memoire-m2

My M2 Memoire on mapping class groups & their representations

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Commit: 7d921b6ee8881a3a4376f6c8daf6e53b007fbc2d
Parent: 6078bdea5d725c15f9ebbc63fdad2d5f3fb3dba3
Author: Pablo <pablo-pie@riseup.net>
Date: Thu, 27 Jun 2024 14:53:39 +0000

Corrected grammar throughout

"cut across" → "cut along"

"so called" → "so-called"

Fixed "in" vs. "on" issues

Diffstat

5 files changed, 56 insertions, 57 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	memoire.tex	16	8	8
Modified	sections/introduction.tex	28	14	14
Modified	sections/presentation.tex	33	17	16
Modified	sections/representations.tex	7	3	4
Modified	sections/twists.tex	29	14	15

diff --git a/memoire.tex b/memoire.tex
@@ -4,7 +4,7 @@
 
 \title{Mapping Class Groups \\ \& \\ their Representations}
 \author{Thiago Brevidelli Garcia}
-\dedication{This master thesis is dedicated to my dear friend Lucas Schiezari. \\ May he rest in peace.}
+\dedication{This Master's Thesis is dedicated to my dear friend Lucas Schiezari. \\ May he rest in peace.}
 
 \begin{document}
 
@@ -12,25 +12,25 @@
 \maketitle
 
 \begin{about}
-  This is my M2 Mémoire (Master Thesis), written in June 2024 under the
-  supervision of professor Maxime Wolff of the Institut of Mathématiques de
+  This is my M2 Mémoire (Master's Thesis), written in June 2024 under the
+  supervision of Professor Maxime Wolff of the Institut of Mathématiques de
   Toulouse (IMT), France. The subject of the dissertation at hand is the notion
   of the mapping class group of a surface, their linear representations and
   some recent developments in the field. Namely, we discuss Korkmaz' proof of
   the triviality of low-dimensional representations.
 
   Throughout these notes we will follow some guiding principles. First, lengthy
-  proofs are favored as opposed to collections of smaller lemmas. This is a
-  deliberate effort to emphasize the relevant results. That said, this
-  dissertation is meant to be concise. Hence numerous results are left
-  unproved. We refer the reader to proofs in other materials when appropriate.
+  proofs are favored over collections of smaller lemmas. This is a deliberate
+  effort to emphasize the relevant results. That said, this dissertation is
+  intended to be concise. Hence numerous results are left unproved. We refer
+  the reader to proofs in other materials when appropriate.
 
   Our main references are the beautiful book by Farb-Margalit
   \cite{farb-margalit}, as well as the 2023 article by Korkmaz \cite{korkmaz}.
   We assume the reader is already familiar with basic topology and group
   theory. Crucially, we \emph{do not} assume any familiarity with
   representation theory. Because of tight deadlines, we have opted to
-  hand-draw all of the figures.
+  hand-draw all the figures.
 
   \section*{Acknowledgments}

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -43,7 +43,7 @@ homeomorphisms.
   orientation-preserving homeomorphism \(\phi : \Sigma \isoto \Sigma\) fixing
   \(\partial \Sigma\) pointwise such that \(\phi(\alpha) = \beta\) with
   orientation. To see this, we consider the surface \(\Sigma_\alpha\) obtained
-  by cutting \(\Sigma\) across \(\alpha\): we subtract the curve \(\alpha\)
+  by cutting \(\Sigma\) along \(\alpha\): we subtract the curve \(\alpha\)
   from \(\Sigma\) and then add one additional boundary component \(\delta_i\)
   in each side of \(\alpha\), as shown in
   Figure~\ref{fig:change-of-coordinates}. By identifying \(\delta_1\) with
@@ -57,7 +57,7 @@ homeomorphisms.
   find an orientation-preserving homeomorphism \(\tilde\phi : \Sigma_\alpha
   \isoto \Sigma_\beta\). Even more so, we can choose \(\tilde\phi\) taking
   \(\delta_i\) to \(\delta_i'\). The homeomorphism \(\tilde\phi\) then descends
-  to a self-homeomorphism \(\phi\) the quotient surface \(\Sigma \cong
+  to a self-homeomorphism \(\phi\) of the quotient surface \(\Sigma \cong
   \mfrac{\Sigma_\alpha}{\sim} \cong \mfrac{\Sigma_\beta}{\sim}\) with
   \(\phi(\alpha) = \beta\), as desired.
 \end{observation}
@@ -70,8 +70,8 @@ homeomorphisms.
   \label{fig:change-of-coordinates}
 \end{figure}
 
