memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: 1b87dd5efb423ad25ae23786afc7942dd938a8dd
Parent: 42a8cad4122167378003901a538b3d55168e2538
Author: Pablo <pablo-pie@riseup.net>
Date: Sat, 22 Jun 2024 12:34:25 +0000

Moved around the section on concrete examples

The section is now split across multiple chapter, with each part of it closer to the relevant discussion

Diffstat

2 files changed, 165 insertions, 168 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	232	79	153
Modified	sections/twists.tex	101	86	15

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -290,6 +290,83 @@ representation.}
   \label{fig:intersection-index}
 \end{minipage}
 
+The symplectic representation already allows us to compute some important
+examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 =
+S_1\) and the once-punctured torus \(S_{1, 1}\).
+
+% TODO: Draw a diagram?
+\begin{example}[Alexander trick]\label{ex:alexander-trick}
+  The group \(\Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) of homeomorphisms of the
+  unit disk \(\mathbb{D}^2 \subset \mathbb{Z}\) is contractible. In particular,
+  \(\Mod(\mathbb{D}^2) = 1\). Indeed, for any \(\phi \in
+  \Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) the isotopy
+  \begin{align*}
+    \phi_t : \mathbb{D}^2 & \to     \mathbb{D}^2 \\
+                   (z, t) & \mapsto
+    \begin{cases}
+      (1 - t) \phi(\sfrac{z}{1 - t}) & \text{if } 0 \le |z| \le 1 - t \\
+      z                              & \text{otherwise}
+    \end{cases}
+  \end{align*}
+  that ``fixes the band \(\{ z \in \mathbb{D}^2 : |z| \ge 1 - t \}\) and does
+  \(\phi\) inside the subdisk \(\{ z \in \mathbb{D}^2 : |z| \le 1 - t\}\)''
+  joins \(\phi = \phi_0\) and \(1 = \phi_1\).
+\end{example}
+
+\begin{example}\label{ex:mdg-once-punctured-disk}
+  By the same token, \(\Mod(\mathbb{D}^2 \setminus \{0\}) = 1\).
+\end{example}
+
+\begin{example}\label{ex:torus-mcg}
+  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, \(\Mod(\mathbb{T}^2) \cong
+  \operatorname{SL}_2(\mathbb{Z})\). To see \(\psi\) is surjective, first
+  observe \(\mathbb{Z}^2 \subset \mathbb{R}^2\) is
+  \(\operatorname{SL}_2(\mathbb{Z})\)-invariant. Hence any matrix \(g \in
+  \operatorname{SL}_2(\mathbb{Z})\) descends to an orientation-preserving
+  homeomorphism \(\phi_g\) of the quotient \(\mathbb{T}^2 =
+  \mfrac{\mathbb{R}^2}{\mathbb{Z}^2}\), which satisfies \(\psi([\phi_g]) = g\).
+  To see \(\psi\) is injective we consider the curves \(\alpha_1\) and
+  \(\beta_1\) from Figure~\ref{fig:homology-basis}. Given \(f = [\phi] \in
+  \Mod(\mathbb{T}^2)\) with \(\psi(f) = 1\), \(f \cdot \vec{[\alpha_1]} =
+  \vec{[\alpha_1]}\) and \(f \cdot \vec{[\beta_1]} = \vec{[\beta_1]}\), so we
+  may choose a representative \(\phi\) of \(f\) fixing \(\alpha_1 \cup
+  \beta_1\) pointwise. Such \(\phi\) determines a homeomorphism \(\tilde \phi\)
+  of the surface \(\mathbb{T}_{\alpha_1 \beta_1}^2 \cong \mathbb{D}^2\)
+  obtained by cutting \(\mathbb{T}^2\) across \(\alpha_1\) and \(\beta_1\), as
+  in Figure~\ref{fig:cut-torus-across}. Now by the Alexander trick from
+  Example~\ref{ex:alexander-trick}, \(\tilde\phi\) must be isotopic to the
+  identity. The isotopy \(\tilde\phi \simeq 1 \in \Homeo^+(\mathbb{D}^2,
+  \mathbb{S}^1)\) then decends to an isotopy \(\phi \simeq 1 \in
+  \Homeo^+(\mathbb{T}^2)\), so \(f = 1 \in \Mod(\mathbb{T}^2\) as desired.
+\end{example}
+
+\begin{figure}[ht]
+  \centering
+  \includegraphics[width=.55\linewidth]{images/torus-cut.eps}
+  \caption{By cutting $\mathbb{T}^2$ across $\alpha_1$ we obtain a cylinder,
+  where $\beta_1$ determines a yellow arc joining the two boundary components.
+  Now by cutting across this yellow arc we obtain a disk.}
+  \label{fig:cut-torus-across}
+\end{figure}
+
+\begin{example}\label{ex:punctured-torus-mcg}
+  By the same token, \(\Mod(S_{1, 1}) \cong \operatorname{SL}_2(\mathbb{Z})\).
+\end{example}
+
+\begin{note}
+  Despite the fact \(\psi : \Mod(\mathbb{T}^2) \to
+  \operatorname{SL}_2(\mathbb{Z})\) is an isomorphism, the symplectic
+  representation is \emph{not} injective for surfaces of genus \(g \ge 2\) --
+  see \cite[Section~6.5]{farb-margalit} for a description of its kernel. In
+  fact, the question of existance of injective representations of \(\Mod(S_g)\)
+  remains wide-open. Recently, Korkmaz \cite[Theomre~3]{korkmaz} established
+  the lower bound of \(3 g - 3\) for the dimension of an injective
+  representation of \(\Mod(S_g)\) in the \(g \ge 3\) case -- if one such
+  representation exists.
+\end{note}
+
 Another fundamental class of examples of representations are the so called
 \emph{TQFT representations}.
 
