memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: 0eb06874f69e5cf392f056f14722be6681ed315b
Parent: 7ef18d837c07281c6da1e8dd763c228c9078c8f3
Author: Pablo <pablo-pie@riseup.net>
Date: Wed, 19 Jun 2024 19:11:22 +0000

Hydrated the 1st chapter

Diffstat

12 files changed, 1589 insertions, 125 deletions

Status	File Name	N° Changes	Insertions	Deletions
Added	images/change-of-coords-cut.svg	265	265	0
Modified	images/cutting-homeo.svg	28	10	18
Added	images/half-twist-disk.svg	91	91	0
Added	images/homology-generators.svg	240	240	0
Added	images/intersection-index.svg	296	296	0
Added	images/torus-cut.svg	151	151	0
Modified	preamble.tex	5	5	0
Modified	references.bib	93	83	10
Modified	sections/introduction.tex	529	440	89
Modified	sections/presentation.tex	4	2	2
Modified	sections/representations.tex	4	2	2
Modified	sections/twists.tex	8	4	4

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diff --git a/preamble.tex b/preamble.tex
@@ -69,6 +69,7 @@
 
 % General linear group
 \DeclareMathOperator{\GL}{GL}
+\DeclareMathOperator{\PGL}{PGL}
 
 % Group Abelianization
 \newcommand{\ab}{{\operatorname{ab}}}
@@ -82,3 +83,7 @@
 
 % A normal subobject in a pointed cathegory
 \newcommand{\normal}{\triangleleft}
+
+% Useful category definitions
+\newcommand{\Cob}{\mathbf{Cob}_3^+}
+\newcommand{\Vect}{\mathbf{Vect}(\mathbb{C})}

diff --git a/references.bib b/references.bib
@@ -1,5 +1,5 @@
 @book{farb-margalit,
-  author    = {Farb, Benson and Margalit, Dan},
+  author    = {Farb,  Benson and Margalit,  Dan},
   isbn      = {978-0-691-14794-9},
   language  = {English},
   publisher = {Princeton University Press},
@@ -10,7 +10,7 @@
 }
 
 @article{korkmaz,
-  author   = {Korkmaz, Mustafa},
+  author   = {Korkmaz,  Mustafa},
   doi      = {10.1112/topo.12305},
   issn     = {1753-8416},
   journal  = {Journal of Topology},
@@ -24,7 +24,7 @@
 }
 
 @article{funar,
-  author    = {Louis Funar},
+  author    = {Funar,  Louis},
   doi       = {10.1090/S0002-9939-2010-10555-5},
   journal   = {Proceedings of the American Mathematical Society},
   language  = {English},
@@ -36,7 +36,7 @@
 }
 
 @article{franks-handel,
-  author    = {Franks, John and Handel, Michael},
+  author    = {Franks,  John and Handel,  Michael},
   doi       = {10.1090/S0002-9939-2013-11556-X},
   issn      = {0002-9939},
   journal   = {Proceedings of the American Mathematical Society},
@@ -62,7 +62,7 @@
 }
 
 @article{korkmaz-mccarthy,
-  author  = {Korkmaz, Mustafa and McCarthy, John D.},
+  author  = {Korkmaz,  Mustafa and McCarthy,  John D.},
   doi     = {10.1017/S0305004199004259},
   journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
   number  = 1,
@@ -87,7 +87,7 @@
 
 @article{birman-hilden,
  ISSN = {0003486X},
- author = {Birman, Joan and Hilden, Hugh},
+ author = {Birman,  Joan and Hilden,  Hugh},
  journal = {Annals of Mathematics},
  number = 3,
  pages = {424--439},
@@ -98,7 +98,7 @@
 }
 
 @article{wajnryb,
-  author  = {Wajnryb, Bronislaw},
+  author  = {Wajnryb,  Bronislaw},
   doi     = {10.1007/bf02774014},
   issn    = {1565-8511},
   journal = {Israel Journal of Mathematics},
@@ -111,7 +111,7 @@
 }
 
 @article{lickorish,
-  author    = {Lickorish, William Bernard Raymond},
+  author    = {Lickorish,  William Bernard Raymond},
   doi       = {10.1017/s030500410003824x},
   issn      = {1469-8064},
   journal   = {Mathematical Proceedings of the Cambridge Philosophical Society},
@@ -124,8 +124,68 @@
   year      = {1964},
 }
 
+@article{witten,
+  author    = {Witten,  Edward},
+  doi       = {10.1007/bf01217730},
+  issn      = {1432-0916},
+  journal   = {Communications in Mathematical Physics},
+  month     = sep,
+  number    = {3},
+  pages     = {351--399},
+  publisher = {Springer Science and Business Media LLC},
+  title     = {Quantum field theory and the Jones polynomial},
+  volume    = {121},
+  year      = {1989},
+}
+
+@article{reshetikhin-turaev,
+  title = {Invariants of $3$-manifolds via link polynomials and quantum groups},
+  volume = {103},
+  issn = {1432-1297},
+  doi = {10.1007/bf01239527},
+  number = {1},
+  journal = {Inventiones Mathematicae},
+  publisher = {Springer Science and Business Media LLC},
+  author = {Reshetikhin,  Nicolai and Turaev,  Vladimir Georgievich},
+  year = {1991},
+  month = dec,
+  pages = {547--597},
+}
+
+@article{rado,
+  author  = {Rad{\'o}, Tibor},
+  journal = {Acta Litt. Sci. Szeged},
+  number  = {101-121},
+  pages   = {10},
+  title   = {{\"U}ber den begriff der riemannschen fl{\"a}che},
+  volume  = {2},
+  year    = {1925},
+}
+
+@article{thomassen,
+  author    = {Thomassen,  Carsten},
+  doi       = {10.2307/2324180},
+  issn      = {0002-9890},
+  journal   = {The American Mathematical Monthly},
+  month     = feb,
+  number    = {2},
+  pages     = {116},
+  publisher = {JSTOR},
+  title     = {The Jordan-Schonflies Theorem and the Classification of Surface},
+  volume    = {99},
+  year      = {1992},
+}
+
+@book{kerekjarto,
+  author    = {Kerékjártó,  Béla},
+  doi       = {10.1007/978-3-642-50825-7},
+  isbn      = {9783642508257},
+  publisher = {Springer},
+  title     = {Vorlesungen \"{u}ber Topologie},
+  year      = {1923},
+}
 @incollection{julien,
-  author    = {Marché, Julien},
+  author    = {Marché,  Julien},
   booktitle = {Topology and geometry. A collection of essays dedicated to Vladimir G. Turaev},
   doi       = {10.4171/IRMA/33-1/7},
   isbn      = {978-3-98547-001-3; 978-3-98547-501-8},
@@ -136,8 +196,21 @@
   year      = {2021},
 }
 