-By splitting \(\Sigma\) across curves \(\alpha, \alpha' \subset \Sigma\)
-crossing once, we can also show the following result.
+By cutting \(\Sigma\) along curves \(\alpha, \alpha' \subset \Sigma\) 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:
+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
@@ -279,7 +279,7 @@ Here we collect a few fundamental examples of linear representations of
 \end{minipage}
 
 The symplectic representation already allows us to compute some important
-examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 =
+examples of mapping class groups: namely, that of the torus \(\mathbb{T}^2 =
 \Sigma_1\) and the once-punctured torus \(\Sigma_{1, 1}\).
 
 \begin{observation}[Alexander trick]\label{ex:alexander-trick}
@@ -321,8 +321,8 @@ examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 =
   may choose a representative \(\phi\) of \(f\) fixing \(\alpha_1 \cup
   \beta_1\) pointwise. Such \(\phi\) determines a homeomorphism \(\tilde \phi\)
   of the surface \(\mathbb{T}_{\alpha_1 \beta_1}^2 \cong \mathbb{D}^2\)
-  obtained by cutting \(\mathbb{T}^2\) across \(\alpha_1\) and \(\beta_1\), as
-  in Figure~\ref{fig:cut-torus-across}. Now by the Alexander trick from
+  obtained by cutting \(\mathbb{T}^2\) along \(\alpha_1\) and \(\beta_1\), as
+  in Figure~\ref{fig:cut-torus-along}. Now by the Alexander trick from
   Observation~\ref{ex:alexander-trick}, \(\tilde\phi\) must be isotopic to the
   identity. The isotopy \(\tilde\phi \simeq 1 \in \Homeo^+(\mathbb{D}^2,
   \mathbb{S}^1)\) then descends to an isotopy \(\phi \simeq 1 \in
@@ -332,10 +332,10 @@ examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 =
 \begin{figure}[ht]
   \centering
   \includegraphics[width=.55\linewidth]{images/torus-cut.eps}
-  \caption{By cutting $\mathbb{T}^2$ across $\alpha_1$ we obtain a cylinder,
+  \caption{By cutting $\mathbb{T}^2$ along $\alpha_1$ we obtain a cylinder,
   where $\beta_1$ determines a yellow arc joining the two boundary components.
-  Now by cutting across this yellow arc we obtain a disk.}
-  \label{fig:cut-torus-across}
+  Now by cutting along this yellow arc we obtain a disk.}
+  \label{fig:cut-torus-along}
 \end{figure}
 
 \begin{observation}\label{ex:punctured-torus-mcg}
@@ -357,7 +357,7 @@ examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 =
   exists.
 \end{remark}
 
-Another fundamental class of examples of representations are the so called
+Another fundamental class of examples of representations are the so-called
 \emph{TQFT representations}.
 
 \begin{definition}
@@ -408,7 +408,7 @@ Another fundamental class of examples of representations are the so called
 \end{definition}
 
 \begin{observation}
-  Given \(\phi \in \Homeo^+(\Sigma_g)\), we may consider the so called
+  Given \(\phi \in \Homeo^+(\Sigma_g)\), we may consider the so-called
   \emph{mapping cylinder} \(C_\phi = (\Sigma_g \times [0, 1], \phi, 1)\), a
   cobordism between \(\Sigma_g\) and itself -- where \(\partial_+ (\Sigma_g
   \times [0, 1]) = \Sigma_g \times 0\) and \(\partial_- (\Sigma_g \times [0,
@@ -443,7 +443,7 @@ Hence constructing an actual functor typically requires \emph{extending}
 linear and projective representations of the \emph{extended mapping class
 groups} \(\Mod(\Sigma_g) \times \mathbb{Z}\). We refer the reader to
 \cite{costantino, julien} for constructions of one such extended TQFT and its
-corresponding representations: the so called \emph{\(\operatorname{SU}_2\) TQFT
+corresponding representations: the so-called \emph{\(\operatorname{SU}_2\) TQFT
 of level \(r\)}, first introduced by Witten and Reshetikhin-Tuarev
 \cite{witten, reshetikhin-turaev} in their foundational papers on quantum
 topology.

