%~Mouliné par MaN_auto v.0.25.0 2021-10-05 12:07:38 \documentclass[AHL,Unicode,longabstracts,published]{cedram} \usepackage{tikz-cd} \newcommand{\cal}[1]{\mathcal{#1}} \newcommand{\bbZ}{\mathbb{Z}} \newcommand{\calC}{\mathcal{C}} \newcommand{\calV}{\mathcal{V}} \newcommand{\calO}{\mathcal{O}} \newcommand{\calG}{\mathcal{G}} \newcommand{\calD}{\mathcal{D}} \newcommand{\calT}{\mathcal{T}} \newcommand{\calS}{\mathcal{S}} \newcommand{\calE}{\mathcal{E}} \newcommand{\calA}{\mathcal{A}} \newcommand{\calN}{\mathcal{N}} \newcommand{\calP}{\mathcal{P}} \newcommand{\calB}{\mathcal{B}} \newcommand{\calH}{\mathcal{H}} \newcommand{\calM}{\mathcal{M}} \newcommand{\calF}{\mathcal{F}} \newcommand{\calU}{\mathcal{U}} \DeclareMathOperator{\Mod}{Mod} \DeclareMathOperator{\Stab}{Stab} \DeclareMathOperator{\spin}{spin} \DeclareMathOperator{\Arf}{Arf} \newcounter{casecount} \newenvironment{case}{\refstepcounter{casecount}\begin{proof}[Case~\thecasecount]}{\let\qed\relax\end{proof}} \newenvironment{lastcase}{\refstepcounter{casecount}\begin{proof}[Case~\thecasecount]}{\end{proof}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \graphicspath{{./figures/}} \newcommand*{\mk}{\mkern -1mu} \newcommand*{\Mk}{\mkern -2mu} \newcommand*{\mK}{\mkern 1mu} \newcommand*{\MK}{\mkern 2mu} %\hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red} \newcommand*{\romanenumi}{\renewcommand*{\theenumi}{\roman{enumi}}} \newcommand*{\Romanenumi}{\renewcommand*{\theenumi}{\Roman{enumi}}} \newcommand*{\alphenumi}{\renewcommand*{\theenumi}{\alph{enumi}}} \newcommand*{\Alphenumi}{\renewcommand*{\theenumi}{\Alph{enumi}}} \let\oldtilde\tilde \renewcommand*{\tilde}[1]{\mathchoice{\widetilde{#1}}{\widetilde{#1}}{\oldtilde{#1}}{\oldtilde{#1}}} \let\oldhat\hat \renewcommand*{\hat}[1]{\mathchoice{\widehat{#1}}{\widehat{#1}}{\oldhat{#1}}{\oldhat{#1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title[Spin mapping class group] {Generating the spin mapping class group by Dehn twists} \alttitle{Comment engendrer le groupe de classes d'homéomorphismes spin par des twists de Dehn} \subjclass{30F30, 30F60, 37B10, 37B40} \keywords{Spin mapping class group, Dehn twists, curve systems, group generators} \author[\initial{U.} \lastname{Hamenst\"adt}]{\firstname{Ursula} \lastname{Hamenst\"adt}} \address{Mathematisches Institut\\ der Universit\"at Bonn\\ Endenicher Allee 60,\\ D-53115 BONN, (Germany)} \email{ursula@math.uni-bonn.de} \thanks{The author is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC-2047/1 390685813. The work was carried out while the author was in residence at the MSRI in Berkeley, California, in the fall semester 2019, supported by the National Science Foundation under Grant No.~DMS-1440140.} \begin{abstract} Let $\phi$ be a $\bbZ/2\bbZ$-spin structure on a closed oriented surface $\Sigma_g$ of genus $g\geq 4$. We determine a generating set of the stabilizer of $\phi$ in the mapping class group of $\Sigma_g$ consisting of Dehn twists about an explicit collection of $2g+1$ curves on $\Sigma_g$. If $g=3$ then we determine a generating set of the stabilizer of an odd $\bbZ/4\bbZ$-spin structure consisting of Dehn twists about a collection of $6$ curves. \end{abstract} \begin{altabstract} Soit $\phi$ une $\bbZ/2\bbZ$-structure de spin sur une surface fermée orientée $\Sigma_g$ de genre $g\geq 4$. Nous déterminons une partie génératice du stabilisateur de $\phi$ dans le groupe de classes d'homéomorphismes de $\Sigma_g$, composée de twists de Dehn autour d'un ensemble explicite de $2g+1$ courbes de $\Sigma_g$. Lorsque $g=3$, nous déterminons une partie génératrice du stabilisateur d'une $\bbZ/4\bbZ$-structure de spin impaire, formée de twist de Dehn autour de $6$ courbes. \end{altabstract} \datereceived{2020-04-04} \daterevised{2021-03-08} \dateaccepted{2021-04-01} \editor{V. Colin} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{DefTralics} \newcommand{\bbZ}{\mathbb{Z}} \end{DefTralics} \dateposted{2021-12-03} \begin{document} \maketitle \section{Introduction} For some $r\geq 2$, a $\bbZ/r\bbZ$-spin structure on a closed surface $\Sigma_g$ of genus $g$ is a cohomology class $\phi\in H^1(UT\Sigma_g,\bbZ/r\bbZ)$ which evaluates to one on the oriented fibre of the unit tangent bundle $UT\Sigma_g\to \Sigma_g$ of $\Sigma_g$. Such a spin structure exists for all $r$ which divide $2g-2$. If $r$ is even, then it reduces to a $\bbZ/2\bbZ$-spin structure on $\Sigma_g$. A $\bbZ/2\bbZ$-spin structure on $\Sigma_g$ has a \emph{parity}, either even or odd. Thus there is a notion of parity for all $\bbZ/r\bbZ$-spin structures with $r$ even. If $\phi,\phi^\prime$ are two $\bbZ/r\bbZ$-spin structures on $\Sigma_g$ so that either $r$ is odd or $r$ is even and the parities of $\phi,\phi^\prime$ coincide, then there exists an element of the mapping class group $\Mod(\Sigma_g)$ of $\Sigma_g$ which maps $\phi$ to $\phi^\prime$. Hence the stabilizers of $\phi$ and $\phi^\prime$ in $\Mod(\Sigma_g)$ are conjugate. Spin structures naturally arise in the context of abelian differentials on $\Sigma_g$. The moduli space of such differentials decomposes into strata of differentials whose zeros are of the same order and multiplicity. Understanding the orbifold fundamental group of such strata requires some understanding of their projection to the mapping class group. If the orders of the zeros of the differentials are all multiples of the same number $r\geq 2$, then this quotient group preserves a $\bbZ/r\bbZ$-spin structure $\phi$ on $\Sigma_g$. Hence the orbifold fundamental groups of components of strata relate to stabilizers $\Mod(\Sigma_g)[\phi]$ of spin structures $\phi$ on $\Sigma_g$. To make such a relation explicit we define \begin{defi}\label{curvesystem} A \emph{curve system} on a closed surface $\Sigma_g$ is a finite collection of smoothly embedded simple closed curves on $\Sigma_g$ which are non-contractible and mutually not freely homotopic, and such that any two curves from this collection intersect transversely in at most one point. \end{defi} A curve system defines a \emph{curve diagram} which is a finite graph whose vertices are the curves from the system and where two such vertices are connected by an edge if the curves intersect. \begin{defi} A curve system on $\Sigma_g$ is \emph{admissible} if it decomposes $\Sigma_g$ into a collection of topological disks and if its curve diagram is a tree. \end{defi} Using a construction of Thurston and Veech (see~\cite{L04} for a comprehensive account), admissible curve systems on $\Sigma_g$ give rise to abelian differentials on $\Sigma_g$, and the component of the stratum and hence the equivalence class of a spin structure (if any) it defines can be read off explicitly from the combinatorics of the curve system. This makes it desirable to investigate the subgroup of the mapping class group generated by Dehn twists about the curves of an admissible curve system. The main goal of this article is to present a systematic study of stabilizers of suitably chosen curves in the spin mapping class group $\Mod(\Sigma_g)[\phi]$ and to use this information to build generators for this group by induction over subsurfaces. As a main application we obtain the following. For $g\geq 3$ let $\calC_g$ and $\calV_g$ be the collections of $2g+1$ nonseparating simple closed curves on a closed surface $\Sigma_g$ of genus $g$ shown in Figure~\ref{fig1}. \pagebreak \begin{figure}[!ht] \begin{center} \includegraphics[width=0.9\textwidth]{spinfigure12.pdf} \end{center} \caption{} \label{fig1} \end{figure} We show \begin{theo}\label{main2}\ \\*[-1.4em] \begin{enumerate} \item\label{theo1.3.1} Let $\phi$ be an odd $\bbZ/2 \bbZ$-spin structure on a closed surface $\Sigma_g$ of genus $g\geq 3$. Then $\Mod(\Sigma_g)[\phi]$ is generated by the Dehn twists about the curves from the curve system $\calC_g$. \item\label{theo1.3.2} Let $\phi$ be an even $\bbZ/2\bbZ$-spin structure on a closed surface $\Sigma_g$ of genus $g\geq 4$. Then $\Mod(\Sigma_g)[\phi]$ is generated