+@inproceedings{costantino,
+  author    = {Costantino,  Francesco},
+  booktitle = {Winter Braids Lecture Notes},
+  doi       = {10.5802/wbln.7},
+  issn      = {2426-0312},
+  month     = feb,
+  pages     = {1--45},
+  publisher = {Cellule MathDoc/CEDRAM},
+  title     = {Notes on Topological Quantum Field Theories},
+  volume    = {2},
+  year      = {2015},
+}
+
 @book{hatcher,
-  author    = {Hatcher, Allen},
+  author    = {Hatcher,  Allen},
   edition   = {1},
   isbn      = {9780521795401; 0521795400},
   language  = {English},

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -1,9 +1,115 @@
 \chapter{Introduction}\label{ch:introduction}
 
-% TODO: Motivation
-% TODO: Talk about the classification of surfaces
+Ever since Mankind first stepped foot on the surface of Earth, humanity has
+been asking what is the shape of the planet we enhabit. 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. 
 
-\begin{definition}
+\begin{theorem}[Classification of surfaces]\label{thm:classification-of-surfaces}
+  Any closed connected orientable surface is homeomorphic to the connected sum
+  \(S_g\) of \(g \ge 0\) copies of the torus \(\mathbb{T}^2 =
+  \mfrac{\mathbb{R}^2}{\mathbb{Z}^2}\). Any compact connected orientable
+  surface \(S\) is isomorphic to the surface \(S_g^p\) obtained from \(S_g\) by
+  removing \(p \ge 0\) open disks with disjoint closures.
+\end{theorem}
+
+The integer  \(g \ge 0\) in Theorem~\ref{thm:classification-of-surfaces} is
+called \emph{the genus of \(S\)}. We also have the noncompact surface \(S_{g,
+r}^p = S_g^p \setminus \{x_1, \ldots, x_r\}\), where \(x_1, \ldots, x_r\) in
+the interior of \(S_g^p\). The points \(x_1, \ldots, x_r\) are called the
+\emph{puctures} of \(S_{g, r}^p\). Throught these notes, all surfaces
+considered will be of the form \(S = S_{g, r}^p\). Any such \(S\) admits a
+natural compactification \(\bar S\) obtained by filling the its punctures. We
+denote \(S_{g, r} = S_{g, r}^0\). All closed curves \(\alpha, \beta \subset S\)
+we consider lie in the interior \(S\degree\) of \(S\) and intersect
+transversily.
+
+It is interesting to remark that, aside from the homeomorphism type of a
+surface \(S\), Theorem~\ref{thm:classification-of-surfaces} also informs the
+geometry of the curves in \(S\) and their intersections. For example\dots
+
+\begin{lemma}[Change of coordinates principle]\label{thm:change-of-coordinates}
+  Given oriented nonseparating simple closed curves \(\alpha, \beta \subset
+  S\), we can find an orientation-presering homeomorphis \(\phi : S \isoto S\)
+  fixing \(\partial S\) pointwise such that \(\phi(\alpha) = \beta\) with
+  orientation. Even more so, if \(\alpha', \beta' \subset S\) 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 \(S = S_{g, r}^b\) and consider the surface \(S_{\alpha \alpha'}\)
+  obtained by cutting \(S\) across \(\alpha\) and \(\alpha'\), as in
+  Figure~\ref{fig:change-of-coordinates}. Since \(\alpha\) and \(\alpha'\) are
+  nonseparating, this surface has genus \(g - 1\) and one additional boundary
+  component \(\delta \subset \partial S_{\alpha \beta}\), so \(S_{\alpha \beta}
+  \cong S_{g-1,r}^{b+1}\). The boundary component \(\delta\) is naturally
+  subdived into the four arcs in Figure~\ref{fig:change-of-coordinates}, each
+  corresponding to one of the curves \(\alpha\) and \(\alpha'\) in \(S\). By
+  identifying the pairs of arcs corresponding to the same curve we obtain the
+  surface \(\mfrac{S_{\alpha \beta}}{\sim} \cong S\).
+
+  Similarly, \(S_{\beta \beta'} \cong S_{g-1, r}^{b+1}\) also has an additional
+  boundary component \(\delta' \subset \partial S_{\beta \beta'}\) subdivided
+  into four arcs. Now by the classification of surfaces we can find an
+  orientation-preserving homemorphism \(\tilde\phi : S_{\alpha \alpha'} \isoto
+  S_{\beta \beta'}\). Even more so, we can choose \(\tilde\phi\) taking each
+  one of the arcs in \(\delta\) to the corresponding arc in \(\delta'\). Now
+  \(\tilde\phi\) descends to a self-homeomorphism \(\phi\) the quotient surface
+  \(S \cong \mfrac{S_{\alpha \alpha'}}{\sim} \cong \mfrac{S_{\beta
+  \beta'}}{\sim}\). Moreover, \(\phi\) is such that \(\phi(\alpha) = \alpha'\)
+  and \(\phi(\beta) = \beta'\), as desired.
+\end{proof}
+
+\begin{figure}[ht]
+  \centering
+  \includegraphics[width=.8\linewidth]{images/change-of-coords-cut.eps}
+  \caption{By cutting $S_{g, r}^b$ across $\alpha$ we obtain $S_{g-1,
+  r}^{b+2}$, where $\alpha'$ deterimines a yellow arc joining the two
+  additional boundary components. Now by cutting $S_{g-1, r}^{b+2}$ across this
+  arc we obtain $S_{g-1,r}^b$, with the added boundary component subdivided
+  into the four arcs corresponding to $\alpha$ and $\alpha'$.}
+  \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
+unawsered questions about surfaces and their
+homeomorphisms. Given a surface \(S\), the group \(\Homeo^+(S, \partial S)\) of
+orientation-preserving homeomorphism of \(S\) fixing its boundary pointwise is
+a topological group\footnote{Here we endow \(\Homeo^+(S, \partial S)\) 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 \(\alpha, \beta \subset S\), can we find \(\phi \in
+    \Homeo^+(S, \partial S)\) with \(\phi(\alpha) = \beta\)?
+
+  \item What are the conjugacy classes of \(\Homeo^+(S, \partial S)\)? What
+    about its connected components?
+
+  \item Does \(\Homeo^+(S, \partial S)\) determine \(S\)? If the answer is
+    \emph{no}, what about in the closed case?
+\end{enumerate}
+
+Unfortunately, however, the algebraic structure \(\Homeo^+(S, \partial)\) is
+tipically too complex to tackle. More importantly, all of this complexity is
+arguably unnecessary for most topological applications, in the sence that
+usually we are only really interested in considering \emph{homeomorphisms up to
+isotopy}. For instance, isotopic homeomorphisms \(\phi \simeq \psi : S \isoto
+S\) determine the same automorphism \(\phi_* = \psi_*\) at the levels of
+homotopy and homology. This leads us to consider the group of connected
+components of \(\Homeo^+(S, \partial S)\), also 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 \(\Mod(S)\) of an orientable surface \(S\)} is
   the group of isotopy classes of orientation-preserving homeomorphisms \(S
   \isoto S\), where both the homeomorphisms and the isotopies are assumed to
@@ -13,10 +119,55 @@
   \]
 \end{definition}
 
