memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: c93e7373196dc96e6b9d65275f85ce3b4dffcc59
Parent: 2dc7ea2f5adfbd9db880ef1b9864a09597635d7d
Author: Pablo <pablo-pie@riseup.net>
Date: Sat, 22 Jun 2024 13:21:52 +0000

Changed the notation for the number of boundary components

b → p

Diffstat

7 files changed, 182 insertions, 177 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	images/lickorish-gens-gen-2.svg	35	18	17
Modified	images/lickorish-gens-korkmaz-proof.svg	33	17	16
Modified	images/lickorish-gens.svg	35	19	16
Modified	sections/introduction.tex	14	7	7
Modified	sections/presentation.tex	92	46	46
Modified	sections/representations.tex	102	51	51
Modified	sections/twists.tex	48	24	24

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diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -46,18 +46,18 @@ geometry of the curves in \(S\) and their intersections. For example\dots
 \end{lemma}
 
 \begin{proof}
-  Let \(S = S_{g, r}^b\) and consider the surface \(S_{\alpha \alpha'}\)
+  Let \(S = S_{g, r}^p\) and consider the surface \(S_{\alpha \alpha'}\)
   obtained by cutting \(S\) across \(\alpha\) and \(\alpha'\), as in
   Figure~\ref{fig:change-of-coordinates}. Since \(\alpha\) and \(\alpha'\) are
   nonseparating, this surface has genus \(g - 1\) and one additional boundary
   component \(\delta \subset \partial S_{\alpha \beta}\), so \(S_{\alpha \beta}
-  \cong S_{g-1,r}^{b+1}\). The boundary component \(\delta\) is naturally
+  \cong S_{g-1,r}^{p+1}\). The boundary component \(\delta\) is naturally
   subdived into the four arcs in Figure~\ref{fig:change-of-coordinates}, each
   corresponding to one of the curves \(\alpha\) and \(\alpha'\) in \(S\). By
   identifying the pairs of arcs corresponding to the same curve we obtain the
   surface \(\mfrac{S_{\alpha \beta}}{\sim} \cong S\).
 
-  Similarly, \(S_{\beta \beta'} \cong S_{g-1, r}^{b+1}\) also has an additional
+  Similarly, \(S_{\beta \beta'} \cong S_{g-1, r}^{p+1}\) also has an additional
   boundary component \(\delta' \subset \partial S_{\beta \beta'}\) subdivided
   into four arcs. Now by the classification of surfaces we can find an
   orientation-preserving homemorphism \(\tilde\phi : S_{\alpha \alpha'} \isoto
@@ -72,10 +72,10 @@ geometry of the curves in \(S\) and their intersections. For example\dots
 \begin{figure}[ht]
   \centering
   \includegraphics[width=.8\linewidth]{images/change-of-coords-cut.eps}
-  \caption{By cutting $S_{g, r}^b$ across $\alpha$ we obtain $S_{g-1,
-  r}^{b+2}$, where $\alpha'$ deterimines a yellow arc joining the two
-  additional boundary components. Now by cutting $S_{g-1, r}^{b+2}$ across this
-  arc we obtain $S_{g-1,r}^b$, with the added boundary component subdivided
+  \caption{By cutting $S_{g, r}^p$ across $\alpha$ we obtain $S_{g-1,
+  r}^{p+2}$, where $\alpha'$ deterimines a yellow arc joining the two
+  additional boundary components. Now by cutting $S_{g-1, r}^{p+2}$ across this
+  arc we obtain $S_{g-1,r}^p$, 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
@@ -38,21 +38,21 @@ interesting relations between the corresponding Dehn twists in \(\Mod(S)\). For
 example\dots
 
 \begin{proposition}\label{thm:trivial-abelianization}
-  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 =
+  The Abelianization \(\Mod(S_g^p)^\ab = \mfrac{\Mod(S_g^p)}{[\Mod(S_g),
+  \Mod(S_g)]}\) is cyclic. Moreover, if \(g \ge 3\) then \(\Mod(S_g^p)^\ab =
   1\). In other words, \(\Mod(S_g)\) is a perfect group for \(g \ge 3\).
 \end{proposition}
 
 \begin{proof}
-  By Theorem~\ref{thm:lickorish-gens}, \(\Mod(S_g^b)^\ab\) is generated by the
+  By Theorem~\ref{thm:lickorish-gens}, \(\Mod(S_g^p)^\ab\) 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
-  \(\Mod(S_g^b)^\ab = \langle [\alpha] \rangle\).
+  \subset S_g^p\) is conjugate to the Lickorish generators too, so
+  \(\Mod(S_g^p)^\ab = \langle [\alpha] \rangle\).
 
