memoire-m2
My M2 Memoire on mapping class groups & their representations
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\chapter{Relations Between Twists}\label{ch:relations} Having found a convenient set of generators for \(\Mod(\Sigma_g)\), it is now natural to ask what the relations between such generators are. In this chapter, we highlight some additional relations between Dehn twists and the geometric intuition behind them, culminating in the statement of a presentation for \(\Mod(\Sigma_g)\) whose relations can be entirely explained in terms of the geometry of curves in \(\Sigma_g\) -- see Theorem~\ref{thm:wajnryb-presentation}. \begin{fundamental-observation}[Lantern relation] Let \(\Sigma_0^4\) be the surface of genus \(0\) with \(4\) boundary components and \(\alpha, \beta, \gamma, \delta_1, \ldots, \delta_4 \subset \Sigma_0^4\) be as in Figure~\ref{fig:latern-relation}. Consider the surfaces \(\Sigma_0^3 = \Sigma_0^4 \cup_{\delta_1} \mathbb{D}^2\) and \(\Sigma_{0,1}^3 = \Sigma_0^4 \cup_{\delta_1} (\mathbb{D}^2 \setminus \{ 0 \})\), as well as the map \(\operatorname{push} : \pi_1(\Sigma_0^3, 0) \to \Mod(\Sigma_{0,1}^3)\). Let \(\eta_1, \eta_2, \eta_3 : \mathbb{S}^1 \to \Sigma_0^3\) be the loops from Figure~\ref{fig:lantern-relation-capped}, so that \([\eta_1] \cdot [\eta_2] = [\eta_3]\) in \(\pi_1(\Sigma_0^3, 0)\). From Observation~\ref{ex:push-simple-loop} we obtain \[ (\tau_{\delta_2} \tau_\alpha^{-1}) (\tau_{\delta_3} \tau_\gamma^{-1}) = \operatorname{push}([\eta_1]) \cdot \operatorname{push}([\eta_2]) = \operatorname{push}([\eta_3]) = \tau_\beta \tau_{\delta_4}^{-1} \in \Mod(\Sigma_{0, 1}^3). \] Using the capping exact sequence from Observation~\ref{ex:capping-seq}, we can then see \(\tau_{\delta_2} \tau_\alpha^{-1} \tau_{\delta_3} \tau_\gamma^{-1}, \tau_\beta \tau_{\delta_4}^{-1} \in \Mod(\Sigma_0^4)\) differ by a power of \(\tau_{\delta_1}\). In fact, one can show \((\tau_{\delta_2} \tau_\alpha^{-1} \tau_{\delta_3} \tau_\gamma^{-1}) (\tau_\beta \tau_{\delta_4}^{-1})^{-1} = \tau_{\delta_1}^{-1} \in \Mod(\Sigma_0^4)\). Now the disjointness relations \([\tau_{\delta_i}, \tau_\alpha] = [\tau_{\delta_i}, \tau_\beta] = [\tau_{\delta_i}, \tau_\gamma] = 1\) give us the \emph{lantern relation} (\ref{eq:lantern-relation}) in \(\Mod(\Sigma_0^4)\). \begin{equation}\label{eq:lantern-relation} \tau_\alpha \tau_\beta \tau_\gamma = \tau_{\delta_1} \tau_{\delta_2} \tau_{\delta_3} \tau_{\delta_4} \end{equation} \end{fundamental-observation} \noindent \begin{minipage}[t]{.5\linewidth} \centering \includegraphics[width=\linewidth]{images/lantern-relation.eps} \captionof{figure}{Two views of $\Sigma_0^4$: on the left-hand side we see the \emph{lantern-like} surface we get by subtracting \(4\) disjoint open disks from \(\mathbb{S}^2\), and on the right-hand side we see the disk with three open disks subtracted from its interior.} \label{fig:latern-relation} \end{minipage} \hspace{.6cm} % \begin{minipage}[t]{.44\linewidth} \centering \includegraphics[width=.45\linewidth]{images/lantern-relation-capped.eps} \captionof{figure}{The curves $\eta_1, \eta_2, \eta_3 \subset \Sigma_0^3$ from the proof of the lantern relation.