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+{"text":"\\section{Introduction}\nThe algebraic $K$-theory of ring spectra encodes subtle and\ninteresting invariants.  It has long been known that the $K$-theory of\nordinary rings contains a great deal of arithmetic information.  On\nthe other hand, Waldhausen showed that there is a deep connection\nbetween the $K$-theory of the sphere spectrum and the geometry of\nhigh-dimensional manifolds (as seen by pseudo-isotopy theory)\n\\cite{waldhausen}.  Waldhausen's ``chromatic'' program for\nanalyzing $K(S)$ in terms of a chromatic tower of $K$-theory spectra\nsuggests a connection between these seemingly disparate bodies of\nwork, as such a tower can be regarded as interpolating from arithmetic\nto geometry \\cite{waldhausen-chrom}.  Recently, Rognes' development of\na Galois theory of $S$-algebras \\cite{rognes-galois} and attendant\ngeneralizations of classical $K$-theoretic descent\n\\cite{ausoni-rognes} along with Lurie's work on derived algebraic\ngeometry \\cite{lurie} have raised the prospect of an arithmetic\ntheory of ring spectra, which would provide a unified viewpoint on\nthese phenomena.  To gain insight into the situation, examples\nprovided by computations of the $K$-theory of ring spectra which do\nnot come from ordinary rings are essential.\n\nOf course, computation of algebraic $K$-theory tends to be extremely\ndifficult.  However, for connective ring spectra, algebraic $K$-theory\nis in principle tractable via ``trace methods'', which relates\n$K$-theory to the more computable topological Hochschild homology\n($THH$) and topological cyclic homology ($TC$).  Specifically, there is a \ntopological lifting of the Dennis trace to a ``cyclotomic trace'' map\n\\cite{bokstedt-hsiang-madsen}, and the fiber of this map is\nwell-understood \\cite{dundas, mccarthy}.  Moreover, $TC(R)$ is built\nas a certain homotopy limit of the fixed-point spectra of $THH(R)$ \nwith regard to the action of subgroups of the circle, and so is\nrelatively computable via the methods of equivariant stable homotopy\ntheory.  One of the major early successes of this methodology was the\nresolution of the ``K-theory Novikov conjecture'' by Bokstedt, Hsiang,\nand Madsen \\cite{bokstedt-hsiang-madsen}.  Central to their results\nwas a computation of the $TC$ and $THH$ associated to the ``group\nring'' $\\Sigma^\\infty (\\Omega X)_+$, for a space $X$.\n\nThom spectra associated to multiplicative classifying maps provide a\nnatural generalization of the suspension spectra of monoids.\nMoreover, many interesting ring spectra arise naturally as Thom\nspectra.  The purpose of this paper is to provide an explicit and\nconceptual description of the $THH$ of Thom spectra which are\n$E_\\infty$ ring spectra.  As the starting point for the calculation of\n$TC$ is the determination of $THH$, this description provides\nnecessary input to ongoing work to understand the $TC$ and $K$-theory\nof such spectra.  This paper is a companion to a joint paper with\nR. Cohen and C. Schlichtkrull \\cite{blumberg-cohen-schlichtkrull}\nwhich uses somewhat different methods to study the $THH$ of Thom\nspectra which are $A_\\infty$ ring spectra.\n\nThe operadic approach to Thom spectra of Lewis and May\n\\cite[7.3]{lms}, \\cite{may-quinn-ray} provides a Thom spectrum functor\n$M$ which yields structured ring spectra when given suitable input.\nSpecifically, for suitable topological groups and monoids $G$, Lewis\nconstructs a Thom spectrum functor \n\\[\nM \\colon {\\catsymbfont{T}} \/ BG  \\to {\\catsymbfont{S}} \\backslash S\n\\]\nfrom the category of based spaces over $BG$ to the category\n${\\catsymbfont{S}} \\backslash S$ of unital spectra.  Furthermore, he shows that if\n$f \\colon X \\rightarrow BG$ is an $E_n$ map then $Mf$ is an $E_n$ ring\nspectrum, where $E_n$ denotes an operad which is augmented over the\nlinear isometries operad $\\scr{L}$ and weakly equivalent to the little $n$-cubes\noperad.  In particular, $M$ takes $E_\\infty$ maps to $E_\\infty$ ring\nspectra.  Since $E_\\infty$ ring spectra can be functorially\nreplaced by commutative $S$-algebras, we can regard $M$ as\nrestricting to a functor\n\\[\nM \\colon {\\catsymbfont{T}}[\\scr{L}]\/BG \\to {\\catsymbfont{C}}{\\catsymbfont{A}}_S.\n\\] \nThus, $M$ produces output which is suitable for the construction of\n$THH$.\n\nThe development of symmetric monoidal categories of spectra has made\npossible direct constructions of topological Hochschild homology\n($THH$) which mimic the classical algebraic descriptions of Hochschild\nhomology, replacing the tensor product with the smash product.  Thus for a\ncofibrant $S$-algebra $R$, $THH(R)$ can be \ncomputed as the realization of the cyclic bar construction $N^{\\ensuremath{\\operatorname{cyc}}}\nR$ with respect to the smash product, where $N^{\\ensuremath{\\operatorname{cyc}}} R$ is the the\nsimplicial spectrum \n\\[[k] \\rightarrow \\underbrace{R \\sma R \\sma \\ldots \\sma R}_{k+1}\\]\nwith the usual Hochschild structure maps \\cite[9.2.1]{ekmm}.\n\nRecall that the category of commutative $S$-algebras is enriched and\ntensored over unbased spaces, and more generally has all indexed\ncolimits \\cite[7.2.9]{ekmm}.  When $R$ is commutative, McClure,\nSchwanzl, and Vogt \\cite{mcclure-schwanzl-vogt} made precise an\ninsight of Bokstedt's that there should be a homeomorphism \\[|N^{cyc}\nR| \\cong R \\otimes S^1.\\] Here $R \\otimes S^1$ denotes the tensor of\nthe commutative $S$-algebra $R$ with the unbased space $S^1$.  Thus,\nwe can describe $THH(Mf)$ by studying $Mf \\otimes S^1$.\n\nThe category of $\\scr{L}$-spaces is also tensored over unbased spaces, and\nthis induces a tensored structure on the category of $\\scr{L}$-maps\n$f \\colon X \\rightarrow BG$.  Our first main theorem, proved in\nSection~\\ref{sec:thomleft}, states that the Thom spectrum functor is\ncompatible with the topologically tensored structures on its domain\nand range categories.\n\n\\begin{thm}\\label{commutation}\nThe Thom spectrum functor\n\\[\nM \\colon {\\catsymbfont{T}}[\\scr{L}]\/BG \\to {\\catsymbfont{C}}{\\catsymbfont{A}}_S\n\\]\npreserves indexed colimits and in fact is a continuous left adjoint.\nIn particular, for an unbased space $A$ and an $\\scr{L}$-map\n$X \\rightarrow BG$, there is a homeomorphism \n\\[M(f \\otimes A) \\cong Mf \\otimes A.\\]\n\\end{thm}\n\nThis theorem follows from an appropriate categorical viewpoint on the\nThom spectrum functor.  The category of $\\scr{L}$-spaces can be regarded\nas the category ${\\catsymbfont{T}}[\\mathbb{K}]$ of algebras over a certain monad $\\mathbb{K}$ on\nthe category ${\\catsymbfont{T}}$ of based spaces.  We can utilize this description\nto describe the category of $\\scr{L}$-maps $X\n\\rightarrow BG$ as the category $({\\catsymbfont{T}}\/BG) [\\mathbb{K}_{BG}]$ of algebras over\na closely related monad $\\mathbb{K}_{BG}$.  Similarly, the category of\n$E_\\infty$-ring spectra can be regarded as the category $({\\catsymbfont{S}}\n\\backslash S)[\\tilde{\\mathbb{C}}]$ of algebras over a monad $\\tilde{\\mathbb{C}}$ on\nthe category ${\\catsymbfont{S}} \\backslash S$ of unital spectra.  Each of these\ncategories admits the structure of a topological model category, by\nwhich we mean a model category structure compatible with an enrichment\nin spaces \\cite[7.2-7.4]{ekmm}.  In particular, each of these\ncategories has tensors with unbased spaces.\n\nFurthermore, work of Lewis \\cite[7]{lms} describes the interaction of\n$M$ with these monads.  Specifically, Lewis shows \\cite[7.7.1]{lms} that\n\\[M \\mathbb{K}_{BG} f \\cong \\tilde{\\mathbb{C}} Mf\\]\nand moreover that in fact $M$ takes the monad $\\mathbb{K}_{BG}$ to the\nmonad $\\tilde{\\mathbb{C}}$ (i.e. that the indicated isomorphism is suitably\ncompatible with the monad structure maps).  In\nSection~\\ref{sec:coltech}, we study this situation more generally and\nprove the following result about the preservation of indexed colimits \nby induced functors on categories of monadic algebras;\nTheorem~\\ref{commutation} is then a straightforward consequence.\n\n\\begin{thm}\\label{lifting}\nLet ${\\catsymbfont{A}}$ and ${\\catsymbfont{B}}$ be categories tensored over unbased spaces, and\nlet $\\mathbb{M}_A$ be a continuous monad on $A$ and $\\mathbb{M}_B$ be a continuous\nmonad on $B$, such that $\\mathbb{M}_A$ and $\\mathbb{M}_B$ preserve reflexive\ncoequalizers.  Let $F \\colon {\\catsymbfont{A}} \\to {\\catsymbfont{B}}$ be a continuous functor such\nthat\n\\begin{itemize}\n\\item $F$ preserves colimits and tensors, and\n\\item There is an isomorphism $F \\mathbb{M}_A X \\cong \\mathbb{M}_B FX$ which is compatible\nwith the monad structure maps.\n\\end{itemize}\nThen $F$ restricts to a functor \n\\[F_{\\mathbb{M}}\\colon {\\catsymbfont{A}}[\\mathbb{M}_A] \\to {\\catsymbfont{B}}[\\mathbb{M}_B]\\]\nwhich preserves colimits and tensors.  If $F$ is a left adjoint, then \n$F_{\\mathbb{M}}$ is also a left adjoint.\n\\end{thm}\n\nIn order to use the formula $M(f \\otimes S^1) \\cong Mf \\otimes S^1$\nprovided by Theorem~\\ref{commutation} to compute $THH(Mf)$, we must\nfirst ensure that we have homotopical control over $Mf$.  Two\ntechnical issues arise.  First, the cyclic bar construction\ndescription of $THH(R)$ only has the correct homotopy type when the\npoint-set smash product $R \\sma R$ represents the derived smash\nproduct (for instance if $R$ is cofibrant as a commutative\n$S$-algebra).  Second, when working over $BF$, Lewis' construction of\nthe Thom spectrum functor we give preserves weak equivalences only for\ncertain classifying maps (``good'' maps), notably Hurewicz fibrations.\n\nWe show in Section~\\ref{sec:compute} that by appropriate cofibrant\nreplacement of $f\\colon X \\to BG$, we can ensure that $Mf$ is suitable\nfor computing the derived smash product.  The second problem can be\nhandled by the classical device of functorial replacement by a\nHurewicz fibration.  Unfortunately, it turns out to be complicated to\nanalyze the interaction of these two replacements.  In the companion\npaper \\cite{blumberg-cohen-schlichtkrull} we discuss the technical\ndetails of the interaction between these processes.  In the present\ncontext, we are able to obtain our main applications without\nconfronting this issue; although with the tools described herein the\nnext result is only practically applicable when $G$ is a group, in\nwhich case all maps are good, the splitting in\nTheorem~\\ref{e2splitting} holds for $BF$ as well.\n\n\\begin{cor}\nLet $f \\colon X \\rightarrow BG$ be a good map of $\\scr{L}$-spaces such\nthat $X$ is a cofibrant $\\scr{L}$-space.  Then $THH(Mf)$ and $M(f \\otimes\nS^1)$ are isomorphic in the derived category.\n\\end{cor}\n\nJust as $R \\otimes S^1$ is the cyclic bar construction in the category\nof commutative $S$-algebras, for an $\\scr{L}$-space $X$ we can similarly\nregard $X \\otimes S^1$ as a cyclic bar construction\n\\cite[6.7]{basterra-mandell}.  Unlike commutative $S$-algebras,\n$\\scr{L}$-spaces are tensored over based spaces and the tensor with an\nunbased space is constructed by adjoining a disjoint basepoint.  Thus,\nfor an $\\scr{L}$-space $X$ it is preferable to think of the unbased tensor\n$X \\otimes S^1$ as the based tensor $X \\otimes S^1_+$.  This\ndescription allows us to construct a natural map to the free loop\nspace \n\\[X \\otimes S^1_+ \\to L(X \\otimes S^1)\\]\nwhich is a weak equivalence when $X$ is group-like.  Note that the\nbased tensor $X \\otimes S^1$ is a model of the classifying space of\n$X$, so that we have recovered the familiar relationship between\n$N^{\\ensuremath{\\operatorname{cyc}}} X$ and $L(BX)$ \\cite{bokstedt-hsiang-madsen}.  Furthermore,\nin Section~\\ref{sec:split} we use the stable splitting of $S^1_+$ to\nprovide an extremely useful splitting of $THH(Mf)$.\n\n\\begin{thm}\\label{einfsplittingthm}\nLet $f \\colon X \\rightarrow BG$ be a good map of $\\scr{L}$-spaces such\nthat $X$ is a cofibrant and group-like $\\scr{L}$-space.  Then there is a\nweak equivalence of commutative $S$-algebras \n\\[THH(Mf) \\htp Mf \\sma BX_+.\\]\n\\end{thm}\n\nThis theorem provides convenient formulas describing $THH$ for various\nbordism spectra, notably \\[THH(MU) \\htp MU \\sma BBU_+.\\]  Furthermore,\nwe show that this splitting theorem holds when $f \\colon X \\rightarrow\nBG$ is only an $E_2$ map, provided that the induced multiplicative\nstructure on $Mf$ ``extends to'' an $E_\\infty$-structure.  In this\ncontext, the result follows from a separate analysis which exploits\nthe multiplicative equivalence \n\\[Mf \\sma Mf \\htp Mf \\sma X_+\\]\ninduced by the Thom isomorphism.  Note that in the statement of the\nfollowing theorem we do not require $X$ to be cofibrant.\n\n\\begin{thm}\\label{e2splitting}\nLet ${\\catsymbfont{C}}_2$ denote an $E_2$-operad augmented over the linear\nisometries operad, and let $f \\colon X \\rightarrow BG$ be a good\n${\\catsymbfont{C}}_2$ map such that $X$ is group-like.  Assume there is a map\n$\\gamma \\colon Mf \\rightarrow M^\\prime$ which is a weak equivalence of\nhomotopy commutative $S$-algebras such that $M^\\prime$ is a\ncommutative $S$-algebra.  Then there is a weak equivalence of\n$S$-modules \n\\[THH(Mf) \\htp Mf \\sma BX_+.\\]\n\\end{thm}\n\nAlthough the hypotheses of this theorem may seem strange, in fact this\nsituation arises in nature.  It has long been known that $H\\ensuremath{\\mathbb{Z}} \/ 2$ is\nthe Thom spectrum of an $E_2$ map $f \\colon \\Omega^2 S^3 \\rightarrow BO$\n\\cite{cohen-may-taylor, mahowald}.  There is a similar construction of\n$H\\ensuremath{\\mathbb{Z}} \/ p$ for odd primes due to Hopkins which is described in\n\\cite{thomified}.  Constructions of $H\\ensuremath{\\mathbb{Z}}$ as a Thom spectrum over\n$\\Omega^2 S^3\\left<3\\right>$ are also well-known\n\\cite{cohen-may-taylor, mahowald}, but these descriptions only yield\nan $H$-space structure on $H\\ensuremath{\\mathbb{Z}}$.  \n\nIn Section~\\ref{S:TEM}, we discuss a construction of $H\\ensuremath{\\mathbb{Z}}$ as the\nThom spectrum associated to an $E_2$ map.  Then\nTheorem~\\ref{e2splitting} allows us to recover the classical\ncomputations of Bokstedt of $THH(\\ensuremath{\\mathbb{Z}}\/2)$, $THH(\\ensuremath{\\mathbb{Z}}\/p)$, and $THH(\\ensuremath{\\mathbb{Z}})$.\n\n\\bigskip\nThese results appeared as part of the author's 2005 University of\nChicago thesis.  I would like to thank Peter May for his support and\nsuggestions throughout the conduct of this research.  I would also\nlike to express my gratitude to Michael Mandell --- this paper could\nnot have been written without his generous assistance.  In addition, I\nwould like thank Christian Schlichtkrull and Ralph Cohen for agreeing\nto join forces in the preparation\nof \\cite{blumberg-cohen-schlichtkrull}.  The paper was improved by\ncomments from Christopher Douglas and Halvard Fausk.\n     \n\\section{Colimit-preserving functors in categories of monadic\nalgebras}\n\\label{sec:coltech}\n\nIn this section, we prove Theorem~\\ref{lifting}.  The theorem is\nessentially a straightforward consequence of categorical results due\nto Kelly describing the construction of colimits and indexed colimits\nin enriched categories of monadic algebras.  We begin by reviewing the\nrelevant background material, largely following the exposition\nof \\cite{ekmm}.\n\nLet ${\\catsymbfont{V}}$ denote a symmetric monoidal category, and let ${\\catsymbfont{C}}$ be a\ncategory enriched over ${\\catsymbfont{V}}$.  In such a context we can define tensors\nand cotensors (and more generally indexed colimits and limits).\n\n\\begin{defn}\nLet ${\\catsymbfont{C}}$ be a category enriched over ${\\catsymbfont{V}}$.  Then ${\\catsymbfont{C}}$ is tensored\nif there exists a functor $\\otimes_{{\\catsymbfont{C}}} \\colon {\\catsymbfont{C}} \\times {\\catsymbfont{V}} \\rightarrow\n{\\catsymbfont{C}}$ which is continuous in each variable and such that there is an\nisomorphism\n\\[{\\catsymbfont{C}}(X \\otimes_{{\\catsymbfont{C}}} A, Y) \\cong {\\catsymbfont{V}}(A, {\\catsymbfont{C}}(X,Y))\\]\nof objects of ${\\catsymbfont{A}}$.  There is a dual notion of cotensors.\n\\end{defn}\n\nFor example, both the category of based spaces and the category of\nspectra are tensored over based spaces.  The tensor of a spectrum $X$\nand a based space $A$ is $X \\sma A$.  The cotensor of a spectrum $X$\nand a based space $A$ is the mapping spectrum $F(A,X)$.  Notice that\nwe can define the tensor of a spectrum $X$ and an unbased space $B$ by\nadjoining a disjoint basepoint to $B$ and taking the tensor with\nrespect to the enrichment in based spaces --- the tensor of a spectrum\n$X$ and an unbased space $B$ is $X \\sma B_+$.\n\nIn an enriched category, there are notions of indexed colimits and\nlimits which take the enrichment into account.  Tensors and cotensors\nare examples of such indexed colimits and limits, and in the\ntopological setting are particularly important as a consequence of the\nfollowing result of Kelly \\cite[7.2.6]{ekmm}.\n\n\\begin{thm}\\label{t:kelly}\nA topological category has all indexed colimits provided that it is\ncocomplete and tensored.  Dually, a topological category has all\nindexed limits provided it is complete and cotensored.\n\\end{thm}\n\nFor our application, we will need to understand the tensor in the\ncategory of commutative $S$-algebras and the tensor in the category of\n$E_\\infty$ spaces.  A priori, it is not clear that either of these\ncategories is tensored.  Unlike in the case of spectra, there is not a\nfamiliar construction which yields the tensor.  For that matter,\nconstruction of colimits in these categories is not obvious either.\nThe key observation is that each of these categories can be regarded\nas a category of algebras over a monad. \n\nLet $\\mathbb{A} \\colon {\\catsymbfont{C}} \\rightarrow {\\catsymbfont{C}}$ be a monad with multiplication $\\mu$\nand unit $\\eta$.  Recall that an object $X$ in ${\\catsymbfont{C}}$ is an algebra\nover $\\mathbb{A}$ if there is an action map $\\psi \\colon \\mathbb{A} X \\rightarrow X$ such\nthat the following diagrams commute :\n\\[\n\\xymatrix{\n\\relax\\mathbb{A} \\relax\\mathbb{A} X \\ar[r]^-{\\psi} \\ar[d]^-{\\mu} & \\relax\\mathbb{A} X \\ar[d]^-{\\psi} & X \\ar[r]^-{\\eta} \\ar[dr]^-{=} & \\relax\\mathbb{M} X \\ar[d]^-{\\mu} \\\\\n\\relax\\mathbb{A} X \\ar[r]^-{\\psi} & X & & X\\\\\n}.\n\\]\n\nThe category of commutative $S$-algebras is precisely the category of\nalgebras over a certain monad in $S$-modules, and the category of\n$\\scr{L}$-spaces is the category of algebras over a certain monad in\nbased spaces; we will define these monads in Section~\\ref{S:einf}.\n\nA key observation of McClure and Hopkins \\cite{hopkins}, further\ndeveloped in \\cite{ekmm}, is that there are general constructions for\nlifting colimits and tensors from a category ${\\catsymbfont{C}}$ to the category\n${\\catsymbfont{C}}[\\mathbb{A}]$ of algebras for a monad $\\mathbb{A}$ on ${\\catsymbfont{C}}$.  That is, colimits\nand tensors in ${\\catsymbfont{C}}[\\mathbb{A}]$ can be constructed in terms of certain\ncolimits and tensors in ${\\catsymbfont{C}}$.  However, in order to utilize these\nresults a technical condition must be satisfied by the monad $\\mathbb{A}$,\nwhich we will now recall \\cite[2.6.5]{ekmm}.\n\n\\begin{defn}\nLet $A$, $B$, and $C$ be objects of a category ${\\catsymbfont{C}}$.  A reflexive\ncoequalizer is a coequalizer diagram\n\\[\n\\xymatrix{\nA \\ar@<1ex>[r]^-e \\ar@<-1ex>[r]_{f} & B \\ar[r]^-g & C \\\\\n}\n\\]\nsuch that there exists a splitting map $h \\colon B \\rightarrow A$ such that\n$e \\circ h = \\id$ and $f \\circ h = \\id$.\n\\end{defn}\n\nIn order for the lifting results to apply, $\\mathbb{A}$ must preserve\nreflexive coequalizers.  In this situation, if $A$ and $B$ are\n$\\mathbb{A}$-algebras, there is a unique structure of $\\mathbb{A}$-algebra on $C$\nand moreover $C$ is the coequalizer of $A$ and $B$ in the category\n${\\catsymbfont{C}}[\\mathbb{A}]$ \\cite[2.6.6]{ekmm}.  That is, we can form the coequalizer\nin the category ${\\catsymbfont{C}}[\\mathbb{A}]$ by taking the coequalizer in ${\\catsymbfont{C}}$.  Now we\ncan state the lifting results.  Recall the following proposition from\nEKMM \\cite[2.7.4]{ekmm}.\n\n\\begin{prop}\nLet $\\mathbb{E}$ be a continuous monad defined on a topologically enriched\ncategory ${\\catsymbfont{C}}$.  If $\\mathbb{E}$ preserves reflexive coequalizers, then the\ncolimit in the category ${\\catsymbfont{C}}[\\mathbb{E}]$ of algebras over $\\mathbb{E}$ is given by\nthe following coequalizer :\n\\[\n\\xymatrix{\n\\relax\\mathbb{E} (\\colim \\mathbb{E} R_i) \\ar@<1ex>[rr]^{\\mathbb{E}(\\colim \\xi_i)} \\ar@<-0.5ex>[rr]_{\\mu \\circ \\mathbb{E} \\alpha} && \\relax\\mathbb{E}(\\colim R_i)\n}\n.\n\\]\nHere $\\mu$ is the composition map for the monad $\\mathbb{E}$, $\\xi_i$ is the\naction map $\\mathbb{E} R_i \\rightarrow R_i$, and \n\\[\\alpha \\colon \\colim \\mathbb{E} R_i \\rightarrow \\mathbb{E} \\colim R_i\\]\nis obtained as follows.  For each $i$ there is a natural map $\\iota_i\n\\colon R_i \\rightarrow \\colim R_i$, and $\\alpha$ is specified as the unique\nmap whose composite with the natural map $\\mathbb{E} R_i \\rightarrow\n\\colim \\mathbb{E} R_i$ is precisely $\\mathbb{E}$ applied to $\\iota_i$.\nThe splitting of the coequalizer is obtained from the unit of the\nmonad. \n\\end{prop}\n\nThere is a related technique for constructing tensors as appropriate\ncoequalizer diagrams via the following proposition from EKMM\n\\cite[7.2.10]{ekmm}.\n\n\\begin{prop}\nLet $\\mathbb{E}$ be a continuous monad defined on a topologically enriched\ncategory ${\\catsymbfont{C}}$.  If $\\mathbb{E}$ preserves reflexive coequalizers, then the\ntensor in the category ${\\catsymbfont{C}}[\\mathbb{E}]$ of algebras over $\\mathbb{E}$ is given by\nthe following coequalizer : \n\n\\[\n\\xymatrix{\n\\relax\\mathbb{E} (\\mathbb{E} X \\otimes A) \\ar@<1ex>[rr]^{\\mathbb{E}(\\xi \\otimes \\id)} \\ar@<-0.5ex>[rr]_{\\mu \\circ \\mathbb{E} \\nu} && \\relax\\mathbb{E} (X \\otimes A)\n}\n,\n\\]\nwhere $\\nu \\colon \\mathbb{E} X \\otimes A \\rightarrow \\mathbb{E}(X \\otimes A)$ is the adjoint of composite\n\\[A \\rightarrow {\\catsymbfont{C}}(X, X \\otimes A) \\rightarrow {\\catsymbfont{C}}(\\mathbb{E} X, \\mathbb{E}(X \\otimes A)).\\]\nHere the first arrow is the adjoint of the identity map.\n\\end{prop}\n\nFor our application, we note that the relevant monads preserve\nreflexive coequalizers and so the preceding theorems construct the\ntensors and colimits in the category of commutative $S$-algebras and\nthe category of $E_\\infty$-spaces.  The limits and cotensors are\ninherited from the base categories of $S$-modules (and hence spectra)\nand based spaces respectively.\n\nWe are now ready to prove Theorem~\\ref{lifting}.  Let\n$F \\colon {\\catsymbfont{C}} \\rightarrow {\\catsymbfont{D}}$ be a functor between topological\ncategories, let $\\mathbb{A} \\colon {\\catsymbfont{C}} \\rightarrow {\\catsymbfont{C}}$ be a monad on ${\\catsymbfont{C}}$,\nand let $\\mathbb{B} \\colon {\\catsymbfont{D}} \\rightarrow {\\catsymbfont{D}}$ be a monad on ${\\catsymbfont{D}}$.  The\nfollowing easy lemma provides a simple condition for $F$ to yield a\nfunctor on the associated categories of algebras,\n$F \\colon {\\catsymbfont{C}}[\\mathbb{A}] \\to {\\catsymbfont{D}}[\\mathbb{B}]$.\n\n\\begin{lem}\\label{l:monad-comm}\nLet $\\phi \\colon  \\mathbb{B} F(X) \\cong F(\\mathbb{A} X)$ be a natural isomorphism such\nthat the following diagrams commute for any object $X$ of ${\\catsymbfont{C}}$.\n\\[\n\\xymatrix{\n\\relax\\mathbb{B} F(X) \\ar[r]^-{\\phi} & F(\\relax\\mathbb{A} X) & \\relax\\mathbb{B} \\relax\\mathbb{B} F(X) \\ar[r]^-{\\mu_B} \\ar[d]^-{\\phi} & \\relax\\mathbb{B} F(X) \\ar[d]^-{\\phi} \\\\\n& F(X) \\ar[ul]^-{\\eta_B} \\ar[u]_-{F(\\eta_A)} & F(\\relax \\mathbb{A} \\relax \\mathbb{A} X) \\ar[r]^-{F(\\mu_A)} & F(\\mathbb{A} X) \\\\\n}\n\\]\n\nThen if $X$ is a $\\mathbb{A}$-algebra in ${\\catsymbfont{C}}$ with action map $\\psi \\colon \\mathbb{A} X\n\\rightarrow X$, $F(X)$ is a $\\mathbb{B}$-algebra in ${\\catsymbfont{D}}$ with action map\n\\[\n\\xymatrix{\n\\relax\\mathbb{B} F(X) \\cong F(\\mathbb{A} X) \\ar[r]^-{F(\\psi)} & F(X)\n}\n.\n\\]\nTherefore $F$ yields a functor from ${\\catsymbfont{C}}[\\mathbb{A}]$ to ${\\catsymbfont{C}}[\\mathbb{B}]$.\n\\end{lem}\n\nNow we prove the main technical result of the section.  Suppose we are\nin the situation described in the preceding lemma, with the additional\nassumption that ${\\catsymbfont{C}}$ and ${\\catsymbfont{D}}$ are topological categories.\n\n\\begin{thm}\\label{t:pres}\nLet ${\\catsymbfont{C}}$ and ${\\catsymbfont{D}}$ be cocomplete topological categories, and $\\mathbb{A} \\colon {\\catsymbfont{C}}\n\\rightarrow {\\catsymbfont{C}}$ and $\\mathbb{B} \\colon {\\catsymbfont{D}} \\rightarrow {\\catsymbfont{D}}$ continuous monads.\nFurther suppose that there is a continuous functor $F \\colon {\\catsymbfont{C}}\n\\rightarrow {\\catsymbfont{D}}$ which satisfies the hypothesis of the preceding lemma\nand therefore yields a functor $F \\colon {\\catsymbfont{C}}[\\mathbb{A}] \\rightarrow {\\catsymbfont{D}}[\\mathbb{B}]$.\n\\begin{enumerate}\n\\item{If $F \\colon {\\catsymbfont{C}} \\rightarrow {\\catsymbfont{D}}$ preserves colimits and tensors, and the monads\n$\\mathbb{A}$ and $\\mathbb{B}$ preserve reflexive coequalizers, then $F \\colon {\\catsymbfont{C}}[\\mathbb{A}]\n\\rightarrow {\\catsymbfont{D}}[\\mathbb{B}]$ preserves colimits and tensors in ${\\catsymbfont{C}}[\\mathbb{A}]$.\nTherefore $F$ preserves all indexed colimits in ${\\catsymbfont{C}}$.}\n\\item{If furthermore $F$ is a left adjoint as a functor from ${\\catsymbfont{C}}$ to\n  ${\\catsymbfont{D}}$, then $F$ induces a left adjoint from ${\\catsymbfont{C}}[\\mathbb{A}]$ to ${\\catsymbfont{D}}[\\mathbb{B}]$.}\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\nFirst, we handle the issue of colimits.  We can apply\n\\cite[2.7.4]{ekmm} to describe colimits in the category ${\\catsymbfont{C}}[\\mathbb{A}]$ of\n$\\mathbb{A}$-algebras.  Given a diagram of $\\{R_i\\}$ of $\\mathbb{A}$-algebras, we\ncan describe $F(\\colim R_i)$ as $F$ applied to the reflexive\ncoequalizer which creates colimits in the category ${\\catsymbfont{C}}[\\mathbb{A}]$.\n\n\\[\nF\\left(\n\\xymatrix{\n\\relax\\mathbb{A}(\\colim \\mathbb{A} R_i) \\ar@<1ex>[rr]^{\\mathbb{A}(\\colim \\xi_i)}\n\\ar@<-0.5ex>[rr]_{\\mu \\circ \\mathbb{A} \\alpha} && \\relax\\mathbb{A}(\\colim R_i)\n}\n\\right)\n.\n\\]\nSince $F$ commutes with colimits in $\\mathbb{A}$, this is isomorphic to the\nreflexive coequalizer :\n\n\\[\n\\xymatrix{\n\\relax\\mathbb{B}(\\colim \\mathbb{B} FR_i)\n\\ar@<1ex>[rr]^{\\mathbb{B}(\\colim F(\\xi_i))} \\ar@<-0.5ex>[rr]_{\\mu\n  \\circ \\mathbb{B} F(\\alpha)} && \\relax\\mathbb{B}(\\colim FR_i))\n}\n.\n\\]\nThis is precisely the colimit of the diagram $\\{FR_i\\}$ in the\ncategory of $\\mathbb{B}$-algebras by \\cite[2.7.4]{ekmm} once again.\n\nNext, we consider tensors.  We can express $F(X \\otimes A)$ as $F$ applied to\nthe reflexive coequalizer which creates the tensors in the category\n${\\catsymbfont{C}}[\\mathbb{A}]$.\n\\[\nF\\left(\n\\xymatrix{\n\\relax\\mathbb{A} (\\mathbb{A} X \\otimes A) \\ar@<1ex>[rr]^{\\mathbb{A}(\\xi \\otimes \\id)} \\ar@<-0.5ex>[rr]_{\\mu \\circ \\mathbb{A} \\nu} && \\relax\\mathbb{A} (X \\otimes A)\n}\n\\right)\n.\n\\]\nWe can rewrite this expression using the fact that $F$ commutes with\ncolimits in ${\\catsymbfont{A}}$, as follows.\n\\[\n\\xymatrix{\n\\relax\\mathbb{B} F(\\mathbb{A} X \\otimes A) \\ar@<1ex>[rr]^{\\mathbb{B} (\\xi \\otimes \\id)} \\ar@<-0.5ex>[rr]_{\\mu \\circ \\mathbb{B} \\nu} && \\relax\\mathbb{B} F(X \\otimes A)\n}\n.\n\\]\nAs $F$ commutes with tensors in ${\\catsymbfont{A}}$, this becomes :\n\\[\n\\xymatrix{\n\\relax\\mathbb{B} (\\mathbb{B} FX \\otimes A) \\ar@<1ex>[rr]^{\\mathbb{B} (\\mathbb{B} (\\xi \\otimes \\id)} \\ar@<-0.5ex>[rr]_{\\mu \\circ \\mathbb{B} \\nu} && \\relax\\mathbb{B} (FX \\otimes A)\n}\n.\n\\]\nThis is precisely the diagram expressing the tensor $FX \\otimes A$\nin the category ${\\catsymbfont{C}}[\\mathbb{B}]$.  It is now a consequence of theorem\n\\ref{t:kelly} that $M$ preserves all indexed colimits.  \n\nFinally, assume that $F \\colon {\\catsymbfont{C}} \\rightarrow {\\catsymbfont{D}}$ is a left adjoint.\nThere is a diagram of categories:\n\\[\n\\xymatrix{\n\\relax{\\catsymbfont{C}}[\\relax\\mathbb{A}] \\ar[r]^-F \\ar@<-0.5ex>[d]_U &\n\\relax{\\catsymbfont{D}}[\\relax\\mathbb{B}] \\ar@<-0.5ex>[d]_V \\\\ \n\\relax{\\catsymbfont{C}} \\ar@<-0.5ex>[u]_G \\ar[r]^-F & \\relax{\\catsymbfont{D}} \\ar@<-0.5ex>[u]_G. \\\\\n}\n\\]\nHere $U$ and $V$ denote forgetful functors, and $G$ denotes the free\nalgebra functors.  The square commutes in the sense that $F \\circ G =\nG \\circ F$ and $F \\circ U = V \\circ F$.  To show that $F \\colon {\\catsymbfont{C}}[\\mathbb{A}]\n\\rightarrow {\\catsymbfont{D}}[\\mathbb{B}]$ is a continuous left adjoint, it suffices to\nshow that $F$ preserves tensors and $F$ is a left adjoint when the\nenrichment is ignored \\cite[6.7.6]{borceux}.  We know that the former\nholds, and since $F \\colon {\\catsymbfont{A}} \\rightarrow {\\catsymbfont{B}}$ is a left adjoint by\nhypothesis and ${\\catsymbfont{C}}[\\mathbb{A}]$ has coequalizers, we can apply the adjoint\nlifting theorem \\cite[4.5.6]{borceux} and conclude the latter.\n\\end{proof}\n\n\\section{Parameterized spaces and operadic algebras}\\label{S:einf}\n\nIn this section, we review the definitions of the domain and range\ncategories of the Lewis-May operadic Thom spectrum functor.  We begin\nby discussing operadic algebras.  \n\n\\subsection{Review of operadic algebras}\n\nLet $\\scr{I}$ be the (unbased) category of finite-dimensional or\ncountably-infinite real inner product spaces and linear isometries.\nThis is a symmetric monoidal category under the direct sum.\n\n\\begin{defn}\nLet $U^j$ be the direct sum of $j$ copies of $U$ (an\ninfinite-dimensional real inner product space), and let $\\scr{L}(j)$ be\nthe mapping space $\\scr{I}(U^j,U)$.  The action of $\\Sigma_j$ on $U^j$ by\npermutation induces an action of $\\Sigma_j$ on $\\scr{L}(j)$.  There are\nmaps\n\\[\\gamma \\colon \\scr{L}(k) \\times \\scr{L}(j_1) \\times \\ldots \\times \\scr{L}(j_k) \\rightarrow \\scr{L}(j_1 + \\ldots + j_k)\\]\ngiven by $\\gamma(g;f_1, \\ldots, f_k) = g \\circ (f_1 \\oplus \\ldots\n\\oplus f_k)$.  The spaces $\\scr{L}(j)$ form an operad, which we will refer\nto as the linear isometries operad.\n\\end{defn}\n\nThe properties of the linear isometries operad have been explored at\nlength, notably in section XI of \\cite{ekmm}.  Recall that $\\scr{L}$ is an\n$E_\\infty$-operad, as $\\scr{L}(j)$ is contractible, $\\scr{L}(1)$ contains the\nidentity, $\\scr{L}(0)$ is a point, and $\\Sigma_n$ acts freely on $\\scr{L}(n)$.\nWe can consider both based spaces and spectra which admit actions of\n$\\scr{L}$.  We will make frequent use of the fact that for any operad ${\\catsymbfont{O}}$,\nthere is an associated monad $\\mathbb{O}$ such that objects $X$ with actions\nby ${\\catsymbfont{O}}$ are precisely algebras over $\\mathbb{O}$ \\cite{may-geom}.\n\nA space $X$ with an action of the operad $\\scr{L}$ is the same as an\nalgebra over a certain monad $\\mathbb{K}$ on the category of based spaces.\nSince the monad $\\mathbb{K}$ preserves reflexive coequalizers, standard\nlifting techniques suffice to show the following theorem\n\\cite{hopkins}, \\cite[6.2]{basterra-mandell}.\n\n\\begin{thm}\nThe category ${\\catsymbfont{T}}[\\mathbb{K}]$ of $\\scr{L}$-spaces admits the structure of a\ntopological model category.  Fibrations and weak equivalences are\ncreated in the category ${\\catsymbfont{T}}$, and cofibrations are defined as having\nthe left-lifting property with respect to acyclic fibrations.\n\\end{thm}\n\nSince $\\scr{L}$ is an $E_\\infty$ operad, we can functorially associate a\nspectrum $Z$ to an $\\scr{L}$-space $X$ such that the map $X \\rightarrow\n\\Omega^\\infty Z$ is a group completion.  When $\\pi_0(X)$ is a group\nand not just a monoid, this map is a weak equivalence.  Such\n$\\scr{L}$-spaces $X$ for which $\\pi_0(X)$ is a group are said to be\ngroup-like.\n\nSimilarly, the category of $E_\\infty$-ring spectra can be described as\na category of algebras over monads, following \\cite[2.4]{ekmm}.  Let\n${\\catsymbfont{S}}$ denote the category of coordinate-free spectra \\cite{lms}.  For\nclarity, we emphasize that ${\\catsymbfont{S}}$ is not a symmetric monoidal category\nof spectra prior to passage to the homotopy category.  An\n$E_\\infty$-ring spectrum structured by the operad $\\scr{L}$ is an algebra\nover a certain monad $\\mathbb{C}$ in ${\\catsymbfont{S}}$.\n\nSince the Thom spectrum associated to an object $f$ of ${\\catsymbfont{T}} \/ BG$ will\nhave a natural unit $S \\rightarrow Mf$ induced by the inclusion of the\nbasepoint, we also consider the category ${\\catsymbfont{S}} \\backslash S$ of\nunital spectra.  In this setting, an $E_\\infty$-ring spectrum $X$ over\nthe operad $\\scr{L}$ is the same as an algebra over the monad\n$\\tilde{\\mathbb{C}}$, where $\\tilde{\\mathbb{C}} X$ is a ``reduced'' version of $\\mathbb{C}$\nquotiented to ensure that the unit provided by the algebra structure\ncoincides with the existing unit.\n\nThere is a close relationship between the category of algebras over\n$\\mathbb{C}$ and algebras over $\\tilde{\\mathbb{C}}$ \\cite[2.4.9]{ekmm}.\nThe category ${\\catsymbfont{S}} \\backslash S$ is itself a category of algebras over\n${\\catsymbfont{S}}$ for the monad $\\mathbb{U}$ which takes $X$ to $X \\vee S$.  The monad\n$\\mathbb{C}$ is precisely the composite monad $\\tilde{\\mathbb{C}} \\mathbb{U}$, and in this\nsituation the categories of algebras are equivalent\n\\cite[2.6.1]{ekmm}.  Therefore the two notions of $E_\\infty$-ring\nspectrum we have described are equivalent.  In the language of\n\\cite{ekmm}, $\\tilde{\\mathbb{C}}$ is the ``reduced'' monad associated to the\nmonad $\\mathbb{C}$.  Both of these monads preserve reflexive coequalizers.\n\nFinally, given an $E_\\infty$-ring spectrum, the functor $S \\sma_{\\scr{L}}\n-$ converts it to a weakly equivalent commutative\n$S$-algebra \\cite[2.3.6,2.4.2]{ekmm}.  Moreover, $S \\sma_{\\scr{L}} -$ is a\ncontinuous left adjoint.\n\n\\subsection{Parametrized operadic algebras}\n\nNow we move on to consider the category of spaces over a fixed base\nspace $B$.  The category ${\\catsymbfont{U}} \/ B$ has objects maps $p \\colon\nX \\rightarrow B$, where $X$ and $B$ are objects of ${\\catsymbfont{U}}$.  A morphism\n$(p_1 \\colon X \\rightarrow B) \\rightarrow (p_2 \\colon Y \\rightarrow\nB)$ is a map $f \\colon X \\rightarrow Y$ such that $p_2 f = p_1$.  The\nproperties of this category have been investigated in a variety of\nplaces \\cite{lewis, intermont-johnson}, \\cite[7.1]{lms}.  In\nparticular, this is a topological category where the tensor of\n$p \\colon X \\rightarrow B$ and an unbased space $A$ is given by the\ncomposite  \n\\[\n\\xymatrix{\nX \\times A \\ar[r]^{\\pi_1} & X \\ar[r]^p & B \\\\\n}\n\\]\n(where $\\pi_1$ is the projection onto the first factor).\n\nSince we will be interested in spaces which admit operad actions, we\nalso consider the related category of based spaces over $B$.  This is\nthe category ${\\catsymbfont{T}} \/ B$, defined in the same fashion as ${\\catsymbfont{U}} \/ B$,\nreplacing spaces with based spaces and requiring that the maps be \nbased.  The category ${\\catsymbfont{T}} \/ B$ inherits the structure of a category\ntensored over unbased spaces from ${\\catsymbfont{U}} \/B$, where the tensor of\n$X \\rightarrow B$ and an unbased space $A$ is given by $X \\sma\nA_+ \\rightarrow B$. \n\nColimits in ${\\catsymbfont{T}} \/ B$ are formed as follows.  Given a diagram $D\n\\rightarrow {\\catsymbfont{T}} \/ B$, via the forgetful functor we obtain a\ndiagram $D \\rightarrow {\\catsymbfont{T}} \/ B \\rightarrow {\\catsymbfont{T}}$.  The colimit over $D\n\\rightarrow {\\catsymbfont{T}} \/ B$ is computed by taking the colimit of this\ndiagram in ${\\catsymbfont{T}}$ and using the induced map to $B$ given by the\nuniversal property of the colimit.\n\n\nWhen $B$ is an $\\scr{L}$-space, there is a subcategory of ${\\catsymbfont{T}} \/ B$ where\nthe objects are $\\scr{L}$-maps $X \\rightarrow B$ and the morphisms are\n$\\scr{L}$-maps over $B$.  In slight abuse of terminology, we will\nsometimes refer to this category as $\\scr{L}$-spaces over $B$.  We can\nregard this category as algebras over a monad on ${\\catsymbfont{T}} \/ B$.  Given a\nmap $f \\colon Y \\rightarrow B$, where $B$ is an $\\scr{L}$-space, the space\n$\\mathbb{K} Y$ admits an $\\scr{L}$-map to $B$ given by the unique extension of\n$f$ \\cite[7.7]{lms}.  This specifies a monad on ${\\catsymbfont{T}} \/ B$, with\nstructure maps inherited from those of $\\mathbb{K}$, which we will refer to\nas $\\mathbb{K}_{B}$.  Denote by $({\\catsymbfont{T}} \/ B)[\\mathbb{K}_{B}]$ the category of\n$\\mathbb{K}_{B}$-algebras.  There is a model structure on this category\ndefined in analogy with the naive model structure on ${\\catsymbfont{T}} \/ B$.  We\nneed to verify the existence of tensors and colimits in $({\\catsymbfont{T}} \/\nB)[\\mathbb{K}_{B}]$.  In order to show that $({\\catsymbfont{T}} \/ B)[\\mathbb{K}_{B}]$ is\ntopologically cocomplete, it will suffice to show that the monad\n$\\mathbb{K}_{B}$ preserves reflexive coequalizers.  This follows immediately\nfrom the fact that $\\mathbb{K}$ preserves reflexive coequalizers, since\ncolimits in ${\\catsymbfont{T}} \/ B$ are constructed by taking the colimit in ${\\catsymbfont{T}}$\nand using the natural map to $B$.\n\n\\begin{prop}\nThe category $({\\catsymbfont{T}} \/ B)[\\mathbb{K}_{B}]$ is topologically cocomplete (and in\nparticular has all colimits and tensors with based spaces). \n\\end{prop}\n\nIt will be useful later on to write out an explicit description of the\ntensor in $({\\catsymbfont{T}} \/ B)[\\mathbb{K}_{B}]$.  We regard the category of\n$\\scr{L}$-spaces as tensored over unbased spaces via the tensor over based\nspaces: for an unbased space $A$ the tensor with an $\\scr{L}$-space $X$ is\nthe based tensor $X \\otimes A_+$.\n\n\\begin{lem}\nThe tensor of an unbased \nspace $A$ and $(X \\rightarrow B)$ \nis given by \n\\[X \\otimes A_+ \\rightarrow X \\otimes S_0 \\cong X \\rightarrow BG,\\]\nwhere the first map is the collapse map which takes $A$ to the\nnon-basepoint of $S^0$. \n\\end{lem}\n\n\\section{The operadic Thom spectrum functor}\n\nIn this section we review the operadic theory of Thom spectra\ndeveloped by Lewis \\cite[7.3]{lms} and May \\cite{may-quinn-ray}.\nOur discussion is updated slightly to take account of more recent\ndevelopments in the theory of diagram spectra \\cite{mandell-may,\n  mmss}.  In particular, our terminology regarding $\\scr{I}$-spaces\nreflects the modern usage and is at variance with the definitions in\nthe original articles.\n\n\\subsection{The definition of $M$}\n\nRecall that $\\scr{I}$ denote the category of finite-dimensional or\ncountably-infinite real inner product spaces and linear isometries.   \n\n\\begin{defn}\nAn $\\scr{I}$-space is a continuous functor $X$ from $\\scr{I}$ to the category\n of based topological spaces.\n\\end{defn}\n\nWe will restrict attention to $\\scr{I}$-spaces with the property that\n$X(V)$ is the colimit of $X(W)$ for the finite-dimensional subspaces\n$W \\subset V$.  This constraint implies that it is sufficient to\nconsider the restriction of $X$ to the full subcategory of $\\scr{I}$\nconsisting of the finite-dimensional real inner product spaces\n\\cite[1.1.8-1.1.9]{may-quinn-ray}.   \n\nThe idea of using $\\scr{I}$ to capture structure about infinite loop\nspaces and operad actions dates back to Boardman and Vogt's original\ntreatment \\cite{boardman-vogt}.  In the context of Thom spectra,\n$\\scr{I}$-spaces first arose in \\cite{may-quinn-ray}.  More recently, May\nhas introduced the terminology of``functors with cartesian product''\n(FCP) to highlight the connection to diagram spectra \\cite{may-fcp},\nin analogy with Bokstedt's ``functors with smash product'' (FSP's).\n\n\\begin{defn}\nA functor with cartesian product over $\\scr{I}$ ($\\scr{I}$-FCP) is a\n$\\scr{I}$-space equipped with a unital and associative ``Whitney sum''\nnatural transformation $\\omega$ from $X \\times X$ to $X \\circ \\oplus$. \n\\end{defn}\n\nA commutative $\\scr{I}$-FCP is a $\\scr{I}$-FCP for which the natural\ntransformation $X \\times X$ to $X \\circ \\oplus$ is commutative.  We\nwill assume in the following that by default $\\scr{I}$-FCP's are\ncommutative.  Commutative $\\scr{I}$-FCP's encode an $E_\\infty$-structure\n\\cite[1.1.6]{may-quinn-ray}; specifically, a commutative $\\scr{I}$-FCP $X$\nyields an $\\scr{L}$-space structure on $X(\\ensuremath{\\mathbb{R}}^{\\infty})$.  The essential\nobservation is that we can use the Whitney sum to obtain a natural map\n$\\scr{L}(j) \\times X(R^{\\infty})^j \\rightarrow X(R^{\\infty})$ specified by\n\\[\n(f, x_1, x_2, \\ldots, x_j) \\mapsto Xf(x_1 \\oplus x_2 \\oplus \\ldots \\oplus x_j).\n\\]\nSimilarly, a noncommutative $\\scr{I}$-FCP yields a\nnon-$\\Sigma$ $\\scr{L}$-space structure on $X(\\ensuremath{\\mathbb{R}}^{\\infty})$.\n\nThere is an obvious product structure on the category of $\\scr{I}$-spaces\nspecified by the levelwise cartesian product.  A monoid $\\scr{I}$-FCP is\nan $\\scr{I}$-FCP such that the levelwise monoid product specifies a\nmorphism of $\\scr{I}$-spaces.  A notable example is the monoid $\\scr{I}$-FCP\n$F$ given by taking $F(V)$ to be the space of based homotopy\nequivalences of $S^V$.  We will always assume that for a monoid\n$\\scr{I}$-FCP $X$, the monoids $X(V)$ are grouplike.  Analogously, we will\nconsider group $\\scr{I}$-FCP's.  Familiar examples include the functor\nspecified by $V \\mapsto O(V)$ and the functor specified by $V \\mapsto\nU(V)$.\n\nFor any monoid $\\scr{I}$-FCP $X$, we can construct a related $\\scr{I}$-FCP $BX$\nvia the two-sided bar construction.  Specifically, define $BX$ as the\nfunctor specified by\n\\[BX(V) = B(*,X(V),*),\\]\nwhere $B(-,-,-)$ denotes the geometric realization of the two-sided bar\nconstruction.  When $X$ is equipped with an augmentation to $F$ which\nis a map of monoid $\\scr{I}$-FCP's, we can construct $EX$ as \\[EX(V) = B(*,X(V),S^V),\\]\nwhere $X(V)$ acts on $S^V$ via the augmentation.  There is a\nprojection map $\\pi \\colon EX \\rightarrow BX$ and a section defined by\nthe basepoint inclusion $* \\lhook\\joinrel\\relbar\\joinrel\\rightarrow S^V$.  This section is a\ncofibration, $\\pi$ is a quasifibration, and $\\pi$ has fiber $S^V$\n\\cite[7.2]{lms}.  If $X$ actually takes values in groups, $\\pi$ is a\nbundle.\n\nWhen $X = F$, this construction provides a model of the universal\nquasifibration with spherical fibers \\cite{may-class}.  More\ngenerally, we obtain universal quasifibrations and fibrations with\nspherical fibers and prescribed structure groups.  Note that we are\nfollowing Lewis in letting $EG(V)$ denote the total space of the\nuniversal spherical quasifibration rather than the associated\nprincipal quasifibration.\n\nMoving on, we now describe the Thom spectrum construction.  Let $G$ be a\nmonoid $\\scr{I}$-FCP which is augmented over $F$.  Abusing notation, we\nwill write $BG$ to denote both the $\\scr{I}$-FCP $BG$ and the space\n$\\colim_V BG(V)$.  We will assume we are given a map of spaces $f\n\\colon Y \\rightarrow BG$.  \n\n\\begin{defn}\nLet $f\\colon Y \\rightarrow BG$ be a map of spaces.  The filtration of\n$BG$ by inner product spaces $V$ induces a filtration on $Y$ defined\nas $Y(V) = f^{-1}(BG(V))$.  The Thom prespectrum associated to $f\n\\colon Y \\rightarrow BG$ is specified as follows.  Set $Tf(V)$ to be\nthe Thom space of the pullback $Z(V)$ in the diagram :\n\\[\n\\xymatrix{\nZ(V) \\ar[r]\\ar[d] & EG(V) \\ar[d] \\\\\nY(V) \\ar[r] & BG(V). \\\\\n}\n\\]\nThat is, the map $Z(V) \\rightarrow Y(V)$ has a section $i$, and $Tf(V)\n= Z(V)\/i(Y(V))$.  $Tf$ is a prespectrum, and we define the Thom\nspectrum in ${\\catsymbfont{S}}$ associated to $f$ as the spectrification $Mf = LTf$.  \n\\end{defn}\n\nOther filtrations can also be used in this construction, but it can be\nshown that the choice of filtration does not matter up to isomorphism\nof spectra \\cite[7.4.4]{lms}.\n\nTo see that $Tf$ is actually a prespectrum, we must describe the\nsuspension maps.  Associated to the inclusion $V \\subset W$ is an\ninclusion $Y(V) \\subset Y(W)$, and this induces a map of pullbacks\n$Q_V \\rightarrow Z_W$ in the following diagrams :\n\\[\n\\xymatrix{\nZ_W \\ar[r] \\ar[d] & EG(W) \\ar[d] && Q_V \\ar[r] \\ar[d] & EG(V) \\ar[r] \\ar[d] & \\ar[d] EG(W) \\\\\nY(W) \\ar[r] & BG(W) && Y(V) \\ar[r] & BG(V) \\ar[r] & BG(W). \\\\\n}\n\\]\nUpon passage to Thom spaces, we can identify the Thom space of $Q_V$\nas the fiberwise suspension $\\Sigma^{W-V}$ of the Thom space of $Z_V$\n\\cite[7.2.2]{lms}, and so the map in question is a suspension map.\nOne checks that these suspension maps are appropriately coherent\n\\cite[7.2.1]{lms}.\n\n\\begin{rem}\nLewis treated only monoid $\\scr{I}$-FCP's $X$ augmented over $F$; this\naugmentation gives an action of $X$ on $S^V$ which allows the\nconstruction of $EX$.  However, we can develop the theory of Thom\nspectra for other choices of fiber, as long as we specify a levelwise\naction of $X$ on the fiber.  Such constructions will be useful for us\nwhen considering models of Eilenberg-Mac Lane spectra as Thom spectra\nin section \\ref{S:TEM}.  We will consider $p$-local and $p$-complete\nspherical fibrations, and employ ``localized'' and ``completed''\nversions of $F$ formed from spaces of based self-equivalences of the\n$p$-local sphere $S^V_{(p)}$ and based self-equivalences of the\n$p$-complete sphere $(S^V)^{\\scriptscriptstyle\\wedge}_{p}$.