by the Dehn twists about the curves from the curve system $\calV_g$. \end{enumerate} \end{theo} That the spin mapping class group can be generated by finitely many finite products of Dehn twists is due to Hirose. In~\cite{Hi02} he found for any genus $g\geq 2$ a generating set for the stabilizer of an even $\bbZ/2\bbZ$-spin structure by finitely many finite products of Dehn twists, and the stabilizer of an odd $\bbZ/2\bbZ$-spin structure is treated in~\cite{Hi05}. For surfaces of genus $g\geq 5$, Calderon~\cite{Cal19} and Calderon and Salter~\cite{CS19} identified the image of the orbifold fundamental group of most components of strata in the mapping class group by constructing a different but equally explicit generating set for the spin mapping class group. Earlier Salter (\cite[Theorem~9.5]{Sa19}) obtained a partial result by identifying for $g\geq 5$ a finite generating set of a finite index subgroup of the spin mapping class group by Dehn twists. Walker~\cite{W09,W10} obtained some information on the image of the orbifold fundamental group of some strata of quadratic differentials in the mapping class group using completely different~tools. Theorem~\ref{main2} does not construct generators for the stabilizer of an even $\bbZ/2\bbZ$-spin structure on a surface of genus $g=2,3$. Namely, in these cases there is no admissible curve system with the property that the Dehn twists about the curves from the system stabilize an even $\bbZ/2\bbZ$-spin structure and such that the Dehn twists about these curves generate a finite index subgroup of the mapping class group. This corresponds to a classification result of Kontsevich and Zorich~\cite{KZ03}: There is no component of a stratum of abelian differentials with a single zero on a surface of genus $2$ and even spin structure. On a surface $\Sigma_3$ of genus 3, the component of the stratum of abelian differentials with two zeros of order two and even spin structure is hyperelliptic and hence the projection of its orbifold fundamental group to $\Mod(\Sigma_3)$ commutes with a hyperelliptic involution and is of infinite index. Our results can be used to construct an explicit finite set of generators of the stabilizer of a $\bbZ/r\bbZ$-spin structure for any $r\leq 2g-2$ and any closed surface $\Sigma_g$, given by Dehn twists, positive powers of Dehn twists and products of Dehn twists about two simple closed curves forming a bounding pair. Potentially they can also be used inductively to find generators by Dehn twists about curves from an admissible curve system. We carry this program only out in a single case, which is the odd $\bbZ/4\bbZ$-spin structure on a surface of genus 3. Consider the system ${\calE}_6$ of simple closed curves on the surface $\Sigma_3$ of genus 3 shown in Figure~\ref{Fig2} which is of particular relevance for the understanding of the stratum of abelian differentials with a single zero on $\Sigma_3$~\cite{LM14}. \begin{figure}[ht] \begin{center} \includegraphics[width=0.8\textwidth]{e6.pdf} \end{center} \caption{} \label{Fig2} \end{figure} We show \begin{theo}\label{main3} The subgroup of $\Mod(\Sigma_3)$ generated by the Dehn twists about the curves from the curve system ${\calE}_6$ equals the stabilizer of an odd $\bbZ/4\bbZ$-spin structure on $\Sigma_3$. \end{theo} The strategy for the proofs of the main results is as follows. For some $r\geq 2$ let us consider an arbitrary $\bbZ/r\bbZ$-spin structure $\phi$ on a compact oriented surface $S$ of genus $g\geq 2$, perhaps with boundary. Following~\cite{HJ89} and~\cite{Sa19}, the spin structure can be viewed as a $\bbZ/r\bbZ$-valued function on oriented closed curves on $S$ which assumes the value one on the oriented boundary of an embedded disk in $S$. Changing the orientation of the curve changes the value of $\phi$ on the curve to its negative~\cite{HJ89,Sa19}. Define a graph ${\calC\calG}_1^+$ as follows. Vertices are nonseparating simple closed curves $c$ on $S$ with $\phi(c)=\pm 1$, and two such vertices $d,e$ are connected by an edge if $d,e$ can be realized disjointly and if furthermore, $S-(d\cup e)$ is connected. Thus ${\calC\calG}_1^+$ is a subgraph of the curve graph of $S$. The stabilizer $\Mod(S)[\phi]$ of $\phi$ in the mapping class group of $S$ acts on ${\calC\calG}_1^+$ as a group of simplicial automorphisms. In Section~\ref{graphsofcurves} we show that for any $g\geq 3$ and $r\leq 2g-2$ the graph ${\calC\calG}_1^+$ is connected. We also note that for an odd $\bbZ/2\bbZ$-spin structure on a surface of genus $g=2$, this is not true. In Section~\ref{theaction} we verify that the action of the group $\Mod(S)[\phi]$ on the graph ${\calC\calG}_1^+$ is transitive on vertices. For a vertex $c$ of ${\calC\calG}_1^+$ we are then led to describing the intersection of $\Mod(S)[\phi]$ with the stabilizer of $c$ in $\Mod(S)$. Most important is the understanding of the intersection of $\Mod(S)[\phi]$ with the so-called \emph{disk pushing subgroup}, namely the kernel of the natural homomorphism of the stabilizer of $c$ to the mapping class group of the surface obtained from $S-c$ by capping off the two distinguished boundary components of $S-c$. This is also carried out in Section~\ref{theaction}. In Section~\ref{structureof} we specialize further to a $\bbZ/2\bbZ$-spin structure $\phi$. We find a presentation of $\Mod(S)[\phi]$ as a quotient of a $\bbZ/2\bbZ$-extension of the free product of two copies of the stabilizer of a vertex of ${\calC\calG}_1^+$, amalgamated over the stabilizer of an edge of ${\calC\calG}_1^+$. This is used to prove Theorem~\ref{main2}$\MK$\eqref{theo1.3.1} with an argument by induction on the genus $g$ of the closed surface $\Sigma_g$. The proof of Theorem~\ref{main2}$\MK$\eqref{theo1.3.2} uses similar methods and is contained in Section~\ref{structureeven}. A variation of these arguments yield the proof of Theorem~\ref{main3} in Section~\ref{special}. The Appendix~\ref{addition} contains a technical variation of the main result of Section~\ref{graphsofcurves} which is used in Section~\ref{structureeven}. Its proof follows along exactly the same line as the proof of the main result of Section~\ref{graphsofcurves}. This work is inspired by the article~\cite{Sa19} of Salter. However, aside from some simple constructions using curves and~\cite[Proposition~4.9]{Sa19}, our approach uses different methods. \subsection*{Acknowledgements} I am grateful to Dawei Chen, Samuel Grushevsky, Martin M\"oller and Nick Salter for useful discussions. Thanks to Susumu Hirose for pointing out the references~\cite{Hi02} and~\cite{Hi05}. Finally I am very indebted to the anonymous referee for careful reading and for suggesting the proof of Proposition~\ref{genus3} which largely simplifies my original argument. \section{Graphs of curves with fixed spin value}\label{graphsofcurves} In this section we consider a compact surface $S$ of genus $g\geq 2$, with or without boundary. For a number $r\geq 2$ we introduce $\bbZ/r\bbZ$-spin structures on $S$ and use these structures to define various subgraphs of the curve graph of $S$. Of primary interest is a graph ${\calG\calG}_1$ whose vertices are nonseparating simple closed curves with spin value $\pm 1$ and where two such curves are connected by an edge if they can be realized disjointly. We then study connectedness of this graph. Small genus of the surface may cause the graph ${\calC\calG}_1$ to have few edges. This problem leads us to proceed in two steps. In Proposition~\ref{genus3} we show connectedness of ${\calC\calG}_1$ for surfaces of genus $g\geq 3$ and $r=2,4$, taking advantage of some special properties of $\bbZ/2\bbZ$ and $\bbZ/4\bbZ$ spin structures. Proposition~\ref{connected6} shows connectedness of ${\calC\calG}_1$ for surfaces of genus $g\geq 4$ and all $r$, taking advantage of sufficiently large complexity of the underlying surface. These results are used in Section~\ref{theaction} to study the stabilizer of a spin structure in the mapping class