memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: 6078bdea5d725c15f9ebbc63fdad2d5f3fb3dba3
Parent: ab471a10bb3bb1227bb0f20142f1a887982387af
Author: Pablo <pablo-pie@riseup.net>
Date: Thu, 27 Jun 2024 13:44:37 +0000

Removed the used of \dots

Diffstat

4 files changed, 19 insertions, 16 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	6	3	3
Modified	sections/presentation.tex	7	4	3
Modified	sections/representations.tex	5	2	3
Modified	sections/twists.tex	17	10	7

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -71,7 +71,7 @@ homeomorphisms.
 \end{figure}
 
 By splitting \(\Sigma\) across curves \(\alpha, \alpha' \subset \Sigma\)
-crossing once, we can also show\dots
+crossing once, we can also show the following result.
 
 \begin{observation}\label{ex:change-of-coordinates-crossing}
   Let \(\alpha, \beta, \alpha', \beta' \subset \Sigma\) be nonseparating curves
@@ -88,7 +88,7 @@ orientation-preserving homeomorphisms of \(\Sigma\) fixing each point in
 rich geometry, but its algebraic structure is often regarded as too complex to
 tackle. More importantly, all of this complexity is arguably unnecessary for
 most topological applications, in the sense that usually we are only really
-interested in considering \emph{homeomorphisms up to isotopy}. For example\dots
+interested in considering \emph{homeomorphisms up to isotopy}. For example:
 \begin{enumerate}
   \item Isotopic \(\phi \simeq \psi \in \Homeo^+(\Sigma, \partial \Sigma)\)
     determine the same application \(\phi_* = \psi_* : \pi_1(\Sigma, x) \to
@@ -116,7 +116,7 @@ hand.
   \]
 \end{definition}
 
-There are many variations of Definition~\ref{def:mcg}. For example\dots
+There are many variations of Definition~\ref{def:mcg}.
 
 \begin{observation}\label{ex:action-on-punctures}
   Any \(\phi \in \Homeo^+(\Sigma, \partial \Sigma)\) extends uniquely to a

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -64,7 +64,8 @@ Theorem~\ref{thm:wajnryb-presentation}.
 We may exploit different embeddings \(\Sigma_0^4 \hookrightarrow \Sigma\) and
 their corresponding inclusion homomorphisms \(\Mod(\Sigma_0^4) \to
 \Mod(\Sigma)\) to obtain interesting relations between the corresponding Dehn
-twists in \(\Mod(\Sigma)\). For example\dots
+twists in \(\Mod(\Sigma)\). For example, the lantern relation can be used to
+compute \(\Mod(\Sigma_g^b)^\ab\) for \(g \ge 3\).
 
 \begin{proposition}\label{thm:trivial-abelianization}
   The Abelianization \(\Mod(\Sigma_g^b)^\ab =
@@ -106,7 +107,7 @@ twists in \(\Mod(\Sigma)\). For example\dots
 To get extra relations we need to investigate certain branched covers \(\Sigma
 \to \mathbb{D}^2 \setminus \{x_1, \ldots, x_r\}\), as well as the relationship
 between \(\Mod(\Sigma)\) and \(\Mod(\mathbb{D}^2 \setminus \{x_1, \ldots,
-x_r\})\). This is what is known as\dots
+x_r\})\). This is what is known as \emph{the Birman-Hilden theorem}.
 
 \section{The Birman-Hilden Theorem}\label{birman-hilden}
 
@@ -174,7 +175,7 @@ sequence
   \end{tikzcd}
 \end{center}
 for \(\Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) is contractible by
-Observation~\ref{ex:alexander-trick}. In other words\dots
+Observation~\ref{ex:alexander-trick}. We thus obtain the following result.
 
 \begin{proposition}
   The map \(\operatorname{push} : B_n \to \Mod(\Sigma_{0, n}^1)\) is a group

diff --git a/sections/representations.tex b/sections/representations.tex
@@ -10,8 +10,7 @@ Historically, these relations have been exploited by Funar \cite{funar},
 Franks-Handel \cite{franks-handel} and others to establish the triviality of
 low-dimensional representations, culminating in Korkmaz' \cite{korkmaz} recent
 classification of representations of dimension \(n \le 2 g\) for \(g \ge 3\).
-The goal of this chapter is to provide a concise account of Korkmaz' results,
-starting by\dots
+The goal of this chapter is to provide a concise account of Korkmaz' results.
 
