- Commit
- 740f8753ea9a218bc87c996d54beefafd91bfa38
- Parent
- b1216255fea0f6fb36117915f1b55f02cb767dbd
- Author
- Pablo <pablo-pie@riseup.net>
- Date
Changed the notation for the number of boundary components
p → b once and for all
memoire-m2
My M2 Memoire on mapping class groups & their representations
Diffstat
7 files changed, 190 insertions, 177 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | images/lickorish-gens-gen-2.svg | 25 | 15 | 10 |
| Modified | images/lickorish-gens-korkmaz-proof.svg | 17 | 11 | 6 |
| Modified | images/lickorish-gens.svg | 15 | 10 | 5 |
| Modified | sections/introduction.tex | 28 | 14 | 14 |
| Modified | sections/presentation.tex | 128 | 63 | 65 |
| Modified | sections/representations.tex | 102 | 51 | 51 |
| Modified | sections/twists.tex | 52 | 26 | 26 |
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diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -17,17 +17,17 @@ proof. \(\Sigma_g\) of the sphere \(\mathbb{S}^2\) with \(g \ge 0\) copies of the torus \(\mathbb{T}^2 = \mfrac{\mathbb{R}^2}{\mathbb{Z}^2}\). Any compact connected orientable surface \(\Sigma\) is isomorphic to the surface - \(\Sigma_g^p\) obtained from \(\Sigma_g\) by removing \(p \ge 0\) open disks + \(\Sigma_g^b\) obtained from \(\Sigma_g\) by removing \(b \ge 0\) open disks with disjoint closures. \end{theorem} The integer \(g \ge 0\) in Theorem~\ref{thm:classification-of-surfaces} is called \emph{the genus of \(\Sigma\)}. We also have the noncompact surface -\(\Sigma_{g, r}^p = \Sigma_g^p \setminus \{x_1, \ldots, x_r\}\), where \(x_1, -\ldots, x_r\) lie in the interior of \(\Sigma_g^p\). The points \(x_1, \ldots, -x_r\) are called the \emph{punctures} of \(\Sigma_{g, r}^p\). Throughout these +\(\Sigma_{g, r}^b = \Sigma_g^b \setminus \{x_1, \ldots, x_r\}\), where \(x_1, +\ldots, x_r\) lie in the interior of \(\Sigma_g^b\). The points \(x_1, \ldots, +x_r\) are called the \emph{punctures} of \(\Sigma_{g, r}^b\). Throughout these notes, all surfaces considered will be of the form \(\Sigma = \Sigma_{g, -r}^p\). Any such \(\Sigma\) admits a natural compactification +r}^b\). Any such \(\Sigma\) admits a natural compactification \(\widebar\Sigma\) obtained by filling its punctures. We denote \(\Sigma_{g, r} = \Sigma_{g, r}^0\). All closed curves \(\alpha, \beta \subset \Sigma\) we consider lie in the interior \(\Sigma\degree\) of \(\Sigma\) and intersect @@ -49,14 +49,14 @@ example\dots \end{lemma} \begin{proof} - Let \(\Sigma = \Sigma_{g, r}^p\) and consider the surface \(\Sigma_\alpha\) + Let \(\Sigma = \Sigma_{g, r}^b\) and consider the surface \(\Sigma_\alpha\) obtained by cutting \(\Sigma\) across \(\alpha\), as in Figure~\ref{fig:change-of-coordinates}. Since \(\alpha\) is nonseparating, this surface has genus \(g - 1\) and two additional boundary component \(\delta_1, \delta_2 \subset \partial \Sigma_\alpha\), so \(\Sigma_\alpha - \cong \Sigma_{g-1,r}^{p+2}\). By identifying \(\delta_1\) and \(\delta_2\) + \cong \Sigma_{g-1,r}^{b+2}\). By identifying \(\delta_1\) and \(\delta_2\) we can see \(\Sigma\) as a quotient of \(\Sigma_\alpha\). Similarly, - \(\Sigma_\beta \cong \Sigma_{g-1, r}^{p+2}\) also has two additional boundary + \(\Sigma_\beta \cong \Sigma_{g-1, r}^{b+2}\) also has two additional boundary components \(\delta_1', \delta_2' \subset \partial \Sigma_\beta\). Now by the classification of surfaces we can find an orientation-preserving homeomorphism \(\tilde\phi : \Sigma_\alpha \isoto \Sigma_\beta\). Even more @@ -68,7 +68,7 @@ example\dots As for the second part of the lemma, we consider the surface \(\Sigma_{\alpha \alpha'}\) obtained by cutting \(\Sigma_\alpha\) across the arc determined by \(\alpha'\). Since \(\alpha'\) is nonseparating, \(\Sigma_{\alpha \alpha'} - \cong \Sigma_{g-1, r}^{p+1}\) has one boundary component more than + \cong \Sigma_{g-1, r}^{b+1}\) has one boundary component more than \(\Sigma\), say \(\partial \Sigma_{\alpha \alpha'} = \delta \amalg \partial \Sigma\). The boundary component \(\delta\) is naturally subdivided into the four arcs in Figure~\ref{fig:change-of-coordinates}, each corresponding to @@ -76,7 +76,7 @@ example\dots the pairs of arcs corresponding to the same curve we obtain the surface \(\mfrac{\Sigma_{\alpha \alpha'}}{\sim} \cong \Sigma\). - Likewise, \(\Sigma_{\beta \beta'} \cong \Sigma_{g-1, r}^{p+1}\) also has a + Likewise, \(\Sigma_{\beta \beta'} \cong \Sigma_{g-1, r}^{b+1}\) also has a boundary component \(\delta' \subset \partial \Sigma_{\beta \beta'}\) subdivided into four arcs. By the classification of surfaces we can find an orientation-preserving homeomorphism \(\tilde\phi : \Sigma_{\alpha \alpha'} @@ -93,10 +93,10 @@ example\dots \begin{figure}[ht] \centering \includegraphics[width=.8\linewidth]{images/change-of-coords-cut.eps} - \caption{By cutting $\Sigma_{g, r}^p$ across $\alpha$ we obtain $\Sigma_{g-1, - r}^{p+2}$, where $\alpha'$ determines a yellow arc joining the two - additional boundary components. Now by cutting $\Sigma_{g-1, r}^{p+2}$ across - this arc we obtain $\Sigma_{g-1,r}^p$, with the added boundary component + \caption{By cutting $\Sigma_{g, r}^b$ across $\alpha$ we obtain $\Sigma_{g-1, + r}^{b+2}$, where $\alpha'$ determines a yellow arc joining the two + additional boundary components. Now by cutting $\Sigma_{g-1, r}^{b+2}$ across + this arc we obtain $\Sigma_{g-1,r}^b$, with the added boundary component subdivided into the four arcs corresponding to $\alpha$ and $\alpha'$.} \label{fig:change-of-coordinates} \end{figure}