} \label{fig:lantern-relation-capped} \end{minipage} We may exploit different embeddings \(\Sigma_0^4 \hookrightarrow \Sigma\) and their corresponding inclusion homomorphisms \(\Mod(\Sigma_0^4) \to \Mod(\Sigma)\) to obtain interesting relations between the corresponding Dehn twists in \(\Mod(\Sigma)\). For example, the lantern relation can be used to compute \(\Mod(\Sigma_g^b)^\ab\) for \(g \ge 3\). \begin{proposition}\label{thm:trivial-abelianization} The Abelianization \(\Mod(\Sigma_g^b)^\ab = \mfrac{\Mod(\Sigma_g^b)}{[\Mod(\Sigma_g), \Mod(\Sigma_g)]}\) is cyclic. Moreover, if \(g \ge 3\) then \(\Mod(\Sigma_g^b)^\ab = 0\). In other words, \(\Mod(\Sigma_g)\) is a perfect group for \(g \ge 3\). \end{proposition} \begin{proof} By Theorem~\ref{thm:lickorish-gens}, \(\Mod(\Sigma_g^b)^\ab\) is generated by the image of the Lickorish generators, which are all conjugate and thus represent the same class in the Abelianization. In fact, any nonseparating \(\alpha \subset \Sigma_g^b\) is conjugate to the Lickorish generators too, so \(\Mod(\Sigma_g^b)^\ab = \langle [\alpha] \rangle\). Now for \(g \ge 3\) we can embed \(\Sigma_0^4\) in \(\Sigma_g^b\) in such a way that all the corresponding curves \(\alpha, \beta, \gamma, \delta_1, \ldots, \delta_4 \subset \Sigma_g^b\) are nonseparating, as shown in Figure~\ref{fig:latern-relation-trivial-abelianization}. The lantern relation (\ref{eq:lantern-relation}) then becomes \[ 3 \cdot [\tau_\alpha] = [\tau_\alpha] + [\tau_\beta] + [\tau_\gamma] = [\tau_{\delta_1}] + [\tau_{\delta_2}] + [\tau_{\delta_3}] + [\tau_{\delta_4}] = 4 \cdot [\tau_\alpha] \] in \(\Mod(\Sigma_g^b)^\ab\). In other words, \([\tau_\alpha] = 0\) and thus \(\Mod(\Sigma_g^b)^\ab = 0\). \end{proof} \begin{figure}[ht] \centering \includegraphics[width=.5\linewidth]{images/lantern-relation-trivial-abelianization.eps} \caption{The embedding of $\Sigma_0^4$ in $\Sigma_g^b$ for $g \ge 3$.} \label{fig:latern-relation-trivial-abelianization} \end{figure} To get extra relations we need to investigate certain branched covers \(\Sigma \to \mathbb{D}^2 \setminus \{x_1, \ldots, x_r\}\), as well as the relationship between \(\Mod(\Sigma)\) and \(\Mod(\mathbb{D}^2 \setminus \{x_1, \ldots, x_r\})\). This is what is known as \emph{the Birman-Hilden theorem}. \section{The Birman-Hilden Theorem}\label{birman-hilden} Let \(\Sigma_{0, r}^1 = \mathbb{D}^2 \setminus \{x_1, \ldots, x_r\}\) be the surface of genus \(0\) with \(r\) punctures and one boundary component. We begin our investigation by providing an alternative description of its mapping class group. Namely, we show that \(\Mod(\Sigma_{0, r}^1)\) is the braid group on \(r\) strands. \begin{definition} The \emph{braid group on \(n\) strands} \(B_n\) is the fundamental group \(\pi_1(C(\mathbb{D}^2, n), *)\) of the configuration space \(C(\mathbb{D}^2, n) = \mfrac{C^{\operatorname{ord}}(\mathbb{D}^2, n)}{S_n}\) of \(n\) points in the interior of the disk. The elements of \(B_n\) are referred to as \emph{braids}. \end{definition} \begin{example} Given \(i = 1, \ldots, n-1\), we define \(\sigma_i \in B_n\) as in Figure~\ref{fig:braid-group-generator}. \end{example} \begin{figure}[ht] \centering \includegraphics[width=.25\linewidth]{images/braid-group-generator.eps} \caption{The braid $\sigma_i$.