diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzndrw" "b/data_all_eng_slimpj/shuffled/split2/finalzzndrw" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzndrw" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nMulti Object fibre Spectroscopy (MOS) is a well-established technique for efficiently carrying out\nspectroscopy on a large number of targets in the field of view. The AAO's 2dF instrument\n\\citep{2002lct+} has facilitated spectroscopic studies of many thousands of objects in the 17 years\nsince it began operation --- e.g. 2dFGRS \\citep{2001cdm+}, 2QZ \\citep{2000bsc+}, WiggleZ\n\\citep{2010djb+}, GAMA \\citep{2011dhk+} and GALAH \\citep{2015dfb+}.\n\nDue to its single robot arm and the sequential nature of its operation, the field configuration time\nof 2dF increases linearly with the number of fibres. \nThis and other limitations (such as diversity of payload and non-planar focal planes) \nare resolved by the AAO's new Starbugs technology, which consists of one independently positionable\nrobot per science fibre. The initial phase of TAIPAN, the first instrument to make use of this\ntechnology, will have 150 science fibres and therefore consist of 150 Starbugs. This will allow the\nfield configuration process to be carried out in parallel, and thereby decrease the configuration\ntime from around 60~min for 2dF to the order of 5 minutes.\n\n\\section{The TAIPAN Instrument and Survey}\nA Starbugs positioner is currently being developed for the TAIPAN instrument \\citep{2014klb+} on the AAO's 1.2m,\n6\u00b0 FoV UK-Schmidt telescope, located at Siding Spring Observatory in New South Wales, Australia, and\nis scheduled to commence on-sky observations in early 2016.\n\nThe TAIPAN survey will obtain visible band spectra for $5\\times10^5$ Southern Sky galaxies (${\\sim}70\\%$\ncompleteness) at R=2200 in $140$, and the overlapping region $\\Omega_o$ is sufficiently large, then the corresponding saddle point problem \\eqref{weak_formula_local}-\\eqref{weak_coupling} is well-posed. However, for the $L^2$-norm coupling \n$(\\kappa_0,\\kappa_1)=(1,0)$, the well-posedness is unclear.\n\\item Since the linear patch test is only weakly imposed via \\eqref{weak_formula_Lagrange}, ghost forces and spurious effects cannot be completely removed in the overlapping region.\n\\item The energy equivalence of this formulation depends on the choice of the weight functions $\\alpha_1$ and $\\alpha_2$, the size of overlapping region ${\\Omega_o}$, and the choice of the Lagrange multiplier space \\cite{Chamoin2010,Belytschko2007a}. In some cases, the Arlequin method is not equivalent neither to the local model nor the nonlocal model, even when homogeneous deformations are assumed. In fact, as pointed out in \\cite{Chamoin2010}, with inappropriate choices of $\\alpha_1$, ${\\Omega_o}$, and $(\\kappa_0,\\kappa_1)$, the Arlequin formulation may be not coercive.\n\\end{itemize}\n\n\\paragraph{The time-dependent problem.}\nThe Arlequin method has been used in many time-dependent applications, and the extension is straightforward, see, e.g., \\cite{Bauman2008,Winker2013,ArlequinWang2019}. \nIn particular, applications to dynamic LtN coupling mechanical problems appear in \\cite{ArlequinWang2019}. \nHowever, such extension consists in solving the saddle point problem \\eqref{weak_formula_local}-\\eqref{weak_formula_Lagrange} at every time step, \nwhich limits its applicability due to high computational cost.\n\n\\subsubsection{Applications and results}\nIn this section, as done in others below, we provide references of applications of the Arlequin method as the reproduction of numerical results is non-trivial.\n\nIn \\cite{HanLubineau2012}, the authors first apply the $H^1$-norm coupling \nand piece-wise linear weight functions to study a two-dimensional cantilever beam \nof isotorpic homogenous material. \nTheir results show that the accuracy of Arlequin solutions is comparable to that of the fully nonlocal elasticity model.\nNext, they test a static cracked square plate using various options of $(\\kappa_0, \\kappa_1)$. When an $H^1$-norm coupling \nwith piece-wise linear weight functions is used, the strain distribution from the Arlequin approach agrees with the strain field, especially near the crack-tip, computed with the fully nonlocal model. \n\nIn \\cite{Prudhomme2008}, the authors use the Arlequin method to investigate a one-dimensional problem that consists of a collection of springs that exhibit a localized defect, resulting in a sudden change in the spring properties. They test the method and study its accuracy for several choices of coupling parameters. They prove that the Arlequin formulation is well-posed with both $H^1$-seminorm and $H^1$-norm couplings. Their numerical results also indicate that the method is sensitive to the location and size of the overlapping region, and they propose to utilize adaptive strategies, based on a posteriori error estimates, to identify them.\n\n\\subsection{Morphing method}\\label{subsec:morphing}\n\nThe morphing method for LtN coupling was developed in \\cite{azdoud2013morphing,azdoud2014morphing,han2016morphing,lubineau2012morphing} based on a blending approach to morph the material properties of local and nonlocal sub-domains.\nThe coupling formulation consists of a single unified model obtained by a transition (morphing) from local to nonlocal descriptions. More specifically, a hybrid model is introduced in the transition region or morphing zone, $\\Omega_t$ (see Figure~\\ref{fig:Omegab-domain}), whose constitutive law changes gradually from a local to a nonlocal response. As a result, a single model with evolving material properties is defined on the whole domain and the equivalence of the energy of the system with the fully nonlocal energy is enforced under homogeneous deformations in the morphing zone \\cite{lubineau2012morphing}.\n\n\\subsubsection{Mathematical formulation}\nWe describe the morphing method for the coupling of the linear isotropic bond-based peridynamic model~\\eqref{eq: bond-based-PD-force-state-linear} and the corresponding classical linear elasticity model \\eqref{eq: CE-Navier-Cauchy}, following \\cite{lubineau2012morphing}. \nWe note, however, that this technique has also been applied to more complex material models, including linear anisotropic bond-based peridynamic models \\cite{azdoud2013morphing} and a state-based peridynamic model \\cite{han2016morphing}.\n\nThe strain energy density \nof the linear bond-based peridynamic model is given by\n\\begin{equation}\\label{morphing_nonlocal}\n\\begin{split}\n W^{nl}(\\mathbf{x})=&\\dfrac{1}{4}\\int_{B_\\delta(\\mathbf{0})}\\lambda(\\|{\\boldsymbol\\xi}\\|) (\\mathbf{u}(\\mathbf{x} + {\\boldsymbol\\xi}) - \\mathbf{u}(\\mathbf{x}))^T ({\\boldsymbol\\xi}\\otimes {\\boldsymbol\\xi} ) (\\mathbf{u}(\\mathbf{x} + {\\boldsymbol\\xi}) - \\mathbf{u}(\\mathbf{x})) d{\\boldsymbol\\xi},\n\\end{split} \n\\end{equation}\nwhereas the corresponding local strain energy density \nis given by\n\\begin{equation}\\label{morphing_local}\n\\begin{split}\nW^l(\\mathbf{x})=&\n{\\frac{4E}{5}\\big\\{\n (\\nabla\\cdot \\mathbf{u}(\\mathbf{x}))\\,(\\nabla\\cdot \\mathbf{u}(\\mathbf{x}))}\\\\\n & {\\qquad\n +\\frac{1}{2}\\left(\\nabla \\mathbf{u}(\\mathbf{x})+\\nabla \\mathbf{u}^T(\\mathbf{x})\\right):\n \\left(\\nabla \\mathbf{u}(\\mathbf{x})+\\nabla \\mathbf{u}^T(\\mathbf{x})\\right)\\big\\} }\\\\\n=&\\dfrac{1}{2}{\\boldsymbol \\varepsilon}(\\mathbf{x}):\\mathcal{C}^l:{\\boldsymbol \\varepsilon}(\\mathbf{x}),\n\\end{split}\n\\end{equation}\nwhere $\\mathcal{C}^l$ is the \nfourth-order elasticity tensor and ${\\boldsymbol \\varepsilon} = \\dfrac{1}{2}(\\nabla \\mathbf{u}+\\nabla\\mathbf{u} ^T)$ is the infinitesimal strain tensor. When considering an infinitesimal homogeneous deformation, we can define a local stiffness tensor $\\mathcal{C}^0$ from the nonlocal model \\eqref{morphing_nonlocal} such that the strain energy density of the resultant local model is equal to the strain energy density of the nonlocal model, i.e.,\n$$\nW^{nl}(\\mathbf{x})\\approx \\dfrac{1}{2}\\epsilon(\\mathbf{x}):\\mathcal{C}^0:\\epsilon(\\mathbf{x}),$$\nwhere $\\mathcal{C}^0$ is given by \\eqref{eq: fourth-order elasticity tensor bond-based}.\nTo define the morphing model, due to consistency requirements on the energy densities, we assume $\\mathcal{C}^0 = \\mathcal{C}^{l}$. \n\nGiven a blending or morphing function $\\beta$, such as the one introduced in \\eqref{general_def_blend_fnc} and illustrated in Figure~\\ref{fig: blending function2}(b), \nthe morphing model is fully defined by the following strain energy density:\n\\begin{align}\n\\nonumber &W^m(\\mathbf{x})=\\dfrac{1}{2}\\epsilon(\\mathbf{x}):\\mathcal{C}(\\mathbf{x}):\\epsilon(\\mathbf{x})\\\\\n\\nonumber &+\\dfrac{1}{4}\\int_{B_\\delta(\\mathbf{0})}\\lambda(\\|{\\boldsymbol\\xi}\\|)\\dfrac{\\beta(\\mathbf{x} + {\\boldsymbol\\xi})+\\beta(\\mathbf{x})}{2} (\\mathbf{u}(\\mathbf{x} + {\\boldsymbol\\xi}) - \\mathbf{u}(\\mathbf{x}))^T ({\\boldsymbol\\xi}\\otimes {\\boldsymbol\\xi} ) (\\mathbf{u}(\\mathbf{x} + {\\boldsymbol\\xi}) - \\mathbf{u}(\\mathbf{x})) d{\\boldsymbol\\xi}, \n\\end{align}\nwhere\n$$\n\\mathcal{C}(\\mathbf{x}):=(1-\\beta(\\mathbf{x}))\\mathcal{C}^l+\\int_{B_\\delta(\\mathbf{0})}\\lambda(\\|{\\boldsymbol\\xi}\\|)\\dfrac{\\beta(\\mathbf{x})-\\beta(\\mathbf{x} + {\\boldsymbol\\xi})}{4} {\\boldsymbol\\xi}\\otimes{\\boldsymbol\\xi}\\otimes{\\boldsymbol\\xi}\\otimes{\\boldsymbol\\xi} d{\\boldsymbol\\xi}.\n$$\nNote that, similarly to \\eqref{eq: fourth-order elasticity tensor bond-based}, $\\mathcal{C}(\\mathbf{x})$ is a fully-symmetric fourth-order tensor. \n\n\\paragraph{Properties.} The morphing method has the following properties:\n\\begin{itemize}\n\n\\item For $\\mathbf{x}\\in {\\Omega_l}$, $\\mathcal{C}(\\mathbf{x})=\\mathcal{C}^l$ and $W^m(\\mathbf{x})=W^l(\\mathbf{x})$.\n\n\\item For $\\mathbf{x}\\in {\\Omega_{nl}}$, $\\mathcal{C}(\\mathbf{x})=0$ and $W^m(\\mathbf{x})=W^{nl}(\\mathbf{x})$.\n\n\\item This method does not pass the {linear patch test}. In fact, $u^{\\rm lin}$ (see Definition \\ref{def:patch-test}) does not satisfy the equilibrium equation throughout the morphing zone and a nonzero ghost force density arises \\cite{lubineau2012morphing}. However, these ghost forces can be approximately corrected using exactly the same deadload correction approach used in AtC coupling methods \\cite{Shenoy1999b}. Besides, the ghost force intensity decreases when using smoother morphing functions $\\beta$ or sufficiently large morphing zones. Moreover, ghost forces are localized to the morphing zone and vanish when $\\delta\\rightarrow 0$. \n\\item For homogeneous deformations, the {strain energy density} is equivalent to {both strain energy densities of the local and nonlocal models}. Therefore, this method is considered { energy preserving} under homogeneous deformations \\cite{lubineau2012morphing}.\n\\item Even though there are a few theoretic studies regarding the morphing method for LtN coupling, this method has been studied as a type of blending for AtC coupling \\cite{li2012positive}. The corresponding operator is coercive with respect to the nonlocal energy norm with smooth morphing function $\\beta$ and sufficiently large morphing zone ${\\Omega}_t$ \\cite{li2012positive}.\n\\end{itemize}\n\\paragraph{The time-dependent problem.} \n\nThe extension of the morphing method to time-dependent problems is straightforward, even though the implementation of the model has been only demonstrated in static\/quasi-static problems. \n\n\n\n\n\\subsubsection{Applications and results}\nIn this section, we provide references to applications of the morphing method as the reproduction of numerical results is non-trivial. In \\cite{lubineau2012morphing} this strategy is applied to couple linear\n bond-based peridynamics with classical linear elasticity for isotropic materials. \nThe authors investigate the ghost force intensity when using different \nmorphing \nfunctions. They perform a one-dimensional analysis followed by numerical studies in one and two dimensions. The results suggest that the ghost forces are localized to the morphing zone and a smoother morphing function $\\beta$ reduces the maximum relative ghost forces \\cite{lubineau2012morphing}. Follow-on two-dimensional simulations for a cracked plate under both traction and shear \ndemonstrate the effectiveness of the method, compared to a fully peridynamic simulation. \nLater on, in \\cite{azdoud2014morphing}, the morphing method is combined with an adaptive algorithm that updates the nonlocal sub-domain based on damage progression. The resulting coupling framework is applied to three-dimensional quasi-static problems. \nIn \\cite{han2016morphing}, the method is further extended to couple a linearized state-based peridynamic model and the corresponding classical linear elasticity model. For anisotropic materials, in \\cite{azdoud2013morphing}, the authors introduce anisotropic nonlocal models based on spherical harmonic descriptions, and present three-dimensional results.\n\n\\subsection{Quasi-nonlocal method}\\label{subsec:quasinonlocal}\nThe quasi-nonlocal (QNL) method is an energy-based coupling approach introduced in the context of AtC coupling. \nThis method redefines the nonlocal energy via a ``geometric reconstruction'' scheme in the transition region and local sub-domain of a LtN coupling configuration, \nin such a way that the method is {linearly} patch-test consistent \\cite{Luskin2013a,Shimokawa:2004,E:2006}. We point out that the idea of ``geometric reconstruction'' is not limited to AtC coupling {of solids;} \nsimilar coupling strategies in the literature have been applied, for example in computational fluid dynamics (see, e.g., the review papers \\cite{AidunClausen2010,LiLiu2002}).\nHere, we focus on the QNL method for \nLtN coupling of one-dimensional diffusion models, following \\cite{DuLiLuTian2018,XHLiLu2017}.\n\n\\subsubsection{Mathematical formulation}\nWe refer to Figure \\ref{fig:blended-domains} (top): without loss of generality, consider the domain $\\wideparen\\Omega=[-1-\\delta,1]$. %\nThe domain is decomposed into four \ndisjoint sub-domains:\n{$\\wideparen\\Omega={\\Omega_p}\\cup{\\Omega}_{nl}\\cup\\Omega_t\\cup{\\Omega_l}$}, which include the left physical nonlocal boundary ${\\Omega_p}=(-1-\\delta, -1)$, the nonlocal sub-domain ${\\Omega_{nl}}=(-1, \\;x^*)$, the transition region ${\\Omega_t}=(x^*,\\; x^*+\\delta)$, and the local sub-domain ${\\Omega_l}=(x^*+\\delta, \\;1)$. \nThe right physical local boundary is $\\Gamma_p=\\{1\\}$. \nNote that the interface between the nonlocal sub-domain and the transition region occurs at $x^*$ satisfying\n$x^*\\in(-1+2\\delta, 1-2\\delta)$, \nand the transition region, ${\\Omega}_t$, has thickness $\\delta$. \n\n\nThe crucial step in the QNL formulation is the ``geometric reconstruction'' \\cite{Shapeev2012a,Shimokawa:2004,E:2006} of the directional distance $u(\\mathbf{x}')-u(\\mathbf{x})$ in the definition of the energy. Because of difficulties arising from reconstructing geometries with corners in high dimensions, we limit the discussion to the one-dimensional case.\n\nRecall that the one-dimensional nonlocal diffusion energy density associated with the bond $\\xi = x'-x$ is \n\\begin{equation}\\label{eq:nl-energy-density}\n\\frac{1}{2}\\gamma_\\delta(x'-x)\\left(u(x')-u(x)\\right)^2,\n\\end{equation}\nwhere we assume a radially symmetric nonlocal diffusion kernel, i.e., $\\gamma_{\\delta}(\\xi) = \\gamma_{\\delta}(|\\xi|)$. \nIn the QNL model, such bond energy density is modified when the \nbond \nis entirely located in the local sub-domain. Specifically, the nonlocal energy is redefined by substituting the directional distance $\\left(u(x')-u(x)\\right)$ with a path integral, such that the local energy density, $\\frac{1}{2}|u'(x)|^2$, is equivalent to the nonlocal one for sufficiently smooth~$u$. Thus, we have that \\eqref{eq:nl-energy-density} is replaced by\n\\begin{equation}\\label{eq:geom_reconstruct}\n{\\frac{1}{2}}\\gamma_{\\delta}(x'-x)\\int_{0}^{1}\\left|u'\\big(x+t(x'-x)\\big)\\right|^2|x'-x|^2dt.\n\\end{equation}\nThe combined total energy of the QNL model with interface at $x^*$ is\n\\begin{align}\\label{eq:qnl_energy}\nE^{\\rm{QNL}}_\\delta(u)\n=& \\frac{1}{4} \n\\iint\\limits_{x\\leqslant {x^*} \\text{ or }x'\\leqslant {x^*}\n} \\gamma_{\\delta}(\\abs{x'-x})\\left({u(x')}-u(x)\\right)^2\\, dx'dx \\\\\n& +\\frac{1}{4}\n\\iint\\limits_{x> {x^*}\\text{ and }x'> {x^*}\n} \\gamma_{\\delta}(\\abs{x'-x} )\\int_{0}^1 {\\abs{u'(x+t(x'-x))}}^2 |x'-x|^2 dt \\, dx'dx.\\nonumber\n\\end{align}\nThe definition of the QNL coupling operator $\\mathcal{L}^{\\rm{QNL}}$ is obtained by taking the negative first variation of the total energy \\eqref{eq:qnl_energy}. We split it into three parts \\cite{DuLiLuTian2018}:\n\\begin{itemize}\n\\item[] {\\bf I.} \nFor $x\\in {\\Omega_{nl}}$: \n \\begin{align}\\label{force_case1}\n \\mathcal{L}^{\\rm{QNL}}u(x)\n =& \\int_{x\n \\in \\wideparen\\Omega\n } \\gamma_{\\delta} (\\abs{x'-x})\\left(u(x')-u(x)\\right) dx'.\n \\end{align}\n\\item[] {\\bf II.} \nFor $x\\in{\\Omega_t}$: \n \\begin{align}\\label{force_case2}\n \\mathcal{L}^{\\rm{QNL}}u(x) =& \\int_{{x', \n\\end{equation}\nwhere $n$ is the spatial dimension, $\\widehat{\\underline{\\bf T}}_1$ is a reference material model, $\\underline{\\bf Y}$ is the deformation state, and $\\underline{\\bf Y}_1$ is the reference deformation state. The hat notation is used to explicitly indicate dependence on the deformation. \n\\end{remark}\n\n\n\nWith the localization of the nonlocal interactions at the boundary $\\Gamma$, it is shown in \\cite{TD17trace} that the nonlocal energy space $\\mathcal{S}^{\\rm{ND}}(\\wideparen\\Omega_{nl})$ has $H^{1\/2}(\\Gamma)$ as the trace space on $\\Gamma$, which is exactly the trace space of $H^1$ functions. As a result of the trace theorem, we can define the combined energy space\n\\[\n\\mathcal{W}(\\wideparen\\Omega)=\\{ u\\in \\mathcal{S}(\\wideparen\\Omega_{nl})\\cap H^1(\\Omega_l)\\, :\\, u_-=u_+ \\text{ on } \\Gamma, u|_{\\mathcal{B}\\Omega} =0 \\}\\,,\n\\]\nwhere $u_-(\\mathbf{x})$ and $u_+(\\mathbf{x})$ are defined as $\\lim_{\\mathbf{y}\\to \\mathbf{x}, \\mathbf{y}\\in\\Omega_{nl}}u(\\mathbf{y})$ and $\\lim_{\\mathbf{y}\\to \\mathbf{x}, \\mathbf{y}\\in\\Omega_{l}}u(\\mathbf{y})$, respectively. The total energy is a combination of the nonlocal and local parts given by\n\\[\n\\begin{split}\nE(u, f)=\\frac{1}{4}\\int_{\\wideparen\\Omega_{nl}}\\int_{\\wideparen\\Omega_{nl}} \\gamma(\\mathbf{x},\\mathbf{x}') ((\\mathcal{D}^\\ast u)(\\mathbf{x},\\mathbf{x}'))^2 d\\mathbf{x} d\\mathbf{x}' & +\\frac{1}{2} \\int_{\\Omega_l}|\\nabla u(\\mathbf{x})|^2d\\mathbf{x} \\\\\n&- \\int_{\\Omega} f(\\mathbf{x})u(\\mathbf{x}) d\\mathbf{x} \\,,\n\\end{split}\n\\]\nfor any $u\\in\\mathcal{W}(\\wideparen\\Omega)$. The well-posedness of the coupled problem is guaranteed by the extension of the nonlocal Poincar\\'e inequality to nonlocal space with variable-horization function $\\gamma$ given by \\eqref{eq:shrink-kernel}. \n\n\\paragraph{Properties}\n\\begin{itemize}\n\\item The shrinking horizon method is mathematically well-posed and energy stable for diffusion problems on general domains in all dimensions. \n\\item There is no overlapping region \n{between nonlocal and local sub-domains}.\nMoreover, since the nonlocal and local energy functionals have the same trace space on the interface, one can use all the {classical} non-overlapping domain-decomposition methods for solving the coupled problem. \n\\item\nThe method does not pass the patch-tests. However, ghost forces can be controlled by using a slowly varying horizon function. \n\\item\nThe order of convergence of the coupled problem to the local problem as $\\delta\\to0$ depends on the choice of the horizon function. For a piece-wise linear horizon function, the solution convergences at a rate of $O(\\delta)$. With a slowly varying horizon function, the optimal order $O(\\delta^2)$ could be achieved. \n\\item\nThe total energy is equivalent to either fully nonlocal or fully local energy up to linear functions. \n\\end{itemize}\n\n\\subsubsection{{Applications and results}} \\label{subsubsec:shrink_num}\nThe shrinking horizon approach produces well-posed coupled models for a general horizon function $\\delta(\\mathbf{x})$ that gets localized at $\\Gamma$, as shown in \\cite{TTD19,tian_thesis}. \nHowever, the particular choices of the horizon function affect the convergence rate of the solutions as $\\delta(=\\max_{\\mathbf{x}} \\delta(\\mathbf{x})) \\to 0$. In \\cite{TTD19},\none-dimensional experiments are performed to illustrate that the solutions converge only at the order $O(\\delta)$ when using a piece-wise linear horizon function, as the one given by \\eqref{deltax:linear}. The authors provide two remedies for increasing the order of the convergence. The first approach is to use a specific auxiliary function, and the second is to use a smooth and slowly varying horizon function.\nHere we only discuss the second approach; this approach was also discussed in \\cite{silling2015variable} as a means to reduce ghost forces. \n\\begin{figure}[H]\n\\begin{center}\n\\subfigure[Horizon functions]{\n\\includegraphics[width =5.5cm]{.\/horizon_function.pdf}}\n\\qquad\n\\subfigure[Ghost forces]{\n\\includegraphics[width= 5.5cm]{.\/ghost_force_shrink.pdf}}\n\\vspace{-0.14 in}\n\\caption{Left: examples of $\\delta(x)$ in one dimension. The orange solid line represents a piece-wise linear horizon function and the blue dashed line represents a $C^2$ horizon function. Right: the ghost forces under linear displacement using the piece-wise linear horizon function and the $C^2$ horizon function given by the left plot. }\n\\label{fig:horizon}\n\\end{center}\n\\end{figure}\n\\begin{figure}[H]\n\\begin{center}\n\\subfigure[Linear Patch Test]{\n\\includegraphics[width =5.5cm]{.\/linear_patch_test_shrink.pdf}}\n\\qquad\n\\subfigure[Quadratic Patch Test]{\n\\includegraphics[width= 5.5cm]{.\/quad_patch_test_shrink.pdf}}\n\\vspace{-0.14 in}\n\\caption{Patch tests for the shrinking horizon method with $C^2$ horizon function $\\delta(x)$: linear (left) and quadratic (right). \nNodes in the nonlocal sub-domain are represented by blue filled circles, whereas nodes in the left nonlocal boundary are empty blue circles. Nodes in the local sub-domain are represented by red filled squares, whereas the node in the right local boundary is an empty red square.\n}\n\\label{fig:patch_tests_shrink}\n\\end{center}\n\\end{figure}\nThe left plot of Figure \\ref{fig:horizon} shows two choices of the horizon function $\\delta(x)$ in one dimension, with one being a piece-wise linear function and another being a $C^2$ function. \nSince the model does not pass the linear patch test exactly, it generates ghost forces when the solution is a linear profile $u^{\\rm lin}$ (see Definition \\ref{def:patch-test}). The right plot of Figure \\ref{fig:horizon} shows the ghost forces over the domain $\\Omega=(-1,1)$ using each of the horizon functions. \nIt is clear that the piece-wise linear horizon function generates a large magnitude of ghost forces around the interface between $\\Omega_{nl}=(-1,0)$ and $\\Omega_l=(0,1)$, while the ghost forces are under control by using the $C^2$ horizon function. Moreover, it is observed in \\cite{TTD19} that the ghost forces converge to zero everywhere as $\\delta\\to0$ if the $C^2$ horizon function is used. \nFigure \\ref{fig:patch_tests_shrink} further shows the linear and quadratic patch tests by using the $C^2$ horizon function. In these tests, the largest horizon is $\\delta =0.1$ and the spatial mesh size is $\\Delta x=0.025$. \nThe FEM \nwith piece-wise linear basis functions is used for computing the numerical solution, since it is shown to be an asymptotically compatible scheme for nonlocal variational problems \\cite{Tian2014}. Although the coupled model does not pass the patch tests in theory, Figure \\ref{fig:patch_tests_shrink} shows that by using the $C^2$ horizon function, the patch-test consistency could be preserved approximately. \n\nFurthermore, numerical examples in \\cite{TTD19} show that by using the $C^2$ horizon function, the optimal order of convergence to the local limits could be achieved. Solutions of the coupled problems converge to the solutions of the local problems in the $L^2$ norm at the rate $O(\\delta^2)$, and the numerical derivative of the solutions converge at the rate $O(\\delta)$. In contrast, using the piece-wise linear horizon function, one can only observe first order convergence in the solutions. \n\n\\subsection{Partial stress method}\n\\label{subsec:partial-stress}\n\n\nAs discussed in Section~\\ref{subsec:shrink}, a shrinking horizon approach is not patch-test consistent, even though deviations from the patch test can be controlled by the regularity of the horizon function. In~\\cite{silling2015variable}, this consideration led to the development of an alternative strategy to spatially vary the horizon, referred to as the partial stress method.