memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: 740f8753ea9a218bc87c996d54beefafd91bfa38
Parent: b1216255fea0f6fb36117915f1b55f02cb767dbd
Author: Pablo <pablo-pie@riseup.net>
Date: Wed, 26 Jun 2024 11:48:00 +0000

Changed the notation for the number of boundary components

p → b once and for all

Diffstat

7 files changed, 190 insertions, 177 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	images/lickorish-gens-gen-2.svg	25	15	10
Modified	images/lickorish-gens-korkmaz-proof.svg	17	11	6
Modified	images/lickorish-gens.svg	15	10	5
Modified	sections/introduction.tex	28	14	14
Modified	sections/presentation.tex	128	63	65
Modified	sections/representations.tex	102	51	51
Modified	sections/twists.tex	52	26	26

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diff --git a/images/lickorish-gens-korkmaz-proof.svg b/images/lickorish-gens-korkmaz-proof.svg
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 </svg>

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -17,17 +17,17 @@ proof.
   \(\Sigma_g\) of the sphere \(\mathbb{S}^2\) with \(g \ge 0\) copies of the
   torus \(\mathbb{T}^2 = \mfrac{\mathbb{R}^2}{\mathbb{Z}^2}\). Any compact
   connected orientable surface \(\Sigma\) is isomorphic to the surface
-  \(\Sigma_g^p\) obtained from \(\Sigma_g\) by removing \(p \ge 0\) open disks
+  \(\Sigma_g^b\) obtained from \(\Sigma_g\) by removing \(b \ge 0\) 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 \(\Sigma\)}. We also have the noncompact surface
-\(\Sigma_{g, r}^p = \Sigma_g^p \setminus \{x_1, \ldots, x_r\}\), where \(x_1,
-\ldots, x_r\) lie in the interior of \(\Sigma_g^p\). The points \(x_1, \ldots,
-x_r\) are called the \emph{punctures} of \(\Sigma_{g, r}^p\). Throughout these
+\(\Sigma_{g, r}^b = \Sigma_g^b \setminus \{x_1, \ldots, x_r\}\), where \(x_1,
+\ldots, x_r\) lie in the interior of \(\Sigma_g^b\). The points \(x_1, \ldots,
+x_r\) are called the \emph{punctures} of \(\Sigma_{g, r}^b\). Throughout these
 notes, all surfaces considered will be of the form \(\Sigma = \Sigma_{g,
-r}^p\). Any such \(\Sigma\) admits a natural compactification
+r}^b\). Any such \(\Sigma\) admits a natural compactification
 \(\widebar\Sigma\) obtained by filling its punctures. We denote \(\Sigma_{g, r}
 = \Sigma_{g, r}^0\). All closed curves \(\alpha, \beta \subset \Sigma\) we
 consider lie in the interior \(\Sigma\degree\) of \(\Sigma\) and intersect
@@ -49,14 +49,14 @@ example\dots
 \end{lemma}
 
 \begin{proof}
-  Let \(\Sigma = \Sigma_{g, r}^p\) and consider the surface \(\Sigma_\alpha\)
+  Let \(\Sigma = \Sigma_{g, r}^b\) and consider the surface \(\Sigma_\alpha\)
   obtained by cutting \(\Sigma\) across \(\alpha\), as in
   Figure~\ref{fig:change-of-coordinates}. Since \(\alpha\) is nonseparating,
   this surface has genus \(g - 1\) and two additional boundary component
   \(\delta_1, \delta_2 \subset \partial \Sigma_\alpha\), so \(\Sigma_\alpha
-  \cong \Sigma_{g-1,r}^{p+2}\). By identifying \(\delta_1\) and \(\delta_2\)
+  \cong \Sigma_{g-1,r}^{b+2}\). By identifying \(\delta_1\) and \(\delta_2\)
   we can see \(\Sigma\) as a quotient of \(\Sigma_\alpha\). Similarly,
-  \(\Sigma_\beta \cong \Sigma_{g-1, r}^{p+2}\) also has two additional boundary
+  \(\Sigma_\beta \cong \Sigma_{g-1, r}^{b+2}\) also has two additional boundary
   components \(\delta_1', \delta_2' \subset \partial \Sigma_\beta\). Now by the
   classification of surfaces we can find an orientation-preserving
   homeomorphism \(\tilde\phi : \Sigma_\alpha \isoto \Sigma_\beta\). Even more
@@ -68,7 +68,7 @@ example\dots
   As for the second part of the lemma, we consider the surface \(\Sigma_{\alpha
   \alpha'}\) obtained by cutting \(\Sigma_\alpha\) across the arc determined by
   \(\alpha'\). Since \(\alpha'\) is nonseparating, \(\Sigma_{\alpha \alpha'}
-  \cong \Sigma_{g-1, r}^{p+1}\) has one boundary component more than
+  \cong \Sigma_{g-1, r}^{b+1}\) has one boundary component more than
   \(\Sigma\), say \(\partial \Sigma_{\alpha \alpha'} = \delta \amalg \partial
   \Sigma\). The boundary component \(\delta\) is naturally subdivided into the
   four arcs in Figure~\ref{fig:change-of-coordinates}, each corresponding to
@@ -76,7 +76,7 @@ example\dots
   the pairs of arcs corresponding to the same curve we obtain the surface
   \(\mfrac{\Sigma_{\alpha \alpha'}}{\sim} \cong \Sigma\).
 
