My M2 Memoire on mapping class groups & their representations
Fixed some TODOs
Added a reference to the figure of the Humphreys generators in the statement of Wajnryb's presentation
Added multiple examples to the introduction
Moved the discussion on the cutting morphism to the introduction
3 files changed, 70 insertions, 47 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/introduction.tex | 48 | 45 | 3 |
| Modified | sections/presentation.tex | 3 | 1 | 2 |
| Modified | sections/twists.tex | 66 | 24 | 42 |
diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -1,5 +1,8 @@ \chapter{Introduction}\label{ch:introduction} +% TODO: Motivation +% TODO: Talk about the classification of surfaces + \begin{definition} The \emph{mapping class group \(\Mod(S)\) of an orientable surface \(S\)} is the group of isotopy classes of orientation-preserving homeomorphisms \(S @@ -10,7 +13,7 @@ \] \end{definition} -\begin{example} +\begin{example}[Inclusion homomorphism] Let \(R \subset S\) be a closed subsurface. Given some \(\phi \in \Homeo^+(R, \partial R)\), we may extend \(\phi\) to \(\tilde{\phi} \in \Homeo^+(S, \partial S)\) by setting \(\tilde{\phi}(p) = p\) for \(p \in S\) outside of @@ -23,14 +26,14 @@ known as \emph{the inclusion homomorphism}. \end{example} -\begin{example} +\begin{example}[Capping homomorphism] Let \(\alpha \subset \partial S\) be a boundary component of \(S\) and fix some orientation of \(\alpha\). We refer to the inclusion homomorphism \(\operatorname{cap} : \Mod(S) \to \Mod(S \cup_\alpha (\mathbb{D}^2 \setminus \{0\}))\) as \emph{the capping homomorphism}. \end{example} -\begin{proposition}\label{ex:capping-seq} +\begin{proposition}[Capping exact sequence]\label{ex:capping-seq} Given some oriented boundary component \(\alpha \subset \partial S\) of \(S\), there is an exact sequence \begin{center} @@ -44,6 +47,40 @@ \end{center} \end{proposition} +\begin{example}[Cutting homomorphism]\label{ex:cutting-morphism} + Given a simple closed curve \(\alpha \subset S\), denote by + \(\Mod(S)_{\vec{[\alpha]}} \subset \Mod(S)\) the subgroup of mapping classes + that fix the isotopy class of \(\alpha\), with orientation. Any \(f \in + \Mod(S_{g+1})_{\vec{[\alpha]}}\) has a representative \(\phi \in + \Homeo^+(S_{g+1})\) fixing \(\alpha\) point-wise, so that \(\phi\) restricts + to a homeomorphism of \(S \setminus \alpha\). This construction yields a + group homomorphism + \begin{align*} + \operatorname{cut} : \Mod(S)_{\vec{[\alpha]}} + & \to \Mod(S\setminus\alpha) \\ + [\phi] & \mapsto [\phi\!\restriction_{S_{g+1} \setminus \alpha}], + \end{align*} + known as \emph{the cutting homomorphism}. Furthermore, \(\ker + \operatorname{cut} = \langle \tau_\alpha \rangle\). +\end{example} + +\begin{example} + Let \(S_{g, r}^b = S_g^b \setminus \{ x_1, \ldots, x_r \}\) be the surface of + genus \(g\) with \(r\) punctures and \(b\) boudary components. Given \(\phi + \in \Homeo^+(S_{g, r}^b, \partial S_{g, r}^b)\), there is some \(\sigma_\phi + \in \mathfrak{S}_r\) with \(\phi(x) \to x_{\sigma(i)}\) as \(x \to x_i\) in + \(S_{g, r}\). It is clear that \(\sigma_\phi = \sigma_\psi\) for \(\phi + \simeq \psi\). Hence \(\Mod(S_{g,r}^b) \leftaction \{ x_1, \ldots, x_r \}\) + via \(f \cdot x_i = x_{\sigma_\phi(i)}\) for \(f = [\phi]\). +\end{example} + +\begin{definition} + Given an orientable surface \(S\) and a puncture \(x\) of \(S\), denote by + \(\Mod(S, x) \subset \Mod(S)\) the subgroup of mapping classes that fix + \(x\). The \emph{pure mapping class group \(\PMod(S) \subset \Mod(S)\) of + \(S\)} is the subgroup of mapping classes that fix every puncture of \(S\). +\end{definition} + \begin{example}\label{ex:inclusion-morphism} \(\Mod(S) \leftaction \{ \vec{[\alpha]} : \alpha \subset S \}\) and \(\Mod(S) \leftaction \{ [\alpha] : \alpha \subset S \}\) via @@ -54,6 +91,11 @@ for \(f = [\phi] \in \Mod(S)\). \end{example} +\begin{example}[Change of coordinates principle]\label{ex:change-of-coordinates} + % Also highlight the fact that Mod(S) acts transitively on the pairs of + % curves crossing once. +\end{example} + \begin{example} \(\Mod(S) \leftaction H_k(S, \mathbb{R})\) \end{example}
diff --git a/sections/presentation.tex b/sections/presentation.tex @@ -425,8 +425,7 @@ which is widely considered to the the standard presentation of \(\Mod(S_g)\) \cite{wajnryb}. \begin{theorem}[Wajnryb]\label{thm:wajnryb-presentation} - If \(\alpha_0, \ldots, \alpha_g\) are as in - % TODO: Reference the drawing of the curves somewhere + If \(\alpha_0, \ldots, \alpha_g\) are as in Figure~\ref{fig:humphreys-gens} and \(a_i = \tau_{\alpha_i} \in \Mod(S_g)\) are the Humphreys generators, then there is a presentation of \(\Mod(S_g)\) with generators \(a_0, \ldots a_{2g}\) subject to the following relations.