\n\nThe proposition of the partial stress approach, which relates to Figure~\\ref{fig:blended-domains}, for LtN coupling is to reformulate the operator~\\eqref{eq: PD operator bond-based bulk} in the transition region connecting local and nonlocal sub-domains, \nin a way that spatially varying the horizon in that region does not give rise to ghost forces under uniform deformations. \n\n\n\\subsubsection{Mathematical formulation}\nThis method introduces a new tensor-valued function called the {\\it partial stress} tensor, \n\\begin{equation}\\label{eq: partial stress tensor}\n{\\boldsymbol \\nu}^{\\rm ps}(\\mathbf{x}) := \\int_{B_\\delta({\\bf 0})} \\underline{\\rm \\bf T}[\\mathbf{x}]\\langle {\\boldsymbol\\xi}\\rangle \\otimes {\\boldsymbol\\xi} \\, d{\\boldsymbol\\xi},\n\\end{equation}\nwhere $\\underline{\\rm \\bf T}$ is the force vector state from \\eqref{eq: PD operator state-based}, \nand defines a corresponding {\\it partial internal force density},\n\\begin{equation}\\label{eq: PS operator bond-based bulk}\n\\mathcal{L}^{\\rm ps} \\mathbf{u} (\\mathbf{x}) := \\nabla \\cdot {\\boldsymbol \\nu}^{\\rm ps}(\\mathbf{x}).\n\\end{equation}\nIt is important to note that the partial stress tensor coincides with the collapse stress tensor, ${\\boldsymbol \\nu}^0$ ({\\it cf.}~\\eqref{eq: PD operator state-based limit}), \nin the case of a uniform deformation of a homogeneous body, which is characterized by a constant $\\delta$.\n\nWe now refer to Figure \\ref{fig:blended-domains}: the domain ${\\Omega}$ is decomposed into {three} disjoint sub-domains: {${\\Omega}={\\Omega}_{nl}\\cup\\Omega_t\\cup{\\Omega_l}$}, i.e., the nonlocal sub-domain, the transition region, and the local sub-domain. The partial stress coupled problem is given by \n\n\\begin{equation} \\label{eq: PS static problem}\n\\left\\{\n\\begin{aligned}\n- \\int_{ B_\\delta(\\mathbf{x})} \\left\\{\n\\underline{\\rm \\bf T}[\\mathbf{x}]\\langle \\mathbf{x}' - \\mathbf{x}\\rangle - \\underline{\\rm \\bf T}[\\mathbf{x}']\\langle \\mathbf{x} - \\mathbf{x}' \\rangle\n \\right\\}d\\mathbf{x}'\n= \\mathbf{b}(\\mathbf{x}) & \\quad\\mathbf{x}\\in{\\Omega_{nl}}, \\\\[2mm]\n-\\nabla \\cdot {\\boldsymbol \\nu}^{\\rm ps}(\\mathbf{x}) = \\mathbf{b}(\\mathbf{x}) & \\quad\\mathbf{x}\\in{\\Omega_t}, \\\\[2mm]\n-\\nabla \\cdot {\\boldsymbol \\nu}^0(\\mathbf{x})= \\mathbf{b}(\\mathbf{x}) & \\quad\\mathbf{x}\\in{\\Omega_l}.\n\\end{aligned}\\right.\n\\end{equation}\n\n\n\\paragraph{Properties.} Not many properties have been discussed for the partial stress method in the literature. Here, we simply summarize two properties from~\\cite{silling2015variable}: \n\\begin{itemize}\n\n\n\\item For a uniform deformation of a homogeneous body,\n\\begin{equation*}\n{\\boldsymbol \\nu}^{\\rm ps} \\equiv{\\boldsymbol \\nu}^0 \n\\qquad \\mbox{and} \\qquad\n\\nabla \\cdot {\\boldsymbol \\nu}^{\\rm ps} \\equiv\\nabla \\cdot {\\boldsymbol \\nu}^{0}\\equiv {\\bf 0}.\n\\end{equation*}\n\n\\item The partial stress tensor \\eqref{eq: partial stress tensor} and partial internal force density \\eqref{eq: PS operator bond-based bulk} converge, under suitable regularity assumptions, to the corresponding classical local counterparts, \nin the limit as $\\delta \\to 0$, as $O(\\delta)$, so the method is asymptotically compatible. \nConsequently, the partial stress tensor \\eqref{eq: partial stress tensor} and partial internal force density \\eqref{eq: PS operator bond-based bulk} are also compatible to the fully nonlocal counterparts with a $\\delta$-order difference.\n\n\\end{itemize}\n\n\n\n\n\\paragraph{The time-dependent problem.} The partial stress method was, in fact, only demonstrated in a dynamic setting in~\\cite{silling2015variable}, where in conjunction with the splice method was applied to the study of a spall initiated by the impact of two brittle elastic plates. \n\n\n\\subsubsection{{Applications and results}} \\label{subsubsec:partial_stress_num}\n\nAs described above, the partial stress method was only applied in~\\cite{silling2015variable} for a dynamic problem. We therefore omit the details here and refer the reader to that work. \n\n\\bigskip\n\n\\section{Conclusions}\\label{sec:conclusion}\nThis paper presents a review of the state-of-the-art of LtN coupling for nonlocal diffusion and nonlocal mechanics, specifically peridynamics, and provides a classification of different LtN coupling approaches (see Figure \\ref{fig:overview-chart}). Following a description of various coupling configurations and a highlight of desired properties of a general LtN coupling strategy, we report different LtN coupling methods from the literature. For each method, we briefly present its mathematical formulation and properties, and we discuss relevant applications and numerical results. \n\nWe observe that, while many features and challenges are shared by all methods, there exist some significant differences in their formulation and implementation. \nFor instance, we find that even though a LtN coupling configuration can generally be divided into a local sub-domain, a transition region, and a nonlocal sub-domain (as illustrated in Figure \\ref{fig:general-domains}), each coupling method treats the transition region in its own specific way. Some methods overlap local and nonlocal descriptions, some employ a hybrid representation, some reduce the transition region to a sharp interface, and some utilize a variable horizon with or without changing the nonlocal operator. \nThis variation in the treatment of the transition region has both analytical and numerical implications. \nFor instance, an overlapping approach is normally non-intrusive, whereas a hybrid technique is typically intrusive; the variable horizon method may or may not be intrusive, depending on the available nonlocal implementation. \nAnother important property largely emphasized in this review is the ability of a coupling method to pass the patch test. Some coupling methods do pass it exactly for up to certain polynomial order, whereas others only pass it approximately; a linear patch test is the most popular one. \nFinally, we recognize two major formulations for LtN coupling, energy-based and force-based. The former provides a natural setting to impose energy preservation; however, because such formulation requires energy minimization, it is more native to static problems and the extension to dynamics settings may not be practical. \nOn the other hand, while force-based approaches normally equally apply to static and dynamics problems, they not always carry a well-defined energy functional. \nIn Table \\ref{table:coupling_summary}, we outline the methods discussed in this review, indicating relevant references and sections, and summarize some of these properties. \n\n\n\n \n %\n \n\n\n\n\n\n\n\nWe conclude by stating that the goal of this review is not to provide a preferred way to perform LtN coupling, but rather to broaden the perspective of the reader that can use this review as a guide for selecting the most appropriate method based on the characteristics \nof the problem at hand, available discretization methods, and accesible data. \n\n\n\n\n\n\n\\begin{table}[H]\n\\centering\n\\small\n\\label{table:coupling_summary}\n\\begin{tabular}{cccccc}\n\\hline\n{\\bf Method} & \\shortstack{\\bf References} & \\shortstack{\\bf Section}& \\shortstack{\\bf Transition}& \\shortstack{\\bf Linear Patch Test}& \\shortstack{\\bf Formulation}\\\\\n\\hline\n\\shortstack{Optimization-based} &\n\\shortstack{\\cite{Bochev_14_INPROC,Delia2019,Bochev_16b_CAMWA}}\n&\\shortstack{ \\ref{subsec:OBM}} & \\shortstack{Overlap} &\n\\shortstack{Exact}& Force-based\\\\\n\\hline\n\\shortstack{Partitioned}\n&\\shortstack{\\cite{you2019coupling,yu2018partitioned}} &\\shortstack{ \\ref{subsec:Partition_Robin}} & \\shortstack{Overlap} &\n\\shortstack{Exact$^*$} &{Force-based}\\\\[-0.02in]\n& & & \\shortstack{or Sharp} & &\\\\\n\\hline\n{Arlequin}&\\shortstack{\\cite{HanLubineau2012,ArlequinWang2019}}&\\shortstack{ \\ref{subsec:Arlequin}} & Hybrid\n&\\shortstack{Approximate} & Energy-based\\\\\n\\hline\n{Morphing}&\\shortstack{\\cite{han2016morphing,lubineau2012morphing}} \n&\\shortstack{ \\ref{subsec:morphing}}\n& Hybrid\n& Approximate &{Force-based}\\\\\n\\hline\n\\shortstack{Quasi-nonlocal}&\\shortstack{\\cite{DuLiLuTian2018,XHLiLu2017}}&\\shortstack{ \\ref{subsec:quasinonlocal}} & Hybrid\n& {Exact}&{Energy-based}\\\\\n\\hline\n\\shortstack{Blending}&\\shortstack{\\cite{Seleson2013CMS,seleson2015concurrent}}& \\shortstack{\\ref{subsec:blending}} & Hybrid\n&{Exact}&{Force-based}\\\\\n\\hline\n\\shortstack{Splice}&\\shortstack{\\cite{silling2015variable,galvanetto2016effective}}& \\shortstack{\\ref{subsec:splice}} \n& Sharp\n&{Exact}&{Force-based}\\\\\n\\hline\n\\shortstack{Shrinking horizon}&\\shortstack{\n\\cite{silling2015variable,TTD19}}&\\shortstack{ \\ref{subsec:shrink}}\n& Variable horizon\n&{Approximate}&{Energy-based}\\\\\n\\hline\n\\shortstack{Partial stress}&\\shortstack{\n\\cite{silling2015variable}}& \\shortstack{\\ref{subsec:partial-stress}} & Variable horizon\n&{Exact}&{Force-based}\\\\[-0.02in]\n& & & \\shortstack{with partial stress} & &\\\\\n\\hline\n\\end{tabular}\n\\caption{Summary of LtN coupling methods. The $^*$ in the table indicates that the patch-test consistency for the partitioned procedure is exact up to certain conditions (see Section \\ref{subsec:Partition_Robin}). \n}\n\\end{table}\n\n\\normalsize\n\n\\bibliographystyle{spmpsci}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Braiding and homological stability for groups}\n\\label{sec:braiding}\nIn order to use the framework of Krannich~\\cite{krannich19}\nto prove homological stability for a sequence of groups,\none needs the structure of an ``$E_1$-module over an $E_2$-algebra''.\nWe give in Proposition~\\ref{prop:YB} below a simple way to construct such a module structure,\nin terms of Yang--Baxter operators. \nCompared to earlier approaches to homological stability such as~\\cite{RWW17},\nwhich Krannich's work generalizes,\nthis has the advantage of being very lightweight.\nInstead of having to provide the structure of a braiding on the monoidal category whose automorphism\ngroups one is interested in,\nit suffices to provide a single morphism satisfying a simple\nequation.\n\nOur main example of a Yang--Baxter operator is the\ninverse Dehn twist $T_1^{-1}\\in \\Aut_{{{\\mathbf M}_2}}(D\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D)$, defined in\nSection~\\ref{sec:braidedaction} and used to prove our main result.\nIn Section~\\ref{sec:notbraided}, we show that this Yang--Baxter operator is not part\nof a braided monoidal structure on the category ${{\\mathbf M}_2}$, but gives\ninstead a twisted version of such a structure.\n\n\n\\subsection{Yang--Baxter operators and braid groupoid actions}\\label{sec:YB}\nLet $\\mathcal X = (\\mathcal X,\\oplus ,\\mathbbm 1)$ be a monoidal category.\nA {\\em Yang--Baxter operator} in $\\mathcal X$\nis a pair $(X,\\tau )$ consisting of an object $X\\in\\mathcal X$\nand a morphism $\\tau\\in \\Aut_{\\mathcal C}(X\\oplus X)$,\nsatisfying the Yang--Baxter equation\n$$\n(\\tau\\oplus 1)(1\\oplus\\tau)(\\tau\\oplus 1)= (1\\oplus\\tau)(\\tau\\oplus 1)(1\\oplus\\tau ) \\in \\Aut_{\\mathcal C}(X\\oplus X\\oplus X),\n$$\nwhere we suppress associators from the notation.\n\nYang--Baxter operators are closely related to the braid groupoid: \nRecall from Section~\\ref{sec:braidedaction} the braid groupoid $\\braidGrpd$, with objects\nthe natural numbers and only non-trivial morphisms\n$\\Aut_\\braidGrpd(n)=B_n$. \nA variant of the coherence theorem for braided\nmonoidal categories says that the category of strong monoidal functors\nfrom the braid groupoid into $\\mathcal X$ is equivalent to a naturally\ndefined category of Yang--Baxter operators in $\\mathcal X$\n\\cite[Prop 2.2]{joyal-street}.\\footnote{\nIn other words, the pair consisting of the braid groupoid $\\braidGrpd$\nand the Yang--Baxter operator $\\sigma_1\\in\\text{Aut}_\\braidGrpd (2)$,\nis the initial monoidal category\nwith a distinguished Yang--Baxter element.}\nTo a Yang--Baxter operator $(X,\\tau )$ in $\\mathcal X$,\nthis equivalence associates\nthe strong monoidal functor $\\Phi_{X,\\tau}\\colon\\braidGrpd\\to\\mathcal X$ given by\n$\\Phi_{X,\\tau} (n) = X^{\\oplus n}$ on objects,\nand on morphisms by letting\n$$\n\\Phi_{X,\\tau}\\colon B_n\\to\\Aut_{\\mathcal X}(X^{\\oplus n})\n$$\nsend the $i$th standard generator $\\sigma_i$ to\n$\\text{id}_{X^{\\oplus i-1}}\\oplus\\tau\\oplus\\text{id}_{X^{\\oplus n-i-1}}$,\nwhere the required maps\n$\\Phi_{X,\\tau} (m)\\oplus \\Phi_{X,\\tau} (n)\\to \\Phi_{X,\\tau} (m+n)$\nare given by the monoidal structure of $\\mathcal X$.\n\n\\smallskip\n\nSuppose now that the monoidal category\n$\\mathcal X$ acts on a category $\\mathcal M$\nvia a functor $\\mathcal M\\times\\mathcal X\\to\\mathcal M$,\nwhich we also denote by $\\oplus$, compatible with the\nmonoidal sum in $\\mathcal X$.\nThe following result shows that the choice of a Yang--Baxter operator\ndefines an action of the braid groupoid $\\braidGrpd$ on $\\mathcal M$, and hence is\nappropriate data to apply the stability framework of \\cite{krannich19}: \n\n\n\n\\begin{proposition}\\label{prop:YB}\n Let $(\\mathcal X,\\oplus ,\\mathbbm 1)$ be a monoidal category\n with $\\tau\\in \\Aut_{\\mathcal X}(X\\oplus X)$\n a Yang--Baxter operator in $\\mathcal X$.\n Suppose $\\mathcal X$ acts on a category $\\mathcal M$.\n Then there is an action of the braid groupoid\n $$\n \\alpha_\\tau\\colon \\mathcal M\\times\\braidGrpd\\to\\mathcal M\n $$\n given on objects by $\\alpha_\\tau(A,n)= A\\oplus X^{\\oplus n}$\n and determined on morphisms by $$\\alpha_\\tau(f,\\sigma_i )=f\\oplus\\text{id}_{X^{\\oplus i-1}}\\oplus\\tau\\oplus\\text{id}_{X^{\\oplus n - i -1}},$$\n for $\\sigma_i$ the $i$th elementary braid in $B_n$. \n Furthermore, taking classifying spaces this endows $B\\mathcal M$\n with the structure of an $E_1$-module over the $E_2$-algebra $B\\braidGrpd$.\n\\end{proposition}\n\nNote that if we are interested in homological stability for\nstabilization by $X$ for the automorphism groups\n$G_n:=\\Aut_{\\mathcal M}(A\\oplus X^{\\oplus n})$\nfor some object $A$ of $\\mathcal M$,\nonly the full subcategory $\\mathcal M_{A,X}\\subseteq\\mathcal M$\n spanned by objects of the form $A\\oplus X^{\\oplus n}$, is relevant. \n So for stability purposes, it is enough to consider the\n subfunctor\n $$\n \\alpha_\\tau\\colon \\mathcal M_{A,X}\\times\\braidGrpd\\to\\mathcal M_{A,X}. \n $$\nIn fact, to make sure that the structure of $E_1$--module over the\n$E_2$--algebra $B\\braidGrpd$ is graded, one can even replace the category\n$\\mathcal M_{A,X}$ by a category with objects the natural\n numbers and setting $\\text{Aut}(n) = \\text{Aut}_{\\mathcal M}(A\\oplus\n X^{\\oplus n})$, avoiding any potential issue coming from unwanted\n equalities $A\\oplus X^{\\oplus n} = A\\oplus X^{\\oplus m}$ for $m\\neq\n n$.\n \n\\begin{proof}\nThe functor\n$\n\\alpha_\\tau\\colon \\mathcal M\\times\\braidGrpd\\to\\mathcal M\n$\nis defined as the composite functor $$\\alpha(-,-)= (-)\\oplus \\Phi_{X,\\tau} (-),$$ for\n$\\Phi_{X,\\tau}\\colon \\braidGrpd\\to \\mathcal X$ as above. \nThe result follows from \\cite[Lem 7.2]{krannich19} because $\\alpha$\nmakes $\\mathcal M$ into a module over $\\braidGrpd$ and $\\braidGrpd$ is braided monoidal.\n\\end{proof}\n\n\n\\begin{example}\\label{ex:braided}\nIf $\\mathcal{X}=(\\mathcal X,\\oplus ,\\mathbbm 1)$ admits a braiding\n$b$, then $\\tau=b_{X,X}\\in \\Aut_{\\mathcal X}(X\\oplus X)$\nis a Yang--Baxter operator for any object $X$. For\n$\\mathcal X$ a groupoid acting on itself\nor $\\mathcal X$ acting on a category $\\mathcal M$,\nthis recovers the basic set-up for homological stability\nof the paper \\cite{RWW17}, or Section 7 of \\cite{krannich19}.\n \\end{example}\n\n\n \\begin{example}[Mapping class groups of surfaces]\\label{exmp:mcg-YB}\n As explained above, a Yang--Baxter operator $\\tau\\in\n \\Aut_{\\mathcal X}(X\\oplus X)$\n gives in particular a collection of homomorphisms\n $\\Phi_{X,\\tau}\\colon B_n\\to\\Aut_{\\mathcal X}(X^{\\oplus n})$\n from the braid groups to the automorphism group of $n$ copies of $X$.\n There are two standard ways to embed braid groups in mapping class\n groups of surfaces, and we explain here how they both come from\n Yang--Baxter elements in appropriate categories of surfaces. \n \\begin{enumerate}\n \\item Let ${{\\mathbf M}_2}$ be the category of bidecorated surfaces of\n Section~\\ref{sec:category}.\n As explained in Section~\\ref{sec:braidedaction}, \n the Dehn twist\n $T\\in \\Aut_{{{\\mathbf M}_2}}(D\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\n D)\\cong\\pi_0\\operatorname{Homeo}_\\partial(S^1\\times I)\\cong \\mathbb{Z}$, or its\n inverse $T^{-1}$, \n is a Yang--Baxter operator.\n The associated map $\\Phi_{D,T}\\colon B_n\\to \\Aut_{{{\\mathbf M}_2}}(D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})$\n is the embedding of braid group in the mapping class\n groups of $S_{g,1}$ (when $n=2g+1$) and of $S_{g,2}$ (when\n $n=2g+2$) associated to Dehn twists along the chain of embedded\n curves in the surfaces described in Lemma~\\ref{lem:chain}. This\n embedding goes back at least to the work of Birman\n and Hilden \\cite{BH1,BH2}.\n \\item\n Let ${\\mathbf M}_1$ denote instead the category of surfaces decorated\n by a single interval, with monoidal structure $\\oplus$ defined just as in\n the case of $\\mathbf M_1$ but gluing only along one interval. Then $\\mathbf M_1$ in\n braided monoidal, see \\cite[Sec 5.6.1]{RWW17}. Hence by\n Example~\\ref{ex:braided}, for any\n object $X$ of $\\mathbf M_1$, we have a Yang--Baxter element $\\tau_X\\in\n \\Aut_{\\mathbf M_1}(X\\oplus X)$. For $X = S_{1,2}$,\n this can be used to prove genus stabilization (albeit with the suboptimal slope $\\rfrac 1 2$),\n and in the case $X=S^1\\times I$ marked\n by an interval in one of its boundary components, we have that\n $X^{\\oplus n}$ has underlying surface an $n$-legged pair of\n pants $D^2\\backslash (\\sqcup_n \\mathring{D}^2)$ and the associated morphism \n $$\n \\Phi_{X,\\tau_X}\\colon\n B_n\\to \\Aut_{\\mathbf M_1}(X^{\\oplus n})\n = \\pi_0\\operatorname{Homeo}_\\partial(D^2\\backslash (\\sqcup_n\\mathring{D}^2))\n $$ \nis the standard embedding of the braid group as the subgroup of the\nmapping class group of the multi-legged pants that does not twist the\nlegs, see e.g.~\\cite[Sec 5.6.1]{RWW17}.\n \\end{enumerate}\n We will show in Proposition~\\ref{prop:nobraiding} below that the\n Yang--Baxter operator $T$ of the first example, in\n the category ${{\\mathbf M}_2}$, does not come from a braiding in ${{\\mathbf M}_2}$. \n \\end{example}\n\n\n\n\n\n\\subsection{Homological stability from Yang--Baxter elements}\n\n \nSuppose we are given the data of a monoidal category\n$(\\mathcal X,\\oplus ,\\mathbbm 1)$ acting on a category $\\mathcal M$,\nalong with a choice of stabilizing object $X\\in\\mathcal X$\nand Yang--Baxter operator $\\tau\\in\\text{Aut}_{\\mathcal X}(X\\oplus X)$.\nProposition~\\ref{prop:YB} above allows to apply \\cite[Thm\nA]{krannich19}, which in this case says that for any\n$A\\in\\mathcal M$, there is a sequence of simplicial spaces $W_n(X,A)_\\bullet$, for $n\\ge\n0$, so that if $W_n(X,A)$ is highly-connected\nfor large $n$, then the sequence\n$$\n\\text{Aut}_{\\mathcal M}(A)\\xrightarrow{{-}\\oplus X}\n\\text{Aut}_{\\mathcal M}(A\\oplus X)\\xrightarrow{{-}\\oplus X}\n\\text{Aut}_{\\mathcal M}(A\\oplus X\\oplus X)\\xrightarrow{{-}\\oplus X}\\cdots\n$$\nsatisfies homological stability. Theorem B of the same paper gives in\naddition a stability statement with twisted coefficients.\nUnder an injectivity assumption of the form of\nProposition~\\ref{prop:Mmonoid}, this simplicial space is homotopy\ndiscrete, and modeled by the space of destabilizations\nas described in Definition~\\ref{def:WS}.\n\n\\begin{remark}\n The fact that $(X,\\tau )$ is a Yang--Baxter operator\n is precisely what is needed\n for the collection of sets $W_n(A,X)_p$ and maps\n $d_i\\colon W_n(A,X)_p\\to W_n(A,X)_{p-1}$,\n defined as in Definition \\ref{def:WS},\n to assemble into a semi-simplicial set;\n indeed, the Yang--Baxter equation implies the necessary\n simplicial identities.\n\\end{remark}\n\n\\smallskip\n\nFor a fixed monoidal category\n$\\mathcal X = (\\mathcal X,\\oplus ,\\mathbbm 1)$\nacting on a category $\\mathcal M$,\nand a stabilizing object $X\\in\\mathcal X$,\nthe choice of Yang--Baxter\nelement will not affect the stabilizing map, but it will affect the\nspaces $W_n(X,A)_\\bullet$. \nThe identity map $1\\in\\text{Aut}_{\\mathcal X}(X\\oplus X)$ is a trivial\nchoice of Yang--Baxter operator.\nBut, as is to be expected, this trivial twist is not useful for proving\nstability:\n\n\\begin{proposition}\n Let $\\mathcal X,\\mathcal M$, $A$ and $X$ be as above. If we choose\n the Yang--Baxter operator $\\tau\\in \\Aut_{\\mathcal X}(X\\oplus X)$\n to be the identity element,\n then the semi-simplicial set $W_n(A,X)_\\bullet$ is connected if and only if\n the map\n $$\n G_{n-1}=\\text{Aut}_{\\mathcal M}(A\\oplus X^{\\oplus n-1})\\xrightarrow{{-}\\oplus X}\n \\text{Aut}_{\\mathcal M}(A\\oplus X^{\\oplus n})=G_n\n $$\n is an isomorphism.\n\\end{proposition}\n\n\\begin{proof}\n If $\\tau$ is the identity element,\n all face maps $d_i$ are equal to the canonical map $G_n\/G_{n-p-1}\\to G_n\/G_{n-p}$.\n In particular, the vertices of any $p$-simplex are all equal,\n so the semi-simplicial set $W_n(A,X)_\\bullet$ is isomorphic to a disjoint union\n of semi-simplicial sets, one for each $0$-simplex. The result\n follows from the fact that the set of $0$-simplices is precisely\nthe quotient $G_n\/G_{n-1}$. \n\\end{proof}\nIn fact,\nBarucco proved in his master thesis a result that translates to the following\nstronger statement (stated in the thesis in the context of a groupoid acting on itself,\ni.e.~$\\mathcal M=\\mathcal X$):\n\n\\begin{lemma}\\cite[Lem 3.1]{Barucco}\nThe space $W_n(A,X)$ is connected if and only if $1^{\\oplus n-2}\\oplus\n\\tau$ and $G_{n-1}\\oplus 1$ together generate $G_n=\\Aut(A\\oplus\nX^{\\oplus n})$. \n\\end{lemma}\n\n\nThe connectivity of the semi-simplicial set $W_n(A,X)$ (or of the associated simplical complex defined in \\cite[Def 2.8]{RWW17}) can be thought of as a measure a form of {\\em higher generation} of the group $G_n$ by the cosets of the subgroups $G_{n-p}$ for $p\\ge 1$ and braid subgroups generated by the chosen Yang--Baxter element $t$,\nin a way similar to the notion of higher generation for a family of subgroups of a group defined in \\cite[2.1]{AbeHol}.