memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: e221d5fd6496104788fee9af632808f7690cfdd9
Parent: 8894c953609e975dc716d9cf7ac375986198093e
Author: Pablo <pablo-pie@riseup.net>
Date: Tue, 18 Jun 2024 12:59:15 +0000

Tweak in notation

Replace 𝔻 with 𝔻²

Replaced 𝕋 with 𝕋²

Also updated a drawing

Diffstat

4 files changed, 65 insertions, 55 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	images/torus-mcg-generators.svg	64	37	27
Modified	sections/introduction.tex	20	10	10
Modified	sections/presentation.tex	18	9	9
Modified	sections/twists.tex	18	9	9

diff --git a/images/torus-mcg-generators.svg b/images/torus-mcg-generators.svg
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diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -26,7 +26,7 @@
 \begin{example}
   Let \(\alpha \subset \partial S\) be a boundary component of \(S\) and fix
   some orientation of \(\alpha\). We refer to the inclusion homomorphism
-  \(\operatorname{cap} : \Mod(S) \to \Mod(S \cup_\alpha (\mathbb{D} \setminus
+  \(\operatorname{cap} : \Mod(S) \to \Mod(S \cup_\alpha (\mathbb{D}^2 \setminus
   \{0\}))\) as \emph{the capping homomorphism}.
 \end{example}
 
@@ -38,7 +38,7 @@
       1 \rar &
       \langle \tau_\alpha \rangle \rar &
       \Mod(S) \rar{\operatorname{cap}} &
-      \Mod(S \cup_\alpha (\mathbb{D} \setminus \{0\})) \rar &
+      \Mod(S \cup_\alpha (\mathbb{D}^2 \setminus \{0\})) \rar &
       1
     \end{tikzcd}
   \end{center}
@@ -70,18 +70,18 @@
 
 % TODO: Explain the Alexander trick
 \begin{example}[Alexander trick]\label{ex:alexander-trick}
-  \(\Mod(\mathbb{D}) = 1\)
+  \(\Mod(\mathbb{D}^2) = 1\)
 \end{example}
 
 \begin{example}
-  By the same token, \(\Mod(\mathbb{D} \setminus \{0\}) = 1\).
+  By the same token, \(\Mod(\mathbb{D}^2 \setminus \{0\}) = 1\).
 \end{example}
 
 % TODO: Explain this
 \begin{example}\label{ex:torus-mcg}
-  The symplectic representation \(\psi : \Mod(\mathbb{T}) \to
+  The symplectic representation \(\psi : \Mod(\mathbb{T}^2) \to
   \operatorname{Sp}_2(\mathbb{Z}) = \operatorname{SL}_2(\mathbb{Z})\) is a
-  group isomorphism. In particular, \(\Mod(\mathbb{T}) \cong
+  group isomorphism. In particular, \(\Mod(\mathbb{T}^2) \cong
   \operatorname{SL}_2(\mathbb{Z})\).
 \end{example}
 
@@ -137,15 +137,15 @@
 
 % TODO: Can we prove this without using braid groups?
 \begin{example}\label{ex:mcg-twice-punctured-disk}
-  The mapping class group \(\Mod(\mathbb{D} \setminus \{-\sfrac{1}{2},
+  The mapping class group \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2},
   \sfrac{1}{2}\})\) of the twice punctured unit disk in \(\mathbb{C}\) is
   freely generated by \(f = [\phi]\), where
   \begin{align*}
-    \phi : \mathbb{D} \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}
-    & \isoto \mathbb{D} \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\} \\
+    \phi : \mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}
+    & \isoto \mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\} \\
     z & \mapsto -z.
   \end{align*}
-  In particular, \(\Mod(\mathbb{D} \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\})
+  In particular, \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\})
   \cong \mathbb{Z}\).
 \end{example}

