\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.