\n\n\\subsection{Braidings and bidecorated surfaces}\\label{sec:notbraided}\nWe show in this section that the Yang--Baxter operator $T$ on the bidecorated\ndisk $D$ in the groupoid ${{\\mathbf M}_2}$ does not come from a braiding on the\nsubcategory of ${{\\mathbf M}_2}$ generated by the disk. In fact, we will show that\nthis subcategory does not admit a braiding. \n\n\n\\medskip\n\nLet $D=(D^2,1,\\operatorname{id})$ be the standard bidecorated disk of Section~\\ref{sec:category},\nwhere we recall that $X_1=D^2$.\nWe define a ``rotated'' bidecorated disk $\\overline D=(D^2,1,r_\\pi)$,\nwhere $r_\\pi$ is the rotation of $\\del X_1 = \\del D^2$ by $\\pi$ radians,\nwhich has the effect of interchanging the intervals $I_0$ and $I_1$.\nRotating all of $D^2$\nby $\\pi$ then induces a morphism $\\iota\\colon D\\to \\overline D$ in ${{\\mathbf M}_2}$,\nand likewise morphisms\n$$\n\\iota^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}\\colon D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}\\longrightarrow\n\\overline D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}\n$$\nfor every $m\\ge 1$, each which we will by abuse of notation also denote by $\\iota$.\nThe morphism $\\iota$ can be identified with the hyperelliptic involution\nof the underlying surface depicted in Figure~\\ref{fig:hyperelliptic-involution}\nfor the two cases $m = 2g$ and $m = 2g+1$,\nwhere in the latter case the boundary components\nare exchanged by $\\iota$.\n\\begin{figure}\n \\def0.8\\textwidth{0.8\\textwidth}\n \\input{hyperelliptic-involution.pdf_tex}\n \\caption{The hyperelliptic involutions $\\iota$ of $S_{g,1}$ and $S_{g,2}$}\\label{fig:hyperelliptic-involution}\n\\end{figure}\nThe morphism $\\iota$ induces an identification\n\\begin{align*}\n \\text{Aut}_{{{\\mathbf M}_2}}(D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m})\n &\\tox{\\cong} \\text{Aut}_{{{\\mathbf M}_2}}(\\overline D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m})\n \\\\\n f &\\mapstox{\\phantom{\\sim}} \\iota \\circ f \\circ \\iota^{-1}\n\\end{align*}\nIn order to precisely state the failure of $T$ to extend to a braiding,\nwe will also need the identification\n\\begin{align*}\n I\\colon \\text{Aut}_{{{\\mathbf M}_2}}(D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m})\n &\\tox{\\cong} \\text{Aut}_{{{\\mathbf M}_2}}(\\overline D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m})\\\\\n f &\\mapstox{\\phantom{\\sim}} f\n\\end{align*}\nthat comes from the fact that an element $f\\in\\text{Aut}_{{{\\mathbf M}_2}}(D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m})$\nis just a mapping class for the underlying surface of $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}$,\nwhich is the same as the underlying surface of $\\overline D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}$,\nso $f$ can just as well be viewed as an element of\n$\\text{Aut}_{{{\\mathbf M}_2}}(\\bar D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m})$.\nIn contrast with the identification induced by $\\iota$,\nthe second identification is ``external'',\nin the sense that it does not come from a morphism in ${{\\mathbf M}_2}$.\n\nViewing $\\iota$ as a diffeomorphism of the underlying surface $X_m$ of\n$D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}$ that does not fix the boundary,\nand specifically exchanges the marked points $b_0 = I_0(\\rfrac 1 2)$\nand $b_1 = I_1(\\rfrac 1 2)$, we see that it takes the isotopy class of\narc $\\rho_i$ of Section~\\ref{sec:disk}\nto the reversed arc $\\overline{\\rho_i}$. \nWe will use in the proof of the following result that the homotopy classes $\\rho_i$\ngenerate the fundamental groupoid of $X_m$ based at the points $b_0,b_1$.\\footnote{\n As a full subgroupoid of the ordinary fundamental groupoid of $X_m$,\n this groupoid is the one spanned by the objects corresponding to the\n points $b_0,b_1\\in X_m$.\n } The mapping class \n$\\iota$ is in fact\ncompletely determined by the fact that $\\iota(\\rho_i) = \\overline{\\rho_i}$.\n\n\\newpage\n\n\\begin{proposition}\\label{prop:nobraiding}\n Let $\\mathbf D\\subset{{\\mathbf M}_2}$ denote the full monoidal subcategory\n generated by $D$.\n \\begin{enumerate}[label=(\\roman*)]\n \\item The monoidal category $\\mathbf D$ does not admit a braiding.\n In particular, the monoidal functor $$\\Phi\\colon (\\braidGrpd,\\oplus)\\to\n (\\mathbf D,\\mathbin{\\text{\\normalfont \\texttt{\\#}}})\\subset ({{\\mathbf M}_2},\\mathbin{\\text{\\normalfont \\texttt{\\#}}})$$ does not come from a braiding on\n $\\mathbf D$.\n \\item Let $f\\in \\Aut_{{\\mathbf M}_2}(D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m})$ and $g\\in \\Aut_{{\\mathbf M}_2}(D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})$, and\n put $\\beta_{m,n}=\\Phi (b_{m,n})$, where the block braid $b_{m,n}$\n is the braid which passes the last $n$ strands over the first $m$ strands.\n Then \n $$\n \\beta_{m,n} \\circ (f\\mathbin{\\text{\\normalfont \\texttt{\\#}}} g) \\circ \\beta_{n,m}^{-1}\n =\n \\begin{cases}\n g \\mathbin{\\text{\\normalfont \\texttt{\\#}}} (\\iota^{-1}\\circ f \\circ \\iota)\n &\\text{if } n \\text{ is odd},\\\\\n g\\mathbin{\\text{\\normalfont \\texttt{\\#}}} f&\\text{else,}\n \\end{cases}\n $$\n for $\\iota\\colon D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}\\to \\overline D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}$ the involution defined\n above, and where $f$ in the rightmost expression is the map $f$ considered as\n an element of $\\Aut_{{\\mathbf M}_2}(\\overline D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m})$ via the isomorphism\n $I$ defined above.\n \\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nWe start by proving (ii). It is enough to check the statement when $f$\nand $g$ are Dehn twists, as those \ngenerate the mapping class groups. \nNote that if $c$ is a curve in the underlying surface $X_{m+n}$\nof $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m + n}$, and $T_c$ denotes the Dehn twist along $c$, then conjugating $T_c$ by a diffeomorphism $\\phi$ of the surface gives \n$$\\phi\\circ T_c\\circ \\phi^{-1}=T_{\\phi(c)}.$$\nRecall further that the isotopy class of a Dehn twist $T_c$ depends only on the free homotopy class of the curve $c$.\nWe are therefore to compute the images of curves in $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}$ and $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}$ under the map $\\beta_{m,n}$, as free homotopy classes. A curve $c$ can be written, up to free homotopy, as a concatenation of the arcs $\\rho_i$ and their inverses $\\overline\\rho_i$,\nas the homotopy classes of these arcs\ngenerate the fundamental groupoid of the surface $X_{m+n}$ based at $b_0,b_1$.\nIn particular, write\n \\begin{equation}\\label{equ:c}\n c \\simeq \\rho_{i_1}*\\overline\\rho_{i_2}*\\rho_{i_3}\\dots *\\overline\\rho_{i_k}.\n \\end{equation}\n The mapping class $\\beta_{m,n}$ can be written as the composition\n $$\n \\beta_{m,n}= (T_n\\circ \\dots \\circ T_{m+n-1})\\circ \\dots\\circ (T_2\\circ \\dots \\circ T_{m+1}) \\circ (T_1\\circ \\dots \\circ T_m)\n $$\n and hence we can compute the image of each $\\rho_i$ using Lemma~\\ref{lem:Trho}.\n For $r>0$, denote by $T_{i, i+r}$ the composition of Dehn twists $T_i\\circ T_{i+1}\\circ \\dots\\circ T_{i+r}$. \n Note first that\n $$T_{i, j}(\\rho_{j+1}) \\simeq T_{i, j-1}(\\rho_{j}) \\simeq \\dots \\simeq \\rho_i.$$\n From this, it follows that for $i\\ge 1$, \n \\begin{align*}\n \\beta_{m,n}(\\rho_{m+i})&\\simeq (T_{n, m+n-1})\\circ \\dots\\circ (T_{1, m})(\\rho_{m+i}) \\\\\n &\\simeq (T_{n, m+n-1})\\circ \\dots\\circ (T_{i, m+i-1})(\\rho_{m+i}) \\\\\n &\\simeq (T_{n, m+n-1})\\circ \\dots\\circ (T_{i+1, m+i})(\\rho_{i}) \\\\\n &\\simeq\\rho_i. \n \\end{align*}\n On the other hand, for $i\\le k\\le j$, we have\n $$T_{i, j}(\\rho_{k})\\simeq T_{i, k}(\\rho_{k})\\simeq T_{i, k-1}(\\rho_k*\\overline\\rho_{k+1}*\\rho_k)\\simeq \\rho_i*\\overline\\rho_{k+1}*\\rho_i,$$\n from which we can deduce that for $i\\le m$, \n \\begin{align*}\n \\beta_{m,n}(\\rho_i)&\\simeq (T_{n, m+n-1})\\circ \\dots\\circ (T_{1, m})(\\rho_{i}) \\\\\n &\\simeq (T_{n, m+n-1})\\circ \\dots\\circ (T_{2, m+1})(\\rho_1*\\overline\\rho_{i+1}*\\rho_1) \\\\\n &\\simeq (T_{n, m+n-1})\\circ \\dots\\circ (T_{3, m+2})(\\rho_1*\\overline\\rho_2*\\rho_{i+2}*\\overline\\rho_{2}*\\rho_1)\\\\\n & \\simeq \\cdots \\\\\n &\\simeq \\rho_1*\\iota(\\rho_2)*\\dots* \\iota^{n-1}(\\rho_{n})*\\iota^n(\\rho_{i+n})*\\iota^{n-1}(\\rho_{n})*\\dots*\\iota(\\rho_{2})*\\rho_1\n \\end{align*}\n since $\\iota^j(\\rho_i)$ is $\\rho_i$ when $j$ is even and $\\overline\\rho_i$ when $j$ is odd.\n\n If the curve $c$ lies in the last $n$ disks $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}$ inside $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m+n}$, it can be written as a product \\eqref{equ:c} with each $i_j>m$. Then the above computation gives that\n $$\n \\beta_{m,n}(c)\\simeq \\rho_{i_1-m}*\\overline\\rho_{i_2-m}*\\rho_{i_3-m}*\\dots *\\overline \\rho_{i_k-m},\n $$\n that is, $c$ is mapped to the corresponding curve in the {\\em first} $n$ disks $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}$ inside $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n+m}=D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m+n}$.\n\n If the curve $c$ instead lies in the first $m$ disks $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}$ inside $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m+n}$, it can be written as a product \\eqref{equ:c} with each $i_j\\le m$. Then the above computation gives that\n \\begin{align*}\n \\beta_{m,n}(c)&\\simeq \\rho_1*\\iota(\\rho_2)*\\dots* \\iota^{n-1}(\\rho_{n})*\\iota^n(\\rho_{i_1+n})*\\iota^{n+1}(\\rho_{i_2+n})*\\dots*\\iota^{n+1}(\\rho_{i_k+n})\\\\\n & \\hspace{6.66cm} *\\iota^{n}(\\rho_{n})*\\dots*\\iota^2(\\rho_{2})*\\iota(\\rho_1)\\\\\n &\\simeq \\iota^n(\\rho_{i_1+n})*\\iota^{n+1}(\\rho_{i_2+n})*\\iota^{n}(\\rho_{i_3+n})\\dots*\\iota^{n+1}(\\rho_{i_k+n}) \\\\\n &\\simeq \\iota^n ( \\rho_{i_1+n}*\\overline\\rho_{i_2+n}*\\rho_{i_3+n}\\dots *\\overline\\rho_{i_k+n})\n \\end{align*}\n Hence $c$ is mapped to the curve $\\iota^n(c)$ in the last $m$ disks $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}$ inside $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n+m}=D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m+n}$, from which the statement follows.\n\n\n \\smallskip\n\n We are left to prove (i). To see that the images $\\beta_{m,n}$ of\n block braids under $\\beta$ do not define a braiding in $\\mathbf D$,\n using (ii) it is enough to find a curve $c$ in $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}$ for some $m$\n so that $\\iota (c)\\not\\simeq c$, and such curves are plentiful. \n\n The same argument shows that the inverses $\\beta_{m,n}^{-1}$\n likewise do not define a braiding.\n\n Now suppose that $\\tilde \\beta$ is a braiding on $\\mathbf D$. The braiding is determined by\n $\\tilde\\beta_{1,1}\\in\\Aut_{{\\mathbf M}_2} (D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2})\\cong \\mathbb Z$, a group\n generated by the Dehn twist $T_1$.\n We have excluded the possibilities $\\tilde\\beta_{1,1} = T_1^{\\pm\n 1}$, and $\\tilde\\beta_{1,1} = \\operatorname{id}$ is similarly ruled out using\n now the fact that curves are not moved at all by the identity.\n So assume that $\\tilde\\beta_{1,1} = T_1^k$, with $|k|> 1$.\n Then $\\tilde\\beta_{2,1} = T_1^kT_2^k$ would have to satisfy\n $T_1^kT_2^k(a_1) = a_2$ in order for naturality to hold, where $T_i$\n is the Dehn twist along the curve $a_i$ as in Section~\\ref{sec:braidedaction}.\n Applying Proposition 3.2 in~\\cite{MCG-primer} twice, we get \n that the intersection number $i(a_2,T_2^k(a_1)) =\n i(T_1^k(T_2^k(a_1)),T_2^k(a_1)) = |k|i(a_1,T_2^k(a_1)^2 =\n |k|^2i(a_1,a_2)^4 = |k|^2$. On the other hand, using Proposition 3.4 in~\\cite{MCG-primer}\n we obtain \n $$\n |k|^2 = i(a_2,T_2^k(a_1)) = |i(T_2^k(a_1),a_2) - |k|i(a_1,a_1)i(a_1,a_2)|\\leq i(a_1,a_2) = 1,\n $$\n where we have also used that $i(a_1,a_1) = 0$. This contradicts our\n assumption of $\\tilde \\beta_{1,1}$. \n\\end{proof}\n\n\n\n\\section{The monoidal category of bidecorated surfaces}\n\\label{sec:category}\nIn this section, we describe a monoidal groupoid $({{\\mathbf M}_2} ,\\mathbin{\\text{\\normalfont \\texttt{\\#}}} )$ of surfaces\ndecorated by two intervals in their boundary, where the monoidal\nstructure glues the intervals in pairs. \nWe show that this groupoid is a module over the braided monoidal\ngroupoid $\\mathbf B$ of braid groups, giving, on classifying spaces,\nthe structure of an $E_1$-module over an $E_2$-algebra in the sense of \\cite{krannich19}.\n\n\\subsection{Bidecorated surfaces and the monoidal structure}\\label{sec:bidecorated}\n\n\n\nThe groupoid ${{\\mathbf M}_2}$ has objects {\\em bidecorated surfaces}, that are, informally, surfaces with two intervals marked in their boundary. To give a precise definition of the objects that is convenient for the\nmonoidal structure, we start by \n constructing a special sequence of bidecorated\n surfaces $X_n$, built out of disks, and defined inductively.\n \nLet $X_1 = D^2\\subset\\C$ denote the unit disk in the complex plane,\nand define the embeddings $\\iota_{1}^0,\\iota_{1}^1\\colon I\\to X_1$ by\n$$\n\\iota_{1}^0(t) = e^{i(\\pi \/4 + t\\pi \/2)}\\quad\\text{and}\\quad\n\\iota_{1}^1(t) = e^{i(5\\pi \/4 + t\\pi \/2)}.\n$$\nWe denote by $\\overline{\\iota_{1}^i}\\colon I\\to X_1$ the reversed map\n$t\\mapsto \\iota_{1}^i(1-t)$ for $i=0,1$.\n\nRecursively, suppose we have defined $(X_m,\\iota^0_{m},\\iota_{m}^{1})$ for some $m\\geq 1$.\nWe construct $X_{m+1}$ from $X_m$ by gluing an additional disk along\ntwo half intervals, with new markings $\\iota_{m+1}^0,\\iota_{m+1}^1$\n coming from the first half of the markings of $X_m$ and\nthe second half of the markings of the attached disk: \n\n \n$$ X_{m+1}\n\\defeq\n \\frac{\n X_m \\sqcup X_1\n }{\n \\substack{\n \\iota^i_m(t) \\sim \\overline{\\iota^i_1}(t), \\ t\\in{[\\rfrac12,1]},\n }\n }\n\\ \\ \\ \\textrm{with}\\ \\ \\ \n\\iota_{m+1}^i(t)\n=\n\\begin{cases}\n \\iota_{m}^i(t), & \\text{if } t\\leq 1\/2,\\\\\n \\iota_{1}^i(t), & \\text{else.}\n\\end{cases}\n $$\nfor $i=0,1$.\n Note that the marked intervals in the boundary of $X_m$ might live in different boundary components (in fact this will happen every other time). Figure~\\ref{fig:glueD} shows what happens when a disk is glued to a surface in the above described manner, in each of these two possible cases. \n\n\n\n\n\\begin{figure}\n \\def0.8\\textwidth{0.8\\textwidth}\n \\input{gluing_disc.pdf_tex}\n \\caption{Gluing a disk $X_1 = D^2$\n to a bidecorated surface $S$ }\\label{fig:glueD}\n\\end{figure}\n\n\\begin{lemma}\\label{lem:surface-type}\n Let $m \\geq 1$.\n Then $X_m\\cong S_{g,r}$ is a surface of genus $g$ with $r$ boundary components, where \n \\[\n (g,r) =\n \\begin{cases}\n (\\frac{m}{2} - 1, 2), & \\text{if $m$ is even},\n \\\\\n (\\frac{m - 1}{ 2}, 1), & \\text{if $m$ is odd}.\n \\end{cases}\n \\]\n\\end{lemma}\n\n\\begin{proof}\n Note first that $X_m$ is a connected surface for each $m$, since $X_1$ is a disk and $X_m$ is obtained from $X_1$ by successively adding disks (or strips), attached along two disjoint intervals in the boundary. \nFor the same reason, we get that the Euler characteristic of $X_m$ is \n $$\n \\chi (X_{m+1}) = \\chi (X_m) - 1 = \\dots = 1 - m. \n $$\nBy the classification of surfaces, we are left to compute the number of boundary components of $X_m$. \nFor this, observing Figure~\\ref{fig:glueD}, we notice that if we glue a disk along two intervals of $S$ that lie in the same boundary component, the new marked intervals given by the above procedure will give new intervals in different boundary components and vice versa, and no boundary component without marked intervals are ever created. It follows that the number of boundary components of $X_m$ alternates between $1$ and $2$. The result follows. \n\\end{proof}\n\nWe are now ready to define the objects of the groupoid ${{\\mathbf M}_2}$. We will use the boundary of the above defined surfaces $X_m$ to parametrize the boundary components of the surfaces that contain the marked intervals, to allow us to work with parametrized boundary components instead of parametrized arcs, in order to simplify some definitions. \n\\begin{definition}\\label{def:bidec-surfaces}\n A \\emph{bidecorated surface} is a tuple $(S,m,\\varphi)$\n where $S$ is a surface,\n $m\\ge 1$ is an integer, and \n $$\n \\varphi\\colon \\del X_m\\sqcup (\\sqcup_kS^1) \\xrightarrow\\sim \\del S\n $$\n is a homeomorphism, giving a parametrization of the boundary of $S$. \n We think of $(S,m,\\varphi)$ as a surface with two parametrized arcs\n $$\n I_0 \\defeq \\varphi\\circ\\iota_{m}^0\\quad\\text{and}\\quad\n I_1 \\defeq \\varphi\\circ\\iota_{m}^1\n $$\n in its boundary, and $k$ additional parametrized boundaries. The\n surface $S$ may also have punctures. \n \\end{definition}\n The monoidal groupoid \n $({{\\mathbf M}_2}, \\mathbin{\\text{\\normalfont \\texttt{\\#}}},U)$ has objects the bidecorated surfaces together with\n a formal unit $U$.\n There are no morphisms between two bidecorated surfaces\n $(S,m ,\\varphi)$ and $(S',m',\\varphi ')$ unless $S$ and $S'$\n are homeomorphic and $m=m'$,\n in which case we define the set of morphisms to be all the mapping\n classes of homeomorphisms\n that preserve the boundary parametrizations\n \\begin{multline*}\n \\Hom_{{\\mathbf M}_2}((S,m,\\varphi), (S',m,\\varphi '))\n \\defeq\n \\pi_0\\f{Homeo}_{\\del}(S,S') = \\pi_0\\lbrace f\\in\\f{Homeo} (S,S')\\mid f\\circ\\varphi = \\varphi '\\rbrace, \n \\end{multline*}\n where $\\f{Homeo} (S,S' )$ is endowed with the compact-open topology,\n and $\\f{Homeo}_{\\del}(S,S')$ with the subspace topology. \n The only morphism involving the unit $U$\n is the identity $\\text{id}_U$.\n\n\n \\begin{remark}\nOur definition of the morphisms in the category ${{\\mathbf M}_2}$ is such that\npunctures in a surfaces $S$ can be permuted by automorphisms of $S$ in\n${{\\mathbf M}_2}$. Our argument works just as well with labeled punctures, that are\nnot permutable by homeomorphims, or both labeled and unlabeled\npunctures, just like we could also have additional boundary components\nthat are only marked up to a permutation. The only changes this would\ncause to the\nargument would be that it would make the notations and conventions more\ncumbersome. \n \\end{remark}\n \n The monoidal structure $\\mathbin{\\text{\\normalfont \\texttt{\\#}}}$ is defined as follows. \n The object $U$ is by definition a unit for $\\mathbin{\\text{\\normalfont \\texttt{\\#}}}$.\n For the remaining objects, the monoidal product $\\mathbin{\\text{\\normalfont \\texttt{\\#}}}$\n is defined by\n \\begin{align*}\n (S,m,\\varphi)\\mathbin{\\text{\\normalfont \\texttt{\\#}}} (S',m',\\varphi ')\n\\defeq\n \\(\\frac{\n S \\sqcup S'\n }{\n \\substack{\n I_i(t) \\sim \\overline{I_i'}(t), t\\in {[\\rfrac12,1]},\n }\n },m+m',\\varphi\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\\varphi ' \\),\n \\end{align*}\n where $i=0,1$, and where \n $$\n \\varphi\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\\varphi '\n \\colon \\del X_{m+m'}\\sqcup (\\sqcup_{k+k'}S^1) \\ \\hookrightarrow\\ \\del (S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} S'),\n $$\n is obtained using the canonical identification $\\del X_{m+m'}\\cong\n (\\del X_n\\backslash \\iota_m(\\frac{1}{2},1))\\cup (\\del\n X_{m'}\\backslash \\iota_{m'}(0,\\frac{1}{2}))$. \n On morphisms, the monoidal product is given by juxtaposition.\n\n \\medskip\n \nThe monoidal category ${{\\mathbf M}_2}$ has the following {\\em injectivity\n property} with respect to gluing a disk, that will be useful in the\nproof of our stability result. \n\n\\begin{proposition}\\label{prop:Mmonoid}\n For any object $S=(S,m,\\varphi)$ of ${{\\mathbf M}_2}$, and any $p\\ge 0$, \n the map\n \\[\n \\Aut_{{\\mathbf M}_2}(S)\n \\tox{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}}{p+1}}}\n \\Aut_{{\\mathbf M}_2}(S \\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} {p+1}})\n \\]\n is injective, where $D=(X_1,1,\\operatorname{id})$ is our chosen disk. \n\\end{proposition}\n\n\n\n\n\n\\begin{proof} Recall that the underlying surface of $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} p+1}$ is the surface $X_{p+1}$ defined above. \n Picking a smooth representative of the underlying surface of $S\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\n X_{p+1}$, with $S$ a smooth subsurface in its interior, we can model\n the map in the statement using the description of the mapping class\n group of surfaces in terms of isotopy classes of diffeomorphisms\n rather than homeomorphisms. (See e.g., \\cite[Thm 1.2]{boldsen2009}\n for a detailed account of the classical isomorphism\n $\\pi_0\\f{Homeo}_{\\del}(S)\\cong \\pi_0\\operatorname{Diff}_\\del(S)$\n when $S$ is compact.)\n Now the result follows by essentially the same argument as \n the case of attaching of surface along a single arc instead of two, as treated in \\cite[Prop 5.18]{RWW17}, using the fibration\n \\begin{align*}\n \\operatorname{Diff}(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} X_{p+1} \\ \\textrm{rel}\\ \\del S\\cup X_{p+1}) &\\longrightarrow\n\\operatorname{Diff}(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} X_{p+1} \\ \\textrm{rel}\\ \\del (S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} X_{p+1}))\\\\\n &\\longrightarrow \\text{Emb}((X_{p+1},I_0|_{[\\frac{1}{2},1]}\\cup I_1|_{[\\frac{1}{2},1]}),(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} X_{p+1},I_0|_{[\\frac{1}{2},1]}\\cup I_1|_{[\\frac{1}{2},1]}))\n \\end{align*}\n where the fiber identifies with $\\operatorname{Diff}(S\\ \\textrm{rel}\\ \\del_0S)$ and where we note that $I_0|_{[\\frac{1}{2},1]}\\cup I_1|_{[\\frac{1}{2},1]}=\\del X_{p+1}\\cap \\del(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} X_{p+1})$.\n Injectivity of the first map on $\\pi_0$ follows if we can show that the base is simply-connected. \n In fact the base can be shown inductively to have contractible components, using that $X_{p+1}$ is built inductively by attaching disks along two intervals, or homotopically attaching arcs, and using the contractibility of the components of embeddings of arcs in a surface, as proved in \\cite[Thm 5]{Gramain}. \n \\end{proof}\n\n\n\n\n\n \\subsection{Braided action}\\label{sec:braidedaction}\n We want to apply the homological stability machine of \\cite{krannich19} to stabilization\n in ${{\\mathbf M}_2}$ with the bidecorated disk\n $$\n D \\defeq (X_1,1,\\text{id}).\n $$\n For this, we need that\n the classifying space of ${{\\mathbf M}_2}$ is an $E_1$--module over an $E_2$--algebra.\n This will follow if we can show on the categorical level that\n ${{\\mathbf M}_2}$ admits an appropriate action of a braided monoidal groupoid.\n We will build such an action in this section,\n using as braided monoidal groupoid the groupoid of braid groups.\n In constrast with most classical examples of homological stability,\n we will show in Section~\\ref{sec:notbraided} that\n this action of the braid groupoid does not come from a braided structure on ${{\\mathbf M}_2}$,\n or the full monoidal subcategory generated by $D$.\n It is instead constructed using a {\\em\n Yang--Baxter element} in ${{\\mathbf M}_2}$, associated to a braid subgroup of\n the mapping class group of $X_m$, that we will describe now.\n\n\\smallskip\n \nWrite\n $$\n D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}\n =\n D_1 \\mathbin{\\text{\\normalfont \\texttt{\\#}}} \\ldots \\mathbin{\\text{\\normalfont \\texttt{\\#}}} (D_i \\mathbin{\\text{\\normalfont \\texttt{\\#}}} D_{i+1}) \\mathbin{\\text{\\normalfont \\texttt{\\#}}} \\ldots \\mathbin{\\text{\\normalfont \\texttt{\\#}}} D_m,\n $$\n where we use subscript to enumerate the disks, and where the\n underlying surface is $X_m$.