memoire-m2

 \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.