memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: 3dcf4f4eaab8cc8674f4c1d03b1cc9c257c5525c
Parent: e9f6ff13f218bc6725abfd19bdc0f918cc353fa0
Author: Pablo <pablo-pie@riseup.net>
Date: Fri, 7 Jun 2024 10:51:54 +0000

Wrote a proof of the fact that PMod is finitely-generated by twists

Diffstat

4 files changed, 586 insertions, 13 deletions

Status	File Name	N° Changes	Insertions	Deletions
Added	images/cutting-homeo.svg	295	295	0
Added	images/torus-mcg-generators.svg	140	140	0
Modified	sections/introduction.tex	6	3	3
Modified	sections/twists.tex	158	148	10

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diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -33,7 +33,7 @@
   \{0\}))\) as \emph{the capping homomorphism}.
 \end{example}
 
-\begin{proposition}
+\begin{proposition}\label{ex:capping-seq}
   Given some oriented boundary component \(\alpha \subset \partial S\) of
   \(S\), there is an exact sequence
   \begin{center}
@@ -81,14 +81,14 @@
 \end{example}
 
 % TODO: Explain this
-\begin{example}
+\begin{example}\label{ex:torus-mcg}
   The symplectic representation \(\psi : \Mod(\mathbb{T}) \to
   \operatorname{Sp}_2(\mathbb{Z}) = \operatorname{SL}_2(\mathbb{Z})\) is a
   group isomorphism. In particular, \(\Mod(\mathbb{T}) \cong
   \operatorname{SL}_2(\mathbb{Z})\).
 \end{example}
 
-\begin{example}
+\begin{example}\label{ex:punctured-torus-mcg}
   By the same token, \(\Mod(S_{1, 1}) \cong \operatorname{SL}_2(\mathbb{Z})\).
 \end{example}

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -28,7 +28,7 @@
   has infinite order.
 \end{proposition}
 
-\begin{fact}
+\begin{fact}\label{thm:dehn-twist-is-uniq}
   \(\tau_\alpha = \tau_\beta \iff [\alpha] = [\beta]\).
 \end{fact}
 
@@ -47,15 +47,16 @@
 \end{fact}
 
 \begin{theorem}\label{thm:mcg-is-fg}
-  Let \(g \ge 1\) and \(n \ge\). Then the pure mapping class group
-  \(\PMod(S_{g, n})\) is generated by finitely many Dehn twists about
-  nonseparating curves.
+  Let \(S\) be an orientable surface of genus \(g \ge 1\), potentially with
+  punctures and boundary components. Then the pure mapping class group
+  \(\PMod(S)\) is generated by finitely many Dehn twists about nonseparating
+  curves or boundary components.
 \end{theorem}
 
 \section{The Birman Exact Sequence}
 
 % TODO: Explain who the fuck are push & forget
-\begin{theorem}[Birman exact sequence]
+\begin{theorem}[Birman exact sequence]\label{thm:birman-exact-seq}
   If \(\chi(S) < 0\) then there is an exact sequence 
   \begin{center}
     \begin{tikzcd}
@@ -93,8 +94,7 @@
 
 \section{The Modified Complex of Curves}
 
-% TODO: Add comments on the proof?
-\begin{lemma}
+\begin{lemma}\label{thm:ggt-lemma}
   Let \(G\) be a group and \(\Gamma\) be a \emph{connected} graph with \(G
   \actson \Gamma\) via graph automorphisms. Suppose that \(G\) acts
   transitively both in \(V(\Gamma)\) and \(\{(v, w) \in V(\Gamma)^2 :
@@ -141,13 +141,151 @@
 