-  Likewise, \(\Sigma_{\beta \beta'} \cong \Sigma_{g-1, r}^{p+1}\) also has a
+  Likewise, \(\Sigma_{\beta \beta'} \cong \Sigma_{g-1, r}^{b+1}\) also has a
   boundary component \(\delta' \subset \partial \Sigma_{\beta \beta'}\)
   subdivided into four arcs. By the classification of surfaces we can find an
   orientation-preserving homeomorphism \(\tilde\phi : \Sigma_{\alpha \alpha'}
@@ -93,10 +93,10 @@ example\dots
 \begin{figure}[ht]
   \centering
   \includegraphics[width=.8\linewidth]{images/change-of-coords-cut.eps}
-  \caption{By cutting $\Sigma_{g, r}^p$ across $\alpha$ we obtain $\Sigma_{g-1,
-  r}^{p+2}$, where $\alpha'$ determines a yellow arc joining the two
-  additional boundary components. Now by cutting $\Sigma_{g-1, r}^{p+2}$ across
-  this arc we obtain $\Sigma_{g-1,r}^p$, with the added boundary component
+  \caption{By cutting $\Sigma_{g, r}^b$ across $\alpha$ we obtain $\Sigma_{g-1,
+  r}^{b+2}$, where $\alpha'$ determines a yellow arc joining the two
+  additional boundary components. Now by cutting $\Sigma_{g-1, r}^{b+2}$ across
+  this arc we obtain $\Sigma_{g-1,r}^b$, with the added boundary component
   subdivided into the four arcs corresponding to $\alpha$ and $\alpha'$.}
   \label{fig:change-of-coordinates}
 \end{figure}

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -67,22 +67,22 @@ their corresponding inclusion homomorphisms \(\Mod(\Sigma_0^4) \to
 twists in \(\Mod(\Sigma)\). For example\dots
 
 \begin{proposition}\label{thm:trivial-abelianization}
-  The Abelianization \(\Mod(\Sigma_g^p)^\ab =
-  \mfrac{\Mod(\Sigma_g^p)}{[\Mod(\Sigma_g), \Mod(\Sigma_g)]}\) is cyclic.
-  Moreover, if \(g \ge 3\) then \(\Mod(\Sigma_g^p)^\ab = 0\). In other words,
+  The Abelianization \(\Mod(\Sigma_g^b)^\ab =
+  \mfrac{\Mod(\Sigma_g^b)}{[\Mod(\Sigma_g), \Mod(\Sigma_g)]}\) is cyclic.
+  Moreover, if \(g \ge 3\) then \(\Mod(\Sigma_g^b)^\ab = 0\). In other words,
   \(\Mod(\Sigma_g)\) is a perfect group for \(g \ge 3\).
 \end{proposition}
 
 \begin{proof}
-  By Theorem~\ref{thm:lickorish-gens}, \(\Mod(\Sigma_g^p)^\ab\) is generated by
+  By Theorem~\ref{thm:lickorish-gens}, \(\Mod(\Sigma_g^b)^\ab\) is generated by
   the image of the Lickorish generators, which are all conjugate and thus
   represent the same class in the Abelianization. In fact, any nonseparating
-  \(\alpha \subset \Sigma_g^p\) is conjugate to the Lickorish generators too,
-  so \(\Mod(\Sigma_g^p)^\ab = \langle [\alpha] \rangle\).
+  \(\alpha \subset \Sigma_g^b\) is conjugate to the Lickorish generators too,
+  so \(\Mod(\Sigma_g^b)^\ab = \langle [\alpha] \rangle\).
 
-  Now for \(g \ge 3\) we can embed \(\Sigma_0^4\) in \(\Sigma_g^p\) in such a
+  Now for \(g \ge 3\) we can embed \(\Sigma_0^4\) in \(\Sigma_g^b\) in such a
   way that all the corresponding curves \(\alpha, \beta, \gamma, \delta_1,
-  \ldots, \delta_4 \subset \Sigma_g^p\) are nonseparating, as shown in
+  \ldots, \delta_4 \subset \Sigma_g^b\) are nonseparating, as shown in
   Figure~\ref{fig:latern-relation-trivial-abelianization}. The lantern relation
   (\ref{eq:lantern-relation}) then becomes
   \[
@@ -92,14 +92,14 @@ twists in \(\Mod(\Sigma)\). For example\dots
     + [\tau_{\delta_3}] + [\tau_{\delta_4}]
     = 4 \cdot [\tau_\alpha]
   \]
-  in \(\Mod(\Sigma_g^p)^\ab\). In other words, \([\tau_\alpha] = 0\) and thus
-  \(\Mod(\Sigma_g^p)^\ab = 0\).
+  in \(\Mod(\Sigma_g^b)^\ab\). In other words, \([\tau_\alpha] = 0\) and thus
+  \(\Mod(\Sigma_g^b)^\ab = 0\).
 \end{proof}
 
 \begin{figure}[ht]
   \centering
   \includegraphics[width=.5\linewidth]{images/lantern-relation-trivial-abelianization.eps}
-  \caption{The embedding of $\Sigma_0^4$ in $\Sigma_g^p$ for $g \ge 3$.}
+  \caption{The embedding of $\Sigma_0^4$ in $\Sigma_g^b$ for $g \ge 3$.}
   \label{fig:latern-relation-trivial-abelianization}
 \end{figure}
 
@@ -209,10 +209,10 @@ get\dots
 To get from \(\Sigma_{0, n}^1\) to surfaces of genus \(g > 0\) we may consider
 the \emph{hyperelliptic involution} \(\iota : \Sigma_g \isoto \Sigma_g\), which
 rotates \(\Sigma_g\) by \(\pi\) around some axis as in
-Figure~\ref{fig:hyperelliptic-involution}. Given \(\ell < g\) and \(p = 1, 2\),
-we can also embed \(\Sigma_\ell^p\) in \(\Sigma_g\) in such way that \(\iota\)
+Figure~\ref{fig:hyperelliptic-involution}. Given \(\ell < g\) and \(b = 1, 2\),
+we can also embed \(\Sigma_\ell^b\) in \(\Sigma_g\) in such way that \(\iota\)
 restricts to an involution\footnote{This involution does not fix $\partial
-\Sigma_\ell^p$ point-wise.} \(\Sigma_\ell^p \isoto \Sigma_\ell^p\).
+\Sigma_\ell^b$ point-wise.} \(\Sigma_\ell^b \isoto \Sigma_\ell^b\).
 
