memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: 7057b04bf6c633bf7df7bc5b26d516eb1fff98cb
Parent: 4127c23b2acd5c54bccffff057640c690cd118eb
Author: Pablo <pablo-pie@riseup.net>
Date: Tue, 11 Jun 2024 21:46:40 +0000

Hydrated the 3rd chapter

Diffstat

15 files changed, 3248 insertions, 47 deletions

Status	File Name	N° Changes	Insertions	Deletions
Added	images/braid-group-center.svg	194	194	0
Added	images/braid-group-generator.svg	290	290	0
Added	images/hyperelliptic-covering.svg	608	608	0
Added	images/hyperelliptic-involution.svg	336	336	0
Added	images/hyperelliptic-relation.svg	213	213	0
Added	images/lantern-relation-disk-filling.svg	109	109	0
Added	images/lantern-relation-trivial-abelianization.svg	275	275	0
Added	images/lantern-relation.svg	249	249	0
Added	images/sphere-rotation.svg	164	164	0
Added	images/wajnryb-presentation-curves.svg	377	377	0
Modified	preamble.tex	4	3	1
Modified	references.bib	49	49	0
Modified	sections/introduction.tex	2	1	1
Modified	sections/presentation.tex	395	362	33
Modified	sections/twists.tex	30	18	12

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diff --git a/preamble.tex b/preamble.tex
@@ -5,7 +5,7 @@
 \usepackage{enumitem, xfrac, xcolor, cancel, multicol, tabularx, relsize}
 \usepackage[normalem]{ulem}
 \usepackage{hyperref, refcount}
-\usepackage{fancyhdr, titling}
+\usepackage{fancyhdr, titling, caption}
 \usepackage[backend=biber, citetracker=true, citestyle=alphabetic, bibstyle=ieee-alphabetic, sorting=nty, sortcites=true]{biblatex}
 \usepackage{pgfplots, tikz, tikz-cd}
 \usepackage{graphicx, wrapfig}
@@ -154,6 +154,7 @@
 % Display long arrows instead of short ones
 \renewcommand{\to}{\longrightarrow}
 \renewcommand{\mapsto}{\longmapsto}
+\renewcommand{\hookrightarrow}{\lhook\joinrel\longrightarrow}
 
 % Isomorphism arrow
 \newcommand{\isoto}{\xlongrightarrow{\sim}}
@@ -171,6 +172,7 @@
 
 % Group of homeomorphisms
 \DeclareMathOperator{\Homeo}{Homeo}
+\DeclareMathOperator{\SHomeo}{SHomeo}
 
 % General linear group
 \DeclareMathOperator{\GL}{GL}

diff --git a/references.bib b/references.bib
@@ -61,6 +61,55 @@
   year      = {2023},
 }
 
+@article{korkmaz-mccarthy,
+  author  = {Korkmaz, Mustafa and McCarthy, John D.},
+  doi     = {10.1017/S0305004199004259},
+  journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
+  number  = 1,
+  pages   = {35--53},
+  title   = {Surface mapping class groups are ultrahopfian},
+  volume  = {129},
+  year    = {2000},
+}
+
+@article{artin,
+  author  = {Artin,  Emil},
+  doi     = {10.2307/1969218},
+  issn    = {0003-486X},
+  journal = {The Annals of Mathematics},
+  month   = jan,
+  number  = 1,
+  pages   = {101--126},
+  title   = {Theory of Braids},
+  volume  = {48},
+  year    = {1947},
+}
+
+@article{birman-hilden,
+ ISSN = {0003486X},
+ author = {Birman, Joan and Hilden, Hugh},
+ journal = {Annals of Mathematics},
+ number = 3,
+ pages = {424--439},
+ publisher = {Annals of Mathematics},
+ title = {On Isotopies of Homeomorphisms of Riemann Surfaces},
+ volume = {97},
+ year = {1973},
+}
+
+@article{wajnryb,
+  author  = {Wajnryb, Bronislaw},
+  doi     = {10.1007/bf02774014},
+  issn    = {1565-8511},
+  journal = {Israel Journal of Mathematics},
+  month   = jun,
+  number  = {2--3},
+  pages   = {157--174},
+  title   = {A simple presentation for the mapping class group of an orientable surface},
+  volume  = 45,
+  year    = {1983},
+}
+
 @incollection{julien,
   author    = {Marché, Julien},
   booktitle = {Topology and geometry. A collection of essays dedicated to Vladimir G. Turaev},

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -72,7 +72,7 @@
 \section{First Computations}
 
