- Commit
- 8894c953609e975dc716d9cf7ac375986198093e
- Parent
- 111f27939792e691654b57cfb6560ce6f31b4633
- Author
- Pablo <pablo-pie@riseup.net>
- Date
Hydrated the 2nd chapter
memoire-m2
My M2 Memoire on mapping class groups & their representations
Diffstat
9 files changed, 1327 insertions, 178 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Added | images/dehn-twist-bitorus.svg | 265 | 265 | 0 |
| Added | images/dehn-twist-cylinder.svg | 202 | 202 | 0 |
| Added | images/humphreys-gens.svg | 273 | 273 | 0 |
| Added | images/push-map.svg | 199 | 199 | 0 |
| Modified | preamble.tex | 3 | 2 | 1 |
| Modified | references.bib | 23 | 23 | 0 |
| Modified | sections/introduction.tex | 26 | 17 | 9 |
| Modified | sections/presentation.tex | 10 | 5 | 5 |
| Modified | sections/twists.tex | 504 | 341 | 163 |
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diff --git a/preamble.tex b/preamble.tex @@ -74,7 +74,8 @@ \newcommand{\ab}{{\operatorname{ab}}} % Group action symbol -\newcommand{\actson}{\,\rotatebox[origin=c]{-90}{$\circlearrowleft$}\,} +\newcommand{\leftaction}{\,\rotatebox[origin=c]{-90}{$\circlearrowleft$}\,} +\newcommand{\rightaction}{\,\rotatebox[origin=c]{90}{$\circlearrowright$}\,} % Quotient object \newcommand{\mfrac}[2]{\mathlarger{\sfrac{#1}{#2}}}
diff --git a/references.bib b/references.bib @@ -110,6 +110,20 @@ year = {1983}, } +@article{lickorish, + author = {Lickorish, William Bernard Raymond}, + doi = {10.1017/s030500410003824x}, + issn = {1469-8064}, + journal = {Mathematical Proceedings of the Cambridge Philosophical Society}, + month = oct, + number = {4}, + pages = {769--778}, + publisher = {Cambridge University Press (CUP)}, + title = {A finite set of generators for the homeotopy group of a $2$-manifold}, + volume = {60}, + year = {1964}, +} + @incollection{julien, author = {Marché, Julien}, booktitle = {Topology and geometry. A collection of essays dedicated to Vladimir G. Turaev}, @@ -122,3 +136,12 @@ year = {2021}, } +@book{hatcher, + author = {Hatcher, Allen}, + edition = {1}, + isbn = {9780521795401; 0521795400}, + language = {English}, + publisher = {Cambridge University Press}, + title = {Algebraic Topology}, + year = {2001}, +}
diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -1,4 +1,4 @@ -\chapter{Introduction} +\chapter{Introduction}\label{ch:introduction} \begin{definition} The \emph{mapping class group \(\Mod(S)\) of an orientable surface \(S\)} is @@ -45,8 +45,8 @@ \end{proposition} \begin{example}\label{ex:inclusion-morphism} - \(\Mod(S) \actson \{ \vec{[\alpha]} : \alpha \subset S \}\) and \(\Mod(S) - \actson \{ [\alpha] : \alpha \subset S \}\) via + \(\Mod(S) \leftaction \{ \vec{[\alpha]} : \alpha \subset S \}\) and \(\Mod(S) + \leftaction \{ [\alpha] : \alpha \subset S \}\) via \begin{align*} f \cdot [\alpha] & = [\phi(\alpha)] & f \cdot \vec{[\alpha]} & = \vec{[\phi(\alpha)]} @@ -55,7 +55,7 @@ \end{example} \begin{example} - \(\Mod(S) \actson H_k(S, \mathbb{R})\) + \(\Mod(S) \leftaction H_k(S, \mathbb{R})\) \end{example} \begin{example}\label{ex:symplectic-rep} @@ -120,18 +120,26 @@ \(f = [\phi]\), 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). + (e^{2 \pi i t}, s) & \mapsto (e^{2 \pi i (t - s)}, s) \end{align*} - In particular, \(\Mod(\mathbb{S}^1 \times [0, 1]) \cong \mathbb{Z}\). + 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} -% TODO: Add a picture of the Dehn twist +\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} % TODO: Can we prove this without using braid groups? \begin{example}\label{ex:mcg-twice-punctured-disk} The mapping class group \(\Mod(\mathbb{D} \setminus \{-\sfrac{1}{2}, - \sfrac{1}{2}\})\) of the twice punctured unit disk \(\mathbb{D} \subset - \mathbb{C}\) is freely generated by \(f = [\phi]\), where + \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}\} \\