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -1,7 +1,7 @@
 \chapter{Relations Between Twists}\label{ch:relations}
 
 Having found a convenient set of generators for \(\Mod(\Sigma_g)\), it is now
-natural to ask what are the relations between such generators. In this chapter,
+natural to ask what the relations between such generators are. In this chapter,
 we highlight some additional relations between Dehn twists and the geometric
 intuition behind them, culminating in the statement of a presentation for
 \(\Mod(\Sigma_g)\) whose relations can be entirely explained in terms of the
@@ -112,7 +112,7 @@ x_r\})\). This is what is known as \emph{the Birman-Hilden theorem}.
 \section{The Birman-Hilden Theorem}\label{birman-hilden}
 
 Let \(\Sigma_{0, r}^1 = \mathbb{D}^2 \setminus \{x_1, \ldots, x_r\}\) be the
-surface of genus \(0\) with \(r\) punctures and \(1\) boundary component. We
+surface of genus \(0\) with \(r\) punctures and one boundary component. We
 begin our investigation by providing an alternative description of its mapping
 class group. Namely, we show that \(\Mod(\Sigma_{0, r}^1)\) is the braid group
 on \(r\) strands.
@@ -137,7 +137,7 @@ on \(r\) strands.
   \label{fig:braid-group-generator}
 \end{figure}
 
-The third Reidemeister move translates to the so called \emph{braid
+The third Reidemeister move translates to the so-called \emph{braid
 relations}
 \[
   \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_1 \sigma_i
@@ -189,7 +189,7 @@ Observation~\ref{ex:alexander-trick}. We thus obtain the following result.
   the Alexander method, one can check that the center \(Z(\Mod(\Sigma_{0,
   n}^1))\) of \(\Mod(\Sigma_{0, n}^1)\) is freely generated by the Dehn twist
   \(\tau_\delta\) about the boundary \(\delta = \partial \Sigma_{0, n}^1\). It
-  is also not very hard to see that \(\operatorname{push} : B_n \to
+  is also not very difficult to see that \(\operatorname{push} : B_n \to
   \Mod(\Sigma_{0, n}^1)\) takes \(\sigma_1 \cdots \sigma_{n-1}\) to the
   rotation by \(\sfrac{2\pi}{n}\) as in Figure~\ref{fig:braid-group-center},
   which is an \(n\)-th root of \(\tau_\delta\). Hence the center \(Z(B_n)\) is
@@ -262,9 +262,9 @@ symmetric mapping classes}.
 \end{definition}
 
 Fix \(b = 1\) or \(2\). It follows from the universal property of quotients
-that any \(\phi \in \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) defines a
-homeomorphism \(\bar \phi : \Sigma_{0, 2\ell+b}^1 \isoto \Sigma_{0, 2\ell+b}^1\). This
-yields a homomorphism of topological groups
+that any \(\phi \in \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) defines
+a homeomorphism \(\bar \phi : \Sigma_{0, 2\ell+b}^1 \isoto \Sigma_{0,
+2\ell+b}^1\). This yields a homomorphism of topological groups
 \begin{align*}
   \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)
   & \to \Homeo^+(\Sigma_{0, 2\ell + b}^1, \partial \Sigma_{0, 2\ell + b}^1) \\
@@ -274,12 +274,13 @@ yields a homomorphism of topological groups
 which is surjective because any \(\psi \in \Homeo^+(\Sigma_{0, 2\ell + b}^1,
 \partial \Sigma_{0, 2\ell + b}^1)\) lifts to \(\Sigma_\ell^b\).
 
