memoire-m2

diff --git a/references.bib b/references.bib
@@ -176,6 +176,18 @@
   year      = {1992},
 }
 
+@article{gervais,
+  title = {Presentation and central extensions of mapping class groups},
+  volume = {348},
+  ISSN = {1088-6850},
+  DOI = {10.1090/s0002-9947-96-01509-7},
+  number = 8,
+  journal = {Transactions of the American Mathematical Society},
+  author = {Gervais,  Sylvain},
+  year = {1996},
+  pages = {3097--3132}
+}
+
 @book{kerekjarto,
   author    = {Kerékjártó,  Béla},
   doi       = {10.1007/978-3-642-50825-7},
@@ -184,6 +196,7 @@
   title     = {Vorlesungen \"{u}ber Topologie},
   year      = {1923},
 }
+
 @incollection{julien,
   author    = {Marché,  Julien},
   booktitle = {Topology and geometry. A collection of essays dedicated to Vladimir G. Turaev},

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -1,24 +1,21 @@
 \chapter{Relations Between Twists}\label{ch:relations}
 
-Having acomplished the milestones of Theorem~\ref{thm:lickorish-gens} and
-Corollary~\ref{thm:humphreys-gens}, we now find ourselves ready to study some
-of the group-theoretic aspects of \(\Mod(S)\). We should note, however, that
-our current understanding of its group structure is quite lacking: even though
-we know the generators of \(\Mod(S)\), we have already seen in
-Observation~\ref{ex:braid-relation} that these must satisfy nontrivial relations.
-
-This poses the question: what relations between Dehn twists are there? The goal
-of this chapter is to highlight some of these relations and the geometric
-intuition behind them. We start by perhaps the simplest of these, known as
-\emph{the lantern relation}.
+Having found a conveniant set of genetors for \(\Mod(S)\), it is now natural to
+ask what are the relations between such generators. In this chapter, we
+highlight some further relations between Dehn twists and the geometric
+intuition behind them, culminating in the statement of a presentation for
+\(\Mod(S_g)\) whose relations can be entirely explained in terms of the
+geometry of curves in \(S_g\) -- see Theorem~\ref{thm:wajnryb-presentation}.
+
+We start by the so called \emph{lantern relation}.
 
 \begin{fundamental-observation}
   Let \(S_0^4\) be the surface the of genus \(0\) with \(4\) boundary
   components -- i.e. the \emph{lantern-like} surface we get by subtracting
   \(4\) disjoint open disks from \(\mathbb{S}^2\). If \(\alpha, \beta, \gamma,
   \delta_1, \ldots, \delta_4 \subset S_0^4\) are as in
-  Figure~\ref{fig:latern-relation} then from the Alexander method we get the so
-  called \emph{lantern relation} in \(\Mod(S_0^4)\).
+  Figure~\ref{fig:latern-relation} then from the Alexander method we get the
+  \emph{lantern relation} (\label{eq:lantern-relation}) in \(\Mod(S_0^4)\).
   \begin{equation}\label{eq:latern-relation}
     \tau_\alpha \tau_\beta \tau_\gamma
     = \tau_{\delta_1} \tau_{\delta_2} \tau_{\delta_3} \tau_{\delta_4}
@@ -73,46 +70,9 @@ example\dots
   \label{fig:latern-relation-trivial-abelianization}
 \end{figure}
 