@@ -379,6 +456,8 @@ of level \(r\)}, first introduced by Witten and Reshetikhin-Tuarev
 \cite{witten, reshetikhin-turaev} in their foundational papers on quantum
 topology.
 
+% TODOO: Add comments on Costantino's idea to get a faithful representation?
+
 Besides Example~\ref{ex:symplectic-rep} and Example~\ref{ex:tqft-reps}, not a
 lot of other linear representations of \(\Mod(S_g)\) are known. Indeed, the
 representation theory of mapping class groups remains at mistery at large. In
@@ -390,156 +469,3 @@ end, in Chapter~\ref{ch:dehn-twists} and Chapter~\ref{ch:relations} we survay
 the group structure of mapping class groups: its relations and known
 presentations.
 
-% TODOO: Move this to the next chapter
-\section{First Computations}
-
-We begin by the simplest, yet perhaps the most fundamental, of computations in
-this entire thesis.
-
-% TODO: Draw a diagram?
-\begin{example}[Alexander trick]\label{ex:alexander-trick}
-  The group \(\Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) of homeomorphisms of the
-  unit disk \(\mathbb{D}^2 \subset \mathbb{Z}\) is contractible. In particular,
-  \(\Mod(\mathbb{D}^2) = 1\). Indeed, for any \(\phi \in
-  \Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) the isotopy
-  \begin{align*}
-    \phi_t : \mathbb{D}^2 & \to     \mathbb{D}^2 \\
-                   (z, t) & \mapsto
-    \begin{cases}
-      (1 - t) \phi(\sfrac{z}{1 - t}) & \text{if } 0 \le |z| \le 1 - t \\
-      z                              & \text{otherwise}
-    \end{cases}
-  \end{align*}
-  that ``fixes the band \(\{ z \in \mathbb{D}^2 : |z| \ge 1 - t \}\) and does
-  \(\phi\) inside the subdisk \(\{ z \in \mathbb{D}^2 : |z| \le 1 - t\}\)''
-  joins \(\phi = \phi_0\) and \(1 = \phi_1\).
-\end{example}
-
-\begin{example}\label{ex:mdg-once-punctured-disk}
-  By the same token, \(\Mod(\mathbb{D}^2 \setminus \{0\}) = 1\).
-\end{example}
-
-\begin{example}\label{ex:torus-mcg}
-  Let \(\mathbb{T}^2 = S_1\) be the torus. 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,
-  \(\Mod(\mathbb{T}^2) \cong \operatorname{SL}_2(\mathbb{Z})\). To see \(\psi\)
-  is surjective, first observe \(\mathbb{Z}^2 \subset \mathbb{R}^2\) is
-  \(\operatorname{SL}_2(\mathbb{Z})\)-invariant. Hence any matrix \(g \in
-  \operatorname{SL}_2(\mathbb{Z})\) descends to an orientation-preserving
-  homeomorphism \(\phi_g\) of the quotient \(\mathbb{T}^2 =
-  \mfrac{\mathbb{R}^2}{\mathbb{Z}^2}\), which satisfies \(\psi([\phi_g]) = g\).
-  To see \(\psi\) is injective we consider the curves \(\alpha_1\) and
-  \(\beta_1\) from Figure~\ref{fig:homology-basis}. Given \(f = [\phi] \in
-  \Mod(\mathbb{T}^2)\) with \(\psi(f) = 1\), \(f \cdot \vec{[\alpha_1]} =
-  \vec{[\alpha_1]}\) and \(f \cdot \vec{[\beta_1]} = \vec{[\beta_1]}\), so we
-  may choose a representative \(\phi\) of \(f\) fixing \(\alpha_1 \cup
-  \beta_1\) pointwise. Such \(\phi\) determines a homeomorphism \(\tilde \phi\)
-  of the surface \(\mathbb{T}_{\alpha_1 \beta_1}^2 \cong \mathbb{D}^2\)
-  obtained by cutting \(\mathbb{T}^2\) across \(\alpha_1\) and \(\beta_1\), as
-  in Figure~\ref{fig:cut-torus-across}. Now by the Alexander trick from
-  Example~\ref{ex:alexander-trick}, \(\tilde\phi\) must be isotopic to the
-  identity. The isotopy \(\tilde\phi \simeq 1 \in \Homeo^+(\mathbb{D}^2,
-  \mathbb{S}^1)\) then decends to an isotopy \(\phi \simeq 1 \in
-  \Homeo^+(\mathbb{T}^2)\), so \(f = 1 \in \Mod(\mathbb{T}^2\) as desired.
-\end{example}
-
-\begin{figure}[ht]
-  \centering
-  \includegraphics[width=.55\linewidth]{images/torus-cut.eps}
-  \caption{By cutting $\mathbb{T}^2$ across $\alpha_1$ we obtain a cylinder,
-  where $\beta_1$ determines a yellow arc joining the two boundary components.
-  Now by cutting across this yellow arc we obtain a disk.}
-  \label{fig:cut-torus-across}
-\end{figure}
-
-\begin{example}\label{ex:punctured-torus-mcg}
-  By the same token, \(\Mod(S_{1, 1}) \cong \operatorname{SL}_2(\mathbb{Z})\).
-\end{example}
-
-% TODOO: Add comments on Costantino's idea?
-\begin{note}