-  Now for \(g \ge 3\) we can embed \(S_0^4\) in \(S_g^b\) in such a way that
+  Now for \(g \ge 3\) we can embed \(S_0^4\) in \(S_g^p\) 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
+  \delta_4 \subset S_g^p\) are nonseparating, as shown in
   Figure~\ref{fig:latern-relation-trivial-abelianization}. The lantern relation
   (\ref{eq:latern-relation}) then becomes
   \[
@@ -62,14 +62,14 @@ example\dots
     + [\tau_{\delta_3}] + [\tau_{\delta_4}]
     = 4 \cdot [\tau_\alpha]
   \]
-  in \(\Mod(S_g^b)^\ab\). In other words, \([\tau_\alpha] = 0\) and thus
-  \(\Mod(S_g^b)^\ab = 0\).
+  in \(\Mod(S_g^p)^\ab\). In other words, \([\tau_\alpha] = 0\) and thus
+  \(\Mod(S_g^p)^\ab = 0\).
 \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 $g \ge 3$.}
+  \caption{The embedding of $S_0^4$ in $S_g^p$ for $g \ge 3$.}
   \label{fig:latern-relation-trivial-abelianization}
 \end{figure}
 
@@ -89,14 +89,14 @@ presentations of \(\Mod(S_g)\) for \(g \le 2\) to show the Abelianization is giv
 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
+eventhough \(\Mod(S_2^p)\) is not perfect, its commutator subgroup is.
+In addition, they also show \([\Mod(S_g^p), \Mod(S_g^p)]\) is normaly generated
 by a single mapping class.
 
 \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 \(\Mod(S_2^b)'\).
+  The commutator subgroup \(\Mod(S_2^p)' = [\Mod(S_2^p), \Mod(S_2^p)]\) is
+  perfect -- i.e. \(\Mod(S_2^p)^{(2)} = [\Mod(S_2^p)', \Mod(S_2^p)']\) is the
+  whole of \(\Mod(S_2^p)'\).
 \end{proposition}
 
 \begin{proposition}\label{thm:commutator-normal-gen}
@@ -217,10 +217,10 @@ we get\dots
 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.} \(S_\ell^b \isoto S_\ell^b\).
+Figure~\ref{fig:hyperelliptic-involution}. Given \(\ell < g\) and \(p = 1, 2\),
+we can also embed \(S_\ell^p\) in \(S_g\) in such way that \(\iota\) restricts
+to an involution\footnote{This involution does not fix $\partial S_\ell^p$
+point-wise.} \(S_\ell^p \isoto S_\ell^p\).
 
 \begin{figure}[ht]
   \centering
@@ -232,13 +232,13 @@ point-wise.} \(S_\ell^b \isoto S_\ell^b\).
 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 \(S_\ell^b
-\to \mfrac{S_\ell^b}{\iota} \cong \mathbb{D}^2\) is a double cover with \(2\ell +
+and \(S_\ell^2\), respectively. Given \(p = 1, 2\), the quotient map \(S_\ell^p
+\to \mfrac{S_\ell^p}{\iota} \cong \mathbb{D}^2\) is a double cover with \(2\ell +
 b\) branch points corresponding to the fixed points of \(\iota\). We may thus
-regard \(\mfrac{S_\ell^b}{\iota}\) as the disk \(S_{0, 2\ell + b}^1\) with
+regard \(\mfrac{S_\ell^p}{\iota}\) 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
+\ldots, \alpha_{2\ell} \subset S_\ell^p\) 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.
@@ -262,36 +262,36 @@ 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{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
-  \(\SMod(S_\ell^1) = \{ [\phi] \in \Mod(S_\ell^b) : \phi \in
-  \SHomeo^+(S_\ell^b, \partial S_\ell^b) \}\).
+  Let \(\ell \ge 0\) and \(p = 1, 2\). The \emph{group of symmetric
+  homeomorphisms of \(S_\ell^p\)} is \(\SHomeo^+(S_\ell^p, \partial S_\ell^p) =
+  \{\phi \in \Homeo^+(S_\ell^p, \partial S_\ell^p) : [\phi, \iota] = 1\}\). The
+  \emph{symmetric mapping class group of \(S_\ell^p\)} is the subgroup
+  \(\SMod(S_\ell^1) = \{ [\phi] \in \Mod(S_\ell^p) : \phi \in
+  \SHomeo^+(S_\ell^p, \partial S_\ell^p) \}\).
 \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
+Fix \(p = 1\) or \(2\). It follows from the universal property of quotients
+that any \(\phi \in \SHomeo^+(S_\ell^p, \partial S_\ell^p)\) 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)
+  \SHomeo^+(S_\ell^p, \partial S_\ell^p)
   & \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 \(S_\ell^b\).
+\partial S_{0, 2\ell + b}^1)\) lifts to \(S_\ell^p\).
 
