diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -1,4 +1,4 @@
-\chapter{Relations Among Twists \& Wajnryb's Presentation}
+\chapter{Relations Between Twists}
Having acomplished the milestones of Theorem~\ref{thm:lickorish-gens} and
Corollary~\ref{thm:humphreys-gens}, we now find ourselves ready to study some
@@ -60,7 +60,7 @@ example\dots
= [\tau_\alpha] + [\tau_\beta] + [\tau_\gamma]
= [\tau_{\delta_1}] + [\tau_{\delta_2}]
+ [\tau_{\delta_3}] + [\tau_{\delta_4}]
- = 4 \cdot [\tau_\alpha],
+ = 4 \cdot [\tau_\alpha]
\]
in \(\Mod(S_g^b)^\ab\). In other words, \([\tau_\alpha] = 0\) and thus
\(\Mod(S_g^b)^\ab = 0\).
@@ -73,14 +73,14 @@ example\dots
\label{fig:latern-relation-trivial-abelianization}
\end{figure}
-% TODO: Fix the spacing of this table
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 }
- \(g\) & \(S_g\) & \(\Mod(S_g)^\ab\) \\
+ \begin{tabular}{r|c|l}
+ \(g\) & \(S_g\) & \(\Mod(S_g)^\ab\) \\[1pt]
\hline
+ & & \\[-10pt]
\(0\) & \(\mathbb{S}^2\) & \(0\) \\
\(1\) & \(\mathbb{T}\) & \(\mathbb{Z}/12\) \\
\(2\) & \(S_2\) & \(\mathbb{Z}/10\) \\
@@ -212,6 +212,7 @@ we get\dots
around the punctures $x_1, \ldots, x_n$ of $S_{0, n}^1$.}
\label{fig:braid-group-center}
\end{minipage}
+\smallskip
To get from \(S_{0, n}^1\) to surfaces of genus \(g > 0\) we may consider the
\emph{hyperelliptic involution} \(\iota : S_g \isoto S_g\) which rotates
@@ -294,7 +295,7 @@ continuous bijective homomorphism we find
& \cong \pi_0(\Homeo^+(S_{0, 2\ell+b}^1, \partial S_{0, 2\ell+b}^1)) \\
& = \mfrac{\Homeo^+(S_{0,2\ell+b}^1, \partial S_{0, 2\ell+b}^1)}{\simeq} \\
& = \Mod(S_{0, 2\ell+b}^1) \\
- & \cong B_{2\ell + b}
+ & \cong B_{2\ell + b}.
\end{split}
\]
@@ -336,17 +337,17 @@ not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden
Example~\ref{ex:braid-group-center} gives us the so called \emph{\(k\)-chain
relations} in \(\SMod(S_\ell^b) \subset \Mod(S_g)\).
\[
- \begin{array}{rcll}
- (\sigma_1 \cdots \sigma_k)^{2k+2} = z^2 \in B_{k + 1} &
- \rightsquigarrow &
- (\tau_{\alpha_1} \cdots \tau_{\alpha_k})^{2k + 2}
- = \tau_\delta &
- \text{for } k = 2 \ell \text{ even} \\
- (\sigma_1 \cdots \sigma_k)^{k+1} = z \in B_{k + 1} &
- \rightsquigarrow &
- (\tau_{\alpha_1} \cdots \tau_{\alpha_k})^{k + 1}
- = \tau_{\delta_1} \tau_{\delta_2} &
- \text{for } k = 2 \ell + 1 \text{ odd}
+ \arraycolsep=1.4pt
+ \begin{array}{rlcrll}
+ (\sigma_1 \cdots \sigma_k)^{2k+2} & = z^2 \in B_{k + 1} &
+ \; \rightsquigarrow &
+ \; (\tau_{\alpha_1} \cdots \tau_{\alpha_k})^{2k + 2} & = \tau_\delta &
+ \; \text{for } k = 2 \ell \text{ even} \\
+ (\sigma_1 \cdots \sigma_k)^{k+1} & = z \in B_{k + 1} &
+ \; \rightsquigarrow &
+ \; (\tau_{\alpha_1} \cdots \tau_{\alpha_k})^{k + 1}
+ & = \tau_{\delta_1} \tau_{\delta_2} &
+ \; \text{for } k = 2 \ell + 1 \text{ odd}
\end{array}
\]
\end{example}
@@ -416,6 +417,7 @@ we get branched double cover \(S_g \to S_{0, 2g+2}\).
axis centered around the punctures of $S_{0, 2g + 1}$.}
\label{fig:hyperelliptic-relation-rotation}
\end{minipage}
+\medskip
Wajnryb used the \(k\)-chain relations and the hyperelliptic relations to
derive a presentation of the mapping class group of a closed surface,
@@ -425,8 +427,8 @@ which is widely considered to the the standard presentation of
\begin{theorem}[Wajnryb]\label{thm:wajnryb-presentation}
If \(\alpha_0, \ldots, \alpha_g\) are as in
% TODO: Reference the drawing of the curves somewhere
- 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
+ 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 \cdot