memoire-m2

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