\n We let $a_i$ denote the isotopy class of a curve in the interior $D_i\n \\mathbin{\\text{\\normalfont \\texttt{\\#}}} D_{i+1}\\cong S^1\\times I$\n that is parallel to its boundary components,\n as shown in Figure~\\ref{fig:rhoiai}.\n\\begin{figure}\n \\def0.8\\textwidth{0.8\\textwidth}\n \\input{rho-and-ai.pdf_tex}\n \\caption{The curve $a_i$ in $D_i\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D_{i+1}$}\\label{fig:rhoiai}\n\\end{figure}\n\n\\begin{lemma}\\label{lem:chain}\n The curves $a_1,\\dots,a_{m-1}$ form a {\\em chain} in $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}$, i.e. $a_i$ and $a_{i+1}$ have intersection number 1 for each $i$, and $a_i\\cap a_j=\\emptyset$ if $|i-j|>1$. \n\\end{lemma}\n\n\\begin{proof}\n The curve $a_i$ lives in the disks $D_i$ and $D_{i+1}$,\n so it can only intersect $a_{i-1}$ and $a_{i+1}$ non-trivially,\n and hence it suffices to consider\n the subsurface of $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}$ corresponding to\n $D_i\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D_{i+1}\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D_{i+2}$.\n Here the claim can be checked by hand, see Figure~\\ref{fig:crossing}.\n\\end{proof}\n\\begin{figure}\n \\def0.8\\textwidth{0.65\\textwidth}\n \\input{intersection-of-ai-curves.pdf_tex}\n \\caption{Intersection of $a_i$ (blue) and $a_{i+1}$ (green) in the\n underlying surface of $D_i\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D_{i+1}\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D_{i+2}$.}\n \\label{fig:crossing}\n\\end{figure}\n\nLet $T_i\\in\\text{Aut}_{{{\\mathbf M}_2}}(D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m})$\ndenote the Dehn twist along the curve $a_i$ in $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}$.\nA classical fact states that the Dehn twists along a chain of embedded\ncurves satisfy the braid relations (see e.g.~\\cite[3.9 and 3.11]{MCG-primer}):\n\\begin{equation}\\label{equ:braidrel}\n \\begin{aligned}\n T_{i}T_{i+1}T_{i} & = T_{i+1}T_{i}T_{i+1} &&\\quad\\text{for all } i, \\\\\n T_iT_j & = T_jT_i &&\\quad\\text{if } |i - j| > 1, \n\\end{aligned}\n\\end{equation}\nNote that the same relations are satisfied by the inverse twists $T_i^{-1}$,\nthat will turn out to be more convenient for us.\nAlso, adding a disk to the right or left of $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}$ gives the relations\n$$\nT_i\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\\text{id}_D = T_i\\quad\\text{and}\\quad\\text{id}_D\\mathbin{\\text{\\normalfont \\texttt{\\#}}} T_i = T_{i+1}\n$$\nin $\\Aut_{{{\\mathbf M}_2}}(D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m+1})$.\nIn particular~\\eqref{equ:braidrel} includes the relation\n$$\n(T_1^{-1}\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\\text{id}_D)\n(\\text{id}_D\\mathbin{\\text{\\normalfont \\texttt{\\#}}} T_1^{-1})\n(T_1^{-1}\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\\text{id}_D)\n=\n(\\text{id}_D\\mathbin{\\text{\\normalfont \\texttt{\\#}}} T_1^{-1})\n(T_1^{-1}\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\\text{id}_D)\n(\\text{id}_D\\mathbin{\\text{\\normalfont \\texttt{\\#}}} T_1^{-1})\n$$\nin $\\Aut_{{\\mathbf M}_2} (D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 3})$,\nso in other words,\nthe inverse Dehn twist $T_1^{-1}\\in \\Aut_{{\\mathbf M}_2} (D{\\mathbin{\\text{\\normalfont \\texttt{\\#}}}} D)$ is a Yang--Baxter operator\nin the sense of Section~\\ref{sec:YB}.\n\nRecall from the introduction that $\\braidGrpd$ denotes the groupoid of braid\ngroups, with objects the natural numbers $\\lbrace 0,1,2,\\dots\\rbrace$,\nautomorphisms of $n$ the\nbraid group $B_n$, and no other non-trivial morphisms. \nIn Section~\\ref{sec:YB} we show that, being a Yang-Baxter operator, $T_1^{-1}$ yields\na strong monoidal functor\n$$\n\\Phi = \\Phi_{D,T_1^{-1}}\\colon (\\braidGrpd ,\\oplus )\\to ({{\\mathbf M}_2} ,\\mathbin{\\text{\\normalfont \\texttt{\\#}}} ),\n$$\nuniquely determined up to monoidal natural isomorphism by the fact\nthat $\\Phi (1) = D$ and, for the standard generator $\\sigma_1\\in B_2= \\text{Aut}_\\braidGrpd (1)$,\n$\\Phi (\\sigma_1) = T_1^{-1}$.\n\n\\smallskip\nSuch a functor $\\Phi$ endows ${{\\mathbf M}_2}$ with the structure\nof an $E_1$-module over $\\braidGrpd$ via the associated functor\n\\begin{align*}\n \\alpha=( - \\mathbin{\\text{\\normalfont \\texttt{\\#}}} \\Phi(-)) \\colon \\ {{\\mathbf M}_2}\\times \\braidGrpd &\\longrightarrow {{\\mathbf M}_2}, \n\\end{align*}\ngiven on objects by $\\alpha(S,n)= S\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\\Phi (n) = S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\n n}$, and likewise for morphisms. \nOn classifying spaces, this yields exactly the kind of input needed in Krannich's homological stability\nframework, see \\cite[Lem 7.2]{krannich19}.\n\n\\begin{remark}\n For each $m$, the restriction of the functor $\\Phi: \\braidGrpd\\to {{\\mathbf M}_2}$ to\n $B_m = \\text{Aut}_\\braidGrpd(m)$ maps the standard generator $\\sigma_i$\n to the inverse Dehn twist $T_i^{-1}\\in \\Aut_{{\\mathbf M}_2} (D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\n m})=\\pi_0\\textrm{Homeo}_\\partial(X_m)$.\n By Birman--Hilden theory~\\cite{BH1,BH2}\n the homomorphisms $\\Phi\\vert_{B_m}\\colon B_m\\to\\text{Aut}_{{\\mathbf M}_2} (D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m})$\n are actually injective. \n\\end{remark}\n\n\n\n\\section{High connectivity of the disordered arc complex}\n\\label{sec:high-cnt}\n\n\nIn this section, we prove that the disordered arc complex is highly\nconnected. It will be defined as a subcomplex of the following simplicial complex of non-separating arcs: \n\n\\begin{definition}\\label{def:nonseparating}\n Let $S$ be an orientable surface\\footnote{\n By {\\em surface} we mean a topological $2$-manifold $S$ which is compact except\n for a finite number of punctures, i.e. there is a compact topological $2$-manifold\n $\\overline S$ and an embedding $i\\colon S\\incl\\overline S$ so that\n $\\overline S\\setminus i(S)$ is a (possibly empty) finite union of points.\n}\n with nonempty boundary, and\n let $b_0, b_1$ be distinct points in $\\del S$.\n The \\emph{complex of non-separating arcs} $\\mathcal B(S, b_0, b_1)$ is the simplicial complex\n whose $p$-simplices are collections of $p + 1$ distinct isotopy classes\n of arcs between $b_0, b_1$\n that admit representatives\n $\n a_0, \\ldots, a_p\n $\n such that\n \\begin{enumerate}\n \\item\\label{def:nonseparating:1}\n $a_i \\cap a_j = \\{b_0, b_1\\}$ for each $i \\neq j$ and\n \\item\\label{def:nonseparating:2}\n $S - (a_0 \\cup \\cdots \\cup a_p)$ is connected.\n \\end{enumerate}\n \\end{definition}\n\nFor convenience, we will add a superscript $\\mathcal\nB^\\nu(S, b_0, b_1)$ to the notation of the complex, with \n $\\nu=1$ indicating that $b_0, b_1$ lie on the same boundary component and\n $\\nu=2$ indicating that they do not. \n\n \\smallskip\n\n Note that the orientation of the surface defines orderings of the arcs $a_0,\\dots,a_p$ representing a simplex at both $b_0$ and $b_1$.\n\n\\begin{definition}\\label{def:disord}\n Let $(S, b_0, b_1)$ be as before.\n The \\emph{disordered arc complex} is\n the subcomplex\n $\n \\mathcal D^\\nu(S_{g, r}, b_0, b_1)\n \\subseteq\n \\mathcal B^\\nu(S, b_0, b_1)\n $\n consisting of those simplices $\\sigma$\n that admit arc representatives $a_0, \\ldots, a_p$, again subject to\n \\cref{def:nonseparating:1},\n \\cref{def:nonseparating:2}, satisfying in addition\n \\begin{itemize}\n \\item[(c)] the ordering of the arcs at $b_0$ agrees with the ordering of the arcs at $b_1$. \n \\end{itemize}\n\\end{definition}\n\nThe name ``disordered'' was chosen to contrast with the pre-existing {\\em ordered arc complex} used by Ivanov \\cite{Iva89} in the case $\\nu=1$ and Randall-Williams \\cite{RW16} in their proofs of homological stability for the mapping class group of surfaces; the ``ordered'' version is also a subcomplex of the $\\mathcal B^\\nu(S, b_0, b_1)$, but with the requirement that the order of the arcs at $b_1$ is reversed compared to the order at $b_0$. \nFixing an ordering condition has the effect that the action of the mapping\nclass group is transitive on the set of $p$-simplices for every $p$,\nsee \\cite[Lem 3.2]{harer85}. The ordered and disordered arc complexes represent the two extremes of how fast the genus of the surface decreases when cutting along larger and larger simplices: for the ordered arc complex, the genus goes down as fast as possible, essentially every time one removes an arc, while for the disordered arc complex, the genus goes down as slow as possible, only every other time: \n\n\n\\begin{proposition}\\label{prop:type-of-cut-surface}\n For a $p$--simplex $\\sigma = \\langle a_0,\\dots ,a_p\\rangle\\in\\mathcal D^\\nu (S_{g,r},b_0,b_1)$, \n the surface $S_{g,r}\\setminus \\sigma$ obtained by removing a tubular neighborhood of $a_i$ for each $i$ has genus $g'$ with $r'$ boundary components for \n $$ g' = g-\\floor{\\frac{p+3-\\nu} 2} \\ \\ \\ \\textrm{and}\\ \\ \\ \n r' =\n \\begin{cases}\n r+(-1)^\\nu,&\\text{if } p \\text{ is even,} \\\\\n r&\\text{else}. \n \\end{cases}\n $$\n\\end{proposition}\n\\begin{proof}\nThe number of boundary components $r'$ can be obtained by a direct\ninductive computation, with the genus $g'$ then deduced using the\nEuler characteristic. The computation is a special case of \\cite[Prop 2.11]{boldsen12}, applied to the case where the permutation $\\alpha$ is the inversion $[p\\,(p-1)\\dots 0]$, once one computes that the genus $S(\\alpha)$ of a neighborhood of the arcs is $\\lfloor\\frac{p+2-\\nu}{2}\\rfloor$, e.g.~using Corollary 2.15 of the same paper. \n\\end{proof}\n\n\n\nThe complex $\\mathcal B^\\nu(S, b_0, b_1)$ is known to be $(2g + \\nu - 3)$-connected. (This was first stated in \\cite{harer85}; see \\cite[Thm 3.2]{Wah08} or \\cite[Thm 4.8]{wahl-handbk} for a complete proof.) \nWe will here use this fact to deduce that $\\mathcal D^\\nu(S_{g, r}, b_0, b_1)$ is also highly-connected.\nWhile the ordered arc complex is $(g-2)$-connected \\cite[Thm A.1]{RW16},\nthe following result shows that the disordered arc complex is only slope\n$\\frac{2}{3}$ connected with respect to the genus,\ndespite being $\\sim 2g$-dimensional. \n\n\n\n\\begin{theorem}\\label{thm:disord-cnt}\n The disordered arc complex $\\mathcal D^\\nu(S_{g,r}, b_0, b_1)$ is $\\(\\frac{2g +\\nu - 5} 3\\)$-connected.\n\\end{theorem}\n\n To prove the result, we use essentially the same argument as the one given in \\cite{RW16} in the ordered case. \n\n\\begin{proof}\n Let $S = S_{g, r}$. In the case $g = 0$,\n the statement for $\\mathcal D^1(S)$ is vacuous,\n and for $\\mathcal D^2(S)$ it states that the complex is $(-1)$-connected, i.e.~nonempty, which holds as any arc in the surface connecting $b_0$ and $b_1$ defines a vertex in $\\mathcal D^2(S)$. \n We prove the remaining cases by induction on $g$.\n\n Let $g > 0$. \n Suppose we are given \n $\n f : \\del D^{k + 1} \\to \\mathcal D^\\nu(S, b_0, b_1)\n $\n for some $k \\leq (2g + \\nu - 5)\/ 3$.\n We wish to exhibit a nullhomotopy of this map.\n Since $(2g + \\nu - 5) \/ 3 \\leq 2g + \\nu - 3$, Theorem 3.2 in \\cite{Wah08} \n enables us to choose a map $\\hat f$\n such that the outer diagram\n \\begin{equation}\\label{lifting-prob}\n \\begin{tikzcd}\n \\del D^{k + 1}\n \\ar[d, hook]\n \\ar[r, \"f\"]\n & \\mathcal D^\\nu (S, b_0, b_1)\n \\ar[d, hook]\n \\\\\n D^{k+1}\n \\ar[r, \"\\hat f\"]\n \\ar[ru, dotted]\n & \\mathcal B^\\nu(S, b_0, b_1),\n \\end{tikzcd}\n \\end{equation}\n commutes.\n Using PL-approximation,\n we may assume that $\\hat f$ and $f$ are simplicial\n with respect to some PL-triangulation of $D^{k+1}$.\n We will repeatedly replace $\\hat f$ until the dotted arrow exists,\n thereby giving the desired nullhomotopy.\n\n Write $<_0$ and $<_1$ for the anti-clockwise orderings at $b_0$ and $b_1$. We call a $p$-simplex $\\sigma$ in $D^{k+1}$ \\emph{regular bad} if \n $\\hat f(\\sigma) = \\$,\n indexed in such a way that $a_0 <_0 \\dots <_0 a_{p'}$ are anticlockwise at $b_0$,\n and \n there is $j > 0$ with $a_j <_1 a_0$ at $b_1$. Here $p'\\le p$ is the dimension of the image simplex $\\hat f(\\sigma)$, and $p'\\geq 1$ if $\\sigma$ is regular bad. \n This condition is \\q{dense} in the sense\n that any simplex $\\sigma$ in $D^{k+1}$ with image not included in $\\mathcal D^\\nu(S, b_0, b_1)$\n must contain a regular bad simplex as a face.\n Thus it suffices to give a procedure for exchanging $\\hat f$ with a map\n having strictly fewer regular bad simplices, \n while maintaining commutativity of the outer diagram~\\cref{lifting-prob}.\n\n \n\n Let $\\sigma$ be a regular bad simplex of $D^{k+1}$ of maximal dimension $p$ and consider its link $\\Lk\\sigma\\subset D^{k+1}$.\n \n Maximality implies that $\\hat f | _{\\Lk \\sigma}$ factors as\n \\[\\hat f | _{\\Lk \\sigma}\\colon \n \\Lk \\sigma\n \\to\n \\mathcal D^\\mu(S \\setminus \\hat f(\\sigma), b_0', b_1')\n \\to\n \\mathcal D^\\nu(S,b_0,b_1)\n \\subseteq \\mathcal B^\\nu(S, b_0,b_1), \n \\]\n where $S \\setminus \\hat f (\\sigma)$ is the closure of the surface obtained from $S$\n by cutting out the collection of arcs $\\hat f(\\sigma)$, and \n $b_0'$ and $b_1'$ in $S \\setminus \\hat f (\\sigma)$ are the first copies of $b_0$ and $b_1$ in the cut surfaces as depicted in Figure~\\ref{fig:regularbad}.\n \\begin{figure}\n \\def0.8\\textwidth{0.8\\textwidth}\n \\input{regular-bad-spx.pdf_tex}\n \\caption{Maximal regular bad simplex $\\{a_0,\\dots,a_p\\}$ and simplex $\\{a'_0,\\dots,a'_q\\}$ in its link.}\\label{fig:regularbad}\n \\end{figure}\nIndeed, suppose that $\\tau \\in \\Lk\\sigma$ and write\n $\n \\hat f(\\tau) = \\. \n $\n If $a_0\\le_0a_i'$ at $b_0$ for any $a_i'$, then the simplex $\\sigma*\\$ is regular bad of a larger dimension, contradicting maximality. So we must have $a_i'<_0a_0$ for each $i$, i.e.~the arcs of $\\tau$ are at $b_0'$ in the cut surface. Now we must also have that each $a_i'<_1a_0$ as otherwise $\\sigma*\\$ would again be regular bad. \n Finally, maximality of $\\sigma$ would also be contradicted if the orderings of the arcs $a_0', \\cdots, a_q'$ does not agree at $b_0$ and $b_1$ as, if $a_i'<_0 a_j'$ with $a_j'<_1a_i'$ for some $i,j$, then $\\sigma*\\$ would again be regular bad, of larger dimension. \n Thus $\\hat f(\\tau)$ must be disordered\n and, after cutting the surface at the arcs of $\\sigma$, can be viewed as a simplex of\n $\n \\mathcal D^\\nu(S \\setminus \\hat f(\\sigma), b_0', b_1'). \n $\n\n\n\nThe link $\\Lk(\\sigma)$ is a simplicial sphere $S^{k-p}\\subset D^{k+1}$. We want to show that the map $\\hat f|_{\\Lk (\\sigma)}$ extends to a simplicial map\n \\begin{equation}\\label{equ:F}\n F :\n D^{k - p + 1}\n \\to\n \\mathcal D^\\mu(S - \\hat f(\\sigma), b_0', b_1')\n \\to\n \\mathcal D^\\nu (S, b_0, b_1)\n \\subseteq\n \\mathcal B(S, b_0, b_1)\n \\end{equation}\n for $D^{k - p + 1}$ a disk with some PL-structure extending that of $\\Lk (\\sigma)$.\nThis will follow if we can show that the complex\n $\n \\mathcal D^\\mu (S\\setminus\\hat f (\\sigma ),b_0',b_1')\n $\n is at least $(k-p)$-connected. Note that necessarily have $g(S\\setminus\\hat f (\\sigma )) < g$ as $f(\\sigma)$ is a non-separating $p'$-simplex with $p'\\ge 1$. Hence we can use our induction hypothesis.\n We consider the cases $\\nu = 1$ and $\\nu = 2$ separately.\n\n\\smallskip\n \n \\para{Case 1: $\\nu=1$}\n We have that \n $\n g(S\\setminus\\hat f (\\sigma ))\\ge g-p'-1\\ge g-p-1,\n $\nas removing $p'+1$ arcs reduces the genus by at most $p'+1\\le p+1$.\n Hence by induction we have that $\\mathcal D^\\mu (S\\setminus\\hat f (\\sigma )\n , b_0',b_1')$ is at least $(\\frac{2(g-p-1)-4}{3})$-connected, using also that $\\mu\\ge 1$.\n If $p\\geq 2$, we have \n $$k-p\\leq \\frac{2g-4}{3} - p = \\frac{2g-3p-4}{3} \\leq \\frac{2(g-p-1)-4}{3}.$$\n For $p=p'=1$, note that $b_0',b_1'$ necessarily lie in different boundary components, so that $\\mu=2$ in that case. (See Figure~\\ref{fig:boundaries}.) Hence\n in that case $\\mathcal D^\\mu (S\\setminus\\hat f (\\sigma ), b_0',b_1')$ is \n $(\\frac{2(g-2)-3}{3})$-connected, and \n $$k-1\\leq \\frac{2g-4}{3} - 1 = \\frac{2g-7}{3} = \\frac{2(g-2)-3}{3}.$$\n so we get the desired extension in both subcases.\n \\begin{figure}\n \\def0.8\\textwidth{0.8\\textwidth}\n \\input{nu-one-reg-bad-1-spx.pdf_tex}\n \\caption{Regular bad 1-simplex with $\\nu=1$.}\\label{fig:boundaries}\n \\end{figure}\n\n \\smallskip\n \n \\para{Case 2: $\\nu=2$} The fact that $b_0,b_1$ lie in different\n components implies that\n $$\n g(S\\setminus\\hat f (\\sigma )) \\geq g-p'\\geq g-p\n $$\n as cutting along the first arc has no effect on the genus. Hence induction here gives that $\\mathcal D^\\mu (S\\setminus\\hat f (\\sigma ), b_0',b_1')$ is at least $(\\frac{2(g-p)-4}{3})$-connected.\n Now for all $p\\geq 1$,\n $$k-p\\leq \\frac{2g-4}{3} - p = \\frac{2g-3p-4}{3} \\leq \\frac{2(g-p)-4}{3}$$\nyielding the desired connectivity. \n\n\\medskip\n\n \nWe will use the map $F$ of \\eqref{equ:F} to modify $\\hat f$ in the star $\\f{St}(\\sigma)$. \nFor this purpose, note that as simplicial subcomplexes of $D^{k+1}$, \n \\begin{align*}\n \\f{St}(\\sigma) &= \\sigma * \\f{Lk}(\\sigma), \\quad \\\\\n \\del \\f{St}(\\sigma) &= \\del \\sigma * \\f{Lk}(\\sigma).\n \\end{align*}\n In particular, we get an identification $\\del(\\del \\sigma * D^{k - p + 1}) \\cong \\del \\f{St}(\\sigma)$ for $D^{k-p+1}$ the simplicial disk that is the source of the map $F$ above.\n\n We replace $\\hat f\\, \\vert\\, _{\\f{St}(\\sigma)}$ by the unique simplicial map\n \\[\n \\hat f * F : \\del \\sigma * D^{k - p + 1}\n \\to\n \\mathcal B^\\nu(S, b_0, b_1). \n \\]\n It remains to show that this has improved the situation.\n Indeed, suppose that $\\tau=\\tau_0*\\tau_1$ is a regular bad simplex in $\\del \\sigma * D^{k - p + 1}$. By construction, $\\tau_1$ has image in\n $\\mathcal D^\\mu(S \\setminus\\hat f(\\sigma), b_0', b_1') \\subset \\mathcal D^\\nu (S, b_0, b_1)$, so the ordering of the arcs of $\\tau$ at $b_0$ and $b_1$ starts with the arcs of $\\tau_1$, all in anti-clockwise order. Hence, if $\\tau$ is regular bad, we must have $\\tau=\\tau_0$ is a strict face of $\\sigma$. In particular, no new regular bad simplices have been added. As the simplex $\\sigma$ has been removed, we have thus reduced the total number of regular bad simplices in the disk. \n Repeating this procedure,\n we will after finitely many stages\n remove every regular bad simplex,\n thus making the dashed arrow exist, which proves the result. \n \\end{proof}\n \n\\begin{remark}\\label{rem:optimal1}\n The connectivity estimate above can be shown to be optimal\n in certain low-genus examples, corresponding to known computations\n of the unstable homology of mapping class groups.\n Indeed, $\\mathcal D^2(S_{1,r})$ is disconnected.\n To see this, consider the spectral sequence associated\n to the action of the mapping class group $\\Gamma (S_{1,r})$ on\n the simplicial complex $\\mathcal D^2(S_{1,r})$.\n This is the spectral sequence arising from the vertical filtration\n of the double complex $\\mathbb{Z}\\mathcal D^2(S_{1,r})_\\bullet\\otimes_{\\Gamma (S_{1,r})}\n F_\\bullet$,\n where $F_\\bullet\\to\\mathbb{Z}$ is a free resolution\n of the trivial $\\Gamma (S_{1,r})$-module.\n By a standard argument using Shapiro's lemma (see e.g.~\\cite[Thm\n 5.1]{HatWah10} or \\cite[Sec 1]{HatVog17}), one finds that\n the first page of this spectral sequence is given by\n $$\n E^1_{p,q} \\cong\n \\begin{cases}\n \\widetilde H_q(\\Gamma (S_{1,r})) &\\text{if } p = -1,\\\\\n \\widetilde H_q(\\Gamma (S_{1,r-1})) &\\text{if } p = 0,\\\\\n \\widetilde H_q(\\Gamma (S_{0,r})) &\\text{if } p = 1,\\\\\n \\widetilde H_q(\\Gamma (S_{0,r-1})) &\\text{if } p = 2,\\\\\n 0 &\\text{otherwise.}\n \\end{cases}\n $$\n Assume for contradiction that $\\mathcal D^2(S_{1,r})$ is connected.\n Then an analysis of the horizontal filtration of the double complex\n $\\mathbb{Z}\\mathcal D^2(S_{1,r})_\\bullet\\otimes_{\\Gamma (S_{1,r})}F_\\bullet$\n shows that $E^\\infty_{p,q} = 0$ for $p+q\\leq 0$,\n so the differential $d^1\\colon H_1(\\Gamma (S_{1,r-1}))\\to H_1(\\Gamma (S_{1,r}))$\n must be surjective. This contradicts the fact that\n $H_1(\\Gamma (S_{1,s}))\\cong\\mathbb{Z}^s$ for $s\\geq 1$ (see \\cite[Thm\n 5.1]{korkmaz}).\n Hence it is not true that $\\mathcal D^\\nu$ is\n $\\left(\\frac{2g+\\nu -4} 3\\right)$-connected when $\\nu = 2$. \n \n Similarly, one finds that $H_1(\\mathcal D^1(S_{3,r}))\\neq 0$\n by considering the spectral sequence associated to the action\n of $\\Gamma (S_{3,r})$ on $\\mathcal D^1(S_{2,r+1})$ and noting that\n the differential $d^1\\colon H_1(\\Gamma (S_{2,r+1}))\\to H_1(\\Gamma (S_{3,r}))$\n cannot be injective since the source identifies with $\\mathbb{Z} \/ 10\\mathbb{Z} $\n and the target is zero (see \\cite[Thm 5.1]{korkmaz}).\n Thus $\\mathcal D^\\nu$ fails to be\n $\\left(\\frac{2g+\\nu -4} 3\\right)$-connected when $\\nu = 1$ also.\n\n Note that these low dimensional computations also show that the\n first and last ranges in\n Theorem~\\ref{thmintro:stab1} cannot be improved by a constant. \n\\end{remark}\n\n\\section{Introduction}\n\\label{sec:intro}\nLet $S_{g,r}^s$ be a surface of genus $g$\nwith $r$ boundary components and $s$ punctures.\nThe mapping class group $\\Gamma(S_{g,r}^s):=\\pi_0\\f{Homeo} (S_{g,r}^s \\textrm{ rel } \\del S)$\nof $S$ satisfies {\\em homological stability}:\nthe homology group $H_i(\\Gamma(S_{g,r}^s);\\mathbb{Z})$\nis independent of $g$ and $r$ when $g$ is large relative to $i$.\nThis stability result was originally proved by Harer in \\cite{harer85},\nand later improved by Ivanov, Boldsen and Randal-Williams\n\\cite{Iva89,boldsen12,RW16},\nsee also \\cite{Har93,wahl-handbk,HatVog17,GKRW19MCG}.\nWe recast the result here as a stability theorem\nin the category of {\\em bidecorated surfaces},\nand give a new proof of the best know stability range using the most\nstraightforward inductive argument originally designed by Quillen, and\nformalized in \\cite{RWW17,krannich19}.\nOur proof at the same time illustrates how little is needed to run the stability\nmachines of these two papers. \n\n\n\\smallskip\n\nOur main stability result is the following, recovering precisely the\nranges of \\cite[Thm 1]{boldsen12} and \\cite[Thm 7.1 (i),(ii)]{RW16}:\n\n\\begin{ThA}\\label{thmintro:stab1}\n Let $S_{g,b}^s$ be a surface of genus $g\\ge 0$, with $r\\ge 1$\n marked boundary components and $s\\ge 0$ punctures, and let \n $\\Gamma(S_{g,r}^s)=\\pi_0\\f{Homeo} (S_{g,r}^s\\textrm{ rel }\\del S)$ denote its mapping class group.\n The map \n $$\n H_i(\\Gamma(S_{g,r}^s);\\mathbb{Z} )\\to H_i(\\Gamma(S^s_{g,r+1}); \\mathbb{Z})\n $$\n induced by gluing a pair or pants along one boundary component is always injective, \n and an isomorphism when $i\\leq\\frac{2g} 3$, and the map \n $$\n H_i(\\Gamma(S_{g,r+1}^s);\\mathbb{Z} )\\to H_i(\\Gamma(S^s_{g+1,r}); \\mathbb{Z}) $$\n induced by gluing a pair of pants along two boundary components is an epimorphism\n when $i\\leq\\frac{2g+1} 3$\n and an isomorphism when $i\\leq\\frac{2g-2} 3$.\n\\end{ThA}\n\nCombining the two maps in the theorem gives a genus stabilization that\nis known to be close to optimal by a computation of\nMorita \\cite{Morita} and low dimensional computations, see\nRemarks~\\ref{rem:optimal1} and~\\ref{rem:optimal2}.\nWhile we do not\nknow whether the two\nranges in the above statement can be individually improved, it is\nremarkable that three rather different proofs (those of\nBoldsen \\cite{boldsen12}, Randal-Williams \\cite{RW16}, and ours) end up with the\nexact same ranges. \n\n\n\\medskip\n\nA particular feature of our proof is that the two maps occurring in\nthe theorem will be for us ``the same map'',\nnamely a disk stabilization\nin the category ${{\\mathbf M}_2}$ of {\\em bidecorated surfaces}.\nA bidecorated surface is\na surface $S$ with two marked intervals $I_0,I_1$ in its boundary. \nThe two intervals may lie on the same or on different boundary components.