} \label{fig:braid-group-generator} \end{figure} The third Reidemeister move translates to the so-called \emph{braid relations} \[ \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_1 \sigma_i \] in \(B_n\), which motivates the name used in Observation~\ref{ex:braid-relation}. In his seminal paper on braid groups, Artin \cite{artin} gave the following finite presentation of \(B_n\). \begin{theorem}[Artin] \[ \arraycolsep=1.2pt B_n = \left\langle \sigma_1, \ldots, \sigma_{n - 1} : \begin{array}{rll} \sigma_i \sigma_{i+1} \sigma_i & = \sigma_{i+1} \sigma_i \sigma_{i+1} & \quad \text{for all} \ i, \\ \sigma_i \sigma_j & = \sigma_j \sigma_i & \quad \text{for} \ j \ne i + 1 \ \text{and} \ j \ne i - 1 \end{array} \right\rangle. \] \end{theorem} As promised, we now show that \(B_n\) coincides with \(\Mod(\Sigma_{0, n}^1)\). Recall from Theorem~\ref{thm:birman-exact-seq} that there is an exact sequence \begin{center} \begin{tikzcd} 1 \rar & B_n \rar{\operatorname{push}} & \Mod(\Sigma_{0, n}^1) \rar & \cancelto{1}{\Mod(\mathbb{D}^2)} \rar & 1, \end{tikzcd} \end{center} for \(\Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) is contractible by Observation~\ref{ex:alexander-trick}. We thus obtain the following result. \begin{proposition} The map \(\operatorname{push} : B_n \to \Mod(\Sigma_{0, n}^1)\) is a group isomorphism. \end{proposition} \noindent \begin{minipage}[b]{.47\linewidth} \begin{observation}\label{ex:braid-group-center} Using the capping exact sequence from Observation~\ref{ex:capping-seq} and the Alexander method, one can check that the center \(Z(\Mod(\Sigma_{0, n}^1))\) of \(\Mod(\Sigma_{0, n}^1)\) is freely generated by the Dehn twist \(\tau_\delta\) about the boundary \(\delta = \partial \Sigma_{0, n}^1\). It is also not very difficult to see that \(\operatorname{push} : B_n \to \Mod(\Sigma_{0, n}^1)\) takes \(\sigma_1 \cdots \sigma_{n-1}\) to the rotation by \(\sfrac{2\pi}{n}\) as in Figure~\ref{fig:braid-group-center}, which is an \(n\)-th root of \(\tau_\delta\). Hence the center \(Z(B_n)\) is freely generated by \(z = (\sigma_1 \cdots \sigma_{n - 1})^n\). \end{observation} \end{minipage} \hspace{.6cm} % \begin{minipage}[b]{.47\textwidth} \centering \includegraphics[width=.4\linewidth]{images/braid-group-center.eps} \captionof{figure}{The clockwise rotation by $\sfrac{2\pi}{n}$ about an axis centered around the punctures $x_1, \ldots, x_n$ of $\Sigma_{0, n}^1$.} \label{fig:braid-group-center} \end{minipage} \smallskip To get from \(\Sigma_{0, n}^1\) to surfaces of genus \(g > 0\) we may consider the \emph{hyperelliptic involution} \(\iota : \Sigma_g \isoto \Sigma_g\), which rotates \(\Sigma_g\) by \(\pi\) around some axis as in Figure~\ref{fig:hyperelliptic-involution}. Given \(\ell < g\) and \(b = 1, 2\), we can also embed \(\Sigma_\ell^b\) in \(\Sigma_g\) in such way that \(\iota\) restricts to an involution\footnote{This involution does not fix $\partial \Sigma_\ell^b$ point-wise.} \(\Sigma_\ell^b \isoto \Sigma_\ell^b\). \begin{figure}[ht] \centering \includegraphics[width=\linewidth]{images/hyperelliptic-involution.eps} \caption{The hyperelliptic involution $\iota$.} \label{fig:hyperelliptic-involution} \end{figure} It is clear from Figure~\ref{fig:hyperelliptic-involution} that the quotients \(\mfrac{\Sigma_\ell^1}{\iota}\) and \(\mfrac{\Sigma_\ell^2}{\iota}\) are both disks, with boundary corresponding to the projection of the boundaries of \(\Sigma_\ell^1\) and \(\Sigma_\ell^2\), respectively. Given \(b = 1, 2\), the quotient map \(\Sigma_\ell^b \to \mfrac{\Sigma_\ell^b}{\iota} \cong \mathbb{D}^2\) is a double cover with \(2\ell + b\) branch points corresponding to the fixed points of \(\iota\). We may thus regard \(\mfrac{\Sigma_\ell^b}{\iota}\) as the disk \(\Sigma_{0, 2\ell + b}^1\) with \(2\ell + b\) punctures in its interior, as shown in Figure~\ref{fig:hyperelliptic-covering}. We also draw the curves \(\alpha_1, \ldots, \alpha_{2\ell} \subset \Sigma_\ell^b\) of the Humphreys generators of \(\Mod(\Sigma_g)\). Since these curves are invariant under the action of \(\iota\), they descend to arcs \(\bar{\alpha}_1, \ldots, \bar{\alpha}_{2\ell + b} \subset \Sigma_{0, 2\ell + b}^1\) joining the punctures of the quotient surface. \begin{figure}[ht] \centering \includegraphics[width=.77\linewidth]{images/hyperelliptic-covering.eps} \caption{The double branched covers given by $\iota$.} \label{fig:hyperelliptic-covering} \end{figure} \begin{observation}\label{ex:push-generators-description} The map \(\operatorname{push} : B_{2\ell + b} \to \Mod(\Sigma_{0, 2\ell + b}^1)\) takes \(\sigma_i\) to the half-twist \(h_{\bar{\alpha}_i}\) about the arc \(\bar{\alpha}_i \subset \Sigma_{0, 2\ell + b}^1\). \end{observation} We now study the homeomorphisms of \(\Sigma_\ell^1\) and \(\Sigma_\ell^2\) that descend to the quotient surfaces and their mapping classes, known as \emph{the symmetric mapping classes}. \begin{definition} Let \(\ell \ge 0\) and \(b = 1, 2\). The \emph{group of symmetric homeomorphisms of \(\Sigma_\ell^b\)} is \(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b) = \{\phi \in \Homeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b) : [\phi, \iota] = 1\}\). The \emph{symmetric mapping class group of \(\Sigma_\ell^b\)} is the subgroup \(\SMod(\Sigma_\ell^1) = \{ [\phi] \in \Mod(\Sigma_\ell^b) : \phi \in \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b) \}\). \end{definition} Fix \(b = 1\) or \(2\). It follows from the universal property of quotients that any \(\phi \in \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) defines a homeomorphism \(\bar \phi : \Sigma_{0, 2\ell+b}^1 \isoto \Sigma_{0, 2\ell+b}^1\). This yields a homomorphism of topological groups \begin{align*} \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b) & \to \Homeo^+(\Sigma_{0, 2\ell + b}^1, \partial \Sigma_{0, 2\ell + b}^1) \\ \phi & \mapsto \bar \phi, \end{align*} which is surjective because any \(\psi \in \Homeo^+(\Sigma_{0, 2\ell + b}^1, \partial \Sigma_{0, 2\ell + b}^1)\) lifts to \(\Sigma_\ell^b\). It is also not difficult to see \(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b) \to \Homeo^+(\Sigma_{0, 2\ell + b}^1, \partial \Sigma_{0, 2\ell + b}^1)\) is injective: the only candidates for elements of its kernel are \(1\) and \(\iota\), but \(\iota\) is not an element of \(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) since it does not fix \(\partial \Sigma_\ell^b\) point-wise. Now since we have a continuous bijective homomorphism we find \[ \begin{split} \pi_0(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)) & \cong \pi_0(\Homeo^+(\Sigma_{0, 2\ell+b}^1, \partial \Sigma_{0, 2\ell+b}^1)) \\ & = \mfrac{\Homeo^+(\Sigma_{0,2\ell+b}^1, \partial \Sigma_{0, 2\ell+b}^1)}{\simeq} \\ & = \Mod(\Sigma_{0, 2\ell+b}^1) \\ & \cong B_{2\ell + b}. \end{split} \] We would like to say \(\pi_0(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)) = \SMod(\Sigma_\ell^b)\), but a priori the story looks a little more complicated: \(\phi, \psi \in \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) define