- Commit
- 3dcf4f4eaab8cc8674f4c1d03b1cc9c257c5525c
- Parent
- e9f6ff13f218bc6725abfd19bdc0f918cc353fa0
- Author
- Pablo <pablo-pie@riseup.net>
- Date
Wrote a proof of the fact that PMod is finitely-generated by twists
memoire-m2
My M2 Memoire on mapping class groups & their representations
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}