group of $S$. This section is divided into 5 subsections. We begin with summarizing some information on spin structures. Each of the remaining subsections is devoted to the investigation of a specific subgraph of the curve graph of $S$ defined by a spin structure $\phi$ on $S$. \subsection{Spin structures} The following is taken from~\cite{HJ89}, see~\cite[Definition~3.1] {Sa19}. For its formulation, denote by $\iota$ the symplectic form on $H_1(S,\bbZ)$. \begin{defi}[Humphries--Johnson]\label{spin} For a number $r\geq 2$, a \emph{$\bbZ/r\bbZ$-spin structure} on $S$ is a $\bbZ/r\bbZ$-valued function $\phi$ on isotopy classes of oriented simple closed curves on $S$ with the following properties. \begin{enumerate} \item \label{defi2.1.1}(Twist linearity) Let $c,d$ be oriented simple closed curves and let $T_c$ be the left Dehn twist about $c$; then \[ \phi(T_c(d))=\phi(d)+\iota(d,c)\phi(c)\quad \text{(mod}\,r). \] \item \label{defi2.1.2}(Normalization) $\phi(\zeta)=1$ for the oriented boundary $\zeta$ of an embedded disk $D\subset S$. \end{enumerate} \end{defi} As an additional property, one obtains that whenever $c^{-1}$ is obtained from $c$ by reversing the orientation, then $\phi(c^{-1})=-\phi(c)$ (\cite[Lemma~2.2]{HJ89}). Humphries and Johnson~\cite{HJ89} (see~\cite[Theorem~3.5]{Sa19}) also give an alternative description of spin structures. Namely, for some choice of a hyperbolic metric on $S$ let $UTS$ be the unit tangent bundle of $S$. It can be viewed as the quotient of the complement of the zero section in the tangent bundle of $S$ by the multiplicative group $(0,\infty)$ and hence it does not depend on the metric. The \emph{Johnson lift} of a smoothly embedded oriented simple closed curve $c$ on $S$ is simply the closed curve in $UTS$ which consists of all unit tangents of $c$ defining the given orientation. The following is~\cite[Theorem~2.1 and Theorem~2.5]{HJ89} as formulated in~\cite[Theorem~3.5]{Sa19}. \begin{theo}[Humphries--Johnson]\label{cohomology} Let $S$ be a compact surface and let $\zeta$ be the oriented fibre of the unit tangent bundle $UTS\to S$. A cohomology class $\psi\in H^1(UTS,\bbZ/r\bbZ)$ with $\psi(\zeta)=1$ determines a $\bbZ/r\bbZ$-spin structure via \[ \alpha\to \psi(\tilde \alpha) \] where $\alpha$ is an oriented simple closed curve on $S$ and $\tilde \alpha$ is its Johnson lift. This determines a 1-1 correspondence between $\bbZ/r\bbZ$-spin structures and \[ \left\{\psi\in H^1(UTS,\bbZ/r\bbZ)\middle| \psi(\zeta)=1\right\}. \] \end{theo} There is another interpretation as follows; we refer to~\cite[p.~131]{H95} for more information on this construction. Given a number $r\geq 2$ which divides $2g-2$, an application of the Gysin sequence for the Euler class of $UTS$ yields a short exact sequence \begin{equation}\label{gysin} 0\to \bbZ/r\bbZ\to H_1(UTS,\bbZ/r\bbZ)\to H_1(S,\bbZ/r\bbZ)\to 0. \end{equation} By covering space theory, an $r^{\rm th}$ root of the tangent bundle of $S$, viewed as a complex line bundle for some fixed complex structure, is determined by a homomorphism $H_1(UTS,\bbZ/r\bbZ)\to \bbZ/r\bbZ$ whose composition with the inclusion $\bbZ/r\bbZ\to H_1(UTS,\bbZ/r\bbZ)$ is the identity and therefore \begin{prop}\label{rspin} There is a natural one-to-one correspondence between the $r^{\rm th}$ roots of the canonical bundle of $S$ and splittings of the sequence~\eqref{gysin}. \end{prop} A $\bbZ/2\bbZ$-spin structure on a compact surface $S$ of genus $g$ with empty or connected boundary has a \emph{parity} which is defined as follows. A \emph{geometric symplectic basis} for $H_1(S,\bbZ)$ is a system $a_1,b_1,\,\dots,\,a_g,b_g$ of simple closed curves on $S$ such that $a_i,b_i$ intersect in a single point and that $a_i\cup b_i$ is disjoint from $a_j\cup b_j$ for $i\not=j$. Then the parity of the spin structure $\phi$ equals \begin{equation}\label{arf} \Arf(\phi) =\sum_i\left(\phi(a_i)+1\right)\left(\phi(b_i)+1\right) \in \bbZ/2\bbZ. \end{equation} This does not depend on the choice of the geometric symplectic basis. \subsection{The graph of nonseparating curves with vanishing spin value} The \emph{curve graph} ${\calC\calG}$ of $S$ is the graph whose vertices are \emph{essential} (that is, neither nullhomotopic nor homotopic into the boundary) simple closed curves in $S$ and where two such curves are connected by an edge if they can be realized disjointly. We can use the spin structure $\phi$ to introduce various subgraphs of ${\calC\calG}$ and study their properties. One of the main technical ingredients to this end is the following result of Salter~\cite[Corollary~4.3]{Sa19}). \begin{lemm}[Salter]\label{torus} Let $\Sigma\subset S$ be an embedded one-holed torus. Then there exists a simple closed curve $c\subset \Sigma$ with $\phi(c)=0$. \end{lemm} Denote by ${\calC\calG}_0\subset {\calC\calG}$ the complete subgraph of the curve graph whose vertex set consists of nonseparating curves $c$ with $\phi(c)=0$. Note that this is well defined, that is, it is independent of the choice of an orientation of $c$. As a fairly easy consequence of Lemma~\ref{torus} we obtain \begin{lemm}\label{connected0} Let $\phi$ be a spin structure on a closed surface of genus $g\geq 3$. Then ${\calC\calG}_0$ is connected. \end{lemm} \begin{proof} We use the following result of Masur--Schleimer~\cite{MS06}, see~\cite[Theorem~1.2]{Put08}. Let ${\calS\calG}\subset {\calC\calG}$ be the complete subgraph whose vertex set consists of \emph{separating} simple closed curves; then ${\calS\calG}$ is connected. Note that this requires that $g\geq 3$. Let $a,b$ be vertices of ${\calC\calG}_0$. Choose simple closed curves $\hat a,\hat b$ which intersect $a,b$ in a single point; such curves exist since $a,b$ are nonseparating. Then the boundary $c,d$ of a tubular neighborhood of $a\cup\hat a$ and $b\cup \hat b$, respectively, is a separating simple closed curve which decomposes $S$ into a one-holed torus containing $a,b$ and a surface of genus $g-1\geq 2$ with boundary. Connect $c$ to $d$ by an edge path $(c_i)_{0\,\leq\,i\,\leq\,k}\subset {\calS\calG}$ (here $c=c_0$ and $d=c_k$). Construct inductively an edge path $(a_i)\subset {\calC\calG}_0$ connecting $a=a_0$ to $b=a_k$ such that for each $i$, $a_{i}$ is disjoint from $c_i$, as follows. Put $a_0=a$ and assume that we constructed already such a path for some $j= 3}} In this subsection we study the graph ${\calC\calG}_1$ for a $\bbZ/r\bbZ$-spin structure on a surface of genus $g\geq 3$ for $r=2,4$. We begin with evoking a result of Salter~\cite{Sa19}. Namely, let $c,d$ be disjoint simple closed curves on the compact surface $S$. Let $\epsilon$ be an embedded arc in $S$ connecting $c$ to $d$ whose interior is disjoint from $c\cup d$. A regular neighborhood $\nu$ of $c\cup \epsilon \cup d$ is homeomorphic to a three-holed sphere. Two of the boundary components of $\nu$ are the curves $c,d$ up to homotopy. We choose an orientation of $c,d$ in such a way that $\nu$ lies to the left. The third boundary component $c +_\epsilon d$, oriented in such a way that $\nu$ is to its right, satisfies $[c +_\epsilon d]=[c]+[d]$ where $[c]$ denotes the homology class of the oriented curve $c$. The following is~\cite[Lemma~3.13]{Sa19}. \begin{lemm}[Salter]\label{add} $\phi(c+_\epsilon d)= \phi(c)+\phi(d)+1$. \end{lemm} As a consequence, if $r=2,4$ then the boundary of any embedded pair of pants $P\subset S$ contains a simple closed curve $c$ with $\phi(c)=\pm 1$. To use this fact for our purpose we introduce another graph related to simple closed curves on surfaces. \begin{defi}\label{nonseparatingpairs} Let $S$ be a compact surface of genus $g\geq 2$. The \emph{graph of non\-separating pairs of pants} ${\calN\calS}$ is the graph whose vertices are pairs of pants in $S$ whose boundary consists of three pairwise distinct nonseparating simple closed curves and where two such pair of pants are connected by an edge if their intersection consists of precisely one boundary component. \end{defi} By the