diff --git a/sections/twists.tex b/sections/twists.tex @@ -49,7 +49,6 @@ classes that permute the punctures of \(S\). \Mod(S)\). \end{definition} -% TODO: Define the intersection number beforehand? It is interesting to study how the geometry of two curves affects the relationship between their corresponding Dehn twists. For instance, by investigating the geometric intersection number @@ -92,7 +91,6 @@ too. \([f, \tau_\alpha] = 1 \iff f \cdot [\alpha] = [\alpha]\). \end{example} -% TODO: Talk about the change of coordinates principle beforehand \begin{example} If \(\alpha, \beta \subset S\) are both nonseparing then \(\tau_\alpha, \tau_\beta \in \Mod(S)\) are conjugate. Indeed, by the change of coordinates @@ -100,9 +98,6 @@ too. and then apply Fact~\ref{thm:conjugate-twists}. \end{example} -%A particularly important class of relations obtained this way are the so called -%\emph{braid relations}. - \begin{example}\label{ex:braid-relation} Given \(\alpha, \beta \subset S\) with \(\#(\alpha \cap \beta) = 1\), it is not hard to check that \(\tau_\beta \tau_\alpha \cdot [\beta] = [\alpha]\). @@ -116,7 +111,6 @@ too. A perhaps less obvious fact about Dehn twists is\dots -% TODO: Define PMod beforehand \begin{theorem}\label{thm:mcg-is-fg} Let \(S_{g, r}^b\) be the orientable surface of genus \(g \ge 1\) with \(r\) punctures and \(b\) boundary components. Then the pure mapping class group @@ -138,7 +132,6 @@ 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 allow us to establish the induction on the number of punctures \(r\). -% TODO: Talk about the action of Mod(S) in the set of punctures beforehand Given an orientable surface \(S\) and \(x_1, \ldots, x_n \in S\degree\), denote by \(\Mod(S \setminus \{x_1, \ldots, x_n\})_{\{x_1, \ldots, x_n\}} \subset \Mod(S \setminus \{x_1, \ldots, x_n\})\) the subgroup of mapping @@ -252,8 +245,7 @@ the graph \(\Gamma\), we consider\dots \(\alpha\) and \(\beta\). \end{definition} -% TODO: Cite the change of coordinates principle -It is clear from the change of coordinates principle that the actions of +It is clear from Example~\ref{ex: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 @@ -283,7 +275,6 @@ cases, \(\mathcal{C}(S_{g, r})\) is connected. connected. \end{theorem} -% TODOO: Explain this a little better? In other words, given \([\alpha], [\beta] \in \mathcal{C}(S_{g, r})\), we can find a path \([\alpha] = [\alpha_1] \text{---} \cdots \text{---} [\alpha_n] = [\beta]\) in \(\mathcal{C}(S_{g, r})\). Now if \(\alpha\) and \(\beta\) are @@ -401,47 +392,38 @@ Theorem~\ref{thm:mcg-is-fg}. \(\Mod(S_{g+1})_{\vec{[\alpha]}}\), \(\tau_\beta \tau_\alpha\) and \(\tau_\beta \tau_\alpha^2 \tau_\beta\). - % TODO: Properly state the cutting exact seq. beforehand? 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 \(\phi \in - \Homeo^+(S_{g+1})\) fixing \(\alpha\) point-wise, so \(\phi\) restricts to - a homeomorphism of \(S_{g+1} \setminus \alpha \cong S_{g, 2}\). We thus - obtain an exact sequence + many twists about nonseparating curves. First observe that \(S_{g+1} + \setminus \alpha \cong S_{g,2}\), as shown in + Figure~\ref{fig:cut-along-nonseparating-adds-two-punctures}. + Example~\ref{ex:cutting-morphism} then gives us 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, + \Mod(S_{g+1})_{\vec{[\alpha]}} \rar{\operatorname{cut}} & + \PMod(S_{g,2}) \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*} - The induction hypothesis now implies \(\PMod(S_{g+1} \setminus \alpha) \cong - \PMod(S_{g, 2})\) 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}[ht] + \centering + \includegraphics[width=.75\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 $S_{g, 2}$.} + \label{fig:cut-along-nonseparating-adds-two-punctures} + \end{figure} -% TODO: Should we really omit this little visual proof? -%\begin{figure}[ht] -% \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 $S_{g, 2}$.} -% \label{fig:cut-along-nonseparating-adds-two-punctures} -%\end{figure} + Recall that, by the induction hypothesis, \(\PMod(S_{g, 2})\) 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, as desired. This concludes the induction step in \(g\). +\end{proof} 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