 % TODO: Explain the Alexander trick
-\begin{example}[Alexander trick]
+\begin{example}[Alexander trick]\label{ex:alexander-trick}
   \(\Mod(\mathbb{D}) = 1\)
 \end{example}

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -1,35 +1,104 @@
 \chapter{Relations Among Twists \& Wajnryb's Presentation}
 
-\begin{example}
-  Lantern relation
+Having acomplished the milestones of Theorem~\ref{thm:lickorish-gens} and
+Corollary~\ref{thm:humphreys-gens}, we now find ourselves ready to study some
+of the group-theoretic aspects of \(\Mod(S)\). We should note, however, that
+our current understanding of its group structure is quite lacking: even though
+we know the generators of \(\Mod(S)\), we have already seen in
+Example~\ref{ex:braid-relation} that these must satisfy nontrivial relations.
+
+This poses the question: what relations between Dehn twists are there? The goal
+of this chapter is to highlight some of these relations and the geometric
+intuition behind them. We start by perhaps the simplest of these, known as
+\emph{the lantern relation}.
+
+\begin{example}\label{ex:lanter-relation}
+  Let \(S_0^4\) be the surface the of genus \(0\) with \(4\) boundary
+  components -- i.e. the \emph{lantern-like} surface we get by subtracting
+  \(4\) disjoint open disks from \(\mathbb{S}^2\). If \(\alpha, \beta, \gamma,
+  \delta_1, \ldots, \delta_4 \subset S_0^4\) are as in
+  Figure~\ref{fig:latern-relation} then from the Alexander method we get the so
+  called \emph{lantern relation} in \(\Mod(S_0^4)\).
+  \begin{equation}\label{eq:latern-relation}
+    \tau_\alpha \tau_\beta \tau_\gamma
+    = \tau_{\delta_1} \tau_{\delta_2} \tau_{\delta_3} \tau_{\delta_4}
+  \end{equation}
 \end{example}
 
+\begin{figure}[ht]
+  \centering
+  \includegraphics[width=.25\linewidth]{images/lantern-relation.eps}
+  \caption{The curves from the latern relation of $S_0^4$.}
+  \label{fig:latern-relation}
+\end{figure}
+
+We may exploit different embedings \(S_0^4 \hookrightarrow S\) and their
+corresponding inclusion homomorphisms \(\Mod(S_0^4) \to \Mod(S)\) to obtain
+interesting relations between the corresponding Dehn twists in \(\Mod(S)\). For
+example\dots
+
 \begin{proposition}\label{thm:trivial-abelianization}
-  If \(g \ge 3\) then the Abelianization \(\Mod(S_g^b)^\ab =
-  \mfrac{\Mod(S_g^b)}{[\Mod(S_g), \Mod(S_g)]}\) is trivial. In other words,
-  \(\Mod(S_g) = [\Mod(S_g), \Mod(S_g)]\) is a perfect group for \(g \ge 3\).
+  The Abelianization \(\Mod(S_g^b)^\ab = \mfrac{\Mod(S_g^b)}{[\Mod(S_g),
+  \Mod(S_g)]}\) is cyclic. Moreover, if \(g \ge 3\) then \(\Mod(S_g^b)^\ab =
+  1\). In other words, \(\Mod(S_g)\) is a perfect group for \(g \ge 3\).
 \end{proposition}
 