-It is also not hard to see \(\SHomeo^+(S_\ell^b, \partial S_\ell^b) \to
+It is also not hard to see \(\SHomeo^+(S_\ell^p, \partial S_\ell^p) \to
 \Homeo^+(S_{0, 2\ell + b}^1, \partial S_{0, 2\ell + b}^1)\) 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
+\(\iota\) is not an element of \(\SHomeo^+(S_\ell^p, \partial S_\ell^p)\) since
+it does not fix \(\partial S_\ell^p\) point-wise. Now since we have a
 continuous bijective homomorphism we find
 \[
   \begin{split}
-    \pi_0(\SHomeo^+(S_\ell^b, \partial S_\ell^b))
+    \pi_0(\SHomeo^+(S_\ell^p, \partial S_\ell^p))
     & \cong \pi_0(\Homeo^+(S_{0, 2\ell+b}^1, \partial S_{0, 2\ell+b}^1))     \\
     & = \mfrac{\Homeo^+(S_{0,2\ell+b}^1, \partial S_{0, 2\ell+b}^1)}{\simeq} \\
     & = \Mod(S_{0, 2\ell+b}^1)                                               \\
@@ -299,27 +299,27 @@ continuous bijective homomorphism we find
   \end{split}
 \]
 
-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
+We would like to say \(\pi_0(\SHomeo^+(S_\ell^p, \partial S_\ell^p)) =
+\SMod(S_\ell^p)\), but a priori the story looks a little more complicated:
+\(\phi, \psi \in \SHomeo^+(S_\ell^p, \partial S_\ell^p)\) define the same class
+in \(\SMod(S_\ell^p)\) if they are isotopic, but they may not lie in same
+connected component of \(\SHomeo^+(S_\ell^p, \partial S_\ell^p)\) 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 \(\phi, \psi \in \SHomeo^+(S_\ell^b, \partial S_\ell^b)\) are isotopic
+  If \(\phi, \psi \in \SHomeo^+(S_\ell^p, \partial S_\ell^p)\) are isotopic
   then \(\phi\) and \(\psi\) are isotopic through symmetric homeomorphisms. In
   particular, there is an isomorphism
   \begin{align*}
-    \SMod(S_\ell^b) & \isoto \Mod(S_{0, 2\ell + b}) \\
+    \SMod(S_\ell^p) & \isoto \Mod(S_{0, 2\ell + b}) \\
              [\phi] & \mapsto [\bar \phi].
   \end{align*}
 \end{theorem}
 
 \begin{example}
   Using the notation of Figure~\ref{fig:hyperelliptic-covering}, the
-  Birman-Hilden isomorphism \(\SMod(S_\ell^b) \isoto \Mod(S_{0, 2g + b})\)
+  Birman-Hilden isomorphism \(\SMod(S_\ell^p) \isoto \Mod(S_{0, 2g + b})\)
   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 \(\alpha_i\) --
@@ -335,7 +335,7 @@ not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden
   \(\tau_{\bar\delta_1} = \tau_{\bar\delta_2}\). 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 \(\SMod(S_\ell^b) \subset \Mod(S_g)\).
+  relations} in \(\SMod(S_\ell^p) \subset \Mod(S_g)\).
   \[
     \arraycolsep=1.4pt
     \begin{array}{rlcrll}

diff --git a/sections/representations.tex b/sections/representations.tex
@@ -16,8 +16,8 @@ goal of this chapter is providing a concise account of Korkmaz' results,
 starting by\dots
 