\n\\end{rem}\n\nWe have constructed the Thom spectrum as a continuous functor from\n${\\catsymbfont{U}} \/ BG$ to coordinate-free spectra ${\\catsymbfont{S}}$.  Working with ${\\catsymbfont{T}} \/ BG$,\nwe obtain a functor to ${\\catsymbfont{S}} \\backslash S$, unital spectra.  Here the\nunit $S \\rightarrow Mf$ is induced by the inclusion $* \\rightarrow X$\nover $BG$.  In abuse of notation, we will refer to both of these\nfunctors as $M$.\n\n\\subsection{Properties of $M$}\n\nLewis proves that the Thom spectrum functor $M$ preserves colimits in\n${\\catsymbfont{U}} \/ BG$ \\cite[7.4.3]{lms}.  It is straightforward to extend this to\nthe functor $M$ from ${\\catsymbfont{T}} \/ BG$ to ${\\catsymbfont{S}} \\backslash S$.\n\n\\begin{lem}\nThe Thom spectrum functor takes colimits in ${\\catsymbfont{T}} \/ BG$ to colimits in\nthe category ${\\catsymbfont{S}} \\backslash S$. \n\\end{lem}\n\n\\begin{proof}\nA colimit over ${\\catsymbfont{D}}$ in ${\\catsymbfont{T}} \/ BG$ is given as the pushout in ${\\catsymbfont{U}} \/ BG$ \n\\[\n\\xymatrix{\n\\relax{\\colim_{{\\catsymbfont{D}}}} * \\ar[r] \\ar[d] & \\ar[d] \\relax{*} \\\\\n\\relax{\\colim_{{\\catsymbfont{D}}}} R_d \\ar[r] & Z \\\\\n}\n\\]\n\\\\\nwhere the indicated colimits are also taken in the category ${\\catsymbfont{U}} \/\nBG$.  Similarly, a colimit over ${\\catsymbfont{D}}$ in ${\\catsymbfont{S}} \\backslash S$ is\nconstructed as the pushout in ${\\catsymbfont{S}}$ \n\\[\n\\xymatrix{\n\\colim_{{\\catsymbfont{D}}} S \\ar[r]\\ar[d] & \\ar[d] S \\\\\n\\colim_{{\\catsymbfont{D}}} R_d \\ar[r] & Z \\\\\n}\n\\]\n\\\\\nwhere the indicated colimits are also taken in ${\\catsymbfont{S}}$.  The result\nfollows from the fact that $M$ takes colimits in ${\\catsymbfont{U}} \/ BG$ to\ncolimits in spectra and $M(*) \\cong S$. \n\\end{proof}\n\nLewis also shows that the functor $M$ also preserves tensors with\nunbased spaces in ${\\catsymbfont{T}} \/ BG$ \\cite[7.4.6]{lms}. \n\n\\begin{prop}\\label{P:ten}\nThe Thom spectrum associated to the composition $X \\sma A_+\n\\rightarrow X \\rightarrow BG$ is naturally isomorphic to $Mf \\sma\nA_+$.\n\\end{prop}\n\nWhen $A = I$, this implies that functor $M$ converts fiberwise\nhomotopy equivalences into homotopy equivalences in ${\\catsymbfont{S}} \\backslash\nS$.  \n\nThe question of invariance under weak equivalence is somewhat more\ndelicate.  Unfortunately, quasifibrations are not preserved under\npullback along arbitrary maps.  This can cause technical difficulty\nwhen working with $BF$, or any other monoid $\\scr{I}$-FCP (which is not a\ngroup $\\scr{I}$-FCP).  Following Lewis \\cite[7.3.4]{lms}, we make the\nfollowing definition.\n\n\\begin{defn}\nDefine a map $f \\colon X \\rightarrow BG$ to be good if the projections $Z_V\n\\rightarrow X(V)$ are quasifibrations and the sections $X(V)\n\\rightarrow Z_V$ are Hurewicz cofibrations.\n\\end{defn}\n\nA map $f \\colon X \\rightarrow BG$ associated to an $\\scr{I}$-monoid $G$ with\nvalues in groups is always good, and all Hurewicz fibrations are good\n\\cite[7.3.4]{lms}.  Therefore, it is sometimes useful to replace\narbitrary maps by Hurewicz fibrations when working over $BF$ via the functor\n$\\Gamma$ \\cite[7.1.11]{lms}.  This is compatible with the linear\nisometries operad --- recall that given an $\\scr{L}$-map $f \\colon\nX \\rightarrow BF$, the map $\\Gamma f \\colon \\Gamma X \\rightarrow BF$\nis also an $\\scr{L}$-map \\cite[1.8]{may-geom}.  Our discussion of $\\Gamma$\nis brief, as we do not use it extensively in this paper.\n\nWhen the maps in question are good, the Thom spectrum functor\npreserves weak equivalences over $BG$ \\cite[7.4.9]{lms}.\n\n\\begin{thm}\nIf $f \\colon X \\rightarrow BG$ and $g \\colon X^{\\prime} \\rightarrow BG$ are good\nmaps such that there is a weak equivalence $h \\colon X \\htp X^{\\prime}$\nover $BG$, then there is a stable equivalence $Mh \\colon Mf \\htp Mg$.\n\\end{thm}\n\nIn this situation, $M$ also takes homotopic maps to stably equivalent\nspectra \\cite[7.4.10]{lms}.  Note however that the stable equivalence\ndepends on the homotopy.\n\n\\begin{thm}\nIf $f \\colon X \\rightarrow BG$ and $g \\colon X \\rightarrow BG$ are good maps\nwhich are homotopic, then there is a stable equivalence $Mf \\htp Mg$.\n\\end{thm}\n\n\\section{The Thom spectrum functor is a left adjoint}\\label{sec:thomleft}\n\nAs discussed previously, spaces with actions by the linear isometries\noperad $\\scr{L}$ can be regarded as the category ${\\catsymbfont{T}}[\\mathbb{K}]$ of algebras\nover the monad $\\mathbb{K}$.  Similarly, spectra in ${\\catsymbfont{S}} \\backslash S$ which\nare $E_\\infty$-ring spectra structured by the linear isometries operad\ncan be regarded as the category $({\\catsymbfont{S}} \\backslash S)[\\mathbb{C}]$ of algebras\nwith respect to the monad $\\tilde{\\mathbb{C}}$.\n\nOne of the main results of Lewis' work is that the Thom spectrum\nfunctor $M$ ``commutes'' with these monads.  Specifically, Lewis\nproves \\cite[7.7.1]{lms}  \n\n\\begin{thm}\n\\hspace{5 pt}\n\\begin{enumerate}\n\\item{Given a map $f \\colon X \\rightarrow BF$, there is an isomorphism\n$\\tilde{\\mathbb{C}} Mf \\cong M(\\mathbb{K}_{BG} f)$, where the map \n\\[\\mathbb{K}_{BG}f \\colon \\mathbb{K}_{BG} X \\rightarrow BG\\]\nis the natural map induced from $X \\rightarrow BG$.} \n\\item{This isomorphism is coherently compatible with the unit and\n  multiplication maps for these monads, in the sense of lemma\n  \\ref{l:monad-comm}.}\n\\end{enumerate}\n\\end{thm}\n\nAs we have observed, a consequence of this result is that the Thom\nspectrum functor induces a functor $M_{E_\\infty}$ from $({\\catsymbfont{T}} \/\nBG)[\\mathbb{K}_{BG}]$ to $E_\\infty$-ring spectra structured by\n$\\tilde{\\mathbb{C}}$.  Composing with the functor $S \\sma_{\\scr{L}} -$, we obtain\na Thom spectrum functor $M_{{\\catsymbfont{C}}{\\catsymbfont{A}}_S}$ from $({\\catsymbfont{T}} \/ BG)[\\mathbb{K}_{BG}]$ to\ncommutative $S$-algebras.  Now employing theorem \\ref{lifting}, we\nobtain the main result.\n\n\\begin{thm}\nThe Thom spectrum functor \n\\[M_{{\\catsymbfont{C}}{\\catsymbfont{A}}_S} \\colon ({\\catsymbfont{T}} \/ BG)[\\mathbb{L}_{BG}] \\to {\\catsymbfont{C}}{\\catsymbfont{A}}_S\\]\ncommutes with indexed colimits.\n\\end{thm}\n\n\\begin{proof}\nWe have verified that the functor $M_{E_\\infty}$ satisfies the\nhypotheses of theorem \\ref{lifting}, and so we can conclude that\n$M_{E_\\infty}$ commutes with indexed colimits.  Since $M_{{\\catsymbfont{C}}{\\catsymbfont{A}}_S}$ is\nobtained from $M_{E_\\infty}$ via composition with a continuous left\nadjoint, the result follows.\n\\end{proof}\n\nSince the Thom spectrum functor $M_{{\\catsymbfont{C}}{\\catsymbfont{A}}_S}$ preserves indexed\ncolimits, one would expect that it should in fact be a continuous left\nadjoint.  We will prove this by showing that the hypotheses of the\nsecond part of theorem \\ref{lifting} are satisfied.  However, our\nmethod of proof does not produce an explicit description of the right\nadjoint and so is somewhat unsatisfying.\n\n\\begin{lem}\nThe Thom spectrum functor from ${\\catsymbfont{T}} \/ BG$ to ${\\catsymbfont{S}} \\backslash S$ is a\nleft adjoint. \n\\end{lem}\n\n\\begin{proof}\nWe know that the Thom spectrum functor preserves colimits in ${\\catsymbfont{T}} \/\nBG$.  Moreover, it is easy to verify that the category ${\\catsymbfont{T}} \/ BG$\nsatisfies the hypotheses of the special adjoint functor theorem, since\n${\\catsymbfont{T}}$ does.  Therefore $M$ is a left adjoint. \n\\end{proof}\n\nNow, we have the following diagram of categories :\n\\[\n\\xymatrix{\n\\relax{\\catsymbfont{T}} \/ BG[\\mathbb{K}_{BG}] \\ar@<-0.5ex>[d]_U \\ar[r]^-{M_{E_\\infty}} & \\relax({\\catsymbfont{S}} \\backslash S)[\\tilde{\\mathbb{C}}] \\ar@<-0.5ex>[d]_V \\\\\n\\relax{\\catsymbfont{T}} \/ BG \\ar@<-0.5ex>[u]_F \\ar[r]^M & \\relax({\\catsymbfont{S}} \\backslash S) \\ar@<-0.5ex>[u]_G. \\\\\n}\n\\]\nHere $U$ and $V$ denote forgetful functors, and $F$ and $G$ denote the\nfree algebra functors.  Recall that $({\\catsymbfont{S}} \\backslash S)[\\tilde{\\mathbb{C}}]$\nis the category of $E_\\infty$-ring spectra \\cite[2.4.5]{ekmm}.  The\nsquare commutes in the sense that $M \\circ U = V \\circ M_{E_\\infty}$\nand $M_{E_\\infty} \\circ F = G \\circ M$.\n\n\\begin{cor}\nThe Thom spectrum functor $M_{{\\catsymbfont{C}}{\\catsymbfont{A}}_S}$ from ${\\catsymbfont{T}} \/ BG[\\mathbb{K}_{BG}]$ to\nthe category of commutative $S$-algebras is a continuous left adjoint.  \n\\end{cor}\n\n\\begin{proof}\nIt follows from theorem \\ref{lifting} that $M_{E_\\infty}$ is a\ncontinuous left adjoint.  Since $S \\sma_{\\scr{L}} -$ is a continuous left\nadjoint, the composite functor to commutative $S$-algebras is a\ncontinuous left adjoint as well. \n\\end{proof}\n\nWhen restricting attention to vector bundles, we can refine this\nresult somewhat.  Recall that the categories of $\\scr{L}$-spaces,\n$E_\\infty$-ring spectra, and commutative $S$-algebras are all\ncategories of algebras over monads.  In each case, a model structure\nis constructed by lifting a cofibrantly generated model structure from\nthe base category.  As a consequence, we have an explicit description\nof the cell objects.  \n\nIn each case, the cell objects are colimits of pushouts of the form\n\\[\n\\xymatrix{\n\\mathbb{Z} A \\ar[d]\\ar[r] & \\ar[d] X_{n-1} \\\\\n\\mathbb{Z} CA \\ar[r] & X_n \\\\ \n}\n\\]\nwhere $\\mathbb{Z}$ is the appropriate monad and where $A$ to $CA$ is a\ngenerating cofibration in the base category.  For instance, in the\ncase of $\\scr{L}$-spaces, $A \\rightarrow CA$ is a map of the form\n\\[\\bigvee_i S^{n_i-1}_+ \\rightarrow \\bigvee_i D^{n_i}_+.\\]\nFor the category of commutative $S$-algebras, $A \\rightarrow CA$ is a\nmap of the form\n\\[\\bigvee_i \\Sigma^\\infty S^{n_i-1}_+ \\rightarrow \\bigvee_i\n\\Sigma^\\infty D^{n_i}_+\\]\nwhere here the suspension spectrum functor takes values in\n$S$-modules.  The description for $E_\\infty$-ring spectra is\nanalogous.\n\n\\begin{cor}\nLet $G$ be a group $\\scr{I}$-monoid.  Then the functor $M_{{\\catsymbfont{C}}{\\catsymbfont{A}}_S}$ is a\nQuillen left adjoint.\n\\end{cor}\n\n\\begin{proof}\nIn these cases all maps are good, and so $M$ preserves weak\nequivalences.  Therefore, it will suffice to show that $M$ takes\nthe generating cofibrations and generating acyclic cofibrations to\ncofibrations.  The generating cofibrations in ${\\catsymbfont{T}}[\\mathbb{K}_{BG}]$ are maps\nof the form $h \\colon \\mathbb{K}_{BG} A \\rightarrow \\mathbb{K}_{BG} CA$, where $A$ is a\nwedge of $S^{n_i-1}_+$ and $CA$ the corresponding wedge of $D^n_+$.\nThe maps from $D^n_+ \\rightarrow BG$ is arbitrary, and these choices\ndetermines the maps $S^{n_i-1} \\rightarrow BG$.  Denote the map\n$\\mathbb{K}_{BG} A \\rightarrow BG$ by $h_1$ and the map $\\mathbb{K}_{BG} CA\n\\rightarrow BG$ by $h_2$.  Recall that $M \\mathbb{K}_{BG} f \\cong \\tilde{\\mathbb{C}}\nMf$.  In addition, a map from a contractible space to $BG$ represents\na bundle which is isomorphic to a trivial bundle.  Therefore, there is\na homeomorphism $Mh_1 \\cong \\tilde{\\mathbb{C}} \\Sigma^\\infty A$ and $Mh_2\n\\cong \\tilde{\\mathbb{C}} \\Sigma^\\infty CA$.  The induced map $Mh \\colon Mh_1\n\\rightarrow Mh_2$ clearly yields a generating cofibration in the\ncategory of $E_\\infty$-ring spectra structured by $\\tilde{\\mathbb{C}}$.  The\nanalysis for the acyclic generating cofibrations is similar. \n\\end{proof}\n\n\\section{Computing $THH$}\\label{sec:compute}\n\nThe formula $M(f \\otimes S^1) \\cong Mf \\otimes S^1$ is a point-set\nresult --- $Mf \\otimes S^1$ is an object in the category of\ncommutative $S$-algebras.  In this section we discuss how to ensure\nthat $Mf \\otimes S^1$ has the correct homotopy type so that it\nrepresents $THH(Mf)$.  \n\nFor an $S$-algebra $R$, in analogy with the classical definition of\nHochschild homology as $\\ensuremath{\\operatorname{Tor}}$ we define \\[THH(R) = R \\sma^{L}_{R \\sma\nR^{\\op}} R.\\]\nIn the algebraic setting, this $\\ensuremath{\\operatorname{Tor}}$ can be computed via the\nHochschild resolution.  In spectra, this leads to a candidate\npoint-set description of $THH(R)$ as the cyclic bar construction\n$N^{\\ensuremath{\\operatorname{cyc}}}(R)$.  The precise relationship between these is studied in\n\\cite[9.2]{ekmm}; the main result is that when $R$ is cofibrant they\nare canonically isomorphic in the derived category of $R$-modules\n\\cite[9.2.2]{ekmm}.\n\nFirst, observe that there is a derived version of the cyclic bar\nconstruction in $\\scr{L}$-spaces.  This is a consequence of the very\nuseful fact that for a simplicial set $A_\\cdot$ and an $\\scr{L}$-space\n$X$, there is a homeomorphism $X \\otimes |A_\\cdot| \\cong |X \\otimes\nA_\\cdot|$ \\cite[6.7]{basterra-mandell}.  When $A_\\cdot$ has finitely many\nnondegenerate simplices in each simplicial degree, this provides a\ntractable description of the tensor with $|A_\\cdot|$ in terms of\ntensors with finite sets --- i.e., finite coproducts.\n\n\\begin{lem}\nLet $g \\colon X \\rightarrow X^{\\prime}$ be a weak equivalence of cofibrant\n$\\scr{L}$-spaces.  Then there is an induced weak equivalence $g \\otimes\nS^1_+ \\colon X \\otimes S^1_+ \\rightarrow X^{\\prime} \\otimes S^1_+$.\n\\end{lem}\n\n\\begin{proof}\nSince $X \\otimes S^1_+$ is a proper simplicial space for any\n$\\scr{L}$-space $X$, the result follows from the fact that the induced map\n$g \\coprod g \\colon X \\coprod X \\rightarrow X^{\\prime} \\coprod X^{\\prime}$\nis a weak equivalence when $X$ and $X^{\\prime}$ are cofibrant.\n\\end{proof}\n\nOne might hope that for cofibrant $X$, $Mf$ is necessarily cofibrant\nas a (commutative) $S$-algebra.  Of course when $M$ is a left Quillen\nadjoint this holds, but in general it turns out that $Mf$ does belong\nto a class of commutative $S$-algebras for which the point-set smash\nproduct has the correct homotopy type.\n\n\\begin{thm}\n\\item{Let $f \\colon X \\rightarrow BG$ be a good $\\scr{L}$-map such that $X$ is\n  a cell $\\scr{L}$-space.  Then $Mf \\sma Mf$ represents the derived smash\n  product.}\n\\end{thm}\n\nRecall the notion of an extended cell module \\cite[9.6]{basterra}.  An\nextended $S$-cell is a pair $(X \\sma B^n_+, X \\sma S^{n-1}_+)$, where\n$X = S \\sma_{\\scr{L}} \\scr{L}(i) \\thp_G K$ for a $G$-spectrum $K$ indexed on\n$U^i$ which has the homotopy type of a $G$-CW-spectrum for some $G\n\\subset \\Sigma^i$.  An extended cell $S$-module is an $S$-module $M =\n\\colim M_i$ where $M_0 = 0$ and $M_n$ is obtained from $M_{n-1}$ by a\npushout of $S$-modules of the form\n\n\\[\n\\xymatrix{\n\\bigvee_j X_j \\sma S^{n_j-1}_+ \\ar[d]\\ar[r] & \\ar[d] M_{n-1} \\\\\n\\bigvee_j X_j \\sma B^{n_j}_+ \\ar[r] & M_n. \\\\\n}\n\\]\n\nExtended cell $S$-modules have the correct homotopy type for the\npurposes of the smash product.  Therefore, it will suffice to show the\nfollowing result.\n\n\\begin{prop}\nLet $f \\colon X \\rightarrow BG$ be a good $E_\\infty$-map over the linear \nisometries operad such that $X$ is a cell $\\scr{L}$-space.  Then the\nunderlying $S$-module of the $S$-algebra $Mf$ has the homotopy type of\nan extended cell $S$-module.\n\\end{prop}\n\n\\begin{proof}\nBy hypothesis, $X = \\colim X_i$ where $X_0 = *$ and $X_i$ is obtained\nfrom $X_{i-1}$ as the pushout \n\\[\n\\xymatrix{\n\\tilde{\\mathbb{K}}A \\ar[r]\\ar[d] & \\ar[d] X_{i-1} \\\\\n\\tilde{\\mathbb{K}}CA \\ar[r] & X_i \\\\\n}\n\\]\nwhere $A$ is a wedge of spheres $S^{n_i-1}_+$ and $CA$ is the associated\nwedge of $D^{n_i}_+$.  Since $M$ commutes with colimits and $M\\mathbb{K} g\n\\cong \\tilde{\\mathbb{C}} Mg$, we have that $Mf = \\colim Mf_i$ where $Mf_0 =\nS$ and $Mf_i$ is obtained from $Mf_{i-1}$ as the pushout  \n\\[\n\\xymatrix{\n\\tilde{\\mathbb{C}}MA \\ar[r]\\ar[d] & \\ar[d] Mf_{i-1} \\\\\n\\tilde{\\mathbb{C}}MCA \\ar[r] & Mf_i. \\\\\n}\n\\]\nAs $CA$ is a contractible space with a disjoint basepoint, $MCA$ is\nhomotopy equivalent to a cell $S$-module.  $MA$ is the wedge of a Thom\nspectrum over a suspension with $S$, and so we know that it is also a\ncell $S$-module \\cite[7.3.8]{lms}.  Temporarily assume that $\\tilde{\\mathbb{C}}MA$\nand $\\tilde{\\mathbb{C}}MCA$ are extended cell $S$-modules.  Then we proceed as in\n\\cite[7.7.5]{ekmm}.  $Mf_i$ is isomorphic under $Mf_{i-1}$ to the\ntwo-sided bar construction $B(\\tilde{\\mathbb{C}}MCA, \\tilde{\\mathbb{C}}MA,\nMX_{i-1})$.  This is a proper simplicial spectrum, and since each\nsimplicial level is an extended cell module and the face and\ndegeneracy maps are cellular, so is the bar construction.  By passage\nto colimits, the result follows.\n\nTo see that $\\tilde{\\mathbb{C}}MA$ and $\\tilde{\\mathbb{C}}MCA$ are extended cell\n$S$-modules, we essentially argue as in \\cite[7.7.5]{ekmm} but must\naccount for the quotients since we are using the reduced monads.\nRecall that there is a standard filtration on the reduced monads\n\\cite[7.3.6]{lms}, which allows us to regard the free $\\tilde{\\mathbb{C}}$\nalgebra as the colimit of spectra formed by pushouts of layers of the\nform $Z^j \/ \\Sigma^j$.  These are extended cell $S$-modules, and then\na similar induction as above allows us to conclude the\nresult.\n\\end{proof}\n\nThere is an additional difficulty that arises when working over $BF$;\nit seems to be difficult to replace an arbitrary map of $\\scr{L}$-spaces\n$X \\to BF$ with a map $X' \\to BF$ which is a Hurewicz fibration and\nsuch that $X'$ is cofibrant as an $\\scr{L}$-space.  However, it turns out\nto suffice to work with the following composite replacement --- given\nan arbitrary map of $\\scr{L}$-spaces $X \\to BF$, we work with $\\Gamma\nX' \\to BF$,  where $X'$ is a cofibrant replacement of $X$.  For a\ndetail analysis of this situation, we refer the reader to the\ncompanion paper \\cite{blumberg-cohen-schlichtkrull}, as it depends on\na description of $\\scr{L}$-spaces as commutative monoids with respect to a\nproduct on the category of $\\scr{L}(1)$-spaces defined in analogy with the\nEKMM smash product.\n\n\n\n\\section{Splitting of $THH(Mf)$}\\label{sec:split}\n\nIn the previous section, we have verified that by appropriate\nmodification of the map $f \\colon X \\rightarrow BG$ we can ensure that we\ncan identify $THH(Mf)$ as $M(f \\otimes S^1)$.  In this section, we\nstudy $M(f \\otimes S^1)$.  In particular, we will discuss briefly a\nconnection to the free loop space $LBX$ and then investigate in detail\nthe splitting result $THH(Mf) \\htp Mf \\sma BX_+$. \n\nThe starting point for our analysis is the observation that the based\ncofiber sequence $S^0 \\rightarrow S^1_+ \\rightarrow S^1$ yields an\nassociated sequence of $\\scr{L}$-spaces \n\\[X \\rightarrow X \\otimes S^1_+ \\rightarrow X \\otimes S^1.\\]\nThe map $X \\rightarrow X \\otimes S^1_+$ is split by the collapse map\n$S^1_+ \\rightarrow S^0$, and this induces a map $\\theta : X \\otimes S^1_+\n\\rightarrow X \\times (X \\otimes S^1)$.\n\n\\begin{rem}\nRecall that $X \\otimes S^1_+$ is the realization of the simplicial\nobject $X \\otimes (S^1_+)_\\bullet$ induced by the standard description\nof $S^1_+$ as a simplicial set.  This is in fact a cyclic object, and\ntherefore $X \\otimes S^1_+$ has an action of $S^1$ induced by the\ncyclic structure.  The adjoint of the action map composed with the\nprojection $X \\otimes S^1_+ \\rightarrow X \\otimes S^1$ yields a map $X\n\\otimes S^1_+ \\rightarrow L(X \\otimes S^1)$ which is a weak\nequivalence for group-like $\\scr{L}$-spaces.  When working over a group\n$\\scr{I}$-FCP, this weak equivalence implies a weak equivalence of Thom\nspectra, and so we obtain a description of $THH(Mf)$ in terms of a map\n$L(BX) \\rightarrow BG$.  This relationship is studied in detail in the\ncompanion paper \\cite{blumberg-cohen-schlichtkrull}, and we do not\ndiscuss it further herein.\n\\end{rem}\n\n\\subsection{The splitting arising from an $E_\\infty$-map}\n\nIn this section, we will assume we have a fixed $\\scr{L}$-map $f \\colon X\n\\rightarrow BG$ such that $X$ is a group-like $\\scr{L}$-space and $G$ is a\ngroup $\\scr{I}$-FCP.  We require this latter hypothesis to ensure that all\nmaps that arise are good.\n\n\\begin{lem}\nLet $X$ be a group-like cofibrant $\\scr{L}$-space.  The map $\\theta : X \\otimes\nS^1_+ \\rightarrow X \\otimes S^1 \\times X \\otimes S^0$ is a weak\nequivalence.\n\\end{lem}\n\n\\begin{proof}\nSince $\\scr{L}$ is an $E_\\infty$-operad, we can functorially associate an\n$\\Omega$-prespectrum $Z$ to $X$ using an ``infinite loop space\nmachine''.  We will show that that $X \\otimes A$ is weakly equivalent\nto $\\Omega^\\infty (Z \\sma A)$.  Assuming this fact, the lemma is now a\nconsequence of the stable splitting of $S^1_+$.  Specifically, there\nis a chain of equivalences $Z \\sma S^1_+ \\htp (Z \\sma S^0) \\vee (Z\n\\sma S^1) \\htp (Z \\sma S^0) \\times (Z \\sma S^1)$.  Applying\n$\\Omega^\\infty$ to this composite yields an equivalence $\\Omega^\\infty\n(Z \\sma S^1_+) \\rightarrow (\\Omega^\\infty Z) \\times (\\Omega^\\infty (Z \\sma\nS^1))$, since $\\Omega^\\infty$ preserves products and weak equivalences\nof spectra.  Under the equivalence between $X$ and $\\Omega^\\infty Z$,\nthis map coincides with the map induced from the splitting and so the\nresult follows.\n\nTo compare $X \\otimes A$ and $Z \\sma A$, we use a technique from\n\\cite{basterra-mandell}.  Let $\\tilde{X}$ denote the functor which\nassigns to a finite set $\\underbar{n}$ the tensor $X \\otimes\n\\underbar{n}$.  Using the folding map, this specifies a\n$\\Gamma$-object in $\\scr{L}$-spaces.  Recall that the construction of a\nprespectrum from a $\\Gamma$-object proceeds by prolonging the\n$\\Gamma$-object to a functor from the category of spaces of the\nhomotopy type of finite $CW$-complexes.  Such a functor is called a\n${\\catsymbfont{W}}$-space, and is an example of a diagram spectrum \\cite{mmss}.  In\nthis situation, the associated ${\\catsymbfont{W}}$-space can be specified simply as\n$A \\mapsto X \\otimes A$.  For any ${\\catsymbfont{W}}$-space $Y$ and based space $A$,\nthere is a stable equivalence between the prespectrum $\\{Y(S^n) \\sma\nA\\}$ and the prespectrum $\\{Y(A \\sma S^n)\\}$ induced by the assembly map\n$Y(S^n) \\sma A \\rightarrow Y(A \\sma S^n)$ \\cite[17.6]{mmss}.  Since\n$X$ was a cofibrant group-like $\\scr{L}$-space, $\\tilde{X}$ is very special\n\\cite[6.8]{basterra-mandell}. Therefore the associated ${\\catsymbfont{W}}$-space\n$\\tilde{X}$ is fibrant, which means that the underlying prespectra\n$\\{\\tilde{X}(S^n \\sma A)\\}$ are $\\Omega$-prespectra for all $A$.\nFinally, this implies that there is an equivalence between\n$\\Omega^\\infty (Z \\sma A)$ and $Z(A)$.  A similar result (with a\ndifferent proof) appears in \\cite{schlichtkrull}.\n\\end{proof}\n\n\\begin{prop}\nLet $f \\colon X \\rightarrow BG$ be an $\\scr{L}$-map where $G$ is a group\n$\\scr{I}$-FCP and $X$ is a cofibrant group-like $\\scr{L}$-space.  Then there\nis a weak equivalence of commutative $S$-algebras\n\\[Mf \\otimes S^1 \\simeq BX_+ \\sma Mf.\\]\n\\end{prop}\n\n\\begin{proof}\nBy inspection of the description of the map \n$f \\otimes S^1_+ : X \\otimes S^1_+ \\rightarrow BG$, we see that it can\nbe factored  \n\\[\n\\xymatrix{\nX \\otimes S^1_+ \\ar[r]^-\\theta & (X \\otimes S^0) \\times (X \\otimes S^1)\n\\ar[r]^-{\\pi_1} & X \\otimes S^0 \\cong X \\ar[r]^-f & BG,\n}\n\\] \nwhere $\\pi_1$ is the projection onto the first factor.  By the\npreceding lemma, the hypotheses imply that the map $\\theta : X \\otimes\nS^1_+ \\rightarrow (X \\otimes S^1) \\times (X \\otimes S^0)$ is a weak\nequivalence.  Therefore, there is an equivalence of Thom spectra\n$M\\theta : M(f \\otimes S^1_+) \\rightarrow M(f \\circ \\pi_1)$.  By the\nstandard description of the Thom spectrum of a projection (proposition\n\\ref{P:ten}), we know that $M(f \\circ \\pi_1) \\cong Mf \\sma (X \\otimes\nS^1)_+$.  Moreover, theorem \\ref{commutation} implies that $M(f\n\\otimes S^1_+) \\cong Mf \\otimes S^1$.  Finally, $X \\otimes S^1$ is a\nmodel of $BX$ --- this follows by considering the $\\Gamma$-space\nassociated to $X$ as in the previous lemma\n\\cite[6.5]{basterra-mandell}. \n\\end{proof}\n\n\\subsection{Splitting arising from an $E_2$-map $f \\colon X \\rightarrow BG$}\n\nIt is sometimes the case that even though $f \\colon X \\rightarrow BG$ is\nnot an $E_\\infty$-map, $Mf$ is equivalent to a commutative\n$S$-algebra.  We will consider the situation in which $f \\colon X\n\\rightarrow BG$ is an $E_2$-map such that the there is an equivalence\nof $E_2$-ring spectra from $Mf$ to an $E_\\infty$-ring spectrum.\nAlthough this may seem at first like an artificial hypothesis, in fact\nthis situation arises when considering the Thom spectra that yield\nEilenberg-Mac Lane spectra.  We will show that the splitting result\nholds here as well.\n\nFix an $E_2$-operad ${\\catsymbfont{C}}_2$ which is augmented over the linear\nisometries operad.  Then $BG$ is a ${\\catsymbfont{C}}_2$-space and Lewis' theorem\n\\cite[7.7.1]{lms} shows that the Thom spectrum associated to an\n${\\catsymbfont{C}}_2$-map $f \\colon X \\rightarrow BG$ is a ${\\catsymbfont{C}}_2$-ring spectrum.\n\nRecall that there is a two-sided bar construction for spectra\n\\cite[4.7.2]{ekmm}.  Let $R$ be a commutative $S$-algebras.  If $A$ is\na left $R$-module and $N$ a right $R$-module, the bar construction\n$B(A,R,N)$ is the realization of a simplicial spectrum in which the\n$k$-simplices are given by $A \\sma R^k \\sma N$ and the faces are given\nby the multiplication.  When $R$ is a cofibrant commutative\n$S$-algebra and $A$ is a cofibrant $R$-module, the bar construction is\nnaturally weakly equivalent to $A \\sma_R N$ and weak equivalences in\neach variable induce weak equivalences of bar constructions.\n\n\\begin{rem}\nA simplicial spectrum $K$ is proper if the ``inclusion'' $sK_q\n\\rightarrow K_q$ is a cofibration, where $sK_q$ is the ``union'' of\nthe subspectra $s_j K_{q-1}, 0 \\leq j < q$ \\cite[10.2.2]{ekmm}.  Of\ncourse, the ``union'' denotes an appropriate pushout, and the ``inclusion''\nassociated maps, but the terms are useful to emphasize the analogy\nwith the situation in spaces.  Maps between proper simplicial spectra\nwhich induce levelwise equivalences produce weak equivalences upon\nrealization \\cite[10.2.4]{ekmm}.  When $R$ is a cofibrant commutative\n$S$-algebra and $A$ is a cofibrant $R$-module, the bar construction is\na proper simplicial spectrum. \n\\end{rem}\n\n\\begin{thm}\\label{T:e2split}\nLet $f \\colon X \\rightarrow BG$ be a good ${\\catsymbfont{C}}_2$-map.  Assume that $Mf$ is\nequivalent as a homotopy commutative $S$-algebra to some (strictly)\ncommutative $S$-algebra $M^{\\prime}$.  Then there is an isomorphism in\nthe derived category \n\\[THH(Mf) \\simeq BX_+ \\sma Mf.\\]\n\\end{thm}\n\n\\begin{proof}\n$THH(A)$ can be described as the derived smash product $A\n\\sma^L_{A \\sma A^{\\op}} A$ \\cite[9.1.1]{ekmm}.  Of course if $A$ is\ncommutative, $A \\sma A^{\\op} \\cong A \\sma A$.  In our situation, this\nspecializes to the derived smash product \n\\[THH(Mf) = Mf \\sma^L_{Mf \\sma Mf^{\\op}} Mf.\\]\nIf $Mf$ were a commutative $S$-algebra, we could use the Thom\nisomorphism to replace $Mf \\sma Mf^{\\op} \\cong Mf \\sma Mf$.  We will\nshow that in fact it suffices for $Mf$ to be weakly equivalent to a\ncommutative $S$-algebra.  We can assume without loss of generality\nthat $Mf$ is cofibrant.  Moreover, the hypotheses provide an\nequivalence of $S$-algebras $Mf \\rightarrow M^\\prime$, where\n$M^\\prime$ can be taken to be a cofibrant commutative $S$-algebra.\n\nThe composite \\[Mf^{\\op} \\rightarrow Mf^{\\op} \\sma S_0 \\rightarrow\nMf^{\\op} \\sma X^{\\op}_+ \\rightarrow (M^{\\prime})^{\\op} \\sma X^{\\op}_+\n\\htp M^{\\prime} \\sma X^{\\op}_+\\] is a map of $S$-algebras, and the map\n$M^{\\prime} \\rightarrow M^{\\prime} \\sma S^0 \\rightarrow M^{\\prime}\n\\sma X^{\\op}_+$ is central \\cite[7.1.2]{ekmm}.  Therefore extension of\nscalars yields an induced map of $M^{\\prime}$-algebras $M^{\\prime}\n\\sma Mf^{\\op} \\rightarrow M^{\\prime} \\sma X^{\\op}_+$, and the Thom\nisomorphism theorem implies this map is a weak equivalence.\n\nWe will model the derived smash product using the two-sided bar\nconstruction.  The preceding discussion implies that the composite\n\\[B(Mf, Mf \\sma Mf^{\\op}, Mf) \\rightarrow B(M^\\prime, Mf \\sma Mf^{\\op},\nM^{\\prime}) \\rightarrow B(M^{\\prime}, M^{\\prime} \\sma X^{\\op}_+,\nM^{\\prime})\\] is a weak equivalence.  Therefore we have an isomorphism \n\\[Mf \\sma^L_{Mf \\sma Mf^{\\op}} Mf \\rightarrow M^\\prime\n\\sma^L_{M^\\prime \\sma X^{\\op}_+} M^\\prime.\\]\n\nThe $k$th simplicial level of $B(M^{\\prime}, M^{\\prime} \\sma\nX^{\\op}_+, M^{\\prime})$ is the product \\[M^{\\prime} \\sma (M^{\\prime}\n\\sma X^{\\op}_+)^k \\sma M^{\\prime},\\] where the actions of $M^{\\prime}\n\\sma X^{\\op}_+$ on $M^{\\prime}$ are given by projecting $M^{\\prime}\n\\sma X^{\\op}_+ \\rightarrow M^{\\prime}$ and then using the\nmultiplication on $M^{\\prime}$.  Clearly, there is an isomorphism\n\\[M^{\\prime} \\sma (M^{\\prime} \\sma X^{\\op}_+)^k \\sma M^{\\prime}\n\\rightarrow (M^{\\prime} \\sma (M^{\\prime})^k \\sma M^{\\prime} \\sma\n(X^{\\op}_+)^k\\]\ngiven by permuting the $X^{\\op}_+$ factors to the right, and this map\ncommutes with the simplicial identities.  Thus, there is an equivalence\n\\[B(M^{\\prime},M^{\\prime} \\sma X^{\\op}_+, M^{\\prime}) \\htp\nB(M^{\\prime},M^{\\prime},M^{\\prime}) \\sma B(S, \\Sigma^\\infty X^{\\op}_+,\nS), \\] \nusing the fact that the smash product commutes with realization.\nHowever, since $\\Sigma^\\infty$ commutes with the bar construction for\nmonoids \\cite{ekmm}, we have weak equivalences \\[B(S, \\Sigma^\\infty\nX^{\\op}_+, S) \\htp \\Sigma^\\infty BX^{\\op}_+ \\htp \\Sigma^\\infty BX_+.\\]\nWe also know that $B(M^{\\prime}, M^{\\prime}, M^{\\prime})$ is homotopic\nto $M^{\\prime}$.\n\\end{proof}\n\nNotice that the preceding proof did not require $X$ to be a cofibrant\n$\\scr{L}$-space, and so we can circumvent issues of the interaction of\n$\\Gamma$ and cofibrant replacement in the applications.\n\n\\section{Calculation of $THH(\\ensuremath{\\mathbb{Z}})$, $THH(\\ensuremath{\\mathbb{Z}}\/p)$, and $THH(MU)$}\n\nIn this section, we use the splitting results of the previous section\nto provide easy calculations of $THH$ for various interesting Thom\nspectra.  First, we recover results of Bokstedt for $H\\ensuremath{\\mathbb{Z}}\/p$ and\n$H\\ensuremath{\\mathbb{Z}}$ \\cite{bokstedt2}.  Next, we compute $THH(MU)$, recovering a\ncalculation of McClure and Staffeldt \\cite{mcclure-staffeldt}.\nFurther calculations of bordism spectra are discussed in the companion\npaper \\cite{blumberg-cohen-schlichtkrull}.\n\n\\subsection{$THH(\\ensuremath{\\mathbb{Z}})$ and $THH(\\ensuremath{\\mathbb{Z}}\/p)$}\n\nThere is an identification due to Mahowald of $H\\ensuremath{\\mathbb{Z}}\/2$ as the Thom\nspectrum associated to a certain map $\\Omega^2 S^3 \\rightarrow BO$\n\\cite{cohen-may-taylor, mahowald}.  A modification of this approach\ndue to Hopkins allows the construction of $H\\ensuremath{\\mathbb{Z}}\/p$ as the Thom spectrum\nassociated to a certain $p$-local bundle over $\\Omega^2 S^3$.\nFinally, $H\\ensuremath{\\mathbb{Z}}$ can be obtained as the Thom spectrum of a map $\\Omega^2\nS^3 \\left<3\\right> \\rightarrow BSF$.  We will discuss these\nconstructions in the following section, in particular verifying that\nall of these Thom spectra are $E_2$ ring spectra associated to\n$E_2$ maps structured by the little 2-cubes operad.  Using standard\n``change of operad'' techniques discussed in Appendix~\\ref{appop}, we\ncan functorially convert these to classifying maps structured by an\n$E_2$ operad augmented over the linear isometries operad.\n\nWe have the following proposition, which will allow us to apply\ntheorem \\ref{T:e2split}.\n\n\\begin{prop}\nFor any connective $E_2$-ring spectrum $R$, there is a map of\n$E_2$-ring spectra from $R$ to $H\\pi_0(R)$, unique up to homotopy,\nwhich induces an isomorphism on $\\pi_0$.  Here $H\\pi_0(R)$ is regarded \nas an $E_2$-ring spectrum by forgetting from the commutative\n$S$-algebra structure.\n\\end{prop}\n\nRecall that $THH(HR)$ for $R$ a commutative ring is a product of\nEilenberg-Mac Lane spectra \\cite{bokstedt2}, \\cite[9.1.3]{ekmm}.  This\nimplies that we can read off the homotopy type from the homotopy\ngroups.  Thus to compute $THH(H\\ensuremath{\\mathbb{Z}}\/2)$, we must compute\n$\\pi_*(B(\\Omega^2 S^3) \\sma H\\ensuremath{\\mathbb{Z}}\/2)$.  This is just the homology of\n$\\Omega S^3$ with $\\ensuremath{\\mathbb{Z}}\/2$ coefficients, which can be easily calculated\nvia inspection of the James construction.  One easily recovers the\nresult\n\\[THH(H\\ensuremath{\\mathbb{Z}}\/2) = \\prod_{i=0}^{\\infty} K(\\ensuremath{\\mathbb{Z}}\/2, 2i).\\] \nA similar argument applies to $THH(H\\ensuremath{\\mathbb{Z}}\/p)$.\n\nFinally, to compute $THH(H\\ensuremath{\\mathbb{Z}})$, we must compute $\\pi_*(B(\\Omega^2\nS^3\\left<3\\right>) \\sma H\\ensuremath{\\mathbb{Z}})$.  Once more, this is just the ordinary\nhomology with integral coefficients of $\\Omega S^3\\left<3\\right>$.\nComputing again, we find \n\\[THH(H\\ensuremath{\\mathbb{Z}}) = K(\\ensuremath{\\mathbb{Z}},0) \\times \\prod_{i=1}^\\infty K(\\ensuremath{\\mathbb{Z}}\/i, 2i-1).\\]\n\n\\subsection{THH(MU)}\n\nThe splitting formula implies that \n\\[THH(MU) \\htp MU \\sma BBU_+ \\htp MU \\sma SU_+.\\]\nWe can compute $MU_*(SU)$ via a standard Atiyah-Hirzebruch spectral\nsequence calculation, and it turns out to be $MU_*(pt) \\otimes\n\\Lambda(x_1, x_2, \\ldots)$, with the generators in odd degrees.  This\nagrees with the answer obtained by McClure and Staffeldt\n\\cite{mcclure-staffeldt}, and as they observe implies that $THH(MU)$\nis a product of suspensions of $MU$.  Other bordism spectra are\nanalogous; see the companion paper \\cite{blumberg-cohen-schlichtkrull}\nfor further discussion.\n\n\\section{Realizing Eilenberg-Mac Lane spectra as Thom spectra}\\label{S:TEM}\n\nIn this section, we review and extend the classical realizations of\nEilenberg-Mac Lane spectra as Thom spectra associated to certain\nbundles over $\\Omega^2 S^3$ and $\\Omega^2 S^3 \\left<3\\right>$.  Our\nmain purpose is to ensure that we can obtain these Thom spectra as\nring spectra which are sufficiently structured so as to permit the\nconstruction of $THH$ and the application of our splitting theorem.\nIn particular, improving on \\cite{cohen-may-taylor} we give a new\ndescription of $H\\ensuremath{\\mathbb{Z}}$, based on a suggestion of Mike Mandell, as the\nThom spectrum associated to a double loop map $\\Omega^2\nS^3\\left<3\\right> \\rightarrow BSF$.\n\n\\subsection{$H\\ensuremath{\\mathbb{Z}}\/2$ as the Thom spectrum of a double loop map}\n\nThe construction of $H\\ensuremath{\\mathbb{Z}}\/2$ as a Thom spectrum was the first to be\nextensively studied \\cite{cohen-may-taylor, mahowald, priddy}.  We\nbriefly review the construction.  Consider the map $\\psi \\colon S^1\n\\rightarrow BO$ representing the nontrivial element of $\\pi_1(BO)$.\nThe Thom spectrum associated to this map is the Moore spectrum\n$M\\ensuremath{\\mathbb{Z}}\/2$.  There is an induced map $\\gamma \\colon \\Omega^2 S^3 \\rightarrow\nBO$, as $BO$ is an infinite loop space (and in particular a double\nloop space).  The Thom spectrum of $\\gamma$ is $H\\ensuremath{\\mathbb{Z}}\/2$.\n\nA sketch of the proof for this is as follows.  There is a map ${\\catsymbfont{A}}\n\\rightarrow H^*(M\\gamma;\\ensuremath{\\mathbb{Z}}\/2)$ given by evaluation on the Thom class\nwhich is a map of modules over the Steenrod algebra.  As $M\\gamma$ is\n$2$-local, it suffices to show that this map is an isomorphism.\nDualizing, we can consider the corresponding map $H_*(M\\gamma; \\ensuremath{\\mathbb{Z}}\/2)\n\\rightarrow {\\catsymbfont{A}}^*$ of comodules over the dual Steenrod algebra\n${\\catsymbfont{A}}^*$.  Next, by the Thom isomorphism we know that\n$H_*(M\\gamma;\\ensuremath{\\mathbb{Z}}\/2) \\cong H_*(\\Omega^2 S^3; \\ensuremath{\\mathbb{Z}}\/2)$.  The homology of\n$\\Omega^2 S^3$ is $P\\{x_n \\mid n \\geq 0\\}$, where $x_0$ comes from the\ninclusion of $H_*(S^1; \\ensuremath{\\mathbb{Z}}\/2)$ and the action of the Dyer-Lashof\noperations is known \\cite{cohen-may-taylor} --- specifically, $x_0$\ngenerates the homology as a module over the Dyer-Lashof algebra.  Now,\nnote that since the dimensions of ${\\catsymbfont{A}}$ and $H^*(\\Omega^2 S^3; \\ensuremath{\\mathbb{Z}}\/2)$\nare the same, it is enough to show that the evaluation map is either\nan injection or a surjection.  \n\nThere are a variety of arguments to establish this fact; we will\nreview the technique used by \\cite{priddy}.  First, we observe that\nboth the Thom isomorphism and the map $\\gamma_* \\colon H_*(\\Omega^2 S^3; \\ensuremath{\\mathbb{Z}}\/2)\n\\rightarrow H_*(BO; \\ensuremath{\\mathbb{Z}}\/2)$ commute with the Dyer-Lashof operations.\nRecall that $H_*(BO; \\ensuremath{\\mathbb{Z}}\/2)$ is generated by the images of the class in\ndegree 1 under the first Dyer-Lashof operation.  Therefore the\nbehavior of $\\gamma_*$ is completely determined by the fact that\n$\\gamma_* (x_0)$ is that generating class in degree 1.  Finally, we\nnote that under the evaluation map $H_*(MO; \\ensuremath{\\mathbb{Z}}\/2) \\rightarrow {\\catsymbfont{A}}$ the\nimages of the iterates of $\\gamma_*(x_0)$ under the Dyer-Lashof\noperation hit all of the generators of ${\\catsymbfont{A}}$.\n\n\\subsection{$H\\ensuremath{\\mathbb{Z}}\/p$ as the Thom spectrum of a double loop map}\n\nUnfortunately, no stable spherical fibration can have $H\\ensuremath{\\mathbb{Z}}\/p$ as its\nassociated Thom spectrum --- $\\pi_0(Mf)$ is either $\\ensuremath{\\mathbb{Z}}$ or $\\ensuremath{\\mathbb{Z}}\/2$,\ndepending on whether $f$ represents an orientable bundle or not.\nNonetheless, in \\cite{thomified} there is a brief discussion of an\nargument due to Hopkins for realizing $H\\ensuremath{\\mathbb{Z}}\/p$ as the Thom spectrum\nassociated to a $p$-local stable spherical fibration.\n\nIn the bulk of this paper, we studied Thom spectra associated to\nmonoid $\\scr{I}$-FCP's which were augmented over $F$.  The map to $X\n\\rightarrow F$ was used to give an action of $X(V)$ on the sphere\n$S^V$, the fiber of the universal quasifibration $B(*,X(V),S^V)\n\\rightarrow B(*,X(V),*)$.  However, as we noted previously, this\ntheory can be carried out with other choices of fiber, in particular\nthe collection of $p$-local spheres $S^V_{(p)}$ or $p$-complete\nspheres $(S^V)^{\\scriptscriptstyle\\wedge}_{p}$.  Rather than an augmentation over $F$, we will\nin this setting require augmentation over the appropriate ``p-local''\nor ``p-complete'' analogue.  We rely on the careful treatment of\nfiberwise localization and completion given by May\n\\cite{may-fiberloc}.\n\n\\begin{defn}\n\\hspace{5 pt}\n\\begin{enumerate}\n\\item{Let $F_{(p)}$ denote the monoid $\\scr{I}$-FCP specified by taking\n  $V$ to the based homotopy self-equivalences of $S^V_{(p)}$.  Denote\n  by $BF_{(p)}$ the $\\scr{I}$-FCP obtained by passing to classifying\n  spaces levelwise.}\n\\item{Let $(F)^{\\scriptscriptstyle\\wedge}_{p}$ denote the $\\scr{I}$-FCP specified by taking $V$ to\n  the based homotopy self-equivalences of $(S^V)^{\\scriptscriptstyle\\wedge}_{p}$.  Denote by\n  $B(F)^{\\scriptscriptstyle\\wedge}_{p}$ the $\\scr{I}$-FCP obtained by passing to classifying spaces\n  levelwise.}\n\\end{enumerate}\n\\end{defn}\n\n$BF_{(p)}(V)$ classifies spherical fibrations with\nfiber $S^V_{(p)}$ and $B(F)^{\\scriptscriptstyle\\wedge}_{p}(V)$ classifies spherical fibrations\nwith fiber $(S^V)^{\\scriptscriptstyle\\wedge}_{p}$ \\cite{may-fiberloc}.  Note that we must use\ncontinuous versions of localization and completion in order to ensure\nwe have continuous functors \\cite{iwase}.\n\n\\begin{rem}\nThe notation we are using is potentially confusing, as the spaces\n$BF_{(p)}(V)$ are not the p-localizations of $BF(V)$ and the spaces\n$B(F)^{\\scriptscriptstyle\\wedge}_{p}$ are not the p-completions of $BF(V)$.  Such equivalences\nare only true after passage to universal covers, as there is an\nevident difference at $\\pi_1$.\n\\end{rem}\n\nIn this setting, we can set up the theory of Thom spectra as discussed\nin previous sections of the paper with minimal modifications.  For\noriented bundles, there is a Thom isomorphism with $\\ensuremath{\\mathbb{Z}}_{(p)}$ or\n$Z^{\\scriptscriptstyle\\wedge}_{p}$ respectively and for unoriented bundles there is a\n$\\ensuremath{\\mathbb{Z}}\/p$ Thom isomorphism \\cite{may-fiberloc}.\n\nNow, $\\pi_1(BF_{(p)})$ is the group of $p$-local units $\\ensuremath{\\mathbb{Z}}_{p}^{\\times}$.\nConsider a map $\\phi \\colon S^1 \\rightarrow BF_{p}$ associated to a\nchoice of unit $u$.  The Thom spectrum associated to\n$\\phi$ is the Moore spectrum obtained as the cofiber of the map $S_{p}\n\\rightarrow S_{p}$ given by multiplication by $u-1$.  This\nidentification follows immediately from the general description of the\nThom spectrum of a bundle over a suspension \\cite[9.3.8]{lms}.  Taking\n$u = p+1$, which is a $p$-local unit, we obtain the Moore spectrum\n$M(\\ensuremath{\\mathbb{Z}}\/p)$.  As before, there is an induced map $\\gamma \\colon \\Omega^2 S^3\n\\rightarrow BF_{(p)}$ since $BF_{(p)}$ in an infinite loop space.  \n\nWe will show that the Thom spectrum associated to this map is $H\\ensuremath{\\mathbb{Z}}\/p$.\nOnce again, the Thom class specifies a map ${\\catsymbfont{A}}_{p} \\rightarrow\nH^*(M\\gamma)$ of modules over the Steenrod algebra.  For odd $p$,\n$H_*(\\Omega^2 S^3;\\ensuremath{\\mathbb{Z}}\/p) = E\\{x_n \\mid n \\geq 0\\} \\otimes P\\{\\beta x_n\n\\mid n \\geq 1\\}$, where $x_0$ comes from the inclusion of $H_*(S^1;\n\\ensuremath{\\mathbb{Z}}\/p)$, and is generated as a module over the Dyer-Lashof algebra by\n$x_0$ \\cite{cohen-may-taylor}.  Again, note that since the dimensions\nof ${\\catsymbfont{A}}$ and $H^*(\\Omega^2 S^3; \\ensuremath{\\mathbb{Z}}\/p)$ are the same, it is enough to\nshow that the evaluation map is either an injection or a surjection.\nThis can be shown by an argument analogous to the one described for\n$p=2$ above.\n\n\\subsection{$H\\ensuremath{\\mathbb{Z}}$ as the Thom spectrum of a double loop map}\n\nFinally, we consider the case of $H\\ensuremath{\\mathbb{Z}}$.  It has long been known that\n$H\\ensuremath{\\mathbb{Z}}$ arises as the Thom spectrum associated to a certain map $\\gamma\n\\colon \\Omega^2 (S^3\\left<3\\right>) \\rightarrow BSF$\n\\cite{cohen-may-taylor, mahowald}.  However, the best published\nresults obtain a description of this map as an $H$-map\n\\cite{cohen-may-taylor}, which is inadequate for construction of\n$THH$.  Moreover, it is not clear how to adapt the existing\nconstruction to improve this --- the map $\\gamma$ is constructed a\nprime at a time, and the localized maps $\\gamma_p$ are seen to be\n$H$-maps because certain obstructions vanish.\n\nTherefore, we give a new construction, based on a suggestion of Mike\nMandell, which enables us to see that there is a suitable map which is\na double loop map.  Both \n$\\Omega^2 S^3 \\left<3\\right>$ and $BSF$ are rationally trivial, and so \nsplit as the product of their completions.  Therefore a map $\\Omega^2\nS^3 \\left<3 \\right> \\rightarrow BSF$ can be specified by the\nconstruction of a collection of maps $\\Omega^2 S^3 \\left<3 \\right> \\rightarrow\n(BSF)^{\\scriptscriptstyle\\wedge}_{p}$.  Note that the $p$-completion of $BSF$ is weakly\nequivalent to $\\colim_V B((SF)^{\\scriptscriptstyle\\wedge}_{p})$, where $(SF)^{\\scriptscriptstyle\\wedge}_{p}$ is the\nmonoid $\\scr{I}$-FCP constructed analogously to $(F)^{\\scriptscriptstyle\\wedge}_{p}$.  The\nfollowing lemma is standard.\n\n\\begin{lem}\nLet $f \\colon \\Omega^2 S^3 \\left<3\\right> \\rightarrow BSF$ be a map\nspecified by a collection of maps $f_p \\colon \\Omega^2 S^3 \\left<3 \\right>\n\\rightarrow (BSF)^{\\scriptscriptstyle\\wedge}_{p}$.  If each $f_p$ is an $n$-fold loop map, then\n$f$ is an $n$-fold loop map.\n\\end{lem}\n\nNext, we observe that it will suffice to show that at each prime, the\nmap given by evaluation on the Thom class induces an equivalence\nbetween the Thom spectrum associated to $\\Omega^2 S^3 \\left<3\n\\right> \\rightarrow B(SF)^{\\scriptscriptstyle\\wedge}_{p}$ and $H\\ensuremath{\\mathbb{Z}}^{\\scriptscriptstyle\\wedge}_{p}$.  The Thom class\nclearly induces an equivalence in integral homology.  Therefore, if\nthe evaluation map induces an equivalence in $\\ensuremath{\\mathbb{Z}}\/p$ cohomology for\neach $p$, by naturality it must induce a stable equivalence of\nspectra.\n\nFor $p = 2$, we can use the map induced by the composite \n\\[\\Omega^2 S^3 \\left<3\\right> \\rightarrow \\Omega^2 S^3 \\rightarrow BO\n\\rightarrow B(SF)^{\\scriptscriptstyle\\wedge}_{p}.\\]\nThis is a double loop map, and the associated Thom spectrum is\n$H\\ensuremath{\\mathbb{Z}}^{\\scriptscriptstyle\\wedge}_{2}$ \\cite{cohen-may-taylor}.  For odd primes, we proceed\nas follows.  We know that $\\pi_1(B(F)^{\\scriptscriptstyle\\wedge}_{p})$ is the group of $p$-adic\nunits $(\\ensuremath{\\mathbb{Z}}^{\\scriptscriptstyle\\wedge}_{p})^\\times$.  Explicitly, for odd primes this is\n$(\\ensuremath{\\mathbb{Z}}^{\\scriptscriptstyle\\wedge}_{p})^\\times \\cong \\ensuremath{\\mathbb{Z}}\/(p-1) \\oplus \\ensuremath{\\mathbb{Z}}^{\\scriptscriptstyle\\wedge}_{p}$.\nTake a map $\\phi$ representing an element of $\\pi_1(B(F)^{\\scriptscriptstyle\\wedge}_{p})$\nwhich is $0$ on the $\\ensuremath{\\mathbb{Z}}\/(p-1)$ factor and induces an isomorphism on\nthe other component.  We can equivalently regard $\\phi$ as a map\n$\\phi \\colon S^3 \\rightarrow B^3(F)^{\\scriptscriptstyle\\wedge}_{p}$.  