-% TODO: Explain how to get the groups in the low-genus settings we use explicit
-% presentations of Mod(S_g)
+\begin{proof}
+  By Theorem~\ref{thm:lickorish-gens}, \(\Mod(S_g^b)^\ab\) is generated by the
+  image of the Lickrish generators, which are all conjugate and thus represent
+  the same class in the Abelianization. In fact, any nonseparating \(\alpha
+  \subset S_g^b\) is conjugate to the Lickorish generators too, so
+  \(\Mod(S_g^b)^\ab = \langle [\alpha] \rangle\).
+
+  Now for \(g \ge 3\) we can embed \(S_0^4\) in \(S_g^b\) in such a way that
+  all the corresponding curves \(\alpha, \beta, \gamma, \delta_1, \ldots,
+  \delta_4 \subset S_g^b\) are nonseparating, as shown in
+  Figure~\ref{fig:latern-relation-trivial-abelianization}. The lantern relation
+  (\ref{eq:latern-relation}) then becomes
+  \[
+    3 \cdot [\tau_\alpha]
+    = [\tau_\alpha] + [\tau_\beta] + [\tau_\gamma]
+    = [\tau_{\delta_1}] + [\tau_{\delta_2}]
+    + [\tau_{\delta_3}] + [\tau_{\delta_4}]
+    = 4 \cdot [\tau_\alpha],
+  \]
+  in \(\Mod(S_g^b)^\ab\). In other words, \([\tau_\alpha] = 0\) and thus
+  \(\Mod(S_g^b)^\ab = 0\).
+\end{proof}
+
+\begin{figure}[ht]
+  \centering
+  \includegraphics[width=.5\linewidth]{images/lantern-relation-trivial-abelianization.eps}
+  \caption{The embedding of $S_0^4$ in $S_g^b$ for $g \ge 3$.}
+  \label{fig:latern-relation-trivial-abelianization}
+\end{figure}
+
+% TODO: Fix the spacing of this table
+We should note that in general \(\Mod(S)^\ab\) needs not be trivial. For
+example, in \cite[Section~5.1.3]{farb-margalit} Farb-Margalit use different
+presentations of \(\Mod(S_g)\) for \(g \le 2\) to show the Abelianization is given by
 \begin{center}
   \begin{tabular}{ r|c|l }
     \(g\) & \(S_g\)          & \(\Mod(S_g)^\ab\) \\
     \hline
     \(0\) & \(\mathbb{S}^2\) & \(0\)             \\
     \(1\) & \(\mathbb{T}\)   & \(\mathbb{Z}/12\) \\
-    \(2\) & \(S_2\)          & \(\mathbb{Z}/10\)
+    \(2\) & \(S_2\)          & \(\mathbb{Z}/10\) \\
   \end{tabular}
 \end{center}
+for closed surfaces with small genus.
+
+In \cite{korkmaz-mccarthy} Korkmaz-McCarthy showed that
+eventhough \(\Mod(S_2^b)\) is not perfect, its commutator subgroup is.
+In addition, they also show \([\Mod(S_g^b), \Mod(S_g^b)]\) is normaly generated
+by a single mapping class.
 
-% TODO: Look for a proof of this? In any case, cite a reference
 \begin{proposition}\label{thm:commutator-is-perfect}
   The commutator subgroup \(\Mod(S_2^b)' = [\Mod(S_2^b), \Mod(S_2^b)]\) is
   perfect -- i.e. \(\Mod(S_2^b)^{(2)} = [\Mod(S_2^b)', \Mod(S_2^b)']\) is the
   whole of \(\Mod(S_2^b)'\).
 \end{proposition}
 
-% TODO: Comment on the proof of this?
 \begin{proposition}\label{thm:commutator-normal-gen}
   If \(g \ge 2\) and \(\alpha, \beta \subset S_g\) are simple closed curves
   with \(\#(\alpha \cap \beta) = 1\) then \(\Mod(S_g)'\) is \emph{normally
@@ -37,15 +106,54 @@
   \tau_\beta^{-1} \in N \normal \Mod(S_g)'\) then \(\Mod(S_g)' \subset N\).
 \end{proposition}
 
+These past few results combined paint a remarkably clear picture of the
+Abelianization \(\Mod(S)^\ab\) and the commutator \(\Mod(S)'\), but this is
+still a far cry from a general undertanding of the structure of \(\Mod(S)\)
+itself. For that, we need to intruduce some extra relations. To that end, we
+study certain branched covers \(S \to \mathbb{D} \setminus \{x_1, \ldots,
+x_r\}\) and how they may be used to relate both mapping class groups. This is
+what is known as\dots
+
 \section{The Birman-Hilden Theorem}
 
+Let \(S_{0, r}^1 = \mathbb{D} \setminus \{x_1, \ldots, x_r\}\) be the surface
+of genus \(0\) with \(r\) punctures and \(1\) boundary component. We begin our
+investigation by providing an alternative description of its mapping class
+group. Namely, we show that \(\Mod(S_{0, r}^1)\) is the braid group on \(r\)
+strands.
+
 \begin{definition}
   The \emph{braid group on \(n\) strands} \(B_n\) is the fundamental group
   \(\pi_1(C(\mathbb{D}, n), *)\) of the unordered configuration space
-  \(C(\mathbb{D}, n)\) of \(n\) distinct points in the disk \(\mathbb{D}\).
+  \(C(\mathbb{D}, n) = \mfrac{C^{\operatorname{ord}}(\mathbb{D},
+  n)}{\mathfrak{S}_n}\) of \(n\) distinct points in the interior of the disk.
+  The elements of \(B_n\) are referred to as \emph{braids}.
 \end{definition}
 