 \begin{theorem}[Korkmaz]\label{thm:low-dim-reps-are-trivial}
-  Let \(S_g^b\) be the surface of genus \(g \ge 1\) and \(b\) boundary
-  components and \(\rho : \Mod(S_g^b) \to \GL_n(\mathbb{C})\) be a linear
+  Let \(S_g^p\) be the surface of genus \(g \ge 1\) and \(b\) boundary
+  components and \(\rho : \Mod(S_g^p) \to \GL_n(\mathbb{C})\) 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}
@@ -28,15 +28,15 @@ 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 \(\rho : \Mod(S_2^b) \to \GL_n(\mathbb{C})\) with \(n \le 3\), the
+  Given \(\rho : \Mod(S_2^p) \to \GL_n(\mathbb{C})\) with \(n \le 3\), the
   image of \(\rho\) is Abelian.
 \end{proposition}
 
 \begin{proof}[Sketch of proof]
-  Given \(\alpha \subset S_2^b\), let \(L_\alpha = \rho(\tau_\alpha)\) and
+  Given \(\alpha \subset S_2^p\), 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 \(\alpha_1, \alpha_2, \beta_1, \beta_2,
-  \gamma, \eta_1, \ldots, \eta_{b - 1} \subset S_2^b\) be the curves of the
+  \gamma, \eta_1, \ldots, \eta_{p-1} \subset S_2^p\) 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}
@@ -46,12 +46,12 @@ 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(S_2^b)) \subset \GL_1(\mathbb{C}) =
+  If \(n = 1\) then \(\rho(\Mod(S_2^p)) \subset \GL_1(\mathbb{C}) =
   \mathbb{C}^\times\) is Abelian. Now if \(n = 2\) or \(3\), by
   Propositon~\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 \(\Mod(S_2^b)' \subset \ker \rho\) -- i.e.
-  \(\rho(\Mod(S_2^b))\) is Abelian. Given the braid relation
+  \rho\) and thus \(\Mod(S_2^p)' \subset \ker \rho\) -- i.e.
+  \(\rho(\Mod(S_2^p))\) 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},
@@ -138,8 +138,8 @@ 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 \(\Mod(S_2^b)\)-invariant. Indeed, by change of
-  coordinates we can always find \(f, g, h_i \in \Mod(S_2^b)\) with
+  \(E_{\alpha_2 = \lambda}\) is \(\Mod(S_2^p)\)-invariant. Indeed, by change of
+  coordinates we can always find \(f, g, h_i \in \Mod(S_2^p)\) with
   \begin{align*}
     f \cdot [\alpha_2]      & = [\alpha_1]
     &
@@ -184,16 +184,16 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
   \end{align*}
   In other words, \(E_{\alpha_1 = \lambda} = E_{\alpha_2 = \lambda} =
   E_{\beta_1 = \lambda} = E_{\beta_2 = \lambda} = E_{\gamma = \lambda} =
-  E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b - 1} = \lambda}\) is invariant
+  E_{\eta_1 = \lambda} = \cdots = E_{\eta_{p-1} = \lambda}\) is invariant
   under the action of all Lickorish generators.
 
-  Hence \(\rho\) restricts to a subrepresentation \(\bar \rho : \Mod(S_2^b) \to
+  Hence \(\rho\) restricts to a subrepresentation \(\bar \rho : \Mod(S_2^p) \to
   \GL(E_{\alpha_2 = \lambda}) = \GL_2(\mathbb{C})\) -- recall \(E_{\alpha_2 =
   \lambda} = \mathbb{C} e_1 \oplus \mathbb{C} e_2\). By case (2), \(\bar
-  \rho(f) = 1\) for all \(f \in \Mod(S_2^b)'\), given that \(\bar
-  \rho(\Mod(S_2^b))\) is Abelian. Thus
+  \rho(f) = 1\) for all \(f \in \Mod(S_2^p)'\), given that \(\bar
+  \rho(\Mod(S_2^p))\) is Abelian. Thus
   \[
-    \rho(\Mod(S_2^b)') \subset
+    \rho(\Mod(S_2^p)') \subset
     \begin{pmatrix}
       1 & 0 & * \\
       0 & 1 & * \\
@@ -202,7 +202,7 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
   \]
   lies inside the group of upper triangular matrices, a solvalbe subgroup of
   \(\GL_3(\mathbb{C})\). Now by Proposition~\ref{thm:commutator-is-perfect} we
-  get \(\rho(\Mod(S_2^b)') = 1\): any homomorphism from a perfect group to a
+  get \(\rho(\Mod(S_2^p)') = 1\): any homomorphism from a perfect group to a
   solvable group is trivial.
 