\nMorphisms in ${{\\mathbf M}_2}$ are mapping classes, i.e.~isotopy classes of homeomorphisms,\nand ${{\\mathbf M}_2}$ admits a monoidal structure $\\mathbin{\\text{\\normalfont \\texttt{\\#}}}$\ndefined by identifying the marked intervals in pairs.\n\nOur main example of a bidecorated surface will be the bidecorated disk $D$.\nAs shown in Lemma~\\ref{lem:surface-type},\ntaking sums of the disk with itself in ${{\\mathbf M}_2}$ produces surfaces of any genus:\n$D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2g+1}$ is a surface $S_{g,1}$ of genus $g$ with a single boundary component,\nwhile $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2g+2}$ is a surface $S_{g,2}$ of genus $g$ with two boundary components,\neach containing a marked interval.\nTo obtain any surface $S_{g,r}^s$ with $r\\ge 1$,\nwe will consider the object $S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2g}$ in ${{\\mathbf M}_2}$,\nfor $S=S_{0,r}^s$ a genus 0 surface with $r$ boundary components and $s$ punctures.\nNow the maps in Theorem~\\ref{thmintro:stab1}\nare precisely the disk stabilization maps in ${{\\mathbf M}_2}$:\n$$\n\\Aut_{{\\mathbf M}_2} (S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2g}) \\xrightarrow{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D} \\Aut_{{\\mathbf M}_2} (S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2g+1}) \\xrightarrow{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D} \\Aut_{{\\mathbf M}_2} (S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2g+2})\n$$\nfor these particular choices of surfaces. \n\nTheorem~\\ref{thmintro:stab1} is thus the statement that disk stabilization $\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D$ in ${{\\mathbf M}_2}$ induces isomorphisms on\nthe homology of these automorphism groups in a range.\n We show in the present paper that this result can be obtained as a\ndirect application of the main result of \\cite{krannich19},\nfrom which an additional stability statement with twisted coefficients\nautomatically follows. We start by stating this additional result. \n\n\n\n\\subsection*{Twisted coefficients}\nFix $r\\ge 1 $ and $s\\ge 0$. In our setting, a coefficient system $F$ for the mapping class group $\\Gamma(S_{g,r}^s)$ is a collection of $\\mathbb{Z}[\\Gamma(S_{g,r}^s)]$-modules\n$F_{2g}$ and $\\mathbb{Z}[\\Gamma(S_{g,r+1}^s)]$-modules $F_{2g+1}$ for each\n$g\\ge 0$, together with maps\n$$\nF_{n}\\to F_{n+1}\n$$\nequivariant with respect to the disk stabilization and satisfying that\na certain Dehn twist\nacts trivially on the image of $F_{n}$ in $F_{n+2}$\nunder double stabilization (see Definition~\\ref{def:coef-system}).\nGiven a coefficient system,\none can define a notion of degree;\na constant coefficient systems has degree 0\nand for example the coefficient system\n$F_{2g+i }=H_1(S_{g,r+i}^s;\\mathbb{Z})^{\\otimes k}$, $i\\in\\lbrace 0,1\\rbrace$, has degree $k$\n(see Example~\\ref{exmp:coef-systems}). \n\nWe obtain the following twisted stability result:\n\n\n \\begin{ThA}\\label{thmintro:stab2}\n Let $\\Gamma(S_{g,r}^s)$ be as in Theorem~\\ref{thmintro:stab1}, and $F$ be a coefficient system of degree $k$.\n The stabilization map\n \\begin{equation*}\n \\label{eq:twisted-stab1}\n H_i(\\Gamma (S_{g,r}^s);F_{2g} )\\to H_i(\\Gamma (S^s_{g,r+1}); F_{2g+1} ) \n \\end{equation*}\n is an epimorphism\n for $i\\leq \\frac{2g-3k-2} 3$\n and an isomorphism for $i\\leq\\frac{2g - 3k - 5} 3$, and\n the map\n \\begin{equation*}\n \\label{eq:twisted-stab2}\n H_i(\\Gamma (S_{g,r+1}^s);F_{2g+1} )\\to H_i(\\Gamma (S^s_{g+1,r}); F_{2g+2} )\n \\end{equation*}\n is an epimorphism\n for $i\\leq\\frac{2g-3k-1}{3}$\n and an isomorphism for $i\\leq\\frac{2g-3k-4} 3$.\n \n \n \n \n \n In these bounds, $3k$ can be replaced by $k$ if $F$ is in addition split in the sense of Definition~\\ref{def:coef-system}. \n \n \\end{ThA}\n\nStability theorems for mapping class groups with twisted coefficients\ncan be found in the work of Ivanov, Boldsen, Randal-Williams--Wahl,\nand Galatius--Kupers--Randal-Williams \\cite{boldsen12,IvanovTwisted,RWW17,GKRW19MCG}. \nThe results are not easy to compare as the types of coefficient\nsystem that are permitted depend on the paper,\nbut some classical examples such as the one described above fit\nall frameworks (see Remarks~\\ref{rmk:different-category}\nand~\\ref{rem:optimal2} for more details).\n\n\n\\subsection*{Braided action and Yang--Baxter operators}\nWe want to obtain Theorems~\\ref{thmintro:stab1}\nand~\\ref{thmintro:stab2} as consequences of \nTheorems A and C of \\cite{krannich19}. For this, we first have to show that disk\nstabilization in the monoidal category $({{\\mathbf M}_2},\\mathbin{\\text{\\normalfont \\texttt{\\#}}})$ comes from an\naction of a braided monoidal groupoid.\n\nLet $\\braidGrpd$ denote the groupoid of braid groups, with\nobject the natural numbers and the braid group $B_n$ as automorphisms\nof $n$. We will construct an action of $\\braidGrpd$ on ${{\\mathbf M}_2}$ using\nan appropriate {\\em Yang--Baxter operator} in ${{\\mathbf M}_2}$: \nThe sum of bidecorated disks $D\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D$ in ${{\\mathbf M}_2}$ is a cylinder, whose\nmapping class group is an infinite cyclic group generated\nby the Dehn twist $T$ along the core circle of the cylinder. It turns\nout that this morphism $T\\in\n\\Aut_{{{\\mathbf M}_2}}(D\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D)$ is a Yang--Baxter operator in ${{\\mathbf M}_2}$,\nin the sense that it satisfies the equation\n$$(T\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 1)(1\\mathbin{\\text{\\normalfont \\texttt{\\#}}} T)(T\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 1)=(1\\mathbin{\\text{\\normalfont \\texttt{\\#}}} T)(T\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 1)(1\\mathbin{\\text{\\normalfont \\texttt{\\#}}} T)$$\nin $\\Aut_{{{\\mathbf M}_2}}(D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 3})$. The same holds for the inverse twist\n$T^{-1}$, that will turn out more convenient for us. \nAs explained in Section~\\ref{sec:YB},\nwe get an associated strong monoidal functor\n$\\braidGrpd\\to{{\\mathbf M}_2}$ taking the object $n$ to $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}$. \nThe corresponding homomorphism $B_n\\to\\text{Aut}_{{\\mathbf M}_2} (D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})$\ncan be identified with the geometric embedding in the sense of \\cite{Waj99},\nassociated to the chain of curves $a_1,\\dots,a_{n-1}$ in\n$$\nD^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n} = D\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D\\mathbin{\\text{\\normalfont \\texttt{\\#}}} \\dots \\mathbin{\\text{\\normalfont \\texttt{\\#}}} D,\n$$\nwhere the $i$th curve $a_i$ is the core circle in the $i$th cylinder\n$D\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D$ in the above sum,\nsee Lemma~\\ref{lem:chain} and Example~\\ref{exmp:mcg-YB}.\n\nThe strong monoidal functor $\\braidGrpd\\to{{\\mathbf M}_2}$ from above endows ${{\\mathbf M}_2}$\nwith the structure of an $E_1$-module over the braid groupoid $\\braidGrpd$,\nand since the latter is braided monoidal,\nwe can apply the results of \\cite{krannich19} to study disk stabilization in ${{\\mathbf M}_2}$.\n\n\\begin{remark}\n Homological stability frameworks such as \\cite{RWW17,krannich19,GKRW19MCG}\n require an $E_2$-algebra, or the weaker structure of $E_1$-module\n over an $E_2$-algebra, as input. \n This is a priori a lot of data, and it may be that the most natural choice\n in a given context simply does not admit an $E_2$-structure.\n This turns out to be the case for the monoidal category of bidecorated surfaces\n ${{\\mathbf M}_2}$: \n In the context of categories,\n $E_2$-structures are given by braided monoidal structures and we show in Section~\\ref{sec:notbraided}\n that even the full monoidal subcategory of ${{\\mathbf M}_2}$ generated by\n our stabilizing object, the disk $D$,\n does not admit a braiding. \n This distinguishes our situation from most previous examples of homological stability.\n\n On the other hand, it does not take much\n to equip a given monoidal category $\\mathcal X$\n with the structure of an $E_1$-module over a braided monoidal category.\n In fact, as shown in Section~\\ref{sec:YB}, any Yang--Baxter operator in $\\mathcal X$ determines\n a strong monoidal functor $\\braidGrpd\\to\\mathcal X$ from the braid groupoid\n $\\braidGrpd$, and thus endows $\\mathcal X$\n with the structure of an $E_1$-module over $\\braidGrpd$.\n \n \n \n \n \n \n This perspective also makes sense if $\\mathcal X$ itself acts\n on a category $\\mathcal M$, and one is interested in the stabilization\n $$\n \\mathcal M\\xrightarrow{\\oplus X}\\mathcal M\\xrightarrow{\\oplus X}\\cdots\n $$\n\ninduced by acting with an object $X$ of $\\mathcal X$ admitting a\nYang-Baxter operator $\\tau\\in\\text{Aut}_{\\mathcal X}(X\\oplus X)$. \n The category $\\mathcal M$ becomes this way likewise a module over $\\braidGrpd$,\n where the object $n$ of $\\braidGrpd$ acts on $A\\in\\mathcal M$ via $A\\oplus n = A\\oplus X^{\\oplus n}$.\n\\end{remark}\n\n\\subsection*{Disordered arcs}\nGiven a category $\\mathcal M$ as above, with the structure of an $E_1$-module over a\nmonoidal category $\\mathcal X$\nwith a distinguished Yang--Baxter operator $(X,\\tau )$,\nsuch that acting by $X$ satisfies a certain injectivity property\n(see Proposition~\\ref{prop:Mmonoid}),\nthe main result of \\cite{krannich19} implies that \nhomological stability for stabilization with $X$\nis controlled by the connectivity of certain {\\em complexes of\n destabilizations}.\nIn the category of bidecorated surfaces ${{\\mathbf M}_2}$,\nstabilizing with the bidecorated disk $D$ corresponds\nhomotopically to attaching an arc,\nand we show in Proposition~\\ref{prop:Wiso}\nthat the relevant complex of destabilizations\nfor stabilizing a surface $S$ with a disk $n$ times \nidentifies with the\n``disordered arc complex''\\footnote{We called those {\\em disordered} arcs because it is the opposite ordering convention than the one used in the ``ordered arc complex'' of \\cite{RW16}.}\nassociated to the surface $S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}$. This is\na simplicial complex whose vertices are isotopy classes of \nnon-separating arcs in the surface\nwith endpoints $b_0 = I_0(\\rfrac 1 2)$\nand $b_1 = I_1(\\rfrac 1 2)$,\nand where a collection of isotopy classes forms a simplex\nif the classes can be represented by arcs that are disjoint away from\nthe endpoints, are jointly non-separating,\nand such that the arcs have the same ordering at $I_0$ and $I_1$.\n\nWriting $\\mathcal D^\\nu(S_{g,r}, b_0, b_1)$ for the disordered arc\ncomplex of a surface $S_{g,r}$ with marked points $b_0$ and $b_1$ in\n$\\nu=1$ or $\\nu=2$ boundary components, the main ingredient of our proof of homological stability is the\nfollowing connectivity result:\n\n\\begin{ThA}(Theorem~\\ref{thm:disord-cnt})\\label{introthm:C}\n The disordered arc complex $\\mathcal D^\\nu(S_{g,r}, b_0, b_1)$ is $\\(\\frac{2g +\\nu - 5} 3\\)$-connected.\n \\end{ThA}\n\n\n \\begin{remark}\n It is conjectured in \\cite[Conj C]{RWW17} that the complex of destabilizations is highly connected if and only\n if stability holds with all appropriate twisted coefficients.\n The slope $2\/3$ bounds in Theorems~\\ref{thmintro:stab1} and\n \\ref{thmintro:stab2} is precisely dictated by \n the same slope $2\/3$ in Theorem~\\ref{introthm:C} in the connectivity of the arc\n complex, which is the complex of destabilizations in that case.\n This connectivity bound is best possible among linear bounds as a\n better bound would prove an incorrect stability statement, see\n Remark~\\ref{rem:optimal1}.\n \\end{remark}\n\n\n \\subsection*{Organization of the paper.}\n In Section~\\ref{sec:high-cnt}\n we prove the high connectivity of the disordered arc complex.\n In Section~\\ref{sec:category} we define the monoidal category of\n bidecorated surfaces $({{\\mathbf M}_2},\\mathbin{\\text{\\normalfont \\texttt{\\#}}})$, as well as the action of the braid groupoid\n $\\braidGrpd$ on this category.\n In Section~\\ref{sec:stab}, we show Theorems A and B by showing that\n the disordered arc complex agrees with the complex of\n destabilizations, and applying the main result of \n \\cite{krannich19}. \n Finally, in Section~\\ref{sec:braiding} we explain the relationship between\n homological stability and Yang--Baxter operators, and show the\n non-braidedness of the category of bidecorated surfaces.\n\n \\subsection*{Acknowledgements} \nThe first and third authors were partially supported by the Danish\nNational Research Foundation through the Copenhagen Centre for\nGeometry and Topology (DRNF151), and the third author by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 772960). \n\n\n\n\n\n\\section{Homological stability}\n\\label{sec:stab}\n\n\n\nGeneralizing the main result of \\cite{RWW17},\nKrannich associates to an $E_1$-module $\\mathcal M$\nover an $E_2$-algebra $\\mathcal X$\nwith a chosen stabilizing object $X\\in\\mathcal X$,\na {\\em space of destabilizations} at every $A\\in\\mathcal M$,\nwhose high connectivity implies homological stability at $A$\nwhen stabilizing by $X$.\nWe are interested in the case\nwhere $\\mathcal M=B{{\\mathbf M}_2}$ is the classifying\nspace of ${{\\mathbf M}_2}$ and $\\mathcal X=B\\braidGrpd$, acting on $B{{\\mathbf M}_2}$\nvia the map $\\alpha\\colon {{\\mathbf M}_2}\\times \\braidGrpd\\to {{\\mathbf M}_2}$ defined in\nSection~\\ref{sec:braidedaction}. \nWe will pick $A=S\\in {{\\mathbf M}_2}$ to be some surface, with $X=1\\in \\braidGrpd$\n modelling stabilization with the disk as $\\alpha(-,X)=-\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D$ is the sum with the bidecorated disk $D=(X_1,1,\\operatorname{id})$ of\nSection~\\ref{sec:bidecorated}. \n\nGenerally,\nthe space of destabilizations is a semi-simplicial space,\nbut in settings such as ours, it\nis actually levelwise homotopy discrete. Indeed, by \\cite[Lem\n7.6]{krannich19}), \nwhen the structure of $E_1$-module\nover an $E_2$-algebra is induced by an action of a braided monoidal\ncategory on a groupoid, and under the injectivity condition given in \nProposition~\\ref{prop:Mmonoid}, the space of destabilizations \nis equivalent to the following semi-simplicial set,\ndefined just as in \\cite{RWW17} in \nthe case of a braided monoidal groupoid acting on itself.\n\n\\begin{definition}(\\cite[Def 7.5]{krannich19})\\label{def:WS}\n Let $(\\mathcal M,\\oplus)$ be a right module\n over a braided monoidal groupoid $(\\mathcal X,\\oplus,b)$,\n where we denote also by $\\oplus$ the module action.\n Let $A$ and $X$ be objects of $\\mathcal M$ and $\\mathcal X$ respectively.\n The {\\em space of destabilizations} $W_n(A,X)_\\bullet$ is the semi-simplicial\n set with set of $p$-simplices \n\\begin{align*}\n W_n(A,X)_p \n =& \\ \\{(B,f)\\ |\\ B\\in \\textrm{Ob}(\\mathcal M) \\ \\textrm{and}\\ f\\colon B\\oplus X^{\\oplus p+1}\\to A\\oplus X^{\\oplus n} \\ \\textrm{in}\\ \\mathcal M\\}\/_\\sim\n\\end{align*}\nwhere $(B,f)\\sim (B',f')$ if there exists an isomorphism $g\\colon B\\to B'$ in $\\mathcal C$ satisfying that $f=f'\\circ (g\\oplus \\operatorname{id}_{X^{\\oplus p+1}})$. \nThe face map $d_i\\colon W_n(A,X)_p\\to W_n(A,X)_{p-1}$ is defined by $d_i[B,f]=[B\\oplus X, d_if]$ for \n$$d_i f\\colon B\\oplus X\\oplus X^{p} \\xrightarrow{\\operatorname{id}_B\\oplus b_{X^{\\oplus i},X}^{-1}\\oplus \\operatorname{id}_{X^{\\oplus p-i}}} B\\oplus X^{\\oplus i}\\oplus X\\oplus X^{\\oplus p-i}\\xrightarrow{\\ f\\ } A\\oplus X^{\\oplus n},$$\nfor $b_{X^{\\oplus i},X}^{-1}\\colon X\\oplus X^{\\oplus i}\\to X^{\\oplus i}\\oplus X$ coming from the braiding in $\\mathcal X$. \n\\end{definition}\n\n\n\n\n\\subsection{Disk destabilizations and disordered arcs}\\label{sec:disk}\n\n\nGiven a bidecorated orientable\\footnote{The definition of the\n disordered arc complex naturally extend to non-orientable\n bidecorated surfaces, ordering the arcs according to the\n orientations of $I_0$ and $I_1$, but we will only consider orientable surfaces\n here} surface $S = (S,m,\\varphi )$, with $I_0,I_1$ compatibly oriented, let $\\mathcal D(S) = \\mathcal D^\\nu(S,b_0,b_1)$ \ndenote the disordered arc complex of $S$ as in Section~\\ref{sec:high-cnt}, where \n$$\nb_0 = I_0(1\/2)\\quad\\text{and}\\quad b_1 = I_1(1\/2)\n$$\nare the midpoints of the marked intervals, and $\\nu=1$ if $I_0$ and\n$I_1$ lie on the same boundary component and $\\nu=2$ otherwise. \nThe vertices of a simplex in $\\mathcal D(S)$ are canonically\nordered by the anti-clockwise ordering at $b_0$ \n(or equivalently at $b_1$).\nHence we can associate to this simplicial complex a semi-simplicial set\nthat we denote $\\mathcal D(S)_\\bullet$,\nwith same set of $p$-simplices and whose $i$th face map is given\nby forgetting the $(i+1)$st arc with respect to that ordering.\nAs $\\mathcal D(S)$ and $\\mathcal D(S)_\\bullet$ have homeomorphic realizations,\nthey have the same connectivity.\n\n\n\\medskip\n\nWrite $W_n(S,D)_\\bullet$ for the space of destabilization of\nDefinition~\\ref{def:WS} associated to the module \n $\\mathcal M={{\\mathbf M}_2}$ over the $E_2$--algebra $\\mathcal X=\\braidGrpd$ acting on ${{\\mathbf M}_2}$ as above,\nwith \n$X=1\\in \\braidGrpd$,\nand $A=S=(S_{g,r}^s,m,\\varphi)$ some bidecorated\norientable surface of small genus \n$g\\ge 0$, with $r$ boundary components and $s$ punctures. \nThe space $W_n(S,D)_\\bullet$ is then the space of destabilizations\nof the stabilization map\n$$\n\\Aut_{{\\mathbf M}_2} (S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n-1}) \\xrightarrow{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D} \\Aut_{{\\mathbf M}_2} (S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})\n$$\nthat attaches an additional disk to the surface along the two marked\nintervals.\n\n\n\n\nWe want to identify $W_n(S,D)_\\bullet$ with $\\mathcal D(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\nD^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})_\\bullet$. \nFor this, we start by constructing a particular disordered collection of arcs in $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}$.\nWrite again\n$$\nD^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n} = D_1\\mathbin{\\text{\\normalfont \\texttt{\\#}}} \\dots \\mathbin{\\text{\\normalfont \\texttt{\\#}}} D_i\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\\dots\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\nD_n, \n$$\nand let $\\rho_i$ denote the unique isotopy class of arc in the $i$th disk $D_i$\ngoing from $b_0 = I_0(1\/2)$ to $b_1 = I_1(1\/2)$.\n\\begin{lemma}\\label{lem:rhoord}\n The arcs $\\rho_1,\\dots,\\rho_m$ are ordered anti-clockwise at both $b_0$ and $b_1$.\n\\end{lemma}\n\n\n\\begin{proof}\n It suffices to show that $\\rho_i$ and $\\rho_{i+1}$ are ordered anti-clockwise\n at $b_0$ and $b_1$ for each $i$.\n Thus we need only consider what happens in the subsurface $D_i\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D_{i+1}$.\n The gluing being defined in exactly the same way at $I_0$ and $I_1$,\n the arcs are ordered in the same way at both endpoints,\n and the particular choice of gluing gives the anti-clockwise ordering,\n see Figure~\\ref{fig:rhoiorder}.\n\\end{proof}\n\\begin{figure}\n \\def0.8\\textwidth{0.8\\textwidth}\n \\input{ordering-of-rhoi.pdf_tex}\n \\caption{Ordering of the arcs $\\rho_i$ at their endpoints}\\label{fig:rhoiorder}\n\\end{figure}\n\n\nRecall from Section~\\ref{sec:braidedaction} the Dehn twist $T_i$ along\nthe curve $a_i$ in $D_i\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D_{i+1}$.\nThe union of the arcs $\\rho_i$ in $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} m}$ define a deformation retract of the surface,\nas each disk $D_i$ retracts onto the corresponding arc $\\rho_i$, \nand we can understand the action of the twists $T_i$ on the surface\nby considering their action on the arcs $\\rho_i$. The action is given\nby the following result, that will be needed to compare the face maps\nin the semi-simplicial sets $W_n(S,D)_\\bullet$ with $\\mathcal D(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\nD^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})_\\bullet$. \n\\begin{lemma}\\label{lem:Trho}\n The action of the Dehn twist $T_i$ along the curve $a_i$ on the homotopy classes of the arcs $\\rho_i$, relative to their endpoints, is \n $$\n T_i(\\rho_j)\n =\n \\begin{cases}\n \\rho_i\\overline{\\rho_{i+1}}\\rho_i &\\text{if } j = i, \\\\\n \\rho_{i} &\\text{if } j = i+1, \\\\\n \\rho_j &\\text{else.}\n \\end{cases}\n $$\n Equivalently, \n $$T_i^{-1}(\\rho_i)=\\rho_{i+1} \\ \\ \\ \\textrm{and} \\ \\ \\ T_i^{-1}(\\rho_{i+1})=\\rho_{i+1}\\overline{\\rho_i}\\rho_{i+1}$$\n and $T_i^{-1}$ leaves the other $\\rho_j$ invariant. \n\\end{lemma}\n\\begin{proof}\n The Dehn twist $T_i$ can only affect $\\rho_i$ and $\\rho_{i+1}$ as the curve $a_i$ only intersects these two arcs, from which the last case in the statement follows.\n The computation for the arcs $\\rho_i$ and $\\rho_{i+1}$ is local to\n $D_i\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D_{i+1}$, where, as shown in Figure~\\ref{fig:twist},\n we have $T_i(\\rho_i)\\simeq\\rho_i\\overline{\\rho_{i+1}}\\rho_i$,\n giving the first case in the statement,\n and $T_i(\\rho_{i+1})\\simeq\\rho_i$, giving the second case.\n\\end{proof}\n\\begin{figure}\n \\def0.8\\textwidth{0.8\\textwidth}\n \\input{action-of-t-on-rho.pdf_tex}\n \\caption{The action of the Dehn twist $T_i$ on the arcs\n $\\rho_i$ (top) and $\\rho_{i+1}$ (bottom)}\\label{fig:twist}\n\\end{figure}\n \n \\begin{proposition}\\label{prop:Wiso}\n Let $S=(S,m,\\varphi)$\n be an object of ${{\\mathbf M}_2}$.\n There is an isomorphism of semi-simplicial sets\n $$W_n(S,D)_\\bullet\\cong \\mathcal D^\\nu(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})_\\bullet$$\n where the marked points $b_0$ and $b_1$ are the midpoints of the\n intervals $I_0$ and $I_1$ in $S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}$ and with \n $\\nu=\\operatorname{parity}(m+n)$, that is $\\nu=1$ if $I_0$ and $I_1$ lie in the same boundary component of $S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}$ and $\\nu=2$ otherwise. \n \\end{proposition}\n\\begin{proof}\n We first show that both $W_n(S,D)_p$\n and $\\mathcal D^\\nu(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})_p$ are isomorphic to\n $\\Aut_{{\\mathbf M}_2}(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})\/\\Aut_{{\\mathbf M}_2}(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n-p-1})$\n for every $p\\ge 0$.\n This holds by definition for the first semi-simplicial set.\n For $\\mathcal D^\\nu(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})_p$,\n it will follow from two facts:\n (1) the natural action of\n $$\n \\Aut_{{\\mathbf M}_2} (S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}) = \\pi_0\\f{Homeo}_\\partial (S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\n n})\n $$\n on this set of $p$-simplices is transitive, and\n (2) the stabilizer of a $p$-simplex is isomorphic to\n $\\Aut_{{\\mathbf M}_2}(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n-p-1})$.\n The first fact follows because the homeomorphism type\n of the complement $S\\setminus \\sigma$ of a collection of\n non-separating arcs $\\sigma=\\$ is determined\n by the orderings of the arcs at the endpoints as this\n determines the number of boundary components of the complement (see\n \\cite[Lem 3.2]{harer85}),\n and the second from the fact that this complement is precisely diffeomorphic\n to $S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n-p-1}$ for any $p$-simplex in the disordered\n arc complex. Indeed, \n this diffeomorphism type does not depend on the simplex by\n transitivity of the action, so it is\n enough to check the claim for any chosen\n simplex. Let $$\\sigma_p=\\<\\rho_{n-p},\\dots,\\rho_n\\>$$\n be the collection of arcs in $S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}$ consisting of the\n cores $\\rho_i$ of the last $p+1$ disks. Recall from Lemma~\\ref{lem:rhoord}\n that this is a disordered simplex, once we note additionally that the arcs are\n also non-separating. Now Figure~\\ref{fig:cutrho} shows\n that the operation of cutting along the core $\\rho$ of a disk exactly undoes\n the gluing operation, which proves the claim in that case.