Now, we can lift to a map\n$S^3\\left<3\\right> \\rightarrow B^3 (SF)^{\\scriptscriptstyle\\wedge}_{p}$.  Since $\\phi$ is\ntrivial on the $\\ensuremath{\\mathbb{Z}}\/(p-1)$ component of $\\pi_3(B^3(F)^{\\scriptscriptstyle\\wedge}_{p})$, we can\nlift the map to the fiber over the map $B^3(F)^{\\scriptscriptstyle\\wedge}_{p} \\rightarrow\nK(\\ensuremath{\\mathbb{Z}}\/(p-1),3)$.  The induced map is an isomorphism on $\\pi_3$ by\nconstruction, and so now we can pass to fibers over $K((Z)^{\\scriptscriptstyle\\wedge}_{p},3)$\nto obtain the desired map.  Looping twice, denote by $\\gamma$ the\nresulting map $\\Omega^2 S^3 \\rightarrow B(F)^{\\scriptscriptstyle\\wedge}_{p}$ and\n$\\gamma^\\prime$ the resulting map $\\Omega^2 S^3\\left<3\\right>\n\\rightarrow BS(F)^{\\scriptscriptstyle\\wedge}_{p}$.\n\nFirst, let us identify the Thom spectrum $M\\gamma$.  This proceeds\nessentially as in the previous examples.  Specifically, the Thom\nspectrum associated to the map $\\phi$ is the Moore spectrum obtained\nas the cofiber of the map which is multiplication by $u - 1$, where\n$u$ is the chosen $p$-adic unit.  This Moore spectrum is determined by\nthe $p$-adic valuation of $u-1$.  To compute this, let us recall the\nidentification of the $p$-adic units.  A unit in $(Z)^{\\scriptscriptstyle\\wedge}_{p}$ is a\n$p$-adic integer with an expansion such that the first digit is\nnonzero.  The projection onto the units of $\\ensuremath{\\mathbb{Z}}\/p$ induces the first\ncomponent of the identification.  In our case, we are requiring a\nchoice where the first component is $1$.  Subtracting $1$ from this,\nwe find that the first component must be $0$ and the later components\nare arbitrary.  Combining with the constraint that the projection of\n$u$ generates the $(Z)^{\\scriptscriptstyle\\wedge}_{p}$, we find that we have the Moore spectra\n$M(\\ensuremath{\\mathbb{Z}}\/p)$.  A similar argument to the the one employed above implies\nthat $M\\gamma$ is $H\\ensuremath{\\mathbb{Z}}\/p$.\n\nFinally, we will use this identification to determine the Thom\nspectrum $M\\gamma^{\\prime}$.  Let us first consider the case of $p$ an\nodd prime.  Essentially by construction, there is a commutative\ndiagram of Thom spectra\n\\[\n\\xymatrix{\nMf \\ar[r] \\ar[d] & M((SF)^{\\scriptscriptstyle\\wedge}_{p}) \\ar[d] \\\\\nH\\ensuremath{\\mathbb{Z}}\/p \\ar[r] & M((F)^{\\scriptscriptstyle\\wedge}_{p}) \\\\\n}\n\\]\nassociated to the commutative diagram of spaces\n\\[\n\\xymatrix{\n\\Omega^2 S^3\\left<3\\right> \\ar[r]\\ar[d] & \\ar[d] B((SF)^{\\scriptscriptstyle\\wedge}_{p}) \\\\\n\\Omega^2 S^3 \\ar[r] & B((F)^{\\scriptscriptstyle\\wedge}_{p}). \\\\\n}\n\\]\n\nBy the naturality of the Thom isomorphism, this implies that we have a\ncommutative diagram of modules over the Steenrod algebra\n\\[\n\\xymatrix{\n{\\catsymbfont{A}} \\ar[r] \\ar[d] & H^*(M\\gamma) \\ar[d] \\\\\n{\\catsymbfont{A}}\/ \\beta {\\catsymbfont{A}} \\ar[r] & H^*(M\\gamma^\\prime) \\\\\n}\n\\]\n\nThe map ${\\catsymbfont{A}} \\rightarrow {\\catsymbfont{A}} \/ \\beta {\\catsymbfont{A}}$ is a surjection, we have\nseen that the map ${\\catsymbfont{A}} \\rightarrow H^*(M\\gamma)$ is an isomorphism,\nand $\\Omega^2 S^3 \\left<3\\right> \\rightarrow \\Omega^2 S^3$ induces a\nsurjection on cohomology (and on homology a map of comodules over the\ndual Steenrod algebra).  This implies that the top horizontal map must\nbe a surjection.  Since the dimension of ${\\catsymbfont{A}} \/ \\beta {\\catsymbfont{A}}$ and\n$H^*(\\Omega^2 S^3 \\left<3\\right>; \\ensuremath{\\mathbb{Z}}\/p)$ are the same, this map must\nin fact be an isomorphism.  \n\n\\begin{rem}\nIf we work at the prime $2$, we have that $\\pi_1$ is\n$(\\ensuremath{\\mathbb{Z}}^{\\scriptscriptstyle\\wedge}_{2})^\\times = \\ensuremath{\\mathbb{Z}}\/2 \\oplus \\ensuremath{\\mathbb{Z}}^{\\scriptscriptstyle\\wedge}_{2}$.  Following the outline\nabove, we would like to identify the Thom spectrum associated to\n$\\phi$.  The projection onto the units of $\\ensuremath{\\mathbb{Z}}\/4$ induces the first\ncomponent of the identification of $(\\ensuremath{\\mathbb{Z}}^{\\scriptscriptstyle\\wedge}_{2})^\\times$.  The two\nchoices are expansions which begin $1, 1, \\ldots$ and $1, 0, \\ldots$.\nSince we want something which projects to $0$, we must have the\nlatter.  Subtracting $1$ from this, we find we end up with a $p$-adic\nnumber which begins $0, 0, \\ldots$ and therefore has $p$-adic\nvaluation $2$ or higher.  Therefore the associated Thom spectrum is\nthe Moore spectrum $M(\\ensuremath{\\mathbb{Z}}\/4)$.\n\nHowever, consideration of the Dyer-Lashof operations tells us that the\nThom spectrum of $\\gamma$ is not $H(\\ensuremath{\\mathbb{Z}}\/4)$.  In general, we cannot\nobtain $H(\\ensuremath{\\mathbb{Z}}\/p^n)$ as a Thom spectrum over $\\Omega^2 S^3$.  This can\nbe seen by considering the element $x_0$ in $H_1(\\Omega^2 S^3)$.  The\nlast Dyer-Lashof operation takes this to $Q_2 x_0$, but since the\nclassifying map takes $x_0$ to $0$ it must take $Q_2 x_0$ to zero and\nthus must be $0$ on $H^3$ as well, which implies that the Thom\nspectrum cannot be the Eilenberg-Mac Lane spectrum.  It is also\npossible to deduce the impossibility of realizing $H(\\ensuremath{\\mathbb{Z}}\/p^n)$ as such\na Thom spectrum by observing that the computations of \\cite{brun} are\nincompatible with our splitting results.\n\\end{rem}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n \nTransit events offer the opportunity to characterise the atmospheres of extrasolar planets. During a transit, a fraction of the star light shines through the outmost layers of the planetary atmosphere. Depending on the opacity of its chemical constitution, the star light transmits or becomes scattered or absorbed. Therefore, the altitude of optical depth equaling unity (effective planetary radius) is a function of wavelength and depends on the atmospheric composition. The measurement of the effective planetary radius as a function of wavelength is called transmission spectroscopy and is often accomplished by spectrophotometric observations of the transit event. The observer obtains photometric transit light curves at multiple wavelengths, either simultaneously by low-resolution spectroscopic observations \\citep[e.\\,g.,][]{Bean2010,Gibson2013,MallonnH19} or  in different broad-band filters both simultaneously \\citep[e.\\,g.,][]{Nascimbeni2013,Mancini2013} or at multiple transit epochs \\citep[e.\\,g.,][]{\ndeMooij2012,Dragomir2015,MallonnH12}. \n\nA value of the planetary radius in relation to the (wavelength-independent) stellar radius is derived by a model fit to the transit light curve. This method was successfully applied to measure the spectroscopic absorption feature of, e.\\,g., sodium, potassium, and water \\citep{Sing2011b,Huitson2013,Nikolov2014}. In other target spectra the spectral features predicted by cloud-free models could be ruled out \\citep[e.\\,g.,][]{Gibson2013,Pont2013,MallonnH19,Lendl2016}. These spectra are either simply flat in the probed wavelength region, or show a trend of increasing opacity toward blue wavelengths explainable by Rayleigh- or Mie-scattering \\citep{Jordan2013,Stevenson2014}. Interestingly, at wavelengths shorter than about 500~nm currently all gas giants on close orbits (the so-called hot Jupiters) seem to show an increase in opacity when measured with sufficient precision \\citep[uncertainty of the effective planetary size about one scale height,][]{Sing2016}. \n\nOne target of special interest is the hot Jupiter HAT-P-32\\,b. It was discovered by \\cite{Hartman2011} and is one of the best targets for transmission spectroscopy because of its large transit depth of more than two percent, its large planetary scale height of about 1000~km, and a relatively bright host star (V\\,=\\,11.4~mag). HAT-P-32\\,b's atmospheric transmission spectrum lacks the predicted cloud-free absorption of Na, K and H$_2$O \\citep[assuming solar composition,][hereafter G13]{GibsonH32}. This result was recently confirmed by transmission spectroscopy of \\cite{Nortmann2016} (hereafter N16). However, \\cite{MS2016} (hereafter Paper~I) found indications for a slope of increasing effective planetary radius toward the blue by measuring the spectrum from 330 to 1000~nm. The amplitude of this increase of only two atmospheric pressure scale heights is small compared to other hot Jupiters with measured scattering signatures, e.\\,g. HD189733\\,b \\citep{Sing2011,Pont2013}. In this work, we attempt a verification \nof this blueward increase of effective planetary radius of HAT-P-32 by multi-epoch and multi-colour observations using broad-band photometry. \n\nWhile in principle the broad-band filters allow for low photon-noise in the photometry even with meter-sized telescopes, it has proven to be a demanding task to reach sufficient precision to discriminate between different models \\citep[e.\\,g.,][]{Teske2013,Fukui2013,MallonnH12}. Furthermore, for single observations per filter a potential effect of correlated noise is not always obvious \\citep[e.\\,g.,][]{Southworth2012}. One way to lower such potential effect is to observe multiple transit light curves per filter under the assumption that the correlated noise does not repeat because observing conditions change from night to night for ground-based observations \\citep{Lendl2013,MallonnH12}. Therefore, we collected the largest sample of light curves analysed for broad-band spectrophotometry of an exoplanet so far. We present 21 new observations and analyse them homogeneously together with 36 already published light curves. Section \\ref{chap_obs} gives an overview about the observations and data reduction and \nSection \\ref{chap_anal} describes the analysis. The results are presented in Section \\ref{chap_res} and discussed in Section \\ref{chap_disc}. The conclusions follow in Section \\ref{chap_concl}.\n\n\\section{Observations and data reduction}\n\\label{chap_obs}\nTransit light curves were taken with the robotic 1.2\\,m STELLA telescope, with the 2.5\\,m Nordic Optical Telescope (NOT), the 0.8\\,m IAC80 telescope, the 2.2\\,m telescope of Calar Alto, the 70\\,cm telescope of the Leibniz Institute for Astrophysics Potsdam (AIP), and the 4.2\\,m William Herschel Telescope (WHT).\n\nSTELLA and its wide field imager WiFSIP \\citep{Strassmeier2004,Weber2012} observed in total 11 transits in five observing seasons using the filters Johnson B and Sloan r'. The first seven transits were observed with both filters alternating. These r' band light curves were already published by \\cite{Seeliger2014}. The four light curves from 2014 to 2016 were taken only in Johnson B. WiFSIP holds a back-illuminated 4k\\,$\\times$\\,4k 15~$\\mu$m pixel CCD and offers a field of view (FoV) of 22$'\\times$22$'$. To minimise the read-out time, a windowing of the CCD was used reducing the FoV to about 15$'\\times$15$'$. A small defocus was applied to spread the PSF to an artificial FWHM of about 3$''$.\n\nOne transit was observed with the NOT as a Fast Track program using ALFOSC in imaging mode. ALFOSC contains a 2k\\,$\\times$\\,2k E2V CCD providing a FoV of 6.4$'\\times$6.4$'$. \n\nOne transit was observed with BUSCA, the four-channel imager at the Calar Alto Observatory 2.2\\,m telescope \\citep{Reif1999}. The instrument performs simultaneous photometry in four different bandpasses with a FoV of 11$'\\times$11$'$. For the bandpass of shortest wavelength we used a white glass filter and the beam splitter defined the limit of $\\lambda < 430$~nm. For the other bandpasses from blue to red we used a Thuan-Gunn~g, Thuan-Gunn~r, and Bessel I filter. Unfortunately, the observation of a pre-ingress baseline was lost due to weather, and the observing conditions remained to be non-photometric.\nWe discarded the light curve of shortest wavelength because it exhibited significantly larger correlated noise than the other three light curves.\n\nOne transit was observed with the IAC80 telescope, owned and operated by the Instituto de Astrof\\'{\\i}sica de Canarias (IAC), using its wide field imager CAMELOT. It is equipped with a CCD-E2V detector of 2k\\,$\\times$\\,2k, with a pixel scale of about 0.3$''$\/pixel providing a FoV of about 10.5\\,$'\\times$\\,10.5\\,$'$.\n\nTwo transits were observed with the 70\\,cm telescope of the AIP, located in the city of Potsdam at the Babelsberg Observatory. The telescope is equipped with a cryogenic cooled 1k\\,$\\times$\\,1k TEK-CCD providing a FoV of 8$'\\times$8$'$. The first transit was observed in Johnson B and Johnson V quasi-simultaneously with alternating filters, the second one in Johnson V only. We applied a 3$\\times$3 pixel binning to reduce the detector read-out time.\n\nData of an additional transit were taken with the triple beam, frame-transfer CCD camera ULTRACAM \\citep{Dhillon2007} mounted at the WHT. The instrument optics allow for the simultaneous photometry in three different bandpasses. We chose the filters Sloan u', Sloan g' and a filter centred on the sodium doublet at 591.2~nm with a width (FWHM) of 31.2~nm. To avoid saturation of the brightest stars, the exposure time was extremely short with 0.3~s. However, the overheads are negligible owing to the frame-transfer technique. The detectors were read out in windowed mode. For the u' channel, ten exposures were co-added on chip before read-out. In the analysis, we used the light curves binned in time. The g' band light curve showed correlated noise of $\\sim$\\,3~mmag on short time scales and was excluded from the analysis.\n\nWe extended our sample of broad-band transit data by the five light curves published by \\cite{Hartman2011} and 26 light curves published by \\cite{Seeliger2014}. Furthermore, we searched the Exoplanet Transit Database\\footnote{http:\/\/var2.astro.cz\/ETD\/} \\citep[ETD,][]{Poddany2010} for available amateur observations of sufficient quality to be helpful in the recent work. ETD performs online a simple transit fit, and we used its derived transit depth uncertainty and a visual inspection for selecting five light curves for our purposes. The characteristics of all light curves are summarised in Table \\ref{tab_overview}.\n\nThe data reduction of all new light curves except the ULTRACAM data was done as described in our previous work on HAT-P-12\\,b \\citep{MallonnH12}. Bias and flat-field correction was done in the standard way, with the bias value extracted from the overscan regions. We performed aperture photometry with the publicly available software SExtractor using the option of a fixed circular aperture MAG\\_APER. The set of comparison stars (flux sum) was chosen to minimise the root mean square (rms) of the light curve residuals after subtraction of a second order polynomial over time plus transit model using literature transit parameter. Using the same criterion of a minimised rms, we also determined and applied the best aperture width. A significant fraction of our light curves nearly reached photon-noise limited precision. The transit light curves of 2012 Oct 12, observed simultaneously with the NOT and WHT, suffered from clouds moving through. We discarded all data points with flux levels below 50\\% of its mean.\n\nThe ULTRACAM data were reduced in the same way as described in Kirk et al. (in prep), using the ULTRACAM data reduction pipeline\\footnote{http:\/\/deneb.astro.warwick.ac.uk\/phsaap\/\\\\software\/ultracam\/html\/index.html} with bias frames and flatfields used in the standard way. Again, the aperture size used to perform aperture photometry was optimised to deliver the most stable photometry. Only two useful comparison stars were observed in the FoV, and a simple flux sum of both as reference was found to give the differential light curve of lowest scatter. \n\nThe 21 new light curves are shown in Figure \\ref{plot_lcs}, and the 36 re-analysed literature light curves are presented in Figure \\ref{plot_lc_lit}.\n\n\\begin{table*}\n\\small\n\\caption{Overview of the analysed transit observations of HAT-P-32\\,b. The columns give the observing date, the telescope used, the chosen filter, the airmass range of the observation, the exposure time, the observing cadence, the number of observed data points in-transit, the number of observed data points out-of-transit, the dispersion of the data points as root mean square (rms) of the observations after subtracting a transit model and a detrending function, the $\\beta$ factor (see Section \\ref{chap_anal}), and the reference of the light curves.}\n\\label{tab_overview}\n\\begin{center}\n\\begin{tabular}{p{19mm}p{24mm}p{10mm}p{9mm}p{9mm}p{9mm}p{9mm}p{8mm}p{8mm}l}\n\\hline\n\\hline\n\\noalign{\\smallskip}\n\nDate        &   Telescope   &   Filter  &   $t_{\\mathrm{exp}}$ (s) &  Cadence (s) & $N_{\\mathrm{it}}$ & $N_{\\mathrm{oot}}$ & rms (mmag)  &  $\\beta$ & Reference \\\\\n\\hline\n\\noalign{\\smallskip}\n\n2007 Sep 24 &   FLWO 1.2\\,m        & z'     &           &   33      & 330  & 273  &   1.93 &  1.13 &  \\cite{Hartman2011}  \\\\\n2007 Oct 22 &   FLWO 1.2\\,m        & z'     &           &   53      & 205  & 284  &   2.51 &  1.00 &  \\cite{Hartman2011}  \\\\\n2007 Nov 06 &   FLWO 1.2\\,m        & z'     &           &   28      & 386  & 373  &   2.06 &  1.00 &  \\cite{Hartman2011}  \\\\\n2007 Nov 19 &   FLWO 1.2\\,m        & z'     &           &   38      & 292  & 373  &   1.68 &  1.27 &  \\cite{Hartman2011}  \\\\\n2007 Dec 04 &   FLWO 1.2\\,m        & g'     &           &   33      & 322  & 274  &   1.85 &  1.00 &  \\cite{Hartman2011}  \\\\                                            \n2011 Nov 01 &   STELLA             & B      &    30     &   115     & 95   & 45   &   1.96 &  1.04 &  this work           \\\\\n2011 Nov 01 &   STELLA             & r'     &    15     &   115     & 92   & 41   &   1.90 &  1.00 &  \\cite{Seeliger2014} \\\\\n2011 Nov 14 &   Jena 0.6\/0.9\\,m    & R      &    40     &   63      & 174  & 126  &   1.76 &  1.04 &  \\cite{Seeliger2014} \\\\\n2011 Nov 29 &   STELLA             & B      &    30     &   115     & 94   & 22   &   1.21 &  1.00 &  this work           \\\\\n2011 Nov 29 &   STELLA             & r'     &    15     &   115     & 95   & 21   &   1.30 &  1.01 &  \\cite{Seeliger2014} \\\\\n2011 Nov 29 &   Babelsberg 70\\,cm  & B      &    30     &   75      & 149  & 224  &   2.65 &  1.00 &  this work           \\\\\n2011 Nov 29 &   Babelsberg 70\\,cm  & V      &    30     &   75      & 151  & 222  &   2.44 &  1.38 &  this work           \\\\\n2011 Nov 29 &   Rozhen 2.0\\,m      & V      &    20     &   23      & 461  & 172  &   0.99 &  1.00 &  \\cite{Seeliger2014} \\\\\n2011 Dec 01 &   Rozhen 2.0\\,m      & R      &    20     &   23      & 470  & 218  &   1.38 &  1.99 &  \\cite{Seeliger2014} \\\\\n2011 Dec 14 &   Rozhen 0.6\\,m      & R      &    60     &   63      & 177  & 78   &   1.87 &  1.09 &  \\cite{Seeliger2014} \\\\\n2011 Dec 27 &   Rozhen 2.0\\,m      & R      &    20     &   39      & 280  & 66   &   1.05 &  1.81 &  \\cite{Seeliger2014} \\\\\n2012 Jan 15 &   Swarthmore 0.6\\,m  & R      &    50     &   59      & 184  & 169  &   3.21 &  1.00 &  \\cite{Seeliger2014} \\\\\n2012 Aug 15 &   Rozhen 2.0\\,m      & R      &    25     &   44      & 252  & 93   &   1.24 &  1.00 &  \\cite{Seeliger2014} \\\\\n2012 Aug 17 &   STELLA             & B      &    30     &   115     & 97   & 56   &   1.61 &  1.48 &  this work           \\\\\n2012 Aug 17 &   STELLA             & r'     &    15     &   115     & 96   & 58   &   1.72 &  1.00 &  \\cite{Seeliger2014} \\\\\n2012 Sep 12 &   OSN 1.5\\,m         & R      &    30     &   39      & 269  & 63   &   3.58 &  1.09 &  \\cite{Seeliger2014} \\\\\n2012 Sep 12 &   Trebur 1.2\\,m      & R      &    50     &   59      & 182  & 84   &   2.02 &  1.23 &  \\cite{Seeliger2014} \\\\\n2012 Sep 14 &   OSN 1.5\\,m         & R      &    30     &   40      & 278  & 181  &   1.23 &  1.58 &  \\cite{Seeliger2014} \\\\\n2012 Oct 12 &   WHT                & NaI    &    14.7   &   14.7    & 678  & 928  &   0.99 &  1.77 &  this work           \\\\\n2012 Oct 12 &   WHT                & u'     &    19.5   &   19.5    & 509  & 712  &   1.39 &  1.58 &  this work           \\\\\n2012 Oct 12 &   NOT                & B      &    7.0    &   14.9    & 657  & 266  &   1.82 &  1.00 &  this work           \\\\\n2012 Oct 25 &   STELLA             & B      &    25     &   105     & 104  & 55   &   2.35 &  1.00 &  this work           \\\\\n2012 Oct 25 &   STELLA             & r'     &    25     &   105     & 105  & 54   &   2.48 &  1.03 &  \\cite{Seeliger2014} \\\\\n2012 Oct 31 &   SON 0.4\\,m         & R      &           &   42      & 261  & 150  &   3.07 &  1.00 &  ETD, P. Kehusmaa    \\\\\n2012 Nov 07 &   0.25\\,m            & V      &    180    &   190     & 53   & 24   &   2.23 &  1.00 &  ETD, J. Gaitan      \\\\\n2012 Nov 22 &   OSN 1.5\\,m         & R      &    30     &   35      & 317  & 193  &   1.05 &  1.14 &  \\cite{Seeliger2014} \\\\\n2012 Nov 24 &   STELLA             & B      &    40     &   136     & 79   & 60   &   2.57 &  1.00 &  this work           \\\\\n2012 Nov 24 &   STELLA             & r'     &    25     &   136     & 77   & 62   &   2.60 &  1.00 &  \\cite{Seeliger2014} \\\\\n2012 Dec 05 &   Babelsberg 70\\,cm  & V      &    30     &   36      & 315  & 303  &   2.41 &  1.53 &  this work           \\\\\n2012 Dec 22 &   STELLA             & B      &    40     &   136     & 81   & 76   &   2.22 &  1.00 &  this work           \\\\\n2012 Dec 22 &   STELLA             & r'     &    25     &   136     & 82   & 72   &   1.76 &  1.13 &  \\cite{Seeliger2014} \\\\\n2013 Jan 05 &   STELLA             & B      &    40     &   136     & 81   & 77   &   1.48 &  1.23 &  this work           \\\\\n2013 Jan 05 &   STELLA             & r'     &    25     &   136     & 82   & 76   &   1.32 &  1.12 &  \\cite{Seeliger2014} \\\\\n2013 Aug 22 &   0.35\\,m            & R      &    70     &   87      & 127  & 84   &   2.03 &  1.04 &  ETD, V.P. Hentunen  \\\\\n2013 Sep 06 &   Rozhen 2.0\\,m      & R      &    30     &   33      & 334  & 55   &   0.87 &  1.34 &  \\cite{Seeliger2014} \\\\\n2013 Sep 06 &   Jena 0.6\/0.9\\,m    & R      &    40     &   58      & 191  & 124  &   1.49 &  1.03 &  \\cite{Seeliger2014} \\\\\n2013 Sep 06 &   Torun 0.6\\,m       & R      &    10     &   13      & 829  & 580  &   3.27 &  1.65 &  \\cite{Seeliger2014} \\\\\n2013 Oct 06 &   OSN 1.5\\,m         & R      &    30     &   32      & 348  & 236  &   0.93 &  1.00 &  \\cite{Seeliger2014} \\\\\n2013 Nov 01 &   Rozhen 2.0\\,m      & R      &    25     &   44      & 249  & 60   &   0.73 &  1.07 &  \\cite{Seeliger2014} \\\\\n2013 Nov 03 &   OSN 1.5\\,m         & R      &    30     &   35      & 316  & 195  &   1.20 &  1.29 &  \\cite{Seeliger2014} \\\\\n2013 Dec 01 &   OSN 1.5\\,m         & R      &    30     &   35      & 315  & 218  &   1.64 &  1.69 &  \\cite{Seeliger2014} \\\\\n2013 Dec 14 &   0.30\\,m            & V      &           &   80      & 135  & 173  &   4.01 &  1.17 &  ETD, R. Naves       \\\\\n2013 Dec 29 &   Trebur 1.2\\,m      & R      &    50     &   58      & 187  & 105  &   2.40 &  1.00 &  \\cite{Seeliger2014} \\\\\n2014 Nov 25 &   STELLA             & B      &    60     &   94      & 117  & 61   &   1.44 &  1.25 &  this work           \\\\\n2014 Dec 21 &   Calar Alto 2.2\\,m  & Gunn g &    20     &   65      & 157  & 106  &   1.84 &  1.05 &  this work           \\\\\n2014 Dec 21 &   Calar Alto 2.2\\,m  & Gunn r &    20     &   65      & 160  & 107  &   1.39 &  1.10 &  this work           \\\\\n2014 Dec 21 &   Calar Alto 2.2\\,m  & I      &    20     &   65      & 160  & 107  &   1.33 &  1.00 &  this work           \\\\\n2014 Dec 21 &   0.30\\,m            & V      &           &   84      & 130  & 77   &   2.90 &  1.00 &  ETD, F. Campos      \\\\\n2015 Jan 05 &   STELLA             & B      &    45     &   79      & 142  & 130  &   0.98 &  1.23 &  this work           \\\\\n2015 Dec 17 &   STELLA             & B      &    45     &   79      & 139  & 77   &   1.73 &  1.47 &  this work           \\\\\n2015 Dec 30 &   IAC80              & B      &    60     &   68      & 163  & 103  &   2.59 &  1.18 &  this work           \\\\\n2016 Jan 27 &   STELLA             & B      &    45     &   79      & 142  & 67   &   1.89 &  1.56 &  this work           \\\\\n\n\\hline                                                                                                     \n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\n  \n  \n  \n  \n  \n  \n  \n  \n\n   \\begin{figure*}\n   \\centering\n  \n   \n   \\includegraphics[width=14.5cm,height=18cm]{plot_all_lcs_4col.eps}\n   \n  \n       \\caption{All light curves newly presented in this work. The red solid line denotes the individual best fit model, the resulting parameter values are given in Table \\ref{tab_indiv}. Below the light curves, their corresponding residuals are shown. The scale of all panels is identical with the tickmarks labeled in the lower left.}\n         \\label{plot_lcs}\n   \\end{figure*}\n\n\n\n\\section{Light curve analysis}\n\\label{chap_anal}\n\nIn this work, we model all transit light curves with the publicly available software \\mbox{JKTEBOP}\\footnote{http:\/\/www.astro.keele.ac.uk\/jkt\/codes\/jktebop.html} \\citep{Southworth04,Southworth08} in version 34. It allows for a simultaneous fit of the transit model and a detrending function. Throughout this analysis, we use a second-order polynomial over time to detrend the individual light curves. The transit fit parameters consist of the sum of the fractional planetary and stellar radius, $r_{\\star} + r_p$, and their ratio $k=r_p\/r_{\\star}$, the orbital inclination $i$, the transit midtime $T_0$, the host-star limb-darkening coefficients (LDC) $u$ and $v$ of the quadratic limb darkening law, and the coefficients $c_{0,1,2}$ of the polynomial over time. The index ``$\\star$'' refers to the host star and ``p'' refers to the planet. The dimensionless fractional radius is the absolute radius in units of the orbital semi-major axis $a$, $r_{\\star} = R_{\\star}\/a$, and $r_p = R_p\/a$. The planetary eccentricity \nis fixed to zero following \\cite{Zhao2014} and the orbital period $P_{\\mathrm{orb}}$ to 2.15000825 days according to \\cite{Seeliger2014}.\n\nWe modeled the stellar limb darkening with the quadratic, two-parameter law. Theoretical values for the LDC have been obtained by \\cite{Claret2013} using the stellar parameter from \\cite{Hartman2011} for the case of a circular planetary orbit. Following \\cite{Southworth08} and our previous work on HAT-P-32\\,b in Paper~I, we fitted for the linear LDC $u$ and kept the quadratic LDC $v$ fixed to its theoretical values, perturbing it by $\\pm$0.1 on a flat distribution during the error estimation.\n\nFollowing the same procedure as in \\cite{MallonnH12}, we begin the analysis of each light curve with an initial fit followed by a 3.5\\,$\\sigma$ rejection of outliers. After a new transit model fit, we re-scale the photometric uncertainties derived by SExtractor to yield a reduced $\\chi^2$ of unity for the light curve residuals. Furthermore, we calculate the so-called $\\beta$ factor, a concept introduced by \\cite{Gillon06} and \\cite{Winn08} to include the contribution of correlated noise in the light curve analysis. It describes the evolution of the standard deviation $\\sigma $ of the light curve residuals when they become binned in comparison to Poisson noise. In the presence of correlated noise, $\\sigma $ of the binned residuals is larger by the factor $\\beta$ than with pure uncorrelated (white) noise. The value used here for each light curve is the average of the values for a binning from 10 to 30 minutes in 2 minute steps. In the final transit fit, we fix the $i$ and $r_{\\star}$ to the \nvalues used in G13 and Paper~I to achieve directly comparable values of $k$. We assume their uncertainties to be a common source of noise to all bandpasses, negligible in the search for relative variations of $k$ over wavelength. Therefore, the derived errors on $k$ are relative uncertainties. The free parameters per light curve have been $r_p$, $u$, and $c_{0,1,2}$. For the bandpasses with more than one light curve, we perform an individual fit per light curve, summarised in Table \\ref{tab_indiv}, and a joint run of all light curves per bandpass fitted simultaneously, summarised in Table \\ref{tab_joint}. In the joint fit, the free parameters are $r_p$, $u$, and a set of detrending coefficients per involved light curve. Here, we added the Thuan-Gunn~r light curve to the sample of r' data, and Thuan-Gunn~g to the g' data because of significant overlap in the filter transmission curves. The $k$ values of the individual and the joint fits are shown in Figure \\ref{plot_ind}.\n\n   \\begin{figure*}\n   \\centering\n  \n   \\includegraphics[height=16cm,width=10cm,angle=270]{plot_k_ind.eps}\n      \\caption{Planet-star radius ratio $k$ for the 57 individual transit light curves in the same order as in Table \\ref{tab_indiv}. The horizontal lines mark the $k$ value for the joint fit of all light curves per filter with their uncertainties given by the horizontal dashed lines.}\n         \\label{plot_ind}\n \n   \\end{figure*}\n\n\nThe estimation of the transit parameter uncertainties was done with  ``task 8'' in \\mbox{JKTEBOP} \\citep{Southworth2005}, which is a Monte Carlo simulation, and with ``task 9'' \\citep{Southworth08}, which is a residual-permutation algorithm that takes correlated noise into account. We run the Monte Carlo simulation with 5000 steps. As final parameter uncertainties we adopted the larger value of both methods.\n\nThe time stamps from all light curves analysed in this work were transferred to BJD$_{\\mathrm{TDB}}$ following the recommendation of \\cite{Eastman2010}. All individually derived transit midtimes of the light curves of our sample are in agreement with the ephemeris of \\cite{Seeliger2014}.\n\n\\cite{Adams2013} found a companion object only 2.9\\,$''$ to HAT-P-32, which was classified as a mid-M dwarf by \\cite{Zhao2014}. The flux of this object is fully included in the aperture used for the aperture photometry for all our light curves and dilutes the transit depth as third light contribution to the star-planet system. All values of $k$ derived from r', R, I, and z' band light curves were corrected using Eq.~4 in \\cite{Sing2011} and the third light as a function of wavelength measured in Paper~I.\n\nIn Paper~I we also measured and analysed the photometric variability of the host star and concluded that HAT-P-32 is photometrically constant without significant influence of stellar activity. Therefore, we assume there is no time dependence of $k$ in the interval of our observations. We computed the reduced $\\chi^2$ value of $k$ of the individual transits per filter versus the $k$ value of the joint fit per filter and obtain the values of 4.2, 1.5, 1.7, and 2.4 for the bandpasses B, V, r', and R, respectively. These values are larger than unity and indicate underestimated individual error bars potentially caused by undetected correlated noise \\citep{Southworth2009,Southworth2011}. Therefore, we conservatively inflate the uncertainties of $k$ of all individual light curves by the factor 1.5, which leads to an average reduced $\\chi^2$ of unity. \nThe error bars of $k$ of the joint fits per filter are in all cases the outcome of the residual-permutation algorithm, whose value decreases more conservatively with increasing number of light curves per joint fit than the uncertainty value derived by the Monte Carlo simulation.\n\n\\section{Results}\n\\label{chap_res}\n\n   \\begin{figure}\n   \\centering\n   \\includegraphics[height=\\hsize,angle=270]{plot_transmspec_1pan_v2_sma.eps}\n  \n      \\caption{Broad-band transmission spectrum of HAT-P-32\\,b. This work is shown in black circles, horizontal bars indicate the width of the corresponding filter curve. Spectra published by G13 (green rectangular), N16 (blue triangle), and Paper~I (red diamond) are shown for comparison. The average uncertainties of G13, N16, and Paper~I are presented in the lower left. The dotted horizontal lines indicate plus and minus two scale heights of the mean value.}\n         \\label{plot_1pan}\n \n   \\end{figure}\n\n\n\\subsection{Comparison to published spectra of HAT-P-32\\,b}\nThe broad-band transmission spectrum of this work is shown in Figure \\ref{plot_1pan} together with previously published data of HAT-P-32\\,b. The results presented here are in very good agreement to the transmission spectra derived by G13 and N16 (note that N16 uses slightly different values for $i$ and $r_{\\star}$). There are small discrepancies to the results of Paper~I. The B band measurement differs by about 2\\,$\\sigma$, and the R band value deviates by 2.5\\,$\\sigma$, whereas at all other wavelength there is an agreement within 1\\,$\\sigma$. All four investigations, G13, N16, Paper~I, and this work show good consistency shortwards of $\\sim$\\,720~nm. At redder wavelengths of the Cousins I and Sloan z' band, Paper~I deviates from the other three in exhibiting lower values of $k$ of about one scale height. We experimented again with the LDC by fixing $u$ and $v$ in the analysis to the values derived in the three previous publications and found that the deviation of $k$ in Paper~I cannot be explained by the \nslight discrepancies in the LDC. Instead, a potential origin of the deviations are systematics in the light curves of Paper~I.\n\n\\subsection{Treatment of stellar limb darkening}\nThe treatment of limb darkening in the modelling of exoplanet transit light curves is a debated topic. Multiple stellar limb darkening laws exist in varying complexity with the two-parameter, quadratic law as the one with broadest application. The limb darkening coefficients $u$ and $v$ can be either included in the fitting process as free parameters \\citep[as suggested by, e.\\,g.,][]{Csizmadia2013} or fixed to theoretical values  \\citep[e.\\,g.,][]{Claret2013}. Both options include advantages and disadvantages \\citep[see, e.\\,g.,][]{Mueller2013}. Therefore, \\cite{Southworth08} introduced an intermediate solution by fitting for the linear LDC $u$ while keeping the quadratic LDC $v$ fixed to theoretical values for the quadratic limb darkening law. We used this option here and in Paper~I, finding in both a significant discrepancy between the fitted value of $u$ and its theoretical value from \\cite{Claret2013}. For the majority of transiting exoplanet host stars, the measured LDC agree reasonably well to their \ntheoretical predictions \\citep{Mueller2013}. However, another known exception is HD\\,209458 \\citep{Knutson2007,Claret2009}. \n\nWe repeated our analysis of the light curves of HAT-P-32\\,b by \\textit{i)} keeping both LDC fixed to theoretical values of \\cite{Claret2013} and \\textit{ii)} fitting for both $u$ and $v$ in the light curve fit. The results are shown in Figure \\ref{plot_ldc}. While option i) results in a poorer fit and a general offset in planet-star radius ratio $k$ towards lower values, there is a very good agreement when both $u$ and $v$ were free-to-fit compared to $u$ free and $v$ fix. However, the relative variation of $k$ over wavelength remains nearly the same, irrespective of the treatment of the limb darkening. We verified this result further by the usage of the logarithmic limb darkening law suggested by \\cite{Espinoza2016} for the stellar temperature of HAT-P-32 and by the usage of the four-parameter law introduced by \\cite{Claret2000}. \nThe resulting values of $k$ over wavelength remain nearly unaffected by the choice of the limb darkening law, the same holds for the discrepancy when the LDC are free or fixed in the fit. Nevertheless, the treatment of the LDC affects the spectral slope. A fit of a linear regression results in a slight increase of $k$ towards shorter wavelengths for free LDC, while the slope is nearly zero in the versions of fixed LDC (Figure \\ref{plot_ldc}). However, in agreement to G13, N16, and Paper~I, we consider the version of free LDC more reliable because of a better fit for all bandpasses. We note that using the theoretical LDC would reduce the agreement of the data with atmosphere models containing an enhanced opacity at short wavelengths (see Section \\ref{chap_theo_mod}). Therefore, it would not contradict the outcome of this work. We continue with the results obtained with free LDC.\n\n\n\\subsection{Comparison to theoretical models}\n\\label{chap_theo_mod}\nWe compared the derived spectrophotometric transmission spectrum of this work with theoretical models that were supplied by \\cite{Fortney2010} and calculated for HAT-P-32\\,b (see Fig. \\ref{plot_spec}). The only fitted parameter is a vertical offset. For each model we calculated the $\\chi^2$ value and the probability $P$ of the $\\chi^2$ test, i.\\,e. the probability that the measurements could result by chance if the model represented the true planetary spectrum. The values are summarised in Table \\ref{tab_chi2}. A cloud-free solar-composition model dominated by TiO absorption and a solar-composition model with TiO artificially removed are ruled out by the nine data points of this work, in agreement to results of G13, N16 and Paper~I. Looking specifically at the theoretically predicted, prominent sodium D-line, the data point from the ULTRACAM filter shows no extra absorption in agreement to the three previous studies. A model of solar-composition without TiO including a Rayleigh scattering component with a \ncross section 100$\\times$ that of molecular hydrogen yields $\\chi^2 = 7.3$ for eight degrees of freedom (DOF), indicating good agreement between model and data. Similarly, a wavelength-independent planet-star radius ratio (flat spectrum) also yields an acceptable fit. Therefore, the measurements of this work cannot distinguish between the Rayleigh model and the flat model. \n\nWe attempt a combined comparison of the results derived here and in Paper~I to the theoretical models, justified by the homogeneity of both analyses. Because of the deviating red data points of Paper~I mentioned above, we restrict the wavelength range of these data to $\\lambda<720$~nm. The values of $\\chi^2$ and $P$ are given in Table \\ref{tab_chi2}. This combined data set shows only a very low probability for a Rayleigh slope of 0.017, whereas a flat line is significantly favoured. The low-amplitude scattering slope proposed in Paper~I also yields a good fit ($\\chi^2=50.7$, $\\mathrm{DOF} = 48$), though of lower quality than the flat line.\n\nFor another test, we merge the measurements of all four studies G13, N16, Paper~I, and this work, and compute a regression line. For simplicity and to account for inhomogeneities in the analyses, all data points were given equal weight. The resulting slope is $(-3.28\\,\\pm\\,0.65)\\,\\times\\,10^{-6}$~nm$^{-1}$. However, excluding the data points $\\lambda>720$~nm of Paper~I results in a weaker, insignificant slope of $(-1.16\\,\\pm\\,0.60)\\,\\times\\,10^{-6}$~nm$^{-1}$, which means a consistency of the planet-star radius ratio between 350 and 1000~nm to within half of a scale height. A homogeneous re-analysis of all available data sets of HAT-P-32\\,b would be needed to reliably explore the potential slope at this sub-scale height precision.\n\n\n\n\\begin{table*}\n\\small\n\\caption{Planet-to-star radius ratio $k$ per observation with relative uncertainties. In Column 4, $k$ is derived with $u$ as free parameter, with the $u$ value given in Column 5. The LDC $v$ is fixed in the analysis to its theoretical value given in Column 7. The theoretical of $u$ is given in Column 6 for comparison. Column 8 gives the applied third-light correction of $k$.}\n\\label{tab_indiv}\n\\begin{center}\n\\begin{tabular}{p{19mm}p{21mm}p{11mm}lcccl}\nDate        &   Telescope   &   Filter  &    $k$  &  $u_{\\mathrm{fit}}$  & $u_{\\mathrm{theo}}$  & $v_{\\mathrm{theo}}$  &  $\\Delta k_{\\mathrm{third\\ light}}$ \\\\\n\\hline\n2012 Oct 12 &   WHT              & u'    &  0.1520 $\\pm$ 0.0012  & 0.543 $\\pm$ 0.023   &   0.696 &  0.112  & 0         \\\\\n\\hline\n2011 Nov 01 &   STELLA           & B     &  0.1484 $\\pm$ 0.0035 & 0.559 $\\pm$ 0.038   &   0.583 &  0.208  & 0         \\\\\n2011 Nov 29 &   STELLA           & B     &  0.1539 $\\pm$ 0.0035 & 0.512 $\\pm$ 0.017   &   0.583 &  0.208  & 0         \\\\\n2011 Nov 29 &   Babelsberg       & B     &  0.1499 $\\pm$ 0.0023 & 0.495 $\\pm$ 0.022   &   0.583 &  0.208  & 0         \\\\\n2012 Aug 17 &   STELLA           & B     &  0.1589 $^{+\\,\\,0.0033}_{-\\,0.0035}$ & 0.438 $\\pm$ 0.058   &   0.583 &  0.208  & 0         \\\\\n2012 Oct 12 &   NOT              & B     &  0.1556 $\\pm$ 0.0019 & 0.486 $\\pm$ 0.014   &   0.583 &  0.208  & 0         \\\\\n2012 Oct 25 &   STELLA           & B     &  0.1404 $\\pm$ 0.0043 & 0.535 $\\pm$ 0.020   &   0.583 &  0.208  & 0         \\\\\n2012 Nov 24 &   STELLA           & B     &  0.1559 $\\pm$ 0.0040 & 0.435 $\\pm$ 0.059   &   0.583 &  0.208  & 0         \\\\\n2012 Dec 22 &   STELLA           & B     &  0.1515 $^{+\\,\\,0.0031}_{-\\,0.0029}$ & 0.479 $\\pm$ 0.041   &   0.583 &  0.208  & 0         \\\\\n2013 Jan 05 &   STELLA           & B     &  0.1501 $^{+\\,\\,0.0026}_{-\\,0.0028}$ & 0.430 $\\pm$ 0.053   &   0.583 &  0.208  & 0         \\\\\n2014 Nov 25 &   STELLA           & B     &  0.1554 $^{+\\,\\,0.0026}_{-\\,0.0025}$ & 0.400 $\\pm$ 0.040   &   0.583 &  0.208  & 0         \\\\\n2015 Jan 05 &   STELLA           & B     &  0.1556 $\\pm$ 0.0013 & 0.484 $\\pm$ 0.017   &   0.583 &  0.208  & 0         \\\\\n2015 Dec 17 &   STELLA           & B     &  0.1465 $\\pm$ 0.0039 & 0.514 $\\pm$ 0.051   &   0.583 &  0.208  & 0         \\\\\n2015 Dec 30 &   IAC80            & B     &  0.1601 $\\pm$ 0.0032 & 0.440 $\\pm$ 0.037   &   0.583 &  0.208  & 0         \\\\\n2016 Jan 27 &   STELLA           & B     &  0.1526 $\\pm$ 0.0056 & 0.492 $\\pm$ 0.063   &   0.583 &  0.208  & 0         \\\\\n\\hline\n2007 Dec 04 &   FLWO 1.2\\,m      & g'    &  0.1536 $^{+\\,\\,0.0015}_{-\\,0.0014}$ & 0.465 $\\pm$ 0.018   &   0.545 &  0.205  & 0         \\\\\n2014 Dec 21 &   Calar Alto       & Gunn g&  0.1524 $^{+\\,\\,0.0025}_{-\\,0.0023}$ & 0.400 $\\pm$ 0.021   &   0.518 &  0.221  & 0         \\\\\n\\hline\n2011 Nov 29 &   Rozhen 2.0\\,m    & V     &  0.1527 $\\pm$ 0.0010 & 0.395 $\\pm$ 0.014   &   0.458 &  0.229  & 0         \\\\\n2011 Nov 29 &   Babelsberg       & V     &  0.1490 $^{+\\,\\,0.0029}_{-\\,0.0027}$ & 0.463 $\\pm$ 0.035   &   0.458 &  0.229  & 0         \\\\\n2012 Nov 07 &   0.25\\,m          & V     &  0.1513 $\\pm$ 0.0058 & 0.397 $\\pm$ 0.046   &   0.458 &  0.229  & 0         \\\\\n2012 Dec 05 &   Babelsberg       & V     &  0.1498 $\\pm$ 0.0028 & 0.476 $\\pm$ 0.049   &   0.458 &  0.229  & 0         \\\\\n2013 Dec 14 &   0.30\\,m          & V     &  0.1471 $\\pm$ 0.0058 & 0.240 $\\pm$ 0.070   &   0.458 &  0.229  & 0         \\\\\n2014 Dec 21 &   0.30\\,m          & V     &  0.1548 $^{+\\,\\,0.0042}_{-\\,0.0038}$ & 0.298 $\\pm$ 0.029   &   0.458 &  0.229  & 0         \\\\\n\\hline\n2012 Oct 12 &   WHT              & NaI   &  0.1514 $\\pm$ 0.0011 & 0.297 $\\pm$ 0.021   &   0.421 &  0.237  & 0.0001    \\\\\n\\hline\n2011 Nov 01 &   STELLA           & r'    &  0.1502 $\\pm$ 0.0033 & 0.331 $\\pm$ 0.045   &   0.401 &  0.227  & 0.0001    \\\\\n2011 Nov 29 &   STELLA           & r'    &  0.1561 $\\pm$ 0.0037 & 0.293 $\\pm$ 0.031   &   0.401 &  0.227  & 0.0001    \\\\\n2012 Aug 17 &   STELLA           & r'    &  0.1534 $\\pm$ 0.0023 & 0.361 $\\pm$ 0.024   &   0.401 &  0.227  & 0.0001    \\\\\n2012 Oct 25 &   STELLA           & r'    &  0.1460 $^{+\\,\\,0.0046}_{-\\,0.0048}$ & 0.315 $\\pm$ 0.064   &   0.401 &  0.227  & 0.0001    \\\\\n2012 Nov 24 &   STELLA           & r'    &  0.1518 $\\pm$ 0.0042 & 0.296 $\\pm$ 0.067   &   0.401 &  0.227  & 0.0001    \\\\\n2012 Dec 22 &   STELLA           & r'    &  0.1550 $\\pm$ 0.0039 & 0.274 $\\pm$ 0.055   &   0.401 &  0.227  & 0.0001    \\\\\n2013 Jan 05 &   STELLA           & r'    &  0.1537 $\\pm$ 0.0022 & 0.319 $\\pm$ 0.043   &   0.401 &  0.227  & 0.0001    \\\\\n2014 Dec 21 &   Calar Alto       & Gunn r&  0.1502 $\\pm$ 0.0017 & 0.310 $\\pm$ 0.020   &   0.361 &  0.238  & 0.0001    \\\\\n\\hline\n2011 Nov 14 &   Jena 0.6\/0.9\\,m  & R     &  0.1528 $\\pm$ 0.0022 & 0.349 $\\pm$ 0.021   &   0.387 &  0.219  & 0.0002    \\\\\n2011 Dec 01 &   Rozhen 2.0\\,m    & R     &  0.1521 $\\pm$ 0.0036 & 0.377 $\\pm$ 0.036   &   0.387 &  0.219  & 0.0002    \\\\\n2011 Dec 14 &   Rozhen 0.6\\,m    & R     &  0.1546 $\\pm$ 0.0035 & 0.376 $\\pm$ 0.014   &   0.387 &  0.219  & 0.0002    \\\\\n2011 Dec 27 &   Rozhen 2.0\\,m    & R     &  0.1546 $^{+\\,\\,0.0039}_{-\\,0.0042}$ & 0.371 $\\pm$ 0.032   &   0.387 &  0.219  & 0.0002    \\\\\n2012 Jan 15 &   Swarthmore       & R     &  0.1554 $^{+\\,\\,0.0027}_{-\\,0.0025}$ & 0.267 $\\pm$ 0.032   &   0.387 &  0.219  & 0.0002    \\\\\n2012 Aug 15 &   Rozhen 2.0\\,m    & R     &  0.1499 $\\pm$ 0.0021 & 0.315 $\\pm$ 0.013   &   0.387 &  0.219  & 0.0002    \\\\\n2012 Sep 12 &   OSN 1.5\\,m       & R     &  0.1602 $\\pm$ 0.0088 & 0.294 $\\pm$ 0.019   &   0.387 &  0.219  & 0.0002    \\\\\n2012 Sep 12 &   Trebur 1.2\\,m    & R     &  0.1580 $\\pm$ 0.0035 & 0.325 $\\pm$ 0.039   &   0.387 &  0.219  & 0.0002    \\\\\n2012 Sep 14 &   OSN 1.5\\,m       & R     &  0.1523 $^{+\\,\\,0.0020}_{-\\,0.0021}$ & 0.342 $\\pm$ 0.018   &   0.387 &  0.219  & 0.0002    \\\\\n2012 Oct 31 &   SON 0.4\\,m       & R     &  0.1527 $^{+\\,\\,0.0034}_{-\\,0.0031}$ & 0.365 $\\pm$ 0.036   &   0.387 &  0.219  & 0.0002    \\\\\n2012 Nov 22 &   OSN 1.5\\,m       & R     &  0.1493 $\\pm$ 0.0013 & 0.291 $\\pm$ 0.010   &   0.387 &  0.219  & 0.0002    \\\\\n2013 Aug 22 &   0.35\\,m          & R     &  0.1488 $^{+\\,\\,0.0029}_{-\\,0.0033}$ & 0.412 $\\pm$ 0.033   &   0.387 &  0.219  & 0.0002    \\\\\n2013 Sep 06 &   Rozhen 2.0\\,m    & R     &  0.1508 $\\pm$ 0.0029 & 0.277 $\\pm$ 0.022   &   0.387 &  0.219  & 0.0002    \\\\\n2013 Sep 06 &   Jena 0.6\/0.9\\,m  & R     &  0.1514 $\\pm$ 0.0018 & 0.312 $\\pm$ 0.016   &   0.387 &  0.219  & 0.0002    \\\\\n2013 Sep 06 &   Torun 0.6\\,m     & R     &  0.1484 $\\pm$ 0.0030 & 0.278 $\\pm$ 0.037   &   0.387 &  0.219  & 0.0002    \\\\\n2013 Oct 06 &   OSN 1.5\\,m       & R     &  0.1492 $\\pm$ 0.0009 & 0.344 $\\pm$ 0.012   &   0.387 &  0.219  & 0.0002    \\\\\n2013 Nov 01 &   Rozhen 2.0\\,m    & R     &  0.1534 $\\pm$ 0.0015 & 0.309 $\\pm$ 0.012   &   0.387 &  0.219  & 0.0002    \\\\\n2013 Nov 03 &   OSN 1.5\\,m       & R     &  0.1519 $^{+\\,\\,0.0014}_{-\\,0.0016}$ & 0.276 $\\pm$ 0.027   &   0.387 &  0.219  & 0.0002    \\\\\n2013 Dec 01 &   OSN 1.5\\,m       & R     &  0.1498 $\\pm$ 0.0025 & 0.287 $\\pm$ 0.027   &   0.387 &  0.219  & 0.0002    \\\\\n2013 Dec 29 &   Trebur 1.2\\,m    & R     &  0.1489 $\\pm$ 0.0032 & 0.330 $\\pm$ 0.031   &   0.387 &  0.219  & 0.0002    \\\\\n\\hline\n2014 Dec 21 &   Calar Alto       & I     &  0.1514 $^{+\\,\\,0.0016}_{-\\,0.0015}$ & 0.227 $\\pm$ 0.017   &   0.306 &  0.212  & 0.0007    \\\\\n\\hline\n2007 Sep 24 &   FLWO 1.2\\,m      & z'    &  0.1530 $^{+\\,\\,0.0018}_{-\\,0.0016}$ & 0.188 $\\pm$ 0.021   &   0.265 &  0.211  & 0.0012    \\\\\n2007 Oct 22 &   FLWO 1.2\\,m      & z'    &  0.1481 $^{+\\,\\,0.0019}_{-\\,0.0018}$ & 0.219 $\\pm$ 0.025   &   0.265 &  0.211  & 0.0012    \\\\\n2007 Nov 06 &   FLWO 1.2\\,m      & z'    &  0.1533 $^{+\\,\\,0.0012}_{-\\,0.0013}$ & 0.188 $\\pm$ 0.017   &   0.265 &  0.211  & 0.0012    \\\\\n2007 Nov 19 &   FLWO 1.2\\,m      & z'    &  0.1523 $\\pm$ 0.0015 & 0.231 $\\pm$ 0.036   &   0.265 &  0.211  & 0.0012    \\\\\n\n\\hline                                                                                                     \n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\\begin{table*}\n\\caption{Planet-to-star radius ratio $k$ per filter with relative uncertainties. }\n\\label{tab_joint}\n\\begin{center}\n\\begin{tabular}{lccccl}\n\\hline\n\\hline\n\\noalign{\\smallskip}\n\nFilter  &    $k$  &  $u_{\\mathrm{fit}}$  & $u_{\\mathrm{theo}}$  & $v_{\\mathrm{theo}}$  &  $\\Delta k_{\\mathrm{third\\ light}}$ \\\\ \n\\hline\n\\noalign{\\smallskip}\nu'           &   0.15204 $^{+\\,\\,0.00118}_{-\\,0.00123}$ &  0.543 $\\pm$ 0.023 &  0.696 &  0.112  &  0         \\\\\nB            &   0.15338 $^{+\\,\\,0.00073}_{-\\,0.00082}$ &  0.502 $\\pm$ 0.013 &  0.583 &  0.208  &  0         \\\\\ng' + Gunn g  &   0.15304 $^{+\\,\\,0.00111}_{-\\,0.00101}$ &  0.448 $\\pm$ 0.022 &  0.535 &  0.214  &  0         \\\\\nV            &   0.15223 $^{+\\,\\,0.00123}_{-\\,0.00128}$ &  0.398 $\\pm$ 0.014 &  0.458 &  0.229  &  0         \\\\\nNaI          &   0.15142 $^{+\\,\\,0.00108}_{-\\,0.00105}$ &  0.297 $\\pm$ 0.021 &  0.421 &  0.237  &  0.00006    \\\\      \nr'+ Gunn r   &   0.15216 $^{+\\,\\,0.00080}_{-\\,0.00078}$ &  0.317 $\\pm$ 0.016 &  0.392 &  0.230  &  0.00007   \\\\     \nR            &   0.15071 $^{+\\,\\,0.00053}_{-\\,0.00057}$ &  0.315 $\\pm$ 0.010 &  0.387 &  0.219  &  0.00021    \\\\    \nI            &   0.15141 $^{+\\,\\,0.00155}_{-\\,0.00147}$ &  0.227 $\\pm$ 0.017 &  0.306 &  0.212  &  0.00069    \\\\     \nz'           &   0.15149 $^{+\\,\\,0.00073}_{-\\,0.00078}$ &  0.221 $\\pm$ 0.017 &  0.265 &  0.211  &  0.00115    \\\\                                                                                           \n                                                                                                              \n\\hline                                                                                                     \n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\n   \\begin{figure}\n   \\centering\n   \\includegraphics[width=\\hsize]{plot_ldc_test.eps}\n  \n      \\caption{Planet-star radius ratio over wavelength of HAT-P-32\\,b derived by the usage of different stellar limb darkening laws. Different colours denote different treatment of the LDC. Upper panel: quadratic stellar limb darkening law. Black shows the $k$ values for both LDC fixed to theoretical values, red denotes both LDC fitted, and green linear LDC fitted, but quadratic LDC fixed. Middle panel: logarithmic stellar limb darkening law. Black denotes both LDC fixed, red implies both LDC fitted. Lower panel: four-parameter stellar limb darkening law. Black denotes all four LDC fixed, red implies first LDC fitted, the other three fixed. All dotted lines show a linear regression corresponding to the data points of same colour. }\n         \\label{plot_ldc}\n \n   \\end{figure}\n\n   \\begin{figure}\n   \\centering\n   \\includegraphics[height=\\hsize,angle=270]{plot_transmspec_models.eps}\n      \\caption{Broad-band transmission spectrum of HAT-P-32\\,b. The measured values of this work are given in black, horizontal bars show the width of the corresponding filter curve. Overplotted are a cloud-free solar-composition model of 1750~K (blue line), a cloud-free solar composition model of 1750~K without TiO (green line), and a solar composition model of 1750~K including H$_2$ Rayleigh scattering increased by a factor of 100 \\citep[red line,][]{Fortney2010}.The colour-coded open circles show the bandpass-integrated theoretical values.