-% TODO: Cite the paper by Artin
+\begin{example}
+  Given \(i = 1, \ldots, n-1\), we define \(\sigma_i \in B_n\) as in
+  Figure~\ref{fig:braid-group-generator}.
+\end{example}
+
+\begin{figure}[ht]
+  \centering
+  \includegraphics[width=.25\linewidth]{images/braid-group-generator.eps}
+  \caption{The braid $\sigma_i$.}
+  \label{fig:braid-group-generator}
+\end{figure}
+
+\begin{note}
+  We would like to point out that to the third Reidemeister move translates to
+  the so called \emph{braid relations}
+  \[
+    \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_1 \sigma_i,
+  \]
+  in \(B_n\), which motivates the name used in Example~\ref{ex:braid-relation}.
+\end{note}
+
+In his seminal paper on braid groups, Artin \cite{artin} gave the following
+finite presentation of \(B_n\).
+
 \begin{theorem}[Artin]
   \[
     B_n =
@@ -55,40 +163,201 @@
       \sigma_i \sigma_{i+1} \sigma_i & = \sigma_{i+1} \sigma_i \sigma_{i+1}
       \ \text{for all} \ i, \\
       \sigma_i \sigma_j & = \sigma_j \sigma_i
-        \ \text{for} \ j \ne i + 1 \ \text{and} \ j \ne i - 1
+      \ \text{for} \ j \ne i + 1 \ \text{and} \ j \ne i - 1
     \end{aligned}
     \right\rangle
   \]
 \end{theorem}
 
-\begin{corollary}
+As promised, we now show that \(B_n\) coincides with \(\Mod(S_{0, n}^1)\).
+Recall from Theorem~\ref{thm:generalized-birman-seq} that there is an exact
+sequence
+\begin{center}
+  \begin{tikzcd}
+      1 \rar
+      & B_n \rar{\operatorname{push}}
+      & \Mod(S_{0, n}^1) \rar
+      & \Mod(\mathbb{D}) \rar
+      & 1,
+  \end{tikzcd}
+\end{center}
+given that \(\Homeo^+(\mathbb{D}, \mathbb{S}^1)\) is contractible -- see
+Example~\ref{ex:alexander-trick}. But \(\Mod(\mathbb{D}) = 1\). Hence
+we get\dots
+
+\begin{proposition}
   The map \(\operatorname{push} : B_n \to \Mod(S_{0, n}^1)\) is a group
   isomorphism.
-\end{corollary}
+\end{proposition}
 
-\begin{example}
-  The center \(Z(B_n)\) is freely generated by \(z = (\sigma_1 \cdots \sigma_{n
-  - 1})^n\).
+
+\begin{minipage}[b]{.45\linewidth}
+\begin{example}\label{ex:braid-group-center}
+  Using the capping exact sequence from Proposition~\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
+  to see that \(\operatorname{push} : B_n \to \Mod(S_{0, n}^1)\) 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 freely
+  generated by \(z = (\sigma_1 \cdots \sigma_{n - 1})^n\).
+\end{example}
+\end{minipage}
+\hspace{.5cm} %
+\begin{minipage}[b]{.45\textwidth}
+  \centering
+  \includegraphics[width=.18\linewidth]{images/braid-group-center.eps}
+  \captionof{figure}{The clockwise rotation by $\sfrac{2\pi}{n}$ about an axis center
+  around the punctures $x_1, \ldots, x_n$ of $S_{0, n}^1$.}
+  \label{fig:braid-group-center}
+\end{minipage}
+
+To get from \(S_{0, n}^1\) to surfaces of genus \(g > 0\) we may consider the
+\emph{hyperelliptic involution} \(\iota : S_g \isoto S_g\) which rotates
+\(S_g\) by \(\pi\) around some axis, as shown in
+Figure~\ref{fig:hyperelliptic-involution}. Given \(\ell < g\) and \(b = 1, 2\),
+we can also embed \(S_\ell^b\) in \(S_g\) in such way that \(\iota\) restricts
+to an involution\footnote{This involution does not fix $\partial S_\ell^b$
+point-wise.} \(S_\ell^b \isoto S_\ell^b\).
+
+\begin{figure}[ht]
+  \centering
+  \includegraphics[width=\linewidth]{images/hyperelliptic-involution.eps}
+  \caption{The hyperellipctic involution $\iota$.}
+  \label{fig:hyperelliptic-involution}
+\end{figure}
+
+It is clear from Figure~\ref{fig:hyperelliptic-involution} that the quotients
+\(\mfrac{S_\ell^1}{\iota}\) and \(\mfrac{S_\ell^2}{\iota}\) are both disks,
+with boundary corresponding to the projection of the boundaries of \(S_\ell^1\)
+and \(S_\ell^2\), respectively. Given \(b = 1, 2\), the quotient map \(S_\ell^b
+\to \mfrac{S_\ell^b}{\iota} \cong \mathbb{D}\) is a double cover with \(2\ell +
+b\) branch points corresponding to the fixed points of \(\iota\). We may thus
+regard \(\mfrac{S_\ell^b}{\iota}\) as the disk \(S_{0, 2\ell + b}^1\) with
+\(2\ell + b\) punctures in its interior, as shown in
+Figure~\ref{fig:hyperelliptic-covering}. We also draw the curves \(\alpha_1,
+\ldots, \alpha_{2\ell} \subset S_\ell^b\) of the Humphreys generators of
+\(\Mod(S_g)\). Since these curves are invariant under the action of \(\iota\),
+they descend to arcs \(\bar{\alpha}_1, \ldots, \bar{\alpha}_{2\ell + b} \subset
+S_{0, 2\ell + b}^1\) joining the punctures of the quotient surface.
+
+\begin{figure}[ht]
+  \centering
+  \includegraphics[width=.77\linewidth]{images/hyperelliptic-covering.eps}
+  \caption{The double branched covers given by $\iota$.}
+  \label{fig:hyperelliptic-covering}
+\end{figure}
+
+\begin{example}\label{ex:push-generators-description}
+  The map \(\operatorname{push} : B_{2\ell + b} \to \Mod(S_{0, 2\ell + b}^1)\)
+  takes \(\sigma_i\) to the half-twist
+  \(h_{\bar{\alpha}_i}\) about the arc \(\bar{\alpha}_i \subset S_{0, 2\ell +
+  b}^1\).
 \end{example}
 