   Finally, if \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\) and
@@ -235,18 +235,18 @@ 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 \(g \ge 1\), \(b \ge 0\) and fix \(\rho : \Mod(S_g^b) \to
+  Let \(g \ge 1\), \(p \ge 0\) and fix \(\rho : \Mod(S_g^p) \to
   \GL_n(\mathbb{C})\) 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 \(S_{g -
-  1}^{b'}\) has Abelian image for \(m < 2(g - 1)\). Let us show \(\rho\) has
+  1}^q\) has Abelian image for \(m < 2(g - 1)\). Let us show \(\rho\) has
   Abelian image.
 
   Let \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g, \gamma_1, \ldots,
-  \gamma_{g - 1}, \eta_1, \ldots, \eta_{b - 1} \subset S_g^b\) be the curves
-  from the Lickorish generators of \(\Mod(S_g^b)\), as in
+  \gamma_{g - 1}, \eta_1, \ldots, \eta_{p-1} \subset S_g^p\) be the curves
+  from the Lickorish generators of \(\Mod(S_g^p)\), 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 \(R \cong S_{g -
@@ -256,14 +256,14 @@ representations.
   \begin{figure}[ht]
     \centering
     \includegraphics[width=.35\linewidth]{images/lickorish-gens-korkmaz-proof.eps}
-    \caption{The subsurface $R \subset S_g^b$.}
+    \caption{The subsurface $R \subset S_g^p$.}
     \label{fig:korkmaz-proof-subsurface}
   \end{figure}
 
   % TODO: Add more comments on the injectivity of this map?
   We claim that it suffices to find a \(m\)-dimensional
   \(\Mod(R)\)-invariant\footnote{Here we view $\Mod(R)$ as a subgroup of
-  $\Mod(S_g^b)$ via the inclusion homomorphism $\Mod(R) \to \Mod(S_g^b)$ from
+  $\Mod(S_g^p)$ via the inclusion homomorphism $\Mod(R) \to \Mod(S_g^p)$ 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
@@ -291,9 +291,9 @@ representations.
   follows from Proposition~\ref{thm:commutator-is-perfect} that \(\rho\)
   annihilates all of \(\Mod(R)'\) and, in particular, \(\tau_{\alpha_1}
   \tau_{\beta_1}^{-1} \in \ker \rho\). But recall from
-  Proposition~\ref{thm:commutator-normal-gen} that \(\Mod(S_g^b)'\) is normally
+  Proposition~\ref{thm:commutator-normal-gen} that \(\Mod(S_g^p)'\) is normally
   generated by \(\tau_{\alpha_1} \tau_{\beta_1}^{-1}\), from which we conclude
-  \(\rho(\Mod(S_g^b)') = 1\), as desired.
+  \(\rho(\Mod(S_g^p)') = 1\), as desired.
 
   As before, we exhaustively analyse all possible Jordan forms of
   \(L_{\alpha_g}\). First, consider the case where we can find eigenvalues
@@ -354,9 +354,9 @@ representations.
 
   For case (1), 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 \(\Mod(S_g^b)\)
+  \(L_{\alpha_g} = \lambda\), so all Lickorish generators of \(\Mod(S_g^p)\)
   act on \(\mathbb{C}^n\) as scalar multiplication by \(\lambda\) as well.
-  Hence \(\rho(\Mod(S_g^b))\) is cyclic and thus Abelian. In case (2), \(W =
+  Hence \(\rho(\Mod(S_g^p))\) is cyclic and thus Abelian. In case (2), \(W =
   \ker (L_{\alpha_g} - \lambda)^2\) is a \(2\)-dimensional
   \(\Mod(R)\)-invariant subspace.
 
@@ -372,7 +372,7 @@ representations.
   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 \(f_i, g_i, h_i \in
-  \Mod(S_g^b)\) with
+  \Mod(S_g^p)\) with
   \begin{align*}
     f_i \tau_{\alpha_g}    f_i^{-1} & = \tau_{\alpha_i}
     &
@@ -391,7 +391,7 @@ 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_{b - 1} = \lambda}.
+    = E_{\eta_1 = \lambda} = \cdots = E_{\eta_{p-1} = \lambda}.
   \]
 