-It is also not hard to see \(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b) \to
-\Homeo^+(\Sigma_{0, 2\ell + b}^1, \partial \Sigma_{0, 2\ell + b}^1)\) is injective: the
-only candidates for elements of its kernel are \(1\) and \(\iota\), but
-\(\iota\) is not an element of \(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) since
-it does not fix \(\partial \Sigma_\ell^b\) point-wise. Now since we have a
-continuous bijective homomorphism we find
+It is also not difficult to see \(\SHomeo^+(\Sigma_\ell^b, \partial
+\Sigma_\ell^b) \to \Homeo^+(\Sigma_{0, 2\ell + b}^1, \partial \Sigma_{0, 2\ell
++ b}^1)\) is injective: the only candidates for elements of its kernel are
+\(1\) and \(\iota\), but \(\iota\) is not an element of
+\(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) since it does not fix
+\(\partial \Sigma_\ell^b\) point-wise. Now since we have a continuous bijective
+homomorphism we find
 \[
   \begin{split}
     \pi_0(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b))
@@ -324,7 +325,7 @@ not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden
   \Mod(\Sigma_{0, 2\ell+2})\) takes \(\tau_{\delta_1} \tau_{\delta_2} \in
   \SMod(\Sigma_\ell^2)\) to \(\tau_{\bar\delta_1} = \tau_{\bar\delta_2}\). In
   light of Observation~\ref{ex:push-generators-description},
-  Observation~\ref{ex:braid-group-center} translates into the so called
+  Observation~\ref{ex:braid-group-center} translates into the so-called
   \emph{\(k\)-chain relations} in \(\SMod(\Sigma_\ell^b) \subset
   \Mod(\Sigma_g)\).
   \[
@@ -387,8 +388,8 @@ to obtain other relations. Since \(\iota\) has \(2g + 2\) fixed points in
   \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta\) inverts the
   orientation of \(\alpha_1\), so \(\tau_\delta \tau_{\alpha_{2g}} \cdots
   \tau_{\alpha_1} \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta \ne 1\)
-  and (\ref{eq:hyperelliptic-eq}) follows. In particular, we obtain the so
-  called \emph{hyperelliptic relations} (\ref{eq:hyperelliptic-rel-1}) and
+  and (\ref{eq:hyperelliptic-eq}) follows. In particular, we obtain the
+  so-called \emph{hyperelliptic relations} (\ref{eq:hyperelliptic-rel-1}) and
   (\ref{eq:hyperelliptic-rel-2}) in \(\Mod(\Sigma_g)\).
   \begin{align}\label{eq:hyperelliptic-rel-1}
     (\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1}

diff --git a/sections/representations.tex b/sections/representations.tex
@@ -263,11 +263,10 @@ We are now ready to establish the triviality of low-dimensional
 representations.
 
 \begin{proof}[Proof of Theorem~\ref{thm:low-dim-reps-are-trivial}]
-  Let \(g \ge 1\), \(b \ge 0\) and fix \(\rho : \Mod(\Sigma_g^b) \to
+  Let \(g \ge 1\), \(b \ge 0\), and fix \(\rho : \Mod(\Sigma_g^b) \to
   \GL_n(\mathbb{C})\) with \(n < 2g\). We want to show
   \(\rho(\Mod(\Sigma_g^b))\) is Abelian. As promised, we proceed by induction
-  on \(g\).
-  The base case \(g = 1\) is again clear from the fact \(n = 1\) and
+  on \(g\). The base case \(g = 1\) is again clear from the fact \(n = 1\) and
   \(\GL_1(\mathbb{C}) = \mathbb{C}^\times\). The case \(g = 2\) was also
   established in Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}.
 
@@ -319,7 +318,7 @@ representations.
   identity matrix. Since the group of upper triangular matrices is solvable, it
   follows from Proposition~\ref{thm:commutator-is-perfect} that \(\rho\)
   annihilates all of \(\Mod(\Sigma)'\) and, in particular, \(\tau_{\alpha_1}
-  \tau_{\beta_1}^{-1} \in \ker \rho\). But recall from
+  \tau_{\beta_1}^{-1} \in \ker \rho\). Now recall from
   Proposition~\ref{thm:commutator-normal-gen} that \(\Mod(\Sigma_g^b)'\) is
   normally generated by \(\tau_{\alpha_1} \tau_{\beta_1}^{-1}\), from which we
   conclude \(\rho(\Mod(\Sigma_g^b)') = 1\), as desired.