diff --git a/sections/presentation.tex b/sections/presentation.tex @@ -67,22 +67,22 @@ their corresponding inclusion homomorphisms \(\Mod(\Sigma_0^4) \to twists in \(\Mod(\Sigma)\). For example\dots \begin{proposition}\label{thm:trivial-abelianization} - The Abelianization \(\Mod(\Sigma_g^p)^\ab = - \mfrac{\Mod(\Sigma_g^p)}{[\Mod(\Sigma_g), \Mod(\Sigma_g)]}\) is cyclic. - Moreover, if \(g \ge 3\) then \(\Mod(\Sigma_g^p)^\ab = 0\). In other words, + The Abelianization \(\Mod(\Sigma_g^b)^\ab = + \mfrac{\Mod(\Sigma_g^b)}{[\Mod(\Sigma_g), \Mod(\Sigma_g)]}\) is cyclic. + Moreover, if \(g \ge 3\) then \(\Mod(\Sigma_g^b)^\ab = 0\). In other words, \(\Mod(\Sigma_g)\) is a perfect group for \(g \ge 3\). \end{proposition} \begin{proof} - By Theorem~\ref{thm:lickorish-gens}, \(\Mod(\Sigma_g^p)^\ab\) is generated by + By Theorem~\ref{thm:lickorish-gens}, \(\Mod(\Sigma_g^b)^\ab\) is generated by the image of the Lickorish generators, which are all conjugate and thus represent the same class in the Abelianization. In fact, any nonseparating - \(\alpha \subset \Sigma_g^p\) is conjugate to the Lickorish generators too, - so \(\Mod(\Sigma_g^p)^\ab = \langle [\alpha] \rangle\). + \(\alpha \subset \Sigma_g^b\) is conjugate to the Lickorish generators too, + so \(\Mod(\Sigma_g^b)^\ab = \langle [\alpha] \rangle\). - Now for \(g \ge 3\) we can embed \(\Sigma_0^4\) in \(\Sigma_g^p\) in such a + Now for \(g \ge 3\) we can embed \(\Sigma_0^4\) in \(\Sigma_g^b\) in such a way that all the corresponding curves \(\alpha, \beta, \gamma, \delta_1, - \ldots, \delta_4 \subset \Sigma_g^p\) are nonseparating, as shown in + \ldots, \delta_4 \subset \Sigma_g^b\) are nonseparating, as shown in Figure~\ref{fig:latern-relation-trivial-abelianization}. The lantern relation (\ref{eq:lantern-relation}) then becomes \[ @@ -92,14 +92,14 @@ twists in \(\Mod(\Sigma)\). For example\dots + [\tau_{\delta_3}] + [\tau_{\delta_4}] = 4 \cdot [\tau_\alpha] \] - in \(\Mod(\Sigma_g^p)^\ab\). In other words, \([\tau_\alpha] = 0\) and thus - \(\Mod(\Sigma_g^p)^\ab = 0\). + in \(\Mod(\Sigma_g^b)^\ab\). In other words, \([\tau_\alpha] = 0\) and thus + \(\Mod(\Sigma_g^b)^\ab = 0\). \end{proof} \begin{figure}[ht] \centering \includegraphics[width=.5\linewidth]{images/lantern-relation-trivial-abelianization.eps} - \caption{The embedding of $\Sigma_0^4$ in $\Sigma_g^p$ for $g \ge 3$.} + \caption{The embedding of $\Sigma_0^4$ in $\Sigma_g^b$ for $g \ge 3$.} \label{fig:latern-relation-trivial-abelianization} \end{figure} @@ -209,10 +209,10 @@ get\dots To get from \(\Sigma_{0, n}^1\) to surfaces of genus \(g > 0\) we may consider the \emph{hyperelliptic involution} \(\iota : \Sigma_g \isoto \Sigma_g\), which rotates \(\Sigma_g\) by \(\pi\) around some axis as in -Figure~\ref{fig:hyperelliptic-involution}. Given \(\ell < g\) and \(p = 1, 2\), -we can also embed \(\Sigma_\ell^p\) in \(\Sigma_g\) in such way that \(\iota\) +Figure~\ref{fig:hyperelliptic-involution}. Given \(\ell < g\) and \(b = 1, 2\), +we can also embed \(\Sigma_\ell^b\) in \(\Sigma_g\) in such way that \(\iota\) restricts to an involution\footnote{This involution does not fix $\partial -\Sigma_\ell^p$ point-wise.} \(\Sigma_\ell^p \isoto \Sigma_\ell^p\). +\Sigma_\ell^b$ point-wise.} \(\Sigma_\ell^b \isoto \Sigma_\ell^b\). \begin{figure}[ht] \centering @@ -224,16 +224,16 @@ restricts to an involution\footnote{This involution does not fix $\partial It is clear from Figure~\ref{fig:hyperelliptic-involution} that the quotients \(\mfrac{\Sigma_\ell^1}{\iota}\) and \(\mfrac{\Sigma_\ell^2}{\iota}\) are both disks, with boundary corresponding to the projection of the boundaries of -\(\Sigma_\ell^1\) and \(\Sigma_\ell^2\), respectively. Given \(p = 1, 2\), the -quotient map \(\Sigma_\ell^p \to \mfrac{\Sigma_\ell^p}{\iota} \cong -\mathbb{D}^2\) is a double cover with \(2\ell + p\) branch points corresponding +\(\Sigma_\ell^1\) and \(\Sigma_\ell^2\), respectively. Given \(b = 1, 2\), the +quotient map \(\Sigma_\ell^b \to \mfrac{\Sigma_\ell^b}{\iota} \cong +\mathbb{D}^2\) is a double cover with \(2\ell + b\) branch points corresponding to the fixed points of \(\iota\). We may thus regard -\(\mfrac{\Sigma_\ell^p}{\iota}\) as the disk \(\Sigma_{0, 2\ell + p}^1\) with -\(2\ell + p\) punctures in its interior, as shown in +\(\mfrac{\Sigma_\ell^b}{\iota}\) as the disk \(\Sigma_{0, 2\ell + b}^1\) with +\(2\ell + b\) punctures in its interior, as shown in Figure~\ref{fig:hyperelliptic-covering}. We also draw the curves \(\alpha_1, -\ldots, \alpha_{2\ell} \subset \Sigma_\ell^p\) of the Humphreys generators of +\ldots, \alpha_{2\ell} \subset \Sigma_\ell^b\) of the Humphreys generators of \(\Mod(\Sigma_g)\). Since these curves are invariant under the action of -\(\iota\), they descend to arcs \(\bar{\alpha}_1, \ldots, \bar{\alpha}_{2\ell + p} \subset \Sigma_{0, 2\ell + p}^1\) joining the punctures of the quotient +\(\iota\), they descend to arcs \(\bar{\alpha}_1, \ldots, \bar{\alpha}_{2\ell + b} \subset \Sigma_{0, 2\ell + b}^1\) joining the punctures of the quotient surface. \begin{figure}[ht] @@ -244,9 +244,8 @@ surface. \end{figure} \begin{observation}\label{ex:push-generators-description} - The map \(\operatorname{push} : B_{2\ell + p} \to \Mod(\Sigma_{0, 2\ell + - p}^1)\) takes \(\sigma_i\) to the half-twist \(h_{\bar{\alpha}_i}\) about - the arc \(\bar{\alpha}_i \subset \Sigma_{0, 2\ell + p}^1\). + The map \(\operatorname{push} : B_{2\ell + b} \to \Mod(\Sigma_{0, 2\ell + b}^1)\) takes \(\sigma_i\) to the half-twist \(h_{\bar{\alpha}_i}\) about + the arc \(\bar{\alpha}_i \subset \Sigma_{0, 2\ell + b}^1\). \end{observation} We now study the homeomorphisms of \(\Sigma_\ell^1\) and \(\Sigma_\ell^2\) that @@ -254,66 +253,65 @@ descend to the quotient surfaces and their mapping classes, known as \emph{the symmetric mapping classes}. \begin{definition} - Let \(\ell \ge 0\) and \(p = 1, 2\). The \emph{group of symmetric - homeomorphisms of \(\Sigma_\ell^p\)} is \(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p) = - \{\phi \in \Homeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p) : [\phi, \iota] = 1\}\). The - \emph{symmetric mapping class group of \(\Sigma_\ell^p\)} is the subgroup - \(\SMod(\Sigma_\ell^1) = \{ [\phi] \in \Mod(\Sigma_\ell^p) : \phi \in - \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p) \}\). + Let \(\ell \ge 0\) and \(b = 1, 2\). The \emph{group of symmetric + homeomorphisms of \(\Sigma_\ell^b\)} is \(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b) = + \{\phi \in \Homeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b) : [\phi, \iota] = 1\}\). The + \emph{symmetric mapping class group of \(\Sigma_\ell^b\)} is the subgroup + \(\SMod(\Sigma_\ell^1) = \{ [\phi] \in \Mod(\Sigma_\ell^b) : \phi \in + \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b) \}\). \end{definition} -Fix \(p = 1\) or \(2\). It follows from the universal property of quotients -that any \(\phi \in \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)\) defines a -homeomorphism \(\bar \phi : \Sigma_{0, 2\ell+p}^1 \isoto \Sigma_{0, 2\ell+p}^1\). This +Fix \(b = 1\) or \(2\). It follows from the universal property of quotients +that any \(\phi \in \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) defines a +homeomorphism \(\bar \phi : \Sigma_{0, 2\ell+b}^1 \isoto \Sigma_{0, 2\ell+b}^1\). This yields a homomorphism of topological groups \begin{align*} - \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p) - & \to \Homeo^+(\Sigma_{0, 2\ell + p}^1, \partial \Sigma_{0, 2\ell + p}^1) \\ + \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b) + & \to \Homeo^+(\Sigma_{0, 2\ell + b}^1, \partial \Sigma_{0, 2\ell + b}^1) \\ \phi & \mapsto \bar \phi, \end{align*} -which is surjective because any \(\psi \in \Homeo^+(\Sigma_{0, 2\ell + p}^1, -\partial \Sigma_{0, 2\ell + p}^1)\) lifts to \(\Sigma_\ell^p\). +which is surjective because any \(\psi \in \Homeo^+(\Sigma_{0, 2\ell + b}^1, +\partial \Sigma_{0, 2\ell + b}^1)\) lifts to \(\Sigma_\ell^b\). -It is also not hard to see \(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p) \to -\Homeo^+(\Sigma_{0, 2\ell + p}^1, \partial \Sigma_{0, 2\ell + p}^1)\) is injective: the +It is also not hard to see \(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b) \to +\Homeo^+(\Sigma_{0, 2\ell + b}^1, \partial \Sigma_{0, 2\ell + b}^1)\) is injective: the only candidates for elements of its kernel are \(1\) and \(\iota\), but -\(\iota\) is not an element of \(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)\) since -it does not fix \(\partial \Sigma_\ell^p\) point-wise. Now since we have a +\(\iota\) is not an element of \(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) since +it does not fix \(\partial \Sigma_\ell^b\) point-wise. Now since we have a continuous bijective homomorphism we find \[ \begin{split} - \pi_0(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)) - & \cong \pi_0(\Homeo^+(\Sigma_{0, 2\ell+p}^1, \partial \Sigma_{0, 2\ell+p}^1)) \\ - & = \mfrac{\Homeo^+(\Sigma_{0,2\ell+p}^1, \partial \Sigma_{0, 2\ell+p}^1)}{\simeq} \\ - & = \Mod(\Sigma_{0, 2\ell+p}^1) \\ - & \cong B_{2\ell + p}. + \pi_0(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)) + & \cong \pi_0(\Homeo^+(\Sigma_{0, 2\ell+b}^1, \partial \Sigma_{0, 2\ell+b}^1)) \\ + & = \mfrac{\Homeo^+(\Sigma_{0,2\ell+b}^1, \partial \Sigma_{0, 2\ell+b}^1)}{\simeq} \\ + & = \Mod(\Sigma_{0, 2\ell+b}^1) \\ + & \cong B_{2\ell + b}. \end{split} \] -We would like to say \(\pi_0(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)) = -\SMod(\Sigma_\ell^p)\), but a priori the story looks a little more complicated: -\(\phi, \psi \in \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)\) define the same class -in \(\SMod(\Sigma_\ell^p)\) if they are isotopic, but they may not lie in same -connected component of \(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)\) if they are +We would like to say \(\pi_0(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)) = +\SMod(\Sigma_\ell^b)\), but a priori the story looks a little more complicated: +\(\phi, \psi \in \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) define the same class +in \(\SMod(\Sigma_\ell^b)\) if they are isotopic, but they may not lie in same +connected component of \(\SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) if they are not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden \cite{birman-hilden} showed that this is never the case. \begin{theorem}[Birman-Hilden] - If \(\phi, \psi \in \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)\) are isotopic + If \(\phi, \psi \in \SHomeo^+(\Sigma_\ell^b, \partial \Sigma_\ell^b)\) are isotopic then \(\phi\) and \(\psi\) are isotopic through symmetric homeomorphisms. In particular, there is an isomorphism \begin{align*} - \SMod(\Sigma_\ell^p) & \isoto \Mod(\Sigma_{0, 2\ell + p}) \\ + \SMod(\Sigma_\ell^b) & \isoto \Mod(\Sigma_{0, 2\ell + b}) \\ [\phi] & \mapsto [\bar \phi]. \end{align*} \end{theorem} \begin{observation} Using the notation of Figure~\ref{fig:hyperelliptic-covering}, the - Birman-Hilden isomorphism \(\SMod(\Sigma_\ell^p) \isoto \Mod(\Sigma_{0, 2g + - p})\) takes \(\tau_{\alpha_i}\) to the half twist \(h_{\bar{\alpha}_i} \in - \Mod(\Sigma_{0, 2g + p})\). This can be checked by looking at + Birman-Hilden isomorphism \(\SMod(\Sigma_\ell^b) \isoto \Mod(\Sigma_{0, 2g + b})\) takes \(\tau_{\alpha_i}\) to the half twist \(h_{\bar{\alpha}_i} \in + \Mod(\Sigma_{0, 2g + b})\). This can be checked by looking at \(\iota\)-invariant annular neighborhoods of the curves \(\alpha_i\) -- \cite[Section~9.4]{farb-margalit}. \end{observation} @@ -327,7 +325,7 @@ not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden \SMod(\Sigma_\ell^2)\) to \(\tau_{\bar\delta_1} = \tau_{\bar\delta_2}\). In light of Observation~\ref{ex:push-generators-description}, Observation~\ref{ex:braid-group-center} gives us the so called - \emph{\(k\)-chain relations} in \(\SMod(\Sigma_\ell^p) \subset + \emph{\(k\)-chain relations} in \(\SMod(\Sigma_\ell^b) \subset \Mod(\Sigma_g)\). \[ \arraycolsep=1.4pt @@ -420,7 +418,7 @@ to obtain other relations. Since \(\iota\) has \(2g + 2\) fixed points in \section{Presentations of Mapping Class Groups} -There are numerous known presentations of \(\Mod(\Sigma_{g, r}^p)\), such as +There are numerous known presentations of \(\Mod(\Sigma_{g, r}^b)\), such as the ones due to Birman-Hilden \cite{birman-hilden}, Gervais \cite{gervais} and many others. Wajnryb \cite{wajnryb} derived a presentation of \(\Mod(\Sigma_g)\) using the relations discussed in @@ -500,15 +498,15 @@ given by \end{tabular} \end{center} for closed surfaces of small genus. In \cite{korkmaz-mccarthy} Korkmaz-McCarthy -showed that even though \(\Mod(\Sigma_2^p)\) is not perfect, its commutator -subgroup is. In addition, they also show \([\Mod(\Sigma_g^p), -\Mod(\Sigma_g^p)]\) is normally generated by a single mapping class. +showed that even though \(\Mod(\Sigma_2^b)\) is not perfect, its commutator +subgroup is. In addition, they also show \([\Mod(\Sigma_g^b), +\Mod(\Sigma_g^b)]\) is normally generated by a single mapping class. \begin{proposition}\label{thm:commutator-is-perfect} - The commutator subgroup \(\Mod(\Sigma_2^p)' = [\Mod(\Sigma_2^p), - \Mod(\Sigma_2^p)]\) is perfect -- i.e. \(\Mod(\Sigma_2^p)^{(2)} = - [\Mod(\Sigma_2^p)', \Mod(\Sigma_2^p)']\) is the whole of - \(\Mod(\Sigma_2^p)'\). + The commutator subgroup \(\Mod(\Sigma_2^b)' = [\Mod(\Sigma_2^b), + \Mod(\Sigma_2^b)]\) is perfect -- i.e. \(\Mod(\Sigma_2^b)^{(2)} = + [\Mod(\Sigma_2^b)', \Mod(\Sigma_2^b)']\) is the whole of + \(\Mod(\Sigma_2^b)'\). \end{proposition} \begin{proposition}\label{thm:commutator-normal-gen}
