memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: 6e770a9a47d97ebc97b35a86e9547953558876ba
Parent: b14978121a73375c6426854aa0e772cf3af1344a
Author: Pablo <pablo-pie@riseup.net>
Date: Wed, 26 Jun 2024 13:12:55 +0000

Restructured the introduction

Fixed the historical comments on the classification of surfaces to account for Gauss' fundamental contributions

Fixed some important typos

Structured the motivation for mapping class groups, focusing on the example of mapping tori instead of in the random-ass questions about homeomorphism groups

Diffstat

4 files changed, 217 insertions, 280 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	images/change-of-coords-cut.svg	258	116	142
Modified	preamble.tex	2	1	1
Modified	sections/introduction.tex	234	98	136
Modified	sections/twists.tex	3	2	1

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diff --git a/preamble.tex b/preamble.tex
@@ -78,7 +78,7 @@
 \newcommand{\rightaction}{\,\rotatebox[origin=c]{90}{$\circlearrowright$}\,}
 
 % Quotient object
-\newcommand{\mfrac}[2]{\mathlarger{\mathlarger{\mathlarger{\sfrac{#1}{#2}}}}}
+\newcommand{\mfrac}[2]{\mathlarger{\mathlarger{\sfrac{#1}{#2}}}}
 
 % A normal subobject in a pointed cathegory
 \newcommand{\normal}{\triangleleft}

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -1,16 +1,12 @@
 \chapter{Introduction}\label{ch:introduction}
 
-Ever since Mankind first stepped foot on the surface of Earth, humanity has
-been asking what is the shape of the planet we inhabit. More recently,
-mathematicians have spent the past centuries trying to understand the topology
-of manifolds and, in particular, surfaces. Orientable compact surfaces were
-first classified in the 1920s by Radò\footnote{The classification of closed
-orientable surfaces admitting a triangulation was already known in the
-mid-nineteenth century and is often attributed to Möbius, but at that time it
-was not yet known that all closed surfaces admit a triangulation. Radò
-\cite{rado} would go on to establish this fact in 1925.}, Kerékjártó and others
-\cite{rado, kerekjarto}. We refer the reader to \cite{thomassen} for a complete
-proof.
+Ever since ancestral humans first stepped foot on the surface of Earth, Mankind
+has pondered the shape of the planet we inhabit. More recently, mathematicians
+have spent the past centuries trying to understand the topology of manifolds
+and, in particular, surfaces. Orientable compact surfaces were perhaps first
+classified by Gauss in the early 19th century. The proof of the following
+formulation of the classification, often attributed to Möbius, was completed in
+the 1920s with the work of Radò and others.
 
 \begin{theorem}[Classification of surfaces]\label{thm:classification-of-surfaces}
   Any closed connected orientable surface is homeomorphic to the connected sum
@@ -30,107 +26,73 @@ notes, all surfaces considered will be of the form \(\Sigma = \Sigma_{g,
 r}^b\). Any such \(\Sigma\) admits a natural compactification
 \(\widebar\Sigma\) obtained by filling its punctures. We denote \(\Sigma_{g, r}
 = \Sigma_{g, r}^0\). All closed curves \(\alpha, \beta \subset \Sigma\) we
-consider lie in the interior \(\Sigma\degree\) of \(\Sigma\) and intersect
-transversely.
+consider lie in the interior of \(\Sigma\) and intersect transversely.
 
-It is interesting to remark that, aside from the homeomorphism type of a
-surface \(\Sigma\), Theorem~\ref{thm:classification-of-surfaces} also informs
-the geometry of the curves in \(\Sigma\) and their intersections. For
-example\dots
+Despite the apparent clarity of the picture painted by
+Theorem~\ref{thm:classification-of-surfaces}, there are still plenty of
+interesting, sometimes unanswered, questions about surfaces and their
+homeomorphisms. For instance, it is interesting to consider how the
+classification of surfaces informs the geometry of the curves in \(\Sigma\).
 