 \begin{figure}[ht]
   \centering
@@ -224,16 +224,16 @@ restricts to an involution\footnote{This involution does not fix $\partial
 It is clear from Figure~\ref{fig:hyperelliptic-involution} that the quotients
 \(\mfrac{\Sigma_\ell^1}{\iota}\) and \(\mfrac{\Sigma_\ell^2}{\iota}\) are both
 disks, with boundary corresponding to the projection of the boundaries of
-\(\Sigma_\ell^1\) and \(\Sigma_\ell^2\), respectively. Given \(p = 1, 2\), the
-quotient map \(\Sigma_\ell^p \to \mfrac{\Sigma_\ell^p}{\iota} \cong
-\mathbb{D}^2\) is a double cover with \(2\ell + p\) branch points corresponding
+\(\Sigma_\ell^1\) and \(\Sigma_\ell^2\), respectively. Given \(b = 1, 2\), the
+quotient map \(\Sigma_\ell^b \to \mfrac{\Sigma_\ell^b}{\iota} \cong
+\mathbb{D}^2\) is a double cover with \(2\ell + b\) branch points corresponding
 to the fixed points of \(\iota\). We may thus regard
-\(\mfrac{\Sigma_\ell^p}{\iota}\) as the disk \(\Sigma_{0, 2\ell + p}^1\) with
-\(2\ell + p\) punctures in its interior, as shown in
+\(\mfrac{\Sigma_\ell^b}{\iota}\) as the disk \(\Sigma_{0, 2\ell + b}^1\) with
+\(2\ell + b\) punctures in its interior, as shown in
 Figure~\ref{fig:hyperelliptic-covering}. We also draw the curves \(\alpha_1,
-\ldots, \alpha_{2\ell} \subset \Sigma_\ell^p\) of the Humphreys generators of
+\ldots, \alpha_{2\ell} \subset \Sigma_\ell^b\) of the Humphreys generators of
 \(\Mod(\Sigma_g)\). Since these curves are invariant under the action of
-\(\iota\), they descend to arcs \(\bar{\alpha}_1, \ldots, \bar{\alpha}_{2\ell + p} \subset \Sigma_{0, 2\ell + p}^1\) joining the punctures of the quotient
+\(\iota\), they descend to arcs \(\bar{\alpha}_1, \ldots, \bar{\alpha}_{2\ell + b} \subset \Sigma_{0, 2\ell + b}^1\) joining the punctures of the quotient
 surface.
 
 \begin{figure}[ht]
@@ -244,9 +244,8 @@ surface.
 \end{figure}
 
 \begin{observation}\label{ex:push-generators-description}
-  The map \(\operatorname{push} : B_{2\ell + p} \to \Mod(\Sigma_{0, 2\ell +
-  p}^1)\) takes \(\sigma_i\) to the half-twist \(h_{\bar{\alpha}_i}\) about
-  the arc \(\bar{\alpha}_i \subset \Sigma_{0, 2\ell + p}^1\).
+  The map \(\operatorname{push} : B_{2\ell + b} \to \Mod(\Sigma_{0, 2\ell + b}^1)\) takes \(\sigma_i\) to the half-twist \(h_{\bar{\alpha}_i}\) about
+  the arc \(\bar{\alpha}_i \subset \Sigma_{0, 2\ell + b}^1\).
 \end{observation}
 
 We now study the homeomorphisms of \(\Sigma_\ell^1\) and \(\Sigma_\ell^2\) that
@@ -254,66 +253,65 @@ descend to the quotient surfaces and their mapping classes, known as \emph{the
 symmetric mapping classes}.
 
 \begin{definition}
-  Let \(\ell \ge 0\) and \(p = 1, 2\). The \emph{group of symmetric
-  homeomorphisms of \(\Sigma_\ell^p\)} is \(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p) =
-  \{\phi \in \Homeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p) : [\phi, \iota] = 1\}\). The
-  \emph{symmetric mapping class group of \(\Sigma_\ell^p\)} is the subgroup
-  \(\SMod(\Sigma_\ell^1) = \{ [\phi] \in \Mod(\Sigma_\ell^p) : \phi \in
-  \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p) \}\).
+  Let \(\ell \ge 0\) and \(b = 1, 2\). The \emph{group of symmetric
+  homeomorphisms of \(\Sigma_\ell^b\)} is \(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b) =
+  \{\phi \in \Homeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b) : [\phi, \iota] = 1\}\). The
+  \emph{symmetric mapping class group of \(\Sigma_\ell^b\)} is the subgroup
+  \(\SMod(\Sigma_\ell^1) = \{ [\phi] \in \Mod(\Sigma_\ell^b) : \phi \in
+  \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b) \}\).
 \end{definition}
 
-Fix \(p = 1\) or \(2\). It follows from the universal property of quotients
-that any \(\phi \in \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)\) defines a
-homeomorphism \(\bar \phi : \Sigma_{0, 2\ell+p}^1 \isoto \Sigma_{0, 2\ell+p}^1\). This
+Fix \(b = 1\) or \(2\). It follows from the universal property of quotients
+that any \(\phi \in \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) defines a
+homeomorphism \(\bar \phi : \Sigma_{0, 2\ell+b}^1 \isoto \Sigma_{0, 2\ell+b}^1\). This
 yields a homomorphism of topological groups
 \begin{align*}
-  \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)
-  & \to \Homeo^+(\Sigma_{0, 2\ell + p}^1, \partial \Sigma_{0, 2\ell + p}^1) \\
+  \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)
+  & \to \Homeo^+(\Sigma_{0, 2\ell + b}^1, \partial \Sigma_{0, 2\ell + b}^1) \\
   \phi
   & \mapsto \bar \phi,
 \end{align*}
-which is surjective because any \(\psi \in \Homeo^+(\Sigma_{0, 2\ell + p}^1,
-\partial \Sigma_{0, 2\ell + p}^1)\) lifts to \(\Sigma_\ell^p\).
+which is surjective because any \(\psi \in \Homeo^+(\Sigma_{0, 2\ell + b}^1,
+\partial \Sigma_{0, 2\ell + b}^1)\) lifts to \(\Sigma_\ell^b\).
 