diff --git a/sections/representations.tex b/sections/representations.tex @@ -14,8 +14,8 @@ The goal of this chapter is to provide a concise account of Korkmaz' results, starting by\dots \begin{theorem}[Korkmaz]\label{thm:low-dim-reps-are-trivial} - Let \(\Sigma_g^p\) be the compact surface of genus \(g \ge 1\) with \(p\) - boundary components and \(\rho : \Mod(\Sigma_g^p) \to \GL_n(\mathbb{C})\) be + Let \(\Sigma_g^b\) be the compact surface of genus \(g \ge 1\) with \(b\) + boundary components and \(\rho : \Mod(\Sigma_g^b) \to \GL_n(\mathbb{C})\) be a linear representation with \(n < 2 g\). Then the image of \(\rho\) is Abelian. In particular, if \(g \ge 3\) then \(\rho\) is trivial. \end{theorem} @@ -26,15 +26,15 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g = 2\). \begin{proposition}\label{thm:low-dim-reps-are-trivial-base-case} - Given \(\rho : \Mod(\Sigma_2^p) \to \GL_n(\mathbb{C})\) with \(n \le 3\), the + Given \(\rho : \Mod(\Sigma_2^b) \to \GL_n(\mathbb{C})\) with \(n \le 3\), the image of \(\rho\) is Abelian. \end{proposition} \begin{proof}[Sketch of proof] - Given \(\alpha \subset \Sigma_2^p\), let \(L_\alpha = \rho(\tau_\alpha)\) and + Given \(\alpha \subset \Sigma_2^b\), let \(L_\alpha = \rho(\tau_\alpha)\) and denote by \(E_{\alpha = \lambda} = \{ v \in \mathbb{C}^n : L_\alpha v = \lambda v \}\) its eigenspaces. Let \(\alpha_1, \alpha_2, \beta_1, \beta_2, - \gamma, \eta_1, \ldots, \eta_{p-1} \subset \Sigma_2^p\) be the curves of the + \gamma, \eta_1, \ldots, \eta_{b-1} \subset \Sigma_2^b\) be the curves of the Lickorish generators from Theorem~\ref{thm:lickorish-gens}, as shown in Figure~\ref{fig:lickorish-gens-genus-2}. \begin{figure} @@ -44,12 +44,12 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g \label{fig:lickorish-gens-genus-2} \end{figure} - If \(n = 1\) then \(\rho(\Mod(\Sigma_2^p)) \subset \GL_1(\mathbb{C}) = + If \(n = 1\) then \(\rho(\Mod(\Sigma_2^b)) \subset \GL_1(\mathbb{C}) = \mathbb{C}^\times\) is Abelian. Now if \(n = 2\) or \(3\), by Proposition~\ref{thm:commutator-normal-gen} it suffices to show \(L_{\alpha_1} = L_{\beta_1}\), so that \(\tau_{\alpha_1} \tau_{\beta_1}^{-1} \in \ker - \rho\) and thus \(\Mod(\Sigma_2^p)' \subset \ker \rho\) -- i.e. - \(\rho(\Mod(\Sigma_2^p))\) is Abelian. Given the braid relation + \rho\) and thus \(\Mod(\Sigma_2^b)' \subset \ker \rho\) -- i.e. + \(\rho(\Mod(\Sigma_2^b))\) is Abelian. Given the braid relation \begin{equation}\label{eq:braid-rel-induction-basis} L_{\alpha_1} L_{\beta_1} L_{\alpha_1} = L_{\beta_1} L_{\alpha_1} L_{\beta_1}, @@ -165,8 +165,8 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g case. We claim that if \(E_{\alpha_2 = \lambda} = E_{\beta_2 = \lambda}\) then - \(E_{\alpha_2 = \lambda}\) is \(\Mod(\Sigma_2^p)\)-invariant. Indeed, by - change of coordinates we can always find \(f, g, h_i \in \Mod(\Sigma_2^p)\) + \(E_{\alpha_2 = \lambda}\) is \(\Mod(\Sigma_2^b)\)-invariant. Indeed, by + change of coordinates we can always find \(f, g, h_i \in \Mod(\Sigma_2^b)\) with \begin{align*} f \cdot [\alpha_2] & = [\alpha_1] @@ -212,16 +212,16 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g \end{align*} In other words, \(E_{\alpha_1 = \lambda} = E_{\alpha_2 = \lambda} = E_{\beta_1 = \lambda} = E_{\beta_2 = \lambda} = E_{\gamma = \lambda} = - E_{\eta_1 = \lambda} = \cdots = E_{\eta_{p-1} = \lambda}\) is invariant + E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b-1} = \lambda}\) is invariant under the action of all Lickorish generators. Hence \(\rho\) restricts to a subrepresentation \(\bar \rho : - \Mod(\Sigma_2^p) \to \GL(E_{\alpha_2 = \lambda}) = \GL_2(\mathbb{C})\) -- + \Mod(\Sigma_2^b) \to \GL(E_{\alpha_2 = \lambda}) = \GL_2(\mathbb{C})\) -- recall \(E_{\alpha_2 = \lambda} = \mathbb{C} e_1 \oplus \mathbb{C} e_2\). By - case (2), \(\bar \rho(f) = 1\) for all \(f \in \Mod(\Sigma_2^p)'\), given - that \(\bar \rho(\Mod(\Sigma_2^p))\) is Abelian. Thus + case (2), \(\bar \rho(f) = 1\) for all \(f \in \Mod(\Sigma_2^b)'\), given + that \(\bar \rho(\Mod(\Sigma_2^b))\) is Abelian. Thus \[ - \rho(\Mod(\Sigma_2^p)') \subset + \rho(\Mod(\Sigma_2^b)') \subset \begin{pmatrix} 1 & 0 & * \\ 0 & 1 & * \\ @@ -230,7 +230,7 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g \] lies inside the group of upper triangular matrices, a solvable subgroup of \(\GL_3(\mathbb{C})\). Now by Proposition~\ref{thm:commutator-is-perfect} we - get \(\rho(\Mod(\Sigma_2^p)') = 1\): any homomorphism from a perfect group to + get \(\rho(\Mod(\Sigma_2^b)') = 1\): any homomorphism from a perfect group to a solvable group is trivial. Finally, if \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\) and @@ -263,18 +263,18 @@ We are now ready to establish the triviality of low-dimensional representations. \begin{proof}[Proof of Theorem~\ref{thm:low-dim-reps-are-trivial}] - Let \(g \ge 1\), \(p \ge 0\) and fix \(\rho : \Mod(\Sigma_g^p) \to + Let \(g \ge 1\), \(b \ge 0\) and fix \(\rho : \Mod(\Sigma_g^b) \to \GL_n(\mathbb{C})\) with \(n < 2g\). As promised, we proceed by induction on \(g\). The base case \(g = 1\) is again clear from the fact \(n = 1\) and \(\GL_1(\mathbb{C}) = \mathbb{C}^\times\). The case \(g = 2\) was also established in Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}. Now suppose \(g \ge 3\) and every \(m\)-dimensional representation of \(\Sigma_{g - - 1}^q\) has Abelian image for \(m < 2(g - 1)\). Let us show \(\rho\) has + - 1}^{b'}\) has Abelian image for \(m < 2(g - 1)\). Let us show \(\rho\) has Abelian image. Let \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g, \gamma_1, \ldots, - \gamma_{g - 1}, \eta_1, \ldots, \eta_{p-1} \subset \Sigma_g^p\) be the curves - from the Lickorish generators of \(\Mod(\Sigma_g^p)\), as in + \gamma_{g - 1}, \eta_1, \ldots, \eta_{b-1} \subset \Sigma_g^b\) be the curves + from the Lickorish generators of \(\Mod(\Sigma_g^b)\), as in Figure~\ref{fig:lickorish-gens}. Once again, let \(L_\alpha = \rho(\tau_\alpha)\) and denote by \(E_{\alpha = \lambda}\) the eigenspace of \(L_\alpha\) associated to \(\lambda \in \mathbb{C}\). Let \(\Sigma \cong @@ -284,14 +284,14 @@ representations. \begin{figure}[ht] \centering \includegraphics[width=.35\linewidth]{images/lickorish-gens-korkmaz-proof.eps} - \caption{The subsurface $\Sigma \subset \Sigma_g^p$.} + \caption{The subsurface $\Sigma \subset \Sigma_g^b$.} \label{fig:korkmaz-proof-subsurface} \end{figure} We claim that it suffices to find a \(m\)-dimensional \(\Mod(\Sigma)\)-invariant\footnote{Here we view $\Mod(\Sigma)$ as a subgroup - of $\Mod(\Sigma_g^p)$ via the inclusion homomorphism $\Mod(\Sigma) \to - \Mod(\Sigma_g^p)$ from Example~\ref{ex:inclusion-morphism}, which can be + of $\Mod(\Sigma_g^b)$ via the inclusion homomorphism $\Mod(\Sigma) \to + \Mod(\Sigma_g^b)$ from Example~\ref{ex:inclusion-morphism}, which can be shown to be injective in this particular case.} subspace \(W \subset \mathbb{C}^n\) with \(2 \le m \le n - 2\). Indeed, in this case \(m < 2(g - 1)\) and \(\dim \mfrac{\mathbb{C}^n}{W} = n - m < 2(g - 1)\). Thus both @@ -319,9 +319,9 @@ representations. follows from Proposition~\ref{thm:commutator-is-perfect} that \(\rho\) annihilates all of \(\Mod(\Sigma)'\) and, in particular, \(\tau_{\alpha_1} \tau_{\beta_1}^{-1} \in \ker \rho\). But recall from - Proposition~\ref{thm:commutator-normal-gen} that \(\Mod(\Sigma_g^p)'\) is + Proposition~\ref{thm:commutator-normal-gen} that \(\Mod(\Sigma_g^b)'\) is normally generated by \(\tau_{\alpha_1} \tau_{\beta_1}^{-1}\), from which we - conclude \(\rho(\Mod(\Sigma_g^p)') = 1\), as desired. + conclude \(\rho(\Mod(\Sigma_g^b)') = 1\), as desired. As before, we exhaustively analyze all possible Jordan forms of \(L_{\alpha_g}\). First, consider the case where we can find eigenvalues @@ -385,9 +385,9 @@ representations. \item[\bfseries\color{highlight}(1)] Here we use the change of coordinates principle: each \(L_{\alpha_i}, L_{\beta_i}, L_{\gamma_i}, L_{\eta_i}\) is conjugate to \(L_{\alpha_g} = - \lambda\), so all Lickorish generators of \(\Mod(\Sigma_g^p)\) act on + \lambda\), so all Lickorish generators of \(\Mod(\Sigma_g^b)\) act on \(\mathbb{C}^n\) as scalar multiplication by \(\lambda\) as well. Hence - \(\rho(\Mod(\Sigma_g^p)) = \langle \lambda \rangle\) is Abelian. + \(\rho(\Mod(\Sigma_g^b)) = \langle \lambda \rangle\) is Abelian. \item[\bfseries\color{highlight}(2)] In this case, \(W = \ker (L_{\alpha_g} - \lambda)^2\) is a @@ -407,7 +407,7 @@ representations. = \lambda}\). In this situation, as in the proof of Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}, it follows from the change of coordinates principle that there are \(f_i, g_i, h_i \in - \Mod(\Sigma_g^p)\) with + \Mod(\Sigma_g^b)\) with \begin{align*} f_i \tau_{\alpha_g} f_i^{-1} & = \tau_{\alpha_i} & @@ -426,7 +426,7 @@ representations. E_{\alpha_1 = \lambda} = \cdots = E_{\alpha_g = \lambda} = E_{\beta_1 = \lambda} = \cdots = E_{\beta_g = \lambda} = E_{\gamma_1 = \lambda} = \cdots = E_{\gamma_{g - 1} = \lambda} - = E_{\eta_1 = \lambda} = \cdots = E_{\eta_{p-1} = \lambda}. + = E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b-1} = \lambda}. \] In particular, we can find a basis for \(\mathbb{C}^n\) with respect to @@ -441,20 +441,20 @@ representations. \end{pmatrix}. \] Since the group of upper triangular matrices is solvable and - \(\Mod(\Sigma_g^p)\) is perfect, it follows that \(\rho(\Mod(\Sigma_g^p))\) is - trivial. This concludes the proof \(\rho(\Mod(\Sigma_g^p))\) is Abelian. - - To see that \(\rho(\Mod(\Sigma_g^p)) = 1\) for \(g \ge 3\) we note that, since - \(\rho(\Mod(\Sigma_g^p))\) is Abelian, \(\rho\) factors though the Abelianization - map \(\Mod(\Sigma_g^p) \to \Mod(\Sigma_g^p)^\ab = \mfrac{\Mod(\Sigma_g^p)}{[\Mod(\Sigma_g^p), - \Mod(\Sigma_g^p)]}\). Now recall from Proposition~\ref{thm:trivial-abelianization} - that \(\Mod(\Sigma_g^p)^\ab = 0\) for \(g \ge 3\). In other words, \(\rho\) + \(\Mod(\Sigma_g^b)\) is perfect, it follows that \(\rho(\Mod(\Sigma_g^b))\) is + trivial. This concludes the proof \(\rho(\Mod(\Sigma_g^b))\) is