\n \\begin{figure}\n \\def0.8\\textwidth{0.8\\textwidth}\n \\input{cutting-along-disc.pdf_tex}\n \\caption{Cutting along the core of a disk}\\label{fig:cutrho}\n \\end{figure}\n\n \n\nNote that the actions on both sets of simplices are given by post-composition with mapping classes, where we think here of an arc as an isotopy class of embedding. \nThere is then a unique equivariant \nisomorphism\n$\\phi_p\\colon W_n(S,D)_p\\xrightarrow{\\cong} \\mathcal D^\\nu(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})_p$\ntaking the $p$-simplex \n$$\nf_p=(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n-p-1}, \\,\\operatorname{id}_{S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}}) \n$$\nof $W_n(S,D)$ to the $p$-simplex \n$\\sigma_p=\\<\\rho_{n-p},\\dots,\\rho_n\\>$ of the target already considered above. \n\n\nWe are left to check that the face maps $d_i$ correspond to each other under the isomorphisms $\\phi_p$.\nBecause the face maps are equivariant with respect to the\n$\\Aut_{{\\mathbf M}_2} (S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})$-action in both cases,\nand the actions are transitive,\nit is enough to check that the face maps agree for the simplices\n$f_p$ and $\\sigma_p=\\phi_p(f_p)$. By definition, \n$$\nd_if_p\n=((S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n-p-1})\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D,\\,\n\\operatorname{id}_{S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n-p-1}}\\mathbin{\\text{\\normalfont \\texttt{\\#}}} b_{D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} i},D}^{-1}\\mathbin{\\text{\\normalfont \\texttt{\\#}}} \\operatorname{id}_{D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} p-i}})\n$$\nwhile\n$$\nd_i\\sigma_p=\\<\n\\rho_{n-p},\\dots,\\widehat{\\rho_{n-p+i}},\\dots,\\rho_n\\>$$ is the simplex obtained by forgetting the $(i+1)$st arc.\nIn particular, we immediately have that $d_0(f_p)=f_{p-1}$ and $d_0(\\sigma_p)=\\sigma_{p-1}=\\phi_{p-1}(f_{p-1})$ giving that the face maps agree in that case.\n\nFor the remaining face maps, note that $$\\operatorname{id}_{S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n-p-1}}\\mathbin{\\text{\\normalfont \\texttt{\\#}}} b_{D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} i},D}^{-1}\\oplus \\operatorname{id}_{D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} p-i}}=T_{n-p+i-1}\\circ \\dots\\circ T_{n-p}\\colon S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}\\longrightarrow S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}$$\nas composition of Dehn twists $T_i$ of\nSection~\\ref{sec:braidedaction}.\nWe need to compute the image of $\\rho_{n-p+1},\\dots,\\rho_n$ under this map. \nBy Lemma~\\ref{lem:Trho}, we have that for $1\\le j\\le i$, \n\\begin{align*}\n T_{n-p+i-1}\\circ \\dots\\circ T_{n-p}(\\rho_{n-p+j})&=T_{n-p+i-1}\\circ \\dots\\circ T_{n-p+j-1}(\\rho_{n-p+j})\\\\\n &=T_{n-p+i-1}\\circ \\dots\\circ T_{n-p+j}(\\rho_{n-p+j-1}) \\\\\n &=\\rho_{n-p+j-1}\n\\end{align*}\nwhile for $i+1\\le j\\le p$, \n$$ T_{n-p+i-1}\\circ \\dots\\circ T_{n-p}(\\rho_{n-p+j})=\\rho_{n-p+j}.$$\nHence $d_i(f_p)$ takes the arcs $\\rho_{n-p+1},\\dots,\\rho_n$ to the\narcs $$\\rho_{n-p},\\dots,\\rho_{n-p+i-1},\\rho_{n-p+i+1},\\dots,\\rho_n,$$\ni.e.~precisely to the arcs of $d_i(\\sigma_p)$. So we indeed have that\n$\\phi_{p-1}(d_i(f_p))=d_i(\\phi_{p-1}(f_p))$, which finishes the\nproof. \n \\end{proof}\n\n\n\n\n\\subsection{Coefficient systems}\n\nHaving identified the space of destabilizations with the semi-simplicial\nset of disordered arcs in Proposition~\\ref{prop:Wiso},\nwe can now input the connectivity computation of the disordered arc complex\nof Section~\\ref{sec:high-cnt}\ninto the general stability theorem of \\cite{krannich19}.\nTo state the resulting stability theorem in full generality,\nwe need to introduce the notions of (split) finite degree coefficient systems.\nWe follow \\cite[Sec 4]{krannich19}, which generalizes \\cite[4.1-4]{RWW17} that unify the earlier definitions of Dwyer for the\ngeneral linear groups \\cite{Dwyer} and Ivanov for the mapping class groups\n\\cite{IvanovTwisted}.\n(The papers \\cite{krannich19,RWW17} consider in addition abelian coefficient\nsystems, but these are not relevant here, because the abelianization of the\nmapping class group of surfaces of large enough genus is trivial by a theorem\nof Mumford--Birman--Powell, see Lemma 1.1 in~\\cite{harer83}.) \n\n\\medskip\n\nFix a bidecorated surface $S=(S,m,\\varphi)$,\nand let $D$ be the bidecorated disk as above.\nDefinition~4.1 of\n\\cite{krannich19} becomes in our case: \n\\begin{definition}\n A {\\em coefficient system} for the groups\n $\\Aut_{{\\mathbf M}_2}(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})$ with respect to the stabilization by $D$ is\na collection of $\\mathbb{Z}[\\Aut(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})]$-modules $M_n$ for\n$n\\ge 0$, together with maps $s_n\\colon M_n\\to M_{n+1}$ that are\nequivariant with respect to the stabilization map\n$\\Aut(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})\\xrightarrow{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D} \\Aut(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n+1})$, \nsatisfying the following condition:\n\\begin{equation}\\label{coef-cond}\n T_{n+1}\\in \\Aut(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n+2})\\ \\textrm{acts trivially on the\n image of } M_n \\xrightarrow{s_{n+1}\\circ s_n} M_{n+2}\n\\end{equation}\nfor $T_{n+1}$ the Dehn twist of Section~\\ref{sec:braidedaction} with support the last\ntwo disks in $S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n+2}$. \n\\end{definition}\nWe will encode the data of a coefficient system as a pair $(F,\\sigma^F)$\nwith \n$$F\\colon {{\\mathbf M}_2}|_{S,D} \\longrightarrow \\operatorname{Mod}_\\mathbb{Z}$$\na functor from the full subcategory of ${{\\mathbf M}_2}$ on the objects $S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}$\nfor $n\\ge 0$ to abelian groups, where $M_n=F(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}}\n n})$ with its $\\Aut(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})$-action induced by $F$, and \n$$\\sigma^F\\colon F(-) \\longrightarrow F(-\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D)$$\nis a natural transformation encoding the suspension maps $s_n$,\nwhere we assume that $F(\\operatorname{id} \\mathbin{\\text{\\normalfont \\texttt{\\#}}} T)$ acts trivially on the image of\n$(\\sigma^F)^2\\colon F(-) \\to F(-\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2})$ for $T$ the Dehn twist\nsupported on the added disks $D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2}$.\n\n\n\\medskip\n\nGiven a coefficient system $F$, we define its {\\em suspension} $\\Sigma F\\colon {{\\mathbf M}_2}|_{S,D} \\longrightarrow \\operatorname{Mod}_\\mathbb{Z}$ by $\\Sigma F(-)=F(-\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D)$ with\n$$\\sigma^{\\Sigma F}\\colon \\Sigma F(-)=F(-\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D) \\xrightarrow{\\sigma^F} F(-\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2}) \\xrightarrow{\\operatorname{id}\\mathbin{\\text{\\normalfont \\texttt{\\#}}} T} F(-\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2})=\\Sigma F(-\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D), $$\nwhere one checks that the triviality condition \\ref{coef-cond} is\nsatisfied with this choice of structure map $\\sigma^{\\Sigma F}$. (See\n\\cite[Def 4.4]{krannich19}.)\n\nThe structure map $\\sigma^F$ induces a natural transformation $F\\to \\Sigma F$, called the {\\em suspension map}. We define the {\\em kernel} $\\ker F$ and {\\em cokernel} $\\coker F$ to be the kernel and cokernel functors of that natural transformation. \nWe call $F$ {\\em split} if the suspension map is split injective in the category of coefficient systems. \n\n\\begin{definition}\\label{def:coef-system}\\cite[Def 4.10]{RWW17} A coefficient system $F$ is\n\\begin{enumerate}\n\\item of {\\em (split) degree $-1$ at $N$} if $F(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}}(D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}))=0$ for all $n\\ge N$; \n\\item of {\\em degree $k\\ge 0$ at $N$} if $\\ker(F)$ has degree $-1$ at $N$ and $\\coker(F)$ has degree $(k-1)$ at $(N-1)$; \n\\item of {\\em split degree $k\\ge 0$ at $N$} if $F$ is split and $\\coker(F)$ is of split degree $(k-1)$ at $(N-1)$.\n \\end{enumerate}\n\\end{definition}\n\\begin{example}\n \\leavevmode\n \\begin{enumerate}\\label{exmp:coef-systems}\n \\item A coefficient system $F$ is of degree $0$ at $0$\n if and only if $\\sigma^F$ is a natural isomorphism. This is in\n particular the case for constant coefficient systems. \n \\item The functor $F_k\\colon{{\\mathbf M}_2}\\to \\operatorname{Mod}_\\mathbb{Z}$ defined by\n$$F_k(S)=H_1(S,\\mathbb{Z})^{\\otimes k}$$\nis a split coefficient system of degree $k$ at $0$. \n(This is essentially a result of Ivanov \\cite[Sec 2.8]{IvanovTwisted},\nwho considers a version of the composite stabilization $\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2}$. \nSee also \\cite[Ex 4.3]{boldsen12} for the case $k=1$, and \\cite[Lem 2.9]{Soulie} that proves this in a very general\nset-up, though in the case of a braided groupoid acting over itself only.)\n\\item Given a $k$-connected space $X$,\n the coefficient system $F_n^k\\colon{{\\mathbf M}_2}\\to\\text{Mod}_\\mathbb{Z}$ defined by\n $$\n F_n^k(S) = H_n(\\operatorname{Map}(S\/\\partial S),X),\n $$\n which appears in the work of Cohen--Madsen~\\cite{CohMad},\n is a coefficient system of degree $\\floor{n\/k}$ (see \\cite[Ex 4.3]{boldsen12}).\n \\end{enumerate}\n\\end{example}\n\n\\begin{remark}\\label{rmk:different-category}\nAlthough the above examples all makes sense in the different set-ups\nconsidered in the literature, one should keep in mind that there are\nvariations in what precisely a finite degree coefficient system for\nthe mapping class groups of surfaces means in e.g.~the papers\n\\cite{IvanovTwisted,CohMad,boldsen12,RWW17} and \\cite{krannich19}.\nThis is due to two facts: first, the definition of the coefficient\nsystem depends on the category of surfaces considered and on the stabilization map(s) one works with,\nand second, the triviality\ncondition \\eqref{coef-cond} arising from Krannich's framework is\nactually weaker than the one used in earlier frameworks, see\ne.g.~\\cite[Rem 7.9]{krannich19}.\n\nIn addition, the paper \n\\cite{GKRW19MCG} uses a homological\ncondition instead of a finite degree condition (see 5.5.1 in that\npaper). The relationship between that condition and finite degree\nconditions is discussed in \\cite[Rem 19.11]{GKRW18cell}.\n \\end{remark}\n\n\n\n\n \\subsection{The stability theorem}\n\nWe are now ready to state our main theorem: \n \\begin{theorem}\\label{thm:stab} Let $S=(S,m,\\phi)$ be an object of\n ${{\\mathbf M}_2}$ with $m$ odd, i.e.~such that $I_0,I_1$ are in the same boundary component. \n Let $F\\colon{{\\mathbf M}_2}|_{S,D}\\to\\operatorname{Mod}_\\mathbb{Z}$ be a coefficient system\n and write $F_n = F(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})$.\n The map\n $$\n H_i(\\Aut_{{\\mathbf M}_2}(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n});F_n)\n \\longrightarrow H_i(\\Aut_{{\\mathbf M}_2}(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n+1});F_{n+1})\n $$ \n is\n \\begin{enumerate}\n \\item an epimorphism\n for $i\\leq\\frac{n} 3$\n and an isomorphism for $i\\leq\\frac{n-3} 3$ if $F$ is constant.\n \\item an epimorphism\n for $i\\leq \\frac{n-3k-2} 3$\n and an isomorphism for $i\\leq\\frac{n - 3k - 5} 3$\n if $F$ has degree $k$\n at $N\\geq 0$ and $n>N$.\n \\item an epimorphism\n for $i\\leq\\frac{n-k-2} 3 $\n and an isomorphism for $i\\leq\\frac{n-k-5} 3$\n if $F$ has split degree $k$\n at $N\\geq 0$ and $n>N$.\n \\end{enumerate}\n \\end{theorem}\n\n \\begin{remark}\nWe have stated the theorem in the case of an initial surface $S$ with\n$I_0$ and $I_1$ in the same boundary component for simplicity. The\ncase of a surface $S'$ where the two intervals lie in different components is actually\nalso included in the statement, by writing $S'=S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D$ for $S$ of\nthe previous type, or considering $S'\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D$ if $S'$ does not admit\nsuch a decomposition. Indeed, as we have already seen in\nSection~\\ref{sec:category} (see Figure~\\ref{fig:glueD}),\ngluing in a disk exactly changes whether\n$I_0$ and $I_1$ are in the same boundary or not. \n \\end{remark}\n\n\nWe will first show that the above results implies the two main\ntheorems stated in the introduction.\n\n\\begin{proof}[Proof of Theorems~\\ref{thmintro:stab1}\n and~\\ref{thmintro:stab2} from Theorem~\\ref{thm:stab}]\nLet $S_{0,r}^s$ be a surface of genus 0 with $r\\ge 1$ boundary components and\n$s$ punctures, and consider the associated object\n$S=(S_{0,r}^s,1,\\phi)$ of ${{\\mathbf M}_2}$, with two marked intervals in the first\nboundary component. \nThen $S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2g}$ has the form $(S_{g,r}^s,1+2g,\\phi)$ while\n$S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2g+1}$ has the form\n$(S_{g,r+1}^s,2+2g,\\phi)$. Moreover, the maps\n $$S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2g} \\xrightarrow{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D} S \\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2g+1}\n \\xrightarrow{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D}S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2g+2}$$\n precisely induce on automorphism groups in ${{\\mathbf M}_2}$ the two maps appearing in Theorems~\\ref{thmintro:stab1}\n and~\\ref{thmintro:stab2}.\n \nThe fact that the first map is always injective in homology follows from the fact\nthat postcomposing the map $S_{g,r}^s\\to S_{g,r+1}^s$,\ndefined by the sum $\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D$, with the map $S_{g,r+1}^s \\to\nS_{g,r+1}^s\\cup_{S^1}D^2\\simeq S_{g,r}^s$ filling in one of the newly created\nboundary component, is homotopic to the identity. \nNow Theorem~\\ref{thm:stab}(a) gives that the map\n$$H_i(\\Aut_{{{\\mathbf M}_2}}(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2g})) \\xrightarrow{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D} H_i(\\Aut_{{{\\mathbf M}_2}}(S \\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} 2g+1}))$$\nis surjective for\n$i\\leq\\frac{2g} 3$ in homology with constant coefficients. Given that the map is always injective, we get an isomorphism \nin that same range, proving the first part of\nTheorem~\\ref{thmintro:stab1}. Applying (b) and (c) instead gives\nTheorem~\\ref{thmintro:stab2} for the first map.\n\nFor the second map, we now apply Theorem~\\ref{thm:stab} in the case $n=2g+1$, but\nin that case, there is no additional argument for injectivity, so the\nbounds translate directly to surjectivity and isomorphism bounds. \n \\end{proof}\n \n \n\n \n \n \\begin{proof}[Proof of Theorem~\\ref{thm:stab}]\n Proposition~\\ref{prop:Wiso} together with Theorem~\\ref{thm:disord-cnt}\n give that $W_n(S,D)_\\bullet$ is $\\left(\\frac{2g+\\nu-5} 3\\right)$-connected, for $g$ the genus of $S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}$ and $\\nu=1$ if $I_0$ and $I_1$ are in the same boundary component of $S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}$, which is the case precisely when $n$ is even, and $\\nu=2$ otherwise. \n The surface $S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}$ has genus greater than or equal to the genus of $D\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n}$, that is $\\frac{n}{2} $ if $n$ is even and $\\frac{n-1}{2}$ if $n$ is odd (see Lemma~\\ref{lem:surface-type}).\nHence $2g+\\nu\\ge n+1$ is both cases, and $W_n(S,D)_\\bullet$ \nis at least $\\left(\\frac{n-4} 3\\right)$-connected. \n\n\nNow $W_n(S,D)_\\bullet$ is the semi-simplicial set denoted $W^{\\text{RW}}(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})_\\bullet$ in \\cite{krannich19} (see Definition~7.5 in that paper). By Lemma 7.6 in the same paper, using Proposition~\\ref{prop:Mmonoid}, this semi-simplicial set has the same connectivity as the semi-simplicial space $W(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})_\\bullet$ of \\cite{krannich19}, which by Remark~2.7 of that paper determines the connectivity assumption of Theorem A in that paper: the canonical resolution of the assumption of the theorem is $m$-connected, if and only if the space $W(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})_\\bullet$ is $(m-1)$-connected.\nGiven that $W(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})_\\bullet$ is $\\left(\\frac{n-4}\n 3\\right)$--connected, we have that the canonical resolution of\n is $\\left(\\frac{n-4+3} 3\\right)$--connected. Hence we can apply\n \\cite[Thm A]{krannich19} with $k=3$ and grading $g_{{{\\mathbf M}_2}}:{{\\mathbf M}_2}|_{S,D}\\to\n \\N$ given by $g_{{{\\mathbf M}_2}}(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n})=n-2$; see also \\cite[Rem\n 2.24]{krannich19}, where we can take $m=4$.\nThe theorem, with the improvement given by (i) in the remark, then gives that\n $$\n H_i(\\text{Aut}_{{{\\mathbf M}_2}}(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n});\\mathbb{Z} )\\longrightarrow\n H_i(\\text{Aut}_{{{\\mathbf M}_2}}(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n+1});\\mathbb{Z} )\n $$\n is an isomorphism for $i\\leq\\frac{n-3} 3$ and\n an epimorphism for $i\\leq \\frac{n}3$, giving the stated result\n in the case of constant coefficients. \n For a coefficient system $F$ of degree $k$ at $N$,\n \\cite[Thm C]{krannich19} gives that\n $$\n H_i(\\text{Aut}_{{{\\mathbf M}_2}}(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n});F_n )\\longrightarrow\n H_i(\\text{Aut}_{{{\\mathbf M}_2}}(S\\mathbin{\\text{\\normalfont \\texttt{\\#}}} D^{\\mathbin{\\text{\\normalfont \\texttt{\\#}}} n+1});F_{n+1} )\n $$\n is an isomorphism for $i\\leq\\frac{n-3k-5} 3$ and\n an epimorphism for $i\\leq \\frac{n-3k -2} 3 $\n for $n>N$, improved to an isomorphism for $i\\leq\\frac{n-k-5} 3$ and\n an epimorphism for $i\\leq \\frac{n-k -2} 3 $ if $F$ is split. \n \\end{proof}\n\n\n \n \\begin{remark}[Optimality of the stability bounds]\\label{rem:optimal2}\nCombining the two maps in Theorem~\\ref{thmintro:stab1}, \nwe obtain that the genus stabilization \n\\begin{equation*}\n H_i(\\Gamma(S_{g,r}^s);\\mathbb{Z} )\\to H_i(\\Gamma(S^s_{g+1,r}); \\mathbb{Z}) \n\\end{equation*}\nis an epimorphism when $i\\leq\\frac{2g} 3$\nand an isomorphism when $i\\leq\\frac{2g-2} 3$.\nThe slope $\\frac{2}{3}$ is known to be optimal by a computation of\nMorita \\cite{Morita}, with optimal isomorphism range\nsince for instance $H_1(\\Gamma (S_{2,r});\\mathbb{Z} )\\to H_1(\\Gamma (S_{3,r});\\mathbb{Z} )$\nis not injective as the source is isomorphic to $\\mathbb{Z} \/12$ and\nthe target is trivial, see e.g.~\\cite[Theorem 5.1]{korkmaz}.\nOur combined genus epimorphism range, on the other hand, falls short\nof the range $i\\leq\\frac{2g+1} 3$, as given in\n\\cite{GKRW19MCG}, a range that is optimal by Morita's computation\n(see Theorem B (i) of \\cite{GKRW19MCG}).\n \n\n\n Our results for twisted coefficients are most easily compared with those\n of Boldsen \\cite[Thm 3]{boldsen12}, whose coefficient systems are \n coefficient systems of finite split degree in our sense, \n though with a stricter triviality condition upon double stabilization.\n For these coefficient systems, he obtains slightly better\n ranges, with improvement $+\\rfrac{2}{3}$ for the first\n map and $+\\rfrac{5}{3}$ for the second.\nThe papers \\cite{RWW17,GKRW19MCG} only consider genus stability. \n In \\cite{RWW17}, the stability slope obtained is only\n $\\frac{1}{2}$, while in \\cite[Sec 5.5.1]{GKRW19MCG}, the\n finite degree condition is replaced by a more general homological\n condition that applies to some finite coefficient systems\n \\cite[Sec 19.2]{GKRW18cell}. In the particular\n case of the $k$th tensor\n power of the first homology of the surface, they do however\n only get the epimorphism range $i\\leq\\frac{ 2g-2k+1} 3$\n and isomorphism range $i\\leq\\frac{ 2g-2k-2} 3$, see Example 5.22\n in that paper. \n \\end{remark}\n\n\n\n\n \n\n\n\n\n\n\n\n\n \n\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Calculation of the Quadrupole Asymmetry parameter}\n\nThe quadrupole asymmetry parameter $\\Delta Q$ in the main text, is defined as\n\\be\n \\Delta Q \\equiv \\int\\!\\! \\text{d}S\\, \\varrho(\\rr) \\pare{z^2-x^2},\n\\ee\nwhere $\\varrho(\\rr)$ is the surface charge density, and $\\rr=(x,y,z)$ is the coordinate of a point on the surface of the spheroid with respect to the body-fixed frame, and the integral is taken on the surface of the spheroid. \nAssuming a uniformly charged particle $\\varrho(\\rr)=q\/S$ and introducing the coordinates $\\rr=(a\\cos\\phi\\sin\\theta,a\\sin\\phi\\sin\\theta,b\\cos\\theta)^T$, we obtain\n\\be\\label{eq:DeltaQ_EllipsCoord}\n \\Delta Q = q\\frac{ab^3}{S} \\int_0^{2\\pi}\\!\\!\\text{d}\\phi \\int_{-1}^{1}\\!\\!\\text{d}\\xi\\, \\sqrt{\\pare{1-\\xi^2}+\\pare{\\frac{a}{b}}^2\\xi^2}\\spare{\\xi^2-\\pare{\\frac{a}{b}}^2\\pare{1-\\xi^2}\\cos^2\\phi}.\n\\ee\nFor a prolate spheroid for which $a\/b\\ll1$, we expand \\eqnref{eq:DeltaQ_EllipsCoord} to second order in $a\/b$, and obtain\n\\be\n \\Delta Q \\simeq \\frac{q}{4}b^2\\spare{1+2\\pare{\\frac{a}{b}}^2}.\n\\ee\n\n\\section{Definition of Euler Angles and Canonical Angular Momenta}\n\nWe define the transformation between the laboratory-fixed frame $O\\bold{e}_1\\bold{e}_2\\bold{e}_3$ and the body-fixed frame $O\\nn_1\\nn_2\\nn_3$ as $\\nn_k= {\\sf R}(\\W)\\bold{e}_k$ according to the $zy'z''$ convention for the Euler angles, namely\n\\be\\label{eq:R_euler}\n {\\sf R}(\\W)\\equiv \\begin{pmatrix}\n\t\t\\cos \\alpha & -\\sin \\alpha & 0\\\\\n\t\t\\sin \\alpha & \\cos \\alpha & 0\\\\\n\t\t0 & 0 & 1\n\t\\end{pmatrix}\n\t\\begin{pmatrix}\n\t\t\\cos \\beta & 0 & \\sin \\beta \\\\\n\t\t0 & 1 & 0\\\\\n\t\t-\\sin \\beta & 0 & \\cos \\beta \\\\\n\t\\end{pmatrix}\\\\\n\t\\begin{pmatrix}\n\t\t\\cos \\gamma & -\\sin \\gamma & 0\\\\\n\t\t\\sin \\gamma & \\cos \\gamma & 0\\\\\n\t\t0 & 0 & 1\n\t\\end{pmatrix}.\n\\ee\nWithin this convention the components of the total angular momentum operators $\\hbar\\Jop_i$ ($i=1,2,3$) along the body-fixed principal axes are represented in orientation space by the following differential operators\n\\be\\label{eq:Jvec_euler}\n\t\\vect{\\Jop_1}{\\Jop_2}{\\Jop_3} =\\frac{-\\im}{\\sin\\beta}\n\t\\begin{pmatrix}\n\t-\\cos\\gamma\t& \\sin\\gamma \\sin\\beta & \\cos\\gamma \\cos\\beta\\\\\n\t\\sin\\gamma & \\cos\\gamma \\sin\\beta & -\\sin\\gamma \\cos\\beta\\\\\n\t0 & 0 & \\sin\\beta\n\t\\end{pmatrix}\n\t\\vect{\\pa{\\alpha}}{\\pa{\\beta}}{\\pa{\\gamma}}.\n\\ee\nThe canonical momenta $\\pop_\\alpha,\\pop_\\beta,$ and $\\pop_\\gamma$ are then given by the operators\n\\bea\n\t\\pop_\\alpha &\\equiv&-\\im \\hbar\\frac{\\partial}{\\partial \\alpha},\\label{eq:pop_alpha}\\\\\n\t\\pop_\\beta &\\equiv& -\\im\\hbar\\pare{\\frac{\\partial}{\\partial\\beta}+\\inv{2}\\cot\\beta},\\label{eq:pop_beta}\\\\\n \\pop_\\gamma &\\equiv& -i \\hbar \\frac{\\partial}{\\partial\\gamma},\\label{eq:pop_gamma}\n\\eea\nwhere the definition \\eqnref{eq:pop_beta} stems from standard canonical quantisation in curved space~\\cite{Gneiting2013Supp,DeWitt1952Supp}.