(M,\\omega)$ as a discrete group determines the conformal symplectomorphism type of $(M,\\omega)$. However, the ways topological properties of $\\Symp(M,\\omega)$ relate to the geometry of $(M,\\omega)$ remain largely unknown. To measure our level of ignorance, it suffices to observe that the following two questions are almost completely open in all dimensions $2n \\geq 4$:\n\n\\begin{enumerate}[label=\\textbf{Q\\arabic*.}]\n\\item At the most basic level, how does the homotopy type of $\\Symp(M,\\omega)$ depend on $\\omega\\,$? \n\\item Similarly, suppose $(M,\\omega)$ admits a symplectic or Hamiltonian actions by a compact Lie group $G$ (possibly finite), viewed here as a Lie subgroup $G\\subset\\Symp(M,\\omega)$. What are the homotopy types of the centralizer and of the normalizer $N(G)$ in $C(G)\\subset\\Symp(M,\\omega)\\,$? Can we understand the inclusions \n\\[G\\hookrightarrow \\Symp(M,\\omega)\\quad \\text{~and~}\\quad C(G)\\hookrightarrow N(G)\\hookrightarrow \\Symp(M,\\omega)\\]\nfrom a homotopy theoretic point of view?\n\n\\end{enumerate}\n\n\n\nRegarding the first question, most efforts as been devoted to the study of symplectomorphism groups of rational $4$-manifolds. Following the seminal work of M. Gromov~\\cite{Gr} who showed that the group of compactly supported symplectomorphisms of $\\mathbb{R}^4$ is contractible, the homotopical properties of the group of symplectomorphisms of $\\mathbb{C}P^2$, $S^2 \\times S^2$ and of the $k$-fold symplectic blow-ups $\\mathbb{C}P^2\\#k\\overline{\\mathbb{C}P}^2$, $k\\leq 5$, were studied in several papers such as \\cite{abreu}, \\cite{AGK}, \\cite{MR1775741}, \\cite{P-com}, \\cite{AG}, \\cite{AP}, \\cite{AE}, and~\\cite{LiLiWu2022}. In particular, for $\\mathbb{C}P^2$, $S^2 \\times S^2$, and $\\mathbb{C}P^2\\#k\\overline{\\mathbb{C}P^2}$, $k\\leq 3$, the rational homotopy type of $\\Symp(M,\\omega)$ can be described precisely in terms of the cohomology class $[\\omega]$. For $k\\geq 4$, partial results are known, mostly for $\\pi_0(\\Symp(M,\\omega))$ and $\\pi_1(\\Symp(M,\\omega))$.\nIt is worth pointing out that all these results rely on  special properties of $J$-holomorphic curves in symplectic $4$-manifolds and, as such, do not generalize readily to higher dimensions.\\\\ \n\nThe second question can be partially answered in the case of Hamiltonian toric actions using moment map techniques, see~\\cite{P-MaxTori}. If a torus $\\mathbb{T}^n\\subset\\Ham(M^{2n},\\omega)$ acts effectively on $(M^{2n},\\omega)$ with moment map $\\mu:M^{2n}\\to\\mathbb{R}^n$, then \n\n\\begin{enumerate}\n\\item the centralizer $C(\\mathbb{T}^n)$ is equal to the group of all symplectomorphisms $\\phi$ that preserve the moment map, that is, such that $\\mu \\circ \\phi=\\mu$.\n\\item $C(\\mathbb{T}^n)$ is a maximal torus in $\\Symp(M^{2n},\\omega)$, that is, a maximal, connected, and abelian subgroup of $\\Symp(M^{2n},\\omega)$. In particular, since toric manifolds are simply-connected, $C(\\mathbb{T}^n) \\subset\\Ham(M^{2n},\\omega)$. \n\\item $C(\\mathbb{T}^n)$ deformation retracts onto $\\mathbb{T}^{n}$. In particular, the homotopy type of the centraliser of a toric action is independent of the action.\n\\item Moreover, the Weyl group $W(\\mathbb{T}^n):=N(\\mathbb{T}^n)\/C(\\mathbb{T}^n)$ is always finite\\footnote{Futhermore, as for maximal tori in compact Lie groups, it can be shown that the number of conjugacy classes of toric centralizers is finite, and that each   $C(\\mathbb{T}^n)$ is flat and totally geodesic in $\\Symp(M^{2n},\\omega)$ for the $L^2$  metric}.\n\\end{enumerate}\nOn the other hand, even for toric actions, very little is known about the homotopy theoretic properties of the inclusions $G\\hookrightarrow\\Symp(M,\\omega)$ or $C(G)\\hookrightarrow N(G)\\hookrightarrow\\Symp(M,\\omega)$. Two noteworthy exceptions are i) the works of McDuff-Slimowitz~\\cite{McDS} and McDuff-Tolman~\\cite{McDT} who show that under some rather mild conditions, Hamiltonian $G$ actions induce injective maps $\\pi_1(G)\\hookrightarrow\\pi_1(\\Ham(M^{2n},\\omega))$, and ii) the results on symplectomorphism groups of rational surface mentionned above that, as a byproduct, allow one to understand the subrings of $\\pi_*(\\Symp(M,\\omega))$ or $H_*(\\Symp(M,\\omega))$ that the various Hamiltonian actions on $(M,\\omega)$ generate.\\\\\n\n\n\nIn this paper we combine pseudo-holomorphic curve techniques with moment map techniques to determine the homotopy type of equivariant symplectomorphisms of $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ under the presence of a Hamiltonian circle action. Our main result are Theorem~\\ref{full_homo} and Theorem~\\ref{full_homo_CCC} that give the full homotopy type of the centralizer $C(S^1)\\subset\\Symp(M,\\omega)$ for all choices of symplectic forms and of Hamiltonian circle actions on these two $4$-manifolds. Contrary to the toric case, the homotopy type of the stabilizer $C(S^1)$ is not constant and depends, essentially, on whether the circle action extends to a single toric action or to two distinct toric actions. In the former case, apart from a few exceptional circle actions that must be treated separately, $C(S^1)$ retracts onto the unique torus $\\mathbb{T}^2$ the circle extends to. In the latter case, the two toric actions $\\mathbb{T}^2_1$, $\\mathbb{T}^2_2$ that extend the circle action to do not commute, even up to homotopy. We show that the Pontryagin products of the generators of $H_1(\\mathbb{T}^2_1)$ and $H_1(\\mathbb{T}^2_2)$ generate a subalgebra $P^{alg}\\subset H_*(C(S^1))$ that contains classes of arbitrary large degrees. In particular, $C(S^1)$ does not have the homotopy type of a finite dimensional $H$-space. Moreover, looking at the action of the centralizer $C(S^1)$ on the space of invariant almost complex structures, we prove that there is an equality of Pontryagin rings $P^{alg}= H_*(C(S^1))$. This homology equivalence implies that $C(S^1)$ has the homotopy type of a pushout $\\mathbb{T}^2_1 \\leftarrow S^1\\to \\mathbb{T}^2_2$ in the category of topological groups. Finally, when viewed as a bare topological space, we show that $C(S^1)$ is homotopy equivalent to the product $\\Omega S^3 \\times S^1 \\times S^1 \\times S^1$.\\\\\n\nIt seems likely that our results on centralizers of Hamiltonian circle actions could be proven using moment map techniques alone. The main advantage of introducing pseudo-holomorphic curve techniques is that our setting generalises to actions of any compact, possibly finite, abelian group $A\\subset\\Ham(M,\\omega)$. For instance, in the companion paper~\\cite{Zn_symp}, we use the same framework to determine the homotopy type of the centralizers of most finite cyclic subgroups $\\mathbb{Z}_n$ acting on $(S^2 \\times S^2,\\omega_\\lambda)$ and $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ by Hamiltonian diffeomorphisms.\\\\\n\nIn an effort to make the paper understandable to both equivariant geometers and symplectic topologists, we include more details than would be needed in a document aimed at either audience alone. The paper is structured as follows:\\\\\n\nIn Chapter 2, we recall T. Delzant's equivariant classification of toric actions in terms of moment polytopes, and Y. Karshon's equivariant classifications of Hamiltonian circle actions on $4$-manifolds in terms of labelled graphs. We then determine all possible toric extensions, up to equivalences, of an arbitrary Hamiltonian circle actions on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$.\\\\ \n\nThe crux of the paper lies in Chapters 3 and 4 in which we adapt the framework of \\cite{MR1775741} to study groups of equivariant symplectomorphisms in the presence of an $S^1$ action. In particular we show that the space of invariant almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ decomposes into disjoint strata, each of them being homotopy equivalent to an orbit of the centralizer, with stabilizer homotopy equivalent to $S^1$ equivariant K\\\"ahler isometries (see Theorems~\\ref{homogenous} and \\ref{homog}).  In Chapter 3, we show that the number of invariant strata in the decomposition corresponds to the number of toric extensions of the given circle action (Proposition~\\ref{prop:ToricExtensionCorrespondance}). In particular we prove that $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ decomposes into either one or two strata, and that the later case occurs only for an exceptional family of circle actions on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ (Corollaries~\\ref{cor:lambda=1_IntersectingOnlyOneStratum},\\ref{cor:IntersectingTwoStrata} and \\ref{cor:IntersectingOnlyOneStratum}). In Chapter 4, using techniques similar to the ones developed in~\\cite{AG} we compute the homotopy type of $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ for all Hamiltonian circle actions on $(S^2 \\times S^2,\\omega_\\lambda)$. \n\\\\\n\nIn Chapter 5, we prove $S^1$ equivariant analogues of some technical lemmas of~\\cite{AGK} involving deformation theory. We use these results to prove that in the case the circle admits two distinct toric extensions, the stratum with positive codimension in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ is always of codimension two. \n\\\\\n\nFinally in Chapter 6, we carry out a similar analysis on the manifold $\\CP^2\\# \\overline{\\CP^2}$ and obtain the homotopy type of $\\Symp^{S^1}(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ for all Hamiltonian circle actions on $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$. \n\\\\\n\n\n\n\n\n\n\\chapter{Hamiltonian torus actions on ruled $4$-manifolds}\n\n\\section{Preliminaries on Hamiltonian actions}\nLet $G$ be a compact Lie group, possibly finite, acting effectively and symplectically on a symplectic manifold $(M,\\omega)$. Every such action $\\rho:G\\times M\\to M$ induces an injective homomorphism $G\\hookrightarrow\\Symp(M,\\omega)$ and, in particular, defines a subgroup $G\\subset\\Symp(M,\\omega)$. Let $\\mathfrak{g}$ denote the Lie algebra of $G$ and $\\mathfrak{g}^*$ be it's dual. Given $Y\\in \\mathfrak{g}$, we denote by $\\overline{Y}$ the corresponding fundamental vector field on~$M$.  \n\n\\begin{defn}\\label{def:HamiltonianActions} The action of $G$ is called Hamiltonian if\n\\begin{itemize}\n\\item $G$ is finite and the induced homomorphism $G\\to\\Symp(M,\\omega)$ takes values in the subgroup of Hamiltonian diffeomorphisms $\\Ham(M,\\omega)$, or\n\\item $\\dim G\\geq 1$ and there exists a moment map, that is, a smooth map $\\mu:M \\to \\mathfrak{g}^*$ such that \n\\begin{enumerate}\n    \\item $d\\mu_p(X_p)(Y) = \\omega(X,\\overline{Y})$ for all $X \\in T_pM$ and $Y \\in \\mathfrak{g}$, and \n    \\item the map $\\mu$ is equivariant with respect to the $G$ action on $M$ and the coadjoint action on~$\\mathfrak{g}^*$.\n\\end{enumerate}\nIn particular, $G\\subset\\Ham(M,\\omega)\\subset\\Symp(M,\\omega)$. \n\\end{itemize} \n\\end{defn}\n\n\\begin{remark}\\label{rmk:UniquenessMomentMap}\nWe note that in this definition, the moment map $\\mu:M\\to\\mathfrak{g}^{*}$ is an auxiliary structure whose sole existence is required. There is no claim about its unicity. For instance, the moment map of a Hamiltonian torus action is only defined up to adding a constant $c\\in\\mathfrak{t}^*$. However, there are situations where the choice of a specific moment map is needed. For this reason, we will always distinguish between the structure $(M,\\omega,\\rho)$ given by a Hamiltonian action alone, from the structure $(M,\\omega,\\rho,\\mu)$ in which a moment map $\\mu$ is chosen.\n\\end{remark}\n\n\\begin{defn}\\label{defn:EquivalenceActions}\nWe will say that two actions $\\rho_{i}:G\\to\\Symp(M,\\omega)$ are symplectically equivalent if the corresponding subgroups $\\rho_{i}(G)$ belong to the same conjugacy class with respect to the action of $\\Symp(M,\\omega)$.\n\\end{defn}\n\nGiven a symplectic action $\\rho:G\\times M\\to M$, let $C(G)$ be the centralizer of the corresponding subgroup $G\\subset\\Symp(M,\\omega)$, that is,\n\\[C(G)=\\{\\phi\\in\\Symp(M,\\omega)~|~\\phi\\circ g\\circ \\phi^{-1} = g,~\\forall g\\in G\\}\\]\nand let $N(G)$ be its normalizer, namely,\n\\[N(G)=\\{\\phi\\in\\Symp(M,\\omega)~|~\\phi\\circ G\\circ \\phi^{-1} = G\\}\\]\nClearly, equivalent actions have isomorphic centralizers and normalizers. \n\nIn this paper, we will investigate the homotopy type of the centralizers $C(T)$ of Hamiltonian torus actions on some $4$-manifolds. We start by giving a simple characterization of symplectomorphisms commuting with a Hamiltonian torus action.\n\\begin{lemma}\\label{lemma:CharacterizationCentralizer}\nLet $(M, \\omega)$ be a compact symplectic manifold equipped with a Hamiltonian torus action $\\rho:T\\times M\\to M$ with moment map $\\mu$. Then $\\phi \\in C(T)$ if, and only if, $\\mu \\circ \\phi = \\mu$ and $\\phi \\in \\Symp(M,\\omega)$.\n\\end{lemma}\n\\begin{proof}\n$(\\Leftarrow)$ Suppose $\\phi \\in \\Symp(M,\\omega)$ satisfies $\\mu \\circ \\phi = \\mu$. Let $X \\in\\mathfrak{t}$ and let $\\overline{X}$ denote the associated fundamental vector field. We then have\n\\[\\omega(d\\phi^{-1}(\\overline{X}),Y) = \\phi^*\\omega(\\overline{X}, d\\phi(Y)) = \\omega(\\overline{X}, d\\phi(Y) =  d\\mu(d\\phi(Y))=d\\mu(Y) = \\omega(\\overline{X}, Y)\\]\nfor any vector field $Y$, which implies $d\\phi(\\overline{X}) = \\overline{X}$ for all $X \\in \\mathfrak{t}$. Consequently, $\\phi$ commutes with the action.\n\\\\\n\n$(\\Rightarrow)$ If the action $\\rho$ has moment map $\\mu$, then $\\mu \\circ \\phi^{-1}$ is a moment map for the conjugate action $\\phi^{-1} \\circ \\rho \\circ \\phi$. If $\\phi^{-1} \\circ \\rho \\circ \\phi = \\rho$, this implies $\\mu \\circ \\phi^{-1} = \\mu + C$ for some constant $C\\in\\mathfrak{t}$. Since the moment images $\\mu(M)$ and $\\mu\\circ \\phi(M)$ are compact and coincide, this constant must be $0$. \n\\end{proof}\n\nThe image of the moment map of a torus action have special convexity properties.\n\n\\begin{thm}(Atiyah-Guillemin-Sternberg)\nLet $(M,\\omega)$ be a symplectic manifold on which a torus $\\mathbb{T}^d$ is acting with moment map $\\mu$. Then the image of $\\mu$ is a convex polytope of $\\mathfrak{t}^*$ whose vertices are images of the fixed points of the torus action.\n\\end{thm}\n\nAs usual, we call an effective Hamiltonian torus action \\emph{toric} if the torus acting is half the dimension of the manifold $M$. By a theorem of Delzant, the moment map image $\\Delta:=\\mu(M)$ of a toric action determines the symplectic manifold $(M,\\omega)$, the action $\\rho$, and the moment map $\\mu$ up to equivariant symplectomorphisms. \n\\begin{thm}(Delzant \\cite{Delzant}) \nLet $(M_{i},\\omega_{i},\\rho_{i},\\mu_{i})$, $i=1,2$, be two toric manifolds of dimension $2n$ with moment maps $\\mu_{i}:M_{i}\\to\\mathfrak{t}^{*}$. If the two moment polytopes $\\mu_{i}(M_{i})$ coincide, then there exists a $\\mathbb{T}^{n}$-equivariant symplectomorphism $\\phi:(M_{1},\\omega_{1})\\to (M_{2},\\omega_{2})$ such that $\\phi^{*}\\mu_{2}=\\mu_{1}$. Conversely, the moment map images of two equivariantly symplectomorphic toric structures $(M_{i},\\omega_{i},\\rho_{i},\\mu_{i})$, $i=1,2$, coincide.\n\\end{thm}\nChoose an identification $\\mathfrak{t}\\simeq\\mathbb{R}^{n}$ such that $\\ker(\\exp:\\mathfrak{t}\\to \\mathbb{T}^{n})\\simeq \\mathbb{Z}^{n}$. The moment polytopes of toric actions on manifolds of dimension $2n$ -- called Delzant polytopes of dimension $n$ -- are completely characterized by the following three properties, see~\\cite{Delzant}:  \n\\begin{itemize}\n\\item (simplicity) there are exactly $n$ edges meeting at each vertex; \n\\item (rationality) the edges meeting at any vertex $p$ are of the form $ p + t u_{i} $, $ t \\in [0,\\ell_i] $, where $ \\ell_i \\in \\mathbb{R}$ and $ u_{i} \\in \\mathbb{Z}^{n} $;\n\\item (smoothness) for each vertex $p$, the corresponding vectors $u_{1},\\ldots,u_{n}$ can be chosen to be a basis of $\\mathbb{Z}^{n}$ over $\\mathbb{Z}$.\n\\end{itemize}\nThis provides a purely combinatorial description of toric structures. Finally, if we are only interested in the subgroup $\\mathbb{T}^n\\subset\\Ham(M^{2n},\\omega)$ associated to a toric action, that is, if we disregard both the pa\\-ra\\-me\\-tri\\-za\\-tion $\\mathbb{T}^n\\hookrightarrow\\Ham(M^{2n},\\omega)$ and the moment map, Delzant's classification theorem yields a bijection\n\\begin{gather*}\n\\{\\text{Inequivalent toric actions on~} 2n\\text{-manifolds}\\}\\\\\n\\updownarrow\\\\\n\\{\\text{Delzant polytopes in~}\\mathbb{R}^n \\text{~up to~} \\AGL(n;\\mathbb{Z}) \\text{~action}\\}\n\\end{gather*}\nAs the next proposition shows, the centralizer and normalizer of symplectic toric actions are easy to describe in terms of moment maps. In particular, the homotopy type of $C(T)$ does not depend on the toric structure.\n\\begin{prop}\\label{prop:CentralizersToricActions}\nConsider an effective toric action $\\rho:\\mathbb{T}^{n}\\to\\Symp(M^{2n},\\omega)$\nwith moment map $\\mu:M\\to\\mathfrak{t}$. Denote by $T$ the corresponding torus\nin $\\Symp(M,\\omega)$, by $C(T)$ its centralizer, by $N(T)$ its normalizer, and by $W(T) =N(T)\/C(T)$ its Weyl group.\n\\begin{enumerate}\n\\item The normalizer $C(T)$ deformation retracts onto $\\mathbb{T}^{n}$. In particular, $C(T)\\subset\\Ham(M,\\omega)$.\n\\item The normalizer $N(T)$ is equal to the group of all symplectomorphisms $\\psi$ such that $\\mu\\circ\\psi = \\Lambda\\circ\\mu$ for some $\\Lambda\\in\\AGL(n;\\mathbb{Z})$. In particular, the Weyl group $W(T)$ is finite.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nThis is a slightly stronger version of Proposition 3.21 in~\\cite{P-MaxTori}, and only the first statement requires some additional justification. By Lemma~\\ref{lemma:CharacterizationCentralizer}, $\\phi\\in C(T)$ iff $\\phi\\in\\Symp(M,\\omega)$ and $\\mu=\\mu\\circ\\phi$. Since the action is toric, each fiber of the moment map consists in a single orbit. It follows that there is a unique map $g_{\\phi}:\\mu(M)=\\Delta\\to\\mathbb{T}^{n}$ such that $\\phi$ can be written as\n\\[\\phi(m) = g_{\\phi}(\\mu(m))\\cdot m\\]\nPick a base point $b\\in\\Delta$ and consider the evaluation fibration\n\\begin{align*}\nC_{b}(T)\\to C(T)&\\to \\mathbb{T}^{n}\\\\\n\\phantom{C_{b}(T)\\to} \\phi&\\mapsto g_{\\phi}(b) \n\\end{align*} \nwhose fiber is the subgroup\n\\[C_{b}(T)=\\{\\phi\\in C(T)~|~\\phi=\\id\\text{~on~}\\mu^{-1}(b)\\}\\]\nConsider $\\phi\\in C_{b}(T)$. Since $\\Delta$ is contractible, the function $g_{\\phi}$ has a unique lift $\\tilde{g}_{\\phi}:\\Delta\\to\\mathfrak{t}$ such that $\\tilde{g}_{\\phi}(b)=0$, and $g_{\\phi}=\\exp\\circ\\tilde{g}_{\\phi}$. Setting $\\tilde{g}_{\\phi,s}=s\\tilde{g}_{\\phi}$ for $s\\in[0,1]$, we obtain a retraction of $C_{b}(T)$ onto $\\{\\id\\}$ through diffeomorphisms $\\phi_{s}$ leaving $\\mu$ invariant and such that $\\phi_{s}=\\id$ on $\\mu^{-1}(b)$. Applying an equivariant Moser isotopy to the family $\\phi_{s}$, this retraction is seen to be homotopic to a retraction in $C_{b}(T)$. This shows that $C_{b}(T)$ is contractible, which in turns implies the result.\n\\end{proof}\nFor Hamiltonian torus actions $T^{d}\\to\\Ham(M^{2n},\\omega)$ for which the torus $T^{d}$ has dimension $d<n$, there is no simple description of the homotopy type of the symplectic centralizer $C(T)$. As we will see, this already happens in the simplest case, namely, for Hamiltonian circle actions on $4$-manifolds. \n\nIn \\cite{Karshon}, Y. Karshon established an equivariant classification of Hamiltonian circle actions on 4-dimensional symplectic manifolds, similar to Delzant classification, in which the moment map image is replaced by labelled graphs. More precisely, given any Hamiltonian $S^1$ action on a 4-manifold $(M,\\omega)$ with moment map $\\mu$, one can construct a labelled graph as follows:\n\\begin{itemize}\n\\item Each component of the fixed point set corresponds to a unique vertex of the graph.\n\\item Each vertex is labeled by the value of the moment map on the corresponding\nfixed point component. \n\\item An extremal vertex that corresponds to a fixed symplectic\nsurface S is said to be \"fat\". Two additional labels are attached to fat vertices: the genus of that surface,\nand its normalized symplectic area. \n\\item Two vertices are connected by an edge if and only if the corresponding isolated\nfixed points are connected by a $\\mathbb{Z}_k$-sphere i.e by a $S^1$ invariant sphere on which  $S^1$ acts with a global stabilizer $\\mathbb{Z}_k$, $k \\geq 2$.\n\\item  Each edge is labelled by the isotropy weight $k \\geq 2$ of the corresponding $\\mathbb{Z}_k$ sphere.\n\\end{itemize}\nLabeled graphs associated to effective Hamiltonian circle actions are characterized by the following properties. If we order the vertices according to their moment map labels, then\n\\begin{itemize}\n\\item there are exactly two extremal vertices;\n\\item fat vertices are extremal, and if the graph contains two fat vertices, then their genus label must coincide;\n\\item the area label of any fat vertex must be strictly positive;\n\\item a vertex is connected to no more than two edges, and no edge is connected to a fat vertex;\n\\item the moment map labels must be strictly monotone along each chain of edges;\n\\item if $e_{1},\\ldots,e_{\\ell}$ is a chain of edges, and if $k_{1},\\ldots,k_{\\ell}$ are the orders of their stabilizers, then $\\gcd(k_{i}, k_{i+1}) = 1$ for $i = 1,\\ldots\\ell-1$, and $(k_{i-1} + k_{i+1})\/k_{i}$ is an integer for $i= 2,\\ldots,\\ell-1$.\n\\end{itemize}\nWe call such graphs \\emph{admissible}. Just as Delzant polytopes classify toric structures up to symplectomorphisms, admissible labelled graphs classify Hamiltonian $S^1$ actions with moment maps.\n\n\\begin{thm}(Karshon \\cite{Karshon})\\label{graphiso}\nEach admissible labelled graph determines a Hamiltonian circle action $\\rho$ with moment map $\\mu$ on a symplectic $4$ manifold $(M,\\omega)$. The tuple $(M,\\omega,\\rho,\\mu)$ is unique up to $S^1$-equivariant symplectomorphisms preserving the moment map. Conversely, the labelled graphs of two equivariantly symplectomorphic Hamiltonian circle actions with moment maps are isomorphic. \n\\end{thm}\nAgain, if one is only interested in conjugacy classes of circles in Hamiltonian groups, Karshon's classification implies that there is a bijection\n\\begin{gather*}\n\\{\\text{Inequivalent Hamiltonian circle actions on~} 4\\text{-manifolds}\\}\\\\\n\\updownarrow\\\\\n\\{\\text{Admissible labelled graphs up to~} \\AGL(1;\\mathbb{Z}) \\text{~action on moment map labels}\\}\n\\end{gather*}\n\n\\begin{remark}\\label{rmk:NormalizationMomentMap}~\n\\begin{itemize}\n\\item Note that when $M$ is compact, it is always possible to canonically assign a moment map to an Hamiltonian circle action by imposing one of the normalizations $\\min \\mu = 0$ or $\\int_{M}\\mu = 0$. \n\\item Note also that in Karshon's classification, it is not necessary to keep track of invariant spheres on which the restricted action is effective. In particular, these spheres do not appear in the labelled graphs. This point will be important in our analysis.\n\\end{itemize}\n\\end{remark}\n\nThe symplectic slice theorem implies that a Hamiltonian circle action on a $4$-manifold has a simple local description near any fixed point.\n\n\\begin{prop}\\label{prop:DefinitionOfWeights}\n[see~\\cite{Karshon} Corollary A.7 or~\\cite{Lerman-Tolman} Lemma A.1] Let $p$ be a fixed point of a Hamiltonian circle action on a $4$-manifold $(M,\\omega)$. Then there exist complex coordinates $(z,w)$ on a neighborhood of $p$ in $M$, and unique integers $\\alpha$ and $\\beta$, called the isotropy weights at $p$, such that\n\\begin{enumerate}\n\\item the circle action is  $t\\cdot (z, w) = (t^{\\alpha}z, t^{\\beta}w)$,\n\\item the symplectic form is $\\omega = \\frac{i}{2}(dz\\wedge d\\bar z + dw \\wedge d\\bar w)$,\n\\item the moment map is $\\mu(z,w) = \\mu(p) + \\frac{\\alpha}{2}|z|^{2} + \\frac{\\beta}{2}|w|^{2}$.\n\\end{enumerate}\n\\end{prop}\nAlthough complex coordinates are used in the definition of weights, the unordered set $\\{\\alpha,\\beta\\}$ is a symplectic invariant of the circle action at the fixed point $p$. This already imposes some restrictions on equivariant symplectomorphisms.\n\n\\begin{cor}\\label{cor:ActionPreservesWeights}\nLet $\\phi\\in C(S^{1})$ be an $S^1$-equivariant symplectomorphism. Then $\\phi$ acts on the fixed point set preserving their weights as unordered sets. In particular, equivalent actions have the same set of weights.\\qed\n\\end{cor}\n\nAs explained in \\cite{Karshon} pages 6-7, the isotropy weights of a Hamiltonian circle action can be read from its labelled graph as follows:\n\\begin{itemize}\n    \\item for $k \\geq 2$, a fixed point has an isotropy weight $-k$ if and only if it is the north pole of a $\\mathbb{Z}_{k}$ -sphere, and it has an isotropy weight $k$ if and only if it is the south pole of a $\\mathbb{Z}_{k}$ -sphere;\n    \\item an interior fixed point has one positive weight and one negative weight, a maximal fixed point has both weights non-positive, a minimal fixed point has both weights non-negative;\n    \\item a fixed point has a weight $0$ if and only if it lies on a fixed surface.\n\\end{itemize}\nWe end this section by stating two simple lemmata that will be useful in computing homology classes of invariant spheres directly from labelled graphs.\n\\begin{lemma}[Lemma 5.4 in~\\cite{Karshon}]\\label{weight}\nLet $S^1$  act on $S^2$ by rotating it $k$ times while fixing the\nnorth and south poles. Suppose that the action lifts to a complex line\nbundle $E$ over $S^2$. Then $S^1$ acts linearly on the fibers over the north and south\npoles; let $\\alpha$ and $\\beta$ be the weights for these actions. Then\n$$\\alpha - \\beta = -ek$$\nwhere $e$ is the self intersection of the zero section.\\qed\n\\end{lemma}\n\n\\begin{lemma}\\label{Lemma:SymplecticArea}\nLet $S^1$  act on $S^2$ by rotating it $k$ times while fixing the\nnorth and south poles. Let $\\mu(n)$ denote the value of the momentum map at the north pole and $\\mu(s)$ the value of the momentum map at the south pole. Then the symplectic area of the $S^2$ is given by $\\frac{\\mu(n) - \\mu(s)}{k}$.\\qed\n\\end{lemma}\n\n\n\n\n\\section{Torus actions on \\texorpdfstring{$S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$}{S2xS2 and CP2\\#CP2}}\nWe now combine the \\emph{equivariant} classification theorems of Delzant and Karshon, together with the Lalonde-McDuff classification of symplectic forms on ruled surfaces, to describe all inequivalent Hamiltonian circle actions on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ and their relations to toric actions. \n\n\\subsection{Symplectic forms on \\texorpdfstring{$S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$}{S2xS2 and CP2\\#CP2}}\nWe equip $S^2 \\times S^2$ with the product symplectic form $\\omega_\\lambda=\\lambda \\sigma \\oplus \\sigma$, where $\\sigma$ is the standard area form on $S^2$ normalized such that $\\sigma(S^2)=1$. If we think of $S^2 \\times S^2$ as a trivial fiber bundle, $\\omega_\\lambda$ gives area $1$ to the fibers, while the area of horizontal sections is $\\lambda$. Similarly, if we view $\\CP^2\\# \\overline{\\CP^2}$ as the non-trivial $S^2$ bundle over $S^2$, we define an analogous form $\\omega_\\lambda$ which gives area $1$ to the fibers and area $\\lambda-1$ to symplectic sections of self-intersection $-1$, that is, to sections homologous to the exceptional divisor. From an homological point of view, if $F$ denotes the homology class of a fiber in either $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$, if $B$ denotes the class of a section of self-intersection $0$ in $S^2 \\times S^2$, if $E$ denotes the class of the exceptional divisor in $\\CP^2\\# \\overline{\\CP^2}$, and if $L$ denotes the class of a line in $\\CP^2\\# \\overline{\\CP^2}$, then $[\\omega_\\lambda]F=1$, $[\\omega_\\lambda]B=\\lambda$, $[\\omega_\\lambda]L=\\lambda$ and $[\\omega_\\lambda]E=\\lambda-1$. Note that with our conventions on $\\omega_\\lambda$ for the non-trivial bundle $\\CP^2\\# \\overline{\\CP^2}$, $\\lambda$ must be strictly greater than 1 for the symplectic curve in class $E$ to have positive area. We can now state the Lalonde-McDuff classification theorem.\n\n\\begin{thm}[Lalonde-McDuff~\\cite{MR1426534}]\\label{L-McD-Classification} Any symplectic form on $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$ is conformally diffeomorphic to a form $\\omega_\\lambda$ with $\\lambda\\geq 1$. Moreover, any two cohomologous forms are diffeomorphic.\n\\end{thm}\n\n\\subsection{Toric actions on \\texorpdfstring{$S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$}{S2xS2 and CP2\\#CP2}} It is well known that, up to symplectic equivalence, the only toric actions on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ are given by the standard torus actions on the Hirzebruch surfaces. We now review the main points of the argument. \n \nRecall that the $m^{\\text{th}}$ Hirzebruch surface $W_m$ is the complex submanifold of $\\mathbb{C}P^1 \\times \\mathbb{C}P^2$ defined by the equation\n\\[\nW_m:=  \\left\\{ \\left(\\left[x_1,x_2\\right],\\left[y_1,y_2, y_3\\right]\\right) \\in \\mathbb{C}P^1 \\times \\mathbb{C}P^2 ~|~  x^m_1y_2 - x_2^my_1 = 0 \\right\\}\n\\]\nThe projection map $\\mathbb{C}P^1 \\times \\mathbb{C}P^2 \\longrightarrow \\mathbb{C}P^1$ gives $W_m$ the structure of a $\\mathbb{C}P^1$ bundle over $\\mathbb{C}P^1$ which is diffeomorphic (but not complex isomorphic) to $S^2 \\times S^2$ if $m$ is even and diffeomorphic to the non-trivial $S^2$ bundle over $S^2$ i.e $\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2}$ if $m$ is odd. Thus, for each $m$ we have an integrable complex structure $J_m$ induced on $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$.  When $m=2k$ even, choose $\\lambda > k$, then we can endow  $\\mathbb{C}P^1 \\times \\mathbb{C}P^2$ with the symplectic form $(\\lambda -k) \\sigma_1 \\oplus \\sigma_2$, where $\\sigma_1$ is   the standard symplectic form on  $\\mathbb{C}P^1$ with area 1 and $\\sigma_2$ is the standard symplectic form on $\\mathbb{C}P^2$ such that $\\sigma_2(L) =1$, where $L$ is the class of the line in $\\mathbb{C}P^2$.  Restricting this symplectic form to $W_m$ makes it a symplectic manifold. Similarly when $m=2k+1$,  choose $\\lambda > k+1$, then we can analogously define the form $(\\lambda - (k+1)) \\sigma_1 \\oplus \\sigma_2$ when $m$ is odd. With these choices of symplectic forms, the Lalonde-McDuff classification theorem~\\ref{L-McD-Classification} implies that $W_m$ is symplectomorphic  to $(S^2 \\times S^2,\\omega_\\lambda)$ if $m$ is even and to $(\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2},\\omega_\\lambda)$ when $m$ is odd.\\\\\n\nGiven an integer $m\\geq 0$, the torus $\\mathbb{T}^2$ acts on $\\mathbb{C}P^1 \\times \\mathbb{C}P^2$ by setting\n\\[\n \\left(u,v\\right) \\cdot  \\left(\\left[x_1,x_2\\right],\\left[y_1,y_2, y_3\\right]\\right) = \\left(\\left[ux_1,x_2\\right],\\left[u^my_1,y_2,vy_3\\right]\\right)\n\\]\nThis action leaves $W_m$ invariant and preserves both the complex and the symplectic structures. Its restriction to $W_m$ defines a toric action that we denote $\\mathbb{T}^2_m$. \n\nWhen $m$ is even, the image of the moment map is the polytope of Figure~\\ref{hirz}\n\\begin{figure}[H]\n\\centering       \n\\begin{tikzpicture}\n\\node[left] at (0,2) {$Q=(0,1)$};\n\\node[left] at (0,0) {$P=(0,0)$};\n\\node[right] at (4,2) {$R= (\\lambda - \\frac{m}{2} ,1)$};\n\\node[right] at (6,0) {$S=(\\lambda + \\frac{m}{2} ,0)$};\n\\node[above] at (2,2) {$D_m=B-\\frac{m}{2}F$};\n\\node[right] at (5.15,1) {$F$};\n\\node[left] at (0,1) {$F$};\n\\node[below] at (3,0) {$B+ \\frac{m}{2}F$};\n\\draw (0,2) -- (4,2) ;\n\\draw (0,0) -- (0,2) ;\n\\draw (0,0) -- (6,0) ;\n\\draw (4,2) -- (6,0) ;\n\\end{tikzpicture}\n\\caption{Even Hirzebruch polygon}\n\\label{hirz}\n\\end{figure}\n\n\\noindent where the labels along the edges refer to the homology classes of the  $\\mathbb{T}^2$ invariant spheres in $S^2 \\times S^2$, and where the vertices $P$,$Q$,$R$,$S$ are the fixed points of the torus action. \nSimilarly, when $m$ is odd, the moment map image is given in Figure~\\ref{fig:OddHirzebruch}\n\\begin{figure}[H]\n\\centering   \n\\label{hirz0}\n\\begin{tikzpicture}\n\\node[left] at (0,2) {$Q=(0,1)$};\n\\node[left] at (0,0) {$P=(0,0)$};\n\\node[right] at (4,2) {$R= (\\lambda - \\frac{m+1}{2} ,1)$};\n\\node[right] at (6,0) {$S=(\\lambda + \\frac{m-1}{2} ,0)$};\n\\node[above] at (2,2) {$D_m=B-\\frac{m+1}{2}F$};\n\\node[right] at (5.15,1) {$F$};\n\\node[left] at (0,1) {$F$};\n\\node[below] at (3,0) {$B+ \\frac{m-1}{2}F$};\n\\draw (0,2) -- (4,2) ;\n\\draw (0,0) -- (0,2) ;\n\\draw (0,0) -- (6,0) ;\n\\draw (4,2) -- (6,0) ;\n\\end{tikzpicture}\n \\caption{Odd Hirzebruch polygon}\\label{fig:OddHirzebruch}\n\\end{figure}\n\\noindent where $B$ now refers to the homology class of a line $L$ in $\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2}$, $E$ is the class of the exceptional divisor, and $F$ is the fiber class $L - E$.\\\\\n\nWe define the zero-section $s_0$ to be\n\\begin{align*}\n   s_0: \\mathbb{C}P^1 &\\to W_m \\\\\n    [x_1;x_2]  &\\mapsto \\{[x_1,x_2], [0;0;1]\\}\n\\end{align*}\nand the section at infinity $s_\\infty$ to be \n\\begin{align*}\n     s_\\infty: \\mathbb{C}P^1 &\\to W_m \\\\\n    [x_1;x_2]  &\\mapsto \\{[x_1,x_2], [x^m_1;x^m_2;0]\\}\n\\end{align*}\nThe image of $s_0$ is an invariant and holomorphic sphere homologous to either $D_m:=B-\\frac{m}{2}F$ in $S^2 \\times S^2$ or to $D_m:=B- \\frac{m+1}{2}F$ in $\\CP^2\\# \\overline{\\CP^2}$, depending on the parity of $m$. Similarly, $s_\\infty$ defines an invariant and holomorphic sphere that represents either $B + \\frac{m}{2}F$ in $S^2 \\times S^2$ or $B+\\frac{m+1}{2}F$ in $\\CP^2\\# \\overline{\\CP^2}$. Finally, the homology class $F$ can be represented by an invariant fibre such as $\\{[1,0], [y_1,0,y_3]\\}$.\\\\\n\nIt is well known (see~\\cite{Audin}) that a Delzant polygon $\\Delta\\subset\\mathbb{R}^{2}$ with $e\\geq 3$ edges defines a toric $4$-manifold $M_{\\Delta}$ whose second Betti number is $b_{2}(M_{\\Delta})=e-2$. It follows that the moment polytope of any toric action on either $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$ is a quadrilateral. It is easy to see that any Delzant quadrilateral is equivalent to one of the above Hirzebruch trapezoids up to $\\AGL(2,\\mathbb{Z})$ action and up to rescaling. It follows from Delzant's classification that any toric action on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ is equivalent to an action of the above form. In particular, the equivalence classes of toric actions on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ are completely characterized by the existence of invariant spheres in specific homology classes.\n\n\n\n\\begin{lemma}\\label{lemma_torus_action_-vecurve}\nUp to equivalence, the toric action $\\mathbb{T}^2_m$, $m\\geq 1$, is characterised by the existence of an invariant, embedded, symplectic sphere $C_m$ with self intersection $-m$. The toric action $\\mathbb{T}^2_{0}$ is characterized by the existence of invariant, embedded, symplectic spheres representing the classes $B$ and~$F$.\\qed\n\\end{lemma}\n\n\n\\begin{lemma}\\label{lemma_number_of_torus_actions}\nWrite $\\lambda=\\ell+\\delta$ with $\\ell$ an integer and $0<\\delta\\leq 1$. Then, up to symplectomorphisms and reparametrizations, \n\\begin{itemize}\n\\item if $\\lambda \\geq 1$, there are exactly $\\ell+1$ inequivalent toric actions on $(S^2 \\times S^2,\\omega_\\lambda)$ given by the even Hirzebruch actions $\\mathbb{T}^2_{2k}$ with $0\\leq k\\leq \\ell$, and\n\\item if $\\lambda >1$, there are exactly $\\ell$ inequivalent toric actions on $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ given by the odd Hirzebruch actions $\\mathbb{T}^2_{2k+1}$ with $0\\leq k\\leq \\ell-1$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nWrite $m=2k$ or $m=2k+1$ with $k\\geq 0$. As seen above, any toric action on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ is $\\mathbb{T}^2$-equivariantly symplectomorphic to one of the actions $\\mathbb{T}^2_m$. The invariant symplectic sphere $D_k$ in class $B-kF$ on $S^2 \\times S^2$ or $L - (k+1)F$ in $\\CP^2\\# \\overline{\\CP^2}$ must have positive area, that is, \n\\[0<\\omega_\\lambda(B-kF) = \\lambda - k = \\ell+\\delta-k\\quad\\text{~or~}\\quad 0<\\omega_\\lambda(L-(k+1)F) =\\lambda-(k+1)F = \\ell+\\delta-(k+1)  \\]\nThe result follows.\n\\end{proof}\n\n\\begin{prop}\\label{prop:NormalizersHirzebruch}\nThe Weyl group $W(\\mathbb{T}^2_{m})=N(\\mathbb{T}^2_{m})\/C(\\mathbb{T}^2_{m})$ is isomorphic to\n\\[W(\\mathbb{T}^2_{m})\\simeq\n\\begin{cases}\n\\text{the dihedral group~} D_{8}, & \\text{~when~} m=0 \\text{~and~} \\lambda=1;\\\\\n\\text{the dihedral group~} D_{2}, & \\text{~when~} m=0 \\text{~and~} \\lambda>1;\\\\\n\\text{the dihedral group~} D_{1}\\simeq\\mathbb{Z}_{2},& \\text{~when~} m\\geq 1.\n\\end{cases}\\]\n\\end{prop}\n\\begin{proof}\nThis follows directly from Proposition~\\ref{prop:CentralizersToricActions} and from the description of Hirzebruch trapezoids. Indeed, for $m=0$ and $\\lambda=1$, the moment map polygon is a unit square whose symmetries can be realized by elements of $\\AGL(2,\\mathbb{Z})$. The same holds for $m=0$ and $\\lambda>1$, with the only difference that the moment polygone is now a rectangle with sides of lengths $1$ and $\\lambda$. Finaly, for $m\\geq 1$, if we write $m=2k$ or $m=2k+1$, the only non-trivial element of  $\\AGL(2,\\mathbb{Z})$ that leaves the standard Hirzebruch trapezoid invariant is\n\\[\\Lambda \n=\\begin{pmatrix}-1 & -m\\\\ 0 & 1\\end{pmatrix}\n+\\begin{pmatrix}\\lambda+k,0\\end{pmatrix}\\]\nwhich is an element of order 2.\n\\end{proof}\n\n\\subsection{Hamiltonian circle actions on \\texorpdfstring{$S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$}{S2xS2 and CP2\\#CP2}}\nA list of all possible Hamiltonian circle action on $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$ comes easily from an extension theorem due to Y. Karshon.\n\\begin{thm}[Karshon~\\cite{finTori}, Theorem 1]\nAny symplectic $S^1$ action on $(S^2 \\times S^2,\\omega_\\lambda)$ and $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ extends to an Hamiltonian toric action. \n\\end{thm} \n\nConsequently, every symplectic $S^1$ action on $S^2 \\times S^2$ and $\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2}$ is given by an embedding \n\\begin{align*}\n    S^1 &\\hookrightarrow \\mathbb{T}^2_m \\\\\n    t &\\mapsto (t^a, t^b)\n\\end{align*}\nwhich is itself characterized by a unique triple of numbers $(a,b;m)\\in \\mathbb{Z} \\times \\mathbb{Z} \\times \\mathbb{Z}_{\\geq 0}$. Since we are only interested in effective actions, this translates numerically into the condition  $\\gcd(a,b) = 1$. We shall always assume this unless otherwise stated. \n\n\n\nIn order to construct the graphs of the circle action $S^1(a,b;m)$, we claim that it is enough to compute the isotropy weights at every fixed points. Indeed, given an Hirzebruch trapezoid, the pre-image of a vertex under the moment map is a toric fixed point, and the pre-image of an edge is an invariant embedded two-sphere connecting two fixed points. When we view the space as a Hamiltonian $S^1$-space, such a two-sphere is either fixed by the action, or is a $\\mathbb{Z}_k$ sphere, or has trivial global isotropy. These three possibilities are completely determined by the weights of the $S^{1}(a,b;m)$ action at its two fixed points. We can thus construct the graphs starting from the fixed points and adding edges according to the weights. If there is a fixed surface, its area label can be read from the Hirzebruch trapezoid. Finally, the moment map labels can be added starting with the minimal vertex (characterised by having two positive weights) that we label $\\mu=0$, and then using Lemma~\\ref{Lemma:SymplecticArea} to compute the moment labels for the remaining interior fixed points, and for the maximal fixed point (characterised by having two negative weights).\n\nNow, for the Hirzebruch actions $\\mathbb{T}^2_{m}=(S^{1}\\times S^{1})_{m}$, each $S^{1}$ factor defines two weights at each of the four fixed points $P,Q,R,S$. These weights are listed in table~\\ref{table_weights}. Further, if the the $\\mathbb{T}^2_m$ action has weights $\\{\\alpha_1,\\beta_{1}\\}$ and $\\{\\alpha_{2},\\beta_2\\}$ at a fixed point $x$, then the restricted $S^1(a,b;m)$ action has weights\n\\[\\left\\{a\\alpha_1 + b\\alpha_{2}, a\\beta_{1} + b\\beta_2 \\right\\} \\]\nat $x$. This gives the weights of the $S^{1}(a,b;m)$ actions at the fixed points $P,Q,R,S$.\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{ |p{2cm}||p{4cm}|p{6cm}|  }\n \\hline\n Vertex & Weights for $\\mathbb{T}^2_m$ action & Weights for the $S^1(a,b;m)$ action \\\\\n \\hline\n P & $\\big\\{\\{1,0\\},\\{0,1\\}\\big\\}$   & $\\{a,b\\}$ \\rule{0pt}{10pt}\\\\\n Q & $\\big\\{\\{1,0\\},\\{0,-1\\}\\big\\}$  & $\\{a,-b\\}$ \\rule{0pt}{10pt}\\\\\n R & $\\big\\{\\{-1,m\\},\\{0,-1\\}\\big\\}$ & $\\{-a,am-b\\}$ \\rule{0pt}{10pt}\\\\\n S & $\\big\\{\\{-1,-m\\},\\{0,1\\}\\big\\}$ & $\\{-a,-am+b\\}$ \\rule{0pt}{10pt}\\\\\n \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Weights of $\\mathbb{T}^2_{m}$ and $S^{1}(a,b;m)$ actions}\n\\label{table_weights}\n\\end{table}\n\n\n\nIn Figures~\\ref{fig:GraphsWithFixedSurfaces} and~\\ref{fig:GraphsIsolatedFixedPoints}, we present the graphs of circles actions $S^{1}(a,b;m)$ on $(S^2 \\times S^2,\\omega_\\lambda)$ and $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$. As before, we write $m=2k$ or $m=2k+1$, and in order to deal with even and odd Hirzebruch surfaces simultaneously, we introduce the symbol $\\epsilon_{m} := m\\mod 2$. Each label $\\mu$ represents the value of the moment map normalized by setting $\\min \\mu=0$, while $A$ is the area label of a fixed surface. All fixed surfaces have genus 0. Note that when the isotropy label on an edge is 1, then the circle action on the corresponding invariant sphere has no isotropy, and we erase that edge from the graph. Note also that the identification of the fixed points with the vertices $P,Q,R,S$ of the Hirzebruch trapezoid is there for convenience only and is not part of the data.  \n\n\n\n\nFor actions with fixed surfaces, one of the weights $a$, $b$, or $am-b$ must be zero. Since $\\gcd(a,b)=1$, the possible triples are $(\\pm1,0;m)$, $(0,\\pm1;m)$, and $(\\pm1,\\pm m;m)$. \n\\begin{figure}[H]\n\\centering\n\\subcaptionbox{Graph of $S^{1}(1,0;m)$\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,0) ellipse (0.3cm and 0.1cm);\n\\draw [fill] (0,1.2) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,2.8) ellipse (0.1cm and 0.1cm);\n\\draw (0,1.2) -- (0,2.8);\n\\node[right] at (0,2) {m};\n\\node[right] at (0,1.2) {R};\n\\node[right] at (0,2.8) {S};\n\\node[right] at (0.5,2.8){$\\mu= \\lambda +k$};\n\\node[right] at (0.5,0){$\\mu=0$};\n\\node[left] at (-0.4,0) {$A= 1$};\n\\node[right] at (0.5,1.2) {$\\mu=\\lambda - k - \\epsilon_{m}$};\n\\end{tikzpicture}}\n\n\\subcaptionbox{Graph of $S^{1}(-1,0;m)$\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,0) ellipse (0.3cm and 0.1cm);\n\\draw [fill] (0,-1.2) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-2.8) ellipse (0.1cm and 0.1cm);\n\\draw (0,-1.2) -- (0,-2.8);\n\\node[right] at (0,-2.2) {m};\n\\node[right] at (0,-2.8) {S};\n\\node[right] at (0,-1.2) {R};\n\\node[right] at (0.5,0){$\\mu= \\lambda +k$};\n\\node[right] at (0.5,-2.8){$\\mu=0$};\n\\node[left] at (-0.5,0) {$A= 1$};\n\\node[right] at (0.5,-1.2) {$\\mu=m$};\n\\end{tikzpicture}}\n\n\\subcaptionbox{Graph of $S^{1}(0,1;m)$\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,0) ellipse (0.3cm and 0.1cm); \n\\draw [fill] (0,2.8) ellipse (0.3cm and 0.1cm); \n\\node[left] at (-0.3,2.8) {$\\mu = 1$};\n\\node[left] at (-0.3,0){$\\mu=0$};\n\\node[right] at (0.3,2.8){$A= \\lambda-k-\\epsilon_{m}$};\n\\node[right] at (0.3,0){$A= \\lambda+k$};\n\\node[above] at (0,2.8) {\\rule{0em}{3em}}; \n\\end{tikzpicture}}\n\n\\subcaptionbox{Graph of $S^{1}(0,-1;m)$\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,0) ellipse (0.3cm and 0.1cm); \n\\draw [fill] (0,2.8) ellipse (0.3cm and 0.1cm); \n\\node[left] at (-0.3,2.8) {$\\mu = 1$};\n\\node[left] at (-0.3,0){$\\mu=0$};\n\\node[right] at (0.3,2.8){$A= \\lambda+k$};\n\\node[right] at (0.3,0){$A= \\lambda-k-\\epsilon_{m}$};\n\\end{tikzpicture}}\n\\subcaptionbox{Graph of $S^{1}(1,m;m)$\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,0) ellipse (0.3cm and 0.1cm);\n\\draw [fill] (0,-2) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-3.5) ellipse (0.1cm and 0.1cm);\n\\draw (0,-2) -- (0,-3.5);\n\\node[above] at (0,0) {\\rule{0em}{3em}}; \n\\node[right] at (0,-2.75) {m};\n\\node[right] at (0,-3.5) {P};\n\\node[right] at (0,-2) {Q};\n\\node[right] at (0.5,0){$\\mu= \\lambda +k$};\n\\node[right] at (0.5,-3.5){$\\mu=0$};\n\\node[left] at (-0.5,0) {$A= 1$};\n\\node[right] at (0.5,-2) {$\\mu=m$};\n\\end{tikzpicture}}\n\\subcaptionbox{Graph of $S^{1}(-1,-m;m)$\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,-3.5) ellipse (0.3cm and 0.1cm);\n\\draw [fill] (0,-2) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (0,-2);\n\\node[right] at (0,-1) {m};\n\\node[right] at (0,0) {P};\n\\node[right] at (0,-2) {Q};\n\\node[right] at (0.5,0){$\\mu= \\lambda +k$};\n\\node[right] at (0.5,-3.5){$\\mu=0$};\n\\node[left] at (-0.5,-3.5) {$A= 1$};\n\\node[right] at (0.5,-2) {$\\mu=\\lambda - k-\\epsilon_{m}$};\n\\end{tikzpicture}}\n\\caption{Graphs of circle actions with non-isolated fixed points}\\label{fig:GraphsWithFixedSurfaces} \n\\end{figure}\n\n\nFor actions whose fixed points are isolated, none of the weights are zero. The graphs then depend on the signs of $a$, $b$, and $am-b$, as these signs determine which fixed point is minimal and which one is maximal.\n\n\\begin{figure}[H]\n\\centering\n~\n\n\\subcaptionbox*{}{} \n\\setcounter{subfigure}{6} \n\\subcaptionbox{When $a>0,~b>0$, and $am-b>0$\\label{fig:a>0|b>0|am-b>0}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-1.5) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2.5) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-1.5);\n\\draw (1,-1.5) -- (-1,-2.5);\n\\draw (0,0) .. controls (-4,-1.5) and (-2,-3.5) .. (0,-4);\n\\draw (-1,-2.5) -- (0,-4);\n\\node[right] at (-0.35,-3.2) {$b$};\n\\node[right] at (- 0.4,-1.8) {$a$};\n\\node[left] at (-2.2,-1.5) {$a$};\n\\node[right] at (0.7,-0.5) {$am-b$};\n\\node[above] at (0,0.2) {$\\mu = a(\\lambda + k)$};\n\\node[right] at (-0.6,-2.5) {$\\mu = b$};\n\\node[right] at (1,-1.8) {$\\mu = b + a(\\lambda - k-\\epsilon_{m}) $};\n\\node[below] at (0, -4.2) {$\\mu = 0$};\n\\node[left] at (-1,-2.5) {Q};\n\\node[left] at (0.9,-1.4) {R};\n\\node[right] at (0,0) {S};\n\\node[right] at (0,-4) {P};\n\\end{tikzpicture}}\n\\subcaptionbox{When $a>0,~b>0$ and $am-b<0\n}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (1,-2) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (-1,-3) ellipse (0.1cm and 0.1cm); \n\\draw (0,0) -- (1,-2);\n\\draw (-1,-3) -- (0,-4);\n\\draw (0,0) -- (-1,-3) ;\n\\draw (1,-2) -- (0,-4);\n\\node[right] at (-0.9,-3) {Q};\n\\node[left] at (0.9,-2) {S};\n\\node[right] at (0,0) {R};\n\\node[above] at (0,0.2) {$\\mu = b + a (\\lambda -k-\\epsilon_{m})$};\n\\node[right] at (1.3,-2) {$\\mu = a(\\lambda + k) $};\n\\node[left] at (-1.4,-3) {$\\mu = b$};\n\\node[right] at (0,-4) {P};\n\\node[right] at (-1,-3.7) {$b$};\n\\node[right] at (0.5, -3) {$a$};\n\\node[left] at (-0.5,-1.5) {$a$};\n\\node[right] at (0.7,-1) {$b-am$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\end{tikzpicture}}\n\\subcaptionbox{When $a>0,~b<0$\\label{fig:a<0|b>0}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-3) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-3);\n\\draw (-1,-2) -- (0,-4);\n\\draw (0,0) -- (-1,-2) ;\n\\draw (1,-3) -- (0,-4);\n\\node[above] at (0,1) {\\rule{0em}{1em}};\n\\node[above] at (0,0.2) {$\\mu = a(\\lambda +k)-b$};\n\\node[right] at (1.4,-3) {$\\mu = -b $};\n\\node[left] at (-1.4,-2) {$\\mu = a(\\lambda-k-\\epsilon_{m})$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\node[right] at (-1,-3.2) {$a$};\n\\node[right] at (0.5, -3.6) {$-b$};\n\\node[left] at (-0.7,-1.0) {$am-b$};\n\\node[right] at (0.6,-1.3) {$a$};\n\\node[left] at (0.9,-3) {P};\n\\node[right] at (-1,-2) {R};\n\\node[right] at (0,0) {S};\n\\node[right] at (0,-4) {Q};\n\\end{tikzpicture}}\n\\subcaptionbox{When $a<0,~b>0$\\label{fig:a<0|b>0}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-3) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-3);\n\\draw (-1,-2) -- (0,-4);\n\\draw (0,0) -- (-1,-2) ;\n\\draw (1,-3) -- (0,-4);\n\\node[above] at (0,1) {\\rule{0em}{1em}}; \n\\node[above] at (0,0.2) {$\\mu = b - a (\\lambda +k)$};\n\\node[right] at (1.4,-3) {$\\mu = b-am $};\n\\node[left] at (-1.4,-2) {$\\mu = -a(\\lambda+k)$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\node[right] at (-1.2,-3.2) {$-a$};\n\\node[right] at (0.5, -3.6) {$b-am$};\n\\node[left] at (-0.7,-1.0) {$b$};\n\\node[right] at (0.6,-1.3) {$-a$};\n\\node[left] at (0.9,-3) {R};\n\\node[right] at (-1,-2) {P};\n\\node[right] at (0,0) {Q};\n\\node[right] at (0,-4) {S};\n\\end{tikzpicture}}\n\\subcaptionbox{When $a<0,~b<0$, and $am-b>0$\\label{fig:a<0|b<0|am-b>0}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-3) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-3);\n\\draw (-1,-2) -- (0,-4);\n\\draw (0,0) -- (-1,-2) ;\n\\draw (1,-3) -- (0,-4);\n\\node[above] at (0,1) {\\rule{0em}{1em}}; \n\\node[above] at (0,0.2) {$\\mu = -b - a (\\lambda -k-\\epsilon_{m})$};\n\\node[right] at (1.4,-3) {$\\mu = am-b $};\n\\node[left] at (-1.4,-2) {$\\mu = -a(\\lambda-k-\\epsilon_{m})$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\node[right] at (-1.2,-3.2) {$-a$};\n\\node[right] at (0.5, -3.6) {$am-b$};\n\\node[left] at (-0.7,-1.0) {$-b$};\n\\node[right] at (0.6,-1.3) {$-a$};\n\\node[left] at (0.9,-3) {S};\n\\node[right] at (-1,-2) {Q};\n\\node[right] at (0,0) {P};\n\\node[right] at (0,-4) {R};\n\\end{tikzpicture}}\n\\subcaptionbox{When $a<0,~b<0$, and $am-b<0$\\label{fig:a<0|b<0|am-b<0}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-1.5) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2.5) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-1.5);\n\\draw (1,-1.5) -- (-1,-2.5);\n\\draw (0,0) .. controls (-4,-1.5) and (-2,-3.5) .. (0,-4);\n\\draw (-1,-2.5) -- (0,-4);\n\\node[right] at (-0.35,-3.2) {$b-am$};\n\\node[right] at (- 0.4,-1.8) {$-a$};\n\\node[left] at (-2.2,-1.5) {$-a$};\n\\node[right] at (0.7,-0.5) {$-b$};\n\\node[above] at (0,0.2) {$\\mu = -a(\\lambda + k)$};\n\\node[right] at (-0.6,-2.5) {$\\mu = b-am$};\n\\node[right] at (1,-1.8) {$\\mu = b-a(\\lambda + k-\\epsilon_{m}) $};\n\\node[below] at (0, -4.2) {$\\mu = 0$};\n\\node[right] at (0,0) {P};\n\\node[left] at (0.9,-1.4) {Q};\n\\node[left] at (-1,-2.5) {R};\n\\node[right] at (0,-4) {S};\n\\end{tikzpicture}}\n\\caption{Graphs of circle actions whose fixed points are all isolated}\\label{fig:GraphsIsolatedFixedPoints}\n\\end{figure}\n\n\n\\section{Equivalent circle actions}\n\nA single equivalence class of Hamiltonian circle actions on $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$ may be represented by more than one triple $(a,b;m)$, and this happens whenever the associated labelled graphs are the same up to the action of $\\AGL(1,\\mathbb{Z})$ on the moment map labels. \n\n\\subsection{Equivalent circle actions in \\texorpdfstring{$\\mathbb{T}^2_{m}$}{Tm}}\nThe action $S^{1}(-a,-b;m)$ is equivalent to $S^{1}(a,b;m)$ since they only differ by a reparametrization of the circle. It follows that in \nFigure~\\ref{fig:GraphsWithFixedSurfaces}, the graphs (A), (C), and (E) are $\\AGL(1,\\mathbb{Z})$-equivalent, respectively, to the graphs (B), (D), and (F). Similarly, the graphs (G), (H), and (I) in Figure~\\ref{fig:GraphsIsolatedFixedPoints} are $\\AGL(1,\\mathbb{Z})$-equivalent, respectively, to the graphs (L), (K), and (J). Other examples of equivalent subcircles in $\\mathbb{T}^2_{m}$ are obtained by letting the normalizer $N(\\mathbb{T}^2_{m})$ act on the circle subgroups of $\\mathbb{T}^2_{m}$. For instance, $S^{1}(a,b;m)$ is conjugated to $S^{1}(-a,b-am;m)$ by the adjoint of the element $\\Lambda\\in N(\\mathbb{T}^2_{m})$ of Proposition~\\ref{prop:NormalizersHirzebruch}.\n\n\\begin{prop}\\label{prop:EquivalentSubcircles}\nThe only subcircles of $\\mathbb{T}^2_{m}$ that are equivalent to $S^{1}(a,b;m)$ are the elements of the orbit of $S^{1}(a,b;m)$ under the action of the Weyl group $W(\\mathbb{T}^2_{m})$ and their reparametrizations. \n\\end{prop}\n\\begin{proof}\nWe have to show that if $S^{1}(a,b;m)$ and $S^{1}(c,d;m)$ are in the same conjugacy class, then there is an element of $N(\\mathbb{T}^2_{m})$ taking one circle to the other. As conjugation of actions corresponds to  uniform translation of the moment map labels, this will follow from a systematic inspection of the possible labelled graphs.\n\nThe first cases to consider are actions $S^{1}(a,b;m)$ whose fixed points are all isolated, which means that the weights $a$, $b$, and $am-b$ are all non-zero. Two equivalent graphs arising from normalized moment maps must have the same moment map values at the maximum, and their respective weights at the maximum and at the minimum must also coincide. \n\n\\begin{table}[H]\n\\begin{tabular}{|l||c|c|c|}\n\\hline\nType of graph & Weights at max & Weights at min & Moment label at max\\\\\n\\hline\n\\hline\nG: $a>0$, $b>0$, $am-b>0$ & $-a$, $b-am$ & $a$, $b$ & $a(\\lambda+k)$\\\\\n\\hline\nH: $a>0$, $b>0$, $am-b<0$ & $-a$, $am-b$ & $a$, $b$ & $b+a(\\lambda-k-\\epsilon_{m})$ \\\\\n\\hline\nI: $a>0$, $b<0$ & $-a$, $b-am$ & $a$, $-b$ & $a(\\lambda+k)-b$\\\\\n\\hline\nJ: $a<0$, $b>0$ & $a$, $-b$ & $-a$, $b-am$ & $b-a(\\lambda+k)$\\\\\n\\hline\nK: $a<0$, $b<0$, $am-b>0$ & $a$, $b$ & $-a$, $am-b$ & $-b-a(\\lambda-k-\\epsilon_{m})$\\\\\n\\hline\nL: $a<0$, $b<0$, $am-b<0$ & $a$, $b$ & $-a$, $b-am$ & $-a(\\lambda+k)$\\\\\n\\hline\n\\end{tabular}\n\\caption{Weights and moment map values for graphs (G) -- (L)}\\label{table:WeightsMomentMapValuesGL}\n\\end{table}\nWe start by assuming $a>0$, $b>0$, and $m>0$, which implies that the graph of $S^{1}(a,b;m)$ is of type (G) or (H).\n\nAssume the graph of $S^{1}(a,b;m)$ is of type (G), that is, $am-b>0$, and suppose $S^{1}(c,d;m)$, $(c,d)\\neq (a,b)$, is in the same conjugacy class.\n\n\\begin{itemize}\n\\item Suppose the graph of $S^{1}(c,d;m)$ is also of type (G). Looking at the weights at the minimum, we see that $(c,d)=(b,a)$ is the only non-trivial possibility. Looking at the moment map values at the maximum, we conclude that $a(\\lambda+k)=b(\\lambda+k)$, that is, $a=b$, contradicting the fact that $(a,b)\\neq(c,d)$.\n\\item Suppose the graph of $S^{1}(c,d;m)$ is of type (H). Looking again at the weights at the minimum, we see that $c=b$, and $d=a$, and that $a\\neq b$ as before. The sets of weights at the maximum must also be equal, that is, $\\{-a,b-am\\}=\\{-c, cm-d\\}=\\{-b,bm-a\\}$, which implies that $a=a-bm$, contradicting the fact that $bm\\neq 0$.\n\\item Suppose the graph of $S^{1}(c,d;m)$ is of type (I), which implies $c>0$, $d<0$. Looking at the weights at the minimum, we must have either $(c,d)=(a,-b)$ or $(c,d)=(b,-a)$. In the former case, looking at the moment map values at the maximum, we conclude that $a(\\lambda+k)=c(\\lambda+k)-d=a(\\lambda+k)+b$, that is, $b=0$, which is a contradiction. In the later case, the weights at the maximum become $\\{-c, d-cm\\} = \\{-b, -a-bm\\}$, and we must have $\\{-b, -a-bm\\}=\\{-a, b-am\\}$ as sets. Since $a+bm\\neq a$, we must have $b=a$, which implies $a=b=1$, $c=1$, and $d=-1$. The moment map values at the maximum are then $a(\\lambda+k)=(\\lambda+k)$ and $c(\\lambda+k)-d=(\\lambda+k)+1$, which are not equal. \n\\item Suppose the graph of $S^{1}(c,d;m)$ is of type (J), which implies $c<0$, $d>0$. Looking at the weights at the minimum, we must have $\\{a,b\\}=\\{-c,d-cm\\}$ as sets. So, either $(c,d)=(-a,b-am)$ or $(c,d)=(-b,a-bm)$. In the first case, $b-am<0$, which is impossible at the minimum. In the second case, looking at the moment map values at the maximum, we conclude that $a(\\lambda+k)=d-c(\\lambda+k)=a-bm+b(\\lambda+k)=a+b(\\lambda-k)$. This implies $a(\\lambda+k-1)\/(\\lambda+k)=b$. As $a$ and $b$ are non-zero integers, this is impossible.