preceding remark, the graph ${\calN\calS}$ can be used to find paths in the graph ${\calC\calG}_1$ provided we can show that it is connected. To this end we evoke an observation of Putman (\cite[Lemma~2.1]{Put08}) which we refer to as the \emph{Putman trick} in the sequel. \begin{lemm}[Putman]\label{putmantrick} Let $G$ be a graph which admits a vertex transitive isometric action of a finitely generated group $\Gamma$ and let $v$ be a vertex of $G$. If for each element $s$ of a finite generating set ${\calS}$ of $\Gamma$, the vertex $v$ can be connected to $sv$ by an edge path in $G$, then $G$ is connected. \end{lemm} We apply the Putman trick to show \begin{lemm}\label{connectedpants} The graph of nonseparating pairs of pants is connected. \end{lemm} \begin{proof} If $P\subset S$ is a nonseparating pair of pants, then $S-P$ is a connected surface of genus $g-2$ with three distinguished boundary components. Thus the pure mapping class group $P\Mod(S)$ of $S$ acts transitively on the vertices of ${\calN\calS}$. As a consequence, it suffices to show that there exists a generating set ${\calS}$ of $P\Mod(S)$ and a nonseparating pair of pants $P\in {\calN\calS}$ which can be connected to its image $\psi(P)$ by an edge path in ${\calN\calS}$ for every element $\psi\in {\calS}$. Now $P\Mod(S)$ can be generated by Dehn twists $T_{c_i}$ about the collection of simple closed curves $c_0,\,\dots,\,c_k$ shown in Figure~\ref{fig3} (see~\cite[Section~4.4]{FM12}). \begin{figure}[ht] \begin{center} \includegraphics[width=0.9\textwidth]{humphries.pdf} \caption{} \label{fig3} \end{center} \end{figure} Furthermore, the simple closed curves $c_0,c_1,c_3$ are nonseparating and bound a pair of pants $P$. This pair of pants is stabilized by all elements of ${\calS}$ with the exception of the Dehn twists about the simple closed curves $c_2$ and $c_4$. As the genus of $S$ is at least $3$, for $i=2,4$ the complement $\hat S$ in $S$ of the union of $P$ with $c_i$ is a surface of genus $g-2\geq 1$ with two distinguished boundary components, one of which, say the curve $c$, is a boundary component of $P$. The surface $\hat S$ contains a nonseparating simple closed curve $d$. As in Lemma~\ref{add}, choose an embedded arc $\epsilon\subset \hat S$ connecting $c$ to $d$. A small neighborhood of $c\cup \epsilon \cup d$ is a nonseparating pair of pants $\hat P$ whose intersection with $P$ equals the curve $c$. As $\hat P$ is disjoint from $c_i$, it is left fixed by the Dehn twist $\psi_i$ about $c_i$ and hence $P,\hat P,\psi_i(P)$ is a path of length three in ${\calN\calS}$ connecting $P$ to $\psi_i(P)$. Lemma~\ref{connectedpants} now is an immediate consequence of Lemma~\ref{putmantrick}. \end{proof} The following is now an easy consequence of Lemma~\ref{connectedpants}. \begin{prop}\label{genus3} Let $r=2,4$ and let $\phi$ be a $\bbZ/r\bbZ$ spin structure on a compact surface $S$ of genus $g\geq 3$, with or without boundary. Then the graph ${\calC\calG}_1$ is connected. \end{prop} \begin{proof} Let $c,d$ be nonseparating simple closed curves with $\phi(c)=\pm 1, \phi(d)=\pm 1$. Choose nonseparating pairs of pants $P,Q$ containing $c,d$ in their boundary. By Lemma~\ref{connectedpants}, we can connect $P$ to $Q$ by a path in the graph ${\calN\calS}$, say the path $(P_i)$ with $P_0=P$ and $P_k=Q$. For each $i$ let $c_i$ be a boundary component of $P_i$ with $\phi(c_i)=\pm 1$ and such that $c_0=c,c_k=d$. Then for each $i$, either $c_i=c_{i+1}$ or $c_i$ and $c_{i+1}$ are disjoint. Thus $(c_i)$ is a path in ${\calC\calG}_1$ connecting $c$ to $d$. This shows the proposition. \end{proof} \subsection{\texorpdfstring{$\bbZ/r\bbZ$}{Z/rZ}-spin structures on a surface of genus \texorpdfstring{$g\geq 4$}{g>= 4}} \label{allr} In this subsection we investigate the graph ${\calC\calG}_1$ on a surface of genus $g\geq 4$ for an arbitrary $r\geq 2$. To show connectedness we use the following auxiliary graph ${\calP\calS}$. The vertices of ${\calP\calS}$ are pairs of disjoint separating curves $(c,d)$ which each decompose $S$ into a surface of genus $g-1$ and a one-holed torus. Thus $S-(c\cup d)$ is the disjoint union of two one-holed tori and a surface of genus $g-2$. Two such pairs $(c_1,d_1)$ and $(c_2,d_2)$ are connected by an edge if up to renaming, $c_1=c_2$ and $d_2$ is disjoint from $c_1,d_1$. Then $S-(c_1\cup d_1\cup d_2)$ is the disjoint union of a surface of genus $g-3$ with at least three holes and three one-holed tori. In particular, the graph ${\calP\calS}$ is only defined if the genus of $S$ is at least three. We use the Putman trick to show \begin{lemm}\label{connected5} For a compact surface $S$ of genus $g\geq 4$, perhaps with boundary, the graph ${\calP\calS}$ is a connected $\Mod(S)$-graph. \end{lemm} \begin{proof} The mapping class group $\Mod(S)$ of the surface $S$ clearly acts on ${\calP\calS}$, furthermore this action is vertex transitive. Namely, for any two vertices $(a_1,b_1)$ and $(a_2,b_2)$ of ${\calP\calS}$, the complement $S-(a_i\cup b_i)$ is the union of two one-holed tori and a surface of genus $g-2$ with $k+2$ boundary components where $k\geq 0$ is the number of boundary components of $S$. Hence there exists $\phi\in \Mod(S)$ with $\phi(a_1,b_1)=(a_2,b_2)$. Consider again the curve system ${\calH}$ shown in Figure~\ref{fig3} with the property that the Dehn twists about these curves generate the mapping class group. Choose a pair of disjoint separating simple closed curves $(a,b)$ which decompose $S$ into a surface of genus $g-1$ and a one-holed torus $X(a), X(b)$ and such that a curve $c\in {\calH}$ intersects at most one of the curves $a,b$. If it intersects one of the curves $a,b$, then this intersection consists of precisely two points. For example, we can choose $a$ to be the boundary of a small neighborhood of $c_1\cup c_2$, and $b$ to be the boundary of a small neighborhood of $c_5\cup c_6$. Now let $c\in {\calH}$ and let $T_c$ be the left Dehn twist about $c$. If $c$ is disjoint from $a\cup b$, then $T_c(a,b)=(a,b)$ and there is nothing to show. Thus assume that $c$ intersects $a$. The image $T_c(a)$ of $a$ is a separating simple closed curve contained in a small neighborhood $Y$ of $X(a)\cup c$. By assumption on $c$, this surface is a two-holed torus disjoint from $b$. As $g\geq 4$, the genus of $S-(Y\cup X(b))$ is at least one and hence there is a separating curve $e\subset S-(Y\cup X(b))$ which decomposes $S-(Y\cup X(b))$ into a one-holed torus and a surface $S^\prime$. But this means that $(a,b)$ can be connected to $T_c(a,b)=(T_ca,b)$ by the edge path $(a,b)\to (e,b)\to (T_ca,b)$. As the roles of $a$ and $b$ can be exchanged, the lemma now follows from the Putman trick. \end{proof} We shall use another auxiliary graph which is defined as follows. \begin{defi}\label{nonseparc} Let $S$ be a compact surface of genus $g\geq 1$ with two distinguished boundary components $A_1,A_2$. The \emph{nonseparating arc graph} is the graph whose vertices are isotopy classes of embedded arcs in $S$ connecting $A_1$ to $A_2$. The endpoints of an arc may move freely along the boundary circles $A_1,A_2$ in such an isotopy class. Two such arcs $\epsilon_1,\epsilon_2$ are connected by an edge if $\epsilon_1,\epsilon_2$ are disjoint and $S-(\epsilon_1\cup \epsilon_2)$ is connected. \end{defi} We apply the Putman trick to show \begin{lemm}\label{connectedarc} The nonseparating arc graph ${\calA}(A_1,A_2)$ on a compact surface $S$ of genus $g\geq 1$ with two distinguished boundary components $A_1,A_2$ is connected. \end{lemm} \begin{proof} Clearly the pure mapping class group $P\Mod(S)$ of $S$ acts transitively on the vertices of ${\calA}(A_1,A_2)$, so it suffices to show that there exists a generating set ${\calS}$ of $P\Mod(S)$ and an arc $\epsilon\in {\calA}(A_1,A_2)$ which can be connected to its image $\psi(\epsilon)$ by an edge path in ${\calA}(A_1,A_2)$ for every element $\psi\in {\calS}$. There exists two disjoint arcs $\epsilon_1,\epsilon_2$ connecting $A_1$ to $A_2$ such that $\epsilon_1\cup \epsilon_2$ projects to an essential nonseparating simple closed curve in the surface obtained from $S$ by capping off the boundary components $A_1,A_2$. Furthermore, we may assume that $\epsilon_1$ intersects one of the curves shown in Figure~\ref{fig3}, say the curve $c_1$, in a single point and is disjoint from the remaining curves, and $c_1$ is disjoint from $\epsilon_2$. Then $T_{c_i}\epsilon_1=\epsilon_1$ for $i\geq 2$, and $\epsilon_1$ can be connected to $T_{c_1}(\epsilon_1)$ by the edge path $\epsilon_1,\epsilon_2,T_{c_1}\epsilon_1$. By the Putman trick this implies that ${\calA}(A_1,A_2)$ is connected. \end{proof} We are now ready to show \begin{prop}\label{connected6} Let $\phi$ be an $r$-spin structure $(r\geq 2)$ on a compact surface $S$ of genus $g\geq 4$. Then the graph ${\calC\calG}_1$ is connected. \end{prop} \begin{proof} Let $S$ be a compact surface of genus $g\geq 2$ and consider the graph ${\calP\calS}$. To each of its vertices, viewed as a disjoint pair $(c,d)$ of separating simple closed curves, we associate in a non-deterministic way a vertex $\Lambda(c,d)$ of ${\calC\calG}_1$ as follows. Denote by $\Sigma_c, \Sigma_d$ the one-holed torus bounded by $c,d$. If one of the tori $\Sigma_c,\Sigma_d$ contains a simple closed curve $a$ with $\phi(a)=\pm 1$ then define $\Lambda(c,d)=a$. Now assume that none of the tori $\Sigma_c,\Sigma_d$ contains a simple closed curve $a$ with $\phi(a)=\pm 1$. By Lemma~\ref{torus}, there are simple closed nonseparating curves $a\subset \Sigma_c,b\subset \Sigma_d$ so that $\phi(a)=0=\phi(b)$. Since the tori $\Sigma_c,\Sigma_d$ are disjoint, the pair $(a,b)$ is nonseparating, that is, $S-(a\cup b)$ is connected. Choose an embedded arc $\epsilon$ in $S$ connecting $a$ to $b$. By Lemma~\ref{add}, the curve $\Lambda(c,d)=a+_\epsilon b$ satisfies $\phi(a+_\epsilon b)=\pm 1$, furthermore it is nonseparating. Let $a$ be any vertex of ${\calC\calG}_1$ and let $b$ be any simple closed curve which intersects $a$ in a single point. Such a curve exists since $a$ is nonseparating. Then a tubular neighborhood of $a\cup b$ is a torus containing $a$. Let $c$ be the boundary curve of this torus and choose a second separating simple closed curve $d$ so that $(c,d)\in {\calP\calS}$. Let $e\in {\calC\calG}_1$ be another vertex. Construct as above a vertex $(p,q)\in {\calP\calS}$ so that $e$ is contained in the one-holed torus cut out by $p$. Connect $(c,d)$ to $(p,q)$ by an edge path $(c_i,d_i)_{0\,\leq\,i\,\leq\,k}$ in ${\calP\calS}$. We use this edge path to construct an edge path $(a_j)\subset {\calC\calG}_1$ connecting $a$ to $e$ which passes through suitable choices $a_{j_i}$ $(i\leq k)$ of the curves $\Lambda(c_i,d_i)$. Define $a_0=a$ and by induction, let us assume that we constructed already the path $(a_j)_{0\,\leq\,j\,\leq\,j_i}$ for some $i\geq 0$. We distinguish two cases. %%Cases à introduire ici !! %{\sl Case 1:} \begin{case} One of the tori $\Sigma_{c_i},\Sigma_{d_i}$ contains a curve $f$ with $\phi(f)=\pm 1$. By construction, in this case we may assume by renaming that $f=a_{j_i}\subset \Sigma_{c_i}$. If $c_i\in \{c_{i+1},d_{i+1}\}$ then define $a_{j_i+1}=a_{j_{i+1}}=a_{j_i}=\Lambda(c_{i+1},d_{i+1})$ and note that this is consistent with the requirements for the induction step. Thus we may assume now that $c_i\not\in \{c_{i+1},d_{i+1}\}$. If one of the tori $\Sigma_{c_{i+1}},\Sigma_{d_{i+1}}$, say the torus $\Sigma_{c_{i+1}}$, contains a curve $h$ with $\phi(h)=\pm 1$, then as $\Sigma_{c_i}$ is disjoint from $\Sigma_{c_{i+1}}$, the curve $h$ is disjoint from $a_{j_i}$ and we can define $a_{j_i+1}=h= a_{j_{i+1}}=\Lambda(c_{i+1},d_{i+1})$. Thus assume that neither $\Sigma_{c_{i+1}}$ nor $\Sigma_{d_{i+1}}$ contains such a curve. Since $\Sigma_{c_i}$ and $\Sigma_{c_{i+1}},\Sigma_{d_{i+1}}$ are pairwise disjoint, we can find an embedded arc $\epsilon$ in $S-\Sigma_{c_i}$ connecting a simple closed curve $u\subset \Sigma_{c_{i+1}}$ with $\phi(u)=0$ to a curve $h\subset \Sigma_{d_{i+1}}$ with $\phi(h)=0$. We then can define $a_{j_i+1}=u+_\epsilon h=\Lambda(c_{i+1},d_{i+1}) =a_{j_{i+1}}$. \end{case} %{\sl Case 2:} \begin{lastcase} None of the tori $\Sigma_{c_i},\Sigma_{d_i}$ contains a curve $f$ with $\phi(f)=\pm 1$. In this case there are simple closed curves $f\subset \Sigma_{c_i},h\subset \Sigma_{d_i}$ with $\phi(f)=\phi(h)=0$, and there is an embedded arc $\epsilon$ connecting $f$ to $h$ so that \[ a_{j_i}=\Lambda\left(c_{i},d_{i}\right)=f+_\epsilon h. \] Assume without loss of generality that $d_i=d_{i+1}$. Let us in addition assume for the moment that the arc $\epsilon$ is disjoint from $c_{i+1}$. If furthermore there exists a simple closed curve $u\subset \Sigma_{c_{i+1}}$ with $\phi(u)=\pm 1$, then this curve is a choice for $\Lambda(c_{i+1},d_{i+1})$ which is disjoint from $a_{j_i}$ and we are done. Otherwise cut $S$ open along the simple closed curve $h\subset \Sigma_{d_i}=\Sigma_{d_{i+1}}$ and let $H_1,H_2$ be the two boundary components of $S-h$. By renaming, assume without loss of generality that $\epsilon$ connects the boundary component $H_1$ to the curve $f$, i.e. it leaves the curve $h$ from the side corresponding to $H_1$. Now note that $M=S-h-\epsilon-\Sigma_{c_i}$ is a connected surface of genus $g-2\geq 2$ with two distinguished boundary circles, one of which is the curve $H_2$, and $M\supset \Sigma_{c_{i+1}}$. Therefore there exists an embedded arc $\epsilon^\prime\subset M$ connecting $H_2$ to a simple closed curve $u\subset \Sigma_{c_{i+1}}$ with $\phi(u)=0$. Define $a_{j_i+1}=h+_{\epsilon^\prime}u$ and note that this definition is consistent with all requirements. This construction completes the induction step under the additional assumption that the arc~$\epsilon$ is disjoint from $c_{i+1}$. We are left with the case that $\epsilon$ is \emph{not} disjoint from $\Sigma_{c_{i+1}}$. Cut $S$ open along $f\cup h$ and note that the resulting surface $Z$ has genus $g-2\geq 2$ and four distinguished boundary components, say the components $F_1,F_2,H_1,H_2$. Assume that $\epsilon$ connects $F_1$ to $H_1$. Consider the nonseparating arc graph ${\calA}(F_1,H_1)$ in $Z$ of arcs connecting $F_1$ to $H_1$. By Lemma~\ref{connectedarc}, this graph is connected. Let $\epsilon_i$ be a path in ${\calA}(F_1,H_1)$ which connects $\epsilon$ to an arc $\epsilon^\prime$ disjoint from $\Sigma_{c_{i+1}}$. For any two consecutive of such arcs, say the arcs $\epsilon_j,\epsilon_{j+1}$, the surface $Z-(\epsilon_1\cup \epsilon_2)$ is connected and hence we can find a disjoint arc $\delta_j$ connecting $F_2$ to $H_2$. The curves $f+_{\epsilon_j}h,f+_{\delta_j}h, f+_{\epsilon_{j+1}}h$ are disjoint and yield a path in ${\calC\calG}_1$ connecting $f+_\epsilon h$ to a curve $f+_{\epsilon^\prime} h$ which is disjoint from $\Sigma_{c_{i+1}}$. We then can apply the construction for the case that the arc connecting $f$ to $h$ is disjoint from $\Sigma_{c_{i+1}}$. This completes the proof of the Proposition~\ref{connected6}. \end{lastcase}\let\qed\relax \end{proof} For technical reasons we need a stronger version of Proposition~\ref{genus3} and Proposition~\ref{connected6}. Consider a $\bbZ/r\bbZ$-spin structure $\phi$ on a compact surface $S$ of genus $g$ (with or without boundary) for an arbitrary number $r\geq 2$. We introduce another graph ${\calC\calG}_1^+$ as follows. The vertices of ${\calC\calG}_1^+$ coincide with the vertices of ${\calC\calG}_1$. Any two such vertices $c,d$ are connected by an edge if $c,d$ are disjoint and if furthermore $S-(c\cup d)$ is connected. Thus ${\calC\calG}_1^+$ is obtained from ${\calC\calG}_1$ by removing some of the edges. In particular, if ${\calC\calG}_1^+$ is connected then the same holds true for ${\calC\calG}_1$. We use connectedness of ${\calC\calG}_1$ to establish connectedness of ${\calC\calG}_1^+$. \begin{lemm}\label{next} If the genus $g$ of $S$ is at least 3 then the graph ${\calC\calG}_1^+$ is connected provided that ${\calC\calG}_1$ is connected. \end{lemm} \begin{proof} Let $c,d\in {\calC\calG}_1$ be two vertices which are connected by an edge in ${\calC\calG}_1$ and which are not connected by an edge in ${\calC\calG}_1^+$. This means that $c,d$ are disjoint, and $S-(c\cup d)$ is disconnected. We have to show that $c,d$ can be connected in ${\calC\calG}_1^+$ by an edge path. To this end recall that $c,d$ are nonseparating and therefore the disconnected surface $S-(c\cup d)$ has two connected components $S_1,S_2$. The surface $S_1$ has genus $g_1\geq 1$ and at least two boundary components, and the surface $S_2$ has genus $g_2=g-g_1-1\geq 0$ and at least two boundary components. Choose a simple closed curve $d_i\subset S_i$ $(i=1,2)$ which bounds with $c\cup d$ a pair of pants $P_i$. Write $\Sigma_i=S_i-P_i$; the genus of $\Sigma_i$ equals $g_i$. Glue $P_1$ to $P_2$ along $c\cup d$ so that the resulting surface $\Sigma_0$ is a two-holed torus containing $c\cup d$ in its interior. Choose a nonseparating simple closed curve $e\subset \Sigma_0$ which intersects both $c,d$ in a single point. Since $\phi(c)=\pm 1$ we have $\phi(T_ce)=\phi(e)\pm 1$ where $T_c$ is the left Dehn twist about $c$. Thus via replacing $e$ by $T_c^ke$ for a suitable choice of $k\in \bbZ$ we may assume that $\phi(e)=1$. In other words, we may assume that $e$ is a vertex of ${\calC\calG}_1$. Assume for the moment that $g_2\geq 1$. By Lemma~\ref{torus}, there exist simple closed curves $a\subset \Sigma_1, b\subset \Sigma_2$ with $\phi(a)=\phi(b)=0$. Connect $a$ to $b$ by an embedded arc $\epsilon$ which is disjoint from $c\cup e$ (and crosses through the curve $d$). The curve $a+_\epsilon b$ satisfies $\phi(a+_\epsilon b)=1$, and it is disjoint from both $c$ and $e$. Moreover, the surfaces $S-(c\cup a+_\epsilon b)$ and $S-(e\cup a+_\epsilon b)$ are connected. As a consequence, $c$ can be connected to $e$ by an edge path in ${\calC\calG}_1^+$ of length two which passes through $a+_\epsilon b$. By symmetry of this construction, $e$ can also be connected to $d$ by an edge path in ${\calC\calG}_1^+$ and hence $c$ can be connected to $d$ by such a path. This completes the proof in the case that the genus $g_2$ of $S_2$ is positive. If the genus of $S_2$ vanishes then the genus of $S_1$ equals $g_1=g-1\geq 2$. Any nonseparating simple closed curve in $S_1$ forms with both $c,d$ a nonseparating pair. To find such a curve $e$ with $\phi(e)=1$, note that $S_1$ contains two disjoint one-holed tori $T_1,T_2$, and by Lemma~\ref{torus}, there are embedded simple closed curves $a_i\in T_i$ which satisfy $\phi(a_i)=0$. Then for any arc $\epsilon$ in $S_1$ connecting $a_1$ to $a_2$, the curve $e=a_1 +_\epsilon a_2$ is nonseparating, and it is connected with both $c,d$ by an edge in ${\calC\calG}_1^+$. This is what we wanted to show. \end{proof} Proposition~\ref{genus3}, Proposition~\ref{connected6} and Lemma~\ref{next} together show \begin{coro}\label{connected} Let $\phi$ be a $\bbZ/r\bbZ$-spin structure on a closed surface $\Sigma$ of genus $g\geq 3$. Then the graph ${\calC\calG}_1^+$ is connected. \end{coro} \section{The action of \texorpdfstring{$\Mod(S)[\phi]$}{Mod(S)[phi]} on geometrically defined graphs}\label{theaction} In this section we consider an arbitrary $\bbZ/r\bbZ$-spin structure $\phi$ on a compact surface $S$ of genus $g\geq 3$, possibly with boundary, for some number $r\geq 2$. Our goal is to gain some information on the stabilizer $\Mod(S)[\phi]$ of $\phi$ through its action on the graph ${\calC\calG}_1^+$ introduced in Section~\ref{graphsofcurves}. We begin with some information on the stabilizer of a spin structure $\phi$ on a compact surface $S$ with boundary. Fix a boundary component $C$ of $S$. Denote by $P_C\Mod(S)$ the subgroup of the mapping class group $\Mod(S)$ of $S$ which fixes the boundary component $C$. Note that as we allow that a mapping class in $P_C\Mod(S)$ exchanges boundary components of $S$ different from $C$, the group $P_C\Mod(S)$ coincides with the pure mapping class group of $S$ only if the boundary of $S$ consists of one or two components. Write $P_C\Mod(S)[\phi]$ to denote the stabilizer of $\phi$ in $P_C\Mod(S)$. This is a subgroup of $P_C\Mod(S)$ of finite index. Let $\Sigma$ be the surface obtained from $S$ by attaching a disk to $C$. There is an embedding $S\to \Sigma$ which induces a surjective homomorphism \[ \Pi:P_C\Mod(S)\to \Mod(\Sigma). \] By a result of Johnson, extending earlier work of Birman (see~\cite[Section~4.2.5]{FM12}), there is an exact sequence \begin{equation}\label{birman1} 1\to \bbZ\to {\ker}(\Pi)\xrightarrow{\Upsilon} \pi_1(\Sigma)\to 1 \end{equation} where $\bbZ$ is the infinite cyclic central subgroup of $P_C\Mod(S)$ generated by the Dehn twists about $C$ and where $\pi_1(\Sigma)$ is a so-called point pushing group. For the formulation of the following Lemma~\ref{pointpush1}, recall that the integral homology $H_1(\Sigma,\bbZ)$ of a compact surface $\Sigma$ of genus $g\geq 2$, possibly with boundary, is a free abelian group $\bbZ^{h}$ for some $h\geq 4$. In fact, $h=2g$ if the boundary of $\Sigma$ is empty or connected, and in this case this group is generated by the homology classes of nonseparating simple closed curves on $\Sigma$. If the boundary of $\Sigma$ is disconnected, then it is still true that $H_1(\Sigma,\bbZ)$ is generated by simple closed possibly peripheral curves. For $m\geq 1$ let $\Lambda_m\subset \pi_1(S)$ be the subgroup defined by the exact sequence \[ 0\to \Lambda_m\to \pi_1(S)\to H_1\left(S,\bbZ/m\bbZ\right)\to 0. \] If $\zeta:\pi_1(S)\to H_1(S,\bbZ)$ denotes the natural surjective projection, then $\Lambda_m$ is the preimage under $\zeta$ of the lattice in $H_1(S,\bbZ)$ generated by $m$ times the simple loop generators, and it is a subgroup of $\pi_1(S)$ of finite index. Using the notations from the previous paragraph we have \begin{lemm}\label{pointpush1} Assume that the boundary circle $C$ is equipped with the orientation induced from the orientation of $S$. \begin{enumerate} \item \label{lemm3.1.1}If $\phi(C)=-1$ then $\Upsilon({\ker}\,\Pi\cap P_C\Mod(S)[\phi])= \pi_1(\Sigma)$. \item \label{lemm3.1.2}If $\phi(C)=1$, then $\Upsilon({\ker}\, \Pi\cap P_C\Mod(S)[\phi])=\Lambda_{m}$ where $m=r/2$ if $r$ is even, and $m=r$ otherwise. \end{enumerate} \end{lemm} \begin{proof} Choose a basepoint $p$ for $\pi_1(\Sigma)$ in the interior of the attached disk. Let $\alpha\subset \Sigma$ be a simple nonseparating loop through the basepoint $p$. Up to homotopy, the oriented boundary of a tubular neighborhood of $\alpha$ consists of two simple closed curves $c_1,c_2$ which enclose the circle $C$. In other words, together with $C$ the curves $c_1,c_2$ bound a pair of pants $P$ in $S$. We equip the curves $c_i$ with the orientation as boundary curves of $P$. By~\cite[Proposition~3.8]{Sa19}, we have \begin{equation}\label{pairofpants} \phi(C)+\phi(c_1)+\phi(c_2)=-1 \end{equation} and hence if $\phi(C)=-1$ then $\phi(c_1)+\phi(c_2)=0$. Let as before $T_d$ be the left Dehn twist about a simple closed curve $d$. Let $\beta\subset S$ be an oriented simple closed curve which crosses through the pair of pants $P$. As $c_1,c_2$ are disjoint, we have $\iota(T_{c_2}^{-1}(\beta),c_1)=\iota(\beta,c_1)$ and therefore Definition~\ref{spin} shows that \begin{align}\label{intersectcom} \phi\left(T_{c_1}T_{c_2}^{-1}(\beta)\right) &=\phi\left(T_{c_2}^{-1}(\beta)\right)+ \iota(\beta,c_1)\phi(c_1)\\ &= \phi(\beta)+\iota(\beta,c_1)\phi(c_1)- \iota(\beta,c_2)\phi(c_2).