}\n         \\label{plot_spec}\n \n   \\end{figure}\n\n\n\\begin{table*}\n\\caption{Fit statistics of theoretical models against the derived transmission spectrum.}\n\\label{tab_chi2}\n\\begin{center}\n\\begin{tabular}{l|ccc|ccc}\n\\hline\n\\hline\n\\noalign{\\smallskip}\n\n  &    \\multicolumn{3}{|c|}{This work}   & \\multicolumn{3}{c}{This work \\& Paper~I, $\\lambda<720$~nm} \\\\\nModels  &    $\\chi^2$  &  N,\\ DOF  & $P$  &  $\\chi^2$  &  N,\\ DOF  & $P$\\\\\n\\hline\n\\noalign{\\smallskip}\nSolar composition, clear           &  62.9  & 9,\\ 8  & $\\ll0.001$  &  219.6  &  49,\\ 48  &  $\\ll0.001$     \\\\\nSolar comp., without TiO, clear    &  21.9  & 9,\\ 8  & 0.005       &  106.2  &  49,\\ 48  &  $\\ll0.001$     \\\\\nRayleigh                           &  7.6   & 9,\\ 8  & 0.50        &  70.9   &  49,\\ 48  &  0.017     \\\\\nFlat                               &  10.3  & 9,\\ 8  & 0.24        &  43.8   &  49,\\ 48  &  0.65     \\\\\n                                                                                 \n                                                                                                              \n\\hline                                                                                                     \n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\\section{Discussion}\n\\label{chap_disc}\nA flat spectrum of a hot Jupiter exoplanet could be explained by a thick cloud layer at high altitudes acting as a grey absorber. A comparable case of a cloudy atmosphere is the hot Jupiter WASP-31\\,b with a cloud deck at only 1~mbar \\citep{Sing2015}, and the super-Earth GJ1214b, whose transmission spectrum is featureless \\citep{Kreidberg2014} without measurable indications for scattering at blue wavelengths \\citep{deMooij2013,Nascimbeni2015}. In the presence of brightness inhomogeneities on the stellar surface, a flat spectrum could also be produced by the interplay of scattering in the planetary atmosphere and bright plage regions not occulted by the  transiting planet \\citep{Oshagh2014}. However, in paper I we discuss this scenario to be very unlikely.\n\nThe results of this work indicate a flat spectrum with a spectral gradient of lower amplitude than two scale heights. Interestingly, the near-UV\/optical measurements of \\cite{Sing2016} made with HST\/STIS show no hot Jupiter spectrum without a blueward gradient. Each of the ten investigated objects exhibit an increase in $k$ from the z' band to the u' band of at least $\\sim$\\,two, and up to $\\sim$\\,six scale heights. Therefore, HAT-P-32\\,b could be the hot Jupiter with the flattest spectrum measured so far.\nOne way to confirm this result would be a homogeneous re-analysis of the published GEMINI, GTC, and LBT ground-based transit observations or new HST measurements at near-UV and optical wavelengths with STIS. Extrapolating from the optical spectrum, we would expect a muted or absent water feature at 1.4~$\\mu$m. It would be very interesting to verify this prediction with HST\/WFC3 measurements. If confirmed, HAT-P-32\\,b would strengthen the tentative correlation between cloud occurrence at high altitudes and a low planetary surface gravity found by \\cite{Stevenson2016}. \n\n\\section{Conclusions}\n\\label{chap_concl}\nHAT-P-32\\,b is one of the most favourable targets for transmission spectroscopy in terms of host star brightness and predicted amplitude of the potential transmission signal. In Paper~I, we obtained a transmission spectrum from 330 to 1000~nm showing a tentative increase of the planet-star radius ratio towards blue wavelength of low amplitude. In this work, we collected the largest sample of broad-band transit photometry used for spectrophotometry so far to follow-up on this slope in the planetary spectrum. The light curves were taken in nine different filters from Sloan u' over several Johnson bands and one specific filter centred at the NaI D line to the Sloan z' band. The resulting spectrum was independent of the choice of the stellar limb darkening law in the analysis. However, the spectral gradient was dependent on the treatment of the limb darkening coefficients in the fit. We advocate the inclusion of the linear coefficient as a free parameter in the light curve modelling because it resulted in a \nbetter fit in all bandpasses.\n\nWhile the new measurements of this work were of sufficient precision to rule out clear atmosphere models with solar metallicity, they could not distinguish between a Rayleigh-slope model and a wavelength-independent planet-star radius ratio. However, the new data helped to verify the proposed spectral gradient in showing that the $k$ values redward of 720~nm of Paper~I might be too low by about one scale height, potentially caused by systematics in the measurements. Excluding these data favoured a spectrum that is flat over the entire wavelength range. HAT-P-32\\,b might be the hot Jupiter with the flattest spectrum observed so far. However, we suggest the combination of the currently available ground-based data by a homogeneous re-analysis, and the performance of transit observations with HST\/STIS to confirm this peculiar result.\n\n\n\\section*{Acknowledgements}\nThis article is based on observations made with the STELLA robotic telescopes in Tenerife, an AIP facility jointly operated by AIP and IAC, the IAC80 telescope operated on the island of Tenerife by IAC in the Spanish Observatorio del Teide, the Nordic Optical Telescope, operated by the Nordic Optical Telescope Scientific Association at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the IAC, the William Herschel Telescope, operated by the Isaac Newton Group and run by the Royal Greenwich Observatory at the Spanish Roque de los Muchachos Observatory in La Palma, and the 70cm telescope operated at the Babelsberg Observatory, Potsdam, Germany, by the AIP. The data presented here were obtained in part with ALFOSC, which is provided by the Instituto de Astrofisica de Andalucia (IAA) under a joint agreement with the University of Copenhagen and NOTSA. EH acknowledges support from the Spanish MINECO through grant ESP2014-57495-C2-2-R. VSD and ULTRACAM are supported by STFC. This research has made \nuse of the SIMBAD data base and VizieR catalogue access tool, operated at CDS, Strasbourg, France, and of the NASA Astrophysics Data System (ADS). We thank the anonymous referee for a constructive review.\n\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe study of the class of compact spaces that admit a retractional skeleton was initiated in \\cite{KM06}, where the authors proved that a compact space is Valdivia if and only if it admits a commutative retractional skeleton. Later, in \\cite{kubis09} a notion similar to retractional skeletons in the context of Banach spaces was introduced; namely, the notion of projectional skeletons. In some sense, those notions are dual to each other. More precisely, if a compact space $K$ admits a retractional skeleton, then $\\C(K)$ admits a projectional skeleton and if a Banach space $X$ admits a projectional skeleton, then $(B_{X^*},w^*)$ admits a retractional skeleton. The class of Banach (compact) spaces with a projectional (retractional) skeleton was deeply investigated from various perspectives and nowadays we have quite a rich family of natural examples and interesting results related to various fields of mathematics such as topology \\cite{So20}, Banach space theory \\cite{FM18}, theory of von Neumann algebras \\cite{BHK16} or $JBW^*$-triples \\cite{BHKPP18}. Let us note that, quite surprisingly, there was independently introduced also the notion of monotonically retractable topological spaces which turned out to be very closely related to the study of compact spaces that admit a retractional skeleton, see \\cite{CK15}, and from there on, several results and modifications of the corresponding notions were considered, see e.g. \\cite{CGR17, GFRH16, GFYA20}.\n\nOne of the recent streams in the area is to describe some classes of Banach (compact) spaces using the notion of projectional (retractional) skeletons, see e.g. \\cite{KM06, CF17, FM18, kubis09} where the characterizations of Plichko spaces (and Valdivia compacta), WLD spaces (and Corson compacta), Asplund spaces, WLD+Asplund spaces and WCG spaces were given.\n\nThe main two results of this paper (Theorems \\ref{thm:Intro1} and \\ref{thm:Intro2}) are characterizations of Eberlein and semi-Eberlein compacta, respectively, using the notion of retractional skeletons. \nLet us recall that given a set $I$ we define\n\\[\nc_0(I):=\\{x\\in \\er^I\\colon (\\forall \\varepsilon > 0) |\\{i\\in I\\colon |x(i)|>\\varepsilon\\}| < \\omega\\}\\subset \\er^I\n\\]\nand that a compact space $K$ is \\emph{Eberlein} if it homeomophically embeds into $c_0(I)$, for some set $I$. This is a central concept in Banach space theory, as it is known that a compact space is Eberlein if and only if it is homeomorphic to a weakly compact set of a Banach space, see \\cite{AL68} or \\cite[Corollary 13.19]{FHHMZ}. For the notion of shrinkingness we refer the reader to Definition~\\ref{def:shrinking}.\n\n\\begin{thmx}\\label{thm:Intro1}\nLet $K$ be a compact space. Then the following conditions are equivalent:\n\\begin{enumerate}\n    \\item $K$ is Eberlein.\n    \\item There exist a bounded set $\\A\\subset \\C(K)$ separating the points of $K$ and a retractional skeleton $\\mathfrak{s} = (r_s)_{s\\in\\Gamma}$ on $K$ such that $\\mathfrak{s}$ is $\\A$-shrinking.\n    \\item There exist a countable family $\\A$ of subsets of $B_{\\C(K)}$ and a full retractional skeleton $\\mathfrak{s} = (r_s)_{s\\in\\Gamma}$ on $K$ such that\n\\begin{enumerate}[label=(\\alph*)]\n    \\item For every $A\\in\\A$ there exists $\\varepsilon_A>0$ such that $\\mathfrak{s}$ is $(A,\\varepsilon_A)$-shrinking, and\n    \\item\\label{cond:fullInBall0} for every $\\varepsilon>0$ we have $B_{\\C(K)} = \\bigcup\\{A\\in \\A\\colon \\varepsilon_A < \\varepsilon\\}$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{thmx}\n\nRecall that a compact space $K$ is Eberlein if and only if $\\C(K)$ is WCG if and only if $\\C(K)$ is a subspace of a WCG space, thus Theorem \\ref{thm:Intro1} is naturally connected to the characterization of WCG Banach spaces and their subspaces presented in \\cite{FM18}. Moreover, from Theorem~\\ref{thm:Intro1} one may deduce that continuous images of Eberlein compacta are Eberlein, see Remark \\ref{rem:contImageEberlein} below. Quite many steps of our proof seem to be much more flexible and we believe that those may be used in order to find characterizations of other natural subclasses of Valdivia compacta (the most important in this respect is probably Theorem~\\ref{thm:Intro3} mentioned below). This is witnessed by the characterization of semi-Eberlein compacta presented in Theorem \\ref{thm:Intro2}. Recall that, following \\cite{KL04}, we say a compact space $K$ is \\emph{semi-Eberlein} if there exists a homeomorphic embedding $h:K\\to\\er^I$ such that $h^{-1}[c_0(I)]$ is dense in $K$. We denote by $D(\\mathfrak s)$ the set induced by a retractional skeleton $\\mathfrak s$ (see Definition \\ref{def: r-skeleton}).\n\n\\begin{thmx}\\label{thm:Intro2}\nLet $K$ be a compact space. Then the following conditions are equivalent:\n\\begin{enumerate}\n    \\item\\label{it: semi-Eberlein} $K$ is semi-Eberlein.\n    \\item\\label{it: semi-EberleinRetractionalSkeleton} There exist a dense subset $D\\subset K$, a bounded set $\\A\\subset \\C(K)$ separating the points of $K$ and a retractional skeleton $\\mathfrak{s} = (r_s)_{s\\in\\Gamma}$ on $K$ with $D\\subset D(\\mathfrak{s})$ such that\n\\begin{enumerate}[label=(\\alph*)]\n    \\item $\\mathfrak{s}$ is $\\A$-shrinking with respect to $D$, and\n        \\item $\\lim_{s\\in\\Gamma'} r_s(x)\\in D$, for every $x\\in D$ and every up-directed subset $\\Gamma'$ of $\\Gamma$.\n\\end{enumerate}\n\\item  There exist a dense set $D\\subset K$, a countable family $\\A$ of subsets of $B_{\\C(K)}$ and a retractional skeleton $\\mathfrak{s} = (r_s)_{s\\in\\Gamma}$ on $K$ with $D\\subset D(\\mathfrak{s})$ such that\n\\begin{enumerate}[label=(\\alph*)]\n    \\item For every $A\\in\\A$ there exists $\\varepsilon_A>0$ such that $\\mathfrak{s}$ is $(A,\\varepsilon_A)$-shrinking with respect to $D$,\n    \\item\\label{cond:fullInBall2} for every $\\varepsilon>0$ we have $B_{\\C(K)} = \\bigcup\\{A\\in \\A\\colon \\varepsilon_A < \\varepsilon\\}$, and\n    \\item\\label{it:inSpecialCaseEquivalentToCommutativity0} $\\lim_{s\\in\\Gamma'} r_s(x)\\in D$, for every $x\\in D$ and every up-directed subset $\\Gamma'$ of $\\Gamma$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{thmx}\n\nFinally, using Theorem~\\ref{thm:Intro2} we provide new structural results for the class of semi-Eberlein compacta, answering in particular the second part of \\cite[Question 6.6]{KL04} in positive. The most important new stability results are summarized below.\n\\begin{thmx}\\label{thm:Intro4}\nLet $K$ be a semi-Eberlein compact space.\n\\begin{itemize}\n    \\item If $L$ is an open continuous image of $K$ and it has densely many $G_\\delta$-points, then $L$ is semi-Eberlein.\n    \\item If $K$ is moreover Corson and $L$ is a continuous image of $K$, then $L$ is semi-Eberlein.\n\\end{itemize}\n\\end{thmx}\n\n\nAs mentioned previously, many steps of the proofs of Theorem~\\ref{thm:Intro1} and Theorem~\\ref{thm:Intro2} are of independent interest and we believe those could be used when trying to characterize other subclasses of Valdivia compacta, which opens quite a wide area of potential further research. This is outlined in Section~\\ref{sec:questions}.\n\n\nLet us now briefly describe the content of each section, emphasizing the general steps mentioned above.\n\nSection~\\ref{sec:prelim} contains basic notations and some preliminary results.\n\nIn Section~\\ref{sec:models} we consider retractions associated to (not necessary countable) suitable models. The most important outcome is Theorem~\\ref{thm:canonicalRetraction}, where we summarize the properties of canonical retractions associated to suitable models. As an easy consequence, in Proposition~\\ref{prop:rri} we show a very general method of obtaining a continuous chain of retractions on a compact space admitting a retractional skeleton. This part is essentially known as similar results were obtained e.g. in \\cite[Lemma 2.5]{CGR17} (using other methods than suitable models), but our approach is in a certain sense much more flexible (most importantly, because it may be combined with other statements involving suitable models) and we actually use this flexibility later. As a corollary of our investigations we show in Theorem~\\ref{thm:subskeletons} that we may in a certain way combine properties of countably many retractional skeletons.\n\nIn Section~\\ref{sec:valdivia}, inspired by the proof of \\cite[Theorem 2.6]{CGR17}, we aim at seeing as concretely as possible the ``Valdivia embedding'' of compact spaces with a commutative retractional skeleton. As a consequence we obtain the following result which might be thought of as the fourth main result of the whole paper. The most important part which we use later is the implication \\ref{it:inducedByCommutative}$\\Rightarrow$\\ref{it:sigmaEmbedding}.\n\n\n\n\\begin{thmx}\\label{thm:Intro3}\n    Let $K$ be a compact space and $\\mathfrak{s}=(r_s)_{s\\in\\Gamma}$ be a retractional skeleton on $K$. \n    Then the following conditions are equivalent.\n    \\begin{enumerate}[label = (\\roman*)]  \n     \\newcounter{saveenum2}\n     \n     \n     \\item\\label{it:inducedByCommutative}$D(\\mathfrak{s})$ is induced by a commutative retractional skeleton. \n    \\item\\label{it:subsetOfComutativeInducedSubset} There exists a subskeleton of $\\mathfrak{s}$ which is commutative.\n    \n    \\item\\label{it:skeletonInvariantSubset} There exist a subskeleton $\\mathfrak{s}_2 = (r_s)_{ s\\in\\Gamma'}$ of $\\mathfrak{s}$ and a dense set $D\\subset D(\\mathfrak{s})$ such that for every up-directed set $\\Gamma''\\subset \\Gamma'$ and every $x\\in D$ we have $\\lim_{s\\in\\Gamma''}r_s(x)\\in D$.\n    \n    \\setcounter{saveenum2}{\\value{enumi}}\n    \\end{enumerate}\n Moreover, if $\\lambda \\geq 1$ and $\\A\\subset \\lambda B_{\\C(K)}$ is a closed, symmetric and convex set separating the points of $K$ such that $f\\circ r_s\\in\\A$, for every $f\\in\\A$ and $s\\in\\Gamma$, then those conditions are also equivalent to the following one.\n \\begin{enumerate}[label = (\\roman*)]\n\\setcounter{enumi}{\\value{saveenum2}}\n \\item\\label{it:sigmaEmbedding} There exists $\\HH\\subset \\A$ such that the mapping $\\varphi:K\\to [-1,1]^\\HH$ defined as $\\varphi(x)(h):=\\tfrac{h}{\\lambda}(x)$, for every $h\\in\\HH$ and $x \\in K$, is a homeomorphic embedding and $\\varphi[D(\\mathfrak{s})]\\subset \\Sigma(\\HH)$.\n \\end{enumerate}\n\\end{thmx}\n\nNote that Theorem \\ref{thm:Intro3} provides a characterization of Valdivia compacta, since a compact space is Valdivia if and only if it admits a commutative retractional skeleton. \n\nIn Section~\\ref{sec:semiEberlein} we prove (slightly more general versions of) Theorem~\\ref{thm:Intro1} and Theorem~\\ref{thm:Intro2}. Section~\\ref{sec:contImages} is devoted to applications (in particular to the proof of Theorem~\\ref{thm:Intro4}) and  Section~\\ref{sec:questions} is devoted to open problems and remarks.\n\n\\section{Notation and preliminary results}\\label{sec:prelim}\n\nWe use standard notations from topology and Banach space theory as can be found in \\cite{Eng} and \\cite{FHHMZ}.\n\nFor a set $I$, we define\n\\[\\Sigma(I):=\\{x\\in \\er^{I}\\colon |\\suppt(x)|\\leq \\omega\\},\n\\]\nwhere $\\suppt(x)=\\{i\\in I\\colon x(i)\\neq 0\\}$ is the \\emph{support} of $x$. Given a subset $S$ of $I$ we denote the characteristic   function of $S$ by $1_S$.\n\nAll topological spaces are assumed to be Tychonoff. Let $T$ be a topological space.  A subset $S\\subset T$ is said to be \\emph{countably closed} if $\\overline{C}\\subset S$, for every countable subset $C\\subset S$. We denote by $\\w(T)$ the weight of $T$, by $\\C(T,T)$ the set of continuous functions from $T$ to $T$ and by  $\\beta T$ the \\v{C}ech-Stone compactification of $T$. If $T$ is compact, then as usual $\\C(T)$ denotes the Banach algebra of real-valued continuous functions defined on $T$, endowed with the supremum norm. Moreover, if $\\A\\subset \\C(T)$, we denote by $\\alg(\\A)$ the algebraic hull of $\\A$ in the algebra $\\C(T)$.\nRecall that a compact space $T$  is said to be \\emph{Valdivia} if there is a homeomorphic embedding $h:T\\to\\er^I$ such that $h^{-1}[\\Sigma(I)]$ is dense in $T$, we refer to \\cite{K00} for a survey in this subject.\n\nLet $(\\Gamma,\\leq)$ be an up-directed partially ordered set. We say that a sequence $(s_n)_{n\\in\\omega}$ of elements of $\\Gamma$ is increasing if $s_n\\leq s_{n+1}$, for every $n\\in\\omega$. We say that  $\\Gamma$ is \\emph{$\\sigma$-complete} if for every increasing sequence $(s_n)_{n\\in\\omega}$ in $\\Gamma$ there exists $\\sup_n s_n$ in $\\Gamma$. We say that $\\Gamma'\\subset\\Gamma$ is \\emph{cofinal} in $\\Gamma$ if for every $s_0\\in \\Gamma$ there is $s\\in\\Gamma'$ with $s\\geq s_0$. If $\\Gamma$ is $\\sigma$-complete and $A\\subset \\Gamma$, we denote by $A_{\\sigma}$ the smallest $\\sigma$-closed subset of $\\Gamma$ containing $A$. Notice that, by \\cite[Proposition 2.3]{K20}, if $A$ is up-directed, then $A_{\\sigma}$ is up-directed.\n\n\\begin{defin}\\label{def: r-skeleton}\nFollowing \\cite{CK15}, a \\emph{retractional skeleton} in a countably compact space $K$ is a family of continuous retractions $\\mathfrak s=(r_{s})_{s\\in\\Gamma}$ on $K$ indexed by an up-directed, $\\sigma$-complete partially ordered set $\\Gamma$, such that:\n\\begin{enumerate}[label = (\\roman*)]\n\\item $r_{s}[K]$ is a metrizable compact space for each $s\\in\\Gamma$,\n\\item $s,t\\in\\Gamma$, $s\\leq t$ then $r_{s}=r_t\\circ r_s=r_s\\circ r_t$,\n\\item given an increasing sequence $(s_n)_{n\\in\\omega}$ in $\\Gamma$, if $s=\\sup_{n\\in\\omega}s_{n}\\in\\Gamma$, then $r_{s}(x)=\\lim_{n\\to \\infty}r_{s_{n}}(x)$, for every $x\\in K$,\n\\item for every $x\\in K$, $x=\\lim_{s\\in\\Gamma}r_{s}(x)$.\n\\end{enumerate}\nWe say that $\\bigcup_{s\\in\\Gamma}r_{s}[K]$ is the set \\emph{induced} by the retractional skeleton $\\mathfrak s$ and we denote it by $D(\\mathfrak{s})$. We say that $\\mathfrak{s}$ is \\emph{commutative} if we have $r_s\\circ r_t = r_t \\circ r_s$ for every $s,t\\in\\Gamma$. We say that $\\mathfrak{s}$ is \\emph{full} if $D(\\mathfrak{s}) = K$.\n\\end{defin}\n\nThe following preliminary result will be used in what follows quite frequently. It seems to be new even though it could be known to some experts as well.\n\n\\begin{lemma}\\label{l: retraction from up-directed subset}\nLet $K$ be a compact space. Suppose that $K$ has a retractional skeleton $\\mathfrak{s}=(r_s)_{s\\in \\Gamma}$. Let $\\Gamma^{'}\\subset \\Gamma$ be an up-directed subset, then the mapping $R_{\\Gamma^{'}}:K \\to K$ defined by $R_{\\Gamma^{'}}(x)=\\lim_{s\\in \\Gamma^{'}}r_{s}(x)$ is a continuous retraction and $R_{\\Gamma^{'}}[K] = \\overline{\\bigcup_{s\\in\\Gamma'}r_s[K]}$.\nMoreover, the following holds.\n\\begin{enumerate}[label = (\\roman*)]\n\\item\\label{it:countableUpDirected} If $\\Gamma'$ is countable, then $s=\\sup\\Gamma'$ exists and we have $R_{\\Gamma'} = r_s$.\n\\item\\label{it:generalUpDirected} If $\\M$ is an up-directed subset of $\\mathcal{P}(\\Gamma)$ such that each $M\\in\\M$ is up-directed. Then $\\lim_{M\\in\\M}R_{M}(x) = R_{\\bigcup \\M}(x)$, $x\\in K$.\n\\item\\label{it:sigmaClosure} For every $s\\in (\\Gamma')_\\sigma$ we have that $r_s[K]\\subset R_{\\Gamma'}[K]$ and $r_s\\circ R_{\\Gamma'} = R_{\\Gamma'}\\circ r_s$.\n\\item\\label{it:skeletonOnTheSubspace} $(r_s|_{R_{\\Gamma'}[K]})_{s\\in(\\Gamma')_\\sigma}$ is a retractional skeleton on $R_{\\Gamma'}[K]$ with induced set $D(\\mathfrak{s})\\cap R_{\\Gamma'}[K]$.\n\\item\\label{it:commutativeCase} If $\\mathfrak{s}$ is commutative, then $D(\\mathfrak{s})\\cap R_{\\Gamma'}[K] = R_{\\Gamma'}[D(\\mathfrak{s})]$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}Let us start by proving that the mapping $R_{\\Gamma'}$ is well-defined. In order to do that fix $x\\in K$ and suppose that $(r_{s}(x))_{s\\in \\Gamma^{'}}$ is an infinite set (otherwise the assertion would be trivial). Since $K$ is compact, there exists a cluster point $x_1\\in K$ for the net $(r_{s}(x))_{s\\in \\Gamma^{'}}$. Let us show that such a cluster point $x_1$ is unique. Indeed, let $x_{1}\\neq x_{2}$ be two cluster points of $(r_{s}(x))_{s\\in \\Gamma^{'}}$. Let  $U_{1},U_{2}\\subset K$ be two open subsets such that $x_{1}\\in U_{1}$, $x_{2}\\in U_{2}$ and $\\overline{U_{1}}\\cap \\overline{U_{2}}=\\emptyset$. Let $(s_n)_{n<\\omega}, (t_{n})_{n<\\omega}\\subset \\Gamma'$ be two increasing sequences of indexes\nsuch that $s_n\\leq t_n\\leq s_{n+1}$, $r_{s_{n}}(x)\\in U_{1},$ and $r_{t_{n}}(x)\\in U_{2}$, for every $n\\in \\omega$. Since $\\Gamma$ is $\\sigma$-complete, we have that $\\sup_{n\\in\\omega}s_{n}=\\sup_{n\\in\\omega}t_{n}=s\\in \\Gamma$. Then $r_{s}(x)\\in \\overline{U_{1}}\\cap \\overline{U_{2}}$, a contradiction. Therefore $R_{\\Gamma'}$ is well-defined.\\\\\nThe map $R_{\\Gamma^{'}}$ is continuous. Indeed, let $(x_{\\lambda})_{\\lambda\\in\\Lambda}$ be a net converging to $x\\in K$. Up to taking a subnet we may assume without loss of generality that $R_{\\Gamma^{'}}(x_{\\lambda})$ converges to $y$. Suppose by contradiction $y\\neq R_{\\Gamma^{'}}(x)$, then there are two open subsets $U,V\\subset K$ with $y\\in U$ and $ R_{\\Gamma^{'}}(x)\\in V$, such that $\\overline{U}\\cap\\overline{V}=\\emptyset$. We find recursively two increasing sequences of indexes $(s_n)_{n<\\omega}$ in $\\Gamma^{'}$ and $(\\lambda_n)_{n<\\omega}$ in $\\Lambda$ such that $r_{s_k}(x_{\\lambda_i})\\in V \\mbox{ if } i\\geq k$ and $r_{s_k}(x_{\\lambda_i})\\in U \\mbox{ if } i< k$.\\\\\nLet us sketch the recursion here. Since $R_{\\Gamma^{'}}(x_{\\lambda})\\to y$, there exists $\\lambda_0\\in\\Lambda$ such that $R_{\\Gamma^{'}}(x_{\\lambda})\\in U$ for every $\\lambda\\geq \\lambda_0$.  Since $r_s(x)\\stackrel{\\Gamma'}{\\to}R_{\\Gamma' (x)}$, there exists $s_0\\in \\Gamma^{'}$ such that $r_{t}(x)\\in V$ for every $t\\geq s_0$. Since $r_s(x_{\\lambda_0})\\stackrel{\\Gamma'}{\\to}R_{\\Gamma^{'}}(x_{\\lambda_0})\\in U$, there exists $s_1\\geq s_0$ such that $r_t(x_{\\lambda_0})\\in U$ for every $t\\geq s_1$. By the continuity of $r_{s_1}$, we have $r_{s_1}(x_{\\lambda})\\to r_{s_1}(x)\\in V$; hence there exists $\\lambda_1\\geq\\lambda_0$ such that $r_{s_1}(x_{\\lambda})\\in V$ for every $\\lambda\\geq\\lambda_1$. We proceed recursively in an obvious way.\\\\\nSince $\\Gamma$ is $\\sigma$-complete, $s=\\sup_{k\\in \\omega}s_k$ belongs to $\\Gamma$. Hence $r_{s_k}(x_{\\lambda_i})$ converges to $r_{s}(x_{\\lambda_i})\\in\\overline{U}$ for every $i\\in \\omega$. Moreover, by compactness we have $\\bigcap_{k\\in\\omega}\\overline{(x_{\\lambda_i})_{i\\geq k}}\\neq \\emptyset$, so we may pick $\\tilde{x}\\in\\bigcap_{k\\in\\omega}\\overline{(x_{\\lambda_i})_{i\\geq k}}$. \nWe observe that $r_{s_k}(\\tilde{x})\\in \\overline{V}$ for every $k\\in\\omega$, hence $r_s(\\tilde{x})\\in \\overline{V}$. On the other hand $r_{s_k}(x_{\\lambda_i})\\to r_s(x_{\\lambda_i})\\in \\overline{U}$ for every $i\\in\\omega$; therefore $r_s(\\tilde{x})\\in\\overline{U}$, a contradiction. Thus, $R_{\\Gamma'}$ is continuous.\\\\ \nLet us check that $R_{\\Gamma'}$ is a retraction. Indeed, pick $x\\in K$. Then\n\\begin{equation*}\n\\begin{split}\nR_{\\Gamma^{'}}(R_{\\Gamma^{'}}(x))&=\\lim_{t\\in\\Gamma^{'}}r_t(\\lim_{s\\in\\Gamma^{'}}r_s(x))\n=\\lim_{t\\in\\Gamma^{'}}\\lim_{s\\in\\Gamma^{'}}r_t(r_s(x))\\\\\n&=\\lim_{t\\in\\Gamma'}\\lim_{s\\in\\Gamma', s\\geq t}r_t(r_s(x))\n=\\lim_{t\\in\\Gamma^{'}}r_t(x)=R_{\\Gamma^{'}}(x).\n\\end{split}\n\\end{equation*}\nFinally, for every $s\\in\\Gamma'$ and $x\\in r_s[K]$ we have $R_{\\Gamma'}(x)=\\lim_{t\\in\\Gamma', t\\geq s} r_t(r_{s}(x))=x$ so we obtain $\\overline{\\bigcup_{s\\in\\Gamma'}r_{s}[K]}\\subset R_{\\Gamma'}[K]$ and the other inclusion follows from the definition of $R_{\\Gamma'}$.\\\\\nIt remains to prove the ``Moreover'' part. We first observe (see the proof of \\cite[Proposition 2.3]{K20} for more details) that $(\\Gamma')_{\\sigma}$ is directed, $\\sigma$-closed and $(\\Gamma')_{\\sigma}=\\bigcup_{\\alpha<\\omega_1}B_{\\alpha}$, where\n\\begin{itemize}\n    \\item $B_0=\\Gamma'$;\n    \\item $B_{\\alpha+1}=B_{\\alpha}\\cup \\{\\sup t_n : \\; (t_n) \\mbox{ is an increasing sequence in $B_{\\alpha}$}\\};$\n    \\item $B_{\\lambda}=\\bigcup_{\\alpha<\\lambda}B_{\\alpha}$, if $\\lambda<\\omega_1$ is a limit ordinal.\n\\end{itemize}\n\\noindent\\ref{it:countableUpDirected}: If $\\Gamma'$ is countable, then we can find an increasing sequence $(s_n)_{n\\in\\omega}$ from $\\Gamma$ with $\\sup_n s_n = s = \\sup \\Gamma'$. Then, using that the sequence $\\{s_n\\colon n\\in\\omega\\}$ is cofinal in $\\Gamma'$, we obtain $R_{\\Gamma'} = R_{\\{s_n\\colon n\\in\\omega\\}} = r_s$.\\\\\n\\ref{it:generalUpDirected}: Suppose that $\\M\\subset \\mathcal{P}(\\Gamma)$ is up-directed and that each $M\\in\\M$ is up-directed. Put $M_\\infty:=\\bigcup_{M\\in \\M} M$, fix $x\\in K$ and open set $U$ such that $R_{M_{\\infty}}(x) \\in U$. Let $V$ be an open neighborhood of $R_{M_{\\infty}}(x)$ such that $\\overline{V} \\subset U$. Then, there exists $s_0 \\in M_{\\infty}$ such that $r_s(x) \\in V$, for every $s\\in M_\\infty$ with $s\\geq s_0$.\nBy the definition of $M_{\\infty}$, there exists $M_0\\in \\M$ such that $s_0 \\in M_0$. If $M \\in \\M$ and  $M_0 \\subset M$, then $s_0 \\in M$. This implies that the set $\\{s\\in M\\colon s\\geq s_0\\}$ is cofinal in $M$ and so we have \n\\[R_M(x)=\\lim_{s \\in M}r_s(x)= \\lim_{s \\in M, s \\ge s_0}r_s(x)\\in \\overline{V} \\subset U.\\]\nThis shows that $\\lim_{M\\in\\M} R_M(x) = R_{M_\\infty}(x)$.\\\\\n\\ref{it:sigmaClosure}:  We prove inductively that for every $\\alpha<\\omega_1$ and $s\\in B_\\alpha$, it holds that $r_s[K]\\subset R_{\\Gamma'}[K]$ and $r_s\\circ R_{\\Gamma'} = R_{\\Gamma'}\\circ r_s$. Pick $s\\in B_0 = \\Gamma'$. Then $r_s[K]\\subset R_{\\Gamma'}[K]$, since $R_{\\Gamma^{'}}[K] = \\overline{\\bigcup_{s\\in\\Gamma'}r_s[K]}$. Moreover, for $x\\in K$ we have \n\\[\nr_s\\big(R_{\\Gamma'}(x)\\big)=\\lim_{t \\in \\Gamma', t \\ge s} r_s\\big(r_t(x)\\big)=\\lim_{t \\in \\Gamma', t \\ge s} r_t\\big(r_s(x)\\big)=R_{\\Gamma'}\\big(r_s(x)\\big).\\]\nNow, fix $\\alpha<\\omega_1$ and suppose that the result holds for every $\\gamma<\\alpha$. If $\\alpha$ is a limit ordinal, then it follows easily from the induction hypothesis that the result also holds for $\\alpha$. Suppose that $\\alpha=\\gamma+1$. Let $s\\in B_{\\alpha}$, $x\\in r_{s}[K]$ and $(s_n)_{n\\in\\omega}\\subset B_{\\gamma}$ such that $\\sup s_n=s$. By the induction hypothesis, we have that $R_{\\Gamma'}(r_{s_n}(x))=r_{s_n}(x)$, for every $n \\in \\omega$ and therefore:\n\\[R_{\\Gamma'}(x)=\\lim_{n\\in\\omega}R_{\\Gamma'}(r_{s_n}( x))=\\lim_{n\\in\\omega}r_{s_n}(x)=r_s(x)=x.\\]\nWith a similar argument, we also conclude that $r_s\\circ R_{\\Gamma'} = R_{\\Gamma'}\\circ r_s$.\\\\\n\\ref{it:skeletonOnTheSubspace}: First, we \\emph{claim} that for every $x\\in R_{\\Gamma'}[K]$ we have $\\lim_{s\\in (\\Gamma')_\\sigma} r_s(x) = x$. Indeed, since $(\\Gamma')_{\\sigma}$ is up-directed, it holds that $R_{(\\Gamma')_{\\sigma}}[K]=\\overline{\\bigcup_{s \\in (\\Gamma')_{\\sigma}}r_s[K]}$, which implies that $R_{\\Gamma'}[K] \\subset R_{(\\Gamma')_{\\sigma}}[K]$ and therefore if $x \\in  R_{\\Gamma'}[K]$, then $x=R_{(\\Gamma')_\\sigma}(x)=\\lim_{s\\in (\\Gamma')_\\sigma} r_s(x)$.\n\nUsing \\ref{it:sigmaClosure} and the previous claim, it is easy to see that $\\mathfrak{s'}:=(r_s|_{R_{\\Gamma'}[K]})_ {s\\in(\\Gamma')_\\sigma}$ is a retractional skeleton on $R_{\\Gamma'}[K]$ with $D(\\mathfrak{s'}) = \\bigcup_{s\\in(\\Gamma')_\\sigma} r_s[R_{\\Gamma'}[K]]\\subset D(\\mathfrak{s})\\cap R_{\\Gamma'}[K]$. On the other hand, since $D(\\mathfrak{s})$ is Fr\\'echet-Urysohn (see \\cite[Theorem 32]{kubis09}), for every $x\\in D(\\mathfrak{s})\\cap R_{\\Gamma'}[K]$ there is a sequence $(s_n)_{n\\in\\omega}$ in $\\Gamma'$ with $r_{s_n}(x)\\to x$ and therefore $x\\in D(\\mathfrak{s'})$, because $D(\\mathfrak{s'})$ is a countably closed set. Thus, we have that $D(\\mathfrak{s'}) = D(\\mathfrak{s})\\cap R_{\\Gamma'}[K]$.\\\\\n\\ref{it:commutativeCase}: If $(r_s)_{s\\in\\Gamma}$ is commutative, then for every $s\\in\\Gamma$ and $x\\in K$ we have\n\\[R_{\\Gamma'}(r_s(x)) = \\lim_{t\\in\\Gamma'} r_t(r_s(x)) = r_s(\\lim_{t\\in\\Gamma'} r_t(x))\\in D(\\mathfrak{s}),\\]\nwhich implies $R_{\\Gamma'}[D(\\mathfrak{s})]\\subset D(\\mathfrak{s})$ and so $R_{\\Gamma'}[D(\\mathfrak{s})] = D(\\mathfrak{s})\\cap R_{\\Gamma'}[K]$.\n\\end{proof}\n\n\\section{Retractions associated to suitable models}\\label{sec:models}\n\nThe most important results concerning projectional skeletons were originally proved in \\cite{kubis09} using the so-called ``method of suitable countable models'' which replaces inductive constructions by ``suitable countable models''. The presentation of this method was further simplified in \\cite{C12} and later it was also used in the context of spaces admitting retractional skeletons, see e.g. \\cite{C14} or \\cite{CK15}. Here we further generalize and deeply investigate this method. The main difference of our approach is that we do not consider only countable models. The main outcome of this section is that for every (not necessarily countable) suitable model we can define a canonical retraction associated to this model. Those canonical retractions will be deeply used in the remainder of the paper.\n\nProperties of retractions associated to suitable models are summarized in Theorem~\\ref{thm:canonicalRetraction} and, consequently, in Proposition~\\ref{prop:rri} we obtain a continuous chain of retractions associated to suitable models with very pleasant properties. As an example of an application we show in Theorem~\\ref{thm:subskeletons} that we may in a certain way combine properties of countably many retractional skeletons.\n\n\\subsection{Preliminaries}\n\nHere we settle the notation and give some basic observations concerning suitable models. We refer the interested reader to \\cite{C12} and \\cite{CK15}, where more details about this method may be found (warning: in \\cite{C12, CK15} only \\textbf{countable} models were considered, while here we consider suitable models which are not necessarily countable).\n\nAny formula in the set theory can be written using symbols $\\in,=,\\wedge,\\vee,\\neg,\\rightarrow,\\leftrightarrow,\\exists,(,),[,]$ and symbols for variables. On the other hand, it would be very laborious and pointless to use only the basic language of the set theory. For example, we often write $x < y$ and we know, that in fact this is a shortcut for a formula $\\varphi(x,y,<)$ with all free variables shown. Thus, in what follows we will use this extended language of the set theory as we are used to, having in mind that the formulas we work with are actually sequences of symbols from the list mentioned above.\n\nLet $N$ be a fixed set and $\\phi$ be a formula. Then the {\\em relativization of $\\phi$ to $N$} is the formula $\\phi^N$ which is obtained from $\\phi$ by replacing each quantifier of the form ``$\\exists x$'' by ``$\\exists x\\in N$'' (and if we extend our language of  set theory by the symbol ``$\\forall$'' then we replace also each quantifier of the form ``$\\forall x$'' by ``$\\forall x\\in N$'').\n\nIf $\\phi(x_1,\\ldots,x_n)$ is a formula with all free variables shown, then {\\em $\\phi$ is absolute for $N$} if\n\\[\n\\forall a_1,\\ldots,a_n\\in N\\quad (\\phi^N(a_1,\\ldots,a_n) \\leftrightarrow \\phi(a_1,\\ldots,a_n)).\n\\]\n\n\\begin{defin}\n\\rm Let $\\Phi$ be a finite list of formulas and $X$ be any set.\nLet $M \\supset X$ be a set such that each $\\phi$ from $\\Phi$ is absolute for $M$.\nThen we say that $M$ \\emph{is a suitable model for $\\Phi$ containing $X$}.\nThis is denoted by $M \\prec (\\Phi; X)$.\n\\end{defin}\n\nNote that suitable models do exist.\n\n  \\begin{thm}[see Theorem IV.7.8 in \\cite{kunen}]\\label{thm:modelExists}\n Let $\\Phi$ be a finite list of formulas and $X$ be any set. Then there exists a set $R$ such that $R\\prec (\\Phi; X)$ and $|R|\\leq \\max(\\omega,|X|))$ and moreover, for every countable set $Z\\subset R$ there exists $M\\subset R$ such that $M\\prec(\\Phi;\\; Z)$ and $M$ is countable.\n\\end{thm}\n\n\n\nThe fact that certain formula is absolute for $M$ will always be used exclusively in order to satisfy the assumption of the following lemma. Using this lemma we can force the model $M$ to contain all the needed objects created (uniquely) from elements of $M$. We give here the well-known proof for the convenience of the reader.\n\n\\begin{lemma}\\label{l:unique-M}\nLet $\\phi(y,x_1,\\ldots,x_n)$ be a formula with all free variables shown and let $M$ be a set that is absolute for $\\phi$ and for $\\exists y \\phi(y,x_1, \\ldots, x_n)$. If $a_1, \\ldots, a_n\\in M$ are such that there exists a set $u$ satisfying $\\phi(u, a_1, \\ldots, a_n)$, then there exists a set $v \\in M$ satisfying $\\phi(v, a_1, \\ldots, a_n)$. Moreover, if there exists a unique set $u$ such that $\\phi(u, a_1, \\ldots, a_n)$, then $u \\in M$.\n\\end{lemma}\n\\begin{proof}\nIt follows from the absoluteness of the formula $\\exists y \\phi(y,x_1, \\ldots, x_n)$, that there exists $v \\in M$ such that $\\phi^M(v, a_1, \\ldots, a_n)$. Therefore the absoluteness of the formula $\\phi(y,x_1, \\ldots, x_n)$ implies that $\\phi(v, a_1, \\ldots, a_n)$ holds. Moreover, if $u$ is the only set such that $\\phi(u, a_1, \\ldots, a_n)$, then $v=u$ and thus $u \\in M$.\n\\end{proof}\n\n\\begin{convention}\nWhenever we say ``\\emph{for any suitable model $M$ (the following holds \\dots)}''\nwe mean that  ``\\emph{there exists a finite list of formulas $\\Phi$ and a countable set $Y$ such that for every $M \\prec (\\Phi;Y)$ (the following holds \\dots)}''.\n\\end{convention}\n\nIf $M$ is a suitable model and $\\langle X,\\tau\\rangle$ is a topological space (or is $\\langle X,d\\rangle$ a metric space or is $\\langle X,+,\\cdot,\\|\\cdot\\|\\rangle$ a normed linear space) then we say that \\emph{$M$ contains $X$} if $\\langle X,\\tau\\rangle\\in M$, $\\langle X,d\\rangle\\in M$ and $\\langle X,+,\\cdot,\\|\\cdot\\|\\rangle\\in M$, respectively.\n\nThe following summarizes certain easy observations. For the proofs we refer the reader to \\cite[Sections 2 and 3]{C12}, where it is assumed that $M$ is countable but this fact is not used in proofs. \n\n\\begin{lemma}\\label{l:basics-in-M}\nFor any suitable model $M$ the following holds:\n\\begin{enumerate}\n    \\item \\label{number-sets} $\\mathbb{Q}, \\omega, \\er \\in M$ and $M$ contains the usual operations and relations on $\\er$.\n    \\item\\label{it: Dom-Rng-in-M} For every function $f\\in M$ we have $\\dom f\\in M$, $\\rng f\\in M$ and $f[M\\cap \\dom f]\\subset M$.\n    \\item \\label{finite-in-M} For every finite set $A$ we have $A\\in M$ if and only if $A\\subset M$.\n    \\item \\label{belongIsContained}For every countable set $A\\in M$ we have $A\\subset M$. Moreover, if $\\kappa\\in M$ is a cardinal and $\\kappa\\subset M$ then for every $A\\in M$ with $|A|\\leq \\kappa$ we have $A\\subset M$.\n    \\item For every natural number $n>0$ and sets $a_1,\\ldots,a_n$ we have $\\{a_1,\\ldots,a_n\\}\\subset M$ if and only if $\\langle a_1,\\ldots,a_n\\rangle \\in M$.\n    \\item \\label{operations-sets-in-M} If $A,B\\in M$, then $A\\cap B\\in M$, $B\\setminus A\\in M$ and $A\\cup B\\in M$.\n    \\item \\label{linearSubspaceM}If $M$ contains a normed linear space $X$, then $\\overline{X\\cap M}$ is a linear subspace of $X$.\n\\end{enumerate}\n\\end{lemma}\n\nSome more easy observations are summarized in the following. \n\n\\begin{lemma}\\label{l:basics-in-M-2}\nFor any suitable model $M$ the following holds:\n\\begin{enumerate}\n    \\item\\label{it:upDirected} If $(\\Gamma,\\leq)$ is up-directed and $(\\Gamma,\\leq)\\in M$, then $\\Gamma\\cap M$ is up-directed.\n    \\item\\label{it:composition} If $f,g\\in M$ are functions and $f\\circ g$ is well-defined, then $f\\circ g\\in M$.\n    \\item \\label{it:inverse-in-M} If $f \\in M$ is a function which is one-to-one then $f^{-1}\\in M$.\n    \\item\\label{it:one-to-one-imageOfM} If $f\\in M$ is a function and $X\\in M$ is a subset of $\\dom f$, then $f[M\\cap X] = M\\cap f[X]$.\n    \\item\\label{it:exp-in-M} If $A$ and $B$ are sets and $A, B \\in M$, then $B^A \\in M$ and $A\\times B \\in M$.\n    \\item \\label{it:Pi-in-M} For every set $I\\in M$ and $X\\subset \\er^I$ with $X\\in M$ we have $\\pi\\in M$, where $\\pi:I\\to {\\er^X}$ is the mapping given for $i\\in I$ and $x\\in X$ as $\\pi(i)(x):=x(i)$.\n    \\item\\label{it:supportSubsetOfM} Let $X\\subset \\Sigma(I)$ be such that $I\\in M$. Then $\\suppt(x)\\subset M$ for every $x\\in X\\cap M$.\n    \\item \\label{it:containsspacecontinuousfunctions}If $(X,\\tau)$ is a topological space with $\\{X,\\tau\\}\\subset M$, then $\\{\\C(X),+,\\cdot,\\otimes\\}\\subset M$ (where $\\cdot$ is a multiplication by real numbers and $\\otimes$ pointwise multiplication of functions). Moreover, if $X$ is a compact space then $M$ contains the normed linear space $\\C(X)$, $\\overline{\\C(X)\\cap M}$ is a closed subalgebra of $\\C(X)$ and $1\\in\\C(X)\\cap M$.  \n    \\item\\label{it:algebraDenseInFunctionsFromM} If $(K,\\tau)$ is a compact space, $\\A\\subset \\C(K)$ separates the points of $K$ and  $\\{K,\\tau,\\A\\}\\subset M$, then $\\overline{\\alg((\\A\\cup \\{1\\})\\cap M)} = \\overline{\\C(K)\\cap M}$.\n    \\item\\label{it:metrizableSubspace} If $(K,\\tau)$ is a compact space , $K'\\subset K$ is closed and metrizable with $\\{K',\\tau,K\\}\\in M$ then $\\C(K)\\cap M$ separates the points of $K'$ and $K'\\subset \\overline{K'\\cap M}$.\n    \\item\\label{it:norm} If $(K,\\tau)$ is a compact space, $D\\subset K$ a dense subset with $\\{K,D,\\tau\\}\\subset M$ and $f\\in\\C(K)\\cap M$, then $\\|f\\| = \\|f|_{D\\cap M}\\|$\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nLet $S$ and $\\Phi$ be the countable set and the list of formulas from the statement of Lemma~\\ref{l:basics-in-M}, where $\\Phi$ is enriched by formulas (and their subformulas) marked by $(*)$ in the proof below. Let $M\\prec (\\Phi; S)$. Then $M$ satisfies \\eqref{it:upDirected}, \\eqref{it:composition}, \\eqref{it:inverse-in-M}, \\eqref{it:exp-in-M}, and \\eqref{it:Pi-in-M}. Indeed those items follow easily using Lemma~\\ref{l:unique-M} and the absoluteness of the following formulas (and their subformulas)\n\\[\n\\forall u,v \\in \\Gamma\\,\\exists w\\in\\Gamma\\, w\\geq u,v,\\eqno{(*)}\n\\]\n\\[\n\\exists h \\quad (h= f\\circ g).