\n\n\\item Suppose the graph of $S^{1}(c,d;m)$ is of type (K), which implies $c<0$, $d<0$, and $cm-d>0$. Comparing the weights at the minimum, we see that either $(c,d)=(-a, -am-b)$ or $(c,d)=(-b,-bm-a)$ with $a\\neq b$. In the former case, comparing the weights at the maximum yields $\\{-a,b-am\\}=\\{c,d\\}=\\{-a,-am-b\\}$. This implies $b=0$, which is excluded. In the former case, we must have $a\\neq b$ and $\\{-a,b-am\\}=\\{c,d\\}=\\{-b,-bm-a\\}$, which again implies $b=0$.\n\\item Finally, suppose the graph of $S^{1}(c,d;m)$ is of type (K), which implies $c<0$, $d<0$, and $cm-d<0$. Comparing the weights at the minimum, we see that either $(c,d)=(-a, b-am)$ or $(c,d)=(-b,a-bm)$. In the former case, we get the conjugate circle $S^{1}(-a,b-am;m)$. In the later case, the moment map values at the maximum gives $a(\\lambda+k)=-c(\\lambda+k)=b(\\lambda+k)$, that is, $a=b$. Then $(c,d)=(-a, b-am)$ and $S^{1}(c,d;m)=S^{1}(-a,b-am;m)$ as before.\n\\end{itemize}\nWe conclude that the only circle action $S^{1}(c,d;m)$ conjugated to an action $S^{1}(a,b;m)$ of type (G) is $S^{1}(-a,b-am;m)$.\n\nAssume now that the graph of $S^{1}(a,b;m)$ is of type (H), that is, $a>0$, $b>0$, and $am-b<0$. By transitivity, we already know that an action $S^{1}(c,d;m)$ in the same conjugacy class cannot be of type (G) or (L).\n\n\\begin{itemize}\n\\item Suppose the graph of $S^{1}(c,d;m)$ is also of type (H). Comparing the weights at the minimum, we see that we must have $(c,d)=(b,a)$. Looking at the weights at the maximum, we must have $\\{-a,am-b\\}=\\{-c,cm-d\\}$ as sets. Since $(c,d)\\neq(a,b)$, the only possibility is $a=d-cm=a-bm$, which implies $b=0$.\n\\item Suppose the graph of $S^{1}(c,d;m)$ is of type (I). Comparing the weights at the minimum, we see that we must have either $(c,d)=(a,-b)$ or $(c,d)=(b,-a)$ with $a\\neq b$. In the first case, comparing the weights at the maximum, we should have $\\{-a,am-b\\}=\\{-c,d-cm\\}=\\{-a,-b-am\\}$. This implies $a=0$, which is excluded. Similarly, in the second case, we get $\\{-a,am-b\\}=\\{-c,d-cm\\}=\\{-b,-b-am\\}$ with $a\\neq b$. This again implies  $a=0$.\n\\item Suppose the graph of $S^{1}(c,d;m)$ is of type (J). Looking at the weights at the minimum, we must have $\\{a,b\\}=\\{-c,d-cm\\}$ as sets. So, either $(c,d)=(-a,b-am)$ or $(c,d)=(-b,a-bm)$. In the first case, we obtain the conjugate action $S^{1}(-a,b-am;m)$. In the second case, looking at the weights at the maximum, we must have $\\{-a,am-b\\}=\\{c,-d\\}=\\{-b, bm-a\\}$. If $a=b=1$, this gives\nus the conjugate action $S^{1}(-1,1-m;m)$. If $a=a-bm$, then $b=0$, which is excluded.\n\\item Finally, suppose the graph of $S^{1}(c,d;m)$ is of type (K). Looking at the weights at the minimum, we must have $\\{a,b\\}=\\{-c,cm-d\\}$ as sets. So, either $(c,d)=(-a,-b-am)$ or $(c,d)=(-b,-a-bm)$. Looking at the weights at the maximum, we must also have $\\{-a,am-b\\}=\\{c,d\\}$. If $(c,d)=(-a,-b-am)$, then $am-b=-am-b$, which implies $a=0$. Instead, if $(c,d)=(-b,-a-bm)$, then either $a=b=1$, which is equivalent to the previous case, or $a=a+bm$ which forces $b=0$.\n\\end{itemize}\nWe conclude that $S^{1}(-a,b-am;m)$ is the only subcircle of $\\mathbb{T}^2_{m}$ conjugated to an action $S^{1}(a,b;m)$ of type (H).\n\n\nAssume now that the graph of $S^{1}(a,b;m)$ is of type (I), that is, $a>0$, and $b<0$. By transitivity, we already know that an action $S^{1}(c,d;m)$ in the same conjugacy class cannot be of type (G), (H), (J), or (L). By symmetry of Table~\\ref{table:WeightsMomentMapValuesGL} under a change of sign of the pair $(a,b)$, and comparing with actions of types (H) and (J), we see that the only action $S^{1}(c,d;m)$ of type (K) conjugated to $S^{1}(a,b;m)$ is $S^{1}(-a,b-am;m)$. \n\n\\begin{itemize}\n\\item Suppose the graph of $S^{1}(c,d;m)$ is also of type (I). Comparing the weights at the minimum, we see that we must have $(c,d)=(-b,-a)$. Looking at the weights at the maximum, we must also have $\\{-a,b-am\\}=\\{-c, d-cm\\}=\\{b,-a+bm\\}$. If $b=-a$, then $(a,b)=(1,-1)=(c,d)$. If $b=b-am$, then we must have $a=0$, which is impossible.\n\n\\end{itemize}\nThis shows that for an action $S^{1}(a,b;m)$ of type (I), the only other action $S^{1}(c,d;m)$ in the same conjugacy class is $S^{1}(-a,b-am;m)$.\n\nThis concludes the proof of the proposition for actions $S^{1}(a,b;m)$ whose fixed points are all isolated, in the case $m>0$. When $m=0$ and $\\lambda>1$, the graphs (G) and (L) no longer exist, and the four remaining graphs (H)--(K) only differ by the action of $D_{2}$ on the pair $(a,b)$, that is, by a change of sign of $a$ or $b$. When $\\lambda=1$, we can also interchange $a$ and $b$ to obtain an equivalent graph, which defines an action of $D_{4}$. Finally, the case of actions with non-isolated surfaces involves the graphs (A)--(F) and is simpler. The details are left to the reader. \n\\end{proof}\n\n\n\n\\subsection{Toric extensions of circle actions}\n\\begin{defn}\nWe shall say a circle action $S^1(a,b; m)$ \\emph{extends} to a toric action $\\mathbb{T}^2_{n}$ if it is $S^1$-equivariantly symplectomorphic to a circle action of the form $S^1(c,d; n)$.\n\\end{defn}\n\nIn this section, we determine all possible toric extensions of $S^{1}(a,b;m)$. We begin with the exceptional case $\\lambda=1$.\n\\begin{prop}\nConsider $(S^2 \\times S^2,\\omega_\\lambda)$ with $\\lambda=1$. Then the only Hamiltonian circle actions are of the form $S^1(a,b; 0)$. In particular, they can only extend to the torus $\\mathbb{T}^2_0$. \n\\end{prop}\n\\begin{proof}\nThis follows directly from Theorem~\\ref{lemma_number_of_torus_actions}.\n\\end{proof}\n\n\n\nBy Lemma~\\ref{lemma_torus_action_-vecurve}, a toric extension of $S^{1}(a,b;m)$ to $\\mathbb{T}^2_{n}$, $n\\geq 1$, implies the existence of an invariant sphere of self-intersection $-n$ and of positive symplectic area. These two numerical invariants can be read from labelled graphs using Lemma~\\ref{weight} and Lemma~\\ref{Lemma:SymplecticArea}. As we will see, this imposes enough conditions on the triple $(a,b;m)$ to determine all possible embeddings $S^{1}(a,b;m)\\hookrightarrow \\mathbb{T}^2_{n}$, $n\\neq m$.\n\n\n\n\\begin{prop}\\label{prop:AtMostTwoExtensions}\nConsider a Hamiltonian circle action $S^1(a,b;m)$ with $\\lambda >1$. Under the following numerical conditions on $a,b,m,\\lambda$, the circle action only extends to the toric action~$\\mathbb{T}^2_m$:\n\\begin{itemize}\n    \\item when $a\\neq\\pm 1$;\n    \\item when $b=0$ or $b=am$\n    \\item when $a = \\pm 1$ and $2 \\lambda \\leq |2b-am|+\\epsilon_{m}$.\n\\end{itemize}\nIn all other cases, the circle action may extend to at most two inequivalent toric actions, namely,\n\\begin{itemize}\n    \\item when $a=\\pm 1$, $2\\lambda > |2b-am|+\\epsilon_{m}$, and $b \\not\\in \\{0,am\\}$, the circle action $S^1(a,b;m)$ only extends to the toric action $\\mathbb{T}^2_{m}$ and, possibly, to $\\mathbb{T}^2_{|2b-am|}$.\n\n\\end{itemize}\n\\end{prop}\n\\begin{proof}\nSuppose $S^{1}(a,b;m)$ is equivariantly symplectomorphic to $S^{1}(c,d;n)$ for some $n\\neq m$. Note that we necessarily have $m=n\\mod 2$. By assumption, the two actions share the same normalized labelled graph\n\nWe first consider an action $S^{1}(a,b;m)$ whose fixed points are all isolated. For the moment, let's also assume that $m\\neq 0$ and $n\\neq 0$. We observe that if the $\\mathbb{T}^2_{n}$ invariant curve $C_{n}$ of self-intersection $-n$ has non-trivial isotropy, then it must appear as some edge in the graph of $S^{1}(a,b;m)$. As the edge connecting the vertices $Q$ and $R$ is the only one that corresponds to an invariant sphere of negative self-intersection, namely $-m$, we conclude that $m=n$. This contradiction shows that $C_{n}$ must have trivial isotropy. By symmetry, the same is true of the invariant curve $C_{m}$ of self-intersection $-m$. It follows that $a=\\pm 1$ and $c=\\pm 1$. \n\nAssume $a=1$. Because there are no fixed surfaces, we know that $b\\neq 0$ and $m-b\\neq 0$. Figure~\\ref{fig:PossibleGraphsToricExtensions} shows the three possible graphs for $S^{1}(1,b;m)$, which are then of types (G), (H), or (I). The dashed edges represent the possible locations of the $\\mathbb{T}^2_{n}$ invariant curves $C_{n}$ and $C_{-n}$. We can compute the self-intersection of $C_{n}$ and $C_{-n}$ by applying Lemma~\\ref{weight} to the normal bundle of these invariant spheres, and using the configurations of weights shown in Figure~\\ref{fig:Self-Intersection}. The self-intersection of the curve between $P$ and $R$ is $2b-m$, while the self-intersection of the curve between $Q$ and $S$ is $m-2b$. Set $n=|2b-m|$. For the toric action $\\mathbb{T}^2_{n}$ to exist, we must also have $0<2\\omega_\\lambda (C_{n}) = 2\\lambda-|2b-m|-\\epsilon_{m}$. We conclude that the action $S^{1}(1,b;m)$, $m\\neq 0$, may extend to $\\mathbb{T}^2_{|2b-m|}$ whenever $2\\lambda>|2b-m|+\\epsilon_{m}$, and that this is the only other possible toric extension.\n\n\n\\begin{figure}\n\\centering\n~\n\\subcaptionbox*{}{}\n\\setcounter{subfigure}{6}\n\\subcaptionbox{When $a=1,~b>0$, and $m-b>0$\\label{fig:a=1|b>0|m-b>0}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-1.5) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2.5) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-1.5);\n\\draw [dashed, blue] (0,0) -- (-1,-2.5);\n\\draw [dashed, blue] (1,-1.5) -- (0,-4);\n\\draw (-1,-2.5) -- (0,-4);\n\\node[left] at (-0.6,-3.2) {$b$};\n\\node[right] at (0.7,-0.5) {$m-b$};\n\\node[above] at (0,0.2) {$\\mu = (\\lambda + k)$};\n\\node[right] at (-0.6,-2.5) {$\\mu = b$};\n\\node[right] at (1,-1.8) {$\\mu = b + (\\lambda - k-\\epsilon_{m}) $};\n\\node[below] at (0, -4.2) {$\\mu = 0$};\n\\node[left] at (-1,-2.5) {Q};\n\\node[left] at (0.9,-1.4) {R};\n\\node[right] at (0,0) {S};\n\\node[right] at (0,-4) {P};\n\\end{tikzpicture}}\n\\subcaptionbox{When $a=1,~b>0$ and $m-b<0$\n}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (1,-2) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (-1,-3) ellipse (0.1cm and 0.1cm); \n\\draw [dashed, blue] (0,0) .. controls (-4,-1.5) and (-2,-3.5) .. (0,-4);\n\\draw [dashed, blue] (1,-2) -- (-1,-3);\n\\draw (0,0) -- (1,-2);\n\\draw (-1,-3) -- (0,-4);\n\\node[right] at (0,-4) {P};\n\\node[left] at (-1,-3) {Q};\n\\node[left] at (0.9,-2) {S};\n\\node[right] at (0,0) {R};\n\\node[above] at (0,0.2) {$\\mu = b +  (\\lambda -k-\\epsilon_{m})$};\n\\node[right] at (1.2,-2) {$\\mu = (\\lambda + k) $};\n\\node[right] at (-0.7,-3) {$\\mu = b$};\n\\node[right] at (-0.4,-3.5) {$b$};\n\\node[right] at (0.7,-1) {$b-m$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\end{tikzpicture}\n}\n\\subcaptionbox{When $a=1,~b<0$\\label{fig:a=1|b>0}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-3) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2) ellipse (0.1cm and 0.1cm);\n\\draw [dashed, blue] (0,0) .. controls (-4,-1.5) and (-2,-3.5) .. (0,-4);\n\\draw [dashed, blue] (-1,-2) -- (1,-3);\n\\draw (0,0) -- (-1,-2) ;\n\\draw (1,-3) -- (0,-4);\n\\node[above] at (0,1) {\\rule{0em}{1em}};\n\\node[above] at (0,0.2) {$\\mu = (\\lambda +k)-b$};\n\\node[right] at (1.4,-3) {$\\mu = -b $};\n\\node[right] at (-0.8,-1.9) {$\\mu = (\\lambda-k-\\epsilon_{m})$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\node[right] at (0.5, -3.6) {$-b$};\n\\node[right] at (-0.4,-1.0) {$m-b$};\n\\node[right] at (1,-3) {P};\n\\node[left] at (-1,-2) {R};\n\\node[right] at (0,0) {S};\n\\node[right] at (0,-4) {Q};\n\\end{tikzpicture}}\n\\caption{Graphs of $S^{1}(1,b;m)$ with possible locations of $C_{n}$ and $C_{-n}$}\n\\label{fig:PossibleGraphsToricExtensions}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n   \n\\begin{tikzpicture}\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (4,0) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (7,0) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (11,0) ellipse (0.1cm and 0.1cm);\n\\draw [dashed, blue] (0,0) -- (4,0); \n\\draw [dashed, blue] (7,0) -- (11,0);\n\\draw (0,-0.5) -- (0,0.5);\n\\draw (4,-0.5) -- (4,0.5);\n\\draw (7,-0.5) -- (7,0.5);\n\\draw (11,-0.5) -- (11,0.5);\n\\node[left] at (0,0) {$1\\,$}; \n\\node[above] at (0,0.5) {$b$}; \n\\node[right] at (4,0) {$\\,-1$}; \n\\node[above] at (4,0.5) {$m-b$}; \n\\node[right] at (0,-0.25) {$P$};\n\\node[left] at (4,-0.25) {$R$};\n\\node[above,blue] at (2,0) {$1$};\n\\node[left] at (7,0) {$1\\,$}; \n\\node[above] at (7,0.5) {$-b$}; \n\\node[right] at (11,0) {$\\,-1$};\n\\node[above] at (11,0.5) {$b-m$}; \n\\node[right] at (7,-0.25) {$Q$};\n\\node[left] at (11,-0.25) {$S$};\n\\node[above,blue] at (9,0) {$1$};\n\\end{tikzpicture}\n\\caption{Configurations of weights along $C_{\\pm n}$ when $a=1$}\n\\label{fig:Self-Intersection}\n\\end{figure}\n\nWhen $a=-1$, the same arguments apply to $S^{1}(-1,b;m)$ whose possible graphs are now of types (J), (K), and (L). The self-intersections of the curves $C_{\\pm 1}$ are now $\\pm |2b+m|$. We conclude that the action $S^{1}(-1,b;m)$ may extend to $\\mathbb{T}^2_{|2b+m|}$ whenever $m\\neq 0$ and $2\\lambda>|2b+m|+\\epsilon_{m}$.\n\nIn the special case $m=0$, any invariant sphere appearing in the graph of $S^{1}(a,b;0)$ has zero self-intersection. If $S^{1}(a,b;0)$ extends to $\\mathbb{T}^2_{n}$ with $n\\geq 2$, it again follows that the invariant curve $C_{n}$ of self-intersection $-n$ must have trivial isotropy. Consequently, $c=\\pm 1$, and at least one of the weights $a$ or $b$ must be $\\pm1$. If $a=1$, the possible graphs of $S^{1}(1,b;0)$ are the graphs (H) and (I) of Figure~\\ref{fig:PossibleGraphsToricExtensions}. Then, the same argument as before shows that the self-intersection of invariant curves are $0$ and $\\pm 2b$. Consequently, the action $S^{1}(1,b;0)$ may extend to $\\mathbb{T}^2_{|2b|}$ provided $2\\lambda>|2b|$. Similarly, when $a=-1$, the action $S^{1}(-1,b;0)$ may extend to $\\mathbb{T}^2_{|2b|}$ whenever $2\\lambda>|2b|$.\n\nWhen $m=0$ and $b=1$, then the possible self-intersections of invariant curves are $0$ and $\\pm 2a$. However, the two possible graphs are now of type (H) and (J). In both cases, the area of the tentative invariant curve $C_{\\pm n}$ connecting the two interior fixed points is negative. It follows that it is impossible to have $m=0$ and $b=1$. Similarly, if $b=-1$, the possible graphs are of type (I) and (K) and, as before, the area of the tentative invariant curve $C_{\\pm n}$ connecting the two interior fixed points is negative. Consequently, we cannot have $m=0$ and $b=\\pm 1$. This concludes the proof of the statement for actions whose fixed points are all isolated.\n\n\\begin{figure}\n\\centering\n~\n\\subcaptionbox*{}{}\n\\setcounter{subfigure}{7}\n\\subcaptionbox{When $m=0$, $a>0,~b=1$}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (1,-2) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (-1,-3) ellipse (0.1cm and 0.1cm); \n\\draw [dashed, blue] (0,0) .. controls (-4,-1.5) and (-2,-3.5) .. (0,-4);\n\\draw[dashed, blue] (-1,-3) -- (1,-2);\n\\draw (0,0) -- (-1,-3) ;\n\\draw (1,-2) -- (0,-4);\n\\node[left] at (-1.1,-3) {Q};\n\\node[left] at (0.9,-1.9) {S};\n\\node[right] at (0,0) {R};\n\\node[above] at (0,0.2) {$\\mu = 1 + a (\\lambda -k-\\epsilon_{m})$};\n\\node[right] at (1.2,-2) {$\\mu = a(\\lambda + k) $};\n\\node[right] at (-0.9,-3) {$\\mu = 1$};\n\\node[right] at (0,-4) {P};\n\\node[right] at (0.5, -3) {$a$};\n\\node[left] at (-0.5,-1.5) {$a$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\end{tikzpicture}}\n\\setcounter{subfigure}{9}\n\\subcaptionbox{When $m=0$, $a<0,~b=1$\\label{fig:a<0|b=1}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-3) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-3);\n\\draw (-1,-2) -- (0,-4);\n\\draw [dashed, blue] (0,0) .. controls (-4,-1.5) and (-2,-3.5) .. (0,-4);\n\\draw[dashed, blue] (-1,-2) -- (1,-3);\n\\node[above] at (0,1) {\\rule{0em}{1em}};\n\\node[above] at (0,0.2) {$\\mu = 1 - a (\\lambda +k)$};\n\\node[right] at (1.4,-3) {$\\mu = 1 $};\n\\node[above] at (-1,-2) {$\\mu = -a(\\lambda+k)$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\node[right] at (-1.2,-3.2) {$-a$};\n\\node[right] at (0.6,-1.3) {$-a$};\n\\node[left] at (0.9,-3) {R};\n\\node[left] at (-1,-2.2) {P};\n\\node[right] at (0,0) {Q};\n\\node[right] at (0,-4) {S};\n\\end{tikzpicture}}\n\\caption{Graphs of $S^{1}(a,1;0)$ with impossible locations of $C_{n}$ and $C_{-n}$}\n\\label{fig:ImpossibleGraphsToricExtensions}\n\\end{figure}\n\nNow suppose $S^{1}(a,b;m)$ has non-isolated fixed points, that is, $a=0$, or $b=0$, or $b=am$. As the moment map values of graphs (A), (B), (E), and (F) depend only on $m$, it is impossible to get the same graphs from another action $S^{1}(c,d;n)$ with $n\\neq m$. The same remark applies to the area labels of graphs (C) and (D), showing that $S^{1}(a,b;m)$ only extends to $\\mathbb{T}^2_{m}$.\n\\end{proof}\n\n\nIt remains to investigate whether the action $S^{1}(\\pm 1,b;m)$ extends to $\\mathbb{T}^2_{|2b-m|}$ when $2\\lambda > |2b-am|+\\epsilon_{m}$ and $b \\not\\in \\{0,am\\}$. A straightforward but lengthy comparison of the possible graphs of $S^{1}(\\pm1,b;m)$ and $S^{1}(\\pm,d;|2b-am|)$ shows that this is always the case and yields the equivalences stated in the following two corollaries. The graphs giving the first equivalence are shown in Figure~\\ref{fig:ExampleTwoToricExtensions}. The other cases are left to the reader.\n\n\\begin{cor} \\label{cor:CircleExtensionsWith_a=1}\nConsider the action $S^1(1,b;m)$ and suppose $2\\lambda > |2b-m|+\\epsilon_{m}$. Then under the following numerical conditions on $b$ and $m$, the  $S^1(1,b;m)$ action extends to the toric action $\\mathbb{T}^2_{|2b-m|}$ and is equivariantly symplectomorphic to the following subcircle in $\\mathbb{T}^2_{|2b-m|}$\\,:\n\\begin{enumerate}\n    \\item if $b>0$ and $b>m$, then $S^1(1,b; m)$ is equivalent to $S^1(1,b; |2b-m|)$;\n    \\item if $b>0$, $m>b$, and $2b-m < 0$, then $S^1(1,b; m)$ is equivalent to $S^1(1,-b; |2b-m|)$;\n    \\item if $b>0$, $m>b$, and $2b-m > 0$, then $S^1(1,b; m)$ is equivalent to $S^1(1,b;|2b-m|)$;\n    \\item finally, if $b<0$, then $S^1(1,b; m)$ is equivalent to $S^1(1,-b;|2b-m|)$.\\qed\n\\end{enumerate}\n\\end{cor}\n\n\n\\begin{cor}\\label{cor:CircleExtensionsWith_a=-1}\nConsider the $S^1$ actions $S^1(-1,b;m)$ on $(S^2 \\times S^2,\\omega_\\lambda)$  and suppose $2\\lambda > |2b+m|$. Then under the following numerical conditions on $b$ and $m$, the  $S^1(-1,b;m)$ action extends to the toric action $\\mathbb{T}^2_{|2b+m|}$ and is equivariantly symplectomorphic to the following subcircle in $\\mathbb{T}^2_{|2b+m|}$\\,:\n\\begin{enumerate}\n\\item if $b<0$ and $m>-2b$, then $S^1(-1,b;m)$ is equivalent to  $S^1(-1,-b; |2b+m|)$;\n\\item if $b<0$, $m>-b$, and $-2b>m$, then $S^1(-1,b;m)$ is equivalent to \n$S^1(-1,b; |2b+m|)$;\n\\item if $b<0$ and $-b>m$, then $S^1(-1,b;m)$ is equivalent to $S^1(-1,b; |2b+m|)$;\n\\item if $b>0$, then $S^1(-1,b; m)$ is equivalent to $S^1(-1,-b; |2b+m|)$.\\qed\n\\end{enumerate}\n\\end{cor}\n\n\\begin{figure}[H]\n\\centering\n\\subcaptionbox*{$S^1(1,b;m)$ of type (H)\\label{fig:ExtendedGraph1}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (1,-2) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (-1,-3) ellipse (0.1cm and 0.1cm); \n\\draw (0,0) -- (1,-2);\n\\draw (-1,-3) -- (0,-4);\n\\draw[dashed, blue] (0,0) -- (-1,-3) ;\n\\draw[dashed, blue] (1,-2) -- (0,-4);\n\\node[right] at (-0.9,-3) {Q};\n\\node[left] at (0.9,-2) {S};\n\\node[right] at (0,0) {R};\n\\node[above] at (0,0.2) {$\\mu = b +  (\\lambda-k-\\epsilon_{k})$};\n\\node[right] at (1.3,-2) {$\\mu = (\\lambda+k) $};\n\\node[left] at (-1.4,-3) {$\\mu = b$};\n\\node[right] at (0,-4) {P};\n\\node[right] at (-1,-3.7) {$b$};\n\\node[right] at (0.5, -3) {$a=1$};\n\\node[left] at (-0.5,-1.5) {$a=1$};\n\\node[right] at (0.7,-1) {$b-m$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\end{tikzpicture}\n}\n~\n\\subcaptionbox*{$S^1(1,b;2b-m)$ of type (G)\\label{sndstrata}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-1.5) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2.5) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-1.5);\n\\draw[dashed, blue] (1,-1.5) -- (-1,-2.5);\n\\draw[dashed, blue] (0,0) .. controls (-4,-1.5) and (-2,-3.5) .. (0,-4);\n\\draw (-1,-2.5) -- (0,-4);\n\\node[right] at (-0.35,-3.2) {$b$};\n\\node[right] at (- 1,-1.8) {$a=1$};\n\\node[left] at (-2.2,-1.5) {$a=1$};\n\\node[right] at (0.7,-0.5) {$(2b-m)-b=b-m$};\n\\node[above] at (0,0.2) {$\\mu = \\lambda+\\frac{(2b-m)-\\epsilon_{k}}{2}= b+\\lambda - k-\\epsilon_{k}$};\n\\node[right] at (-0.6,-2.5) {$\\mu = b$};\n\\node[right] at (0.6,-1.8) {$\\mu=b+\\lambda-\\frac{(2b-m)-\\epsilon_{k}}{2}-\\epsilon_{k}$};\n\\node[right] at (0.91,-2.25) {$=\\lambda+k$};\n\\node[below] at (0, -4.2) {$\\mu = 0$};\n\\node[left] at (-1,-2.5) {Q};\n\\node[left] at (0.9,-1.4) {R};\n\\node[right] at (0,0) {S};\n\\node[right] at (0,-4) {P};\n\\end{tikzpicture}}\n\\caption{The equivalence $S^{1}(1,b;m)\\sim S^{1}(1,b;2b-m)$ when $b>m$}\n\\label{fig:ExampleTwoToricExtensions}\n\\end{figure}\n\n\n\n\n\n\\chapter{Action of \\texorpdfstring{$\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$}{Symp(S2xS2)} on \\texorpdfstring{$\\mathcal{J}^{S^1}_{\\om_\\lambda}$}{J\\hat{}S1}}\\label{ChapterActionOfSymp}\nIn this chapter we show that the space of $S^1$ invariant compatible almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ can be decomposed into strata each of which being homotopy equivalent to a homogeneous space under the action of the equivariant symplectomorphism group. \n\n\n\n\n\n\n\\section{\\texorpdfstring{$J$}{J}-Holomorphic Preliminaries}\n\nWe first recall a few facts about compatible almost complex structures and associated $J$-holomorphic curves.\n\n\n\\begin{defn}[Compatible almost complex structures] \nAn almost complex structure $J$ on a symplectic manifold $(M,\\omega)$ is said to be compatible with $\\omega$ if $\\omega(u,Ju)>0$ and $\\omega(Ju, Jv)=\\omega(u,v)$ for all non-zero $u,v\\in T_* M$.\n\\end{defn}\n\n\\begin{lemma}\nThe space $\\mathcal{J}_\\omega = \\mathcal{J}(M,\\omega)$ of all compatible almost complex structures on a symplectic manifold $(M,\\omega)$ is non-empty and contractible.\n\\end{lemma}\n\n\\begin{defn}{\\textit{$J$-holomorphic spheres}:}\nLet $(M,\\omega)$ be a symplectic manifold endowed with a compatible almost complex structure $J$. A rational $J$-holomorphic map, also called a parametrized $J$-holomorphic sphere, is a $C^\\infty$ map\n\\[\nu: ({S}^2, j) \\longrightarrow (M,\\omega,J)\n\\]\nsatisfying the Cauchy-Riemann equation\n\\[\n\\bar\\partial_J(u)=\\frac{1}{2}(du\\circ j - J\\circ du) = 0\n\\]\nwhere $j$ is the usual complex structure on the sphere. The image of a $J$-holomorphic rational map is called a rational $J$-holomorphic curve or simply a $J$-curve.\n\\end{defn}\n\n\n\n\\begin{defn}[Multi-covered and simple maps]\nWe say that a $J$-holomorphic map $u:\\mathbb{C}P^1 \\longrightarrow (M,J)$ is multi-covered if $u = {u}^\\prime \\circ f$, where $f:\\mathbb{C}P^1 \\to \\mathbb{C}P^1$ is a holomorphic map of degree greater than one and where $u':\\mathbb{C}P^1 \\to (M,J)$ is a $J$-holomorphic map. We call a $J$-holomorphic map simple if it is not multi-covered.\n\\end{defn}\n\n\\begin{remark}\nWe usually assume that a $J$-holomorphic map is somewhere injective, meaning that $\\exists z\\in {S}^2$ such that $du_z \\neq 0$ and $u^{-1}u(z) = z$. In particular, somewhere injective maps do not factor through multiple covers $h:S^2\\to S^2$. \n\\end{remark}\n\n\\begin{defn}[Moduli spaces of $J$-holomorphic maps or curves]\nLet $(M,\\omega)$ be a symplectic manifold and let $J \\in \\mathcal{J}_\\omega$. Given $A \\in H_2(M, \\mathbb{Z})$ we denote by $\\widetilde{\\mathcal{M}}(A,J)$ the space of all $J$-holomorphic, somewhere injective maps representing the homology class $A$. The Mobius group $G =\\PSL(2,\\mathbb{C})$ acts freely on this space by reparametrization and the quotient space $\\mathcal{M}(A,J):=\\widetilde{\\mathcal{M}}(A,J)\/G$ is called the moduli space of (unparametrised) $J$-curves in class $A$.\n\\end{defn}\n\nIn dimension 4, the geometric properties of $J$-holomorphic curves are, to a large extend, controlled by homological data. As a result, many properties of complex algebraic curves in complex algebraic surfaces extend to $J$-holomorphic curves in $4$-dimensional symplectic manifolds. Below we list the key properties of $J$-holomorphic curves we will be relying on.\n\n\\begin{thm}[Positivity]\\label{thm_Positivity}Let $(M,\\omega)$ be a 4-dimensional symplectic manifold. If a homology class A $\\in H_2(M,\\mathbb{Z})$ is represented by a nonconstant J-curves for some $J \\in \\mathcal{J_\\omega}$ then $\\omega(A) > 0$.\n\\end{thm}\n\n\\begin{thm}[Fredholm property and automatic regularity.]\\label{thm_Regularity}Let $(M,\\omega)$ be a 4-dimensional symplectic manifold.  \nThen the universal moduli space\n\\[\n\\widetilde{\\mathcal{M}}(A,\\mathcal{J}_{\\omega}) := \\bigcup_{J \\in {\\mathcal{J}_{\\omega}}} \\widetilde{\\mathcal{M}}(A,J)\n\\]\nwith $C^l$-topology ($l \\geq 2$) is a smooth Banach manifold and the projection map\n\\[\n\\pi_A: \\widetilde{\\mathcal{M}}(A,\\mathcal{J}_{\\omega}) \\longrightarrow \\mathcal{J}_{\\omega}\n\\]\nis a Fredholm map of index $2(c_1(A) + 2)$ where $c_1 \\in H^2(M,\\mathbb{Z})$ is the first chern class of $(TM, J)$ (note that the Chern class is independent of choice of $J \\in \\mathcal{J}_{\\omega}$).  An almost complex structure is said to be regular for the class $A$ if it is a regular value for the projection $\\pi_A$.  If this is the case then the moduli spaces $\\widetilde{\\mathcal{M}}(A,J)$ and $\\mathcal{M}(A,J)$ are smooth manifolds of dimensions $2(c_1(A)+2)$ and $2(c_1(A)-1)$ respectively.  The set of regular values $\\mathcal{J} \\in \\mathcal{J}_\\omega$ is a subset of second category and is denoted by $\\mathcal{J}_{\\omega}^{\\textrm{reg}}(A)$. If $J \\in  J_\\omega$ is integrable and $S$ is an embedded $J$-holomorphic sphere with self-intersection number $[S]\\cdot [S] \\geq -1$, then $J$ is regular for the class $[S]$. In dimension $4$, the same conclusion holds without the integrability assumption.\n\\end{thm}\n\n\\begin{defn}[Cusp Curves] \\label{defn_cusp}\nLet $(M,\\omega)$ be a symplectic manifold. Let $J \\in J_\\omega$. A $J$-holomorphic cusp curve $C$ is a connected finite union of $J$-holomorphic curves \n\\[C = C_1 \\cup C_2 \\ldots \\cup C_k\\] where $C_i = u_i(\\mathbb{C}P^1)$ and $u_i: \\mathbb{C}P^1 \\to (M,J)$ is a (possibly multi-covered) $J$-holomorphic map.\n\\end{defn}\n\n\\begin{thm}[Gromov's compactness theorem]\\label{thm_Compactness} Let $(M,\\omega)$ be a compact symplectic manifold. Let $J_n \\in \\mathcal{J}_{\\omega}$ be a sequence converging to $J$ in the $C^\\infty$ topology and let $S_i$ be $J_i$-holomorphic spheres of bounded symplectic area $\\omega(S_i)$. Then there is a subsequence of the $S_i$ which converges weakly to a $J$-holomorphic curve or cusp-curve $S$. In particular if all the $S_i$'s belong to the class $A$, then $S$ also belongs the class $A$, and any cusp curve defines a homological decomposition of $A=\\sum_i A_i$ such that $\\omega(A_i)>0$.\n\\end{thm}\n\n\\begin{thm}[Positivity of intersections]\\label{thm_PositivityIntersections} Let $J \\in \\J_{\\om_\\lambda}$ and $A$, $B$ be two distinct $J$-holomorphic curves in a $4$-dimensional manifold. Then they intersect at only finitely many points and each point contributes positively to the intersection multiplicity $[A]\\cdot[B]$. Moreover,  $[A]\\cdot[B]=1$ iff the curves intersect transversally at exactly one point, while $[A]\\cdot[B]= 0$ iff the curves are disjoint.\n\\end{thm}\n\nAs a corollary of Positivity of intersections we have the following result under the presence of a group action. \n\\begin{cor}\\label{cor_pos}\nLet $(M,\\omega)$ be a symplectic 4-manifold and let $G$ be a compact Lie group acting symplectically on $M$. Suppose that $G$ acts trivially on homology. Let $\\mathcal{J}^{G}$ denote the space of $\\omega$ tame (or compatible) almost complex structures and let $C$ be a $J$ holomorphic curve for some $J \\in \\mathcal{J}^G$. \nThen,\n\\begin{enumerate}\n    \\item if C has  negative self intersection, then $g \\cdot C = C$ for all $g \\in G$.\n    \\item If C has zero self intersection, then $g \\cdot C = C$ or $g\\cdot C \\cap C = \\emptyset$ for all $g \\in G$.\n\\end{enumerate} \n\\end{cor}\n\n\\begin{thm}[Adjunction formula]\\label{thm_Adjunction} Let $u: ({S}^2, j) \\longrightarrow (M^4,J)$ be a somewhere injective $J$-holomorphic map representing the homology class $A$ in a $4$-dimensional manifold. Define the virtual genus of $A$ as\n\\[\ng_v(A) = 1+\\frac{1}{2}([A]\\cdot[A]- c_1(A))\n\\]\nwhere $c_1(A) = \\langle c_1(TM,J), A \\rangle$. Then $g_v(A)\\geq 0$ with equality if, and only if, the map $u$ is an embedding.\n\\end{thm}\n\n\n\\section{\\texorpdfstring{$J$}{J}-holomorphic spheres in \\texorpdfstring{$S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$}{S2xS2 and CP2\\#CP2}}\n\nIn this section, will show how the existence of certain $J$-holomorphic spheres in ruled $4$-manifolds induces a natural partition of the space $\\mathcal{J}_\\omega$. We shall state all the relevant results, and refer the reader to Chapter 9 of \\cite{McD} for more details.\n\n\\begin{cor}\\label{FSimple}\nLet $\\lambda = l + \\delta$ where $l \\in \\mathbb{N}$ and $0<\\delta\\leq 1$.\nThen we have \n\\begin{enumerate}\n    \\item Any $J$-holomorphic representative of the class $F$ is a simple curve.\n    \\item The only $J$-holomorphic decomposition of the class $B$ are of the form $B = (B-kF) + kF$, where $0\\leq k\\leq\\ell$. In this decomposition, the $J$-holomorphic representative of the class $(B-kF)$ is an embedded sphere, while the class $kF$ may be represented by a collection of (possibly multiply covered) spheres representing multiples of the class $F$.\\qed\n\\end{enumerate}\n\\end{cor}\n\n\n\n\\begin{prop}\\label{prop_FIndecomposable}\nThe moduli space $\\widetilde{\\mathcal{M}}(F,J) \\neq \\emptyset$ for all $J \\in  J_\\omega$. In particular, for every compatible almost complex structure $J\\in\\mathcal{J}_{\\omega_\\lambda}$, and for any given point $p\\in S^2\\times S^2$, there is a unique embedded $J$-holomorphic sphere representing the class $F$ passing through~$p$.\\qed\n\\end{prop}\n\n\nWe also have an analogous result for $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$.\n\\begin{thm}\nGiven any point $p \\in \\CP^2\\# \\overline{\\CP^2}$, and any $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda}$, there is a $J$-holomorphic curve in the class $F:=L-E$ passing through $p$. \\qed\n\\end{thm}\n\nThe following  Theorem due to Abreu and McDuff~\\cite{MR1775741} tells us about the decomposition of the space of compatible almost complex structures on $(S^2 \\times S^2,\\omega_\\lambda)$ into finitely many strata. \n\\begin{thm} \\label{strata}\nLet $\\mathcal{J}_{\\omega_\\lambda}$ denote the space of all compatible almost complex structures (not necessarily invariant) for the form $\\omega_\\lambda$, on $S^2 \\times S^2$ then the space $\\mathcal{J}_{\\omega_\\lambda}$ admits a finite decomposition into disjoint Fr\u00e9chet manifolds of finite codimensions\n\\[\n\\mathcal{J}_{\\omega_\\lambda} = U_0 \\sqcup U_2 \\sqcup U_4 \\ldots \\sqcup U_{2n}\n\\]\nwhere $2n= \\lceil 2\\lambda \\rceil -1$ and $\\lceil \\lambda \\rceil$ is the unique integer $l$ such that $l < \\lambda \\leq l+1$ and where\n\\[\nU_{2i} := \\left\\{ J \\in \\mathcal{J}_{\\omega_\\lambda}~|~ D_{2i}=B-iF \\in H_2(S^2 \\times S^2,\\mathbb{Z})\\text{~is represented by a $J$-holomorphic sphere}\\right\\}\n\\]\n\\end{thm}\n\nA completely analogous description holds true for $\\CP^2\\# \\overline{\\CP^2}$.\n\n\\begin{thm}\\label{strata-CCC}\nLet $\\mathcal{J}_{\\omega_\\lambda}$ denote the space of all compatible almost complex structures (not necessarily invariant) for the form $\\omega_\\lambda$, then the space $\\mathcal{J}_{\\omega_\\lambda}$ admits a finite decomposition into disjoint Fr\u00e9chet manifolds of finite codimensions\n\\[\n\\mathcal{J}_{\\omega_\\lambda} = U_1 \\sqcup U_3 \\sqcup U_5\\ldots \\sqcup U_{2n-1}\n\\]\nwhere $2n  = \\lceil 2\\lambda \\rceil -1$, and $\\lceil \\lambda \\rceil$ is the unique integer $l$ such that $l < \\lambda \\leq l+1$ and where\n\\[ \nU_{2i-1} := \\left\\{ J \\in \\mathcal{J}_{\\omega_\\lambda}~|~ D_{2i-1}=B-iF \\in H_2(\\CP^2\\# \\overline{\\CP^2},\\mathbb{Z})\\text{~is represented by a $J$-holomorphic sphere}\\right\\}\n\\]\n\\end{thm}\n\n\\begin{remark}\nWe label the strata in $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ by the homological self-intersection of the classes $D_s$.\n\n\\end{remark}\n\n\\begin{remark}\\label{remark-starta-toric}\nWe note that for both $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$,  there is a canonical integrable almost complex structure $J_s$ in the strata $U_s$ coming from realizing $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ as the $s^{\\text{th}}$- Hirzebruch surface $W_s$ of section 1.2. Further recall that associated to each $J_s$ we have a unique $J_s$-holomorphic Hamiltonian toric action $\\mathbb{T}_s$. Thus the set of possible equivalence classes of toric actions on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ are in one-to-one correspondence with the strata in the decomposition of $\\mathcal{J}_{\\omega_\\lambda}$. This fact will be crucial in our later analysis of centralizer subgroups. \n\\end{remark}\n\n\n\n\\section{Intersection of \\texorpdfstring{$\\mathcal{J}^{S^1}_{\\om_\\lambda}$}{J\\hat{}S1} with the strata}\\label{section:intersection}\nIn this section we shall answer the question as to which strata the space of invariant almost complex structures for a given action $(S^2 \\times S^2,\\omega_\\lambda)$ or $(\\CP^2\\# \\overline{\\CP^2}, \\omega_\\lambda)$ intersects.\\\\\n\nIn what follows, we will use the following simple observation several times. Let $(M,\\omega)$ be a simply connected symplectic $4$-manifold. There is a left-exact sequence\n\\begin{equation}\n1 \\to \\Symp_h(M,\\omega) \\to \\Symp(M,\\omega) \\to \\Aut_{c_1,\\omega}\\left(H_2(M,\\mathbb{Z})\\right)\\label{Sequence:ActionOnHomology}\n\\end{equation}\nwhere $\\Symp_h(M,\\omega)$ is the subgroup of symplectomorphisms acting trivially on homology, and where $\\Aut_{c_1,\\omega_\\lambda}\\left(H_2(M,\\mathbb{Z})\\right)$ is the group of automorphisms of $H_2(M,\\mathbb{Z})$ that preserve the intersection product and the Poincar\u00e9 duals of the cohomology classes $c_1(TM)$ and $[\\omega_\\lambda]$. This later group is the identity for $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ with $\\lambda \\geq 1$ and for $(S^2 \\times S^2,\\omega_\\lambda)$ with $\\lambda > 1$. In the case of $(S^2 \\times S^2,\\omega_\\lambda)$ with $\\lambda = 1$, the group $\\Aut_{c_1,\\omega_\\lambda}\\left(H_2(M,\\mathbb{Z})\\right)$ is equal to $\\mathbb{Z}_2$ and is generated by the symplectomorphism that swaps the two $S^2$ factors. Consequently, for any symplectically ruled rational surface, the above sequence is also right-exact and splits.\n\n \n\n\\begin{lemma}\\label{lemma:ActionOnHomology} We have the following equalities among symplectomorphism groups:\n\\begin{itemize}\n\\item $\\Symp(S^2 \\times S^2,\\omega_\\lambda) = \\Symp_h(S^2 \\times S^2,\\omega_\\lambda)\\ltimes\\mathbb{Z}_2$ when $\\lambda=1$,\n\\item $\\Symp(S^2 \\times S^2,\\omega_\\lambda) = \\Symp_h(S^2 \\times S^2,\\omega_\\lambda)$ when $\\lambda>1$, and\n\\item $\\Symp_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda) = \\Symp(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ for all $\\lambda\\geq 1$.\n\\end{itemize}\\qed\n\\end{lemma}\n\n\n\\begin{prop}\\label{prop:ToricExtensionCorrespondance}\nFix a circle action $S^1(a,b;m)$. Then the space of invariant almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects the stratum $U_n$ iff the circle action $S^1(a,b; m)$ extends to the toric action $\\mathbb{T}^2_n$. \n\\end{prop}\n\\begin{proof}\nIf $\\lambda = 1$, the result is immediate as the only toric action is $\\mathbb{T}^2_0$. \n\nLet $\\lambda > 1$ and suppose the action $S^1(a,b;m)$ extends to $\\mathbb{T}^2_n$. This means that there exists a circle action $S^1(c,d; n) \\subset \\mathbb{T}^2_n$ which is equivariantly symplectomorphic to the action $S^1(a,b;m)$ via some  symplectomorphism $\\phi$. Let $J_{n}$ denote the standard complex structure in the stratum $U_n$. By Lemma \\ref{lemma:ActionOnHomology}, when $\\lambda > 1$, the group $\\Symp_h(S^2 \\times S^2,\\omega_\\lambda)$ is equal to $\\Symp(S^2 \\times S^2,\\omega_\\lambda)$ so that $\\phi$ preserves homology. Consequently, $\\phi^*J_{n}$  belongs to the stratum $U_n$ and is invariant with respect to the $S^1(a,b;m)$ action, showing that the space $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects the stratum $U_n$.\n\nConversely, suppose that $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_n$. If $n\\geq 1$, then there exists an invariant, embedded, symplectic sphere $C_{n}$ of self-intersection $-n$. Arguing as in Proposition~\\ref{prop:AtMostTwoExtensions} and Corollaries~\\ref{cor:CircleExtensionsWith_a=1} and~\\ref{cor:CircleExtensionsWith_a=-1}, we conclude that $S^1(a,b;m)$ must extend to the torus $\\mathbb{T}^2_n$. If $n=0$, there there exist invariant spheres representing the homology classes $B$ and $F$ passing to a common fixed point. Again, this implies that $S^1(a,b;m)$ extends to the torus $\\mathbb{T}^2_0$. Alternatively, we can adapt the proofs of Lemma~\\ref{lemma:EvaluationFibrationConfigurations} in the present document and of Proposition~4.7 in~\\cite{Liat} to show that if an invariant $J$ is in the stratum $U_n$ the circle action $S^1(a,b; m)$ extends to the toric action $\\mathbb{T}^2_n$.\n\\end{proof}\n\n\nThe following corollaries immediately follow from Proposition~\\ref{prop:AtMostTwoExtensions} and Corollaries~\\ref{cor:CircleExtensionsWith_a=1} and~\\ref{cor:CircleExtensionsWith_a=-1}.\n\n\\begin{cor}\\label{cor:lambda=1_IntersectingOnlyOneStratum}\nSuppose $\\lambda=1$. Then the space $\\mathcal{J}_{\\omega_1}$ of compatible, almost-complex structures on $(S^2 \\times S^2,\\omega_1)$ is made of only one stratum. In particular, any Hamiltonian circle action extends to the toric action $\\mathbb{T}^2_0$ and the subspace $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ of $S^1$ invariant almost-complex structures is contractible. \\qed\n\\end{cor}\n\n\\begin{cor}\\label{cor:IntersectingOnlyOneStratum}\nSuppose $\\lambda >1$. Consider an $S^1(a,b;m)$ action $(S^2 \\times S^2,\\omega_\\lambda)$ or $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ depending on whether $m$ is even or odd. Under the following numerical conditions on $a,b,m,\\lambda$, the space $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects only the stratum $U_{m}$:\n\\begin{itemize}\n    \\item when $a\\neq\\pm 1$;\n    \\item when $b=0$ or  $b=am$;\n    \\item when $a = \\pm 1$ and $2 \\lambda \\leq |2b-m|+\\epsilon_{m}$.\\qed\n\\end{itemize}\n\\end{cor}\n\n\\begin{cor} \\label{cor:IntersectingTwoStrata}\nConsider the $S^1(\\pm 1,b,m)$ action on $(S^2 \\times S^2,\\omega_\\lambda)$ or $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$, then $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects exactly two strata. More precisely,\n\\begin{enumerate}\n    \\item if $a=1$, $b \\neq\\{0,m\\}$, $\\lambda >1$, and $2\\lambda > |2b-m|+\\epsilon_{m}$, then the space of $S^1(1,b;m)$-equivariant almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects the two strata $U_m$ and $U_{|2b-m|}$.\n    \\item If $a=-1$, $b \\neq\\{0,-m\\}$, $\\lambda >1$, and $2\\lambda > |2b+m|+\\epsilon_{m}$, then the space of $S^1(-1,b;m)$-equivariant almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects the two strata $U_m$ and $U_{|2b+m|}$.\\qed\n\\end{enumerate} \n\\end{cor}\n\n\n\n\n\n\n\\section{Symplectic actions of compact abelian groups on~\\texorpdfstring{$\\mathbb{R}^4$}{R4}}\n\nIn order to study the action of the the equivariant symplectomorphism group $\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ on each invariant stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_k$, we will need to understand the equivariant topology of linearised symplectic  actions. The main result is Theorem~\\ref{thm:EqGr} which is an equivariant version of Gromov's Theorem on the contractibility of the group of compactly supported symplectomorphisms of star-shaped domains of $\\mathbb{R}^{4}$. This relies on Lemma~\\ref{lemma:remove} and Proposition~\\ref{prop:linearize} that were proven by W. Chen in the unpublished manuscript \\cite{ChenUnpub}. For completeness, we shall reproduce their proof here.\n\nLet $G$ be a compact group acting effectively and symplectically  on $ \\mathbb{C}^2 = \\mathbb{R}^4$ with the symplectic form $\\omega_0:= dx_1 \\wedge dy_1 +  (dx_2 \\wedge dy_2)$. We say it acts linearly if it acts as a subgroup of $\\U(2) \\subset \\Sp(4)$.\n\n\\begin{lemma}[Chen]\\label{lemma:remove}\nLet $G$ be a compact group acting linearly on $(\\mathbb{R}^4, \\omega_0)$. Suppose $V$ is a $G$ invariant, compact, and star-shaped neighbourhood of $0$. Let $f: \\mathbb{R}^4 \\setminus V \\to \\mathbb{R}^4$ be a $G$-equivariant symplectic embedding which is the identity near infinity. Then, for every $G$-invariant neighbourhood $W$ of $V$, there exists a $G$-equivariant symplectomorphism $g:\\mathbb{R}^4 \\to \\mathbb{R}^4$ such that $g|_{\\mathbb{R}^4\\setminus W} = f$.  \n\\end{lemma}\n\n\\begin{proof}\nAs $0 \\in int(V)$ and as $f$ is the identity near infinity, there exist $T>0$ such that $f(Tx) = Tx$ for all $x \\in \\mathbb{R}^4 \\setminus~ V$. Define $f_t(x) = \\frac{f(tx)}{t}$ for $1 \\leq t \\leq T$. As the $G$ action is linear, we observe that $f_t$ is equivariant for all $t \\in [1,T]$. By construction, $f_1 = f$, $f_T = id$  and $f_t^*\\omega_0 = \\omega_0$ for all $t$. Thus there are compact sets $V_t = f_t(V)$ and open neighbourhoods $W_t= f_t(W)$ of $V_t$ such that the restriction  $f_t: \\mathbb{R}^4 \\setminus V \\to \\mathbb{R}^4 \\setminus V_t$ and $f_t: \\mathbb{R}^4 \\setminus W \\to \\mathbb{R}^4 \\setminus W_t$  are diffeomorphic. As $G$ acts linearly, each of the sets $W_t$ and $V_t$ are $G$-invariant. \n\nDefine $X_t$ as the vector field that satisfies $\\frac{d}{dt} f_t = X_t \\circ f_t$ and consider the one form $\\alpha_t = i_{X_t}\\omega_0$. As $f_t$ is $G$ equivariant and symplectic, both $X_t$ and $\\alpha_t$ are $G$-invariant. Let $H_t: \\mathbb{R}^4 \\setminus V_t \\to \\mathbb{R}$ be a one parameter family of Hamiltonians that are $G$-invariant and that satisfy $\\alpha_t= dH_t$. Since $f_t$ is the identity near infinity, this implies that $H_t$ is constant near infinity and we can take this constant to be 0. \n\nFinally, we can take a family of $G$-invariant bump functions $\\rho_t: \\mathbb{R}^4 \\to [0,1]$ such that $\\rho_t \\equiv 0$ in a neighbourhood of $V_t$ and $\\rho_t \\equiv 1$ on $\\mathbb{R}^4 \\setminus W_t$. Then the Hamiltonian $\\rho_t H_t: \\mathbb{R}^4 \\to \\mathbb{R}$ is defined on the whole $\\mathbb{R}^4$ and is also $G$-invariant. The Hamiltonian isotopy $g_t$ generated by $\\rho_t H_t$ is $G$ equivariant for all $1\\leq t\\leq T$ and satisfies the properties $g_T = \\id$ and $g_1|_{\\mathbb{R}^4 \\setminus W} = f$. Consequently, $g_1$ is the symplectomorphism $g$ we were looking for.\n\\end{proof}\n\nAny unitary representation of a compact abelian group $G$ on $\\mathbb{C}^2$ induces a splitting into eigenspaces $\\mathbb{C}^2 = \\mathbb{C}_1 \\bigoplus \\mathbb{C}_2$. This simple fact turns out to be essential in our treatment of the equivariant Gromov's Theorem. Consequently, from now on, we assume that the group $G$ is abelian.\n\n\\begin{prop}[Chen]\\label{prop:linearize}\nLet $V\\subset\\mathbb{R}^{4}$ be a compact star-shaped neighbourhood of $\\{0\\}$ and let $\\omega$ be a symplectic form on $\\mathbb{R}^{4}$ such that $\\omega = \\omega_0$ outside some smaller open neighbourhood of the origin $U\\subset V$. Let $G$ be a compact abelian group acting on $(V,\\omega)$ via symplectomorphisms that are linear near the boundary of V. Then the G action is conjugate to a linear symplectic action of $G$ on $(V,\\omega_0)$ by a diffeomorphism $\\Phi$ which is the identity near the boundary and which satisfies $\\Phi^*\\omega = \\omega_0$.\n\\end{prop}\n\n\\begin{proof}\nIdentify $\\mathbb{R}^4$ with $\\mathbb{C}^2$. The linear action near $\\partial V$ extends to a unitary action on $\\mathbb{R}^4$. As $G$ is abelian this linear action splits into two eigenspaces  $\\mathbb{C}_{1} \\oplus \\mathbb{C}_{2}$. We can then compactify $\\mathbb{C}^{2}$ along these eigenspaces to obtain $\\mathbb{C}P^{1}\\times \\mathbb{C}P^{1}$ (diffeomorphic to $S^2 \\times S^2$) equipped with two symplectic forms $\\tilde\\omega$ and $\\tilde\\omega_{0}$ induced from $\\omega$ and $\\omega_{0}$. The compactified space inherits two actions of $G$, namely, $\\rho: G \\hookrightarrow \\Symp(S^2 \\times S^2,\\tilde\\omega)$ coming from extending the $G$ action on $\\mathbb{R}^{4}$ and $\\rho_{\\lin}:G \\hookrightarrow \\Symp(S^2 \\times S^2,\\tilde\\omega)$ which extends the linear action near $\\partial V$.  \n\\\\\n\n\n\nBy construction, there exists a star-shaped subset $V_1 \\subset V$ such that both the action $\\rho$ and $\\rho_{\\lin}$ agree on $\\mathbb{C}^2 \\setminus V_1$. Note that the point at infinity $p:=(\\infty,\\infty)$ is a fixed point for both the action $\\rho$ and $\\rho_{lin}$. Choose a $\\tilde\\omega$ compatible $G$-invariant almost complex structure $J$ on $S^2 \\times S^2$ which is standard in some neighborhood $X_\\epsilon:= (S^2 \\times D_\\epsilon) \\cup (S^2 \\times D_\\epsilon)$ of the wedge $(S^2\\times\\{\\infty\\})\\vee (\\{\\infty\\}\\times S^2)$ (where $D_\\epsilon$ is a small disk of radius $\\epsilon$ at $\\infty$). As $(S^2\\times\\{\\infty\\})$ and $(\\{\\infty\\}\\times S^2)$ are $J$-holomorphic spheres representing the classes $B$ and $F$ respectively, we conclude that $J$ belongs to the stratum $U_{0}$ so that there exists foliations $\\mathcal{B}_J$ and $\\mathcal{F}_J$ by embedded $J$-holomorphic spheres in the classes $B$ and $F$. Given any $q=(z,w) \\in S^2 \\times S^2$, let $u_w$ denote the unique curve in the class B  passing through $(0,w)$ and, similarly, let $v_z$ be the curve in class F  passing through $(z,0)$. We can define a self-map of $S^2 \\times S^2$ by setting\n\\begin{align*}\n\\Psi_J: S^2 \\times S^2 &\\longrightarrow S^2 \\times S^2  \\\\\n(z,w) &\\longmapsto u_w \\cap v_z\n\\end{align*}\nAs explain in~\\cite{McD} Chapter 9, this map $\\Psi_J$ is a diffeomorphism. Moreover, $\\Psi_J$ is equivariant with respect to the linear action $\\rho_{\\lin}$ on the domain  and the action $\\rho$ on the codomain. Furthermore, as $J$ is the standard complex structure in a neighbourhood $X_\\epsilon$ of the wedge, $\\Psi_J$ is the identity near the base point $p$. We now modify $\\Psi_J$ in order to make it the identity in the neighbourhood $X_\\epsilon$. As $J$ is the standard complex structure on $X_\\epsilon$ we can write\n\\[\n\\Psi_J (z,w)= \n\\begin{cases} \n(z,\\phi_2(z,w)) & \\text{~if~} z \\in    D_\\epsilon \\subset S^2 \\times S^2\\\\\n(\\phi_1(z,w),w) & \\text{~if~} w \\in   D_\\epsilon \\subset S^2 \\times S^2\n\\end{cases}\n\\]\nwhere $\\phi_1(z,0)=z$ and $\\phi_2(0,w)=w$ for all $z,w \\in S^2$. Choose a $G$ equivariant (for the $G$ action on $\\{\\infty\\}\\times S^2)$) smooth map $\\beta_1:S^2 \\longrightarrow S^2$ such that $\\beta_1(z) = z$ for all $z \\in D_\\epsilon$ and $\\beta_1 =\\infty$ in a neighbourhood $D_\\delta$ contained in $D_\\epsilon$ and such that $\\det(d\\beta_{1}(z)) \\geq 0 ~\\forall ~z \\in S^2$.\nSimilarly define a $G$ equivariant map (for the $G$ action on $S^2\\times\\{\\infty\\}$) $\\beta_2: S^2 \\to S^2$ satisfying analogous conditions as $\\beta_1$. Then we define a modification of $\\Psi_{J}$ by setting\n\\[\n\\Psi^{\\prime}_J(z,w) = \n\\begin{cases}\n\\Psi_J   &\\text{~if~}  z \\in  (S^2 \\times S^2) \\setminus X_\\epsilon \\\\\n(z,(\\phi_2(\\beta_1(z),w)) &\\text{~if~} z \\in D_\\epsilon \\\\\n(\\phi_1(z,\\beta_2(w)),w) &\\text{~if~}  w \\in D_\\epsilon\n\\end{cases}\n\\]\nThis modification $\\Psi^{\\prime}_J$ is the identity in a smaller neighbourhood $X_\\delta:=(S^2 \\times D_\\delta) \\cup (S^2 \\times D_\\delta)\\subset X_{\\epsilon} $.\\\\\n\nThe submanifolds $\\{z\\} \\times S^2$ and $S^2 \\times \\{w\\}$ for all $z,w \\in S^2$ are symplectic for the form ${\\Psi'_{\\!J}}^* \\tilde\\omega $ and hence $\\tilde\\omega_0 \\wedge {\\Psi'_{\\!J}}^* \\tilde\\omega>0$. Thus the path $\\omega_t:= t\\tilde\\omega +(1-t){\\Psi'_{\\!J}}^* \\tilde\\omega$ is a path of non-degenerate symplectic forms for all $t \\in [0,1]$. We can then apply an equivariant Moser isotopy to get an equivariant diffeomorphism $\\alpha$ of $S^2 \\times S^2$ such that $\\alpha^* {\\Psi'_{\\!J}}^* \\tilde\\omega = \\tilde\\omega_0$. Further, as ${\\Psi'_{\\!J}}^* \\tilde\\omega= \\tilde \\omega_0$ on $X_\\delta$, the restriction of $\\alpha$ to $X_\\delta$ is the identity. We then set $\\tilde \\Psi_J:= \\Psi'_{\\!J} \\circ \\alpha$.\n\\\\\n\nThe restriction of $\\tilde\\Psi_J: \\mathbb{C}^2 \\to \\mathbb{C}^2$ gives us a map which is $G$-equivariant with respect to the action $\\rho_{\\lin}$ on the domain and $\\rho$ on the codomain. As noted before there exists a star-shaped subset $V_1 \\subset V$ such that both the action $\\rho$ and $\\rho_{\\lin}$ agree on $\\mathbb{C}^2 \\setminus V_1$. We can choose $V_1$ such that $0 \\in int(V_1)$. We now apply Theorem \\ref{lemma:remove} to the map $f:={\\tilde\\Psi_J}^{-1}|_{\\mathbb{C}^2 \\setminus V_1}$ and we choose W in Theorem \\ref{lemma:remove} to be a $G$-invariant open subset of $V$ which contains $V_1$. Let $g:\\mathbb{C}^2 \\to \\mathbb{C}^2$ be an equivariant symplectomorphism as in Theorem \\ref{lemma:remove} such that $g|_{\\mathbb{R}^4\\setminus W} = f$, then the map $\\Phi:= \\left(\\tilde\\Psi_J \\circ g\\right)\\Big|_V: V \\to V$ is identity near the boundary and satisfies $\\Phi^*\\omega = \\omega_0$ and is\n$G$-equivariant where the action of G on the domain of $\\Phi$ is the linear action $\\rho_{\\lin}$ while on the range of $\\Phi$ it is the $G$ action $\\rho$ on V that we started out with. \nThus $\\Phi$ is the required equivariant symplectomorphism that linearizes the given $G$ action and takes the form $\\omega$ to $\\omega_0$.\n\\end{proof}\n\nConsider a polydisk $D^2 \\times D^2$ whose product structure is compatible with the eigenspace decomposition $\\mathbb{R}^{4}=\\mathbb{C}^{2}=\\mathbb{C}_{1}\\oplus \\mathbb{C}_{2}$. Consider the symplectic form $\\omega$ such that $\\omega = \\omega_0$ outside of some smaller polydisk of the form  $D_r \\times D_r \\subset D^2 \\times D^2$  for some radius $r$. \n\n\\begin{thm}[Equivariant Gromov's Theorem for Polydisks]\\label{thm:EquivariantGromovForPolydisks}\nLet $G$ be an abelian group. Let $\\omega$ be a symplectic form on $D^2 \\times D^2$ which is equal to $\\omega_0$ near the boundary. Let $G$ act symplectically on $(D^2 \\times D^2 ,\\omega)$ and suppose the action is linear near the boundary. Then the group $\\Symp_c^{G}(D^2\\times D^2, \\omega)$ of equivariant symplectomorphisms that are equal to the identity near the boundary of $D^2 \\times D^2$ is non-empty and contractible.\n\\end{thm}\n\\begin{proof}\nAs the $G$ action outside of $D_r \\times D_r \\subset \\mathbb{R}^4$  is linear, we can extend this $G$ action to the whole of $\\mathbb{R}^4$. Then we can compactify each eigenspace of $\\mathbb{C}$ to an $S^2$ and hence this $G$ action extends to a symplectic action $S^2 \\times S^2$ with respect to the form $\\tilde\\omega$ induced by $\\omega$. \n\\\\\n\nBy Proposition~\\ref{prop:linearize} we can conjugate our $G$ action by a symplectomorphism which is identity near the boundary to get a linear $G$ action on the whole of $V$. As any two conjugate topological subgroups are  homeomorphic, we shall just study the homotopy type of the compactly supported equivariant symplectomorphism group $\\Symp_{c,\\lin}^G(D^2 \\times D^2,\\omega)$ for the linear $G$ action on $V$.\n\\\\\n\nLet $\\mathcal{J}^G_{\\omega}$  be the non-empty and contractible space of all equivariant almost complex structures on $D^2 \\times D^2$ that are compatible with $\\omega$ and are the standard split almost complex structure $J_0$ outside of $D_r \\times D_r$. As they are the standard almost complex structures outside of a neighbourhood these almost complex structure extend to $S^2 \\times S^2$ and are compatible with $\\tilde\\omega_0$. Further once we pick a base point $p=(\\infty,\\infty)\\in S^2\\times S^2$ and identify $D^2\\times D^2$ with the complement of a standard neighborhood $X_\\epsilon:= (S^2 \\times D_\\epsilon) \\cup (S^2 \\times D_\\epsilon)$  of the wedge $(S^2\\times\\{\\infty\\})\\vee (\\{\\infty\\}\\times S^2)\\subset S^2\\times S^2$ (note that the wedge point $(\\infty,\\infty)$ is a fixed point for the extended action of G on $S^2 \\times S^2$),  then any element $J\\in  \\mathcal{J}^G_{\\sigma}$ extends to a equivariant almost complex structure of $S^2 \\times S^2$ which is the standard product complex structure on $S^2 \\times S^2$ on a neighbourhood $X_\\epsilon$ of the wedge $(S^2\\times\\{y\\})\\vee (\\{x\\}\\times S^2)\\subset S^2\\times S^2$. Conversely any such equivariant almost complex structure compatible with $\\tilde\\omega$ that is  standard in some neighbourhood of the wedge $(S^2\\times\\{\\infty\\})\\vee (\\{\\infty\\}\\times S^2)\\subset S^2\\times S^2$ gives us an element of $\\mathcal{J}^G_{\\omega}$ .\n\\\\\n\nIn order to show that $\\Symp_{c,\\lin}^G(D^2 \\times D^2,\\omega)$ is contractible, we shall prove that it is homotopy equivalent to the contractible space $\\mathcal{J}^G_{\\omega}$. \n\\\\\n\n\n\n\n\nDefine the map $\\tilde \\Psi_J$ as in the proof of Proposition~\\ref{prop:linearize}. Thus we have a map \n\\begin{align*}\n\\tau: \\mathcal{J}^G_\\omega  &\\longrightarrow \\Symp_{c,\\lin}^G(D^2 \\times D^2,\\omega) \\\\ \nJ &\\longmapsto \\tilde\\Psi_J \n\\end{align*}\n\nTo prove that $\\tau$ is a\nhomotopy equivalence we construct a homotopy inverse as follows.  Fix a $J' \\in \\mathcal{J}^G_\\omega$, then the homotopy inverse $\\beta$ is defined as,\n\n\\begin{align*}\n\\beta: \\Symp_{c,\\lin}^G(D^2 \\times D^2,\\omega)  &\\longrightarrow \\mathcal{J}^G_\\omega \\\\ \n\\phi &\\longmapsto \\phi_{*}J'\n\\end{align*}\n\nBy construction we see that $\\tau(\\beta(\\phi)) = \\id$ and the other direction is homotopic to the identity as $J^G_\\omega$ is contractible.\n\\end{proof}\n\nWe shall repeatedly use  the following theorem in our analysis of the homotopy type of the equivariant symplectomorphism groups of $S^2 \\times S^2$.\n\n\n\\begin{thm}(Equivariant Gromov's Theorem)\\label{thm:EqGr}\nLet $(V,\\omega)$ be an  compact star-shaped symplectic domain of $\\mathbb{R}^4$ such that $0 \\in int(V)$ and let $\\omega$ be such that $\\omega = \\omega_0$ near the boundary of $V$. Let $G$ be a compact abelian group that acts symplectically and linearly near the boundary and that sends the boundary to itself, then the space of equivariant symplectomorphisms that act as identity near the boundary (denoted by $\\Symp_c^G(V,\\omega)$) is non-empty and contractible.\n\\end{thm}\n\n\\begin{proof}\nBy Proposition~\\ref{prop:linearize} we can conjugate our $G$ action by a symplectomorphism which is identity near the boundary to get a linear $G$ action on the whole of $V$ and such that it takes the form $\\omega$ to $\\omega_0$. As the homotopy type of the two conjugate equivariant symplectomorphism group is the same (they are in fact homeomorphic), we shall just study the homotopy type of the compactly supported equivariant symplectomorphism group  for the linear $G$ action on $(V,\\omega_0)$. We denote this group by $\\Symp_{c,\\lin}^G(V,\\omega_0)$.\n\\\\\n\nChoose real numbers $r>0$ and $T>1$ , $D_r \\times D_r$ is a polydisk of radius $r$, such that $\\frac{1}{T}V \\subset D_r \\times D_r \\subset int(V)$,\nand consider the family of maps $F_t : \\Symp_{c,\\lin}^G(V,\\omega_0) \\to \\Symp_{c,\\lin}^G(V,\\omega_0)$ for $1 \\leq t \\leq T$ \ndefined by $F_t(\\phi)(x) = \\frac{\\phi(tx)}{t} $ for all $x \\in V$.