+There are many variations of the Definition~\ref{def:mcg}. For example\dots
+
+\begin{example}\label{ex:action-on-punctures}
+  Any \(\phi \in \Homeo^+(S, \partial S)\) extends to a homomorphism
+  \(\tilde\phi\) of \(\bar{S}\) that permutes the set \(\{x_1, \ldots, x_r\} =
+  \bar{S} \setminus S\) of punctures of \(S\). We may thus define an action
+  \(\Mod(S) \leftaction \{x_1, \ldots, x_r\}\) via \(f \cdot x_i =
+  \tilde\phi(x_i)\) for \(f = [\phi] \in \Mod(S)\) -- which is independant of
+  the choice of representative \(\phi\) of \(f\).
+\end{example}
+
+\begin{definition}
+  Given an orientable surface \(S\) and a puncture \(x \subset \bar{S}\) of
+  \(S\), denote by \(\Mod(S, x) \subset \Mod(S)\) the subgroup of mapping
+  classes that fix \(x\). The \emph{pure mapping class group \(\PMod(S) \subset
+  \Mod(S)\) of \(S\)} is the subgroup of mapping classes that fix every
+  puncture of \(S\).
+\end{definition}
+
+\begin{example}\label{ex:action-on-curves}
+  Given a simple closed curve \(\alpha \subset S\), denote by
+  \(\vec{[\alpha]}\) and \([\alpha]\) the isotopy classes of \(\alpha\) with
+  and without orientation, respectively. There are natural actions \(\Mod(S)
+  \leftaction \{ \vec{[\alpha]} : \alpha \subset S \}\) and \(\Mod(S)
+  \leftaction \{ [\alpha] : \alpha \subset S \}\) given by
+  \begin{align*}
+    f \cdot \vec{[\alpha]} & = \vec{[\phi(\alpha)]} &
+    f \cdot [\alpha]       & = [\phi(\alpha)]
+  \end{align*}
+  for \(f = [\phi] \in \Mod(S)\).
+\end{example}
+
+\begin{definition}
+  Given a simple closed curve \(\alpha \subset S\), we denote by
+  \(\Mod(S)_{\vec{[\alpha]}} = \{ f \in \Mod(S) : f \cdot \vec{[\alpha]} =
+  \vec{[\alpha]} \}\) and \(\Mod(S)_{[\alpha]} = \{ f \in \Mod(S) : f \cdot
+  [\alpha] = [\alpha] \}\) the subgroups of mapping classes that fix the
+  isotopy classes of \(\alpha\).
+\end{definition}
+
+While trying to understand the mapping class group of \(S\), it is interesting
+to consider how the geometric relationship between \(S\) and other surfaces
+affects \(\Mod(S)\). Indeed, different embeddings \(R \hookrightarrow S\)
+translate to homomorphisms at the level of mapping class groups.
+
 \begin{example}[Inclusion homomorphism]
   Let \(R \subset S\) be a closed subsurface. Given some \(\phi \in \Homeo^+(R,
   \partial R)\), we may extend \(\phi\) to \(\tilde{\phi} \in \Homeo^+(S,
-  \partial S)\) by setting \(\tilde{\phi}(p) = p\) for \(p \in S\) outside of
+  \partial S)\) by setting \(\tilde{\phi}(x) = x\) for \(x \in S\) outside of
   \(R\) -- which is well defined since \(\phi\) fixes every point in \(\partial
   R\). This contruction yields a group homomorphism
   \begin{align*}
@@ -26,26 +177,23 @@
   known as \emph{the inclusion homomorphism}.
 \end{example}
 
