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.