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -1,16 +1,15 @@
 \chapter{Dehn Twists}\label{ch:dehn-twists}
 
 With the goal of studying the linear representations of mapping class groups in
-mind, in this chapter we start investigating the group structure of
-\(\Mod(\Sigma)\). We begin by computing some fundamental examples. We then
-explore how we can use these examples to understand the structure of the
-mapping class groups of other surfaces. Namely, we compute \(\Mod(\mathbb{S}^1
-\times [0, 1]) \cong \mathbb{Z}\), and discuss how its generator gives rise to
-a convenient generating set for \(\Mod(\Sigma)\), known as the set of
-\emph{Dehn twists}.
+mind, we now start investigating the group structure of \(\Mod(\Sigma)\). We
+begin by computing some fundamental examples and then explore how we can use
+these examples to understand the structure of the mapping class groups of other
+surfaces. Namely, we compute \(\Mod(\mathbb{S}^1 \times [0, 1]) \cong
+\mathbb{Z}\), and discuss how its generator gives rise to a convenient
+generating set for \(\Mod(\Sigma)\), known as the set of \emph{Dehn twists}.
 
 The idea here is to reproduce the proof of injectivity in
-Observation~\ref{ex:torus-mcg}: by cutting across curves and arcs, we can always
+Observation~\ref{ex:torus-mcg}: by cutting along 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
@@ -25,7 +24,7 @@ following result.
     \item Each pair \((\alpha_i, \alpha_j)\) crosses at most once.
     \item Given distinct \(i, j, k\), at least one of \(\alpha_i \cap \alpha_j,
       \alpha_i \cap \alpha_k, \alpha_j \cap \alpha_k\) is empty.
-    \item The surface obtained by cutting \(\Sigma\) across the \(\alpha_i\) is a
+    \item The surface obtained by cutting \(\Sigma\) along the \(\alpha_i\) is a
       disjoint union of disks and once-punctured disks.
   \end{enumerate}
   Suppose \(f \in \Mod(\Sigma)\) is such that \(f \cdot \vec{[\alpha_i]} =
@@ -66,7 +65,7 @@ applications of the Alexander method.
   \centering
   \includegraphics[width=.7\linewidth]{images/dehn-twist-cylinder.eps}
   \captionof{figure}{The generator $f$ of $\Mod(\mathbb{S}^1 \times [0, 1])
-  \cong \mathbb{Z}$ takes the yellow arc in the left-hand side to the arc on
+  \cong \mathbb{Z}$ takes the yellow arc on the left-hand side to the arc on
   the right-hand side that winds about the curve $\alpha$.}
   \label{fig:dehn-twist-cylinder}
 \end{minipage}
@@ -101,14 +100,14 @@ illustrated in Figure~\ref{fig:dehn-twist-bitorus} in the case of the bitorus
   \centering
   \includegraphics[width=.6\linewidth]{images/dehn-twist-bitorus.eps}
   \caption{The Dehn twist about the curve $\alpha$ takes the peanut-shaped curve
-  on the left-hand side to the yellow curve in the right-hand side.}
+  on the left-hand side to the yellow curve on the right-hand side.}
   \label{fig:dehn-twist-bitorus}
 \end{figure}
 
 Similarly, using the description of the mapping class group of the
 twice-puncture disk derived in Example~\ref{ex:mcg-twice-punctured-disk}, the
 generator of \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\})\)
-gives rise the so called \emph{half-twists}. These are examples of mapping
+gives rise the so-called \emph{half-twists}. These are examples of mapping
 classes that permute the punctures of \(\Sigma\).
 
 \begin{definition}
@@ -328,8 +327,8 @@ twists.
   \Mod(\Sigma)$ maps a simple loop $\alpha : \mathbb{S}^1 \to \Sigma$ to the
   mapping class supported at an annular neighborhood $A$ of $\alpha$. Inside
   this neighborhood, $\operatorname{push}([\alpha])$ takes the arc joining the
-  boundary components $\delta_i \subset \partial A$ in the left-hand side to
-  the yellow arc in the right-hand side.}
+  boundary components $\delta_i \subset \partial A$ on the left-hand side to
+  the yellow arc on the right-hand side.}
   \label{fig:push-map}
 \end{figure}
 
@@ -569,7 +568,7 @@ generators}.
 
 In the boundaryless case \(b = 0\), we can write \(\tau_{\alpha_3}, \ldots,
 \tau_{\alpha_g} \in \Mod(\Sigma_g)\) as products of the twists about the
-remaining curves, from which we get the so called \emph{Humphreys generators}.
+remaining curves, from which we get the so-called \emph{Humphreys generators}.
 
 \begin{corollary}[Humphreys generators]\label{thm:humphreys-gens}
   If \(g \ge 2\) then \(\Mod(\Sigma_g)\) is generated by the Dehn twists about the