memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: ab471a10bb3bb1227bb0f20142f1a887982387af
Parent: 5cdf0ad6cc81f7a81a9ed7472007b5e5bc48fb87
Author: Pablo <pablo-pie@riseup.net>
Date: Thu, 27 Jun 2024 13:24:54 +0000

Revised all of the chapters

Diffstat

4 files changed, 238 insertions, 215 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	15	9	6
Modified	sections/presentation.tex	85	45	40
Modified	sections/representations.tex	132	68	64
Modified	sections/twists.tex	221	116	105

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -47,9 +47,10 @@ homeomorphisms.
   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
+  \(\delta_2\) 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
@@ -127,7 +128,6 @@ There are many variations of Definition~\ref{def:mcg}. For example\dots
   \(\phi\) of \(f\).
 \end{observation}
 
-% TODO: Change this notation?
 \begin{definition}
   Given an orientable surface \(\Sigma\) and a puncture \(x \in
   \widebar\Sigma\) of \(\Sigma\), denote by \(\Mod(\Sigma, x) \subset
@@ -191,8 +191,11 @@ mapping class groups.
   Given a simple closed curve \(\alpha \subset \Sigma\), any \(f \in
   \Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\) has a representative \(\phi \in
   \Homeo^+(\Sigma, \partial \Sigma)\) fixing \(\alpha\) point-wise -- so that
-  \(\phi\) restricts to a homeomorphism of \(\Sigma \setminus \alpha\). There
-  is a group homomorphism
+  \(\phi\) restricts to a homeomorphism of \(\Sigma \setminus \alpha\).
+  Furthermore, if \(\phi\!\restriction_{\Sigma \setminus \alpha} \simeq 1\) in
+  \(\Sigma \setminus \alpha\) then \(\phi \simeq 1 \in \Homeo^+(\Sigma,
+  \partial \Sigma)\) -- see \cite[Proposition~3.20]{farb-margalit}. There is
+  thus a group homomorphism
   \begin{align*}
     \operatorname{cut} : \Mod(\Sigma)_{\vec{[\alpha]}}
     & \to \Mod(\Sigma\setminus\alpha) \\

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -15,10 +15,10 @@ Theorem~\ref{thm:wajnryb-presentation}.
   \(\Sigma_0^3 = \Sigma_0^4 \cup_{\delta_1} \mathbb{D}^2\) and \(\Sigma_{0,1}^3
   = \Sigma_0^4 \cup_{\delta_1} (\mathbb{D}^2 \setminus \{ 0 \})\), as well as
   the map \(\operatorname{push} : \pi_1(\Sigma_0^3, 0) \to
-  \Mod(\Sigma_{0,1}^3)\). Let \(\eta_1, \eta_2, \eta_3 \subset \Sigma_0^3\) be
-  the loops from Figure~\ref{fig:lantern-relation-capped}, so that \([\eta_1]
-  \cdot [\eta_2] = [\eta_3]\). From Observation~\ref{ex:push-simple-loop} we
-  obtain
+  \Mod(\Sigma_{0,1}^3)\). Let \(\eta_1, \eta_2, \eta_3 : \mathbb{S}^1 \to
+  \Sigma_0^3\) be the loops from Figure~\ref{fig:lantern-relation-capped}, so
+  that \([\eta_1] \cdot [\eta_2] = [\eta_3]\) in \(\pi_1(\Sigma_0^3, 0)\). From
+  Observation~\ref{ex:push-simple-loop} we obtain
   \[
     (\tau_{\delta_2} \tau_\alpha^{-1}) (\tau_{\delta_3} \tau_\gamma^{-1})
     = \operatorname{push}([\eta_1]) \cdot \operatorname{push}([\eta_2])
@@ -29,13 +29,13 @@ Theorem~\ref{thm:wajnryb-presentation}.
   Using the capping exact sequence from Observation~\ref{ex:capping-seq}, we
   can then see \(\tau_{\delta_2} \tau_\alpha^{-1} \tau_{\delta_3}
   \tau_\gamma^{-1}, \tau_\beta \tau_{\delta_4}^{-1} \in \Mod(\Sigma_0^4)\)
-  differ by a power of \(\tau_{\delta_1}\). In fact, it follows from the
-  Alexander method that \((\tau_{\delta_2} \tau_\alpha^{-1} \tau_{\delta_3}
-  \tau_\gamma^{-1}) (\tau_\beta \tau_{\delta_4}^{-1})^{-1} =
-  \tau_{\delta_1}^{-1} \in \Mod(\Sigma_0^4)\). Now the disjointness relations
-  \([\tau_{\delta_i}, \tau_\alpha] = [\tau_{\delta_i}, \tau_\beta] =
-  [\tau_{\delta_i}, \tau_\gamma] = 1\) give us the \emph{lantern relation}
-  (\ref{eq:lantern-relation}) in \(\Mod(\Sigma_0^4)\).
+  differ by a power of \(\tau_{\delta_1}\). In fact, one can show
+  \((\tau_{\delta_2} \tau_\alpha^{-1} \tau_{\delta_3} \tau_\gamma^{-1})
+  (\tau_\beta \tau_{\delta_4}^{-1})^{-1} = \tau_{\delta_1}^{-1} \in
+  \Mod(\Sigma_0^4)\). Now the disjointness relations \([\tau_{\delta_i},
+  \tau_\alpha] = [\tau_{\delta_i}, \tau_\beta] = [\tau_{\delta_i}, \tau_\gamma]
+  = 1\) give us the \emph{lantern relation} (\ref{eq:lantern-relation}) in
+  \(\Mod(\Sigma_0^4)\).
   \begin{equation}\label{eq:lantern-relation}
     \tau_\alpha \tau_\beta \tau_\gamma
     = \tau_{\delta_1} \tau_{\delta_2} \tau_{\delta_3} \tau_{\delta_4}
@@ -169,13 +169,12 @@ sequence
       1 \rar
       & B_n \rar{\operatorname{push}}
       & \Mod(\Sigma_{0, n}^1) \rar
-      & \Mod(\mathbb{D}^2) \rar
+      & \cancelto{1}{\Mod(\mathbb{D}^2)} \rar
       & 1,
   \end{tikzcd}
 \end{center}
-given that \(\Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) is contractible by
-Observation~\ref{ex:alexander-trick}. But \(\Mod(\mathbb{D}^2) = 1\). Hence we
-get\dots
+for \(\Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) is contractible by
+Observation~\ref{ex:alexander-trick}. In other words\dots
 