\n\\\\\n\nThen we have that $F_t$ is $G$ equivariant for all $1 \\leq t \\leq T$, $F_1(\\phi) = \\phi$ for all $\\phi \\in \\Symp_{c,\\lin}^G(V,\\omega_0)$,  $F_t(id) = id$ for all t, and $F_T \\left(\\Symp_{c,\\lin}^G(V,\\omega_0)\\right) \\subset \\Symp_{c,\\lin}^G(D_r \\times D_r,\\omega)$.\n\\\\\n\nThe proof of Theorem \\ref{thm:EquivariantGromovForPolydisks} tells us that the inclusion $i:\\Symp_{c,\\lin}^G(D_r \\times D_r,\\omega) \\hookrightarrow \\Symp_{c,\\lin}^G(V,\\omega_0)$ is contractible. Hence we can fix a contraction $\\alpha_t$ for $T \\leq t \\leq T+1$ such that $\\alpha_T = i$ and $\\alpha_{T+1}(\\phi) = id$ for all $\\phi \\in \\Symp_{c,\\lin}^G(D_r \\times D_r,\\omega)$. Then the concatenation \n$$\\Tilde{F_t}:=\\begin{cases}\nF_t ~~1 \\leq t\\leq T\\\\\nF_T \\circ \\alpha_t ~~ T\\leq t \\leq T+1\n\\end{cases}$$ \ngives  us a retraction of  $\\Symp_{c,\\lin}^G(V,\\omega_0)$  to $id$ and hence $\\Symp_{c}^G(V,\\omega)$ is contractible.\n\\end{proof}\n\n\\begin{remark}\nTo our knowledge, it is not known whether an equivariant version of Gromov's Theorem on holds true for non-abelian compact groups.\n\\end{remark}\n\n\\section{Homotopical description of \\texorpdfstring{$\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_k$}{J\\hat{}S1\\_k}} \nWe now consider the action of the group of equivariant symplectomorphisms $\\Symp_{h}^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ on the contractible space $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ of invariant, compatible, almost-complex structures, and we investigate the orbit-type stratification of this action up to homotopy. In the case of the nontrivial bundle $\\CP^2\\# \\overline{\\CP^2}$, the analysis of the action of the centralizer on the stratification will be postponed to Section~\\ref{Chapter-CCC}.\n\n\n\\subsection{Notation}\\label{not}\n\n\nWe shall use the following notation in the rest of the document. Let $M$ denote the manifolds $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$. Let $G$ be a compact abelian group acting symplectically on $(M,\\omega_\\lambda)$. Let $p_0$ be a fixed point for the group action. Given a $G$ invariant symplectic  curve C, and a $\\omega_\\lambda$ orthogonal $G$ invariant sphere $\\overline{F}$ in the homology class $F$ that intersects C at a point $p_0$, we define the following spaces: \n\n\\begin{itemize}\n\\item $N(C)$:= The symplectic normal bundle to a symplectic submanifold $C$. \n\\item $\\Symp^{G}_h(M,\\omega_\\lambda)$ := The group of $G$ equivariant symplectomorphisms on $(M,\\omega_\\lambda)$ that acts trivially on homology.\n    \\item $\\Stab^{G}(C)$ := The group of all  $\\phi \\in \\Symp^{G}_h(M,\\omega_\\lambda)$ such that $\\phi(C) = C$, that is, such that $\\phi$ \\emph{stabilises} $C$ but does not necessarily act as the identity on $C$.\n    \\item $\\Fix^{G}(C)$ := The group of all  $\\phi \\in \\Symp^{G}_h(M,\\omega_\\lambda)$ such that $\\phi|_C = id$, that is, such that \\emph{fixes $C$ pointwise}. \n    \\item $\\Fix^{G}(N(C))$:= The group of all $\\phi \\in \\Symp^{G}_h(M,\\omega_\\lambda)$ such that $\\phi|_C = \\id$ and  $d\\phi|_{N(C)}: N(C) \\to N(C)$ is the identity on $N(C)$.\n    \\item $\\Gauge1^{G}(N(C))$:= The group of $G$-equivariant symplectic gauge automorphisms of the symplectic normal bundle of $C$.\n    \\item $\\Gauge1^{G}(N(C \\vee \\overline{F}))$ := The group of $G$-equivariant symplectic gauge automorphism of the symplectic normal bundle of the crossing divisor $C \\vee \\overline{F}$ that are identity in a neighbourhood of the wedge point. \n    \\item $\\mathcal{S}^{G}_{K}$ := The space of unparametrized $G$-invariant symplectic embedded spheres in the homology class $K$. \n    \\item $\\mathcal{S}^{G}_{K,p_0}$:= The space of unparametrized $G$-invariant symplectic embedded spheres in the homology class $K$ passing through $p_0$.\n    \\item $\\mathcal{J}_{\\omega_\\lambda}^{G}(C)$ := The space of $G$-equivariant $\\omega_\\lambda$ compatible almost complex structures s.t the curve $C$ is holomorphic.\n    \\item $\\Symp^{G}(C)$:= The space of all $G$-equivariant symplectomorphisms of the curve C.\n    \\item $\\Fix^{G}(N(C \\vee {\\overline{F}}))$ := The space of all $G$-equivariant symplectomorphisms that are the identity in the neighbourhood of $C \\vee {\\overline{F}}$.\n    \\item $\\Symp^{S^1}({\\overline{F}}, N(p_0))$ := equivariant symplectomorphism of the sphere $\\overline{F}$ that are the identity in an open set of $\\overline{F}$ around $p_0$. \n    \\item $\\overline{\\mathcal{S}^{G}_{F,p_0}}$:= The space of unparametrized $G$-invariant symplectic spheres in the homology class $F$ that are equal to a fixed curve ${\\overline{F}}$ in a neighbourhood of $p_0$.\n    \\item $\\Symp^{G}_{p_0,h}(M,\\omega_\\lambda)$:=  The group  of all  $\\phi \\in \\Symp^{G}_{h}(M,\\omega_\\lambda)$ fixing $p_0$.\n    \\item $\\Stab^{G}_{p_0}(C)$:=  The group of all  $\\phi \\in \\Stab^G(C)$ such that $\\phi(p_0) = p_0$.  \n\\end{itemize}\n\nAll the above spaces are equipped with the $C^\\infty$ topology.\n\n\\subsection{Case 1: \\texorpdfstring{$\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$}{Symp(S2xS2)} action on \\texorpdfstring{$\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$}{J\\hat{}S1\\_2k} with \\texorpdfstring{$s \\neq 0$}{s>0}}\\label{section:ActionOnU_2k}\n\nLet $\\lambda > 1$ and consider a $S^1$-action $S^1(a,b;m)$ on $(S^2 \\times S^2,\\omega_\\lambda)$ with $m=2k$. Let  ${\\mathcal{S}^{S^1}_{D_{2s}}}$ denote the space of all $S^1$ invariant symplectic embedded spheres in the class $D_{2s}=B-sF$. We shall assume that ${\\mathcal{S}^{S^1}_{D_{2s}}}$ is non-empty which, by Theorems~\\ref{cor:IntersectingOnlyOneStratum} and~\\ref{cor:IntersectingTwoStrata}, means that $2s=m$ or $2s=|2b\\pm m|$ depending on $a$. \n\nWe first show that ${\\mathcal{S}^{S^1}_{D_{2s}}}$ is a homogeneous space under the natural action of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$. To this end, we observe that among the two fixed points on the curve in class $D_{2s}$, Table~\\ref{table_weights} shows that at least one fixed point, say $p$, has weights $(w_1,w_2)$ with $w_1\\neq w_2$. It follows that if an invariant curve in the class of the fiber $F$ intersects a curve in the class $D_{2s}$ at $p$, then the two curves intersect $\\omega_\\lambda$-orthogonally. Let $\\mathcal{C}(D_{2s}\\vee F,p)^{S^1}$ be the space of invariant configurations made of curves in classes $D_{2s}$ and $F$ intersecting orthogonally at $p$. \n\n\\begin{lemma}\\label{lemma:EvaluationFibrationConfigurations} \nThe evaluation map at a standard configuration $\\overline{D}_{2s}\\vee \\overline{F}$ through $p$\n\\[\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda) \\twoheadrightarrow \\mathcal{C}(D_{2s}\\vee F,p)^{S^1}\\]\nis a serre fibration.\n\\end{lemma}\n\\begin{proof}\nWe first show that the action is transitive. Given an invariant configuration $C\\vee A \\in \\overline{D}_{2s}\\vee \\overline{F}$, \nthe equivariant symplectic neighbourhood theorem implies that we can find an invariant neighbourhood V of $C \\vee A$, an invariant neighbourhood $V'$ of $\\overline{D} \\cup \\overline{F}$, and an equivariant symplectomorphism $\\alpha:V \\to V'$. We claim that $\\alpha$ can be extended to an ambient diffeomorphism $\\beta$ of $S^2 \\times S^2$. Assume this for the moment. By construction, the pullback form $\\omega_\\beta:=\\beta^*\\omega_\\lambda$ is invariant under the the conjugate action $\\beta^{-1}\\rho\\beta$.\n\nWe observe that the complement of the standard configuration $\\overline{D} \\cup \\overline{F}$ in $W_{2s}$ is symplectomorphic to $\\mathbb{C}^2$ with the symplectic form $\\omega_f:=\\frac{1}{2\\pi}\\partial\\bar\\partial f$ where $f=\\log\\left(\\left(1+||w||^2\\right)^{\\lambda}\\left(1+||w||^{4k}+||z||^2\\right)\\right)$, see~\\cite[Lemma~3.5]{abreu}. Under this identification, the standard $\\mathbb{T}^2_{2s}$ action on $W_{2s}$ become linear on $\\mathbb{C}^2$. It follows that near infinity, the form $\\omega_\\beta$ is equal to $\\omega_f$ and that the action $\\beta^{-1}\\rho\\beta$ is linear. By Proposition~\\ref{prop:linearize} we get that there is an equivariant symplectomorphism $\\gamma$ that is equal to the identity near infinity, and that identifies $(\\mathbb{C}^2,\\omega_f, \\rho)$ with $(\\mathbb{C}^2,\\beta^*\\omega_\\lambda,\\beta^{-1}\\rho\\beta)$. By construction, the equivariant symplectomorphism $\\phi := \\gamma \\circ \\beta$ takes the configuration $C\\vee A$ to $\\overline{D}\\vee\\overline{F}$. \n\nIt remains to see that the local diffeomorphism $\\alpha$ can be extended to an ambient diffeomorphism $\\beta$ of $S^2 \\times S^2$. By the Isotopy Extension Theorem (see~\\cite[Theorem~1.4, p.180]{Hi}), it suffices to show that any two configurations of embedded spheres in classes $\\overline{D}\\cup F$ intersecting transversely and positively are isotopic. In turns, this follows from the fact that any two $F$-foliations corresponding to two almost-complex structures $J$ and $J'$ are diffeotopic, and that any two sections of the product $S^2 \\times S^2$ are diffeotopic through sections iff they are homotopic. This shows that $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ acts transitively on $\\mathcal{C}(D_{2s}\\vee F,p)^{S^1}$.\n\nTo prove the homotopy lifting property, consider any family of maps $\\gamma: D^n \\times [0,1] \\rightarrow \\mathcal{C}(D_{2s}\\vee F,p)^{S^1}$ from a $n$ dimensional disk $D^n$ to $\\mathcal{C}(D_{2s}\\vee F,p)^{S^1}$, and choose a lift $\\overline{\\gamma_0}:D^n \\rightarrow \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ of $\\gamma_0$. Since the complement of a configuration is contractible, the equivariant version of Banyaga's Extension Theorem for families implies that there exists a lift $\\overline{\\gamma}: D^n \\times [0,1] \\rightarrow \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda) $ extending $\\overline{\\gamma_0}$. \n\\[\n\\begin{tikzcd}\nD^n \\times \\{0\\} \\arrow[d,hookrightarrow] \\arrow[r,\"\\overline{\\gamma_0}\"]    &\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda) \\arrow[d,\"\\theta\"] \\\\\n    D^n \\times [0,1]\\arrow[r,\"\\gamma\"] \\arrow[ur,dashrightarrow,\"\\exists ~ \\overline{\\gamma}\"] &\\mathcal{C}(D_{2s}\\vee F,p)^{S^1}\n\\end{tikzcd}\n\\]\nAlternatively, one can apply the equivariant Gromov-Auroux Lemma~\\ref{Au} to show the existence of the lift  $\\overline{\\gamma}$. In both cases, this concludes the proof.\n\\end{proof}\n\n\\begin{cor}\\label{trans} \nFix the action $S^1(a,b;m)$. Then $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ acts transitively on ${\\mathcal{S}^{S^1}_{D_{2s}}}$.\\qed\n\\end{cor}\n\n\n\n\n\\begin{lemma}\\label{first}\nLet $\\overline{D}$ be an invariant symplectic sphere in the class $B - sF$ for which $\\mathcal{S}^{S^1}_{D_{2s}}$ is nonempty. Then the evaluation map\n\\begin{align*}\n    \\theta: \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda) &\\twoheadrightarrow {\\mathcal{S}^{S^1}_{D_{2s}}} \\\\\n    \\phi &\\mapsto \\phi(\\overline{D})\n\\end{align*}\nis a Serre fibration with fibre over $\\overline{D}$ given by\n\\[\\Stab(\\overline{D}):= \\left\\{ \\phi \\in \\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)~|~ \\phi(\\overline{D}) = \\overline{D}\\right\\}\\]\n\\end{lemma}\n\\begin{proof}\nThe evaluation map is transitive by Corollary~\\ref{trans}. The homotopy lifting property follows from Lemma~\\ref{Au} as in the proof of Lemma~\\ref{lemma:EvaluationFibrationConfigurations}. Alternatively, one can also note that the action map factors through the restriction map \n\\[\\mathcal{C}(D_{2s}\\vee F,p)^{S^1}\\to\\mathcal{S}^{S^1}_{D_{2s}}\\]\nwhich is itself a fibration. To see this, note that the restriction maps fits into a commuting diagram\n\\[\n\\begin{tikzcd}\n\\mathcal{J}^{S^1}_{\\om_\\lambda}\\arrow[r,\"f_1\"]\\arrow[rd,\"f_2\"] & \\mathcal{C}(D_{2s}\\vee F,p)^{S^1} \\arrow[d]\\\\\n& \\mathcal{S}^{S^1}_{D_{2s}}\n\\end{tikzcd}\n\\]\nwhere the maps $f_1$ and $f_2$ are fibrations. Observe that the map $f_1$ is well defined because the weights at the chosen fixed point $p$ are not equal. Hence, for any choice of invariant almost complex structure $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda}$, the unique invariant $J$-holomorphic curve $C$ in class $D_{2s}$ intersects the unique invariant $J$-holomorphic fiber through $\\omega_\\lambda$-orthogonally at~$p$. \n\\end{proof}\n\n\n\n\n\n\\begin{remark}\nAs both $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ and $\\mathcal{S}^{S^1}_{D_{2s}}$ can be shown to be CW-complexes, we see from Theorem 1 in \\cite{Serrefib} (with proof corrected in \\cite{error}), that a Serre fibration in which the total space and base space are both CW complexes is necessarily a Hurewicz fibration. Thus the map $\\theta: \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda) \\twoheadrightarrow {\\mathcal{S}^{S^1}_{D_{2s}}}$ is in fact a Hurewicz fibration and hence the fibre over any arbitrary $D \\in \\mathcal{S}^{S^1}_{D_{2s}}$ is homotopy equivalent to $\\Stab(\\overline{D})$. \n\\end{remark}\n\nThe homotopy type of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ is related to the strata $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$ through the following sequence of fibrations. We use the symbol ``$\\simeq$\" to mean ``weakly homotopy equivalent\" throughout the rest of the document. In the fibrations below, we use the notation established in Section~\\ref{not}.\n\n\n\\[\\Stab^{S^1}(\\overline{D}) \\longrightarrow \\Symp^{S^1}_{h,p_0}(S^2 \\times S^2,\\omega_\\lambda) \\longtwoheadrightarrow {\\mathcal{S}^{S^1}_{D_{2s}}} \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}\\rule{0em}{2em}\\]\n\n\\[\\Fix^{S^1}(\\overline{D}) \\longrightarrow \\Stab^{S^1}(\\overline{D}) \\longtwoheadrightarrow  \\Symp^{S^1}(\\overline{D}) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} S^1 ~\\text{or}~ \\SO(3)\\rule{0em}{2em}\\]\n\n\\[\\Fix^{S^1} (N(\\overline{D})) \\longrightarrow \\Fix^{S^1}(\\overline{D}) \\longtwoheadrightarrow  \\Gauge1^{S^1}(N(\\overline{D})) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} S^1\\rule{0em}{2em}\\]\n \n\\[\\Stab^{S^1}(\\overline{F}) \\cap \\Fix^{S^1}(N(\\overline{D})) \\longrightarrow \\Fix^{S^1}(N(\\overline{D})) \\longtwoheadrightarrow  \\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\mathcal{J}^{S^1}(\\overline{D})\\simeq \\{*\\}\\rule{0em}{2em}\\]\n \n\\[\\Fix^{S^1}(\\overline{F})  \\longrightarrow \\Stab^{S^1}(\\overline{F}) \\cap \\Fix^{S^1}(N(\\overline{D})) \\longtwoheadrightarrow  \\Symp^{S^1}(\\overline{F}, N(p_0))  \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\left\\{*\\right\\}\\rule{0em}{2em}\\]\n\n\\[\\left\\{*\\right\\} \\mathbin{\\textcolor{blue}{\\xleftarrow{\\text{~~~$\\simeq$~~~}}}} \\Fix^{S^1}(N(\\overline{D} \\vee \\overline{F})) \\longrightarrow \\Fix^{S^1}(\\overline{F}) \\longtwoheadrightarrow  \\Gauge1^{S^1}(N(\\overline{D} \\vee \\overline{F}))  \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\left\\{*\\right\\}\\rule{0em}{2em}\\]\nHere $\\overline{F}$ and $\\overline{D}$ are the $\\omega_\\lambda$-orthogonally intersecting invariant curves in the $2s^{\\,\\text{th}}$-Hirzebruch surface $W_{2s}$ and whose moment map images are depicted below in red. We denote by $\\overline{F}$ the unique curve in class $F$ intersecting the given curve $\\overline{D} \\in {\\mathcal{S}^{S^1}_{D_{2s}}}$ $\\omega_\\lambda$-orthogonally at $p_0$. In the second fibration, the group $\\Symp^{S^1}(\\overline{D})$ is homotopy equivalent to $\\SO(3)$ when the $S^1$ action fixes the curve $\\overline{D}$ pointwise. Otherwise, it is homotopy equivalent to~$S^1$. \n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}\n\\draw[red] (0,1) -- (3,1) ;\n\\draw (0,1) -- (0,0) ;\n\\draw (0,0) -- (4,0) ;\n\\draw[red] (3,1) -- (4,0) ;\n\\node[above] at (1.5, 1.0) {$\\overline{D}$};\n\\node[right] at (3.55, 0.75) {$\\overline{F}$};\n\\node[above] at (3,1) {$p_0$};\n\\end{tikzpicture}\n\\caption{The isolated fixed point $p_0$}\\label{fig:IsolatedFixedPoint}\n\\end{figure}\n\n\nAssuming  the homotopy equivalence in the first fibration, we immediately get\n\\begin{thm}\nConsider the $S^1(a,b;m)$ action on $(S^2 \\times S^2,\\omega_\\lambda)$ with $\\lambda >1$. If $ \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s} \\neq \\phi$, then $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\Stab^{S^1}(\\overline{D})  \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$.\n\\end{thm}\nFurthermore, tracking down the various homotopy equivalences in the other fibrations, we will prove that the equivariant stabilizer of the curve $\\overline{D}$, namely  $\\Stab^{S^1}(\\overline{D})$, is homotopy equivalent to the equivariant stabilizer of the corresponding complex structure under the natural action of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$. More precisely, \n\\begin{itemize}\n    \\item $\\Stab^{S^1}(\\overline{D}) \\simeq \\mathbb{T}^2_{2s}$ when $(a,b) \\neq (0,\\pm1)$;\n    \\item  $\\Stab^{S^1}(\\overline{D}) \\simeq SO(3) \\times S^1$ when $(a,b) = (0,\\pm1)$.\n\\end{itemize}\n \n\n\nWe shall now justify each of the homotopy equivalences in the above fibrations. \n\n\n\n\nNow that we know that the action map\n\\begin{align*}\n \\Stab^{S^1}(\\overline{D}) \\to  \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda) &\\twoheadrightarrow {\\mathcal{S}^{S^1}_{D_{2s}}}\n\\end{align*}\nis a fibration, we show that ${\\mathcal{S}^{S^1}_{D_{2s}}}$ is weakly homotopic to $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$. \n\\begin{lemma}\\label{first2}\nThe natural map $\\alpha: \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s} \\to {\\mathcal{S}^{S^1}_{D_{2s}}}$ defined by sending an almost complex structure $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$ to the unique $J$-holomorphic curve in class $D_{2s}$ is a weak homotopic equivalence. \n\\end{lemma}\n\\begin{proof}\nTo show that $\\alpha$ is a weak homotopy equivalence, we first show that the map is Serre fibration.  To do so, consider an arbitrary element ${D} \\in {\\mathcal{S}^{S^1}_{D_{2s}}}$. As in the proof of Lemma \\ref{first}, it suffices to show that given a family of map $\\gamma_t$ from a n-dimensional disk $D^n$, such that $\\gamma_0(0) = {D}$, and a lift $g_0^\\prime:D^n: \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$ lifting $\\gamma_0$, then there exists a lift $\\gamma_t^\\prime$ to $ \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$.\n\\[\n\\begin{tikzcd}\n    D^n \\times \\{0\\} \\arrow[d,hookrightarrow] \\arrow[r,\"\\gamma_0^\\prime\"] &\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s} \\arrow[d,\"\\alpha\"] \\\\\n    D^n \\times [0,1] \\arrow[r,\"\\gamma\"] \\arrow[ur,dashrightarrow,\"\\exists ~ \\gamma^\\prime\"] &\\mathcal{S}^{S^1}_{D_{2s}}\n\\end{tikzcd}\n\\]\nAs in the proof of Lemma \\ref{first}, we have that there exists a lift $\\overline{\\gamma}: D^n \\times [0,1] \\to \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ of $\\gamma$. Pick an element $ J \\in \\alpha^{-1}({D})$ and define $\\gamma^\\prime(s) := \\overline{\\gamma}^*J$. This defines a lift $\\gamma^\\prime$ of $\\gamma$. Hence $\\alpha$ is a fibration, with fibre begin contractible. Thus we get the required result that $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s} \\simeq {\\mathcal{S}^{S^1}_{D_{2s}}}$.\n\\end{proof}\n\n\n\n\n\n\n\\begin{lemma}{\\label{stab}}\nThe restriction map $\\Stab^{S^1}(\\overline{D}) \\twoheadrightarrow \\Symp^{S^1}(\\overline{D})$ is a fibration.  \n\\end{lemma}\n\n\\begin{proof}\nTo show that the restriction map is a fibration we use Theorem \\ref{palais}, in which we set $X= \\Symp^{S^1}(\\overline{D})$, $G=\\Stab^{S^1}(\\overline{D})$ and the action  is given by \n\\begin{align*}\n    G \\times X &\\to X \\\\\n    (\\phi, \\psi) &\\to \\phi|_{\\overline{D}} \\circ \\psi\n\\end{align*}\nHence in order to show that the restriction map $r :\\Stab^{S^1}(\\overline{D}) \\to \\Symp^{S^1}(\\overline{D})$ is a fibration, we only need to show that the action described above  admits local cross sections. Suppose we only  show that a neighbourhood of identity admits local cross sections and that  $\\Stab^{S^1}(\\overline{D})$ acts transitively on $\\Symp^{S^1}(\\overline{D})$ this would suffice to show that $r$ is a fibration as by Theorem \\ref{palais}, its a local fibration near the identity and the map $r$ is equivariant with respect to the action of $\\Stab^{S^1}(\\overline{D})$, thus completing the proof.\n\\\\\n\nConsider the identity $\\id \\in \\Symp^{S^1}(\\overline{D})$. As Let $\\alpha:N(\\overline{D}) \\to U$ be a equivariant diffeomorphism between the symplectic normal bundle $N(\\overline{D})$ and a neighbourhood $U$ of $\\overline{D}$. As $\\Symp^{S^1}(\\overline{D})$ is locally contractible (this can be seen for example by noticing that the proof of Prop 3.3.14 in \\cite{MS} can be made equivariant) we can find a neighbourhood V of $\\id$, and a fixed retraction $\\beta_t$ of the neighbourhood $V$ onto the identity. Hence given any $\\psi \\in \\Symp^{S^1}(\\overline{D})$, we get a one parameter family $\\beta_t(\\psi)$ of symplectomorphisms.  As $\\pi_1(\\overline{D}) = 0$, $\\beta_t(\\psi)$ is Hamiltonian and is generated by  a function $H_t$. Let $\\pi:N(\\overline{D}) \\to \\overline{D}$ be the projection of the normal bundle. Define $\\Tilde{H_t}:= \\alpha \\circ \\pi^*H_t$. Thus $\\Tilde{H_t}$  defines an invariant function on a U. Fix an invariant bump function $\\rho$ with support in $U$ and is 1 in a small neighbourhood around $\\overline{D}$, then $\\rho \\Tilde{H_t}$ is an invariant function and the corresponding symplectomorphism it generates $\\Tilde{\\psi}$ belong to $\\Stab^{S^1}(\\overline{D})$ and extends $\\psi$.  Note that if we fix the neighbourhood $U$, the bump function and the retraction of the neighbourhood in $\\Symp^{S^1}(\\overline{D})$ then this  procedure gives us a lift of $\\psi$ near id in $\\Symp^{S^1}(\\overline{D}) $ to $\\Stab^{S^1}(\\overline{D})$. By Theorem \\ref{palais} this shows $r$ is fibration. \n\\end{proof}\n\n\nThe above proof also shows that the action is transitive. As given $\\gamma \\in \\Symp^{S^1}(\\overline{D})$ we give a procedure to construct an element $\\tilde\\gamma \\in \\Stab^{S^1}(\\overline{D})$ such that the restriction to $\\overline{D}$ is $\\gamma$, thus showing the action is transitive.\n\n\n\n\\vspace{3mm}\n\n\\begin{lemma}\n$\\Symp^{S^1}(\\overline{D})$ is homotopic to $\\SO(3)$ for the circle action $S^1(0,\\pm 1,m)$  and $\\Symp^{S^1}(\\overline{D})$ is homotopic to $S^1$ for all other circle actions.\n\\end{lemma}\n\n\\begin{proof}\nConsider the circle action induced on $\\overline{D}$. The action $S^1(0,\\pm 1,m)$ fixes $\\overline{D}$ pointwise. Hence $\\Symp^{S^1}(\\overline{D}) = \\Symp(\\overline{D})$. By Smale's theorem we know that $\\Symp(\\overline{D})$ is homotopy equivalent to $\\SO(3)$. \n\\\\\n\nThe restriction for all other actions not equal to $S^1(0,\\pm 1,m)$, do not point wise fix the curve $\\overline{D}$. For all other actions, we have the following two subcases. Assume that the action is effective. Let $\\mu: \\overline{D} \\to \\mathbb{R}$ be it's moment map. Then as explained in the proof of Proposition \\ref{prop:CentralizersToricActions} we have that  $\\Symp^{S^1}(\\overline{D}) \\simeq C^\\infty(\\mu(\\overline{D}),S^1)$, where $C^\\infty(\\mu(\\overline{D}),S^1)$ denotes the space of smooth maps from the image of the moment map to $S^1$. As the image of the moment map is an interval, and as the space of smooth maps from an interval to $S^1$ is homotopy equivalent to $S^1$, we have the required result that $ \\Symp^{S^1}(\\overline{D}) \\simeq S^1$.\n\\\\\n\n\nFinally if the induced symplectic $S^1$ action on $\\overline{D}$ is not effective and has $\\mathbb{Z}_k$ stabilizer, the action of $S^1\/\\mathbb{Z}_k \\cong S^1$, is effective and the space of symplectomorphisms equivariant with respect to this quotient  effective action is the same as space of symplectomorphisms equivariant with respect to the non-effective $S^1$ action. Thus the homotopy type of $\\Symp^{S^1}(\\overline{D}) \\simeq S^1$.\n\\end{proof}\n\n\n\n\n\n\n\\begin{lemma}\\label{gauge}\nThe map\n\\begin{align*} \n   \\alpha: \\Fix^{S^1}(\\overline{D}) &\\twoheadrightarrow \\Gauge1^{S^1}(N(\\overline{D})) \\\\\n     \\phi &\\mapsto d\\phi|_{N(\\overline{D})}\n\\end{align*}\n is a Serre fibration with fibre homotopic to $\\Fix^{S^1}(N(\\overline{D}))$. The base space  $\\Gauge1^{S^1}(N(\\overline{D}))$  is homotopy equivalent to $S^1$. \n\\end{lemma}\n\n\\begin{proof}\nThe fact that $\\Gauge1^{S^1}(N(\\overline{D})) \\simeq S^1$ is explained in Appendix A. Thus we only need to prove that the restriction of the derivative indeed is a  fibration and the fibre is homotopic to $\\Fix^{S^1}(N(\\overline{D}))$.\n\\\\\n\nConsider the action \n \n\\begin{align*}\n     \\Fix^{S^1}(\\overline{D}) \\times \\Gauge1^{S^1}(N(\\overline{D})) &\\to\\Gauge1^{S^1}(N(\\overline{D})) \\\\\n     (\\phi, \\psi) &\\to d\\phi|_{N(D)} \\circ \\psi\n\\end{align*}\n\n\nAgain by Theorem \\ref{palais} it suffices to show that the there is a local section to above action. Such a local section is produced by Lemma \\ref{EqSymN}.\n\nThe fibre is apriori given by all equivariant symplectomorphisms that act as identity on the normal bundle of $\\overline{D}$. The claim is that this is in fact homotopy equivalent to the space $\\Fix^{S^1}(N(\\overline{D}))$.  This follows from lemma \\ref{germ}.\n\\end{proof}\n\nLet $\\overline{\\mathcal{S}^{S^1}_{F,p_0}}$ be the space of unparametrized $S^1$-invariant symplectic spheres in the homology class $F$ that are equal to a fixed invariant curve ${\\overline{F}}$ in a neighbourhood of $p_0$.\n\n\\begin{lemma}\nThe map \\begin{align*}\n    \\Fix^{S^1}(N(\\overline{D})) &\\to \\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\\\\n    \\phi \\mapsto \\phi(\\overline{F})\n    \\end{align*} \nis a fibration and $\\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D}) \\simeq \\{*\\}$\n\\end{lemma}\n\n\\begin{proof}\n The proof for this is exactly the same as the proof of Corollary \\ref{trans}. We only note that if $F' \\in \\overline{\\mathcal{S}^{S^1}_{F,p_0}}$ then the map $\\phi$ constructed in the proof of Corollary~\\ref{trans} belongs to $\\Fix^{S^1}(N(\\overline{D}))$. Thus we  have that $\\Fix^{S^1}(N(\\overline{D}))$ acts transitively on $\\overline{\\mathcal{S}^{S^1}_{F,p_0}}$, and thus the action map induces a fibration. \n \\\\\n \n\n \n \n \\textit{Proof that $\\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D}) \\simeq \\{*\\}$}:\n This follows from the equivariant version of the standard proof of considering the homeomorphic space of equivariant compatible metrics and noting that this space of metrics is contractible. \n \\\\\n \n\\textit{Proof that $\\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D}) \\simeq \\{*\\}$}:\n Let $\\mathcal{S}^\\perp_{F,p_0}$ denote the space of all $S^1$ invariant symplectically embedded spheres S in class F such that $S \\cap \\overline{D} =p_0$ and $S$ and $\\overline{D}$ intersect $\\omega_\\lambda$-orthogonally at $p_0$. By Lemma \\ref{transverse} we see that for every $S \\in  \\mathcal{S}^\\perp_{F,p_0}$ there exists a $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D})$ such that the configuration $S \\vee \\overline{D}$ is $J$-holomorphic.  We now have  the following fibration\n \n \\begin{equation*}\n     \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D}) \\longtwoheadrightarrow  \\mathcal{S}^\\perp_{F,p_0}\n \\end{equation*}\n\nwhere the map $\\gamma: \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D}) \\to  \\mathcal{S}^\\perp_{F,p_0}$ is just sending $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D})$ to the corresponding curve in class $F$ passing through $p_0$. Note that the above map is well defined because the fixed $p_0$ was chosen such that the weights at $p_0$ were distinct. Hence any $S^1$ invariant $F$ curve passing through $p_0$ must be $\\omega_\\lambda$-orthogonal to $\\overline{D}$. Now we show that $\\gamma$ is a homotopy equivalence. To do that we consider the following commutative diagram\n \n\\[ \n\\begin{tikzcd}\n    &T \\arrow[d,\"\\pi_2\"] \\arrow[r,\"\\pi_1\"] & \\mathcal{S}^\\perp_{F,p_0} \\\\\n    &\\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D}) \\arrow[ur, \"\\gamma\"]\n\\end{tikzcd} \n \\]\n\nwhere $T:= \\left\\{ (A,J) \\in  \\mathcal{S}^\\perp_{F,p_0} \\times \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D})~|~  A ~~\\text{is J-holomorphic} \\right\\}$. Both the maps $\\pi_1$ and $\\pi_2$ are fibrations (this can be argued as in Lemma \\ref{first}) with contractible fibres. As the diagram commutes, the map $\\gamma$ must be a homotopy equivalence. The proof then follows by showing that $\\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\simeq \\mathcal{S}^\\perp_{F,p_0}$, which is a consequence of Theorem~\\ref{gmpf}.\n\\end{proof}\n\n\n\\begin{lemma}\n      \\begin{align*}\n          \\Stab^{S^1}(\\overline{F}) \\cap \\Fix^{S^1}(N(\\overline{D})) &\\to \\Symp^{S^1}(\\overline{F}, N(p_0)) \\\\\n          \\phi &\\mapsto  \\phi|_{\\overline{F}}\n      \\end{align*}\n      \n      is a fibration and $\\Symp^{S^1}(\\overline{F}, N(p_0)) \\simeq \\{*\\}$\n \\end{lemma}\n \n \\begin{proof}\n The fact that this is a fibration follows from applying the proof of  Lemma \\ref{stab} mutatis mutandis. The proof that $\\Symp^{S^1}(\\overline{F}, N(p_0)) \\simeq \\{*\\}$, is similar to Lemma \\ref{stab}.  $\\Symp^{S^1}(\\overline{F}, N(p_0))$ is homotopy equivalent to maps from the interval $[0,1]$ to $S^1$ that is identity near 0. The space of such maps is contractible thus giving the result. \n \\end{proof}\n\n\n\n\\begin{lemma}\\label{ngauge}\n\n\n\\begin{align*}\n    \\Fix^{S^1}(\\overline{F}) &\\to \\Gauge1^{S^1}(N(\\overline{D} \\vee \\overline{F})) \\\\\n    \\phi &\\mapsto d\\phi|_{N(\\overline{D} \\vee \\overline{F})}\n\\end{align*}\nis a fibration and $\\Gauge1^{S^1}(N(\\overline{D} \\vee \\overline{F})) \\simeq \\{*\\}$ and the fibre $\\Fix^{S^1}(N(\\overline{D} \\vee \\overline{F})) \\simeq \\{*\\}$\n\\end{lemma}\n\n\\begin{proof}\nThe proof that this is a fibration is similar to the proof of Lemma \\ref{gauge}. The fact that $\\Gauge1^{S^1}(N(\\overline{D} \\vee \\overline{F})) \\simeq \\{*\\}$ follows from by Lemma \\ref{Gauge(N(D))}. The fact that $\\Fix_{S^1}(N(\\overline{D} \\vee \\overline{F})) \\simeq \\{*\\}$ follows from Theorem \\ref{thm:EqGr}.\n\\end{proof}\n\n\nPutting all the fibrations together gives the following theorem.\n\\begin{thm}\\label{homogenous}\nConsider the $S^1(a,b;m)$ action on $(S^2 \\times S^2,\\omega_\\lambda)$ with $\\lambda >1$. If $ \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s} \\neq \\phi$, then we have the following homotopy equivalences:\n\\begin{enumerate}\n    \\item when $(a,b) \\neq (0,\\pm1)$, we have $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_{2s} \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$;\n    \\item when $(a,b) = (0,\\pm1)$, we have $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/(SO(3) \\times S^1) \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$.\n\\end{enumerate} \n\\end{thm} \n\\begin{proof}\n\n\nWhen $(a,b;m) \\neq (0,\\pm 1;0)$ we have a commutative diagram of fibrations\n\\[\n\\begin{tikzcd}\n     &\\Fix^{S^1}(\\overline{D}) \\arrow[r] &\\Stab^{S^1}(\\overline{D}) \\arrow[r,twoheadrightarrow] &\\Symp^{S^1}(\\overline{D}) \\\\\n     &S^1 \\arrow[u,hookrightarrow] \\arrow[r] &\\mathbb{T}^2_{2s} \\arrow[u,hookrightarrow] \\arrow[r] &S^1 \\arrow[u,hookrightarrow]\n\\end{tikzcd}\n\\]\nwhile in the case $(a,b) = (0,\\pm1)$, we have the diagram\n\\[\n\\begin{tikzcd}\n     &\\Fix^{S^1}(\\overline{D}) \\arrow[r] &\\Stab^{S^1}(\\overline{D}) \\arrow[r,twoheadrightarrow] &\\Symp^{S^1}(\\overline{D}) \\\\\n     &S^1 \\arrow[u,hookrightarrow] \\arrow[r] & S^1 \\times SO(3) \\arrow[u,hookrightarrow] \\arrow[r] &SO(3) \\arrow[u,hookrightarrow]\n\\end{tikzcd}\n\\]\nFrom the discussion above, in both the diagrams the leftmost and the rightmost arrows are homotopy equivalences.  As the diagram commutes, the 5 lemma implies that the middle inclusion $\\mathbb{T}^2 \\hookrightarrow \\Stab^{S^1}(\\overline{D})$  or $\\left(S^1 \\times SO(3)\\right) \\hookrightarrow \\Stab^{S^1}(\\overline{D})$ are also homotopy equivalences. This gives us the required result.\n\\end{proof}\n\n\\begin{remark}\\label{ActionOnJsom}\nLet $J_{2s}$ be the standard complex structure on $W_{2s}$. We note that for the  action $S^1(0,\\pm1,;m)$ the  stabiliser of $J_{2s}$ under the natural action of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ on $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$ is the group of K\\\"ahler isometries $S^1 \\times \\SO(3)$. For all other circle actions $S^1(a,b;m)$ with $(a,b) \\neq (0,\\pm1)$, the stabiliser of $J_{2s}$ is the maximal torus $\\mathbb{T}^2_{2s}\\subset S^1 \\times \\SO(3)$.\n\\end{remark}\n\n\n\n\n\n\\subsection{Case 2: \\texorpdfstring{$\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$}{Symp(S2xS2)} action on \\texorpdfstring{$\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$}{J\\hat{}S1\\_0}}\\label{IntwithU0}\nIn order to describe the action of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ on the open stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$, we need to  modify slightly the setting introduced in the previous section. The main difference comes from the fact that for an almost-complex structure $J\\in\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_0$, there is no invariant curve with negative self-intersection  representing a class $B-kF$, $k\\geq 1$. Instead, each such $J$ determines a regular 2-dimensional foliation of $J$-holomorphic curves in the class $B$. Consequently, there is no natural map between the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$ and the space $\\mathcal{S}^{S^1}_{B}$ of invariant curves in the class $B$. However, once we choose a fixed point $p_0$, given any $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$, there is a unique invariant $J$-holomorphic curve in the class $B$ passing through $p_0$. This defines a map $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0 \\to \\mathcal{S}^{S^1}_{B,p_0}$ that can be used to prove that the space $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$ is homotopy equivalent to an orbit of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$. To do so, because the fixed point $p_0$ is not unique, we must also investigate how the group $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ acts on the fixed point set of the circle action. This is done in Lemma~\\ref{lemma:SymphPreservesAnIsolatedFixedPoint}. Before we proceed to prove this lemma we first describe the action of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ on $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$. Note that by Theorem~\\ref{cor:IntersectingOnlyOneStratum}, Corollary~\\ref{cor:CircleExtensionsWith_a=1} and Corollary~\\ref{cor:CircleExtensionsWith_a=-1}, the space $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$ is non-empty only for the following circle actions:\n\\begin{itemize}\n    \\item $S^1(a,b;0)$, or\n    \\item  $S^1(1,b;m)$ with $|2b-m|=0$ and $2\\lambda > |2b-m|$, or\n    \\item $S^1(-1,b;m)$ with $|2b+m|=0$ and $2\\lambda > |2b+m|$.\n\\end{itemize}\nSecondly, we observe that all these actions have at least one isolated fixed point \\emph{except} the actions of the forms\n\\begin{itemize}\n    \\item $S^1(\\pm 1,0;0)$ and\n    \\item $S^1(0,\\pm 1;0)$\n\\end{itemize}\n\\subsubsection{Actions with an isolated fixed point} We now consider actions $S^1(a,b;m)$ with an isolated fixed point $p_0$. We can choose $p_0$ to correspond to the vertex $R$ in the Hirzerbruch surface $W_m$ shown in Figure~\\ref{hirz}. Given $J\\in\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_0$, there is a unique $J$-holomorphic curve $B_{p_0,J}$ in class $B$ that passes through $p_0$. Because $p_0$ is fixed, $J$ is invariant, and $B\\cdot B=0$, positivity of intersection implies that $B_{p_0,J}$ is $S^1$-invariant. We thus get a well-defined map \n\\[\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0 \\to \\mathcal{S}^{S^1}_{B,p_0}\\]\nwhere $\\mathcal{S}^{S^1}_{B,p_0}$ denotes the space of invariant, embedded, symplectic spheres representing the class $B$ and containing the point $p_0$. \n\\begin{lemma}\\label{projection_surjective}\nConsider any $S^1(a,b; m)$ action on $(S^2 \\times S^2,\\omega_\\lambda)$. Let $p_0$ and $p_1$ be two fixed points such that there exists an invariant fibre $\\{*\\} \\times S^2$ passing through them . Then there exists no $S^1$ invariant curve in the class $B-kF$ for $k \\geq 0$ passing through $p_0$ and $p_1$.\n\\end{lemma}\n\\begin{proof}\nSuppose not, let $\\overline{D_{2s}}$ be a $S^1$ invariant curve in the class $B-kF$ with $k \\geq 0$ passing through $p_0$ and $p_1$. Then the projection onto the first factor\n$$ \\pi_1 : \\overline{D_{2s}} \\rightarrow S^2 \\times \\{0\\} \\subset S^2 \\times S^2$$\nis surjective. Hence the curve $\\overline{D_{2s}}$ passes through a third  fixed point $p_2$. As the symplectic $S^1$ action on $\\overline{D_{2s}}$ has three fixed points, it has to fix $\\overline{D_{2s}}$ pointwise. This is a contradiction as all fixed surfaces for $S^1$ actions must be either a maximum or minimum for the moment map, but the fixed points $p_2$, $p_1$ and $p_0$ cannot have the same moment map value.\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{lemma:SymphPreservesAnIsolatedFixedPoint}\nLet $S^1(a,b;m)$ be a circle action for which the space $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$ is non-empty. Assume there is an isolated fixed point $p_0$ corresponding to the vertex $R$ in Figure~\\ref{hirz}.  Then any equivariant symplectomorphism that preserves homology $\\phi\\in \\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda)$ fixes~$p_0$.\n\\end{lemma}\n\\begin{proof}\n\\emph{Case 1: $\\lambda >1$:}\nBy Lemma~\\ref{lemma:CharacterizationCentralizer} and Corollary~\\ref{cor:ActionPreservesWeights} any such $\\phi$ must preserve the moment values and the weights of the fixed points (up to change of order of the tuples). These weights are given in Table~\\ref{table_weights} and  the moment map values are given in the graphs~\\ref{fig:GraphsWithFixedSurfaces} and \\ref{fig:GraphsIsolatedFixedPoints}. The two conditions on the circle action imply that either $m=0$, $|2b-m|=0$, or $|2b+m|=0$. It is now easy to see that under any of these three numerical conditions, the weights and moment map values at $R$ differ from the weights at all other fixed points. The result follows.\n\\\\\n\n\\emph{Case 2: $\\lambda = 1$:}\nIf the actions are \\emph{not} of  the form $S^1(1,1; 0)$ or $S^1(-1,-1;0)$ with  $\\lambda = 1$, then an argument similar to Case 1 holds. \nThe only case left are the actions of the form $S^1(1,1; 0)$ or $S^1(-1,-1;0)$ with  $\\lambda = 1$. In this case, the  homology classes $F$, $B$ have the same area and the fixed points $R$ and $Q$ have the same weights (up to change of order of tuples) and the same moment map values. We again argue by contradiction in this case. Let $\\overline{B}$ denote a fixed curve in class $B$ passing through $R$ and $P$. Suppose $\\phi \\in \\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda)$ doesn't fix the point ~$p_0 = R$. Then $\\phi$ has to take the point $R$ to the point $Q$. Further by Lemma~\\ref{lemma:CharacterizationCentralizer}, $\\phi$ fixes the maximum and minimum and hence $\\phi(P) = P$. As $\\phi$ preserves homology, the curve $\\phi(\\overline{B})$ has homology class $B$ and has as to pass through $Q$ and $P$ which contradicts  Lemma~\\ref{projection_surjective}. \n\\end{proof}\n\nLet $J_0\\in U_0$ be the complex structure of the Hirzebruch surface $W_0$ and let $B_{p_0}$ be the unique $J_0$-holomorphic curve containing $p_0$ and representing the homology class $B$.\n\\begin{cor}\nLet $S^1(a,b;m)$ be a circle action with an isolated fixed point and for which the structure $J_0\\in U_0$ is invariant. Then the group $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ acts transitively on the space $\\mathcal{S}^{S^1}_{B,p_0}$, and the action map \n\\begin{align*}\n\\Symp^{S^1}_{h}(S^2 \\times S^2,\\omega_\\lambda) &\\longtwoheadrightarrow \\mathcal{S}^{S^1}_{B,p_0}\\\\\n\\phi & \\mapsto \\phi(B_{p_0})\n\\end{align*}\nis a Serre fibration.\n\\end{cor}\n\\begin{proof}\n\nSince any element of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ fixes $p_0$, it follows that this group acts on $\\mathcal{S}^{S^1}_{B,p_0}$. Let $p_1$ be the other fixed point corresponding to the point $Q$ in Figure~\\ref{hirz}. All curves in  $\\mathcal{S}^{S^1}_{B,p_0}$ pass through $p_1$ and $p_0$. Since the weights at one of $p_0$ or $p_1$ are always distinct, we can use the fixed point with distinct weights and proceed as in the proofs of Corollary ~\\ref{trans} and Lemma~\\ref{first} to show that the action defines a fibration.\n\\end{proof}\nAs before, we can now show that the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}$ is homotopy equivalent to a space of invariant curves.\n\\begin{lemma}\nThe natural map $\\alpha: \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0} \\to {\\mathcal{S}^{S^1}_{B,p_0}}$ defined by sending an almost complex structure $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}$ to the unique $J$-holomorphic curve in class $B$ passing through $p_0$ is a weak homotopic equivalence. \n\\end{lemma}\n\\begin{proof}\nThe argument is identical to the proof of Lemma~\\ref{first2}\n\\end{proof}\n\nFrom now on, we can determine the homotopy type of $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}$ by going through a similar sequence of fibrations and homotopy equivalences as in Section~\\ref{section:ActionOnU_2k}, namely,\n\\[\\Stab^{S^1}(B_{p_0}) \\to \\Symp^{S^1}_{h}(S^2 \\times S^2,\\omega_\\lambda) \\longtwoheadrightarrow \\mathcal{S}^{S^1}_{B,p_0} \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}}  \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}\\rule{0em}{2em}\\]\n    \n\\[\\Fix^{S^1}(B_{p_0}) \\to \\Stab^{S^1}_{p_0}(B_{p_0}) \\longtwoheadrightarrow  \\Symp^{S^1}(B_{p_0}) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} S^1\\rule{0em}{2em}\\]\n\n\\[\\Fix^{S^1} (N(B_{p_0})) \\to \\Fix^{S^1}(B_{p_0}) \\longtwoheadrightarrow  \\Gauge1^{S^1}(N(B_{p_0})) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} S^1\\rule{0em}{2em} \\]\n\n\\[\\Stab^{S^1}(\\overline{F}) \\cap \\Fix^{S^1}(N(B_{p_0})) \\to \\Fix^{S^1}(N(B_{p_0})) \\longtwoheadrightarrow  \\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\mathcal{J}^{S^1}(B_{p_0})\\rule{0em}{2em}\\]\n   \n\\[\\Fix^{S^1}(\\overline{F}) \\to \\Stab^{S^1}(\\overline{F}) \\cap \\Fix^{S^1}(N(B_{p_0})) \\longtwoheadrightarrow  \\Symp^{S^1}(\\overline{F}, N(p_0))  \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\left\\{*\\right\\}\\rule{0em}{2em}\\]\n\n\\[\\left\\{*\\right\\} \\mathbin{\\textcolor{blue}{\\xleftarrow{\\text{~~~$\\simeq$~~~}}}} \\Fix^{S^1}(N(B_{p_0} \\vee \\overline{F})) \\to \\Fix^{S^1}(\\overline{F}) \\longtwoheadrightarrow  \\Gauge1^{S^1}(N(B_{p_0} \\vee \\overline{F}))  \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\left\\{*\\right\\}\\rule{0em}{2em}\\]\nwhere $\\overline{\\mathcal{S}^{S^1}_{F,p_0}}$ denotes the space of all symplectically embedded curve in the class $F$ that pass through $p_0$ and agree with a standard curve $F_{p_0}$ in a neighbourhood of $p_0$. The proofs that these maps are fibrations, and the proofs of the homotopy equivalences are exactly the same as before.\nConsequently, we obtain the following homotopical description of $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}$.\n\\begin{thm}\\label{Thm_isolatedfixedpt}\nConsider one of the following circle actions on $(S^2 \\times S^2,\\omega_\\lambda)$\n\\begin{itemize}\n    \\item $S^1(a,b;0)$ with $(a,b)\\neq (\\pm1,0)$ and $(a,b)\\neq(0,\\pm1)$, or\n    \\item  $S^1(1,b;m)$ with $|2b-m|=0$ and $2\\lambda > |2b-m|$, or\n    \\item $S^1(-1,b;m)$ with $|2b+m|=0$ and $2\\lambda > |2b+m|$.\n\\end{itemize}\nThen the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$ is non-empty and \n\\[\\Symp^{S^1}_{h,p_0}(S^2 \\times S^2,\\omega_\\lambda)\/ \\mathbb{T}^2_0 \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}\\]\n\\end{thm}\\qed\n\n\n\\subsubsection{Actions without isolated fixed points} \n\nWe now turn our attention to the action of the group of equivariant symplectomorphisms $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ on the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}$ when the circle action is either\n\\begin{enumerate}\n    \\item $S^1(\\pm 1,0;0)$ or\n    \\item $S^1(0,\\pm 1;0)$.\n\\end{enumerate}\n\n\nThese actions has no isolated fixed points and the associated graphs are of the form\n\\begin{figure}[H]\n    \\centering\n   \n\n\n\\subcaptionbox{Subcase 1: $S^1(\\pm 1,0;0)$  }\n[.45\\linewidth]\n{\\begin{tikzpicture}\n[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,0) ellipse (0.3cm and 0.1cm); \n\\draw [fill] (0,2.8) ellipse (0.3cm and 0.1cm); \n\\node[above] at (0,2.9) {$F_{max}$}; \n\\node[below] at (0,-0.1) {$F_{min}$}; \n\\node[left] at (-0.3,2.8) {$\\mu = \\lambda$};\n\\node[left] at (-0.3,0){$\\mu=0$};\n\\node[right] at (0.3,2.8){$A= 1$};\n\\node[right] at (0.3,0){$A= 1$};\n\\end{tikzpicture}\n}\n\\quad\n\\subcaptionbox{Subcase 2: $S^1(0,\\pm 1;0)$}\n[.45\\linewidth]\n{\\begin{tikzpicture}\n[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,0) ellipse (0.3cm and 0.1cm); \n\\draw [fill] (0,2.8) ellipse (0.3cm and 0.1cm); \n\\node[above] at (0,2.9) {$B_{max}$}; \n\\node[below] at (0,-0.1) {$B_{min}$}; \n\\node[left] at (-0.3,2.8) {$\\mu = 1$};\n\\node[left] at (-0.3,0){$\\mu=0$};\n\\node[right] at (0.3,2.8){$A= \\lambda$};\n\\node[right] at (0.3,0){$A= \\lambda$};\n\\end{tikzpicture}\n}\n\\end{figure}\\label{subcase2:Fixedsurface}\n\\noindent where $\\mu$ denotes the value of the moment map and $A$ denotes the area of the fixed surface. We notice that there are pointwise fixed curves in the class $F$ for the circle action $S^1(\\pm 1,0;0)$ and pointwise fixed curves in class $B$ for the action $S^1(0,\\pm 1;0)$. We denote the fixed surface which is a minimum for the moment map as $F_{min}$, $B_{min}$ respectively and the maximum by $F_{max}$, $B_{max}$.\n\\\\\n\nConsider the action $S^1(0,\\pm 1;0)$. By Lemma~\\ref{lemma:CharacterizationCentralizer} we note that any $\\phi \\in \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ must send $B_{max}$ to itself. Then, given $p_0 \\in B_{max}$, we define the following sequence of fibrations and homotopy equivalences:\n\\[\\Fix^{S^1}(B_{max}) \\longrightarrow \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)  \\longtwoheadrightarrow \\Symp(B_{max}) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} SO(3)\\rule{0em}{2em}\\]\n\\[\\Stab^{S^1}(F_{p_0}) \\longrightarrow \\Fix^{S^1}(B_{max}) \\longtwoheadrightarrow \\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\mathcal{J}^{S^1}_{\\om_\\lambda} \\simeq \\{*\\}\\rule{0em}{2em}\\]\n\\[\\Fix^{S^1} (F_{p_0}) \\longrightarrow \\Stab^{S^1}(F_{p_0}) \\longtwoheadrightarrow \\Symp^{S^1}(F_{p_0}) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} S^1\\rule{0em}{2em}\\]\n\\[\\left\\{*\\right\\}\\mathbin{\\textcolor{blue}{\\xleftarrow{\\text{~~~$\\simeq$~~~}}}} \\Fix^{S^1}(N(B_{max} \\vee F_{p_0})) \\longrightarrow  \\Fix^{S^1}(F_{p_0})\\longtwoheadrightarrow \\Gauge1^{S^1}(N(B_{max} \\vee F_{p_0}))\n\\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}}  \\left\\{*\\right\\}\\rule{0em}{2em}\\]\n\\\\\n\nFor the other circle action $S^1(\\pm 1,0;0)$, we obtain a similar sequence of fibrations and homotopy equivalences in which $B_{\\max}$ is replaced by the curve $F_{\\max}$. As before, putting all the homotopy equivalences together, we obtain the following theorem:\n\\begin{thm}\\label{Thm_fixedsurfaces}\nConsider the following two circle actions on $(S^2 \\times S^2,\\omega_\\lambda)$  \n\\begin{itemize}\n\\item $S^1(\\pm 1,0;0)$ or\n\\item  $S^1(0,\\pm 1;0)$\n\\end{itemize}\nThen there is a homotopy equivalence\n\\[\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/(S^1 \\times \\SO(3)) \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0\\]\n\\end{thm}\\qed\n\nFor convenience, we collect together the two main results of this section in the theorem below. \n\\begin{thm}{\\label{homog}}\nConsider the action $S^1(a,b;m)$ on $(S^2 \\times S^2,\\omega_\\lambda)$ such that  one of the following  hold:\n\\begin{itemize}\n    \\item $S^1(a,b;0)$ with $(a,b)\\neq (\\pm1,0)$ and $(a,b)\\neq(0,\\pm1)$, or\n    \\item  $S^1(1,b;m)$ with $|2b-m|=0$ and $2\\lambda > |2b-m|$, or\n    \\item $S^1(-1,b;m)$ with $|2b+m|=0$ and $2\\lambda > |2b+m|$.\n\\end{itemize}\nThen the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$ is non-empty and \n\\[\\Symp^{S^1}_{h,p_0}(S^2 \\times S^2,\\omega_\\lambda)\/ \\mathbb{T}^2_0 \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}\\]\nIf instead the $S^1(a,b;m)$ action satisfies \n\\begin{itemize}\n\\item $(a,b;m) = (\\pm 1,0;0)$ or\n\\item  $(a,b;m) = (0,\\pm 1;0)$\n\\end{itemize} then we have that \\[\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/(S^1 \\times \\SO(3)) \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0\\]\nand $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects only the strata $U_0$.\n\\end{thm}\\qed \n\n\n\n\n\n\n\\chapter{The homotopy type of the symplectic centralisers of \\texorpdfstring{$S^1(a,b;m)$}{S1(a,b;m)}}\nGiven any Hamiltonian circle action on $(S^2 \\times S^2,\\omega_\\lambda)$, the two Theorems~\\ref{cor:IntersectingOnlyOneStratum} and~\\ref{cor:IntersectingTwoStrata} give us a complete understanding of which strata the space $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects. Together with Theorems~\\ref{homogenous}, and~\\ref{homog} describing the strata as homogeneous spaces, this allows us to compute the homotopy type of the group of equivariant symplectomorphisms.\n\n\\section{When \\texorpdfstring{$\\mathcal{J}^{S^1}_{\\om_\\lambda}$}{J\\hat S1} is homotopy equivalent to a single symplectic orbit}\n\n\n\\begin{thm}\\label{table}\nConsider the  circle action $S^1(a,b;m)$ on $(S^2 \\times S^2,\\omega_\\lambda)$. Under the following numerical conditions on $a,b,m,\\lambda$, the homotopy type of $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ is given by the  table below.\n\\\\\n\n\\noindent{\n\\begin{tabular}{|p{5cm}||p{3.5cm}|p{2cm}|p{3.3cm}|}\n\\hline\n $S^1$ action $(a,b;m)$ & $\\lambda$ &Number of strata $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects &Homotopy type of $\\Symp^{S^1}(S^2 \\times S^2)$\\\\\n\\hline\n {$(0,\\pm 1;m)$ ~~ $m\\neq 0$}  &$\\lambda > 1$ & 1 & $S^1 \\times SO(3)$ \\\\\n\\hline\n\\multirow{2}{10em}{$(0,\\pm 1;0)$ or $(\\pm 1,0;0)$} &$\\lambda = 1$  &1  &$S^1 \\times SO(3)$ \\\\\n&$\\lambda >1$ &1  &$S^1 \\times SO(3)$ \\\\\n\\hline\n$(\\pm 1,\\pm1,0)$&$\\lambda = 1$ &1 & $\\mathbb{T}^2 \\times \\mathbb{Z}_2$ \\\\\n\\hline\n $(\\pm 1,0;m) ~ m\\neq0$ &$\\lambda >1$ &1 &$\\mathbb{T}^2$\\\\\n \\hline\n$(\\pm 1,\\pm m;m)$ $m \\neq 0$ & $\\lambda > 1$   &1 & $\\mathbb{T}^2$\\\\\n\\hline\n$( 1,b;m)$ $b \\neq \\{ m,0\\}$ &$|2b-m| \\geq2 \\lambda \\geq 1$ &1 &$\\mathbb{T}^2$ \\\\\n\\hline\n$(-1,b;m), b \\neq \\{ -m,0\\}$ &$|2b+m| \\geq2 \\lambda \\geq 1$ &1 &$\\mathbb{T}^2$ \\\\\\cline{2-4}\n\\hline\nAll other values of $(a,b;m)$ except $(\\pm 1,b;m)$ &$\\forall \\lambda$  &1   &$\\mathbb{T}^2$ \\\\\n\\hline\n\\end{tabular}\n}\n\\end{thm}\n \n \\begin{proof}\n By Theorem~\\ref{cor:IntersectingOnlyOneStratum}, in each of the above $S^1(a,b;m)$ actions, the space of $S^1$ invariant compatible almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects only the stratum $U_m$. Consequently,\n \\[\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)\/\\Stab(J_m) \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m = \\mathcal{J}^{S^1}_{\\om_\\lambda} \\simeq \\{*\\}\\]\n where $\\Stab(J_m)$ denotes the stabiliser of the standard complex structure $J_m \\in U_m$. Thus, for all the actions in the table, we have that $\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\simeq \\Stab(J_m)$. For the $S^1$ action given by the triples $(0,\\pm 1,m)$, $(\\pm1,0,0)$ or the circle action $S^1(0,\\pm1,0)$ when $\\lambda =1$, Theorems~\\ref{homogenous} and~\\ref{homog} imply that $\\Stab(J_m) \\simeq S^1 \\times \\SO(3)$. For all other $S^1$ actions in the table, the stabilizers are homotopy equivalent to $\\mathbb{T}^2$. \n\\\\\n\nWe now show how to recover the homotopy type of the full group $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda) $ from the homotopy type of the subgroup $\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$. When $\\lambda > 1$, we have the equality $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda) = \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ as stated in Lemma~\\ref{lemma:ActionOnHomology}.\n\\\\\n\n\n\nWhen $\\lambda = 1$ and $a\\neq b$, \nthere exists standard $S^1(a,b;m)$ invariant curves in classes $B$ and $F$ such that the isotropy weight of the action on the curve in class $B$ is $a$ and the isotropy weight of the $S^1$ action on the curve in the class $F$ is $b$. Hence, as $\\phi$ is an equivariant symplectomorphism, Lemma~\\ref{lemma:ActionOnHomology} implies that must have $\\phi_*[F] = [F]$ and $\\phi_*[B] = [B]$. Consequently, $\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda) = \\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$.\n\\\\\n\nIn the special case when $\\lambda =1$ and $a=b = \\pm 1$, then we have an equivariant version of the exact sequence~\\ref{Sequence:ActionOnHomology}\n\\begin{equation*}\n   1 \\longrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\longrightarrow  \\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\longrightarrow \\Aut_{c_1,\\omega_\\lambda}(H^2(S^2 \\times S^2)) \\longrightarrow 1\n\\end{equation*}\n\nwhere $\\Aut_{c_1,\\omega_\\lambda}(H^2(S^2 \\times S^2)) \\cong \\mathbb{Z}_2$. The map \\begin{align*}\n    \\phi: S^2 \\times S^2 \\to S^2 \\times S^2 \\\\\n    (z,w) \\mapsto (w,z)\n\\end{align*} \nis a $S^1$ equivariant symplectomorphism (for the action $S^1(1,1,0)$ or $ (-1,-1,0)$) and gives a section from $\\mathbb{Z}_2 \\cong \\Aut_{c_1,\\omega_\\lambda}(H^2(S^2 \\times S^2))$ to $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$. Thus we have $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\cong  \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\rtimes \\mathbb{Z}_2$. As the semidirect product of two topological groups is homotopy equivalent(as a topological space) to the direct product of the groups,  we have that  $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\cong  \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\rtimes \\mathbb{Z}_2 \\simeq  \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\times \\mathbb{Z}_2 \\simeq \\mathbb{T}^2 \\times \\mathbb{Z}_2$. This completes the proof.\n\n\\end{proof}\n\n\\section[The two orbits case]{When \\texorpdfstring{$\\mathcal{J}^{S^1}_{\\om_\\lambda}$}{J\\hat{}S1} is homotopy equivalent to the union of two symplectic orbits}\\label{section:TwoOrbits}\n\n\nTheorem~\\ref{table} gives the homotopy type of the group of equivariant symplectomorphisms for all circle actions on $S^2 \\times S^2$ apart from the following two families of actions:\n\\begin{itemize}\n\\item (i) $a=1$, $b \\neq \\{0, m\\}$,  and $2\\lambda > |2b-m|$; or\n\\item (ii) $a=-1$, $b \\neq \\{0, -m\\}$, and $2 \\lambda > |2b+m|$.\n\\end{itemize}\nFor convenience, we will write $m'$ for either $|2b-m|$ or $|2b+m|$ depending on which of the above families we consider. Up to swapping $m$ and $m'$, we will also assume $m'>m$. The goal of this section is to show that the symplectic stabilizers of any of these circle actions is homotopy equivalent to the pushout of the two tori $\\mathbb{T}^2_m$ and $\\mathbb{T}^2_{m'}$ along the common $S^1$ in the category of topological groups.