\nWe remark that Eq.~(\\ref{eq:pop_alpha}-\\ref{eq:pop_gamma}) guarantee the standard canonical commutation relations.\nWith the definitions Eq.~(\\ref{eq:R_euler}-\\ref{eq:pop_gamma}) the rotational kinetic energy of the symmetric rotor reads\n\\be\\label{eq:H_rot}\n\\begin{split}\n\t\\Hop_\\text{rot} = \\frac{(\\pop_\\alpha - \\cos \\hat{\\eulb} \\pop_\\gamma)^2}{2I\\sin^2\\beta} + \\frac{\\pop_\\beta^2 }{2I} + \\frac{\\pop_\\gamma^2}{2I_3}-\\frac{\\hbar^2}{2I}\\pare{1+\\frac{1}{\\sin^2\\hat{\\eulb}}}.\n\\end{split}\n\\ee\nHere the last term is the so-called quantum potential~\\cite{Gneiting2013Supp}.\n\n\\section{Derivation of the dispersive Hamiltonian}\n\nWe start from \\eqnref{eq:Ham0} and derive the dispersive Hamiltonian of the system \\eqnref{eq:Hdisp}.\nFirst, we separate the macromotion from the micromotion similarly to what is done for the center-of-mass motion for trapped ions~\\cite{Cook1985Supp}.\nStarting from the Schr\\\"odinger equation $\\pa{t}\\ket{\\psi(t)}=\\Hop(t)\\ket{\\psi(t)}$, we define\n\\be\\label{eq:Ansatz_psi_t} \n\t\\ket{\\psi(t)} \\equiv e^{-\\im\\hat{\\mathcal{A}}(t)}\\ket{\\psi_\\text{sec}},\n\\ee\nwhere\n\\be\\label{eq:A_t}\n\\begin{split}\n \\hat{\\mathcal{A}}(t) \\equiv& \\frac{1}{\\hbar \\omega_0} \\bigg\\{ \\spare{\\frac{I \\w_0^2}{2} (1 - \\delta) \\sin^2 \\hat{\\eulb}} \\sin(\\omega_0t) - \\frac{1}{2}\\spare{ \\varepsilon\\frac{I\\w_0^2}{4} \\pare{1 + \\frac{\\delta}{3}} \\sin^2 \\hat{\\eulb} \\cos (2 \\hat{\\alpha})} \\sin (2\\omega_0t)\\\\\n &- \\frac{1}{2}\\spare{ \\varepsilon\\frac{I\\w_0^2}{4} \\pare{1 + \\frac{\\delta}{3} } \\sin^2 \\hat{\\eulb} \\sin (2 \\hat{\\alpha})} \\cos(2\\omega_0t)\\bigg\\}.\n\\end{split}\n\\ee\nThe micromotion (macromotion) is encoded in the rapidly (slowly) evolving phase $e^{\\im \\Aop(t)}$ (state $\\ket{\\psi_\\text{sec}(t)}$). Substituting this definition in the Schr\\\"odinger equation one obtains\n\\be\\label{eq:Secular_Schroedinger_Eq}\n \\im\\hbar \\pa{t} \\ket{\\psi_\\text{sec}} =\\pare{\\Hop(t)+\\pa{t}\\hat{\\mathcal{A}}(t)+ V_I(t)} \\ket{\\psi_\\text{sec}},\n\\ee\nwhere we defined\n\\be \\label{eq:VI}\n \\Vop_{I}(t) \\equiv e^{\\im \\hat{\\mathcal{A}}(t)} \\left [ \\Hop_\\text{rot} - \\frac{\\omega_0}{2}\\pop_\\alpha, e^{-\\im \\hat{\\mathcal{A}}(t)} \\right ].\n\\ee\nLet us note that $\\pa{t}\\hat{\\mathcal{A}}(t)$ cancels exactly the time-dependent potential $\\Vop(t)$ inside $\\Hop(t)$ in \\eqnref{eq:Ham0}. So far no approximation has been introduced and the solution of \\eqnref{eq:Secular_Schroedinger_Eq} is equivalent to the solution of the Sch\\\"odinger equation for $\\ket{\\psi(t)}$.\nThe secular approximation consists in replacing the rapidly oscillating potential \\eqnref{eq:VI} by its time average, $\\Vop_{I}(t) \\simeq \\avg{\\Vop_I(t)} = \\Vop_{\\rm sec}$. Here, $\\langle \\cdot \\rangle$ denotes the time-average over one Paul trap period $2 \\pi\/\\omega_0$.\nThe secular Hamiltonian is thus given by the time-independent Hamiltonian $\\Hop_\\text{sec} = \\Hop_\\text{rot} + \\Vop_{\\rm sec}$. One then proceeds as described in the main text and obtains\nthe secular Hamiltonian in the linear approximation and in the two-level approximation, namely\n\\be\\label{Seq:Hsec_lin}\n\t\\Hop = \\frac{\\pop_\\beta^2}{2I} + \\frac{I}{2}\\w_\\beta^2\\hat{\\eulb}^2 + \\frac{\\pop_\\gamma^2}{2I_3}+\\frac{I_3}{2}\\pare{\\frac{\\w_\\gamma}{\\sqrt{2}}}^2\\hat{\\gamma}^2 + \\hbar\\hat{F}_x \\sx + \\hbar\\hat{F}_y\\sy + \\hbar\\hat{F}_z\\sz.\n\\ee\nHere, we introduced the operators $\\hat{F}_x \\equiv - \\hat{\\gamma}\\w_\\text{L}\/\\sqrt{2}$, $F_y \\equiv - \\hat{\\eulb} \\w_\\text{L}\/\\sqrt{2}$, and $\\hat{F}_z\\equiv \\id\\Delta\/2+\\hat{\\eulb}^2\\w_\\text{L}\/2+\\hat{\\gamma}^2I_3\\w_\\gamma^2\/4$. Note that $\\omega_\\gamma$ is thus the trapping frequency if the NV spin is prepared in the state $\\ket{\\uparrow}$. In \\eqnref{Seq:Hsec_lin}, we have also neglected the quantum potential as it arises from the curvature of the support of $\\beta$, which is neglected in the linear regime.\n\nWe proceed to diagonalize the spin-oscillator's interaction \\eqnref{Seq:Hsec_lin} with the unitary transformation\n\\be\\label{eq:U_2}\n\t\\Uop_2 = \\exp\\pare{\\im \\frac{\\pi}{2} \\bold{m}\\cdot \\hat{\\boldsymbol{\\sigma}}},\n\\ee\nwhere $\\bold{m}$ is a function of $\\hat{\\eulb}$ and $\\hat{\\gamma}$ and it can be understood geometrically as the unit vector which bisects the angle between the local direction of $\\bold{F}\\equiv(F_x,F_y,F_z)$ and $\\bold{e}_3$.\nThe transformed Hamiltonian according to \\eqnref{eq:U_2} reads \n\\be\\label{Seq:Hprimed}\n\\Hop' = \\Uop_2 \\Hop_\\text{sr}\\Udop_2 = \\Hop_\\text{disp} + \\Hop_\\text{na}.\n\\ee\nThe first term reads\n\\be\\label{Seq:H_disp_full}\n\t\\Hop_\\text{disp} = \\frac{\\pop_\\beta^2}{2I}+\\frac{I}{2}\\w_\\beta^2\\hat{\\eulb}^2+\\frac{\\pop_\\gamma^2}{2I_3}+\\frac{I_3}{2}\\pare{\\frac{\\w_\\gamma}{\\sqrt{2}}}^2\\hat{\\gamma}^2 + \\hbar |\\hat{\\bold{F}}|\\,\\sz\n\\ee\nand represents the dispersive dynamics of the system. \nThe last term in \\eqnref{Seq:H_disp_full} provides a spin dependent potential for the oscillators. Let us approximate the spin dependent potential as\n\\be\\label{Seq:F_approx}\n\t |\\hat{\\bold{F}}|\\simeq \\frac{\\Delta}{2} +\\hat{\\eulb}^2\\pare{\\frac{\\w_\\text{L}}{2}+\\frac{\\w_\\text{L}^2}{2\\Delta}}+\\hat{\\gamma}^2\\spare{\\frac{I_3}{2}\\pare{\\frac{\\w_\\gamma}{2}}^2+\\frac{\\w_\\text{L}^2}{2\\Delta}},\n\\ee\nwhich holds when the following conditions are satisfied\n\\be\\label{Seq:Adiabatic_Condition}\n\t\\frac{\\w_\\text{L}}{\\Delta}\\avg{\\hat{\\eulb}^2}\\ll1 \\quad \\frac{\\w_\\text{L}^2}{\\Delta^2}\\avg{\\hat{\\gamma}^2}\\ll1, \\quad \\frac{I_3}{2\\Delta}\\pare{\\frac{\\w_\\gamma}{\\sqrt{2}}}^2\\avg{\\hat{\\gamma}^2}\\ll 1.\n\\ee\n\\eqnref{Seq:Adiabatic_Condition} are the conditions for the validity of the dispersive regime. In \\figref{fig:FigSM1}, we show the range of validity of the dispersive approximation for the paramters considered in the main text. In general the dispersive approximation breaks down for an interval of magnetic field from $B_{c1}$ to $B_{c2}$ as shown in the right panel of \\figref{fig:FigSM1}.\n\\begin{figure}\n\t\\includegraphics[width=0.8\\columnwidth]{.\/FigS1}\n\t\\caption{Plot of $\\w_\\text{L}\\beta_0^2\/\\Delta$ and $(\\w_\\text{L}\\gamma_0\/\\Delta)^2$ as functions of the applied field $B_0$ (left panel) and detail of the region around the breakdown of the condition \\eqnref{Seq:Adiabatic_Condition} represented by the hatched region (right panel). The horizontal dotted line corresponds to the value $0.1$, which we define as the critical value to compute the hatched region. Note that $I_3\\w_\\gamma^2\\gamma_0^2\/2\\Delta$ is much smaller than the others in the regime considered and thus it is not shown in the plot. Other parameters as in the caption of \\figref{fig:Fig1}.}\\label{fig:FigSM1}\n\\end{figure}\nSubstituting \\eqnref{Seq:F_approx} into \\eqnref{Seq:H_disp_full} we obtain\n\\be\\label{Seq:H_disp_quadrature}\n\t\\Hop_\\text{disp} = \\frac{\\pop_\\beta^2}{2I}+\\frac{I}{2}\\spare{\\w_\\beta^2+\\frac{\\hbar \\w_\\text{L}}{I}\\pare{1+\\frac{\\w_\\text{L}}{\\Delta}}\\sz}\\hat{\\eulb}^2 +\\frac{\\pop_\\gamma^2}{2I_3} +\\frac{I_3}{2}\\spare{\\w_\\gamma^2\\pare{\\frac{\\id+\\sz}{2}}+\\frac{\\hbar\\w_\\text{L}^2}{I_3\\Delta}\\sz}\\hat{\\gamma}^2 + \\frac{\\hbar\\Delta}{2}\\sz.\n\\ee\nThe last term in \\eqnref{Seq:Hprimed} represents non-adiabatic corrections to the dynamics generated by \\eqnref{Seq:H_disp_quadrature} and reads\n\\be\\label{eq:H_na}\n\t\\Hop_\\text{na} = \\frac{\\pop_\\beta \\Aop_\\beta+\\Aop_\\beta\\pop_\\beta}{2I}+\\frac{\\Aop_\\beta^2}{2I}+\\frac{\\pop_\\gamma \\Aop_\\gamma+\\Aop_\\gamma\\pop_\\gamma}{2I_3}+\\frac{\\Aop_\\gamma^2}{2I_3},\n\\ee\nwhere $\\Aop_{\\beta,\\gamma} \\equiv \\hbar (\\pa{\\beta,\\gamma}\\bold{m}\\times \\bold{m})\\cdot\\hat{\\boldsymbol{\\sigma}}$. \n\\eqnref{eq:H_na} describes spin-flip transitions which leads to heating of the particle libration dynamics.\nIn the dispersive regime of \\eqnref{Seq:Adiabatic_Condition}, the probability of spin-flip transitions is exponentially suppressed as $\\exp(-\\Delta\/\\w_\\text{L}\\avg{\\hat{\\eulb}})$ and $\\exp(-\\Delta^2\/\\w_\\text{L}^2\\avg{\\hat{\\gamma}})$~\\cite{Sukumar1997Supp}. When \\eqnref{Seq:Adiabatic_Condition} holds, \\eqnref{eq:H_na} can thus be neglected.\n\nIt is important to distinguish two regimes depending on the sign of $\\Delta$.\n(i) For $\\Delta>0$, both modes are harmonically trapped when the spin is in $\\ket{\\uparrow}$. The Hamiltonian of the system thus reads\n\\be\\label{Seq:Hdisp_Blow}\n\\begin{split}\n\t\\frac{\\Hop'}{\\hbar} =& \\pare{ \\tilde{\\w}_{\\beta} \\bdop\\bop + \\tilde{\\w}_\\gamma \\cop^\\dag\\hat{c}+\\frac{\\Delta}{2}\\id} \\otimes\\ketbra{\\uparrow}{\\uparrow} + \\spare{\\tilde{\\w}_{\\beta} \\bdop\\bop - \\frac{\\chi_\\beta}{2} \\pare{\\bdop+\\bop}^2+\\tilde{\\w}_\\gamma \\cop^\\dag\\hat{c}-\\frac{\\chi_\\gamma}{2}\\pare{\\cop^\\dag+\\hat{c}}^2-\\frac{\\Delta}{2}\\id}\\otimes \\ketbra{\\downarrow}{\\downarrow}.\n\\end{split}\n\\ee\nHere, we defined $\\tilde{\\w}_{\\beta}\\equiv \\sqrt{\\w_\\beta^2+\\hbar\\w_\\text{L}(1+\\w_\\text{L}\/\\Delta)\/I}$, $\\chi_\\beta \\equiv \\hbar \\w_\\text{L}(1+\\w_\\text{L}\/\\Delta)\/I\\tilde{\\w}_\\beta$, $\\tilde{\\w}_\\gamma \\equiv \\sqrt{\\hbar\\w_\\text{L}(1+\\w_\\text{L}\/\\Delta)\/I_3}$, and $\\chi_\\gamma \\equiv \\hbar \\w_\\text{L}(1+2\\w_\\text{L}\/\\Delta)\/(2I_3\\tilde{\\w}_\\gamma)$. We have also introduced the bosonic operators $\\hat{c}$ and $\\bop$ according to $\\hat{\\eulb} \\equiv \\sqrt{\\hbar\/(2I\\tilde{\\w}_\\beta)}(\\bdop+\\bop)$ and $\\hat{\\gamma} \\equiv \\sqrt{\\hbar\/2I_3\\tilde{\\w}_\\gamma}(\\cop^\\dag+\\hat{c})$.\nIn \\figref{fig:FigS2}.a we plot the characteristic rates appearing in \\eqnref{Seq:Hdisp_Blow} as a function of the applied field $B_0$ ranging from $0.1~\\text{mT}$ up to $100\\text{mT}$. We note that for a given value of $\\w_0\/2\\pi$ there exists a critical field $B^\\star$ such that for $B_0>B^\\star$, the mode $\\beta$ becomes unstable, \\ie it experience a repulsive potential, when the spin is in $\\ket{\\downarrow}$ since in this case $\\w_\\beta<2\\chi_\\beta$. In~\\figref{fig:FigS2}.b, we plot $B^\\star$ as a function of the Paul trap frequency $\\w_0\/2\\pi$.\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{.\/FigS2}\n\t\\caption{Frequencies and coupling rate of the dispersive Hamiltonian. a) Characteristic rates in the dispersive regime as a function of $B_0$ such that $\\Delta>0$ (weak field). b) Critical field $B^\\star$ at which the $\\beta$-mode becomes unstable, \\ie the frequency of $\\beta$ vanishes in the state $\\ket{\\downarrow}$ in \\eqnref{Seq:Hdisp_Blow}. c) Characteristic rates in the dispersive regime as a function of $B_0$ such that $\\Delta>0$ (strong field). c) Ratio $\\W_\\gamma\/\\W_\\beta$ as a function of the applied field for the same interval shown in panel b). Other parameters (when needed) are as in the caption of~Fig.1.}\\label{fig:FigS2}\n\\end{figure}\n(ii) For $\\Delta<0$, the two libration mode are harmonically trapped when the spin is in the state $\\ket{\\downarrow}$, while they both experience an inverted potential when the spin is in $\\ket{\\uparrow}$. In this case we write \\eqnref{Seq:H_disp_quadrature} as\n\\be\\label{Seq:Hdisp_Bhigh}\n\t\\frac{\\Hop'}{\\hbar} = \\spare{ \\W_{\\beta} \\bdop\\bop - \\frac{\\tilde{\\chi}_\\beta}{2}\\pare{\\bdop+\\bop}^2 + \\W_\\gamma \\cop^\\dag\\hat{c}-\\frac{\\tilde{\\chi}_\\gamma}{2}\\pare{\\cop^\\dag+\\hat{c}}^2+\\frac{\\Delta}{2}\\id} \\otimes\\ketbra{\\uparrow}{\\uparrow} + \\pare{\\W_{\\beta} \\bdop\\bop +\\W_\\gamma \\cop^\\dag\\hat{c}-\\frac{\\Delta}{2}\\id}\\otimes \\ketbra{\\downarrow}{\\downarrow},\n\\ee\nwhere we defined $\\W_{\\beta}\\equiv \\sqrt{\\w_\\beta^2+\\hbar \\w_\\text{L}(\\w_\\text{L}\/|\\Delta|-1)\/I}$, $\\W_\\gamma = \\sqrt{\\hbar\\w_\\text{L}^2\/I_3|\\Delta|}$, $\\tilde{\\chi}_\\beta \\equiv (\\W_\\beta^2-\\w_\\beta^2)\/ \\W_\\beta$, and $\\tilde{\\chi}_\\gamma \\equiv \\sqrt{2\\hbar \\w_\\text{L}^2\/(I_3|\\Delta|)}(1-|\\Delta|\/2\\w_\\text{L})$. The bosonic modes appearing in \\eqnref{Seq:Hdisp_Bhigh} are defined as $\\hat{\\eulb} = \\sqrt{\\hbar\/2I\\W_\\beta}(\\bdop+\\bop)$ and $\\hat{\\gamma} \\equiv \\sqrt{\\hbar\/2I_3 \\W_\\gamma}(\\cop^\\dag+\\hat{c})$.\nIn \\figref{fig:FigS2}.c we plot the characteristic rates appearing in \\eqnref{Seq:Hdisp_Bhigh} as a function of the applied field $B_0$ ranging from $B_{c2}$ up to $200~\\text{mT}$.\nWe note that for the spin in $\\ket{\\uparrow}$ both modes feel a repulsive potential since $\\chi_{\\beta,\\gamma}>\\W_{\\beta,\\gamma}\/2$.\n\\figref{fig:FigS2}.c shows the the ratio $\\W_\\gamma\/\\W_\\beta$ as function of the applied field. We note that $\\W_\\gamma\/\\W_\\beta= 1$ for $B_0\\simeq 118~\\text{mT}$.\n\nThe superposition protocol can be applied to the case of \\eqnref{Seq:Hdisp_Bhigh} however for the probability to rephase the protocol duration $\\tau$ should be such that that both oscillator's evolve for an integer multiple of their half period. As evidenced in \\figref{fig:FigS2}.c, $\\W_\\gamma\/\\W_\\beta = 3$ for $B_0\\simeq 140~\\text{mT}$. \nIn this case the rephasing time $\\pi\/\\W_\\gamma$ is slightly smaller than the rephasing time $\\pi\/\\tilde{\\w}_\\gamma$ leading to slightly better performance of the protocol in the presence of qubit dephasing. This however requires precise tuning of the magnetic field to ensure that $\\W_\\gamma$ is an integer multiple of $\\W_\\beta$.\n\n\n\\section{Interference Protocol}\n\nWe consider the spin-oscillators Hamiltonian given in \\eqnref{Seq:Hdisp_Blow}.\nFor later convenience we introduce the evolution operators for the oscillator when the spin is in the state $\\ket{\\uparrow(\\downarrow)}$. They read respectively,\n\\bea\n\t\\Uop_\\uparrow &\\equiv& \\exp\\pare{-\\im t\\tilde{\\w}_{\\beta} \\bdop\\bop} \\exp\\pare{-\\im t\\tilde{\\w}_\\gamma \\cop^\\dag\\hat{c}} \\equiv \\Uop_{\\beta\\uparrow}\\Uop_{\\gamma\\uparrow},\\label{eq:Uup}\\\\\n\t\\Uop_\\downarrow &\\equiv& \\exp\\spare{-\\im t\\tilde{\\w}_{\\beta} \\bdop\\bop+\\im t\\frac{\\chi_\\beta}{2}\\pare{\\bdop+\\bop}^2} \\exp\\spare{-\\im \\tilde{\\w}_\\gamma t \\cop^\\dag\\hat{c}+\\im\\frac{\\chi_\\gamma t}{2}\\pare{\\cop^\\dag+\\hat{c}}^2}\\equiv \\Uop_{\\beta\\downarrow}\\Uop_{\\gamma\\downarrow}.\\label{eq:Udw}\n\\eea\nLet us now consider the following protocol:\n\\begin{itemize}\n\t\\item[0.] Prepare the system in the product state $\\hat{\\rho}_0=\\hat{\\rho}_\\text{th}\\otimes \\ketbra{\\uparrow}{\\uparrow}$ of \\eqnref{Seq:Hsec_lin}, where $\\hat{\\rho}_\\text{th}$ is the thermal state of the oscillator. Note that the corresponding product state between the spin and the thermal state of \\eqnref{Seq:Hdisp_Blow} is obtained as $\\hat{\\rho}_0' = \\Uop_2\\hat{\\rho}_0\\Udop_2$. In the dispersive regime of \\eqnref{Seq:Adiabatic_Condition}, however, $\\hat{\\rho}_0'\\simeq \\hat{\\rho}_0$, and the oscillator thermal state for $\\ket{\\uparrow}$ well approximate the thermal state of Eq.(4b).\n\t\\item[1.] Apply a $\\pi\/2$-microwave pulse to the spin, thus preparing the state\n\t\\be\n\t\\hat{\\rho}_1= \\hat{\\rho}_\\text{th}\\otimes \\inv{2} \\pare{\\ketbra{\\uparrow}{\\uparrow}+\\ketbra{\\downarrow}{\\uparrow}+\\ketbra{\\uparrow}{\\downarrow}+\\ketbra{\\downarrow}{\\downarrow}}.\n\t\\ee\n\t We assume the microwave pulse to have a duration much smaller than the oscillator's evolution time scale, $\\tilde{\\w}_{\\beta}^{-1}$, $\\chi_\\beta^{-1}$, $\\w_\\gamma^{-1}$, $\\chi_\\gamma^{-1}$, such that the evolution of the oscillator on the time-scale of the pulse can be neglected.\n\t\\item[2.] Let the state evolve for a time $\\tau$. At the end of this stage the state reads\n\t\\be\n\t\t\\hat{\\rho}_2 = \\frac{1}{2}\\spare{\\Uop_\\uparrow\\hat{\\rho}_{\\text{th}}\\Udop_\\uparrow\\otimes \\ketbra{\\uparrow}{\\uparrow}+e^{\\im \\Delta \\tau}\\Uop_\\downarrow\\hat{\\rho}_{\\text{th}}\\Udop_\\uparrow\\otimes \\ketbra{\\downarrow}{\\uparrow}+e^{-\\im \\Delta \\tau}\\Uop_\\uparrow\\hat{\\rho}_{\\text{th}}\\Udop_\\downarrow\\otimes \\ketbra{\\uparrow}{\\downarrow}+\\Uop_\\downarrow\\hat{\\rho}_{\\text{th}}\\Udop_\\downarrow\\otimes \\ketbra{\\downarrow}{\\downarrow}}.\n\t\\ee\n\t\\item[3.] Apply a $\\pi$-microwave pulse to the spin such that $\\ket{\\uparrow(\\downarrow)}\\rightarrow\\ket{\\downarrow(\\uparrow)}$ and let the system evolve for another time $\\tau$. At the end of this stage the system is in the state\n\t\\be\n\t\t\\hat{\\rho}_3 =\\frac{1}{2}\\spare{\\Uop_\\downarrow\\Uop_\\uparrow\\hat{\\rho}_{\\text{th}}\\Udop_\\uparrow\\Udop_\\downarrow\\otimes \\ketbra{\\downarrow}{\\downarrow}+\\Uop_\\uparrow\\Uop_\\downarrow\\hat{\\rho}_{\\text{th}}\\Udop_\\uparrow\\Udop_\\downarrow\\otimes \\ketbra{\\uparrow}{\\downarrow}+\\Uop_\\downarrow\\Uop_\\uparrow\\hat{\\rho}_{\\text{th}}\\Udop_\\downarrow\\Udop_\\uparrow\\otimes \\ketbra{\\downarrow}{\\uparrow}+\\Uop_\\uparrow\\Uop_\\downarrow\\hat{\\rho}_{\\text{th}}\\Udop_\\downarrow\\Udop_\\uparrow\\otimes \\ketbra{\\uparrow}{\\uparrow}}.\n\t\\ee\n\t\\item[4.] Apply a $\\pi\/2$-microwave pulse such that $\\ket{\\uparrow(\\downarrow)}\\rightarrow(\\ket{\\uparrow}\\pm\\ket{\\downarrow})\/\\sqrt{2}$ and perform a spin measurement. The final probability to find the spin in the state $\\ket{\\uparrow(\\downarrow)}$ reads\n\t\\be\\label{eq:Prob_spin}\n\t\tP_{\\uparrow\\downarrow}(\\tau) =\\frac{1}{2}\\pm \\frac{1}{4}\\Tr\\spare{\\Udop_\\downarrow\\Udop_\\uparrow\\Uop_\\downarrow\\Uop_\\uparrow\\hat{\\rho}_\\text{th}+\\Udop_\\uparrow\\Udop_\\downarrow\\Uop_\\uparrow\\Uop_\\downarrow\\hat{\\rho}_\\text{th}}\n\t\\ee\n\\end{itemize}\nWe note that substituting \\eqnref{eq:Uup} and \\eqnref{eq:Udw} into \\eqnref{eq:Prob_spin} we obtain\n\\be\\label{eq:Prob_final_spin}\n\\begin{split}\n\t\tP_{\\uparrow\\downarrow}(\\tau)\n\t\t=& \\frac{1}{2}\\pm \\frac{1}{2}\\int\\!\\!\\text{d}^2\\xi_\\beta \\text{d}^2\\xi_\\gamma\\, \\text{P}_\\text{th}(\\xi_\\beta)\\text{P}_\\text{th}(\\xi_\\gamma) \\Re\\pare{ \\bra{\\xi_\\beta}\\Udop_{\\beta\\downarrow}\\Udop_{\\beta\\uparrow}\\Uop_{\\beta\\downarrow}\\Uop_{\\beta\\uparrow}\\ket{\\xi_\\beta}\\bra{\\xi_\\gamma}\\Udop_{\\gamma\\downarrow}\\Udop_{\\gamma\\uparrow}\\Uop_{\\gamma\\downarrow}\\Uop_{\\gamma\\uparrow}\\ket{\\xi_\\gamma}},\n\\end{split}\n\\ee\nwhere we introduced the thermal state for the the $\\nu=\\beta,\\gamma$ oscillator,\n\\be\n\t\\hat{\\rho}_{\\nu} \\equiv \\int\\!\\!\\text{d}^2\\xi_\\nu\\, \\frac{e^{-|\\xi_\\nu|^2\/n_\\nu}}{\\pi n_\\nu} \\ketbra{\\xi_\\nu}{\\xi_\\nu} \\equiv \\int\\!\\!\\text{d}^2\\xi\\, \\text{P}_\\text{th}(\\xi_\\nu) \\ketbra{\\xi_\\nu}{\\xi_\\nu}.\n\\ee\nHere, $\\bop\\ket{\\xi_\\beta}=\\xi_\\beta\\ket{\\xi_\\beta}$, $\\hat{c}\\ket{\\xi_\\gamma}=\\xi_\\gamma\\ket{\\xi_\\gamma}$, $n_\\nu\\equiv 1\/(e^{\\beta_\\text{th}\\hbar \\w_\\nu}-1)$ is the average thermal occupation number, $\\beta_\\text{th} \\equiv 1\/k_\\text{b} T$, and $k_\\text{b}$ is the Boltzmann constant.\n\nLet us now evaluate the two expectation values in \\eqnref{eq:Prob_final_spin}. We first consider the expectation values for the $\\gamma$-oscillator. We write $\\Uop_{\\gamma\\downarrow}$ as\n\\be\\label{eq:Udw_gamma}\n\\begin{split}\n\t\\Uop_{\\gamma\\downarrow} \\equiv& \\exp\\bigg\\{\\underbrace{-\\im 2t\\pare{\\tilde{\\w}_\\gamma-\\chi_\\gamma}}_{=\\lambda_{\\eulc0}} \\frac{\\cop^\\dag\\hat{c}+\\hat{c}\\cop^\\dag}{4}+\\Big[\\underbrace{\\im\\chi_\\gamma t}_{=\\lambda_\\gamma}\\frac{\\hat{c}^{\\dag2}}{2}-\\underbrace{(-\\im \\chi_\\gamma t)}_{=\\lambda_\\gamma^*}\\frac{\\hat{c}^2}{2}\\Big]\\bigg\\}\\\\\n\t=& \\exp\\pare{\\eta_\\gamma \\hat{c}^{\\dag2}}\\exp\\spare{\\log(\\eta_{\\eulc0})\\frac{\\cop^\\dag\\hat{c}+\\hat{c}\\cop^\\dag}{4}}\\exp\\pare{\\eta_\\gamma \\hat{c}^2},\n\\end{split}\n\\ee\nwhere in the last passage we have used the Baker-Campbell-Hausdorf formula of SU(1,1)~\\cite{Ban1992Supp} and we introduced the parameters\n\\begin{subequations}\n\\bea\n\t\\eta_\\gamma &\\equiv& \\frac{\\lambda \\sinh \\zeta_\\gamma}{\\zeta_\\gamma \\cosh\\zeta_\\gamma - (\\lambda_0\/2)\\sinh\\zeta_\\gamma},\\label{eq:eta}\\\\\n\t\\eta_{\\eulc0} &\\equiv& \\spare{\\frac{\\zeta_\\gamma}{\\zeta_\\gamma \\cosh\\zeta_\\gamma - (\\lambda_0\/2)\\sinh\\zeta_\\gamma}}^2,\\label{eq:eta_0}\\\\\n\t\\zeta_\\gamma^2 &\\equiv& \\pare{\\frac{\\lambda_{\\eulc0}}{2}}^2 - \\lambda_\\gamma^2.\n\\eea\n\\end{subequations}\nUsing the expression in \\eqnref{eq:Udw_gamma}, we can write the product of unitary operators for the $\\gamma$-oscillator appearing in \\eqnref{eq:Prob_final_spin} as\n\\be\n\\begin{split}\n\t\\Udop_{\\gamma\\downarrow}\\Udop_{\\gamma\\uparrow}\\Uop_{\\gamma\\downarrow}\\Uop_{\\gamma\\uparrow} =& e^{\\eta_\\gamma^* \\hat{c}^{\\dag2}}e^{\\log(\\eta_{\\eulc0}^*)(\\cop^\\dag\\hat{c}+\\hat{c}\\cop^\\dag)\/4}e^{-\\eta_\\gamma^* \\hat{c}^2}e^{-\\im \\tilde{\\w}_\\gamma t \\cop^\\dag\\hat{c}}e^{\\eta_\\gamma \\hat{c}^{\\dag2}}e^{\\log(\\eta_{\\eulc0})(\\cop^\\dag\\hat{c}+\\hat{c}\\cop^\\dag)\/4}e^{\\eta_\\gamma \\hat{c}^2}e^{-\\im \\tilde{\\w}_\\gamma t \\cop^\\dag\\hat{c}} \\\\\n\t=& \\exp\\pare{\\eta_\\gamma^* \\hat{c}^{\\dag2}}\\!\\exp\\!\\spare{\\log(\\eta_{_\\eulc0}^*)\\frac{\\cop^\\dag\\hat{c}+\\hat{c}\\cop^\\dag}{4}}\\!\\exp\\pare{\\eta_\\gamma^* \\hat{c}^2}\\exp\\pare{\\eta_\\gamma e^{\\im 2\\tilde{\\w}_\\gamma t} \\hat{c}^{\\dag2}}\\exp\\spare{\\log(\\eta_{\\eulc0})\\frac{\\cop^\\dag\\hat{c}+\\hat{c}\\cop^\\dag}{4}}\\\\\n &\\times\\exp\\pare{\\eta_\\gamma e^{-\\im 2\\tilde{\\w}_\\gamma t} \\hat{c}^2}.\n\\end{split}\n\\ee\nAfter some work we arrive at~\\cite{Tibaduiza2020Supp}\n\\be\\label{eq:Product_S1S2}\n\t\\Udop_{\\gamma\\downarrow}\\Udop_{\\gamma\\uparrow}\\Uop_{\\gamma\\downarrow}\\Uop_{\\gamma\\uparrow} = \\exp\\pare{\\frac{\\phi_\\gamma}{2} \\hat{c}^{\\dag2}}\\exp\\spare{\\log(\\theta_\\gamma)\\frac{\\cop^\\dag\\hat{c}+\\hat{c}\\cop^\\dag}{4}} \\exp\\pare{\\frac{\\psi_\\gamma}{2} \\hat{c}^{2}},\n\\ee\nwhere we defined the c-numbers\n\\begin{subequations}\n\\bea\n\t\\phi_\\gamma &\\equiv& \\eta_\\gamma^* + \\frac{\\eta_{\\eulc0}^*\\eta_\\gamma e^{\\im2\\tilde{\\w}_\\gamma t}}{1-|\\eta|^2 e^{\\im2\\tilde{\\w}_\\gamma t}},\\label{eq:phi_gamma}\\\\\n\t\\theta_\\gamma &\\equiv& \\frac{|\\eta_{\\eulc0}|^2}{(1-|\\eta_\\gamma|^2 e^{\\im2\\tilde{\\w}_\\gamma t})^2},\\\\\n\t\\psi_\\gamma &\\equiv&\t\\eta_\\gamma e^{-\\im 2\\tilde{\\w}_\\gamma t} + \\frac{\\eta_\\gamma^*\\eta_{\\eulc0}}{1-|\\eta_\\gamma|^2e^{\\im 2\\tilde{\\w}_\\gamma t}}.\\label{eq:psi_gamma}\n\\eea\n\\end{subequations}\nProceeding in the same way one can prove that for the $\\beta$-oscillator\n\\be\n\t\\Udop_{\\beta\\downarrow}\\Udop_{\\beta\\uparrow}\\Uop_{\\beta\\downarrow}\\Uop_{\\beta\\uparrow} = \\exp\\pare{\\frac{\\phi_\\beta}{2}\\bop^{\\dag2}}\\exp\\spare{\\log(\\theta_\\beta)\\frac{\\bdop\\bop+\\bop\\bdop}{4}}\\exp\\pare{\\frac{\\psi_\\beta}{2}\\aop^2}\n\\ee\nwhere $\\phi_\\beta$, $\\theta_\\beta$, and $\\psi_\\beta$ are define as in Eq.~(\\ref{eq:phi_gamma}-\\ref{eq:psi_gamma}) and in \\eqnref{eq:eta} and \\eqnref{eq:eta_0}, with the obvious modifications.\n\nLet us now evaluate the integral over the thermal distribution. We first notice that since the expectation values appearing in \\eqnref{eq:Prob_final_spin} for the $\\beta$ and $\\gamma$ oscillators are factorised, namely $P_{\\uparrow\\downarrow}(\\tau) = 1\/2 \\pm \\mathcal{I}_\\gamma \\mathcal{I}_\\beta \/2$. We can thus separately evaluate the two integrals over the coherent state basis. Let us evaluate $\\mathcal{I}_\\gamma$ and $\\mathcal{I}_\\beta$. Substituting \\eqnref{eq:Product_S1S2} back into \\eqnref{eq:Prob_final_spin} we obtain\n\\be\\label{eq:Int_gamma_step1}\n\\begin{split}\n\t\\mathcal{I}_\\gamma =& \\int\\!