 \begin{proposition}
   The map \(\operatorname{push} : B_n \to \Mod(\Sigma_{0, n}^1)\) is a group
@@ -324,7 +323,7 @@ not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden
   \Mod(\Sigma_{0, 2\ell+2})\) takes \(\tau_{\delta_1} \tau_{\delta_2} \in
   \SMod(\Sigma_\ell^2)\) to \(\tau_{\bar\delta_1} = \tau_{\bar\delta_2}\). In
   light of Observation~\ref{ex:push-generators-description},
-  Observation~\ref{ex:braid-group-center} gives us the so called
+  Observation~\ref{ex:braid-group-center} translates into the so called
   \emph{\(k\)-chain relations} in \(\SMod(\Sigma_\ell^b) \subset
   \Mod(\Sigma_g)\).
   \[
@@ -376,16 +375,20 @@ to obtain other relations. Since \(\iota\) has \(2g + 2\) fixed points in
   \(\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1}\) 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}, \(\tau_\delta
-  \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots
-  \tau_{\alpha_{2g}} \tau_\delta \in \ker (C_{\Mod(\Sigma_g)}([\iota]) \to
-  \Mod(\Sigma_{0, 2g+2})) = \langle [\iota] \rangle \cong \mathbb{Z}/2\). Given
-  the fact \(\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1}
+  inverse. By Theorem~\ref{thm:boundaryless-birman-hilden},
+  \[
+    \tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1}
+    \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta
+    \in \ker (C_{\Mod(\Sigma_g)}([\iota]) \to \Mod(\Sigma_{0, 2g+2}))
+    = \langle [\iota] \rangle \cong \mathbb{Z}/2.
+  \]
+  One can then show \(\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1}
   \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta\) inverts the
-  orientation of \(\alpha_1\), (\ref{eq:hyperelliptic-eq}) follows. In
-  particular, we obtain the so called \emph{hyperelliptic relations}
-  (\ref{eq:hyperelliptic-rel-1}) and (\ref{eq:hyperelliptic-rel-2}) in
-  \(\Mod(\Sigma_g)\).
+  orientation of \(\alpha_1\), so \(\tau_\delta \tau_{\alpha_{2g}} \cdots
+  \tau_{\alpha_1} \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta \ne 1\)
+  and (\ref{eq:hyperelliptic-eq}) follows. In particular, we obtain the so
+  called \emph{hyperelliptic relations} (\ref{eq:hyperelliptic-rel-1}) and
+  (\ref{eq:hyperelliptic-rel-2}) in \(\Mod(\Sigma_g)\).
   \begin{align}\label{eq:hyperelliptic-rel-1}
     (\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1}
     \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta)^2
@@ -418,13 +421,15 @@ to obtain other relations. Since \(\iota\) has \(2g + 2\) fixed points in
 
 \section{Presentations of Mapping Class Groups}
 
-There are numerous known presentations of \(\Mod(\Sigma_{g, r}^b)\), such as
-the ones due to Birman-Hilden \cite{birman-hilden}, Gervais \cite{gervais} and
-many others. Wajnryb \cite{wajnryb} derived a presentation of
-\(\Mod(\Sigma_g)\) using the relations discussed in
-Chapter~\ref{ch:dehn-twists} and Section~\ref{birman-hilden}. This is a
-particularly satisfactory presentation, since all of its relations can be
-explained in terms of the geometry of curves in \(\Sigma_g\).
+Having explored some of the relations in \(\Mod(\Sigma)\), it is natural to ask
+if these relations are enough to completely describe the structure of
+\(\Mod(\Sigma)\). Different presentations of mapping class groups are due to
+the work of Birman-Hilden \cite{birman-hilden}, Gervais \cite{gervais} and many
+others. Wajnryb \cite{wajnryb} derived a presentation of \(\Mod(\Sigma_g)\)
+only using the relations discussed in Chapter~\ref{ch:dehn-twists} and
+Section~\ref{birman-hilden}. This is quite a satisfactory result, for we have
+seen that all of these relations can be explained in terms of the topology of
+\(\Sigma_g\).
 
 \begin{theorem}[Wajnryb]\label{thm:wajnryb-presentation}
   Suppose \(g \ge 3\). If \(\alpha_0, \ldots, \alpha_g\) are as in
@@ -433,19 +438,19 @@ explained in terms of the geometry of curves in \(\Sigma_g\).
   of \(\Mod(\Sigma_g)\) with generators \(a_0, \ldots a_{2g}\) subject to the
   following relations.
   \begin{enumerate}
-    \item The disjointness relations \([a_i, a_j] = 1\) for \(\alpha_i\) and
-      \(\alpha_j\) disjoint.
+    \item The \emph{disjointness relations} \([a_i, a_j] = 1\) for \(\alpha_i\)
+      and \(\alpha_j\) disjoint.
 
-    \item The braid relations \(a_i a_j a_i = a_j a_i a_j\) for \(\alpha_i\)
-      and \(\alpha_j\) crossing once.
+    \item The \emph{braid relations} \(a_i a_j a_i = a_j a_i a_j\) for
+      \(\alpha_i\) and \(\alpha_j\) crossing once.
 
-    \item The \(3\)-chain relation \((a_1 a_2 a_3)^4 = a_0 b_0\), where
+    \item The \emph{\(3\)-chain relation} \((a_1 a_2 a_3)^4 = a_0 b_0\), where
       \[
         b_0 = (a_4 a_3 a_2 a_1 a_1 a_2 a_3 a_4)
         a_0 (a_4 a_3 a_2 a_1 a_1 a_2 a_3 a_4)^{-1}.
       \]
 
-    \item The lantern relation \(a_0 b_2 b_1 = a_1 a_3 a_5 b_3\), where
+    \item The \emph{lantern relation} \(a_0 b_2 b_1 = a_1 a_3 a_5 b_3\), where
       \begin{align*}
         b_1 & = (a_4 a_5 a_3 a_4)^{-1} a_0 (a_4 a_5 a_3 a_4) \\
         b_2 & = (a_2 a_3 a_1 a_2)^{-1} b_1 (a_2 a_3 a_1 a_2) \\
@@ -454,8 +459,8 @@ explained in terms of the geometry of curves in \(\Sigma_g\).
               (a_4 a_3 a_2)^{-1}.
       \end{align*}
 
-    \item The hyperelliptic relation \([a_{2g} \cdots a_1 a_1 \cdots a_{2g}, d]
-      = 1\), where \(d = n_g\) for \(n_1 = a_1\), \(n_2 = b_0\) and
+    \item The \emph{hyperelliptic relation} \([a_{2g} \cdots a_1 a_1 \cdots
+      a_{2g}, d] = 1\), where \(d = n_g\) for \(n_1 = a_1\), \(n_2 = b_0\) and
       \begin{align*}
         n_{i + 2} & = w_i n_i w_i^{-1} \\
         w_i & = (a_{2i + 4} a_{2i + 3} a_{2i + 2} n_{i + 1})

diff --git a/sections/representations.tex b/sections/representations.tex
@@ -4,11 +4,11 @@ Having built a solid understanding of the combinatorics of Dehn twists, we are
 now ready to attack the problem of classifying the representations of
 \(\Mod(\Sigma_g)\) of sufficiently small dimension. As promised, our strategy
 is to make use of the \emph{geometrically-motivated} relations derived in
-Chapter~\ref{ch:relations}.
+Chapter~\ref{ch:dehn-twists} and Chapter~\ref{ch:relations}.
 