\\eqno{(*)}\n\\]\n\\[\n\\exists h \\quad (h=f^{-1}).\\eqno{(*)}\n\\]\n\\[\n\\exists W \\quad (W = B^A).\\eqno{(*)}\n\\]\n\\[\n\\exists W \\quad (W = B\\times A).\\eqno{(*)}\n\\]\n\\[\n\\exists \\pi\\in ({\\er^X})^{I} \\quad (\\forall i\\in I \\ \\forall x\\in X: \\pi(i)(x) = x(i)).\\eqno{(*)}\n\\]\n\n\\eqref{it:one-to-one-imageOfM}: By Lemma~\\ref{l:basics-in-M} \\eqref{it: Dom-Rng-in-M}, we have that $f[M\\cap \\dom f]\\subset M$ so in particular $f[M\\cap X]\\subset M\\cap f[X]$. For the other inclusion pick $x\\in f[X]\\cap M$. Using Lemma~\\ref{l:unique-M} and the absoluteness of the following formula (and its subformulas)\n\\[\n\\exists y\\in X\\quad  (f(y)=x),\\eqno{(*)}\n\\]\nthere exists $y\\in M\\cap X$ with $f(y)=x$ and so $x\\in f[M\\cap X]$.\\\\\n\\eqref{it:supportSubsetOfM}: Pick $x\\in X\\cap M$. Using Lemma~\\ref{l:unique-M} and absoluteness of the following formula (and its subformulas)\n\\[\n\\exists D\\subset I\\quad  (i\\in D \\Leftrightarrow x(i)\\neq 0),\\eqno{(*)}\n\\]\nwe obtain that $\\suppt(x)\\in M$. Since $\\suppt(x)$ is a countable set, by Lemma~\\ref{l:basics-in-M} \\eqref{belongIsContained} we obtain that $\\suppt(x)\\subset M$.\n\n\\eqref{it:containsspacecontinuousfunctions}: Using Lemma~\\ref{l:unique-M} and absoluteness of the following formulas (and their subformulas) \n\\[\n\\exists \\C(X)\\in \\er^X (\\forall f\\in \\er^X: f\\in \\C(X)\\Leftrightarrow f \\text{ is continuous}),\\eqno{(*)}\n\\]\n\\[\n\\exists +\\in \\C(X)^{\\C(X)\\times \\C(X)} (\\forall f, g\\in \\C(X)\\;\\forall x\\in X: + (f,g)(x) = f(x) + g(x)),\\eqno{(*)}\n\\]\n\\[\n\\exists \\cdot\\in \\C(X)^{\\er\\times \\C(X)} (\\forall \\alpha\\in\\er \\forall f\\in \\C(X)\\;\\forall x\\in X: \\cdot (\\alpha,f)(x) = \\alpha f(x)),\\eqno{(*)}\n\\]\n\\[\n\\exists \\otimes\\in \\C(X)^{\\C(X)\\times \\C(X)} (\\forall f, g\\in \\C(X)\\;\\forall x\\in X: \\otimes (f,g)(x) = f(x)g(x)),\\eqno{(*)}\n\\]\nwe obtain that $\\C(X)\\in M$ and that $\\{+,\\cdot,\\otimes\\}\\subset M$. Morevoer, if $X$ is a compact space, then using Lemma~\\ref{l:unique-M} and the absoluteness of the following formula (and its subformulas)\n\\[\n\\exists \\|\\cdot\\|_\\infty\\in \\er^{\\C(X)}\\quad (\\forall f\\in\\C(X):\\; \\|\\cdot\\|(f) = \\sup_{x\\in X} |f(x)|),\\eqno{(*)}\n\\]\nwe obtain that $M$ contains the normed linear space $\\C(X)$. Thus, by Lemma~\\ref{l:basics-in-M} \\eqref{linearSubspaceM}, $\\overline{\\C(X)\\cap M}$ is a closed subspace of $\\C(X)$ and, since $\\otimes\\in M$, $\\C(X)\\cap M$ is closed under multiplication and therefore $\\overline{\\C(X)\\cap M}$ is closed under multiplication as well. Finally, using Lemma~\\ref{l:unique-M} and absoluteness of the following formula (and its subformulas)\n\\[\\exists f\\in\\C(X)\\quad (\\forall x\\in X\\;f(x)=1),\\eqno{(*)}\\]\nwe obtain that $1\\in\\C(X)\\cap M$.\\\\\n\\eqref{it:algebraDenseInFunctionsFromM}: By \\eqref{it:containsspacecontinuousfunctions}, $\\overline{\\C(K)\\cap M}$ is a closed subalgebra of $C(K)$ that contains $(\\A\\cup \\{1\\})\\cap M$, so we have $\\overline{\\alg((\\A\\cup \\{1\\})\\cap M)} \\subset \\overline{\\C(K)\\cap M}$. For the other inclusion, pick $f\\in\\C(K)\\cap M$. By Lemma~\\ref{l:unique-M} and absoluteness of the following formula (and its subformulas)\n\\[\n\\exists A\\subset \\A\\; \\quad(A\\text{ is countable and }f\\in \\overline{\\operatorname{alg}(A\\cup \\{1\\})}),\\eqno{(*)}\n\\]\nthere is a countable set $A\\subset \\A$ with $A\\in M$ and $f\\in \\overline{\\operatorname{alg}(A\\cup \\{1\\})}$. By Lemma~\\ref{l:basics-in-M} \\eqref{belongIsContained}, we have that $A\\subset \\A\\cap M$. Therefore, using that $1\\in M$, we obtain\n\\[\\overline{\\alg((\\A\\cup \\{1\\})\\cap M)} = \\overline{\\operatorname{alg}((\\A\\cap M)\\cup \\{1\\})} \\supset \\overline{\\C(K)\\cap M}.\\]\n\\eqref{it:metrizableSubspace}: By \\eqref{it:containsspacecontinuousfunctions}, Lemma~\\ref{l:unique-M} and the absoluteness of the following formula (and its subformulas)\n\\[\n\\exists A\\subset \\C(K)\\quad (A\\text{ is countable and separates the points of $K'$}),\\eqno{(*)}\n\\]\nthere is a countable set $A\\subset \\C(K)$ with $A\\in M$ which separates the points of $K'$. By Lemma~\\ref{l:basics-in-M} \\eqref{belongIsContained}, we have that $A\\subset \\C(K)\\cap M$ so $\\C(K)\\cap M$ separates the points of $K'$. Therefore, since by \\eqref{it:containsspacecontinuousfunctions} the set $\\overline{\\C(K)\\cap M}$ is a closed algebra containing constant functions, Stone-Weierstrass theorem ensures that the set $\\{f|_{K'}\\colon f\\in \\overline{\\C(K)\\cap M}\\}$ is dense in $\\C(K')$, which implies that $\\{f|_{K'}\\colon f\\in \\C(K)\\cap M\\}$ is dense in $\\C(K')$ and therefore $\\{f^{-1}(-1\/2,1\/2)\\cap K'\\colon f\\in \\C(K)\\cap M\\}$ is an open basis of $K'$. Moreover, for every $f\\in\\C(K)\\cap M$ using Lemma~\\ref{l:unique-M} and the absoluteness of the following formula (and its subformulas)\n\\[\n\\exists x\\in f^{-1}(-1\/2,1\/2)\\cap K',\\eqno{(*)}\n\\]\nwe have that $f^{-1}(-1\/2,1\/2)\\cap (K'\\cap M)\\neq \\emptyset$ for every $f\\in \\C(K)\\cap M$ and therefore the set $K'\\cap M$ is dense in $K'$.\\\\\n\\eqref{it:norm}: Since $D\\subset K$ is a dense set, we have that $\\|f\\| = \\|f|_D\\|$. It follows from \\eqref{it:containsspacecontinuousfunctions} that $\\|\\cdot\\|\\in M$ and so $\\|f\\|\\in M$. Therefore, using Lemma~\\ref{l:unique-M} and the absoluteness of the following formula (and its subformulas) \n\\[\n\\forall n\\in\\omega\\;\\exists x\\in D\\quad (\\|f\\|-1\/n<|f(x)|<\\|f\\|+1\/n),\\eqno{(*)}\n\\]\nwe obtain that for every $n \\in \\omega$, there exists $x_n\\in D\\cap M$ such that $|f(x_n)|\\to \\|f\\|$. \n\\end{proof}\n\n\\subsection{Retractions associated to suitable models} Here we show that in a compact space with a retractional skeleton, for every suitable model there is a canonical retraction associated to it (see Definition~\\ref{def:canonicalRetraction}). The main outcome of this subsection is Theorem~\\ref{thm:canonicalRetraction}, where the properties of a canonical retraction are summarized.\n\nLemma~\\ref{l:abstraction} and Lemma~\\ref{l:retraction} are inspired by \\cite[Lemma 4.7]{CK15}, where something similar was proved for suitable models which are countable.\n\n\\begin{lemma}\\label{l:abstraction}\nFor every suitable model $M$ the following holds: Let $X$ be a set and $\\A\\subset \\er^X$ such that $\\{X\\}\\cup \\A\\subset M$. Consider the mapping $q_M:X\\to \\er^\\A$ defined for $x\\in X$ as $q_M(x)(f):=f(x)$, $f\\in\\A$. Then for every $B\\subset X$ with $B\\in M$ we have $q_M[B]\\subset \\overline{q_M[B\\cap M]}$.\n\\end{lemma}\n\\begin{proof}\nIn this proof we will use the identification of any $n\\in\\omega$ with the set $\\{0, ..., n-1\\}$. Further, denote by $\\B$ the set of all the open intervals with rational endpoints and by $\\B^{<\\omega}$ the set of all the functions whose domain is some $n\\in\\omega$ and whose values are in $\\B$.\n\nLet $S$ be the countable set from the statement of Lemma~\\ref{l:basics-in-M} enriched by $\\{\\B,\\B^{<\\omega}\\}$ and let $\\Phi$ be the list of formulas from the statement of Lemma~\\ref{l:basics-in-M} enriched by formulas (and their subformulas) marked by $(*)$ in the proof below. Let $M\\prec (\\Phi; S\\cup\\{X\\}\\cup \\A)$.\n\nFix $B\\subset X$ with $B\\in M$, a point $x\\in B$ and a basic neighborhood of a point $q_M(x)$; that is, let us pick finitely many functions $F\\subset \\A$ and a sequence of rational intervals such that $f(x)\\in I_f$, $f\\in F$ and consider the neighborhood \n\\[\n\tN:= \\{y\\in\\er^{\\A}\\setsep y(f)\\in I_f\\text{ for every }f\\in F\\}.\n\\]\nBy Lemma~\\ref{l:basics-in-M} \\eqref{finite-in-M}, we have that $F\\in M$ and by absoluteness of the formula\n\\[\n\\exists n\\in\\omega \\exists\\eta\\quad (\\eta\\text{ is a bijection between $n$ and $F$}),\\eqno{(*)}\n\\]\nand its subformulas, there is $n\\in\\omega$ and a bijection $\\eta\\in M$ between $n$ and $F$. \n\nLet us further define the mapping $\\xi:n\\to\\B$ by $\\xi(i) =I_{\\eta(i)}$. Since $\\xi\\in \\B^{<\\omega}\\in M$, it follows from Lemma~\\ref{l:basics-in-M} \\eqref{belongIsContained} that $\\xi\\in M$.\nBy Lemma~\\ref{l:unique-M} and the absoluteness of the formula (and its subformulas)\n\\[\n\\exists x\\in B \\quad(\\forall i<n: \\eta(i)(x)\\in \\xi(i)),\\eqno{(*)}\n\\]\nthere is a point $x_0\\in B\\cap M$ such that $q_M(x_0)\\in N$; hence, $q_M[B\\cap M]$ is dense in $q_M[B]$.\n\\end{proof}\n\n\\begin{lemma}\\label{l:retraction}For every suitable model $M$ the following holds: Let $(K, \\tau)$ be a compact space and $D\\subset K$ be a dense subset with $\\{K,D,\\tau\\}\\subset M$.\n\nIf $\\C(K)\\cap M$ separates the points of $\\overline{D\\cap M}$, then there exists a unique retraction $r_M:K\\to \\overline{D\\cap M}$ such that $f = f\\circ r_M$, for every $f\\in\\C(K)\\cap M$.\n\nMoreover, in this case\n\\begin{enumerate}\n    \\item for every $x\\in K$ and $\\A\\subset \\C(K)$ separating the points of $K$ with $\\A\\in M$, $r_M(x)$ is the unique point from $\\overline{D\\cap M}$ satisfying $f(r_M(x)) = f(x)$, for every $f\\in \\A\\cap M$. \n    \\item\\label{it: imageofclosure} if $B\\subset D$ and $B\\in M$, then $r_M[\\overline{B}] = \\overline{B\\cap M}$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof} Let $S$ and $\\Phi$ be the union of sets and the lists of formulas from the statements of Lemma~\\ref{l:basics-in-M}, Lemma~\\ref{l:basics-in-M-2} and Lemma~\\ref{l:abstraction}. Let $M\\prec (\\Phi; S\\cup\\{K,D,\\tau\\})$ be such that $\\C(K)\\cap M$ separates the points of $\\overline{D\\cap M}$. By Lemma~\\ref{l:basics-in-M-2} \\eqref{it:containsspacecontinuousfunctions}, we have that $\\C(K)\\in M$.\n\nLet us consider the mapping $q_M:K\\to \\er^{\\C(K)\\cap M}$ given by $q_M(x) = (f(x))_{f\\in\\C(K)\\cap M}$, $x\\in K$. Then $q_M$ is continuous and, by the assumption, $q_M|_{\\overline{D\\cap M}}$ is one-to-one; hence, $q_M|_{\\overline{D\\cap M}}$ is a homeomorphic embedding. Moreover, whenever $B\\subset D$ is such that $B\\in M$, then by Lemma~\\ref{l:abstraction} we have $\\overline{q_M[B\\cap M]}\\supset q_M[B]$ which implies $q_M[\\overline{B}] = q_M[\\overline{B\\cap M}]$.\n\nNow, put $r_M:= (q_M|_{\\overline{D\\cap M}})^{-1}\\circ q_M$. Then it is a continuous retraction with $r_M[K] = \\overline{D\\cap M}$. Moreover, for every $x\\in K$\n\\[\nr_M(x) = (q_M|_{\\overline{D\\cap M}})^{-1}\\circ q_M(x) = y,\n\\]\nwhere $y\\in K$ is the unique point such that $y\\in\\overline{D\\cap M}$ and $g(y) = g(x)$, for every $g\\in \\C(K)\\cap M$. Hence, for $f\\in \\C(K)\\cap M$ we have\n\\[\nf(r_M(x)) = f(y) = f(x).\n\\]\n\nIn order to see that $r_M$ is unique, let us consider another retraction $r':K \\to \\overline{D \\cap M}$ satisfying that $f = f\\circ r'$ for every $f\\in\\C(K)\\cap M$. Then, for every $x\\in K$, and every $f\\in \\C(K)\\cap M$ we have $f(r_M(x)) = f(x) = f(r'(x))$; hence, since $\\C(K)\\cap M$ separates the points of $r_M[K]$, it holds that $r_M(x) = r'(x)$. Since $x\\in K$ was arbitrary, we have $r_M = r'$.\nMoreover, given $y\\in \\overline{D\\cap M}$ such that $f(y)=f(x)$ for every $f\\in\\A\\cap M$, where $\\A\\subset \\C(K)$ is a set separating the points of $K$ with $\\A\\in M$, we obtain that $f(y)=f(x)$ for $f\\in \\overline{\\alg((\\A\\cup \\{1\\})\\cap M)} = \\overline{\\C(K)\\cap M}$ (the last equality follows from Lemma~\\ref{l:basics-in-M-2} \\eqref{it:algebraDenseInFunctionsFromM}) and so $y=r_M(x)$.\n\n\nFinally, if $B\\subset D$ is such that $B\\in M$ then by the above we have $q_M[\\overline{B}] = q_M[\\overline{B\\cap M}]$ and so $r_M[\\overline{B}] = r_M[\\overline{B\\cap M}] = \\overline{B\\cap M}$.\n\\end{proof}\n\nLet us note that a compact space $K$ admits a retractional skeleton if and only if there exists a dense set $D\\subset K$ such that for every suitable model $M$ which is moreover countable, the set $\\C(K)\\cap M$ separates the points of $\\overline{D\\cap M}$, see e.g. \\cite[Theorem 4.9]{C14} or \\cite[Theorem 19.16]{KKLbook} (that is, the assumption of Lemma~\\ref{l:retraction} is satisfied for suitable models which are countable). The following shows that we do not need to assume countability of the model.\n\n\\begin{prop}\\label{p:characterizationRSkeletonNotCountableModels}\nFor every suitable model $M$ the following holds: If $(K,\\tau)$ is a compact space and $D\\subset K$ is a subset of a set induced by a retractional skeleton with $\\{D,K,\\tau\\}\\subset M$, then $\\C(K)\\cap M$ separates the points of $\\overline{D\\cap M}$.\n\\end{prop}\n\\begin{proof}\nLet $S$ be the union of sets from the statements of Lemma~\\ref{l:basics-in-M} and Lemma~\\ref{l:basics-in-M-2} and let $\\Phi$ be the union of lists of formulas from the statements of Lemma~\\ref{l:basics-in-M} and Lemma~\\ref{l:basics-in-M-2} enriched by formulas (and their subformulas) marked by $(*)$ in the proof below. Let $M\\prec (\\Phi; S\\cup \\{K,D,\\tau\\})$.\n\nBy Lemma~\\ref{l:unique-M} and the absoluteness of the following formula (and its subformulas)\n\\[\\begin{split}\n\\exists \\Gamma\\; \\exists \\le\\;\\exists r\\; \\big(& D \\text{ is a subset of a set induced by}\\\\ & \\text{the retractional skeleton }\\{r(s)\\colon s\\in\\Gamma\\}\\big),\\qquad (*)\n\\end{split}\\]\nthere exist $\\Gamma, \\le, r\\in M$ such that $\\{r(s)\\colon s\\in \\Gamma\\}$ is a retractional skeleton on $K$ inducing a set containing $D$. For $s\\in\\Gamma$ we will write below $r_s$ instead of $r(s)$. By Lemma~\\ref{l:basics-in-M-2} \\eqref{it:upDirected}, the set $\\Gamma\\cap M$ is up-directed. Hence by Lemma~\\ref{l: retraction from up-directed subset} there exists a continuous retraction $R_M\\colon K\\to K$ defined by $R_{M}(x):=\\lim_{s\\in \\Gamma\\cap M}r_{s}(x)$, for every $x\\in K$. Using the absoluteness of the following formula (and its subformulas)\n\\[\n\\forall u \\in D\\,\\exists s\\in\\Gamma\\, u\\in r_s[K],\\eqno{(*)}\n\\]\nwe obtain that $D\\cap M\\subset R_{M}[K]$ and therefore $\\overline{D\\cap M}\\subset R_{M}[K]$. Now fix $x,y\\in\\overline{D\\cap M}$ with $x\\neq y$. Since $x=\\lim_{s\\in \\Gamma\\cap M}r_s(x)$ and $y=\\lim_{s\\in\\Gamma\\cap M}r_{s}(y)$ there exists $s\\in \\Gamma\\cap M$ such that $r_{s}(x)\\neq r_s(y)$. By Lemma~\\ref{l:basics-in-M} \\eqref{it: Dom-Rng-in-M}, we have that $r(s)=r_s\\in M$ and $r_s[K]\\in M$. Thus, by Lemma~\\ref{l:basics-in-M-2} \\eqref{it:metrizableSubspace}, there exists $f\\in \\C(K)\\cap M$ such that $f(r_s(x))\\neq f(r_s(y))$. Now using Lemma~\\ref{l:basics-in-M-2} \\eqref{it:composition}, we obtain that $g=f\\circ r_s\\in \\C(K)\\cap M$ and $g(x)\\neq g(y)$. Thus, $\\C(K)\\cap M$ separates the points of $\\overline{D\\cap M}$\n\\end{proof}\n\nThe retraction constructed in Lemma~\\ref{l:retraction} (whose assumption is satisfied by Proposition~\\ref{p:characterizationRSkeletonNotCountableModels} in compact spaces admitting a retractional skeleton) will be the key to our considerations. Let us give it a name.\n\n\\begin{defin}\\label{def:canonicalRetraction}\nLet $K$ be a compact space and let $D\\subset K$ be a dense subset that is contained in the set induced by a retractional skeleton. Given a set $M$, we say that $r_M$ is the \\emph{canonical retraction associated to $M$, $K$ and $D$} if it is the unique retraction on $K$ satisfying $r_M[K]=\\overline{D\\cap M}$ and $f = f\\circ r_M$, for every $f\\in \\C(K)\\cap M$. We say that a set $M$ \\emph{admits canonical retraction} if there exists the canonical retraction associated to $M$, $K$ and $D$.\n\nIn the case when $D=K$ we say that $M$ \\emph{admits canonical retraction $r_M$ associated to $M$ and $K$}.\n\\end{defin}\n\nThe properties of canonical retractions associated to suitable models are summarized in Theorem~\\ref{thm:canonicalRetraction}. We need two lemmas first.\n\n\\begin{lemma}\\label{l:separation}For every suitable model $M$ the following holds: Let $(K,\\tau)$ be a compact space and $\\A\\subset\\C(K)$ be a set separating the points of $K$ with $\\{\\A,K,\\tau\\}\\subset M$. Then for every compact set $K'\\subset K$, $\\A\\cap M$ separates the points of $K'$ if and only if $\\C(K)\\cap M$ separates the points of $K'$.\n\\end{lemma}\n\\begin{proof}Let $S$ and $\\Phi$ be the set and the list of formulas from the statements of Lemma~\\ref{l:basics-in-M} and Lemma~\\ref{l:basics-in-M-2}. Let $M\\prec (\\Phi; S\\cup\\{K,\\tau,\\A\\})$. \nIn order to get a contradiction, let us assume that $\\C(K)\\cap M$ separates the points of $K'$ but $\\A\\cap M$ does not separate the points of $K'$. Then also $\\overline{\\operatorname{alg}((\\A\\cap M)\\cup \\{1\\})}$ does not separate the points of $K'$ (because if there are $x\\neq y$ with $f(x) = f(y)$ for every $f\\in \\A\\cap M$, then also $g(x) = g(y)$ for every $g$ of the form $g = a_0 + \\sum_{i=1}^n a_i\\Pi_{j=1}^m f_{i,j}$). But this is a contradiction, because using Lemma~\\ref{l:basics-in-M-2} \\eqref{it:containsspacecontinuousfunctions} and \\eqref{it:algebraDenseInFunctionsFromM} we conclude that $\\overline{\\operatorname{alg}((\\A\\cap M)\\cup \\{1\\})} = \\overline{\\C(K)\\cap M}$.\n\\end{proof}\n\n\\begin{lemma}\\label{l:dense}For every suitable model $M$ the following holds: Let $(K,\\tau)$ be a compact space and let $D\\subset K$ be a dense subset that is contained in the set induced by a retractional skeleton such that $\\{K,D,\\tau\\}\\subset M$. Then the mapping $\\Phi:\\overline{\\C(K)\\cap M}\\to \\C(\\overline{D\\cap M})$ defined by $\\Phi(f):=f|_{\\overline{D\\cap M}}$, for every $f\\in \\overline{\\C(K)\\cap M}$, is a surjective isometry.\n\\end{lemma}\n\\begin{proof}\nLet $S$ and $\\Phi$ be the union of countable sets and finite lists of formulas from the statements of Lemma~\\ref{l:basics-in-M-2} and Proposition~\\ref{p:characterizationRSkeletonNotCountableModels}. Let $M\\prec (\\Phi; S\\cup \\{K,D,\\tau\\})$. By Lemma~\\ref{l:basics-in-M-2} \\eqref{it:norm}, we have that $\\|f\\| = \\|f|_{D\\cap M}\\|$, for every $f\\in \\C(K)\\cap M$, so the mapping $\\Phi|_{\\C(K)\\cap M}$ is an isometry which implies that $\\Phi$ is also an isometry. It remains to show that it is surjective. By Lemma~\\ref{l:basics-in-M-2} \\eqref{it:containsspacecontinuousfunctions}, $\\overline{\\C(K)\\cap M}$ is a closed subalgebra of $\\C(K)$ and so the image of $\\Phi$ is a closed subalgebra of $\\C(\\overline{D\\cap M})$ which, by Proposition~\\ref{p:characterizationRSkeletonNotCountableModels} separates the points of $\\overline{D\\cap M}$. Therefore, it follows from Stone-Weierstrass theorem that $\\Phi[\\overline{\\C(K)\\cap M}] = \\C(\\overline{D\\cap M})$.\n\\end{proof}\n\n\\begin{thm}\\label{thm:canonicalRetraction}\nFor every suitable model $M$ the following holds: Let $(K,\\tau)$ be a compact space and let $D\\subset K$ be a dense subset that is contained in the set induced by a retractional skeleton with $\\{K,D,\\tau\\}\\subset M$. Then there exists a unique retraction $r_M:K\\to \\overline{D\\cap M}$ with $r_M[K] = \\overline{D\\cap M}$ and $f=f\\circ r_M$, for every $f\\in\\C(K)\\cap M$. Moreover, for this retraction $r_M$ the following holds:\n\\begin{enumerate}[label = (\\roman*)]\n    \\item\\label{it:characterizationFirst} Whenever $\\A\\subset \\C(K)$ separates the points of $K$ and $\\A\\in M$, then for every $x\\in K$, $r_M(x)$ is the unique point from $\\overline{D\\cap M}$ satisfying $f(r_M(x)) = f(x)$, for every $f\\in \\A\\cap M$. \n    \\item\\label{it:characterizationSecond} Whenever $(\\Gamma,\\leq)$ is up-directed and $\\sigma$-complete and $r:\\Gamma\\to\\C(K,K)$ is a mapping such that $\\mathfrak{s} = \\{r(s)\\colon s\\in\\Gamma\\}$ is a retractional skeleton on $K$ with $D\\subset D(\\mathfrak{s})$ and $\\{r,\\Gamma,\\leq\\}\\subset M$, then the following holds.\n    \\begin{enumerate}[label=(\\alph*)]\n        \\item\\label{it:imageOfRs} For every $s\\in \\Gamma\\cap M$, $r(s)[K]\\subset \\overline{D\\cap M}$.\n        \\item\\label{it:formulaUsingSkeleton} \\[r_M(x) = \\lim_{s\\in\\Gamma\\cap M} r(s)(x),\\quad x\\in K.\n    \\]\n    \\item\\label{it:countableModelsAreInSkeleton} If $M$ is countable, then $r_M = r(s)$ for $s = \\sup \\Gamma\\cap M$.\n    \\item\\label{it:skeletonOnTheInducedSubspace} $\\{r(s)|_{r_M[K]}\\colon s \\in (\\Gamma\\cap M)_\\sigma\\}$ is a retractional skeleton on $r_M[K]$ with induced set $D(\\mathfrak{s})\\cap r_M[K]$.\n    \\item\\label{it:commutativeCaseModels} If $\\mathfrak{s}$ is commutative, then $r_M[D(\\mathfrak{s})] = D(\\mathfrak{s})\\cap r_M[K]$.\n        \\end{enumerate}\n    \\item\\label{it:homeomorphicImageOfCanonicalRetraction} Whenever $h:K\\to L$ is a surjective homeomorphism with $h\\in M$, then $r:=h\\circ r_M\\circ h^{-1}$ is the unique retraction on $L$ such that $r[L] = \\overline{h[D]\\cap M}$ and $f\\circ r = f$, for every $f\\in \\C(L)\\cap M$.\n    \\item\\label{it:weight} $\\w(r_M[K])\\leq |M|$.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof}Denote by $\\B$ the set of all open intervals with rational endpoints. Let $S$ be the union of sets from the statements of Lemma~\\ref{l:basics-in-M}, Lemma~\\ref{l:basics-in-M-2}, Lemma~\\ref{l:retraction}, Lemma~\\ref{l:dense} and Proposition~\\ref{p:characterizationRSkeletonNotCountableModels} and let $\\Phi$ be the union of lists of formulas from the statements of Lemma~\\ref{l:basics-in-M}, Lemma~\\ref{l:basics-in-M-2}, Lemma~\\ref{l:retraction}, Lemma~\\ref{l:dense} and Proposition~\\ref{p:characterizationRSkeletonNotCountableModels} enriched by the formula (and its subformulas) marked by $(*)$ in the proof below. Let $M\\prec (\\Phi; S\\cup \\{K,D,\\tau\\})$. It follows directly from Proposition~\\ref{p:characterizationRSkeletonNotCountableModels} and Lemma~\\ref{l:retraction} that the retraction $r_M$ exists and that it satisfies \\ref{it:characterizationFirst}.\n\nLet $\\{r,\\Gamma,\\leq\\}\\subset M$ be as in \\ref{it:characterizationSecond}. For $s\\in\\Gamma\\cap M$, by Lemma~\\ref{l:basics-in-M} \\eqref{it: Dom-Rng-in-M} we have that $r(s)[K] \\in M$ and thus Lemma~\\ref{l:basics-in-M-2} \\eqref{it:metrizableSubspace} ensures that $r(s)[K]\\subset \\overline{r(s)[K]\\cap M}$. Moreover, for every $x\\in r(s)[K]\\cap M$, using the countable tightness of $D(\\mathfrak{s})$ the following formula holds\n\\[\\exists C\\subset D\\;\\quad ( \\text{$C$ is countable and }x\\in \\overline{C}).\\eqno{(*)}\\] Thus, by Lemma~\\ref{l:unique-M}  there exists a countable set $C\\subset D$ with $C\\in M$ (which implies $C\\subset M$) such that $x\\in \\overline{C}$, which implies that $x\\in \\overline{D\\cap M}$, so $r(s)[K]\\subset \\overline{r(s)[K]\\cap M}\\subset \\overline{D\\cap M}$ and \\ref{it:imageOfRs} holds. By Lemma~\\ref{l:basics-in-M-2} \\eqref{it:upDirected}, $\\Gamma\\cap M$ is up-directed and so, using Lemma~\\ref{l: retraction from up-directed subset} the limit $\\lim_{s\\in\\Gamma\\cap M} r(s)(x) =:R_M(x)$ exists for every $x\\in K$.\n\nWe \\emph{claim} that $R_M=r_M$. Note that due to the uniqueness of $r_M$, it is enough to show that $R_M[K] \\subset \\overline{D \\cap M}$ and that $f\\circ R_M = f$, for every $f\\in\\C(K)\\cap M$. By the definition of $R_M$, using \\ref{it:imageOfRs}, we observe that $R_M[K] \\subset \\overline{D \\cap M}$.   Moreover, for every $x\\in D\\cap M$, by Lemma~\\ref{l:unique-M} and the absoluteness of the following formula (and its subformulas)\n\\[\\exists s\\in\\Gamma\\; \\big(r_s(x)=x\\big),\\eqno{(*)}\\]\nthere is $s\\in\\Gamma\\cap M$ with $r_s(x)=x$ and so $x\\in R_M[K]$. Thus, we have that $R_M[K] = \\overline{D\\cap M}$. Pick $f\\in \\C(K)\\cap M$. Since $(r_s)_{s\\in\\Gamma\\cap M}$ converges pointwise to $R_M$, using \\cite[Lemma 5.2]{K20} we conclude that $(f\\circ r_s)_{s\\in\\Gamma\\cap M}$ converges in norm to $f\\circ R_M$. Therefore $f\\circ R_M\\in \\overline{\\C(K)\\cap M}$, since it follows from Lemma~\\ref{l:basics-in-M} \\eqref{it: Dom-Rng-in-M} and Lemma~\\ref{l:basics-in-M-2} \\eqref{it:composition} that $f\\circ r_s\\in \\C(K) \\cap M$, for every $s\\in\\Gamma\\cap M$.\nThus $f$ and $f\\circ R_M$ are two functions from $\\overline{\\C(K)\\cap M}$ which have the same values on $\\overline{D\\cap M}$ and so, by Lemma~\\ref{l:dense}, $f = f\\circ R_M$. This proves the claim and establishes \\ref{it:formulaUsingSkeleton} which, using Lemma~\\ref{l: retraction from up-directed subset}, implies \\ref{it:countableModelsAreInSkeleton}, \\ref{it:skeletonOnTheInducedSubspace} and \\ref{it:commutativeCaseModels}.\n\nThe proof of \\ref{it:homeomorphicImageOfCanonicalRetraction} is easy once we realize that by Lemma~\\ref{l:basics-in-M-2} \\eqref{it:one-to-one-imageOfM} we have that $h[D\\cap M] = h[D]\\cap M$ and that $f\\circ h\\in \\C(K)\\cap M$, for every $f\\in \\C(L)\\cap M$. We omit the straightforward details. \n\nFinally, \\ref{it:weight} follows from Proposition~\\ref{p:characterizationRSkeletonNotCountableModels}, since $r_M[K]=\\overline{D\\cap M}$.\n\\end{proof}\n\n\\subsection{Families of canonical retractions} Here we study families of canonical retractions associated to suitable models. Those are more-or-less straightforward consequences of Theorem~\\ref{thm:canonicalRetraction}. The most important for what follows is Proposition~\\ref{prop:rri} which will be repeatedly used further.\n\n\\begin{lemma}\\label{l:propertiesOfCanonicalRetractions}There are a countable set $S$ and a finite list of formulas $\\Phi$ such that the following holds: Let $(K, \\tau)$ be a compact space, $(\\Gamma,\\leq)$ be an up-directed set, $r:\\Gamma\\to\\C(K,K)$ be a mapping such that $\\mathfrak{s}:=\\{r(s)\\colon s\\in\\Gamma\\}$ is a retractional skeleton on $K$ and let $D\\subset D(\\mathfrak{s})$ be dense in $K$. Put $S'=S\\cup\\{K, \\tau, D,\\Gamma,\\leq,r\\}$. Then every $M\\prec(\\Phi; S')$ admits the canonical retraction $r_M$ associated to $M$, $K$ and $D$.\n\nMoreover, we have the following.\n\\begin{enumerate}\n    \\item\\label{it:commutativity} If $M, N\\prec(\\Phi; S')$ and $M\\subset N$ then $r_M\\circ r_N = r_N\\circ r_M = r_M$.\n    \\item\\label{it:increasingSequence} Let $\\M$ be an up-directed set with $M\\prec(\\Phi; S')$, for every $M\\in \\M$ and let $M_\\infty:=\\bigcup_{M\\in\\M} M$. Then $M_\\infty\\prec (\\Phi; S')$ and $\\lim_{M\\in\\M}r_{M}(x) = r_{M_\\infty}(x)$, $x\\in K$.\n    \\item\\label{it:toIdentity} If $\\U$ is a basis of $\\tau$, $\\M$ is an up-directed set with $M\\prec(\\Phi; S')$, for every $M\\in \\M$ and $\\U\\subset\\bigcup_{M\\in\\M} M$, then $\\lim_{M\\in\\M}r_{M}(x) = x$, for every $x\\in K$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nThe existence of $S$ and $\\Phi$ follows from Theorem~\\ref{thm:canonicalRetraction}. Let us prove the moreover part using the additional properties of canonical retractions established in Theorem~\\ref{thm:canonicalRetraction}.\\\\\n\\eqref{it:commutativity} Since $r_M[K] = \\overline{D\\cap M}\\subset \\overline{D\\cap N}=r_N[K]$, we have $r_M = r_N\\circ r_M$. Moreover, for every $f\\in\\C(K)\\cap M\\subset \\C(K)\\cap N$ we have \n    \\[f(r_{M}(x)) = f(x) = f(r_N(x)) = f(r_M(r_N(x))),\\quad x\\in K\\]\n    and so $r_M(x) = r_M(r_N(x))$, for every $x\\in K$.\\\\\n\\eqref{it:increasingSequence} Since $\\M$ is up-directed, it follows from \\cite[Lemma~2.1]{C12} and Lemma~\\ref{l:unique-M} that $M_\\infty\\prec (S';\\Phi)$.  Now, combining Theorem~\\ref{thm:canonicalRetraction} \\ref{it:characterizationSecond} with Lemma \\ref{l:basics-in-M-2} \\eqref{it:upDirected} and Lemma~\\ref{l: retraction from up-directed subset} \\ref{it:generalUpDirected} we obtain that $\\lim_{M\\in\\M}r_{M}(x) = r_{M_\\infty}(x)$, $x\\in K$.\\\\\n\\eqref{it:toIdentity} Pick $x\\in K$, $U\\in\\U$ such that $x\\in U$ and find $V \\in\\U$ with $x\\in V\\subset \\overline{V}\\subset U$. Since $V \\in \\U$ and $\\U \\subset \\bigcup_{M \\in \\M} M$, there exists $M_0 \\in \\M$ such that $V \\in M_0$. Now fix $M \\in \\M$ with $M_0 \\subset M$. It follows from Lemma~\\ref{l:basics-in-M} \\eqref{operations-sets-in-M} that $V \\cap D \\in M$. Therefore Lemma~\\ref{l:retraction} ensures that $r_M(x)\\in r_M[\\overline{V\\cap D}]\\subset \\overline{V\\cap D}\\subset U$, since $x\\in \\overline{V\\cap D}$.\n\\end{proof}\n\n\\begin{prop}\\label{prop:rri}There exist a countable set $S$ and a finite list of formulas $\\Phi$ such that the following holds:\\\\\nLet $(K,\\tau)$ be a compact space, $(\\Gamma,\\leq)$ be an up-directed set and $r:\\Gamma\\to\\C(K,K)$ be a mapping such that $\\mathfrak{s}=\\{r(s)\\colon s\\in\\Gamma\\}$ is a retractional skeleton on $K$. Let $\\kappa:=\\w(K)$ and $\\U:\\kappa\\to\\tau$ be such that $\\{\\U(i)\\colon i<\\kappa\\}$ is an open basis of $\\tau$. Put $S':=S\\cup \\{K,D(\\mathfrak{s}),\\Gamma,\\leq,r,\\tau,\\U\\}$. Let $(M_\\alpha)_{\\alpha\\leq \\kappa}$ be a sequence of sets satisfying\n\\begin{enumerate}[label=(R\\alph*)]\n    \\item\\label{it:areModels} $M_\\alpha\\prec (\\Phi;S')$, for every $\\alpha\\in [0,\\kappa]$,\n    \\item\\label{it:cardinalityMalpha} $\\vert M_\\alpha \\vert \\le \\max(\\omega, \\vert \\alpha \\vert)$, for every $\\alpha\\in [0,\\kappa]$,\n    \\item\\label{it:areIncreasing} $M_{\\alpha+1}\\supset M_\\alpha\\cup\\{\\alpha\\}$, for every $\\alpha\\in [0,\\kappa)$,\n    \\item\\label{it:areContinuous} $M_\\alpha = \\bigcup_{\\beta<\\alpha} M_\\beta$, if $\\alpha\\in(0,\\kappa]$ is a limit ordinal.\n\\end{enumerate}\nThen for every $\\alpha\\in[0,\\kappa]$ there exists a canonical retraction $r_\\alpha$ associated to $M_\\alpha$, $K$ and $D(\\mathfrak{s})$ and the following holds.\n\n\n\\begin{enumerate}[label=(R\\arabic*)]\n\\newcounter{saveenum}\n    \\item\\label{it:comutative} For every $\\alpha<\\beta$, we have that $r_\\alpha\\circ r_\\beta = r_\\beta\\circ r_\\alpha = r_\\alpha$.\n    \\item\\label{it:separatePoints} $r_\\alpha(x)\\to x$, for every $x\\in K$.\n     \\item\\label{it:upDirectedLimit} Let $\\alpha\\le\\kappa$, let $\\eta:[0,\\alpha)\\to \\kappa$ be an increasing function and let $\\xi \\le\\kappa$ be a limit ordinal with $\\sup_{\\beta<\\alpha} \\eta(\\beta) = \\xi$. Then $\\lim_{\\beta < \\alpha} r_{\\eta(\\beta)}(x) = r_\\xi(x)$, for every $x\\in K$.\n     \\item\\label{it:identity} $r_\\kappa = id$.\n     \\item\\label{it:skeletonOnSubset} For every $\\alpha\\in[0,\\kappa]$, we have that $\\w(r_\\alpha[K])\\leq max(\\omega, |\\alpha|)$ and $(r_s|_{r_\\alpha[K]})_{s\\in (\\Gamma\\cap M_\\alpha)_\\sigma}$ is a retractional skeleton on $r_\\alpha[K]$ with induced set $D(\\mathfrak{s})\\cap r_\\alpha[K]$.\n     \\item\\label{it:compositionWithRa} If $\\A$ is a closed subset of $C(K)$ and $f \\circ r_s \\in \\A$, for every $f \\in \\A$ and $s\\in\\Gamma$, then $\\{f \\circ r_\\alpha\\colon f\\in\\A, \\alpha\\le\\kappa\\}\\subset \\A$.\n      \\item\\label{it:ordinalSeparatingPoints} If $x, y$ are distinct points in $K$ then $\\beta:=\\min\\{\\alpha< \\kappa\\colon r_\\alpha(x)\\neq r_\\alpha(y)\\}$ exists and it is a successor ordinal or $\\beta=0$.\n      \\item\\label{it:secondFormula} For every $\\alpha\\leq \\kappa$, the set $\\Gamma\\cap M_\\alpha$ is up-directed and $r_\\alpha(x) = \\lim_{s\\in (M_\\alpha\\cap \\Gamma)} r_s(x)$, for every $x \\in K$.\n      \\item \\label{comutaralpha3} For every $\\alpha \\le \\kappa$ and $t \\in (\\Gamma \\cap M_\\alpha)_\\sigma$, it holds that $r_t \\circ r_\\alpha=r_\\alpha \\circ r_t$.\n      \\item \\label{Model-compositionRalpha} Let $\\A \\subset \\C(K)$ be a set that separates the points of $K$ and $\\alpha < \\kappa$. If $\\A \\in M_\\alpha$,  then $\\A \\cap M_\\alpha$ separates the points of $r_\\alpha[K]$ and $f \\circ r_\\alpha=f$, for every $f \\in \\A \\cap M_\\alpha$.\n      \\setcounter{saveenum}{\\value{enumi}}\n\\end{enumerate}\nMoreover, if $\\mathfrak{s}$ is full or commutative, then $r_\\alpha[D(\\mathfrak{s})]\\subset D(\\mathfrak{s})$, for every $\\alpha\\in [0,\\kappa]$ and if $D\\subset D(\\mathfrak{s})$ is such that $r_\\alpha[D]\\subset D$ for every $\\alpha\\in [0,\\kappa]$, then we also have the following.\n\\begin{enumerate}[label=(R\\arabic*)]\n \\setcounter{enumi}{\\value{saveenum}}\n    \\item\\label{it:countableSupport2} The sets $\\{r_\\alpha(x)\\colon \\alpha<\\kappa\\}$ and $\\{\\alpha<\\kappa: r_{\\alpha}(x)\\neq r_{\\alpha+1}(x)\\}$ are countable, for every $x\\in D$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nLet $S$ and $\\Phi$ be the union of sets and lists of formulas from the statements of Theorem~\\ref{thm:canonicalRetraction} and Lemma~\\ref{l:propertiesOfCanonicalRetractions}. Then the existence of $r_\\alpha$, $\\alpha\\in [0,\\kappa]$ follows from Theorem~\\ref{thm:canonicalRetraction}. Now, \\ref{it:comutative} follows immediately from Lemma~\\ref{l:propertiesOfCanonicalRetractions}. \\ref{it:separatePoints} and \\ref{it:upDirectedLimit} follow from Lemma~\\ref{l:propertiesOfCanonicalRetractions} as well (using for \\ref{it:separatePoints} the fact that $\\{\\U(i)\\colon i<\\kappa\\}\\subset \\bigcup_{\\alpha<\\kappa} M_\\alpha$ and for \\ref{it:upDirectedLimit} the fact that $\\bigcup_{\\beta<\\alpha} M_{\\eta(\\beta)} = M_\\xi$). \\ref{it:identity} follows from \\ref{it:separatePoints} and \\ref{it:upDirectedLimit} applied to $\\eta(i):=i$, $i<\\kappa$ and $\\xi = \\kappa$. \\ref{it:skeletonOnSubset} and \\ref{Model-compositionRalpha} follow from Theorem~\\ref{thm:canonicalRetraction}. \nFor \\ref{it:compositionWithRa} we observe that by Theorem~\\ref{thm:canonicalRetraction} \\ref{it:characterizationSecond} the net of continuous retractions $(r_s)_{s \\in \\Gamma \\cap M_\\alpha}$ converges pointwise to the continuous retraction $r_\\alpha$ and so \\cite[Lemma~5.2]{K20} ensures that the net $(f \\circ r_s)_{s \\in \\Gamma \\cap M_\\alpha}$ converges in norm to $f \\circ r_\\alpha$, for every $f \\in \\C(K)$, which implies \\ref{it:compositionWithRa}. For \\ref{it:ordinalSeparatingPoints} we observe that by \\ref{it:separatePoints} there is $i<\\kappa$ such that $r_i(x)\\neq r_i(y)$ so $\\beta$ is well defined and if $\\beta\\neq 0$ then it is a successor ordinal by \\ref{it:upDirectedLimit}. For \\ref{it:secondFormula} is suffices to apply Lemma \\ref{l:basics-in-M-2} \\eqref{it:upDirected} and Theorem~\\ref{thm:canonicalRetraction} \\ref{it:characterizationSecond}. \\ref{comutaralpha3} follows from Lemma~\\ref{l: retraction from up-directed subset} \\ref{it:sigmaClosure}.\n\nMoreover, if $\\mathfrak{s}$ is full then we obviously have $r_\\alpha[D(\\mathfrak{s})] = r_\\alpha[K]\\subset K=D(\\mathfrak{s})$ and if it is commutative then Theorem~\\ref{thm:canonicalRetraction} \\ref{it:characterizationSecond} ensures that $r_\\alpha[D(\\mathfrak{s})]\\subset D(\\mathfrak{s})$.\\\\\n\\ref{it:countableSupport2}: Pick $x\\in D$ and note that in order to prove that $\\{r_\\alpha(x)\\colon \\alpha<\\kappa\\}$ is countable, it suffices to show that for every strictly increasing function $\\eta:[0,\\omega_1)\\to\\kappa$, there is $\\zeta<\\omega_1$ with $r_{\\eta(\\zeta)}(x)=r_{\\eta(\\beta)}(x)$, for every $\\zeta<\\beta<\\omega_1$. \nLet $\\eta:[0,\\omega_1)\\to\\kappa$ be a strictly increasing function and set $\\xi:=\\sup_{\\beta<\\omega_1} \\eta(\\beta)$. By \\ref{it:upDirectedLimit}, we have $r_\\xi(x) = \\lim_{\\beta < \\omega_1} r_{\\eta(\\beta)}(x)$. Hence, since $r_\\xi[D]\\subset D$ and $D$ has countable tightness (see \\cite[Theorem 32]{kubis09}), there is a $\\zeta<\\omega_1$ with $r_\\xi(x) \\in r_{\\eta(\\zeta)}[K]$; so, for $\\zeta\\leq\\beta<\\omega_1$, using \\ref{it:comutative}, we obtain $r_{\\eta(\\beta)}(x) = r_{\\eta(\\beta)}(r_\\xi(x)) = r_\\xi(x)$. Finally, to conclude that the set $\\{\\alpha<\\kappa: r_{\\alpha}(x)\\neq r_{\\alpha+1}(x)\\}$ is countable, note that the mapping $\\varphi(\\alpha)=r_\\alpha(x)$ is an injection from this set into $\\{r_\\alpha(x)\\colon \\alpha<\\kappa\\}$. Indeed, suppose by contradiction that there exist $\\alpha, \\beta<\\kappa$ with $\\alpha \\ne \\beta$, $r_\\alpha(x)\\neq r_{\\alpha+1}(x)$ and $r_\\beta(x)\\neq r_{\\beta+1}(x)$ such that $r_\\alpha(x)=r_\\beta(x)$. Without loss of generality, we may assume that $\\alpha<\\beta$. Then applying the map $r_{\\alpha+1}$, by \\ref{it:comutative}, we obtain $r_{\\alpha}(x)=r_{\\alpha+1}(r_{\\alpha}(x))= r_{\\alpha+1}(r_\\beta(x))=r_{\\alpha+1}(x)$, which is a contradiction.\n\\end{proof}\n\n\\subsection{Application - passing to a subskeleton} Here we introduce the notion of a (weak) subskeleton and show that for a countable family of retractional skeletons inducing the same set there is a common weak subskeleton, see Theorem~\\ref{thm:subskeletons}.\n\n\\begin{defin}\\label{def:subskeleton}\n    Let $K$ be a compact space and let $\\mathfrak{s} = (r_s)_{s\\in \\Gamma}$ be a retractional skeleton on $K$. We say that $(r_s)_{s\\in\\Gamma'}$ is a \\emph{subskeleton} of $\\mathfrak{s}$, if $\\Gamma'\\subset \\Gamma$ is a $\\sigma$-closed and cofinal subset.\n    \\end{defin}\nIt is easy to see that every subskeleton is a retractional skeleton.\n\n\\begin{defin}\\label{def:weakSubskeleton}\nLet $(r_s)_{s \\in \\Gamma}$ be a retractional skeleton on a compact space $K$. We say that $(R_i)_{i\\in\\Lambda}$ is a \\emph{weak subskeleton of $(r_s)_{s \\in \\Gamma}$} if $(R_i)_{i\\in\\Lambda}$ is a retractional skeleton on $K$ and there exists a mapping $\\phi:\\Lambda\\to \\Gamma$ such that\n\\begin{itemize}\n    \\item $R_i = r_{\\phi(i)}$, for every $i\\in\\Lambda$;\n    \\item $\\phi$ is $\\omega$-monotone, that is, if $i,j\\in\\Lambda$ with $i\\leq j$, then $\\phi(i)\\leq\\phi(j)$ and if $(i_n)_{n\\in\\omega}$ is an increasing sequence from $\\Lambda$, then $\\sup_n \\phi(i_n) = \\phi(\\sup_n i_n)$;\n    \\item $\\{\\phi(i)\\colon i\\in\\Lambda\\}$ is cofinal in $\\Gamma$.\n\\end{itemize}\n\\end{defin}\n\nClearly, every subskeleton of a retractional skeleton is also a weak subskeleton. Some basic properties of weak subskeletons are summarized below.\n\n\\begin{fact}\nLet $(r_s)_{s \\in \\Gamma}$ be a retractional skeleton on a compact space $K$ and $(R_i)_{i\\in\\Lambda}$ be a weak subskeleton of $(r_s)_{s \\in \\Gamma}$. Then\n\\begin{itemize}\n    \\item $(R_i)_{i\\in\\Lambda}$ induces the same subset as $(r_s)_{s\\in\\Gamma}$;\n    \\item if $(S_j)_{j\\in\\Delta}$ is a weak subskeleton of $(R_i)_{i\\in\\Lambda}$, then it is a weak subskeleton of $(r_s)_{s \\in \\Gamma}$.\n\\end{itemize}\n\\end{fact}\n\nThe following result shows that we may concentrate properties of countably many retractional skeletons into one skeleton, which is moreover generated by suitable models.\n\n\\begin{thm}\\label{thm:subskeletons}\nLet $K$ be a compact space and let $(r_s^n)_{s\\in\\Gamma_n}$, $n\\in\\omega$ be a sequence of retractional skeletons on $K$ inducing the same set $D$. Then there exists a retractional skeleton which is a weak subskeleton of $(r_s^n)_{s\\in\\Gamma_n}$, for every $n\\in\\omega$.\n\nMoreover, for every countable set $S$ and every finite list of formulas $\\Phi$, there exists a family $\\M$ consisting of countable suitable models for $\\Phi$ containing $S$ such that every $M\\in\\M$ admits canonical retraction $r_M$ associated to $M$, $K$ and $D$ and $(r_M)_{M\\in\\M}$ is a weak subskeleton of $(r_s^n)_{s\\in\\Gamma_n}$, for every $n\\in\\omega$, where the ordering on $\\M$ is given by inclusion. \n\\end{thm}\n\\begin{proof}Let $\\Gamma_n = (\\Gamma_n,\\leq_n)$ and $r_n:\\Gamma_n\\to \\C(K,K)$ be such that $r_n(s):=r_s^n$, for every $n\\in\\omega$ and $s\\in\\Gamma_n$. Let $\\tau$ be the topology on $K$. Let $S'$ be the union of $S$ and the countable set from the statement of Theorem~\\ref{thm:canonicalRetraction} enriched by $\\{K,D,\\tau,r_n,\\Gamma_n,\\leq_n\\colon n\\in\\omega\\}$ and let $\\Phi'$ be the union of $\\Phi$ and the list of formulas from the statement of Theorem~\\ref{thm:canonicalRetraction}.\nBy Theorem~\\ref{thm:modelExists}, there is a set $R\\supset S'\\cup \\tau\\cup \\bigcup_{n\\in\\omega}\\Gamma_n$ such that $R\\prec (\\Phi'; S')$ and for every countable set $Z\\subset R$ there is a countable set $M(Z)\\subset R$ satisfying $M(Z)\\prec (\\Phi';Z)$. Set\n\\[\\M=\\{M\\in[R]^{\\omega}\\colon M\\prec(\\Phi',S')\\},\\]\nordered by inclusion. The $\\sigma$-completeness of $\\M$ follows from \\cite[Lemma 2.4]{C12}. To see that $\\M$ is up-directed, let $N_1,N_2\\in \\M$, then  the set $M(N_1\\cup N_2)\\in \\M$ and satisfies $N_1\\cup N_2\\subset M(N_1\\cup N_2)$.\nBy Theorem~\\ref{thm:canonicalRetraction}, every $M\\in\\M$ admits canonical retraction $r_M$ associated to $M$, $K$ and $D$. \nNote that for every $U\\in\\tau$ there is $M\\in\\M$ with $U\\in M$ (it suffices to put $M=M(\\{U\\}\\cup S')$) and so $\\tau\\cap (\\bigcup \\M) = \\tau$ is indeed a basis of the topology $\\tau$. Therefore it follows from Theorem \\ref{thm:canonicalRetraction} \\ref{it:weight} and Lemma \\ref{l:propertiesOfCanonicalRetractions} that $(r_M)_{M\\in\\M}$ is a retractional skeleton on $K$. \nNow for every $n\\in\\omega$, let $\\phi_n:\\M\\to \\Gamma_n$ be the mapping defined by $\\phi_n(M):=\\sup (\\Gamma_n\\cap M)$. By Theorem \\ref{thm:canonicalRetraction} \\ref{it:characterizationSecond}, we have that $r_M=r^n_{\\phi_n(M)}$, for every $n \\in \\omega$. Now, let $(M_k)_{k\\in\\omega}\\subset \\M$ be an increasing sequence, then it is easy to see that $\\sup_k \\phi_n(M_k) = \\phi_n(M_\\infty)$, where $M_{\\infty}=\\bigcup_{k\\in\\omega}M_k$.\nIt remains to prove that the set $\\{\\phi_n(M):M\\in\\M\\}$ is cofinal in $\\Gamma_n$, for each $n\\in\\omega$. Let $n\\in\\omega$ and $s\\in \\Gamma_n$, then there exists $M\\in\\M$ such that $s\\in M$ (it suffices to put $M=M(\\{s\\}\\cup S')$). Therefore $\\phi_n(M)=\\sup(\\Gamma_n\\cap M)\\geq s$. This concludes the proof.\n\\end{proof}\n\n\\section{Valdivia embedding of compact spaces admitting a commutative skeleton}\\label{sec:valdivia}\n\nGiving a compact space $K$ that admits a commutative retractional skeleton $\\mathfrak{s}$, it is known that there exists a homeomorphic embedding $h:K\\to [-1,1]^I$ such that $h[D(\\mathfrak{s})]\\subset \\Sigma(I)$ (that is, $K$ is Valdivia and $D(\\mathfrak{s})$ is a $\\Sigma$-subset of $K$). The main aim of this section is to have a very concrete and very flexible way of understanding the mapping $h$, which is the topic handled in Subsection~\\ref{subsec:valdiviaEmbedding}, where the proof of Theorem~\\ref{thm:Intro3} is given. Apart from Theorem~\\ref{thm:Intro3}, we would like to highlight Theorem~\\ref{thm:valdiviaThroughModels} which gives a new characterization of Valdivia compacta using suitable models.\n\n\\subsection{Canonical retractions associated to suitable models in Valdivia compact spaces} The goal here is to obtain the following concrete description of the canonical retractions associated to suitable models in Valdivia compact spaces.\n\n\\begin{lemma}\\label{l:canonicalRetractionInSigmaProduct}\nFor every suitable model $M$ the following holds: Let $D\\subset \\Sigma(I)$ be such that $K=\\overline{D}$ is compact and $\\{K,D,I,\\tau\\}\\subset M$ (where $\\tau$ is the topology on $K$). Then the mapping $r:K\\to K$ defined as $r(x) = x|_{I\\cap M}$, for every $x\\in K$, is the canonical retraction associated to $M$, $K$ and $D$.\n\\end{lemma}\n\nThis was in a certain sense most probably well-known for countable models (see e.g. \\cite[Lemma 2.4]{KM06}), here we show that the situation is the same for uncountable models as well. The remainder of this subsection is more-or-less devoted to the proof of Lemma~\\ref{l:canonicalRetractionInSigmaProduct}. We start with two preliminary results.\n\n\\begin{lemma}\\label{lem:restrictionToIndicesFromModels}\nFor every suitable model $M$ the following holds: Let $X\\subset\\Sigma(I)$ be such that $\\{X,I\\}\\subset M$. Then we have\n\\[\\overline{X\\cap M} = \\overline{\\{x|_{I\\cap M}\\colon x\\in X\\}}.\\]\n\\end{lemma}\n\\begin{proof}\nLet $S$ and $\\Phi$ be the union of sets and lists of formulas from the statements of Lemma~\\ref{l:basics-in-M}, Lemma~\\ref{l:basics-in-M-2} and Lemma~\\ref{l:abstraction}\nand let $M\\prec (\\Phi; S\\cup\\{X,I\\})$. Let $\\pi:I\\to \\er^X$ be the mapping given by $\\pi_i(x)=x(i)$, for every $x\\in X$ and $i\\in I$. By Lemma~\\ref{l:basics-in-M} \\eqref{it: Dom-Rng-in-M} and Lemma~\\ref{l:basics-in-M-2} \\eqref{it:Pi-in-M}, we have that $\\pi\\in M$ and $\\A:=\\pi[I\\cap M]\\subset M$. Let $q_M:X\\to \\er^{\\A}$ be the mapping from Lemma~\\ref{l:abstraction}, that is, for $x\\in X$ we have $q_M(x)(\\pi_i) = x(i)$, $i\\in I\\cap M$. \nConsider the mapping $\\phi:\\er^{\\A}\\to \\er^I$ given for $x\\in \\er^{\\A}$ by $\\phi(x)(i):=x(\\pi_i)$, if $i\\in I\\cap M$ and $\\phi(x)(i):=0$, if $i\\in I\\setminus M$. It is easy to see that $\\phi$ is continuous. \nBy Lemma~\\ref{l:abstraction}, we have $q_M[X]\\subset \\overline{q_M[X\\cap M]}$ which implies that \\[\\{x|_{I\\cap M}\\colon x\\in X\\} = \\phi(q_M[X])\\subset \\overline{\\phi(q_M[X\\cap M])} = \\overline{\\{x|_{I\\cap M}\\colon x\\in X\\cap M\\}}.\\] \nThus, it suffices to note that for every $x\\in X\\cap M$ we have $x|_{I\\cap M} = x$, which follows from the fact that the support of every $x\\in X\\cap M$ is contained in $M$, see Lemma~\\ref{l:basics-in-M-2} \\eqref{it:supportSubsetOfM}.