-\begin{lemma}[Change of coordinates principle]\label{thm:change-of-coordinates}
+\begin{observation}[Change of coordinates principle]
   Given oriented nonseparating simple closed curves \(\alpha, \beta \subset
-  \Sigma\), we can find an orientation-preserving homeomorphism \(\phi : \Sigma
-  \isoto \Sigma\) fixing \(\partial \Sigma\) pointwise such that \(\phi(\alpha)
-  = \beta\) with orientation. Even more so, if \(\alpha', \beta' \subset
-  \Sigma\) are nonseparating curve such that each pair \((\alpha, \alpha'),
-  (\beta, \beta')\) crosses only once, then we can choose \(\phi\) with
-  \(\phi(\alpha) = \beta\) and \(\phi(\alpha') = \beta'\).
-\end{lemma}
-
-\begin{proof}
-  Let \(\Sigma = \Sigma_{g, r}^b\) and consider the surface \(\Sigma_\alpha\)
-  obtained by cutting \(\Sigma\) across \(\alpha\), as in
-  Figure~\ref{fig:change-of-coordinates}. Since \(\alpha\) is nonseparating,
-  this surface has genus \(g - 1\) and two additional boundary component
-  \(\delta_1, \delta_2 \subset \partial \Sigma_\alpha\), so \(\Sigma_\alpha
-  \cong \Sigma_{g-1,r}^{b+2}\). By identifying \(\delta_1\) and \(\delta_2\)
-  we can see \(\Sigma\) as a quotient of \(\Sigma_\alpha\). Similarly,
-  \(\Sigma_\beta \cong \Sigma_{g-1, r}^{b+2}\) also has two additional boundary
-  components \(\delta_1', \delta_2' \subset \partial \Sigma_\beta\). Now by the
+  \Sigma = \Sigma_{g, r}^b\), we can find an 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 \(\alpha\) 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 \(\delta_2\) we can see \(\Sigma\) as a
+  quotient of \(\Sigma_\alpha\). Since \(\alpha\) is nonseparating,
+  \(\Sigma_\alpha\) is a connected surface of genus \(g - 1\). In other words,
+  \(\Sigma_\alpha \cong \Sigma_{g-1,r}^{b+2}\). Similarly, \(\Sigma_\beta \cong
+  \Sigma_{g-1, r}^{b+2}\) also has two additional boundary components
+  \(\delta_1', \delta_2' \subset \partial \Sigma_\beta\). Now by the
   classification of surfaces we can 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 \(\Sigma \cong \mfrac{\Sigma_\alpha}{\sim} \cong
-  \mfrac{\Sigma_\beta}{\sim}\) with \(\phi(\alpha) = \beta\).
-
-  As for the second part of the lemma, we consider the surface \(\Sigma_{\alpha
-  \alpha'}\) obtained by cutting \(\Sigma_\alpha\) across the arc determined by
-  \(\alpha'\). Since \(\alpha'\) is nonseparating, \(\Sigma_{\alpha \alpha'}
-  \cong \Sigma_{g-1, r}^{b+1}\) has one boundary component more than
-  \(\Sigma\), say \(\partial \Sigma_{\alpha \alpha'} = \delta \amalg \partial
-  \Sigma\). The boundary component \(\delta\) is naturally subdivided into the
-  four arcs in Figure~\ref{fig:change-of-coordinates}, each corresponding to
-  one of the curves \(\alpha\) and \(\alpha'\) in \(\Sigma\). By identifying
-  the pairs of arcs corresponding to the same curve we obtain the surface
-  \(\mfrac{\Sigma_{\alpha \alpha'}}{\sim} \cong \Sigma\).
-
-  Likewise, \(\Sigma_{\beta \beta'} \cong \Sigma_{g-1, r}^{b+1}\) also has a
-  boundary component \(\delta' \subset \partial \Sigma_{\beta \beta'}\)
-  subdivided into four arcs. By the classification of surfaces we can find an
-  orientation-preserving homeomorphism \(\tilde\phi : \Sigma_{\alpha \alpha'}
-  \isoto \Sigma_{\beta \beta'}\) taking each one of the arcs in \(\delta\) to
-  the corresponding arc in \(\delta'\). Hence \(\tilde\phi\) descends to a
-  self-homeomorphism \(\phi\) of the quotient \(\Sigma
-  \cong\mfrac{\Sigma_{\alpha \alpha'}}{\sim} \cong \mfrac{\Sigma_{\beta
-  \beta'}}{\sim}\). Finally, since \(\tilde\phi\) takes the arcs corresponding
-  to \(\alpha\) to the arcs corresponding to \(\beta\) and the arcs
-  corresponding to \(\alpha'\) to the arcs corresponding to \(\beta'\),
-  \(\phi(\alpha) = \alpha'\) and \(\phi(\beta) = \beta'\), as desired.
-\end{proof}
+  \mfrac{\Sigma_\beta}{\sim}\) with \(\phi(\alpha) = \beta\), as desired.
+\end{observation}
 