\\! \\text{d}^2\\xi_\\gamma \\text{P}_\\text{th}(\\xi_\\gamma)\\Re\\pare{\\bra{\\xi_\\gamma}\\Udop_{\\gamma\\downarrow}\\Udop_{\\gamma\\uparrow}\\Uop_{\\gamma\\downarrow}\\Uop_{\\gamma\\uparrow}\\ket{\\xi_\\gamma}}\\\\\n\t =& \\frac{1}{2\\pi n_\\gamma} \\sqrt{\\frac{|\\eta_{\\eulc0}|}{1-|\\eta_\\gamma|^2e^{\\im 2\\tilde{\\w}_\\gamma\\tau}}}\\int\\!\\!\\text{d}^2\\xi_\\gamma \\exp\\spare{-|\\xi_\\gamma|^2\\pare{1+\\frac{1}{n_\\gamma}-\\frac{|\\eta_{\\eulc0}|}{1-|\\eta_\\gamma|^2e^{\\im 2\\tilde{\\w}_\\gamma\\tau}}}}\\exp\\pare{\\frac{\\phi_\\gamma \\xi_\\gamma^{*2}+\\psi_\\gamma \\xi_\\gamma^{2}}{2}}.\n\\end{split}\n\\ee\nThe integral in \\eqnref{eq:Int_gamma_step1} can be evaluated by expressing $\\xi_\\gamma$ in polar coordinate. Carrying out the polar integral first one obtains\n\\be\n\t\\mathcal{I}_\\gamma = \\frac{1}{n_\\gamma} \\Re\\cpare{ \\sqrt{\\frac{|\\eta_{\\eulc0}|}{1-|\\eta_\\gamma|^2e^{\\im 2\\tilde{\\w}_\\gamma\\tau}}}\\int_0^{\\infty}\\!\\!\\text{d}r\\, r \\exp\\spare{-r^2\\pare{1+\\frac{1}{n_\\gamma}-\\frac{|\\eta_{\\eulc0}|}{1-|\\eta_\\gamma|^2e^{\\im 2\\tilde{\\w}_\\gamma\\tau}}}}I_0\\pare{2r^2\\sqrt{\\phi_\\gamma\\psi_\\gamma}}},\n\\ee\nwhere $I_0(x)$ is the zero-order modified Bessel function of the first kind.\nThe radial integral is tabulated (see for instance~\\citep[Eq.~(6.611.4)]{Gradshteyn1994Supp}), and we finally obtain\n\\be\\label{eq:I_gamma}\n\t\\mathcal{I}_\\gamma =\\frac{1}{2n_\\gamma}\\sqrt{\\frac{|\\eta_{\\eulc0}|}{1-|\\eta_\\gamma|^2e^{\\im 2\\tilde{\\w}_\\gamma \\tau}}\\spare{\\pare{1+\\frac{1}{n_\\gamma}-\\frac{|\\eta_{\\eulc0}|}{1-|\\eta_\\gamma|^2e^{\\im 2\\tilde{\\w}_\\gamma\\tau}}}^2-\\phi_\\gamma\\psi_\\gamma}^{-1}}.\n\\ee\nThe intrgral $\\mathcal{I}_\\beta$ is evaluated following identical steps. Substituting these results back into the definition \\eqnref{eq:Prob_final_spin} we obtain\n\\be\\label{eq:Pth}\n\tP_\\pm^{(\\text{th})}(\\tau) = \\inv{2}\\pm \\inv{2 n_\\beta n_\\gamma}\\Re\\pare{\\prod_{\\nu=\\beta,\\gamma}\\sqrt{\\frac{|\\eta_{0\\nu}|}{(1-|\\eta_\\nu|^2e^{\\im 2\\tilde{\\w}_\\nu \\tau})}\\spare{\\pare{1+\\frac{1}{n_\\nu}-g_\\nu(t)}^2-\\phi_\\nu'\\psi_\\nu'}^{-1}}}.\n\\ee\nWe note that the choice of the phase for the square root appearing in \\eqnref{eq:Pth} is fixed by the initial state of the protocol. For the case we considered the phase should be chosen such that $\\lim_{\\tau\\rightarrow0}P_\\uparrow(\\tau)=1$.\nLet us conclude by noting that the qubit dephasing can be straightforwardly included leading to a factor $\\exp(-2\\tau\/T_2^*)$ multpling the second term in \\eqnref{eq:Pth} as shown in the main text.\n\n\nIn \\eqnref{eq:Pth} appear both the thermal occupation of the thermal state for the $\\gamma$ and $\\beta$ oscillator. The two oscillators in $\\Hop_\\uparrow$ have generally different frequencies and thus different thermal occupations for a fixed value of the temperature $T$. To conclude let us notice that if for a given temperature $T$ the mean thermal occupation $n_\\gamma$ of $\\rho_\\gamma$, we obtain the mean thermal occupation for $\\rho_\\beta$ as\n\\be\\label{eq:ny}\n\tn_\\beta = \\spare{\\pare{\\inv{n_\\gamma}+1}^{\\tilde{\\w}_\\beta\/\\tilde{\\w}_\\gamma}-1}^{-1}.\n\\ee\n\\eqnref{eq:ny} is how we calculate the thermal occupation $n_\\beta$ for a given thermal occupation $n_\\gamma$.\n\n\n\nPerfect rephasing of $P_\\uparrow(\\tau)$ occurs if $T_\\gamma = n T_\\beta $ for $n\\in \\mathbb{N}$ where $T_\\nu \\equiv \\pi\/\\tilde{\\w}_\\nu$ ($\\nu=\\beta,\\gamma$), as it can be easily checked in \\eqnref{eq:Pth}. However, this occurs only for particular values of the parameters of the system.\nIn the most general case $\\tilde{\\w}_\\beta\\neq \\tilde{\\w}_\\gamma$, thus a perfect constructive interference of the $\\gamma$ superposition does not coincide with constructive interference of the $\\beta$-superposition. \nLet us now discuss under which condition it is possible to observe a rephasing of $P_\\uparrow(\\tau)$ at $\\tau=\\pi\/\\tilde{\\w}_\\gamma \\neq n T_\\beta$. In this case perfect rephasing is limited by two main factors, (i) the amount of squeezing of the $\\beta$-oscillator during the protocol and (ii) the thermal occupation of the initial states of the oscillators.\nWhile the squeezing parameter of the $\\gamma$-oscillator grows exponential with time due to the repulsive potential, squeezing of the $\\beta$-oscillator is fixed by the ratio of the frequencies corresponding to the $\\ket{\\uparrow}$ and $\\ket{\\downarrow}$ states~\\cite{Rashid2016Supp}. When this frequency change is negligible the generated squeezing is negligible and the overlap between the two superposition states of the $\\beta$ oscillator is large even in the absence of perfect rephasing. In~\\figref{fig:ConditionRecoherence}.a we plot the ratio $\\delta\\w_\\beta\/\\w_\\beta$, where $\\delta \\w_\\beta \\equiv \\sqrt{\\hbar \\w_\\text{L}(1+\\w_\\text{L}\/\\Delta)\/I}$ is the frequency difference between the two branches, as function of the applied field and the Paul trap driving frequency.\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{.\/FigS3}\n \\caption{a) Value of $\\delta\\w_\\beta\/\\w_\\beta$ as a function of both $B_0$ and $\\w_0\/2\\pi$. The hatched region corresponds to the regime of instability for the $\\beta$-oscillator, nameyl $\\delta\\w_\\beta>\\w_\\beta$. b) Density plot of $P^*_\\uparrow$ as a function of both $B_0$ and $\\w_0\/2\\pi$ for $n_\\gamma=10^3$ and $T_2=0.5~\\text{ms}$. c) Plot of $P^*_\\uparrow$ as a function of $B_0$ for different values of $T_2$ for the case $n_\\gamma=10^2$. In all panels, unless otherwise specified we used the same parameters as given in the caption of Fig.1 in the main text.}\n \\label{fig:ConditionRecoherence}\n\\end{figure}\nIt is shown that to reduce $\\delta\\w_\\beta\/\\w_\\beta$ it is advantageous to work at $\\w_0\/2\\pi> 1\\text{MHz}$.\n\nThe initial temperature of $\\rho_\\text{th}$ also has an impact on the rephasing. The width of the rephasing peak in \\eqnref{eq:Pth} decreases with temperature because a larger number of states participate in the evolution and thus set a tighter requirement on the rephasing. In particular, even for $\\chi_\\beta\\ll\\tilde{\\w}_\\beta$ the suppression of rephasing can be significant for larger initial temperature. Intuitively this is due to the fact that highly excited states of the oscillator are more susceptible to frequency changes~\\footnote{To understand this point it is advantageous to think of thermal state decomposition into the coherent state basis. For high temperature coherent states $\\ket{\\varphi}$ with large $|\\varphi|$ are occupied. These states oscillates to region away from the center where a small change in frequency significantly change the slope of the harmonic potential.}.\nThe impact of thermal population on the rephasing is shown in~\\figref{fig:Fig2}.d in the main text where $P_\\uparrow^*\\equiv P_\\uparrow(\\tau = \\pi\/\\tilde{\\w}_\\gamma)$ is plotted for different values of $n_\\gamma$ as a function of $B_0$ for $T^*_2=500~\\mu \\text{s}$. We see that $P^*_\\uparrow$ always assumes the maximum value set by the spin dephasing time $T^*_2$ whenever $B_0$ takes values $\\tilde{B}_n$ such that $\\tilde{\\w}_\\beta\/\\tilde{\\w}_\\gamma =n \\in\\mathbb{N}$. Furthermore, $P^*_\\uparrow$ seems to be robust to changes in the magnetic field with near optimal rephasing being achieved even for values of $B_0$ around $\\tilde{B}_n$.\nThe interval of values of $B_0$ around $\\tilde{B}_n$ over which $P^*_\\uparrow$ achieve its optimal value depends on $\\w_0\/2\\pi$: for larger ac potential frequency the $\\beta$-squeezing is smaller and thus the rephasing more robust to fluctuations in $B_0$.\nIn \\figref{fig:ConditionRecoherence}.b, we plot the dependence of $P^*_\\uparrow$ on both $B_0$ and $\\w_0\/2\\pi$ for large termal occupation $n_\\gamma=10^3$ and $n_\\beta$ calculated according to \\eqnref{eq:ny}. Finally in \\figref{fig:ConditionRecoherence}.c we show that for a given temperature the maximum value achievable by $P_\\uparrow^\\star$ is set by the spin dephasing time.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nSoft deformable objects, such as liquid droplets, vesicles, cells, and synthetic capsules\nexhibit a complex behavior under flows. \nFor example, in capillary flow,\nfluid vesicles~\\cite{vitk04}, red blood cells (RBCs)~\\cite{fung97,fung04,skal69,gaeh80,nogu05b,mcwh09,pozr05}, \nand synthetic capsules~\\cite{lefe08} \ndeform to parachute shapes, and RBCs also deform to slipper shapes~\\cite{skal69,gaeh80,nogu05b,mcwh09}.\nShape transitions of fluid vesicles occur in simple shear flow~\\cite{nogu04,nogu05,nogu09}.\nMembrane wrinkling appears\nfor fluid vesicles after inversion \nof an elongational flow~\\cite{kant07,turi08} and for synthetic capsules in simple shear flow~\\cite{walt01,fink06}.\nAmong these soft objects, RBC have received a great deal of attention,\nsince they are important for both fundamental research and medical \napplications. In microcirculation,\nthe deformation of RBCs reduces the flow resistance of microvessels.\nIn patients with diseases such as diabetes mellitus and sickle cell anemia, the RBCs have\na reduced deformability and often block the microvascular flow~\\cite{fung97,tran84,nash83,tsuk01,higg07}.\n\nIn a simple shear flow with flow velocity ${\\bf v}=\\dot\\gamma y {\\bf e}_x$, \nfluid vesicles and RBCs show\na transition from a tank-treading (TT) mode with a constant inclination angle $\\theta$ \nto a tumbling (TB) mode with increasing viscosity \nof the internal fluid $\\eta_{\\rm {in}}$~\\cite{kell82,beau04,made06,kant06}\nor membrane viscosity $\\eta_{\\rm {mb}}$~\\cite{nogu04,nogu05}.\nThis transition \nis described well by the theory of Keller and Skalak \n(KS)~\\cite{kell82}, which assumes a fixed ellipsoidal vesicle shape.\nExperimentally, synthetic capsules and RBCs \nshow the oscillation of their lengths and $\\theta$, called swinging (SW) \\cite{chan93,walt01,abka07},\nduring TT motion, and RBCs\nalso transit from TB to TT with increasing $\\dot\\gamma$ \\cite{gold72,abka07}.\nRecently, this dynamics was explained by the KS theory with the \naddition of an energy barrier for the TT rotation caused by the \nmembrane shear elasticity~\\cite{skot06,abka07}.\nMore recently, this transition was also obtained by simulations~\\cite{kess08,sui08}.\nHowever, the detailed dynamics has not yet been investigated.\n\nFor fluid vesicles in high shear flow,\nshape transitions \\cite{nogu04,nogu05,nogu09} occur,\nand a swinging phase \\cite{kant06,nogu07b,misb06,lebe07,lebe08}, \nwhere the shape and $\\theta$ oscillate around $\\theta \\simeq 0$,\nappears between the TT and TB phases.\nThis SW mode is also called trembling \\cite{kant06,lebe07,lebe08} or vacillating-breathing \\cite{misb06};\nit is explained by\nthe KS theory extended to a deformable ellipsoidal vesicle \\cite{nogu07b}\nand the perturbation theory for a quasi-spherical vesicle \\cite{misb06,lebe07,lebe08}.\nShape deformation plays an essential role in the SW of fluid vesicles.\nThe deformation is not necessary to explain\nthe SW of elastic capsules \\cite{abka07,chan93,walt01,skot06,kess08,sui08,navo98,rama98}\nbut is required for quantitative analysis.\nIn this letter, we extend the theory in Ref.~\\cite{skot06} to include the shape deformation of RBCs\nand investigate the dynamics of deformable RBCs.\n\n\nThe internal fluid of RBCs behaves as a Newtonian fluid since\nRBCs do not have a nucleus and other intracellular organelles.\nThe RBC membrane consists of a lipid bilayer with an \nattached spectrin network as cytoskeleton. \nThe lipid bilayer is an area-incompressible fluid membrane. \nThe shear elasticity of the composite membrane is induced by \nthe spectrin network.\nUnder physiological conditions, \nan RBC has a constant volume $V = 94\\mu{\\rm m}^3$, surface area $S= 135\\mu{\\rm m}^2$,\n $\\eta_{\\rm {in}}=0.01$Pa$\\cdot$s, $\\eta_{\\rm {mb}}\\sim 10^{-7}-10^{-6}$Ns\/m,\n membrane shear elasticity $\\mu=6\\times 10^{-6}$N\/m, and \nbending rigidity $\\kappa=2 \\times 10^{-19}$J \\cite{nogu09,fung04,moha94,tran84,dao06}.\n\n\nThe models and results are presented with dimensionless quantities (denoted by a superscript $*$).\nThe lengths and energies are normalized by $R_0=\\sqrt{S\/4\\pi}$\nand $\\mu R_0^2$, respectively.\nFor RBCs, they are $R_0=3.3$ $\\mu$m\nand $\\mu R_0^2=6.5\\times 10^{-17}$J.\nThere are two intrinsic time units:\nthe shape relaxation time $\\tau=\\eta_0 R_0\/\\mu$ by the shear elasticity $\\mu$,\nand the time of shear flow $1\/\\dot\\gamma$;\n the reduced shear rate is defined as $\\dot\\gamma^*=\\dot\\gamma \\tau$.\nThe relative viscosities are\n$\\eta_{\\rm {in}}^*=\\eta_{\\rm {in}}\/\\eta_0$\nand $\\eta_{\\rm {mb}}^*=\\eta_{\\rm {mb}}\/\\eta_0R_0$, where \n$\\eta_0$ is the viscosity of the outside fluid. \nIn typical experimental conditions, the Reynolds number is low, Re$<1$; hence,\nthe effects of the inertia are neglected.\n\nIn Sec.~\\ref{sec:fix}, we describe the extended KS theory~\\cite{skot06} \nfor an elastic capsule with a fixed ellipsoidal shape, and the phase behavior of the capsule.\nIn Sec.~\\ref{sec:def}. we introduce the shape equation for deformable RBCs\nand present the dynamics of deformed RBCs.\nThe dependence of the function shape of the RBC free-energy potential\nis described in Sec.~\\ref{sec:pot}.\nDiscussion and summary are given in Sec.~\\ref{sec:dis} and Sec.~\\ref{sec:sum},\nrespectively. \nThe comparison with experimental results is presented in Sec.~\\ref{sec:dis}.\n\n\\section{Dynamics of elastic capsules with fixed shape}~\\label{sec:fix}\n\n\\subsection{Models}\n\n\\subsubsection{Keller-Skalak Theory}\n\nKeller and Skalak (KS) \\cite{kell82}\n analytically derived the equation of the motion of vesicles or capsules\nbased on Jeffery's theory \\cite{jeff22}.\nIn the KS theory,\nthe vesicles are assumed to have a fixed ellipsoidal shape, \n\\begin{eqnarray}\n\\Big(\\frac{x_1}{a_1}\\Big)^2 +\\Big(\\frac{x_2}{a_2}\\Big)^2 +\\Big(\\frac{x_3}{a_3}\\Big)^2 =1, \n\\end{eqnarray}\nwhere $a_i$ are the semi-axes of the ellipsoid, and\nthe coordinate axes $x_i$ point along its principal directions. \nThe $x_1$ and $x_2$ axes, with $a_1>a_2$, are on the vorticity ($xy$) plane,\nand the $x_3$ axis is in the vorticity ($z$) direction.\nThe maximum lengths in three directions are $L_1=2a_1$, $L_2=2a_2$, and $L_3=2a_3$.\nThe velocity field on the membrane is assumed to be \n\\begin{eqnarray}\n{\\bf v}^{\\rm {m}}=\\omega {\\bf u}^{\\rm {m}}= \n \\omega \\Big(-\\frac{a_1}{a_2}x_2,\\frac{a_2}{a_1}x_1,0\\Big).\n\\label{eq:KS-vel}\n\\end{eqnarray}\nThe energy $W_{\\rm {ex}}$ supplied from the external fluid \nhas to be balanced with the energy dissipated in \nthe vesicle, $W_{\\rm {ex}}=D_{\\rm {in}}+D_{\\rm {mb}}$,\nwhere $D_{\\rm {in}}$ and $D_{\\rm {mb}}$ are the energies dissipated\ninside the vesicle and on the membrane, respectively. The motion of \nthe vesicle is derived from this energy balance.\nThen the motion of the inclination angle $\\theta$ is given by\n\\begin{eqnarray}\n\\frac{d\\theta}{dt} &=& \\frac{\\dot\\gamma}{2}\\big\\{-1+f_0 f_1 \\cos(2\\theta)\\big\\} - f_0 \\omega \\nonumber \\\\\n &=& \\frac{\\dot\\gamma}{2}\\{-1+B\\cos(2\\theta)\\} \n\\label{eq:thetb} \\\\\n\\label{eq:KS-B}\nB &=& f_0\\left\\{f_1+ \\frac{f_1^{-1}}\n {1+f_2(\\eta_{\\rm {in}}^* -1)\n + f_2f_3 \\eta_{\\rm {mb}}^*}\\right\\}\\\\\n\\label{eq:KS-omega}\n\\omega &=& -\\frac{\\dot\\gamma \\cos(2\\theta) }\n {2f_1\\{1+f_2(\\eta_{\\rm {in}}^* -1) \n + f_2f_3 \\eta_{\\rm {mb}}^*\\}}.\n\\end{eqnarray}\nThe membrane-viscosity term has been derived by Tran-Son-Tay {\\it et al.} \n\\cite{tran84}.\nThe factors appearing in Eqs.~(\\ref{eq:thetb}-\\ref{eq:KS-omega}) are \ngiven by\n\\begin{eqnarray*}\nf_0 &=& 2\/(a_1\/a_2+a_2\/a_1),\\\\\nf_1 &=& 0.5(a_1\/a_2-a_2\/a_1),\\\\\nf_2 &=& 0.5g(\\alpha_1^2+\\alpha_2^2),\\\\\nf_3 &=& 0.5E_{\\rm s}R_0\/(f_1^2V),\\\\\ng &=& \\int_0^\\infty (\\alpha_1^2+s)^{-3\/2}(\\alpha_2^2+s)^{-3\/2}\n (\\alpha_3^2+s)^{-1\/2}ds,\\\\\n\\alpha_i &=& a_i\/(a_1a_2a_3)^{1\/3},\\\\\nE_{\\rm s} &=& \\oint \\tilde{e}_{ij}\\tilde{e}_{ij}dS,\\\\\n\\tilde{e}_{ij} &=& e_{ij}-0.5\\Theta P_{ij},\\\\\ne_{ij} &=& 0.5P_{ik}P_{jl}(\\partial u^{\\rm {m}}_k\/\\partial x_l\n +\\partial u^{\\rm {m}}_l\/\\partial x_k),\\\\\n\\Theta &=& P_{ij} \\partial u^{\\rm {m}}_i\/\\partial x_j,\\\\\nP_{ij} &=& \\delta_{ij}-n_in_j,\n\\end{eqnarray*}\nwhere $E_{\\rm s}$ is an integral over the membrane surface, and\n${\\bf n}$ is the normal vector of the surface.\n\nFor $B>1$, a stable fixed point $\\theta=0.5\\arccos(1\/B)$ exists,\nand TT motion occurs, while \nfor $B<1$, there is no fixed point, and \nthe angle $\\theta$ periodically rotates (TB).\nAs $\\eta_{\\rm {in}}^*$ or $\\eta_{\\rm {mb}}^*$ increases,\nthe transition from TT to TB motion occurs,\nwhere $B$ decreases from $B>1$ to $B<1$.\nThe membrane viscosity \n$\\eta_{\\rm {mb}}$ and the internal viscosity $\\eta_{\\rm {in}}^*$\nhave a similar effect; hence,\n an effective internal viscosity can be defined as\n$\\eta_{\\rm {eff}}^*=\\eta_{\\rm {in}}^* + f_3 \\eta_{\\rm {mb}}^*$.\nThe factor $f_3$ in $\\eta_{\\rm {eff}}^*$ depends on \nthe vesicle shape and can give different dynamics for\ndeformable vesicles, in particular for shape transformations between\nprolate and oblate vesicles \\cite{nogu04,nogu05}.\n\nThe KS theory quantitatively predicts the TT-TB transition with increasing $\\eta_{\\rm {eff}}^*$.\nHowever, it cannot explain the TB-TT transition with increasing $\\dot\\gamma$.\nIn the KS theory, \nvesicle motion does not depend on $\\dot\\gamma$ except that\nthe TT or TB rotation velocity increases linearly with $\\dot\\gamma$.\n\n\\begin{figure}\n\\includegraphics{fig1.eps}\n\\caption{ \\label{fig:skot_phase}\n(Color online)\nDynamic phase diagram of elastic capsules \nwith fixed shapes calculated from Eqs.~(\\ref{eq:qks}) and (\\ref{eq:phiks}) with the potential $F_0=E_0\\sin^2(\\phi)$.\n(a) Viscosity $\\eta_{\\rm {eff}}^*$ dependence for the RBC-like oblate shape with $L_2\/L_1=0.25$ and $L_3\/L_1=1$.\n(b) Aspect ratio $L_2\/L_1$ dependence at the TT-TB transition viscosity \n$\\eta_{\\rm {eff}}^*=\\eta_{\\rm {c}}^*$ of the KS theory ($U_{\\rm e}=0$).\nThe boundary lines of the TB (brown), \nTT (violet), and synchronization regions \n[$f_{\\rm {rot}}^{\\theta}:f_{\\rm {rot}}^{\\phi}=1:1$ (red), $2:1$ (blue), and $3:1$ (green)] are shown.\n}\n\\end{figure}\n\n\\subsubsection{KS Theory with an Energy Barrier}\n\nSkotheim and Secomb extended the KS theory to take into account an energy barrier during TT membrane rotation~\\cite{skot06}.\nFor RBCs and synthetic capsules with non-spherical rest shape, their membranes are locally deformed\nduring the TT rotation.\nFischer experimentally demonstrated that the RBC membrane rotates back to the original position \nwhen the shear flow is switched off~\\cite{fisc04}.\nTo describe the energy barrier, \na phase angle $\\phi$ and free energy potential $F(\\phi)$ are introduced; see inset of Fig.\\ref{fig:skot_phase}.\nThe potential is periodic, $F(\\phi+ n\\pi)=F(\\phi)$ and $\\phi=0$ at the rest shape.\nThus, the motions of the inclination angle\n$\\theta$ and phase angle $\\phi$ are given by\n\\begin{eqnarray}\n\\label{eq:qks}\n\\frac{\\ \\ d \\theta}{\\dot\\gamma dt} &=& \\frac{1}{2}\\big\\{-1+f_0 f_1 \\cos(2\\theta)\\big\\} - \\frac{f_0 d \\phi}{\\dot\\gamma dt},\\\\\n\\label{eq:phiks}\n\\frac{\\ \\ d \\phi}{\\dot\\gamma dt} &=& -\\frac{(c_0\/\\dot\\gamma^*V^*) \\partial F^*\/\\partial \\phi + \\cos(2\\theta) }\n {2f_1\\{1+f_2(\\eta_{\\rm {in}}^* -1) \n + f_2f_3 \\eta_{\\rm {mb}}^*\\}},\n\\end{eqnarray}\nwhere $c_0=3f_2\/8\\pi f_1$.\nThe equations of the original KS theory are recovered\n in the absence of barriers of the free energy $F$,\ni.e., $\\partial F^*\/\\partial \\phi=0$,\nwhere $\\omega=d\\phi\/dt$ is independent of $\\phi$.\n\nSkotheim and Secomb used a simple potential $F_0(\\phi)=E_0 \\sin^2(\\phi)$\nand a reduced energy $U_{\\rm e}=f_2E_0\/2f_1\\eta\\dot\\gamma V = E_0^* c_0\/\\dot\\gamma^*V^*$.\nWe employ the potential $F_0$ in this section\nand describe the dependence on the potential shape in Sec.~\\ref{sec:pot}.\nEqs.~(\\ref{eq:qks}) and (\\ref{eq:phiks}) are numerically integrated\nusing the fourth-order Runge-Kutta method.\nAn oblate capsule with $L_2\/L_1=0.25$ and $L_3\/L_1=1$\n is used as a model RBC.\n\n\\begin{figure}\n\\includegraphics{fig2.eps}\n\\caption{ \\label{fig:skot_frot}\n(Color online)\nRotation frequency $f_{\\rm {rot}}$ of \nthe inclination angle $\\theta$ and phase angle $\\phi$\nfor the fixed oblate shape with $L_2\/L_1=0.25$ \nat $\\eta_{\\rm {eff}}^*=1$.\nNumbers represent $f_{\\rm {rot}}^{\\theta}:f_{\\rm {rot}}^{\\phi}$.\n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics{fig3.eps}\n\\caption{ \\label{fig:skot_det}\n(Color online)\nDynamics of oblate capsules for the fixed shape.\nTop panel: time development of angles $\\theta$ and $\\phi$ at $U_{\\rm e}=0.78$.\nMiddle and bottom panels: trajectories on the phase space ($\\theta$, $\\phi$).\nThe capsules exhibit TT and TB rotations at $U_{\\rm e}=0.76$ and $0.81$, respectively.\nSynchronized rotations with $f_{\\rm {rot}}^{\\theta}:f_{\\rm {rot}}^{\\phi}=1:1$\nand $2:1$, and intermittent rotation are observed at $U_{\\rm e}=0.78$, $0.793$, and $0.7782$, \nrespectively.\nThe other parameters are the same as in Fig.~\\ref{fig:skot_frot}.\n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics{fig4.eps}\n\\caption{ \\label{fig:skot_rmap}\n(Color online)\nReturn map of oblate capsules for the fixed shape.\nThe angle $\\phi$ at $\\theta=\\pi\/2- n \\pi$ is plotted.\nSynchronized rotation with $f_{\\rm {rot}}^{\\theta}:f_{\\rm {rot}}^{\\phi}=1:1$\noccurs at $0.77823 1$)\nin the generalized KS theory~\\cite{nogu07b}.\nPreviously, we distinguished that\nin the SW oscillation of the elastic capsules, $\\theta$ is always positive and the shape deformation is negligibly small,\nwhile in SW of fluid vesicles, $\\theta$ changes its sign and the shape shows large deformation.\nHowever, we know now that the condition for $\\theta$ is not always true.\nThe clear difference is the dependence on $\\eta_{\\rm {in}}^*$ or $\\eta_{\\rm {mb}}^*$.\nSW induced by the shape deformation appears only in a narrow range of the viscosity,\nwhereas SW induced by the membrane shear elasticity appears at a wide range of the viscosities with no lower viscosity limit.\nIn the future, it will be interesting to investigate the coupling of different \noscillation mechanisms in elastic capsules.\n\n\\section{Summary}~\\label{sec:sum}\n\nIn summary, we described the dynamics of RBCs in simple shear flow using a simple theory.\nThe phase diagram of RBCs is divided into three regions:\ntank-treading, tumbling, and intermediate regions.\nIn the intermediate regions,\nRBCs exhibit intermittent or synchronized rotations of the inclination angle $\\theta$ and phase angle $\\phi$. \nSynchronized rotations, in particular with $f_{\\rm {rot}}^{\\theta}:f_{\\rm {rot}}^{\\phi}=1:1$,\nwould be much easier to experimentally observe than intermittent rotations.\nIn the TT (TB) phase, the shape and $\\theta$ ($\\phi$) oscillate with the frequency of $\\phi$ ($\\theta$) rotation.\nThe coexistence of two synchronized rotations can appear when the potential function of $\\phi$ has a sharp peak.\nThe other dynamic properties are not sensitive to the function shape of the free-energy potential.\nWe focused on the dynamics of RBCs in this paper, but \n the resulting dynamics would be generally applicable to other elastic capsules.\n\n\n\\begin{acknowledgments}\nWe would like to thank \nG. Gompper (J{\\\"u}lich) for the helpful discussion.\nThis study is partially supported by a Grant-in-Aid for Scientific Research on Priority Area ``Soft Matter Physics'' from\nthe Ministry of Education, Culture, Sports, Science, and Technology of Japan.\n\\end{acknowledgments}\n\n\\bibliographystyle{apsrev}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}