-It is also not hard to see \(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p) \to
-\Homeo^+(\Sigma_{0, 2\ell + p}^1, \partial \Sigma_{0, 2\ell + p}^1)\) is injective: the
+It is also not hard to see \(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b) \to
+\Homeo^+(\Sigma_{0, 2\ell + b}^1, \partial \Sigma_{0, 2\ell + b}^1)\) is injective: the
 only candidates for elements of its kernel are \(1\) and \(\iota\), but
-\(\iota\) is not an element of \(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)\) since
-it does not fix \(\partial \Sigma_\ell^p\) point-wise. Now since we have a
+\(\iota\) is not an element of \(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) since
+it does not fix \(\partial \Sigma_\ell^b\) point-wise. Now since we have a
 continuous bijective homomorphism we find
 \[
   \begin{split}
-    \pi_0(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p))
-    & \cong \pi_0(\Homeo^+(\Sigma_{0, 2\ell+p}^1, \partial \Sigma_{0, 2\ell+p}^1))     \\
-    & = \mfrac{\Homeo^+(\Sigma_{0,2\ell+p}^1, \partial \Sigma_{0, 2\ell+p}^1)}{\simeq} \\
-    & = \Mod(\Sigma_{0, 2\ell+p}^1)                                               \\
-    & \cong B_{2\ell + p}.
+    \pi_0(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b))
+    & \cong \pi_0(\Homeo^+(\Sigma_{0, 2\ell+b}^1, \partial \Sigma_{0, 2\ell+b}^1))     \\
+    & = \mfrac{\Homeo^+(\Sigma_{0,2\ell+b}^1, \partial \Sigma_{0, 2\ell+b}^1)}{\simeq} \\
+    & = \Mod(\Sigma_{0, 2\ell+b}^1)                                               \\
+    & \cong B_{2\ell + b}.
   \end{split}
 \]
 
-We would like to say \(\pi_0(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)) =
-\SMod(\Sigma_\ell^p)\), but a priori the story looks a little more complicated:
-\(\phi, \psi \in \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)\) define the same class
-in \(\SMod(\Sigma_\ell^p)\) if they are isotopic, but they may not lie in same
-connected component of \(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)\) if they are
+We would like to say \(\pi_0(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)) =
+\SMod(\Sigma_\ell^b)\), but a priori the story looks a little more complicated:
+\(\phi, \psi \in \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) define the same class
+in \(\SMod(\Sigma_\ell^b)\) if they are isotopic, but they may not lie in same
+connected component of \(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) if they are
 not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden
 \cite{birman-hilden} showed that this is never the case.
 
 \begin{theorem}[Birman-Hilden]
-  If \(\phi, \psi \in \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)\) are isotopic
+  If \(\phi, \psi \in \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) are isotopic
   then \(\phi\) and \(\psi\) are isotopic through symmetric homeomorphisms. In
   particular, there is an isomorphism
   \begin{align*}
-    \SMod(\Sigma_\ell^p) & \isoto \Mod(\Sigma_{0, 2\ell + p}) \\
+    \SMod(\Sigma_\ell^b) & \isoto \Mod(\Sigma_{0, 2\ell + b}) \\
              [\phi] & \mapsto [\bar \phi].
   \end{align*}
 \end{theorem}
 
 \begin{observation}
   Using the notation of Figure~\ref{fig:hyperelliptic-covering}, the
-  Birman-Hilden isomorphism \(\SMod(\Sigma_\ell^p) \isoto \Mod(\Sigma_{0, 2g +
-  p})\) takes \(\tau_{\alpha_i}\) to the half twist \(h_{\bar{\alpha}_i} \in
-  \Mod(\Sigma_{0, 2g + p})\). This can be checked by looking at
+  Birman-Hilden isomorphism \(\SMod(\Sigma_\ell^b) \isoto \Mod(\Sigma_{0, 2g + b})\) takes \(\tau_{\alpha_i}\) to the half twist \(h_{\bar{\alpha}_i} \in
+  \Mod(\Sigma_{0, 2g + b})\). This can be checked by looking at
   \(\iota\)-invariant annular neighborhoods of the curves \(\alpha_i\) --
   \cite[Section~9.4]{farb-margalit}.
 \end{observation}
@@ -327,7 +325,7 @@ not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden
   \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
-  \emph{\(k\)-chain relations} in \(\SMod(\Sigma_\ell^p) \subset
+  \emph{\(k\)-chain relations} in \(\SMod(\Sigma_\ell^b) \subset
   \Mod(\Sigma_g)\).
   \[
     \arraycolsep=1.4pt
@@ -420,7 +418,7 @@ to obtain other relations. Since \(\iota\) has \(2g + 2\) fixed points in
 
 \section{Presentations of Mapping Class Groups}
 
-There are numerous known presentations of \(\Mod(\Sigma_{g, r}^p)\), such as
+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
@@ -500,15 +498,15 @@ given by
   \end{tabular}
 \end{center}
 for closed surfaces of small genus. In \cite{korkmaz-mccarthy} Korkmaz-McCarthy
-showed that even though \(\Mod(\Sigma_2^p)\) is not perfect, its commutator
-subgroup is. In addition, they also show \([\Mod(\Sigma_g^p),
-\Mod(\Sigma_g^p)]\) is normally generated by a single mapping class.
+showed that even though \(\Mod(\Sigma_2^b)\) is not perfect, its commutator
+subgroup is. In addition, they also show \([\Mod(\Sigma_g^b),
+\Mod(\Sigma_g^b)]\) is normally generated by a single mapping class.
 