 \begin{figure}[ht]
   \centering
-  \includegraphics[width=.8\linewidth]{images/change-of-coords-cut.eps}
-  \caption{By cutting $\Sigma_{g, r}^b$ across $\alpha$ we obtain $\Sigma_{g-1,
-  r}^{b+2}$, where $\alpha'$ determines a yellow arc joining the two
-  additional boundary components. Now by cutting $\Sigma_{g-1, r}^{b+2}$ across
-  this arc we obtain $\Sigma_{g-1,r}^b$, with the added boundary component
-  subdivided into the four arcs corresponding to $\alpha$ and $\alpha'$.}
+  \includegraphics[width=.5\linewidth]{images/change-of-coords-cut.eps}
+  \caption{The surface $\Sigma_\alpha \cong \Sigma_{g-1, r}^{b+2}$ for a
+  certain $\alpha \subset \Sigma$.}
   \label{fig:change-of-coordinates}
 \end{figure}
 
-More generally, despite the apparent clarity of the picture painted by
-Theorem~\ref{thm:classification-of-surfaces}, there are still plenty of
-unanswered questions about surfaces and their homeomorphisms. Given a surface
-\(\Sigma\), the group \(\Homeo^+(\Sigma, \partial \Sigma)\) of
-orientation-preserving homeomorphism of \(\Sigma\) fixing its boundary
-pointwise is a topological group\footnote{Here we endow \(\Homeo^+(\Sigma,
-\partial \Sigma)\) with the compact-open topology.} with a rich geometry. It is
-not hard to come up with interesting questions about such group. For example,
-\begin{enumerate}
-  \item Given closed curves \(\alpha, \beta \subset \Sigma\), can we find
-    \(\phi \in \Homeo^+(\Sigma, \partial \Sigma)\) with \(\phi(\alpha) =
-    \beta\)?
-
-  \item What are the conjugacy classes of \(\Homeo^+(\Sigma, \partial
-    \Sigma)\)? What about its connected components?
-
-  \item Does \(\Homeo^+(\Sigma, \partial \Sigma)\) determine \(\Sigma\)? If the
-    answer is \emph{no}, what about in the closed case?
-\end{enumerate}
-
-Unfortunately, however, the algebraic structure of \(\Homeo^+(\Sigma, \partial
-\Sigma)\) is typically 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 instance, isotopic homeomorphisms
-\(\phi \simeq \psi : \Sigma \isoto \Sigma\) determine the same automorphism
-\(\phi_* = \psi_*\) at the levels of homotopy and homology. This leads us to
-consider the group of connected components of \(\Homeo^+(\Sigma, \partial
-\Sigma)\), also known as \emph{the mapping class group}. This will be the focus
-of the dissertation at hand.
+A very similar argument goes to show\dots
+
+\begin{observation}\label{ex:change-of-coordinates-crossing}
+  Let \(\alpha, \beta, \alpha', \beta' \subset \Sigma\) be nonseparating curve
+  such that each pair \((\alpha, \alpha'), (\beta, \beta')\) crosses exactly
+  once. Then we can find an orientation-preserving \(\phi : \Sigma \isoto
+  \Sigma\) fixing \(\partial \Sigma\) poitwise such that \(\phi(\alpha) =
+  \beta\) and \(\phi(\alpha') = \beta'\).
+\end{observation}
+
+Given a surface \(\Sigma\), the group \(\Homeo^+(\Sigma, \partial \Sigma)\) of
+orientation-preserving homeomorphism of \(\Sigma\) fixing each point in
+\(\partial \Sigma\) is a topological group\footnote{Here we endow
+\(\Homeo^+(\Sigma, \partial \Sigma)\) with the compact-open topology.} with a
+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, given \(\phi \in \Homeo^+(\Sigma, \partial \Sigma)\), it is well
+known that the diffeomorphism class of the mapping torus \(M_\phi =
+\mfrac{\Sigma \times [0, 1]}{(x, 0) \sim (\phi(x), 1)}\) -- a fundamental
+construction in low-dimensional topology -- is invariant under isotopy. This
+fact underspins some of the steps in Thurston's geometrization of
+\(3\)-manifolds. It is thus more natural to consider the group of connected
+components of \(\Homeo^+(\Sigma, \partial \Sigma)\), a countable discrete group
+known as \emph{the mapping class group}. This will be the focus of the
+dissertation at hand.
 