-\begin{example}[Capping homomorphism]
-  Let \(\alpha \subset \partial S\) be a boundary component of \(S\) and fix
-  some orientation of \(\alpha\). We refer to the inclusion homomorphism
-  \(\operatorname{cap} : \Mod(S) \to \Mod(S \cup_\alpha (\mathbb{D}^2 \setminus
-  \{0\}))\) as \emph{the capping homomorphism}.
-\end{example}
-
-\begin{proposition}[Capping exact sequence]\label{ex:capping-seq}
-  Given some oriented boundary component \(\alpha \subset \partial S\) of
-  \(S\), there is an exact sequence
+\begin{example}[Capping exact sequence]\label{ex:capping-seq}
+  Let \(\delta \subset \partial S\) be an oriented boundary component of \(S\).
+  We refer to the inclusion homomorphism \(\operatorname{cap} : \Mod(S) \to
+  \Mod(S \cup_\delta (\mathbb{D}^2 \setminus \{0\}))\) as \emph{the capping
+  homomorphism}. There is an exact sequence
   \begin{center}
     \begin{tikzcd}
       1 \rar &
       \langle \tau_\alpha \rangle \rar &
       \Mod(S) \rar{\operatorname{cap}} &
-      \Mod(S \cup_\alpha (\mathbb{D}^2 \setminus \{0\})) \rar &
-      1
+      \Mod(S \cup_\delta (\mathbb{D}^2 \setminus \{0\}), 0) \rar &
+      1,
     \end{tikzcd}
   \end{center}
-\end{proposition}
+  known as \emph{the capping exact sequence} -- see
+  \cite[Proposition~3.19]{farb-margalit} for a proof.
+\end{example}
 
 \begin{example}[Cutting homomorphism]\label{ex:cutting-morphism}
   Given a simple closed curve \(\alpha \subset S\), denote by
@@ -61,102 +209,298 @@
     [\phi] & \mapsto [\phi\!\restriction_{S_{g+1} \setminus \alpha}],
   \end{align*}
   known as \emph{the cutting homomorphism}. Furthermore, \(\ker
-  \operatorname{cut} = \langle \tau_\alpha \rangle\).
+  \operatorname{cut} = \langle \tau_\alpha \rangle\) -- see
+  \cite[Propostion~3.20]{farb-margalit} for a proof.
 \end{example}
 
+As goes for most groups, another approach to understanding the mapping class
+group of a given surface \(S\) 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 \(\Mod(S)\) are its \emph{linear representations} -- i.e. the group
+homomorphisms \(\Mod(S) \to \GL_n(\mathbb{C})\). These may be seen as actions
+\(\Mod(S) \leftaction \mathbb{C}^n\) where each \(f \in \Mod(S)\) acts via some
+\(\mathbb{C}\)-linear isomorphism \(\mathbb{C}^n \isoto \mathbb{C}^n\).
+
+\section{Representations}
+
+Here we collect a few fundamental examples of linear representations of
+\(\Mod(S)\).
+
+\begin{example}
+  Given \(k \ge 0\) and \(f = [\phi] \in \Mod(S)\), we may consider the map
+  \(\phi_* : H_k(S, \mathbb{Z}) \to H_k(S, \mathbb{Z})\) induced at the level
+  of singular homology. By homotopy invariance, the map \(\phi_*\) is
+  independant of the choice of representative \(\phi\) of \(f\). By the
+  functoriality of homology groups we then get a \(\mathbb{Z}\)-linear action
+  \(\Mod(S) \leftaction H_k(S, R)\), given by \(f \cdot \xi = \phi_*(\xi)\) for
+  \(f = [\phi] \in \Mod(S)\).
+\end{example}
+
+Now by choosing \(k = 1\) we obtain the so called \emph{symplectic
+representation.} 
+
 \begin{example}
-  Let \(S_{g, r}^b = S_g^b \setminus \{ x_1, \ldots, x_r \}\) be the surface of
-  genus \(g\) with \(r\) punctures and \(b\) boudary components. Given \(\phi
-  \in \Homeo^+(S_{g, r}^b, \partial S_{g, r}^b)\), there is some \(\sigma_\phi
-  \in \mathfrak{S}_r\) with \(\phi(x) \to x_{\sigma(i)}\) as \(x \to x_i\) in
-  \(S_{g, r}\). It is clear that \(\sigma_\phi = \sigma_\psi\) for \(\phi
-  \simeq \psi\). Hence \(\Mod(S_{g,r}^b) \leftaction \{ x_1, \ldots, x_r \}\)
-  via \(f \cdot x_i = x_{\sigma_\phi(i)}\) for \(f = [\phi]\).
+  Recall \(H_1(S_g, \mathbb{Z}) \cong \mathbb{Z}^{2g}\), with standard basis
+  given by \([\alpha_1], [\beta_1], \ldots, [\alpha_g], [\beta_g] \in H_1(S_g,
+  \mathbb{Z})\) for \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g\) as
+  in Figure~\ref{fig:homology-basis}. The Abelian group \(H_1(S_g,
+  \mathbb{Z})\) is endowed with a natural \(\mathbb{Z}\)-bilinear alternating
+  form given by the \emph{algebraic intersection number} \([\alpha] \cdot
+  [\beta] = \sum_{x \in \alpha \cap \beta} \operatorname{ind}\,x\) -- where the
+  index \(\operatorname{ind}\,x = \pm 1\) of an intersection point is given by
+  Figure~\ref{fig:intersection-index}. In terms of the standard basis of
+  \(H_1(S_g, \mathbb{Z})\), this form is given by
+  \begin{align}\label{eq:symplectic-form}
+    [\alpha_i] \cdot [\beta_j]  & = \delta_{i j} &
+    [\alpha_i] \cdot [\alpha_j] & = 0            &
+    [\beta_i]  \cdot [\beta_j]  & = 0
+  \end{align}
+  and thus coincides with the pullback of the standard \(\mathbb{Z}\)-bilinear
+  symplectic form in \(\mathbb{Z}^{2g}\).
 \end{example}
 