 Historically, these relations have been exploited by Funar \cite{funar},
 Franks-Handel \cite{franks-handel} and others to establish the triviality of
-low-dimensional representations, culminating Korkmaz' \cite{korkmaz} recent
+low-dimensional representations, culminating in Korkmaz' \cite{korkmaz} recent
 classification of representations of dimension \(n \le 2 g\) for \(g \ge 3\).
 The goal of this chapter is to provide a concise account of Korkmaz' results,
 starting by\dots
@@ -20,7 +20,7 @@ starting by\dots
   Abelian. In particular, if \(g \ge 3\) then \(\rho\) is trivial.
 \end{theorem}
 
-Like so many of the results we have encountered so far, the proof of
+Like some of the results we have encountered so far, the proof of
 Theorem~\ref{thm:low-dim-reps-are-trivial} is elementary in nature: we proceed
 by induction on \(g\) and tedious case analysis. We begin by the base case \(g
 = 2\).
@@ -166,8 +166,8 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
 
   We claim that if \(E_{\alpha_2 = \lambda} = E_{\beta_2 = \lambda}\) then
   \(E_{\alpha_2 = \lambda}\) is \(\Mod(\Sigma_2^b)\)-invariant. Indeed, by
-  change of coordinates we can always find \(f, g, h_i \in \Mod(\Sigma_2^b)\)
-  with
+  Observation~\ref{ex:change-of-coordinates-crossing} we can always find \(f,
+  g, h_i \in \Mod(\Sigma_2^b)\) with
   \begin{align*}
     f \cdot [\alpha_2]      & = [\alpha_1]
     &
@@ -234,29 +234,30 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
   a solvable group is trivial.
 
   Finally, if \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\) and
-  the Jordan form of \(L_{\alpha_2}\) is given by (8) then
+  the Jordan form of \(L_{\alpha_2}\) is given by (8) then the disjointness
+  relations \([\tau_{\alpha_2}, \tau_{\alpha_1}] = [\tau_{\alpha_2},
+  \tau_{\beta_1}] = [\tau_{\beta_2}, \tau_{\alpha_1}] = [\tau_{\beta_2},
+  \tau_{\beta_1}] = 1\) implies that \(L_{\alpha_1}\) and \(L_{\beta_1}\)
+  preserve the eigenspaces of both \(L_{\alpha_2}\) and \(L_{\beta_2}\), so
   \[
     0
     \subsetneq E_{\alpha_2 = \lambda} \cap E_{\beta_2 = \lambda}
     \subsetneq E_{\alpha_2 = \lambda}
     \subsetneq V
   \]
-  is a flag of subspaces invariant under \(L_{\alpha_1}\) and \(L_{\beta_1}\),
-  for \(\alpha_2\) and \(\beta_2\) are disjoint from \(\alpha_1 \cup \beta_1\)
-  and thus \([\tau_{\alpha_2}, \tau_{\alpha_1}] = [\tau_{\alpha_2},
-  \tau_{\beta_1}] = [\tau_{\beta_2}, \tau_{\alpha_1}] = [\tau_{\beta_2},
-  \tau_{\beta_1}] = 1\). In this case we can find a basis for \(\mathbb{C}^3\)
-  with respect to which the matrices of \(L_{\alpha_1}\) and \(L_{\beta_1}\) are
-  both upper triangular with \(\lambda\) along the diagonal: take \(v_1, v_2,
-  v_3 \in \mathbb{C}^3\) with \(v_1 \in E_{\alpha_2 = \lambda} \cap E_{\beta_2
-  = \lambda}\), \(v_2 \in V_{L_{\alpha_2}}\) and adjust \(v_3\) to get the
-  desired diagonal entry. Any such pair of matrices satisfying the braid
-  relation (\ref{eq:braid-rel-induction-basis}) commute.
-
-  Similarly, if \(L_{\alpha_2}\) has Jordan form (9) and \(E_{\alpha_2 = \lambda}
-  \ne E_{\beta_2 = \lambda}\) we use (\ref{eq:braid-rel-induction-basis})
-  to conclude \(L_{\alpha_1}\) and \(L_{\beta_1}\) commute -- again, see
-  \cite[Proposition~5.1]{korkmaz}. We are done.
+  is a flag of subspaces invariant under \(L_{\alpha_1}\) and \(L_{\beta_1}\).
+  In this case we can find a basis for \(\mathbb{C}^3\) with respect to which
+  the matrices of \(L_{\alpha_1}\) and \(L_{\beta_1}\) are both upper
+  triangular with \(\lambda\) along the diagonal: take \(v_1, v_2, v_3 \in
+  \mathbb{C}^3\) with \(v_1 \in E_{\alpha_2 = \lambda} \cap E_{\beta_2 =
+  \lambda}\) and \(v_2 \in V_{L_{\alpha_2}}\). Any such pair of matrices
+  satisfying the braid relation (\ref{eq:braid-rel-induction-basis}) commute.
+
+  Similarly, if \(L_{\alpha_2}\) has Jordan form (9) and \(E_{\alpha_2 =
+  \lambda} \ne E_{\beta_2 = \lambda}\) we use
+  (\ref{eq:braid-rel-induction-basis}) to conclude \(L_{\alpha_1}\) and
+  \(L_{\beta_1}\) commute -- again, see \cite[Proposition~5.1]{korkmaz}. We are
+  done.
 \end{proof}
 
 We are now ready to establish the triviality of low-dimensional
@@ -264,14 +265,15 @@ representations.
 
 \begin{proof}[Proof of Theorem~\ref{thm:low-dim-reps-are-trivial}]
   Let \(g \ge 1\), \(b \ge 0\) and fix \(\rho : \Mod(\Sigma_g^b) \to
-  \GL_n(\mathbb{C})\) with \(n < 2g\). As promised, we proceed by induction on
-  \(g\). The base case \(g = 1\) is again clear from the fact \(n = 1\) and
+  \GL_n(\mathbb{C})\) with \(n < 2g\). We want to show
+  \(\rho(\Mod(\Sigma_g^b))\) is Abelian. As promised, we proceed by induction
+  on \(g\).
+  The base case \(g = 1\) is again clear from the fact \(n = 1\) and
   \(\GL_1(\mathbb{C}) = \mathbb{C}^\times\). The case \(g = 2\) was also
-  established in Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}. Now
-  suppose \(g \ge 3\) and every \(m\)-dimensional representation of \(\Sigma_{g
-  - 1}^{b'}\) has Abelian image for \(m < 2(g - 1)\). Let us show \(\rho\) has
-  Abelian image.
+  established in Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}.
 
+  Now suppose \(g \ge 3\) and every \(m\)-dimensional representation of
+  \(\Sigma_{g - 1}^{b'}\) has Abelian image for \(m < 2(g - 1)\).
   Let \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g, \gamma_1, \ldots,
   \gamma_{g - 1}, \eta_1, \ldots, \eta_{b-1} \subset \Sigma_g^b\) be the curves
   from the Lickorish generators of \(\Mod(\Sigma_g^b)\), as in
@@ -338,8 +340,8 @@ representations.
 