 \begin{definition}\label{def:mcg}
   The \emph{mapping class group \(\Mod(\Sigma)\) of an orientable surface
@@ -142,11 +104,11 @@ of the dissertation at hand.
   \]
 \end{definition}
 
-There are many variations of the Definition~\ref{def:mcg}. For example\dots
+There are many variations of Definition~\ref{def:mcg}. For example\dots
 
 \begin{example}\label{ex:action-on-punctures}
   Any \(\phi \in \Homeo^+(\Sigma, \partial \Sigma)\) extends uniquely to a
-  homomorphism \(\tilde\phi\) of \(\widebar\Sigma\) that permutes the set
+  homeomorphism \(\tilde\phi\) of \(\widebar\Sigma\) that permutes the set
   \(\{x_1, \ldots, x_r\} = \widebar\Sigma \setminus \Sigma\) of punctures of
   \(\Sigma\). We may thus define an action \(\Mod(\Sigma) \leftaction \{x_1,
   \ldots, x_r\}\) via \(f \cdot x_i = \tilde\phi(x_i)\) for \(f = [\phi] \in
@@ -181,17 +143,16 @@ There are many variations of the Definition~\ref{def:mcg}. For example\dots
 
 \begin{definition}
   Given a simple closed curve \(\alpha \subset \Sigma\), we denote by
-  \(\Mod(\Sigma)_{\vec{[\alpha]}} = \{ f \in \Mod(\Sigma) : f \cdot
-  \vec{[\alpha]} = \vec{[\alpha]} \}\) and \(\Mod(\Sigma)_{[\alpha]} = \{ f \in
-  \Mod(\Sigma) : f \cdot [\alpha] = [\alpha] \}\) the subgroups of mapping
-  classes that fix the isotopy classes of \(\alpha\).
+  \(\Mod(\Sigma)_{\vec{[\alpha]}}\) and \(\Mod(\Sigma)_{[\alpha]}\) the
+  subgroups of mapping classes that fix \(\vec{[\alpha]}\) and \([\alpha]\),
+  respectively.
 \end{definition}
 
-While trying to understand the mapping class group of \(\Sigma\), it is
-interesting to consider how the geometric relationship between \(\Sigma\) and
-other surfaces affects \(\Mod(\Sigma)\). Indeed, different embeddings \(\Sigma'
-\hookrightarrow \Sigma\) translate to homomorphisms at the level of mapping
-class groups.
+While trying to understand the mapping class group of some surface \(\Sigma\),
+it is interesting to consider how the geometric relationship between \(\Sigma\)
+and other surfaces affects \(\Mod(\Sigma)\). Indeed, different embeddings
+\(\Sigma' \hookrightarrow \Sigma\) translate to homomorphisms at the level of
+mapping class groups.
 
 \begin{example}[Inclusion homomorphism]\label{ex:inclusion-morphism}
   Let \(\Sigma' \subset \Sigma\) be a closed subsurface. Given \(\phi \in
@@ -207,7 +168,7 @@ class groups.
   known as \emph{the inclusion homomorphism}.
 \end{example}
 
-\begin{example}[Capping exact homomorphism]\label{ex:capping-morphism}
+\begin{example}[Capping homomorphism]\label{ex:capping-morphism}
   Let \(\delta \subset \partial \Sigma\) be an oriented boundary component of
   \(\Sigma\). We refer to the inclusion homomorphism \(\operatorname{cap} :
   \Mod(\Sigma) \to \Mod(\Sigma \cup_\delta (\mathbb{D}^2 \setminus \{0\}))\) as
@@ -229,13 +190,14 @@ class groups.
 \end{example}
 