-  Despite the fact \(\psi : \Mod(\mathbb{T}^2) \to
-  \operatorname{SL}_2(\mathbb{Z})\) is an isomorphism, the symplectic
-  representation is \emph{not} injective for surfaces of genus \(g \ge 2\) --
-  see \cite[Section~6.5]{farb-margalit} for a description of its kernel. In
-  fact, the question of existance of injective representations of \(\Mod(S_g)\)
-  remains wide-open. Recently, Korkmaz \cite[Theomre~3]{korkmaz} established a
-  lower bound for the dimension of an injective representation of \(\Mod(S_g)\)
-  in the \(g \ge 3\) case -- if one such representation exists.
-\end{note}
-
-By cutting across curves and arcs as in the proof of injectivity in
-Example~\ref{ex:torus-mcg}, we can always decompose a surface into copies of
-\(\mathbb{D}^2\) and \(\mathbb{D}^2 \setminus \{0\}\).
-Example~\ref{ex:alexander-trick} and Example~\ref{ex:mdg-once-punctured-disk}
-then imply\dots
-
-\begin{proposition}[Alexander method]\label{thm:alexander-method}
-  Let \(\alpha_1, \ldots, \alpha_n \subset S\) be essencial simple closed
-  curves or proper arcs satisfying the following conditions.
-  \begin{enumerate}
-    \item \([\alpha_i] \ne [\alpha_j]\) for \(i \ne j\).
-    \item Each pair \((\alpha_i, \alpha_j)\) crosses at most once.
-    \item Given distinct \(i, j, k\), at least one of \(\alpha_i \cap \alpha_j,
-      \alpha_i \cap \alpha_k, \alpha_j \cap \alpha_k\) is empty. 
-    \item The surface obtained by cutting \(S\) across the \(\alpha_i\) is a
-      disjoint union of disks and once-punctured disks.
-  \end{enumerate}
-  Let \(f \in \Mod(S)\). If there is \(\sigma \in \mathfrak{S}_n\) such that
-  \(f \cdot \vec{[\alpha_i]} = \vec{[\alpha_{\sigma(i)}]}\) for all \(i\), then
-  \(f^{\operatorname{ord}(\sigma)} = 1 \in \Mod(S)\). In particular, if
-  \(\sigma = 1\) then \(f = 1 \in \Mod(S)\).
-\end{proposition}
-
-See \cite[Proposition~2.8]{farb-margalit} for a proof of
-Proposition~\ref{thm:alexander-method}. The Alexander method can also be used
-to show two mapping classes \(f, g \in \Mod(S)\) are the same: if we decompose
-\(S\) into disks and once-punctured disks with \(\alpha_1, \ldots, \alpha_n\)
-such that \(f \cdot [\alpha_i] = g \cdot [\alpha_i]\), then \(f g^{-1} \cdot
-[\alpha_i] = [\alpha_i]\) and so we may apply
-Proposition~\ref{thm:alexander-method} to the class \(f g^{-1}\). We now
-collect some important applications of the method.
-
-\begin{example}\label{ex:mcg-annulus}
-  \(\Mod(\mathbb{S}^1 \times [0, 1]) \cong \mathbb{Z}\) is freely generated by
-  \(f = [\phi]\), where
-  \begin{align*}
-    \phi : \mathbb{S}^1 \times [0, 1] & \isoto  \mathbb{S}^1 \times [0, 1] \\
-                   (e^{2 \pi i t}, s) & \mapsto (e^{2 \pi i (t - s)}, s)
-  \end{align*}
-  is the map illustrated in Figure~\ref{fig:dehn-twist-cylinder}. In
-  particular, \(\Mod(\mathbb{S}^1 \times [0, 1]) \cong \mathbb{Z}\).
-\end{example}
-
-\begin{example}\label{ex:mcg-twice-punctured-disk}
-  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}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}
-    & \isoto \mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\} \\
-    z & \mapsto -z
-  \end{align*}
-  is the map from Figure~\ref{fig:hald-twist-disk}. In particular,
-  \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}) \cong
-  \mathbb{Z}\).
-\end{example}
-
-\begin{figure}[ht]
-  \centering
-  \includegraphics[width=.3\linewidth]{images/dehn-twist-cylinder.eps}
-  \caption{The generator $f$ of $\Mod(\mathbb{S}^1 \times [0, 1]) \cong
-  \mathbb{Z}$ takes the yellow arc in the left-hand side to the arc on the
-  right-hand side that winds about the curve $\alpha$.}
-  \label{fig:dehn-twist-cylinder}
-\end{figure}
-
-\begin{figure}[ht]
-  \centering
-  \includegraphics[width=.18\linewidth]{images/half-twist-disk.eps}
-  \caption{The generator $f$ of $\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2},
-  \sfrac{1}{2}) \cong \mathbb{Z}$ corresponds to the cclockwise rotation by
-  $\pi$ about the origen.}
-  \label{fig:hald-twist-disk}
-\end{figure}