 \begin{proposition}\label{thm:commutator-is-perfect}
-  The commutator subgroup \(\Mod(\Sigma_2^p)' = [\Mod(\Sigma_2^p),
-  \Mod(\Sigma_2^p)]\) is perfect -- i.e. \(\Mod(\Sigma_2^p)^{(2)} =
-  [\Mod(\Sigma_2^p)', \Mod(\Sigma_2^p)']\) is the whole of
-  \(\Mod(\Sigma_2^p)'\).
+  The commutator subgroup \(\Mod(\Sigma_2^b)' = [\Mod(\Sigma_2^b),
+  \Mod(\Sigma_2^b)]\) is perfect -- i.e. \(\Mod(\Sigma_2^b)^{(2)} =
+  [\Mod(\Sigma_2^b)', \Mod(\Sigma_2^b)']\) is the whole of
+  \(\Mod(\Sigma_2^b)'\).
 \end{proposition}
 
 \begin{proposition}\label{thm:commutator-normal-gen}

diff --git a/sections/representations.tex b/sections/representations.tex
@@ -14,8 +14,8 @@ The goal of this chapter is to provide a concise account of Korkmaz' results,
 starting by\dots
 
 \begin{theorem}[Korkmaz]\label{thm:low-dim-reps-are-trivial}
-  Let \(\Sigma_g^p\) be the compact surface of genus \(g \ge 1\) with \(p\)
-  boundary components and \(\rho : \Mod(\Sigma_g^p) \to \GL_n(\mathbb{C})\) be
+  Let \(\Sigma_g^b\) be the compact surface of genus \(g \ge 1\) with \(b\)
+  boundary components and \(\rho : \Mod(\Sigma_g^b) \to \GL_n(\mathbb{C})\) be
   a linear representation with \(n < 2 g\). Then the image of \(\rho\) is
   Abelian. In particular, if \(g \ge 3\) then \(\rho\) is trivial.
 \end{theorem}
@@ -26,15 +26,15 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
 = 2\).
 
 \begin{proposition}\label{thm:low-dim-reps-are-trivial-base-case}
-  Given \(\rho : \Mod(\Sigma_2^p) \to \GL_n(\mathbb{C})\) with \(n \le 3\), the
+  Given \(\rho : \Mod(\Sigma_2^b) \to \GL_n(\mathbb{C})\) with \(n \le 3\), the
   image of \(\rho\) is Abelian.
 \end{proposition}
 
 \begin{proof}[Sketch of proof]
-  Given \(\alpha \subset \Sigma_2^p\), let \(L_\alpha = \rho(\tau_\alpha)\) and
+  Given \(\alpha \subset \Sigma_2^b\), let \(L_\alpha = \rho(\tau_\alpha)\) and
   denote by \(E_{\alpha = \lambda} = \{ v \in \mathbb{C}^n : L_\alpha v =
   \lambda v \}\) its eigenspaces. Let \(\alpha_1, \alpha_2, \beta_1, \beta_2,
-  \gamma, \eta_1, \ldots, \eta_{p-1} \subset \Sigma_2^p\) be the curves of the
+  \gamma, \eta_1, \ldots, \eta_{b-1} \subset \Sigma_2^b\) be the curves of the
   Lickorish generators from Theorem~\ref{thm:lickorish-gens}, as shown in
   Figure~\ref{fig:lickorish-gens-genus-2}.
   \begin{figure}
@@ -44,12 +44,12 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
     \label{fig:lickorish-gens-genus-2}
   \end{figure}
 
-  If \(n = 1\) then \(\rho(\Mod(\Sigma_2^p)) \subset \GL_1(\mathbb{C}) =
+  If \(n = 1\) then \(\rho(\Mod(\Sigma_2^b)) \subset \GL_1(\mathbb{C}) =
   \mathbb{C}^\times\) is Abelian. Now if \(n = 2\) or \(3\), by
   Proposition~\ref{thm:commutator-normal-gen} it suffices to show \(L_{\alpha_1}
   = L_{\beta_1}\), so that \(\tau_{\alpha_1} \tau_{\beta_1}^{-1} \in \ker
-  \rho\) and thus \(\Mod(\Sigma_2^p)' \subset \ker \rho\) -- i.e.
-  \(\rho(\Mod(\Sigma_2^p))\) is Abelian. Given the braid relation
+  \rho\) and thus \(\Mod(\Sigma_2^b)' \subset \ker \rho\) -- i.e.
+  \(\rho(\Mod(\Sigma_2^b))\) is Abelian. Given the braid relation
   \begin{equation}\label{eq:braid-rel-induction-basis}
     L_{\alpha_1} L_{\beta_1} L_{\alpha_1}
     = L_{\beta_1} L_{\alpha_1} L_{\beta_1},
@@ -165,8 +165,8 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
   case.
 
   We claim that if \(E_{\alpha_2 = \lambda} = E_{\beta_2 = \lambda}\) then
-  \(E_{\alpha_2 = \lambda}\) is \(\Mod(\Sigma_2^p)\)-invariant. Indeed, by
-  change of coordinates we can always find \(f, g, h_i \in \Mod(\Sigma_2^p)\)
+  \(E_{\alpha_2 = \lambda}\) 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
   \begin{align*}
     f \cdot [\alpha_2]      & = [\alpha_1]
@@ -212,16 +212,16 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
   \end{align*}
   In other words, \(E_{\alpha_1 = \lambda} = E_{\alpha_2 = \lambda} =
   E_{\beta_1 = \lambda} = E_{\beta_2 = \lambda} = E_{\gamma = \lambda} =
-  E_{\eta_1 = \lambda} = \cdots = E_{\eta_{p-1} = \lambda}\) is invariant
+  E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b-1} = \lambda}\) is invariant
   under the action of all Lickorish generators.
 