+\begin{example}[Symplectic representation]\label{ex:symplectic-rep}
+  Consider the \(\mathbb{Z}\)-linear action \(\Mod(S_g) \leftaction H_1(S_g,
+  \mathbb{Z}) \cong \mathbb{Z}^{2g}\). Since pushforwards by
+  orientation-preserving homeomorphisms preserve the index of intersection
+  points, \((f \cdot [\alpha]) \cdot (f \cdot [\beta]) = [\alpha] \cdot
+  [\beta]\) for all \(\alpha, \beta \subset S_g\) and \(f \in \Mod(S_g)\). In
+  light of (\ref{eq:symplectic-form}), this implies \(\Mod(S_g)\) acts on
+  \(\mathbb{Z}^{2g}\) via \(\mathbb{Z}\)-linear symplectomorphisms. We thus
+  obtain a group homomorphism \(\psi : \Mod(S_g) \to
+  \operatorname{Sp}_{2g}(\mathbb{Z}) \subset \GL_{2g}(\mathbb{C})\), known as
+  \emph{the symplectic representation of \(\Mod(S_g)\)}.
+\end{example}
+
+\begin{minipage}[b]{.45\linewidth}
+  \centering
+  \includegraphics[width=\linewidth]{images/homology-generators.eps}
+  \captionof{figure}{The curves $\alpha_1, \beta_1, \ldots, \alpha_g, \beta_g
+  \subset S_g$ that generate its first singular homology group.}
+  \label{fig:homology-basis}
+\end{minipage}
+\hspace{.5cm} %
+\begin{minipage}[b]{.45\linewidth}
+  \centering
+  \includegraphics[width=\linewidth]{images/intersection-index.eps}
+  \vspace*{.4cm}
+  \captionof{figure}{The index of an intersection point $x \in \alpha \cap
+  \beta$.}
+  \label{fig:intersection-index}
+\end{minipage}
+
+Another fundamental class of examples of representations are the so called
+\emph{TQFT representations}.
+
 \begin{definition}
-  Given an orientable surface \(S\) and a puncture \(x\) of \(S\), denote by
-  \(\Mod(S, x) \subset \Mod(S)\) the subgroup of mapping classes that fix
-  \(x\). The \emph{pure mapping class group \(\PMod(S) \subset \Mod(S)\) of
-  \(S\)} is the subgroup of mapping classes that fix every puncture of \(S\).
+  A \emph{cobordism} between closed oriented surfaces \(R\) and \(S\) is a
+  triple \((W, \phi_+, \phi_-)\) where \(W\) is a smooth oriented compact
+  \(3\)-manifold with \(\partial W = \partial_+ W \amalg \partial_- W\),
+  \(\phi_+ : S \isoto \partial_+ W\) is an orientation preserving
+  diffeomorphism and \(\phi_- : R \isoto \partial_- W\) is an
+  orientation-reversing diffeomorphism.
 \end{definition}
 
-\begin{example}\label{ex:inclusion-morphism}
-  \(\Mod(S) \leftaction \{ \vec{[\alpha]} : \alpha \subset S \}\) and \(\Mod(S)
-  \leftaction \{ [\alpha] : \alpha \subset S \}\) via
+\begin{definition}
+  We denote by \(\Cob\) the category whose objects are (possibly disconnected)
+  closed oriented surfaces and whose morphisms \(R \to S\) are diffeomorphism
+  classes\footnote{Here we only consider orientation-preserving diffeomorphisms
+  $\varphi : W \isoto W'$ that are compatible with the boundary identifications
+  in the sence that $\varphi(\partial_\pm W) = \partial_\pm W'$ and $\psi_\pm =
+  \varphi \circ \phi_\pm$.} of cobordisms between \(R\) and \(S\), with
+  composition given by
+  \[
+    [W, \phi_-, \phi_+] \circ [W', \psi_-, \psi_+]
+    = [W \cup_{\psi_- \circ \phi_+^{-1}} W', \phi_-, \psi_+]
+  \]
+  for \([W, \phi_-, \phi_+] : R \to S\) and \([W', \psi_-, \phi_+] : S \to L\).
+  We endow \(\Cob\) with the monoidal structure given by
   \begin{align*}
-    f \cdot [\alpha]       & = [\phi(\alpha)] &
-    f \cdot \vec{[\alpha]} & = \vec{[\phi(\alpha)]}
+    S \otimes R
+    & = S \amalg R &
+    [W,\phi_+,\phi_-] \otimes [W',\psi_+,\psi_-]
+    & = [W \amalg W', \phi_+ \amalg \psi_+, \phi_- \amalg \psi_-].
   \end{align*}
-  for \(f = [\phi] \in \Mod(S)\).
-\end{example}
+\end{definition}
 