Abelian. + + To see that \(\rho(\Mod(\Sigma_g^b)) = 1\) for \(g \ge 3\) we note that, since + \(\rho(\Mod(\Sigma_g^b))\) is Abelian, \(\rho\) factors though the Abelianization + map \(\Mod(\Sigma_g^b) \to \Mod(\Sigma_g^b)^\ab = \mfrac{\Mod(\Sigma_g^b)}{[\Mod(\Sigma_g^b), + \Mod(\Sigma_g^b)]}\). Now recall from Proposition~\ref{thm:trivial-abelianization} + that \(\Mod(\Sigma_g^b)^\ab = 0\) for \(g \ge 3\). In other words, \(\rho\) factors though the homomorphism \(1 \to \GL_n(\mathbb{C})\). We are done. \end{proof} Having established the triviality of the low-dimensional representations \(\rho -: \Mod(\Sigma_g^p) \to \GL_n(\mathbb{C})\), all that remains for us is to understand -the \(2g\)-dimensional representations of \(\Mod(\Sigma_g^p)\). We certainly know a +: \Mod(\Sigma_g^b) \to \GL_n(\mathbb{C})\), all that remains for us is to understand +the \(2g\)-dimensional representations of \(\Mod(\Sigma_g^b)\). We certainly know a nontrivial example of such, namely the symplectic representation \(\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) from Example~\ref{ex:symplectic-rep}. Surprisingly, this turns out to be @@ -462,13 +462,13 @@ Example~\ref{ex:symplectic-rep}. Surprisingly, this turns out to be representation in the compact case. More precisely, \begin{theorem}[Korkmaz]\label{thm:reps-of-dim-2g-are-symplectic} - Let \(g \ge 3\) and \(\rho : \Mod(\Sigma_g^p) \to \GL_{2g}(\mathbb{C})\). Then + Let \(g \ge 3\) and \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\). Then \(\rho\) is either trivial or conjugate to the symplectic - representation\footnote{Here the map $\Mod(\Sigma_g^p) \to + representation\footnote{Here the map $\Mod(\Sigma_g^b) \to \operatorname{Sp}_{2g}(\mathbb{Z})$ is given by the composition of the - inclusion morphism $\Mod(\Sigma_g^p) \to \Mod(\Sigma_g)$ with the usual symplectic + inclusion morphism $\Mod(\Sigma_g^b) \to \Mod(\Sigma_g)$ with the usual symplectic representation $\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})$.} - \(\Mod(\Sigma_g^p) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(\Sigma_g^p)\). + \(\Mod(\Sigma_g^b) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(\Sigma_g^b)\). \end{theorem} Unfortunately, the limited scope of these master thesis does not allow us to @@ -518,7 +518,7 @@ main lemma}. Namely\dots This is proved in \cite[Lemma 7.6]{korkmaz} using the braid relations. Notice that for \(n = g\) and \(m = 2g\) the matrices in Lemma~\ref{thm:main-lemma} coincide with the action of the Lickorish generators \(\tau_{\alpha_1}, \ldots, -\tau_{\alpha_g}, \tau_{\beta_1}, \ldots, \tau_{\beta_g} \in \Mod(\Sigma_g^p)\) on +\tau_{\alpha_g}, \tau_{\beta_1}, \ldots, \tau_{\beta_g} \in \Mod(\Sigma_g^b)\) on \(H_1(\Sigma_g, \mathbb{C}) \cong \mathbb{C}^{2g}\) -- represented in the standard basis \([\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]\) for \(H_1(\Sigma_g, \mathbb{C})\). @@ -541,28 +541,28 @@ basis \([\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]\) for \right) \end{align*} -Hence by embedding \(B_3^g\) in \(\Mod(\Sigma_g^p)\) via +Hence by embedding \(B_3^g\) in \(\Mod(\Sigma_g^b)\) via \begin{align*} - B_3^g & \to \Mod(\Sigma_g^p) \\ + B_3^g & \to \Mod(\Sigma_g^b) \\ a_i & \mapsto \tau_{\alpha_i} \\ b_i & \mapsto \tau_{\beta_i} \end{align*} -we can see that any \(\rho : \Mod(\Sigma_g^p) \to \GL_{2g}(\mathbb{C})\) in a +we can see that any \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\) in a certain class of representation satisfying some technical conditions must be -conjugate to the symplectic representation \(\Mod(\Sigma_g^p) \to +conjugate to the symplectic representation \(\Mod(\Sigma_g^b) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) when restricted to \(B_3^g\). Korkmaz then goes on to show that such technical conditions are met for any -nontrivial \(\rho : \Mod(\Sigma_g^p) \to \GL_{2g}(\mathbb{C})\). Furthermore, +nontrivial \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\). Furthermore, Korkmaz also argues that we can find a basis for \(\mathbb{C}^{2g}\) with respect to which the matrices of \(\rho(\tau_{\gamma_1}), \ldots, \rho(\tau_{\gamma_{g - 1}}), \rho(\tau_{\eta_1}), \ldots, -\rho(\tau_{\eta_{p-1}})\) also agrees with the action of \(\Mod(\Sigma_g^p)\) on +\rho(\tau_{\eta_{b-1}})\) also agrees with the action of \(\Mod(\Sigma_g^b)\) on \(H_1(\Sigma_g, \mathbb{C})\), concluding the classification of \(2g\)-dimensional representations. % TODO: Add some final comments about how the rest of the landscape of % representations is generally unknown and how there is a lot to study in here Recently, Kasahara \cite{kasahara} also classified the \((2g+1)\)-dimensional -representations of \(\Mod(\Sigma_g^p)\) for \(g \ge 7\) in terms of certain twisted +representations of \(\Mod(\Sigma_g^b)\) for \(g \ge 7\) in terms of certain twisted \(1\)-cohomology groups.