+We now study the homemorphisms of \(S_\ell^1\) and \(S_\ell^2\) that descend to
+the quotient surfaces and their mapping classes, known as \emph{the symmetric
+mapping clases}.
+
 \begin{definition}
-  Let \(\ell \ge 0\) and \(b = 1, 2\). The \emph{symmetric mapping class group
-  of \(S_\ell^1\)} is the subgroup \(\SMod(S_\ell^1) = \{ [\phi] \in
-  \Mod(S_\ell^1) : [\phi, \iota] = 1 \}\).
+  Let \(\ell \ge 0\) and \(b = 1, 2\). The \emph{group of symmetric
+  homeomorphisms of \(S_\ell^b\)} is \(\SHomeo^+(S_\ell^b, \partial S_\ell^b) =
+  \{\phi \in \Homeo^+(S_\ell^b, \partial S_\ell^b) : [\phi, \iota] = 1\}\). The
+  \emph{symmetric mapping class group of \(S_\ell^b\)} is the subgroup
+  \(\SMod(S_\ell^1) = \{ [\phi] \in \Mod(S_\ell^b) : \phi \in
+  \SHomeo^+(S_\ell^b, \partial S_\ell^b) \}\).
 \end{definition}
 
+Fix \(b = 1\) or \(2\). It follows from the universal property of quotients
+that any \(\phi \in \SHomeo^+(S_\ell^b, \partial S_\ell^b)\) defines a
+homeomorphism \(\bar \phi : S_{0, 2\ell+b}^1 \isoto S_{0, 2\ell+b}^1\). This
+yeilds a homomorphism of topological groups
+\begin{align*}
+  \SHomeo^+(S_\ell^b, \partial S_\ell^b)
+  & \to \Homeo^+(S_{0, 2\ell + b}^1, \partial S_{0, 2\ell + b}^1) \\
+  \phi
+  & \mapsto \bar \phi,
+\end{align*}
+which is surjective because any \(\psi \in \Homeo^+(S_{0, 2\ell + b}^1,
+\partial S_{0, 2\ell + b}^1)\) lifts to \(S_\ell^b\).
+
+It is also not hard to see \(\SHomeo^+(S_\ell^b, \partial S_\ell^b) \to
+\Homeo^+(S_{0, 2\ell + b}^1, \partial S_{0, 2\ell + b}^1)\) is injective: the
+only cadidates for elements of its kernel are \(1\) and \(\iota\), but
+\(\iota\) is not an element of \(\SHomeo^+(S_\ell^b, \partial S_\ell^b)\) since
+it does not fix \(\partial S_\ell^b\) point-wise. Now since we have a
+continuous bijective homomorphism we find
+\[
+  \begin{split}
+    \pi_0(\SHomeo^+(S_\ell^b, \partial S_\ell^b))
+    & \cong \pi_0(\Homeo^+(S_{0, 2\ell+b}^1, \partial S_{0, 2\ell+b}^1))     \\
+    & = \mfrac{\Homeo^+(S_{0,2\ell+b}^1, \partial S_{0, 2\ell+b}^1)}{\simeq} \\
+    & = \Mod(S_{0, 2\ell+b}^1)                                               \\
+    & \cong B_{2\ell + b}
+  \end{split}
+\]
+
+We would like to say \(\pi_0(\SHomeo^+(S_\ell^b, \partial S_\ell^b)) =
+\SMod(S_\ell^b)\), but a priori the story looks a little more complicated:
+\(\phi, \psi \in \SHomeo^+(S_\ell^b, \partial S_\ell^b)\) define the same class
+in \(\SMod(S_\ell^b)\) if they are isotopic, but they may not lie in same
+connected component of \(\SHomeo^+(S_\ell^b, \partial S_\ell^b)\) if they are
+not isotopic \emph{through homeomorphisms symmetric homeomorphisms}.
+Birman-Hilden \cite{birman-hilden} showed that this is never the case.
+
 \begin{theorem}[Birman-Hilden]
-  If \(\phi \simeq \psi \in \Homeo^+(S_\ell^1, \partial S_\ell^1)\) both