the same class in \(\SMod(\Sigma_\ell^b)\) if they are isotopic, but they may not lie in same connected component of \(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) if they are not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden \cite{birman-hilden} showed that this is never the case. \begin{theorem}[Birman-Hilden] If \(\phi, \psi \in \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) are isotopic then \(\phi\) and \(\psi\) are isotopic through symmetric homeomorphisms. In particular, there is an isomorphism \begin{align*} \SMod(\Sigma_\ell^b) & \isoto \Mod(\Sigma_{0, 2\ell + b}) \\ [\phi] & \mapsto [\bar \phi]. \end{align*} \end{theorem} \begin{observation} Using the notation of Figure~\ref{fig:hyperelliptic-covering}, the Birman-Hilden isomorphism \(\SMod(\Sigma_\ell^b) \isoto \Mod(\Sigma_{0, 2g + b})\) takes \(\tau_{\alpha_i}\) to the half twist \(h_{\bar{\alpha}_i} \in \Mod(\Sigma_{0, 2g + b})\). This can be checked by looking at \(\iota\)-invariant annular neighborhoods of the curves \(\alpha_i\) -- \cite[Section~9.4]{farb-margalit}. \end{observation} \begin{fundamental-observation}[$k$-chain relations] The Birman-Hilden isomorphism \(\SMod(\Sigma_\ell^1) \isoto \Mod(\Sigma_{0, 2\ell+1}^1)\) takes the twists \(\tau_\delta \in \SMod(\Sigma_\ell^1)\) about the boundary \(\delta = \partial \Sigma_\ell^1\) to \(\tau_{\bar\delta}^2 \in \Mod(\Sigma_{0, 2\ell+1}^1)\). Similarly, \(\SMod(\Sigma_\ell^2) \isoto \Mod(\Sigma_{0, 2\ell+2})\) takes \(\tau_{\delta_1} \tau_{\delta_2} \in \SMod(\Sigma_\ell^2)\) to \(\tau_{\bar\delta_1} = \tau_{\bar\delta_2}\). In light of Observation~\ref{ex:push-generators-description}, Observation~\ref{ex:braid-group-center} translates into the so-called \emph{\(k\)-chain relations} in \(\SMod(\Sigma_\ell^b) \subset \Mod(\Sigma_g)\). \[ \arraycolsep=1.4pt \begin{array}{rlcrll} (\sigma_1 \cdots \sigma_k)^{2k+2} & = z^2 \in B_{k + 1} & \; \rightsquigarrow & \; (\tau_{\alpha_1} \cdots \tau_{\alpha_k})^{2k + 2} & = \tau_\delta & \; \text{for } k = 2 \ell \text{ even} \\ (\sigma_1 \cdots \sigma_k)^{k+1} & = z \in B_{k + 1} & \; \rightsquigarrow & \; (\tau_{\alpha_1} \cdots \tau_{\alpha_k})^{k + 1} & = \tau_{\delta_1} \tau_{\delta_2} & \; \text{for } k = 2 \ell + 1 \text{ odd} \end{array} \] \end{fundamental-observation} We may also exploit the quotient \(\mfrac{\Sigma_g}{\iota} \cong \mathbb{S}^2\) to obtain other relations. Since \(\iota\) has \(2g + 2\) fixed points in \(\Sigma_g\), we get branched double cover \(\Sigma_g \to \Sigma_{0, 2g+2}\). \begin{theorem}[Birman-Hilden without boundary]\label{thm:boundaryless-birman-hilden} If \(g \ge 2\) then we have an exact sequence \begin{center} \begin{tikzcd} 1 \rar & \langle [\iota] \rangle \rar & C_{\Mod(\Sigma_g)}([\iota]) \rar & \Mod(\Sigma_{0, 2g + 2}) \rar & 1, \end{tikzcd} \end{center} where \(C_{\Mod(\Sigma_g)}([\iota]) \subset \Mod(\Sigma_g)\) is the commutator subgroup of \([\iota]\) and the right map takes \([\phi] \in C_{\Mod(\Sigma_g)}([\iota])\) to \([\bar \phi] \in \Mod(\Sigma_{0, 2g + 2})\). \end{theorem} \begin{fundamental-observation}[Hyperelliptic relations] Let \(\alpha_1, \ldots, \alpha_{2g}, \delta \subset \Sigma_g\) be as in Figure~\ref{fig:hyperellipitic-relations}. Then \begin{equation}\label{eq:hyperelliptic-eq} [\iota] = \tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta. \end{equation} Indeed, \(C_{\Mod(\Sigma_g)}([\iota]) \to \Mod(\Sigma_{0, 2g+2})\) takes \(\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1}\) to the rotation from Figure~\ref{fig:hyperelliptic-relation-rotation}, while \(\tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta\) is taken to its inverse. By Theorem~\ref{thm:boundaryless-birman-hilden}, \[ \tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta \in \ker (C_{\Mod(\Sigma_g)}([\iota]) \to \Mod(\Sigma_{0, 2g+2})) = \langle [\iota] \rangle \cong \mathbb{Z}/2. \] One can then show \(\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta\) inverts the orientation of \(\alpha_1\), so \(\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta \ne 1\) and (\ref{eq:hyperelliptic-eq}) follows. In particular, we obtain the so-called \emph{hyperelliptic relations} (\ref{eq:hyperelliptic-rel-1}) and (\ref{eq:hyperelliptic-rel-2}) in \(\Mod(\Sigma_g)\). \begin{align}\label{eq:hyperelliptic-rel-1} (\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta)^2 & = 1 \\ \label{eq:hyperelliptic-rel-2} [\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta, \tau_\delta] & = 1 \end{align} \end{fundamental-observation} \noindent \begin{minipage}[b]{.47\textwidth} \centering \includegraphics[width=.7\linewidth]{images/hyperelliptic-relation.eps} \vspace*{.5cm} \captionof{figure}{The curves from the Humphreys generators of $\Mod(\Sigma_g)$ and the curve $\delta$ from the hyperelliptic relations.} \label{fig:hyperellipitic-relations} \end{minipage} \hspace{.6cm} % \begin{minipage}[b]{.47\textwidth} \centering \includegraphics[width=.33\linewidth]{images/sphere-rotation.eps} \captionof{figure}{The clockwise rotation by $\sfrac{\pi}{g + 1}$ about an axis centered around the punctures of $\Sigma_{0, 2g + 1}$.} \label{fig:hyperelliptic-relation-rotation} \end{minipage} \medskip \section{Presentations of Mapping Class Groups} Having explored some of the relations in \(\Mod(\Sigma)\), it is natural to ask if these relations are enough to completely describe the structure of \(\Mod(\Sigma)\). Different presentations of mapping class groups are due to the work of Birman-Hilden \cite{birman-hilden}, Gervais \cite{gervais} and many others. Wajnryb \cite{wajnryb} derived a presentation of \(\Mod(\Sigma_g)\) only using the relations discussed in Chapter~\ref{ch:dehn-twists} and Section~\ref{birman-hilden}. This is quite a satisfactory result, for we have seen that all of these relations can be explained in terms of the topology of \(\Sigma_g\). \begin{theorem}[Wajnryb]\label{thm:wajnryb-presentation} Suppose \(g \ge 3\). If \(\alpha_0, \ldots, \alpha_g\) are as in Figure~\ref{fig:humphreys-gens} and \(a_i = \tau_{\alpha_i} \in \Mod(\Sigma_g)\) are the Humphreys generators, then there is a presentation of \(\Mod(\Sigma_g)\) with generators \(a_0, \ldots a_{2g}\) subject to the following relations. \begin{enumerate} \item The \emph{disjointness relations} \([a_i, a_j] = 1\) for \(\alpha_i\) and \(\alpha_j\) disjoint. \item The \emph{braid relations} \(a_i a_j a_i = a_j a_i a_j\) for \(\alpha_i\) and \(\alpha_j\) crossing once. \item The \emph{\(3\)-chain relation} \((a_1 a_2 a_3)^4 = a_0 b_0\), where \[ b_0 = (a_4 a_3 a_2 a_1 a_1 a_2 a_3 a_4) a_0 (a_4 a_3 a_2 a_1 a_1 a_2 a_3 a_4)^{-1}. \] \item The \emph{lantern relation} \(a_0 b_2 b_1 = a_1 a_3 a_5 b_3\), where \begin{align*} b_1 & = (a_4 a_5 