diff --git a/sections/twists.tex b/sections/twists.tex @@ -218,9 +218,9 @@ too. A perhaps less obvious fact about Dehn twists is\dots \begin{theorem}\label{thm:mcg-is-fg} - Let \(\Sigma_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with - \(r\) punctures and \(p\) boundary components. Then the pure mapping class - group \(\PMod(\Sigma_{g, r}^p)\) is generated by finitely many Dehn twists + Let \(\Sigma_{g, r}^b\) be the orientable surface of genus \(g \ge 1\) with + \(r\) punctures and \(b\) boundary components. Then the pure mapping class + group \(\PMod(\Sigma_{g, r}^b)\) is generated by finitely many Dehn twists about nonseparating curves or boundary components. \end{theorem} @@ -233,9 +233,9 @@ curves}. \section{The Birman Exact Sequence} 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 \(\Sigma_{g, r}^p\) and -that of \(\Sigma_{g, r+1}^p = \Sigma_{g, r}^p \setminus \{ x \}\) for some \(x\) in the -interior \((\Sigma_{g, r}^p)\degree\) of \(\Sigma_{g, r}^p\). Indeed, this will later +consider the relationship between the mapping class group of \(\Sigma_{g, r}^b\) and +that of \(\Sigma_{g, r+1}^b = \Sigma_{g, r}^b \setminus \{ x \}\) for some \(x\) in the +interior \((\Sigma_{g, r}^b)\degree\) of \(\Sigma_{g, r}^b\). Indeed, this will later allow us to establish the induction on the number of punctures \(r\). Given an orientable surface \(\Sigma\) and \(x_1, \ldots, x_n \in \Sigma\degree\), @@ -324,7 +324,7 @@ show\dots \section{The Modified Graph of Curves} Having established Theorem~\ref{thm:birman-exact-seq}, we now need to address -the induction step in the genus \(g\) of \(\Sigma_{g, r}^p\). Our strategy is to +the induction step in the genus \(g\) of \(\Sigma_{g, r}^b\). Our strategy is to apply the following lemma from geometric group theory. \begin{lemma}\label{thm:ggt-lemma} @@ -336,7 +336,7 @@ apply the following lemma from geometric group theory. \(G\) is generated by \(G_v\) and \(g\). \end{lemma} -We are interested, of course, in the group \(G = \PMod(\Sigma_{g, r}^p)\). As for +We are interested, of course, in the group \(G = \PMod(\Sigma_{g, r}^b)\). As for the graph \(\Gamma\), we consider\dots \begin{definition} @@ -352,10 +352,10 @@ the graph \(\Gamma\), we consider\dots \end{definition} It is clear from Lemma~\ref{thm:change-of-coordinates} that the actions of -\(\Mod(\Sigma_{g, r}^p)\) on \(V(\hat{\mathcal{N}}(\Sigma_{g, r}^p))\) and -\(\{([\alpha], [\beta]) \in V(\hat{\mathcal{N}}(\Sigma_{g, r}^p))^2 : \#(\alpha +\(\Mod(\Sigma_{g, r}^b)\) on \(V(\hat{\mathcal{N}}(\Sigma_{g, r}^b))\) and +\(\{([\alpha], [\beta]) \in V(\hat{\mathcal{N}}(\Sigma_{g, r}^b))^2 : \#(\alpha \cap \beta) = 1 \}\) are both transitive. But why should -\(\hat{\mathcal{N}}(\Sigma_{g, r}^p)\) be connected? +\(\hat{\mathcal{N}}(\Sigma_{g, r}^b)\) be connected? Historically, the modified graph of nonseparating curves first arose as a \emph{modified} version of another graph, known as\dots @@ -398,31 +398,31 @@ Corollary~\ref{thm:mofied-graph-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 \(\Sigma_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with - \(r\) punctures and \(p\) boundary components. We want to establish that - \(\PMod(\Sigma_{g, r}^p)\) is generated by a finite number of Dehn twists + Let \(\Sigma_{g, r}^b\) be the orientable surface of genus \(g \ge 1\) with + \(r\) punctures and \(b\) boundary components. We want to establish that + \(\PMod(\Sigma_{g, r}^b)\) is generated by a finite number of Dehn twists about nonseparating simple closed curves or boundary components. - First, observe that if \(p \ge 1\) and \(\partial \Sigma_{g, r}^p = \delta_1 + First, observe that if \(b \ge 1\) and \(\partial \Sigma_{g, r}^b = \delta_1 \cup \cdots \cup \delta_p\) then, by recursively applying the capping exact sequence \begin{center} \begin{tikzcd} 1 \rar & \langle \tau_{\delta_1} \rangle \rar & - \PMod(\Sigma_{g, r}^p) \rar{\operatorname{cap}} & - \PMod(\Sigma_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\})) \rar & + \PMod(\Sigma_{g, r}^b) \rar{\operatorname{cap}} & + \PMod(\Sigma_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\})) \rar & 1 \end{tikzcd} \end{center} from Observation~\ref{ex:capping-seq}, it suffices to show that \(\Sigma_{g, n}\) is finitely generated by twists about nonseparating simple closed - curves. Indeed, if \(\PMod(\Sigma_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 - \setminus \{0\})) \cong \PMod(\Sigma_{g, r+1}^{p-1})\) is finitely generated + curves. Indeed, if \(\PMod(\Sigma_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 + \setminus \{0\})) \cong \PMod(\Sigma_{g, r+1}^{b-1})\) is finitely generated by twists about nonseparating curves or boundary components, then we may lift - the generators of \(\PMod(\Sigma_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 + the generators of \(\PMod(\Sigma_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\) to Dehn twists about the corresponding curves in - \(\Sigma_{g, r}^p\) and add \(\tau_{\delta_1}\) to the generating set. + \(\Sigma_{g, r}^b\) and add \(\tau_{\delta_1}\) to the generating set. It thus suffices to consider the boundaryless case \(\Sigma_{g, r}\). As promised, we proceed by double induction on \(r\) and \(g\). For the base case, it is @@ -538,18 +538,18 @@ Theorem~\ref{thm:mcg-is-fg}. 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 \(\Mod(\Sigma_g^p)\) by adapting the induction steps in the +generators of \(\Mod(\Sigma_g^b)\) by adapting 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 \(\Mod(\Sigma_g^p)\) is generated by the Dehn twists + If \(g \ge 1\) then \(\Mod(\Sigma_g^b)\) is generated by the Dehn twists about the curves \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g, - \gamma_1, \ldots, \gamma_{g - 1}, \eta_1, \ldots, \eta_{p-1}\) as in + \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 \(p = 0\), we can write \(\tau_{\alpha_3}, \ldots, +In the boundaryless case \(b = 0\), we can write \(\tau_{\alpha_3}, \ldots, \tau_{\alpha_g} \in \Mod(\Sigma_g)\) as products of the twists about the remaining curves, from which we get the so called \emph{Humphreys generators}. @@ -564,7 +564,7 @@ remaining curves, from which we get the so called \emph{Humphreys generators}. \centering \includegraphics[width=\linewidth]{images/lickorish-gens.eps} \captionof{figure}{The curves from Lickorish generators of - $\Mod(\Sigma_g^p)$.} + $\Mod(\Sigma_g^b)$.} \label{fig:lickorish-gens} \end{minipage} \hspace{.6cm} %