   If no sum of the form \(\bigoplus_i E_{\alpha_g = \lambda_i}\) has dimension
   lying between \(2\) and \(n - 2\), then there must be at most \(2\) distinct
-  eigenvalues and \(\dim E_{\alpha_g = \lambda} = 1, n - 1, n\) for all
-  eigenvalues \(\lambda\) of \(L_{\alpha_g}\). Hence the Jordan form of
+  eigenvalues \(\lambda\) of \(L_{\alpha_g}\), and \(\dim E_{\alpha_g =
+  \lambda} = 1, n - 1, n\) for all such \(\lambda\). Hence the Jordan form of
   \(L_{\alpha_g}\) has to be one of
   \begin{align*}
     \begin{pmatrix}
@@ -398,16 +400,16 @@ representations.
   \ne E_{\beta_g = \lambda}\) or \(E_{\alpha_g = \lambda} = E_{\beta_g =
   \lambda}\). If \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\), then \(W
   = E_{\alpha_g = \lambda} \cap E_{\beta_g = \lambda}\) is a \((n -
-  2)\)-dimensional \(\Mod(\Sigma)\)-invariant subspace: since \(L_{\alpha_g}\)
-  and \(L_{\beta_g}\) are conjugate and \(\beta_g\) lies outside of \(\Sigma\),
-  both \(E_{\alpha_g = \lambda}\) and \(E_{\beta_g = \lambda}\) are
+  2)\)-dimensional \(\Mod(\Sigma)\)-invariant subspace: since \(\beta_g\) lies
+  outside of \(\Sigma\) and \(L_{\alpha_g}, L_{\beta_g}\) are conjugate, both
+  \(E_{\alpha_g = \lambda}\) and \(E_{\beta_g = \lambda}\) are
   \(\Mod(\Sigma)\)-invariant \((n - 1)\)-dimensional subspaces.
 
   Finally, we consider the case where \(E_{\alpha_g = \lambda} = E_{\beta_g
   = \lambda}\). In this situation, as in the proof of
   Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}, it follows from
-  the change of coordinates principle that there are \(f_i, g_i, h_i \in
-  \Mod(\Sigma_g^b)\) with
+  Observation~\ref{ex:change-of-coordinates-crossing} that there are \(f_i,
+  g_i, h_i \in \Mod(\Sigma_g^b)\) with
   \begin{align*}
     f_i \tau_{\alpha_g}    f_i^{-1} & = \tau_{\alpha_i}
     &
@@ -422,12 +424,10 @@ representations.
     h_i \tau_{\beta_g} h_i^{-1} & = \tau_{\eta_i}
   \end{align*}
   and thus
-  \[
-    E_{\alpha_1 = \lambda} = \cdots = E_{\alpha_g = \lambda}
-    = E_{\beta_1 = \lambda} = \cdots = E_{\beta_g = \lambda}
-    = E_{\gamma_1 = \lambda} = \cdots = E_{\gamma_{g - 1} = \lambda}
-    = E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b-1} = \lambda}.
-  \]
+  \(E_{\alpha_1 = \lambda} = \cdots = E_{\alpha_g = \lambda} = E_{\beta_1 =
+  \lambda} = \cdots = E_{\beta_g = \lambda} = E_{\gamma_1 = \lambda} = \cdots =
+  E_{\gamma_{g - 1} = \lambda} = E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b-1}
+  = \lambda}\).
 
   In particular, we can find a basis for \(\mathbb{C}^n\) with respect to
   which the matrix of any Lickorish generator has the form
@@ -444,31 +444,32 @@ representations.
   \(\Mod(\Sigma_g^b)\) is perfect, it follows that \(\rho(\Mod(\Sigma_g^b))\) is
   trivial. This concludes the proof \(\rho(\Mod(\Sigma_g^b))\) is Abelian.
 
-  To see that \(\rho(\Mod(\Sigma_g^b)) = 1\) for \(g \ge 3\) we note that, since
-  \(\rho(\Mod(\Sigma_g^b))\) is Abelian, \(\rho\) factors though the Abelianization
-  map \(\Mod(\Sigma_g^b) \to \Mod(\Sigma_g^b)^\ab = \mfrac{\Mod(\Sigma_g^b)}{[\Mod(\Sigma_g^b),
-  \Mod(\Sigma_g^b)]}\). Now recall from Proposition~\ref{thm:trivial-abelianization}
-  that \(\Mod(\Sigma_g^b)^\ab = 0\) for \(g \ge 3\). In other words, \(\rho\)
-  factors though the homomorphism \(1 \to \GL_n(\mathbb{C})\). We are done.
+  To see that \(\rho(\Mod(\Sigma_g^b)) = 1\) for \(g \ge 3\) we note that,
+  since \(\rho\) has Abelian image and thus factors though the Abelianization
+  map \(\Mod(\Sigma_g^b) \to \Mod(\Sigma_g^b)^\ab =
+  \mfrac{\Mod(\Sigma_g^b)}{[\Mod(\Sigma_g^b), \Mod(\Sigma_g^b)]}\). Now recall
+  from Proposition~\ref{thm:trivial-abelianization} that \(\Mod(\Sigma_g^b)^\ab
+  = 0\) for \(g \ge 3\). We are done.
 \end{proof}
 
 Having established the triviality of the low-dimensional representations \(\rho
-: \Mod(\Sigma_g^b) \to \GL_n(\mathbb{C})\), all that remains for us is to understand
-the \(2g\)-dimensional representations of \(\Mod(\Sigma_g^b)\). We certainly know a
-nontrivial example of such, namely the symplectic representation \(\psi :
-\Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) from
-Example~\ref{ex:symplectic-rep}. Surprisingly, this turns out to be
+: \Mod(\Sigma_g^b) \to \GL_n(\mathbb{C})\), all that remains for us is to
+understand the \(2g\)-dimensional representations of \(\Mod(\Sigma_g^b)\). We
+certainly know a nontrivial example of such, namely the symplectic
+representation \(\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\)
+from Example~\ref{ex:symplectic-rep}. Surprisingly, this turns out to be
 \emph{essentially} the only example of a nontrivial \(2g\)-dimensional
 representation in the compact case. More precisely,
 
 \begin{theorem}[Korkmaz]\label{thm:reps-of-dim-2g-are-symplectic}
-  Let \(g \ge 3\) and \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\). Then
-  \(\rho\) is either trivial or conjugate to the symplectic
+  Let \(g \ge 3\) and \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\).
+  Then \(\rho\) is either trivial or conjugate to the symplectic
   representation\footnote{Here the map $\Mod(\Sigma_g^b) \to
   \operatorname{Sp}_{2g}(\mathbb{Z})$ is given by the composition of the
-  inclusion morphism $\Mod(\Sigma_g^b) \to \Mod(\Sigma_g)$ with the usual symplectic
-  representation $\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})$.}
-  \(\Mod(\Sigma_g^b) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(\Sigma_g^b)\).
+  inclusion morphism $\Mod(\Sigma_g^b) \to \Mod(\Sigma_g)$ with the usual
+  symplectic representation $\psi : \Mod(\Sigma_g) \to
+  \operatorname{Sp}_{2g}(\mathbb{Z})$.} \(\Mod(\Sigma_g^b) \to
+  \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(\Sigma_g^b)\).
 \end{theorem}
 