-We should note that in general \(\Mod(S)^\ab\) needs not be trivial. For
-example, in \cite[Section~5.1.3]{farb-margalit} Farb-Margalit use different
-presentations of \(\Mod(S_g)\) for \(g \le 2\) to show the Abelianization is given by
-\begin{center}
-  \begin{tabular}{r|c|l}
-    \(g\) & \(S_g\)          & \(\Mod(S_g)^\ab\) \\[1pt]
-    \hline
-          &                  &                   \\[-10pt]
-    \(0\) & \(\mathbb{S}^2\) & \(0\)             \\
-    \(1\) & \(\mathbb{T}^2\)   & \(\mathbb{Z}/12\) \\
-    \(2\) & \(S_2\)          & \(\mathbb{Z}/10\) \\
-  \end{tabular}
-\end{center}
-for closed surfaces with small genus.
-
-In \cite{korkmaz-mccarthy} Korkmaz-McCarthy showed that
-eventhough \(\Mod(S_2^p)\) is not perfect, its commutator subgroup is.
-In addition, they also show \([\Mod(S_g^p), \Mod(S_g^p)]\) is normaly generated
-by a single mapping class.
-
-\begin{proposition}\label{thm:commutator-is-perfect}
-  The commutator subgroup \(\Mod(S_2^p)' = [\Mod(S_2^p), \Mod(S_2^p)]\) is
-  perfect -- i.e. \(\Mod(S_2^p)^{(2)} = [\Mod(S_2^p)', \Mod(S_2^p)']\) is the
-  whole of \(\Mod(S_2^p)'\).
-\end{proposition}
-
-\begin{proposition}\label{thm:commutator-normal-gen}
-  If \(g \ge 2\) and \(\alpha, \beta \subset S_g\) are simple closed crossing
-  only once, then \(\Mod(S_g)'\) is \emph{normally generated} by \(\tau_\alpha
-  \tau_\beta^{-1}\) -- i.e. if \(\tau_\alpha \tau_\beta^{-1} \in N \normal
-  \Mod(S_g)'\) then \(\Mod(S_g)' \subset N\).
-\end{proposition}
-
-These past few results combined paint a remarkably clear picture of the
-Abelianization \(\Mod(S)^\ab\) and the commutator \(\Mod(S)'\), but this is
-still a far cry from a general undertanding of the structure of \(\Mod(S)\)
-itself. For that, we need to intruduce some extra relations. To that end, we
-study certain branched covers \(S \to \mathbb{D}^2 \setminus \{x_1, \ldots,
-x_r\}\) and how they may be used to relate both mapping class groups. This is
-what is known as\dots
+To get extra relations we need to consider certain branched covers \(S \to
+\mathbb{D}^2 \setminus \{x_1, \ldots, x_r\}\) and how they may be used to
+relate both mapping class groups. This is what is known as\dots
 
 \section{The Birman-Hilden Theorem}
 
@@ -406,6 +366,7 @@ we get branched double cover \(S_g \to S_{0, 2g+2}\).
 \begin{minipage}[b]{.45\textwidth}
   \centering
   \includegraphics[width=.7\linewidth]{images/hyperelliptic-relation.eps}
+  \vspace*{.5cm}
   \captionof{figure}{The curves from the Humphreys generators of $\Mod(S_g)$
   and the curve $\delta$ from the hyperelliptic relations.}
   \label{fig:hyperellipitic-relations}
@@ -420,15 +381,21 @@ we get branched double cover \(S_g \to S_{0, 2g+2}\).
 \end{minipage}
 \medskip
 
-Wajnryb \cite{wajnryb} used the \(k\)-chain relations and the hyperelliptic
-relations to derive a presentation of the mapping class group of a closed
-surface.
+\section{Presentations of Mapping Class Groups}
+
+There are numerous known presentations of \(\Mod(S_{g, r}^p)\), such as the
+ones due to Birman-Hilden \cite{birman-hilden} and Gervais \cite{gervais}.
+Wajnryb \cite{wajnryb} derived a presentation of \(\Mod(S_g)\) using the
+relations discussed in Chapter~\ref{ch:dehn-twists} and
+Chapter~\ref{ch:relations}. This is a particularly satisfactory presentation,
+since all of its relations can be explained in terms of the geometry of curves
+in \(S_g\).
 
 \begin{theorem}[Wajnryb]\label{thm:wajnryb-presentation}
-  If \(\alpha_0, \ldots, \alpha_g\) are as in Figure~\ref{fig:humphreys-gens}
-  and \(a_i = \tau_{\alpha_i} \in \Mod(S_g)\) are the Humphreys generators,
-  then there is a presentation of \(\Mod(S_g)\) with generators \(a_0, \ldots
-  a_{2g}\) subject to the following relations.
+  Suppose \(g \ge 3\). If \(\alpha_0, \ldots, \alpha_g\) are as in
+  Figure~\ref{fig:humphreys-gens} and \(a_i = \tau_{\alpha_i} \in \Mod(S_g)\)
+  are the Humphreys generators, then there is a presentation of \(\Mod(S_g)\)
+  with generators \(a_0, \ldots a_{2g}\) subject to the following relations.
   \begin{enumerate}
     \item The disjointness relations \([a_i, a_j] = 1\) for \(\alpha_i\) and
       \(\alpha_j\) disjoint.
@@ -478,7 +445,45 @@ surface.
   \label{fig:wajnryb-presentation-curves}
 \end{figure}
 