\n\\end{proof}\n\nThe following is well-known. We did not find a suitable reference, but it follows e.g. from the proof of \\cite[Theorem 6.1]{KM06} (for the key step see also \\cite[Lemma 1.2]{AMN88}). For the convenience of the reader we show a short argument based on our previous considerations. \n\n\\begin{lemma}\\label{lem:existenceOfTheSkeleton}\nLet $K\\subset \\er^I$ be a compact space and let $D:=\\Sigma(I)\\cap K$ be dense in $K$. Put\n\\[\\Gamma:=\\{A\\in [I]^{\\leq \\omega}\\colon x|_A\\in K\\text{ for every }x\\in K\\}\\]\nand for every $A\\in\\Gamma$ define $r_A:K\\to K$ by $r_A(x):=x|_A$, $x\\in K$. Then $(r_A)_{A\\in\\Gamma}$ is a commutative retractional skeleton on $K$ inducing the set $D$.\n\\end{lemma}\n\\begin{proof}\nIt is obvious that each $r_A$ is a continuous retraction with $r_A[K]$ metrizable. For every $A,B\\in\\Gamma$, we have $A\\cap B\\in\\Gamma$ and $r_A\\circ r_B = r_{A\\cap B}$ which implies that $r_A\\circ r_B = r_B\\circ r_A$. Having an increasing sequence $(A_n)_{n\\in\\omega}$ from $\\Gamma$ and $x\\in K$, we have $r_{A_n}x\\to x|_{\\bigcup A_n}$ and so $A_\\infty:=\\bigcup A_n\\in\\Gamma$ and $r_{A_n}x\\to r_{A_\\infty}x$. Let us now observe that for every $x\\in D$ there is $A\\in\\Gamma$ with $r_Ax = x$. Indeed, any $x\\in D$ has a countable support, so it suffices to see that for every countable $E\\subset I$ there is $A\\in\\Gamma$ with $E\\subset A$. Indeed, by Theorem~\\ref{thm:modelExists} and Lemma~\\ref{lem:restrictionToIndicesFromModels} (applied to $X=D$), there exists a countable set $M$ such that $E\\subset M\\cap I$ and $\\overline{\\{x|_{I\\cap M}\\colon x\\in D\\}}=\\overline{D\\cap M}$, which implies that $M\\cap I\\in \\Gamma$.\nFinally, note that the cofinality of $\\Gamma$ in $[I]^{\\leq \\omega}$ implies that the net $(r_A)_{A \\in \\Gamma}$ converges pointwise to the identity in $K$.\nThus, $\\Gamma$ is cofinal and $\\sigma$-closed in $[I]^{\\leq \\omega}$ (in particular $\\Gamma$ is a $\\sigma$-complete up-directed set) and $(r_A)_{A\\in\\Gamma}$ is a commutative retractional skeleton on $K$ inducing the the set $D$.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{l:canonicalRetractionInSigmaProduct}]\nLet $S$ and $\\Phi$ be the union of the countable sets and finite lists of formulas from the statements of Lemma~\\ref{l:basics-in-M}, Lemma~\\ref{l:basics-in-M-2}, Theorem~\\ref{thm:canonicalRetraction} and Lemma~\\ref{lem:restrictionToIndicesFromModels}. Pick $M\\prec (\\Phi; S\\cup \\{K,D,I,\\tau\\})$. Since $D$ is dense in $K$ and contained in the set induced by a retractional skeleton (see e.g. Lemma~\\ref{lem:existenceOfTheSkeleton}), by Theorem~\\ref{thm:canonicalRetraction}, $M$ admits canonical retraction $r_M$ associated to $M$, $K$ and $D$.\n\nBy Lemma~\\ref{lem:restrictionToIndicesFromModels}, using the continuity of the mapping $K\\ni x\\mapsto x|_{I\\cap M}$ and compactness of $K$, we have $\\overline{D\\cap M} = \\{x|_{I\\cap M}\\colon x\\in K\\}$ and so the retraction $r$ is well-defined, continuous and $r[K]=\\overline{D\\cap M}$.\nLet $\\pi:I\\to {\\er^K}$ be the mapping given for $i\\in I$ and $x\\in K$ as $\\pi(i)(x):=x(i)$. By Lemma~\\ref{l:basics-in-M-2} \\eqref{it:one-to-one-imageOfM} and \\eqref{it:Pi-in-M}, we have $\\pi\\in M$ and $\\pi[I]\\cap M = \\pi[I\\cap M]$. Since we obviously have $f\\circ r = f$, for every $f\\in \\pi[I\\cap M] = \\pi[I]\\cap M$ and $\\pi[I]\\in M$ separates the points of $K$, using Theorem~\\ref{thm:canonicalRetraction} we obtain that $r=r_M$.\n\\end{proof}\n\n\\subsection{Valdivia embedding}\\label{subsec:valdiviaEmbedding}\n\nThis subsection is devoted to the proof of  Theorem~\\ref{thm:Intro3} whose proof is based on the proof of \\cite[Theorem 2.6]{CGR17}. Quite surprisingly, the inductive argument does not give us only the ``Valdivia embedding'' (that is, Theorem~\\ref{thm:Intro3} \\ref{it:inducedByCommutative}$\\Rightarrow$\\ref{it:sigmaEmbedding}), but it also provides us with a new characterization of Valdivia compacta (that is, Theorem~\\ref{thm:Intro3} \\ref{it:inducedByCommutative}$\\Leftrightarrow$\\ref{it:skeletonInvariantSubset}) which we use later. Let us also highlight that there is an analogy of this new characterization in the language of suitable models, see Theorem~\\ref{thm:valdiviaThroughModels}. We start with a lemma.\n\n\\begin{lemma}\\label{lem:subsetSeparating}\nLet $K$ be a compact space and let $\\A\\subset \\C(K)$ be a set separating the points of $K$. Then there exists $\\A'\\subset\\A$ with $|\\A'|=\\w(K)$ which separates the points of $K$.\n\\end{lemma}\n\\begin{proof}\nFirst, since by the Stone-Weierstrass theorem $\\operatorname{alg}(\\A \\cup \\{1\\})$ is dense in $\\C(K)$, we easily observe that $\\{f^{-1}(-1\/2,1\/2)\\colon f\\in\\operatorname{alg}(\\A \\cup \\{1\\})\\}$ is a basis for the topology of $K$. Thus, by \\cite[Theorem 1.1.15]{Eng}, there is $\\F\\subset \\operatorname{alg}(\\A \\cup \\{1\\})$ with $|\\F| = \\w(K)$ such that $\\{f^{-1}(-1\/2,1\/2)\\colon f\\in\\F\\}$ is a basis for the topology of $K$. Pick $\\A'\\subset \\A$ such that $|\\A'| = \\w(K)$ and $\\F\\subset \\operatorname{alg}(\\A' \\cup \\{1\\})$. Then $\\A'$ separates the points of $K$, because otherwise $\\operatorname{alg}(\\A' \\cup \\{1\\})$ would not separate the points of $K$, a contradiction with the fact that $\\{f^{-1}(-1\/2,1\/2)\\colon f\\in\\F\\}$ is a basis of the topology.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:Intro3}]\n    \\ref{it:inducedByCommutative}$\\Rightarrow$\\ref{it:subsetOfComutativeInducedSubset} Let $\\mathfrak{s}_2$ be the commutative retractional skeleton on $K$ inducing $D(\\mathfrak{s})$. Then by Theorem~\\ref{thm:subskeletons} there is a weak subskeleton of both $\\mathfrak{s}$ and $\\mathfrak{s_2}$, which easily implies that there is a cofinal subset $\\Gamma''\\subset \\Gamma$ such that $r_s\\circ r_t = r_t\\circ r_s$, for every $s,t\\in\\Gamma''$. Thus, it suffices to let $\\Gamma'=(\\Gamma'')_{\\sigma}$. \\\\\n    \\ref{it:subsetOfComutativeInducedSubset}$\\Rightarrow$\\ref{it:skeletonInvariantSubset}: Let $\\mathfrak{s}_2=(r_s)_{s\\in\\Gamma'}$ be a commutative subskeleton of $\\mathfrak{s}$. Pick an up-directed set $\\Gamma''\\subset \\Gamma'$ and $x\\in D(\\mathfrak{s}) = D(\\mathfrak{s}_2)$. By Lemma~\\ref{l: retraction from up-directed subset} the limit $\\lim_{s\\in\\Gamma''}r_s(x)$ exists. Since there exists $s_0\\in\\Gamma'$ such that $x = r_{s_0} x$, using the commutativity we obtain\n    \\[\\lim_{s\\in\\Gamma''}r_s(x) = \\lim_{s\\in\\Gamma''}r_s(r_{s_0}x) = r_{s_0}(\\lim_{s\\in\\Gamma''}r_s(x)) \\in D(\\mathfrak{s}).\\]\n    Thus, $\\mathfrak{s}_2$ satisfies \\ref{it:skeletonInvariantSubset} with $D=D(\\mathfrak{s})$.\\\\\n    \\ref{it:skeletonInvariantSubset}$\\Rightarrow$\\ref{it:sigmaEmbedding}: First, we may without loss of generality assume that $\\mathfrak{s}_2 = \\mathfrak{s}$. We will prove the result by induction on $\\kappa:=\\w(K)$. We may without loss of generality assume that $\\lambda = 1$. If $\\kappa=\\omega$, then by Lemma~\\ref{lem:subsetSeparating}, there exists a countable set $\\HH\\subset \\A$ which separates the points of $K$ and this set does the job.\n    So let us assume that the result holds for every compact space of weight strictly smaller than $\\kappa$. Proposition \\ref{prop:rri} together with Theorem~\\ref{thm:modelExists} imply the existence of sets $(M_\\alpha)_{\\alpha \\le \\kappa}$ satisfying \\ref{it:areModels}-\\ref{it:areContinuous} and retractions $(r_\\alpha)_{\\alpha \\le \\kappa}$ satisfying \\ref{it:comutative}-\\ref{it:countableSupport2}. Note that using \\ref{it:secondFormula}, we obtain that $r_\\alpha[D] \\subset D$. For every $\\alpha < \\kappa$, define $\\A_\\alpha:=\\{f\\in\\C(r_\\alpha[K])\\colon f\\circ r_\\alpha\\in\\A\\}$. It is easy to see that, for every $\\alpha < \\kappa$, the set $\\A_\\alpha$ is symmetric, closed, convex and bounded. The fact that $\\A_\\alpha$ separates the points of $r_\\alpha[K]$ follows from \\ref{it:compositionWithRa} and \\ref{comutaralpha3} implies that $f \\circ r_s \\in \\A_\\alpha$, for every $f \\in \\A_\\alpha$ and every $s \\in (\\Gamma \\cap M_\\alpha)_\\sigma$. For every $\\alpha < \\kappa$, define $D_\\alpha:=D \\cap r_\\alpha[K]\\subset D(\\mathfrak s) \\cap r_\\alpha[K]$. Since $r_{\\alpha}[D]$ is dense in $r_{\\alpha}[K]$ and $r_{\\alpha}[D]\\subset  D_{\\alpha}$, we have that $D_\\alpha$  is dense in $r_\\alpha[K]$. \n    Therefore the induction hypothesis and \\ref{it:skeletonOnSubset} imply that there are sets $T_\\alpha\\subset \\A_{\\alpha}$ such that the mapping $\\varphi_\\alpha:r_\\alpha[K]\\to [-1,1]^{T_\\alpha}$ given by $\\varphi_\\alpha(x)(t):=t(x)$, $t\\in T_{\\alpha}$ and $x \\in r_\\alpha[K]$ is a homeomorphic embedding and $\\varphi_\\alpha[D_\\alpha]\\subset \\Sigma(T_\\alpha)$, for every $\\alpha<\\kappa$. We may without loss of generality assume that $T_\\alpha\\cap T_\\beta = \\emptyset$ for $\\alpha\\neq \\beta$. Now, we put $T=T_0\\cup \\bigcup_{\\alpha<\\kappa} T_{\\alpha+1}$ and define $\\varphi:K\\to [-1,1]^T$ by\n    \\[\n\\varphi(x)(t):=\\begin{cases}\n\\tfrac{1}{2}\\big(\\varphi_{\\alpha+1}(r_{\\alpha+1}(x))(t) - \\varphi_{\\alpha+1}(r_{\\alpha}(x))(t)\\big), & t\\in T_{\\alpha+1},\\\\\n\\varphi_0(r_0(x))(t), & t\\in T_0.\n\\end{cases}\n\\]\nThen $\\varphi$ is of course continuous. Let us verify that it is one-to-one. Indeed, if $x, y$ are distinct points from $K$ then by \\ref{it:ordinalSeparatingPoints} there is a minimal ordinal $\\alpha_0<\\kappa$ for which $r_{\\alpha_0}(x)\\neq r_{\\alpha_0}(y)$ and $\\alpha_0=0$ or it is a successor ordinal. If $\\alpha_0=0$, then there exists $t \\in T_0$ such that $\\varphi_0\\big(r_0(x)\\big)(t) \\ne \\varphi_0\\big(r_0(y)\\big)(t)$ and so we have $\\varphi(x)(t) \\ne \\varphi(y)(t)$. Otherwise, $\\alpha_0=\\beta_0+1$ for some $\\beta_0<\\kappa$ and there is $t\\in T_{\\alpha_0}$ such that $\\varphi_{\\alpha_0}(r_{\\alpha_0}(x))(t)\\neq \\varphi_{\\alpha_0}(r_{\\alpha_0}(y))(t)$. Moreover, since $\\alpha_0$ is minimal, we have $r_{\\beta_0}(x)=r_{\\beta_0}(y)$, hence we obtain $\\varphi(x)(t) \\neq \\varphi(y)(t)$ and so $\\varphi(x)\\neq \\varphi(y)$.\nThus, $\\varphi$ is a homeomorphic embedding.\n\nLet us show that $\\varphi[D(\\mathfrak{s})]\\subset \\Sigma(T)$. Indeed, by \\ref{it:countableSupport2}, for every $x\\in D$ the set $\\{\\alpha<\\kappa\\colon r_{\\alpha+1}(x)\\neq r_\\alpha(x)\\}$ is countable. Moreover, since $r_\\alpha[D] \\subset D_\\alpha$, the induction hypothesis ensures that the supports of $\\varphi_0\\big(r_0(x)\\big)$, $\\varphi_{\\alpha+1}(r_{\\alpha+1}(x))$ and $\\varphi_{\\alpha+1}(r_{\\alpha}(x))$ are countable. Therefore the support of $\\varphi(x)$ is countable and we obtain $\\varphi[D]\\subset \\Sigma(T)$. Moreover, since $D$ is dense in $K$, by Lemma~\\ref{lem:existenceOfTheSkeleton} there is a commutative retractional skeleton $\\mathfrak{s}_2$ on $\\varphi[K]$ such that $D(\\mathfrak{s}_2) = \\varphi[K]\\cap \\Sigma(T)$. Since $\\varphi[D(\\mathfrak{s})]\\cap D(\\mathfrak{s}_2)\\supset \\varphi[D]$, by \\cite[Lemma 3.2]{C14} we have that $\\varphi[D(\\mathfrak{s})] = D(\\mathfrak{s}_2)\\subset \\Sigma(T)$.\n\nNow, let us show that $\\pi_t\\circ \\varphi\\in\\A$, for every $t\\in T$. Firstly, note that $\\pi_t\\circ \\varphi_\\alpha\\in \\A_\\alpha$, for every $t\\in T_\\alpha$ and every $\\alpha < \\kappa$. If $t\\in T_0$, then we have that $\\pi_t\\circ \\varphi = \\pi_t\\circ \\varphi_0\\circ r_0\\in\\A$. Pick $\\alpha<\\kappa$ and $t\\in T_{\\alpha+1}$. Then, similarly as above, $\\pi_t\\circ \\varphi_{\\alpha+1}\\circ r_{\\alpha+1}\\in \\A$ and therefore using \\ref{it:compositionWithRa}, we obtain\n\\[\n\\pi_t\\circ \\varphi = \\tfrac{1}{2}\\big(\\pi_t\\circ \\varphi_{\\alpha+1}\\circ r_{\\alpha+1} - \\pi_t\\circ \\varphi_{\\alpha+1}\\circ r_{\\alpha+1}\\circ r_\\alpha\\big)\\in \\A.\n\\]\nOmitting some indices, we may without loss of generality assume that the mapping $T\\ni t\\mapsto f_t:=\\pi_t\\circ \\varphi\\in \\A$ is one-to-one and so $\\HH:=\\{f_t\\colon t\\in T\\}$ does the job.\n\\\\\n    \\ref{it:sigmaEmbedding}$\\Rightarrow$\\ref{it:inducedByCommutative}: By Lemma~\\ref{lem:existenceOfTheSkeleton} there is a commutative retractional skeleton $\\mathfrak{s}_2$ on $\\varphi[K]$ such that $D(\\mathfrak{s}_2) = \\varphi[K]\\cap \\Sigma(I)$. Since $\\varphi[D(\\mathfrak{s})]\\subset D(\\mathfrak{s}_2)$, by \\cite[Lemma 3.2]{C14} we have that $\\varphi[D(\\mathfrak{s})] = D(\\mathfrak{s}_2)$ and so the set $D(\\mathfrak{s})$ is induced by a commutative retractional skeleton.\\\\\n    \\ref{it:skeletonInvariantSubset}$\\Rightarrow$\\ref{it:inducedByCommutative}: follows from \\ref{it:skeletonInvariantSubset}$\\Rightarrow$\\ref{it:sigmaEmbedding}$\\Rightarrow$\\ref{it:inducedByCommutative} applied to the set $\\A:=B_{\\C(K)}$.\n    \\end{proof}\n\n    The following corollary might be well-known, but let us mention it for future reference.\n    \n    \\begin{cor}\\label{cor:fullIsCommutative}\n    Let $K$ be a compact space and let $\\mathfrak{s}$ be a full retractional skeleton on $K$. Then there exists a commutative subskeleton of $\\mathfrak{s}$.\n    \\end{cor}\n    \\begin{proof}\n    Apply Theorem~\\ref{thm:Intro3} \\ref{it:skeletonInvariantSubset}$\\implies$\\ref{it:subsetOfComutativeInducedSubset} to $D:=K$.\n    \\end{proof}\n    \n    The proof of Theorem~\\ref{thm:Intro3} gives us also the following.\n    \n    \\begin{thm}\\label{thm:valdiviaThroughModels}\n    Let $K$ be a compact space and let $D$ be a set induced by a retractional skeleton on $K$. Then the following are equivalent.\n    \\begin{enumerate}[label = (\\alph*)]\n        \\item\\label{it:DinducedByCommutative} $D$ is induced by a commutative retractional skeleton.\n        \\item\\label{it:equivUsingSuitableModels} For every suitable model $M$ the following holds:\n        \\[\\forall x\\in D\\; \\exists y\\in D\\cap \\overline{D\\cap M}\\; \\forall f\\in\\C(K)\\cap M:\\quad f(x)=f(y).\\]\n    \\end{enumerate}\n    \\end{thm}\n    \\begin{proof}\n    \\ref{it:DinducedByCommutative}$\\Rightarrow$\\ref{it:equivUsingSuitableModels}: By Theorem~\\ref{thm:canonicalRetraction}, there is a finite list of formulas $\\Phi$ and a countable set $S$ (depending on the compact space $K$ and the set $D$) such that for any $M\\prec(\\Phi,S)$, $M$ admits canonical retraction $r_M$ and we have $r_M[D]\\subset D$. Then \\ref{it:equivUsingSuitableModels} follows from Theorem~\\ref{thm:canonicalRetraction} \\ref{it:characterizationFirst} applied to $\\A:=\\C(K)$.\\\\\n    \\ref{it:equivUsingSuitableModels}$\\Rightarrow$\\ref{it:DinducedByCommutative}: Follows from the fact that in the proof of Theorem~\\ref{thm:Intro3} \\ref{it:skeletonInvariantSubset}$\\Rightarrow$\\ref{it:sigmaEmbedding} we used condition \\ref{it:skeletonInvariantSubset} only to ensure that for a suitable model $M_\\alpha$ we have $r_{\\alpha}[D]\\subset D$, which by Theorem~\\ref{thm:canonicalRetraction} \\ref{it:characterizationFirst} follows from the condition \\ref{it:equivUsingSuitableModels} above.\n    \\end{proof}\n \n\\section{Characterization of (semi-)Eberlein compacta}\\label{sec:semiEberlein}\n\nHere, we apply the results of the preceding sections and characterize (semi)-Eberlein compacta using the notion of an $\\A$-shrinking retractional skeleton.\n\n\\begin{defin}\\label{def:shrinking}\nLet $K$ be a countably compact space. Let $\\emptyset\\neq \\A\\subset \\C(K)$ be a bounded set. The pseudometric $\\rho_{\\A}$ on $K$ is given as \n\\[\n\\rho_{\\A}(k,l):=\\sup_{f\\in\\A} |f(k)-f(l)|,\\quad k,l\\in K.\n\\]\nIf $(r_s)_{s \\in \\Gamma}$ is a retractional skeleton on $K$ and $D\\subset K$, we say that $(r_s)_{s \\in \\Gamma}$ is {\\it $\\A$-shrinking with respect to $D$} if for every $x \\in D$ and every increasing sequence $(s_n)_{n \\in \\omega}$ in $\\Gamma$ with $s:=\\sup_{n\\in \\omega}s_{n}$, we have that $\\lim_{n\\in \\omega}\\rho_{\\A}(r_{s_n}(x),r_s(x))=0$. If $(r_s)_{s\\in\\Gamma}$ is $\\A$-shrinking with respect to $K$, then we just write that $(r_s)_{s\\in\\Gamma}$ is \\emph{$\\A$-shrinking}.\n\n\nFinally, given $\\varepsilon> 0$ we say that $(r_s)_{s \\in \\Gamma}$ is  {\\it $(\\A,\\varepsilon)$-shrinking with respect to $D$} if for every $x \\in D$ and every increasing sequence $(s_n)_{n \\in \\omega}$ in $\\Gamma$ with $s:=\\sup_{n\\in \\omega}s_{n}$, we have that $\\limsup_{n\\in \\omega}\\rho_{\\A}(r_{s_n}(x),r_s(x))\\leq \\varepsilon$.\n\\end{defin}\n\nNote that if the nonempty and bounded set $\\A$ separates the points of $K$, then $\\rho_{\\A}$ is a metric on $K$.\n\nThe aim of this section is to prove the following result from which Theorem~\\ref{thm:Intro1} and Theorem~\\ref{thm:Intro2} easily follow.\n\n\n\\begin{thm}\\label{t:mainEberleinThroughSkeleton2}\nLet $K$ be a compact space and let $D\\subset K$ be a dense set. Consider the following conditions.\n\\begin{enumerate}[label = (\\roman*)]\n\\item\\label{it:defSemiEberlein} There exists a homeomorphic embedding $h:K\\to [-1,1]^I$ such that $h[D] = c_0(I)\\cap h[K]$.\n\\item\\label{it:eqdefSemiEberleinThroughSkeleton} There exist a bounded set $\\A\\subset \\C(K)$ separating the points of $K$ and a retractional skeleton $\\mathfrak{s} = (r_s)_{s\\in\\Gamma}$ on $K$ with $D\\subset D(\\mathfrak{s})$ such that\n\\begin{enumerate}[label=(\\alph*)]\n    \\item $\\mathfrak{s}$ is $\\A$-shrinking with respect to $D$,\n    \\item\\label{it:inSpecialCaseEquivalentToCommutativity2} $\\lim_{s\\in\\Gamma'} r_s(x)\\in D$, for every $x\\in D$ and every up-directed subset $\\Gamma'$ of $\\Gamma$.\n\\end{enumerate}\n\n\\item\\label{it:eqdefSemiEberleinThroughSkeletonAvailableForContImage} There exist a countable family $\\A$ of subsets of $B_{\\C(K)}$ and a retractional skeleton $\\mathfrak{s} = (r_s)_{s\\in\\Gamma}$ on $K$ with $D\\subset D(\\mathfrak{s})$ such that\n\\begin{enumerate}[label=(\\alph*)]\n    \\item For every $A\\in\\A$ there exists $\\varepsilon_A>0$ such that $\\mathfrak{s}$ is $(A,\\varepsilon_A)$-shrinking with respect to $D$,\n    \\item\\label{cond:fullInBall3} for every $\\varepsilon>0$ we have $B_{\\C(K)} = \\bigcup\\{A\\in \\A\\colon \\varepsilon_A < \\varepsilon\\}$, and\n    \\item\\label{it:inSpecialCaseEquivalentToCommutativity3} $\\lim_{s\\in\\Gamma'} r_s(x)\\in D$, for every $x\\in D$ and every up-directed subset $\\Gamma'$ of $\\Gamma$.\n\\end{enumerate}\n\\item\\label{it:subsetCondition} There exists a homeomorphic embedding $h:K\\to [-1,1]^J$ such that $h[D] \\subset c_0(J)$.\n\\end{enumerate}\nThen (i)$\\Rightarrow$(ii)$\\Rightarrow$(iii)$\\Rightarrow$(iv).\n\\end{thm}\n\nLet us first give some comments.\n\n\\begin{remark}\\label{rem:comparison}\nThe notion of a shrinking retractional skeleton is inspired by \\cite{FM18}, where the definition of a shrinking projectional skeleton was given and WCG Banach spaces were characterized using this notion.\n\nGiven a retractional skeleton $(r_s)_{s\\in\\Gamma}$ on a compact space $K$, it is well known that $(P_s)_{s\\in\\Gamma}$ given by $P_s(f):=f\\circ r_s$, $s\\in\\Gamma$ is a projectional skeleton on $\\C(K)$, see e.g. \\cite[Proposition 5.3]{K20}. Moreover, if $\\emptyset\\neq \\A\\subset \\C(K)$ is a bounded set and $(P_s)_{s\\in\\Gamma}$ is $\\A$-shrinking in the sense of \\cite[Definition 16]{FM18}, it is not very difficult to observe that $(r_s)_{s\\in\\Gamma}$ is $\\A$-shrinking in the sense of Definition~\\ref{def:shrinking}. It is not clear whether the converse holds as it is (at least formally) a stronger condition. Thus, Theorem~\\ref{thm:Intro1} allows us in a certain sense to strengthen implication (ii)$\\Rightarrow$(i) from \\cite[Theorem 21]{FM18}. Since the other implication is easier, Theorem~\\ref{thm:Intro1} may be thought of as a topological counterpart and in a certain sense also strengthening of \\cite[Theorem 21]{FM18} in the context of $\\C(K)$ spaces.\n\\end{remark}\n\nNotice that the shrinkingness of a retractional skeleton is not a specific property of one particular skeleton. First, observe that any $\\A$-shrinking retractional skeleton is also full, whenever $\\A\\subset \\C(K)$ separates the points of $K$, this is generalized in the following.\n\n    \\begin{lemma}\\label{lem:shrinkingImpliesFull}\n    Let $K$ be a compact space, $\\A\\subset \\C(K)$ be a bounded set separating the point of $K$ and let $\\mathfrak{s} = (r_s)_{s\\in\\Gamma}$ be a retractional skeleton on $K$ which is $\\A$-shrinking with respect to a set $D$ with $D\\supset D(\\mathfrak{s})$. Then $D=D(\\mathfrak{s})$ and $\\lim_{s\\in\\Gamma'} r_s(x)\\in D$, for every $x\\in D$ and every up-directed subset $\\Gamma'$ of $\\Gamma$.\n    \\end{lemma}\n    \\begin{proof}\n    Fix $x\\in D$ and an up-directed set $\\Gamma'\\subset\\Gamma$. Since $\\mathfrak{s}$ is $\\A$-shrinking with respect to $D$, it is not very difficult to observe (see e.g. \\cite[Proposition 20]{FM18}) that there exists an increasing sequence $(s_n)_{n\\in\\omega}$ in $\\Gamma'$ with $s = \\sup_n s_n\\in\\Gamma$ such that $\\rho_\\A-\\lim_{t\\in\\Gamma'} r_t(x) = r_s(x)$. Therefore, since the limit $\\lim_{t\\in\\Gamma'} r_t(x)$ exists, we obtain $f(\\lim_{t\\in\\Gamma'} r_t(x)) = f(r_s(x))$, for every $f\\in \\A$. Since $\\A$ separates the points of $K$, we deduce that $\\lim_{t\\in\\Gamma'} r_t(x) = r_s(x)\\in D(\\mathfrak{s})\\subset D$. Finally, for $\\Gamma'=\\Gamma$ we obtain $x = \\lim_{s\\in \\Gamma} r_s(x)\\in D(\\mathfrak{s})$ and so $D\\subset D(\\mathfrak{s})$.\n    \\end{proof}\n    \n\\begin{lemma}\\label{ShrinkingCommutativeSkeleton}\nLet $K$ be a compact space and $\\A\\subset \\C(K)$ be a bounded set separating the points of $K$. If there exists an $\\A$-shrinking retractional skeleton on $K$, then every full retractional skeleton on $K$ admits a weak subskeleton which is $\\A$-shrinking and commutative. \n\\end{lemma}\n\\begin{proof}\nBy Lemma \\ref{lem:shrinkingImpliesFull}, there exists an $\\A$-shrinking and full retractional skeleton $(\\widetilde{r}_i)_{i\\in I}$ on $K$. Moreover, by Corollary~\\ref{cor:fullIsCommutative} we may without loss of generality assume that $(\\widetilde{r}_i)_{i\\in I}$ is commutative.\nNow, let $(r_{s})_{s\\in\\Gamma}$ be a full retractional skeleton on $K$. By Theorem \\ref{thm:subskeletons}, there exists a retractional skeleton $(R_i)_{i\\in\\Lambda}$ which is a weak subskeleton of both $(r_s)_{s\\in\\Gamma}$ and $(\\widetilde{r}_i)_{i\\in I}$. It is easy to see that $(R_i)_{i \\in \\Lambda}$ is commutative and $\\A$-shrinking.\n\\end{proof}\n\nIn the remainder of this section we provide the proof of  Theorem~\\ref{t:mainEberleinThroughSkeleton2}. Let us start with the proof of the implication \\ref{it:defSemiEberlein}$\\Rightarrow$\\ref{it:eqdefSemiEberleinThroughSkeleton}.\n\n \\begin{lemma}\\label{lem:semiEbImpliesShrinkingSkeleton}\nLet $K$ be a compact space and $D\\subset K$ be a dense subset. If there exists a homeomorphic embedding $h:K\\to [-1,1]^I$ such that $h[D] = h[K]\\cap c_0(I)$, then there are $\n\\A\\subset B_{\\C(K)}$, $\\mathfrak{s} = (r_s)_{s\\in\\Gamma}$ such that \\ref{it:eqdefSemiEberleinThroughSkeleton} in Theorem~\\ref{t:mainEberleinThroughSkeleton2} holds with $\n\\A$, $\\mathfrak{s} = (r_s)_{s\\in\\Gamma}$ and we moreover have $f\\circ r_s\\in \\A$ for every $f\\in\\A$ and $s\\in\\Gamma$.\n\\end{lemma}\n\\begin{proof}\nWe may without loss of generality assume that $K\\subset [-1,1]^I$, $D = K\\cap c_0(I)$. Pick the commutative retractional skeleton $(r_A)_{A\\in\\Gamma}$  from Lemma~\\ref{lem:existenceOfTheSkeleton} and put $\\SSS:=\\{\\pi_i|_K\\colon i\\in I\\}\\cup \\{0\\}\\subset \\C(K)$. Clearly $\\SSS$ is bounded and separating. Moreover $D$ is obviously contained in the set $K\\cap \\Sigma(I)$ (which is the set induced by $(r_A)_{A\\in\\Gamma}$) and it is easy to observe that if $A\\in\\Gamma$ and $f\\in \\mathcal{S}$, then $f\\circ r_A\\in \\SSS$.\n\nNow, let us show that $(r_A)_{A \\in \\Gamma}$ is $\\mathcal{S}$-shrinking with respect to $D$. Pick $x\\in D$, an increasing sequence $(A_n)_{n\\in\\omega}$ in $\\Gamma$ and put $A = \\sup_n A_n= \\bigcup_{n\\in\\omega} A_n$. Fix $\\varepsilon>0$ and let $n_0\\in \\omega$ be such that $\\{i\\in A\\colon |x(i)|>\\varepsilon\\}\\subset A_{n_0}$. Then for every $n\\geq n_0$ we obtain $r_{A_n}x(i) - r_Ax(i) = 1_{A\\setminus A_n}\\cdot x(i)$, therefore for every $i\\in I$ we have $|r_{A_n}x(i) - r_Ax(i)| <\\varepsilon$; hence $\\sup_{f\\in \\mathcal{S}} |f(r_{A_n}x) - f(r_{A}x)| < \\varepsilon$.\n\nFinally, we note that whenever $\\Gamma'\\subset \\Gamma$ is up-directed and $x\\in D$, then $y:=\\lim_{A\\in\\Gamma'} r_Ax$ exists by Lemma~\\ref{l: retraction from up-directed subset} and moreover if $i \\in \\suppt y$, then $y(i)=x(i)$. Therefore, we have that $y \\in K \\cap c_0(I)=D$.\n\\end{proof}\n\n\n\nThe most demanding is the proof of the implication \\ref{it:eqdefSemiEberleinThroughSkeletonAvailableForContImage}$\\Rightarrow$\\ref{it:subsetCondition} in Theorem~\\ref{t:mainEberleinThroughSkeleton2}. We start with an easy observation.\n\n\\begin{lemma}\\label{l: retraction from up-directed subset semi-eberlein caseImproved}\nLet $K$ be a compact space, $\\A\\subset \\C(K)$ be a bounded set, $\\varepsilon>0$ and $D$ be a subset of $K$. Suppose that $(r_s)_{s\\in \\Gamma}$ is a retractional skeleton on $K$ that is $(\\A,\\varepsilon)$-shrinking with respect to $D$. For an up-directed set $\\Gamma^{'}\\subset \\Gamma$ let $R_{\\Gamma^{'}}$ be as in Lemma~\\ref{l: retraction from up-directed subset}. Then for every $x\\in D$ we have the following.\n\\begin{enumerate}[label=(S\\alph*)]\n    \\item\\label{it:shrinkingCanonicalRetraction} If $\\Gamma^{'}\\subset\\Gamma$ is up-directed, then \n    there exists $s_0\\in \\Gamma'$ such that we have\n    \\[\\rho_\\A(R_{\\Gamma^{'}}(x),r_s(x)) \\leq 7\\varepsilon,\\qquad s\\geq s_0, s\\in\\Gamma'\\] \n    \\item\\label{it:shrinkingUpDirectedCanonicalRetractions} If $\\M\\subset \\mathcal{P}(\\Gamma)$ is such that $\\M$ is up-directed and each $M\\in\\M$ is up-directed, then there exists $M_0\\in \\M$ such that for every $M\\in\\M$ with $M\\supset M_0$ we have\n    \\[\n    \\rho_\\A(R_M(x),R_{\\bigcup \\M}(x)) \\leq 14\\varepsilon.\n    \\]\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}Pick $x\\in D$.\\\\\n\\ref{it:shrinkingCanonicalRetraction} First, let us observe that there exists $s_0\\in\\Gamma'$ such that\n\\begin{equation}\\label{eq:cauchyShrinkingness}\n\\rho_\\A(r_s(x),r_{s_0}(x)) < 3\\varepsilon,\\qquad s\\geq s_0,s\\in\\Gamma'.\n\\end{equation}\nIndeed, if this is not the case we inductively construct an increasing sequence $(s_n)$ in $\\Gamma'$ with $\\rho_\\A(r_{s_n}(x),r_{s_{n+1}}(x))\\geq 3\\varepsilon$, $n\\in\\en$ which is in contradiction with $(\\A,\\varepsilon)$-shrinkingness.\n\nPick $f\\in \\A$ and $s_1\\geq s_0$, $s_1\\in\\Gamma'$. Since $\\lim_{s\\in\\Gamma'} r_s(x) = R_{\\Gamma'}(x)$, there exists $s_2\\geq s_1$, $s_2\\in\\Gamma'$ with $|f(r_{s_2}(x)) - f(R_{\\Gamma'}(x))|<\\varepsilon$. Therefore, by \\eqref{eq:cauchyShrinkingness} we obtain\n\\[\n|f(r_{s_1}(x)) - f(R_{\\Gamma'}(x))|\\leq \\rho_\\A(r_{s_1}(x),r_{s_0}(x)) + \\rho_\\A(r_{s_0}(x),r_{s_2}(x)) + \\varepsilon < 7\\varepsilon.\n\\]\nSince $f\\in A$ was arbitrary, this proves \\ref{it:shrinkingCanonicalRetraction}.\\\\ \n\\ref{it:shrinkingUpDirectedCanonicalRetractions} By \\ref{it:shrinkingCanonicalRetraction}, there exists $s_0\\in \\bigcup \\M$ such that $\\rho_\\A(r_s(x),R_{\\bigcup \\M}(x))\\leq 7\\varepsilon$, for every $s\\geq s_0$, $s\\in \\bigcup \\M$. Let $M_0\\in \\M$ be such that $s_0\\in M_0$. Then, for every $M\\in \\M$ with $M\\supset M_0$ by \\ref{it:shrinkingCanonicalRetraction} there exists $s_M\\geq s_0$, $s_M\\in M$ with $\\rho_\\A(r_{s_M}(x),R_M(x))\\leq 7\\varepsilon$, which implies that\n\\[\\rho_\\A(R_M(x),R_{\\bigcup \\M}(x)) \\leq 14\\varepsilon.\\qedhere\\]\n\\end{proof}\n\n\nThe following proposition together with Theorem~\\ref{thm:Intro3} is the core of our argument. The idea to use such a result is related to a characterization of Eberlein compacta by Farmaki \\cite[Theorem 2.9]{F87} (see also \\cite[Theorem 10]{FGZ}). Note however, that our methods enable us to present a self-contained proof.\n\n\n\\begin{prop}\\label{prop:semiEberleinFarmakiImproved}\nLet $K \\subset [-1,1]^I$ be a compact space and for $I'\\subset I$ define\n\\[\\mathcal{S}_{K,I'}=\\{\\pi_i|_K: i \\in I'\\}.\\]\nSuppose that $K$ admits a retractional skeleton $\\mathfrak{s}=(r_s)_{s \\in \\Gamma}$ such that $D(\\mathfrak{s})\\subset \\Sigma(I)$ and let $D \\subset D(\\mathfrak{s})$. Assume that there is a countable family $\\A$ consisting of subsets of $I$ such that\n\\begin{enumerate}\n    \\item\\label{it:epsilon-shrinking} For every $A \\in \\A$, there exists $\\varepsilon_A>0$ such that $(r_s)_{s \\in \\Gamma}$ is $(\\mathcal{S}_{K,A}, \\varepsilon_A)$-shrinking with respect to $D$;\n    \\item\\label{it:covers-ball} For every $\\varepsilon> 0$, it holds that $I=\\bigcup \\{A \\in \\A: \\varepsilon_A< \\varepsilon\\}$;\n    \\item $\\lim_{s\\in\\Gamma'}r_s(x)\\in D$, for every $x\\in D$ and every up-directed subset $\\Gamma'$ of $\\Gamma$.\n\\end{enumerate}\nThen for every $\\varepsilon>0$ there is a decomposition $I = \\bigcup_{n=0}^\\infty I_n^\\varepsilon$ such that \\[\\forall n\\;\\forall x\\in D:\\quad |\\{i\\in I_n^\\varepsilon\\colon |x(i)| > \\varepsilon\\}|<\\omega.\\]\n\\end{prop}\n\n\\begin{proof}\nBy \\cite[Proposition 19.5]{KKLbook}, we may pick a set $J\\subset I$ such that $|J|=\\w(K)$ and $\\suppt{x}\\subset J$, for every $x\\in K$. By \\cite[Lemma 3.2]{C14}, we have that $D(\\mathfrak{s})=\\Sigma(I)\\cap K$ and hence Lemma~\\ref{lem:existenceOfTheSkeleton} ensures that $D(\\mathfrak{s})$ is induced by a commutative retractional skeleton.\nTherefore it follows from Theorem \\ref{thm:Intro3} that we may assume the retractional skeleton $\\mathfrak{s}$ is commutative. Now, let us prove the result by induction on the weight of $K$. If $K$ has countable weight, then the set $J$ is countable and we may enumerates it as $J:=(j_n)_{n\\ge 1}$. For each $\\varepsilon>0$, let $I_{0}^{\\varepsilon}= I\\setminus J$, and $I_{n}^{\\varepsilon}=\\{j_n\\}$, for every $n \\ge 1$. Then $I=\\bigcup_{n=0}^{\\infty}I_{n}^{\\varepsilon}$ and \\[\\forall n\\;\\forall x\\in K:\\quad |\\{i\\in I_n^\\varepsilon\\colon |x(i)| > \\varepsilon\\}|\\leq 1.\\]\nNow suppose that $\\w(K)=\\kappa \\ge \\omega_1$ and that the result holds for compact spaces of weight less than $\\kappa$. \nProposition \\ref{prop:rri} together with Theorem~\\ref{thm:modelExists} imply the existence of sets $(M_\\alpha)_{\\alpha \\le \\kappa}$ satisfying \\ref{it:areModels}-\\ref{it:areContinuous} and retractions $(r_\\alpha)_{\\alpha \\le \\kappa}$ satisfying \\ref{it:comutative}-\\ref{it:countableSupport2}. Note that we can assume that $J \\subset \\bigcup_{\\alpha < \\kappa} M_\\alpha$, by replacing \\ref{it:areIncreasing} by the following (stronger) condition:\n\\[M_{\\alpha+1}\\prec (\\Phi;\\{j_\\alpha,\\alpha\\}\\cup M_\\alpha), \\ \\forall \\alpha < \\kappa,\\]\nwhere $J=\\{j_\\alpha: \\alpha < \\kappa\\}$.\nNote that, by Lemma~\\ref{l:canonicalRetractionInSigmaProduct}, we may assume that $r_\\alpha(x) = x|_{I\\cap M_\\alpha}$, for every $x\\in K$ and $\\alpha<\\kappa$. \nFor each $\\alpha < \\kappa$, it is easy to see that for every $A\\in\\A$ the retractional skeleton $(r_s|_{r_\\alpha[K]})_{s \\in (\\Gamma \\cap M_\\alpha)_\\sigma}$ given by \\ref{it:skeletonOnSubset} is $(\\mathcal S_{r_\\alpha[K],A},\\varepsilon_A)$-shrinking with respect to the set  $D\\cap r_\\alpha[K]\\subset D(\\mathfrak{s})\\cap r_{\\alpha}[K] \\subset \\Sigma(I) \\cap [-1,1]^{I}$. Moreover, if $\\Gamma'\\subset (\\Gamma \\cap M_\\alpha)_\\sigma$ is up-directed and $x \\in D \\cap r_\\alpha[K]$, then using \\ref{comutaralpha3} we conclude that $\\lim_{s\\in\\Gamma'} r_s(x) \\in D\\cap r_{\\alpha}[K]$.\nNow fix $\\varepsilon>0$ and let $I=\\bigcup_{n \\ge 1}I^\\varepsilon_{n,0}$ be the decomposition given by induction hypothesis applied to $r_0[K]$ (using that by \\ref{it:skeletonOnSubset} we may apply the inductive hypothesis to $r_0[K]$), that is, for every $y \\in D\\cap r_0[K]$ and $n \\ge 1$ the set\n\\[\\{i \\in I^\\varepsilon_{n,0}: \\vert y(i) \\vert >\\varepsilon\\}\\]\nis finite. Fix $\\alpha < \\kappa$, similarly let $I=\\bigcup_{n \\ge 1}I^\\varepsilon_{n,\\alpha+1}$ be the decomposition given by the induction hypothesis applied to $r_{\\alpha+1}[K]$, that is, for every $y \\in D\\cap r_{\\alpha+1}[K]$ and $n \\ge 1$ the set\n\\[\\{i \\in I^\\varepsilon_{n,\\alpha+1}: \\vert y(i) \\vert >\\varepsilon\\}\\]\nis finite. For $A \\in \\A$ with $\\varepsilon_A< \\varepsilon\/14$ define $I^\\varepsilon_{(0,A)}=I \\setminus J$ and for every $n \\ge 1$ put\n\\[I^\\varepsilon_{(n,A)}=(A\\cap J \\cap I^\\varepsilon_{n,0} \\cap M_0) \\cup \\bigcup_{\\alpha < \\kappa} \\big(A\\cap J \\cap I^\\varepsilon_{n, \\alpha+1} \\cap (M_{\\alpha+1} \\setminus M_{\\alpha})\\big).\\]\nNote that $I=\\bigcup \\{I^\\varepsilon_{(n,A)}\\colon n \\ge 0, A\\in \\A, 14\\varepsilon_A<\\varepsilon\\}$, since $J \\subset \\bigcup_{\\alpha < \\kappa}M_\\alpha$ and $I=\\bigcup \\{A \\in \\A: 14\\varepsilon_A< \\varepsilon\\}$.\nFixed $x \\in D$, $A \\in \\A$ with $\\varepsilon_A< \\varepsilon\/14$ and $n \\ge 0$, let us show that the set\n\\[S_{(n,A)}=\\{i \\in I^\\varepsilon_{(n,A)}: \\vert x(i) \\vert >\\varepsilon\\}\\]\nis finite. Since $\\suppt(x) \\subset J$, we have that $S_{(0,A)}$ is empty.\nFixed $n \\ge 1$, note that in order to conclude that $S_{(n,A)}$ is finite it suffices to prove that the set \\[\\Lambda_{(n,A)}=\\{\\alpha<\\kappa\\colon |x(i)|>\\varepsilon\\text{ for some }i\\in A\\cap J \\cap I^\\varepsilon_{n,\\alpha+1} \\cap (M_{\\alpha+1} \\setminus M_\\alpha)\\}\\]\nis finite. Indeed, using that $r_0(x) = x|_{I\\cap M_0} = x|_{J\\cap M_0}$ we obtain that:\n\\[S_{(n,A)} \\cap (A\\cap J \\cap I^\\varepsilon_{n,0} \\cap M_0) \\subset \\{i \\in I^\\varepsilon_{n,0}: \\vert r_0(x)(i) \\vert >\\varepsilon\\}\\]\nand therefore, since $r_0(x) = \\lim_{s\\in(\\Gamma\\cap M_0)} r_s(x)\\in D\\cap r_0[K]$, we conclude that $S_{(n,A)} \\cap (A\\cap J \\cap I^\\varepsilon_{n,0} \\cap M_0)$ is finite. Similarly, for $\\alpha < \\kappa$ we have\n\\[\\begin{split}\nS_{(n,A)} \\cap (A\\cap J \\cap I^\\varepsilon_{n,\\alpha+1} \\cap M_{\\alpha+1} \\setminus M_\\alpha) & \\subset \\{i \\in I^\\varepsilon_{n,\\alpha+1}: \\vert r_{\\alpha+1}(x)|_{I \\setminus M_\\alpha}(i)\\vert  >\\varepsilon\\}\\\\\n& \\subset \\{i \\in I^\\varepsilon_{n,\\alpha+1}: \\vert r_{\\alpha+1}(x)(i)\\vert  >\\varepsilon\\}\\end{split}\\]\nand therefore $S_{(n,A)} \\cap (A\\cap J \\cap I^\\varepsilon_{n,\\alpha+1} \\cap M_{\\alpha+1} \\setminus M_\\alpha)$ is finite.\nIt remains to prove that $\\Lambda_{(n,A)}$ is finite. In order to do that suppose by contradiction that $\\Lambda_{(n,A)}$ is infinite, so there is a strictly increasing sequence $(\\alpha_k)_{k \\ge 1}$ of elements of $\\kappa$ and a sequence $(i_k)_{k \\ge 1}$ such that $i_k\\in A\\cap J \\cap I^\\varepsilon_{n,\\alpha_k+1} \\cap (M_{\\alpha_k+1} \\setminus M_{\\alpha_k})$ and  $|x(i_k)|>\\varepsilon$, for every $k \\ge 1$.\nPut $\\alpha = \\sup_k \\alpha_k$. Then we have (because $i_k\\in M_{\\alpha_k+1}\\setminus M_{\\alpha_k}\\subset M_\\alpha\\setminus M_{\\alpha_k}$):\n \\[\\varepsilon < |x(i_k)| = |r_\\alpha(x)(i_k) - r_{\\alpha_k}(x)(i_k)|\\leq \\rho_{\\mathcal{S}_{(K,A)}} (r_{\\alpha}(x),r_{\\alpha_k}(x)),\\]\nfor every $k \\ge 1$. \nThis is a contradiction, because using \\ref{it:secondFormula} and Lemma \\ref{l: retraction from up-directed subset semi-eberlein caseImproved}  \\ref{it:shrinkingUpDirectedCanonicalRetractions} applied to $\\M=\\{M_{\\alpha_k} \\cap \\Gamma: k \\ge 1\\}$, we conclude that \\[\\limsup_{k \\to \\infty}\\rho_{\\mathcal{S}_{(K,A)}} (r_{\\alpha}(x),r_{\\alpha_k}(x))\\leq 14\\varepsilon_A < \\varepsilon,\\] since $\\mathfrak{s}$ is $(\\mathcal{S}_{K,A},\\varepsilon_A)$-shrinking with respect to $D$.\n\\end{proof}\n\n\nThe following is based on \\cite[Theorem 10]{FGZ}.\n\n\\begin{prop}\\label{prop:decompImpliesSemiEberlein}\nLet $K \\subset [-1,1]^I$ be a compact space and $D$ be a subset of $K$. If for every $\\varepsilon>0$, there exists a decomposition $I=\\bigcup_{n \\in \\omega} I^\\varepsilon_n$ such that for every $x \\in D$ and every $n \\in \\omega$ the set\n\\[\\{i \\in I^\\varepsilon_n: \\vert x(i) \\vert> \\varepsilon\\}\\]\nis finite, then there is a homeomorphic embedding $\\Phi:K\\to [-1,1]^{I\\times\\omega}$ such that $\\Phi[D]\\subset c_0(I\\times \\omega)$.\n\\end{prop}\n\n\\begin{proof}\nLet $k\\in\\omega$ and define the function $\\tau_k:\\mathbb{R}\\to \\mathbb{R}$ as\n\\begin{equation*}\n\\tau_k(t)=\\begin{cases}\nt + \\frac{1}{k}, & \\mbox{if } t\\leq -\\frac{1}{k},\\\\\n0, & \\mbox{if } -\\frac{1}{k}\\leq t \\leq \\frac{1}{k},\\\\\nt-\\frac{1}{k}, & \\mbox{if } t\\geq \\frac{1}{k}.\n\\end{cases}\n\\end{equation*}\nDefine then $\\Phi:K\\to [-1,1]^{I \\times\\omega}$ by\n\\begin{equation*}\n    \\Phi(x)(i,k)=\\frac{1}{nk}\\tau_k(x(i)),\n\\end{equation*}\nif $i\\in I^{1\/k}_{n}$, $n\\in\\omega$ and $k\\in\\omega$. Since the map $\\pi_{(i,k)}\\circ\\Phi:K\\to \\mathbb{R}$ is continuous, for every $(i,k)\\in I\\times \\omega$, the map $\\Phi$ is continuous as well.\nThe map is also one-to-one. Indeed, for distinct $x_1,x_2 \\in K$ there exists an $i \\in I$ with $x_1(i)\\neq x_2(i)$. Let $k\\in\\omega$ be such that $1\/k<\\max\\{|x_1(i)|,|x_2(i)|\\}$ and pick $n\\in\\omega$ with $i\\in I_n^{1\/k}$. Then $\\tau_k(x_1(i))\\neq\\tau_k(x_2(i))$, therefore $\\Phi(x_1)(i,k)\\neq\\Phi(x_2)(i,k)$.\n\nIt remains to prove that $\\Phi[D]$ is contained in $c_{0}(I\\times \\omega)$. In order to do that, let $x\\in D$ and fix $\\varepsilon>0$. If $n,k\\in\\omega$, and $n>1\/\\varepsilon$ or $k>1\/\\varepsilon$, then $|\\Phi(x)(i,k)|<\\varepsilon$, for any choice of $i\\in I^{1\/k}_n$. Let $n,k<1\/\\varepsilon$ (observe that there are only finitely many $n$ and $k$ such that this inequality holds). Then we have\n\\begin{equation*}\n    \\begin{split}\n        \\{i\\in I_{n}^{1\/k}:|\\Phi(x)(i,k)| & >\\varepsilon\\}  \\subseteq  \\{i\\in I_{n}^{1\/k}:\\tau_k(x(i))\\neq 0\\}\\\\\n        & =\\{i\\in I_{n}^{1\/k}:x(i)>1\/k\\}\\cup \\{i\\in I^{1\/k}_n: x(i)<-1\/k\\}\\\\\n        & =\\{i \\in I_{n}^{1\/k}: \\vert x(i) \\vert>1\/k\\}.\n    \\end{split}\n\\end{equation*}\nTherefore, the set $\\{i\\in I_{n}^{1\/k}:|\\Phi(x)(i,k)|>\\varepsilon\\}$ is finite and thus we conclude that $\\Phi(x)\\in c_0(I\\times \\omega)$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{t:mainEberleinThroughSkeleton2}]\nLemma~\\ref{lem:semiEbImpliesShrinkingSkeleton} ensures that $\\ref{it:defSemiEberlein} \\Rightarrow \\ref{it:eqdefSemiEberleinThroughSkeleton}$.\n\n\nNow let us prove that \\ref{it:eqdefSemiEberleinThroughSkeleton}$\\Rightarrow$\\ref{it:eqdefSemiEberleinThroughSkeletonAvailableForContImage}. Let $\\A$ and $\\mathfrak{s}=(r_s)_{s\\in\\Gamma}$ be as in the assumption and let $\\lambda \\ge 1$ be such that $\\A \\subset \\lambda B_{\\C(K)}$. We may without loss of generality assume that the constant $1$ function is  member of $\\A$. For every $n\\in\\en$ put\n\\[\n\\A_n:=\\left\\{\\sum_{i=1}^k a_i\\Pi_{j=1}^n f_{i,j}\\colon f_{i,j}\\in\\A,\\; k\\in\\en\\;, \\sum_{i=1}^k |a_i|\\leq n\\right\\}\n\\]\nand for $m\\in\\en$ we further put\n\\[\n\\A_{n,m}:=(\\A_n + \\tfrac{1}{2m}B_{\\C(K)})\\cap B_{\\C(K)}.\n\\]\nNow, we claim that the family $\\widetilde{\\A}:=\\{\\A_{n,m}\\colon n,m\\in\\en\\}$ and the retractional skeleton $\\mathfrak{s}$ satisfy the condition from \\ref{it:eqdefSemiEberleinThroughSkeletonAvailableForContImage}. Pick $n,m\\in\\en$. Then $\\mathfrak{s}$ is $(\\A_{n,m},\\tfrac{1}{m})$-shrinking with respect to $D$. Indeed, given $x\\in D$ and an increasing sequence $(s_k)$ in $\\Gamma$ with $s = \\sup s_k$, we have\n\\[\n\\rho_{\\A_{n,m}}(r_{s_k}(x),r_s(x))\\leq \\rho_{\\A_{n}}(r_{s_k}(x),r_s(x)) + \\tfrac{1}{m}\\leq n^2\\lambda^{n+1}\\rho_{\\A}(r_{s_k}(x),r_s(x)) + \\tfrac{1}{m},\n\\]\nso using that $\\mathfrak{s}$ is $\\A$-shrinking with respect to $D$, we obtain \\[\\limsup_k \\rho_{\\A_{n,m}}(r_{s_k}(x),r_s(x))\\leq \\tfrac{1}{m}.\\]\nFinally, since $\\bigcup_{n\\in\\en} \\A_n = \\alg(\\A)$ is norm-dense in $\\C(K)$ we easily observe that $B_{\\C(K)} = \\bigcup_{n\\in\\en} \\A_{n,m}$ for every $m\\in\\en$ from which the condition \\ref{cond:fullInBall3} follows.\n\nNow let us prove that \\ref{it:eqdefSemiEberleinThroughSkeletonAvailableForContImage}$\\Rightarrow$\\ref{it:subsetCondition}. Let $\\A$ and $\\mathfrak{s}=(r_s)_{s\\in\\Gamma}$ be as in the assumption. By Theorem~\\ref{thm:Intro3}, there exists $\\HH\\subset B_{\\C(K)}$ such that the mapping $\\varphi:K\\to [-1,1]^\\HH$ given by $\\varphi(x)(h):=h(x)$, for $h\\in\\HH$ and $x\\in K$, is a homeomorphic embedding and $\\varphi[D(\\mathfrak s)]\\subset \\Sigma(\\HH)$.\nFor every $s \\in \\Gamma$, define $q_s=\\varphi \\circ r_s \\circ \\varphi^{-1}:\\varphi[K] \\to \\varphi[K]$ and note that the retractional skeleton $(q_s)_{s \\in \\Gamma}$ is $(\\mathcal{S}_{(\\varphi[K],\\HH\\cap A)},\\varepsilon_A)$-shrinking with respect to the set $\\varphi[D]\\subset\\varphi[D(\\mathfrak s)]$ for every $A\\in\\A$. Indeed, fix $x \\in D$ and an increasing sequence $(s_n)_{n \\ge 1}$ of elements of $\\Gamma$ with $s=\\sup_n s_n$. Then we have\n\\[ \\rho_{\\mathcal{S}_{(\\varphi[K],\\HH\\cap A)}}\\Big(q_{s_n}\\big(\\varphi(x)\\big), q_{s}\\big(\\varphi(x)\\big)\\Big)=\\sup_{h \\in \\HH\\cap A} \\Big\\vert \\varphi \\big(r_{s_n}(x)\\big)(h)-\\varphi \\big(r_{s}(x)\\big)(h)\\Big \\vert=\\]\n\\[= \\sup_{h \\in \\HH\\cap A} \\Big\\vert h\\big(r_{s_n}(x)\\big)-h\\big(r_{s}(x)\\big)\\Big \\vert \\le  \\rho_A(r_{s_n}(x),r_{s}(x)),\\]\nso since $\\mathfrak{s}$ is $(A,\\varepsilon_A)$-shrinking with respect to the set $D$, $(q_s)$ is $(\\mathcal{S}_{(\\varphi[K],\\HH\\cap A)},\\varepsilon_A)$-shrinking with respect to the set $\\varphi[D]$. Obviously, we have $\\HH = \\bigcup\\{A\\cap\\HH\\colon \\varepsilon_A<\\varepsilon\\}$ for every $\\varepsilon>0$. Finally, for every up-directed set $\\Gamma'\\subset \\Gamma$ and every $x\\in D$ we have that $\\lim_{s\\in\\Gamma'} q_s(\\varphi(x)) = \\varphi (\\lim_{s\\in\\Gamma'} r_s(x)) \\in \\varphi[D]$. Therefore, the result follows from Proposition~\\ref{prop:semiEberleinFarmakiImproved} and Proposition \\ref{prop:decompImpliesSemiEberlein}.\n\\end{proof}\n\n\\section{Applications to the structure of (semi)-Eberlein compacta}\\label{sec:contImages}\n\nWe collect our applications to the structure of (semi-)Eberlein compacta. Most importantly, we prove Theorem~\\ref{thm:Intro4}.\n\n\\subsection{Eberlein compacta}\n\nAs mentioned above, using Theorem~\\ref{thm:Intro1} it is not very difficult to show that any continuous image of Eberlein compacta is Eberlein. The reason is that for continuous images of Eberlein compacta it is quite standard to verify the condition \\ref{it:eqdefSemiEberleinThroughSkeletonAvailableForContImage} from Theorem~\\ref{t:mainEberleinThroughSkeleton2}. We will not provide here the full argument as it is possible to further generalize this observation, see Remark~\\ref{rem:contImageEberlein} below. The remainder of this subsection is devoted to the proof of Theorem~\\ref{thm:countableCompactEberlein}, which is a generalization of Theorem~\\ref{thm:Intro1}, where instead of compactness we assume countable compactness. In order to show the argument, we need a lemma first.\nRecall that every real-valued continuous function defined on a countably compact space $D$ is bounded so we may consider the supremum norm on $\\C(D)$.\n\\begin{lemma}\\label{l:separatePoints}\nLet $D$ be a countably compact space. Suppose that there exist a bounded set $\\A\\subset \\C(D)$ separating the points of $D$ and a full retractional skeleton $\\mathfrak{s} = (r_s)_{s\\in\\Gamma}$ on $D$ such that $f\\circ r_s\\in\\A$, for every $f\\in\\A$ and $s\\in\\Gamma$. Then $\\A'=\\{\\beta f\\colon f\\in \\A\\}$ separates the points of $\\beta D$.\n\\end{lemma}\n\\begin{proof}\nBy \\cite[Proposition 4.5]{CK15}, there exists a retractional skeleton $\\mathfrak{S}=(R_s)_{s\\in\\Gamma}$ on $\\beta D$ such that $D(\\mathfrak{S}) = D$ and $R_s|_D = r_s$, for every $s\\in\\Gamma$. Let $x,y \\in \\beta D$ be distinct points. Since $\\lim_{s\\in\\Gamma}R_s(x)=x$ and $\\lim_{s\\in\\Gamma}R_s(y)=y$, there exists $s\\in \\Gamma$ such that $R_s(x)\\neq R_s (y)$. Since $R_s(x), R_s(y)\\in D$, there exists a function $f\\in \\A$ such that $f(R_s(x))\\neq f(R_s(y))$. Therefore we have $\\beta f(R_s(x))\\neq\\beta f(R_s(y))$. It is easy to see that $(\\beta f \\circ R_s)|_D= f \\circ r_s$, which implies that $\\beta f \\circ R_s= \\beta(f \\circ r_s) \\in \\A'$, since $f \\circ r_s\\in\\A$.\n\\end{proof}\n\n\\begin{remark}\nNote that the assumption ``$f\\circ r_s\\in \\A$, for every $f\\in \\A$ and $s\\in\\Gamma$'' in Lemma~\\ref{l:separatePoints} is essential. Indeed, consider $D=[0,\\omega_1)$ and \\[\\A = \\{1_{\\{0\\}\\cup [\\alpha+1,\\omega_1)}\\colon \\alpha < \\omega_1\\}.\\]\nThen it is easy to see that $\\A$ separates the points of $D$ and that $D$ admits the full retractional skeleton $(r_\\alpha)_{\\alpha<\\omega_1}$ given by the formula\n\\begin{equation*}\n   r_{\\alpha}(\\beta)= \\begin{cases} \\beta & \\beta\\leq\\alpha \\\\ \\alpha+1 & \\alpha<\\beta<\\omega_1,\\end{cases}\n\\end{equation*}\nfor every $\\alpha<\\omega_1$.\nHowever, we have $\\beta D = [0,\\omega_1]$ and the set \\[\\A' = \\{\\beta 1_{\\{0\\}\\cup [\\alpha+1,\\omega_1)}\\colon \\alpha<\\omega_1\\} = \\{1_{\\{0\\}\\cup [\\alpha+1,\\omega_1]}\\colon \\alpha<\\omega_1\\}\\]\ndoes not separate $0$ from $\\omega_1$.\n\\end{remark}\n\n\n\\begin{thm}\\label{thm:countableCompactEberlein}\nLet $D$ be a countably compact space. \nThen the following conditions are equivalent.\n\\begin{enumerate}[label = (\\roman*)]\n    \\item There exists a set $I$ such that $D$ embeds homeomorphically into $(c_0(I),w)$.\n    \\item $D$ is an Eberlein compact space.