\\\\ \n\nBefore delving into the technicalities, it may be useful to outline the proof, which is an adaptation of the Anjos-Granja argument used in~\\cite{AG} to compute the homotopy type of the full group of symplectomorphisms of $S^2 \\times S^2$ for $1<\\lambda\\leq 2$. The first step is to show that the two inclusions \\[\\mathbb{T}^2_{m} \\hookrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)\\quad \\text{~and~}\\quad \\mathbb{T}^2_{m'} \\hookrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)\\]\ninduce injective maps in homology. By the Leray-Hirsch theorem, it follows that the cohomology modules of the total space of the fibrations\n\\[\\mathbb{T}^2_m\\to\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\to \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_m \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_m\\]\n\\[\\mathbb{T}^2_{m'}\\to\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\to\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_{m'} \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m'}\\]\nsplit (with coefficients in an arbitrary field $k$). Using the fact that the contractible space of invariant compatible almost-complex structures decomposes as the disjoint union\n\\[\\mathcal{J}^{S^1}_{\\om_\\lambda}=(\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_m)\\sqcup (\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m'})\\]\nthe rank of $H^i(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda);k)$ can be computed inductively from Alexander-Eells duality. We then compute the cohomology algebra and the Pontryagin algebra of the pushout\n\\[P = \\pushout(\\mathbb{T}^2_m\\leftarrow S^1\\to \\mathbb{T}^2_{m'})\\]\nin the category of topological groups. We then show that the natural map\n\\[\\Upsilon:P\\to \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\]\nis a homotopy equivalence in the category of topological groups. We further prove that $P$ is weakly homotopy equivalent, as a topological space, to the product $\\Omega S^{3}\\times S^{1}\\times S^{1}\\times S^{1}$.\n\n\n\n\n\\subsection{Homological injectivity}\nWe first show that the two inclusions $\\mathbb{T}^2_{m} \\hookrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ and $\\mathbb{T}^2_{m'} \\hookrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ induce injective maps in homology. As the argument does not depend on $m$, we shall only provide the details for the inclusion $\\mathbb{T}^2_{m} \\hookrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$.\\\\\n\nFix a symplectomorphism $\\phi_m: W_m\\to (S^2 \\times S^2,\\omega_\\lambda)$ compatible with the fibration structures. Let $\\{*\\}$ be the $S^1(1,b,m)$ fixed point $\\left([0,1][0,0,1]\\right)$ in $W_m$, and let $\\mathcal{E}(W_m, *)$ denote the space of orientation preserving, pointed, homotopy self-equivalences of $(W_m,*)$. Similarly, define $\\mathcal{E}(S^2, *)$ to be the space of all orientation preserving homotopy self-equivalences of the sphere preserving a base point $\\{*\\}$.\n\\\\\n\nWe now observe that for the above two families of circle actions (i) and (ii), the same argument as in Lemma~\\ref{lemma:SymphPreservesAnIsolatedFixedPoint} shows that any $\\phi\\in\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)= \\Symp_h^{S^1}(W_{m})$ fixes the base point $\\{*\\}$.\\\\\n\nNow, recall that the zero section $s_0$ of $W_m$ is given by\n\\begin{align*}\n    s_{0}: S^2 &\\to W_m \\\\\n    \\left[z_{0}, z_{1}\\right] &\\mapsto\\left(\\left[z_{0}, z_{1}\\right],[0,0,1]\\right)\n\\end{align*} \nand the projection to the first factor is\n\\begin{align*}\n    \\pi_1: W_m &\\to S^2 \\\\\n    \\left(\\left[z_{0}, z_{1}\\right],\\left[w_{0}, w_{1}, w_{2}\\right]\\right) &\\mapsto\\left[z_{0}, z_{1}\\right]\n\\end{align*}\nWe define a continuous map $h_1:\\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda) \\to \\mathcal{E}\\left(S^{2}, *\\right)$ by setting\n\\begin{equation*}\n  \\begin{aligned}\nh_1:\\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda) &\\to \\mathcal{E}\\left(S^{2}, *\\right) \\\\\n\\psi &\\mapsto \\psi_1:= \\pi_{1} \\circ \\psi \\circ s_{0}\n\\end{aligned}\n\\end{equation*}\nSimilarly, using the inclusion of $S^{2}$ as the fiber \n\\begin{align*}\n    f: S^2 &\\to W_m \\\\\n    \\left[z_{0}, z_{1}\\right] &\\mapsto\\left([0,1],\\left[ 0,z_{0}, z_{1}\\right]\\right)\n\\end{align*}\nand the projection to the second factor $\\pi_{2}: S^2 \\times S^2 \\to S^{2}$, we can define a map \n\\begin{align*}\n   h_2: \\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda) &\\to \\mathcal{E}\\left(S^{2}, *\\right) \\\\\n \\psi &\\mapsto \\psi_2:= \\pi_{2} \\circ \\psi \\circ f\n\\end{align*}\n\nWe thus get a continuous map \n\\begin{align*}\n    h: \\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda) &\\to \\mathcal{E}(S^2, *) \\times \\mathcal{E}(S^2,*) \\\\\n    \\psi &\\mapsto \\left(h_1(\\psi),h_2(\\psi)\\right) \\\\\n\\end{align*}\n\\begin{lemma} \\label{inj}\nThe inclusion $i_m:\\mathbb{T}^2_m \\hookrightarrow \\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda) $ induces a map which is injective in homology with coefficients in any field $k$.\n\\end{lemma}\n\\begin{proof}\nAs $\\mathbb{T}^2$ is connected, $i_m: H_0(\\mathbb{T}^2_m; k) \\to H_0(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda); k)$ is injective. To show that the inclusion map induces an injection at the $H_1$ level, we consider the composition $\\alpha: \\mathbb{T}^2_m \\to \\mathcal{E}(S^2,*) \\times \\mathcal{E}(S^2,*)$ given by\n\\[\n\\begin{tikzcd}\n    &\\mathbb{T}^2_m \\arrow[r,hookrightarrow]  &\\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda)  \\arrow[r,rightarrow,\"h\"]  &\\mathcal{E}(S^2, *) \\times \\mathcal{E}(S^2, *) \n\\end{tikzcd}\n\\]\nand show that $\\alpha$ induces a map which is injective in homology. \n\nWe claim that $H_1(\\mathcal{E}(S^2,*);\\mathbb{Z})\\simeq \\mathbb{Z}$. Indeed, the standard action of $\\SO(3)$ on $S^2$ gives rise to a diagram of fibrations\n\\[\n\\begin{tikzcd}\n        & \\mathcal{E}(S^2,*)\\arrow[r] &\\mathcal{E}(S^2) \\arrow[r,twoheadrightarrow,\"\\ev\"] &S^2 \\\\\n     &S^1= \\SO(2) \\arrow[u,hookrightarrow] \\arrow[r] &\\SO(3) \\arrow[u,hookrightarrow] \\arrow[r,\"\\ev\",twoheadrightarrow] &S^2 \\arrow[u,equal] \\\\\n\\end{tikzcd}\n\\]\n\nwhere the maps $\\ev$ are evaluations at the base point $\\{*\\}$. This induces a long exact ladder of homotopy groups\n\\[\n\\begin{tikzcd}\n     &\\cdots \\arrow[r] &\\cancelto{\\mathbb{Z}}{\\pi_2(S^2)} \\arrow[r] &\\pi_1(\\mathcal{E}(S^2,*)) \\arrow[r] &\\pi_1(SO(3)) \\times \\cancelto{0}\\pi_1(\\Omega) \\arrow[r] &\\cancelto{0}{\\pi_1(S^2)} \\\\\n     &\\cdots \\arrow[r] &\\mathbb{Z} \\arrow[u,equal] \\arrow[r]&\\cancelto{\\mathbb{Z}}{\\pi_1(S^1)} \\arrow[r] \\arrow[u,\"\\beta\"] &{\\pi_1(SO(3))}\\arrow[u] \\arrow[r] &\\cancelto{0}{\\pi_1(S^2)} \\arrow[u,equal]\n\\end{tikzcd}\n\\]\nwhere we have used the fact, proven by Hansen in~\\cite{Hans}, that $\\mathcal{E}(S^2) \\simeq SO(3) \\times \\widetilde{\\Omega^2}$, where $\\widetilde{\\Omega^2}$ denotes the universal covering space for the connected component of the double loop space of $S^2$ containing the constant based map, and where the $\\SO(3)$ component is just the inclusion. Consequently, $\\pi_1(\\widetilde{\\Omega^2})= 0 $ and the map $\\pi_1(SO(3)) \\to \\pi_1(SO(3)) \\times \\pi_1(\\widetilde{\\Omega^2})$ is an isomorphism. From the commutativity of the middle square, it follows that $\\beta:\\pi_1(S^1) \\to \\pi_1(\\mathcal{E}(S^2, *))$ is also an isomorphism. As the spaces we consider are topological groups, $\\pi_1$ is abelian and hence $\\pi_1 = H_1$, proving the claim.\\\\\n\nNow, the classes $a$, $b$, of the subcircles $(0,1)$ and $(1,0)$ form a basis for $H_1(\\mathbb{T}^2_m; k)$. We claim that $\\alpha_*[0,1]$ and $\\alpha_*[1,0]$ generate a subgroup of rank 2. To see this, let write $\\alpha_*^1$ and $\\alpha_*^2$ for the components of $\\alpha_*$. Then, $\\alpha^1_*(0,1) = 0$ as the circle $(0,1)$ fixes the zero section $\\left([x_1,x_2],[0,0,1]\\right) \\subset W_m$ pointwise, while $\\alpha^2_*[0,1] \\neq 0$ by the reasoning in the previous paragraph. Similarly, $\\alpha^1_*[1,0] \\neq 0$ and $\\alpha^2_*[1,0]= 0$, proving our claim. We conclude that $\\alpha$ is injective on $H_1(\\mathbb{T}^2_m; k)$. \n\\\\\n\nFinally, to show that $i_*$ is injective on $H_2(\\mathbb{T}^2_m;k)$, we will prove the dual statement, namely, that the map $i^*:H^2(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda);k) \\to H^2(\\mathbb{T}^2_m;k)$ is surjective. A generator of $H^2(\\mathbb{T}^2_m;k) \\cong k$ is given by $a \\cup b$. Because $i_*$ is injective at the $H_1$ level, $i^*: H^1(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda);k) \\to H^1(\\mathbb{T}^2;k)$ is surjective, hence there exists elements $a^\\prime$, $b^\\prime \\in H^1(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda);k)$ such that $i^*(a^\\prime) = a$ and $i^*(b^\\prime) = b$. Since $i^*(a^\\prime) \\cup i^*(b^\\prime) = a \\cup b$, it follows that $i^*:H^2(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda);k) \\to H^2(\\mathbb{T}^2_m;k)$ is surjective.\n\\end{proof}\n\\subsection[Cohomology module of the centralizer]{Cohomology module of the centralizer of $S^1(\\pm1,b;m)$}\\label{subsection:CohomologyModule}\nWe are now ready to compute the cohomology module of the centralizer of $S^1(\\pm1,b;m)$ with coefficients in in a field $k$. By duality, this is equivalent to determining the homology module.\n\nRecall that the contractible space of invariant compatible almost-complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ decomposes as the disjoint union\n\\[\\mathcal{J}^{S^1}_{\\om_\\lambda}=(\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_m)\\sqcup (\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m'})=:U_{m}^{S^1} \\sqcup U_{m'}^{S^1}\\]\nwhere, for convenience, we set $U_m^{S^1}=\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_m$ and $U_{m'}^{S^1}=\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m'}$. We will show in Chapter~\\ref{Chapter-codimension} the following two important facts:\n\\begin{itemize}\n\\item the strata $U_{m}^{S^1}$ and $U_{m'}^{S^1}$ are submanifolds of $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ (see Corollary~\\ref{cor:StrataAreSubmanifolds}), and \n\\item the stratum $U_{m}^{S^1}$ is open in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$, while $U_{m'}^{S^1}$ is of codimension $2$ (see Theorem~\\ref{codimension_calc}). \n\\end{itemize}\nIn particular, it follows that $U_{m}^{S^1}=\\mathcal{J}^{S^1}_{\\om_\\lambda}-U_{m'}^{S^1}$ is connected. As explained in Appendix~\\ref{Appendix-Alexander-Eells},  Proposition~\\ref{prop:AlexanderEellsGeometric},\n\nthe Alexander-Eells duality induces an isomorphism of homology groups\n\\begin{equation}\\label{eq:AlexanderEellsIsomorphismHomology}\n\\lambda_{*}:H_{p}(U_{m'}^{S^1};k)\\to H_{p+1}(U_{m}^{S^1};k)\n\\end{equation}\n\nNow recall that we also have fibrations\n\\begin{align}\\label{eq:TheTwoMainFibrations}\n\\mathbb{T}^2_m\\to\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\xrightarrow{p_m} \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_m \\simeq U_{m}^{S^1}\\\\\n\\mathbb{T}^2_{m'}\\to\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\xrightarrow{p_{m'}}\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_{m'} \\simeq U_{m'}^{S^1}\\notag\n\\end{align}\nFrom the first fibration, the connectedness of the open stratum $U_{m}^{S^1}$ implies that the group $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ is connected. In turns, the second fibration implies that the codimension 2 stratum $U_{m'}^{S^1}$ is also connected.\nBecause the two inclusions \\[\\mathbb{T}^2_{m} \\hookrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)\\quad \\text{~and~}\\quad \\mathbb{T}^2_{m'} \\hookrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)\\]\ninduce surjective maps in cohomology, the Leray-Hirsch theorem implies that the cohomology module of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ splits as\n\\begin{align}\\label{eq:SplittingCohomology}\nH^*(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda),k) \\cong H^*(U_{m}^{S^1};k) \\otimes H^*(\\mathbb{T}^2_m;k)\\\\\nH^*(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda),k) \\cong H^*(U_{m'}^{S^1};k) \\otimes H^*(\\mathbb{T}^2_{m'};k)\\notag\n\\end{align}\nBy duality, we have corresponding splittings in homology, namely,\n\\begin{align}\\label{eq:SplittingHomology}\nH_*(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda),k) \\cong H_*(U_{m}^{S^1};k) \\otimes H_*(\\mathbb{T}^2_m;k)\\\\\nH_*(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda),k) \\cong H_*(U_{m'}^{S^1};k) \\otimes H_*(\\mathbb{T}^2_{m'};k)\\notag\n\\end{align}\nIt follows that \n\\[H_{p}(U_{m};k)\\simeq H_{p}(U_{m'};k)\\text{~for all~}p\\geq 0\\]\nTogether with the Alexander-Eells isomorphism~(\\ref{eq:AlexanderEellsIsomorphismHomology}) and the connectedness of $U_{m'}$, this implies that\n\\[H_{p}(U_{m};k)\\simeq k\\text{~for all~}p\\geq 0\\]\nUsing the splitting~\\ref{eq:SplittingHomology} and dualizing, we can finally compute the cohomology module of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$.\n\n\\begin{thm}{\\label{cohom}} Consider any of the following circle actions:\n\\begin{itemize}\n\\item (i) $a=1$, $b \\neq \\{0, m\\}$,  and $2\\lambda > |2b-m|$; or\n\\item (ii) $a=-1$, $b \\neq \\{0, -m\\}$, and $2 \\lambda > |2b+m|$.\n\\end{itemize} Then, the cohomology groups of the symplectic centralizer are\n$$H^p\\left(\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda); k\\right) \\simeq \\begin{cases}\nk^4 ~~p \\geq 2\\\\\nk^3 ~~p =1 \\\\\nk ~~ p=0\\\\\n\\end{cases}$$\nfor any field $k$. In particular, the topological group $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ is of finite type.\n\\end{thm}\n\n\n\n\\subsection{The homotopy pushout \\texorpdfstring{$T_m\\leftarrow S^1(\\pm1,b;m)\\to T_{m'}$}{the two inclusions}}\nAs explained in Corollary~\\ref{cor:CircleExtensionsWith_a=1} and Corollary~\\ref{cor:CircleExtensionsWith_a=-1}, the circle actions $S^1(\\pm1,b;m)$ we are considering in this section extend to exactly two toric actions $\\mathbb{T}^2_m$ and $\\mathbb{T}^2_{m'}$. Geometrically, this means that the two tori $\\mathbb{T}^2_m$ and $\\mathbb{T}^2_{m'}$ intersect in $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ along the circle $S^1(\\pm1,b;m)$ and, in particular, that we have two inclusions of Lie groups\n\\[\n\\begin{tikzcd}\nS^{1} \\arrow{r}{(1,b')} \\arrow[swap]{d}{(1,b)} & T^{2}_{m'} \\\\\nT^{2}_{m}  & \n\\end{tikzcd}\n\\]\nIn this section we consider the homotopy pushout of these two inclusions, namely,\n\\[P:=\\pushout(T_m\\leftarrow S^1\\to T_{m'})\\]\nThis pushout is to be understood in the category of topological groups. As we will show later, the topological group $P$ turns out to be a model for the homotopy type of the centralizer $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$.\n\n\\subsubsection*{The Pontryagin algebra of the pushout}\nIn what follows, all k algebras are graded, and the commutator of two elements is given by\n\\[[a,b] = ab - (-1)^{|a|\\cdot|b|}ba\\]\nFor any field $k$, and for any abelian group $A$, the Pontryagin algebra $H_{*}(A;k)$ is isomorphic to the cohomology algebra $H^{*}(A;k)$. It follows that $H_{*}(S^{1})$ is isomorphic to $\\Lambda(t)$, where $t$ is of degree $1$. Similarly, the Pontryagin algebra $H_{*}(T^{2};k)$ is isomorphic to the to an exterior algebra $\\Lambda(z_{1},z_{2})$ generated by two elements of degree one. The pushout diagram of topological groups \n\\[\n\\begin{tikzcd}\nS^{1} \\arrow{r}{(1,b')} \\arrow[swap]{d}{(1,b)} & T^{2}_{m'} \\arrow{d}{} \\\\\nT^{2}_{m} \\arrow{r}{} & P\n\\end{tikzcd}\n\\]\nis homologically free (see Definition~3.1 in~\\cite{AG}). As before, $P$ denotes the pushout in the category of topological groups. By Theorem~3.8 of~\\cite{AG}, the Pontryagin algebra of $P$ is the pushout of $k$ algebras\n\\[\n\\begin{tikzcd}\nH_{*}(S^{1};k) \\arrow{r}{H(1,b')} \\arrow[swap]{d}{H(1,b)} & H_{*}(T^{2}_{m'};k) \\arrow{d}{} \\\\\nH_{*}(T^{2}_{m};k) \\arrow{r}{} & H_{*}(P;k)\n\\end{tikzcd}\n\\]\nwhich is isomorphic to\n\\[\n\\begin{tikzcd}\n\\Lambda(t) \\arrow{r}{(1,b')} \\arrow[swap]{d}{(1,b)} & \\Lambda(y_{1},y_{2}) \\arrow{d}{} \\\\\n\\Lambda(x_{1},x_{2}) \\arrow{r}{} & P^{alg}_{*}\n\\end{tikzcd}\n\\]\nwhere $P^{alg}_{*}\\simeq H_{*}(P;k)$. By the description of the pushout of $k$ algebras as amalgamated products (see \\cite{AG} for more details), the $k$ algebra $P^{alg}_{*}$ can be identified with equivalence classes of finite linear combinations of words in the letters $\\{x_{1},x_{2},y_{1},y_{2}\\}$ under the relations $x_{i}x_{i}=0$, $y_{i}y_{i}=0$, $[x_{1},x_{2}]=0$, $[y_{1},y_{2}]=0$, and $x_{1}+bx_{2}=y_{1}+b'y_{2}$.  From the last equality, we can write $y_{1}=(x_{1}+bx_{2})-b'y_{2}$, which means that we can choose, as generators, the elements \\[\\{t=x_{1}+bx_{2}, ~x_{2}, ~y_{2}\\}\\]\nwith the relations $t^{2}=x_{2}^{2}=y_{2}^{2}=0$, $[t,x_{2}]=[t,y_{2}]=0$. The remaining commutator $w=[x_{2},y_{2}]$ is nonzero and commutes with $t$, $x_{2}$ and $y_{2}$. It follows that any word in $t,x_{2},y_{2}$ is equivalent to a linear combination of words of the form\n\\[w^{\\alpha}x_{2}^{\\beta}y_{2}^{\\gamma}t^{\\delta}\\]\nwith $\\alpha\\in\\mathbb{N}\\cup\\{0\\}$, and $\\beta,\\gamma,\\delta\\in\\{0,1\\}$. Hence, there is an isomorphism of graded algebras\n\\[P^{alg}_{*}\\cong \\frac{F(x_{2},y_{2})}{\\langle x_{2}^{2},y_{2}^{2}\\rangle}\\otimes \\Lambda(t)\\]\nwhere $F(x_{2},y_{2})$ denotes the free graded algebra over $k$ generated by the elements $x_{2}$ and $y_{2}$, and where $x_{2},y_{2},t$ are of degree one. In particular,\n\\[P^{alg}_{n}\\simeq\n\\begin{cases}\nk & n=0\\\\\nk^{3} & n=1\\\\\nk^{4} & n\\geq 2\n\\end{cases}\\]\nand the words $w^{\\alpha}x_{2}^{\\beta}y_{2}^{\\gamma}t^{\\delta}$ form an additive basis of the homology module $P^{alg}_{*}$.\n\nBy duality, the cohomology modules $P^{alg,*}$ are $P^{alg,0} \\simeq k$, $P^{alg,1}\\simeq k^3$, and $P^{alg,n}\\simeq k^4$ for all $n\\geq 2$. The algebra structure of $P^{alg,*}$ can be determined as follows. Let $\\hat t$, $\\hat x_{2}$, and $\\hat y_{2}$ be the duals of the generators of degree $1$, and let $\\hat w$ be the dual of the generator $w=[x_2,y_2]$ of degree $2$.\n\\\\\n\nLet us now recall the Hopf-Borel theorem (see~\\cite{McCleary} Theorem 6.36).\n\\begin{thm}(Hopf-Borel)\nLet k be a field of characteristic p where p may be zero or a prime. A connected Hopf algebra $H$ over $k$ is said to be monogenic if $H$ is generated as an algebra by 1 and one homogeneous element $x$ of degree strictly greater than 0. If $H$ is a monogenic Hopf algebra, then\n\\begin{enumerate}\n    \\item if $p \\neq 2$ and degree $x$ is odd, then $H \\cong \\Lambda(x)$,\n    \\item if $p \\neq 2$ and degree $x$ is even, then $H \\cong k[x] \/\\left\\langle x^{s}\\right\\rangle$ where $s$ is a power of p or is infinite i.e $H \\cong k[x]$,\n    \\item if $p=2$, then $H \\cong k[x] \/\\left\\langle x^{s}\\right\\rangle$ where $s$ is a power of 2 or is infinite.\n\\end{enumerate}\n\\end{thm}\n\nAs $P^{alg,*}$ is an associative, graded commutative Hopf algebra of finite type, the Hopf-Borel theorem (see~\\cite{McCleary} Theorem 6.36) implies that $P^{alg,*}$ is a tensor product of monogenic Hopf algebras. For a field $k$ of characteristic $p$ different from $2$, including $p=0$, $P^{alg,*}$ contains a subalgebra of the form \n\\[A^*=\\Lambda(\\hat t,\\hat x_2,\\hat y_2)\\otimes k[\\hat w]\/\\langle \\hat w^s\\rangle\\]\nwhere $s$ is a power of $p$ or is infinite. Suppose $s=p^n\\geq 3$ is finite. Then, the rank of $A^i$ would coincide with the rank of $P^{alg,i}$ up to degree $i=2s-1$, and we would have $A^i=0$ for $i\\geq 2s$. Therefore, we would need $4$ more generators of degree $2s$ to account for the rank of $P^{alg,2s}$, and their pairwise products would imply that $\\rk P^{alg,4s}>4$. This contradiction shows that $s$ must be infinite and that the rank of $A^i$ equals the rank of $P^{alg,i}$ for all $i\\geq 0$. Consequently, for a field $k$ of characteristic $p\\neq 2$, the $k$-algebra $P^{alg,*}$ is isomorphic to\n\\[P^{alg,*}\\cong \\Lambda(\\hat t,\\hat x_{2},\\hat y_{2})\\otimes S(\\hat w)\\]\nIn characteristic $p=2$, $P^{alg,*}$ is the tensor product of truncated polynomial algebras $k[z_i]\/z^{s_i}_{i}$ where $s_i$ is a power of $2$. As before, it contains a subalgebra of the form\n\\[A^*=k[\\hat t,\\hat x_2,\\hat y_2]\/ \\langle \\hat t^2,\\hat x_2^2,\\hat y_2^2 \\rangle \\otimes k[\\hat w]\/\\langle \\hat w^s\\rangle\\]\nAgain, assuming $s$ is finite forces the existence of $4$ new generators in degree $2s$ whose products would yield too many generators in degree $4s$. Therefore, in characteristic $p=2$, the cohomology algebra of $P$ is isomorphic to\n\\[P^{alg,*}\\cong k[\\hat t,\\hat x_{2},\\hat y_{2}]\/ \\langle \\hat t^2,\\hat x_2^2,\\hat y_2^2 \\rangle \\otimes k[\\hat w]\\]\nIn characteristic zero, the computation of the cohomology ring yields the minimal model of $H^{*}(P)\\otimes\\mathbb{Q}$. As $P$ is a H-space, it is a nilpotent space (see Exercise~1.13 in \\cite{dgcRationalHomotopy}), so that the main theorem of dgc rational homotopy theory applies (see \\cite{dgcRationalHomotopy}, Theorem~2.50) namely, the dimension $\\pi_{p}(P)\\otimes\\mathbb{Q}$ for $p\\geq 2$ is equal to the number of generators of degree $p$ in the minimal model. For $p=1$, as $P$ is a topological group, the dimension of $\\pi_1(P) \\otimes \\mathbb{Q}$ is same as the rank of $H_1(P,\\mathbb{Q})$. Consequently,\n\\[\\pi_{p}(P)\\otimes\\mathbb{Q}\\simeq\n\\begin{cases}\n\\mathbb{Q} & p=0\\\\\n\\mathbb{Q}^{3} & p=1\\\\\n\\mathbb{Q} & p= 2\\\\\n0 & p\\geq 3\n\\end{cases}\\]\n\n\\subsubsection*{The homotopy type of $P$}\nWe want to better understand the homotopy type of the space $P$. To this end, consider the embeddings\n\\begin{align}\nf_{m}:T^{2}_{m}&\\to S^{1}\\times S^{1}\\times S^{1}\\\\\n(x_{1},x_{2})&\\mapsto (x_{1},x_{2},b'x_{1})\\notag\\\\\n&\\notag\\\\\nf_{m'}:T^{2}_{m'}&\\to S^{1}\\times S^{1}\\times S^{1}\\\\\n(y_{1},y_{2})&\\mapsto (y_{1},by_{1},y_{2})\\notag\n\\end{align}\nThe universal property of pushouts implies that there is a unique map $f_{P}:P\\to S^{1}\\times S^{1}\\times S^{1}$ making the following diagram commutative\n\\[\n\\begin{tikzcd}\nBS^{1} \\arrow{r}{B(1,b')} \\arrow[swap]{d}{B(1,b)} & BT^{2}_{m'} \\arrow{d}{} \\arrow[bend left=10]{ddr}{Bf_{m'}} & \\\\\nBT^{2}_{m} \\arrow{r}{}\\arrow[bend right=10,swap]{drr}{Bf_{m}} & BP \\arrow[dotted]{rd}{Bf_{P}} & \\\\\n & & BS^{1}\\times BS^{1}\\times BS^{1}\n\\end{tikzcd}\n\\]\nBy Theorem~3.9 of~\\cite{AG}, the homotopy fiber of $Bf_{P}$ is the pushout of the homotopy fibers of the other maps in the diagram. To determine this fiber, we first replace the maps in the diagram of groups by homotopy equivalent fibrations\n\\[\n\\begin{tikzcd}[column sep=huge]\n\\mathbb{Z}\\ar[swap]{d}{(1,1,a_{1})} & \\mathbb{Z}\\times\\mathbb{Z} \\ar{d}{(1,a_{1},a_{2})}\\ar[swap]{l}{a_{2}} \\ar{r}{a_{1}}& \\mathbb{Z} \\ar{d}{(1,1,a_{1})}\\\\\nT^{2}_{m}\\times\\mathbb{R} \\ar[swap]{d}{(a_{1},a_{2},b'a_{1}e({a_{3}}))} & S^{1}\\times\\mathbb{R}\\times\\mathbb{R} \\ar[swap]{l}{(a_{1},ba_{1}e({a_{2}}),a_{3})} \\ar{r}{(a_{1},b'a_{1}e({a_{3}}),a_{2})} \\ar{d}{(a_{1},ba_{1}e({a_{2}}),b'a_{1} e({a_{3}}))}& T^{2}_{m'}\\times\\mathbb{R}\\ar{d}{(a_{1},ba_{1}e({a_{3}}),a_{2})}\\\\\nS^{1}\\times S^{1}\\times S^{1}\\ar{r}{=} & S^{1}\\times S^{1}\\times S^{1} &S^{1}\\times S^{1}\\times S^{1}\\ar[swap]{l}{=}\n\\end{tikzcd}\n\\]\n\nwhere $a_{i}$ denote the $i^{\\text{th}}$ coordinate function and $e(a_j) = e^{2\\pi i a_j}$. Applying the classifying space functor, this gives\n\\[\n\\begin{tikzcd}[column sep=normal]\nS^{1}\\ar[swap]{d}{} & S^{1}\\times S^{1}\\ar{d}{}\\ar[swap]{l}{\\text{pr}_{2}} \\ar{r}{\\text{pr}_{1}}& S^{1} \\ar{d}{}\\\\\nBT^{2}_{m} \\ar[swap]{d}{} & BS^{1} \\ar[swap]{l}{} \\ar{r}{} \\ar{d}{}& BT^{2}_{m'}\\ar{d}{}\\\\\nBS^{1}\\times BS^{1}\\times BS^{1}\\ar{r}{=} & BS^{1}\\times BS^{1}\\times BS^{1} &BS^{1}\\times BS^{1}\\times BS^{1}\\ar[swap]{l}{=}\n\\end{tikzcd}\n\\]\nwhich shows that the homotopy fiber of the canonical map $BP\\to BS^{1}\\times BS^{1}\\times BS^{1}$ is homotopy equivalent to\n\\[\\hocolim\\{S^{1}\\xleftarrow{\\text{pr}_{2}}S^{1}\\times S^{1}\\xrightarrow{\\text{pr}_{1}}S^{1}\\}\\simeq S^{1}*S^{1}\\simeq S^{3}\\]\nConsequently, $BP$ is the total space of a fibration\n\\[S^{3}\\to BP\\to BS^{1}\\times BS^{1}\\times BS^{1}\\]\nthat, after looping, becomes\n\\[\n\\begin{tikzcd}[column sep=normal]\n & T^{2}_{m'}\\ar[swap]{d}{j_{m'}} \\ar{rd}{f_{m'}=(a_{1},ba_{1},a_{2})}& \\\\\n\\Omega S^{3}\\ar{r} & P\\ar{r}{f_{P}} & S^{1}\\times S^{1}\\times S^{1}\\\\\n & T^{2}_{m}\\ar{u}{j_{m}} \\ar[swap]{ru}{f_{m}=(a_{1},a_{2},b'a_{1})}& \n\\end{tikzcd}\n\\]\nThe map $f_{P}$ admits a section given by \n\\[s(a_{1},a_{2},a_{3})= j_{m'}(a_1, {b'}^{-1}a_3)j_{m}(1,{b}^{-1}a_1^{-1}a_{2})\\]\nIt follows that, as a space, $P$ is weakly homotopically equivalent to the product\n\\[P\\simeq \\Omega S^{3}\\times S^{1}\\times S^{1}\\times S^{1}\\]\nwhich is consistent with the algebraic computations of the previous section.\n\n\\subsection{Homotopy type of \\texorpdfstring{$S^1(\\pm1,b;m)$}{S1(1,b;m)} equivariant symplectomorphisms}\n\nWe are now able to determine the homotopy type of the group $\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ for the circle actions\n\\begin{itemize}\n\\item $S^1(1,b,m)$ when $2\\lambda > |2b-m|$, and\n\\item $S^1(-1,b,m)$ when $2\\lambda > |2b+m|$.\n\\end{itemize}\nSince the arguments are identical in the two cases, we will only discuss the first one. Again, in order to keep the notation simple, we write $\\mathbb{T}^2_m$ and $\\mathbb{T}^2_{m'}$ for the two tori the circle extends to, assuming $m'>m$, and we write $(1,b):S^1\\to\\mathbb{T}^2_m$ and  $(1,b'):S^1\\to\\mathbb{T}^2_{m'}$ for the two inclusions.\\\\\n \n\nFrom the universal property of pushouts, there is a canonical map \n\\[\\Upsilon:P^{alg}_{*}\\to H_{*}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)\\]\nmaking the following diagram commutative\n\\[\n\\begin{tikzcd}\n\\Lambda(t) \\arrow{r}{(1,b')} \\arrow[swap]{d}{(1,b)} & \\Lambda(y_{1},y_{2}) \\arrow{d}{} \\arrow[bend left=10]{ddr}{i_{m'}} & \\\\\n\\Lambda(x_{1},x_{2}) \\arrow{r}{}\\arrow[bend right=10,swap]{drr}{i_{m}} & P^{alg}_{*} \\arrow[dotted]{rd}{\\Upsilon} & \\\\\n & & H_{*}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)\n\\end{tikzcd}\n\\]\n\n\\begin{prop}\nFor every field $k$, the map $\\Upsilon:P^{alg}_{*}\\to H_{*}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)$ is an isomorphism of $k$-algebras.\n\\end{prop}\n\\begin{proof}\nBy definition, the map $\\Upsilon$ is an homomorphism of $k$-algebras. Since $P^{alg}_{i}\\cong H_{i}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)$ for each $i$, it is sufficient to show that $\\Upsilon$ is surjective.\\\\\n\nLet $R$ be the image of $\\Upsilon$. Since the maps $i_{m}$ and $i_{m'}$ are injective, $R$ is the subring generated by the classes $t,x_{2},y_{2}$ viewed as elements in $H_{*}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)$. Consider the two fibrations induced by the action maps\n\\begin{align*\n\\mathbb{T}^2_m\\to\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\xrightarrow{p_m} \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_m \\simeq U_{m}^{S^1}\\\\\n\\mathbb{T}^2_{m'}\\to\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\xrightarrow{p_{m'}}\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_{m'} \\simeq U_{m'}^{S^1}\n\\end{align*}\nObserve that $p_{m}(t)=0$, $p_{m}(x_{2})=0$, $p_{m'}(t)=0$, and $p_{m'}(y_{2})=0$. Now suppose there is an element $z\\in H_{*}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)$, not in $R$, and of minimal degree $d$. Since\n\\begin{multline}\nH_{d}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)\\cong\\\\\nH_{d}(U_{m}^{S^1};k)\\otimes H_{0}(T^{2}_{m};k)\n~\\oplus~ H_{d-1}(U_{m}^{S^1};k)\\otimes H_{1}(T^{2}_{m};k)\n~\\oplus~ H_{d-2}(U_{m}^{S^1};k)\\otimes H_{2}(T^{2}_{m};k)\n\\end{multline}\nwe would have a decomposition\n\\[z=c_{1}\\otimes\\mathbf{1}~\\oplus~ c_{t}\\otimes t~\\oplus~ c_{x_{2}}\\otimes x_{2}+c_{T}\\otimes [T^{2}_{m}]\\]\nwith at least one coefficient $c_{j}$ which is not a polynomial in the classes $p_{m}(w)$ and $p_{m}(y_{2})$. Let $c_{\\ell}$ be such coefficient of minimal degree $d-2\\leq\\ell\\leq d$. The inverse of the Alexander-Eells isomorphism of Proposition~\\ref{prop:AlexanderEellsGeometric}\n\\[\\lambda_{*}^{-1}:H_{p+1}(U_{m}^{S^1})\\to H_{p}(U_{m'}^{S^1})\\]\nwould map $c_{\\ell}$ to a class $c_{\\ell-1}'\\in H_{\\ell-1}(U_{m'}^{S^1};k)$. This class could not be a polynomial in $p_{m'}(w)$ and $p_{m'}(x_{2})$ since, otherwise,  \n\\[c_{\\ell} = \\lambda_{*}(c_{\\ell-1}')=p_{m}\\big([y_{2}\\otimes c_{\\ell-1}']\\big)\\]\nwould be a polynomial in the classes $p_{m}(w)$ and $p_{m}(y_{2})$. \nIn turn, this class $c_{\\ell-1}'$ would have to be the image of some element in $H_{\\ell-1}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)$ not in $R$, contradicting the minimality of~$z$.\n\\end{proof}\n\n\\begin{cor}\nThe map $\\Upsilon:P^{alg}_{*}\\to H_{*}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);\\mathbb{Z})$ is an isomorphism of Pontryagin algebras over the ring of integers.\n\\end{cor}\n\\begin{proof}\nThis follows from the well known fact that a map induces isomorphisms on homology with $\\mathbb{Z}$ coefficients iff it induces isomorphisms on homology with $\\mathbb{Q}$ and $\\mathbb{Z}_{p}$ coefficients for all primes $p$, see~\\cite{Ha}, Corollary~3A.7~(b).\n\\end{proof}\n\n\\begin{thm}\\label{full_homo}\nThe map $\\Upsilon:P\\to\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ is an homotopy equivalence.\n\\end{thm}\n\\begin{proof}\nThe map $\\Upsilon$ is a homology equivalence on integral homology. Because $P$ and $\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ are topological groups, it follows that it is a weak equivalence, see~\\cite{Dror-WhiteheadTheorem}, Example~4.2. Because both spaces are homotopy equivalent to CW-complexes, this weak equivalence is a homotopy equivalence (See~\\cite{Ha}, Proposition~4.74).\n\\end{proof}\n\n\\section{Centralizers of Hamiltonian \\texorpdfstring{$S^1$}{circle} actions on \\texorpdfstring{$S^2 \\times S^2$}{the product}}\n\nWe summarise all the results we have obtained in this chapter in the following theorem. \n\\begin{thm} \\label{circle}\nConsider any Hamiltonian circle action $S^1(a,b;m)$ on $(S^2 \\times S^2, \\omega_\\lambda)$. The homotopy type of the symplectic stabilizer $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ is given in the table below:\n\\begin{center}\n\\begin{tabular}{|p{4.5cm}|p{3.5cm}|p{2cm}|p{4cm}|}\n \\hline\n Values of $(a,b ;m)$ & $\\lambda$ &Number of strata $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ intersects &Homotopy type of $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$\\\\\n\\hline\n {$(0,\\pm 1;m)$, $m\\neq 0$}  &$\\lambda > 1$ & 1 & $S^1 \\times SO(3)$ \\\\\n\\hline\n\\multirow{2}{10em}{$(0,\\pm 1;0)$ or $(\\pm 1,0;0)$} &$\\lambda = 1$  &1  &$S^1 \\times SO(3)$ \\\\\\cline{2-4}\n&$\\lambda >1$ &1  &$S^1 \\times SO(3)$ \\\\\n\\hline\n$(\\pm 1,\\pm1;0)$&$\\lambda = 1$ &1 & $\\mathbb{T}^2 \\times \\mathbb{Z}_2$ \\\\\n\\hline\n $(\\pm 1,0;m), m\\neq0$ &$\\lambda >1$ &1 &$\\mathbb{T}^2$\\\\\n \\hline\n$(\\pm1,\\pm m;m), m \\neq 0$ &$\\lambda > 1$   &1 & $\\mathbb{T}^2$\\\\\n\\hline\n\\multirow{2}{10em}{$(1,b;m), b \\neq \\{ m,0\\}$} \n&$|2b-m| \\geq2 \\lambda \\geq 1$ &1 &$\\mathbb{T}^2$ \\\\\\cline{2-4}\n&$2 \\lambda >|2b-m| \\geq 0$  &2 &$\\Omega S^3 \\times S^1 \\times S^1 \\times S^1$ \\\\\n\\hline\n\\multirow{2}{10em}{$(-1,b;m), b \\neq \\{ -m,0\\}$} \n&$|2b+m| \\geq2 \\lambda \\geq 1$ &1 &$\\mathbb{T}^2$ \\\\\\cline{2-4}\n&$2 \\lambda >|2b+m| \\geq 0$  &2 &$\\Omega S^3 \\times S^1 \\times S^1 \\times S^1$ \\\\\n \\hline\nAll other values of $(a,b;m)$ &$\\forall \\lambda$  &1   &$\\mathbb{T}^2$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\nwhere $\\Omega S^3$ denotes the based loop space of $S^3$. \\qed\n\\end{thm}\n\n\n\n\\chapter{Partition of the space of invariant almost-complex structures}\\label{Chapter-codimension}\nIn the previous section, we calculated the homotopy type of the group of $S^1(\\pm1,b;m)$ equivariant symplectomorphisms assuming that the codimension of the the invariant strata $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{m^\\prime}$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ was 2. In this section, we use deformation theory to show that the invariant strata $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{m^\\prime}$ is a submanifold of $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ and to characterize its normal bundle. We then calculate its codimension. \\\\\n\nWe mimic the techniques in \\cite{AGK} in the equivariant setting. Fix a K\\\"ahler 4-manifold $(M,\\omega,J)$ and an $S^1$ action on $(M,\\omega,J)$ such that $g^*\\omega=\\omega$ and $g^*J= J$ $\\forall g \\in S^1$.\nThe holomorphic $S^1$ action on the base manifold $M$ induces a natural action on the various tensor spaces such as $T^{1,0}M$ or $\\Omega^{0,k}_{J}(M, TM)$. We write $T^{1,0}M^{S^1}$, $\\Omega^{0,k}_{J}(M, TM)^{S^1}$, to denote the $S^1$ invariant elements of these tensor spaces.\n\n\n\\section{Space of invariant complex structures}\nLet $\\mathcal{J}_l$ be the space of almost complex structures of regularity $C^l$ on $M$, endowed with $C^l$ topology. Being a space of sections, $\\mathcal{J}_l$ is a smooth Banach manifold. An explicit atlas can be constructed using the Cayley transform, see for instance~\\cite{Smolentsev}. Given $J\\in\\mathcal{J}_l$, let $\\Omega^{0,1}_{J,l}(M,TM) \\subset \\End_l(T M)$ be the space of  endomorphisms of the tangent bundle of regularity $C^l$ that anticommute with $J$, that is,\n$$\n\\Omega^{0,1}_{J,l}(M,TM)=\\left\\{A  \\in \\End_l(T M) \\mid A J+J A=0\\right\\} \n$$\nThe map $\\phi_{J}: \\Omega^{0,1}_{J,l}(M,TM) \\rightarrow \\mathcal{J}_l$ given by\n$$\n\\phi_{J}(A)= J e^{A}\n$$\nis a local diffeomorphism sending $C^k$ endomorphisms ($k \\geq l$) to $C^k$ almost complex structures. If $J$ is $S^1$ invariant, then $\\phi$ gives a bijection between invariant endomorphisms near $0$ in $\\Omega^{0,1}_{J,l}(M,TM)$ and invariant almost complex structures in a neighborhood of $J$. This shows that the space $\\mathcal{J}^{S^1}_l$ of invariant almost complex structures is a Banach submanifold of $\\mathcal{J}_l$ whose tangent space $T_J\\mathcal{J}^{S^1}_l$ at $J$ is naturally identified with the linear subspace $\\Omega^{0,1}_{J,l}(M,TM)^{S^1}$.\\\\\n\n\n\n\nLet $I^{S^1}_l$ denote the space of invariant and integrable almost complex structures of $M$ with regularity $C^l$. We now show  that $I^{S^1}_l$is a Banach submanifold of $\\mathcal{J}^{S^1}_l$. To this end, let  $N_J(X,Y) = [X,Y] + J\\left([JX,Y] + [X,JY]\\right) - [JX,JY]$ denote the Nijenhuis tensor with respect to $J$. By the Newlander-Nirenberg theorem, we know that $J \\in \\mathcal{J}^{S^1}_l$ is integrable iff $N(J)=0$.\n\\\\\n\nConsider the vector bundle $\\Omega^{0,2}_{l-1}(M,TM)^{S^1}$ over $\\mathcal{J}^{S^1}_l$ whose fibre over $J$ is the space $\\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}$ of $S^1$-invariant $(0,2)$ forms of regularity $C^{l-1}$ with values in the holomorphic tangent bundle $TM$.  \nThe Nijenhuis tensor can be interpreted as a section $N:\\mathcal{J}_l\\to\\Omega^{0,2}_{l-1}(M,TM)$. This section is equivariant since, for all $g \\in S^1$,\n\\begin{align*}\n\\begin{split}\ng \\cdot N_J(X,Y) &:= g \\cdot [X,Y] + g \\cdot J\\left([JX,Y] + [X,JY]\\right) - g \\cdot [JX,JY] \\\\ \n&= g_* [g_*^{-1}X,g_*^{-1} Y]  + g_*J\\left([Jg_*^{-1}X,g_*^{-1}Y] + [g_*^{-1}X,Jg_*^{-1}Y]\\right) - g_* \\cdot [Jg_*^{-1}X,Jg_*^{-1}Y]  \\\\ \n&= [X,Y] + J\\left([JX,Y] + [X,JY]\\right) - [JX,JY] \n\\end{split}\n\\end{align*}\nwhere the last equality follows from the facts that $g_*[X,Y] = [g_*X,g_*Y]$ and that $J$ is invariant. In particular, $N$ takes invariant tensors to invariant tensors, that is,\n\\begin{align*}\n    N: \\mathcal{J}^{S^1}_l &\\to \\Omega^{0,2}_{l-1}(M,TM)^{S^1} \\\\\n      J &\\mapsto N_J\n\\end{align*}\nTo show that $I^{S^1}_l$ is a Banach submanifold, it suffices to show that Nijenhuis tensor intersects the 0-section of the bundle transversally. This is equivalent to showing that, for an integrable $J$, the projection of the derivative to the vertical tangent bundle is surjective. We denote this projection of the derivative of $N$ to the vertical tangent bundle as $\\nabla N$. A priori, $\\nabla N$ depends on a choice of connection on $\\Omega^{0,2}_{l-1}(M,TM)^{S^1}$. However, as shown in Appendix A of \\cite{AGK}, given an arbitrary almost complex structure $J$, we can extend the usual $\\bar\\partial_J$ operator to an operator $\\overline{\\partial}_J:\\Omega^{0,1}_{J,l}(M, TM) \\to \\Omega^{0,2}_{J,l-1}(M, TM)$ so that $\\nabla N_J:=\\nabla N(J)$ is given by the following composition. \n\\[\n\\begin{tikzcd}\n     &\\Omega^{0,1}_{J,l}(M, TM)^{S^1} \\arrow[r,\"dN_J\"] \\arrow[rr,bend right=15,\"\\nabla N_J\"]&\\left(\\Omega^{2}_{l-1}(M, TM \\otimes \\mathbb{C})\\right)^{S^1} \\arrow[r,\"\\pi\"] &\\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}\n\\end{tikzcd} \n\\]\n\\noindent where $\\pi$ is the canonical projection of \n\\[\n\\Omega^{2}_{J,l-1}(M, TM \\otimes \\mathbb{C})^{S^1} \n\\cong \\Omega^{2,0}_{J,l-1}(M,TM)^{S^1} \\oplus \\Omega^{1,1}_{J,l-1}(M,TM)^{S^1} \\oplus \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}\n\\]\nonto the last summand. \n\n\\begin{thm}[\\cite{AGK}, Corollary A.9]\n$\\nabla N(J) = -2 J \\overline \\partial_J$. \\qed\n\\end{thm}\nWe are lead to show that $\\overline \\partial_J: \\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}$ is surjective. This is trivially true  whenever the manifold  as $M$ is 4 dimensional and $H_J^{0,2}(M,TM)^{S^1} = 0$. \n\n\\begin{lemma}{\\label{Lemma:averaging_surj}}\nConsider a complex manifold $(M,J)$ with a holomorphic $S^1$ action. Then the averaging map \n\\begin{align*}\n    \\rho: H_J^{0,2}(M,TM) &\\rightarrow H_J^{0,2}(M,TM)^{S^1} \\\\\n             [\\beta] &\\mapsto \\left[\\int_{S^1}g^*\\beta ~dg\\right]\n\\end{align*}\nis surjective.\n\\end{lemma}\n\\begin{proof}\nThe fact that the above map is well defined follows by noting that the $\\overline \\partial_J$ operator commutes with the averaging operator. The surjectivity follows as the averaging operator is the identity on invariant forms.\n\\end{proof}\n\nNote that the above theorem works for any compact group. By the discussion in the previous paragraph and Lemma~\\ref{Lemma:averaging_surj} we can conclude that $\\overline \\partial_J: \\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}$ is surjective for any holomorphic $S^1$ action on a complex 4-manifold satisfying $H_J^{0,2}(M,TM)=0$.\n\n\n\n\\begin{thm}\nLet $(M,J)$ be a $4$-manifold endowed with an integrable complex structure $J$, and with a holomorphic $S^1$ action. Suppose $H_J^{0,2}(M,TM) = 0$. Then the space $I^{S^1}_l$ of invariant complex structures is a Banach submanifold of $\\mathcal{J}^{S^1}_l$ in a neighbourhood of $J$ with tangent space at $J$ identified with $ \\ker \\overline \\partial_J:\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1} $. Equivalently,\n\\[\nT_J I_l \\cong \\left(\\im  \\overline \\partial_J:\\left(\\Omega^{0,0}_{J,l}(M,TM)\\right)^{S^1} \\to \\Omega^{0,1}_{J,l}(M,TM)^{S^1}\\right) \\oplus H^{0,1}_J(M,TM)^{S^1}\n\\]\n\\end{thm}\n\nLet us now assume that $M$ is symplectic. Let $\\mathcal{J}^{S^1}_{\\omega,l}$ denote the space of all $S^1$ equivariant \\emph{compatible} almost complex structures of regularity $C^l$ endowed with the $C^l$-topology. Our next goal is to show that under some cohomological restrictions, the space of equivariant integrable \\emph{compatible} almost complex structures of regularity $C^l$ denoted by $I^{S^1}_{\\omega,l}$ is a Banach submanifold of $\\mathcal{J}^{S^1}_{\\omega,l}$. We first note that given $J \\in \\mathcal{J}^{S^1}_{\\omega,l}$, the equivariant metric \n$h_J (\\cdot, \\cdot) := \\omega(\\cdot, J\\cdot) - i \\omega(\\cdot, \\cdot)$\n induced by the pair $(\\omega, J)$ identifies \n$T_J \\mathcal{J}^{S^1}_{l} = \\Omega^{0,1}_l(M,TM)^{S^1}$ \nwith the space  $\\left(T^{0,2}\\right)^{S^1}:= \\left(\\Omega^{0,2}(M)\\right)^{S^1} \\otimes \\left(\\Omega^{0,2}(M)\\right)^{S^1}$ of complex equivariant $(0,2)$-tensors via the map\n\\begin{align*}\n\\theta:\\left(T^{0,2}\\right)^{S^1} &\\to \\Omega^{0,1}_l(M,TM)^{S^1} \\\\\nA &\\mapsto \\theta(A):= h_J(A \\cdot, \\cdot)\n\\end{align*}\n\nLet us denote by $S\\Omega^{0,1}_{J,l}(M,TM)^{S^1}$ the tangent space of $T_J \\mathcal{J}^{S^1}_{\\omega,l} \\subset T_J \\mathcal{J}^{S^1}_{l}$ of all equivariant compatible almost complex structures. More explicitly, the tangent space consists of elements $A \\in \\Omega^{0,1}_{J,l}(M,TM)^{S^1} $ such that $\\omega(A\\cdot, \\cdot) = - \\omega(\\cdot, A\\cdot)$. Under the above identification, we can check that $S\\Omega^{0,1}_{J,l}(M,TM)^{S^1}$ gets mapped to the space of symmetric $S^1$ invariant $(0,2)$-tensors which we denote by $\\left(S^{0,2}\\right)^{S^1}$. \n\n\nFurther, the quotient $T_{J} \\mathcal{J}_l^{S^1} \/ T_{J} \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ may be identified with the space of invariant $(0,2)$ forms on $M$ since\n\\[T_{J} \\mathcal{J}_l^{S^1} \/ T_{J} \\mathcal{J}^{S^1}_{\\om_\\lambda, l} = \\Omega^{0,1}_{J,l}(M,TM)^{S^1} \/ S \\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\cong T_{J}^{0,2}(M) \/ \\left(S^{0,2}\\right)^{S^1} = \\Omega_{J}^{0,2}(M)^{S^1}.\\]\nAs before, the Nijenhuis tensor defines a map \n\\[N:\\mathcal{J}^{S^1}_{\\omega,l} \\to \\Omega^{0,2}_{l-1}(M,TM)^{S^1}\\]\nwhose kernel is precisely the subspace $I^{S^1}_{\\omega,l}$. We want to show that the derivative $\\nabla N$ is surjective at all $J\\in I^{S^1}_{\\omega,l}$. As we know that $\\nabla N(J) = -2 J \\overline \\partial_J $, we would need to show that $\\overline\\partial_J:S\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to  \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}$ is surjective. As $M$ is a 4-manifold, all forms in $\\Omega^{0,2}_{l-1}(M,TM)^{S^1}$ are closed, hence to show that the restriction of $\\overline\\partial_J$ to $S\\Omega^{0,1}_{J,l}(M,TM)^{S^1}$ is surjective, it would suffice to show that the vector space $SH_{J}^{0,2}(TM)^{S^1}$ defined below is 0.\n\\begin{align*}\nSH_{J}^{0,2}(TM)^{S^1}\n&:= \\frac{\\ker \\overline\\partial:\\Omega^{0,2}_{J,l-1}(M,TM)^{S^1} \\to \\Omega^{0,3}_{J,l-2}(M,TM)^{S^1}}{\\im \\overline\\partial:S\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to  \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}}\\\\\n&=\\frac{\\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}}{\\im \\overline\\partial:S\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to  \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}}\n\\end{align*}\nAs the above condition is not easy to check directly, we consider the following commutative diagram \n\\[\n\\begin{tikzcd}\n     &0 \\arrow[d] \\arrow[r] &S\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\arrow[r, \"\\overline\\partial\"] \\arrow[d] &\\Omega^{0,2}_{J,l-1}(M,TM)^{S^1} \\arrow[d] \\arrow[r] &0 \\\\\n     &\\Omega^{0,0}_{J,l+1}(M,TM)^{S^1} \\arrow[r,\"\\overline\\partial\"] \\arrow[d,\"\\alpha\"] &\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\arrow[r,\"\\overline\\partial\"] \\arrow[d] &\\Omega^{0,2}_{J,l-1}(M,TM)^{S^1} \\arrow[d] \\arrow[r] &0 \\\\\n     &\\Omega^{0,1}_{J,l+1}(M)^{S^1} \\arrow[r, \"\\overline\\partial\"] &\\Omega^{0,2}_{J,l}(M)^{S^1} \\arrow[r, \"\\overline\\partial\"] &0 \\arrow[r] &0\n\\end{tikzcd}  \n\\]\nwhere the map $S\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to \\Omega^{0,1}_{J,l}(M,TM)^{S^1}$ is just the inclusion and the where the map $\\Omega^{0,1}_{J,l}(TM)^{S^1} \\to \\Omega^{0,2}_{J,l}(M)^{S^1}$ is the quotient \\[\\Omega^{0,1}_{J,l}(TM)^{S^1} \\to \\Omega^{0,1}_{J,l}(TM)^{S^1}\/S\\Omega^{0,1}_{J,l}(M,TM)^{S^1}\\]\nfollowed by identifying $\\Omega^{0,1}_{J,l}(TM)^{S^1}\/S\\Omega^{0,1}_{J,l}(M,TM)^{S^1}$ with\n$\\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}$ (see\\cite{AGK} p. 548 for more details about the identification). The map $\\alpha$ is defined as\n\\begin{align*}\n    \\alpha: \\Omega^{0,0}_{J,l+1}(M,TM)^{S^1} &\\to \\Omega^{0,1}_{J,l+1}(M)^{S^1} \\\\\n    X &\\mapsto \\alpha(X)(Y):= \\omega(X,JY)-i\\omega(X,Y)\n\\end{align*} \nwhere $J \\in I^{S^1}_{\\omega,l}$ and $X,Y \\in \\Omega^{0,0}_{J,l+1}(M,TM)^{S^1}$. We refer the reader to  Appendix B in \\cite{AGK} for  the proof of commutativity of the diagram in the non-equivariant case, and we note that it still holds in the equivariant setting due to the fact that $\\overline\\partial_J$ is equivariant. The above diagram gives rise to a long exact sequence is cohomology\n\\\\\n\n\n\\begin{equation}\\label{les}\n\\begin{aligned} 0 \\longrightarrow H_{J}^{0}(T M)^{S^1} & \\longrightarrow cl\\Omega_{J}^{0,1}(M)^{S^l} \\stackrel{\\delta}{\\longrightarrow} cl S\\Omega_{J}^{0,1}(M,TM)^{S^1} \\stackrel{q}\\longrightarrow H_{J}^{0,1}(T M)^{S^1} \\longrightarrow \\\\ & \\longrightarrow H_{J}^{0,2}(M)^{S^1} \\longrightarrow SH_{J}^{0,2}(TM)^{S^1} \\longrightarrow H_{J}^{0,2}(T M)^{S^1} \\longrightarrow 0 \n\\end{aligned}\n\\end{equation}\n\n\n\n\\noindent where $cl\\Omega_{J}^{0,1}(M)^{S^l}$ denotes the kernel of $\\overline\\partial_J$ in $\\Omega_{J}^{0,1}(M)^{S^l}$ and similarly  $cl S\\Omega_{J}^{0,1}(M,TM)^{S^1}$ is the kernel of $\\overline\\partial_J: S\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to  \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}$. Thus if we had a 4-manifold $M$ with an $S^1$ invariant compatible integrable almost complex structure $J$ such that  $H_J^{0,2}(M) =0$ and $H_J^{0,2}(TM)=0$, noting that $\\overline{\\partial}_J$ takes $S^1$ invariant elements to $S^1$ invariant elements we can conclude that $H_J^{0,2}(M)^{S^1} =0$ and $H_J^{0,2}(M,TM)^{S^1}=0$. Further, it follows from equation \\ref{les} for such a manifold $(M,\\omega,J)$ as above, that $SH_{J}^{0,2}(TM)^{S^1}=0$ and hence $I^{S^1}_{\\omega,l}$ would indeed be a manifold in a neighbourhood of such a $J$. Thus $H_J^{0,2}(M) =0$ and $H^{0,2}_J(TM)=0$ gives us a simpler condition for when $I^{S^1}_{\\omega,l}$ would indeed be a manifold in a neighbourhood of $J$ as required.\n\\\\\n\nAdditionally as  the averaging operator commutes with the $\\overline{\\partial}_J$ operator,  $H_J^{0,2}(M) =0$ implies that $H_J^{0,2}(M)^{S^1} =0$. This tells us that $q:cl S\\Omega_{J}^{0,1}(M,TM)^{S^1} \\to H_{J}^{0,1}(T M)^{S^1} $ is surjective and hence by the first isomorphism theorem we have $\\frac{cl S\\Omega_{J}^{0,1}(M,TM)^{S^1}}{ker ~q:cl S\\Omega_{J}^{0,1}(M,TM)^{S^1} \\to H_{J}^{0,1}(T M)^{S^1}}$ is isomorphic to $H_{J}^{0,1}(T M)^{S^1}$.\nThen the above long exact sequence gives us \n\\[\\frac{cl S\\Omega_{J}^{0,1}(M,TM)^{S^1}}{ker~ q} \\cong \\frac{cl S\\Omega_{J}^{0,1}(M,TM)^{S^1}}{\\im \\delta} \\cong H_{J}^{0,1}(M,TM)^{S^1}\\]\nPutting all this together we obtain the following local description of $I^{S^1}_{\\omega,l}$.\n\\begin{thm}\\label{integrable}\nLet $(M,\\omega,J)$ be a  K\\\"ahler 4-manifold with a K\\\"ahler $S^1$ action. Suppose that $H_J^{0,2}(M)=0$ and $H_J^{0,2}(TM)=0$. Then  $I^{S^1}_{\\omega,l}$ is a Banach submanifold of $\\mathcal{J}^{S^1}_{\\omega,l}$ in a neighbourhood of $J$ with tangent space at $J \\in I^{S^1}_{\\omega,l}$ identified with\n\\[\nT_J I^{S^1}_{\\omega,l} = cl S\\Omega_{J}^{0,1}(M,TM)^{S^1} = \\ker \\overline\\partial_J: S\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\longrightarrow  \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}.\n\\]\nEquivalently,\n\\[T_J I^{S^1}_{\\omega,l} \\cong \\im \\delta \\bigoplus H_{J}^{0,1}(T M)^{S^1}.\\]\n\\end{thm}\n\n\\begin{prop}\nThe conditions $H_J^{0,2}(M) =0$ and $H^{0,2}_J(M,TM)=0$ are satisfied for all the Hirzebruch surfaces.\n\\end{prop}\n\\begin{proof}\nTo prove $H^{0,2}_J(M,TM)=0$ for all Hirzebruch surfaces see the computation in Example 6.2b) p.312 in \\cite{Ko}. To prove, $H_J^{0,2}(M) =0$ we note that the the rank of $H_J^{0,2}(M) =0$ (usually called the geometric genus $p_g$) is a birational invariant. As all Hirzebruch surfaces are birationally equivalent, the result follows from the computation on p.220 in \\cite{Ko}. \n\\end{proof}\nFinally, we would like to show that the strata $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\omega,l}$\nis  a Banach submanifold of $\\mathcal{J}^{S^1}_{\\omega,l}$. The most naive method to try to prove this would be to consider the universal moduli space $\\mathcal{M}(D_s,\\J_{\\om_\\lambda})$ of curves in the class $D_s$ (where $D_s$ is defined to be the class $B -\\frac{s}{2}F$ if $M =S^2 \\times S^2$ or the class $B- \\frac{s+1}{2}F$ if $M= \\CP^2\\# \\overline{\\CP^2}$) and try to prove that the inclusion of $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ is transverse to the projection of $\\mathcal{M}(D_s,\\J_{\\om_\\lambda})$ to the space of all compatible almost complex structures of regularity $C^l$:\n\\[\n\\begin{tikzcd}\n&\\mathcal{M}(D_s,\\J_{\\om_\\lambda})\\arrow[d,\"\\pi\"]\\\\\n\\mathcal{J}^{S^1}_{\\om_\\lambda, l} \\arrow[r,\"i\"] &\\J_{\\om_\\lambda}\n\\end{tikzcd} \n\\] \nHowever, this approach is flawed as the two maps are never transversal. An alternative method is to try to define an equivariant universal moduli space $\\mathcal{M}^{S^1}(D_s,\\mathcal{J}^{S^1}_{\\om_\\lambda, l})$ and argue that the image under the projection to $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ is a Banach submanifold of $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$. This is the approach we implement in the following section.\n\n\\section{Construction of Equivariant moduli spaces}\n\nIn this section we construct moduli spaces of $S^1$ invariant $J$-holomorphic maps into $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$. Recall that $J_m$ is the standard complex structure on the $m^\\text{th}$ Hizerbruch surface $W_m$, where $m=2k$ or $m=2k+1$. Let $D_s$ denote the homology class $B-\\frac{s}{2}F$ in $S^2 \\times S^2$ and let it denote the class $B -\\frac{s+1}{2}F$ in $\\CP^2\\# \\overline{\\CP^2}$. As seen in Chapter~2, there is a $\\mathbb{T}^2_m$ invariant, $J_m$-holomorphic curve $\\overline{D}$ in $W_m$ in the homology class $D_s$. Consider the $S^1(a,b;m)$ action on $(S^2 \\times S^2,\\omega_\\lambda)$ or $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$. From the graph for the circle action $S^1(a,b;m)$ we see that $S^1$ acts on $\\overline{D}$ in a non-effective manner with global stabilizer $\\mathbb{Z}_{a}$. The following theorem is useful in our analysis. \n\n\\begin{lemma}\\label{WellDefinedModuli}\nConsider the $S^1(a,b;m)$ action on $(S^2 \\times S^2,\\omega_\\lambda)$ or $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$. Let $S$ be any $S^1(a,b;m)$-invariant symplectic embedded sphere in the homology class $D_s$ with $s>0$. Then the $S^1$ action on $S$ has global stabilizer isomorphic to~$\\mathbb{Z}_a$. \n\\end{lemma}\n\n\\begin{proof}\nThis follows \\ref{weight} from noting that any other $S^1$ invariant curve passes through the same set of fixed points as $\\overline{D}$ , and hence by \\ref{weight} we that the global stabilizer is the same. \n\\end{proof}\n\nThus we can fix an action on a base sphere, namely the standard $S^1$ action that agrees with the action of $S^1$ on $\\overline{D}$ and consider the moduli space of all equivariant maps $u: (S^2,j_0) \\to (M,J)$ for some $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$. We define $\\mathcal{M}^{S^1}(D_s,\\mathcal{J}^{S^1}_{\\om_\\lambda, l})$ as follows\n\\[\n\\begin{aligned}\n\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}): = \\{(&u,J) ~|~ \\text{$u:S^2 \\to M$ is  equivariant, somewhere injective, J-holomorphic and}  \\\\\n&\\text{represents the class $D_s$}\\}\n\\end{aligned}\n\\]\n\n\n\n\\begin{remark}\nAs we are only interested in the case when $s>0$, the curves in class $D_s$ have negative self intersection and the adjunction formula tells us that these curves are embedded. Thus all somewhere injective curves in class $D_s$ for $s>0$ are embedded. \n\\end{remark}\nAs in the non-equivariant case we now wish to prove that this moduli space is a smooth Banach manifold. To prove this we recall the non-equivariant set up as in Chapter 3 in \\cite{McD} and reformulate it in the equivariant setting.\n\\\\\n\nWe have a bundle \n\\[\n\\begin{tikzcd}\n     &\\mathscr{E}^{S^1}_{q-1,p} \\arrow[d, \"\\pi\"] \\\\\n     & B^{S^1}_{q,p}\\times \\mathcal{J}^{S^1}_l \n\\end{tikzcd}\n\\]\n\\noindent where $\\mathscr{E}^{S^1}_{q-1,p}$ is a vector bundle over $B^{S^1}_{q,p}\\times \\mathcal{J}^{S^1}_l $ with fibre over $(u,J)$ consisting of $S^1$ invariant sections of $\\Omega^{0,1}_J(S^2, u^*TM)$ of Sobolev  regularity $W^{q-1,p}$ i.e \n\\[ \\pi^{-1}(u,J) = \\Gamma(S^2, \\Omega^{0,1}_J(S^2, u^*TM)^{S^1}) \\] and the space $B^{S^1}_{q,p}$ is defined as follows: \\[B^{S^1}_{q,p}:= \\{ u \\in \\left(W^{q,p}(S^2,M)\\right)^{S^1} ~|~ [u]=D_s\\}\\]\nwhere $W^{q,p}(S^2,M))^{S^1}$ denotes the space of equivariant maps  of  Sobolev regularity $W^{q,p}$ from $S^2$ to $M$. \n\\\\\n\nWe would like to show that the  section $\\mathscr{F}^{S^1}(u,J):= (u,\\overline \\partial_J u) :B_{q,p}^{S^1} \\times \\mathcal{J}^{S^1}_l \\to \\mathscr{E}^{S^1}_{q-1,p}$  (where $\\overline \\partial_J u = \\frac{1}{2}(du + J \\circ du \\circ j_{S^2})$) is transversal to the zero section. Note that   $\\left({\\mathscr{F}^{S^1}}\\right)^{-1}(0) = \\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l})$, thus giving it a smooth structure as in the non-equivariant case (See \\cite{McD} Lemma 3.2.1) \n\\\\\n\n In order to show trasnversality,  we consider the projection  of the derivative map $d\\mathscr{F}^{S^1}$ to the vertical tangent bundle and show that this map is surjective at $(u,J)$ when $u$ is a simple equivariant curve. We shall denote this projection by $D\\mathscr{F}^{S^1}$. More explicitly, we need to show that the map \n\n\\[\\begin{aligned}\nD\\mathscr{F}_{u,J}^{S^1}: \\mathscr{W}^{q,p}(S^2, u^*TM)^{S^1} \\times &C^l(M, \\End(TM, J, \\omega))^{S^1} \\\\\n\\to &\\mathscr{W}^{q,p}(S^2, \\Omega^{0,1}_J(S^2, u^*TM))^{S^1}\n\\end{aligned}\n\\]\nis surjective. But by Lemma 3.2.1 in \\cite{McD} we know that the analogously defined linearized derivative $D\\mathscr{F}_{u,J}$ in the non-equivariant case \n\\[\\begin{aligned}\nD\\mathscr{F}_{u,J}: \\mathscr{W}^{q,p}(S^2, u^*TM) \\times &C^l(M, \\End(TM, J, \\omega)) \\\\\n\\to &\\mathscr{W}^{q,p}\\left(S^2, \\Omega^{0,1}_J\\left(S^2, u^*TM\\right)\\right)\n\\end{aligned}\n\\]\nis surjective. As $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda}$, the $\\bar\\partial_J$ operator commutes with the averaging operator with respect to the $S^1$ action.  Averaging the above non-equivariant derivative $D\\mathscr{F}_{u,J}$ by the $S^1$ action then shows that $D\\mathscr{F}_{u,J}^{S^1}$ is surjective as well.\n\n\\begin{thm}\n $\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l})$ is a smooth Banach manifold. \\qed\n\\end{thm}\n\n\n\nWe now consider the projection map \n\\[\n\\begin{tikzcd}\n     &\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}) \\arrow[d,\"\\pi\"] \\\\\n     &\\mathcal{J}^{S^1}_{\\om_\\lambda, l}\n\\end{tikzcd}\n\\]\nTo conclude that the image of $\\pi$ is a submanifold of $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ we need the following theorem whose proof can be found in  \\cite{mardsen}.\n\\\\\n\n\n\n\n\\begin{thm}\\label{mrdsn}(Theorem 3.5.18 in \\cite{mardsen})\nLet $f:M \\to N$ be a smooth map between Banach manifolds such that \n\\begin{enumerate}\n    \\item $\\ker Tf$ is a sub-bundle of $TM$\n    \\item For each $m \\in M$, $f_*(T_m M)$ is closed and splits in $T_{f(m)}N$ \n    \\item f is open or closed onto it's image\n\\end{enumerate}    \nThen $f(M)$ is a smooth Banach submanifold of N.