 Unfortunately, the limited scope of these master thesis does not allow us to
@@ -478,8 +479,11 @@ heart of this proof lies in a result about representations of the product
 main lemma}. Namely\dots
 
 \begin{lemma}[Korkmaz' main lemma]\label{thm:main-lemma}
-  Given \(i = 1, \ldots, n\), denote by \newline \(a_i = (1, \ldots, 1,
-  \sigma_1, 1, \ldots 1)\) and \(b_i = (1, \ldots, 1, \sigma_2, 1, \ldots, 1)\)
+  Given \(i = 1, \ldots, n\), denote by 
+  \begin{align*}
+    a_i & = (1, \ldots, 1, \sigma_1, 1, \ldots, 1) &
+    b_i & = (1, \ldots, 1, \sigma_2, 1, \ldots, 1)
+  \end{align*}
   the \(n\)-tuples in \(B_3^n\) whose \(i\)-th coordinates are \(\sigma_1\) and
   \(\sigma_2\), respectively, and with all other coordinates equal to \(1\).
   Let \(m \ge 2n\) and \(\rho : B_3^n \to \GL_m(\mathbb{C})\) be a
@@ -557,12 +561,12 @@ nontrivial \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\). Furthermore,
 Korkmaz also argues that we can find a basis for \(\mathbb{C}^{2g}\) with
 respect to which the matrices of \(\rho(\tau_{\gamma_1}), \ldots,
 \rho(\tau_{\gamma_{g - 1}}), \rho(\tau_{\eta_1}), \ldots,
-\rho(\tau_{\eta_{b-1}})\) also agrees with the action of \(\Mod(\Sigma_g^b)\) on
-\(H_1(\Sigma_g, \mathbb{C})\), concluding the classification of \(2g\)-dimensional
-representations.
+\rho(\tau_{\eta_{b-1}})\) also agree with the action of \(\Mod(\Sigma_g^b)\) on
+\(H_1(\Sigma_g, \mathbb{C})\), concluding the classification of
+\(2g\)-dimensional representations.
 
 % TODO: Add some final comments about how the rest of the landscape of
 % representations is generally unknown and how there is a lot to study in here
 Recently, Kasahara \cite{kasahara} also classified the \((2g+1)\)-dimensional
-representations of \(\Mod(\Sigma_g^b)\) for \(g \ge 7\) in terms of certain twisted
-\(1\)-cohomology groups.
+representations of \(\Mod(\Sigma_g^b)\) for \(g \ge 7\) in terms of certain
+twisted \(1\)-cohomology groups.

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -10,7 +10,7 @@ a convenient generating set for \(\Mod(\Sigma)\), known as the set of
 \emph{Dehn twists}.
 
 The idea here is to reproduce the proof of injectivity in
-Example~\ref{ex:torus-mcg}: by cutting across curves and arcs, we can always
+Observation~\ref{ex:torus-mcg}: by cutting across curves and arcs, we can always
 decompose a surface into copies of \(\mathbb{D}^2\) and \(\mathbb{D}^2
 \setminus \{0\}\). Observation~\ref{ex:alexander-trick} and
 Observation~\ref{ex:mdg-once-punctured-disk} then imply the triviality of
@@ -194,7 +194,8 @@ too.
 \begin{observation}[Disjointness relations]
   Given \(f \in \Mod(\Sigma)\), \([f, \tau_\alpha] = 1 \iff f \cdot [\alpha] =
   [\alpha]\). In particular, \([\tau_\alpha, \tau_\beta] = 1\) for \(\alpha\)
-  and \(\beta\) disjoint.
+  and \(\beta\) disjoint, for we can choose a representative of \(\tau_\beta\)
+  whose support is disjoint from \(\alpha\).
 \end{observation}
 
 \begin{observation}
@@ -204,7 +205,7 @@ too.
   [\alpha] = [\beta]\) and then apply Observation~\ref{ex:conjugate-twists}.
 \end{observation}
 
-\begin{fundamental-observation}[Braid relations]\label{ex:braid-relation}
+\begin{observation}[Braid relations]\label{ex:braid-relation}
   Given \(\alpha, \beta \subset \Sigma\) with \(\#(\alpha \cap \beta) = 1\), it
   is not hard to check that \(\tau_\beta \tau_\alpha \cdot [\beta] =
   [\alpha]\). From Observation~\ref{ex:conjugate-twists} we then get
@@ -213,7 +214,7 @@ too.
   \[
     \tau_\alpha \tau_\beta \tau_\alpha = \tau_\beta \tau_\alpha \tau_\beta.
   \]
-\end{fundamental-observation}
+\end{observation}
 
 A perhaps less obvious fact about Dehn twists is\dots
 
@@ -225,36 +226,42 @@ A perhaps less obvious fact about Dehn twists is\dots
 \end{theorem}
 
 The proof of Theorem~\ref{thm:mcg-is-fg} is simple in nature: we proceed by
-induction in \(g\) and \(r\). On the other hand, the induction steps are
-somewhat involved and require two ingredients we have not encountered so far,
-namely the \emph{Birman exact sequence} and the \emph{modified graph of
-curves}.
+induction in \(g\), \(b\) and \(r\). On the other hand, the induction steps
+require two ingredients we have not encountered so far, namely the \emph{Birman
+exact sequence} and the \emph{modified graph of curves}. We now provide a
+concise account of these ingredients.
 
 \section{The Birman Exact Sequence}
 
 Having the proof of Theorem~\ref{thm:mcg-is-fg} in mind, it is interesting to
-consider the relationship between the mapping class group of \(\Sigma_{g, r}^b\) and
-that of \(\Sigma_{g, r+1}^b = \Sigma_{g, r}^b \setminus \{ x \}\) for some \(x\) in the
-interior \((\Sigma_{g, r}^b)\degree\) of \(\Sigma_{g, r}^b\). Indeed, this will later
-allow us to establish the induction on the number of punctures \(r\).
-
-Given an orientable surface \(\Sigma\) and \(x_1, \ldots, x_n \in \Sigma\degree\),
-denote by \(\Mod(\Sigma \setminus \{x_1, \ldots, x_n\})_{\{x_1, \ldots,
-x_n\}} \subset \Mod(\Sigma \setminus \{x_1, \ldots, x_n\})\) the subgroup of mapping
-classes \(f\) that permute \(x_1, \ldots, x_n\) -- i.e. \(f \cdot x_i =
-x_{\sigma(i)}\) for some \(\sigma \in S_n\). We certainly have a
-surjective homomorphism \(\operatorname{forget} : \Mod(\Sigma \setminus \{x_1,
-\ldots, x_n\})_{\{x_1, \ldots, x_n\}} \to \Mod(\Sigma)\) which ``\emph{forgets} the
-additional punctures \(x_1, \ldots, x_n\) of \(\Sigma \setminus \{x_1, \ldots,
-x_n\}\),'' but what is its kernel?
+consider the relationship between the mapping class group of \(\Sigma_{g,
+r}^b\) and that of \(\Sigma_{g, r+1}^b = \Sigma_{g, r}^b \setminus \{ x \}\)
+for some \(x\) in the interior \((\Sigma_{g, r}^b)\degree\) of \(\Sigma_{g,
+r}^b\). Indeed, this will later allow us to establish the induction step on the
+number of punctures \(r\).
+
+Given an orientable surface \(\Sigma\) and \(x_1, \ldots, x_n \in
+\Sigma\degree\), denote by \(\Mod(\Sigma \setminus \{x_1, \ldots,
+x_n\})_{\{x_1, \ldots, x_n\}} \subset \Mod(\Sigma \setminus \{x_1, \ldots,
+x_n\})\) the subgroup of mapping classes \(f\) that permute \(x_1, \ldots,
+x_n\) -- i.e. \(f \cdot x_i = x_{\sigma(i)}\) for some permutation \(\sigma \in
+S_n\). We certainly have a surjective homomorphism
+\begin{align*}
+  \operatorname{forget} :
+  \Mod(\Sigma \setminus \{x_1, \ldots, x_n\})_{\{x_1, \ldots, x_n\}}
+  & \to \Mod(\Sigma) \\
+  [\phi] & \mapsto [\tilde\phi]
+\end{align*}
+which ``\emph{forgets} the additional punctures \(x_1, \ldots,
+x_n\) of \(\Sigma \setminus \{x_1, \ldots, x_n\}\),'' but what is its kernel?
 