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -82,7 +82,7 @@ presentations of \(\Mod(S_g)\) for \(g \le 2\) to show the Abelianization is giv
     \hline
           &                  &                   \\[-10pt]
     \(0\) & \(\mathbb{S}^2\) & \(0\)             \\
-    \(1\) & \(\mathbb{T}\)   & \(\mathbb{Z}/12\) \\
+    \(1\) & \(\mathbb{T}^2\)   & \(\mathbb{Z}/12\) \\
     \(2\) & \(S_2\)          & \(\mathbb{Z}/10\) \\
   \end{tabular}
 \end{center}
@@ -110,13 +110,13 @@ These past few results combined paint a remarkably clear picture of the
 Abelianization \(\Mod(S)^\ab\) and the commutator \(\Mod(S)'\), but this is
 still a far cry from a general undertanding of the structure of \(\Mod(S)\)
 itself. For that, we need to intruduce some extra relations. To that end, we
-study certain branched covers \(S \to \mathbb{D} \setminus \{x_1, \ldots,
+study certain branched covers \(S \to \mathbb{D}^2 \setminus \{x_1, \ldots,
 x_r\}\) and how they may be used to relate both mapping class groups. This is
 what is known as\dots
 
 \section{The Birman-Hilden Theorem}
 
-Let \(S_{0, r}^1 = \mathbb{D} \setminus \{x_1, \ldots, x_r\}\) be the surface
+Let \(S_{0, r}^1 = \mathbb{D}^2 \setminus \{x_1, \ldots, x_r\}\) be the surface
 of genus \(0\) with \(r\) punctures and \(1\) boundary component. We begin our
 investigation by providing an alternative description of its mapping class
 group. Namely, we show that \(\Mod(S_{0, r}^1)\) is the braid group on \(r\)
@@ -124,8 +124,8 @@ strands.
 
 \begin{definition}
   The \emph{braid group on \(n\) strands} \(B_n\) is the fundamental group
-  \(\pi_1(C(\mathbb{D}, n), *)\) of the unordered configuration space
-  \(C(\mathbb{D}, n) = \mfrac{C^{\operatorname{ord}}(\mathbb{D},
+  \(\pi_1(C(\mathbb{D}^2, n), *)\) of the unordered configuration space
+  \(C(\mathbb{D}^2, n) = \mfrac{C^{\operatorname{ord}}(\mathbb{D}^2,
   n)}{\mathfrak{S}_n}\) of \(n\) distinct points in the interior of the disk.
   The elements of \(B_n\) are referred to as \emph{braids}.
 \end{definition}
@@ -177,12 +177,12 @@ sequence
       1 \rar
       & B_n \rar{\operatorname{push}}
       & \Mod(S_{0, n}^1) \rar
-      & \Mod(\mathbb{D}) \rar
+      & \Mod(\mathbb{D}^2) \rar
       & 1,
   \end{tikzcd}
 \end{center}
-given that \(\Homeo^+(\mathbb{D}, \mathbb{S}^1)\) is contractible -- see
-Example~\ref{ex:alexander-trick}. But \(\Mod(\mathbb{D}) = 1\). Hence
+given that \(\Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) is contractible -- see
+Example~\ref{ex:alexander-trick}. But \(\Mod(\mathbb{D}^2) = 1\). Hence
 we get\dots
 
 \begin{proposition}
@@ -233,7 +233,7 @@ It is clear from Figure~\ref{fig:hyperelliptic-involution} that the quotients
 \(\mfrac{S_\ell^1}{\iota}\) and \(\mfrac{S_\ell^2}{\iota}\) are both disks,
 with boundary corresponding to the projection of the boundaries of \(S_\ell^1\)
 and \(S_\ell^2\), respectively. Given \(b = 1, 2\), the quotient map \(S_\ell^b
-\to \mfrac{S_\ell^b}{\iota} \cong \mathbb{D}\) is a double cover with \(2\ell +
+\to \mfrac{S_\ell^b}{\iota} \cong \mathbb{D}^2\) is a double cover with \(2\ell +
 b\) branch points corresponding to the fixed points of \(\iota\). We may thus
 regard \(\mfrac{S_\ell^b}{\iota}\) as the disk \(S_{0, 2\ell + b}^1\) with
 \(2\ell + b\) punctures in its interior, as shown in

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -34,14 +34,14 @@ Figure~\ref{fig:dehn-twist-bitorus} in the case of the bitorus \(S_2\).
 