\n    \\item There exist a bounded set $\\A\\subset \\C(D)$ separating the points of $D$ and a full retractional skeleton $\\mathfrak{s} = (r_s)_{s\\in\\Gamma}$ on $D$ such that\n    \\begin{enumerate}[label=(\\alph*)]\n        \\item $\\mathfrak{s}$ is $\\A$-shrinking, \n        \\item\\label{it: compositionassumption} $f\\circ r_s\\in\\A$, for every $f\\in\\A$ and $s\\in\\Gamma$.\n    \\end{enumerate}\n   \\end{enumerate}\n\\end{thm}\n\\begin{proof}\n$(i)\\Rightarrow(ii)$ follows from the classical Eberlein-\\v{S}mulian theorem and $(ii)\\Rightarrow(iii)$ follows from Theorem~\\ref{t:mainEberleinThroughSkeleton2} and Lemma \\ref{lem:shrinkingImpliesFull}. If (iii) holds, pick the corresponding set $\\A$ and the full retractional skeleton $(r_s)_{s\\in\\Gamma}$ on $D$. By \\cite[Proposition 4.5]{CK15}, there exists a retractional skeleton $\\mathfrak{S}=(R_s)_{s\\in\\Gamma}$ on $\\beta D$ such that $D(\\mathfrak{S}) = D$ and $R_s|_D = r_s$, for every $s\\in\\Gamma$.\nConsider now the set $\\A':=\\{\\beta f\\colon f\\in \\A\\}\\subset \\C(\\beta D)$ and the retractional skeleton $\\mathfrak{S}$. By Lemma~\\ref{l:separatePoints}, $\\A'$ separates the points of $\\beta D$. Obviously, $\\mathfrak{S}$ is $\\A'$-shrinking with respect to $D$ and it is easy to see that for every $f\\in \\A$ and $s\\in\\Gamma$ we have $\\beta f\\circ R_s = \\beta(f\\circ r_s) \\in \\A'$. By Lemma \\ref{lem:shrinkingImpliesFull} we have that $\\lim_{s\\in \\Gamma'}R_s(x)\\in D$, for every $x\\in D$ and every up-directed subset $\\Gamma'$ of $\\Gamma$. Therefore, Theorem~\\ref{t:mainEberleinThroughSkeleton2} ensures that (i) holds.\n\\end{proof}\n\n\\subsection{Semi-Eberlein compacta}\n In this subsection we provide new stability results for the class of semi-Eberlein compacta. The most important in this respect is probably Corollary~\\ref{cor:images} which implies Theorem~\\ref{thm:Intro4}.\n \n\\begin{lemma}\\label{l:contImageConnectionOfSkeletons}\nFor every suitable model $M$ the following holds: Let $(K,\\tau)$ and $(L,\\tau')$ be compact spaces, $D\\subset K$ a dense subset that is contained in the set induced by a retractional skeleton, and $\\varphi:K\\to L$ a continuous map such that $\\varphi[D]\\subset L$ a dense subset that is contained in the set induced by a retractional skeleton. If $\\{K,L,\\tau,\\tau',D,\\varphi\\}\\subset M$, then there are canonical retractions $r_M$ and $R_M$ associated to $M$, $K$ and $D$ and to $M$, $L$ and $\\varphi[D]$, respectively, and we have $R_M\\circ \\varphi = \\varphi\\circ r_M$.\n\\end{lemma}\n\n\\begin{proof}Let $S$ and $\\Phi$ be the union of countable sets and finite lists of formulas from the statements of Lemma~\\ref{l:basics-in-M-2} and Theorem~\\ref{thm:canonicalRetraction}. Let $M\\prec (\\Phi; S\\cup\\{K,L,\\tau,\\tau',D,\\varphi\\})$.\n\nThe existence of $r_M$ and $R_M$ follows from Theorem~\\ref{thm:canonicalRetraction}. Moreover, given $x\\in K$ and $f\\in \\C(L)\\cap M$, by Lemma~\\ref{l:basics-in-M-2} we have $f\\circ \\varphi\\in \\C(K)\\cap M$ and so by the definition of $r_M$, $f\\circ \\varphi\\circ r_M = f\\circ \\varphi$ which implies that $y=\\varphi(r_M(x))\\in\\varphi[\\overline{D\\cap M}]\\subset \\overline{\\varphi[D]\\cap M}$ is a point satisfying $f(y) = f(\\varphi(x))$ for every $f\\in \\C(L)\\cap M$ and so by the uniqueness property of $R_M(\\varphi(x))$ (see Theorem~\\ref{thm:canonicalRetraction} \\ref{it:characterizationFirst}) we obtain that $R_M(\\varphi(x)) = \\varphi(r_M(x))$.\n\\end{proof}\n\n\\begin{thm}\\label{t:mainContImage}\nLet $K$ be a compact space and $D\\subset K$ be a dense subset such that there exists a homeomorphic embedding $h:K\\to [-1,1]^J$ such that $h[D] = c_0(J)\\cap h[K]$. Let us suppose that $\\varphi:K\\to L$ is a continuous surjection and $\\varphi[D]$ is subset of the set induced by a retractional skeleton on $L$.\n\nThen there is a homeomorphic embedding $H:L\\to[-1,1]^I$ with $H[\\varphi[D]]\\subset c_0(I)$. In particular, $L$ is semi-Eberlein.\n\\end{thm}\n\\begin{proof}\nBy Lemma~\\ref{lem:semiEbImpliesShrinkingSkeleton}, there exists a set $\\A\\subset  B_{\\C(K)}$ separating the points of $K$ and a retractional skeleton $\\mathfrak{s} = (r_s)_{s\\in\\Gamma}$ on $K$ with $D\\subset D(\\mathfrak{s})$ such that $\\mathfrak{s}$ is $\\A$-shrinking with respect to $D$ and $\\lim_{s\\in\\Gamma'}r_s(x)\\in D$ for every $x\\in D$ and every up-directed subset $\\Gamma'$ of $\\Gamma$. Using Lemma~\\ref{l:contImageConnectionOfSkeletons} and an argument similar to the one presented in the proof of Theorem~\\ref{thm:subskeletons}, we conclude that there are countable set $S$, finite list of formulas $\\Phi$ and a set $R$ such that\n\\[\n\\M = \\{M\\in[R]^\\omega\\colon M\\prec(\\Phi,S)\\}\n\\]\nordered by inclusion is an up-directed and $\\sigma$-complete set and moreover we have\n\\begin{itemize}\n    \\item every $M\\in\\M$ admits canonical retractions $r_M$ and $R_M$ associated to $M$, $K$ and $D$ and to $M$, $L$ and $\\varphi[D]$, respectively;\n    \\item $\\mathfrak{s}_K:=(r_M)_{M\\in\\M}$ is a weak subskeleton of $\\mathfrak{s}$ and $\\mathfrak{s}_L:=(R_M)_{M\\in\\M}$ is a retractional skeleton on $L$;\n    \\item for every $M\\in\\M$ we have $R_M\\circ \\varphi = \\varphi\\circ r_M$.\n\\end{itemize}\nIn particular, we have that $D(\\mathfrak{s}) = D(\\mathfrak{s}_K)$, $\\mathfrak{s}_K$ is $\\A$-shrinking with respect to $D$ and $\\lim_{M\\in\\M'}r_M(x)\\in D$ for every $x\\in D$ and every up-directed subset $\\M'$ of $\\M$. We obviously have $\\varphi[D]\\subset D(\\mathfrak{s}_L)$. Consider now the isometric embedding $\\varphi^*:\\C(L)\\to\\C(K)$ given by the formula $\\varphi^*f:=f\\circ \\varphi$, $f\\in\\C(L)$. Further, similarly as in the proof of  \\ref{it:eqdefSemiEberleinThroughSkeleton}$\\Rightarrow$\\ref{it:eqdefSemiEberleinThroughSkeletonAvailableForContImage} of Theorem~\\ref{t:mainEberleinThroughSkeleton2} for every $n\\in\\en$ put\n\\[\n\\A_n:=\\left\\{\\sum_{i=1}^k a_i\\Pi_{j=1}^n f_{i,j}\\colon f_{i,j}\\in\\A,\\; k\\in\\en\\;, \\sum_{i=1}^k |a_i|\\leq n\\right\\},\n\\]\nfor $m\\in\\en$ we further put\n\\[\n\\A_{n,m}:=(\\A_n + \\tfrac{1}{2m}B_{\\C(K)})\\cap B_{\\varphi^*\\C(L)},\\quad \\B_{n,m}:=(\\varphi^*)^{-1}(\\A_{n,m})\n\\]\nand we observe that $B_{\\C(L)} = \\bigcup_{n\\in\\en} \\B_{n,m}$ for every $m\\in\\en$. Moreover, observe that for every $n,m\\in\\en$ the retractional skeleton $\\mathfrak{s}_L$ is $\\big(\\B_{n,m},\\tfrac{1}{m}\\big)$-shrinking with respect to $\\varphi[D]$. Indeed, given $x\\in D$ and an increasing sequence $(M_k)_{k \\in \\en}$ in $\\M$ with $M = \\sup_{k \\in \\en} M_k$, we have\n\\[\\begin{split}\n\\rho_{\\B_{n,m}} & (R_{M_k}(\\varphi(x)),R_M(\\varphi(x))) = \\sup_{f\\in\\C(L), f\\circ \\varphi\\in\\A_{n,m}}\\big|f(\\varphi(r_{M_k}(x))) - f(\\varphi(r_{M}(x)))\\big|\\\\ & \\leq \\rho_{\\A_{n,m}}(r_{M_k}(x),r_M(x)) \\leq \\rho_{\\A_{n}}(r_{M_k}(x),r_M(x)) + \\tfrac{1}{m}\\\\  & \\leq n^2\\rho_{\\A}(r_{M_k}(x),r_M(x)) + \\tfrac{1}{m},\n\\end{split}\\]\nso using that $\\mathfrak{s}$ is $\\A$-shrinking with respect to $D$, we obtain \\[\\limsup_k \\rho_{\\B_{n,m}}(R_{M_k}(x),R_M(x))\\leq \\tfrac{1}{m}.\\]\nFinally, for every $x\\in D$ and up-directed subset $\\M'$ of $\\M$ we have \\[\\lim_{M\\in\\M'}R_M(\\varphi(x)) = \\varphi(\\lim_{M\\in\\M'} r_M(x))\\in \\varphi[D].\\]\nHence, application of  Theorem~\\ref{t:mainEberleinThroughSkeleton2} \\ref{it:eqdefSemiEberleinThroughSkeletonAvailableForContImage}$\\implies$\\ref{it:subsetCondition} finishes the proof.\n\\end{proof}\n\n\\begin{cor}\\label{cor:firstContImage}\nLet $K$ be a semi-Eberlein compact space, $\\varphi:K\\to L$ be a continuous surjection and $D\\subset K$ be a dense subset such that there exists a homeomorphic embedding $h:K\\to [-1,1]^J$ with $h[D]= c_0(J)\\cap h[K]$. Then $S:=h^{-1}[\\Sigma(J)]$ is the unique set induced by a retractional skeleton in $K$ with $D\\subset S$. Assume that one of the following conditions holds:\n\\begin{enumerate}\n    \\item\\label{it:firstSufficientCondition} $\\varphi^*\\C(L) = \\{f\\circ \\varphi\\colon f\\in\\C(L)\\}$ is $\\tau_p(S)$-closed in $\\C(K)$;\n    \\item\\label{it:secondSufficientCondition} The set $\\{(x,y)\\in S\\times S\\colon \\varphi(x)=\\varphi(y)\\}$ is dense in $\\{(x,y)\\in K\\times K\\colon \\varphi(x)=\\varphi(y)\\}$.\n\\end{enumerate}\nThen there is a homeomorphic embedding $H:L\\to[-1,1]^I$ with $H[\\varphi[D]]\\subset c_0(I)$. In particular, $L$ is semi-Eberlein.\n\\end{cor}\n\\begin{proof}\nBy Lemma~\\ref{lem:existenceOfTheSkeleton}, $\\Sigma(J)\\cap h[K]$ is induced by a retractional skeleton and so its preimage $S$ is induced by a retractional skeleton as well. The uniqueness of $S$ follows from \\cite[Lemma 3.2]{C14}. If \\eqref{it:firstSufficientCondition} holds, then by \\cite[Theorem 4.5]{cuthSimul} the set $\\varphi[S]$ is induced by a retractional skeleton and so we may apply Theorem~\\ref{t:mainContImage}. Finally, by \\cite[Lemma 2.8]{K00} condition \\eqref{it:secondSufficientCondition} implies \\eqref{it:firstSufficientCondition}.\n\\end{proof}\n\n\\begin{remark}\\label{rem:contImageEberlein}\nNote that as a very particular case we obtain that a continuous image of an Eberlein compact space is Eberlein, since in this case we have $D=K$ and thus condition \\eqref{it:secondSufficientCondition} in Corollary \\ref{cor:firstContImage} is trivially satisfied.\n\\end{remark}\n\nThe following answers the second part of \\cite[Question 6.6]{KL04}.\n\n\\begin{cor}\\label{cor:images}\nLet $K$ be a semi-Eberlein compact space, $\\varphi:K\\to L$ be a continuous surjection. Assume that one of the following conditions holds:\n\\begin{enumerate}\n    \\item $K$ is Corson.\n    \\item $\\varphi$ is open and $K$ or $L$ has a dense set of $G_\\delta$ points.\n\\end{enumerate}\nThen $L$ is semi-Eberlein.\n\\end{cor}\n\\begin{proof}\nLet $D\\subset K$ be a dense subset such that there exists a homeomorphic embedding $h:K\\to [-1,1]^J$ with $h[D]= c_0(J)\\cap h[K]$ and put $S:=h^{-1}[\\Sigma(J)]$. Note that $S$ is a dense $\\Sigma$-subset of $K$ and, by Corollary~\\ref{cor:firstContImage}, it is the unique set induced by a retractional skeleton with $D\\subset S$.\n\nIf $K$ is Corson then it admits a full retractional skeleton, so by the uniqueness of $S$ we have that $S=K$ and thus condition \\eqref{it:secondSufficientCondition} from Corollary~\\ref{cor:firstContImage} is obviously satisfied.\n\nIf $\\varphi$ is open and $L$ has dense set of $G_\\delta$ points, then by \\cite[Lemma 6.1]{cuthSimul} the set $\\varphi[S]$ is induced by a retractional skeleton and thus we may apply directly Theorem~\\ref{t:mainContImage} (note that inspecting the proof of \\cite[Lemma 6.1]{cuthSimul} one can observe that even the condition \\eqref{it:secondSufficientCondition} from Corollary~\\ref{cor:firstContImage} is satisfied).\n\nFinally, if $\\varphi$ is open and $K$ has dense set of $G_\\delta$ points then it is easy to see that $L$ has dense set of $G_\\delta$ points and we may apply the above.\n\\end{proof}\n\n\n\nLet us note that if $D$ is a dense $\\Sigma$-subset of $K$ and $\\varphi:K\\to L$ is a continuous retract, it may happen that $\\varphi[D]$ is not $\\Sigma$-subset of $K$, see \\cite[Remark 3.25]{K00}. Thus, it is not possible to apply directly Theorem~\\ref{t:mainContImage} for the case when $\\varphi$ is a retraction and this is basically the reason why we do not know how to answer also the first part of \\cite[Question 6.6]{KL04} using our methods.\n\n\\section{Open questions and remarks}\\label{sec:questions}\n    \nIn Section~\\ref{sec:models} we obtained as an application of our methods that for a countable family of retractional skeletons inducing the same set there is a common weak subskeleton, see Theorem \\ref{thm:subskeletons}. It would be interesting to know whether we can find even a subskeleton (not only a weak one).\n\n\\begin{question}\\label{q: weaksubtosub}\nIn Theorem \\ref{thm:subskeletons}, is it possible to obtain a subskeleton instead of a weak subskeleton?\n\\end{question}\n\nWhen working with retractional skeletons, their index sets are quite mysterious. For Banach spaces with a projectional skeleton, the index set may be chosen to consist of the ranges of the involved projections (ordered by inclusion), see \\cite[Theorem 4.1]{CK14}. We wonder whether something similiar holds for spaces with a retractional skeleton.\n\n\\begin{question}\nLet $K$ be a compact space and let $D\\subset K$ be induced by a retractional skeleton on $K$. Does there exist a family of retractions $\\{r_F\\colon F\\in\\F\\}$ indexed by a family of compact spaces $\\F$ ordered by inclusion satisfying the following conditions?\n    \\begin{enumerate}[label = (\\roman*)]\n        \\item\\label{it:sigma} whenever $(F_n)_{n\\in\\omega}$ is an increasing sequence from $\\F$, then $\\sup_n F_n = \\overline{\\bigcup_n F_n}\\in\\F$,\n        \\item\\label{it:range} for every $F\\in\\F$ we have $r_F[K] = F$,\n        \\item\\label{it:sub} $(r_F)_{F\\in\\F}$ is a retractional skeleton on $K$ inducing the set $D$.\n    \\end{enumerate}\n\\end{question}\n\nNote that in Proposition~\\ref{prop:semiEberleinFarmakiImproved} we proved a result in a sense very similar to the characterization of Eberlein compacta from \\cite[Theorem 2.9]{F87} (see also \\cite[Theorem 10]{FGZ}). We wonder whether an analogoue of \\cite[Theorem 10]{FGZ} holds also in the context semi-Eberlein compacta. Note that one implication follows from  Proposition~\\ref{prop:decompImpliesSemiEberlein}, so a positive answer to the following question would give a characterization of semi-Eberlein compact subspaces of $[-1,1]^I$.\n\n\\begin{question}\nLet $K\\subset [-1,1]^I$ be a compact space such that $\\Sigma(I)\\cap K$ is dense in $K$. Let $K$ be semi-Eberlein. Does there exist $D\\subset \\Sigma(I)\\cap K$ which is dense in $K$ such that for every $\\varepsilon>0$ there exists a decomposition $I = \\bigcup_{n=0}^\\infty I_n^\\varepsilon$ satisfying\n\\[\\forall n\\in\\omega\\; \\forall x\\in D:\\quad |\\{i\\in I_n^\\varepsilon\\colon |x(i)|>\\varepsilon\\}|<\\omega?\\]\n\\end{question}\n\nFinally, let us emphasize that we believe that Theorem~\\ref{thm:Intro3} gives quite a big flexibility to consider other subclasses of Valdivia compact spaces and characterize them using the notion of retractional skeletons. The reason why we believe so is, that by Theorem~\\ref{thm:Intro3} we may consider any set induced by a retractional skeleton to be a subset of $\\Sigma(I)$ (where $\\A\\subset \\C(K)$ plays the role of the set $I$); moreover, for subsets of $\\Sigma(I)$ several classes of compact spaces were characterized using their evaluations on the set $I$, see \\cite{FGZ}. Thus, there is enough room for further possible research by considering those classes of compacta and try to develop the right notion which would give a characterization using retractional skeletons.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently, there has been renewed interest in basic hypergeometric integrals because of their connection with various branches of mathematical physics, such as supersymmetric field theory, 3-manifold invariants and integrable systems.\nThe purpose of this paper is to state and prove new basic hypergeometric integral identities and give their physical interpretations in terms of superconformal indices. \n\nThere is an interesting connection between partition functions of supersymmetric gauge theories on different curved manifolds and certain classes of hypergeometric functions. The first observation of this relation was made by Dolan and Osborn \\cite{Dolan:2008qi}. They found that the  superconformal index  of four-dimensional $\\mathcal N=1$ supersymmetric gauge theory can be written via elliptic hypergeometric integrals. Similarly,   three-dimensional superconformal indices can be expressed in terms of basic hypergeometric integrals (see e.g. \\cite{Krattenthaler:2011da,Kapustin:2011jm,Gahramanov:2013rda,Gahramanov:2014ona}).\n\nThe superconformal index for a three-dimensional ${\\mathcal N}=2$ supersymmetric field theory is defined as \n\\begin{equation}\n\\text{Tr} \\left[ (-1)^\\text{F} e^{-\\beta\\{Q, Q^\\dagger \\}} q^{\\frac12 (\\Delta+j_3)}\\prod_i\nt_i^{F_i} \\right] \\;,\n\\end{equation}\nwhere the trace is taken over the Hilbert space of the theory,   $Q$ and ${Q}^\\dagger$ are supercharges, $\\Delta$, $j_3$ are Cartan elements of the superconformal group and the fugacities $t_i$ are associated with the flavor symmetry group. \n\nStudying the relation between basic hypergeometric integrals and  superconformal indices is an important field of research from different points of view (see e.g. \\cite{Gahramanov:2015tta}). Non-trivial mathematical identities for superconformal indices provide a very powerful tool to check known supersymmetric dualities and to establish  new ones. Such identities are also important for  better understanding the structure of the moduli of three-dimensional supersymmetric theories and supersymmetric dualities. On the other hand, there is an interesting relationship between three-dimensional ${\\mathcal N}=2$ supersymmetric gauge theories and geometry of triangulated 3-manifolds. The independence of a certain topological invariant of 3-manifolds on the choice of triangulation corresponds to equality of superconformal indices of three-dimensional ${\\mathcal N}=2$ supersymmetric dual theories. \n\nBesides their appearance in  supersymmetric field theory, basic hypergeometric integrals discussed in this paper recently appeared in the theory of exactly solvable two-dimensional statistical models \\cite{Gahramanov:2015cva,Kels:2015bda}. \n\nIn this paper we extend the results of our previous work \\cite{Gahramanov:2013rda,Gahramanov:2014ona} on superconformal indices to a number of three-dimensional dualities. We provide explicit expressions  for the generalized superconformal indices of some three-dimensional ${\\mathcal N}=2$ supersymmetric electrodynamics and quantum chromodynamics in terms of basic hypergeometric integrals.\n\nWe will only consider confining theories, which means that the duality leads to a closed form evaluation of a sum of  integrals (rather than a transformation between two such expressions). As an example, one of the resulting identities is\n\\begin{multline}\\label{wp}\\sum_{m=-\\infty}^\\infty\\oint \\prod_{j=1}^6\\frac{(q^{1+m\/2}\/a_jz,q^{1-m\/2}z\/a_j;q)_\\infty}{(q^{N_j+m\/2}a_jz,q^{N_j-m\/2}a_j\/z;q)_\\infty}\\frac{(1-q^mz^2)(1-q^mz^{-2})}{q^mz^{6m}}\\frac{dz}{2\\pi\\mathrm i z}\\\\\n=\\frac{2}{\\prod_{j=1}^6q^{\\binom {N_j}2}a_j^{N_j}}\\prod_{1\\leq j<k\\leq 6}\\frac{(q\/a_ja_k;q)_\\infty}{(a_ja_kq^{N_j+N_k};q)_\\infty} \\;,\n\\end{multline}\nwhere $|q|<1$, the parameters $a_j$ are generic  and $N_j$ are integers, subject to\n the balancing conditions $\\prod_{i=1}^6 a_i=q$ and $N_1+\\dots+N_6=0$.\nHere, we use the standard notation\n$$(a;q)_\\infty=\\prod_{j=0}^\\infty(1-aq^j)\\;, $$\n$$(a_1,\\dots,a_m;q)_\\infty=(a_1;q)_\\infty\\dotsm(a_m;q)_\\infty$$\nand the integration is over a positively oriented contour separating sequences of poles going to infinity from sequences going to zero.\n\nThe organization of the paper is as follows.\n\n\\begin{itemize}\n\\item In Section \\ref{sis} we outline the superconformal index technique for three-dimensional ${\\mathcal N}=2$ supersymmetric gauge theories.\n\n\\item In Section \\ref{iis} we discuss supersymmetric dualities  and present  explicit expressions of superconformal indices for certain supersymmetric dual theories in terms of basic hypergeometric integrals. \nWe present four examples, each leading to an integral evaluation similar\nto \\eqref{wp}. Some of these evaluations generalize identities\npreviously obtained in \\cite{Krattenthaler:2011da, Kapustin:2011jm, Gahramanov:2013rda}.\n\n\\item \n In Section \\ref{mps} we give mathematical proofs of the four integral evaluations that were derived using non-rigorous methods in Section \\ref{iis}.\n This gives a consistency check of the corresponding supersymmetric dualities. \n\n\\item We review the basic aspects of three-dimensional $\\mathcal N = 2$ supersymmetric gauge theories with focus on the necessary elements for the superconformal index computations and give some details of index computation in Appendices.\n\\end{itemize}\n\n \n\n\\section{3d superconformal index}\n\\label{sis}\n\n\nIn this section, we recall basic facts related to the superconformal index technique. The presentation closely follows that in \\cite{Imamura:2011su,Krattenthaler:2011da,Kapustin:2011jm}.\n\nThe concept of the superconformal index was first introduced for four-di\\-men\\-sio\\-nal theories in \\cite{Romelsberger:2005eg,Kinney:2005ej} and later extended to other dimensions. The superconformal index of three-dimensional ${\\mathcal N}=2$ superconformal field theory is a twisted partition\nfunction defined on $S^2 \\times S^1$ as follows \\cite{Bhattacharya:2008bja,Kim:2009wb,Imamura:2011su}\n\\begin{equation}\n{I}(q,\\{ t_i \\})=\\text{Tr} \\left[ (-1)^\\text{F} e^{-\\beta\\{Q, Q^\\dagger \\}} q^{\\frac12 (\\Delta+j_3)}\\prod_i\nt_i^{F_i} \\right] \\;,\n\\end{equation}\nwhere \n\\begin{itemize}\n\n\\item the trace is taken over the Hilbert space of the theory on $S^2$,\n\n\\item $\\text{F}$ plays the role of the fermion number which takes value zero on bosons and one on fermions. In presence of monopoles one needs to refine this number by shifting it by $e \\times m$, where $e$ and $m$ are electric charge and magnetic monopole charge, respectively. See \\cite{Dimofte:2011py,Aharony:2013dha} for a discussion of this issue.\n\n\\item $\\Delta$ is the energy (or conformal dimension via radial quantization), $j_3$ is the third component of the angular momentum on $S^2$,\n\n\\item $F_i$ is the charge of global symmetry with fugacity $t_i$,  \n\n\\item $Q$ is a certain supersymmetric charge in three-dimensional ${\\mathcal N}=2$ superconformal algebra with quantum numbers $\\Delta=\\frac12$ and $j_3=-\\frac12$ and $R$-charge $R=1$. The supercharges $Q^\\dagger = S$ and $Q$ satisfy the  anti-commutation relation\\footnote{ The full algebra\ncan be found in many places, see e.g., \\cite{Dolan:2008vc}.}\n\\begin{equation}\\label{qs}\n\\frac12 \\{Q, S\\}=\\Delta-R-j_3 \\;.\n\\end{equation}\n\\end{itemize}\n\nOnly BPS states with $\\Delta-R-j_3 =0 $ contribute to the superconformal index.\nConsequently,  the index is $\\beta$-independent but depends non-trivially on the fugacities $t_i$ and $q$. The superconformal index counts the number of BPS states weighted by their quantum numbers.\n\nThe superconformal index can be evaluated by a path integral on $S^2 \\times S^1$  via the localization technique \\cite{Pestun:2007rz}, leading to the  matrix integral \\cite{Kim:2009wb,Imamura:2011su} \n\\begin{align} \\nonumber\nI(q,\\{t_i\\})& = \\sum_{m\\in \\mathbb{Z}^{\\text{rank}\\,G}} \\int \\frac{1}{|W_m|}\ne^{-S^{(0)}_{CS}}e^{\\mathrm i b_0} q^{\\frac12 \\epsilon_0} \\prod_{j=1}^{\\text{rank}\\, F} t_{j}^{q_{0j}} \\\\ \\label{theindex}\n& \\qquad \\qquad \\times \\exp\\left[\\sum^\\infty_{n=1}\\frac{1}{n} \\text{ind}(z_i^{n},\nt^n, q^n; m)\\right] \\; \\prod_{i=1}^{\\text{rank}\\,G} \\frac{dz_i}{2 \\pi \\mathrm i z_i} \\;.\n\\end{align}\nThe sum in the formula is to be understood as follows. It is a sum over magnetic fluxes $m=(m_1, \\ldots, m_{\\text{rank}\\,G})$ on the two-sphere with\n\\begin{equation}\nm_i \\ = \\ \\frac{1}{2 \\pi} \\int_{S^2} F_i \\;,\n\\end{equation}\nwhere $m_i$ parametrizes the GNO charge of the monopole configuration\\footnote{The operators creating magnetic fluxes are not completely understood yet, for details, see e.g. \\cite{Kim:2009wb}.}, in the examples we consider it runs over the integers. The prefactor $|W_m|= \\prod_{i=1}^k ( \\text{rank}\\,G_i)!$ is the order of the Weyl group of $G$ which is ``broken'' by the monopoles into the product $G_1 \\times G_2 \\times \\dots\\times G_k$. For instance, in case of $U(N)$ gauge group $|W_m|=\\prod N_k!$.\n\nThe term \n\\begin{align} \\nonumber\nS^{(0)}_{CS} & \\ = \\ \\frac{\\mathrm i k}{4 \\pi} \\int \\text{tr}_{CS} (A^{(0)} dA^{(0)}\n-\\frac{2 \\mathrm i} {3} A^{(0)} A^{(0)} A^{(0)}) \\\\ \\label{CScont}\n& \\ = \\ 2 \\mathrm i \\text{tr}_{CS} (g m)\\;,\n\\end{align}\nis the contribution of the Chern--Simons term if the action contains such term and \n\\begin{equation} \\label{oneloopcont}\nb_0 \\ = \\ -\\frac{1}{2} \\; \\sum_\\Phi\\sum_{\\rho\\in R_\\Phi}|\\rho(m)|\\rho(g)\n\\end{equation}\nis the 1-loop correction to the Chern--Simons term. The $\\text{tr}_{CS}$ stands for the trace containing the Chern--Simons levels, $k$ is the Chern--Simons level and $\\sum_\\Phi$ and $\\sum_{\\rho\\in R_\\Phi}$ are sums over all chiral multiplets and all weights of the representation $R_\\Phi$, respectively. We give the contribution (\\ref{CScont}) for completeness; in all our examples we will consider theories without the Chern--Simons term\\footnote{Note that even in this case the term $b_0$ is not absent since the gauge fields generate the one-loop correction to the Chern-Simons term, see Appendixe B.}.\n\n\nThe term $q_{0j}$ in (\\ref{theindex}) is the zero-point contribution to the energy,\n\\begin{equation}\nq_{0j}(m) = -\\frac{1}{2} \\sum_\\Phi \\sum_{\\rho\\in R_\\Phi} |\\rho(m)| f_j (\\Phi) \\;.\n\\end{equation}\nIn addition, there is the contribution from the Casimir energy of the vacuum state on the two-sphere with magnetic flux $m$,\n\\begin{equation}\n\\epsilon_0(m) = \\frac{1}{2} \\sum_\\Phi (1-\\Delta_\\Phi) \\sum_{\\rho\\in\nR_\\Phi} |\\rho(m)|\n- \\frac{1}{2} \\sum_{\\alpha \\in G} |\\alpha(m)| \\;,\n\\end{equation}\nwhere $\\sum_{\\alpha\\in G}$ represents summation over all roots of $G$, $\\Delta_\\Phi$ is the superconformal $R$-charge of the chiral multiplet $\\Phi$. \n\nOne can calculate the single letter index\n\\begin{align}\n& \\text{ind}(z_j=e^{ig_j},t_j,q;m_j)  = -\\sum_{\\alpha\\in G} e^{\\mathrm i\\alpha(g)} q^{\\frac12|\\alpha(m)|}\\\\ \\nonumber\n& \\qquad + \\sum_\\Phi \\sum_{\\rho\\in R_\\Phi} \\left[\ne^{\\mathrm i\\rho(g)}  \\prod_{j} t_j^{f_j}\n\\frac{q^{\\frac12 |\\rho(m)|+\\frac12 \\Delta_\\Phi}}{1-q}  -  e^{-\\mathrm i\\rho(g)} \n\\prod_{j} t_j^{-f_j} \\frac{q^{\\frac12 |\\rho(m)|+1-\\frac12 \\Delta_\\Phi}}{1-q}\n\\right] \\;.\n\\end{align}\nHere, the first term is the contribution of the vector multiplets and the second line is the contribution of matter multiplets, labeled by $\\Phi$, where $j$ runs over the rank of the flavor symmetry group. Given the single letter index it is a combinatorial problem \\cite{Benvenuti:2006qr,Feng:2007ur} to compute the full multi-letter index. The result is given by the so-called ``plethystic'' exponential\n\\begin{equation}\n\\exp \\bigg( \\sum_{n=1}^\\infty \\frac{1}{n} \\text{ind} ( z^n, t^n, q^n; m) \\bigg) \\;.\n\\end{equation}\nFor instance, let us consider the ${\\mathcal N}=2$ theory with $U(N)$ gauge group. Then, the chiral multiplet $\\Phi$ with $R$-charge $r$ in the fundamental representation of the gauge group contributes to the single-letter index as\n\\begin{equation}\n\\sum_{i=1}^{N} \\left[ z_i t^{f(\\Phi)} \\frac{q^{\\frac{r}{2}+\\frac{|m_i|}{2}}}{1-q}-z_i^{-1}t^{-f(\\Phi)} \\frac{q^{1-\\frac{r}{2}+\\frac{|m_i|}{2}}}{1-q} \\right] \\;.\n\\end{equation}\nAfter the ``plethystic'' exponential one obtains the contribution of the chiral multiplet to the index\n\\begin{equation} \\label{contmul}\n\\prod_{i=1}^{N} \\frac{(q^{1-\\frac{r}{2}+\\frac{|m_i|}{2}} t^{-f(\\Phi)}z_i^{-1}; q)_{\\infty}}\n{(q^{\\frac{r}{2}+\\frac{|m_i|}{2}} t^{f(\\Phi)}z_i; q)_{\\infty}} \\;.\n\\end{equation}\nSimilarly the contribution of the vector multiplet to the single-letter index is\n\\begin{equation}\n-\\sum_{i,j=1,\\ldots, N,\\, i\\neq j} q^{\\frac14 |m_i-m_j|} \\frac{z_i}{z_j} \\;,\n\\end{equation}\nand the multi-letter index gets the  form\n\\begin{equation}\nq^{-\\sum_{1\\leq i<j\\leq N} \\frac{|m_i-m_j|}{2}} \\prod_{i,j=1,\\ldots, N,\\, i\\neq j}\n\\left(1-\\frac{z_i}{z_j} q^{\\frac{|m_i-m_j|}{2}}\\right) \\;.\n\\end{equation}\n\nOur main interest is the so-called generalized superconformal index which includes integer parameters corresponding to global symmetries. In \\cite{Kapustin:2011jm} Kapustin and Willett pointed out that one can generalize the superconformal index of three-dimensional supersymmetric gauge theory by considering the theory in a non-trivial background gauge field coupled to the global symmetries of the theory. As a result the superconformal index includes new discrete parameters for global symmetries; we do not sum over these parameters. In case of the generalized superconformal index the contribution (\\ref{contmul}) has the  form \n\\begin{equation}\n\\prod_{i=1}^{N} \\frac{(q^{1-\\frac{r}{2}+\\frac{|m_i+ f(\\Phi) n_\\Phi|}{2}}\nt^{-f(\\Phi)}z_i^{-1}; q)_{\\infty}}{(q^{\\frac{r}{2}+\\frac{|m_i+f(\\Phi) n_\\Phi|}{2}} t^{f(\\Phi)}z_i; q)_{\\infty}} \\;,\n\\end{equation}\nwhere the parameters $n_\\Phi$ are new discrete variables. It is convenient to express the index as a product of contributions from chiral and vector multiplets\n\\begin{align} \\nonumber\nI(q, \\{t_a\\}, \\{n_a\\}) = \\sum_{m_1,\\ldots , m_{\\text{rank}(G)}}\\frac{1}{|W_m|} \\oint \\prod_{j=1}^{\\text{rank}\\,G} \\frac{dz_j}{2 \\pi \\mathrm i z_j} Z_{\\text{gauge}} (z_j,m_j; q) \\\\  \\label{gindex}\n\\times \\prod_{\\Phi} Z_{\\Phi}(z_j, m_j; t_a, n_a; q) \\;,\n\\end{align}\nwhere \n\\begin{equation}\nZ_{\\text{gauge}} (z_j,m_j; q) \\ = \\ \\prod_{\\alpha \\in \\text{ad}(G)} q^{-\\frac12 |\\alpha(m)|} \\left( 1- e^{\\alpha(g)} q^{\\frac{|\\alpha(m)|}{2}} \\right)\n\\end{equation}\nand \n\\begin{align} \\nonumber\nZ_\\Phi \\ = \\ \\prod_{\\rho \\in R_\\Phi} \\left( q^{\\frac{1-r_\\phi}{2}} \\prod_{j} e^{-\\mathrm i \\rho(g)} t(\\Phi)^{-f(\\Phi)} \\right)^{\\frac12 |\\rho(m)+f(\\Phi) n(\\Phi)|} \\\\\n\\times \\frac{(e^{-\\mathrm i\\rho(g)} t(\\Phi)^{-f(\\Phi)} q^{\\frac12 |\\rho(m)+f(\\Phi) n(\\Phi)|+\\frac{1-r_\\Phi}{2}}; q)_\\infty}{(e^{\\mathrm i\\rho(g)} t(\\Phi)^{f(\\Phi)} q^{\\frac12 |\\rho(m)+f(\\Phi) n(\\Phi)|+\\frac{r_\\Phi}{2}}; q)_\\infty} \\;.\n\\end{align}\nHere $\\text{ad}(G)$ stands for the adjoint representation of the gauge group $G$. Note that we do not write the contribution of the Chern--Simons term in (\\ref{gindex}), since as we mentioned before we consider theories without this term. \n\nIt is worth to mention that the three-dimensional superconformal index can be constructed from the so-called holomorphic blocks \\cite{Beem:2012mb} due to its factorization property \\cite{Krattenthaler:2011da,Pasquetti:2011fj,Taki:2013opa,Nieri:2013yra,Hwang:2012jh,Hwang:2015wna}, i.e. the superconformal index can be expressed in terms of two identical $3d$ holomorphic blocks ${\\mathcal B}(x;q)$\nas\\footnote{Geometrically it means that the index can be obtained by gluing two solid tori. In this context $B_c(x;q)$ are partition functions on solid tori.}\n\\begin{equation}\n\\sum_c {\\mathcal B}_c(x;q) {\\mathcal B}_c(\\tilde{x};\\tilde{q}) \\;.\n\\end{equation}\nIt is possible to obtain the factorized superconformal index directly from the localization technique via the so-called Higgs branch localization \\cite{Fujitsuka:2013fga,Benini:2013yva}. \n\n\n\n\\section{Integral identities from 3d dualities}\n\\label{iis}\n\n\nIn \\cite{Seiberg:1994pq} Seiberg found that there exist pairs of different four-dimensional ${\\mathcal N}=1$ supersymmetric gauge theories which describe the same physics in the infrared limit. This is called supersymmetric duality. Since its proposal a large number of dualities in various dimensions have been found. \n\nIn this section, we study three-dimensional ${\\mathcal N} = 2$  supersymmetric dualities \\cite{Intriligator:1996ex,Aharony:1997gp, Karch:1997ux, deBoer:1997ka} and demonstrate the matching of the superconformal index for dual theories. The superconformal index technique is one of the main tools for establishing and checking supersymmetric dualities.\n\nIn this work, we consider only confining theories, i.e.\\ theories whose infrared limit can be described in terms of gauge invariant composites (mesons and baryons) and without dual quarks. There are definitely more confining supersymmetric theories in three dimensions (for recent discussions, see \\cite{Csaki:2014cwa, Amariti:2015kha}). We restrict our attention to samples of theories with $U(1)$ (supersymmetric quantum electrodynamics) and $SU(2)$ (supersymmetric quantum chromodynamics) gauge symmetry. We also limit ourselves to the cases of vanishing Chern-Simons term; however, one can add such a term to the action of the theories considered in the paper.\n\nNote that similar results for ${\\mathcal N}=1$ supersymmetric gauge theories in four dimensions were intensively studied in \\cite{Dolan:2008qi,Spiridonov:2009za,Spiridonov:2011hf}. All $3d$ dualities considered in the next section can be obtained via dimensional reduction from $4d$ dualities. However, obtaining the right duality in three dimensions is more tricky (for details see \\cite{Aharony:2013dha,Niarchos:2012ah}). The main issue is that the reduction procedure and renormalization group flow from ultraviolet to infrared do not commute with each other. This happens because of an anomalous $U(1)$ symmetry in $4d$, which  one needs to break  in $3d$ theory. This can be done by adding a monopole operator to the $3d$ Lagrangians. To be more precise we need to add the effective superpotential $W=\\eta X$ to the Lagrangian of electric theory and $W=\\tilde{\\eta} \\tilde{X}$ to the magnetic theory (dual theory), where $X$ is a monopole operator and $\\eta$ is the $4d$ instanton factor.\n\n\nIn our examples we give only the necessary input to compute the superconformal index and do not discuss other aspects of dual theories. As for many other dualities in physics, systematic proofs of supersymmetric dualities are absent and the superconformal index computations do not constitute a proof of the duality. There are other important arguments for three-dimensional supersymmetric dualities, i.e. study of superpotentials for interactions among chiral multiplets \\cite{Aharony:2013dha}, brane construction (see e.g, \\cite{deBoer:1997ka, Amariti:2015yea}), contact terms (see e.g., \\cite{Closset:2012vp,Amariti:2014lla}) and other powerful methods very much in the spirit of the superconformal index such as study of sphere partition functions \\cite{Kapustin:2009kz,Jafferis:2010un}, ellipsoid partition functions \\cite{Dolan:2011rp,Niarchos:2012ah,Gahramanov:gka}, lens partition functions \\cite{Imamura:2012rq, GahKels}, etc.\n\nThe 't Hooft anomaly matching conditions which played a central role in checking Seiberg dualities for ${\\mathcal N} = 1$ supersymmetric gauge theories become  useless in three dimensions since, unlike four-dimensional gauge theories, in three dimensions there are no chiral anomalies. In three dimensions it is possible to have a classical Chern-Simons term which breaks parity. One can then use the matching condition for the parity anomaly; however, conditions for discrete anomalies are weaker than those for continuous anomalies. \n\nIn what follows, we omit the $R$-charges for chiral multiplets, since the superconformal indices of dual theories match for arbitrary assignment of the $R$-charge \\cite{Imamura:2011su}. The correct $R$-charges for matter fields in the infrared fixed points can be obtained by the so-called $Z$-extremization procedure \\cite{Jafferis:2010un}.\n\nAs a final remark, let us comment that the matching of superconformal indices for dual pairs were studied mainly by expanding in terms of fugacities \\cite{Imamura:2011su, Kapustin:2011vz, Kim:2013cma, Park:2013wta} and only in a few works \\cite{Krattenthaler:2011da,Kapustin:2011jm,Gahramanov:2013rda} authors give rigorous proofs of the index identities. \n\nBelow we give explicit expressions of generalized superconformal indices for some theories.  Equality of indices for dual theories \nleads to integral evaluations, which will be proved rigorously  in Section \\ref{mps}.\n\n\\medskip\n\n{\\textbf{Example 1.}}\n\n\\medskip\n\n\n\nWe first consider a \\textbf{Theory A} and its low-energy description \\textbf{Theory B} which can be described purely in terms of composite gauge singlets. \n\n\\medskip\n\n\\begin{itemize}\n\n\\item \\textbf{Theory A}: Supersymmetric Quantum Chromodynamics with $SU(2)$ gauge group and with $SU(6)$ flavor group, chiral multiplets in the fundamental representation of the gauge group and the flavor group, a vector multiplet in the adjoint representation of the gauge group. Note that in case of $SU(2)$ gauge theories the fundamental and antifundamental representations are equivalent, therefore we have $SU(6)$ flavor group rather than $SU(3)\\times SU(3) \\times U(1)$.\n\n\n\\medskip\n\n\n\\item \\textbf{Theory B}: no gauge symmetry, fifteen chiral multiplets in the totally antisymmetric tensor representation of the flavor group.\n\n\n\\end{itemize}\n\n\\medskip\n\n\nThis duality was considered in \\cite{Teschner:2012em} where the authors presented the sphere partition functions for dual theories. It is analogous to the four-dimensional duality for similar theories \\cite{Dolan:2008qi} and can be obtained by dimensional reduction.\n\nUsing the group-theoretical data (see Appendix B) it is straightforward to compute explicitly the generalized superconformal indices, and due to the supersymmetric duality we find the  basic hypergeometric integral identity\n \n\\begin{align} \\nonumber\n& \\sum_{m\\in\\mathbb{Z}}\\oint \\frac{dz}{4 \\pi \\mathrm i z} q^{-|m|} (1-q^{|m|} z^2)(1-q^{|m|} z^{-2})  \\; (-q^{\\frac 12})^{\\sum_{i=1}^6 (\\frac{|n_i+m|}{2}+\\frac{| n_i-m|}{2})} \\\\ \\nonumber\n& \\qquad \\times z^{-\\sum_{i=1}^6 (\\frac{|n_i+m|}{2}-\\frac{|n_i-m|}{2})} \n \\prod_{j=1}^6 a_j^{-\\frac{|n_j+m|}{2}-\\frac{|n_j-m|}{2}}\n\\frac{(q^{1+\\frac{|n_j+m|}{2} }\/{a_jz},q^{1+\\frac{|n_j-m|}{2} }{z}\/{a_j};q)_\\infty}\n{(q^{\\frac{|n_j+m|}{2} }a_jz,q^{\\frac{|n_j-m|}{2} }{a_j}\/{z};q)_\\infty}\n \\\\  \\label{iden1}\n& \\qquad \\qquad =(- q^{\\frac12})^{\\sum_{1\\leq j<k\\leq 6} \\frac{|n_j+n_k|}{2}}\n\\prod_{1\\leq j<k\\leq 6} (a_j a_k)^{-\\frac{|n_j+n_k|}{2}} \\frac{(q^{1+\\frac{|n_j+n_k|}{2}}a_j^{-1}a_k^{-1};q)_\\infty}\n{(q^{\\frac{|n_j+n_k|}{2}}a_ja_k;q)_\\infty}\n\\end{align}\nwith the balancing conditions \n\\begin{equation} \\label{BC6}\n\\prod_{i=1}^6 a_i=q \\;\\; \\text{and} \\;\\; \\sum_{i=1}^6 n_i=0 \\;.\n\\end{equation} \n\nThis identity describes confinement without breaking of the ``chiral symmetry''. The left side of the expression (\\ref{iden1}) includes the contributions of twelve chirals and a vector multiplet, while the right hand side contains the contribution of fifteen chirals. From the fact that all physical degrees of freedom of Theory B are gauge invariant there is no integration on the right hand side. \n\nIt is worth mentioning that the duality considered in the example is a special case of the duality claimed in \\cite{Aharony:2013dha}, where the theory A is the three-dimensional $SP(2N)$ SQCD with $2N_f$ fundamentals and theory B is the $SP(2N_f-2N-4)$ theory with $2N_f$ fundamentals. Such duality is qualitatively similar to $SU(N)$ duality with matter in the fundamental representation of the gauge group. In case of $N=2$ one can consider the theory A as $SU(2)$ gauge theory since $SP(2) \\simeq SU(2)$.\n\n\nNote that the balancing conditions are imposed by the effective superpotential and the theories described above are dual only in the presence of certain superpotentials. We refer the interested reader to \\cite{Aharony:2013dha} for more details related to the study of superpotentials for three-dimensional dualities.\n\n\n\n\n\nIn (\\ref{iden1}) we used the absolute values of monopole charges as in the definition of the superconformal index. \nIt is possible to eliminate all absolute values using the elementary\nidentity  \\cite{Dimofte:2011py}\n\n\\begin{equation} \\label{absolute}\n\\frac{(q^{1+|m|\/2}\/z;q)_\\infty}{(q^{|m|\/2}z;q)_\\infty}=(-q^{-\\frac 12}z)^{\\frac{|m|-m}2}\\frac{(q^{1+m\/2}\/z;q)_\\infty}{(q^{m\/2}z;q)_\\infty}.\n\\end{equation}\n\nAfter such simplification, \\eqref{iden1} takes the form\n\n\\begin{multline*}\\sum_{m=\\in\\mathbb Z}^\\infty\\oint \\prod_{j=1}^6\\frac{(q^{1+(m+n_j)\/2}\/a_jz,q^{1+(n_j-m)\/2}z\/a_j;q)_\\infty}{(q^{(n_j+m)\/2}a_jz,q^{(n_j-m)\/2}a_j\/z;q)_\\infty}\\frac{(1-q^mz^2)(1-q^mz^{-2})}{q^mz^{6m}}\\frac{dz}{4\\pi\\mathrm i z}\\\\\n=\\frac{1}{\\prod_{j=1}^6a_j^{n_j}}\\prod_{1\\leq j<k\\leq 6}\\frac{(q^{1+(n_j+n_k)\/2}\/a_ja_k;q)_\\infty}{(q^{(n_j+n_k)\/2}a_ja_k;q)_\\infty} \\;.\\end{multline*}\n\nAfter replacing $(a_j,n_j)\\mapsto(a_jq^{N_j\/2},N_j)$, this is \\eqref{wp}.\nWe give a rigorous mathematical proof of this identity in\nTheorem \\ref{awt}.\n\n\nThe most intriguing physical interpretation of the formula (\\ref{iden1}) stems from the role it plays as a star-triangle relation \\cite{Kels:2015bda, Gahramanov:2015cva} for a certain two-dimensional statistical model. \n\nThe integral identity (\\ref{iden1}) can be obtained by reduction \\cite{Yamazaki:2013fva, Benini:2011nc, Kels:2015bda} from the similar  identity for four-dimensional lens indices. In \\cite{Kels:2015bda} such reduction was made in the context of integrable statistical models.  \n\nThe $q \\rightarrow 1$ limit of (\\ref{iden1}) was discussed in \\cite{Kels:2015bda}. This limit also has an interpretation in terms of exactly solvable statistical models \\cite{Kels:2013ola}.  From the viewpoint of supersymmetric dualities such reduction \\cite{Benini:2012ui,Benini:2011nc} gives the equality of the sphere partition functions of dual two-dimensional ${\\mathcal N}=(2,2)$ supersymmetric gauge theories.\n\n\n\\medskip\n\n{\\textbf{Example 2.}}\n\n\\medskip\n\n\n\nOur second example is again a supersymmetric quantum chromodynamics with a weakly coupled magnetic dual.\n\n\n\\medskip\n\n\\begin{itemize}\n\n\\item \\textbf{Theory A}: Supersymmetric Quantum Chromodynamics with $SU(2)$ gauge group and four flavors, chiral multiplets in the fundamental representation of the gauge group and the flavor group, the vector multiplet in the adjoint representation of the gauge group. \n\n\n\n\\medskip \n\n\\item \\textbf{Theory B}: no gauge degrees of freedom, with six mesons and a singlet chiral field.\n\n\\medskip\n\n\n\n\\end{itemize}\n\n\\medskip\n\nAccording to the supersymmetric duality we have the following integral\nidentity for the generalized superconformal indices:\n\n\\begin{align} \\nonumber\n& \\sum_{m\\in\\mathbb{Z}}\\oint \\frac{dz}{4 \\pi \\mathrm i z}\\,q^{-|m|} (1-q^{|m|} z^2)(1-q^{|m|} z^{-2})  \\; ({ -}q^{\\frac 12})^{\\sum_{i=1}^4 (\\frac{|n_i+m|}{2}+\\frac{|n_i-m|}{2}-{ n_i})} \\\\ \\nonumber\n& \\qquad \\times  z^{-\\sum_{i=1}^4 (\\frac{|n_i+m|}{2}-\\frac{|n_i-m|}{2})} \n\\prod_{j=1}^4 a_j^{-\\frac{|n_j+m|}{2} { -}\\frac{|n_j-m|}{2}{ +n_j}}\n\\frac{(q^{1+\\frac{|n_j+m|}{2} }\/{a_jz},q^{1+\\frac{|n_j-m|}{2} }{z}\/{a_j};q)_\\infty}\n{(q^{\\frac{|n_j+m|}{2} }a_jz,q^{\\frac{|n_j-m|}{2} }{a_j}\/{z};q)_\\infty}\n \\\\ \\nonumber\n& \\quad = (-q^{\\frac 12})^{\\sum_{1\\leq j<k\\leq 4} \\frac{|n_j+n_k|}{2} -\\sum_{i=1}^4 n_i-\\frac {| \\sum_{i=1}^4 n_i|}{2}} (a_1 a_2 a_3 a_4)^{\\frac{ |\\sum_{i=1}^4  n_i| -\\sum_{i=1}^4 n_i}{2}}  \\\\ \\label{iden2}\n& \\times \\frac{( q^{\\frac{|\\sum_{i=1}^4 n_i|}{2}}a_1 a_2 a_3 a_4)_{\\infty}}{(q^{1+ \\frac{|\\sum_{i=1}^4 n_i|}{2}}\/a_1 a_2 a_3 a_4)_{\\infty}} \\prod_{1\\leq j<k\\leq 4} (a_j a_k)^{-\\frac{|n_j+n_k| +(n_j+n_k)}{2}} \\frac{(q^{1+\\frac{|n_j+n_k|}{2}}\/a_ja_k;q)_\\infty}\n{(q^{\\frac{|n_j+n_k|}{2}}a_ja_k;q)_\\infty}.\n\\end{align}\n\nThe ordinary index of the theory A was considered in \\cite{Gahramanov:2013xsa} in the context of global symmetry enhancement. It was shown that the superconformal index of the theory has an extended $SO(10)$ flavor symmetry when coupled to $4d$ multiplets with specific boundary conditions.\n\nNote that one can deform dual theories from Example 1 by adding mass terms for some of the quarks. After integrating out one flavor (massive modes) the theory with the remaining four flavors confines with ``chiral symmetry breaking'' \\cite{Gahramanov:2015cva} if we keep a certain superpotential for the theory giving the balancing conditions similar to (\\ref{BC6}). Here the theory A has no superpotential and therefore we obtain the duality ({\\ref{iden2}}).\n\nEliminating the absolute values as before, \\eqref{iden2} can be expressed as\n\\begin{align} \\nonumber\n& \\sum_{m\\in\\mathbb{Z}}\\oint \\frac{dz}{4 \\pi \\mathrm i z}\\frac{(1-q^{m} z^2)(1-q^{m} z^{-2})}{q^m     z^{4m} }\n\\prod_{j=1}^4 \n\\frac{(q^{1+\\frac{n_j+m}{2} }\/{a_jz},q^{1+\\frac{n_j-m}{2} }{z}\/{a_j};q)_\\infty}\n{(q^{\\frac{n_j+m}{2} }a_jz,q^{\\frac{n_j-m}{2} }{a_j}\/{z};q)_\\infty}\n  \\\\ \\label{iden2b}\n&\\quad= \\frac{( q^{\\frac{\\sum_{i=1}^4 n_i}{2}}a_1 a_2 a_3 a_4)_{\\infty}}{(q^{1+ \\frac{\\sum_{i=1}^4 n_i}{2}}\/a_1 a_2 a_3 a_4)_{\\infty}} \\prod_{1\\leq j<k\\leq 4}  \\frac{(q^{1+\\frac{n_j+n_k}{2}}\/a_ja_k;q)_\\infty}\n{(q^{\\frac{n_j+n_k}{2}}a_ja_k;q)_\\infty}.\n\\end{align}\n\nThis can be recognized as a special case of Proposition \\ref{wps}. More precisely, Proposition \\ref{wps} states that \\eqref{iden2b} holds\neven with the integers $n_j$ replaced by generic complex parameters.\n\n\n\n\\medskip\n\nIn contrast to four dimensions, there exist supersymmetric dualities for abelian gauge theories in three dimensions. For details of such dualities see e.g.\\ \\cite{Strassler:1999hy}.  Below we consider two examples of such dualities.\n\n\\medskip\n\n{\\textbf{Example 3.