 % TODO
 \begin{proof}[Proof of Theorem~\ref{thm:mcg-is-fg}]
+  Let \(S\) be an orientable surface of genus \(g \ge 1\), pententially with
+  punctures and boundary components. We want to establish that \(\PMod(S)\) is
+  genetery by a finite number of Dehn twists about nonseparating simple closed
+  curves or boundary components.
+
+  First, observe that if \(S\) has \(b \ge 1\) boundary components \(\alpha_1,
+  \ldots, \alpha_b\) then by recursively applying the capping exact sequence
+  \begin{center}
+    \begin{tikzcd}
+      1 \rar &
+      \langle \tau_{\alpha_1} \rangle \rar &
+      \Mod(S) \rar{\operatorname{cap}} &
+      \Mod(S \cup_{\alpha_1} (\mathbb{D} \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 \(\PMod(S \cup_{\alpha_1} (\mathbb{D} \setminus \{0\}))\) is
+  finitely generated by twists about nonseparing curves or boundary components,
+  then we may lift the generators of \(\PMod(S \cup_{\alpha_1} (\mathbb{D}
+  \setminus \{0\}))\) to Dehn twists about the corresponding curves in \(S\)
+  and add \(\tau_{\alpha_1}\) to the generating set.
+
+  % TODO: Check the matrices here
+  It thus suffices to consider the boudaryless case \(S = S_{g, n}\). We
+  proceed by double induction on \(n\) and \(g\).
+  For the base case, it is clear from Example~\ref{ex:torus-mcg} and
+  Example~\ref{ex:torus-mcg} that \(\Mod(\mathbb{T}) \cong \Mod(S_{1, 1}) \cong
+  \operatorname{SL}_2(\mathbb{Z})\) are generated by the Dehn twists about the
+  two curves \(\alpha\) and \(\beta\) as in
+  \begin{center}
+    \includegraphics[width=.5\linewidth]{images/torus-mcg-generators.eps},
+  \end{center}
+  each corresponding to one of the standard generators
+  \begin{align*}
+    \begin{pmatrix}
+      1 & -1 \\
+      0 &  1
+    \end{pmatrix}
+    &&
+    \begin{pmatrix}
+      1 & 0 \\
+      1 & 1
+    \end{pmatrix}
+  \end{align*}
+  of \(\operatorname{SL}_2(\mathbb{Z})\).
+
+  Now suppose \(\PMod(S_{g, n})\) is finitely-generated by twists about
+  nonseparating curves for \(g \ge 2\) or \(g = 1\) and \(n > 1\). In both
+  case, \(\chi(S_{g, n}) = 2 - 2g - n < 0\) and thus the Birman exact
+  sequence from Theorem~\ref{thm:birman-exact-seq} gives us
+  \begin{center}
+    \begin{tikzcd}
+      1 \rar
+      & \pi_1(S_{g, n}, p_0) \rar{\operatorname{push}}
+      & \PMod(S_{g, n + 1}) \rar{\operatorname{forget}}
+      & \PMod(S_{g, n}) \rar
+      & 1,
+    \end{tikzcd}
+  \end{center}
+  where \(S_{g, n + 1} = S_{g, n} \setminus \{p_0\}\). Since \(g \ge 1\),
+  \(\pi_1(S_{g, n}, p_0)\) is generated by finitely many nonseparating loops.
+  We have seen that \(\operatorname{push} : \pi_1(S_{g, n}, p_0) \to
+  \Mod(S_{g, n+1}, p_0)\) takes simple loops to products of twists about
+  nonseparating simple curves. Furthermore, we may once again lift the
+  generators of \(\PMod(S_{g, n})\) to Dehn twists about nonseparating simple
+  curves in \(S_{g, n + 1}\). This goes to show that \(\PMod(S_{g, n + 1})\) is
+  also generated by finitely many twists about simple curves, concluding the
+  induction step on \(n\).
+
+  As for the induction step on \(g\), fix \(g \ge 2\) and suppose \(\PMod(S_{g,
+  n})\) is finitely generated by twists about nonseparing curves for all \(n
+  \ge 0\). Let us show that the same holds for \(\Mod(S_{g + 1})\). To that
+  end, we consider the action \(\Mod(S_{g + 1}) \actson \hat{\mathcal{N}}(S_{g
+  + 1})\). Since \(g + 1 \ge 2\), \(\hat{\mathcal{N}}(S_{g + 1})\) is
+  connected. It is also clear from the change of coordinates principle that the
+  actions of \(\Mod(S_{g + 1})\) on \(V(\hat{\mathcal{N}}(S_{g + 1}))\) and
+  \(\{([\alpha], [\beta]) \in V(\hat{\mathcal{N}}(S_{g + 1}))^2 : \#(\alpha
+  \cap \beta) = 1 \}\) are both transitive. In other words, the conditions of
+  Lemma~\ref{thm:ggt-lemma} apply.
+
+  Now observe that given nonseparating \(\alpha, \beta \subset S_{g + 1}\)
+  crossing only once, \(\tau_\beta \tau_\alpha \cdot [\beta] = [\alpha]\) --
+  this is equilent to the braid relation \(\tau_\alpha \tau_\beta \tau_\alpha =
+  \tau_\beta \tau_\alpha \tau_\beta\) because of
+  Fact~\ref{thm:dehn-twist-is-uniq}. Hence by Lemma~\ref{thm:ggt-lemma}
+  \(\Mod(S_{g + 1})\) is generated by \(\Mod(S_{g + 1})_{[\alpha]} = \{ f \in
+  \Mod(S_{g + 1}) : f \cdot [\alpha] = [\alpha]\}\) and \(\tau_\beta
+  \tau_\alpha\). In turn, \(\Mod(S_{g + 1})_{[\alpha]}\) has an index
+  \(2\) subgroup \(\Mod(S_{g + 1})_{\vec{[\alpha]}} = \{ f \in \Mod(S_{g + 1})
+  : f \cdot \vec{[\alpha]} = \vec{[\alpha]}\}\). One can check that
+  \(\tau_\beta \tau_\alpha^2 \tau_\beta \in \Mod(S_{g + 1})_{[\alpha]}\)
+  inverts the orientation of \(\alpha\) and is thus a representative of the
+  nontrivial \(\Mod(S_{g+1})_{\vec{[\alpha]}}\)-coset in
+  \(\Mod(S_{g+1})_{[\alpha]}\).
+
+  % TODO: Properly state the cutting exact seq. beforehand?
+  In other words, \(\Mod(S_{g+1})\) is generated by
+  \(\Mod(S_{g+1})_{\vec{[\alpha]}}\), \(\tau_\beta \tau_\alpha\) and
+  \(\tau_\beta \tau_\alpha^2 \tau_\beta\). Finally, we claim
+  \(\Mod(S_{g+1})_{\vec{[\alpha]}}\) is generated by finitely many twists about
+  nonseparating curves. To show this, we first remark that any \(f \in
+  \Mod(S_{g+1})_{\vec{[\alpha]}}\) has a representative \(\phi \in
+  \Homeo^+(S_{g, n})\) fixing \(\alpha\) point-wise. We thus obtain an exact
+  sequence
+  \begin{equation}\label{eq:cutting-seq}
+    \begin{tikzcd}
+      1 \rar &
+      \langle \tau_\alpha \rangle \rar &
+      \Mod(S_{g+1})_{\vec{[\alpha]}} \rar &
+      \PMod(S_{g+1} \setminus \alpha) \rar &
+      1,
+    \end{tikzcd}
+  \end{equation}
+  where
+  \begin{align*}
+    \Mod(S_{g+1})_{\vec{[\alpha]}} & \to \PMod(S_{g+1} \setminus \alpha) \\
+    [\phi] & \mapsto [\phi\!\restriction_{S_{g+1} \setminus \alpha}].
+  \end{align*}
+
+  But \(S_{g+1} \setminus \alpha \cong S_{g, 2}\) -- see
+  Figure~\ref{fig:cut-along-nonseparating-adds-two-punctures} -- and so the
+  induction hypothesis implies \(\PMod(S_{g+1} \setminus \alpha)\) is
+  finitely-generated by twists about nonseparating simple closed curves. As
+  before, these generators may be lifted to appropriate twists in
+  \(\Mod(S_{g+1})_{\vec{[\alpha]}}\). Now by (\ref{eq:cutting-seq}) we get that
+  \(\Mod(S_{g+1})_{\vec{[\alpha]}}\) is finitely generated by twists about
+  nonseparating curves. This concludes the induction step in \(g\).
 \end{proof}
 