-  commute with \(\iota : S_\ell^1 \isoto S_\ell^1\) then \(\phi\) and \(\psi\)
-  are isotopic through homeomorphisms that commute with \(\iota\). In
-  particular, there is an isomorphism
+  If \(\phi, \psi \in \SHomeo^+(S_\ell^b, \partial S_\ell^b)\) are isotopic
+  then \(\phi\) and \(\psi\) are isotopic through symmetric homeomorphisms. In
+  particular, there are isomorphism
   \begin{align*}
-    \SMod(S_\ell^1) & \isoto B_{2 g + b} \\
+    \SMod(S_\ell^b) & \isoto \Mod(S_{0, 2\ell + b}) &
+    \SMod(S_\ell^b) & \isoto B_{2 g + b}            \\
+             [\phi] & \mapsto [\bar \phi]           &
              [\phi] & \mapsto \operatorname{push}^{-1}([\bar \phi]).
   \end{align*}
 \end{theorem}
 
-\begin{theorem}[Birman-Hilden without boundary]
+\begin{example}
+  Using the notation of Figure~\ref{fig:hyperelliptic-covering}, the
+  Birman-Hilden isomorphism \(\SMod(S_\ell^b) \isoto \Mod(S_{0, 2g + b})\)
+  takes \(\tau_{\alpha_i}\) to the half twist \(h_{\bar{\alpha}_i} \in
+  \Mod(S_{0, 2g + b})\). This can be checked by looking at
+  \(\iota\)-invaratiant anular neighborhoods of the curves \(\alpha_i\) --
+  \cite[Section~9.4]{farb-margalit}.
+\end{example}
+
+\begin{example}
+  The Birman-Hilden isomorphism \(\SMod(S_\ell^1) \isoto \Mod(S_{0,
+  2\ell+1}^1)\) takes the twists \(\tau_\delta \in \SMod(S_\ell^1)\) about the
+  boundary \(\delta = \partial S_\ell^1\) to \(\tau_{\bar\delta}^2 \in
+  \Mod(S_{0, 2\ell+1}^1)\). Similarly, \(\SMod(S_\ell^2) \isoto \Mod(S_{0,
+  2\ell+2})\) takes \(\tau_{\delta_1} \tau_{\delta_2} \in \SMod(S_\ell^2)\) to
+  \(\tau_{\bar\delta_1} = \tau_{\bar\delta_2}\). In light of
+  Example~\ref{ex:push-generators-description},
+  Example~\ref{ex:braid-group-center} gives us the so called \emph{\(k\)-chain
+  relations} in \(\SMod(S_\ell^b) \subset \Mod(S_g)\).
+  \[
+    \begin{array}{rcll}
+      (\sigma_1 \cdots \sigma_k)^{2k+2} = z^2 \in B_{k + 1} &
+      \rightsquigarrow &
+      (\tau_{\alpha_1} \cdots \tau_{\alpha_k})^{2k + 2}
+        = \tau_\delta &
+      \text{for } k = 2 \ell \text{ even} \\
+      (\sigma_1 \cdots \sigma_k)^{k+1} = z \in B_{k + 1} &
+      \rightsquigarrow &
+      (\tau_{\alpha_1} \cdots \tau_{\alpha_k})^{k + 1}
+        = \tau_{\delta_1} \tau_{\delta_2} &
+      \text{for } k = 2 \ell + 1 \text{ odd}
+    \end{array}
+  \]
+\end{example}
+
+We may also exploit the quotient \(\mfrac{S_g}{\iota} \cong \mathbb{S}^2\) to
+obtain other relations. Since \(\iota\) has \(2g + 2\) fixed points in \(S_g\),
+we get branched double cover \(S_g \to S_{0, 2g+2}\).
+
+\begin{theorem}[Birman-Hilden without boundary]\label{thm:boundaryless-birman-hilden}
   If \(g \ge 2\) then we have an exact sequence
   \begin{center}
     \begin{tikzcd}
@@ -105,14 +374,56 @@
 \end{theorem}
 