diff --git a/sections/presentation.tex b/sections/presentation.tex @@ -100,10 +100,10 @@ by a single mapping class. \end{proposition} \begin{proposition}\label{thm:commutator-normal-gen} - If \(g \ge 2\) and \(\alpha, \beta \subset S_g\) are simple closed curves - with \(\#(\alpha \cap \beta) = 1\) then \(\Mod(S_g)'\) is \emph{normally - generated} by \(\tau_\alpha \tau_\beta^{-1}\) -- i.e. if \(\tau_\alpha - \tau_\beta^{-1} \in N \normal \Mod(S_g)'\) then \(\Mod(S_g)' \subset N\). + If \(g \ge 2\) and \(\alpha, \beta \subset S_g\) are simple closed crossing + only once, then \(\Mod(S_g)'\) is \emph{normally generated} by \(\tau_\alpha + \tau_\beta^{-1}\) -- i.e. if \(\tau_\alpha \tau_\beta^{-1} \in N \normal + \Mod(S_g)'\) then \(\Mod(S_g)' \subset N\). \end{proposition} These past few results combined paint a remarkably clear picture of the @@ -170,7 +170,7 @@ finite presentation of \(B_n\). \end{theorem} As promised, we now show that \(B_n\) coincides with \(\Mod(S_{0, n}^1)\). -Recall from Theorem~\ref{thm:generalized-birman-seq} that there is an exact +Recall from Theorem~\ref{thm:birman-exact-seq} that there is an exact sequence \begin{center} \begin{tikzcd}
diff --git a/sections/twists.tex b/sections/twists.tex @@ -1,26 +1,69 @@ \chapter{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 \(S_2\). + \begin{definition} Given a simple closed curve \(\alpha \subset S\), fix a closed annular neighborhood \(A \subset S\) of \(\alpha\) -- i.e. \(A \cong \mathbb{S}^1 - \times [0, 1]\). The \emph{Dehn twist \(\tau_\alpha \in \Mod(S)\) about - \(\alpha\)} is the image of the generator \(f \in \Mod(\mathbb{S}^1 \times - [0, 1]) \cong \mathbb{Z}\) -- as in Example~\ref{ex:mcg-annulus} -- under the - inclusion homomorphism \(\Mod(A) \to \Mod(S)\). + \times [0, 1]\). Let \(f \in \Mod(A) \cong \Mod(\mathbb{S}^1 \times [0, 1])\) + be as in Example~\ref{ex:mcg-annulus}. The \emph{Dehn twist \(\tau_\alpha \in + \Mod(S)\) about \(\alpha\)} is defined as the image of the generator \(f \in + \Mod(A) \cong \mathbb{Z}\) under the inclusion homomorphism \(\Mod(A) \to + \Mod(S)\). \end{definition} +\begin{figure}[ht] + \centering + \includegraphics[width=.6\linewidth]{images/dehn-twist-bitorus.eps} + \caption{The Dehn twist about the curve $\alpha$ takes the figure-eight curve + on the left-hand side to the yellow curve in the right-hand side.} + \label{fig:dehn-twist-bitorus} +\end{figure} + +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 \(\Mod(\mathbb{D} \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\})\) +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 \(S\degree\), fix - a closed neighborhood \(F \subset S\) of \(\alpha\) with \(F \cong \mathbb{D} - \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}\). The \emph{half-twist \(h_\alpha - \in \Mod(S)\) about \(\alpha\)} is the image of the generator \(f \in - \Mod(\mathbb{S}^1 \times [0, 1]) \cong \mathbb{Z}\) -- as in - Example~\ref{ex:mcg-twice-punctured-disk} -- under the inclusion homomorphism - \(\Mod(F) \to \Mod(S)\). + 