   In particular, we can find a basis for \(\mathbb{C}^n\) with respect to which
@@ -405,21 +405,21 @@ representations.
       0       & 0       & \cdots & 0       & *
     \end{pmatrix}.
   \]
-  Since the group of upper triangular matrices is solvable and \(\Mod(S_g^b)\)
-  is perfect, it follows that \(\rho(\Mod(S_g^b))\) is trivial. This concludes
-  the proof \(\rho(\Mod(S_g^b))\) is Abelian.
-
-  To see that \(\rho(\Mod(S_g^b)) = 1\) for \(g \ge 3\) we note that, since
-  \(\rho(\Mod(S_g^b))\) is Abelian, \(\rho\) factors though the Abelinization
-  map \(\Mod(S_g^b) \to \Mod(S_g^b)^\ab = \mfrac{\Mod(S_g^b)}{[\Mod(S_g^b),
-  \Mod(S_g^b)]}\). Now recall from Proposition~\ref{thm:trivial-abelianization}
-  that \(\Mod(S_g^b)^\ab = 0\) for \(g \ge 3\). In other words, \(\rho\)
+  Since the group of upper triangular matrices is solvable and \(\Mod(S_g^p)\)
+  is perfect, it follows that \(\rho(\Mod(S_g^p))\) is trivial. This concludes
+  the proof \(\rho(\Mod(S_g^p))\) is Abelian.
+
+  To see that \(\rho(\Mod(S_g^p)) = 1\) for \(g \ge 3\) we note that, since
+  \(\rho(\Mod(S_g^p))\) is Abelian, \(\rho\) factors though the Abelinization
+  map \(\Mod(S_g^p) \to \Mod(S_g^p)^\ab = \mfrac{\Mod(S_g^p)}{[\Mod(S_g^p),
+  \Mod(S_g^p)]}\). Now recall from Proposition~\ref{thm:trivial-abelianization}
+  that \(\Mod(S_g^p)^\ab = 0\) 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 \(\rho
-: \Mod(S_g^b) \to \GL_n(\mathbb{C})\), all that remains for us is to understand
-the \(2g\)-dimensional reprensentations of \(\Mod(S_g^b)\). We certainly know a
+: \Mod(S_g^p) \to \GL_n(\mathbb{C})\), all that remains for us is to understand
+the \(2g\)-dimensional reprensentations of \(\Mod(S_g^p)\). We certainly know a
 nontrivial example of such, namely the symplectic representation \(\psi :
 \Mod(S_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) from
 Example~\ref{ex:symplectic-rep}. Surprinsgly, this turns out to be
@@ -427,13 +427,13 @@ Example~\ref{ex:symplectic-rep}. Surprinsgly, 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 \(\rho : \Mod(S_g^b) \to \GL_{2g}(\mathbb{C})\). Then
+  Let \(g \ge 3\) and \(\rho : \Mod(S_g^p) \to \GL_{2g}(\mathbb{C})\). Then
   \(\rho\) is either trivial or conjugate to the symplectic
-  representation\footnote{Here the map $\Mod(S_g^b) \to
+  representation\footnote{Here the map $\Mod(S_g^p) \to
   \operatorname{Sp}_{2g}(\mathbb{Z})$ is given by the composition of the
-  inclusion morphism $\Mod(S_g^b) \to \Mod(S_g)$ with the usual symplect
+  inclusion morphism $\Mod(S_g^p) \to \Mod(S_g)$ with the usual symplect
   representation $\psi : \Mod(S_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})$.}
-  \(\Mod(S_g^b) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(S_g^b)\).
+  \(\Mod(S_g^p) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(S_g^p)\).
 \end{theorem}
 
 Unfortunately, the limited scope of these master thesis does not allow us to
@@ -483,7 +483,7 @@ to as \emph{the 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 Lickrish generators \(\tau_{\alpha_1}, \ldots,
-\tau_{\alpha_g}, \tau_{\beta_1}, \ldots, \tau_{\beta_g} \in \Mod(S_g^b)\) on
+\tau_{\alpha_g}, \tau_{\beta_1}, \ldots, \tau_{\beta_g} \in \Mod(S_g^p)\) on
 \(H_1(S_g, \mathbb{C}) \cong \mathbb{C}^{2g}\) -- represented in the standard
 basis \([\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]\) for
 \(H_1(S_g, \mathbb{C})\).
@@ -506,28 +506,28 @@ basis \([\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]\) for
   \right)
 \end{align*}
 