 \begin{example}
-  The \(k\)-chain relation.
+  Let \(\alpha_1, \ldots, \alpha_{2g}, \delta \subset S_g\) be  and \(\delta
+  \subset S_g\) be as in Figure~\ref{fig:hyperellipitic-relations}. Then
+  \begin{equation}\label{eq:hyperelliptic-eq}
+    [\iota]
+    = \tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1}
+      \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta.
+  \end{equation}
+  Indeed, \(C_{\Mod(S_g)}([\iota]) \to \Mod(S_{0, 2g+2})\) takes \(\tau_\delta
+  \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1}\) to the rotation from
+  Figure~\ref{fig:hyperelliptic-relation-rotation}, while \(\tau_{\alpha_1}
+  \cdots \tau_{\alpha_{2g}} \tau_\delta\) is taken to its inverse. By
+  Theorem~\ref{thm:boundaryless-birman-hilden}, \(\tau_\delta
+  \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots
+  \tau_{\alpha_{2g}} \tau_\delta \in \ker (C_{\Mod(S_g)}([\iota]) \to
+  \Mod(S_{0, 2g+2})) = \langle [\iota] \rangle \cong \mathbb{Z}/2\). Given the
+  fact \(\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1}
+  \cdots \tau_{\alpha_{2g}} \tau_\delta\) inverts the orientation of
+  \(\alpha_1\), (\ref{eq:hyperelliptic-eq}) follows. In particular, we obtain
+  the so called \emph{hyperelliptic relations} of \(\Mod(S_g)\).
+  \begin{align*}
+    (\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1}
+    \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta)^2
+    & = 1 \\
+    [\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1}
+    \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta, \tau_\delta]
+    & = 1
+  \end{align*}
 \end{example}
 
-\begin{example}
-  The hyperelliptic relations.
-\end{example}
+\begin{minipage}[b]{.45\textwidth}
+  \centering
+  \includegraphics[width=.7\linewidth]{images/hyperelliptic-relation.eps}
+  \captionof{figure}{The curves from the Humphreys generators of $\Mod(S_g)$
+  and the curve $\delta$ from the hyperelliptic relations.}
+  \label{fig:hyperellipitic-relations}
+\end{minipage}
+\hspace{.5cm} %
+\begin{minipage}[b]{.45\textwidth}
+  \centering
+  \includegraphics[width=.33\linewidth]{images/sphere-rotation.eps}
+  \captionof{figure}{The  clockwise rotation by $\sfrac{\pi}{g + 1}$ about an
+  axis centered around the punctures of $S_{0, 2g + 1}$.}
+  \label{fig:hyperelliptic-relation-rotation}
+\end{minipage}
+
+Wajnryb used the \(k\)-chain relations and the hyperelliptic relations to
+derive a presentation of the mapping class group of a closed surface,
+which is widely considered to the the standard presentation of
+\(\Mod(S_g)\) \cite{wajnryb}.
 
-% TODO: Cite the original paper?
 \begin{theorem}[Wajnryb]\label{thm:wajnryb-presentation}
   If \(\alpha_0, \ldots, \alpha_g\) are as in 
   % TODO: Reference the drawing of the curves somewhere
@@ -153,4 +464,22 @@
   \end{enumerate}
 \end{theorem}
 
-% TODO: Add the drawings for the what each one of these curves mean
+\begin{note}
+  The mapping classes \(b_0, \ldots, b_3, d\)  in the statement of
+  Theorem~\ref{thm:wajnryb-presentation} correspond to the Dehn twists about
+  the curves \(\beta_0, \ldots, \beta_3, \delta \subset S_g\) highlighted in
+  Figure~\ref{fig:wajnryb-presentation-curves}, so Wajnryb's presentation is
+  not as intractible as it might look at first glance.
+\end{note}
+
+\begin{figure}[ht]
+  \centering
+  \includegraphics[width=.7\linewidth]{images/wajnryb-presentation-curves.eps}
+  \caption{The curves from Wajnryb's presentation.}
+  \label{fig:wajnryb-presentation-curves}
+\end{figure}
+
+This presentation paints a much clearer picture of the structure of
+\(\Mod(S_g)\). Nevertheless, its linear representations and many other of if
+its group-theoretic aspects remain a mistery. This will be the focus of the
+next chapter.