-\begin{example}[Change of coordinates principle]\label{ex:change-of-coordinates}
-  % Also highlight the fact that Mod(S) acts transitively on the pairs of
-  % curves crossing once.
-\end{example}
+\begin{definition}[TQFT]\label{def:tqft}
+  A \emph{topological quantum field theory} (abreviated by \emph{TQFT})
+  is a functor \(\mathcal{F} : \Cob \to \Vect\) satisfying
+  \begin{gather*}
+    \begin{aligned}
+      \mathcal{F}(\emptyset) & = \mathbb{C} &
+      \mathcal{F}(S \otimes R) & = \mathcal{F}(S) \otimes \mathcal{F}(R)
+    \end{aligned} \\
+    \mathcal{F}([W,\phi_+,\phi_-] \otimes [W',\psi_+,\psi_-])
+    = \mathcal{F}([W,\phi_+,\phi_-]) \otimes \mathcal{F}([W',\psi_+,\psi_-]),
+  \end{gather*}
+  where \(\Vect\) denotes the category of finite-dimensional complex vector
+  spaces.
+\end{definition}
 
 \begin{example}
-  \(\Mod(S) \leftaction H_k(S, \mathbb{R})\)
+  Given \(\phi \in \Homeo^+(S_g)\), we may consider the so called \emph{mapping
+  cylinder} \(M_\phi = (S_g \times [0, 1], \phi, 1)\), a cobordism between
+  \(S_g\) and itself -- where \(\partial_+ (S_g \times [0, 1]) = S_g \times 0\)
+  and \(\partial_- (S_g \times [0, 1]) = S_g \times 1\). The diffeomorphism
+  class of \(M_\phi\) is independant of the choice of representative of \(f =
+  [\phi] \in \Mod(S_g)\), so \(M_f = [M_\phi] : S_g \to S_g\) is a well defined
+  morphism in \(\Cob\).
 \end{example}
 
-\begin{example}\label{ex:symplectic-rep}
-  The symplectic representation.
+\begin{example}[TQFT representations]\label{ex:tqft-reps}
+  It is clear that \(M_1\) is the identity morphism \(S_g \to S_g\) in
+  \(\Cob\). In addition, \(M_{f \cdot g} = M_f \circ M_g\) in \(\Cob\) for all
+  \(f, g \in \Mod(S_g)\) -- 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(S_g) & \to \GL(\mathcal{F}(S_g)) \\
+                                 f & \mapsto \mathcal{F}(M_f).
+  \end{align*}
 \end{example}
 
-\begin{example}
-  TQFT representations.
-\end{example}
+As simple as the construction in Example~\ref{ex:tqft-reps} is, in practice it
+is not that easy to come across functors as the ones in
+Definition~\ref{def:tqft}. This is because, in most interesting examples, we
+are required to attach some extra data to our surfaces to get a well defined
+association \(S_g \mapsto \mathcal{F}(S_g)\). Moreover, the condition
+\(\mathcal{F}([W] \circ [W']) = \mathcal{F}([W]) \circ \mathcal{F}([W'])\) may
+only hold up to multiplication by scalars.
+
+Hence constructing an actual functor tipically requires \emph{extending}
+\(\Cob\) and \emph{tweaking} \(\Vect\). These ``extended TQFTs'' give rise to
+linear and projective representations of the \emph{extended mapping class
+groups} \(\Mod(S_g) \times \mathbb{Z}\). 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
+of level \(r\)}, first introduced by Witten and Reshetikhin-Tuarev
+\cite{witten, reshetikhin-turaev} in their foundational papers on quantum
+topology.
+
+Besides Example~\ref{ex:symplectic-rep} and Example~\ref{ex:tqft-reps}, not a
+lot of other linear representations of \(\Mod(S_g)\) are known. Indeed, the
+representation theory of mapping class groups remains at mistery 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
+classfication of \(2g\)-dimensional representations \cite{korkmaz}. To that
+end, in Chapter~\ref{ch:dehn-twists} and Chapter~\ref{ch:relations} we survay
+the group structure of mapping class groups: its relations and known
+presentations.
 
+% TODOO: Move this to the next chapter
 \section{First Computations}
 
-% TODO: Explain the Alexander trick
+We begin by the simplest, yet perhaps the most fundamental, of computations in
+this entire thesis.
+
+% TODO: Draw a diagram?
 \begin{example}[Alexander trick]\label{ex:alexander-trick}
-  \(\Mod(\mathbb{D}^2) = 1\)
+  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,
+  \(\Mod(\mathbb{D}^2) = 1\). 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
+    \begin{cases}
+      (1 - t) \phi(\sfrac{z}{1 - t}) & \text{if } 0 \le |z| \le 1 - t \\
+      z                              & \text{otherwise}
+    \end{cases}
+  \end{align*}
+  that ``fixes the band \(\{ z \in \mathbb{D}^2 : |z| \ge 1 - t \}\) and does
+  \(\phi\) inside the subdisk \(\{ z \in \mathbb{D}^2 : |z| \le 1 - t\}\)''
+  joins \(\phi = \phi_0\) and \(1 = \phi_1\).
 \end{example}
 
-\begin{example}
+\begin{example}\label{ex:mdg-once-punctured-disk}
   By the same token, \(\Mod(\mathbb{D}^2 \setminus \{0\}) = 1\).
 \end{example}
 