\n\\end{thm}\n\nA map that satisfies the above conditions is called a sub-immersion. \n\n\\begin{lemma}\\label{subimmersion}\nThe projection map $\\pi: \\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}) \\to \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ is a sub-immersion.\n\\end{lemma}\n\\begin{proof}\n\nNote that the $\\ker  d\\pi$ is of constant rank and is the tangent space to the reparametrization group $\\mathbb{C}^*$ is which freely on $\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l})$. Hence $\\ker d\\pi$ is a sub-bundle of $T\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l})$.\n\\\\\n\nNow we show that the image of $d\\pi$ is closed  $T_J\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$. Note first that $T_J\\mathcal{J}^{S^1}_{\\om_\\lambda, l} = S\\Omega_J^{0,1}(M,TM)^{S^1}$, hence $\\pi_* T_{(u,J)} \\mathcal{M}^{S^1}\\left(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}\\right)$ as a subspace of $S\\Omega_J^{0,1}(M,TM)^{S^1}$ can be described as follows: \n\\begin{multline}\\label{proj_tangent}\n \\pi_*T_{(u,J)} \\mathcal{M}^{S^1}\\left(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}\\right)=\n \\left\\{ \\alpha \\in S\\Omega_J^{0,1}(M,TM)^{S^1}~| ~ [\\alpha \\circ du \\circ j_{S^2}]= 0 \\in H_J^{0,1}(S^2, u^*TM)^{S^1}\n\\right\\}\n\\end{multline}\n\n(This follows from noting that the proof of proposition 2.8 in \\cite{AGK} goes through under the presence of a compact group action.)\nLet $\\gamma_n \\in \\pi_* T_{(u,J)} \\mathcal{M}^{S^1}\\left(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}\\right)$ i.e $\\gamma_n \\in \\Omega^{0,1}(M,TM)^{S^1}= T\\mathcal{J}^{S^1}_{\\om_\\lambda, l} $  and satisfies  $[\\gamma_n \\circ du \\circ J_{S^2}]= 0 \\in H_J^{0,1}(S^2, u^*TM)^{S^1}$.  Further assume the sequence $\\gamma_n$ converges to $\\gamma$ in $\\Omega^{0,1}(M,TM)^{S^1}$. Then $[\\gamma \\circ du \\circ j_{S^2}] = 0 \\in H_{j_{S^2}}^{0,1}(S^2, u^*TM)^{S^1}$, thus showing image of $d\\pi$ is closed  $T\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$.\n\\\\\n\nNext to that the image of $d\\pi$ splits in $T\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$, we proceed as follows. We firstly show that the codimension of the image of $d\\pi$ in $T\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ is finite and hence it follows that the image of $d\\pi$ splits. Consider the map \n\\begin{align*}\n    L: S\\Omega_J^{0,1}(M,TM)^{S^1} &\\rightarrow H_{j_{S^2}}^{0,1}\\left(S^2, u^*TM\\right)^{S^1} \\\\\n    \\alpha &\\mapsto [\\alpha \\circ du \\circ j_{S^2}]\n\\end{align*}\n\nBy equation~\\ref{proj_tangent} we see that the kernel of this map is precisely the image of the map $d\\pi$. As $H_{j_{S^2}}^{0,1}\\left(S^2, u^*TM\\right)^{S^1}$ is finite dimensional it follows that the codimension of $d\\pi$ is finite and hence the image of $d\\pi$ in $T\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ splits.\n\\\\\n\nFinally to show that $\\pi$ is open onto it's image we note that, \n\\[\\begin{tikzcd}\n      &\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}) \\arrow[dr,\"\\pi\"] \\arrow[d, \"q\"] \\\\\n      &\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l})\/\\mathbb{C}^* \\arrow[r,\"h\", \"\\cong\"']  & \\text{im} ~\\pi\n\\end{tikzcd}\n\\]\nwhere the map $h:\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l})\/\\mathbb{C}^* \\to \\text{im} \\pi$ is a homeomorphism. As $q$ is a quotient map for a group action, we have that $q$ is an open map and as $\\pi = h \\circ q$, we have the $\\pi$ too is an open map, thus showing that $\\pi$ satisfies all the conditions in the lemma and hence $\\pi$ is a sub-immersion.\n\\end{proof}\n\n\\begin{cor}\\label{cor:StrataAreSubmanifolds}\n$U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ is a Banach submanifold of $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$.\n\\end{cor}\n\\begin{proof}\nThis follows from Lemma~\\ref{subimmersion} and from observing that the image of $\\pi$ is $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$.\n\\end{proof}\n\nWe will now describe the normal bundle of $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ (when $s > 0$) in terms of infinitesimal deformations of complex structures. To this end, we first find a cohomological condition ensuring that the inclusion of complex integrable structures into $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ is transverse to the stratum $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$, in other words, that we have transverse maps\n\\[\n\\begin{tikzcd}\n      &\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}) \\arrow[d,\"\\pi\"] \\\\\n     I^{S^1}_{\\omega,l} \\arrow[r,\"i\"]  &\\mathcal{J}^{S^1}_{\\om_\\lambda, l} \n\\end{tikzcd}\n\\\\\n\\]\n\\begin{lemma} \\label{compliment}\nLet $(M,\\omega_\\lambda,J_s)$  denote any of the Hirzebruch surfaces $(S^2 \\times S^2,\\omega_\\lambda,J_s)$ or $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda,J_s)$, and let $(u,J_s)\\in \\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l})$. Then the induced map $u^*:H_{J_s}^{0,1}\\left(M,TM\\right)^{S^1} \\rightarrow H^{0,1}_{j_{S^2}}\\left(S^2, u^*TM\\right)^{S^1}$ is an isomorphism. \n\\end{lemma}\n\n\\begin{proof}\nFrom  Proposition 3.4 in \\cite{AGK} we know that $u^*:H_{J_s}^{0,1}(M,TM) \\rightarrow H_{j_{S^2}}^{0,1}(S^2, u^*TM) $ is an isomorphism. As $u$ is equivariant this indeed gives us that \\[u^*:H_{J_s}^{0,1}\\left(M,TM\\right)^{S^1} \\longrightarrow H_{j_{S^2}}^{0,1}\\left(S^2, u^*TM\\right)^{S^1}\\]\nis also an isomorphism.\n\\end{proof}\n\n\\begin{lemma}\\label{trans_strata}\nLet $(M,\\omega_\\lambda)$ denote either $(S^2 \\times S^2,\\omega_\\lambda)$ or $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$. Further let $i: I^{S^1}_{\\omega,l} \\hookrightarrow  \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ denote the inclusion and  $\\pi: \\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}) \\to \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ denote the projection. Then,\n\\begin{itemize}\n    \\item $i \\pitchfork \\pi$,\n    \\item the infinitesimal complement (i.e the fibre of the normal bundle)  of $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ at $J_s \\in I^{S^1}_{\\omega,l}$ can be identified with $H^{0,1}_{J_s}(M,TM)^{S^1}$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}Recall by Theorem~\\ref{integrable}, that the tangent space of $I^{S^1}_{\\omega,l}$ was given by \n\\[T_{J_s} I^{S^1}_{\\omega,l} = cl S\\Omega_{J_s}^{0,1}(M,TM)^{S^1} := \\ker \\overline\\partial_{J_s}: S\\Omega^{0,1}_{{J_s},l}\\left(M,TM\\right)^{S^1} \\longrightarrow  \\Omega^{0,2}_{{J_s},l-1}\\left(M,TM\\right)^{S^1}\\]\n\n\nLet $\\gamma \\in T_{J_s} \\mathcal{J}^{S^1}_{\\om_\\lambda, l} = \\left(S\\Omega^{0,1}_{J_s}(M,TM)\\right)^{S^1}$ and define $\\left[\\gamma \\circ du \\circ j_{S^2}\\right] := \\eta \\in H_{j_{S^2}}^{0,1}(S^2, u^*TM)^{S^1}$. To show that $i \\pitchfork \\pi$, we need to produce $\\beta \\in T_{J_s}I^{S^1}_{\\omega,l} = cl S\\Omega_{J_s}^{0,1}(M,TM)^{S^1}$ such that $[\\left(\\gamma - \\beta\\right) \\circ du \\circ j_{S^2}] = 0$. To do so, we consider the following commutative diagram.\n\n\n\\[ \n\\stackinset{l}{13ex}{b}{6ex}{%\n\\scalebox{.8}\n{%\n\\begin{tikzcd}[row sep=9ex, column sep = 13ex, ampersand replacement=\\&]\n[\\alpha] \n    \\arrow[mapsto]{r}\n    \\arrow[mapsto]{d}\n\\& \\left[\\alpha \\circ du \\right]\n    \\arrow[mapsto]{d}  \\\\\n\\left[\\alpha \\circ J_m\\right]\n    \\arrow[mapsto]{r} \n\\& \\begin{array}{c}[\\alpha \\circ du \\circ j_{S^2}] =\\\\ \\left[\\alpha \\circ J_s \\circ du\\right]\\end{array}\n\\end{tikzcd}\n} \n}{%\n\\begin{tikzcd}[row sep = 20ex, column sep = 20ex, ampersand replacement=\\&]\nH_{J_s}^{0,1}(M,TM)^{S^1} \n    \\arrow{r}{u^*} \n    \\arrow[swap]{d}{J_s^*}\n\\& H_{j_{S^2}}^{0,1}(S^2, u^*TM)^{S^1}\n    \\arrow{d}{j_{S^2}^*}  \\\\\nH_{J_s}^{0,1}(M,TM)^{S^1} \n    \\arrow[swap]{r}{u^*} \n\\& H_{j_{S^2}}^{0,1}(S^2, u^*TM)^{S^1}\n\\end{tikzcd}\n}\n\\]\nwhere all the maps $u^*$,$J^*$ and $j_{S^2}^*$ are isomorphisms. Further we have the equality $\\left[\\alpha \\circ du \\circ j_{S^2}\\right] = \\left[\\alpha \\circ J \\circ du\\right]$ as $u$ is $j_{S^2}$-$J_s$ holomorphic. As we know that $H_{J_s}^{0,2}(M)^{S^1} =0$,  from the long exact sequence equation \\ref{les} we see that the quotient map \n\\[ cl S\\Omega_{J_s}^{0,1}(M,TM)^{S^1} \\to H_{J_s}^{0,1}(M,TM)^{S^1}\\] \nis surjective. As both $u^*$ and $J^*$ are isomorphisms, there exists $\\beta \\in cl S\\Omega_{J_s}^{0,1}(M,TM)^{S^1} = T_{J_s} I_{\\omega_\\lambda,l}^{S^1}$ such that $\\left[\\beta \\circ J \\circ du\\right]  =  \\left[\\beta \\circ du \\circ j_{S^2}\\right]= \\eta:= \\left[\\gamma \\circ du \\circ j_{S^2}\\right] $. Hence we indeed have $\\left[\\left(\\gamma -\\beta\\right) \\circ du \\circ j_{S^2} \\right] = 0 $ as required.\n\\\\\n\n\n\nWe now show that the fibre of the normal bundle of $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ at $J_s \\in I^{S^1}_{\\omega,l}$ can be identified with $H^{0,1}_{J_s}(M,TM)^{S^1}$. As seen in the proof of Lemma~\\ref{subimmersion}, we know that there is a map \n\\begin{align*}\n    L: S\\Omega_{J_s}^{0,1}(M,TM)^{S^1} &\\rightarrow H_{J_s}^{0,1}\\left(S^2, u^*TM\\right)^{S^1} \\\\\n    \\alpha &\\mapsto [\\alpha \\circ du \\circ j_{S^2}]\n\\end{align*}\n\nAs the quotient map $ cl S\\Omega_{J_s}^{0,1}(M,TM)^{S^1} \\to H_{J_s}^{0,1}(M,TM)^{S^1}$ is surjective and the maps  $u^*$ and $j_{S^2}$ are isomorphisms, we have that the map $L$ is surjective. As the kernel of L  is the image of $d\\pi$, the cokernel can be identified with $H_{j_{S^2}}^{0,1}\\left(S^2, u^*TM\\right)^{S^1} \\cong H_{J_s}^{0,1}(M,TM)^{S^1}$  Hence the the fibre of the normal bundle of $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ at $J_s \\in I^{S^1}_{\\omega,l}$ can be identified with $H^{0,1}_{J_s}(M,TM)^{S^1}$.\n\\end{proof}\n\n\n\n\\section{Isotropy representations}\n\nAs shown in the previous section, the codimension of $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ inside $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ is equal to the dimension of $H_{J_s}^{0,1}(M,TM)^{S^1}$. This can be calculated using  Lemma~\\ref{trans_strata}. In this section we only perform the calculation for $S^2 \\times S^2$ and we postpone the discussion of $\\CP^2\\# \\overline{\\CP^2}$ to Section~\\ref{Isom_CCC}.\n\\\\\n\n\\subsection{Even Hirzebruch surfaces and their isometry groups}\n\n\n\nLet $2k=m$. By Theorem 4.2 in \\cite{AGK}, the action of the isometry group $K(2k)\\simeq S^{1}\\times \\SO(3)$ on the space $H_{J}^{0,1}(M,TM)$ of infinitesimal deformations is isomorphic to $\\Det\\otimes \\Sym^{2k-2}$, where $\\Det$ is the standard action of $S^{1}=U(1)$ on $\\mathbb{C}^{2}$, and where $\\Sym(\\mathbb{C}^{2})$ is the representation $\\mathscr{W}_{k-1}$ of $\\SO(3)$ induced by the $(2k-2)$-fold symmetric product of the standard representation of $\\SU(2)$ on $\\mathbb{C}^{2}$. We use this fact to calculate the dimension of the $S^1$ invariant subspace of $H_{J}^{0,1}(M,TM)$ and thus obtain the codimension $U_{m,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ inside $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$. \\\\\n\nFollowing \\cite{AGK}, we construct the Hirzebruch surface $\\mathbb{F}_{2k}$ by K\\\"ahler reduction of $\\mathbb{C}^{4}$ under the action of the torus $T^{2}_{2k}$ defined by\n\\[(s,t)\\cdot z = (s^{2k}tz_{1},tz_{2},sz_{3},sz_{4})\\]\nThe moment map is $\\phi(z)=(2k|z_{1}|^{2}+|z_{3}|^{2}+|z_{4}|^{2}, |z_{1}|^{2}+|z_{2}|^{2})$ and the reduced manifold at level $(\\lambda+k,1)$ is symplectomorphic to $(S^{2}\\times S^{2},\\omega_{\\lambda})$ and biholomorphic to the Hirzebruch surface $\\mathbb{F}_{2k}$. In this model, the projection to the base is given by $[(z_{1},\\ldots,z_{4})]\\mapsto [z_{3}:z_{4}]$, the zero section is $[w_{0}:w_{1}]\\mapsto [(w_{0}^{2k},0,w_{0},w_{1})]$, and a fiber is $[w_{0}:w_{1}]\\mapsto [(w_{0}w_{1}^{2k},w_{0}w_{1},0,w_{1})]$. The torus $T^{2}(2k)=T^{4}\/T^{2}_{2k}$ acts on $\\mathbb{F}_{2k}$. This torus is generated by the elements $[(1,e^{it},1,1)]$ and $[(1,1,e^{is},1)]$, and its moment map is $[(z_{1},z_{2},z_{3},z_{4})]\\mapsto(|z_{2}|^{2},|z_{3}|^{2})$. The moment polytope $\\Delta(2k)$ is the convex hull of the vertices $(0,0)$, $(1,0)$, $(1,\\lambda+k)$, and $(0,\\lambda-k)$.\n\n\\[\n\\begin{tikzpicture}\n\\draw (0,0) -- (1,0) ;\n\\draw (0,0) -- (0,3) ;\n\\draw (1,0) -- (1,4) ;\n\\draw (0,3) -- (1,4) ;\n\\node[right]{};\n\\end{tikzpicture}\n\\]\n\nThe isometry group of $\\mathbb{F}_{2k}$ is \n\\[K(2k) = Z_{\\U(4)}(T^{2}_{2k})\/T^{2}_{2k}=(T^{2}\\times \\U(2))\/T^{2}_{2k}\\simeq S^{1}\\times\\PU(2)\\simeq S^{1}\\times\\SO(3)\\]\nwhere the middle isomorphism is given by\n\\[[(s,t), A]\\mapsto (s^{-1}t\\det A^{k}, [A])\\]\nUnder this isomorphism, an element $[(1,a,b,1)]$ of the torus $T(2k)$ is taken to\n\\[\\left(ab^{k},\\begin{bmatrix}b&0\\\\0&1\\end{bmatrix}\\right)=\\left(b^{k}a,\\begin{bmatrix}b^{1\/2}&0\\\\0&b^{-1\/2}\\end{bmatrix}\\right)\\]\nConsequently, at the Lie algebra level of the maximal tori, the map identifying the maximal torus of $K(2k)$ whose lie algebra is denoted by $\\lie{t}^{2}(2k)$ with the maximal torus $S^1 \\times SO(2) \\subset S^1 \\times SO(3)$ whose lie algebra is denoted by $\\lie{t}^{2}$ (where $SO(2)$ is identified with the rotations around the z-axis)  is given by\n\\[\\begin{pmatrix}\n1&k\\\\\n0&1\n\\end{pmatrix}\\]\nThe moment polytope associated to the maximal torus $T^{2}\\subset K(2k)$ is thus  the balanced polytope obtained from $\\Delta(2k)$ by applying the inverse transpose $\\begin{pmatrix}1&0\\\\-k&1\\end{pmatrix}$\nand has the following shape\n\\[\n\\begin{tikzpicture}\n\\draw (0,0) -- (1,-1) ;\n\\draw (0,0) -- (0,1) ;\n\\draw (0,1) -- (1,2) ;\n\\draw (1,2) -- (1,-1) ;\n\\end{tikzpicture}\n\\]\n\n\\subsection{Even isotropy representations and codimension calculation}\n\nLet $J_m$ be the standard $S^1$ invariant integrable almost complex structure in the strata $U_m$, coming from the Hirzebruch surface $W_m$. The action of the isometry group $K(2k)\\simeq S^{1}\\times \\SO(3)$ on the space $H_{J_m}^{0,1}(S^2 \\times S^2,T(S^2 \\times S^2)) \\cong \\mathbb{C}^{m-1}$ (see \\cite{Ko} Example 6.2(b)(4), p.309 for more details about how the isomorphism is obtained) of infinitesimal deformations is isomorphic to $\\Det\\otimes \\Sym^{2k-2}$ , where  $\\Det$ is the standard action of $S^{1}=U(1)$ on $\\mathbb{C}^{2}$, and where $\\Sym(\\mathbb{C}^{2})$ is the representation $\\mathscr{W}_{k-1}$ of $\\SO(3)$ induced by the $(2k-2)$-fold symmetric product of the standard representation  of $\\SU(2)$ on $\\mathbb{C}^{2}$ (see Theorem 4.2 in\\cite{AGK}). We shall denote this $(2k-2)$-fold symmetric product of the standard representation  of $\\SU(2)$ on $\\mathbb{C}^{2}$ as $\\mathscr{V}_{2k-2}$. See \\cite{B-tD} for more details about the representation theory of $\\SO(3)$ and $\\SU(2)$  \\\\\n\n\nThe circle of $\\SO(3)=\\PU(2)=\\U(2)\/\\Delta(S^{1})$\n\\[R(t)=\\begin{pmatrix}1 & 0 & 0\\\\ 0 & \\cos(t) & -\\sin(t)\\\\ 0 & \\sin(t) & \\cos(t)\\end{pmatrix}, ~t\\in[0,2\\pi)\\]\nlifts to \n\\[e(t\/2):=\\begin{pmatrix}e^{it\/2} & 0\\\\ 0 & e^{-it\/2}\\end{pmatrix}\\in\\SU(2)\\]\nAs explained above, the action of $K(2k)$ on  $H^{0,1}_{J_m}(S^2 \\times S^2,T^{1.0}_{J_m}(S^2 \\times S^2)) \\cong \\mathbb{C}^{m-1}$ is isomorphic to $\\Det\\otimes\\Sym^{2k-2}$.Hence to calculate the the codimension we only need to calculate the dimension of the invariant subspace of $H^{0,1}(S^2 \\times S^2,T^{1.0}_{J_m}(S^2 \\times S^2)) \\cong \\mathbb{C}^{m-1}$ under this action. To do so we note that a basis of $\\Sym^{2k-2}$ is given by the homogeneous polynomials $P_{j}=z_{1}^{2k-2-j}z_{2}^{j}$ for $j\\in\\{0,\\ldots,2k-2\\}$. The action of $R(t)$ on $P_{j}$ is\n\\[R(t)\\cdot P_{j}=e(t\/2)\\cdot P_{j}=e^{i\\big(2k-2-2j\\big)t\/2}P_{j}=e^{it(k-1-j)}P_{j}\\]\nso that the action of $(e^{is},R(t))\\subset S^{1}\\times\\SO(3)$ on $P_{j}$ is\n\\[\\left(e^{is},R(t)\\right)\\cdot P_{j}=e^{i\\big(s+t(k-1-j)\\big)}P_{j}\\]\nEach $P_{j}$ generates an eigenspace for the action of the maximal torus $T(2k)$. In particular, the circle $S^{1}(a,b;2k)$ acts trivially on $P_{j}$ if, and only if, \n\\[a+b(k-1-j)=(a,b)\\cdot(1,k-1-j)=0\\]\nfor $j\\in\\{0,2k-2\\}$. Equivalently, we must have\n\\[a+bj=(a,b)\\cdot(1,j)=0\\]\nfor $j\\in\\{1-k,\\ldots,k-1\\}$\n\nHence the dimension of the invariant subspace is given by the number of $j \\in \\{1-k,\\ldots,k-1\\}$ such that $a+bj=0$.\n\\\\\n\n\n\nNote that the above codimension calculation was with respect to the basis of the maximal torus in $K(2n)$. Hence to calculate the codimension for the $S^1(1,b,m) \\subset \\mathbb{T}^2_m$ as in our case, we need to transform the basis by multiplication by the matrix $\\begin{pmatrix}\\frac{m}{2}& -1\\\\1&0\\end{pmatrix}$. Thus it takes the vector $\\begin{pmatrix}1\\\\b\\end{pmatrix}$ in the basis for the standard moment polytope \n\n\\[\n\\begin{tikzpicture}\n\\draw (0,1) -- (3,1) ;\n\\draw (0,1) -- (0,0) ;\n\\draw (0,0) -- (4,0) ;\n\\draw (3,1) -- (4,0) ;\n\n\\end{tikzpicture}\n\\]\n\nto the vector $\\begin{pmatrix}\\frac{m}{2}-b\\\\1\\end{pmatrix} $ in the basis for the balanced polytope (for which we did the above calculations).\n\n\\[\n\\begin{tikzpicture}\n\\draw (0,0) -- (1,-1) ;\n\\draw (0,0) -- (0,1) ;\n\\draw (0,1) -- (1,2) ;\n\\draw (1,2) -- (1,-1) ;\n\n\\end{tikzpicture}\n\\]\n\nTherefore the codimension of $S^1(1,b;m)$ is given by the number of $j \\in \\{1-\\frac{m}{2}, \\cdots, \\frac{m}{2}-1\\}$ such that $(\\frac{m}{2} - b) + j = 0$. Relabelling $j^\\prime$ as $\\frac{m}{2}+j$, we have that the codimension is given by \nthe number of $j^\\prime \\in \\{1, \\cdots , m-1\\}$ such that $j^\\prime=b$.\n \n\\begin{thm}\\label{codimension_calc}\nGiven the circle action $S^1(1,b,m)$ with $2\\lambda > |2b-m|$ and $b \\neq \\{0, m\\}$, the complex codimension of of the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ in given by the number of $j \\in \\{1, \\cdots , m-1\\}$ such that $j=b$.\n\\\\\nSimilarly for the action $S^1(-1,b,m)$ with $2\\lambda > |2b+m|$ and $b \\neq \\{0, -m\\}$, the complex codimension of of the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ in given by the number of $j \\in \\{1, \\cdots , m-1\\}$ such that $j=-b$.\n\\end{thm}\n\\begin{cor}\nFor the circle actions \\begin{itemize}\n  \\item (i) $a=1$, $b \\neq \\{0, m\\}$,  and $2\\lambda > |2b-m|$; or\n\\item (ii) $a=-1$, $b \\neq \\{0, -m\\}$, and $2 \\lambda > |2b+m|$.\n\\end{itemize}The complex codimension of the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ is either 0 or 1.\n\\end{cor} \n\\begin{proof}\nFollows from the calculation and discussion above.\n\\end{proof}\n\n\\begin{remark}\nIn the beginning of the section, we only show that the space $\\mathcal{J}^{S^1}_{\\om_\\lambda, l} \\cap U_{2k,l}$ was a Banach submanifold.  But in order to obtain the topology of the space of $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ with $\\mathbb{C}^\\infty$-topology, we require that the space  $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2k}$ with the $C^\\infty$ topology is a Fr\\'echet manifold and that the codimension of $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2k}$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ is given by the same formula as in Theorem \\ref{codimension_calc}. As this discrepancy exists in the literature even in the non-equivariant case, and as a resolution of this issue is well beyond the scope of this document we do not attempt to resolve this here. \n\\end{remark}\n\n\n\n\\chapter{Odd Hirzebruch surfaces}\\label{Chapter-CCC}\n\n\\section{Homotopy type of \\texorpdfstring{$\\Symp(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$}{Symp(CP2\\#CP2}}\nWe now compute the centralisers for the $S^1$ actions on the odd Hirzebruch surfaces. The theory is extremely analogous to the even Hirzebruch case i.e $S^2 \\times S^2$, hence we shall only point out the key differences. \n\\\\\n\nRecall that  the  odd Hirzebruch surface $W_m$ (where m is odd) is defined as a complex submanifold of $ \\mathbb{C}P^1 \\times \\mathbb{C}P^2$ defined by setting\n\\[\nW_m:=  \\left\\{ \\left(\\left[x_1,x_2\\right],\\left[y_1,y_2, y_3\\right]\\right) \\in \\mathbb{C}P^1 \\times \\mathbb{C}P^2 ~|~  {x^m}_1y_2 - x_2{y^m}_1 = 0 \\right\\}\n\\]\n\nThis manifold is diffeomorphic to $\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2}$. The Torus $\\mathbb{T}^2$ acts on $\\mathbb{C}P^1 \\times \\mathbb{C}P^2$ in the following manner. \n\n\\[\n \\left(u,v\\right) \\cdot  \\left(\\left[x_1,x_2\\right],\\left[y_1,y_2, y_3\\right]\\right) = \\left(\\left[ux_1,x_2\\right],\\left[u^my_1,y_2,vy_3\\right]\\right)\n\\]\n\n(again with m being odd) and the  moment map image looks like\n\n\\[\n\\begin{tikzpicture}\n\\node[left] at (0,2) {$Q=(0,1)$};\n\\node[left] at (0,0) {$P=(0,0)$};\n\\node[right] at (4,2) {$R= (\\lambda - \\frac{m+1}{2} ,1)$};\n\\node[right] at (6,0) {$S=(\\lambda + \\frac{m-1}{2} ,0)$};\n\\node[above] at (2,2) {$B-\\frac{m+1}{2}F$};\n\\node[right] at (5,1) {$F$};\n\\node[left] at (0,1) {$F$};\n\\node[below] at (3,0) {$B+ \\frac{m-1}{2}F$};\n\\draw (0,2) -- (4,2) ;\n\\draw (0,0) -- (0,2) ;\n\\draw (0,0) -- (6,0) ;\n\\draw (4,2) -- (6,0) ;\n\\end{tikzpicture}\\label{oddHirz}\n\\]\nwhere $B$ now refers to the homology class of a line $L$ in $\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2}$ and $F$ refers to the class $L - E$ where $L$ is the class of the line and $E$ is the class of the exceptional divisor. There is a canonical form which we also call $\\omega_\\lambda$ on $\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2}$, which has weight $\\lambda$ on B and 1 on $F$. Note that with our convention, $\\lambda$ must be strictly greater than 1 for the curve in class $E$ to have positive symplectic area.\nAs before all symplectic $S^1$ action on $\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2}$ extend to toric actions. Hence only need to consider sub-circles of the above family of torus actions. The graphs for the different circles are given in Figures \\ref{fig:GraphsWithFixedSurfaces} and \\ref{fig:GraphsIsolatedFixedPoints}. As explained in Theorem~\\ref{strata-CCC}, we have a stratification of the space of almost complex structures i.e the space $\\mathcal{J}_{\\omega_\\lambda}$  of all compatible almost complex structures for the form $\\omega_\\lambda$, decomposes into disjoint Fr\u00e9chet manifolds of finite codimensions\n\\[\n\\mathcal{J}_{\\omega_\\lambda} = U_1 \\sqcup U_3 \\sqcup U_5\\ldots \\sqcup U_{2n-1}\n\\]\nwhere \n\\[ \nU_{2i-1} := \\left\\{ J \\in \\mathcal{J}_{\\omega_\\lambda}~|~ D_{2i-1}:= B-iF \\in H_2(S^2 \\times S^2,\\mathbb{Z})\\text{~is represented by a $J$-holomorphic sphere}\\right\\}.\n\\]\n\n\n\nWe shall now use this stratification to construct fibrations for the action of equivariant symplectomorphism group $\\Symp^{S^1}_{h}(S^2 \\times S^2,\\omega_\\lambda)$ on the space $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s+1}$ of invariant almost complex structures in each stratum. We follow the same notation as in \\ref{not}. The proofs that the following maps are in fact fibrations is exactly the same as the proof given in Section~\\ref{section:ActionOnU_2k}.\n\n\\[\\Stab^{S^1}(\\overline{D}) \\longrightarrow \\Symp^{S^1}_{h}(S^2 \\times S^2,\\omega_\\lambda) \\longtwoheadrightarrow {\\mathcal{S}^{S^1}_{D_{s}}} \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\mathcal{J}^{S^1} \\cap U_{2s+1}\\rule{0em}{2em}\\]\n\n\\[\\Fix^{S^1}(\\overline{D}) \\longrightarrow \\Stab^{S^1}(\\overline{D}) \\longtwoheadrightarrow  \\Symp^{S^1}(\\overline{D}) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} S^1 ~\\text{or}~ SO(3)\\rule{0em}{2em}\\]\n   \n\\[\\Fix^{S^1} (N(\\overline{D})) \\longrightarrow \\Fix^{S^1}(\\overline{D}) \\longtwoheadrightarrow  \\Gauge1^{S^1}(N(\\overline{D})) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} S^1 \\rule{0em}{2em}\\]\n   \n\\[\\Stab^{S^1}(\\overline{F}) \\cap  \\Fix^{S^1}(N(\\overline{D})) \\longrightarrow \\Fix^{S^1}(N(\\overline{D})) \\longtwoheadrightarrow  \\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\mathcal{J}^{S^1}(\\overline{D})\\simeq \\{*\\}\\rule{0em}{2em} \\]\n    \n\\[\\Fix^{S^1}(\\overline{F}) \\longrightarrow \\Stab^{S^1}(\\overline{F}) \\cap  \\Fix^{S^1}(N(\\overline{D})) \\longtwoheadrightarrow  \\Symp^{S^1}(\\overline{F}, N(p_0))  \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\left\\{*\\right\\}\\rule{0em}{2em}\\]\n    \n\\[\\left\\{*\\right\\} \\mathbin{\\textcolor{blue}{\\xleftarrow{\\text{~~~$\\simeq$~~~}}}} \\Fix^{S^1}(N(\\overline{D} \\vee \\overline{F})) \\longrightarrow \\Fix^{S^1}(\\overline{F}) \\longtwoheadrightarrow  \\Gauge1^{S^1}(N(\\overline{D} \\vee \\overline{F}))  \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\left\\{*\\right\\}\\rule{0em}{2em}\\]\n\nWhen the $S^1(a,b;m) \\subset \\mathbb{T}^2_m$ action is of the form $(a,b)= (0,\\pm1)$\n\\[\n\\begin{tikzcd}\n     &\\Fix^{S^1}(\\overline{D}) \\arrow[r] &\\Stab^{S^1}(\\overline{D}) \\arrow[r,twoheadrightarrow] &\\Symp^{S^1}(\\overline{D}) \\\\\n     &S^1 \\arrow[u,hookrightarrow] \\arrow[r] &U(2) \\arrow[u,hookrightarrow] \\arrow[r] &SO(3) \\arrow[u,hookrightarrow] \\\\\n\\end{tikzcd}\n\\]\nFor all other actions with $(a,b) \\neq (0,\\pm1)$ we have \n\\[\n\\begin{tikzcd}\n     &\\Fix^{S^1}(\\overline{D}) \\arrow[r] &\\Stab^{S^1}(\\overline{D}) \\arrow[r,twoheadrightarrow] &\\Symp^{S^1}(\\overline{D}) \\\\\n     &S^1 \\arrow[u,hookrightarrow] \\arrow[r] &\\mathbb{T}^2_{2s+1} \\arrow[u,hookrightarrow] \\arrow[r] &S^1 \\arrow[u,hookrightarrow] \\\\\n\\end{tikzcd}\n\\]\nIn both diagrams, the leftmost and the rightmost arrows are homotopy equivalences. As the diagram above commutes, the 5 lemma implies that the middle inclusions $\\mathbb{T}^2 \\hookrightarrow \\Stab^{S^1}(\\overline{D})$  or $U(2) \\hookrightarrow \\Stab^{S^1}(\\overline{D})$ are also weak homotopy equivalences.\n\n\\begin{thm}\\label{homogenous_CCC}\nConsider the $S^1(a,b;m)$ action on $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ with $\\lambda >1$. If $ \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s+1} \\neq \\phi$, then we have the following homotopy equivalences:\n\\begin{enumerate}\n    \\item when $(a,b) \\neq (0,\\pm1)$, we have $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)\/\\mathbb{T}^2_{2s+1} \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s+1}$;\n    \\item when $(a,b) = (0,\\pm1)$, we have $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)\/U(2) \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s+1}$.\n\\end{enumerate} \\qed\n\\end{thm} \n\n\n\nAs in the $S^2 \\times S^2$ case, to understand the homotopy type of the equivariant symplectomorphism we need to next understand which strata the space of invariant almost complex structures intersect. Consider the circle action $S^1(a,b;m)$ on $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ and $\\lambda >1$, Corollaries~\\ref{cor:IntersectingOnlyOneStratum} and \\ref{cor:IntersectingTwoStrata} imply that\n\\begin{enumerate}\n    \\item if $a=1$, $b \\neq\\{0,m\\}$ and $2\\lambda > |2b-m|+1$, then the space of $S^1(1,b;m)$-equivariant almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects the two strata $U_m$ and $U_{|2b-m|}$.\n    \\item If $a=-1$,$b \\neq\\{0,-m\\}$ and $2\\lambda > |2b+m|+1$, then the space of $S^1(-1,b;m)$-equivariant almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects the two strata $U_m$ and $U_{|2b+m|}$.\n    \\item for all other cases $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects only the stratum $U_m$.\n\\end{enumerate}  \n\n\n\n\n\nAs before, we the homotopy type of the equivariant symplectomorphism group can be easily described whenever $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects only one strata.\n\n\\begin{thm}\\label{CCC1strata}\nConsider the  circle action $S^1(a,b;m)$ on $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$. Under the following numerical conditions on $a,b,m,\\lambda$, the homotopy type of $\\Symp^{S^1}(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ is given by the  table below.\n\\noindent{\\begin{center}\n \\begin{tabular}{|p{4.5cm}|p{3.5cm}|p{2cm}|p{4cm}|}\n\n \\hline\n Values of $(a,b ;m)$ & $\\lambda >1$ &Number of strata $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects &Homotopy type of $\\Symp^{S^1}(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$\\\\\n\\hline\n {$(0,\\pm 1;m)$, $m\\neq 0$}  &$\\forall\\lambda $ & 1 & $\\U(2)$ \\\\\n\\hline\n{$(0,\\pm 1;0)$ or $(\\pm 1,0;0)$} \n&$\\forall\\lambda$ &1  &$\\U(2)$ \\\\\n\\hline\n $(\\pm 1,0;m), m\\neq0$ &$\\forall \\lambda $ &1 &$\\mathbb{T}^2$\\\\\n \\hline\n$(\\pm 1,\\pm m;m), m \\neq 0$ &$\\forall \\lambda$   &1 & $\\mathbb{T}^2$\\\\\n\\hline\n{$(1,b;m), b \\neq \\{m,0\\}$} \n&$|2b-m|+1 \\geq2 \\lambda$ &1 &$\\mathbb{T}^2$ \\\\\n\\hline\n{$(-1,b;m), b \\neq \\{ -m,0\\}$} \n&$|2b+m|+1 \\geq2 \\lambda$ &1 &$\\mathbb{T}^2$ \\\\\n\n\\hline\nAll other values of $(a,b;m)$ except $(\\pm 1,b;m)$&$\\forall \\lambda$  &1   &$\\mathbb{T}^2$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}}\n\\qed\n\\end{thm}\n\n\n\nTheorem~\\ref{CCC1strata} gives the homotopy type of the group of equivariant symplectomorphisms for all circle actions apart from the following two exceptional families:\n\\begin{itemize}\n\\item (i) $a=1$, $b \\neq \\{0, m\\}$,  and $2\\lambda > |2b-m|+1$; or\n\\item (ii) $a=-1$, $b \\neq \\{0, -m\\}$, and $2 \\lambda > |2b+m|+1$.\n\\end{itemize}\n\nAt this point, we proceed as in Chapter 4. Firstly, we show that the map $\\mathbb{T}^2_m \\hookrightarrow \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ induces a map that is injective in homology. \n\\\\\n\nFix a curve $\\overline{F}$ in the homology class $F$ and passing through the fixed points $Q$ and $P$ in figure~\\ref{oddHirz}. Let $\\mathcal{S}^{S^1}_{F,Q}$ denote the space of $S^1$-invariant curves in the class $F$ passing through $Q$ (defined in figure~\\ref{oddHirz}) and let $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\overline{F},\\omega_\\lambda)$ denote the space of $S^1$-equivariant symplectomorphisms of $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ that pointwise fix the curve $\\overline{F}$.\n\\\\\n\nWithout loss of generality, assume that $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{|2b-m|}$ is the strata of positive codimension in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$. As $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ is contractible, $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m = \\mathcal{J}^{S^1}_{\\om_\\lambda} \\setminus \\left(\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{|2b-m|}\\right)$, and the real codimension of $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{|2b-m|}$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ is 2 (See Corollary~\\ref{odd_codim}), it follows that $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ is connected. Further we have that $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda) \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m\/\\mathbb{T}^2_m$ is connected. As the fixed points for the $S^1(\\pm 1,b,m)$ actions are isolated and as $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ is connected, any element $\\phi \\in \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ takes a fixed point for the $S^1$ action to itself. Thus the action of $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ on $\\mathcal{S}^{S^1}_{F,Q}$ is well defined.\n\n\\begin{lemma}\\label{odd_inj}\nThe inclusion $i: \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\overline{F},\\omega_\\lambda) \\hookrightarrow \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ is a homotopy equivalence. \n\\end{lemma}\n\\begin{proof}\nConsider the fibration\n\\begin{equation*}\n    \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\overline{F},\\omega_\\lambda) \\hookrightarrow \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda) \\longtwoheadrightarrow \\mathcal{S}^{S^1}_{F,Q}\n\\end{equation*}\nTo show that the action $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ on $\\mathcal{S}^{S^1}_{F,Q}$ is transitive we note that given $F^\\prime \\in \\mathcal{S}^{S^1}_{F,Q}$, there exists a $J^\\prime \\in \\mathcal{J}^{S^1}_{\\om_\\lambda}$ such that $F^\\prime$ is $J^\\prime$-holomorphic. As $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ is connected, consider a path $J_t$ such that $J_0=J^\\prime$ and $J_1=J_m$ where $J_m$ is the standard complex structure on the $m^\\text{th}$-Hirzebruch surface for which the curve $\\overline{F}$ is holomorphic.\n\\\\\n\nBy Theorem~\\ref{prop_FIndecomposable}, for every $J_t$ we have a family of curves $F_t$ (with $F_0= F^\\prime$ and $F_1 =\\overline{F}$) in class $F$ passing through $Q$  and this curve is $S^1$ invariant as $J_t$ are $S^1$ invariant. By Lemma~\\ref{Au} we have a one parameter family of Hamiltonian symplectomorphisms $\\phi_t \\in \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$  such that $\\phi_t(F_0) = F_t$ for all $t$.\n\nThus it suffices to show that $\\mathcal{S}^{S^1}_{F,Q}$ is contractible to complete the proof. To do this note that,\n\\begin{equation*}\n  \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{F})  \\to \\mathcal{J}^{S^1}_{\\om_\\lambda} \\to \\mathcal{S}^{S^1}_{F,Q}\n\\end{equation*}\n\nwhere $\\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{F})$ denotes the space of $S^1$ invariant almost complex structures for which the curve $\\overline{F}$ is $J$-holomorphic. As both $\\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{F})$ and $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ are contractible, $\\mathcal{S}^{S^1}_{F,Q}$ is contractible as well completing the proof.\n\\end{proof}\n\n Define the following projections just as in the exposition above Theorem~\\ref{inj}. In our case we take the point $\\{*\\}$ to be the point $Q$\nin Figure~\\ref{oddHirz}.\n\n \\begin{align*}\n    s_{0}: S^2 &\\to W_m \\\\\n    \\left[z_{0}, z_{1}\\right] &\\mapsto\\left(\\left[z_{0}, z_{1}\\right],[0,0,1]\\right)\n\\end{align*} \nand the projection to the first factor of $\\mathbb{C}P^1 \\times \\mathbb{C}P^2$ is\n\\begin{align*}\n    \\pi_1: W_m &\\to S^2 \\\\\n    \\left(\\left[z_{0}, z_{1}\\right],\\left[w_{0}, w_{1}, w_{2}\\right]\\right) &\\mapsto\\left[z_{0}, z_{1}\\right]\n\\end{align*}\nWe define a continuous map $h_1:\\Symp_h^{S^1} (\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda) \\to \\mathcal{E}\\left(S^{2}, *\\right)$ by setting\n\\begin{equation*}\n  \\begin{aligned}\nh_1:\\Symp_h^{S^1} (\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda) &\\to \\mathcal{E}\\left(S^{2}, *\\right) \\\\\n\\psi &\\mapsto \\psi_1:= \\pi_{1} \\circ \\psi \\circ s_{0}\n\\end{aligned}\n\\end{equation*}\n\nFurther define the restriction map $r: \\Symp_h^{S^1} (\\CP^2\\# \\overline{\\CP^2},\\overline{F},\\omega_\\lambda) \\to \\mathcal{E}\\left(S^{2}, *\\right)$ by just restricting $\\phi \\in \\Symp_h^{S^1} (\\CP^2\\# \\overline{\\CP^2},\\overline{F},\\omega_\\lambda)$ to the fibre $\\overline{F}$.\n\nThus we have a well defined map \n\\begin{align*}\n    h: \\Symp_h^{S^1} (\\CP^2\\# \\overline{\\CP^2},\\overline{F},\\omega_\\lambda) &\\to \\mathcal{E}\\left(S^{2}, *\\right) \\times \\mathcal{E}\\left(S^{2}, *\\right) \\\\\n    \\phi &\\mapsto (h_1(\\phi), r(\\phi))\n\\end{align*}\n\n\\begin{thm}\nThe inclusion map $i:\\mathbb{T}^2_m \\hookrightarrow \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ induces a map that is injective in homology.\n\\end{thm}\n\n\\begin{proof} By Lemma~\\ref{odd_inj}, it suffices to prove that the inclusion $i:\\mathbb{T}^2_m \\hookrightarrow \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\overline{F},\\omega_\\lambda) $ induces a map that is injective in homology. \n\\\\\n\nComposing with $h$ we have an inclusion of $h \\circ i:\\mathbb{T}^2_m \\hookrightarrow \\mathcal{E}\\left(S^{2}, *\\right) \\times \\mathcal{E}\\left(S^{2}, *\\right)$ and it suffices to show that this map induces a map that is injective in homology. The proof of this claim in analogous to the proof of Theorem~\\ref{inj}.\n\n\\begin{remark}\nAs the argument above doesn't depend on $m$, the same proof as above also shows that for the family of circle actions given by $S^1(1,b,m)$ with $2\\lambda > |2b-m|+1$, the inclusion $\\mathbb{T}^2_{|2b-m|}$ into $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ also induces a map that is injective in homology. Similarly we can also show that for the $S^1(-1,b;m)$ actions with $2\\lambda > |2b+m|+1$, the inclusion $\\mathbb{T}^2_{|2b+m|}$ and $T_m$ into $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ also induces a map that is injective in homology.\n\\end{remark}\n\n\n\\end{proof}\n\nAs in the $S^2 \\times S^2$ case, we have that $i:\\mathbb{T}^2_m \\hookrightarrow \\Symp^{S^1} (\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda) $ induces a map which is injective in homology, From our discussion above we also had that $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)\/\\mathbb{T}^2_m \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ and $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)\/\\mathbb{T}^2_{|2b-m|} \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{|2b-m|}$. Let $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m := P$ and $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{|2b-m|} ;= Q$, as  $i:\\mathbb{T}^2_m \\hookrightarrow \\Symp^{S^1} (\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda) $ induces a map which is injective in homology, further  by Leray-Hirsch theorem we have that \n\\begin{align*}\n    H^*(\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)) \\cong H^*(P) \\bigotimes H^*(\\mathbb{T}^2) \\\\\n     H^*(\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)) \\cong H^*(Q) \\bigotimes H^*(\\mathbb{T}^2)\n\\end{align*}\nAs before we need to compute the codimension of the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{|2b-m|} $in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$. The computation in the section~\\ref{Isom_CCC} shows it to be two (see Corollary~\\ref{odd_codim}). \n\\\\\n\nThus we have the following theorem on the ranks of the homology groups of the space of equivariant symplectomorphisms.\n\n\\begin{thm} Consider the following circle actions on  $\\mathbb{C}P^2 \\# \\overline \\mathbb{C}P^2$ \\begin{itemize}\n\\item (i) $a=1$, $b \\neq \\{0, m\\}$,  and $2\\lambda > |2b-m|+1$; or\n\\item (ii) $a=-1$, $b \\neq \\{0, -m\\}$, and $2 \\lambda > |2b+m|+1$.\n\\end{itemize} \nThen we have \n$$H^p\\left(\\Symp^{S^1}(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda), k\\right) = \\begin{cases}\nk^4 ~~p \\geq 2\\\\\nk^3 ~~p =1 \\\\\nk ~~ p=0\\\\\n\\end{cases}$$\nfor any field $k$.\n\\end{thm}\n\nAs the proof of Theorem~\\ref{full_homo} holds verbatim for the $S^1(a,b;m)$ actions on $\\CP^2\\# \\overline{\\CP^2}$ satisfying the conditions\n\\begin{itemize}\n\\item (i) $a=1$, $b \\neq \\{0, m\\}$,  and $2\\lambda > |2b-m|+1$; or\n\\item (ii) $a=-1$, $b \\neq \\{0, -m\\}$, and $2 \\lambda > |2b+m|+1$.\n\\end{itemize} \nThe above results give us the homotopy type of the centralizer for all circle actions on $\\CP^2\\# \\overline{\\CP^2}$ which we summarise in the table below.\n\n\\begin{thm}\\label{full_homo_CCC}\nFor the $S^1$ action given by the integers $(a,b;m)$, acting on $(\\CP^2\\# \\overline{\\CP^2}, \\omega_\\lambda)$. Under the following numerical conditions on $a,b,m,\\lambda$, the homotopy type of $\\Symp^{S^1}(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ is given by the  table below.\n\\begin{center}\n \\begin{tabular}{|p{4.5cm}|p{3.5cm}|p{2cm}|p{3.75cm}|}\n\n \\hline\n Values of $(a,b ;m)$ & $\\lambda>1$ &Number of strata $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects &Homotopy type of $\\Symp^{S^1}(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$\\\\\n\\hline\n {$(0,\\pm 1;m)$, $m\\neq 0$}  &$\\forall\\lambda$ & 1 & $\\U(2)$ \\\\\n\\hline\n{$(0,\\pm 1;0)$ or $(\\pm 1,0;0)$}\n&$\\forall\\lambda$ &1  &$\\U(2)$ \\\\\n\n\\hline\n $(\\pm 1,0;m), m\\neq0$ &$\\forall\\lambda$ &1 &$\\mathbb{T}^2$\\\\\n \\hline\n$(\\pm 1,\\pm m;m), m \\neq 0$ &$\\forall\\lambda$   &1 & $\\mathbb{T}^2$\\\\\n\\hline\n\\multirow{2}{10em}{$(1,b;m), b \\neq \\{m,0\\}$} \n&$|2b-m|+1 \\geq2 \\lambda$ &1 &$\\mathbb{T}^2$ \\\\\\cline{2-4}\n&$2 \\lambda >|2b-m|+1 $  &2 &$\\Omega S^3 \\times S^1 \\times S^1 \\times S^1$ \\\\\n \\hline\n \\multirow{2}{10em}{$(-1,b;m), b \\neq \\{ -m,0\\}$} \n&$|2b+m|+1 \\geq2 \\lambda $ &1 &$\\mathbb{T}^2$ \\\\\\cline{2-4}\n&$2 \\lambda >|2b+m|+1$  &2 &$\\Omega S^3 \\times S^1 \\times S^1 \\times S^1$ \\\\\n\\hline\nAll other values of $(a,b;m)$ &$\\forall \\lambda$  &1   &$\\mathbb{T}^2$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\qed\n\\end{thm}\n\n\n\\section{Isometry groups of odd Hirzebruch surfaces}\\label{Isom_CCC}\n\nIn this section we calculate the codimension of the smaller strata in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$. Let $M$ denote the manifold $\\CP^2\\# \\overline{\\CP^2}$. By Theorem~\\ref{trans_strata}, we know that the normal bundle to the strata $U_{2s+1} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda}$ can be identified with $H^{0,1}_{J_{2s+1}}(M,T^{1,0}_{J_{2s+1}}M)^{S^1}$. Thus to calculate the codimension of $U_{2s+1} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda}$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$, we need to understand how $S^1$ acts on $H^{0,1}_{J_{2s+1}}(M,T^{1,0}_{J_{2s+1}}M)$ and calculate the dimension of the invariant subspace. We use the convention $m = 2k+1$.\n\\\\\n\n\nThe Hirzebruch surface $\\mathbb{F}_{2k+1}$ is obtained by K\\\"ahler reduction of $\\mathbb{C}^{4}$ under the action of the torus $T^{2}_{2k+1}$ defined by\n\\[(s,t)\\cdot z = (s^{2k+1}tz_{1},tz_{2},sz_{3},sz_{4})\\]\nThe moment map is $\\phi(z)=((2k+1)|z_{1}|^{2}+|z_{3}|^{2}+|z_{4}|^{2}, |z_{1}|^{2}+|z_{2}|^{2})$ and the reduced manifold at level $(\\lambda +k,1)$ is symplectomorphic to $(\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}}P^2,\\omega_{\\lambda})$ and biholomorphic to the Hirzebruch surface $\\mathbb{F}_{2k+1}$. In this model, the projection to the base is given by $[(z_{1},\\ldots,z_{4})]\\mapsto [z_{3}:z_{4}]$, the zero section is $[w_{0}:w_{1}]\\mapsto [(w_{0}^{2k+1},0,w_{0},w_{1})]$, and a fiber is $[w_{0}:w_{1}]\\mapsto [(w_{0}w_{1}^{2k+1},w_{0}w_{1},0,w_{1})]$. The torus $T^{2}(2k+1)=T^{4}\/T^{2}_{2k+1}$ acts on $\\mathbb{F}_{2k+1}$. This torus is generated by the elements $[(1,e^{it},1,1)]$ and $[(1,1,e^{is},1)]$, and its moment map is $[(z_{1},z_{2},z_{3},z_{4})]\\mapsto(|z_{2}|^{2},|z_{3}|^{2})$. The moment polytope $\\Delta(2k+1)$ is the convex hull of the vertices $(0,0)$, $(1,0)$, $(1,\\lambda+k)$, and~$(0,\\lambda-k -1)$.\n\\\\\n\nThe isometry group of $\\mathbb{F}_{2k+1}$ is \n\\[K(2k+1) = Z_{\\U(4)}(T^{2}_{2k+1})\/T^{2}_{2k+1}=(T^{2}\\times \\U(2))\/T^{2}_{2k+1}\\simeq \\U(2)\\]\nwhere the last isomorphism is given by\n\\[[(s,t), A]\\mapsto (s^{-1}t\\det A^{k}) A\\]\nUnder this isomorphism, an element $[(1,a,b,1)]$ of the torus $T(2k+1)$ is taken to\n\\[ab^{k}\\begin{bmatrix}b&0\\\\0&1\\end{bmatrix}\\]\nConsequently, at the Lie algebra level of the maximal tori, the map $\\lie{t}^{2}(2k+1)\\to \\lie{t}^{2}$ is given by\n\\[\\begin{pmatrix}\n1&k+1\\\\\n1&k\n\\end{pmatrix}\\]\nThe moment polytope associated to the maximal torus $T^{2}\\subset K(2k+1)$ is thus  the balanced polytope obtained from $\\Delta(2k+1)$ by applying the inverse transpose $\\begin{pmatrix}-k&1\\\\k+1&-1\\end{pmatrix}$.\n\n\\subsection{Odd isotropy representations and codimension calculation}\n\nThe action of the isometry group $K(2k+1)\\simeq \\U(2)$ on the space $H_{J}^{0,1}(M,TM)$ of infinitesimal deformations is isomorphic to $\\Det^{1-k}\\otimes \\Sym^{2k-1}$, where $\\Det$ is the determinant representation of $\\U(2)$ on $\\mathbb{C}$, and where $\\Sym^{k}(\\mathbb{C}^{2})$ is the $k$-fold symmetric product of the standard representation of $\\U(2)$ on $\\mathbb{C}^{2}$. Using the double covering $S^{1}\\times\\SU(2)\\to\\U(2)$, we see that irreducible representations of $\\U(2)$ correspond to irreducible representations of $S^{1}\\times\\SU(2)$ for which $(-1,-\\id)$ acts trivially. If $A_{n}$ denotes the representation $t\\cdot z=t^{n}z$ of $S^{1}$ on $\\mathbb{C}$, and if $V_{k}$ is the $k$-fold symmetric product of the defining representation of $\\SU(2)$ on $\\mathbb{C}^{2}$, then the irreducible representations of $\\U(2)$ are $A_{n}\\otimes V_{k}$ with $n+k$ even. In this notation, we have the identifications $\\Det = A_{2}$, while $\\Sym=A_{1}\\otimes V_{1}$. Consequently, $\\Det^{1-k}\\otimes \\Sym^{2k-1} = A_{1}\\otimes V_{2k-1}$.\n\nWith respect to the double covering $S^{1}\\times\\SU(2)\\to\\U(2)$, the maximal torus $T^{2}\\subset\\U(2)$ of diagonal matrices $D_{s,t}:=\\begin{pmatrix}e^{is} & 0\\\\ 0 & e^{it}\\end{pmatrix}$ lifts to\n\\[\\left(|D_{s,t}|^{1\/2},\\frac{D}{|D|^{1\/2}}\\right)\n=\\left(e^{i(s+t)\/2},\\begin{pmatrix}e^{i(s-t)\/2} & 0\\\\ 0 & e^{i(t-s)\/2}\\end{pmatrix}\\right)\\]\nAs explained above, the action of $K(2k+1)$ on $H^{0,1}(\\CP^2\\# \\overline{\\CP^2},T^{1.0}_{J_m}(\\CP^2\\# \\overline{\\CP^2})) \\cong \\mathbb{C}^{m-1}$ is isomorphic to  $\\Det^{1-k}\\otimes\\Sym^{2k-1}$. Hence to calculate the the codimension we only need to calculate the dimension of the invariant subspace of the vector space $ H^{0,1}(\\CP^2\\# \\overline{\\CP^2},T^{1.0}_{J_m}(\\CP^2\\# \\overline{\\CP^2})) \\cong \\mathbb{C}^{m-1}$ under the $S^1(1,b;m)$ action. To do so we note that a basis of $\\Sym^{2k-1}$ is given by the homogeneous polynomials $P_{j}=z_{1}^{2k-1-j}z_{2}^{j}$ for $j\\in\\{0,\\ldots,2k-1\\}$. The action of $D_{s,t}$ on $P_{j}$ is\n\\[D_{s,t}\\cdot P_{j}=e^{i\\big((s+t)(1-k)+s(2k-1-j)+tj\\big)}P_{j}\\]\nso that each $P_{j}$ generates an eigenspace for the action of the maximal torus $T(2k+1)$ generated by $D_{s,t}$. In particular, the circle $S^{1}(a,b;2k+1)$ acts trivially on $P_{j}$ if, and only if, \n\\[(a-b)(k-j)+b=(a,b)\\cdot(k-j,j-k+1)=0\\]\n\\\\\nThus the codimension (in the balanced basis of the maximal torus of K(2k+1) is given by the number of $j \\in \\{0,\\ldots,2k-1\\}$ such that \\begin{equation}\\label{formula_CCC}\n    (a-b)(k-j)+b=(a,b)\\cdot(k-j,j-k+1)=0.\n\\end{equation}\n\\\\\n\nNote that just as in the $S^2 \\times S^2$ case, the above codimension calculation was with respect to the basis of the maximal torus in $K(2k+1)$. Hence to calculate the codimension for the $S^1(1,b,m) \\subset \\mathbb{T}^2_m$ as in our case, we need to transform the basis by multiplication by the matrix $\\begin{pmatrix}\\frac{m-1}{2}+1& -1\\\\\\frac{m-1}{2}&-1\\end{pmatrix}$. Thus it takes the vector $\\begin{pmatrix}1\\\\b\\end{pmatrix}$ in the basis for the standard moment polytope \n\n\n\\[\n\\begin{tikzpicture}\n\\draw (0,1) -- (3,1) ;\n\\draw (0,1) -- (0,0) ;\n\\draw (0,0) -- (4,0) ;\n\\draw (3,1) -- (4,0) ;\n\n\\end{tikzpicture}\n\\]\n\nto the vector $\\begin{pmatrix}\\frac{m+1}{2}-b\\\\\\frac{m-1}{2}-b\\end{pmatrix}$ in the basis for the balanced polytope. Hence $a$ and $b$ in equation~\\ref{formula_CCC} above need to be replaced by $\\frac{m+1}{2}-b$ and $\\frac{m-1}{2}-b$ respectively to get the correct codimension for the $S^1(1,b,m)$ action. \n\\\\\n\nThus we have the following theorem. \n\n\n\n\\begin{thm}\\label{codimension_calc2}\nGiven the circle action $S^1(1,b,m)$ on $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ with $2\\lambda > |2b-m|+1$ and $b \\neq \\{0, m\\}$, the complex codimension of of the strata $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ in given by the number of $j \\in \\{1, \\cdots , m-1\\}$ such that $j=b$.\n\\\\\n\nSimilarly for the action $S^1(-1,b,m)$ with $2\\lambda > |2b+m|+1$ and $b \\neq \\{0, -m\\}$, the complex codimension of of the strata $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ in given by the number of $j \\in \\{1, \\cdots , m-1\\}$ such that $j=-b$.\n\\qed\n\\end{thm}\n\n\\begin{cor}\\label{odd_codim}\nFor the circle actions \\begin{itemize}\n  \\item (i) $a=1$, $b \\neq \\{0, m\\}$,  and $2\\lambda > |2b-m|+1$; or\n\\item (ii) $a=-1$, $b \\neq \\{0, -m\\}$, and $2 \\lambda > |2b+m|+1$.\n\\end{itemize}The complex codimension of the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ is either 0 or 1.\n\\qed\n\\end{cor}\n\n\n\\chapter{Equivariant Gauge Groups}\n\nIn this section we basically show how to calculate the homotopy type of the equivariant gauge groups that arise in  lemma \\ref{gauge} and lemma \\ref{ngauge}. \n\\\\\n\nLet $P$ be a principle $G$- bundle over $B$, where $G$ is an abelian group that acts on the right. Let  $H$ be a lie group that act on the base space $B$ and this action lifts to an action on the bundle $P$. We shall denote this action by a left action. Note that $H$ need not act effectively on the space $B$. \n\\\\\n\nLet $\\Gauge1^H(P)$ denote all equivariant (with respect to the action of H) bundle automorphisms i.e equivariant maps $u$ such that the following diagram commutes.\n\\[ \\begin{tikzcd}\n     &P  \\arrow[d,\"\\pi\"] \\arrow[r, \"u\"]  &P \\arrow[dl,\"\\pi\"] \\\\\n     &B \n\\end{tikzcd}\n\\]\n\nGiven $u \\in \\Gauge1^H(P)$, define the map \n\\begin{align*}\n    \\phi_u : P &\\rightarrow G \\\\\nx &\\mapsto \\phi_u(x)\n\\end{align*}\n\nwhere $\\phi_u(x)$ is defined such that \n$$x \\cdot \\phi_u(x) = u(x)$$\n\nLet us now see how the map $\\phi_u$ behaves under the left action of $H$. We have \n\n$$ u(h \\cdot x) = h\\cdot x \\cdot \\phi_u(h \\cdot x) $$ \n\nBut we already have that \n\n$$u(h \\cdot x) = h \\cdot u(x) = h \\cdot x  \\cdot \\phi_u(x)$$\n\nPutting the equalities together and noticing that the $G$ -action is free we get that \n\n$$ \\phi_u(h \\cdot x) = \\phi_u(x) $$ \n\nThat is that the map $\\phi_u$ is invariant under the action of $H$. \n\\\\\n\nAlso from the defintion we can see that $$\\phi_u( x \\cdot g) = g^{-1} \\cdot \\phi_u(x) \\cdot g = \\phi_u(x)$$ \n\nwhere the last equality follows from $G$ being abelian. \n\\\\\n\nDenote by $\\Maps_{H,G}(P,G)$ the space of all $H$ and $G$-invariant smooth functions from P to G equipped with the $C^\\infty$ topology.  Note that as $G$ is abelian, this space of maps is homeomorphic with the space $\\Inv(B,G)$,  of $H$-invariant maps from $B$ to $G$ endowed with the $C^\\infty$-topology. Further we have that the map\n\n\n\\begin{align*}\n    \\rho: \\Gauge1^H (P) &\\rightarrow \\Maps_{H,G}(P,G) \\\\\n    u &\\mapsto \\phi_u\n\\end{align*}\nis an an homeomorphism (with both spaces endowed with the $C^\\infty$-topology) with the inverse being constructed using the definition  $x \\cdot \\phi_u(x) = u(x)$. \n\\\\\n\nThus $\\Gauge1^H(P)$ is homeomorphic to $\\Inv(B,G)$.\n\\\\\n\nLet us now use this to calculate the Homotopy type of the Gauge groups in the fibrations in Chapter 2. \n\\\\\n\nConsider a rank 2 symplectic normal bundle of $\\overline{D}$. Let us fix an equivariant arbitrary compatible fibre wise almost complex structure $J$ on $N(\\overline{D})$.  As this is a rank two bundle, the structure group is $Sp(2)$, and this now can be reduced to $U(1)= S^1$, and  the two bundles are isomorphic. Thus the space of symplectic automorphisms of the original bundle is homeomorphic to the space of symplectic automorphisms of the reduced bundle.\n\\\\\n\nIn our case we have a right group action of $S^1$ on this bundle and we are interested  in the equivariant symplectic automorphisms of this bundle.This is homotopic to the space of Equivariant symplectic automorphisms of the $U(1)$ bundle.(As the reduction of the structure group can be done equivariantly) And as $U(1) = S^1$ the space of  Equivariant symplectic automorphisms of the $U(1)$ bundle is the same as $\\Gauge1^{S^1}(P)$ where $P$ is the associated principal bundle. This is homeomorphic to $\\Inv(S^2,S^1)$ from the above discussion.\n\\\\\n\nFinally note that for any non-trivial $S^1$ action on $S^2$ (possibly non-effective) the quotient space under this action is just the interval. Hence the space $\\Inv(S^2,S^1)$ is just the space of smooth maps from the interval to $S^1$.\n\\\\\n\nBefore we embark on trying to calculate the homotopy type of  $\\Gauge1^{S^1}(N(\\overline{D}))$ (See Section~\\ref{not} for notation), we would need the following technical lemma. Note that in all our calculations above we have used the $C^\\infty$-topology on $\\Gauge1^{S^1}(N(\\overline{D}))$ and $\\Inv_{S^1}(S^2,S^1)$. Let $\\Gauge1_c^{S^1}(N(\\overline{D}))$ denote the space of continuous $S^1$-equivariant gauge transformations of the bundle $N(\\overline{D})$ equipped the $C^0$-topology. Using the same argument at the beginning of the section we can show that $\\Gauge1_c^{S^1}(N(\\overline{D}))$ is homotopic to the space $\\Inv_{S^1,c}(S^2,S^1)$ of continuous $S^1$-invariant maps from $S^2$ to $S^1$. Then we have that,\n\n\n\\begin{lemma}\\label{wolken}\nThe space $\\Gauge1^{S^1}(N(\\overline{D}))$ with the $C^\\infty$-topology is homotopic to the space $\\Gauge1_c^{S^1}(N(\\overline{D}))$ equipped with the $C^0$-topology.\n\\end{lemma}\n\nThe proof of this lemma follows from an equivariant version of the arguments used to prove Theorem 3.2.13 in \\cite{homgauge}. \n\n\\begin{lemma} \\label{Gauge(D)}\n$\\Gauge1^{S^1}(N(\\overline{D})) \\simeq \\Gauge1_c^{S^1}(N(\\overline{D})) \\simeq \\Inv_{S^1,c}(S^2,S^1) \\simeq S^1$\n\\end{lemma}\n\n\\begin{proof}\n$\\Gauge1^{S^1}(N(\\overline{D})) \\simeq \\Gauge1_c^{S^1}(N(\\overline{D})) \\simeq \\Inv_{S^1,c}(S^2,S^1)$ follows from Lemma \\ref{wolken} and the discussion above. \n\\\\\n\nLet $\\Maps(S^2\/S^1,S^1)$ denote the space of continuous maps from $S^2\/S^1$ to $S^1$.Then the space $\\Inv_{S^1,c}(S^2,S^1)$ is homeomorphic to the space $\\Maps(S^2\/S^1,S^1)$. Further we note that as $S^2\/S^1$ is homeomorphic to an interval $[0,1]$, the space $\\Maps(S^2\/S^1,S^1)$  can be identified  the space of continuous maps from the interval $[0,1]$ to $S^1$ which we denote by $\\Maps([0,1],S^1)$ . Let $p_0$ be a fixed point for the $S^1$ action on $S^2$. Then we consider the following fibration, \n\n\n\\[\n\\begin{tikzcd}\n     &\\left\\{*\\right\\} \\simeq \\Inv_{S^1,c}\\left(\\left(S^2,p_0\\right), \\left(S^1,id\\right)\\right) \\arrow[r,hookrightarrow] &\\Inv(S^2,S^1) \\arrow[r,\"ev\"] &S^1 \\\\ \n\\end{tikzcd}\n\\]\n\nWhere the map $ev:\\Inv(S^2,S^1) \\rightarrow S^1 $ is just the evaluation map at the fixed point (for the $S^1$ action on $S^2$) $p_0$ and the space $\\Inv_{S^1,c}\\left(\\left(S^2,p_0\\right), \\left(S^1,id\\right)\\right)$ is the space of all continuous maps from $S^2$ to $S^1$, invariant under the $S^1$ action and send the point $p_0$ to the identity on $S^1$. As above, the space $\\Inv\\left(\\left(S^2,p_0\\right), \\left(S^1,id\\right)\\right)$ can be identified with the space of continuous maps from the the interval $[0,1]$ to $S^1$ that send $0$ to the the identity on $S^1$. This space of pointed maps from $[0,1]$ to $S^1$ is contractible, thus completing the proof.\n\\end{proof}\n\n\\begin{remark}\nWe need the point $p_0$ to be a fixed point as the evaluation map has to be surjective.\n\\end{remark}\n\n\\begin{lemma} \\label{Gauge(N(D))}\n$\\Gauge1^{S^1}(N(\\overline{D} \\vee \\overline{F})) \\simeq \\left\\{*\\right\\}$\n\\end{lemma}\n\n\\begin{proof}\nAnalogous to our method above, we get that $\\Gauge1^{S^1}(N(\\overline{D} \\vee \\overline{F}))$ is just the space of continuous maps from the following configuration to $S^1$ that send some  neighbourhood of the crossing to the identity in $S^1$\n\\begin{figure}[H]\n    \\centering\n   \n   \n   \n\\begin{tikzpicture}[thick, scale=0.6]\n\\draw (0,-3) -- (0,-1);\n\\draw (0,1) --(0,3);\n\\draw[red] (0,1) --(0,-1);\n\\draw[red] (1,0) --(-1,0);\n\\draw (-1,0) -- (-3,0);\n\\draw (1,0) --(3,0);\n\\draw (10,0) circle (2cm);\n\\draw[->] (4,0.5) -- (7,0.5);\n\\node [right] at (12.2,0) {id};\n\\node[below] at (1.3,0) {$\\overline{D}$};\n\\node[right] at (0,2) {$\\overline{F}$};\n\\draw [fill,red] (12,0) circle (0.1cm);\n\\end{tikzpicture}\n\\end{figure}\n\nAnd the space of such maps is indeed contractible thus completing the proof. \n\\end{proof}\n\\begin{comment}\n\n\nWe now want to carry out similar computation but for action of finite abelian groups $\\mathbb{Z}_n$ on the bundle. As discussed before, we have that $\\Gauge1^{Z_n}(N(\\overline{D})) \\simeq \\Inv_{Z_n}(S^2,S^1)$ where  $\\Inv_{Z_n}(S^2,S^1)$ denotes the space of $Z_n$ invariant maps from $S^2$ to $S^1$. Further let $\\Gauge1_c^{Z_n}(N(\\overline{D}))$ denote the space of continuous $\\mathbb{Z}_n$ equivariant gauge transformations. As in Lemma \\ref{wolken} we have that \n$\\Gauge1^{Z_n}(N(\\overline{D})) \\simeq \\Gauge1_c^{Z_n}(N(\\overline{D}))$. Further, just as in the $S^1$ case we may identify  $\\Gauge1_c^{Z_n}(N(\\overline{D}))$ with the space $\\Inv_{Z_n, c}(S^2,S^1)$ of continuous $\\mathbb{Z}_n$ invariant maps from $S^2$ to $S^1$.