   Hence \(\rho\) restricts to a subrepresentation \(\bar \rho :
-  \Mod(\Sigma_2^p) \to \GL(E_{\alpha_2 = \lambda}) = \GL_2(\mathbb{C})\) --
+  \Mod(\Sigma_2^b) \to \GL(E_{\alpha_2 = \lambda}) = \GL_2(\mathbb{C})\) --
   recall \(E_{\alpha_2 = \lambda} = \mathbb{C} e_1 \oplus \mathbb{C} e_2\). By
-  case (2), \(\bar \rho(f) = 1\) for all \(f \in \Mod(\Sigma_2^p)'\), given
-  that \(\bar \rho(\Mod(\Sigma_2^p))\) is Abelian. Thus
+  case (2), \(\bar \rho(f) = 1\) for all \(f \in \Mod(\Sigma_2^b)'\), given
+  that \(\bar \rho(\Mod(\Sigma_2^b))\) is Abelian. Thus
   \[
-    \rho(\Mod(\Sigma_2^p)') \subset
+    \rho(\Mod(\Sigma_2^b)') \subset
     \begin{pmatrix}
       1 & 0 & * \\
       0 & 1 & * \\
@@ -230,7 +230,7 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
   \]
   lies inside the group of upper triangular matrices, a solvable subgroup of
   \(\GL_3(\mathbb{C})\). Now by Proposition~\ref{thm:commutator-is-perfect} we
-  get \(\rho(\Mod(\Sigma_2^p)') = 1\): any homomorphism from a perfect group to
+  get \(\rho(\Mod(\Sigma_2^b)') = 1\): any homomorphism from a perfect group to
   a solvable group is trivial.
 
   Finally, if \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\) and
@@ -263,18 +263,18 @@ We are now ready to establish the triviality of low-dimensional
 representations.
 
 \begin{proof}[Proof of Theorem~\ref{thm:low-dim-reps-are-trivial}]
-  Let \(g \ge 1\), \(p \ge 0\) and fix \(\rho : \Mod(\Sigma_g^p) \to
+  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
   \(\GL_1(\mathbb{C}) = \mathbb{C}^\times\). 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 \(\Sigma_{g
-  - 1}^q\) has Abelian image for \(m < 2(g - 1)\). Let us show \(\rho\) has
+  - 1}^{b'}\) has Abelian image for \(m < 2(g - 1)\). Let us show \(\rho\) has
   Abelian image.
 
   Let \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g, \gamma_1, \ldots,
-  \gamma_{g - 1}, \eta_1, \ldots, \eta_{p-1} \subset \Sigma_g^p\) be the curves
-  from the Lickorish generators of \(\Mod(\Sigma_g^p)\), as in
+  \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
   Figure~\ref{fig:lickorish-gens}. Once again, let \(L_\alpha =
   \rho(\tau_\alpha)\) and denote by \(E_{\alpha = \lambda}\) the eigenspace of
   \(L_\alpha\) associated to \(\lambda \in \mathbb{C}\). Let \(\Sigma \cong
@@ -284,14 +284,14 @@ representations.
   \begin{figure}[ht]
     \centering
     \includegraphics[width=.35\linewidth]{images/lickorish-gens-korkmaz-proof.eps}
-    \caption{The subsurface $\Sigma \subset \Sigma_g^p$.}
+    \caption{The subsurface $\Sigma \subset \Sigma_g^b$.}
     \label{fig:korkmaz-proof-subsurface}
   \end{figure}
 
   We claim that it suffices to find a \(m\)-dimensional
   \(\Mod(\Sigma)\)-invariant\footnote{Here we view $\Mod(\Sigma)$ as a subgroup
-  of $\Mod(\Sigma_g^p)$ via the inclusion homomorphism $\Mod(\Sigma) \to
-  \Mod(\Sigma_g^p)$ from Example~\ref{ex:inclusion-morphism}, which can be
+  of $\Mod(\Sigma_g^b)$ via the inclusion homomorphism $\Mod(\Sigma) \to
+  \Mod(\Sigma_g^b)$ from Example~\ref{ex: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
@@ -319,9 +319,9 @@ representations.
   follows from Proposition~\ref{thm:commutator-is-perfect} that \(\rho\)
   annihilates all of \(\Mod(\Sigma)'\) and, in particular, \(\tau_{\alpha_1}
   \tau_{\beta_1}^{-1} \in \ker \rho\). But recall from
-  Proposition~\ref{thm:commutator-normal-gen} that \(\Mod(\Sigma_g^p)'\) is
+  Proposition~\ref{thm:commutator-normal-gen} that \(\Mod(\Sigma_g^b)'\) is
   normally generated by \(\tau_{\alpha_1} \tau_{\beta_1}^{-1}\), from which we
-  conclude \(\rho(\Mod(\Sigma_g^p)') = 1\), as desired.
+  conclude \(\rho(\Mod(\Sigma_g^b)') = 1\), as desired.
 
   As before, we exhaustively analyze all possible Jordan forms of
   \(L_{\alpha_g}\). First, consider the case where we can find eigenvalues
@@ -385,9 +385,9 @@ representations.
     \item[\bfseries\color{highlight}(1)]
       Here we use the change of coordinates principle: each \(L_{\alpha_i},
       L_{\beta_i}, L_{\gamma_i},  L_{\eta_i}\) is conjugate to \(L_{\alpha_g} =
-      \lambda\), so all Lickorish generators of \(\Mod(\Sigma_g^p)\) act on
+      \lambda\), so all Lickorish generators of \(\Mod(\Sigma_g^b)\) act on
       \(\mathbb{C}^n\) as scalar multiplication by \(\lambda\) as well. Hence
-      \(\rho(\Mod(\Sigma_g^p)) = \langle \lambda \rangle\) is Abelian.
+      \(\rho(\Mod(\Sigma_g^b)) = \langle \lambda \rangle\) is Abelian.
 