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 \(f \in + \Mod(\mathbb{S}^1 \times [0, 1]) \cong \Mod(D)\) 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 \(f + \in \Mod(D) \cong \mathbb{Z}\) under the inclusion homomorphism \(\Mod(D) \to + \Mod(S)\). \end{definition} -% TODO: Maybe explain the intuition? -\begin{proposition} +% 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 +\[ + \#(\alpha \cap \beta) = \min + \left\{ + |\alpha' \cap \beta'| : [\alpha'] = [\alpha], [\beta'] = [\beta], + \alpha' \text{ intersects } \beta' \text{ transversaly} + \right\} +\] +we can distinguish between powers of Dehn twists +\cite[Proposition~3.2]{farb-margalit}. + +\begin{proposition}\label{thm:twist-intersection-number} Let \(\alpha \subset S\) be a simple closed curve and \(T_\alpha\) be a representative of \(\tau_\alpha \in \Mod(S)\). Then \(\# (T_\alpha^k(\beta) \cap \beta) = |k| \cdot \#(\alpha \cap \beta)^2\) for any \(k \in @@ -28,99 +71,196 @@ has infinite order. \end{proposition} -\begin{fact}\label{thm:dehn-twist-is-uniq} - \(\tau_\alpha = \tau_\beta \iff [\alpha] = [\beta]\). -\end{fact} +\begin{example}\label{thm:dehn-twist-is-uniq} + Given \(\alpha, \beta \subset S\), \(\tau_\alpha = \tau_\beta \iff [\alpha] = + [\beta]\). Indeed, if \(\alpha\) and \(\beta\) are non-isotopic, we can find + \(\gamma\) with \(\#(\gamma \cap \alpha) > 0\) and \(\#(\gamma \cap \beta) = + 0\). It thus follows from Proposition~\ref{thm:twist-intersection-number} + that \(\#(T_\alpha(\gamma) \cap \gamma) > \#(T_\beta(\gamma) \cap \gamma)\), + so \(\tau_\alpha \ne \tau_\beta\). +\end{example} + +Many other relations between Dehn twists can derrived be in a geometric fashion +too. -\begin{fact} +\begin{example}\label{thm:conjugate-twists} Given \(f = [\phi] \in \Mod(S)\), \(\tau_{\phi(\alpha)} = f \tau_\alpha f^{-1}\). -\end{fact} +\end{example} -\begin{fact} - \([f, \tau_\alpha] = 1 \iff f \cdot [\alpha] = \alpha\). -\end{fact} +\begin{example} + \([f, \tau_\alpha] = 1 \iff f \cdot [\alpha] = [\alpha]\). +\end{example} -\begin{fact} - If \(\alpha, \beta \subset S\) are both nonseparing simple closed curves then - \(\tau_\alpha, \tau_\beta \in \Mod(S)\) are conjugate. -\end{fact} +% 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 + principle we can find \(f \in \Mod(S)\) with \(f \cdot [\alpha] = [\beta]\) + 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} - Braid relation + Given \(\alpha, \beta \subset S\) with \(\#(\alpha \cap \beta) = 1\), it is + not hard to check that \(\tau_\beta \tau_\alpha \cdot [\beta] = [\alpha]\). + From Fact~\ref{thm:conjugate-twists} we then get \((\tau_\alpha \tau_\beta) + \tau_\alpha (\tau_\alpha \tau_\beta)^{-1} = \tau_\beta\), from which follows + the \emph{braid relation} + \[ + \tau_\alpha \tau_\beta \tau_\alpha = \tau_\beta \tau_\alpha \tau_\alpha. + \] \end{example} +A perhaps less obvious fact about Dehn twists is\dots + +% TODO: Define PMod beforehand \begin{theorem}\label{thm:mcg-is-fg} - 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. + 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 + \(\PMod(S_{g, r}^b)\) is generated by finitely many Dehn twists about + nonseparating curves or boundary components. \end{theorem} +The proof of Theorem~\ref{thm:mcg-is-fg} is simple in nature: we proceed by +indution in \(g\) and \(r\). On the other hand, the indutction steps are +somewhat involved and require two ingrediantes we have