\n\\\\\n\nPutting all this together we have, \n\n\\begin{lemma}\\label{finitegauge}\n$\\Gauge1^{Z_n}(N(\\overline{D})) \\simeq \\Gauge1_c^{Z_n}(N(\\overline{D})) \\simeq  \\Inv_{Z_n, c}(S^2,S^1) \\simeq S^1$\n\\end{lemma}\n\n\\begin{proof}\nThe homotopy equivalences $\\Gauge1^{Z_n}(N(\\overline{D})) \\simeq \\Gauge1_c^{Z_n}(N(\\overline{D})) \\simeq  \\Inv_{Z_n, c}(S^2,S^1)$ are all explained above. Thus we only need to show that $\\Inv_{Z_n, c}(S^2,S^1) \\simeq S^1$. In our case we know that the $Z_n$ action on $S^2$ are in fact restrictions of $S^1$ actions on $S^2$, hence they are rotations about  fixed points of $S^2$. Note that $\\Inv_{Z_n,c}(S^2,S^1)$ is homeomorphic to the space $\\Maps(S^2\/Z_n, S^1)$ of continuous maps from $S^2\/\\mathbb{Z}_n$ to $S^1$. Further,  $S^2\/Z_n$ is homeomorphic to $S^2$ and hence $\\Maps(S^2\/Z_n, S^1) \\cong  \\Maps(S^2, S^1)$. Finally, we note that $\\Maps(S^2, S^1) \\simeq S^1$ thus completing the result.\n\\end{proof}\n\n\\begin{lemma}\\label{wedgegauge}\n$\\Gauge1^{Z_n}(N(\\overline{D} \\vee \\overline{F})) \\simeq \\left\\{*\\right\\}$\n\\end{lemma}\n\n\\begin{proof}\nAnalogous to the proof of Lemmas \\ref{Gauge(N(D))} and \\ref{finitegauge},  we can identify the group $\\Gauge1_{Z_n}(N(\\overline{D} \\vee \\overline{F}))$ with maps from $S^2 \\vee S^2$ that send a neighbourhood of the wedge point to the identity in $S^1$. The space of such maps is contractible. \n\\end{proof}\n\nFinally, we need to understand the homotopy type of $\\mathbb{Z}_n$ equivariant symplectomorphisms $\\Symp^{\\mathbb{Z}_n}(S^2)$, in our analysis in Chapter 6. \n\\begin{lemma}\\label{Wang}\nConsider a symplectic action of $\\mathbb{Z}_n$ on $S^2$, then the space $\\Symp^{\\mathbb{Z}_n}(S^2)$ is homotopic  to  $\\SO(3)$, if $\\mathbb{Z}_n$ fixes $S^2$ pointwise, and is homotopic to $S^1$ otherwise.\n\\end{lemma}\n\\begin{proof}\nLet $\\psi \\in \\Symp^{\\mathbb{Z}_n}(S^2)$, consider the graph $\\tilde \\psi$ of $\\psi$ i.e\n\n\\begin{align*}\n    \\tilde \\psi: S^2 &\\rightarrow S^2 \\times S^2 \\\\\n    z &\\mapsto (z,\\psi(z))\n\\end{align*}\n\nLet $\\SO(3)^{\\mathbb{Z}_n}$ denote the centraliser of $Z_n$ inside $\\SO(3)$. Choose a $\\mathbb{Z}_n$ equivariant metric for the product $\\mathbb{Z}_n$ action on $S^2 \\times S^2$ coming from the $\\mathbb{Z}_n$ action on $S^2$. Then by Theorem C, Corollary C and Corollary 4.1 in \\cite{wang}, the mean curvature flow with respect to this equivariant metric gives us a canonical homotopy of $\\psi$ to an element inside $\\SO(3)^{\\mathbb{Z}_n}$. Further this homotopy is identity on all the elements of $\\SO(3)^{\\mathbb{Z}_n}$. Thus the map we get is in fact a deformation retract of $\\Symp^{\\mathbb{Z}_n}(S^2)$ and $\\SO(3)^{\\mathbb{Z}_n}$. Note $\\SO(3)^{\\mathbb{Z}_n} = \\SO(3)~ or~ S^1$ depending on whether $\\mathbb{Z}_n$ is in the centre of $SO(3)$ or not respectively, thus proving the claim.\n\\end{proof}\n\\end{comment}\n\\chapter[Equivariant Gompf argument]{Holomorphic configurations and equivariant Gompf argument}\n\n\n\\begin{thm}\\label{transverse}\nLet $G$ be a compact group. Let A and B be two $G$-invariant symplectic spheres in a 4-dimensional symplectic manifold $(M,\\omega)$ intersecting $\\omega$-orthogonally at a unique fixed point $p$ for the $G$ action. Then there exists an invariant $J \\in \\mathcal{J}_\\omega^{G}$ such that both A and B are $J$-holomorphic. Here $\\mathcal{J}_\\omega^{G}$ denotes the space of $G$ invariant compatible almost complex structures on $M$. \n\\end{thm}\n\n\\begin{proof}\nThe proof follows from mimicking the proof of Lemma A.1 in \\cite{Evans} under the presence of a group action.\n\\iffalse\nAs $A$ and $B$ are symplectic submanifolds of $M$. There exists a compatible(with respect to the restricted form on $A$ and $B$) $G$- invariant  almost complex $J_A$ and $J_B$ such that $A$ is $J_A$-holomorphic  and $B$ is $J_B$-holomorphic. This gives us  invariant metrics $g_A$ and $g_B$ defined on the tangent bundles $TA$ and $TB$ respectively. \n\\\\\n\nIn a neighbourhood of $p$, choose a chart $U_p$ such that $A$ and $B$ are $\\mathbb{R}^2 \\times \\{0\\}$ and $\\{0\\}\\times \\mathbb{R}^2$ respectively. This is possible as $A$ and $B$ are symplectically orthogonal. Define the metric $\\Tilde{g}:= g_A \\oplus g_B$ on $T_p M$. We note that \n\nNow extend $g_A$ and $g_B$ in a neighbourhood of  the point $p$  as follows. Pick a smooth chart $U$ such that in that chart the submanifold $A$ is $\\{0\\}\\times \\mathbb{R}^2$ and the submanifold $B$ is $\\mathbb{R}^2 \\times \\{0\\}$. We think of a metric as a section of $TM \\otimes T^*M$ that is positive definite. Further we demand that $U$ is a trivialising chart for the bundle $TM \\otimes T^*M$. Then in the neighbourhood $U$ both $g_A$ and $g_B$ are function from $\\mathbb{R}^4$ to $\\mathbb{R}^4$.  We define a smooth extension of $g_A$ and $g_B$ to the whole neighbourhood as follows. We denote the extension by $\\Tilde{g}$.\n\n$$ \\Tilde{g}\\left((v_1,v_2,v_3,v_4)\\right):= g_A\\left((0,0,v_3,v_4\\right) + g_B\\left(v_1,v_2,0,0)\\right) - g_A(0,0,0,0)$$\n\\\\\n\nThe restriction $\\Tilde{g}$ to $\\{0\\} \\times \\mathbb{R}^2$ is $g_A(0,0,v_1,v_2) + {g_B(0,0,0,0)} - g_A(0,0,0,0)$. As $g_A(0,0,0,0)= g_B(0,0,0,0)$, we have $g_A(0,0,v_1,v_2) + {g_B(0,0,0,0)} - g_A(0,0,0,0) = g_A(0,0,v_1,v_2) = g_A$. Similarly we can show that the restriction of $\\Tilde{g}$ to $\\mathbb{R}^2 \\times \\{0\\}$ is $g_B$. Thus showing that we can extend $\\Tilde{g}$ smoothly to a neighbourhood $U$ of $p$. Apriori this extension need not be equivariant, we further choose a smaller neighbourhood $\\Tilde{U} \\subset U$ which is invariant under the $G$ action. And we average $\\Tilde{g}$ on $\\Tilde{U}$ under the $G$ action to get a function $g^\\prime$ which is equivariant and extends $g_A$ and $g_B$. \n\\\\\n\nFinally, given any other point $x \\in M$, $x \\neq p$ we can find a $G$ invariant open set $U_x$, such that either $U_x \\cap A \\neq \\emptyset$ or $U_x \\cap B \\neq \\emptyset$ and a $G$-equivariant metric $\\tilde{g}_{U_x}$ such that $\\tilde{g}_{U_x}(a) = g_A$ for all $a \\in U_x cap A$ and $\\tilde{g}_{U_x}(b) = g_B(b)$ for all $b \\in U_x \\cap B$. We then patch together all the above metrics using a $G$-invariant partition of unity for the open cover $\\bigcup_{x\\in M\\setminus p} U_x \\cup \\Tilde{U}$, thus giving us a metric with the required properties.\n\\fi\n\\iffalse\nwhere the co-ordinates is gotten by realizing $T_pM \\cong T_p A \\oplus T_p B$ i.e $(v_1,v_2)$ and $(u_1,u_2) \\in T_p A $ and similarly $(v_3,v_4)$ and $(u_3,u_4) \\in T_p B $. Further we note that $\\Tilde{g}_p$ is $S^1$-invariant. \n\\\\\n\nNow consider the equivariant symplectic normal bundles $N(A)$ to $A$ and $N(B)$ which is normal to $B$. At $p$ we see that the fibre symplectic normal bundle $N(A)$ agrees with $T_p B$. We already have a metric defined on $N_p(A) = T_p B$, we define an $S^1$-invariant fibrewise metric $\\Tilde{g}_A$ on $N(A)$ such that $\\Tilde{g}_A$ at the point $p$ agree with $g_B$ at the point $p$. Similarly define $\\Tilde{g}_B$ on $N(B)$ such that $\\Tilde{g}_B$ at the point $p$ agree with $g_A$ at the point $p$.\n\\\\\n\nDefine $h_A:= g_A \\oplus \\Tilde{g}_A$ and $h_B := g_B \\oplus \\Tilde{g}_B$ and define the metric on $TM|_{A \\cup B}$\n\\[\nh_x := \\begin{cases}\n{h_A} ~~ x \\in A \\\\\n{h_B} ~~ x \\in B \\\\\n\\end{cases}\n\\]\n\nFinally extend this metric $h$ to the whole of M, thus giving us an invariant $S^1$-metric $\\Tilde{h}$. Finally consider the almost complex structure corresponding to this metric, thus solving the problem.\n\\vspace{5mm}\n\\fi\n\n\\end{proof}\n\n        \n\\begin{thm}{(Equivariant Gompf Argument)} \\label{gmpf}\nLet $G$ be a compact group. Let $p$ be a fixed point for the action. Let $A$,$B$ and $\\overline{A}$ be $G$-invariant symplectic spheres in a 4-dimensional symplectic manifold $(M,\\omega)$ such that \n\\begin{itemize}\n    \\item $A \\cap B = \\{p\\}$ and the intersection at $p$ is transverse\n    \\item $\\overline{A} \\cap B = \\{p\\}$ and the intersection at $p$ is $\\omega$-orthogonal.\n\\end{itemize} \nThen there exists an $S^1$-equivariant isotopy $A_t$ of $A$  such that $A_t$ intersects $B$ transversely at $p$ for all $t$,  $A_1$ = $\\overline{A}$ in a small neighbourhood of $p$ and the curve $A_1$ agrees with $A$ outside some neighbourhood of $p$.\n\\end{thm}\n\\iffalse\nBefore we embark on the proof of this statement, we make a few observations. The first being that due to the symplectic neighbourhood theorem we can in fact work on the symplectic normal bundle and as   this is a local problem,  we can simplify our hardships by working in a trivialising chart i.e in $\\mathbb{R}^4$. The problem thus reformulated translates to the following one:\n\\\\\n\nGiven an $S^1$ action on $\\mathbb{R}^4$ and two $S^1$ invariant symplectic planes $A$ and $B$. There is an isotopy $\\psi_t \\in \\Ham_c(\\mathbb{R}^4)$ such that $\\psi_t(A)$ intersects $B$ transversely at $0$ for all $t$,  and $\\psi_1(A) = B^{\\bot_\\omega}$ in a small neighbourhood around the origin, where $\\omega_0$ is the standard symplectic form on $\\mathbb{R}^4$ and $B^{\\bot_\\omega}$ is the symplectic orthogonal to $B$. Further we may as well assume $B$ to be the two plane in $\\mathbb{R}^4$ given by $(0,0,x,y)$, and $B^{\\bot_\\omega}$ to the given by the plane $(x,y,0,0)$.\n\\\\\n\nNext we observe that given a function $A:=(f,g): \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2$, the graph of $A$ is a symplectic (for the standard form) submanifold of $\\mathbb{R}^4$ iff $\\left\\{f,g\\right\\} > -1$. This can be proven from  a direct computation. We now have enough background to embark on the proof of the above theorem.\n\\fi\n\\begin{proof}Since this is a local problem, we can work in a trivialising chart in $\\mathbb{R}^4$ in which the action is linear. Let $B^{\\bot_\\omega}$ be the symplectic orthogonal to $B$. We can assume the image of  $B$ to be the two plane in $\\mathbb{R}^4$ given by $(0,0,x,y)$, and $B^{\\bot_\\omega}$ to the given by the plane $(x,y,0,0)$. As $\\overline{A}$ is $\\omega$-orthogonal to $B$, we can choose the neighbourhood such that $\\overline{A} = B^{\\bot_\\omega}$ near $p$. As $A$ is transverse to $B$ at $p$, we can assume its image is given by the graph of function (which we also call A) $A:(f,g): \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2$.\n\\\\\n\nNext we observe that given a function $A:=(f,g): \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2$, the graph of $A$ is a symplectic (for the standard form) submanifold of $\\mathbb{R}^4$ iff $\\left\\{f,g\\right\\} > -1$. This can be proven from  a direct computation.  We will construct an isotopy of graphs of function of the form $A_t:= \\alpha_t(r^2) A$ where $\\alpha_t$ is a bump function depending only on the radius squared (for a fixed $G$ invariant  metric) in $\\mathbb{R}^2$, and such that \\begin{itemize}\n    \\item $A_0 = A$,\n    \\item $A_1 = 0$  near (0,0),\n    \\item $A_t = A$ outside of some neighbourhood of the origin,\n    \\item $A_t$ is symplectic for all $t$.\n\\end{itemize}  \nNote that as $\\alpha_t$ is depends on the radius for a fixed $G$ invariant metric, $A_t$ is also $G$ invariant.\n\\\\\n\n\nDefine $E = g\\left\\{f,r^2\\right\\} + f\\left\\{r^2,g\\right\\}$. Using the fact that $r^2(0,0) = 0$ and $(r^2)^\\prime (0,0) = 0$ we see that $E(0,0) = 0$ and $\\frac{\\partial}{\\partial r}E(0,0) = 0$. By the intermediate value theorem, there exists $c > 0$, $\\epsilon > 0$ and $u > 0$ such that on the ball of radius $u + \\epsilon$ around the origin $B(0,u+\\epsilon)$  we have $E(x) \\geq -c r^2(x)$. Choose $\\delta$ such that  on $B(0,u+\\epsilon)$, $1 + \\left\\{f,g\\right\\} > \\delta > 0$\n\\\\\n\n\n\nPick $\\alpha: \\mathbb{R} \\rightarrow \\mathbb{R}$ satisfying the following properties\n\n\n\n   \n         \\begin{itemize}\n             \\item  $\\alpha(r^2) = 1$ for $r^2 \\geq u$ .\n             \\item $\\alpha(r) = 0$ for r near $0$.\n            \n             \\item $\\alpha^\\prime(r^2) \\leq \\frac{\\delta}{2cr^2} < \\frac{1 + \\{f,g\\}}{2cr^2}$ \n         \\end{itemize} \n         \n\nDefine $\\alpha_t:= (1-t) + t \\alpha(r^2)$ and $A_t := \\alpha_t A$. To show that $A_t$ is symplectic for all $0 \\leq t \\leq 1$ we need to check that  $\\left\\{\\alpha_tf,\\alpha_tg\\right\\} > -1$ for all $0 \\leq t \\leq 1$. \nIn the neighbourhood $B(u)$ we have\n\\begin{equation*}\n    1 + \\left\\{\\alpha_t f,\\alpha_t g\\right\\} = \\underbrace{1 + \\alpha_t^2 \\{f,g\\}}_{\\geq \\delta} + \\underbrace{\\alpha_t\\alpha_t^\\prime E}_{\\geq \\frac{-\\delta}{2}} \\geq 0\n\\end{equation*}\n\nThe inequality $1 + \\alpha_t^2 \\{f,g\\} \\geq \\delta$ follows from the definition of $\\delta$ and from noting that $0 \\geq \\alpha_t \\geq 1$.  $\\alpha_t\\alpha_t^\\prime E \\geq \\frac{-\\delta}{2}$ follows from the  inequality \n\n\\begin{align*}\n    \\alpha_t\\alpha_t^\\prime E &\\geq \\alpha_t\\alpha_t^\\prime (-cr^2) \\\\\n    &\\geq -\\alpha_t\\frac{\\delta}{2cr^2} (cr^2) \\\\\n    &\\geq -\\alpha_t \\frac{\\delta}{2} \\\\\n    &\\geq \\frac{-\\delta}{2}\n\\end{align*}\n\nThus in the neighbourhood $B(u)$ we have the inequality $1 + \\left\\{\\alpha_t f,\\alpha_t g\\right\\} > 0$ for all  t. Outside of $B(u)$, the derivative $\\alpha_t^\\prime$ is identically 0 and $\\alpha_t \\equiv 1$. Hence $\\alpha_t\\alpha_t^\\prime E \\equiv 0$ outside $B(u)$ and $1 + \\left\\{\\alpha_t f,\\alpha_t g\\right\\} = 1 + \\alpha_t^2 \\{f,g\\} + \\cancelto{0}{\\alpha_t\\alpha_t^\\prime E} = 1 + \\{f,g\\} > 0$ outside of $B(u)$.\n\\\\\n\nFinally we note that $A_1 =0$ in a neighbourhood of $(0,0)$ and it equals $A$ outside the ball of radius $u$ around the origin, thus proving the claim.\n\\end{proof}\n\n\n\n\\chapter[Equivariant Differential Topology]{Equivariant versions of classical results from Differential Topology}\n\n\\begin{comment}\n\n\n\\begin{lemma}[Relative Poincare Lemma](see~\\cite{KM}, Lemma~43.10)\\label{RelativePoincareLemma} Let $M$ be a smooth finite dimensional manifold and let $S\\subset M$ be a closed submanifold. Let $\\omega$ be a closed $(k+1)$-form on $M$ which vanishes on $S$. Then there exists a $k$-form $\\sigma$ on an open neighborhood $U$ of $S$ in $M$ such that $d\\sigma=\\omega$ on $U$ and $\\sigma=0$ along $S$. If moreover $\\omega=0$ along $S$, then we may choose $\\sigma$ such that the first derivatives of $\\sigma$ vanish on $S$.\n\\end{lemma}\n\n\n\n\\begin{proof}\nBy restricting to a tubular neighborhood of $S$ in $M$, we may assume that\n$M$ is a smooth vector bundle $p:E\\to S$ and that $i:S\\to E$ is the zero section. We consider $\\mu:\\mathbb{R}\\times E\\to E$, given by $\\mu(t,x)=\\mu_{t}(x)=tx$, then\n$\\mu_{1}=\\id_{E}$ and $\\mu_{0}=i\\circ p:E\\to S\\to E$. Let $V\\in\\mathfrak{X}(E)$ be the vertical vector field $V(x)=vl(x,x)= \\frac{d}{dt}(x+tx)$ whose flow is $\\text{Fl}_{t}^{V}=\\mu_{e^{t}}$. Locally, for $t$ in $(0,1]$ we have\n\\[\\frac{d}{dt}\\mu_{t}^{*}\\omega = \\frac{d}{dt}(\\text{Fl}_{\\log t}^{V})^{*}\\omega =\\frac{1}{t}(\\text{Fl}_{\\log t}^{V})^{*}\\mathcal{L}_{V}\\omega =  \\frac{1}{t}\\mu_{t}^{*}(i_{V}d\\omega+di_{V}\\omega)=\\frac{1}{t}d\\mu_{t}^{*}i_{V}\\omega\\]\nFor $x\\in E$ and $X_{1},\\ldots,X_{k}\\in T_{x}E$ we have\n\\begin{align*}\n(\\frac{1}{t}\\mu_{t}^{*}i_{V}\\omega)_{x}(X_{1},\\ldots,X_{k}) &= \\frac{1}{t}(i_{V}\\omega_{tx}(T_{x}\\mu_{t}\\cdot X_{1},\\ldots,T_{x}\\mu_{t}\\cdot X_{k})\\\\\n&= \\frac{1}{t}\\omega_{tx}(V(tx),T_{x}\\mu_{t}\\cdot X_{1},\\ldots,T_{x}\\mu_{t}\\cdot X_{k})\\\\\n&= \\omega_{tx}(vl(tx, tx), T_{x}\\mu_{t}\\cdot X_{1},\\ldots,T_{x}\\mu_{t}\\cdot X_{k})\n\\end{align*}\nSo the $k$-form $\\frac{1}{t}\\mu_{t}^{*}i_{V}\\omega$ is defined and smooth in $(t,x)$ for all $t\\in[0,1]$ and describes a smooth curve in $\\Omega^{k}(E)$. Note that for $x\\in S = 0_{E}$ we have $\\frac{1}{t}\\mu_{t}^{*}i_{V}\\omega=0$, and if $\\omega = 0$ on $T_{S}M$, we also have $0=\\frac{d}{dt}\\mu_{t}^{*}\\omega=\\frac{1}{t}d\\mu_{t}^{*}i_{V}\\omega$, so that all first derivatives of $\\frac{1}{t}\\mu_{t}^{*}i_{V}\\omega$ vanish along $S$. \nSince $\\mu_{0}^{*}\\omega = p^{*}i^{*}\\omega = 0$ and $\\mu_{1}^{*}\\omega=\\omega$, we have\n\\begin{align*}\n\\omega \n&=\\mu_{1}^{*}\\omega-\\mu_{0}^{*}\\omega\\\\\n&=\\int_{0}^{1}\\frac{d}{dt}\\mu_{t}^{*}\\omega\\,dt\\\\\n&=\\int_{0}^{1}d(\\frac{1}{t}\\mu_{t}^{*}i_{V}\\omega)\\,dt\\\\\n&=d\\left(\\int_{0}^{1}(\\frac{1}{t}\\mu_{t}^{*}i_{V}\\omega)\\,dt\\right)\\\\\t\n&=d\\sigma\n\\end{align*}\nIf $x\\in S$, we have $\\sigma= 0$, and all first derivatives of $\\sigma$ vanish along $S$ whenever $\\omega = 0$ on $T_{S}M$.\n\\end{proof}\n\n\\begin{remark} \nIf there is a symplectic action of a compact group $G$ acting on $M$ such that $\\omega$ is $G$ invariant and $S$ is $G$-invariant, then we can constuct $\\sigma$ as above such that in addition to the above conditions $\\sigma$ also is $G$-invariant. This is gotten by noting that \n\\begin{align*}\n    \\omega = \\int_G \\omega = \\int_G d\\sigma = d \\int_G \\sigma\n\\end{align*}\n\nLet $\\tilde\\sigma:= \\int_G \\sigma$ and hence $d\\tilde\\sigma = \\omega$ and $\\tilde\\sigma$ satisfies all the conditions.\n\\end{remark}\n\n\\begin{lemma}[Moser isotopy] Let $(M,\\omega)$ be a symplectic manifold and let $S\\subset M$ be a submanifold. Suppose that $\\omega_{i}$, $i=0,1$, are closed $2$-forms such that at each point $x\\in S$, the forms $\\omega_{0}$ and $\\omega_{1}$ are equal and non-degenerate on $T_{x}S$. Then there exist open neighborhoods $N_{0}$ and $N_{1}$ of $S$ and a diffeomorphism $\\phi:N_{0}\\to N_{1}$ such that $\\phi^{*}\\omega_{1}=\\omega_{0}$, $\\phi|_{S}=\\id$, and $d\\phi|_{S}=\\id$.\n\\end{lemma}\n\n\n\n\\begin{proof}\nConsider the convex linear combination $\\omega_{t}=\\omega_{0}+t(\\omega_{1}-\\omega_{0})$. Since $\\omega_{0}$ and $\\omega_{1}$ are equal along $S$, there exists a neighborhood $U_{1}$ of $S$ on which $\\omega_{t}$ is non-degenerate for all $t\\in[0,1]$. By restricting $U_{1}$ to a possibly smaller neighborhood $U_{2}$, the Relative Poincar\u00e9 Lemma~\\ref{RelativePoincareLemma} implies that there exists a $1$-form $\\sigma$ such that $d\\sigma=(\\omega_{1}-\\omega_{0})$, $\\sigma=0$ on $S$, and all first derivatives of $\\sigma$ vanish along $S$. Define the time-dependent vector field $X_{t}$ on $U_{2}$ by setting\n\\[\\sigma = -i_{X_{t}}\\omega_{t}\\]\nSince $X_{t}=0$ on $S$, by restricting $U_{2}$ to a smaller neighborhood $U_{3}$, we can ensure that the flow $\\psi_{t}$ of $X_{t}$ exists for $t\\in[0,1]$. We then have\n\\[\\frac{d}{dt}\\psi_{t}^{*}\\omega_{t} = \\psi_{t}^{*}\\left(\\frac{d}{dt}\\omega_{t}+\\mathcal{L}_{X_{t}}\\omega_{t}\\right) = \\psi_{t}^{*}\\left(\\frac{d}{dt}\\omega_{t}+di_{X_{t}}\\omega_{t}\\right) = \\psi^{*}(\\omega_{1}-\\omega_{0}-d\\sigma)=0\\]\nso that $\\psi^{*}\\omega_{t}=\\omega_{0}$. Finally, since $\\sigma=0$ on $S$, $\\psi=\\id$ on $S$, and since all first derivatives of $\\sigma$ vanish on $T_{S}M$, $d\\psi=\\id$ on $T_{S}M$.\n\\end{proof}\n\n\n\\begin{remark}\nAs the remark above, when both $\\omega_1$ and $\\omega_2$ are both invariant under a compact group action $G$, and $S$ is an $G$-invariant submanifold, then there is a $G$-equivariant diffeomorphism $\\phi$ that satisfies the conditions as above. \n\\end{remark}\n\\end{comment}\n\n\\begin{lemma}\\label{EqSymN}[Equivariant Symplectic neighborhoods theorem] \nLet $G$ be a compact group, and let $(M_{i},\\omega_{i})$, $i=0,1$, be two symplectic $G$-manifolds. Let $S_{i}\\subset M_{i}$ be two invariant symplectic submanifolds with invariant symplectic normal bundles $N_{i}$. Suppose that there is an equivariant isomorphism $A:N_{0}\\to N_{1}$ covering an equivariant symplectomorphism $\\phi:S_{0}\\to S_{1}$. Then $\\phi$ extends to a equivariant symplectomorphism of neighborhoods $\\Phi:U_{0}\\to U_{1}$ whose derivative along $S_{0}$ is equal to $A$.\n\\end{lemma}\n\\begin{proof}\nWe can extend the automorphism $A$ to a diffeomorphism of neighborhoods $\\psi:U_{0}\\to U_{1}$ by setting\n\\[\\psi = \\exp\\circ A\\circ \\exp^{-1}\\]\nBy construction, $d\\psi = A$ along $S_{0}$, so that $\\omega_{0}$ and $\\psi^{*}\\omega_{1}$ coincides along $S_{0}$. Applying the $G$- equivariant Moser isotopy lemma gives the result.\n\\end{proof}\n\nLet $(M,\\omega)$ be a symplectic manifold. Let $G$ be a compact lie group acting symplectically on $M$. Let $S$ be an invariant submanifold under the $G$ action. Let $\\Op(S)$ be an invariant open neighbourhood of $S$, Further define\n\n\\[\\Symp^G_{\\id,N}(M,S)=\\{\\phi\\in\\Symp^G_{0}(M)~|~\\phi|_{S}=\\id,~d\\phi|_{T_{S}M}=\\id\\}\\]\n\\[\\Symp^G_{\\id,\\Op(S)}(M,S)=\\{\\phi\\in\\Symp^G_{0}(M)~|~\\phi=\\id~\\text{near}~S\\}\\]\n\nthen we would like to show that  $\\Symp^G_{\\id,N}(M,S) =\\Symp^G_{\\id,\\Op(S)}(M,S) $. But before we do that we would need the following lemmas.\n\nFollowing~\\cite{Hi}, we define a invariant tubular neighborhood of a invariant submanifold $\\iota:S\\hookrightarrow M$ as a smooth equivariant embeddings $f:E\\hookrightarrow M$ of a vector bundle $\\pi:E\\to S$ such that \n\\begin{enumerate}\n\\item $f|_{S}=\\iota$ after identifying $S$ with the zero section of $\\pi:E\\to S$.\n\\item $f(E)$ is an open neighborhood of $S$.\n\\end{enumerate}\nIn practice, it is often enough to work with the normal bundle $N\\subset T_{S}M$ defined as the orthogonal of $T_{S}M$ relative to a equivariant Riemannian structure. (See \\cite{Bredon} p. 306 for existence of such invariant tubular neighbourhood.)\n\n\\begin{lemma}[Unicity of tubular neighborhoods]\\label{UnicityTubularNeighborhoods} (See~\\cite{Hi}, Theorem 4.5.3) Let $M$ be a $G$-manifold, let $\\iota:S\\hookrightarrow M$ be a invariant submanifold with normal bundle $N$. Then, \n\\begin{enumerate}\n\\item given any two invariant tubular neighborhoods $f_{i}:N_i \\hookrightarrow M$, $i=0,1$, there is a equivariant gauge transformation $A\\in\\mathcal{G}(N)$ such that $f_{0}$ and $f_{1}\\circ A$ are equivariant isotopic rel. to $S$. \n\\item The space $\\mathcal{T}_{S}$ of all invariant tubular neighborhoods $f:N\\hookrightarrow M$ is homotopy equivalent to the group of equivariant gauge transformations $\\mathcal{G}(N)$.\n\\item The space $\\mathcal{T}_{S,d\\iota}$ of invariant tubular neighborhoods $f:N\\hookrightarrow M$ such that $df|_{S}=d\\iota$ is contractible.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n(1) We construct an equivariant isotopy $F_{t}$ in two steps. Firstly, given an $G$-invariant smooth function $\\delta:S\\to(0,1]$, let $U_{\\delta}\\subset N$ be the invariant disc bundle\n\\[U_{\\delta}=\\{x\\in N~|~|x|<\\delta(\\pi(x))\\}\\]\nLet k is the rank of the bundle and $C(G)$ denote the centraliser of G in $\\SO(k)$. Note that $N$ smoothly retracts onto $U_{\\delta}$ through embeddings $G_{t}:N\\to N$ of the form\n\\[G_{t} = (1-t)\\id + th_{\\delta(x)}\\]\nwhere $h_{r}:\\mathbb{R}^{k}\\to D^{k}(r)$ is a equivariant one-parameter family of contracting, $C(G) \\subset \\SO(k)$-invariant diffeomorphisms, restricting to the identity on $D^{k}(r\/2)$ and varying smoothly with $r$. Then, choosing an appropriate invariant function $\\delta$, and composing $f_{1}$ with $G_{t}$, we can isotope $f_{1}$ to an embedding $f_{\\delta}=f_{1}G_{1}$ satisfying\n\\begin{equation}\\label{inclusion-assumption}\nf_{\\delta}(N)\\subset f_{0}(N)~\\text{and}~f_{\\delta}=f_{1}~\\text{on}~U_{\\delta\/2}\n\\end{equation}\nso that the map $g=f_{0}^{-1}f_{\\delta}:N\\to N$ is well-defined. Secondly, observe that the map $g$ is equivariantly isotopic to its vertical derivative $A_{f_{\\delta}}=dg^{\\text{vert}}\\in\\mathcal{G}(N)$ along $S$ via the canonical smooth isotopy\n\\[H_{0}(x)=\\phi(x),\\quad H_{t}(x)=g(tx)\/t,~0<t\\leq 1\\]\nNote that H is indeed equivariant. An isotopy from $f_{\\delta}$ to $f_{0}\\circ A_{f_{\\delta}}$ is then given by $f_{0}H_{1-t}$. The sought-for equivariant isotopy $F_{t}$ is the concatenation of $f_{1}G_{t}$ with $f_{0}H_{1-t}$.\\\\\n\n(2) Fix a invariant tubular neighborhood $f_{0}:N\\hookrightarrow M$ and choose once for all a smooth family of equivariant diffeomorphisms $h_{r}:\\mathbb{R}^{k}\\to D^{k}(r)$ as in the proof of (1). Given any other invariant tubular neighborhood $f:N\\hookrightarrow M$, the isotopy constructed in (1) only depends on an auxiliary invariant function $\\delta:S\\to\\mathbb{R}_{+}$. Although the choice of this function is not canonical, the set $\\Delta^G(f)$ of all invariant $\\delta$ for which $h_{\\delta}(N)\\subset f^{-1}f_{0}(N)$ is a convex subset of $C^{\\infty,G}(S,\\mathbb{R}_{+})$, where $C^{\\infty,G}(S,\\mathbb{R}_{+})$ denotes the space of $G$ invariant smooth functions to $\\mathbb{R}_+$. Consequently, the projection of the fibre product\n\\[\\mathcal{T}_{\\Delta}=\\{(f,\\delta)~|~f\\in\\mathcal{T}_{S},~\\delta\\in\\Delta^G(f)\\}\\to\\mathcal{T}_{S}\\]\nis a homotopy equivalence. Consider the embedding\n\\begin{align}\n\\phi:\\mathcal{G}(N)\\times C^{\\infty,G}(S,\\mathbb{R}^{+})&\\hookrightarrow \\mathcal{T}_{\\Delta}\\\\\n(A,\\delta)&\\mapsto (f_{0}\\circ A, \\delta)\n\\end{align}\nand the continuous map\n\\begin{align}\n\\psi:\\mathcal{T}_{\\Delta}&\\to \\mathcal{G}(N)\\times C^{\\infty,G}(S,\\mathbb{R}_{+})\\\\\n(f,\\delta)&\\mapsto (A_{f_{\\delta}},\\delta)\n\\end{align}\nThen we have $\\psi\\phi=\\id_{\\mathcal{G}(N)\\times C^{\\infty,G}(S,\\mathbb{R}_{+})}$, while $\\phi\\psi(f,\\delta)=(f_{0}\\circ A_{f_{\\delta}},\\delta)$, the map $f_{0}\\circ A_{f_{\\delta}}$ being the terminal point of the isotopy defined in (1). This shows that $\\phi$ and $\\psi$ are homotopy inverses.\\\\\n\n(3) Choosing $f_{0}$ such that $df_{0}=d\\iota$ along $S$, this immediately follows from the fact that the space $\\mathcal{T}_{S,d\\iota}$ is homotopy equivalent to the subspace of $\\mathcal{T}_{\\Delta}$ that retracts to the contractible subspace $\\{f_{0}\\}\\times C^{\\infty,G}(S,\\mathbb{R}_{+})$ under the isotopy defined in~(1).\n\\end{proof}\n\n\\begin{lemma}{\\label{germ}}\nThe inclusion $\\Symp^G_{\\id,\\Op(S)}(M,S)\\hookrightarrow\\Symp^G_{\\id,N}(M,S)$ is a homotopy equivalence.\n\\end{lemma}\n\\begin{proof}\nWe follow the same ideas as in the non-equivariant case. Consider the short exact sequence\n\\[\\Symp^G_{\\id,\\Op(S)}(M,S)\\hookrightarrow\\Symp^G_{\\id,N}(M,S)\\to\\mathcal{G}_{S,\\omega}\\]\nwhere the group $\\mathcal{G}_{S,\\omega}:=\\Symp^G_{\\id,N}(M,S)\/\\Symp^G_{\\id,\\Op(S)}(M,S)$ is the group of germs along $S$ of equivariant symplectomorphisms $\\phi\\in\\Symp_{\\id,N}(M,S)$. We wish to show that $\\mathcal{G}_{S,\\omega}$ is contractible.\n\\\\\n\nChoose a compatible $G$-invariant almost-complex structure $J$ and let $g$ be the associated equivariant metric. Let $N$ be the symplectic orthogonal complement of $TS$ in $TM$. $N$ is invariant under the $G$ action. Equip $N$ with the minimal coupling form $\\Omega$ and choose $\\epsilon>0$ so that the $\\epsilon$-disk subbundle $V_{\\epsilon}\\subset N$ is symplectomorphic to a tubular neighborhood $U$ of~$S$. Let $\\iota$ denote both inclusions $U\\hookrightarrow M$ and $V\\hookrightarrow N$.\n\\\\\n\nLet $\\Omega_{S}^{\\mathrm{loc},G}$ be the space of germs of $G$ invariant symplectic forms defined near $S$ and agreeing with $\\omega$ along $T_{S}M$. Given any two germs $[\\omega_{0}]$ and $[\\omega_{1}]$, their linear convex combination $\\omega_{t}=(1-t)\\omega_{0}+t\\omega_{1}$ is non-degenerate in some neighborhood of $S$. Consequently, $\\Omega_{S}^{\\mathrm{loc},G}$ is convex, hence contractible. By the Symplectic neighborhood theorem, the group $\\mathcal{G}_{S,\\omega}$ acts transitively on $\\Omega_{S}^{\\mathrm{loc},G}$, giving rise to a fibration\n\\[\\mathcal{G}_{S,\\omega}^{\\mathrm{loc},G}\\stackrel{\\simeq}{\\to}\\mathcal{G}_{S,\\omega}\\to\\Omega_{S}^{\\mathrm{loc},G}\\]\nwhose fiber $\\mathcal{G}_{S,\\omega}^{\\mathrm{loc},G}$ is the group of germs of equivariant diffeomorphisms that are symplectic near $S$. This space is homeomorphic to the space $\\mathcal{E}_{S,\\omega}$ of germs along $S$ of equivariant symplectic embeddings $f:\\Op(S)\\to M$ such that $f|_{S}=\\iota$ and $df|_{S}=d\\iota$. By Lemma~\\ref{UnicityTubularNeighborhoods} (3), we know that $\\mathcal{E}_{S,\\omega}$   is contractible, so that $\\mathcal{G}_{S,\\omega}$ and $\\mathcal{G}_{S,\\omega}^{\\mathrm{loc},G}$   are also contractible, thus completing the proof. \n\\end{proof}\n\n\\begin{lemma}\\label{Au} Let $G$ be a compact group acting symplectically on a compact manifold $(M,\\omega)$. Let $W_t$ be a smooth $k$-parameter family of symplectic submanifolds ($t \\in [0,1]^k$), which are invariant under the $G$ action. Then there exists a  $k$-parameter family of equivariant Hamiltonian symplectomorphisms $\\phi_{t}: M \\rightarrow M$ such that $\\phi_t(W_0) = W_t$\n\\end{lemma}\n\n\\begin{proof}\nThe proof follows by mimicking the proof of Proposition 4 in \\cite{Auroux} under the presence of a group action.\n\\end{proof}\n\nFinally, we present the following theorem due to Palais, that we repeatedly use in Chapters 3 and 6 to justify that our maps between infinite dimensional spaces is a fibration. \n\nLet $X$ be a topological space with an action of a topological group $G$. We say X admits local cross sections at $x_0 \\in X$ if for there is a neighbourhood $U$ containing $x_0$ and a map $\\chi: X \\rightarrow G$ such that $\\chi(u) \\cdot x_0 = u$ for all $u \\in U$.  We say X admits local cross sections if this is true for all $x_0 \\in X$. \n\n\\begin{thm}{(Palais)}\\label{palais}\nLet $X$, $Y$ be a topological spaces with a action of a topological group $G$. Let the $G$ action on $X$ admit local cross sections. Then any equiariant map $f$ from another space $Y$ to $X$ is locally trivial.\n\\end{thm}\n\n\\begin{proof}\nSuppose for every point $x_0 \\in X$ there is a local section $\\chi: U \\rightarrow G$ where $U$ is an open neighbourhood of $x_0$. Then we define a local trivialisation of $f$ as follows. \n\n\\begin{align*}\n    \\rho: U \\times f^{-1}(x_0) &\\rightarrow f^{-1}(U) \\\\\n     (u, \\gamma) &\\mapsto \\chi(u) \\cdot \\gamma\n\\end{align*}\n\nAs $f$ is equivariant we indeed have $f(\\rho(u,\\gamma)) = f( \\chi(u) \\cdot \\gamma) = \\chi(u) \\cdot f(\\gamma) = \\chi(u) \\cdot x_0 = u$, where the last equality follows from the definition of being a local section. Thus $\\rho$ maps $U \\times  f^{-1}(x_0) $ into $ f^{-1}(U)$.\\\\\n\nConversely there is map \n\n\\begin{align*}\n   \\beta: f^{-1}(U) &\\rightarrow U \\times f^{-1}(x_0) \\\\\n   y &\\mapsto \\left(f(y), {\\chi(f(y))}^{-1} \\cdot y\\right)\n\\end{align*}\n\nWe can indeed check that the two maps are inverses of each other. \n\n$\\beta \\circ \\rho (u, \\gamma) = \\beta(\\chi(u) \\cdot \\gamma) = \\left(\\chi(u) \\cdot \\gamma, {\\chi(\\chi(u) \\cdot f(\\gamma))}^{-1} \\cdot \\chi(u) \\cdot \\gamma \\right) = (u, {\\chi(u)}^{-1} \\chi(u) \\cdot \\gamma) = (u,\\gamma)$.\n\nSimilarly we can check that $\\rho \\circ \\beta = id$, thus completing the proof. \n\\end{proof}\n\n\\chapter{Alexander-Eells isomorphism}\\label{Appendix-Alexander-Eells}\n\n\nIn this appendix, we prove the Alexander-Eells isomorphism used in Section~\\ref{subsection:CohomologyModule}. We first recall an isomorphism between the homology of a submanifold $Y\\subset X$ and the homology of its complement $X-Y$ that is reminiscent of the Alexander-Pontryagin duality in the category of oriented, finite dimensional manifolds. This isomorphism, due to J. Eells, exists whenever the submanifold $Y$ is co-oriented and holds, in particular, for infinite dimensional Fr\u00e9chet manifolds. We then give a geometric realization of this isomorphism in the special case $Y$ and $X-Y$ are orbits of a continuous action $G\\times X\\to X$ satisfying some mild assumptions. We closely follow Eells~\\cite{Eells} p. 125--126.\\\\\n\nLet $X$ be a manifold, possibly infinite dimensional, and let $Y$ be a co-oriented submanifold of positive codimension $p$. As explained in~\\cite{Eells}, there exists an isomorphism of singular cohomology groups\n\\[\\phi:H^{i}(Y)\\to H^{i+p}(X, X-Y)\\]\ncalled the Alexander-Eells isomorphism. We define the fundamental class (Thom class) of the pair $(X,Y)$ as $u=\\phi(1)\\in H^{p}(X, X-Y)$.\n\\begin{prop}[Eells, p. 113]\nThe pairing\n\\[\\begin{aligned}\nH^{*}(Y)\\otimes H^{*}(X,X-Y)&\\to H^{*}(X,X-Y)\\\\\ny\\otimes x &\\mapsto y\\cup x\n\\end{aligned}\\]\nmakes $H^{*}(X,X-Y)$ into a free $H^{*}(Y)$-module of rank one, generated by $u$.\n\\end{prop}\n\n \n\nLet $\\phi_{*}:H_{i+p}(X,X-Y)\\to H_{i}(Y)$ be the dual of the Alexander-Eells isomorphism $\\phi^{*}=\\phi$. By definition, we have\n\\[\\phi_{*}(a)=u\\cap a\\]\n\nSuppose a topological group $G$ acts continuously on $X$ (on the left), leaving $Y$ invariant, and in such a way that both $X-Y$ and $Y$ are homotopy equivalent to orbits. We have continuous maps $\\mu:G\\times (X,X-Y)\\to (X,X-Y)$, ~$\\mu:G\\times (X-Y)\\to (X-Y)$, and $\\mu:G\\times Y\\to Y$ inducing $H_{*}(G)$-module structures on $H_{*}(X,X-Y)$, $H_{*}(X-Y)$ and $H_{*}(Y)$. We write $\\mu_{*}(c\\otimes a)=c*a$ for the action of $c\\in H_{i}(G)$.\n\n\\begin{lemma}\\label{lemma:AlexanderEellsGmodule}\nIn this situation, the Alexander-Eells isomorphism preserves the $H_{*}(G)$-module structure, that is, the following diagram is commutative:\n\\[\n\\begin{tikzcd}\nH_{*}(G)\\otimes H_{*}(X,X-Y) \\arrow{r}{\\mu_{*}} \\arrow[swap]{d}{1\\otimes\\phi_{*}} & H_{*}(X,X-Y) \\arrow{d}{\\phi_{*}} \\\\\nH_{*}(G)\\otimes H_{*}(Y) \\arrow{r}{\\mu_{*}} & H_{*}(Y)\n\\end{tikzcd}\n\\]\nThus for any $a\\in H_{i+p}(X,X-Y)$, $c\\in H_{i}(G)$, we have $\\phi_{*}(c*a) = c*\\phi_{*}(a)$.\n\\end{lemma}\n\\begin{proof}\nWe first note that if $u$ is the fundamental class of the pair $(X,Y)$, then $\\mu^{*}(u)=1\\otimes u\\in H^{0}(G)\\otimes H^{p}(X,X-Y)$, because $H^{i}(X,X-Y)=0$ for all $i<p$. The cap product is a bilinear pairing\n\\[\n\\cap: H^{*}\\big(G\\times (X,X-Y)\\big)\\otimes H_{*}\\big(G\\times (X,X-Y)\\big) \\to H_{*}\\big(G\\times X\\big)\n\\]\nthat is adjoint to the cup product. Using the relative form of K\\\"unneth theorem, this bilinear map defines a pairing\n\\[\n\\cap:\\big(H^{*}(G)\\otimes H^{*}(X,X-Y)\\big)\\otimes\\big(H_{*}(G)\\otimes H_{*}(X,X-Y)\\big)\\to H_{*}(G)\\otimes H_{*}(X)\n\\]\nLet's write $j^{*}:H^{*}(X,X-Y)\\to H^{*}(X)$.  Then, for any $c\\otimes x\\in H^{*}(G)\\otimes H^{*}(X,X-Y)$, and any $b\\otimes a\\in H_{*}(G)\\otimes H_{*}(X,X-Y)$ we have\n\\[\\begin{aligned}\n\\big\\langle c\\otimes j^{*}x, \\mu^{*}(u)\\cap (b\\otimes a)\\big\\rangle \n&= \\big\\langle (c\\otimes x)\\cup\\mu^{*}(u),b\\otimes a\\big\\rangle\\\\\n&= \\big\\langle (c\\otimes x)\\cup(1\\otimes u),b\\otimes a\\big\\rangle\\\\\n&= \\big\\langle (c\\cup1)\\otimes (x\\cup u),b\\otimes a\\big\\rangle\\\\\n&= \\big\\langle c, b\\big\\rangle \\big\\langle (x\\cup u),a\\big\\rangle\\\\\n&= \\big\\langle c,b\\big\\rangle \\big\\langle j^{*}x, u\\cap a\\big\\rangle\\\\\n&=\\big\\langle c\\otimes j^{*}x, b\\otimes (u\\cap a)\\big\\rangle\n\\end{aligned}\\]\nIt follows that $\\mu^{*}(u)\\cap (b\\otimes a) = b\\otimes (u\\cap a) = b\\otimes \\phi_{*}(a)$. We then compute\n\\[\\phi_{*}(\\mu_{*}(c\\otimes x)) \n=u\\cap\\mu_{*}(c\\otimes x) \n=\\mu_{*}\\big(\\mu^{*}(u)\\cap(c\\otimes x)\\big)\n=\\mu_{*}\\big(c\\otimes\\phi_{*}(x)\\big)\n=\\mu_{*}\\big((1\\otimes \\phi_{*})(c\\otimes x)\\big)\\]\nwhich is the desired relation.\n\\end{proof}\nSince $H_{i}(X,X-Y)=0$ for $i<p$, the Universal Coefficient Theorem yields a canonical isomorphism $\\beta:H^{p}(X,X-Y)\\to \\Hom(H_{p}(X,X-Y);\\mathbb{Z})$. If $Y$ is connected, $H_{p}(X,X-Y)$ is of rank one. In this case, define $a_{u}\\in H_{p}(X,X-Y)$ as the unique class such that $\\beta(u)a_{u}=1$. Suppose the Leray-Hirsch theorem applies to the evaluation fibration $G\\to Y$. Then, $H_{*}(X,X-Y)$ becomes a $H_{*}(Y)$-module by identifying $H_{*}(Y)$ with $1\\otimes H_{*}(Y)\\subset H_{*}(G)$ and by setting\n\\[b*x := [1\\otimes b]*x\\]\n\\begin{thm}\\label{thm:AlexanderEells}\nUnder the above assumptions, the isomorphism $\\psi_{*}=\\phi_{*}^{-1}:H_{i}(Y)\\to H_{i+p}(X,X-Y)$ is given by $\\psi_{*}(y)=y*a_{u}$. Thus $H_{*}(X,X-Y)$ is generated by $a_{u}$ as a $H_{*}(Y)$-module.\n\\end{thm}\n\\begin{proof}\nSince\n\\[\\langle 1, \\phi_{*}(a_{u})\\rangle\n=\\langle 1, u\\cap a_{u}\\rangle \n=\\langle 1\\cup u, a_{u}\\rangle\n=\\langle u,a_{u}\\rangle=1\\]\nit follows that $\\phi_{*}(a_{u})=1$, which is equivalent to $\\psi_{*}(1)=a_{u}$. Since $\\phi_{*}$ is an isomorphism of $H_{*}(G)$-modules, its inverse is also an isomorphism of $H_{*}(G)$-modules. We can then write\n\\[\\psi_{*}(y) = \\psi_{*}(y*1) = y*\\psi_{*}(1) = y*a_{u}\\]\n\\end{proof}\nFrom the naturality of the connecting homomorphism $\\partial$ in the long exact sequence of the pair $(X, X-Y)$, we get\n\\begin{cor}\\label{cor:AlexanderEells}\nSuppose in addition to the above hypotheses that $X$ is contractible. Then the isomorphism \n\\[\\lambda_{*}=\\partial\\circ\\psi_{*}: H_{i}(Y)\\to H_{i+p-1}(X-Y)\\]\nis given by $\\lambda_{*}(y) = y*x_{u}$, where $x_{u}=\\partial a_{u}$.\n\\end{cor}\n\nWe now apply the Alexander-Eells isomorphism to the situation of Section~\\ref{section:TwoOrbits}. Let $G$ be the centralizer $\\Symp_{h}^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$, let $X=\\mathcal{J}^{S^1}_{\\om_\\lambda}$ be the contractible space of invariant, compatible,  almost-complex structures, and let $Y$ be the codimension 2 stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m'}$. Let $p_m$ denote the map \n\\[p_m: \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\rightarrow \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_m \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{m}\\]\n\nFurther let $y_1$ and $y_2$ denote the generators of the Pontryagin algebra $H_{*}(T^{2}_{m'};k)$.  Note that under the inclusion $i_m: \\mathbb{T}^2_m \\hookrightarrow \\Symp_{h}^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$, the elements $y_1$ and $y_2$ correspond to the circle actions $(1,0;m)$ and $(0,1;m)$ respectively. \nThe connecting isomorphism $\\partial:H_{2}(\\mathcal{J}^{S^1}_{\\om_\\lambda}, \\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m})\\to H_{1}(\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m})$ maps the generator $a_{u}$ to the link of $\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m'}$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m}$, that is, to the loop $p_{m}(y_{2})$ generated by the action of $y_{2}\\subset G$. Consequently, Corollary~\\ref{cor:AlexanderEells} immediately implies the following geometric description of the Alexander-Eells isomorphism.\n\\begin{prop}\\label{prop:AlexanderEellsGeometric}\nThe Alexander-Eells isomorphism\n\\[\\lambda_{*}:H_{p}(\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m'})\\to H_{p+1}(\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m})\\]\nis given by\n\\[\\lambda_{*}(y)= y*p_{m}(y_{2})=y*y_{2}*1=[y_{2}\\otimes y]*1=\\mu_{m}\\big([y_{2}\\otimes y]\\otimes 1\\big)=p_{m}\\big([y_{2}\\otimes y]\\big)\\]\nIn particular, if $p_{m'}(\\tilde y) = y\\in H_{*}(Y)$, then $\\lambda_{*}(y)=\\tilde y*y_{2}*1=p_{m}(\\tilde y* y_{2})$. \n\\end{prop}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:introduction}\n\nQuantum chromodynamics (QCD) predicts that a new form of matter, consisting of\ndeconfined quarks and gluons, is formed at very high temperatures. Calculations\nin lattice QCD~{\\cite{Karsch:2003jg} indicate that the transition to this\nso-called quark-gluon plasma should occur at a critical temperature of around 150--175\\MeV, corresponding to an energy density of about 1~GeV\/fm$^3$.\nExperiments have provided evidence that dense matter at high temperature can be\ncreated in relativistic heavy ion\ncollisions~\\cite{Adcox:2004mh,Adams:2005dq,Back:2004je,Arsene:2004fa}.\n\nStudies of particle production at high transverse momentum (\\pt) are a well-established way to\nprobe the properties of the medium formed in heavy-ion collisions. The yields and correlations of high momentum particles are modified through the ``jet-quenching'' effect,\nresulting from the energy loss suffered\nby hard-scattered partons passing through the medium~\\cite{Bjorken:1982tu}.\nFast parton energy loss provides key information on the thermodynamic and transport properties of the medium traversed~\\cite{CasalderreySolana:2007zz,dEnterria:2009am}. Evidence for jet quenching was first observed in the suppression of inclusive high-$\\pt$ hadron production and the modification of high-$\\pt$ dihadron angular correlations in nucleus-nucleus collisions at the Relativistic Heavy Ion Collider, in comparison to proton-proton collisions ~\\cite{Adams:2005dq,Adcox:2004mh,Arsene:2004fa,Back:2004je}.\n\nRecent results at the LHC~\\cite{Chatrchyan:2011sx,Aad:2010bu,Alice:dihadron,HIN-10-005,Aamodt:2010jd}, using\nfully reconstructed jets, correlations between jets and single particles, and charged particle measurements,\nprovide detailed information on the jet-quenching effect. For central collisions,\na large broadening of the dijet momentum asymmetry distributions is observed, consistent with\ntheoretical calculations that involve differential energy loss of back-to-back hard-scattered\npartons as they traverse the medium~\\cite{He:2011pd,Young:2011qx,Qin:2010mn}.\nAt the same time, angular correlations between the jets are found to be almost unchanged, ruling\nout single-hard-gluon radiation as the leading energy loss mechanism. Studies of jet-hadron correlations,\ninvolving vector summation of charged hadron momenta,\nfind that the energy balance in events with large dijet asymmetry is recovered on average\nby an excess of low-momentum particles in the hemisphere of the away-side jet,\nat large angles relative to the jet axes~\\cite{Chatrchyan:2011sx}. These results constrain the mechanism of parton energy loss~\\cite{CasalderreySolana:2010eh,Chesler:2011nc}.\nFurther understanding of this mechanism requires the measurement of the \\pt dependence of the\nobserved effects.\n\nThe dijet analysis presented in this paper uses a large dataset of PbPb collisions\nat a nucleon-nucleon center-of-mass energy of $\\rootsNN=2.76\\TeV$, collected with the Compact Muon Solenoid\n(CMS) detector in 2011.\nThe CMS detector has a solid-angle acceptance of nearly 4$\\pi$ and is designed to measure jets and energy flow, an excellent feature for studying heavy ion collisions.\nWith a total integrated luminosity of 150$\\pm$8 $\\mu$b$^{-1}$ this\ndataset provides a significantly larger data sample than the 6.8$\\,\\mu$b$^{-1}$ analyzed in 2010, allowing an extension of previous studies to more peripheral collision\nevents and to jets of transverse momenta in excess of 350\\GeVc.\nAdditionally, this paper also presents  a dijet analysis of pp collisions recorded at $\\sqrt{s}=2.76$\\TeV by CMS in 2011, with a total integrated\nluminosity of 231\\nbinv.\n\n\n\n\\section{Experimental method}\n\\label{sec:experimental_method}\n\\subsection{The CMS detector}\n\\label{sec:cms}\n\nThe CMS detector is described in detail\nelsewhere~\\cite{bib_CMS}.  The calorimeters provide hermetic coverage over a large\nrange of pseudorapidity, $|\\eta| < 5.2$, where $\\eta = -\\ln [\n\\tan(\\theta\/2)]$ and $\\theta$ is the polar angle relative to the counterclockwise\nion beam (the $z$ axis).\nA steel and quartz-fiber Cherenkov\ncalorimeter, called hadron forward (HF), covers the high pseudorapidity range\n$3<|\\eta|<5.2$ and is used to determine the centrality of the PbPb collision.\nHadron calorimeter (HCAL) cells are grouped in projective\ntowers of granularity\n$\\Delta \\eta \\times \\Delta \\phi =0.087\\times0.087$ (where $\\phi$ is the azimuthal angle) at central pseudorapidities, having a segmentation about twice as large at forward\npseudorapidities. The electromagnetic calorimeter (ECAL) has a further segmentation of 5$\\times$5 within a tower, and the signals in these cells are clustered together to reconstruct photons. The central calorimeters are embedded in a 3.8\\unit{T} axial magnetic field produced by a superconducting solenoid.\nThe CMS tracking system, located inside the calorimeter, consists of silicon-pixel and silicon-strip layers covering  $|\\eta| <\n2.5$. This analysis uses tracks reconstructed down to transverse momenta of $900$\\MeVc in PbPb collisions, with\na track momentum resolution of about 1\\% at $\\pt = 100$\\GeVc. At high \\pt, the efficiency of the tracking is not strongly \\pt\\ dependent, and for 100\\GeVc tracks it varies from 65\\% in peripheral collisions to 60\\% in central collisions. A set of scintillator tiles, the beam scintillator counters (BSC), used for triggering and beam-halo rejection, is mounted on the inner side of the HF calorimeters.\nThe detailed Monte Carlo (MC) simulation of the CMS detector response is based on {\\GEANTfour}\n\\cite{bib_geant}.\n\n\\subsection{Jet reconstruction in PbPb collisions}\n\\label{sec:jet_reconstruction}\n\nJets are reconstructed using the CMS ``particle-flow\" algorithm~\\cite{CMS-PAS-PFT-10-002,MattPFlow}.\nThis algorithm attempts to identify all stable particles in an\nevent (electrons, muons, photons, charged and neutral hadrons)\nby combining information from all sub-detector systems.\nThe \\ak\\ sequential recombination algorithm, as encoded in the\n{\\textsc{FastJet}} framework, is used to combine the particle-flow candidates\ninto jets using a distance parameter R = 0.3~\\cite{Cacciari:2008gp}.\n\nThe small value of R helps to reduce the deterioration of the jet energy\nresolution in PbPb collisions due to\nfluctuations of the background from soft interactions.\nThe underlying\nbackground from soft collisions is subtracted using the same method\nas employed in~\\cite{Chatrchyan:2011sx} and originally described in~\\cite{Kodolova:2007hd}.\nThis algorithm is a variant of an iterative ``noise\/pedestal subtraction''\ntechnique, where the mean and dispersion of the energies detected in rings of constant $\\eta$ are subtracted from the jet. The jet and background energies are formed from the total energy, determined by particle-flow, within projective towers with the same segmentation as the HCAL, as described in Section~\\ref{sec:cms}.\n\nThe jets reconstructed with the procedure above are then corrected to final state particle jets.\nCMS uses a factorized multi-step approach to correct the jet energies~\\cite{Chatrchyan:2011ds}. For this analysis, jet energy\ncorrections are derived from \\PYTHIA~\\cite{bib_pythia} simulations without PbPb underlying events. Jets reconstructed\nwith the particle-flow algorithm using heavy-ion tracking~\\cite{HIN-10-005} require different corrections\nthan those derived with the tracking algorithm optimized for pp data, due to the difference of tracking efficiencies.\n\nThe pp sample is reconstructed with the same tracking, particle flow and jet algorithms as those used in the analysis of the PbPb data. Although the underlying event and pile-up in pp collisions at $\\sqrt{s}=2.76$\\TeV is small enough not to require a subtraction method, the jet algorithm works successfully in both types of environment.\n\n\n\\subsection{Data samples and triggers}\n\nFor online event selection,\nCMS uses a two-level trigger system:\na hardware-based level-1 trigger and a software-based high level trigger (HLT). The events used in this analysis are\nselected by an inclusive single-jet trigger that required\na calorimeter based jet reconstructed in HLT with  $\\pt > 80$\\GeVc , where the jet $\\pt$ value is\ncorrected for the $\\pt$-dependent calorimeter energy response.\nThe trigger efficiency is defined as the fraction\nof triggered events out of a sample of minimum bias events (described below)\nin bins of offline reconstructed leading-jet $\\pt$.\nThe trigger becomes fully efficient for collisions with a leading particle-flow jet\nwith corrected $\\pt$ greater than 100\\GeVc.\n\nIn addition to the jet data sample, a minimum bias event sample was collected using\ncoincidences between the trigger signals from both the $+z$ and $-z$ sides of either the BSC\nor the HF, which was pre-scaled to record only about 0.1--0.2\\% of the collisions delivered by the LHC.\nIn order to suppress non-collision-related noise, cosmic-ray muons,\nout-of-time triggers, and beam backgrounds, the minimum bias and jet triggers used in this analysis\nwere required to arrive in time with the presence of both colliding ion bunches in the interaction region.\nThe events selected by the jet trigger described above also satisfy all triggers and selections\nimposed for minimum bias events.\n\n\\subsection{Event selection and centrality determination}\n\\label{sec:event_selection}\nA sample of inelastic hadronic collisions is selected offline from the triggered events. Contamination from beam-halo events is removed based upon the timing of the $+z$ and $-z$ BSC signals. A requirement of a reconstructed primary collision vertex based on at least two tracks with transverse momenta above $75$~\\MeVc is imposed. This requirement removes other beam related background events (e.g., beam-gas, ultraperipheral collisions) with large HF energy deposits but very few pixel detector hits. The vertex is required to be compatible with the length of the pixel clusters reconstructed in the event, as a standard method in CMS~\\cite{Khachatryan:2010us}. Finally, an offline HF coincidence is applied, which requires at least three towers on each side of the interaction point in the HF with at least 3~GeV total deposited energy per tower. This event selection, including the minimum bias trigger, has an efficiency of 97\\% with an uncertainty of 3\\% for hadronic inelastic PbPb collisions. This efficiency is\ntaken into account in the centrality determination, and the uncertainty of the efficiency has a negligible effect\non the results of this study.\n\nTable~\\ref{evselcuts} shows the number of events remaining after the various selection criteria are applied. Events with a jet trigger of $\\pt > 80$\\GeVc are selected, followed by the offline event selection for inelastic hadronic collisions (described above). Prior to jet finding on\nthe selected events, a small contamination of\nnoise events from the electromagnetic calorimeter and hadron calorimeter is removed using signal\ntiming, energy distribution, and pulse-shape information \\cite{ref:EGM-10-002,Chatrchyan:2009hy}. The leading and subleading jets are determined among the jets with pseudorapidity $|\\eta| < 2$, which are reconstructed as described in Section~\\ref{sec:jet_reconstruction}.\nEvents are then selected if the corrected jet \\pt is larger than  $120$\\GeVc (corrected\nfor the $\\pt$- and $\\eta$-dependent detector energy response). The subleading jet in the\nevent is required to have a corrected jet $\\pt > 30$\\GeVc.\nThe azimuthal angle between the leading and the subleading jets is required to be at least $2\\pi\/3$.\nFurther jets found in the event, beyond the leading and the subleading ones, are not considered in this analysis.\nIn order to remove events with residual HCAL noise that are missed by the noise-rejection algorithms,\neither the\nleading or subleading jet is required to have at least one track of $\\pt > 4$\\GeVc. For high-\\pt jet events\nthis selection does not introduce any significant bias on the sample and removes only 2\\% of the\nselected dijet events.\n\nThe centrality of the collisions is represented by the number of participating nucleons (\\npart) in a collision, which is correlated with the total transverse energy measured in HF. The minimum bias event sample is divided into constant fractions of total inelastic cross section and for each fraction the average value of \\npart\\ is determined using a Glauber calculation~\\cite{Miller:2007ri}. The dispersion of the \\npart\\ values due to reconstruction effects is based on {\\GEANTfour} simulations of events generated with a multi-phase transport {\\textsc{ampt}} simulation~\\cite{Lin:2004en}.\n\n\n\\begin{table*}[htbp]\n\\begin{center}\n\\caption{The effects of various selections applied to the data sample. In the third column, the\nfractional values are with respect to the line above and in the fourth column they are with respect to the triggered sample. The selections are applied in\nsequence.}\n\\label{evselcuts} \\begin{tabular}{|l|r|r|r|}\n\\hline\nSelections & Events remaining & \\% of previous & \\% of triggered \\\\\n\\hline\nJet triggered events ($\\pt^{\\text{corr}}>80$\\GeVc) & 369\\,938 & 100.00 & 100.00  \\\\\nOffline collision selection  & 310\\,792 & 84.01 & 84.01  \\\\\nHCAL and ECAL noise rejection   & 308\\,453 & 99.25 & 83.38  \\\\\nLeading jet $\\ptlead>120$\\GeVc  & 55\\,911 & 18.13 & 15.11  \\\\\nSubleading jet $\\ptsub>30$\\GeVc    & 52\\,694 & 94.25 & 14.24  \\\\\n\\dphi $>2\\pi\/3$ &  49\\,993 & 94.87 & 13.51  \\\\\nTrack within a jet &   49\\,054 & 98.12 & 13.26  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\\subsection{Simulated data samples}\n\\label{sec:pythia_samples}\n\n\nIn PbPb collisions there is a high multiplicity of soft particles produced, the PbPb underlying event. It is essential to understand how the jet reconstruction is modified in PbPb collisions at different centralities. This is studied with simulations of dijet events in pp collisions with the \\PYTHIA event generator (version 6.423, tune Z2)~\\cite{bib_pythia},\nmodified for the isospin content of the colliding nuclei. A minimum hard-interaction scale ($\\hat{p}_\\mathrm{T}$) selection of 80\\GeVc\\ is used to increase the number of dijet events produced in the momentum range studied. \\PYTHIA simulations at lower $\\hat{p}_\\mathrm{T}$ (discussed in~\\cite{Cacciari:2011tm}) are also investigated and found to agree with the $\\hat{p}_\\mathrm{T} > 80$\\GeVc\\ results within the uncertainties. To model the PbPb background, minimum bias \\PbPb\\ events are simulated with the \\HYDJET event generator~\\cite{Lokhtin:2005px}, version 1.8 (denoted \\PYTHYD in this paper).\nThe parameters of \\HYDJET are tuned to reproduce the total particle multiplicities, charged hadron spectra, and elliptic flow at all centralities, and to approximate\nthe underlying event fluctuations seen in data, differences being within the underlying event systematic uncertainty.\n\nThe full detector simulation and analysis chain is used to process both \\PYTHIA dijet events and \\PYTHIA dijet events embedded into \\HYDJET events. The reconstruction of particle flow jets is studied by using the \\PYTHIA generator jet information in comparison to the same fully reconstructed jet in \\PYTHYD, matched in momentum space. The effects of the PbPb underlying event on jet \\pt\\ and position resolution, jet \\pt\\ scale, and jet-finding efficiency are determined as a function of collision centrality and jet \\pt. These effects do not require corrections on the results but contribute to the systematic uncertainties.\n\\section{Results}\n\\label{sec:results}\n\nThe goal of this analysis is to characterize possible modifications of dijet event properties as a function\nof centrality and leading jet transverse momentum in \\PbPb\\ collisions.\nThe analysis is performed in six bins of collision centrality: 0--10\\%, 10--20\\%, 20--30\\%, 30--50\\%, 50--70\\%, and 70--100\\%, the latter being the most peripheral bin. The 0--20\\% most central events are further analyzed in bins of leading jet $\\pt$: 120--150, 150--180, 180--220, 220--260, 260--300, 300--500\\GeVc.