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -1,18 +1,88 @@
 \chapter{Dehn Twists}\label{ch:dehn-twists}
 
-We have now seen some concrete examples of mapping class groups. In this
-chapter, we will investigate how we can use the anulus \(\mathbb{S}^1 \times
-[0, 1]\) to understand the structure of the mapping class groups of other
-surfaces. Let \(S\) be an orientable surface, possibly with punctures and
-non-empty boundary.
-
-Recall from Example~\ref{ex:mcg-annulus} that \(\Mod(\mathbb{S}^1 \times [0,
-1]) \cong \mathbb{Z}\) is generated by the mapping class that twists the
-cylinder by \(2\pi\) about the the curve \(\alpha = \mathbb{S}^1 \times
-\{\sfrac{1}{2}\}\). Now given some closed \(\alpha \subset S\), we may envision
-doing something similar by looking at anular neighborhoods of \(\alpha\). These
-are the mapping classes known as \emph{Dehn twists}, illustrated in
-Figure~\ref{fig:dehn-twist-bitorus} in the case of the bitorus \(S_2\).
+With the goal of studying the linear representations of mapping class groups in
+mind, in this chapter we start investigating the group structure of
+\(\Mod(S)\). We begin by computing some fundamental examples. We then explore
+how we can use these examples to understand the structure of the mapping class
+groups of other surfaces. Namely, we compute \(\Mod(\mathbb{S}^1 \times [0, 1])
+\cong \mathbb{Z}\), and discuss how its generators gives rise to a conveniant
+generating set for \(\Mod(S)\), 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
+decompose a surface into copies of \(\mathbb{D}^2\) and \(\mathbb{D}^2
+\setminus \{0\}\). Example~\ref{ex:alexander-trick} and
+Example~\ref{ex:mdg-once-punctured-disk} then imply the triviality of mapping
+classes fixing such arcs and curves. Formally, this translates to\dots
+
+\begin{proposition}[Alexander method]\label{thm:alexander-method}
+  Let \(\alpha_1, \ldots, \alpha_n \subset S\) be essencial simple closed
+  curves or proper arcs satisfying the following conditions.
+  \begin{enumerate}
+    \item \([\alpha_i] \ne [\alpha_j]\) for \(i \ne j\).
+    \item Each pair \((\alpha_i, \alpha_j)\) crosses at most once.
+    \item Given distinct \(i, j, k\), at least one of \(\alpha_i \cap \alpha_j,
+      \alpha_i \cap \alpha_k, \alpha_j \cap \alpha_k\) is empty. 
+    \item The surface obtained by cutting \(S\) across the \(\alpha_i\) is a
+      disjoint union of disks and once-punctured disks.
+  \end{enumerate}
+  Suppose \(f \in \Mod(S)\) is such that \(f \cdot \vec{[\alpha_i]} =
+  \vec{[\alpha_i]}\) for all \(i\). Then \(f = 1 \in \Mod(S)\).
+\end{proposition}
+
+See \cite[Proposition~2.8]{farb-margalit} for a proof of