 \begin{theorem}[Korkmaz]\label{thm:low-dim-reps-are-trivial}
   Let \(\Sigma_g^b\) be the compact surface of genus \(g \ge 1\) with \(b\)
@@ -476,7 +475,7 @@ Unfortunately, the limited scope of these master thesis does not allow us to
 dive into the proof of Theorem~\ref{thm:reps-of-dim-2g-are-symplectic}. The
 heart of this proof lies in a result about representations of the product
 \(B_3^n = B_3 \times \cdots \times B_3\), which Korkmaz refers to as \emph{the
-main lemma}. Namely\dots
+main lemma}.
 
 \begin{lemma}[Korkmaz' main lemma]\label{thm:main-lemma}
   Given \(i = 1, \ldots, n\), denote by

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -14,7 +14,8 @@ Observation~\ref{ex:torus-mcg}: by cutting across curves and arcs, we can always
 decompose a surface into copies of \(\mathbb{D}^2\) and \(\mathbb{D}^2
 \setminus \{0\}\). Observation~\ref{ex:alexander-trick} and
 Observation~\ref{ex:mdg-once-punctured-disk} then imply the triviality of
-mapping classes fixing such arcs and curves. Formally, this translates to\dots
+mapping classes fixing such arcs and curves. Formally, this translates to the
+following result.
 
 \begin{proposition}[Alexander method]\label{thm:alexander-method}
   Let \(\alpha_1, \ldots, \alpha_n \subset \Sigma\) be essential simple closed
@@ -216,7 +217,7 @@ too.
   \]
 \end{observation}
 
-A perhaps less obvious fact about Dehn twists is\dots
+A perhaps less obvious fact about Dehn twists is the following.
 
 \begin{theorem}\label{thm:mcg-is-fg}
   Let \(\Sigma_{g, r}^b\) be the orientable surface of genus \(g \ge 1\) with
@@ -271,7 +272,8 @@ fibration\footnote{See \cite[Chapter~4]{hatcher} for a reference.}
     & & \phi & \mapsto & [\phi(x_1), \ldots, \phi(x_n)]
   \end{array}
 \]
-and its long exact sequence in homotopy we then get\dots
+and its long exact sequence in homotopy we then obtain the following
+fundamental result.
 
 \begin{theorem}[Birman exact sequence]\label{thm:birman-exact-seq}
   Suppose \(\pi_1(\Homeo^+(\Sigma, \partial \Sigma), 1) = 1\). Then there is an
@@ -310,7 +312,8 @@ x_{\sigma(i)}\) for some \(\sigma \in S_n\). The element
 mapping class that ``\emph{pushes} a neighborhood of \(x_{\sigma(i)}\) towards
 \(x_i\) along the curve \(\alpha_i^{-1}\),'' as shown in
 Figure~\ref{fig:push-map} for the case \(n = 1\). Indeed, this goes to
-show\dots
+show \(\operatorname{push}([\alpha])\) can be descrived as a product of Dehn
+twists.
 
 \begin{fundamental-observation}\label{ex:push-simple-loop}
   Using the notation of Figure~\ref{fig:push-map},
@@ -346,7 +349,7 @@ to apply the following lemma from geometric group theory.
 \end{lemma}
 
 We are interested, of course, in the group \(G = \PMod(\Sigma_{g, r}^b)\). As
-for the graph \(\Gamma\), we consider\dots
+for the the role of \(\Gamma\), we consider the following graph.
 
 \begin{definition}
   The \emph{modified graph of nonseparating curves
@@ -397,7 +400,7 @@ In other words, given simple closed curves \(\alpha, \beta \subset \Sigma_{g,
 r}\), we can find closed \(\alpha = \alpha_1, \alpha_2, \ldots, \alpha_n =
 \beta\) in \(\Sigma_{g, r}\) with \(\alpha_i\) disjoint from \(\alpha_{i+1}\).
 Now if \(\alpha\) and \(\beta\) are nonseparating, by inductively adjusting
-this sequence of curves we then get\dots
+this sequence of curves we obtain the following corollary.
 
 \begin{corollary}\label{thm:mofied-graph-is-connected}
   If \(g \ge 2\) then both \(\mathcal{N}(\Sigma_{g, r})\) and
@@ -436,7 +439,7 @@ Theorem~\ref{thm:mcg-is-fg}.
 
   \begin{figure}[ht]
     \centering
-    \includegraphics[width=.55\linewidth]{images/torus-mcg-generators.eps}
+    \includegraphics[width=.5\linewidth]{images/torus-mcg-generators.eps}
     \caption{The curves $\alpha$ and $\beta$ whose Dehn twists generate
     $\Mod(\mathbb{T}^2)$ and $\Mod(\Sigma_{1, 1})$.}
     \label{fig:torus-mcg-generators}