-Hence by embeding \(B_3^g\) in \(\Mod(S_g^b)\) via
+Hence by embeding \(B_3^g\) in \(\Mod(S_g^p)\) via
 \begin{align*}
-  B_3^g & \to \Mod(S_g^b)         \\
+  B_3^g & \to \Mod(S_g^p)         \\
   a_i   & \mapsto \tau_{\alpha_i}    \\
   b_i   & \mapsto \tau_{\beta_i}
 \end{align*}
-we can see that any \(\rho : \Mod(S_g^b) \to \GL_{2g}(\mathbb{C})\) in a
+we can see that any \(\rho : \Mod(S_g^p) \to \GL_{2g}(\mathbb{C})\) in a
 certain class of representation satisfying some technical conditions must be
-conjugate to the symplectic representation \(\Mod(S_g^b) \to
+conjugate to the symplectic representation \(\Mod(S_g^p) \to
 \operatorname{Sp}_{2g}(\mathbb{Z})\) when restricted to \(B_3^g\).
 
 Korkmaz then goes on to show that such technical conditions are met for any
-nontrivial \(\rho : \Mod(S_g^b) \to \GL_{2g}(\mathbb{C})\). Furthermore,
+nontrivial \(\rho : \Mod(S_g^p) \to \GL_{2g}(\mathbb{C})\). 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 \(\Mod(S_g^b)\) on
+\rho(\tau_{\eta_{p-1}})\) also agrees with the action of \(\Mod(S_g^p)\) on
 \(H_1(S_g, \mathbb{C})\), 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 \(\Mod(S_g^b)\) for \(g \ge 7\) in terms of certain twisted
+representations of \(\Mod(S_g^p)\) for \(g \ge 7\) in terms of certain twisted
 \(1\)-cohomology groups.

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -183,9 +183,9 @@ too.
 A perhaps less obvious fact about Dehn twists is\dots
 
 \begin{theorem}\label{thm:mcg-is-fg}
-  Let \(S_{g, r}^b\) be the orientable surface of genus \(g \ge 1\) with \(r\)
+  Let \(S_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with \(r\)
   punctures and \(b\) boundary components. Then the pure mapping class group
-  \(\PMod(S_{g, r}^b)\) is generated by finitely many Dehn twists about
+  \(\PMod(S_{g, r}^p)\) is generated by finitely many Dehn twists about
   nonseparating curves or boundary components.
 \end{theorem}
 
@@ -198,9 +198,9 @@ 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 \(S_{g, r}^b\) and
-that of \(S_{g, r+1}^b = S_{g, r}^b \setminus \{ x \}\) for some \(x\) in the
-interior \((S_{g, r}^b)\degree\) of \(S_{g, r}^b\). Indeed, this will later
+consider the relationship between the mapping class group of \(S_{g, r}^p\) and
+that of \(S_{g, r+1}^p = S_{g, r}^p \setminus \{ x \}\) for some \(x\) in the
+interior \((S_{g, r}^p)\degree\) of \(S_{g, r}^p\). Indeed, this will later
 allow us to establish the induction on the number of punctures \(r\).
 
 Given an orientable surface \(S\) and \(x_1, \ldots, x_n \in S\degree\),
@@ -289,7 +289,7 @@ show\dots
 \section{The Modified Graph of Curves}
 
 Having established Theorem~\ref{thm:birman-exact-seq}, we now need to adress
-the induction step in the genus \(g\) of \(S_{g, r}^b\). Our strategy is to
+the induction step in the genus \(g\) of \(S_{g, r}^p\). Our strategy is to
 apply the following lemma from geomtric group theory.
 
 \begin{lemma}\label{thm:ggt-lemma}
@@ -301,7 +301,7 @@ apply the following lemma from geomtric group theory.
   \(G\) is generated by \(G_v\) and \(g\).
 \end{lemma}
 
-We are interested, of course, in the group \(G = \PMod(S_{g, r}^b)\). As for
+We are interested, of course, in the group \(G = \PMod(S_{g, r}^p)\). As for
 the graph \(\Gamma\), we consider\dots
 
 \begin{definition}
@@ -317,9 +317,9 @@ the graph \(\Gamma\), we consider\dots
 \end{definition}
 
 It is clear from Lemma~\ref{thm:change-of-coordinates} that the actions of
-\(\Mod(S_{g, r}^b)\) on \(V(\hat{\mathcal{N}}(S_{g, r}^b))\) and \(\{([\alpha],
-[\beta]) \in V(\hat{\mathcal{N}}(S_{g, r}^b))^2 : \#(\alpha \cap \beta) = 1
-\}\) are both transitive. But why should \(\hat{\mathcal{N}}(S_{g, r}^b)\) be
+\(\Mod(S_{g, r}^p)\) on \(V(\hat{\mathcal{N}}(S_{g, r}^p))\) and \(\{([\alpha],
+[\beta]) \in V(\hat{\mathcal{N}}(S_{g, r}^p))^2 : \#(\alpha \cap \beta) = 1
+\}\) are both transitive. But why should \(\hat{\mathcal{N}}(S_{g, r}^p)\) be
 connected?
 