 As goes for most groups, another approach to understanding the mapping class
-group of a given surface \(\Sigma\) is to study its actions. We have already seen
-simple example of such actions in Example~\ref{ex:action-on-punctures} and
-Example~\ref{ex:action-on-curves}. A particularly important class of actions
-of \(\Mod(\Sigma)\) are its \emph{linear representations} -- i.e. the group
-homomorphisms \(\Mod(\Sigma) \to \GL_n(\mathbb{C})\). These may be seen as actions
-\(\Mod(\Sigma) \leftaction \mathbb{C}^n\) where each \(f \in \Mod(\Sigma)\) acts via some
-\(\mathbb{C}\)-linear isomorphism \(\mathbb{C}^n \isoto \mathbb{C}^n\).
+group of a given surface \(\Sigma\) is to study its actions. We have already
+seen simple example of such actions in Example~\ref{ex:action-on-punctures} and
+Example~\ref{ex:action-on-curves}. A particularly important class of actions of
+\(\Mod(\Sigma)\) are its \emph{linear representations} -- i.e. the group
+homomorphisms \(\Mod(\Sigma) \to \GL_n(\mathbb{C})\). These may be seen as
+actions \(\Mod(\Sigma) \leftaction \mathbb{C}^n\) where each \(f \in
+\Mod(\Sigma)\) acts via some linear isomorphism \(\mathbb{C}^n \isoto
+\mathbb{C}^n\).
 
 \section{Representations}
 
@@ -270,7 +232,7 @@ Here we collect a few fundamental examples of linear representations of
   independent of the choice of representative \(\phi\) of \(f\). By the
   functoriality of homology groups we then get a \(\mathbb{Z}\)-linear action
   \(\Mod(\Sigma_g) \leftaction H_1(\Sigma_g, \mathbb{Z}) \cong
-  \mathbb{Z}^{2g}\), given by \(f \cdot [\alpha] = \phi_*([\alpha]) =
+  \mathbb{Z}^{2g}\) given by \(f \cdot [\alpha] = \phi_*([\alpha]) =
   [\phi(\alpha)]\). Since pushforwards by orientation-preserving homeomorphisms
   preserve the index of intersection points, \((f \cdot [\alpha]) \cdot (f
   \cdot [\beta]) = [\alpha] \cdot [\beta]\) for all \(\alpha, \beta \subset
@@ -306,12 +268,12 @@ examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 =
 
 \begin{example}[Alexander trick]\label{ex:alexander-trick}
   The group \(\Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) of homeomorphisms of the
-  unit disk \(\mathbb{D}^2 \subset \mathbb{Z}\) is contractible. In particular,
+  unit disk \(\mathbb{D}^2 \subset \mathbb{C}\) is contractible. In particular,
   \(\Mod(\mathbb{D}^2) = 1\). Indeed, for any \(\phi \in
   \Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) the isotopy
   \begin{align*}
     \phi_t : \mathbb{D}^2 & \to     \mathbb{D}^2 \\
-                   (z, t) & \mapsto
+                        z & \mapsto
     \begin{cases}
       (1 - t) \phi(\sfrac{z}{1 - t}) & \text{if } 0 \le |z| \le 1 - t \\
       z                              & \text{otherwise}
@@ -326,7 +288,7 @@ examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 =
   By the same token, \(\Mod(\mathbb{D}^2 \setminus \{0\}) = 1\).
 \end{example}
 