 To answer this question, we consider the configuration space \(C(\Sigma, n) =
-\mfrac{C^{\operatorname{ord}}(\Sigma, n)}{S_n}\) of \(n\) (unordered)
-points in the interior of \(\Sigma\) -- where \(C^{\operatorname{ord}}(\Sigma, n) = \{
-(x_1, \ldots, x_n) \in (\Sigma\degree)^n : x_i \ne x_j \ \text{for}\ i \ne j \}\).
-Denote \(\Homeo^+(\Sigma, \partial \Sigma)_{x_1, \ldots, x_n} = \{\phi \in \Homeo^+(\Sigma,
-\partial \Sigma) : \phi(x_i) = x_i \}\). From the fibration\footnote{See
-\cite[Chapter~4]{hatcher} for a reference.}
+\mfrac{C^{\operatorname{ord}}(\Sigma, n)}{S_n}\) of \(n\) (unordered) points in
+the interior of \(\Sigma\) -- where \(C^{\operatorname{ord}}(\Sigma, n) = \{
+  (x_1, \ldots, x_n) \in (\Sigma\degree)^n : x_i \ne x_j \ \text{for}\ i \ne j
+\}\). Denote \(\Homeo^+(\Sigma, \partial \Sigma)_{x_1, \ldots, x_n} = \{\phi
+\in \Homeo^+(\Sigma, \partial \Sigma) : \phi(x_i) = x_i \}\). From the
+fibration\footnote{See \cite[Chapter~4]{hatcher} for a reference.}
 \[
   \arraycolsep=1.4pt
   \begin{array}{ccrcl}
@@ -267,8 +274,8 @@ Denote \(\Homeo^+(\Sigma, \partial \Sigma)_{x_1, \ldots, x_n} = \{\phi \in \Home
 and its long exact sequence in homotopy we then get\dots
 
 \begin{theorem}[Birman exact sequence]\label{thm:birman-exact-seq}
-  Suppose \(\pi_1(\Homeo^+(\Sigma, \partial \Sigma), 1) = 1\). Then there is an exact
-  sequence
+  Suppose \(\pi_1(\Homeo^+(\Sigma, \partial \Sigma), 1) = 1\). Then there is an
+  exact sequence
   \begin{center}
     \begin{tikzcd}[cramped]
       1 \rar
@@ -282,8 +289,8 @@ and its long exact sequence in homotopy we then get\dots
 \end{theorem}
 
 \begin{remark}
-  Notice that \(C(\Sigma, 1) = \Sigma\degree \simeq S\). Hence for \(n = 1\)
-  Theorem~\ref{thm:birman-exact-seq} gives us a sequence
+  Notice that \(C(\Sigma, 1) = \Sigma\degree \simeq \Sigma\). Hence for \(n =
+  1\) Theorem~\ref{thm:birman-exact-seq} gives us a sequence
   \begin{center}
     \begin{tikzcd}
       1 \rar
@@ -295,13 +302,14 @@ and its long exact sequence in homotopy we then get\dots
   \end{center}
 \end{remark}
 
-We may regard a simple loop \(\alpha \subset C(\Sigma, n)\) based at \([x_1, \ldots,
-x_n]\) as \(n\) disjoint curves \(\alpha_1, \ldots, \alpha_n \subset \Sigma\) with
-\(\alpha_i(0) = x_i\) and \(\alpha_i(1) = x_{\sigma(i)}\) for some \(\sigma \in
-S_n\). The element \(\operatorname{push}([\alpha]) \in \Mod(\Sigma)\) can
-then be seen as the mapping class that ``\emph{pushes} a neighborhood of
-\(x_{\sigma(i)}\) towards \(x_i\) along the curve \(\alpha_i^{-1}\),'' as shown
-in Figure~\ref{fig:push-map} for the case \(n = 1\). Indeed, this goes to
+We may regard a simple loop \(\alpha : \mathbb{S}^1 \to C(\Sigma, n)\) based at
+\([x_1, \ldots, x_n]\) as \(n\) disjoint curves \(\alpha_1, \ldots, \alpha_n :
+[0, 1] \to \Sigma\) with \(\alpha_i(0) = x_i\) and \(\alpha_i(1) =
+x_{\sigma(i)}\) for some \(\sigma \in S_n\). The element
+\(\operatorname{push}([\alpha]) \in \Mod(\Sigma)\) can then be seen as the
+mapping class that ``\emph{pushes} a neighborhood of \(x_{\sigma(i)}\) towards
+\(x_i\) along the curve \(\alpha_i^{-1}\),'' as shown in
+Figure~\ref{fig:push-map} for the case \(n = 1\). Indeed, this goes to
 show\dots
 
 \begin{fundamental-observation}\label{ex:push-simple-loop}
@@ -313,19 +321,20 @@ show\dots
 \begin{figure}[ht]
   \centering
   \includegraphics[width=.35\linewidth]{images/push-map.eps}
-  \caption{The inclusion $\operatorname{push} : \pi_1(\Sigma, x) \to \Mod(\Sigma)$ maps
-  a simple loop $\alpha \subset \Sigma$ to the mapping class supported at an annular
-  neighborhood $A$ of $\alpha$ which takes the arc joining the boundary
-  components $\delta_i \subset \partial A$ in the left-hand side to the yellow
-  arc in the right-hand side.}
+  \caption{The inclusion $\operatorname{push} : \pi_1(\Sigma, x) \to
+  \Mod(\Sigma)$ maps a simple loop $\alpha : \mathbb{S}^1 \to \Sigma$ to the
+  mapping class supported at an annular neighborhood $A$ of $\alpha$. Inside
+  this neighborhood, $\operatorname{push}([\alpha])$ takes the arc joining the
+  boundary components $\delta_i \subset \partial A$ in the left-hand side to
+  the yellow arc in the right-hand side.}
   \label{fig:push-map}
 \end{figure}
 