-This presentation paints a much clearer picture of the structure of
-\(\Mod(S_g)\). Nevertheless, its linear representations and many other of its
-group-theoretic aspects remain a mistery. This will be the focus of the next
-chapter.
+Different presentations can be used to compute the Abelianization of
+\(\Mod(S_g)\) 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}
+    \(g\) & \(S_g\)          & \(\Mod(S_g)^\ab\) \\[1pt]
+    \hline
+          &                  &                   \\[-10pt]
+    \(0\) & \(\mathbb{S}^2\) & \(0\)             \\
+    \(1\) & \(\mathbb{T}^2\)   & \(\mathbb{Z}/12\) \\
+    \(2\) & \(S_2\)          & \(\mathbb{Z}/10\) \\
+  \end{tabular}
+\end{center}
+for closed surfaces of small genus. In \cite{korkmaz-mccarthy} Korkmaz-McCarthy
+showed that eventhough \(\Mod(S_2^p)\) is not perfect, its commutator subgroup
+is. In addition, they also show \([\Mod(S_g^p), \Mod(S_g^p)]\) is normaly
+generated by a single mapping class.
+
+\begin{proposition}\label{thm:commutator-is-perfect}
+  The commutator subgroup \(\Mod(S_2^p)' = [\Mod(S_2^p), \Mod(S_2^p)]\) is
+  perfect -- i.e. \(\Mod(S_2^p)^{(2)} = [\Mod(S_2^p)', \Mod(S_2^p)']\) is the
+  whole of \(\Mod(S_2^p)'\).
+\end{proposition}
+
+\begin{proposition}\label{thm:commutator-normal-gen}
+  If \(g \ge 2\) and \(\alpha, \beta \subset S_g\) are simple closed crossing
+  only once, then \(\Mod(S_g)'\) is \emph{normally generated} by \(\tau_\alpha
+  \tau_\beta^{-1}\) -- i.e. if \(\tau_\alpha \tau_\beta^{-1} \in N \normal
+  \Mod(S_g)'\) then \(\Mod(S_g)' \subset N\).
+\end{proposition}
+
+The different presentations of \(\Mod(S_g)\) may also be used to study its
+representations. Indeed, in light of Theorem~\ref{thm:wajnryb-presentation}, a
+representation \(\rho : \Mod(S_g) \to \GL_n(\mathbb{C})\) is nothing other than
+a choice of \(2g + 1\) matrices \(\rho(\tau_{\alpha_1}), \ldots,
+\rho(\tau_{\alpha_{2g}}) \in \GL_n(\mathbb{C})\) satisfying the relations
+\strong{(i)} to \strong{(v)} as above. In the next chapter, we will discuss how
+these relations may be used to derrive obstructions to the existance of
+nontrivial representations of certain dimensions.

diff --git a/sections/representations.tex b/sections/representations.tex
@@ -2,12 +2,9 @@
 
 Having built a solid understanding of the combinatorics of Dehn twists, we are
 now ready to attack the problem of classifying the representations of
-sufficiently small dimension of \(\Mod(S_g)\) . Indeed, in light of the Wajnryb
-presentation, a representation \(\rho : \Mod(S_g) \to \GL_n(\mathbb{C})\) is
-nothing other than a choice of \(2g + 1\) matrices \(\rho(\tau_{\alpha_1}),
-\ldots, \rho(\tau_{\alpha_{2g}}) \in \GL_n(\mathbb{C})\) satisfying the
-relations \strong{(i)} to \strong{(v)} from
-Theorem~\ref{thm:wajnryb-presentation}.
+\(\Mod(S_g)\) of sufficiently small dimension. As promised, our strategy is to
+make us of the \emph{geometrically-motivated} relations derrived in
+Chapter~\ref{ch:relations}.
 
 Historically, these relations have been exploited by Funar \cite{funar},
 Franks-Handel \cite{franks-handel} and others to establish the triviality of