a_3 a_4)^{-1} a_0 (a_4 a_5 a_3 a_4) \\ b_2 & = (a_2 a_3 a_1 a_2)^{-1} b_1 (a_2 a_3 a_1 a_2) \\ b_2 & = u b_1 u^{-1} \\ u & = (a_6 a_5) (a_4 a_3 a_2) (a_6 a_5)^{-1} b_1 (a_6 a_5) a_1^{-1} (a_4 a_3 a_2)^{-1}. \end{align*} \item The \emph{hyperelliptic relation} \([a_{2g} \cdots a_1 a_1 \cdots a_{2g}, d] = 1\), where \(d = n_g\) for \(n_1 = a_1\), \(n_2 = b_0\) and \begin{align*} n_{i + 2} & = w_i n_i w_i^{-1} \\ w_i & = (a_{2i + 4} a_{2i + 3} a_{2i + 2} n_{i + 1}) (a_{2i + 1} a_{2i}^2 a_{2i + 1}) (a_{2i + 3} a_{2i + 2} a_{2i + 4} a_{2i + 3}) (n_1 a_{2i + 2} a_{2i + 1} a_{2i}). \end{align*} \end{enumerate} \end{theorem} \begin{remark} The mapping classes \(b_0, \ldots, b_3, d\) in the statement of Theorem~\ref{thm:wajnryb-presentation} correspond to the Dehn twists about the curves \(\beta_0, \ldots, \beta_3, \delta \subset \Sigma_g\) highlighted in Figure~\ref{fig:wajnryb-presentation-curves}, so Wajnryb's presentation is not as intractable as it might look at first glance. \end{remark} \begin{figure}[ht] \centering \includegraphics[width=.7\linewidth]{images/wajnryb-presentation-curves.eps} \caption{The curves from Wajnryb's presentation.} \label{fig:wajnryb-presentation-curves} \end{figure} Different presentations can be used to compute the Abelianization of \(\Mod(\Sigma_g)\) for \(g \le 2\). Indeed, if \(G = \langle g_1, \ldots, g_n : R \rangle\) is a finitely-presented group, then \(G^\ab = \langle g_1, \ldots, g_n : R, [g_i, g_j] \text{ for all } i, j \rangle\). Using this approach, Farb-Margalit \cite[Section~5.1.3]{farb-margalit} show the Abelianization is given by \begin{center} \begin{tabular}{r|c|l} \(g\) & \(\Sigma_g\) & \(\Mod(\Sigma_g)^\ab\) \\[1pt] \hline & & \\[-10pt] \(0\) & \(\mathbb{S}^2\) & \(0\) \\ \(1\) & \(\mathbb{T}^2\) & \(\mathbb{Z}/12\) \\ \(2\) & \(\Sigma_2\) & \(\mathbb{Z}/10\) \\ \end{tabular} \end{center} for closed surfaces of small genus. In \cite{korkmaz-mccarthy} Korkmaz-McCarthy showed that even though \(\Mod(\Sigma_2^b)\) is not perfect, its commutator subgroup is. In addition, they also show \([\Mod(\Sigma_g^b), \Mod(\Sigma_g^b)]\) is normally generated by a single mapping class. \begin{proposition}\label{thm:commutator-is-perfect} The commutator subgroup \(\Mod(\Sigma_2^b)' = [\Mod(\Sigma_2^b), \Mod(\Sigma_2^b)]\) is perfect -- i.e. \(\Mod(\Sigma_2^b)^{(2)} = [\Mod(\Sigma_2^b)', \Mod(\Sigma_2^b)']\) is the whole of \(\Mod(\Sigma_2^b)'\). \end{proposition} \begin{proposition}\label{thm:commutator-normal-gen} If \(g \ge 2\) and \(\alpha, \beta \subset \Sigma_g\) are simple closed crossing only once, then \(\Mod(\Sigma_g)'\) is \emph{normally generated} by \(\tau_\alpha \tau_\beta^{-1}\) -- i.e. if \(\tau_\alpha \tau_\beta^{-1} \in N \normal \Mod(\Sigma_g)'\) then \(\Mod(\Sigma_g)' \subset N\). \end{proposition} The different presentations of \(\Mod(\Sigma_g)\) may also be used to study its representations. Indeed, in light of Theorem~\ref{thm:wajnryb-presentation}, a representation \(\rho : \Mod(\Sigma_g) \to \GL_n(\mathbb{C})\) is nothing other than a choice of \(2g + 1\) matrices \(\rho(\tau_{\alpha_0}), \ldots, \rho(\tau_{\alpha_{2g}}) \in \GL_n(\mathbb{C})\) satisfying the relations \strong{(i)} to \strong{(v)} as above. In the next chapter, we will discuss how these relations may be used to derive obstructions to the existence of nontrivial representations of certain dimensions.