}}\n\n \n\n\\medskip\n\n\\begin{itemize}\n\n\\item \\textbf{Theory A}: $d=3$ ${\\mathcal N}=2$ supersymmetric electrodynamics with $U(1)$ gauge symmetry and six chiral multiplets, half of them transforming in the fundamental representation of the gauge group and another half transforming in the anti-fundamental representation. \n\n\n\n\\medskip\n \n\\item \\textbf{Theory B}: no gauge degrees of freedom, nine gauge invariant ``mesons'' transforming in the fundamental representation of the flavor group.\n\n\n\\end{itemize}\n\n\\medskip\n\nSupersymmetric duality leads to the following identity for the generalized super\\-conformal indices: \n\n\\begin{align} \\nonumber\n& \\sum_{m \\in \\mathbb Z}  \\oint \\frac{dz}{2\\pi \\mathrm i z}  (-q^{\\frac 12})^{ \\sum_{i=1}^3 (\\frac{|m_i+m|}{2}+\\frac{|n_i-m|}{2}) } z^{- \\sum_{i=1}^3 (\\frac{|m_i+m|}{2}-\\frac{|n_i-m|}{2})}\\\\ \\nonumber\n& \\qquad \\qquad \\times  \\prod_{i=1}^3  a_i^{-\\frac{|m_i+m|}{2}} b_i^{-\\frac{|n_i-m|}{2}} \\frac{(q^{1+\\frac{|m_i+m|}{2}}\/a_i z, q^{1+\\frac{|n_i-m|}{2}} z\/b_i ;q)_{\\infty}}{(q^{\\frac{|m_i+m|}{2}} a_i z ,q^{\\frac{|n_i-m|}{2}} b_i\/z ;q)_{\\infty}} \\\\ \\label{prepentagon}\n& \\qquad \\qquad  = (-q^{\\frac 12})^{ \\sum_{i,j=1}^3\\frac{|m_i+n_j|}{2}} \\prod_{i,j=1}^{3} (a_ib_j)^{-\\frac{|m_i+n_j|}{2}} \\frac{(q^{1+\\frac{|m_i+n_j|}{2}}\/ a_i b_j;q)_{\\infty}}{(q^{\\frac{|m_i+n_j|}{2}} a_i b_j ;q)_{\\infty}}   \\;,\n\\end{align}\nwhere the fugacities $a_i$ and $b_i$ stand for the flavor symmetry $SU(3)\\times SU(3)$, $z$ is the fugacity for the $U(1)$ gauge group and the balancing conditions are \n\\begin{align} \\label{balcond}\n\\prod_{i=1}^3 a_i  = \\prod_{i=1}^3  b_i =q^{\\frac12} \\;\\; \\text{and} \\;\\; \\sum_{i=1}^3 n_i  =\\sum_{i=1}^3 m_i=0 \\;.\n\\end{align}\n\nEliminating the absolute values, this identity takes the form\n\\begin{multline*}\\sum_{m=-\\infty}^\\infty\\oint \\prod_{i=1}^3\\frac{(q^{1+(m+m_i)\/2}\/a_iz,q^{1+(n_i-m)\/2}z\/b_i;q)_\\infty}{(q^{(m+m_i)\/2}a_iz,q^{(n_i-m)\/2}b_j\/z;q)_\\infty}\\frac{1}{z^{3m}}\\frac{dz}{2\\pi\\mathrm i z}\\\\\n=\\frac{1}{\\prod_{i=1}^3a_i^{m_i}b_i^{n_i}}\\prod_{i,j=1}^3\\frac{(q^{1+(m_i+n_j)\/2}\/a_ib_j;q)_\\infty}{(q^{(m_i+n_j)\/2}a_ib_j;q)_\\infty}\\;\n. \\end{multline*}\n\nAfter the change of variables $z\\mapsto -z$, one may check that this is equivalent to Theorem \\ref{t} below. This identity was first \nproved in\n \\cite{Gahramanov:2013rda} in the special case of\n ordinary superconformal indices\\footnote{Note that the identity of sphere partition functions for this duality was presented in \\cite{Kashaev:2012cz,Benvenuti:2016wet}.}, that is, $m_i\\equiv n_i\\equiv 0$.\nThe general case was presented without proof in \\cite{Gahramanov:2014ona} .\n\nThe expression (\\ref{prepentagon}) can be written as an integral pentagon identity. Following \\cite{Gahramanov:2013rda}, we introduce the function\n\n\\begin{align} \\nonumber\n{\\mathcal B}_m[a, n;b, m] & = (-q^{\\frac 12})^{\\frac{|n|}{2}+\\frac{|m|}{2}-\\frac{|n+m|}{2}} a^{-\\frac{|n|}{2}} b^{-\\frac{|m|}{2}} (ab)^{\\frac{|n+m|}{2}} \\\\ \n& \\quad \\times \\frac{(q^{1+\\frac{|n|}{2}}a^{-1},q^{1+\\frac{|m|}{2}}b^{-1},q^{\\frac{|n+m|}{2}}ab;q)_\\infty}{(q^{\\frac{|n|}{2}}a,q^{\\frac{|m|}{2}}b,q^{1+\\frac{|n+m|}{2}}(ab)^{-1};q)_\\infty}\\; ,\n\\end{align}\nand rewrite the equality (\\ref{prepentagon}) in terms of this function. We obtain the following integral pentagon identity in terms of ${\\mathcal B}$ functions:\n\\begin{align} \\nonumber\n \\sum_{m\\in\\mathbb Z} \\oint \\frac{d z}{2 \\pi \\mathrm i z}   \\prod_{i=1}^3 {\\mathcal B}[ a_i z, n_i+m; b_i z^{-1}, m_i-m] \\\\ \\label{pentagon}\n = {\\mathcal B}[a_1 b_2, n_1+m_2; a_3 b_1; n_3+m_1] \\; {\\mathcal B}[a_2 b_1, n_2+m_1; a_3 b_2, n_3+m_2] \\;,\n\\end{align}\nwith the balancing conditions (\\ref{balcond}). \n\nThe integral identity (\\ref{pentagon}) is interesting from the following point of view. There is a recently proposed relation called $3d\/3d$ correspondence between $3d$ ${\\mathcal N}=2$ supersymmetric gauge theories and 3-manifolds \\cite{Terashima:2011qi, Dimofte:2011ju} (see also \\cite{Dimofte:2014ija,Dimofte:2011py} and earlier works \\cite{Terashima:2011xe, Terashima:2012cx}) in similar spirit as the AGT correspondence \\cite{Alday:2009aq}. This correspondence translates the ideal triangulation of the 3-manifold into mirror symmetry for three-dimensional supersymmetric theories. The independence of the corresponding 3-manifold invariant on the choice of triangulation corresponds to the equality of superconformal indices of mirror dual theories \\cite{Dimofte:2011py}. In this context the identity (\\ref{pentagon}) encodes a 3--2 Pachner move for 3-manifolds. \n\n\\medskip\n\n{\\textbf{Example 4.}}\n\n\\medskip\n\nLet us consider another example of abelian duality, namely the well-known XYZ\/SQED mirror symmetry \\cite{Aharony:1997bx,deBoer:1997ka,Intriligator:1996ex}.\n\n\\medskip\n\n\\begin{itemize}\n\n\\item \\textbf{Theory A:} ${\\mathcal N}=2$ supersymmetric quantum electrodynamics,  with a single $U(1)$ vector multiplet and two chiral multiplets charged oppositely under the gauge group.\n\n\\medskip\n\n\\item \\textbf{Theory B:} free Wess--Zumino theory with three chiral multiplets. This theory is often is called the XYZ model in the literature.\n\n\\medskip\n\n\\end{itemize} \n\nIn this example we wish to turn on the contribution to the generalized superconformal index of the topological symmetry $U(1)_J$, which is not explicit in the Lagrangian. This hidden symmetry is generated by the current\n\\begin{equation}\nJ^\\mu \\ = \\  \\varepsilon^{\\mu \\nu \\rho} F_{\\nu \\rho} \\;.\n\\end{equation}\nThe current $J^{\\mu}$ is topologically conserved\\footnote{The corresponding charge is carried by the Abrikosov-Nielsen-Olesen vortices in the Higgs branch of $\\mathcal N=2$ theory.} due to the Bianchi identity. \n\nIn this case we have a special duality called mirror symmetry which exchanges the Coulomb branch of a theory with the Higgs branch of its mirror dual and vice versa. The duality implies the identity\n\\begin{align} \\nonumber\n& \\sum_{s \\in \\mathbb{Z}} \\oint \\frac{dz}{2 \\pi \\mathrm i z} (-1)^{s+m+\\frac{|s+ m|}{2}+\\frac{|s - m|}{2}} z^{n} w^{s} (q^{\\frac14} z \\alpha^{-1})^{\\frac{|s - m|}{2}} (q^{\\frac14} z^{- 1} \\alpha^{-1})^{\\frac{|s+ m|}{2}} \\\\\n\\nonumber&\\qquad\\qquad \\quad\\times\\frac{(z^{\\pm } \\alpha^{-1} q^{\\frac{|s \\mp m|}{2}+\\frac34};q)_\\infty}{(z^{\\pm } \\alpha q^{\\frac{|s \\pm m|}{2}+\\frac14};q)_\\infty} \\\\ \\nonumber\n& \\qquad \\qquad = (-1)^{n+m+\\frac{|n+ m|}{2}+\\frac{|n- m|}{2}} (q^{\\frac14} \\alpha w)^{\\frac{|m - n|}{2}} (q^{\\frac14} \\alpha w^{- 1})^{\\frac{|m+ n|}{2}} \\alpha^{-2|m|} \\\\ \\label{mirror2}\n& \\qquad \\qquad \\quad  \\times \\frac{(\\alpha w^{\\pm } q^{\\frac{|m \\pm n|}{2}+\\frac34},\\alpha^{-2} q^{|m|+\\frac12};q)_\\infty}{(\\alpha^{-1} w^{\\pm } q^{\\frac{|m \\mp n|}{2}+\\frac14},\\alpha^2 q^{|{m}|+\\frac12};q)_\\infty} \\;,\n\\end{align}\nwhere the fugacity $\\alpha$ and the monopole charge $m$ denote the parameters for the axial $U(1)_A$ symmetry, $\\omega$ and $n$ denote the parameters for the topological $U(1)_J$ symmetry and the discrete parameter $s$ stands for the magnetic charge corresponding to the $U(1)$ gauge group. \nThe factors containing $\\pm$ should be interpreted as the product over both choices; for instance,\n$$(z^{\\pm } \\alpha^{-1} q^{\\frac{|s \\mp m|}{2}+\\frac34};q)_\\infty=(z\\alpha^{-1} q^{\\frac{|s- m|}{2}+\\frac34};q)_\\infty(z^{-1} \\alpha^{-1} q^{\\frac{|s+ m|}{2}+\\frac34};q)_\\infty. $$\n\nHere, we explicitly write the $R$-charges of chiral multiplets. Due to the  permutation symmetry of the superpotential $W=\\tilde{q} S q$ for the theory B, where  $q,\\tilde{q}$, $S$ are three chiral multiplets of the theory, one can fix the $R$-charges\\footnote{ In the infrared limit the superpotential $W$ must have the R-charge $2$, then the $R$-charge of chiral multiplets of theory B must be $\\frac23$. }.\n\n\nThe case $m=n=0$ of (\\ref{mirror2}) was presented in \\cite{Imamura:2011su, Krattenthaler:2011da} and proven in \\cite{Krattenthaler:2011da}.\nThe general case was presented, with a slight mistake\\footnote{{ We have an additional phase factor $(-1)^{s+m+\\frac{|s + m|}{2}+\\frac{|s - m|}{2}}$, which is due to the definition of the fermion number operator $F$ in the definition of the superconformal index \\cite{Dimofte:2011py} (see also \\cite{Aharony:2013dha,Hwang:2015wna}). In fact, in general the superconformal indices match for dual theories in presence of this corrected phase factor \\cite{Aharony:2013dha}. Without the phase factor the identity presented by Kapustin and Willett \\cite{Kapustin:2011jm} is incorrect. It is actually a good example where the naive choice of the fermion number as $2J_3$ does not work.}}, in \\cite{Kapustin:2011jm}, where \na proof was given for the special case $m=0$. In Section \\ref{mps} we give an analytic proof of the general case. More precisely, eliminating the absolute values in \\eqref{mirror2} gives\n\\begin{align} \\nonumber\n& \\sum_{s \\in \\mathbb{Z}} \\oint \\frac{dz}{2 \\pi \\mathrm i z} (-w)^s z^{n-s}\n\\frac{(z^{\\pm } \\alpha^{-1} q^{\\frac{m\\mp s}{2}+\\frac34};q)_\\infty}{(z^{\\pm } \\alpha q^{\\frac{m\\pm s}{2}+\\frac14};q)_\\infty} \\\\ \\nonumber\n& \\qquad \\qquad = (-w)^n\\frac{(\\alpha w^{\\pm } q^{\\frac{m \\pm n}{2}+\\frac34},\\alpha^{-2} q^{m+\\frac12};q)_\\infty}{(\\alpha^{-1} w^{\\pm } q^{\\frac{m \\pm n}{2}+\\frac14},\\alpha^2 q^{{m}+\\frac12};q)_\\infty} \\;,\n\\end{align}\nwhich can be recognized as the special case $a=b=q^{\\frac 14-\\frac m2}\\alpha$, $c=d=q^{\\frac 14+\\frac m2}\\alpha$ of Proposition~\\ref{ramp}.\n\nThe identity \\eqref{mirror2} and related identities can also be written as pentagon identities.\nIn fact, introducing the tetrahedron index \\cite{Dimofte:2011py, Dimofte:2011ju}\n$$\\mathcal I_q[m,z] = \\frac{(q^{1-\\frac m2}\/z;q)_\\infty}{(q^{-\\frac m2}z;q)_\\infty},$$\n it follows from Proposition \\ref{ramp} that \n \\begin{multline*}\\sum_{s\\in\\mathbb Z}\\oint(-w)^sz^{N-s}\\,\\mathcal I_q[m-s;q^{1\/4}\\alpha z]\\,\n\\mathcal I_q[n+s;q^{1\/4}\\beta\/z]\\,\\frac{dz}{2\\pi\\mathrm i z}\\\\\n=(-w)^N\\mathcal I_q[m+n;q^{\\frac 12}\\alpha\\beta]\\,\n\\mathcal I_q[n+N;q^{1\/4}w\/\\beta]\\,\\mathcal I_q[m-N;q^{1\/4}\/\\alpha w].\n\\end{multline*}\nSpecial cases with $m=n=N=0$ and $m=n$ (corresponding to \\eqref{mirror2}) were presented earlier in \\cite{Gahramanov:2013rda},\n   \\cite{Gahramanov:2014ona}, respectively.\n\n\nOne can also consider this duality as a mirror symmetry between ${\\mathcal N}=4$ super\\-symmetric electrodynamics with a single flavor and its dual theory with a free hypermultiplet. Then we obtain instead of (\\ref{mirror2}) the mathematically equivalent identity \n\\begin{align} \\nonumber\n& \\alpha^{2|m|} \\frac{(\\alpha^{2} q^{|m|+\\frac12};q)_\\infty}{(\\alpha^{-2} q^{|{m}|+\\frac12};q)_\\infty} \\sum_{s \\in \\mathbb{Z}} \\oint \\frac{dz}{2 \\pi \\mathrm i z} (-1)^{s+m+\\frac{|s+ m|}{2}+\\frac{|s - m|}{2}} \\\\ \\nonumber\n& \\qquad \\qquad \\qquad \\times z^{n} w^{s} (q^{\\frac14} z \\alpha^{-1})^{\\frac{|s - m|}{2}} (q^{\\frac14} z^{- 1} \\alpha^{-1})^{\\frac{|s+ m|}{2}}  \\frac{(z^{\\pm } \\alpha^{-1} q^{\\frac{|s \\mp m|}{2}+\\frac34};q)_\\infty}{(z^{\\pm } \\alpha q^{\\frac{|s \\pm m|}{2}+\\frac14};q)_\\infty} \\\\ \n& = (-1)^{n+m+\\frac{|n+ m|}{2}+\\frac{|n- m|}{2}} (q^{\\frac14} \\alpha w)^{\\frac{|m - n|}{2}} (q^{\\frac14} \\alpha w^{- 1})^{\\frac{|m+ n|}{2}}   \\frac{(\\alpha w^{\\pm } q^{\\frac{|m \\pm n|}{2}+\\frac34} ;q)_\\infty}{(\\alpha^{-1} w^{\\pm } q^{\\frac{|m \\mp n|}{2}+\\frac14} ; q)_\\infty} \\;.\n\\end{align}\n\n\n\n\\section{Mathematical proofs of identities}\n\\label{mps}\n\nIn this section we will use the standard notation of \\cite{gr}. In particular,\nthe basic hypergeometric series is \\cite[Ex.\\ (1.2.22)]{gr}\n\\begin{equation}\n\\qhyp rs{a_1,\\ldots,a_r}{b_1,\\ldots, b_s}{q,z} = \\sum_{j=0}^{\\infty} \\frac{(a_1;q)_j \\ldots (a_r;q)_j}{(b_1;q)_j \\ldots (b_s;q)_j} \\left[(-1)^{j}q^{{\\binom j 2}}\\right]^{1+s-r} z^j \\;,\n\\end{equation}\nand the bilateral basic hypergeometric series is \\cite[Ex.\\ (5.1.1)]{gr}\n\\begin{equation}\n\\bhyp rs{a_1,\\ldots,a_r}{b_1,\\ldots, b_s}{q,z} = \\sum_{j=0}^{\\infty} \\frac{(a_1;q)_j \\ldots (a_r;q)_j}{(b_1;q)_j \\ldots (b_s;q)_j} (-1)^{(s-r)j}q^{(s-r){\\binom j 2}} z^j \\;,\n\\end{equation}\nwhere\n$$(a;q)_n=\\prod_{j=0}^n (1-aq^j)\\;. $$\nThe very-well-poised basic hypergeomeric series is defined as \\cite[Ex.\\ (2.1.11)]{gr}\n\\begin{equation}\n\\whyp {r+1}r{a_1, a_4, a_5, \\ldots,a_{r+1};q,z} = \\qhyp {r+1}r{a_1,qa_1^\\frac12,-qa_1^\\frac12,a_4,\\ldots,a_{r+1}}{a_1^\\frac12, -a_1^\\frac12, q a_1\/a_4,\\ldots, qa_1\/a_{r+1}}{q,z} \\;.\n\\end{equation}\n\n\nWe will assume that $|q|<1$. We will also write\n$$\\theta(z;q)=(z,q\/z;q)_\\infty\\;.$$\nThis theta function satisfies the quasi-periodicity\n\\begin{equation}\\label{qp}\\theta(zq^N;q)=\\frac{(-1)^N}{q^{\\binom N2}z^N}\\theta(z;q)\\;,\\qquad N\\in\\mathbb Z\\;.\\end{equation}\n\n\n\nWe will formulate four fundamental  identities, which evaluate\na combination of a basic hypergeometric integral and sum. \nIn each case, we assume that the parameters are generic, so that the poles of the integrand split naturally\ninto geometric sequences converging to $0$ and to $\\infty$. The integration is over a positively oriented contour separating these two types of poles.\n\nTo prove the first identity, we use the Nasrallah--Rahman integral and the nonterminating Jackson summation, which are  top level results\nfor basic hypergeometric integral evaluations and summations,\nrespectively.\nConsequently, we expect that Theorem \\ref{awt}\nis a top level result for evaluations of the type considered here, \nwith combined integration and summation.\n\n\\begin{theorem}\\label{awt}\nLet $a_j$ be generic numbers and $N_j$ integers satisfying $a_1\\dotsm a_6=q$\nand $N_1+\\dots+N_6=0$. Then,\n\\begin{multline}\\label{awti}\\sum_{m=-\\infty}^\\infty\\oint \\prod_{j=1}^6\\frac{(q^{1+m\/2}\/a_jz,q^{1-m\/2}z\/a_j;q)_\\infty}{(q^{N_j+m\/2}a_jz,q^{N_j-m\/2}a_j\/z;q)_\\infty}\\frac{(1-q^mz^2)(1-q^mz^{-2})}{q^mz^{6m}}\\frac{dz}{2\\pi\\mathrm i z}\\\\\n=\\frac{2}{\\prod_{j=1}^6q^{\\binom {N_j}2}a_j^{N_j}}\\prod_{1\\leq j<k\\leq 6}\\frac{(q\/a_ja_k;q)_\\infty}{(a_ja_kq^{N_j+N_k};q)_\\infty}\\;. \\end{multline}\n\\end{theorem}\n\n\n\n\\begin{proof}\nLet $L$ denote the left-hand side of \\eqref{awti}. \nNote that the poles of the integrand are situated at fixed values\nof $zq^{m\/2}$. Thus, we may replace\n$z$ by $zq^{-m\/2}$ and interchange the sum and the integral. This gives\n\\begin{align}\\nonumber L&=\\oint \\prod_{j=1}^6\\frac{(qz^\\pm\/a_j;q)_\\infty}{(q^{N_j}a_jz^\\pm;q)_\\infty}{(1-z^2)(1-z^{-2})}\\\\\n\\label{l}&\\quad\\times{}_8\\psi_8\\left(\\begin{matrix}q\/z,-q\/z,a_1\/z,\\dots,a_6\/z\\\\ 1\/z,-1\/z,q\/a_1z,\\dots,q\/a_6z\\end{matrix};q,q\\right)\\frac{dz}{2\\pi\\mathrm i z}\\;.\n\\end{align}\nBy \\cite[Eq.\\ (III.38)]{gr}, we may write\n\\begin{multline}\\label{pwt} \\prod_{j=1}^6{(qz^\\pm\/a_j;q)_\\infty}{(1-z^2)(1-z^{-2})}\\,{}_8\\psi_8\\left(\\begin{matrix}q\/z,-q\/z,a_1\/z,\\dots,a_6\/z\\\\ 1\/z,-1\/z,q\/a_1z,\\dots,q\/a_6z\\end{matrix};q,q\\right)\\\\\n=\\frac{(q;q)_\\infty\\prod_{j=1}^4(qa_5^{\\pm}\/a_j;q)_\\infty\\theta(a_6z^{\\pm};q)(z^{\\pm 2};q)_\\infty}{(qa_5^2,a_6a_5^{\\pm};q)_\\infty}\\\\\n\\times{}_8W_7(a_5^2;a_5a_1,a_5a_2,a_5a_3,a_5a_4,a_5a_6;q,q)\n+\\operatorname{idem}(a_5;a_6)\\;,\\end{multline}\nwhere the second term means that the first term is repeated with $a_5$ and $a_6$ interchanged. Using \\eqref{qp} to write\n$$\\theta(a_6z^{\\pm};q)_\\infty=q^{2\\binom{N_6}2}a_6^{2N_6}\\theta(a_6q^{N_6}z^{\\pm};q)_\\infty\\;, $$\nthis leads to\n\\begin{align*}\nL&=q^{2\\binom{N_6}2}a_6^{2N_6}\\frac{(q;q)_\\infty\\prod_{j=1}^4(qa_5^{\\pm}\/a_j;q)_\\infty}{(qa_5^2,a_6a_5^{\\pm};q)_\\infty}\\,{}_8W_7(a_5^2;a_5a_1,a_5a_2,a_5a_3,a_5a_4,a_5a_6;q,q)\\\\\n&\\quad\\times\\oint  \\frac{(z^{\\pm 2},q^{1-N_6}a_6^{-1}z^\\pm;q)_\\infty}{\\prod_{j=1}^5(q^{N_j}a_jz^\\pm;q)_\\infty}\\frac{dz}{2\\pi\\mathrm i z}+\\operatorname{idem}\\big((a_5,N_5);(a_6,N_6)\\big)\\;.\n\\end{align*}\nApplying the Nasrallah--Rahman identity \\cite[Eq.\\ (6.4.1)]{gr}\n$$\\oint\\frac{(z^{\\pm 2}, B z^{\\pm};q)_\\infty}{\\prod_{j=1}^5(b_jz^{\\pm};q)_\\infty}\\frac{dz}{2\\pi\\mathrm i z}= \\frac{2\\prod_{j=1}^5(B\/b_j;q)_\\infty}{(q;q)_\\infty\\prod_{1\\leq j<k\\leq 5}(b_jb_k;q)_\\infty}\\;,\\qquad B=b_1\\dotsm  b_5\\;, $$\nwe conclude that\n\\begin{align*}\nL&=2q^{2\\binom{N_6}2}a_6^{2N_6}\\frac{\\prod_{j=1}^4(qa_5^{\\pm}\/a_j;q)_\\infty\\prod_{j=1}^5(q^{1-N_6-N_j}\/a_ja_6;q)_\\infty}{(qa_5^2,a_6a_5^{\\pm};q)_\\infty\\prod_{1\\leq j<k\\leq 5}(q^{N_j+N_k}a_ja_k;q)_\\infty}\\\\\n&\\quad\\times \\,{}_8W_7(a_5^2;a_5a_1,a_5a_2,a_5a_3,a_5a_4,a_5a_6;q,q)\n+\\operatorname{idem}((a_5,N_5);(a_6,N_6))\\;.\n\\end{align*}\nBy the non-terminating Jackson summation \\cite[Eq.\\ (II.25)]{gr}, this can be simplified to the right-hand side of \\eqref{awti}.\n\\end{proof}\n\nIf one formally replaces $6$ by $4$ in Theorem \\ref{awt}, it is possible to replace the discrete parameters  $N_j$  by generic complex numbers. \nThe proof of the corresponding identity is in fact very easy.\n\n \n\n\\begin{proposition}\\label{wps}\nFor $a_j$ and $b_j$ generic,\n\\begin{multline}\\label{wpsi}\\sum_{m=-\\infty}^\\infty\\oint \\prod_{j=1}^4\\frac{(q^{1+m\/2}\/a_jz,q^{1-m\/2}z\/a_j;q)_\\infty}{(q^{m\/2}b_jz,q^{-m\/2}b_j\/z;q)_\\infty}\\frac{(1-q^mz^2)(1-q^mz^{-2})}{q^mz^{4m}}\\frac{dz}{2\\pi\\mathrm i z}\\\\\n=\\frac{2(b_1b_2b_3b_4;q)_\\infty}{(q\/a_1a_2a_3a_4;q)_\\infty}\\prod_{1\\leq j<k\\leq 4}\\frac{(q\/a_ja_k;q)_\\infty}{(b_jb_k;q)_\\infty}\\;. \\end{multline}\n\\end{proposition} \n\n\n\n\\begin{proof}\nWith $L$ the left-hand side of \\eqref{wpsi},\n the identity \\eqref{l} is replaced by\n\\begin{align*} L&=\\oint \\prod_{j=1}^4\\frac{(qz^\\pm\/a_j;q)_\\infty}{(b_jz^\\pm;q)_\\infty}{(1-z^2)(1-z^{-2})}\\\\\n&\\quad\\times{}_6\\psi_6\\left(\\begin{matrix}q\/z,-q\/z,a_1\/z,\\dots,a_4\/z\\\\ 1\/z,-1\/z,q\/a_1z,\\dots,q\/a_4z\\end{matrix};q,\\frac q{a_1\\dotsm a_4}\\right)\\frac{dz}{2\\pi\\mathrm i z}\\;.\n\\end{align*}\nApplying Bailey's summation \\cite[Eq.\\ (II.33)]{gr} gives\n$$L=\\frac{(q;q)_\\infty\\prod_{1\\leq j<k\\leq 4}(q\/a_ja_k;q)_\\infty}{(q\/a_1a_2a_3a_4;q)_\\infty}\n\\oint\\frac{(z^{\\pm 2};q)_\\infty}{\\prod_{j=1}^4(b_jz^\\pm;q)_\\infty}\\frac{dz}{2\\pi\\mathrm i z}\\;, $$\nwhich reduces the result to the Askey--Wilson integral \\cite[Eq.\\ (6.1.4)]{gr}.\n\\end{proof}\n\n\nThe next result was obtained in \\cite{Gahramanov:2013rda} for $M_j\\equiv N_j\\equiv 0$ and announced in \\cite{Gahramanov:2014ona} in general.\n\n\n\\begin{theorem}\\label{t}\nLet $a_j$, $b_j$ be generic numbers and $M_j$, $N_j$ integers satisfying $a_1a_2 a_3=b_1b_2 b_3=q^{1\/2}$\nand $M_1+M_2+M_3=N_1+N_2+N_3=0$. Then,\n\\begin{multline}\\label{p}\\sum_{m=-\\infty}^\\infty\\oint \\prod_{j=1}^3\\frac{(q^{1+m\/2}\/a_jz,q^{1-m\/2}z\/b_j;q)_\\infty}{(q^{M_j+m\/2}a_jz,q^{N_j-m\/2}b_j\/z;q)_\\infty}\\frac{(-1)^m}{z^{3m}}\\frac{dz}{2\\pi\\mathrm i z}\\\\\n=\\frac{1}{\\prod_{j=1}^3q^ {\\binom{M_j}2+\\binom{N_j}2}a_j^{M_j}b_j^{N_j}}\\prod_{j,k=1}^3\\frac{(q\/a_jb_k;q)_\\infty}{(a_jb_kq^{M_j+N_k};q)_\\infty}\\;\n. \\end{multline}\n\\end{theorem}\n\n\n\n\\begin{proof}\nThis can be proved similarly as the special case treated in\n\\cite{Gahramanov:2013rda}, so we will be very brief. Shrinking the contour of integration to zero, we  pick up residues at the poles\n$$z=q^{k-\\frac m2+N_j}b_j\\;,\\qquad j=1,2,3,\\quad k\\geq\\max(0,m-N_j)\\;. $$\nWorking out the sum of residues explicitly,  the left-hand side\nof \\eqref{p} can be written\n\\begin{align}\\nonumber L&=\\frac{(-1)^{N_1}}{q^{\\frac 32\\, N_1^2}b_1^{3N_1}}\\frac{(qb_1\/b_2,qb_1\/b_3;q)_\\infty}{(q^{N_2-N_1}b_2\/b_1,q^{N_3-N_1}b_3\/b_1;q)_\\infty}\\prod_{j=1}^3\\frac{(q\/a_jb_1;q)_\\infty}{(q^{N_1+M_j}a_jb_1;q)_\\infty}\\\\\n\\nonumber &\\quad\\times {}_3\\phi_2\\left(\\begin{matrix}q^{M_1+N_1}a_1b_1,q^{M_2+N_1}a_2b_1,q^{M_3+N_1}a_3b_1\\\\ q^{1+N_1-N_2}b_1\/b_2,q^{1+N_1-N_3}b_1\/b_3\\end{matrix};q,q\\right)\n{}_3\\phi_2\\left(\\begin{matrix}a_1b_1,a_2b_1,a_3b_1\\\\ qb_1\/b_2,qb_1\/b_3\\end{matrix};q,q\\right)\\\\\n\\label{lxy}&\\quad+\\operatorname{idem}\\big((b_1,N_1);(b_2,N_2),(b_3,N_3)\\big)\\;.\n \\end{align}\n\nLet\n$$x_1=b_1(qb_1\/b_2,qb_1\/b_3;q)_\\infty\\prod_{j=1}^3(a_jb_2,a_jb_3;q)_\\infty\\,{}_3\\phi_2\\left(\\begin{matrix}a_1b_1,a_2b_1,a_3b_1\\\\ qb_1\/b_2,qb_1\/b_3\\end{matrix};q,q\\right)$$\nand let $x_2$ and $x_3$ be defined by the same expression with $b_1$ interchanged by\n$b_2$ and $b_3$, respectively. Then, by the nonterminating $q$-Saalsch\\\"utz summation \\cite[Eq.\\ (II.24)]{gr},\n\\begin{equation}\\label{xd}x_2-x_1=b_2\\theta(b_1\/b_2;q)\\prod_{j=1}^3\\theta(a_jb_3;q)\\;. \n\\end{equation}\nLet $\\tilde x_j$ denote the result of replacing\n $a_j$ by $a_jq^{M_j}$ and $b_j$ by $b_jq^{N_j}$ in $x_j$. By \\eqref{qp},\nunder the same change of variables, the right hand side of \\eqref{xd}\nis  divided by\n$C=\\prod_{j=1}^3q^{\\binom{M_j}2+2\\binom{N_j}2}a_j^{M_j}b_j^{2N_j}$. Thus, if we define\n$y_j=C\\tilde x_j$, then $y_2-y_1=x_2-x_1$. \n By symmetry,  $y_j=x_j+D$, where $D$ is independent of $j$. It follows that\n$$(x_3-x_2)x_1y_1+(x_1-x_3)x_2y_2+(x_2-x_1)x_3y_3=(x_2-x_1)(x_3-x_2)(x_3-x_1)\\;. $$\nAfter simplification, this  identity reduces to the desired result.\n\\end{proof}\n\n\n\n\n\nWe conclude with the following identity. Note that  the\nparameter $t$ can be removed by scaling $z\\mapsto tz$, but it seems useful to keep it.\n\n \\begin{proposition}\\label{ramp}\n For $a$, $b$, $c$, $d$ and $t$ generic parameters and integer $N$, such that\n$|q^{\\frac{N+1}2}a^{-1}|<|t|<|q^{\\frac{N-1}2}b|$,\n\\begin{multline}\\label{rp}\\sum_{m=-\\infty}^\\infty\\oint \\frac{(q^{1+m\/2}\/az,q^{1-m\/2}z\/b;q)_\\infty}{(q^{m\/2}cz,q^{-m\/2}d\/z;q)_\\infty}\\,t^mz^{N-m}\\frac{dz}{2\\pi\\mathrm i z}\\\\\n=t^N\\frac{(q\/ab,-q^{\\frac{1+N}2}ct,-q^{\\frac{1-N}2}d\/t;q)_\\infty}{(cd,-q^{\\frac{1+N}2}\/at,-q^{\\frac{1-N}2}t\/b;q)_\\infty}\\;.\\end{multline}\n \\end{proposition}\n\n\\begin{proof}\nReplacing $z$ by $zq^{-m\/2}$ and changing the order of summation, we\nfind that the left-hand side is given by\n$$L=\\oint \\frac{(q\/az,qz\/b;q)_\\infty}{(cz,d\/z;q)_\\infty}\\,z^{N}\\,{}_1\\psi_1\\left(\\begin{matrix}b\/z\\\\ q\/az\\end{matrix};q,-\\frac{q^{\\frac{1-N}2}t}{b}\\right)\\frac{dz}{2\\pi\\mathrm i z}\\;. $$\nApplying Ramanujan's summation \\cite[Eq.\\ (II.29)]{gr} gives \n$$L=\\frac{(q,q\/ab;q)_\\infty}{(-q^{\\frac{1+N}2}\/at,-q^{\\frac{1-N}2}t\/b;q)_\\infty}\\oint \\frac{\\theta(-q^{\\frac{N+1}2}z\/t;q)}{(cz,d\/z;q)_\\infty}\\,z^{N}\\,\\frac{dz}{2\\pi\\mathrm i z} $$\n(the restriction on $t$ is needed here for convergence). \nIt remains to prove that\n\\begin{equation}\\label{soi}\\oint \\frac{\\theta(-q^{\\frac{N+1}2}z\/t;q)}{(cz,d\/z;q)_\\infty}\\,z^{N}\\,\\frac{dz}{2\\pi\\mathrm i z}=t^N\\frac{(-q^{\\frac{1+N}2}ct,-q^{\\frac{1-N}2}d\/t;q)}{(q,cd;q)_\\infty}\\;.\\end{equation}\nTo  this end, we expand the integral\n as  the sum of residues at the\npoints $z=q^kd$, $k\\geq 0$. By\n\\cite[Eq.\\ (4.10.8)]{gr}, under the additional assumption\n  $|q^{\\frac{1+N}2}t\/d|<1$, the sum of residues\nconverges and can be computed by the\n  $q$-binomial theorem \\cite[Eq.\\ (II.3)]{gr}.\nSince the left-hand side of \\eqref{soi} is  analytic in $t$ for\n$t\\neq 0$, the result holds also without the restriction  $|q^{\\frac{1+N}2}t\/d|<1$.\n\\end{proof}\n\n Using \\eqref{absolute}, it is easy to see that the case $a=b=c=d$, $N=0$ is equivalent to the identity proved in Appendix A1 of\n\\cite{Krattenthaler:2011da}. It may be remarked that our proof of the general case is simpler.\n\n\\section{Conclusions}\n\nSimilarly to four-dimensional dualities \\cite{Spiridonov:2009za,Spiridonov:2011hf}, equality of the superconformal indices for dual theories in three dimensions leads to  new non-trivial integral identities  \\cite{Gahramanov:2013rda,Krattenthaler:2011da,Kapustin:2011jm}. We have presented four new identities for basic hypergeometric integrals. More concretely, we studied the generalized superconformal index of confining theories in three dimensions that has the form of a basic hypergeometric integral. This kind of result is important for better understanding  the structure of three-dimensional supersymmetric dualities. Most dualities discussed in the work are known in the literature, but the verification of these dualities using the superconformal index technique is new.\n\nWe also presented so-called pentagon identities. They are especially interesting from the geometrical point of view, which interprets the pentagon relation as the $3-2$ Pachner move in the context of the $3d-3d$ correspondence. This relates different  decompositions of a polyhedron with five ideal vertices into ideal tetrahedra.\n\nIt would be interesting to study more general $SU(N)$ gauge theories, other gauge groups and other confining theories. \n\n\n\\medskip\n\n\n\n{\\bf Acknowledgements}.\nIG wishes to thank the Chalmers University of Technology for warm hospitality where this work started.  The research of IG is supported in part by the SFB 647 ``Raum-Zeit-Materie. Analytische und Geometrische Strukturen'', the Research Training Group GK 1504 ``Mass, Spectrum, Symmetry'' and the International Max Planck Research School for Geometric Analysis, Gravitation and String Theory. IG is partially supported by an ESF Short Visit Grant 6454 within the framework of the ``Interactions of Low-Dimensional Topology and Geometry with Mathematical Physics (ITGP)'' network. HR is supported by the Swedish Science Research Council. We are particularly grateful to Jonas Pollok for valuable comments on a preliminary version of the paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n       Let $\\f_q$ be the finite field of $q$ elements of characteristic $p$.      Let $\\psi$ be a fixed nontrivial additive character  on $\\f_q$.\n     For a Laurent polynomial $f\\in\\F_{q}\\left[x_1^{\\pm1}, \\ldots, x_n^{\\pm1} \\right]$,  define its associated $L$-function by\n \\[ \\label{Lfunction} L^*(f,T) = \\exp \\left( \\sum_{k=1}^{\\infty} \\sum_{x \\in {(\\f_{q^k})^*}} \\psi ({f(x)}) \\frac{T^k}{k} \\right). \\]\n A celebrated theorem of Dwork-Bombieri-Grothendieck implies that $L^*(f,T)$ is a rational function in $\\Q[\\zeta_p](x)$ \\cite{DW62}, \\cite{GR64}.\n Furthermore,   Deligne's theorem on the Riemann hypothesis \\cite{De81} gives very nice descriptions of the Archimedean absolute values of its zeroes and poles.\n\n\nAdolphson and Sperber \\cite{AS89} show that if the Laurent polynomial $f$ satisfies a non-degenerate geometric condition, then  $L^*(f,T)^{(-1)^{n-1}}$ is indeed a polynomial in\n$\\Z[\\zeta_p][x]$ of degree $n!\\text{V}(\\Delta(f))$.\n Here   $\\Delta(f)$ is the {\\it Newton polytope} (see \\cite{GKZ94} for details) of $f$, defined to be the convex closure in $\\R^n$ generated by the origin and all exponents of the nonzero monomials (viewed as lattice points in $\\R^n$), and $V(\\Delta(f))$ is the volume.\n\n  Since the Archimedean absolute values of the zeroes of $L^*(F,T)^{(-1)^{n-1}}$ are determined explicitly,  it remains to study the most intriguing problem of determining    the non-Archimedean absolute values. Equivalently, it suffices to\n study the slopes of  the Newton polygon of $L^*(f,T)^{(-1)^{n-1}}$,  which is defined as follows.\n\\begin{definition}Let $g(x) =\\sum_{i=0}^d c_i x^i \\in 1 + x \\Z[\\zeta_p]$. The ($q$-adic) Newton polygon of $g(x)$, denoted by $NP(f)$, is defined to be the lower convex hull of the points $(i, \\ord_qc_i)$ in the plane $\\mathbb{R}^2$. An example is given in Figure 1.\n\\end{definition}\n  Unfortunately, the computation of the Newton polygon of a given\n  L-function is very hard in general.\n  However, as shown by the Dwork theory, in many cases the combinatorial geometrical properties of the Newton polytope may give very nice algebraic properties of $f$.\n  For instance, when $f$ is non-degenerate, Adolphson and Sperber \\cite{AS89} show that  the Newton polygon of\n    $L^*(f,T)^{(-1)^{n-1}}$ lies above a topological lower bound\n     called Hodge polygon, which will be briefly introduced as follows.\n\n\n\n\n\n\n Let $\\Delta$ be an $n$-dimensional convex integral polytope in $\\mathbb{R}^n$. Let $C(\\Delta)$ be the cone generated by $\\Delta$ and the origin.  For a vector $u\\in\\R^n$, $w(u)$ is the smallest nonnegative real number $c$ satisfying $u \\in c \\Delta$.  If such $c$ does not exist, then we define $w(u) = \\infty$.\n\n\n\n  For a  co-dimension $1$ face $\\delta$ not containing $0$ in $\\Delta$, let $\\sum_{i=1}^n e_i X_i=1$ be the equation of the hyperplane containipng $\\delta$ in $\\R^n$.  Since the coordinates of vertices spanning $\\Delta$ are integers, the coefficients $e_i$ are uniquely determined rational numbers.  Let $D(\\delta)$ be the least common denominator of $\\{e_i,  1 \\leq i \\leq n\\}$ and $D=D(\\Delta)$ be the least common denominator of all $D(\\delta)$, where $\\delta$ runs over all the co-dimension $1$ faces of $\\Delta$ which do not contain the origin.\n\n  For an integer $k$, let\n$W_{\\Delta}(k) = \\# \\{u \\in \\Z^n\\  | \\ w(u) = \\frac{k}{D}\\}.$\nLet $$H_{\\Delta}(k) = \\sum_{i=0}^{n} (-1)^i {n \\choose i} W_{\\Delta}(k-iD).$$\n\n\\begin{definition}\n\\label{fig3}\nThe Hodge polygon $HP(\\Delta)$ of $\\Delta$ is defined to be the lower convex polygon in $\\R^2$ with vertices\n$$\\left( \\sum_{k=0}^{m} H_{\\Delta}(k), \\frac{1}{D} \\sum_{k=0}^m k H_{\\Delta}(k) \\right), 0\\leq m\\leq n D.$$\n\\end{definition}\nEquivalently, $HP(\\Delta)$ is the polygon starting from the origin with a side of  slope $k\/D$ with horizontal length $H_{\\Delta}(k)$ for each integer $0 \\leq k \\leq nD$.\n\n\nFor a finite integral polytope $\\Delta \\subset \\R^n$, let $\\mathcal{M}_p(\\Delta)$ be the set of non-degenerate $f$ over $\\overline{\\f_p}$ with $\\Delta(f)=f$.\nTo study how $NP(f)$ varies for $f \\in \\mathcal{M}_p(\\Delta)$,\n by the Grothendieck specialization theorem \\cite{WAN00},\nthe generic Newton polygon is given by\n\\[GNP(\\Delta,p): = \\inf_{f \\in \\mathcal{M}_p(\\Delta)} NP(f).\\]\n\n   Some work of computing $GNP(\\Delta,p)$\n   in one-dimensional cases has been done \\cite{ZHU03} and in certain two variable cases in \\cite{NIU12}.\n\nIn \\cite{AS89} Adolphson and Sperber proved that for $f \\in \\mathcal{M}_p(\\Delta)$, the graph of $NP(f)$ lies above the graph of $HP(\\Delta)$ with the same endpoints, which can be stated as the following inequality $NP(f) \\geq HP(\\Delta).$ See Figure 1 for an example.\n\n\n\\begin{figure}\n  \\centering\n\\begin{tikzpicture}[scale=0.6]\n\\draw[step=1cm,gray,very thin] (-2.5, -2.5) grid (9.5,7.5);\n\\draw[->,thick] (-2.5,0) -- (9.5,0);\n\\draw[->,thick] (0,-2.5) -- (0,7.5);\n\\draw [thick, blue](0,0)--(1,0)--(8,7);\n\\draw [thick,red] (0,0)--(1, 0) -- (2, 1\/2) -- (4, 2) -- (5, 3)--(7, 5.5)--(8, 7);\n\\draw (0,0) node[anchor=north east] {$0$};\n\\draw (1,0)+(0.1,-0.5) node {$(1,0)$};\n\\draw (5,3)+(0.7,-0.5) node {$(5,3)$};\n\\draw (8,7)+(0.7,-0.5) node {$(8,7)$};\n\\end{tikzpicture}\n\\caption{The Newton polygon (blue) and the Hodge polygon (red) of $ f(x_1, x_2) = x_1x_2^3+x_1^3x_2+x_1x_2$.}\n\\label{fig4}\n\\end{figure}\n\nCombining these inequalities we have\n\\begin{prop}\nFor every prime $p$ and every $f \\in \\mathcal{M}_p(\\Delta)$,\n$$NP(f) \\geq GNP(\\Delta,p) \\geq HP(\\Delta).$$\n\\end{prop}\n\n  A natural question is then to decide  when the second\ninequality holds. For this purpose, we have the following definitions.\n\n\\begin{definition}\n\tA Laurent polynomial $f$ is called ordinary if NP($f$) = HP($\\Delta$).\n\\end{definition}\n\n\n\n\\begin{definition}\n The family $\\mathcal{M}_p(\\Delta)$ is called {\\it generically ordinary} if $GNP(\\Delta,p)= HP(\\Delta)$.\n\\end{definition}\n\n\n It is known that  a necessary condition for $GNP(\\Delta,p)=HP(\\Delta)$\n is $ p\\equiv 1 \\md{D(\\Delta)}$.  Adolphson and Sperber \\cite{AS87} conjectured that the converse is also true.\n \\begin{conjecture}[Adophson-Sperber]\n\\label{AS}\n If $ p\\equiv 1 \\md{D(\\Delta)}$, then $\\mathcal{M}_p(\\Delta)$ is  { generically ordinary}.\n\\end{conjecture}\nThis is a generalization of a conjecture of Dwork \\cite{DW73} and Mazur \\cite{MA72}.  Wan showed in \\cite{WAN93} that Conjecture \\ref{AS} is true for $n\\leq 3$.  Furthermore,  he proved the following theorem.\n\\begin{thm}[Wan]\n\\label{theoremD}   For any $n\\geq 4$, there are explicitly constructed counterexamples for the A-S conjecture.\nHowever, a weaker version holds: there is an effectively computable integer $D^*(\\Delta)$ such that if $p \\equiv 1 \\md{D^*(\\Delta)}$, then $GNP(\\Delta,p) = HP(\\Delta)$.\n\\end{thm}\n Roughly speaking, one of the most important techniques of Wan's proof is a decomposition theorem which decomposes $\\Delta$ into small nice pieces and reduce the problem to small problems which are easier to be checked (see Figure 2 for an illustration). Then  Wan developed several very useful decomposition theorems including  the star decomposition, the boundary decomposition  \\cite{WAN93} and the collapsing decomposition \\cite{WAN04}.\n\n\nSuppose $\\Delta=\\Delta(f)$ and thus we can write $f$ as\n\\[f=\\sum_{I\\in \\Delta \\cap Z^n} a_I x^I.\\]\nSimilarly, for a face $\\delta\\subseteq\\Delta$, define\n\\[f^{\\delta}=\\sum_{I\\in \\delta \\cap Z^n} a_I x^I.\\]\n\n\n\\begin{thm}[Wan's facial decomposition theorem]\nSuppose $f$ is non-degenerate (See \\cite{AS89} for the definition).\nLet $\\{\\delta_i, 1\\leq i\\leq s\\}$ be the set all co-dimension 1 faces of $\\Delta(f)$.\nThen $f$ is ordinary if and only if each $f^{\\delta_i}$ is ordinary.\n\\end{thm}\n\n\n\\begin{figure}\n  \\centering\n\\begin{tikzpicture}[scale=0.5]\n \\draw [thick] (0, 0) -- (8, 0) -- (7,1)--(0,0);\n   \\draw [thick] (0,0)--(7,7)-- (6, 4) -- (0,0);\n  \\draw [thick] (7,1)-- (6, 4) -- (9,3)--(7,1);\n    \\draw [thick] (6,4)--(7,7) -- (9,3);\n    \\draw [thick] (9, 3) -- (8, 0) -- (7,1)--(9,3);\n\\draw (0,0) node[anchor=east] {$O$};\n\\draw (3,3.5) node[anchor=east] {$\\Delta$};\n \\draw [dashed] (0,0) -- (9,3);\n  \\draw [dashed] (7,7) -- (8,0);\n  \\draw[->,thin] (7, 5.5) -- (8,5.5) node[anchor=west]{$\\delta_i$};\n \\end{tikzpicture}\n\\caption{A facial decomposition for a 3 dimensional polytope.}\n\\label{fig5}\n\\end{figure}\n\nIn particular, it turns out that if all $f^{\\delta_i}$'s are diagonal, then we have effective methods to determine if they are ordinary.\nSince the cases for diagonal Laurent polynomials are very easy to deal with, this decomposition theorem will immediately give good nontrivial applications.\n\n\n\n\n\n Conjecture \\ref{AS} is true in many important cases as showed by Wan \\cite{WAN93} and more interesting examples are yet to be classified. Motivated by this,  Le \\cite{LE12} studied an important class of Laurant polynomials, in which $f$ is given by\n \\begin{equation}f(x_0, x_1, \\ldots, x_n)=x_0 h(x_1, x_2, \\ldots, x_n)+g(x_1, x_2, \\ldots, x_n)+1\/x_0,\n\\label{2.1}\n\\end{equation}\n  where $\\deg h=d$,  $\\deg g<d\/2$, and  $h$ is an $n$-variable Deligne polynomial, which  will be introduced in Section 2.\nWe are interested in the A-S conjecture for this family of polynomials.\n  \\begin{conjecture}\n Let $f$ be a generic polynomial given in \\ref{2.1}  and let $\\Delta=\\Delta(f)$. Then\n$GNP(\\Delta,p) = HP(\\Delta)$ if and only if $p \\equiv 1 \\md{D(\\Delta)}$.\n\\end{conjecture}\n\nRecall $\\mathcal{M}_p(\\Delta)$ is the set of non-degenerate $f$ over $\\overline{\\f_p}$ with $\\Delta(f)=f$.\n Let $\\widetilde{\\mathcal{M}}_p(\\Delta)\\subseteq \\mathcal{M}_p(\\Delta)$ be the set of non-degenerate $f$ of shape \\ref{2.1} with $\\Delta(f)=\\Delta$, and let\n \\[\\widetilde{GNP}(\\Delta,p): = \\inf_{f \\in \\widetilde{\\mathcal{M}}_p(\\Delta)} NP(f).\\]\n Since clearly $\\widetilde{GNP}({\\Delta},p) \\geq {GNP}({\\Delta},p)\\geq HP(\\Delta),$ a slightly stronger conjecture arises.\n\\begin{conjecture}[Le, 2013]\n$\\widetilde{GNP}({\\Delta},p) = HP(\\Delta)$ if and only if $p \\equiv 1 \\md{D(\\Delta))}$.\n\\label{Con1.10}\n\\end{conjecture}\nLe proved this conjecture for odd $d$ and this implies that the A-S conjecture\nfor this family holds for odd $d$.   He then conjectured\nthat it is also true for even $d$.\n\nIn this paper we give a counterexample to this conjecture.\n\\begin{thm}\nLe's conjecture is false in general.\n\\end{thm}\nNote that the reason that Le's conjecture fails is because\nwhen $d$ is even, the family polynomials  $\\widetilde{\\mathcal{M}}_p(\\Delta)$ studied in this paper and the family of polynomials $\\mathcal{M}_p(\\Delta)$ are not identical. In other words, the family Le considered is slightly smaller and thus is not a universal family.\n A natural question is to determine whether  Le's conjecture\nis true when the family of $f$ defined by \\ref{2.1} becomes larger, i.e., the condition\n\"$\\deg g<d\/2$\" is  replaced by \"$\\deg g\\leq d\/2$\".\n\n\n\n\n\n\n\\section{ A counterexample to Le's conjecture}\n\nA polynomial $f(x)=f(x_1, x_2, \\ldots, x_n)$ in $\\f_q[x_1, x_2, \\ldots, x_n]$ is called a  {\\it Deligne polynomial} if\n      its leading homogeneous form $f_d$ defines a smooth projective hypersurface in the projective space $\\mathbb{P}^{n-1}$, where $d$ is the degree of $f$.    Deligne polynomials were extensively studied by Deligne \\cite{De81},  Katz \\cite{KA07}, Browning and Heath-Brown \\cite{BH09},  Le \\cite{LE12}, Fu and Wan \\cite{FW20}.\n\n\n\nDeligne shows that if $f$ is a Deligne polynomial, then  $L^*(f,T)^{(-1)^{n-1}}$ is polynomial\nof degree $(d-1)^n$ and this fact immediately gives a fundamental estimate\n\\[|\\sum_{x \\in {\\f_{q^k}}} \\psi ({f(x)})|  \\leq (d-1)^n q^{\\frac k2}.\\]\n\nMotivated by the study of exponential sums, it is natural to consider a general class of Laurent polynomials of the following form\n\\[f(x_0, x_1, \\ldots, x_n)=x_0 h(x_1, x_2, \\ldots, x_n)+g(x_1, x_2, \\ldots, x_n)+1\/x_0, \\]\n  where $\\deg h=d$,  $\\deg g<d\/2$, and  $h$ is a Deligne polynomial in $n$ variables.\n\n\n\n\n\nFrom now on, $\\Delta$ is always used to denote the Newton Polytope of $f$ and  $e_0, e_1, \\ldots, e_n$ be the standard basis for $\\R^{n+1}$.\n Let $\\delta_d$ be the polytope spanned by $-e_0, e_0+de_1, \\ldots, e_0+de_n$, and  $\\delta_d'$ spanned by $e_0,e_0+de_1, \\ldots, e_0+de_n$.\n  They are the only two  co-dimension $1$ faces of $\\Delta$ that do not contain the origin.  Let $\\Delta_d$($\\Delta_d'$, respectively) be the convex hull of $\\delta_d$($\\delta_d'$, respectively) and the origin.\n  Please see Figure 3 for an example.\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{tikzpicture\n\\draw(0,0) node {$\\bullet$};\n\\node (n2) at (-0.7,-0.7) {$\\bullet$};\n\\node (n1) at (-1.4,-1.4) {$\\bullet$};\n\\node (v5) at (0,5) {$\\bullet$};\n\\node(v0) at (5,0) {$\\bullet$};\n\\draw[blue] (-1.4,-1.4) -- (0,5) -- (5,0) -- (-1.4,-1.4);\n\\draw[thick,dotted] (0,5) -- (-0.7,-0.7) -- (5,0);\n\\draw[thick,dotted]  (-1.4,-1.4) -- (-0.7,-0.7) ;\n\\draw[dotted] (0,0) -- (0,5) -- (5,0)--(0,0);\n\\draw[dotted] (0,0) -- (-1.4,-1.4);\n\\draw (n1)+(0,-0.4) node {$(-1,0,0)$};\n\\draw (n2)+(0.2,-0.2) node {$O$};\n \\draw (0,0)+(0.75, 0.35) node {$(1, 0, 0)$};\n\\draw (v5)+(0.8,0.2) node {$(1, 0, d)$};\n\\draw (v0)+(0.8,0.2) node {$(1,d,0)$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{$\\Delta \\supseteq \\Delta_d \\supseteq \\delta_d$ ($\\delta_d$ is the blue triangle)}\n\\label{fig6}\n\\end{figure}\n\n\n\n\n\n\nSince the defining equation of $\\delta_d$ is\n\\[ X_1-\\frac{2}{d}X_2-\\frac{2}{d}X_3-\\cdots-\\frac{2}{d}X_{n+1}=1,\\]\none computes that $$D(\\Delta_d)=\n\\begin{cases}\nd &  d \\mathrm{~is~odd;}\\\\\n\\frac{d}{2} & d \\mathrm{~is~even}.\n\\end{cases}$$\n\n\n\n\nFor simplicity we denote $D(\\Delta_d)$ by $D$ .\nThe Hodge polygon $HP(\\Delta_d)$ can be determined explicitly. We would like to know if $\\Delta_d$ satisfies Conjecture \\ref{AS}.\n\n\n Let $\\widetilde{\\mathcal{M}}_p(\\Delta_d)\\subseteq \\mathcal{M}_p(\\Delta_d)$ be the set of non-degenerate $f$ of shape \\ref{2.1} with $\\Delta(f)=\\Delta_d$.\n Let\n \\[\\widetilde{GNP}(\\Delta_d,p): = \\inf_{f \\in \\widetilde{\\mathcal{M}}_p(\\Delta_d)} NP(f).\\]\nNote that  $\\widetilde{GNP}({\\Delta_d},p) \\geq {GNP}({\\Delta_d},p)\\geq HP(\\Delta_d)$.\n\n  By Wan's facial decomposition theorem,  Le's conjecture is then equivalent to\n  \\[\\widetilde{GNP}({\\Delta_d},p) = HP(\\Delta_d) \\Longleftrightarrow p \\equiv 1 \\md{D}.\\]\n  Applying the regular decomposition theorem, Le \\cite{LE12} proved that\n\\begin{thm}\nIf $p \\not\\equiv 1 \\md{D}$ then $GNP(\\Delta_d, p)$ lies strictly above $HP(\\Delta_d)$.\n\\label{prop_nonordinary}\nIn particular, for any odd $d$, $p \\equiv 1 \\md{D}$ if and only if  $\\widetilde{GNP}(\\Delta_d,p) = HP(\\Delta_d)$.\n\\end{thm}\nThis implies that A-S conjecture for the family of polynomials (\\ref{2.1}) holds for odd $d$.   Le then conjectured\nthat it also holds for even $d$. We now give a counterexample to this conjecture.\n\n\\begin{thm}\nLe's conjecture is false in general.\n\\end{thm}\n\n\\begin{proof}\n   Recall $\\widetilde{\\mathcal{M}}_p(\\Delta_d)\\subseteq \\mathcal{M}_p(\\Delta_d)$ be the set of non-degenerate $f$ of shape \\ref{2.1} with $\\Delta(f)=\\Delta_d$. It suffices to prove\n   $$\\widetilde{GNP}(\\Delta_d,p) \\ne HP(\\Delta_d).$$\n Let  $f$ be a polynomial in $\\widetilde{\\mathcal{M}}_p(\\Delta_d)$.\n If we define\n\\[f^{\\delta_d}=x_0 h(x_1, x_2, \\ldots, x_n)+1\/x_0, \\]\nand\n\\[f^{\\delta_d'}=x_0 h(x_1, x_2, \\ldots, x_n), \\]\nthen equivalently the facial decomposition theorem implies that\n     $f$ is generically ordinary if and only if both  $f^{\\delta_d}$ and $f^{\\delta_d'}$ are generically ordinary.\nSince Le has already showed that  $f^{\\delta_d'}$ is generically ordinary, his conjecture is then equivalent to the statement that the polynomial $f^{\\delta_d}$ is also generically ordinary.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{tikzpicture\n\\node (v5) at (0,3) {$\\bullet$};\n \\node (v3) at (1.5,0) {$\\bullet$};\n \\node(v0) at (3,3) {$\\bullet$};\n  \\node(v1) at (0.6,3) {$\\bullet$};\n  \\node(v2) at (0.75,1.5) {$\\bullet$};\n  \\node(v4) at (2.25,1.5) {$ \\small\\circ$};\n\\draw[] (0,3) -- (1.5, 0) -- (3,3)--(0,3);\n  \\draw (v5)+(-0.8,0.3) node {$(1, 0, d)$};\n \\draw (v0)+(0.8,0.3) node {$(1,d,0)$};\n    \\draw (v2)+(-0.9,0) node {$(0, 0, \\frac d2)$};\n  \\draw (v3)+(0.3,-0.4) node {$(-1,0,0)$};\n    \\draw (v1)+(0.8,0.3) node {$(1,1,d-1)$};\n   \\draw (v4)+(0.9,0) node {$(0,\\frac d2,0)$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{ A face $\\tau\\subseteq\\delta_d$  for  $n=2$}\n\\label{fig7}\n\\end{figure}\n\n\n\n\nSuppose now $n=2$, $4\\mid d$ and  $p\\equiv \\frac d2+1\\md{d}$.\nLet $\\tau$ be a 1-dimensional face spanned by $(1, d, 0)$ and $(-1, 0, 0)$ on the two dimensional face $\\delta_d$ (see Figure 4 for the example).\nIts matrix $M$ is given by\n\\[M=\\begin{bmatrix}\n-1 & 1 \\\\\n0 & d\n \\end{bmatrix}.\\]\n Clearly $\\det M=-d$.\n\nLet $r=(r_1, r_2)$.\nSolving the system of linear equations \\[\nM\\left( \\begin{matrix} r_1 \\\\ r_2 \\end{matrix}\\right) \\equiv 0 \\md{1}, r_i \\in \\mathbb{Q}\\cap[0, 1),\\]\n we have\n \\[r_1=r_2=\\frac kd (0\\leq k\\leq d-1).\\]\n This gives  $w(r)=\\frac {2k}d$ and  hence $r$ is stable for even $k$.\n\n However, since  $p\\equiv \\frac d2+1\\md{d}$, $\\{pr_1\\}=\\{pr_2\\}=\n \\{\\frac {(1+d\/2)k}{d}\\}$, for odd $k$ and $k<\\frac d2$ we have\n $\\{pr_1\\}=\\{pr_2\\}= \\frac kd+\\frac 12$.  This shows that $w(r)$ is not stable for odd $k$ and $k<\\frac d2$ and hence $\\delta$ is not generically ordinary.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{tikzpicture}\n\\draw [fill=black](-0.7,-0.7) circle (0.05cm);\n\\draw [fill=black](-1.4,-1.4) circle (0.05cm);\n\\node (n1) at (-1.4,-1.4) {};\n\\node(v5) at (1.8,-0.7) {};\n\\draw [fill=black](1.8,-0.7) circle (0.05cm);\n\\node (v4) at (0,5) {};\n\\draw [fill=black](0,5) circle (0.05cm);\n \\node(v3) at (1.25,3.75) {};\n \\draw [fill=black](1.25,3.75) circle (0.05cm);\n\\node(v2) at (2.5,2.5) {};\n\\draw [fill=black](2.5,2.5) circle (0.05cm);\n\\node(v1) at (3.75,1.25) {};\n\\draw [fill=black](3.75,1.25) circle (0.05cm);\n\\node(v0) at (5,0) {};\n\\draw [fill=black](5,0) circle (0.05cm);\n\\draw [thick, dotted](-0.7,-0.7)--(2.5,2.5);\n\\draw (-1.4,-1.4)--(2.5,2.5);\n\\draw [thick, dotted](-0.7,-0.7)--(1.25,3.75);\n\\draw (-1.4,-1.4)--(1.25,3.75);\n\\draw [thick, dotted](-0.7,-0.7)--(3.75,1.25);\n\\draw (-1.4,-1.4)--(3.75,1.25);\n\\draw (-1.4,-1.4) -- (0,5) -- (5,0) -- (-1.4,-1.4);\n\\draw[thick,dotted] (0,5) -- (-0.7,-0.7) -- (5,0);\n\\draw[thick,dotted]  (-1.4,-1.4) -- (-0.7,-0.7) ;\n \\draw (n1)+(0,-0.4) node {$(-1,0,0)$};\n\\draw (v4)+(0.8,0.2) node {$(1,0,4)$};\n\\draw (v3)+(0.8,0.2) node {$(1,1,3)$};\n \\draw (v2)+(0.8,0.2) node {$(1,2,2)$};\n\\draw (v1)+(0.8,0.2) node {$(1,3,1)$};\n\\draw (v0)+(0.8,0.2) node {$(1,4,0)$};\n\\draw[->,thin] (3.5,0.2) -- (4,-0.8) node[anchor=west] {$\\Pi_1$};\n\\draw (v5)+(0.8,0.2) node [anchor=north] {$(0,2,0)$};\n \\end{tikzpicture}\n\\end{center}\n\\caption{A boundary decomposition for the polytope $\\Delta_d$, $d=4, n=2$}\n\\label{fig7}\n\\end{figure}\nLet $\\Pi_1$ be  the sub polytope spanned by $(0, 0, 0), (-1, 0, 0), (1, d, 0)$ and $(1, d-1, 1)$ (please see the figure below for an illustration). Since its face $\\tau$ is not generically ordinary,  $\\Pi_1$ is also  not generically ordinary.\n By the boundary decomposition theorem (\\cite{WAN93}, Theorem 5.1, see also Figure 5 for the graph) $\\Delta_d$ is not generically ordinary. The proof is complete.\n\n\n\\end{proof}\n\n\n{\\bf Acknowledgement.}   The author wishes to thank Professor  Daqing Wan for his instructive suggestions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}