     \item[\bfseries\color{highlight}(2)]
       In this case, \(W = \ker (L_{\alpha_g} - \lambda)^2\) is a
@@ -407,7 +407,7 @@ representations.
   = \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 \(f_i, g_i, h_i \in
-  \Mod(\Sigma_g^p)\) with
+  \Mod(\Sigma_g^b)\) with
   \begin{align*}
     f_i \tau_{\alpha_g}    f_i^{-1} & = \tau_{\alpha_i}
     &
@@ -426,7 +426,7 @@ representations.
     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_{p-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
@@ -441,20 +441,20 @@ representations.
     \end{pmatrix}.
   \]
   Since the group of upper triangular matrices is solvable and
-  \(\Mod(\Sigma_g^p)\) is perfect, it follows that \(\rho(\Mod(\Sigma_g^p))\) is
-  trivial. This concludes the proof \(\rho(\Mod(\Sigma_g^p))\) is Abelian.
-
-  To see that \(\rho(\Mod(\Sigma_g^p)) = 1\) for \(g \ge 3\) we note that, since
-  \(\rho(\Mod(\Sigma_g^p))\) is Abelian, \(\rho\) factors though the Abelianization
-  map \(\Mod(\Sigma_g^p) \to \Mod(\Sigma_g^p)^\ab = \mfrac{\Mod(\Sigma_g^p)}{[\Mod(\Sigma_g^p),
-  \Mod(\Sigma_g^p)]}\). Now recall from Proposition~\ref{thm:trivial-abelianization}
-  that \(\Mod(\Sigma_g^p)^\ab = 0\) for \(g \ge 3\). In other words, \(\rho\)
+  \(\Mod(\Sigma_g^b)\) 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.
 \end{proof}
 
 Having established the triviality of the low-dimensional representations \(\rho
-: \Mod(\Sigma_g^p) \to \GL_n(\mathbb{C})\), all that remains for us is to understand
-the \(2g\)-dimensional representations of \(\Mod(\Sigma_g^p)\). We certainly know a
+: \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
@@ -462,13 +462,13 @@ Example~\ref{ex:symplectic-rep}. Surprisingly, this turns out to be
 representation in the compact case. More precisely,
 
 \begin{theorem}[Korkmaz]\label{thm:reps-of-dim-2g-are-symplectic}
-  Let \(g \ge 3\) and \(\rho : \Mod(\Sigma_g^p) \to \GL_{2g}(\mathbb{C})\). Then
+  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^p) \to
+  representation\footnote{Here the map $\Mod(\Sigma_g^b) \to
   \operatorname{Sp}_{2g}(\mathbb{Z})$ is given by the composition of the
-  inclusion morphism $\Mod(\Sigma_g^p) \to \Mod(\Sigma_g)$ with the usual symplectic
+  inclusion morphism $\Mod(\Sigma_g^b) \to \Mod(\Sigma_g)$ with the usual symplectic
   representation $\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})$.}
-  \(\Mod(\Sigma_g^p) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(\Sigma_g^p)\).
+  \(\Mod(\Sigma_g^b) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(\Sigma_g^b)\).
 \end{theorem}
 
 Unfortunately, the limited scope of these master thesis does not allow us to
@@ -518,7 +518,7 @@ main lemma}. Namely\dots
 This is proved in \cite[Lemma 7.6]{korkmaz} using the braid relations. Notice
 that for \(n = g\) and \(m = 2g\) the matrices in Lemma~\ref{thm:main-lemma}
 coincide with the action of the Lickorish generators \(\tau_{\alpha_1}, \ldots,
-\tau_{\alpha_g}, \tau_{\beta_1}, \ldots, \tau_{\beta_g} \in \Mod(\Sigma_g^p)\) on
+\tau_{\alpha_g}, \tau_{\beta_1}, \ldots, \tau_{\beta_g} \in \Mod(\Sigma_g^b)\) on
 \(H_1(\Sigma_g, \mathbb{C}) \cong \mathbb{C}^{2g}\) -- represented in the standard
 basis \([\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]\) for
 \(H_1(\Sigma_g, \mathbb{C})\).
@@ -541,28 +541,28 @@ basis \([\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]\) for
   \right)
 \end{align*}
 
-Hence by embedding \(B_3^g\) in \(\Mod(\Sigma_g^p)\) via
+Hence by embedding \(B_3^g\) in \(\Mod(\Sigma_g^b)\) via
 \begin{align*}
-  B_3^g & \to \Mod(\Sigma_g^p)         \\
+  B_3^g & \to \Mod(\Sigma_g^b)         \\
   a_i   & \mapsto \tau_{\alpha_i}    \\
   b_i   & \mapsto \tau_{\beta_i}
 \end{align*}
-we can see that any \(\rho : \Mod(\Sigma_g^p) \to \GL_{2g}(\mathbb{C})\) in a
+we can see that any \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\) in a
 certain class of representation satisfying some technical conditions must be
-conjugate to the symplectic representation \(\Mod(\Sigma_g^p) \to
+conjugate to the symplectic representation \(\Mod(\Sigma_g^b) \to
 \operatorname{Sp}_{2g}(\mathbb{Z})\) when restricted to \(B_3^g\).
 
 Korkmaz then goes on to show that such technical conditions are met for any
-nontrivial \(\rho : \Mod(\Sigma_g^p) \to \GL_{2g}(\mathbb{C})\). Furthermore,
+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 \(\rho(\tau_{\gamma_1}), \ldots,
 \rho(\tau_{\gamma_{g - 1}}), \rho(\tau_{\eta_1}), \ldots,
-\rho(\tau_{\eta_{p-1}})\) also agrees with the action of \(\Mod(\Sigma_g^p)\) on
+\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.
 