+Proposition~\ref{thm:alexander-method}. We now state some \emph{fundamental}
+applications of the Alexander method.
+
+\begin{example}\label{ex:mcg-annulus}
+  \(\Mod(\mathbb{S}^1 \times [0, 1]) \cong \mathbb{Z}\) is freely generated by
+  \(f = [\phi]\), where
+  \begin{align*}
+    \phi : \mathbb{S}^1 \times [0, 1] & \isoto  \mathbb{S}^1 \times [0, 1] \\
+                   (e^{2 \pi i t}, s) & \mapsto (e^{2 \pi i (t - s)}, s)
+  \end{align*}
+  is the map illustrated in Figure~\ref{fig:dehn-twist-cylinder}. In
+  particular, \(\Mod(\mathbb{S}^1 \times [0, 1]) \cong \mathbb{Z}\).
+\end{example}
+
+\begin{example}\label{ex:mcg-twice-punctured-disk}
+  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}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}
+    & \isoto \mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\} \\
+    z & \mapsto -z
+  \end{align*}
+  is the map from Figure~\ref{fig:hald-twist-disk}. In particular,
+  \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}) \cong
+  \mathbb{Z}\).
+\end{example}
+
+\begin{minipage}[b]{.45\linewidth}
+  \centering
+  \includegraphics[width=.7\linewidth]{images/dehn-twist-cylinder.eps}
+  \captionof{figure}{The generator $f$ of $\Mod(\mathbb{S}^1 \times [0, 1])
+  \cong \mathbb{Z}$ takes the yellow arc in the left-hand side to the arc on
+  the right-hand side that winds about the curve $\alpha$.}
+  \label{fig:dehn-twist-cylinder}
+\end{minipage}
+\hspace{.5cm} %
+\begin{minipage}[b]{.45\linewidth}
+  \centering
+  \includegraphics[width=.4\linewidth]{images/half-twist-disk.eps}
+  \captionof{figure}{The generator $f$ of $\Mod(\mathbb{D}^2 \setminus
+  \{-\sfrac{1}{2}, \sfrac{1}{2}) \cong \mathbb{Z}$ corresponds to the
+  cclockwise rotation by $\pi$ about the origen.}
+  \label{fig:hald-twist-disk}
+\end{minipage}
+
+Let \(S\) be an orientable surface, possibly with punctures and non-empty
+boundary. Given some simple closed curve \(\alpha \subset S\), we may envision
+doing something similar to Example~\ref{ex:mcg-annulus} in \(S\) by looking at
+anular neighborhoods of \(\alpha\). These are the precisely the \emph{Dehn
+twists}, illustrated in Figure~\ref{fig:dehn-twist-bitorus} in the case of the
+bitorus \(S_2\).
 
 \begin{definition}
   Given a simple closed curve \(\alpha \subset S\), fix a closed annular
@@ -55,8 +125,9 @@ by investigating the geometric intersection number
 \[
   \#(\alpha \cap \beta) = \min
   \left\{
-  |\alpha' \cap \beta'| : [\alpha'] = [\alpha], [\beta'] = [\beta],
-  \alpha' \text{ intersects } \beta' \text{ transversaly}
+    |\alpha' \cap \beta'| : [\alpha'] = [\alpha]
+    \text{ and }
+    [\beta'] = [\beta]
   \right\}
 \]
 we can distinguish between powers of Dehn twists