not encountered so far, +namely the \emph{Birman exact sequence} and the \emph{modified complex of +curves}. + \section{The Birman Exact Sequence} -% TODO: Explain who the fuck are push & forget +Having the proof of Theorem~\ref{thm:mcg-is-fg} in mind, it is interesting to +consider the relationship between the mapping class group of \(S_{g, r}^b\) and +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 +classes \(f\) that permute \(x_1, \ldots, x_n\) -- i.e. \(f \cdot x_i = +x_{\sigma(i)}\) for some \(\sigma \in \mathfrak{S}_n\). We certainly have a +surjective homomorphism \(\operatorname{forget} : \Mod(S \setminus \{x_1, +\ldots, x_n\})_{\{x_1, \ldots, x_n\}} \to \Mod(S)\) which ``\emph{forgets} the +additional punctures \(x_1, \ldots, x_n\) of \(S \setminus \{x_1, \ldots, +x_n\}\),'' but what is its kernel? + +To answer this question, we consider the configuration space \(C(S, n) = +\mfrac{C^{\operatorname{ord}}(S, n)}{\mathfrak{S}_n}\) of \(n\) (unordered) +points in the interior of \(S\) -- where \(C^{\operatorname{ord}}(S, n) = \{ +(x_1, \ldots, x_n) \in (S\degree)^n : x_i \ne x_j \ \text{for}\ i \ne j \}\). +Denote \(\Homeo^+(S, \partial S)_{x_1, \ldots, x_n} = \{\phi \in \Homeo^+(S, +\partial S) : \phi(x_i) = x_i \}\). From the fibration\footnote{See +\cite[Chapter~4]{hatcher} for a reference.} +\[ + \arraycolsep=1.4pt + \begin{array}{ccrcl} + \Homeo^+(S, \partial S)_{x_1, \ldots, x_n} + & \to & \Homeo^+(S, \partial S) + & \to & C(S, n) \\ + & & \phi & \mapsto & [\phi(x_1), \ldots, \phi(x_n)] + \end{array} +\] +and its long exact sequence in homotopy we then get\dots + \begin{theorem}[Birman exact sequence]\label{thm:birman-exact-seq} - If \(\chi(S) < 0\) then there is an exact sequence + Suppose \(\pi_1(\Homeo^+(S, \partial S), 1) = 1\). Then there is an exact + sequence \begin{center} - \begin{tikzcd} + \begin{tikzcd}[cramped] 1 \rar - & \pi_1(S, x_0) \rar{\operatorname{push}} - & \Mod(S \setminus \{x_0\}, x_0) \rar{\operatorname{forget}} + & \pi_1(C(S, n), [x_1, \ldots, x_n]) \rar{\operatorname{push}} + & \Mod(S \setminus \{x_1, \ldots, x_n\})_{\{x_1, \ldots, x_n\}} + \rar{\operatorname{forget}} & \Mod(S) \rar & 1. \end{tikzcd} \end{center} \end{theorem} -% TODO: Explan configuration spaces -\begin{align*} - C^{\operatorname{ord}}(S, n) - & = \{ (x_1, \ldots, x_n) \in \operatorname{int}(S)^n : - x_i \ne x_j \ \text{for}\ i \ne j \} \\ - C(S, n) - & = \mfrac{C^{\operatorname{ord}}(S, n)}{\mathfrak{S}_n} -\end{align*} - -% TODO: Comment on the proof -\begin{theorem}[Generalized Birman exact sequence]\label{thm:generalized-birman-seq} - Suppose \(\pi_1(\Homeo^+(S, \partial S), 1) = 1\). Then there is an exact - sequence +\begin{note} + Notice that \(C(S, 1) = S\degree \simeq S\). Hence for \(n = 1\) + Theorem~\ref{thm:birman-exact-seq} gives us a sequence \begin{center} \begin{tikzcd} 1 \rar - & \pi_1(C(S, n), [x_1, \ldots, x_n]) \rar{\operatorname{push}} - & \Mod(S \setminus \{x_1, \ldots, x_n\}) \rar{\operatorname{forget}} + & \pi_1(S, x) \rar{\operatorname{push}} + & \Mod(S \setminus \{x\}, x) \rar{\operatorname{forget}} & \Mod(S) \rar & 1. \end{tikzcd} \end{center} -\end{theorem} +\end{note} + +We may regard a simple loop \(\alpha \subset C(S, n)\) based at \([x_1, \ldots, +x_n]\) as \(n\) disjoint curves \(\alpha_1, \ldots, \alpha_n \subset S\) with +\(\alpha_i(0) = x_i\) and \(\alpha_i(1) = x_{\sigma(i)}\) for some \(\sigma \in +\mathfrak{S}_n\). The element \(\operatorname{push}([\alpha]) \in \Mod(S)\) can +then be seen as the mapping class that ``\emph{pushes} a neighborhood of +\(x_{\sigma(i)}\) towards \(x_i\) along the curve \(\alpha_i^{-1}\),'' as shown +in Figure~\ref{fig:push-map} for the case \(n = 1\). Indeed, this goes to +show\dots + +\begin{example}\label{ex:push-simple-loop} + Using the notation of Figure~\ref{fig:push-map}, + \(\operatorname{push}([\alpha]) = \tau_{\delta_1} \tau_{\delta_2}^{-1} \in + \Mod(S)\). +\end{example} + +\begin{figure}[ht] + \centering + \includegraphics[width=.35\linewidth]{images/push-map.eps} + \caption{The inclusion $\operatorname{push} : \pi_1(S, x) \to \Mod(S)$ maps + a simple loop $\alpha \subset S$ to the mapping class supported at an anular + neighborhood $A$ of $\alpha$ which takes the arc joining the boundary + components $\delta_i \subset \partial A$ in the left-hand side to the yellow + arc in the right-hand side.} + \label{fig:push-map} +\end{figure} \section{The Modified Complex of Curves} +Having established Theorem~\ref{thm:birman-exact-seq}, we now need to adress +the induction step in the genus \(g\) of \(S_{g, r}^b\). Our strategy is to +apply the following lemma from geomtric group theory. + \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 + \leftaction \Gamma\) via graph automorphisms. Suppose that \(G\) acts transitively both in \(V(\Gamma)\) and \(\{(v, w) \in V(\Gamma)^2 : v \text{ --- } w \text{ in } \Gamma \}\). If \(v, w \in V(\Gamma)\) are connected by an edge and \(g \in G\) is such that \(g \cdot w = v\) then \(G\) is generated by \(G_v\) and \(g\). \end{lemma} +We are interested, of course, in the group \(G = \PMod(S_{g, r}^b)\). As for +the graph \(\Gamma\), we consider\dots + \begin{definition} The \emph{modified complex of nonseparating curves \(\hat{\mathcal{N}}(S)\) of a surface \(S\)} is the graph whose vertices are (un-oriented) isotopy classes of nonseparating simple closed curves in \(S\) and \[ - \text{\([\alpha]\) --- \([\beta]\) in \(\mathcal{C}(S)\)} - \iff \#(\alpha \cap \beta) = 1. + \text{\([\alpha]\) --- \([\beta]\) in \(\hat{\mathcal{N}}(S)\)} + \iff \#(\alpha \cap \beta) = 1, \] + where \(\#(\alpha \cap \beta)\) is the geometric intersection number of + \(\alpha\) and \(\beta\). \end{definition} -% TODO: Comment on the fact that the complex of curves per say is actually the -% clique complex of this graph +% TODO: Cite the change of coordinates principle +It is clear from the change of coordinates principle 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 +connected? + +Historically, the modified complex of nonseparating curves first arised as a +\emph{modified} version of another graph of curves, known as\dots \begin{definition} Given a surface \(S\), the \emph{complex of curves \(\mathcal{C}(S)\) of @@ -134,54 +274,65 @@ of \(\mathcal{C}(S)\) whose vertices consist of nonseparating curves. \end{definition} +Lickorish \cite{lickorish} showed that, appart from a small number of sporadic +cases, \(\mathcal{C}(S_{g, r})\) is connected. + \begin{theorem}[Lickorish] - If \(S_{g, n}\) is not one \(S_0 = \mathbb{S}^2, S_{0, 1}, \ldots, S_{0, 4}, - S_1 = \mathbb{T}\) and \(S_{1, 1}\) then \(\mathcal{C}(S)\) is connected. + 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 + connected. \end{theorem} -% TODO: Comment on the proof -\begin{corollary} - If \(g \ge 2\) then both \(\mathcal{N}(S_{g, n})\) and - \(\hat{\mathcal{N}}(S_{g, n})\) are