 Similarly, using the description of the mapping class group of the
 twice-puncture disk derived in Example~\ref{ex:mcg-twice-punctured-disk}, the
-generator of \(\Mod(\mathbb{D} \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\})\)
+generator of \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\})\)
 gives rise the so called \emph{half-twists}. These are examples of mapping
 classes that permute the punctures of \(S\).
 
 \begin{definition}
   Given an arc \(\alpha \subset S\) joining two punctures in the interior of
   \(S\), fix a closed neighborhood \(D \subset S\) of \(\alpha\) with \(D \cong
-  \mathbb{D} \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}\). Let \(f \in
+  \mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}\). Let \(f \in
   \Mod(\mathbb{S}^1 \times [0, 1]) \cong \Mod(D)\) be as in
   Example~\ref{ex:mcg-twice-punctured-disk}. The \emph{half-twist \(h_\alpha
   \in \Mod(S)\) about \(\alpha\)} is defined as the image of the generator \(f
@@ -279,7 +279,7 @@ cases, \(\mathcal{C}(S_{g, r})\) is connected.
 
 \begin{theorem}[Lickorish]
   If \(S_{g, r}\) is not one \(S_0 = \mathbb{S}^2, S_{0, 1}, \ldots, S_{0, 4},
-  S_1 = \mathbb{T}\) and \(S_{1, 1}\) then \(\mathcal{C}(S_{g, r})\) is
+  S_1 = \mathbb{T}^2\) and \(S_{1, 1}\) then \(\mathcal{C}(S_{g, r})\) is
   connected.
 \end{theorem}
 
@@ -312,23 +312,23 @@ Theorem~\ref{thm:mcg-is-fg}.
       1 \rar &
       \langle \tau_{\delta_1} \rangle \rar &
       \Mod(S_{g, r}^b) \rar{\operatorname{cap}} &
-      \Mod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D} \setminus \{0\})) \rar &
+      \Mod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\})) \rar &
       1
     \end{tikzcd}
   \end{center}
   from Proposition~\ref{ex:capping-seq}, it suffices to show that \(S_{g, n}\)
   is finitely generated by twists about nonseparating simple closed curves.
-  Indeed, if \(\PMod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D} \setminus \{0\}))\)
+  Indeed, if \(\PMod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\)
   is finitely generated by twists about nonseparing curves or boundary
   components, then we may lift the generators of \(\PMod(S_{g, r}^b
-  \cup_{\delta_1} (\mathbb{D} \setminus \{0\}))\) to Dehn twists about the
+  \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
   generating set.
 
   It thus suffices to consider the boudaryless case \(S_{g, r}\). As promised,
   we proceed by double induction on \(r\) and \(g\). For the base case, it is
   clear from Example~\ref{ex:torus-mcg} and Example~\ref{ex:torus-mcg} that
-  \(\Mod(\mathbb{T}) \cong \Mod(S_{1, 1}) \cong
+  \(\Mod(\mathbb{T}^2) \cong \Mod(S_{1, 1}) \cong
   \operatorname{SL}_2(\mathbb{Z})\) are generated by the Dehn twists about the
   curves \(\alpha\) and \(\beta\) from
   Figure~\ref{fig:torus-mcg-generators}, each corresponding to one of the
@@ -348,9 +348,9 @@ Theorem~\ref{thm:mcg-is-fg}.
 
   \begin{figure}[ht]
     \centering
-    \includegraphics[width=.33\linewidth]{images/torus-mcg-generators.eps}
+    \includegraphics[width=.55\linewidth]{images/torus-mcg-generators.eps}
     \caption{The curves $\alpha$ and $\beta$ whose Dehn twists generate
-    $\Mod(\mathbb{T})$ and $\Mod(S_{1, 1})$.}
+    $\Mod(\mathbb{T}^2)$ and $\Mod(S_{1, 1})$.}
     \label{fig:torus-mcg-generators}
   \end{figure}