+\begin{figure}[h]\label{fig:cut-along-nonseparating-adds-two-punctures}
+  \centering
+  \includegraphics[width=.8\linewidth]{images/cutting-homeo.eps}
+  \caption{The homeomorphism $S_{g + 1} \setminus \alpha \cong S_{g, 2}$:
+  removing the curve $\alpha$ has the same effect as cutting along $\alpha$ and
+  then capping the two resulting boundary components with once-punctured disks,
+  which gives us $S_{g, 2}$.}
+\end{figure}
+
 \begin{theorem}[Lickorish generators]\label{thm:lickorish-gens}
   If \(g \ge 1\) then \(\Mod(S_g^b)\) is generated by \(\tau_{\alpha_1},
-  \ldots, \tau_{\alpha_g}, \tau_{\mu_1}, \ldots, \tau_{\mu_g}, \tau_{\gamma_1},
-  \ldots, \tau_{\gamma_{g - 1}}, \tau_{\eta_1}, \ldots, \tau_{\eta_{b - 1}}\) as
-  in the following diagram.
+  \ldots, \tau_{\alpha_g}\), \newline \(\tau_{\mu_1}, \ldots, \tau_{\mu_g}\),
+  \(\tau_{\gamma_1}, \ldots, \tau_{\gamma_{g - 1}},\), \(\tau_{\eta_1}, \ldots,
+  \tau_{\eta_{b - 1}}\) as in the following diagram.
   \begin{center}
     \includegraphics[width=.5\linewidth]{images/lickorish-gens.eps}
   \end{center}