connected. +% 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 +nonseparating, by inductively adjusting this path we then get\dots + +\begin{corollary}\label{thm:mofied-complex-is-connected} + If \(g \ge 2\) then both \(\mathcal{N}(S_{g, r})\) and + \(\hat{\mathcal{N}}(S_{g, r})\) are connected. \end{corollary} -% TODO +See \cite[Section~4.1]{farb-margalit} for a proof of +Corollary~\ref{thm:mofied-complex-is-connected}. We are now ready to show +Theorem~\ref{thm:mcg-is-fg}. + \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. + Let \(S_{g, r}^b\) be the orientable surface of genus \(g \ge 1\) with \(r\) + punctures and \(b\) boundary components. We want to establish that + \(\PMod(S_{g, r}^b)\) 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 + First, observe that if \(b \ge 1\) and \(\partial S_{g, r}^b = \delta_1 \cup + \cdots \cup \delta_b\) 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 & + \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 & 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 + Indeed, if \(\PMod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D} \setminus \{0\}))\) + is finitely generated by twists about nonseparing curves or boundary + components, then we may lift the generators of \(\PMod(S_{g, r}^b + \cup_{\delta_1} (\mathbb{D} \setminus \{0\}))\) 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 + \(\Mod(\mathbb{T}) \cong \Mod(S_{1, 1}) \cong \operatorname{SL}_2(\mathbb{Z})\) 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 + curves \(\alpha\) and \(\beta\) from + Figure~\ref{fig:torus-mcg-generators}, each corresponding to one of the + standard generators \begin{align*} \begin{pmatrix} 1 & 1 \\ @@ -195,64 +346,68 @@ \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{figure}[ht] + \centering + \includegraphics[width=.33\linewidth]{images/torus-mcg-generators.eps} + \caption{The curves $\alpha$ and $\beta$ whose Dehn twists generate + $\Mod(\mathbb{T})$ and $\Mod(S_{1, 1})$.} + \label{fig:torus-mcg-generators} + \end{figure} + + Now suppose \(\PMod(S_{g, r})\) is finitely-generated by twists about + nonseparating curves for \(g \ge 2\) or \(g = 1\) and \(r > 1\). In both + case, \(\chi(S_{g, r}) = 2 - 2g - r < 0\) and thus \(\pi_1(\Homeo^+(S_{g, + r})) = 1\) -- see \cite[Theorem~1.14]{farb-margalit}. The Birman exact + sequence from Theorem~\ref{thm:birman-exact-seq} then gives us \begin{center} \begin{tikzcd} 1 \rar - & \pi_1(S_{g, n}, x_0) \rar{\operatorname{push}} - & \PMod(S_{g, n + 1}) \rar{\operatorname{forget}} - & \PMod(S_{g, n}) \rar + & \pi_1(S_{g, r}, x) \rar{\operatorname{push}} + & \PMod(S_{g, r + 1}) \rar{\operatorname{forget}} + & \PMod(S_{g, r}) \rar & 1, \end{tikzcd} \end{center} - where \(S_{g, n + 1} = S_{g, n} \setminus \{x_0\}\). Since \(g \ge 1\), - \(\pi_1(S_{g, n}, x_0)\) is generated by finitely many nonseparating loops. - We have seen that \(\operatorname{push} : \pi_1(S_{g, n}, x_0) \to - \Mod(S_{g, n+1}, x_0)\) 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 - \(\Mod(S_{g+1})_{[\alpha]}\). + where \(S_{g, r + 1} = S_{g, r} \setminus \{x\}\). Since \(g \ge 1\), + \(\pi_1(S_{g, r}, x)\) is generated by finitely many nonseparating loops. + We have seen in Example~\ref{ex:push-simple-loop} that \(\operatorname{push} + : \pi_1(S_{g, r}, x) \to \Mod(S_{g, r+1}, x)\) takes nonseparation simple + loops to products of twists about nonseparating simple