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -46,6 +46,11 @@
   \(\tau_\alpha, \tau_\beta \in \Mod(S)\) are conjugate.
 \end{fact}
 
+
+\begin{example}\label{ex:braid-relation}
+  Braid relation
+\end{example}
+
 \begin{theorem}\label{thm:mcg-is-fg}
   Let \(S\) be an orientable surface of genus \(g \ge 1\), potentially with
   punctures and boundary components. Then the pure mapping class group
@@ -61,8 +66,8 @@
   \begin{center}
     \begin{tikzcd}
       1 \rar
-      & \pi_1(S, p_0) \rar{\operatorname{push}}
-      & \Mod(S \setminus \{p_0\}, p_0) \rar{\operatorname{forget}}
+      & \pi_1(S, x_0) \rar{\operatorname{push}}
+      & \Mod(S \setminus \{x_0\}, x_0) \rar{\operatorname{forget}}
       & \Mod(S) \rar
       & 1.
     \end{tikzcd}
@@ -72,20 +77,21 @@
 % TODO: Explan configuration spaces
 \begin{align*}
   C^{\operatorname{ord}}(S, n)
-  & = \{ (p_1, \ldots, p_n) \in S^n : p_i \ne p_j \forall i \ne j \} \\
+  & = \{ (x_1, \ldots, x_n) \in \operatorname{int}(S)^n :
+          x_i \ne x_j \ \text{for}\ i \ne j \} \\
   C(S, n)
   & = \mfrac{C^{\operatorname{ord}}(S, n)}{\mathfrak{S}_n}
 \end{align*}
 
 % TODO: Comment on the proof
-\begin{theorem}[Generalized Birman exact sequence]
+\begin{theorem}[Generalized Birman exact sequence]\label{thm:generalized-birman-seq}
   Suppose \(\pi_1(\Homeo^+(S, \partial S), 1) = 1\). Then there is an exact
   sequence
   \begin{center}
     \begin{tikzcd}
       1 \rar
-      & \pi_1(C(S, n), [p_1, \ldots, p_n]) \rar{\operatorname{push}}
-      & \Mod(S \setminus \{p_1, \ldots, p_n\}) \rar{\operatorname{forget}}
+      & \pi_1(C(S, n), [x_1, \ldots, x_n]) \rar{\operatorname{push}}
+      & \Mod(S \setminus \{x_1, \ldots, x_n\}) \rar{\operatorname{forget}}
       & \Mod(S) \rar
       & 1.
     \end{tikzcd}
@@ -196,16 +202,16 @@
   \begin{center}
     \begin{tikzcd}
       1 \rar
-      & \pi_1(S_{g, n}, p_0) \rar{\operatorname{push}}
+      & \pi_1(S_{g, n}, x_0) \rar{\operatorname{push}}
       & \PMod(S_{g, n + 1}) \rar{\operatorname{forget}}
       & \PMod(S_{g, n}) \rar
       & 1,
     \end{tikzcd}
   \end{center}
-  where \(S_{g, n + 1} = S_{g, n} \setminus \{p_0\}\). Since \(g \ge 1\),
-  \(\pi_1(S_{g, n}, p_0)\) is generated by finitely many nonseparating loops.
-  We have seen that \(\operatorname{push} : \pi_1(S_{g, n}, p_0) \to
-  \Mod(S_{g, n+1}, p_0)\) takes simple loops to products of twists about
+  where \(S_{g, n + 1} = S_{g, n} \setminus \{x_0\}\). Since \(g \ge 1\),
+  \(\pi_1(S_{g, n}, x_0)\) is generated by finitely many nonseparating loops.
+  We have seen that \(\operatorname{push} : \pi_1(S_{g, n}, x_0) \to
+  \Mod(S_{g, n+1}, x_0)\) takes simple loops to products of twists about
   nonseparating simple curves. Furthermore, we may once again lift the
   generators of \(\PMod(S_{g, n})\) to Dehn twists about nonseparating simple
   curves in \(S_{g, n + 1}\). This goes to show that \(\PMod(S_{g, n + 1})\) is
@@ -291,7 +297,7 @@
   \end{center}
 \end{theorem}
 
-\begin{corollary}[Humphreys generators]
+\begin{corollary}[Humphreys generators]\label{thm:humphreys-gens}
   If \(g \ge 2\) then \(\Mod(S_g)\) is generated by the twists
   \(\tau_{\alpha_0}, \ldots, \tau_{\alpha_{2g}} \in \Mod(S_g)\).
 \end{corollary}