\notag \end{align} On the other hand, as $c_1+c_2$ is homologous to the boundary curve $C$, the homological intersection number fulfills $\iota(\beta,c_1+c_2)=0$. Hence from~\eqref{pairofpants} we conclude that if $\phi(C)=-1$ then $\phi(T_{c_1}T_{c_2}^{-1}(\beta))= \phi(\beta)$. Since $\beta$ was an arbitrary simple closed curve, this shows that $T_{c_1}T_{c_2}^{-1}\in P_C\Mod(S)[\phi]$. But $T_{c_1}T_{c_2}^{-1}\in P_C\Mod(S)$ is just the point-pushing map about $\alpha$ and therefore $\alpha$ is contained in $\Upsilon({\ker}\,\Pi\cap P_C\Mod(S)[\phi])$. We refer to~\cite{FM12} for a comprehensive discussion of the various versions of the Birman exact sequence. As the point pushing group $\pi_1(\Sigma)$ is generated by point pushing maps along simple nonseparating loops, this shows the part~\eqref{lemm3.1.1} of the Lemma~\ref{pointpush1}. To show the part~\eqref{lemm3.1.2} of the Lemma~\ref{pointpush1}, assume now that $\phi(C)=1$. Equation~\eqref{pairofpants} shows that $\phi(c_1)+\phi(c_2)=-2$ and hence by formula~\eqref{intersectcom} we have \[ \phi\left(T_{c_1}T_{c_2}^{-1}(\beta)\right)=\phi(\beta) +\iota(\beta,c_1)\phi(c_1)+\iota(\beta,c_2)(\phi(c_1)+2). \] Now let us assume that the oriented simple closed curve $\beta$ crosses a single time through $c_1$, say when it enters $P$. Then $\iota(\beta,c_1)=-1,\iota(\beta,c_2)=1$ and hence \begin{equation}\label{add5} \phi\left(T_{c_1}T_{c_2}^{-1}(\beta)\right)=\phi(\beta)-\phi(c_1)+\phi(c_1)+2= \phi(\beta)+2. \end{equation} Using this formula $r/2$ times if $r$ is even, and $r$ times if $r$ is odd, we conclude that the point pushing map about $\alpha$ is not contained in $\Mod(S)[\phi]$, but it is the case for its $r/2^{\rm th}$ power or $r^{\rm th}$ power, respectively. Namely, putting $m=r/2$ if $r$ is even and $m=r$ otherwise, it follows from the above discussion that we have $\phi((T_{c_1}T_{c_2}^{-1})^{m}(\beta))=\phi(\beta)$ for every simple closed curve $\beta$ which either is disjoint from $P$ or which crosses through $P$ precisely once. As such curves span the first homology of $S$, we conclude that the pull-back of $\phi$ under $(T_{c_1}T_{c_2}^{-1})^{m}$ coincides with $\phi$ on a collection of simple closed curves which span $H_1(S,\bbZ)$. \cite[Corollary~2.6]{HJ89} then shows that indeed, $(T_{c_1}T_{c_2}^{-1})^{m}\in P_C\Mod(S)[\phi]$. Moreover, by equation~\eqref{add5}, we know that $(T_{c_1}T_{c_2}^{-1})^k\not\in P_C\Mod(S)[\phi]$ if $k$ is not a multiple of $m$. On the other hand, by~\cite[Lemma~3.15]{Sa19}, Dehn twists about separating simple closed curves in $S$ are contained in $\Mod(S)[\phi]$. As the commutator subgroup of $\pi_1(\Sigma)$ is generated by simple closed separating curves, and for each such curve $\alpha$ both Dehn twists $T_{c_1}, T_{c_2}$ about the boundary curves of a tubular neighborhood of $\alpha$ as above are contained in $P_C\Mod(S)[\phi]$, this yields the part~\eqref{lemm3.1.2} of Lemma~\ref{pointpush1}. \end{proof} Consider again an arbitrary compact surface $S$ of genus $g\geq 2$, equipped with a $\bbZ/r\bbZ$-spin structure $\phi$ for some $r\geq 2$. We use Lemma~\ref{pointpush1} to analyze the action of $\Mod(S)[\phi]$ on the graph ${\calC\calG}_1^+$. We begin with the investigation of the stabilizer of a vertex $c$ of ${\calC\calG}_1^+$ in $\Mod(S)[\phi]$. As $\Mod(S)[\phi]$ is a subgroup of $\Mod(S)$ of finite index, the stabilizer $\Stab(c)[\phi]$ of $c$ in $\Mod(S)[\phi]$ is a subgroup of finite index of the stabilizer $\Stab(c)$ of $c$ in $\Mod(S)$. The group $\Stab(c)$ can be described as follows. Cut $S$ open along $c$. The result is a surface $\Sigma^2$ of genus $g-1$ with two distinguished boundary components $C_1,C_2$. These components are equipped with an orientation as subsets of the oriented boundary of $\Sigma^2$. To simplify notations, let $\Mod(\Sigma^2)$ be the subgroup of the mapping class group of $\Sigma^2$ which preserves the subset $C_1\cup C_2$ of the boundary. We allow that an element of $\Mod(\Sigma^2)$ exchanges $C_1$ and $C_2$. The stabilizer $\Stab(c)$ of $c$ in the mapping class group $\Mod(S)$ of $S$ can be identified with the quotient of the group $\Mod(\Sigma^2)$ by the relation $T_{C_1}T_{C_2}^{-1}=1$ where $T_{C_i}$ denotes the left Dehn twist about the boundary circle $C_i$ (\cite[Theorem~3.18]{FM12}). In short, we have \[ \Stab(c)=\Mod\left(\Sigma^2\right)/\bbZ. \] The infinite cyclic subgroup of $\Stab(c)$ generated by the Dehn twist about $c$ is central. The quotient group $\Stab(c)/\bbZ$ can naturally be identified with the mapping class group $\Mod(\Sigma_2)$ of a surface $\Sigma_2$ of genus $g-1$ with two punctures and perhaps with boundary if the boundary of $S$ is non-trivial. We refer to~\cite{FM12} for a comprehensive discussion of these facts. Let $\Sigma$ be the surface obtained from $\Sigma_2$ by forgetting the punctures. Alternatively, $\Sigma$ is obtained from $\Sigma^2$ by attaching a disk to each boundary component. The group $\Mod(\Sigma_2)=\Stab(c)/\bbZ$ fits into the \emph{Birman exact sequence} \begin{equation}\label{birman3} 1\to \pi_1\left(C(\Sigma,2)\right)\xrightarrow{\rho} \Stab(c)/\bbZ\to \Mod(\Sigma)\to 1 \end{equation} where $\pi_1(C(\Sigma,2))$ is the \emph{surface braid group}, that is, the fundamental group of the configuration space of two unordered distinct points in $\Sigma$. In particular, $\pi_1(C(\Sigma,2))$ is a normal subgroup of $\Stab(c)/\bbZ=\Mod(\Sigma_2)$. The surjective homomorphism \[ \theta:\Stab(c)\to \Stab(c)/\bbZ=\Mod(\Sigma_2) \] restricts to a homomorphism $\Stab(c)[\phi]\to \Mod(\Sigma_2)$. The next proposition gives some first information on its image under the assumption that $\phi$ is a $\bbZ/2\bbZ$-spin structure and $\phi(c)=1$. \begin{prop}\label{surject} Let $\phi$ be a $\bbZ/2\bbZ$-spin structure on $S$ and let $c$ be a simple closed curve with $\phi(c)=1$. Then $\rho(\pi_1(C(\Sigma,2)))\subset \theta(\Stab(c)[\phi])$. \end{prop} \begin{proof} Let $\pi_1(PC(\Sigma,2))$ be the intersection of the fibre of the Birman exact sequence~\eqref{birman3} with the subgroup of $\Mod(\Sigma_2)$ which fixes each of the two distinguished punctures. Following~\cite[Section~4.2.5]{FM12}, the group $\pi_1(PC(\Sigma,2))$ can be described as follows. Let $C_1,C_2$ be the distinguished boundary components of the surface $\Sigma^2=S-c$. Let $\Sigma^1$ be the surface obtained from $\Sigma^2$ by attaching a disk to the boundary circle $C_1$. Let $P\Stab(c)$ and $P\Mod(\Sigma^2)$ be the index two subgroup of $\Stab(c)$ and $\Mod(\Sigma^2)$ which preserves each of the two boundary components $C_1,C_2$ of $S-c$. The inclusion $\Sigma^2\to \Sigma^1$ induces a surjective homomorphism \[ \Xi:P\Stab(c)/\bbZ\to P\!c_2\Mod\left(\Sigma^1\right)/\bbZ \] where as before $P\!c_2\Mod(\Sigma^1)$ is required to fix the boundary component $C_2$ of $\Sigma^1$ and where the group $\bbZ$ acts as the group of Dehn twists about $c$ and about $C_2$. The kernel ${\ker}(\Xi)$ of this homomorphism is isomorphic to $\pi_1(\Sigma^1)$ (see~\cite{FM12} for more information on this version of the Birman exact sequence). The spin structure $\phi$ pulls back to a spin structure $\hat \phi$ on $\Sigma^2$. Since $\phi$ is a $\bbZ/2\bbZ$-spin structure on $S$ and $\phi(c)=1$, the value of $\hat \phi$ on each of the two boundary circles $C_1,C_2$ coincides with the value of a spin structure on the boundary of an embedded disk. This implies that $\hat \phi$ induces a spin structure $\phi^\prime$ on $\Sigma^1$. Or, equivalently, $\hat \phi$ is the pull-back of a spin structure $\phi^\prime$ on $\Sigma^1$ via the inclusion $\Sigma^2\to \Sigma^1$. By Lemma~\ref{pointpush1}, the group ${\ker}(\Xi)= \pi_1(\Sigma^1)$ stabilizes $\hat \phi$, that is, we have ${\ker}(\Xi)\subset \Mod(\Sigma^2)[\hat \phi]$. Apply Lemma~\ref{pointpush1} a second time to the homomorphism $P\!c_2\Mod(\Sigma^1)/\bbZ\to \Mod(\Sigma)$ where $\Sigma$ is obtained from $\Sigma^1$ by attaching a disk to $C_2$. As the group $\pi_1(PC(\Sigma,2))$ can be described as the quotient by its center $\bbZ^2$ of the kernel of the homomorphism $P\Mod(\Sigma^2)\to \Mod(\Sigma)$ which is obtained by applying the Birman exact sequence twice, first to a map which caps off the boundary component $C_1$, followed by the map which caps off $C_2$, this shows that $\pi_1(PC(\Sigma,2))\subset \theta (\Stab(c)[\phi])$. As exchanging $C_1$ and $C_2$ also preserves $\hat \phi$ the Proposition~\ref{surject} follows. \end{proof} We