memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: f87f15c570cf70f02a35641091e7fd7e2f1f78e3
Parent: 3944f17785f55710dea6c745f8fdc27a245573d5
Author: Pablo <pablo-pie@riseup.net>
Date: Mon, 24 Jun 2024 07:24:33 +0000

Fixed the formatting of double columns

Made it so that double columns are not indented

Diffstat

3 files changed, 25 insertions, 20 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	13	7	6
Modified	sections/presentation.tex	15	8	7
Modified	sections/twists.tex	17	10	7

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -288,18 +288,19 @@ representation.}
   \emph{the symplectic representation of \(\Mod(\Sigma_g)\)}.
 \end{example}
 
-\begin{minipage}[b]{.45\linewidth}
+\noindent
+\begin{minipage}[b]{.47\linewidth}
   \centering
-  \includegraphics[width=\linewidth]{images/homology-generators.eps}
+  \includegraphics[width=.9\linewidth]{images/homology-generators.eps}
   \captionof{figure}{The curves $\alpha_1, \beta_1, \ldots, \alpha_g, \beta_g
   \subset \Sigma_g$ that generate its first homology group.}
   \label{fig:homology-basis}
 \end{minipage}
-\hspace{.5cm} %
-\begin{minipage}[b]{.45\linewidth}
+\hspace{.6cm} %
+\begin{minipage}[b]{.47\linewidth}
   \centering
-  \includegraphics[width=\linewidth]{images/intersection-index.eps}
-  \vspace*{.4cm}
+  \includegraphics[width=.9\linewidth]{images/intersection-index.eps}
+  \vspace*{.75cm}
   \captionof{figure}{The index of an intersection point $x \in \alpha \cap
   \beta$.}
   \label{fig:intersection-index}

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -152,8 +152,8 @@ get\dots
   isomorphism.
 \end{proposition}
 
-
-\begin{minipage}[b]{.45\linewidth}
+\noindent
+\begin{minipage}[b]{.47\linewidth}
 \begin{observation}\label{ex:braid-group-center}
   Using the capping exact sequence from Example~\ref{ex:capping-seq} and
   the Alexander method, one can check that the center \(Z(\Mod(\Sigma_{0, n}^1))\)
@@ -166,8 +166,8 @@ get\dots
   generated by \(z = (\sigma_1 \cdots \sigma_{n - 1})^n\).
 \end{observation}
 \end{minipage}
-\hspace{.5cm} %
-\begin{minipage}[b]{.45\textwidth}
+\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
@@ -369,7 +369,8 @@ to obtain other relations. Since \(\iota\) has \(2g + 2\) fixed points in
   \end{align}
 \end{fundamental-observation}
 
-\begin{minipage}[b]{.45\textwidth}
+\noindent
+\begin{minipage}[b]{.47\textwidth}
   \centering
   \includegraphics[width=.7\linewidth]{images/hyperelliptic-relation.eps}
   \vspace*{.5cm}
@@ -377,8 +378,8 @@ to obtain other relations. Since \(\iota\) has \(2g + 2\) fixed points in
   $\Mod(\Sigma_g)$ and the curve $\delta$ from the hyperelliptic relations.}
   \label{fig:hyperellipitic-relations}
 \end{minipage}
-\hspace{.5cm} %
-\begin{minipage}[b]{.45\textwidth}
+\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

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -60,7 +60,8 @@ applications of the Alexander method.
   \mathbb{Z}\).
 \end{example}
 
-\begin{minipage}[b]{.45\linewidth}
+\noindent
+\begin{minipage}[b]{.47\linewidth}
   \centering
   \includegraphics[width=.7\linewidth]{images/dehn-twist-cylinder.eps}
   \captionof{figure}{The generator $f$ of $\Mod(\mathbb{S}^1 \times [0, 1])
@@ -68,8 +69,8 @@ applications of the Alexander method.
   the right-hand side that winds about the curve $\alpha$.}
   \label{fig:dehn-twist-cylinder}
 \end{minipage}
-\hspace{.5cm} %
-\begin{minipage}[b]{.45\linewidth}
+\hspace{.6cm} %
+\begin{minipage}[b]{.47\linewidth}
   \centering
   \includegraphics[width=.4\linewidth]{images/half-twist-disk.eps}
   \captionof{figure}{The generator $f$ of $\Mod(\mathbb{D}^2 \setminus
@@ -528,14 +529,16 @@ remaining curves, from which we get the so called \emph{Humphreys generators}.
   Figure~\ref{fig:humphreys-gens}.
 \end{corollary}
 
-\begin{minipage}[b]{.45\linewidth}
+\noindent
+\begin{minipage}[b]{.47\linewidth}
   \centering
   \includegraphics[width=\linewidth]{images/lickorish-gens.eps}
-  \captionof{figure}{The curves from Lickorish generators of $\Mod(\Sigma_g^p)$.}
+  \captionof{figure}{The curves from Lickorish generators of
+  $\Mod(\Sigma_g^p)$.}
   \label{fig:lickorish-gens}
 \end{minipage}
-\hspace{.5cm} %
-\begin{minipage}[b]{.45\textwidth}
+\hspace{.6cm} %
+\begin{minipage}[b]{.47\textwidth}
   \centering
   \includegraphics[width=\linewidth]{images/humphreys-gens.eps}
   \captionof{figure}{The curves from Humphreys generators of $\Mod(\Sigma_g)$.}