-% TODO: Explain this
 \begin{example}\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, \(\Mod(\mathbb{T}^2) \cong
-  \operatorname{SL}_2(\mathbb{Z})\).
+  Let \(\mathbb{T}^2 = S_1\) be the torus. 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,
+  \(\Mod(\mathbb{T}^2) \cong \operatorname{SL}_2(\mathbb{Z})\). To see \(\psi\)
+  is surjective, first observe \(\mathbb{Z}^2 \subset \mathbb{R}^2\) is
+  \(\operatorname{SL}_2(\mathbb{Z})\)-invariant. Hence any matrix \(g \in
+  \operatorname{SL}_2(\mathbb{Z})\) descends to an orientation-preserving
+  homeomorphism \(\phi_g\) of the quotient \(\mathbb{T}^2 =
+  \mfrac{\mathbb{R}^2}{\mathbb{Z}^2}\), which satisfies \(\psi([\phi_g]) = g\).
+  To see \(\psi\) is injective we consider the curves \(\alpha_1\) and
+  \(\beta_1\) from Figure~\ref{fig:homology-basis}. Given \(f = [\phi] \in
+  \Mod(\mathbb{T}^2)\) with \(\psi(f) = 1\), \(f \cdot \vec{[\alpha_1]} =
+  \vec{[\alpha_1]}\) and \(f \cdot \vec{[\beta_1]} = \vec{[\beta_1]}\), so we
+  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
+  Example~\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 decends to an isotopy \(\phi \simeq 1 \in
+  \Homeo^+(\mathbb{T}^2)\), so \(f = 1 \in \Mod(\mathbb{T}^2\) as desired.
 \end{example}
 
+\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,
+  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}
+\end{figure}
+
 \begin{example}\label{ex:punctured-torus-mcg}
   By the same token, \(\Mod(S_{1, 1}) \cong \operatorname{SL}_2(\mathbb{Z})\).
 \end{example}
 
-\begin{proposition}[Alexander method]
+% TODOO: Add comments on Costantino's idea?
+\begin{note}
+  Despite the fact \(\psi : \Mod(\mathbb{T}^2) \to
+  \operatorname{SL}_2(\mathbb{Z})\) is an isomorphism, the symplectic
+  representation is \emph{not} injective for surfaces of genus \(g \ge 2\) --
+  see \cite[Section~6.5]{farb-margalit} for a description of its kernel. In
+  fact, the question of existance of injective representations of \(\Mod(S_g)\)
+  remains wide-open. Recently, Korkmaz \cite[Theomre~3]{korkmaz} established a
+  lower bound for the dimension of an injective representation of \(\Mod(S_g)\)
+  in the \(g \ge 3\) case -- if one such representation exists.
+\end{note}
+
+By cutting across curves and arcs as in the proof of injectivity in
+Example~\ref{ex:torus-mcg}, we can always decompose a surface into copies of
+\(\mathbb{D}^2\) and \(\mathbb{D}^2 \setminus \{0\}\).
+Example~\ref{ex:alexander-trick} and Example~\ref{ex:mdg-once-punctured-disk}
+then imply\dots
+
+\begin{proposition}[Alexander method]\label{thm:alexander-method}
   Let \(\alpha_1, \ldots, \alpha_n \subset S\) be essencial simple closed
-  curves or proper arcs pair-wise in minimal position. Suppose \([\alpha_i] \ne
-  [\alpha_j]\) for \(i \ne j\) and that there are no triple intersections among
-  the \(\alpha_i\) -- i.e. 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. Fix \(\phi \in \Homeo^+(S, \partial S)\) and \(f =
-  [\phi] \in \Mod(S)\). Then
+  curves or proper arcs satisfying the following conditions.
   \begin{enumerate}
-    \item If there is \(\sigma \in \mathfrak{S}_n\) such that \(f \cdot
-      [\alpha_i] = [\alpha_{\sigma(i)}]\) for all \(i\) then there is \(\psi
-      \in \Homeo^+(S, \partial S)\) isotopic to \(1\) taking \(\cup_i
-      \alpha_i\) to \(\cup_i \phi(\alpha_i)\).
-
-    % TODO: Explain the meaning of "fill"
-    \item Suppose \(\alpha_1, \ldots, \alpha_n\) fill \(S\) and \(\#(\alpha_i
-      \cap \alpha_j) \le 1\) for all \(i, j\). If there is \(\sigma \in
-      \mathfrak{S}_n\) such that \(f \cdot \vec{[\alpha_i]} =
-      \vec{[\alpha_{\sigma(i)}]}\) then \(f^{\operatorname{ord}(\sigma)} = 1
-      \in \Mod(S)\). In particular, if \(\sigma = 1\) then \(f = 1 \in
-      \Mod(S)\).
+    \item \([\alpha_i] \ne [\alpha_j]\) for \(i \ne j\).
+    \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 \(S\) across the \(\alpha_i\) is a
+      disjoint union of disks and once-punctured disks.
   \end{enumerate}
+  Let \(f \in \Mod(S)\). If there is \(\sigma \in \mathfrak{S}_n\) such that
+  \(f \cdot \vec{[\alpha_i]} = \vec{[\alpha_{\sigma(i)}]}\) for all \(i\), then
+  \(f^{\operatorname{ord}(\sigma)} = 1 \in \Mod(S)\). In particular, if
+  \(\sigma = 1\) then \(f = 1 \in \Mod(S)\).
 \end{proposition}
 