\nThroughout the paper, the results obtained\nfrom \\PbPb\\ data are compared to references based on the \\PYTHYD\nsamples described in Section~\\ref{sec:pythia_samples}. The subscripts $1$ and $2$ in the kinematical quantities always refer to the leading and subleading jets, respectively.\n\n\n\\subsection{Dijet azimuthal correlations}\n\\label{sec:dphi}\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=\\cmsFigWidth]{dijet_dphi_all_pt_0to20_20120104}\n\\end{center}\n\\caption{Distribution of the angle $\\dphi$ between the leading and subleading jets\nin bins of leading\njet transverse momentum from\n120 $ < \\ptlead < 150$\\GeVc\\ to $\\ptlead > 300$\\GeVc\\ for\n  subleading jets of $\\ptsub> 30$\\GeVc.\nResults for 0--20\\% central PbPb events are shown as points while the histogram\nshows the results for\n{\\sc {pythia}} dijets embedded into \\HYDJET PbPb simulated events. The error bars represent the statistical uncertainties.}\n\\label{fig:dphiPt}\n\\end{figure*}\n\nEarlier studies of the dijet events in heavy-ion collisions~\\cite{Chatrchyan:2011sx,Aad:2010bu} have shown persistency in dijet azimuthal correlations despite the asymmetry in dijet momenta. This aspect is crucial\nin the interpretation of energy loss observations~\\cite{CasalderreySolana:2011rq}.\nTo understand the momentum dependence of the quenching effects, this study investigates the angular correlation, \\ie, the opening azimuthal angle, \\dphi, between the leading and subleading jets of the events, in bins of leading jet $\\ptlead$.\n\nFor events with 0--20\\% centrality, two features are visible in the \\dphi\\ distributions shown in Fig.~\\ref{fig:dphiPt}: a peaking structure at $\\dphi = \\pi$, and a constant offset from zero in the overall distribution. The distribution around the $\\dphi = \\pi$ peak reflects the back-to-back dijet production and although this distribution changes across the various leading-jet \\pt\\ bins, there is no significant difference between PbPb data and the \\PYTHYD sample.\nThis observation confirms the conclusions of earlier studies~\\cite{Chatrchyan:2011sx,Aad:2010bu}, extending the analysis to differential leading-jet \\pt bins.\nThe event fraction that extends to small \\dphi\\ values is likely due to the matching of the leading jet with a random underlying event fluctuation instead of the true subleading jet partner. The difference in the rate of such events between the PbPb data and the \\PYTHYD sample is compatible with the effect of quenching, which makes it easier for a background fluctuation to supersede a genuine low \\pt jet.\nThe fraction of these background events strongly depends on the centrality and leading jet \\pt. For the purposes of the study presented in this paper, the contribution of these background events to the results is subtracted by using the events at small \\dphi.\n\n\\subsection{Dijet momentum balance}\n\\label{sec:asymmetry}\nTo characterize the dijet momentum balance (or imbalance) quantitatively, we use the asymmetry ratio\n\\begin{equation}\n\\label{eq:aj}\nA_J = \\frac{\\ptlead-\\ptsub}{\\ptlead+\\ptsub}~.\n\\end{equation}\nDijets are selected with $\\dphi > 2\\pi\/3$.\nIt is important to note that the subleading jet $\\ptsub > 30$\\GeVc\\ selection imposes\na $\\ptlead$-dependent limit on the magnitude of \\AJ. The distributions are normalized to\nthe number of selected dijet events.\n\nAs discussed in Section~\\ref{sec:dphi}, the contribution of background fluctuations is estimated from\nthe events with dijets of $\\dphi < \\pi\/3$, and the distributions obtained from these events are subtracted from the results.\nThe estimated fraction of background events, as a function of both leading jet \\pt\\ and centrality, is shown in the bottom row of Fig.~\\ref{fig:SubRate}.\nThe fraction of dijet events in which the subleading jet is found within the acceptance, after the subtraction of\nbackground events, is shown in the top row of Fig.~\\ref{fig:SubRate}. The events in which the subleading jet is not found should be taken into account when comparing the asymmetry distributions, although the bias is negligible for bins of leading jet $\\pt > 180$\\GeVc.\n\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics[width=\\cmsFigWidth]{SubLeadingRate_d20120103}\n\\caption{\nFraction of events with a genuine subleading jet with $\\dphi > 2\\pi\/3$, as a function of leading jet $\\ptlead$ (left)\nand \\npart\\ (right). The background due to underlying event\nfluctuations is estimated from $\\dphi < \\pi\/3$ events and subtracted from the number of dijets.\nThe fraction of the estimated background is shown in the bottom panels.\nThe error bars represent the statistical uncertainties.}\n\\label{fig:SubRate}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=\\cmsFigWidth]{dijet_imbalance3_lead120_sub30_all_cent_20120103_subt}\n\\caption{Dijet asymmetry ratio, $A_{J}$, for leading jets of $\\ptlead> 120$\\GeVc\\ and\n  subleading jets of $\\ptsub> 30$\\GeVc\\ with a selection of $\\dphi>2\\pi\/3$ between the two jets.\nResults are shown for six bins of collision centrality, corresponding to selections of 70--100\\% to 0--10\\%  of the total inelastic cross section.\nResults from data are shown as points, while the histogram shows the results\nfor\n\\PYTHIA dijets embedded into \\HYDJET PbPb simulated events.\nData from pp collisions at 2.76\\TeV are shown as open points in comparison to PbPb results of 70--100\\% centrality.\nThe error bars represent the statistical uncertainties.}\n\\label{fig:JetAsymm}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=\\cmsFigWidth]{dijet_imbalance3_0to20_pt_20120103_subt}\n\\caption{Dijet asymmetry ratio, $A_{J}$, in bins of leading jet transverse momentum from\n120 $ < \\ptlead < 150$\\GeVc\\ to $\\ptlead > 300$\\GeVc\\ for\n  subleading jets of $\\ptsub> 30$\\GeVc\\\nand $\\dphi>2\\pi\/3$ between leading and subleading jets.\nResults for 0--20\\% central PbPb events are shown as points, while the histogram\nshows the results for\n\\PYTHIA dijets embedded into \\HYDJET PbPb simulated events. The error bars represent the statistical uncertainties.}\n\\label{fig:JetAsymmPt}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=\\cmsFigWidth]{dijet_imbalance5_0to20_pt_20120103_subt}\n\\caption{Subleading jet transverse momentum fraction ($\\ptsub\/\\ptlead$), in bins of leading\njet transverse momentum from\n120 $ < \\ptlead < 150$\\GeVc\\ to $\\ptlead > 300$\\GeVc\\ for\n  subleading jets of $\\ptsub> 30$\\GeVc\\\nand $\\dphi>2\\pi\/3$ between leading and subleading jets.\nResults for 0--20\\% central PbPb events are shown as points, while the histogram\nshows the results for\n\\PYTHIA dijets embedded into \\HYDJET PbPb simulated events.\nThe arrows show the mean values of the distributions and the error bars represent the statistical uncertainties.}\n\\label{fig:JetFractionPt}\n\\end{center}\n\\end{figure*}\n\nThe centrality dependence of $A_J$ for \\PbPb\\ collisions is shown in\nFig.~\\ref{fig:JetAsymm}, in comparison to results from \\PYTHYD simulations.\nThe most peripheral events are also compared\nto results from \\pp\\ collisions at $\\sqrt{s} = 2.76\\TeV$, where the same jet algorithm is used.\nThis comparison supports the use of the\n\\PYTHYD sample as a reference for the dijet asymmetry, which also takes into account underlying event\neffects when comparing with PbPb data.\nThe shape of the dijet momentum balance distribution experiences a gradual change with collision centrality,\ntowards more imbalance. In contrast, the \\PYTHIA simulations only\nexhibit a modest broadening, even when embedded in the highest multiplicity\n\\PbPb\\ events.\n\nTo study the momentum dependence of the amount of energy loss,\nFig.~\\ref{fig:JetAsymmPt} presents the distributions of $A_J$ in different bins of leading jet \\pt,\nfor 0--20\\% central events. One observes a strong evolution in the shape of the distribution across the\nvarious \\pt\\ bins, while a significant difference between PbPb data and \\PYTHYD simulations persists in\neach \\pt\\ bin. The distributions of the \\ptrat\\ ratio, shown in Fig.~\\ref{fig:JetFractionPt}, provide a more intuitive way of quantifying the energy loss.\nBoth the $A_J$ and \\ptrat\\ distributions are affected by the cut on the subleading jet $\\pt$, which should be taken into account in the interpretation of the average value. However, in the bins with leading jet $\\pt> 180\\GeVc$, more than 95\\% of the leading jets are correlated with a subleading jet, indicating that the bias due to dijet selection is very small.\n\n\\subsection{The dependence of dijet momentum imbalance on the \\texorpdfstring{\\pt}{pt} of the leading jet }\n\nThe dependence of the energy loss on the leading jet momentum can be studied using the jet transverse momentum ratio\n$\\ptsub\/\\ptlead$.\nThe mean value of this ratio is presented as a function of $\\ptlead$\nin Fig.~\\ref{deltaPt} for three bins of collision centrality, 50--100\\%, 20--50\\%, and 0--20\\%.\nThe \\PYTHYD simulations are shown as squares and the\nPbPb data are shown as points. Statistical and systematic uncertainties are plotted as error bars and brackets, respectively.\nThe main contributions to the systematic uncertainty in $\\ptsub\/\\ptlead$\nare the uncertainties in the $\\pt$-dependent residual energy scale and the effects of the underlying event on the jet energy resolution.\nEarlier studies of jet-track correlations~\\cite{Chatrchyan:2011sx} have shown that the energy composition of the quenched jets was not significantly different, which\nputs a constraint on the energy scale uncertainty. The uncertainty on the energy scale is derived from\nthree sources: the uncertainty evaluated in the pp studies \\cite{Chatrchyan:2011ds},\nthe energy scale difference in pp data and MC, and the energy scale and its parton type dependence~\\cite{MattPFlow} in simulations of PbPb events (see Section~\\ref{sec:pythia_samples}). These contributions are added in quadrature to assign the total uncertainty on the jet energy scale.  Using this value as a boundary, the uncertainty in the $\\ptsub\/\\ptlead$\nresults is then estimated by varying the jet response at low \\pt\\ and at high \\pt\\ independently.\nThe uncertainty on the underlying event effects is estimated from the full\ndifference between \\pp and \\PYTHYD.\nThese effects add up to 6\\% in the most central events.\nFor the low leading-jet \\pt\\ bins, jet reconstruction efficiency also introduces a minor uncertainty on the order of 1\\%.\nUncertainties due to additional misreconstructed jets, calorimeter noise, and the track requirement are negligible compared\nto the dominating sources of uncertainty.\nFor the centrality bins of 50--100\\%, 20--50\\% and 0--20\\%, the sources of systematic uncertainty are summarized in Table~\\ref{RatioSystematics}.\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=\\cmsFigWidth]{deltaPtOverPt5_lead120_sub30_diff_20120103}\n\\caption{\nAverage dijet momentum ratio $\\ptsub\/\\ptlead$ as a function of\nleading jet \\pt for three bins of collision centrality, from peripheral to central collisions,\ncorresponding to selections of 50--100\\%,  30--50\\% and 0--20\\%  of the total inelastic cross section.\nResults for \\PbPb\\ data are shown as points with vertical bars and brackets indicating\nthe statistical and systematic uncertainties, respectively.  Results for \\PYTHYD are shown as squares. In the 50--100\\% centrality bin,\nresults are also compared with pp data, which is shown as the open circles.\nThe difference between the \\PbPb\\ measurement and the \\PYTHYD expectations is shown in the bottom panels. }\n\\label{deltaPt}\n\\end{center}\n\\end{figure*}\n\nAs shown in Fig.~\\ref{deltaPt}, both the PbPb data and the \\PYTHYD  samples reveal an increasing trend for the mean value of the\n jet transverse momentum ratio, as a function of the leading jet $\\ptlead$. This can be understood\nby the reduction in the effects of jet splitting and energy resolution as one goes to higher jet momenta.\nHowever, the central \\PbPb\\ data points lie consistently below the \\PYTHYD trend. The difference between the pp data and the \\PYTHYD reference is of the order of the systematic uncertainty of the measurement, whereas the difference between \\PbPb\\ data and the reference is more than twice larger. This difference is related to the parton energy loss and for central PbPb collisions it is of significant magnitude across the whole \\pt\\ range explored in this study.\n\n\\begin{table}[htbp]\n\\begin{center}\n\\caption{Summary of the \\ptrat\\ systematic uncertainties. The range of values represent the\nvariation from low ($\\ptlead<140\\GeVc$) to high\n($\\ptlead>300\\GeVc$) leading jet \\pt. }\n\\label{RatioSystematics} \\begin{tabular}{|l|c|c|c|c|}\n\\hline\nSource &  50--100\\% & 20--50\\% & 0--20\\% \\\\\n\\hline\nUnderlying event & 1\\% &  3\\% & 5\\% \\\\\nJet energy scale & 3\\% &3\\% & 3\\%  \\\\\nJet efficiency  & 1--0.1\\% & 1--0.1\\% & 1--0.1\\% \\\\\nJet misidentification & $<0.1$\\%  & $<0.1$\\% & 1--0.1\\%\\\\\nCalorimeter noise & $<0.1$\\% & $<0.1$\\% & $<0.1$\\%  \\\\\nJet identification & $<0.1$\\% & $<0.1$\\% & $<0.1$\\%\\\\\n\\hline\nTotal   & 3.5\\% & 4.5\\% & 6\\% \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\n\n\\section{Summary}\n\\label{sec:summary}\n\nDijet production in PbPb collisions at\n$\\rootsNN=2.76\\TeV$ was studied with the CMS detector in a data sample\ncorresponding to an integrated luminosity of \\lum. The \\ak\\ algorithm was\nused to reconstruct jets based on combined tracker and calorimeter\ninformation. Events containing a leading jet with $\\ptlead > 120$\\GeVc\\ and\na subleading jet with $\\ptsub > 30$\\GeVc\\ in the pseudorapidity range\n$|\\eta| < 2$ were analyzed.\nData were compared to \\PYTHYD dijet simulations,\ntuned to reproduce the observed underlying event fluctuations.\nFor the most peripheral collisions, good agreement between data and simulations is\nobserved. For more central collisions, the dijet momentum imbalance in the data\nis significantly larger than seen in the simulation. Across the entire range of jet momenta studied, no significant broadening of the dijet angular correlations is observed with respect to the reference distributions.\n\nThe dijet momentum imbalance was studied as a function of the leading jet $\\ptlead$\nfor different centrality ranges in comparison to the \\PYTHYD simulation.\nFor leading jet momenta\n$\\ptlead > 180$\\GeVc\\ the dijet balance distributions are found to be essentially\nunbiased by the subleading jet threshold of $\\ptsub> 30 $\\GeVc.\nFor mid-central (30--50\\%) and more central PbPb event selections, a significantly\nlower average dijet momentum ratio $\\langle \\ptsub\/\\ptlead \\rangle$\nis observed than in the pp data and in the dijet embedded simulations. The downward shift in\n$\\langle \\ptsub\/\\ptlead \\rangle$, with respect to the \\PYTHYD reference, is seen to increase\nmonotonically with  increasing collision centrality,\nand to be largely independent of the leading jet $\\ptlead$,\nup to $\\ptlead$ values in excess of 350\\GeVc.\n\nIn summary, the results presented in this paper confirm previous observations based on\na smaller dataset and extend the measurements of jet-quenching effects to wider centrality and\nleading jet transverse momentum ranges, as well as to lower subleading jet transverse momentum.\n\n\n\\section*{Acknowledgments}\n\\hyphenation{Bundes-ministerium Forschungs-gemeinschaft Forschungs-zentren} We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC machine. We thank the technical and administrative staff at CERN and other CMS institutes. This work was supported by the Austrian Federal Ministry of Science and Research; the Belgium Fonds de la Recherche Scientifique, and Fonds voor Wetenschappelijk Onderzoek; the Brazilian Funding Agencies (CNPq, CAPES, FAPERJ, and FAPESP); the Bulgarian Ministry of Education and Science; CERN; the Chinese Academy of Sciences, Ministry of Science and Technology, and National Natural Science Foundation of China; the Colombian Funding Agency (COLCIENCIAS); the Croatian Ministry of Science, Education and Sport; the Research Promotion Foundation, Cyprus; the Ministry of Education and Research, Recurrent financing contract SF0690030s09 and European Regional Development Fund, Estonia; the Academy of Finland, Finnish Ministry of Education and Culture, and Helsinki Institute of Physics; the Institut National de Physique Nucl\\'eaire et de Physique des Particules~\/~CNRS, and Commissariat \\`a l'\\'Energie Atomique et aux \\'Energies Alternatives~\/~CEA, France; the Bundesministerium f\\\"ur Bildung und Forschung, Deutsche Forschungsgemeinschaft, and Helmholtz-Gemeinschaft Deutscher Forschungszentren, Germany; the General Secretariat for Research and Technology, Greece; the National Scientific Research Foundation, and National Office for Research and Technology, Hungary; the Department of Atomic Energy and the Department of Science and Technology, India; the Institute for Studies in Theoretical Physics and Mathematics, Iran; the Science Foundation, Ireland; the Istituto Nazionale di Fisica Nucleare, Italy; the Korean Ministry of Education, Science and Technology and the World Class University program of NRF, Korea; the Lithuanian Academy of Sciences; the Mexican Funding Agencies (CINVESTAV, CONACYT, SEP, and UASLP-FAI); the Ministry of Science and Innovation, New Zealand; the Pakistan Atomic Energy Commission; the Ministry of Science and Higher Education and the National Science Centre, Poland; the Funda\\c{c}\\~ao para a Ci\\^encia e a Tecnologia, Portugal; JINR (Armenia, Belarus, Georgia, Ukraine, Uzbekistan); the Ministry of Education and Science of the Russian Federation, the Federal Agency of Atomic Energy of the Russian Federation, Russian Academy of Sciences, and the Russian Foundation for Basic Research; the Ministry of Science and Technological Development of Serbia; the Ministerio de Ciencia e Innovaci\\'on, and Programa Consolider-Ingenio 2010, Spain; the Swiss Funding Agencies (ETH Board, ETH Zurich, PSI, SNF, UniZH, Canton Zurich, and SER); the National Science Council, Taipei; the Scientific and Technical Research Council of Turkey, and Turkish Atomic Energy Authority; the Science and Technology Facilities Council, UK; the US Department of Energy, and the US National Science Foundation.\n\nIndividuals have received support from the Marie-Curie programme and the European Research Council (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation \\`a la Recherche dans l'Industrie et dans l'Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the Council of Science and Industrial Research, India; and the HOMING PLUS programme of Foundation for Polish Science, cofinanced from European Union, Regional Development Fund.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRecently, assistive robotics became an increasingly popular research field as it provides use-case applications for solutions from a wide range of disciplines. Whereas special-purposed service robots such as autonomous cleaning systems or lawn mowers already found their way into many households, more general-task directed robotic assistants still entail challenges for the safe and reliable use in unconstrained environments. In general, these systems are aimed at providing various services in the household and workspace of humans that are assigned in direct and human-oriented interactions. This requires the assistive robot to enhance the mostly geometric perception as solely needed for cleaning tasks to a more general semantic understanding of its environment. In particular, a common task for a service robot is to find and bring back a specific kind of object. In order to execute such  a command the robot must be able to interact with the human by understanding its needs and subsequently having the capabilities to navigate and find the dedicated object autonomously.\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{info3.png}\n\t%\n\t\\caption{Example image of the TIAGo robot in front of a mapped office environment. Multiple objects on the desk, such as screen, cup, book, mouse and keyboard, are additionally incorporated as projected spheres into the point cloud map.}\n\t\t\\label{info}\n\\end{figure}\nIn this paper we focus on the perceptual semantics for processing meaning-oriented information for objects and humans.\nMultiple approaches \\cite{hobbit, martinez, lee} have been presented targeting the detection and recognition of semantic data for mobile robots. Common methods harness model-based solution approaches where each object is identified by its shape or color using handcrafted descriptors. Given that the target object has an unique appearance, these methods can be very performant for recognition tasks. On the other side, low image resolution, small objects, an ordinary appearance and multiple instances of similar looking objects can impair the detection and recognition of such systems. In addition, the target models have to be taught beforehand with training images of the object showing it from multiple different perspectives.\n\nWe tackle this challenge by using neural networks and combine them with geometric constraints provided by the robot. Our system can be divided in two separate modules. The first one targets the enrichment of geometric maps with semantic information by detecting and localizing objects in a robust manner (Fig. \\ref{info}). Additionally, we make sure to constantly correct and extend currently mapped objects based on updated perception data from the robot. This makes our system also suitable for localization methods that provide retrospective optimization capabilities. The procedure is carried out on the fly in real-time, when the robot is exploring the surroundings making it a well benefiting add-on for geometric-based mapping algorithms.  \nThe second module detects and analyzes potential human interaction partners where we aim to additionally predict cooperation willingness. This provides a first step of interactions with humans in a proactive manner while at the same time semantic objects can be adhoc incorporated in resulting tasks.\nWe implemented our system on the TIAGo, a mobile humanoid robot platform.\nThe rest of this paper is organized as follows: Section \\ref{sec:system} describes our proposed system. In section \\ref{implementation}, we give details of the current state of implementation. Section \\ref{conclusion} concludes and gives an insight about our future work.\n\\begin{figure*}[th]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{system3.pdf}\n\t\n\n\t\\caption{Overview of the proposed method. We use RGB-D images to feed our process lines for object registration and human condition estimation. The object registration consist of 2D object detections that are mapped and associated to 3D objects. The human condition estimation starts with multi task detection process for prediction persons and faces that are in a subsequent step further analyzed for interaction willingness. }\n\t\\label{system}\n\\end{figure*}\n\n\\section{system components} \n\\label{sec:system}\nOur system (Fig. \\ref{system}) consists of two main pipelines for extracting and evaluating semantic information from visual data. The first one is dedicated to detect and register volumetric objects. The second one classifies and estimates the attention related behavior of humans. In the following subsections, we describe each component in detail.\n\n\\subsection{Object registration}       \nThe first step of the object registration is the classification of image regions in the image stream provided by the camera in the head of the robot. For this matter deep neural networks have been proven to reliably detect wide ranges of different types of objects. Popular architectures are the RCNN model family \\cite{rcnn, FasterRCNN} and the YOLO model family \\cite{yolo, yolo9000}. Due to it outstanding speed we chose YOLOv4 \\cite{yolov4} for our framework. The network is pretrained on the COCO database\\cite{coco} and able to predict up to 80 different classes. However, when testing it outside of the database we experienced a quite decreased accuracy rate and repeating occurrences of false positive detections. Accordingly, the unfiltered processing of the provided predictions will result with false labeled map parts in later stages. To overcome this problem, we apply an adapted Intersection over Union (IOU) tracker\\cite{iou} that associates similarly located and equally labeled bounding boxes from consecutive images. Associated detections form a so called \\textit{track}, where the confidence of a true positive object classification increases with the tracks length.\nAfter reaching a specific track length threshold the 2D detection is projected into the 3D space using the intrinsic camera parameters and the extrinsics provided by the robots localization algorithm. For this purpose the depth estimation from the camera active stereo module is used together with the RGB images to generate metric scaled point clouds. Depending on the bounding box sizes we then cut out object cuboids that are afterwards projected into the robots 3D point cloud map. While this method for registering of semantic objects can be performed in a very efficient way, it is combined with supportive solutions to overcome the following challenges: \n \\linebreak\n\\subsubsection{Object Recognition}\nWhen the robot is moving around it may detect objects in the image stream that already has been registered earlier to the map. Processing it a second time will lead to multiple mapped instances of the same physical objects and must be therefore prevented. Consequently, the object mapping must be able to recognize objects that has been seen and registered before. In general, 3D object recognition has been extensively researched in the recent time, but it commonly requires huge computational efforts. In addition, objects from the same class can resemble each other in a way that taking unique visual fingerprints fails. We approach the issue by comparing the projected localization of a new object \\textit{candidate} with the localization of earlier registered objects of the same class. Is the candidate significantly overlaying with its counterpart it is likely that both belong to the same origin. This approach is implemented using a \\textit{nearest neighbor} association process, where we compare the average distance between the entire point clouds instead of their centroids. This way, we incorporate the objects size in the association decision as point clouds of large objects can contain centroids that are wide apart, while the clouds themselves heavily overlap. \n \\linebreak\n\\subsubsection{Object Optimization}\nThe probabilistic localization of the robot involves uncertainties and will only be a convergence of its real pose. This effect is further enhanced when navigating in an unknown environment and, thus, a simultaneous mapping process is required. For localization methods with optimization capabilities the detection of distinctive or familiar map areas is exploited to re-evaluate the past trajectory and, where appropriate, to correct drifts. Concurrently, we use this mechanism to recalculate the poses of our semantic objects derived from the corrected robot trajectory. At the same time, we analyze if this leads to any strong overlaps between their updated point clouds. This provides us the opportunity to even correct mapping errors induced from the erroneous trajectory. Objects with salient overlap are assumed to belong to the same physical instance and are therefore merged whereby the points clouds are concatenated and the localization and object dimensions are recalculated accordingly. This way, we can not only maintain and correct the perception of semantic objects around us, but with the aid of point cloud merging we are able to build up more complete point cloud appearance models of our objects.\n\\subsection{Human Behavior Estimation}\nTo create optimal preconditions for a successful human-robot interaction we apply a dedicated process pipeline to search for and analyze human interaction partners. Instead of using one network for person detection and one for face detection, we utilize a custom neural network that is able to predict person and face bounding boxes simultaneously.\n \\linebreak \n\\subsubsection{Face-Person Detection}\nThe network architecture is based on the SSD approach~\\cite{ssd} that provides a higher inference rate than two-stages detectors. Moreover, both detectors for persons and faces are combined using multi-task learning which enables them to share their first layers of feature extraction. This equally boost the efficiency and the generalization capabilities.  \n\nThe main difficulty for the combination of the two tasks face and person detection in a single neural network is the fact that publicly available databases contain only ground truths for one of the two tasks. For this purpose, we developed a custom multi-task loss function and designed an architecture consisting of a shared backbone and separate detection layers for each detection task. During training, we alternate between batches of person annotations and batches of face annotations, which are taken from different databases.\nThe prediction of the respective class with non-existing ground truth is assumed to be correctly determined and only the gradients of the detection layers with existing ground truth information are adjusted. Thus, a completely end-to-end trainable framework could be created.\n  \nWhile the predicted bounding boxes for persons give us a good estimation of its localization, the face predictions are further used for behavior examination. In particular, we want to know if the human is interested in an interaction with the robot. A good indication for general interest is the gaze or head pose pointing towards the robot.\nAs the gaze estimation is error-prone at low image resolutions, we focus on the head pose to evaluate general interaction willingness.\n \\linebreak\n\\subsubsection{Interaction Willingness Estimation}\nTo estimate the head pose we first determine the position of facial landmarks in the face images provided by the face predictor. These facial keypoints are distinctive spots in the face (e.g. the corners of the eyes, the sides and top of the nose) that provide us information about the current formation of the face. We estimate the landmark positions by applying an ensemble of regression trees~\\cite{faciallm} that has proven to provide very fast and reliable predictions. In the following step we project these 2D landmarks into the 3D space. This is achieved by using a default 3D model that contains the same facial landmarks and aligning it with the previously predicted 2D counterpart. As this represents a minimization problem we apply the commonly used Levenberg-Marquardt optimization to estimate a matching 3D mask. The rotation and translation for the projection that provides the smallest re-projection error implies the head pose.\n\nThe prediction is performed for every single image that contains a detected face. However, the pose information from a single frame is not sufficient to derive assumptions about an underlying general interaction willingness. Brief views in the direction of the robot can be caused by arbitrary intentions including behavior estimation (e.g. when the robot is moving) or even causal glances without deeper conscious intentions. We therefore track a humans head pose over multiple images and gain confidence about interaction intentions the longer the persons focus is directed at the robot. We assume that a viewing direction towards the robot of a duration of 3~ms represents a reliable threshold to determine interaction willingness. However, short distractions are common that result in brief interruptions of the attention towards the robot. We therefore use a solution based on our previous idea \\cite{hempel} and calculate the interaction willingness in a dynamic manner. We apply a progress bar that loads faster when the attention is directed to the robot and unloads slower in case of distractions. In this way, showing interaction willingness can be resumed in natural way even though it has been abandoned for a short time.   \n\n\\section{Implementation}\n\\label{implementation}\nWe implemented our system in form of multiple Robot Operating System (ROS) compatible modules for seamless intercommunication with other (ROS) components and deployed it on the TIAGo robot. For mapping and localization we use the RTAB-Map~\\cite{rtabmap} as its graph-based approach suits our semantic data update and correction process. The robots IMU is used as guess to perform a laser-based ICP-SLAM that runs along with other ROS nodes for path planning, collision avoidance and motion planning on the robots onboard i7 computer. Our image processing focused methods are deployed on an external mobile system that is placed on the robots shoulders. It contains a Quadro RTX 5000 that is able to process the neural networks more efficiently than CPUs.\n\n\n\\section{Conclusion}\n\\label{conclusion}\nIn this work, we addressed the problem of semantic meaningful perception for mobile assistive robots which constitutes a fundamental requirement for solving complex tasks. \n\nWe propose a neural network enhanced approach for successively mapping and maintaining 3D objects of the robots environment that can run alongside other geometrical mapping modules. \nSimilarly, we process humans by utilizing a custom single-shot detector that simultaneously provides person and face predictions in the image stream. The latter are furtherly used to estimates the persons interaction willingness to enable proactive collaboration behavior on the robots side. All modules are implemented on a mobile humanoid robot platform and are compatible for intercommunication with other ROS modules such as path and motion planning.    \nIn future works we will exploit this advantage to incorporate additional text-to-speech and speech recognition modules. First, we will search for potential interaction partners and proactively call for tasks when interaction willingness is predicted. Afterwards the tasks can be provided orally, processed and executed. Typical tasks will be the localization and bringing of specific objects that have either already been mapped or have to be found in a dedicated exploration drive.\nThe behavior will be evaluated in real world scenarios.  \n\n\\bibliographystyle{IEEEtranDOI}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAccording to core accretion models \\citep{mizuno-78}, giant gas\nplanets such as Jupiter and Saturn formed via the accumulation of an\nprotocore of rock and ice which gained solid material until it reached\nsufficient size to begin accreting the gaseous component of the\nprotosolar nebula. The existence of solid ice in the outer solar\nsystem promotes the rapid growth of the more massive protocores\nallowing the accretion of large quantities of gas necessary for\nJupiter-sized planets. Giant planets thus have a dense core of rock\nand ice surrounded by a H-He envelope. It is not known, however,\nwhether the initial dense core remains stable following the accretion\nof the H-He outer layer or whether the core erodes into the fluid\nhydrogen-rich layers above \\citep{stevenson-pss-82,guillot-book}.\n\nThe gravitational moments of Jupiter and Saturn, which have been\nmeasured by prior planetary missions and will be determined for\nJupiter with unprecedented accuracy by the upcoming Juno mission, may\nbe used in combination with interior models\n\\citep{militzer-apj-08,guillot-book,hubbard-ass-05,saumon-apj-04,nettelmann} to\nestimate the mass of the present-day core, but it is unclear whether\nthese masses correspond to the primordial core mass. It has been\nsuggested \\citep{guillot-book,saumon-apj-04} that the present-day core\nmass of Jupiter may be too small to explain its formation by core\naccretion within the relatively short lifetime of the protosolar\nnebula \\citep{pollack-icarus-96}, although a more recent Jupiter model\n\\citep{militzer-apj-08} predicted a larger core of 14--18 Earth masses\nwhich is consistent with core accretion. Furthermore, direct\nmeasurements of Jupiter's atmosphere suggest a significant enhancement\nin the concentration of heavy $(Z > 3)$ elements\n\\citep{niemann-science-96}, but it is unknown to what extent this\nshould be attributed to a large flux of late-arriving planetesimals\nversus the upwelling of core material. Determining the extent of core\nerosion is thus a major priority for understanding the interiors of\ngiant planets and the process by which they were formed.\n\nIn this work we focus on water ice, presumed to be a major constituent\nof the core, and consider the question of whether it has significant\nsolubility in fluid metallic hydrogen at conditions corresponding to\nthe core-mantle boundaries of giant gas planets. Water ice is the most\nprevalent of the planetary ices (water, methane and ammonia) which may\nbe assumed to make up the outermost layers of a differentiated\nrock-ice core \\citep{hubbard-science-81}. At the conditions of\ntemperature and pressure prevalent at giant planet cores, water ice is\npredicted \\citep{cavazzoni,french-prb-09} to be in either in a fully\natomic fluid phase in which oxygen and hydrogen migrate freely and\nindependently, or in a superionic phase in which oxygen atoms vibrate\naround defined lattice sites while hydrogen atoms migrate\nfreely. Assuming the existence of a core-mantle boundary at which\nwater ice and the fluid H-He phase are in direct contact, the relevant\nquestion is the extent to which the system may lower its Gibbs free\nenergy by the redistribution of the atoms of the ice phase into the\nfluid hydrogen. The extreme pressure and temperature conditions\nprevalent at giant planet core-mantle boundaries (8000--12000K and\n8--18 Mbar for Saturn, 18000--21000K and 35--45 Mbar for Jupiter) are\nnot yet obtainable in the laboratory, thus \\emph{ab initio}\nsimulations provide the best available guide to determining the extent\nof core solubility.\n\n\\section{Theory and Methodology}\n\nWe used density functional molecular dynamics (DFT-MD) calculations\nand coupling constant integration (CCI) techniques to compute the\nGibbs free energy of solvation, $\\Delta G_{sol}$, of H$_2$O in fluid\nmetallic hydrogen, i.e. the change in Gibbs free energy when an H$_2$O\nmolecule is removed from the pure ice phase and dissolved in H.  The\nfree energy of solubility is computed from the free energies of three\nsystems: pure ice, pure fluid H, and a mixed system in which the atoms\nof one water molecule are dissolved in $n$ atoms of hydrogen,\n\n\\begin{equation}\n\\Delta G_{sol} = G \\left(\\mbox{O} \\mbox{H}_{n+2}\\right) - \\left[G \\left( {\\mbox{H}_2 \\mbox{O}} \\right) + G\\left( \\mbox{H}_{n} \\right) \\right]\n\\label{deltag}\n\\end{equation}\n\nwhere $G \\left( {\\mbox{H}_2 \\mbox{O}} \\right)$ is the energy per\n$\\mbox{H}_2\\mbox{O}$ stoichiometric unit of the ice phase, and\n$G\\left(\\mbox{H}_{n}\\right)$ is obtained from an appropriately-scaled\nsimulation of 128 H atoms. This quantity becomes more negative as\nsolubility increases. A $\\Delta G_{sol}$ of zero implies a saturation\nconcentration of exactly one H$_2$O to $n$ H. In order to span the\nrange of likely conditions for the core-mantle boundary of Jupiter and\nSaturn, we considered pressures of 10, 20 and 40 Mbar, at a range of\ntemperatures from 2000 to 20000~K.\n\nComputation of free energies from MD simulations is difficult since\nthe entropy term is not directly accessible. Here we use a two-step\nCCI approach as previously applied by several authors\n\\citep{alfe-nature-99,morales-pnas-09,wilson-prl-10} to compute free\nenergies. The CCI method provides a general scheme for computing\n$\\Delta F$ between systems governed by potential energy functions\n$U_1$ and $U_2$. We construct an artificial system $U_\\lambda$ whose\nforces are derived from a linear combination of the potential energies\nof the two systems $U_\\lambda = (1-\\lambda)U_1 + \\lambda U_2$. The\ndifference in Helmholtz free energy between the two systems is then\n\n\\begin{equation}\n\\Delta F = \\int_0^1 \\langle U_2 - U_1 \\rangle_\\lambda \\: d \\lambda,\n\\end{equation}\n\nwhere the average is taken over the trajectories governed by the\npotential $U_\\lambda$.  We perform two CCIs for each $G$ calculation:\nfirst from the DFT-MD system to a system governed by a classical pair\npotential which we fit to the DFT dynamics of the system via a\nforce-matching approach \\citep{izvekov-jcp-04}, and then from the\nclassical system to a reference system whose free energy is known\nanalytically.\n\n\\subsection{Material phases}\n\nThe material phases in question must be established prior to the Gibbs\nfree energy calculations. At the pressure and temperature conditions\nof interest, hydrogen is a metallic fluid of H atoms in which\nmolecular bonds are not stable. Water dissolved within hydrogen\nlikewise is non-molecular, with a free O atom in an atomic H\nfluid. For water ice, the phase diagram at giant planet core pressures\nis divided into three regimes\n\\citep{cavazzoni,goldman,mattsson-prl-06,french-prb-09}: a\nlow-temperature ($<$~2000~K) crystalline regime, an intermediate\nsuperionic regime in which oxygen atoms vibrate around fixed lattice\nsites while hydrogen atoms migrate freely, and a higher-temperature\nfully fluid regime in which both hydrogen and oxygen atoms are\nmobile. The transition between the crystalline and superionic regimes\nhas not yet been studied in detail at high pressures but our\nsimulations find that it occurs below 2000~K. The transition from\nsuperionic to fully fluid is found to occur in simulation at\ntemperatures ranging from 8000~K for 10 Mbar and 13000~K for 50 Mbar\n\\citep{french-prb-09}. Consequently our study includes the superionic\nand fully fluid ice phases.\n\nPrevious studies of superionic ice \\citep{cavazzoni,french-prb-09}\nhave used an \\emph{bcc} arrangement of atoms for the oxygen\nsublattice. We found that such a lattice was stable at all points\nstudied in the superionic regime except for 10 Mbar at 2000 and\n3000~K. At these conditions we found that the oxygen sublattice was\nstable in the \\emph{Pbca} geometry which we recently reported to be\nthe most stable zero-temperature structure for ice at 10 Mbar\n\\citep{militzer-prl-10}. At 10 Mbar and 5000~K, superionic ice with a\n\\emph{bcc} sublattice was stable but the \\emph{Pbca} was not. The\n\\emph{bcc} oxygen sublattice was found to be stable for 20 and 40 Mbar\npressures at all temperatures studied in the superionic regime.\nAttempts to perform a superionic simulation in the \\emph{Cmcm}\ngeometry reported by \\cite{militzer-prl-10} to be the stable\nzero-temperature structure at these pressures resulted in an unstable\nsystem with a large anisotropic strain. We thus used a \\emph{bcc}\noxygen sublattice for all superionic ice simulations, except the\n2000~K and 3000~K simulations at 10 Mbar which used the oxygen\nsublattice from the \\emph{Pbca} phase, as indicated in Figure 1. While\nwe cannot yet exclude the possibility of the existence of yet another\nsuperionic ice structure, we note that the differences between\ndifferent ice phase energies are found to be on the order 0.1~eV per\nH$_2$O, and thus will not significantly affect our results about the\nstability of ice in giant planet cores.\n\n\\subsection{Computation of Gibbs free energies}\n\nOur goal in this paper is to study the solubility of water ice in\nfluid hydrogen. Due to considerations arising from the entropy of\nmixing, the solubility of one material in another is never zero,\nhowever solubility at trace quantities is not sufficient for core\nerosion. In particular, we wish to know whether solubility is\nthermodynamically favored at concentrations significantly greater\nthan the background concentration of oxygen in the fluid envelope of\nJupiter or Saturn -- this is equal to approximately one O atom to 1000\nH atoms if we assume solar concentrations for the Jovian envelope and\napproximately one part in 300 if we assume a threefold enrichment for\noxygen as observed for most other heavy elements \\citep{mahaffy}. We\nbegin by computing the Gibbs free energies of solubility for\ndissolving H$_2$O in pure H at one part in 125, and generalize later.\n\nThe coupling constant integration approach requires, as an integration\nend point, a reference system whose free energy may be computed\nanalytically. It is important to ensure that the system does not\nundergo a phase change along the integration pathway since this may\ncause numerical difficulties in the integration. For the fluid\nsystems, being pure hydrogen, hydrogen with oxygen, and ice at\ntemperatures above the superionic-to-fluid transition, we used an\nideal atomic gas as the reference system\n\\citep{alfe-nature-99,morales-pnas-09,wilson-prl-10}. For superionic\nice, we use as a reference system a combination of an ideal gas system\nfor the hydrogen atoms with an Einstein crystal of oxygen atoms each\noxygen tethered to its ideal lattice site with a harmonic potential of\nspring constant 30~eV\/$\\mbox{\\AA}^2$.\n\nThe DFT-MD simulations in this work used the Vienna Ab Initio\nSimulation Package (VASP) \\citep{vasp} with pseudopotentials of the\nprojector augmented wave type \\citep{paw} and the exchange-correlation\nfunctional of \\citet{pbe}. The pseudopotentials used had a core radius of 0.8~{\\AA} for hydrogen and 1.1~{\\AA} for oxygen. Wavefunctions were expanded in a basis set\nof plane waves with a 900~eV cutoff and the\nBrillouin Zone was sampled with $2 \\times 2 \\times 2$ $k$-points. The\nelectronic temperature effects were taken into account via Fermi-Dirac\nsmearing. A new set of force-matched potentials was fitted for each\npressure-temperature conditions for each stoichiometry. All MD\nsimulations used a 0.2~fs timestep. In the classical potential under\nsuperionic conditions, an additional harmonic potential term was added\nto ensure that oxygen atoms remained in the appropriate lattice sites,\nhowever, we found that in most cases the fitted pair potential alone\nwas sufficient to stabilize the superionic state.\n\nThe first step of the free energy calculations was the determination\nof the appropriate supercell volumes for each system for each set of\npressure and temperature conditions. This was accomplished via\nconstant-pressure MD simulations\n\\citep{hernandez-jcp-01,hernandez-prl-10} with a duration of 1.6~ps\n(0.6~ps for ice). DFT-MD trajectories were then computed in a fixed\ncell geometry in order to fit classical potentials. A run of 0.4~ps\nwas found to be sufficient for fitting suitable potential. We then\nperformed molecular dynamics runs 600 fs long (400 fs for ice) at five\n$\\lambda$ values to integrate between the DFT and classical\nsystems. Finally, we performed classical Metropolis Monte Carlo at 24\n$\\lambda$ values to integrate from the classical to the reference\nsystem.\n\n\\section{Simulation results}\n\nTable I lists the total Gibbs free energies for each simulated system\n(H$_{128}$, OH$_{127}$ and ice) for each set of temperature and\npressure conditions. Ice was confirmed to remain superionic except at\nthe 20~Mbar\/12000~K and 40~Mbar\/20000~K conditions where it was fully\nfluid. The error bars on the computed $G$ values are dominated primarily by the uncertainty in the computed volume at the desired pressure, and secondarily by uncertainty in the $\\langle U_{DFT} - U_{classical} \\rangle_\\lambda$ term in the coupling constant integration. These free energies are combined using Equation \\ref{deltag} to\ngive $\\Delta G_{sol}$ representing the free energy change associated\nwith removing an H$_2$O from the ice phase and dissolving it in the\nhydrogen fluid at a concentration of molecule per 125 solute H\natoms. $\\Delta G_{sol}$ increases strongly with temperature, but shows\nonly a weak dependence on pressure within the 10--40 Mbar range under\nconsideration. The $\\Delta G$ values were found to be well converged with respect to wavefunction cutoff, k-point sampling to within the available error bars. The effect of the electronic entropy term on $\\Delta\nG_{sol}$ was found to be less than 0.1 eV in all cases.  From a linear\ninterpolation through the adjacent data points we estimated the\ntemperature at which $\\Delta G_{sol}$ passes through zero as 2400~K\n$\\pm$ 200~K at 40 Mbar, 2800~K $\\pm$ 200~K at 20 Mbar, and 3400~K\n$\\pm$ 600~K at 10 Mbar.  As shown in Figure 1, this is clearly far\nlower than any reasonable estimate for the core-mantle boundaries for\neither Jupiter or Saturn. The onset of high solubility occurs within\nthe portion of the phase diagram where ice is superionic, and does not\ncoincide with either the superionic-to-fluid transition or the\ncrystalline-to-superionic transition in ice.\n\nThe total $\\Delta G$ of solubility may be broken down into three\ncomponents: a $\\Delta U$ term from the potential energy, a $P \\Delta\nV$ term from the volume difference, and a $-T \\Delta S$ entropic\nterm. Figure 2 shows this breakdown as a function of temperature for\n20~Mbar.  The breakdown for other pressures looks similar. The $\\Delta\nU$ term, representing the difference in chemical binding energy,\nprovides approximately a 2 eV per molecule preference for ice\nformation at all temperatures where ice is superionic, but a much\nsmaller preference for the 12000~K case where ice is fluid.  The $PV$\nterm is indistinguishable from zero within the error bars, suggesting\nthat this is not a pressure-driven transition, in contrast to recent\nresults on the partitioning of noble gases between hydrogen and helium\nin giant planet interiors in which volume effects were found to be the\ndominant term \\citep{wilson-prl-10}. The $-T \\Delta S$ term dominates\nthe temperature-dependent behavior, underlining that ice dissolution\nis indeed an entropy-driven process.\n\nGiven the Gibbs free energy of solubility for the insertion of one\nH$_2$O into 125 H atoms we can determine an approximate Gibbs free\nenergy of solubility at other concentrations, by neglecting the\ncontribution of the oxygen-oxygen interaction and including only the\nentropic term arising from the mixing.  Under these approximations, we\nobtain the expression\n\n\\begin{eqnarray}\n\\frac{\\Delta G_{sol}[m] - \\Delta G_{sol}[n]}{k_BT} &=& (m+2) \\log \\left( \\frac{(m+2)V_H + V_O}{(m+2)V_H}\\right)\\nonumber \\\\ \n&-& (n+2) \\log \\left( \\frac{(n+2)V_H + V_O}{(n+2)V_H} \\right)\\nonumber \\\\\n&-& \\log \\left( \\frac{m+2}{n+2} \\right) ,\n\\label{conc}\n\\end{eqnarray}\n\nwhere $V_H$ and $V_O$ are the effective volumes of each H and O atom\nin the fluid. This approximation becomes invalid as the oxygen-oxygen\ninteraction term in the fluid oxygen phase becomes significant,\nhowever for our purposes it is sufficient to know that the saturation\nconcentration is significantly higher than the background\nconcentration. If a saturation concentration of oxygen in hydrogen\ndoes indeed exist then this may be expected to have a retarding effect\non the erosion of the core.\n\nWe must also consider possibility that hydrogen and oxygen could\ndissolve separately from the H$_2$O mixture, leaving behind a\ncondensed phase with stoichiometry other than H$_2$O. We tested two\ncases explicitly, computing the free energies of pure oxygen and\none-to-one HO phases at the Jupiter-like 40 Mbar, 20,000~K set of\nconditions. Both O and HO were found to be in a fully fluid state at these temperature\/pressure conditions. We found that HO had a free energy of solubility of -11.2\neV per oxygen, while pure oxygen had -22.8 eV per oxygen. Comparing to\nthe -8.9 eV per oxygen solubility of H$_2$O, this suggests oxygen-rich\ncondensed phases are less thermodynamically stable than the H$_2$O\nphase, and certainly far less favourable than dissolution of the dense\nphase into metallic hydrogen. We have also neglected the possibility\nof hydrogen-enriched dense phases such as H$_3$O. While it possible\nsuch phases may be somewhat energetically preferred to H$_2$O, it is\nextremely unlikely that the preference will be strong compared to the\n$8-12$ eV per O unit preference for solubility.\n\nIn Figure 3 we plot the estimated $\\Delta G_{sol}$ as a function of\nconcentration for various computed temperatures at a pressure of\n40~Mbar. A negative value of $\\Delta G_{sol}$ means that it is\nthermodynamically favorable to dissolve further ice into the hydrogen\nphase at a given hydrogen-phase ice concentration, and the point at\nwhich $\\Delta G_{sol}$ is zero is the saturation concentration.  For\n2000~K and 3000~K the saturation concentrations are estimated to be on\nthe order of 1:500 and 1:20 respectively, while for higher\ntemperatures the saturation concentration is much higher. Given that\nwe neglect O--O interactions in the fluid hydrogen phase, it is\ndifficult to precisely determine the saturation concentrations for\nhigher temperatures. However, we may safely say that the saturation\nconcentration for temperatures in excess of 3000~K is very much\ngreater than the background oxygen concentration, and hence that\nsolubility of ice into hydrogen at the core-mantle boundaries of\nJupiter and Saturn is expected to be strongly thermodynamically\nfavored.\n\n\\section{Discussion}\n\nThe consequences of core erosion for planetary evolution models have\nbeen previously considered by \\citet{stevenson-pss-82} and later\n\\citet{guillot-book}. The effects of core erosion can potentially be\ndetected either by orbital probes such as Juno or by atmospheric entry\nprobes, since the redistribution of core material throughout the\nplanet will manifest itself both by a smaller core (detectable from\ngravitational moments) and a higher concentration of heavy elements in\nthe atmosphere than would be expected in a planet without core\nerosion. Once core material has dissolved into the metallic H layers,\nthe rate at which core material can be redistributed throughout the\nplanet is expected to be limited by double diffusive convection\n\\citep{turner,huppert}. Since the higher density due to compositional\ngradients of the lower material interferes with the convection\nprocess, convection may be slowed significantly. \\citet{guillot-book}\nmodelled the effect of core erosion under the assumptions of fully\nsoluble 30 Mbar core in each planet. Under their assumptions up to 19\nEarth masses could have been redistributed from Jupiter's core but\nonly 2 Earth masses from Saturn's, the difference being Jupiter's\nhigher temperatures. While this prediction is subject to significant\nuncertainty in many aspects of the model, it does suggest that a\nredistribution of a significant fraction of the initial protocore is\npossible, at least in Jupiter.  Further refinement of models for the\nupconvection of core material and its observable consequences may be\nfruitful. The effect of core erosion on the heat transport and mass\ndistribution properties of Jupiter and Saturn should also be taken\ninto account in future static models of these planets' interiors.\n\nThese calculations can be expanded in several ways. We have neglected\nthe presence of helium in the hydrogen-rich mantle, however due to the\nlarge magnitude of $\\Delta G_{sol}$ and the chemical inertness of\nhelium we do not expect the presence of helium in the mantle to\nsignificantly affect the solubility behavior. We have explicitly made\nthe assumption that ice and hydrogen are in direct contact, an\nassumption which might fail in one of two ways. If hydrogen and helium\nare immiscible at the base of the atmosphere then the core may make\ndirect contact with a helium-rich layer. This, however, is unlikely in\nthe context of the calculations of \\citet{morales-pnas-09} who predict\nhydrogen-helium immiscibility only far away from the cores of Jupiter\nand Saturn.  