 % 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 \(\Mod(\Sigma_g^p)\) for \(g \ge 7\) in terms of certain twisted
+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
@@ -218,9 +218,9 @@ too.
 A perhaps less obvious fact about Dehn twists is\dots
 
 \begin{theorem}\label{thm:mcg-is-fg}
-  Let \(\Sigma_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with
-  \(r\) punctures and \(p\) boundary components. Then the pure mapping class
-  group \(\PMod(\Sigma_{g, r}^p)\) is generated by finitely many Dehn twists
+  Let \(\Sigma_{g, r}^b\) be the orientable surface of genus \(g \ge 1\) with
+  \(r\) punctures and \(b\) boundary components. Then the pure mapping class
+  group \(\PMod(\Sigma_{g, r}^b)\) is generated by finitely many Dehn twists
   about nonseparating curves or boundary components.
 \end{theorem}
 
@@ -233,9 +233,9 @@ curves}.
 \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 \(\Sigma_{g, r}^p\) and
-that of \(\Sigma_{g, r+1}^p = \Sigma_{g, r}^p \setminus \{ x \}\) for some \(x\) in the
-interior \((\Sigma_{g, r}^p)\degree\) of \(\Sigma_{g, r}^p\). Indeed, this will later
+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 \(x_1, \ldots, x_n \in \Sigma\degree\),
@@ -324,7 +324,7 @@ show\dots
 \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 \(\Sigma_{g, r}^p\). Our strategy is to
+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}
@@ -336,7 +336,7 @@ apply the following lemma from geometric group theory.
   \(G\) is generated by \(G_v\) and \(g\).
 \end{lemma}
 
-We are interested, of course, in the group \(G = \PMod(\Sigma_{g, r}^p)\). As for
+We are interested, of course, in the group \(G = \PMod(\Sigma_{g, r}^b)\). As for
 the graph \(\Gamma\), we consider\dots
 
 \begin{definition}
@@ -352,10 +352,10 @@ the graph \(\Gamma\), we consider\dots
 \end{definition}
 
 It is clear from Lemma~\ref{thm:change-of-coordinates} that the actions of
-\(\Mod(\Sigma_{g, r}^p)\) on \(V(\hat{\mathcal{N}}(\Sigma_{g, r}^p))\) and
-\(\{([\alpha], [\beta]) \in V(\hat{\mathcal{N}}(\Sigma_{g, r}^p))^2 : \#(\alpha
+\(\Mod(\Sigma_{g, r}^b)\) on \(V(\hat{\mathcal{N}}(\Sigma_{g, r}^b))\) and
+\(\{([\alpha], [\beta]) \in V(\hat{\mathcal{N}}(\Sigma_{g, r}^b))^2 : \#(\alpha
 \cap \beta) = 1 \}\) are both transitive. But why should
-\(\hat{\mathcal{N}}(\Sigma_{g, r}^p)\) be connected?
+\(\hat{\mathcal{N}}(\Sigma_{g, r}^b)\) be connected?
 
 Historically, the modified graph of nonseparating curves first arose as a
 \emph{modified} version of another graph, known as\dots
@@ -398,31 +398,31 @@ Corollary~\ref{thm:mofied-graph-is-connected}. We are now ready to show
 Theorem~\ref{thm:mcg-is-fg}.
 
 \begin{proof}[Proof of Theorem~\ref{thm:mcg-is-fg}]
-  Let \(\Sigma_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with
-  \(r\) punctures and \(p\) boundary components. We want to establish that
-  \(\PMod(\Sigma_{g, r}^p)\) is generated by a finite number of Dehn twists
+  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.
 
-  First, observe that if \(p \ge 1\) and \(\partial \Sigma_{g, r}^p = \delta_1
+  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}^p) \rar{\operatorname{cap}} &
-      \PMod(\Sigma_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\})) \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 \(\PMod(\Sigma_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2
-  \setminus \{0\})) \cong \PMod(\Sigma_{g, r+1}^{p-1})\) is finitely generated
+  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}^p \cup_{\delta_1} (\mathbb{D}^2
+  the generators of \(\PMod(\Sigma_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2
   \setminus \{0\}))\) to Dehn twists about the corresponding curves in
-  \(\Sigma_{g, r}^p\) and add \(\tau_{\delta_1}\) to the generating set.
+  \(\Sigma_{g, r}^b\) 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
@@ -538,18 +538,18 @@ Theorem~\ref{thm:mcg-is-fg}.
 
 There are many possible improvements to this last result. For instance, in
 \cite[Section~4.4]{farb-margalit} Farb-Margalit exhibit an explicit set of
-generators of \(\Mod(\Sigma_g^p)\) by adapting the induction steps in the
+generators of \(\Mod(\Sigma_g^b)\) by adapting the induction steps in the
 proof of Theorem~\ref{thm:mcg-is-fg}. These are known as the \emph{Lickorish
 generators}.
 
 \begin{theorem}[Lickorish generators]\label{thm:lickorish-gens}
-  If \(g \ge 1\) then \(\Mod(\Sigma_g^p)\) is generated by the Dehn twists
+  If \(g \ge 1\) then \(\Mod(\Sigma_g^b)\) is generated by the Dehn twists
   about the curves \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g,
-  \gamma_1, \ldots, \gamma_{g - 1}, \eta_1, \ldots, \eta_{p-1}\) as in
+  \gamma_1, \ldots, \gamma_{g - 1}, \eta_1, \ldots, \eta_{b-1}\) as in
   Figure~\ref{fig:lickorish-gens}
 \end{theorem}
 
-In the boundaryless case \(p = 0\), we can write \(\tau_{\alpha_3}, \ldots,
+In the boundaryless case \(b = 0\), we can write \(\tau_{\alpha_3}, \ldots,
 \tau_{\alpha_g} \in \Mod(\Sigma_g)\) as products of the twists about the
 remaining curves, from which we get the so called \emph{Humphreys generators}.
 
@@ -564,7 +564,7 @@ remaining curves, from which we get the so called \emph{Humphreys generators}.
   \centering
   \includegraphics[width=\linewidth]{images/lickorish-gens.eps}
   \captionof{figure}{The curves from Lickorish generators of
-  $\Mod(\Sigma_g^p)$.}
+  $\Mod(\Sigma_g^b)$.}
   \label{fig:lickorish-gens}
 \end{minipage}
 \hspace{.6cm} %