curves. Furthermore, + we may once again lift the generators of \(\PMod(S_{g, r})\) to Dehn twists + about nonseparating simple curves in \(S_{g, r + 1}\). This goes to show that + \(\PMod(S_{g, r + 1})\) is also generated by finitely many twists about + simple curves, concluding the induction step on \(r\). + + As for the induction step on \(g\), fix \(g \ge 1\) and suppose that, for + each \(r \ge 0\), \(\PMod(S_{g, r})\) is finitely generated by twists about + nonseparing curves or boundary components. Let us show that the same holds + for \(\Mod(S_{g + 1})\). To that end, we consider the action \(\Mod(S_{g + + 1}) \leftaction \hat{\mathcal{N}}(S_{g + 1})\). Since \(g + 1 \ge 2\), + \(\hat{\mathcal{N}}(S_{g + 1})\) is connected and the conditions of + Lemma~\ref{thm:ggt-lemma} are met. Now recall from + Example~\ref{ex:braid-relation} that, given nonseparating \(\alpha, \beta + \subset S_{g + 1}\) crossing once, \(\tau_\beta \tau_\alpha \cdot [\beta] = + [\alpha]\). 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 \(\Mod(S_{g+1})_{[\alpha]}\). In + particular, \(\Mod(S_{g+1})\) is generated by + \(\Mod(S_{g+1})_{\vec{[\alpha]}}\), \(\tau_\beta \tau_\alpha\) and + \(\tau_\beta \tau_\alpha^2 \tau_\beta\). % 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 + 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 \begin{equation}\label{eq:cutting-seq} \begin{tikzcd} 1 \rar & @@ -268,36 +423,59 @@ [\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 \(g\). + 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}[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} +% 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} + +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 +generators of \(\Mod(S_g^b)\) by addapting the induction steps in the proof of +Theorem~\ref{thm:mcg-is-fg}. These are known as the \emph{Lickorish +generators}. \begin{theorem}[Lickorish generators]\label{thm:lickorish-gens} - If \(g \ge 1\) then \(\Mod(S_g^b)\) is generated by \(\tau_{\alpha_1}, - \ldots, \tau_{\alpha_g}\), \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} + If \(g \ge 1\) then \(\Mod(S_g^b)\) is generated by the Dehn twists about the + curves \(\alpha_1, \ldots, \alpha_g, \mu_1, \ldots, \mu_g, \gamma_1, + \ldots, \gamma_{g - 1}, \eta_1, \ldots, \eta_{b - 1}\) as in + Figure~\ref{fig:lickorish-gens} \end{theorem} +In the boundaryless case \(b = 0\), we can write \(\tau_{\mu_3}, \ldots, +\tau_{\mu_g} \in \Mod(S_g)\) as products of the twists about the remaining +curves, from which we get the so called \emph{Humphreys generators}. + \begin{corollary}[Humphreys generators]\label{thm:humphreys-gens} - If \(g \ge 2\) then \(\Mod(S_g)\) is generated by the twists - \(\tau_{\alpha_0}, \ldots, \tau_{\alpha_{2g}} \in \Mod(S_g)\). + If \(g \ge 2\) then \(\Mod(S_g)\) is generated by the Dehn twists aboud the + curves \(\alpha_0, \ldots, \alpha_{2g}\) as in + Figure~\ref{fig:humphreys-gens}. \end{corollary} + +\begin{minipage}[b]{.45\linewidth} + \centering + \includegraphics[width=\linewidth]{images/lickorish-gens.eps} + \captionof{figure}{The curves from Lickorish generators of $\Mod(S_g^b)$.} + \label{fig:lickorish-gens} +\end{minipage} +\hspace{.5cm} % +\begin{minipage}[b]{.45\textwidth} + \centering + \includegraphics[width=\linewidth]{images/humphreys-gens.eps} + \captionof{figure}{The curves from Humphreys generators of $\Mod(S_g)$.} + \label{fig:humphreys-gens} +\end{minipage}