 \section{The Modified Graph of Curves}
 
 Having established Theorem~\ref{thm:birman-exact-seq}, we now need to address
-the induction step in the genus \(g\) of \(\Sigma_{g, r}^b\). Our strategy is to
-apply the following lemma from geometric group theory.
+the induction step in the genus \(g\) of \(\Sigma_{g, r}^b\). Our strategy is
+to apply the following lemma from geometric group theory.
 
 \begin{lemma}\label{thm:ggt-lemma}
   Let \(G\) be a group and \(\Gamma\) be a \emph{connected} graph with \(G
@@ -333,16 +342,17 @@ apply the following lemma from geometric group theory.
   transitively on both \(V(\Gamma)\) and \(\{(v, w) \in V(\Gamma)^2 :
   v \text{ --- } w \text{ in } \Gamma \}\). If \(v, w \in V(\Gamma)\) are
   connected by an edge and \(g \in G\) is such that \(g \cdot w = v\) then
-  \(G\) is generated by \(G_v\) and \(g\).
+  \(G\) is generated by \(g\) and the stabilizer \(G_v\).
 \end{lemma}
 
-We are interested, of course, in the group \(G = \PMod(\Sigma_{g, r}^b)\). As for
-the graph \(\Gamma\), we consider\dots
+We are interested, of course, in the group \(G = \PMod(\Sigma_{g, r}^b)\). As
+for the graph \(\Gamma\), we consider\dots
 
 \begin{definition}
-  The \emph{modified graph of nonseparating curves \(\hat{\mathcal{N}}(\Sigma)\)
-  of a surface \(\Sigma\)} is the graph whose vertices are (unoriented) isotopy
-  classes of nonseparating simple closed curves in \(\Sigma\) and
+  The \emph{modified graph of nonseparating curves
+  \(\hat{\mathcal{N}}(\Sigma)\) of a surface \(\Sigma\)} is the graph whose
+  vertices are (unoriented) isotopy classes of nonseparating simple closed
+  curves in \(\Sigma\) and
   \[
     \text{\([\alpha]\) --- \([\beta]\) in \(\hat{\mathcal{N}}(\Sigma)\)}
     \iff \#(\alpha \cap \beta) = 1,
@@ -357,9 +367,9 @@ Observation~\ref{ex:change-of-coordinates-crossing} that the actions of
 \(\{([\alpha], [\beta]) \in V(\hat{\mathcal{N}}(\Sigma_{g, r}^b))^2 : \#(\alpha
 \cap \beta) = 1 \}\) are both transitive. But why should
 \(\hat{\mathcal{N}}(\Sigma_{g, r}^b)\) be connected?
-
 Historically, the modified graph of nonseparating curves first arose as a
-\emph{modified} version of another graph, known as\dots
+\emph{modified} version of another graph, known as \emph{the graph of of
+curves}.
 
 \begin{definition}
   Given a surface \(\Sigma\), the \emph{graph of curves \(\mathcal{C}(\Sigma)\)
@@ -385,7 +395,7 @@ of sporadic cases, \(\mathcal{C}(\Sigma_{g, r})\) is connected.
 
 In other words, given simple closed curves \(\alpha, \beta \subset \Sigma_{g,
 r}\), we can find closed \(\alpha = \alpha_1, \alpha_2, \ldots, \alpha_n =
-\beta\) in \(\Sigma_{g, r}\) with \(\alpha_i\) and \(\alpha_{i+1}\) disjoint.
+\beta\) in \(\Sigma_{g, r}\) with \(\alpha_i\) disjoint from \(\alpha_{i+1}\).
 Now if \(\alpha\) and \(\beta\) are nonseparating, by inductively adjusting
 this sequence of curves we then get\dots
 
@@ -402,32 +412,10 @@ Theorem~\ref{thm:mcg-is-fg}.
   Let \(\Sigma_{g, r}^b\) be the orientable surface of genus \(g \ge 1\) with
   \(r\) punctures and \(b\) boundary components. We want to establish that
   \(\PMod(\Sigma_{g, r}^b)\) is generated by a finite number of Dehn twists
-  about nonseparating simple closed curves or boundary components.
+  about nonseparating simple closed curves or boundary components. As promised,
+  we proceed by triple induction on \(r\), \(g\) and \(b\).
 
-  First, observe that if \(b \ge 1\) and \(\partial \Sigma_{g, r}^b = \delta_1
-  \cup \cdots \cup \delta_p\) then, by recursively applying the capping exact
-  sequence
-  \begin{center}
-    \begin{tikzcd}
-      1 \rar &
-      \langle \tau_{\delta_1} \rangle \rar &
-      \PMod(\Sigma_{g, r}^b) \rar{\operatorname{cap}} &
-      \PMod(\Sigma_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\})) \rar &
-      1
-    \end{tikzcd}
-  \end{center}
-  from Observation~\ref{ex:capping-seq}, it suffices to show that \(\Sigma_{g,
-  n}\) is finitely generated by twists about nonseparating simple closed
-  curves. Indeed, if \(\PMod(\Sigma_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2
-  \setminus \{0\})) \cong \PMod(\Sigma_{g, r+1}^{b-1})\) is finitely generated
-  by twists about nonseparating curves or boundary components, then we may lift
-  the generators of \(\PMod(\Sigma_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2
-  \setminus \{0\}))\) to Dehn twists about the corresponding curves in
-  \(\Sigma_{g, r}^b\) and add \(\tau_{\delta_1}\) to the generating set.
-
-  It thus suffices to consider the boundaryless case \(\Sigma_{g, r}\). As
-  promised, we proceed by double induction on \(r\) and \(g\). For the base
-  case, it is clear from Observation~\ref{ex:torus-mcg} and
+  For the base case, it is clear from Observation~\ref{ex:torus-mcg} and
   Observation~\ref{ex:punctured-torus-mcg} that \(\Mod(\mathbb{T}^2) \cong
   \Mod(\Sigma_{1, 1}) \cong \operatorname{SL}_2(\mathbb{Z})\) are generated by
   the Dehn twists about the curves \(\alpha\) and \(\beta\) from
@@ -454,8 +442,6 @@ Theorem~\ref{thm:mcg-is-fg}.
     \label{fig:torus-mcg-generators}
   \end{figure}
 