 Historically, the modified graph of nonseparating curves first arised as a
@@ -361,30 +361,30 @@ 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 \(S_{g, r}^b\) be the orientable surface of genus \(g \ge 1\) with \(r\)
+  Let \(S_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with \(r\)
   punctures and \(b\) boundary components. We want to establish that
-  \(\PMod(S_{g, r}^b)\) is genetery by a finite number of Dehn twists about
+  \(\PMod(S_{g, r}^p)\) is genetery by a finite number of Dehn twists about
   nonseparating simple closed curves or boundary components.
 
-  First, observe that if \(b \ge 1\) and \(\partial S_{g, r}^b = \delta_1 \cup
-  \cdots \cup \delta_b\) then, by recursively applying the capping exact
+  First, observe that if \(p \ge 1\) and \(\partial S_{g, r}^p = \delta_1 \cup
+  \cdots \cup \delta_p\) then, by recursively applying the capping exact
   sequence
   \begin{center}
     \begin{tikzcd}
       1 \rar &
       \langle \tau_{\delta_1} \rangle \rar &
-      \PMod(S_{g, r}^b) \rar{\operatorname{cap}} &
-      \PMod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\})) \rar &
+      \PMod(S_{g, r}^p) \rar{\operatorname{cap}} &
+      \PMod(S_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\})) \rar &
       1
     \end{tikzcd}
   \end{center}
   from Example~\ref{ex:capping-seq}, it suffices to show that \(S_{g, n}\)
   is finitely generated by twists about nonseparating simple closed curves.
-  Indeed, if \(\PMod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\)
+  Indeed, if \(\PMod(S_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\)
   is finitely generated by twists about nonseparing curves or boundary
-  components, then we may lift the generators of \(\PMod(S_{g, r}^b
+  components, then we may lift the generators of \(\PMod(S_{g, r}^p
   \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\) to Dehn twists about the
-  corresponding curves in \(S_{g, r}^b\) and add \(\tau_{\delta_1}\) to the
+  corresponding curves in \(S_{g, r}^p\) and add \(\tau_{\delta_1}\) to the
   generating set.
 
   It thus suffices to consider the boudaryless case \(S_{g, r}\). As promised,
@@ -498,18 +498,18 @@ 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 \(\Mod(S_g^b)\) by addapting the induction steps in the proof of
+generators of \(\Mod(S_g^p)\) by addapting 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 \(\Mod(S_g^b)\) is generated by the Dehn twists about the
+  If \(g \ge 1\) then \(\Mod(S_g^p)\) is generated by the Dehn twists about the
   curves \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g, \gamma_1,
-  \ldots, \gamma_{g - 1}, \eta_1, \ldots, \eta_{b - 1}\) as in
+  \ldots, \gamma_{g - 1}, \eta_1, \ldots, \eta_{p-1}\) as in
   Figure~\ref{fig:lickorish-gens}
 \end{theorem}
 
-In the boundaryless case \(b = 0\), we can write \(\tau_{\mu_3}, \ldots,
+In the boundaryless case \(p = 0\), we can write \(\tau_{\mu_3}, \ldots,
 \tau_{\mu_g} \in \Mod(S_g)\) as products of the twists about the remaining
 curves, from which we get the so called \emph{Humphreys generators}.
 
@@ -522,7 +522,7 @@ curves, from which we get the so called \emph{Humphreys generators}.
 \begin{minipage}[b]{.45\linewidth}
   \centering
   \includegraphics[width=\linewidth]{images/lickorish-gens.eps}
-  \captionof{figure}{The curves from Lickorish generators of $\Mod(S_g^b)$.}
+  \captionof{figure}{The curves from Lickorish generators of $\Mod(S_g^p)$.}
   \label{fig:lickorish-gens}
 \end{minipage}
 \hspace{.5cm} %