The other possibility is that the ice layers of the core\nmay be gravitationally differentiated, leaving ice beneath layers of\nthe less dense planetary ices methane and ammonia. Since the bonding\nin these is similar to that in water ice one could assume that they\nshow a similar solubility behavior, but this analysis is the subject\nof future work.\n\nWe have also considered only the structure of the present-day\nplanet. As suggested by \\citet{slav}, a proper consideration of core\nsolubility must also include solubility during the formation process,\nas dissolution of the icy parts of the core into the accreting\nhydrogen during the formation may result in the amount of ice on the\ncore itself being small by the time the planet reaches its final\nsize. A treatment of the formation processes for Jupiter and Saturn\nusing ice solubilities derived from \\emph{ab initio} calculations may\nbe valuable.\n\nOur calculations strongly suggest that icy core components are highly\nsoluble in the fluid mantle under the conditions prevalent at the\ncore-mantle boundaries of Jupiter and Saturn. Since many\nrecently-discovered exoplanets are more massive and hence internally\nhotter than Jupiter, it can be expected that any initial icy cores in\nthese exoplanets will also dissolve. The presence of core erosion may allow models predicting a small present-day Jovian core to be made consistent with the large initial core required by core erosion, however models of the interior mass distribution of the planet will need to be revised to take the inhomogeneous composition of the lower layers implied by convection-limited core redistribution into account. Improved models which\ninclude core redistribution processes, combined with the\ndata from the Juno probe, may assist in understanding the history and\npresent structure of Jupiter and other planets in our own and in other\nsolar systems.\n\n\n\n\\acknowledgments\n\n This work was supported by NASA and NSF. Computational resources were supplied in part by TAC, NCCS and NERSC. We thank D. Stevenson for discussions.\n\n\n\\clearpage\n\n\n\\bibliographystyle{apj}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nBoth long and short GRB jets have to cross a significant amount of matter (the stellar atmosphere for long GRBs and the merger's ejecta in short ones) before producing the observed $\\gamma$-rays.  \nThis understanding has lead to great interest in jet propagation within surrounding matter and the question was explored both analytically \\citep[e.g.,][]{Blandford_Rees1974, Begelman_Cioffi1989, Meszaros_Waxman2001, Matzner2003, Lazzati_Begelman2005, Bromberg2011} and numerically \\citep[e.g.,][]{Marti+1995, Marti+1997, Aloy+2000, macfadyen_supernovae_2001, Reynolds+2001, Zhang+2004, Mizuta+2006, Morsony+2007, Wang+2008, Lazzati+2009, Mizuta+2009, Morsony+2010, Nagakura+2011, Lopez+2013, Ito+2015, Lopez+2016, Harrison2018}.\nThis, naturally, raises the possibility that some jets are ``choked\" during their propagation and are unable to break out of the surrounding dense medium. \nThe observed temporal distributions of both long \\citep{Bromberg+2012} and short \\citep{Moharan_Piran2017} GRBs suggest that this happens in both types of events and there are indications that this also happens in some Supernovae \\citep{Piran2019}.\n\nIn cases that the jet does not emerge we may still observed the signature of the cocoon that forms. \nFirst, the breakout of the shock driven by the cocoon produces a bright flash. \nFor example, a cocoon breakout is most likely the origin of low-luminosity GRBs ({\\it ll}GRBs) \\citep{Kulkarni1998, macfadyen_supernovae_2001, Tan+2001, campana_association_2006, Wang+2007, waxman_grb_2007, katz_fast_2010, Nakar_Sari2012,Nakar2015}. \nThese type of GRBs are rarely observed, however, when their low luminosity is taken into account it was realized that they are more numerous than regular LGRBs \\citep{Soderberg+2006}. \nAnother signature arises from the fast cocoon material that engulfs the star once the hot cocoon material breaks out and spreads. \nSpecifically, this material leads to very broad absorption lines that are visible as long as it is optically thick \\citep{Piran2019}. \nSuch lines have been observed in several SNe some accompanied by {\\it ll}GRBs  \\citep{Galama+1998, Iwamoto+1998,   Modjaz+2006, Mazzali+2008, Bufano+2012, Xu+2013, Ashall+2019,Izzo_et_al_2019} and others without \\citep{Mazzali+2000, Mazzali+2002, Mazzali+2009}. \nFinally, the cooling emission of the cocoon will also generate a potentially detectable UV-optical transient on time scale of hours to days \\citep{nakar_piran2017}.\n\nThe important signature that helps determining the origin of the broad absorption lines is the energy-velocity distribution of the fast moving material. \nRegular spherical explosion result in a very steep  distribution  with roughly $\\mathrm{d} E(v)\/\\mathrm{d} \\ln v \\propto v^{-5}$ \\citep[e.g.,][]{nakar_sari2010}. \nHowever, when a jet is involved in the explosion this distribution is expected to be much shallower with much more energy at high velocities.\nRecently, \\cite{eisenberg2022} have shown that when the jet is successful the cocoon generates a unique energy-velocity flat distribution with $\\mathrm{d} E\/\\mathrm{d} \\ln \\Gamma\\beta \\propto {\\rm const.}$ over a wide range of velocities from sub to mildly relativistic, where $\\beta=v\/c$ and $\\Gamma$ is the corresponding Lorentz factor. \nThey also found that, when the jet is choked, it leaves a unique signature of a flat energy-velocity distribution. \nHowever, in the case of choked jets the flat distribution covers a range of velocities that is narrower than that of outflows driven by successful jets. \nMotivated by these results we examine here in detail the energy-velocity distributions of different chocked jet, focusing on the relation between the properties of the choked jet and the final energy-velocity distribution of the outflow after it becomes homologous.\n\nFor our study we use a large set of 2D relativistic hydrodynamical simulations. \nWe consider explosions that are driven by choked jets in which we vary the opening angle and the engine working time of the jet as well as the structure of the progenitor. \nWe follow the simulations until the entire outflow becomes homologous and examine what is the relation between these properties and the outflow energy-velocity distribution.\n\nThe paper is structured as follows. \nIn Section \\ref{sec: methodology} we describe the numerical procedure adopted for the simulations. \nIn Section \\ref{subsec: sim_setup} we describe the code choice and the composite mesh structure adopted, while in Section \\ref{subsec: ics} we report in detail the setup for the stellar and interstellar environment and the initial conditions for the relativistic jet. \nNumerical aspects are discussed in two appendixes: a resolution study is described in Appendix \\ref{sec: appendix A} and in Appendix~\\ref{sec: appendix B} we explore the different use of a numerical smoothing function for the stellar density profile. \nIn Section \\ref{sec: results} we explore the results of our set of simulations.\nWe summarize our findings and consider the implications to  observations in Section \\ref{sec: conclusions}. \n\n\n\n\\section{Methodology}\n\\label{sec: methodology}\n\n\\subsection{Simulation Setup}\n\\label{subsec: sim_setup}\n\nOur simulations are performed using  the open source  massively parallel multidimensional relativistic magneto-hydrodynamic code {\\textsc{pluto}} (v4.3) \\citep{Mignone2007}. \nThe code uses a finite-volume, shock-capturing scheme designed to integrate a system of conservation laws where the flow quantities are discretized on a logically rectangular computational grid enclosed by a boundary. \nWe use the special relativistic hydrodynamics module in 2D cylindrical coordinates. \nWe perform our calculations using a parabolic reconstruction scheme combined with a third-order Runge-Kutta time stepping. \nWe also force the code to reconstruct the 4-velocity vectors at each time step. \n\nThe  2D  simulations enables us to reach  high resolution  with reasonable computational resources. \n3D simulations carried by \\citet{Harrison2018} suggest a similar generic evolution of the jet for the same parameters. \nThe main difference arises in the morphology of the  jet head in 2D simulations that is affected by a plug at the head front. \nThis plug  diverts some of the jet elements sideways to dissipated their energy in oblique shocks. \nThis difference is not significant for our purposes.\n\nWe chose the equation of state of the fluid to be ideal and with a constant relativistic polytropic index of $4\/3$. \nThis equation of state is applicable for a relativistic gas (as in the jet) as well to a radiation dominated Newtonian gas, such as the shocked stellar envelope.\n\nTo study the long term evolution of the jet and the cocoon from the star, we use a large grid spanning for several orders of magnitude. \nThis  allows us to track the evolution of the system for at least two minutes after the breakout.  \nAt that time the entire stellar envelope is shocked by the cocoon and it expands enough to become homologous. \nWe use a grid of size $4736 \\times 4636$ cells, with the radial cylindrical coordinate\\footnote{Throughout the paper $r$ is used for the 2D cylindrical radius while $R$ stands for the 3D radius.} \nextending within the range $r = [0,350] \\times 10^{10} \\cm$ and the vertical coordinate extending within the range $ z= [0.1 , 360] \\times 10^{10} \\cm$. \n\nWe use a combination of a  uniform and two  non-uniform mesh grids in $r-z$ coordinates with a decreasing resolution from the inner region of the simulation box to the outer boundaries. \nThe grid mesh is uniform in the inner part to maintain a high resolution of the jet injection and the formation of the resulting high pressure cocoon.\nThe uniform mesh has  $1000 \\times 900$ grid points extending in the ranges $r = [0, 1] \\times 10^{10} \\cm$ and $ z=[0.1, 1] \\times 10^{10} \\cm$ with a resolution along both coordinates of $\\Delta (r,z)_\\mathrm{unif.} = 10^{7} \\cm $.\nNext to the uniform mesh we placed a stretched mesh with $1278^2$ grid points extending along both coordinates within the range $(r,z) = [1,6] \\times 10^{10} \\cm$ with a stretching ratio of $\\sim1.0018$ \nThe number of grid points for this mesh is chosen such that its initial grid spacing is the same of to the adjacent uniform mesh $\\Delta (r,z)_\\mathrm{s, init} = \\Delta(r,z)_\\mathrm{unif.} = 10^7 \\cm$ and its final grid spacing is $\\Delta(r,z)_\\mathrm{s, final} = 10^8 \\cm $. \nWe cover the remaining grid  at larger distances with a logarithmic spaced mesh with $2458^2$ grid points extending within the range $(r,z) = [6, 360] \\times 10^{10}\\cm$. \nThe number of grid points is chosen such that the  grid spacing of the mesh at $(r,z) = 6 \\times 10^{10} \\cm$  coincides to the resolution of the stretched mesh, such that $\\Delta(r,z)_\\mathrm{log, init} = \\Delta(r,z)_\\mathrm{s, final} = 10^8 \\cm $. \nIn this way we ensure a smooth increase of the resolution without jumps for the entire simulation grid.\nA detailed resolution study for these simulations is reported in Appendix \\ref{sec: appendix A}.\n\nWe inject the jet along  the inner  lower $z$ boundary, denoted $z_0$ (see \\ref{sec:jet}). \nOtherwise, we impose a reflective  boundary condition at this boundary as it approximates the equatorial plane of the system. \nWe impose  axial-symmetric conditions for the inner vertical boundary. \nBoth outer boundaries are set to outflow.\n\n\n\\subsection{Initial conditions}\n\\label{subsec: ics}\n\n\\subsubsection{The star}\n\nWe approximate the stellar density profile  as a continuous power law that mimics the sharp decline of density in radius near the stellar edge\\footnote{This profile diverges at the origin but this region does not influence the jet propagation and it is not included in our computational domain.}\n\\begin{equation}\n\\label{eq: rho_profile}\n \\rho(R) = \\begin{cases}\n \\rho_* \\left( \\dfrac{R_*}{R} - 1\\right)^2 + \\rho_0 ,& \\mathrm{for} ~  R \\leq  R_* \\ , \\\\ \n \\rho_0, & \\mathrm{for} ~ ~ R > R_* \\ .\n \\end{cases}\n\\end{equation}\nHere we choose $\\rho_* = 100 ~ \\g ~ \\cm^{-3}$ and $R_* = 3\\times 10^{10} \\cm$. \nThe total integrated mass of the star is $M_* = (9 \\pi \/ 5) ~ M_\\odot$ (see \\ref{sec:scale} for scaling of these parameters to other values.)\n\nFor this  density profile  the local slope, $\\alpha \\equiv \\mathrm{d} \\log \\rho(R)\/\\mathrm{d} \\log R = {-2}\/{(1-R\/R_*)} $. \nThe slope, $\\alpha$, reaches the critical value of 3 for $R = R_*\/3$. Beyond this value a spherical blast wave  accelerates and eventually looses causality. \nWe present our results for this specific density profile. \nHowever, in Sec.~\\ref{sec: diff_profiles} we show that the results for different stellar density profiles (both inner and outer) are qualitatively similar.\n\nSurrounding the star we have an external CSM density of $\\rho_0 = 1.67 \\times 10^{-21} ~ \\g ~\\cm^{-3}$.\nThis exact value is unimportant as it is added just to avoid a numerical vacuum. \nThe interaction of the jet or the cocoon outflow with this CSM is insignificant. \nTo avoid numerical artifacts arising from the sudden drop in density at the edge of the star  we smooth the density of the outer edge of the star with a power law: \n\\begin{equation}\n\\label{eq: smooth}\n    \\rho_\\mathrm{smooth} (R) = \\rho_\\mathrm{s} \\left( \\dfrac{R}{R_\\mathrm{s}} + 1\\right)^{-8} \\ , \n\\end{equation}\nwith $ \\rho_\\mathrm{s} = 0.05 ~ \\g ~ \\cm^{-3}$ and a gradient scale of $R_\\mathrm{s} = 5 \\times 10^8 \\cm$. \nWe verified that this arbitrary choice of the smoothing function does not affect our results (see Appendix ~\\ref{sec: appendix B}).\nIn order to avoid any initial random motion we set a uniform and low ambient pressure of $P = 3.5 ~ \\keV ~ \\cm^{-3}$ within the simulation grid. \n\n\\subsubsection{The jet}\n\\label{sec:jet}\n\nWe inject a collimated jet with a constant luminosity $L_\\mathrm{j}$, operating for $t_\\mathrm{e}$ so that the total injected energy is $E_{0} = L_\\mathrm{j} \\times t_\\mathrm{e} =  10^{51} \\erg$. \nA uniform jet is injected through a nozzle with a velocity in the $z$ direction with an initial bulk Lorentz factor $\\Gamma_{0,\\mathrm{j}}$ a  density $\\rho_\\mathrm{j} $ and a specific enthalpy $h_\\mathrm{j} \\gg 1$. \nBeing relativistically hot the jet spreads quickly to form an initial opening angle  $\\theta_\\mathrm{j} \\simeq 1\/(1.4 \\Gamma_{0,\\mathrm{j}})$ (see details on this injection method at \\citealt{Mizuta2013} and \\citealt{Harrison2018}).\n\nThe jet is numerically initialized by the injection of density, pressure, and momentum along the $z-$direction through a nozzle parallel to the $z-$axis with a radius $r_\\mathrm{j}$ at an initial height $z = z_0$. \nThe head cross section is then  $\\Sigma_\\mathrm{j} = \\pi r_\\mathrm{j}^2$. \nFor an initial opening angle $\\theta_\\mathrm{j} > 0.1~\\rad$, we set up $r_\\mathrm{j} = 10^8~ \\cm$, allowing a sufficient mesh coverage over the nozzle and we set the initial injection height at $z_0 = 10^9~\\cm$. \n\nWe consider a constant jet luminosity $L_\\mathrm{j}$.\nThis determines the product $\\rho_\\mathrm{j} h_\\mathrm{j}$ as:\n\\begin{equation}\n    \\label{eq: rho_j}\n    \\rho_\\mathrm{j} h_\\mathrm{j} = \\dfrac{L_\\mathrm{j}}{\\Sigma_\\mathrm{j}  \\Gamma_{0,\\mathrm{j}}^2 c^3} \\ .\n\\end{equation}\nWe choose $h_\\mathrm{j} = 100$.  \nThis choice of the enthalpy is arbitrary, as long as $h_\\mathrm{j} \\gg 1$.\nThe jet's pressure is given by $  P_\\mathrm{j} = (h_\\mathrm{j} - 1) {\\rho_\\mathrm{j} c^2}\/{4}$.\n\nWe explored the parameter space running simulations for different initial values of $L_\\mathrm{j} $ at steps of $2.5 \\times 10^{50}~\\erg~\\s^{-1}$ from $2.5 \\times 10^{50}~\\erg~\\s^{-1}$ to $2 \\times 10^{51}~\\erg~\\s^{-1}$ for a total of 9 different luminosities. \nFor each value of the luminosity, we run simulations for a set of different opening angles $\\theta_\\mathrm{j} = [0.05, 0.1, 0.2, 0.4, 0.6]~ \\rad$. \nAs we keep the total jet energy fixed these conditions translate to different engine working times $t_\\mathrm{e} = 10^{51}\\erg \/ L_\\mathrm{j}$. \nFor each of the 9 values of the luminosity we run 5 different values of the opening angle for a total of 45 simulations.\n\n\n\\subsubsection{Scaling  relations}\n\\label{sec:scale}\nWhile we consider specific numerical values for the stellar and jet parameters, our solutions can be scaled to other values. \nThe equation of motion of the forward shock speed $\\beta_\\head$, is regulated by  $\\Tilde{L}$ \\citep{Matzner2003,Bromberg2011}:\n\\begin{equation}\n\\label{eq: tilde_L2}\n    \\Tilde{L} \\simeq  \\dfrac{L_\\mathrm{j}}{ \\Sigma_\\mathrm{j} \\rho(R) c^3} \\ ,\n\\end{equation}  \nwith\n\\begin{equation}\n\\label{eq: beta_h}\n    \\beta_\\head = \\dfrac{1}{1+{\\Tilde{L}}^{-1\/2}} \\ .  \n\\end{equation}\nThe stellar size $R_*$, is the  scale  length of t the system. \nThe scalings $\\Sigma_\\mathrm{j} \\propto R_*^2$ and $\\rho(R\/R_*) \\propto \\rho_*$ we can express $\\tilde{L}$ as\n\\begin{equation} \n\\Tilde{L} \\propto \\dfrac{E_0}{t_\\mathrm{e} \\rho_* R_*^2} \\ . \n\\end{equation}\nIf we scale the  stellar  radius as $R_* = \\lambda R_*'$ we have to scale the density and the jet luminosity accordingly in order to maintain $\\Tilde{L}$ and $\\beta_\\head$ unchanged. \n\nAs we show later the location where the jet is choked (i.e. where the last element launched by the jet reaches the head) with respect to the stellar radius has also to be constant. \nThe choking location $z_\\mathrm{ch}$ is roughly proportional to the engine time $t_\\mathrm{e}$ \\citep{Nakar2015}:\n\\begin{equation}\n\\label{eq: zchoke}\n    z_\\mathrm{ch} = \\int_0^{t_\\mathrm{ch}} \\beta_\\head c \\mathrm{d} t \\simeq \\beta_\\head c t_\\mathrm{ch} = \\dfrac{\\beta_\\head c }{1-\\beta_\\head} t_\\mathrm{e}  \\ .  \n\\end{equation}\nwhere $  t_\\mathrm{ch} =  {t_\\mathrm{e}}\/{(1 - \\beta_\\head)}$ is the choking time. \nIf $\\beta_\\head$ is  kept constant than any transformation on $R_*$ will leave $z_\\mathrm{ch} \/ R_*$ unchanged if $t_\\mathrm{e} = \\lambda t_\\mathrm{e}' \\propto R_*$. \nThus, scaling $E_0 = \\eta E_0'$ and $R_* = \\lambda R_*'$ we require that and $\\rho_* = \\eta \\lambda^{-3} \\rho_*'$.\n\nBecause $M_* \\propto \\rho_* R_*^3$, when keeping this scaling of $t_\\mathrm{e}$ and $R_*$, we can rewrite Eq.~\\ref{eq: tilde_L2} as $ \\Tilde{L} \\propto E_0\/M_*$, which means that since $\\Tilde{L}$ is kept  constant the typical  velocity of the system $v_0 = (2E_0\/M_*)^{1\/2}$ is also conserved under these transformations.\n\nTurning to the jet parameters we recall that the only relevant quantities are $L_\\mathrm{j}$, $t_\\mathrm{e}$ and $\\theta_\\mathrm{j}$.\nThe first two determine $E_0$ and the latter determined $\\Gamma_{0,\\mathrm{j}}$. \nThe luminosity, together with the stellar parameters determine the produce $\\rho_\\mathrm{j} h_\\mathrm{j}$ with the condition that $h_\\mathrm{j}$ while arbitrary should be much larger than one. \n\nIn summary, given the physical scales $R_*$, $\\rho_*$, $E_0$, $v_0$, $t_\\mathrm{e}$ and the scalings $R_* = \\lambda R_*'$, $E_0 = \\eta E_0'$, the parameters defining the physics of our system, i.e. $\\tilde{L}$, $z_\\mathrm{ch}\/R_*$, will not change if  $t_\\mathrm{e} = \\lambda t'_\\mathrm{e}$ and $\\rho_* = \\eta \\lambda^{-3} \\rho'_*$.\n\n\\section{Results}\n\\label{sec: results}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.352]{figures\/frame_different_times.pdf}\n    \\caption{A canonical jet ($\\theta = 0.2$ rad; $L_\\mathrm{j} = 10^{51} \\erg \\ \\s^{-1}$ and $t_e=1 \\s$)  launched in a test star with a density profile given by Eq.~\\ref{eq: rho_profile}. \n    The four panels show, from left to right, the relativistic $\\Gamma\\beta$ factor, the density $\\rho$, the pressure $P$ and a scalar tracer of the ejected jet material (an unitary value implies pure jet material with no mixing). \n    All the quantities but the $\\Gamma\\beta$ factor are normalized to the respective maximum values in order to increase the color contrast. The scale for the velocity of our system is dictated by $v_0$. With the above parameters  $\\beta_0 =v_0\/c =  0.014$ and the relativistic regime begins when $\\beta \\Gamma\/\\beta_0\\approx 50$.\n    The different rows show the evolution of the jet at $t = t_\\mathrm{e} = 1~\\s$ (when the engine stops, first row) - the jet is clearly seen here surrounded by a cocoon, $t=1.3~\\s$ (choking time, second row) - the jet has disappeared as its tail reached the head, $t=14.2~\\s$ (shortly after the breakout, third row), and $t=16.0~\\s$ (sideways spreading outside the star, fourth row). \n    To enhance the color contrast we change the normalization scale of $\\rho\/\\rho_\\max$ and $P\/P_\\max$ in the third and fourth row in order to capture the tenuous material and pressure spilling outside the star. }\n    \\label{fig: jet_t01}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.35]{figures\/frame_040.pdf}\n    \\caption{4-panel figure of the breakout of an unchoked jet at $t=t_\\mathrm{e} = $4 s with an opening angle of $\\theta_\\mathrm{j} = 0.1$ rad with a luminosity of $L_\\mathrm{j}=2.5 \\times 10^{50}~\\erg~\\s^{-1}$. \n    In this case the jet tail did not catch up the jet head, resulting in an unchoked breakout and with a head propagation operating as the jet engine were still active. We use the same color scale of the fourth row of Fig.~\\ref{fig: jet_t01} for comparison.}\n    \\label{fig: jet_t4}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.38]{figures\/spreading_angle_evolution_theta_02_alternate_ratio.pdf}\n    \\caption{\\emph{Left}: A schematic figure of how the aspect ratio $\\theta(t)$ is defined (Eq.~\\ref{eq: theta_of_t}) superposed on the density map of the jet cocoon while it is still inside the star. \n    The dashed line represents the maximal width of the cocoon, while the dashed-dotted line represents the maximal height. \n    \\emph{Centre and right}: Evolution of the aspect ratio of the jet as a function of time (central panel) and of the cocoon\/jet head location $z_\\head$ (right panel). \n    The square dots represent the choking time of the jet while the triangular dots represent the breakout time of the cocoon. \n    In both frames the thick dashed horizontal line represents the condition for an isotropic blast wave ($r_\\mathrm{c} = z_\\head$), while the horizontal dotted line represent the original opening angle of the launched jet ($0.2~\\rad$ for the figure). \n    On the right-hand side the dashed-dotted vertical line represents the star edge, located at $R_* = 3 \\times 10^{10}~\\cm$.}\n    \\label{fig: spreading_angle}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.35]{figures\/curves_divided_by_volume_bo_final.pdf}\n    \\caption{Classification of the energy-velocity distribution grouped according to their different cocoon volume $V_\\bo$ at $t=t_\\bo$. \n    Each curve is marked by a different shade of color and a triplet of numbers indicating, in order: the opening angle in radians, $t_\\mathrm{e}$ in seconds, and then $\\sqrt{V_*\/V_\\bo}$. The dashed black line represents the isotropic case.}\n    \\label{fig: choking_height}\n\\end{figure*}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.34]{figures\/correlation_volume_cutoff.pdf}\n    \\caption{Correlation between $\\sqrt{V_*\/V_\\bo}$ and the cutoff of the energy-velocity distribution for the simulated set of jets for a  cutoff value of  $0.25$ of the maximum of each differential energy distribution in the plots of Fig.~\\ref{fig: choking_height}. \n    The red-colored area represents 1-standard deviation error of the red fit curve. \n    From the fitting formula we see that the distribution cutoff corresponds to 4 times the the square root of the volume ratio at the breakout for values of $(\\Gamma\\beta)_\\mathrm{cut}$ above 3. \n    The black dashed line represents the linear limit for high cutoffs.}\n    \\label{fig: volume_cutoff}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.34]{figures\/V_bo_z_ch.pdf}\n    \\caption{Correlation between the cocoon volume at the breakout and the choking height of the jet for different values of the initial opening angle $\\theta_\\mathrm{j}$. \n    The gray-shaded region represents the 1$\\sigma$ error for the fits. }\n    \\label{fig: v_bo_z_ch}\n\\end{figure}\n\n\\subsection{The jet-cocoon system}\n\\label{subsec: analysis}\n\nWe start analyzing our simulation set considering a jet with our \\emph{canonical} parameters of 1-sided luminosity of $L_\\mathrm{j} = 10^{51} \\erg\/\\s$ and $\\theta_\\mathrm{j} = 0.2~\\rad \\simeq 10^\\circ$.\n\nWhile advancing through the stellar atmosphere the interaction of the relativistic jet with the stellar material results in a forward-reverse shock structure that is called the head of the jet \\citep{Blandford_Rees1974, Begelman_Cioffi1989, Meszaros_Waxman2001, Matzner2003, Lazzati_Begelman2005, Bromberg2011}. \nThe jet head propagation  velocity, $\\beta_\\head$, is much lower than the jet velocity before it reaches the head and for typical GRB jets it is Newtonian. \nThe shock-heated jet and stellar material that enters the head flow sideways because of the high head pressure and form a pressurized  cocoon which enshrines the jet.  \nThe contact discontinuity between the material shocked in forward and the reverse shocks divides the cocoon to  inner and outer parts. \nThe inner cocoon is composed of tenuous jet material which has crossed the reverse shock while the outer cocoon is composed of denser shocked stellar material. \nThe cocoon exerts a pressure on the jet such that, if sufficiently high,  collimates it, thus reducing its opening angle and consequently reducing the jet cross section compared to the uncollimated jet. \n\nWithin our chosen stellar structure model the jet head moves at a constant velocity at the inner region of the core where the local density slope $\\alpha = 2$.  \nIf the jet reaches outer regions where $\\alpha > 2$  it starts accelerating.\n\nThe jet is \\emph{choked} if the engine stops while the jet is propagating within the stellar envelope and the last jet element launched by the engine (\\emph{tail}, hereafter) catches up with the jet head before the latter breaks out of the star.\nIn this case all the engine's energy goes into the cocoon. \nClearly, the choking height  (Eq.~\\ref{eq: zchoke}) satisfies $z_\\mathrm{ch} < R_*$. \nOtherwise we define the jet as unchoked or successful. \nThroughout the following analysis we focus mostly on jets choked at various depths inside the star. \nFor comparison we show also the case of an unchoked jet breaking out of the star.\nFor a detailed study on the energy-velocity distribution of stellar explosions which are driven by successful jets see \\cite{eisenberg2022}.\n\nWe divide the evolution of the jet to three different phases: 1) the injection phase and choking phase $t < t_\\mathrm{ch}$, 2) the cocoon expansion phase $t_\\mathrm{ch} < t < t_\\bo$ and 3) the cocoon breakout phase $t > t_\\bo$.  \nThe different phases of a choked jet are shown in Fig.~\\ref{fig: jet_t01}.\n\n\\subsubsection{Injection and Choking: $t \\leq t_\\mathrm{ch}$}\nThe engine operates for $t_\\mathrm{e}$ producing a jet. \nThis is clearly seen in the first row of Fig.~\\ref{fig: jet_t01} in both the rightmost panel showing $\\beta \\Gamma$ and the leftmost one showing that the tracer of the jet material is concentrated mostly within a narrow cylinder along the symmetry axis $z$ with radius $r \\simeq r_\\mathrm{j}$ (color dark red).\nThis behaviour is typical of the collimated regime \\citep{Bromberg2011}.\n\nAfter the jet engine stops the last jet material launched at the injection nozzle propagates upwards. \nAt $t_\\mathrm{e}$ the jet head is still unaware that the engine stopped and the jet head continues to propagate with $\\beta_\\head$ (in this specific simulation $\\beta_\\head \\simeq 0.2)$ after the central jet engine is switched off.\nHowever, as $\\beta_\\head < 1$ while the jet material moves at $\\beta_\\mathrm{t} \\simeq 1$ the jet tail catches up with the head.\nOnly at that time the information that the engine stopped reaches the head and the reverse shock within the jet disappears. \nThis is the time where the jet is choked.  \n\nAs the head propagates with a velocity of $\\beta_\\mathrm{h} c$ and the tail propagates at $\\beta_\\mathrm{t} \\simeq 1$, we can estimate the the time that the jet tail will catch up with the head  at  $t_\\mathrm{ch}$, as defined in Eq.~\\ref{eq: zchoke}.\nUntil $t < t_\\mathrm{ch}$ the jet continues to drive the head forward through the stellar atmosphere. \nThe second row of Fig.~\\ref{fig: jet_t01} shows the  system  at $t = t_\\mathrm{ch}$, roughly 0.3 seconds after the end of the engine activity, in this specific simulation. \nOne can see that the very fast jet material around the core disappeared. At this stage all the jet's energy has been dissipated and given to the surrounding cocoon. \n\n\\subsubsection{Cocoon expansion: $ t_\\mathrm{ch} < t < t_\\bo$}\n\nAfter the jet choking, the cocoon becomes less and less collimated  and proceeds spreading sideways while the forward shock decelerates when it is deep within the envelope and accelerates as it reaches the steep density gradient near the stellar edge \\citep{Irwin_Nakar_Piran2019}.\nDuring the propagation the inner cocoon transfers energy to the freshly shocked material (via PdV work).\n\n\\subsubsection{Brekout: $t> t_\\bo$}\nAfter the breakout the cocoon material spreads both radially and tangentially to engulfs the stellar surface, quickly shrouding the breakout point from most observers (see  the last row of Fig.~\\ref{fig: jet_t01}). \nThe star is blanketed by the ejecta in a time equal to $t_\\mathrm{wrap} \\simeq \\pi R_* \/2 v_\\mathrm{bo}$, where $v_\\mathrm{bo}$ is the breakout velocity of the cocoon near the pole.\nThe shock driven by the cocoon also moves tangentially towards the equator at a slower pace until the entire stellar envelope is shocked at $t_\\mathrm{shock} \\simeq \\pi R_* \/(2 v_\\mathrm{p})$ where $v_\\mathrm{p}$ is the pattern velocity at which the spilled material travels along the stellar surface \\citep{Irwin_et_al2021}. \nShortly after reaching  the equator the shocked material propagates almost radially and outwards and it becomes homologous once the outflow reaches $\\sim 2R_*$. \n\n\\subsubsection{Successful jets} \nJets whose engine operates long enough break out from the stellar envelope before the end of the activity of the central engine. \nThese jets are not choked and can preserve an ultra-relativistic velocity once they get out of the star. \nWe show an example of a successful jet in Fig.~\\ref{fig: jet_t4}. \nBecause the jet broke out without being choked the cocoon structure inside the star is mostly collimated along the vertical axis. \nFrom the first and fourth panel which show $\\Gamma\\beta$ and the jet tracer respectively, it is evident how the innermost region is still dominated  by tenuous, highly relativistic jet material. \nComparing the last row of Fig.~\\ref{fig: jet_t01} and Fig.~\\ref{fig: jet_t4} we notice that for the same normalized density and normalized pressure scale, the longer duration of the unchoked jet results in expulsion of denser and faster stellar material with respect to the choked jet. \n\n\\subsection{The spreading angle and the cocoon volume}\n\nTo describe quantitatively the geometry of the jet-cocoon during its propagation within the stellar envelope we use the aspect ratio, defined as\n\\begin{equation}\n\\label{eq: theta_of_t}\n    \\theta(t) = \\dfrac{\\max(r_\\mathrm{c}(t))}{z_\\head(t)}\\ ,\n\\end{equation}\nwhere $r_\\mathrm{c}$ is the cocoon cylindrical radius and $z_\\mathrm{h}$ is the head position. \nFor $\\theta \\ll 1$ the aspect ratio is a good approximation of the cocoon spreading angle.\nThe expanded cocoon at the moment of the breakout is shown in the third row of  Fig.~\\ref{fig: jet_t01}. \nThe steep density transition results in an elongation and acceleration of the cocoon and the ejection of low-density material from the star, which rapidly engulfs the star's external layers. \nWe define the breakout angle $\\theta_\\bo$ as the geometric opening angle measured at the breakout time $t=t_\\bo$, namely \n\\begin{equation}\n\\label{eq: theta_bo}\n    \\theta_\\bo = \\theta (t_\\bo) = \\dfrac{\\max(r_\\mathrm{c}(t_\\bo))}{R_*}\\ .\n\\end{equation}\n\nThe evolution of $\\theta(t)$ for $\\theta_\\mathrm{j} = 0.2~\\rad$ and several values of $t_\\mathrm{e}$ is reported in Fig.~\\ref{fig: spreading_angle}. \nAt first, immediately after injection, the aspect ratio starts growing. \nThe growth continues until $z_\\head$ is roughly twice the injection radius, $z_0$, at which point the aspect ratio starts decreasing, approaching the point where the cocoon opening angle is comparable to $\\theta_\\mathrm{j}$. \nThis evolution reflects the time it takes the pressure in the cocoon to build up to the point that it starts collimating the jet effectively (see \\citealt{Harrison2018} for details). \nThe evolution of the aspect ratio changes dramatically immediately as the jet is fully choked. \nSince there is no more fresh jet material to drive the head its velocity drops sharply. \nAt the same time the cocoon pressure, and thus its sideways expansion, is not affected. \nThe result is that the aspect ratio grows continuously after $t_{\\mathrm{ch}}$. \nThere is a short episode, just before and after the breakout when the aspect ratio decreases, as the head accelerates near the edge of the star and after the breakout. \nSoon after that the aspect ratio starts increasing rapidly as some of the material that broke out of the star spreads sideways at speed that is close to the speed of light. \nOne clear property that is seen in the figure is that jet that are choked more deeply have longer time to expand before they breakout and therefore a deeper choking results in a wider cocoon with a larger volume at the time of breakout. \nAs we show next this fact has important implications for the energy-velocity distribution of the outflow.  \n\nThe volume of the cocoon at breakout, $V_\\bo$  is another parameter that describes the  properties of the jet-cocoon system. \nAs the energy of a choked jet is given to the cocoon, for a given energy, the cocoon mass (and hence volume) at breakout, will corresponds to a typical expansion velocity of the cocoon material. \nAs the volume-averaged density in the shocked cocoon material and the volume-averaged density of the star are roughly the same, we can define a characteristic velocity at the breakout linked with the breakout volume, namely:\n\\begin{equation}\n\\label{eq: volume_bo}\n    \\beta_\\bo \\simeq \\beta_0 \\sqrt{\\dfrac{V_*}{V_\\bo}} \\ . \n\\end{equation}\n\n\\subsection{The Energy-velocity distribution}\n\nFig.~\\ref{fig: choking_height} depicts the energy-velocity distribution of the entire set of simulations for different values of the engine working time $t_\\mathrm{e}$ and different initial opening angles $\\theta_\\mathrm{j}$ at $t=120 \\s$ (when the outflow is homologous and kinetic energy dominates). \nThe $x$-axis is normalized by $\\beta_0$ and the $y$-axis by $E_0$. \nEach curve is differentiated by color and labeled by a triplet of numbers describing, respectively, $\\theta_\\mathrm{j}$, $t_\\mathrm{e}$, and $\\sqrt{V_*\/V_\\bo}$.  \nWe  grouped the different curves according to $\\sqrt{V_*\/V_\\bo}$. \nFor a comparison we superposed the energy-velocity distribution of an isotropic spherically symmetric explosion (black-dashed line) for all panels. \n\nFig.~\\ref{fig: choking_height} shows, first, that in all cases the energy-velocity distribution exhibits a roughly constant energy per logarithmic scale of $\\Gamma\\beta$ over a range of velocities. \nThe distribution rises quickly before this rough plateau starts and decays sharply after it ends. \nThe rough plateau always starts at $\\beta_0$ and its highest velocity is determined almost entirely by $V_{\\bo}$, with a weak dependence on the jet opening angle. \nTo estimate the highest velocity of the flat part of the distribution we define $\\beta_{\\rm cut}$ as the velocity obtained when the energy-velocity distribution drops to 1\/4 of its maximum value. \nThis arbitrary definition provides a velocity that is slightly larger than the end of the plateau (e.g., in the spherical case $\\beta_{\\rm cut}=2\\beta_0$).\nFig~\\ref{fig: volume_cutoff} shows that there is a strong positive correlation between $\\beta_{\\rm cut}$  and $\\sqrt{V_*\/V_{\\bo}}$. \nFor small values of $\\sqrt{V_*\/V_{\\bo}}<3$, where typically $z_{\\mathrm{ch}} \\ll R_*$, we see that $\\beta_{\\rm cut}\\approx \\beta_{\\bo}$. \nHowever, for larger values of $\\sqrt{V_*\/V_{\\bo}}$ where the choking takes place not very deep within the stellar envelope, $\\beta_{\\rm cut} > \\beta_{\\bo}$. \nThe origin of the material faster than $\\beta_{\\bo}$ in these cases is the inner cocoon, which retain a significant fraction of its energy at the time of the breakout and outer cocoon material that is close to the edge of the star, where the forward shock is faster than $\\beta_{\\bo}$.\n\nThe value of $V_*\/V_{\\bo}$ is expected to depend  on the jet opening angle and the choking depth. The jet opening angle determines the aspect ratio of the cocoon as long as the head that is pushed ahead by the jet is feeding the cocoon ($\\theta\\approx \\theta_\\mathrm{j}$), while the choking depth determines by how much this aspect ratio increases until the breakout.  \nFig.~\\ref{fig: v_bo_z_ch}  depicts the correlation between $V_\\bo$ and $z_\\mathrm{ch}$ for different values of the initial $\\theta_\\mathrm{j}$. \nAs expected, $V_\\bo$ is a function of $z_\\mathrm{ch}$ and $\\theta_\\mathrm{j}$. \nA deeper choking height and a wider jet correspond to a larger  cocoon volume upon breakout.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.64]{figures\/scheme.pdf}\n    \\caption{A sketch of the division of the stellar volume for the analysis of the distribution of the stellar material. \n    The cocoon shape taken at $t=t_\\bo$ is overlaid. We associate a scalar tracer to each of the four sectors: I) internal-axis, II) external-axis, III) internal-equatorial, and IV) external-equatorial. }\n    \\label{fig: scheme}\n\\end{figure}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.34]{figures\/frame_195.pdf}\n    \\caption{Maps of the four tracers I, II, III and IV (from left to right) associated with the four sectors of the stellar material (see Fig.~ \\ref{fig: scheme}). \n    The maps show the distribution of the stellar material at $t=60~\\s$ resulting  from  a jet with canonical parameters (see Sec~\\ref{subsec: analysis}). }\n    \\label{fig: 4_tracers}\n\\end{figure*}\n\n\\subsubsection{The origin of ejecta with different final velocities}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.51]{figures\/energy_vs_velocity_distribution_tracer3_195_single.pdf}\n    \\caption{The energy-velocity distribution of the four different matter tracers associated with the sectors depicted in Fig.~\\ref{fig: 4_tracers}.\n    Matter from region II (external along the axis) dominates the highest velocity region. \n    Matter from region III (internal equatorial) dominates the low velocity regime. \n    Matter from region I (internal along the axis) dominates the intermediate region and the plateau. \n    Matter from IV (external equatorial) is always subdominant.}\n    \\label{fig: e_vs_v_tracers}\n\\end{figure}\n\nTo understand the origin of the various components of the outflow, we tracked the distribution of the ejected material using four different scalar tracers associated with four distinct regions of the star. \nThe division to the different regions is determined at the time of the breakout and it is shown in Fig.~\\ref{fig: scheme}. \nThe tracers follow the mass in each of this region (at the time of breakout): I) internal-axis, II) external-axis, III) internal-equatorial, IV) external-equatorial.\n\nFig.~\\ref{fig: 4_tracers} shows the distribution of the stellar material from each of the regions at $t=60~\\s$, roughly $46~\\s$ after the breakout.\nFig.~\\ref{fig: e_vs_v_tracers} shows the energy-velocity distributions of the four sectors.\n\nWe see that the  quasi-spherical outflow that leads the ejecta is made only of tenuous material coming from the on-axis, external layers of the progenitor directly above the expanding jet cocoon (region II). \nThis component contains only around $2\\%$ of the total stellar mass but it contains 11\\% of the total ejecta energy. \nEvidently, the fastest ejecta is dominated by this sector. \nThe  material associated with the stellar core part that is along the axis (region I; first panel in Fig.~\\ref{fig: 4_tracers}) is much more concentrated than that of the external-axis region (II) but much more extended than the two equatorial sectors. \nIt contains $30\\%$ of the total stellar mass and 46\\% of the outflow energy. \nThis section dominates the energy distribution over a wide range of velocities. \nAlmost all the rest of the mass and the energy are contained in the internal-equatorial sector (III) which carries $60\\%$ of the mass and 32\\% of the energy. \nIt dominates the energy at low velocities $\\lesssim \\beta_0$. \nFinally, the outer-equatorial section carries  $5\\%$ of the ejecta mass and 11\\% of its energy. \nAll its material is moving at intermediate velocities and it is subdominant at all velocities. \n\n\\subsection{The effect of the stellar density profile}\n\\label{sec: diff_profiles}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.43]{figures\/different_profile.pdf}\n    \\caption{\\emph{Left}: Energy-velocity distributions of  jet simulations with canonical parameters for four different stellar density profiles. \n    The blue line ($\\rho \\propto R^{-2} (R_*-R)^2 $) represents the profile used in most of the previous simulations. \\emph{Right}: A comparison of the  energy distributions of the cases $\\alpha=2,n=3$ and $\\alpha=2.5,n=3$ with those arising from from two jets choked in a star with a canonical density profile at the same heights $z_\\mathrm{ch}\/R_*$, respectively.}\n    \\label{fig: density_profiles_s}\n\\end{figure*}\n\n\\begin{center}\n\\begin{table*}\n\\label{tab: different_profiles}\n\\begin{tabular}{|c||ccccc|}\n\\hline \nJets & $t_\\mathrm{e}$ [s] & $\\theta_\\mathrm{j}$ [rad] & $\\rho(r)$ & $z_\\mathrm{ch}\/R_*$ & $t_\\bo$ [s] \\\\ \n\\hline \nCanonical & 1 & 0.2 & $\\propto R^{-2} (R_*-R)^2 $ & 0.21 & 13.9 \\\\ \n$\\alpha2$\\_$n2.5$ & 1 & 0.2 & $\\propto R^{-2} (R_*-R)^{2.5} $ & 0.25 & 11.2 \\\\ \n$\\alpha2$\\_$n3$ & 1 & 0.2 & $\\propto R^{-2} (R_*-R)^3 $ & 0.25 & 9.5 \\\\ \n$\\alpha2.5$\\_$n2$ & 1 & 0.2 & $\\propto R^{-2.5} (R_*-R)^2 $ & 0.28 & 6.8 \\\\ \n$\\alpha2.5$\\_$n3$ & 1 & 0.2 & $\\propto R^{-2.5} (R_*-R)^3 $ & 0.37 & 4.0 \\\\ \n\\hline \nCanonical\\_t1.33 & 1.33 & 0.2 & $\\propto R^{-2} (R_*-R)^2 $ & 0.25 & 11.5 \\\\ \nCanonical\\_t2 & 2 & 0.2 & $\\propto R^{-2} (R_*-R)^2 $ & 0.37 & 8.3 \\\\ \n\\hline\n\\end{tabular}\n\\caption{Properties of the jets injected in different density profiles. The table lists the engine working time $t_\\mathrm{e}$, the initial opening angle $\\theta_\\mathrm{j}$, the density profile $\\rho(r)$ used in the run, the choking height relative to the star radius $z_\\mathrm{ch}\/R_*$, and the breakout time $t_\\bo$.}\n\\end{table*}\n\\end{center}\n\nTo study the effect of different stellar density profiles we consider stellar density profiles that can be written as:\n\\begin{equation}\n\\label{eq: rhogen}\n    \\rho (R) = \\rho_*\\left(\\dfrac{R_*}{R}\\right)^\\alpha  \\left(1-\\dfrac{R}{R_*}\\right)^n \\ ,\n\\end{equation}\nwhere $n$ is the outer slope at the edge, and $\\alpha$ is the inner slope, with $\\alpha <3$. \nThe density profile described by Eq.~\\ref{eq: rho_profile}, which is used through the rest of the paper, is roughly equivalent to the case of $\\alpha=2,~ n=2$ and it will be referred as the \\emph{canonical profile} hereafter. \nThe profiles that we consider are listed in Table.~\\ref{tab: different_profiles}. For each profile we run a simulation with our canonical jet parameters, $\\theta_\\mathrm{j} = 0.2~\\rad$, $L_\\mathrm{j} = 10^{51}~\\erg~\\s^{-1}$, and we inject the jets from the same initial height ($z_0 = 10^9~\\cm$). \n\nFig.~\\ref{fig: density_profiles_s} shows a comparison of the energy-velocity distributions from different stellar profiles. \nFirst, it shows that the distributions are all flat over a range of velocities, implying that them main feature of the outflow from an explosion driven by a choked jet is independent of exact stellar profile (a similar result was found by \\citealt{eisenberg2022} in the case of explosions that are driven by successful jets). \nWhen looking in more detail, the right-hand side shows two pairs of simulation. \nEach pair shows the results of different stellar profiles with similar $\\theta_\\mathrm{j}$ and $z_\\mathrm{ch}$ (which dominates $V_{\\bo}$). \nThe distributions found in the two simulations of each pair are very similar, implying that when the cocoon properties are similar the stellar profile has a minor effect on the outflow energy-velocity distribution. \n\nOn the left-hand side of Fig.~\\ref{fig: density_profiles_s} we compare the energy-velocity distributions of jets with the exact same parameters (including $t_\\mathrm{e}$) but different envelope density profiles. \nIt shows that the stellar profile affects the velocity of the head (as was found previously by \\citealt{Bromberg2011,Harrison2018}) and therefore jets with the same properties are choked at different heights when propagating in different density profiles. \nSince the energy-velocity profile depends strongly on $z_\\mathrm{ch}$ two jets with the same properties that propagate at different stellar profiles will result in outflows with different energy-velocity distributions, as shown on the right-hand panel of this figure.   \n\n\\subsection{The energy velocity distribution at different viewing angles}\n\\label{sec: profiles_different_angles}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.39]{figures\/comparison_same_choking_height_255_four_plots.pdf}\n    \\caption{Energy-velocity distributions for six different viewing angles (chosen so that the corresponding wedges have the same volumes). \\emph{Top}: Two jets with the same choking height $ (V_* \/ V_\\bo)^{1\/2}= 1.95 $  but different initial opening angles. \n    \\emph{Bottom}: Two jets choked at $ (V_* \/ V_\\bo)^{1\/2} \\sim 4 $ (bottom panels) and with the same opening angles of the jets on the top row, respectively. \n    The black dashed line represents the isotropic energy-velocity distribution for $1\/6$ of the total volume. The profiles are taken at $t=120~\\s$.}\n    \\label{fig: energy_profiles_volumes}\n\\end{figure*}\n\nSince jet driven explosions are aspherical, one expects that the outflow will not be isotropic. \nFig.~\\ref{fig: energy_profiles_volumes} depicts the energy-velocity distribution of four simulations. \nFor each simulation we show the distributions at six different sections, where each section is the sum of the ejecta within a range of polar direction. \nTo see the dependence on the initial conditions we show simulations with two different jet opening angles ($0.2$ and $0.6$ rad) and two different values of breakout volume $\\sqrt{V_*\/V_\\bo}$. \nAs expected, the outflow is aspherical. \nA common, also expected, property of all four simulations is that the maximal velocity of the outflow is around the jet axis at lower polar angles. \nThis result was found also for jet-driven explosions of successful jets \\citep{eisenberg2022}. \nIn the two simulations with the large value of $\\sqrt{V_*\/V_\\bo}$ (i.e., low $z_\\mathrm{ch}$ and\/or wide $\\theta_\\mathrm{j}$; top panels) the energy-velocity distribution of the equatorial outflow ($\\theta \\gtrsim 60^\\circ$) is similar to that of a spherical explosion, with a typical velocity $\\beta_0$. The faster outflow is confined to lower angles. \nIn the two simulations with the small value of $\\sqrt{V_*\/V_\\bo}$ (i.e., high $z_\\mathrm{ch}$ and narrow $\\theta_\\mathrm{j}$; bottom panels) a large range of velocities was seen in all directions, but still faster velocities are observed closer to the jet axis.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.42]{figures\/density_profile_175e49_255_with_fit_four_eqvol.pdf}\n    \\caption{Density profiles at different viewing angles of two jets with a similar breakout volume $ (V_* \/ V_\\bo)^{1\/2}= 1.95 $ (top row) but different initial opening angle compared to the density profile of jets choked at $ (V_* \/ V_\\bo)^{1\/2} \\sim 4 $ (bottom row) with the same opening angles, respectively. \n    The solid thick black line in each panel represent a power-law fit of $\\rho(r) \\propto v^{-5}$, corresponding to a flat distribution of energy per logarithmic bin of the velocity. \n    The profiles are taken at $t=120~\\s$.}\n    \\label{fig: density_profiles_1}\n\\end{figure*}\n\nFor clarity we present in Fig.~\\ref{fig: density_profiles_1} the radial density distribution profiles, $\\rho(R)$, for different viewing angles of four different simulations. \nThese are the same simulations and the same divisions to angular sections as in Fig.~\\ref{fig: energy_profiles_volumes}. \nThis presentation is often used in studies of SN ejecta and at sub-relativistic velocity  $\\rho(R) \\propto \\beta^{-5} \\frac{\\mathrm{d} E}{\\mathrm{d} \\log\\beta}$.\n\n\n\\section{Conclusions and implications to observations}\n\\label{sec: conclusions}\n\nWe carried relativistic hydrodynamical simulations in 2D cylindrical coordinates of stellar explosions driven by jets, focusing on configurations of choked relativistic jets and exploring  how those can lead to different realizations of the velocity distribution of the outflow in its homologous expansion phase.  \nWe followed the evolution of a relativistic jet from the injection deep inside the star to the point where it is choked and then continued to follow the cocoon as it emerges from the envelope and ultimately unbinds it up to the point that the outflow becomes homologous. \nWe scrutinized the various stages of the jet inside the star and analyzed what happens during the choking process and the adiabatic cocoon expansion.  \nWhile the results are given for a specific set of parameters we provided scaling relation for the physical parameters of the jet and the star in order to facilitate a dimensionless treatment of the problem.\n\nWe summarize our findings as follows:\n\n\\begin{itemize}\n    \\item All jet driven explosions in which the jet is not choked too deep within the star generate an outflow with a unique feature: a significant range of velocities over which the outflow carries a roughly constant amount of energy per logarithmic scale of the proper velocity ($\\Gamma\\beta$). \n    This is a universal property of jet driven explosions. \n    The main difference between different setups is the range of velocities over which the energy is constant.\n    \n    \\item The plateau of the energy-velocity distribution starts in all cases at $v_0=\\sqrt{E_0\/M_*}$. \n    The maximal velocity of the plateau depends mostly on the cocoon volume upon breakout and the corresponding velocity is  $\\beta_\\bo=\\beta_0\\sqrt{V_*\/V_\\bo}$. \n    For  $\\sqrt{V_*\/V_\\bo} < 3$ the maximal velocity is comparable to $\\beta_\\bo$, while for larger values of $\\sqrt{V_*\/V_\\bo}$ the maximal velocity is larger than $\\beta_\\bo$ and it can become mildly relativistic. \n    \n    \\item The volume of the cocoon upon breakout, $V_\\bo$, depends on the choking height, $z_\\mathrm{ch}$, and on opening angle of the jet upon launching. \n    A higher $z_\\mathrm{ch}$ and narrower opening angle leads to a smaller $V_\\bo$ and thus to an outflow that extends to higher velocities.\n    \n    \\item The outflow from an explosion driven by a choked jet is not isotropic.\n    In general, the material along the poles (that is along the jet direction) is faster while the material along the equator is slower.\n\\end{itemize}\n\nA spherical explosions accelerate only a negligible fraction of the stellar mass to very high velocities. \nWe have shown here that the situation is drastically different when there is a jet that breaks the symmetry. \nSuch a jet can deposit a significant amount of energy at high velocity matter, even in case that the jet is choked within the envelope. \nThis excess in high velocity outflow (compared to a spherical explosion) is certainly expected when the entire stellar explosion is driven by a jet, but it is also expected if the jet is accompanied by a simultaneous more spherical explosion  (see e.g., \\citealt{eisenberg2022}).\nIf sufficiently optically thick such a high velocity material that surrounds a SN would produce a very broad  absorption lines (with typical width corresponding to 0.1-0.2c) in the observed spectrum. \nIt will be observed in the early spectra but will disappear later when this outer envelope that is rapidly expanding becomes optically thin. \n\nLines that show an excess of high velocity material have been observed in several SNe \\citep{Galama+1998, Iwamoto+1998, Mazzali+2000,  Mazzali+2002, Modjaz+2006, Mazzali+2008, Bufano+2012, Xu+2013, Ashall+2019,Izzo_et_al_2019}.\nOur result show that, as suggested by \\cite{Piran2019},  a chocked jet can lead to that high velocity material. \nHowever, we have found that some conditions are needed to observe the corresponding broad absorption lines. \nFirst, the jet must be chocked at sufficiently large distance at the stellar atmosphere. \nThe signature of jets that are chocked too deep will not be so significant.\nSecond, as there is less fast moving material in directions far from the jet direction, the fast moving matter will become optically thin earlier in these directions. \nAs the broad absorption line will fade faster, this implies that observers at such viewing angles are less likely to observe the broad absorption line signature. \nThese last two facts imply that we may not observer broad emission line in all SNe that harbour relativistic jets. \n\nThe excess in fast material was observed in various types of stripped envelope SNe. \nThis include SNe that are associated with long GRBs, SNE that are associated with {\\it ll}GRBs and SNe that are not associated with GRBs at all. \nLong GRBs must contain successful relativistic jets. \n{\\it ll}GRBs contain jets which may very well be choked \\citep{Kulkarni1998, macfadyen_supernovae_2001, Tan+2001, campana_association_2006, Wang+2007, waxman_grb_2007, katz_fast_2010, Nakar_Sari2012,Nakar2015}. \nWe do not know if SNe that are not associated with GRBs harbour jets, but if they are then these jets must be choked ones. \nA previous study by \\cite{eisenberg2022} have shown that successful jets can generate the energy-velocity distribution which is observed in SNe that are associated with long GRBs. \nOur finding here show that choked jets can explain the energy-velocity distribution seen in SNe that are associated with {\\it ll}GRBs and in SNe that are not associated with any type of GRBs. \nThis provides further support for the interpretation of the ``disappearing'' early very broad absorption lines in some SNe as arising from choked jets. \nThese findings also show that such lines may not be detected in all SNe that harbor choked jets. \nFurther exploration of this model, including estimates of the observed spectra and the fraction of events in which these lines will be observed will be carried out in future work. \n\n\n\\section*{Acknowledgments}\nWe kindly thank Christopher Irwin for the stimulating discussions and suggestions.\nThis work is supported by the ERC grants TReX (TP and MP) and JetNS and an ISF grant 1995\/21 (EN).\n\\section*{Data Availability}\nThe data underlying this article will be shared on reasonable\nrequest to the corresponding author.\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}