-% TODO: Add comments on the proof
-% TODO: Check the sign in the formula for ϕ: is it t + s or t - s
+See \cite[Proposition~2.8]{farb-margalit} for a proof of
+Proposition~\ref{thm:alexander-method}. The Alexander method can also be used
+to show two mapping classes \(f, g \in \Mod(S)\) are the same: if we decompose
+\(S\) into disks and once-punctured disks with \(\alpha_1, \ldots, \alpha_n\)
+such that \(f \cdot [\alpha_i] = g \cdot [\alpha_i]\), then \(f g^{-1} \cdot
+[\alpha_i] = [\alpha_i]\) and so we may apply
+Proposition~\ref{thm:alexander-method} to the class \(f g^{-1}\). We now
+collect some important applications of the method.
+
 \begin{example}\label{ex:mcg-annulus}
   \(\Mod(\mathbb{S}^1 \times [0, 1]) \cong \mathbb{Z}\) is freely generated by
   \(f = [\phi]\), where
@@ -168,16 +512,6 @@
   particular, \(\Mod(\mathbb{S}^1 \times [0, 1]) \cong \mathbb{Z}\).
 \end{example}
 
-\begin{figure}[ht]
-  \centering
-  \includegraphics[width=.3\linewidth]{images/dehn-twist-cylinder.eps}
-  \caption{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 the
-  right-hand side that winds about the curve $\alpha$.}
-  \label{fig:dehn-twist-cylinder}
-\end{figure}
-
-% TODO: Can we prove this without using braid groups?
 \begin{example}\label{ex:mcg-twice-punctured-disk}
   The mapping class group \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2},
   \sfrac{1}{2}\})\) of the twice punctured unit disk in \(\mathbb{C}\) is
@@ -185,10 +519,27 @@
   \begin{align*}
     \phi : \mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}
     & \isoto \mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\} \\
-    z & \mapsto -z.
+    z & \mapsto -z
   \end{align*}
-  In particular, \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\})
-  \cong \mathbb{Z}\).
+  is the map from Figure~\ref{fig:hald-twist-disk}. In particular,
+  \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}) \cong
+  \mathbb{Z}\).
 \end{example}
 
-% TODO: Add a picture explaining that ϕ is counter-clockwise rotation by π
+\begin{figure}[ht]
+  \centering
+  \includegraphics[width=.3\linewidth]{images/dehn-twist-cylinder.eps}
+  \caption{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 the
+  right-hand side that winds about the curve $\alpha$.}
+  \label{fig:dehn-twist-cylinder}
+\end{figure}
+
+\begin{figure}[ht]
+  \centering
+  \includegraphics[width=.18\linewidth]{images/half-twist-disk.eps}
+  \caption{The generator $f$ of $\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2},
+  \sfrac{1}{2}) \cong \mathbb{Z}$ corresponds to the cclockwise rotation by
+  $\pi$ about the origen.}
+  \label{fig:hald-twist-disk}
+\end{figure}

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -1,4 +1,4 @@
-\chapter{Relations Between Twists}
+\chapter{Relations Between Twists}\label{ch:relations}
 
 Having acomplished the milestones of Theorem~\ref{thm:lickorish-gens} and
 Corollary~\ref{thm:humphreys-gens}, we now find ourselves ready to study some
@@ -193,7 +193,7 @@ we get\dots
 
 \begin{minipage}[b]{.45\linewidth}
 \begin{example}\label{ex:braid-group-center}
-  Using the capping exact sequence from Proposition~\ref{ex:capping-seq} and
+  Using the capping exact sequence from Example~\ref{ex:capping-seq} and
   the Alexander method, one can check that the center \(Z(\Mod(S_{0, n}^1))\)
   of \(\Mod(S_{0, n}^1)\) is freely generated by the Dehn twist \(\tau_\delta\)
   about the boundary \(\delta = \partial S_{0, n}^1\). It is also not very hard

diff --git a/sections/representations.tex b/sections/representations.tex
@@ -1,4 +1,4 @@
-\chapter{Low-Dimensional Representations}
+\chapter{Low-Dimensional Representations}\label{ch:representations}
 
 Having built a solid understanding of the combinatorics of Dehn twists, we are
 now ready to attack the problem of classifying the representations of
@@ -264,7 +264,7 @@ representations.
   We claim that it suffices to find a \(m\)-dimensional
   \(\Mod(R)\)-invariant\footnote{Here we view $\Mod(R)$ as a subgroup of
   $\Mod(S_g^b)$ via the inclusion homomorphism $\Mod(R) \to \Mod(S_g^b)$ from
-  Example~\ref{ex:inclusion-morphism}, which can be shown to be injective in
+  Lemma~\ref{thm:inclusion-morphism}, which can be shown to be injective in
   this particular case.} subspace \(W \subset \mathbb{C}^n\) with \(2 \le m \le
   n - 2\). Indeed, in this case \(m < 2(g - 1)\) and \(\dim
   \mfrac{\mathbb{C}^n}{W} = n - m < 2(g - 1)\). Thus both representations

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -1,4 +1,4 @@
-\chapter{Dehn Twists}
+\chapter{Dehn Twists}\label{ch:dehn-twists}
 
 We have now seen some concrete examples of mapping class groups. In this
 chapter, we will investigate how we can use the anulus \(\mathbb{S}^1 \times
@@ -302,12 +302,12 @@ Theorem~\ref{thm:mcg-is-fg}.
     \begin{tikzcd}
       1 \rar &
       \langle \tau_{\delta_1} \rangle \rar &
-      \Mod(S_{g, r}^b) \rar{\operatorname{cap}} &
-      \Mod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\})) \rar &
+      \PMod(S_{g, r}^b) \rar{\operatorname{cap}} &
+      \PMod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\})) \rar &
       1
     \end{tikzcd}
   \end{center}
-  from Proposition~\ref{ex:capping-seq}, it suffices to show that \(S_{g, n}\)
+  from Example~\ref{ex:capping-seq}, it suffices to show that \(S_{g, n}\)
   is finitely generated by twists about nonseparating simple closed curves.
   Indeed, if \(\PMod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\)
   is finitely generated by twists about nonseparing curves or boundary