-  \newpage
-
   Now suppose \(\PMod(\Sigma_{g, r})\) is finitely-generated by twists about
   nonseparating curves for \(g \ge 2\) or \(g = 1\) and \(r > 1\). In both
   case, \(\chi(\Sigma_{g, r}) = 2 - 2g - r < 0\) and thus
@@ -471,13 +457,14 @@ Theorem~\ref{thm:mcg-is-fg}.
       & 1,
     \end{tikzcd}
   \end{center}
-  where \(\Sigma_{g, r + 1} = \Sigma_{g, r} \setminus \{x\}\). Since \(g \ge 1\),
-  \(\pi_1(\Sigma_{g, r}, x)\) is generated by finitely many nonseparating loops.
-  We have seen in Observation~\ref{ex:push-simple-loop} that \(\operatorname{push}
-  : \pi_1(\Sigma_{g, r}, x) \to \Mod(\Sigma_{g, r+1}, x)\) takes nonseparating simple
-  loops to products of twists about nonseparating simple curves. Furthermore,
-  we may once again lift the generators of \(\PMod(\Sigma_{g, r})\) to Dehn twists
-  about nonseparating simple curves in \(\Sigma_{g, r + 1}\). This goes to show that
+  where \(\Sigma_{g, r + 1} = \Sigma_{g, r} \setminus \{x\}\). Since \(g \ge
+  1\), \(\pi_1(\Sigma_{g, r}, x)\) is generated by finitely many nonseparating
+  loops. We have seen in Observation~\ref{ex:push-simple-loop} that
+  \(\operatorname{push} : \pi_1(\Sigma_{g, r}, x) \to \Mod(\Sigma_{g, r+1},
+  x)\) takes nonseparating simple loops to products of twists about
+  nonseparating simple curves. Furthermore, we may lift the
+  generators of \(\PMod(\Sigma_{g, r})\) to Dehn twists about the corresponding
+  curves in \(\Sigma_{g, r + 1}\). This goes to show that
   \(\PMod(\Sigma_{g, r + 1})\) is also generated by finitely many twists about
   simple curves, concluding the induction step on \(r\).
 
@@ -491,22 +478,25 @@ Theorem~\ref{thm:mcg-is-fg}.
   Observation~\ref{ex:braid-relation} that, given nonseparating \(\alpha, \beta
   \subset \Sigma_{g + 1}\) crossing once, \(\tau_\beta \tau_\alpha \cdot
   [\beta] = [\alpha]\). It thus follows from Lemma~\ref{thm:ggt-lemma} that
-  \(\Mod(\Sigma_{g + 1})\) is generated by \(\Mod(\Sigma_{g + 1})_{[\alpha]} =
-  \{ f \in \Mod(\Sigma_{g + 1}) : f \cdot [\alpha] = [\alpha]\}\) and
-  \(\tau_\beta \tau_\alpha\).
+  \(\Mod(\Sigma_{g + 1})\) is generated by \(\tau_\beta \tau_\alpha\) and
+  \(\Mod(\Sigma_{g + 1})_{[\alpha]} = \{ f \in \Mod(\Sigma_{g + 1}) : f \cdot
+  [\alpha] = [\alpha]\}\).
 
   In turn, \(\Mod(\Sigma_{g + 1})_{[\alpha]}\) has its index \(2\) subgroup
-  \(\Mod(\Sigma_{g + 1})_{\vec{[\alpha]}} = \{ f \in \Mod(\Sigma_{g + 1}) : f
-  \cdot \vec{[\alpha]} = \vec{[\alpha]}\}\) of mapping classes fixing the
-  orientation of \(\alpha\). One can check that \(\tau_\beta
-  \tau_\alpha^2 \tau_\beta \in \Mod(\Sigma_{g + 1})_{[\alpha]}\) inverts the
-  orientation of \(\alpha\) and is thus a representative of the nontrivial
+  \[
+    \Mod(\Sigma_{g + 1})_{\vec{[\alpha]}}
+    = \{ f \in \Mod(\Sigma_{g + 1}) : f \cdot \vec{[\alpha]} = \vec{[\alpha]}\}
+  \]
+  of mapping classes fixing any given choice of orientation of \(\alpha\). One
+  can check that \(\tau_\beta \tau_\alpha^2 \tau_\beta \in \Mod(\Sigma_{g +
+  1})_{[\alpha]}\) inverts the orientation of \(\alpha\) and is thus a
+  representative of the nontrivial
   \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\)-coset in
   \(\Mod(\Sigma_{g+1})_{[\alpha]}\). In particular, \(\Mod(\Sigma_{g+1})\) is
   generated by \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\), \(\tau_\beta
   \tau_\alpha\) and \(\tau_\beta \tau_\alpha^2 \tau_\beta\).
 
-  Finally, we claim \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\) is generated by
+  We now claim \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\) is generated by
   finitely many twists about nonseparating curves. First observe that
   \(\Sigma_{g+1} \setminus \alpha \cong \Sigma_{g,2}\), as shown in
   Figure~\ref{fig:cut-along-nonseparating-adds-two-punctures}.
@@ -520,6 +510,13 @@ Theorem~\ref{thm:mcg-is-fg}.
       1.
     \end{tikzcd}
   \end{equation}
+  But by the induction hypothesis, \(\PMod(\Sigma_{g, 2})\) is
+  finitely-generated by twists about nonseparating simple closed curves. As
+  before, these generators may be lifted to appropriate twists in
+  \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\). Now by (\ref{eq:cutting-seq}) we get
+  that \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\) is finitely generated by twists
+  about nonseparating curves, as desired. This concludes the induction step in
+  \(g\).
 
   \begin{figure}[ht]
     \centering
@@ -531,13 +528,27 @@ Theorem~\ref{thm:mcg-is-fg}.
     \label{fig:cut-along-nonseparating-adds-two-punctures}
   \end{figure}
 
-  Recall that, by the induction hypothesis, \(\PMod(\Sigma_{g, 2})\) is
-  finitely-generated by twists about nonseparating simple closed curves. As
-  before, these generators may be lifted to appropriate twists in
-  \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\). Now by (\ref{eq:cutting-seq}) we get
-  that \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\) is finitely generated by twists
-  about nonseparating curves, as desired. This concludes the induction step in
-  \(g\).
+  Finally, we handle the induction in \(b\). The boundaryless case \(b = 0\)
+  was already dealt with before. Now suppose \(\PMod(\Sigma_{g, s}^b)\) is
+  finitely generated by twists about simple closed curves or boundary
+  components for all \(g\) and \(s\). Fix some boundary component \(\delta
+  \subset \partial \Sigma_{g, r}^{b+1}\). From the homeomorphism \(\Sigma_{g,
+  r+1}^b \cong \Sigma_{g, r}^{b+1} \cup_\delta (\mathbb{D}^2 \setminus \{ 0
+  \})\) and the capping exact sequence from Observation~\ref{ex:capping-seq}
+  we obtain a sequence
+  \begin{center}
+    \begin{tikzcd}
+      1 \rar                                              &
+      \langle \tau_\delta \rangle \rar                    &
+      \PMod(\Sigma_{g, r}^{b+1}) \rar{\operatorname{cap}} &
+      \PMod(\Sigma_{g, r+1}^b) \rar                       &
+      1.
+    \end{tikzcd}
+  \end{center}
+  Now by induction hypothesis we may once again lift the generators of
+  \(\PMod(\Sigma_{g, r+1}^b)\) to Dehn twists about the corresponding curves in
+  \(\Sigma_{g, r}^{b+1}\) and add \(\tau_\delta\) to the generating set,
+  concluding the induction in \(b \ge 0\). We are done.
 \end{proof}
 
 There are many possible improvements to this last result. For instance, in