-\begin{example}\label{ex:torus-mcg}
+\begin{example}[$\Mod(\mathbb{T}^2)$]\label{ex:torus-mcg}
   The symplectic representation \(\psi : \Mod(\mathbb{T}^2) \to
   \operatorname{Sp}_2(\mathbb{Z}) = \operatorname{SL}_2(\mathbb{Z})\) is a
   group isomorphism. In particular, \(\Mod(\mathbb{T}^2) \cong
@@ -431,24 +393,24 @@ Another fundamental class of examples of representations are the so called
 
 \begin{observation}
   Given \(\phi \in \Homeo^+(\Sigma_g)\), we may consider the so called
-  \emph{mapping cylinder} \(M_\phi = (\Sigma_g \times [0, 1], \phi, 1)\), a
+  \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 \(\partial_- (\Sigma_g \times [0,
-  1]) = \Sigma_g \times 1\). The diffeomorphism class of \(M_\phi\) is
+  1]) = \Sigma_g \times 1\). The diffeomorphism class of \(C_\phi\) is
   independent of the choice of representative of \(f = [\phi] \in
-  \Mod(\Sigma_g)\), so \(M_f = [M_\phi] : \Sigma_g \to \Sigma_g\) is a well
+  \Mod(\Sigma_g)\), so \(C_f = [C_\phi] : \Sigma_g \to \Sigma_g\) is a well
   defined morphism in \(\Cob\).
 \end{observation}
 
 \begin{example}[TQFT representations]\label{ex:tqft-reps}
-  It is clear that \(M_1\) is the identity morphism \(\Sigma_g \to \Sigma_g\)
-  in \(\Cob\). In addition, \(M_{f \cdot g} = M_f \circ M_g\) in \(\Cob\) for
-  all \(f, g \in \Mod(\Sigma_g)\) -- see \cite[Lemma~2.5]{costantino}. Now
-  given a TQFT \(\mathcal{F} : \Cob \to \Vect\), by functoriality we obtain a
-  linear representation
+  It is clear that \(C_1\) is the identity morphism \(\Sigma_g \to \Sigma_g\)
+  in \(\Cob\). In addition, \(C_{f \cdot g} = C_f \circ C_g\) for all \(f, g
+  \in \Mod(\Sigma_g)\) -- see \cite[Lemma~2.5]{costantino}. Now given a TQFT
+  \(\mathcal{F} : \Cob \to \Vect\), by functoriality we obtain a linear
+  representation
   \begin{align*}
     \rho_{\mathcal{F}} : \Mod(\Sigma_g) & \to \GL(\mathcal{F}(\Sigma_g)) \\
-                                      f & \mapsto \mathcal{F}(M_f).
+                                      f & \mapsto \mathcal{F}(C_f).
   \end{align*}
 \end{example}
 
@@ -471,12 +433,12 @@ of level \(r\)}, first introduced by Witten and Reshetikhin-Tuarev
 topology.
 
 Besides Example~\ref{ex:symplectic-rep} and Example~\ref{ex:tqft-reps}, not a
-lot of other linear representations of \(\Mod(\Sigma_g)\) are known. Indeed, the
-representation theory of mapping class groups remains at mystery at large. In
-Chapter~\ref{ch:representations} we provide a brief overview of the field, as
-well as some recent developments. More specifically, we highlight Korkmaz'
-proof of the triviality of low-dimensional representations and comment on his
-classification of \(2g\)-dimensional representations \cite{korkmaz}. To that
+lot of other linear representations of \(\Mod(\Sigma_g)\) are known. Indeed,
+the representation theory of mapping class groups remains a mystery at large.
+In Chapter~\ref{ch:representations} we provide a brief overview of the field,
+as well as some recent developments. More specifically, we highlight Korkmaz'
+\cite{korkmaz} proof of the triviality of low-dimensional representations and
+comment on his classification of \(2g\)-dimensional representations. To that
 end, in Chapter~\ref{ch:dehn-twists} and Chapter~\ref{ch:relations} we survey
 the group structure of mapping class groups: its relations and known
 presentations.

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -351,7 +351,8 @@ the graph \(\Gamma\), we consider\dots
   \(\alpha\) and \(\beta\).
 \end{definition}
 
-It is clear from Lemma~\ref{thm:change-of-coordinates} that the actions of
+It is clear from the change of coordinates principle and
+Observation~\ref{ex:change-of-coordinates-crossing} that the actions of
 \(\Mod(\Sigma_{g, r}^b)\) on \(V(\hat{\mathcal{N}}(\Sigma_{g, r}^b))\) and
 \(\{([\alpha], [\beta]) \in V(\hat{\mathcal{N}}(\Sigma_{g, r}^b))^2 : \#(\alpha
 \cap \beta) = 1 \}\) are both transitive. But why should