are now ready to give a complete description of the stabilizer in $\Mod(S)[\phi]$ of a nonseparating simple closed curve $c$ on $S$ with $\phi(c)=1$ where as before, $\phi$ is a $\bbZ/2\bbZ$-spin structure on a compact surface $S$ of genus $g\geq 3$, with empty or connected boundary. Cut $S$ open along $c$ and write $\Sigma^2=S-c$. The spin structure $\phi$ of $S$ pulls back to a $\bbZ/2\bbZ$-spin structure $\hat \phi$ on $\Sigma^2$. Denote as before by $\Sigma$ the surface of genus $g-1$ with empty or connected boundary obtained from $\Sigma^2$ by capping off the two distinguished boundary components. We have %\pagebreak \begin{prop}\label{parity} The $\bbZ/2\bbZ$-spin structure $\phi$ on $S$ induces a $\bbZ/2\bbZ$-spin structure $\phi_c$ on $\Sigma$ whose parity coincides with the parity of $\phi$. If $\Pi:\Stab(c)/\bbZ\to \Mod(\Sigma)$ denotes the surjective homomorphism induced by the inclusion $S-c\to \Sigma$ then \[ \Pi^{-1}\Mod(\Sigma)[\phi_c]=\Stab(c)[\phi]/\bbZ. \] \end{prop} \begin{proof} As $\phi$ is a $\bbZ/2\bbZ$-spin structure, the value of $\phi$ on a boundary circle of $S-c$ corresponding to a copy of $c$ coincides with the value of a $\bbZ/2\bbZ$-spin structure on the boundary of a disk. Thus $\phi$ induces a spin structure $\phi_c$ on $\Sigma$. To compare the parities of the spin structures $\phi$ and $\phi_c$, assume that $\Sigma$ is obtained from $S-c$ by attaching disks $D_1,D_2$ to the two boundary components of $S$ which correspond to the two copies of $c$. Choose a geometric symplectic basis $a_1,b_1,\,\dots,\,a_{g-1},b_{g-1}$ for $\Sigma$, consisting of simple closed oriented curves which do not intersect the disks $D_1,D_2$. Then $a_1,b_1,\,\dots,\,a_{g-1},b_{g-1}$ can be viewed as a system of curves in $\Sigma^2=\Sigma-(D_1\cup D_2)$ which maps to a curve system with the same properties in $S$ by the map $\Sigma^2\to S$. This curve system can be extended to a geometric symplectic basis for $S$ containing the curve $c$, equipped with any orientation. As $\phi(c)=1$ we have $\phi(c)+1=0$. The claim now follows from the fact that $\phi_c(u)=\phi(\hat u)$ for $u\in \{a_1,b_1,\,\dots,\,a_{g-1},b_{g-1}\}$ where $\hat u$ is the image of $u$ under the inclusion $\Sigma^2\to S$, together with the formula~\eqref{arf} for the Arf invariant. We are left with showing that $\Stab(c)[\phi]/\bbZ= \Pi^{-1}\Mod(\Sigma)[\phi_c]$. Observe first that as $\phi_c$ is induced from $\phi$, we have $\Pi\Stab(c)[\phi]/\bbZ\subset \Mod(\Sigma)[\phi_c]$. To show that in fact equality holds let $\Sigma_2$ be the surface obtained from $S-c$ by replacing the boundary components by punctures. The group $\Stab(c)[\phi]/\bbZ$ can be identified with a subgroup $\Gamma_c$ of $\Mod(\Sigma_2)$. We view the punctures of $\Sigma_2$ as marked points $p_1,p_2$ in $\Sigma$. Let $\theta$ be any diffeomorphism of $\Sigma$ which preserves $\phi_c$. Then $\theta$ is isotopic to a diffeomorphism of $\Sigma$ which equals the identity on a disk $D\subset \Sigma$ containing both points $p_1,p_2$. Thus $\theta$ lifts to a diffeomorphism $\theta^\prime$ of $\Sigma_2$ which preserves the pull-back of $\phi_c$ to a spin structure on $\Sigma_2$. The boundary circle $\partial D$ of $D$ can be viewed as a simple closed curve in $S-c$. Via the projection $S-c\to S$ which identifies the two distinguished boundary components of $S-c$, the curve $\partial D$ projects to a separating simple closed curve in $S$ which decomposes $S$ into a one-holed torus $T$ containing $c$ and a surface of genus $g-1$ with connected boundary. The diffeomorphism $\theta^\prime$ lifts to a diffeomorphism $\Theta$ of $S$ which is the identity on $T$. Then $\Theta^*\phi$ is a spin structure on $S$ which defines the same function on $H_1(S,\bbZ)$ as $\phi$. Using once more the result of Humphries and Johnson~\cite{HJ89} (see~\cite[Theorem~3.9]{Sa19}), this implies that $\Theta$ stabilizes $\phi$. As $\Theta$ projects to the mapping class of $\Sigma$ defined by the diffeomorphism $\theta$, this shows surjectivity of the homomorphism $\Pi:\Stab(c)[\phi]/\bbZ\to \Mod(\Sigma)[\phi_c]$. On the other hand, by Proposition~\ref{surject} the kernel of the homomorphism $\Pi$ also is contained in $\Stab(c)[\phi]/\bbZ$. Together this completes the proof of the Proposition~\ref{parity}. \end{proof} The next observation uses~\cite[Proposition~4.9]{Sa19}. For its formulation, recall from Section~\ref{graphsofcurves} the definition of the graph ${\calC\calG}_1^+$. Its vertices are nonseparating simple closed curves with prescribed value $\pm 1$ of the spin structure. The graph ${\calC\calG_1}^+$ is well defined if the genus $g$ of $S$ is at least two although it may not have edges. Note that in the statement of Proposition~\ref{connect}, we allow that the surface $S$ has non-empty boundary, and we consider $\bbZ/r\bbZ$-spin structures where $r$ may be larger than $2g-2$. This is crucial for an inductive approach towards higher spin structures via cutting surfaces open along separating simple closed curves, and it is used in the proof of Theorem~\ref{main3}. \begin{prop}\label{connect} Let $\phi$ be a $\bbZ/r\bbZ$-spin structure on a compact surface $S$ of genus $g\geq 2$ with empty or connected boundary. Then for any two directed edges $e_1,e_2$ of the graph ${\calC\calG}_1^+$ there exists a mapping class $\zeta\in \Mod(S)[\phi]$ with $\zeta(e_1)=e_2$. In particular, the action of $\Mod(S)[\phi]$ on ${\calC\calG}_1^+$ is vertex transitive. \end{prop} \begin{proof} The proof consists of an adjustment of the argument in the proof of~\cite[Proposition~4.9]{Sa19}. Recall that a geometric symplectic basis for $S$ is a set $\{a_1,b_1,\,\dots,\,a_{2g},b_{2g}\}$ of simple closed curves on $S$ such that $a_i,b_i$ intersect in a single point, and $a_i\cup b_i$ is disjoint from $a_j\cup b_j$ for $j\not=i$. A vertex of ${\calC\calG}_1^+$ is a simple closed curve $c$ on $S$ with $\phi(c)=\pm 1$. In the sequel we always orient such a vertex $c$ in such a way that $\phi(c)=1$. For a given directed edge $e$ of ${\calC\calG}_1^+$ with ordered endpoints $c,d$, we aim at constructing a geometric symplectic basis ${\calB}(e)$ such that $a_1=c,a_2=d, \phi(a_i)=0$ for $i\geq 3$, $\phi(b_i)=0$ for $i\leq g-1$ and $\phi(b_g)=0$ or $1$ as predicted by the parity of $\phi$. If such a basis ${\calB}(e_1),{\calB}(e_2)$ can be found for any two directed edges $e_1,e_2$ of ${\calC\calG}_1^+$ with ordered endpoints $c_1,d_1$ and $c_2,d_2$, then there exists a diffeomorphism $\zeta$ of $S$ which maps ${\calB}(e_1)$ to ${\calB}(e_2)$ and maps $c_1,d_1$ to $c_2,d_2$. The pullback $\zeta^*\phi$ of $\phi$ is a spin structure on $S$ whose values on ${\calB}(e_1)$ coincide with the values of $\phi$. By a result of Humphries and Johnson~\cite{HJ89}, see~\cite[Theorem~3.9]{Sa19}, this implies that $\zeta^*\phi=\phi$ and hence the isotopy class of $\zeta$ is contained in $\Mod(S)[\phi]$ and maps the directed edge $e_1$ to the directed edge $e_2$. To simplify further, choose any geometric symplectic basis \[ {\calB}=\left\{\alpha_1,\beta_1,\,\dots,\,\alpha_g,\beta_g\right\} \] for $S$ with $\alpha_1=c$, $\alpha_2=d$. A small tubular neighborhood of $\alpha_i\cup \beta_i$ is a one-holed torus $T_i$ embedded in $S$. By Lemma~\ref{torus}, for all $i\geq 3$ we may replace $\alpha_i$ by an oriented simple closed curve in $T_i$, again denoted by $\alpha_i$, which satisfies $\phi(\alpha_i)=0$. Assume that $\beta_i$ $(i=1,2)$ is oriented in such a way that $\iota(\beta_i,\alpha_i)=1$ where $\iota$ is the symplectic form. As $\phi(T_{\alpha_i}(\beta_i))= \phi(\beta_i)+1$, via perhaps replacing $\beta_i$ by its image under a suitably chosen power of a Dehn twist about $\alpha_i$ we may assume that $\phi(\beta_i)=0$. Therefore for the construction of a geometric symplectic basis ${\calB}(e)$ with the required properties, it suffices to modify successively the curves $\beta_i$ $(i\geq 3)$ while keeping $\alpha_j$ $(j\geq 1)$ and $\beta_k$ for $k