- Commit
- 7fb6fb434e092881be904180d6a92a69e9d6e764
- Parent
- 95fbf362def57dc0b4c800cc01662c44210fd63e
- Author
- Pablo <pablo-pie@riseup.net>
- Date
Changed the notation for surfaces
S → Σ
memoire-m2
My M2 Memoire on mapping class groups & their representations
Diffstat
8 files changed, 582 insertions, 612 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | images/cutting-homeo.svg | 44 | 26 | 18 |
| Modified | images/hyperelliptic-involution.svg | 94 | 29 | 65 |
| Modified | images/lickorish-gens-korkmaz-proof.svg | 43 | 18 | 25 |
| Modified | images/torus-mcg-generators.svg | 47 | 26 | 21 |
| Modified | sections/introduction.tex | 298 | 148 | 150 |
| Modified | sections/presentation.tex | 236 | 118 | 118 |
| Modified | sections/representations.tex | 160 | 81 | 79 |
| Modified | sections/twists.tex | 272 | 136 | 136 |
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diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -14,57 +14,56 @@ proof. \begin{theorem}[Classification of surfaces]\label{thm:classification-of-surfaces} Any closed connected orientable surface is homeomorphic to the connected sum - \(S_g\) of \(g \ge 0\) copies of the torus \(\mathbb{T}^2 = + \(\Sigma_g\) of \(g \ge 0\) copies of the torus \(\mathbb{T}^2 = \mfrac{\mathbb{R}^2}{\mathbb{Z}^2}\). Any compact connected orientable - surface \(S\) is isomorphic to the surface \(S_g^p\) obtained from \(S_g\) by + surface \(\Sigma\) is isomorphic to the surface \(\Sigma_g^p\) obtained from \(\Sigma_g\) by removing \(p \ge 0\) 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 \(S\)}. We also have the noncompact surface \(S_{g, -r}^p = S_g^p \setminus \{x_1, \ldots, x_r\}\), where \(x_1, \ldots, x_r\) in -the interior of \(S_g^p\). The points \(x_1, \ldots, x_r\) are called the -\emph{puctures} of \(S_{g, r}^p\). Throught these notes, all surfaces -considered will be of the form \(S = S_{g, r}^p\). Any such \(S\) admits a -natural compactification \(\bar S\) obtained by filling the its punctures. We -denote \(S_{g, r} = S_{g, r}^0\). All closed curves \(\alpha, \beta \subset S\) -we consider lie in the interior \(S\degree\) of \(S\) and intersect +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 \(x_1, \ldots, x_r\) in +the interior of \(\Sigma_g^p\). The points \(x_1, \ldots, x_r\) are called the +\emph{puctures} of \(\Sigma_{g, r}^p\). Throught these notes, all surfaces +considered will be of the form \(\Sigma = \Sigma_{g, r}^p\). Any such \(\Sigma\) admits a +natural compactification \(\widebar\Sigma\) obtained by filling the 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 transversily. It is interesting to remark that, aside from the homeomorphism type of a -surface \(S\), Theorem~\ref{thm:classification-of-surfaces} also informs the -geometry of the curves in \(S\) and their intersections. For example\dots +surface \(\Sigma\), Theorem~\ref{thm:classification-of-surfaces} also informs the +geometry of the curves in \(\Sigma\) and their intersections. For example\dots \begin{lemma}[Change of coordinates principle]\label{thm:change-of-coordinates} - Given oriented nonseparating simple closed curves \(\alpha, \beta \subset - S\), we can find an orientation-presering homeomorphis \(\phi : S \isoto S\) - fixing \(\partial S\) pointwise such that \(\phi(\alpha) = \beta\) with - orientation. Even more so, if \(\alpha', \beta' \subset S\) are nonseparating + Given oriented nonseparating simple closed curves \(\alpha, \beta \subset \Sigma\), we can find an orientation-presering homeomorphis \(\phi : \Sigma \isoto \Sigma\) + fixing \(\partial \Sigma\) pointwise such that \(\phi(\alpha) = \beta\) with + orientation. Even more so, if \(\alpha', \beta' \subset \Sigma\) are nonseparating curve such that each pair \((\alpha, \alpha'), (\beta, \beta')\) crosses only once, then we can choose \(\phi\) with \(\phi(\alpha) = \beta'\) and \(\phi(\alpha') = \beta'\). \end{lemma} \begin{proof} - Let \(S = S_{g, r}^p\) and consider the surface \(S_{\alpha \alpha'}\) - obtained by cutting \(S\) across \(\alpha\) and \(\alpha'\), as in + Let \(\Sigma = \Sigma_{g, r}^p\) and consider the surface \(\Sigma_{\alpha \alpha'}\) + obtained by cutting \(\Sigma\) across \(\alpha\) and \(\alpha'\), as in Figure~\ref{fig:change-of-coordinates}. Since \(\alpha\) and \(\alpha'\) are nonseparating, this surface has genus \(g - 1\) and one additional boundary - component \(\delta \subset \partial S_{\alpha \beta}\), so \(S_{\alpha \beta} - \cong S_{g-1,r}^{p+1}\). The boundary component \(\delta\) is naturally + component \(\delta \subset \partial \Sigma_{\alpha \beta}\), so \(\Sigma_{\alpha \beta} + \cong \Sigma_{g-1,r}^{p+1}\). The boundary component \(\delta\) is naturally subdived into the four arcs in Figure~\ref{fig:change-of-coordinates}, each - corresponding to one of the curves \(\alpha\) and \(\alpha'\) in \(S\). By + corresponding to one of the curves \(\alpha\) and \(\alpha'\) in \(\Sigma\). By identifying the pairs of arcs corresponding to the same curve we obtain the - surface \(\mfrac{S_{\alpha \beta}}{\sim} \cong S\). + surface \(\mfrac{\Sigma_{\alpha \beta}}{\sim} \cong S\). - Similarly, \(S_{\beta \beta'} \cong S_{g-1, r}^{p+1}\) also has an additional - boundary component \(\delta' \subset \partial S_{\beta \beta'}\) subdivided + Similarly, \(\Sigma_{\beta \beta'} \cong \Sigma_{g-1, r}^{p+1}\) also has an additional + boundary component \(\delta' \subset \partial \Sigma_{\beta \beta'}\) subdivided into four arcs. Now by the classification of surfaces we can find an - orientation-preserving homemorphism \(\tilde\phi : S_{\alpha \alpha'} \isoto - S_{\beta \beta'}\). Even more so, we can choose \(\tilde\phi\) taking each + orientation-preserving homemorphism \(\tilde\phi : \Sigma_{\alpha \alpha'} \isoto + \Sigma_{\beta \beta'}\). Even more so, we can choose \(\tilde\phi\) taking each one of the arcs in \(\delta\) to the corresponding arc in \(\delta'\). Now \(\tilde\phi\) descends to a self-homeomorphism \(\phi\) the quotient surface - \(S \cong \mfrac{S_{\alpha \alpha'}}{\sim} \cong \mfrac{S_{\beta + \(S \cong \mfrac{\Sigma_{\alpha \alpha'}}{\sim} \cong \mfrac{\Sigma_{\beta \beta'}}{\sim}\). Moreover, \(\phi\) is such that \(\phi(\alpha) = \alpha'\) and \(\phi(\beta) = \beta'\), as desired. \end{proof} @@ -72,10 +71,10 @@ geometry of the curves in \(S\) and their intersections. For example\dots \begin{figure}[ht] \centering \includegraphics[width=.8\linewidth]{images/change-of-coords-cut.eps} - \caption{By cutting $S_{g, r}^p$ across $\alpha$ we obtain $S_{g-1, + \caption{By cutting $\Sigma_{g, r}^p$ across $\alpha$ we obtain $\Sigma_{g-1, r}^{p+2}$, where $\alpha'$ deterimines a yellow arc joining the two - additional boundary components. Now by cutting $S_{g-1, r}^{p+2}$ across this - arc we obtain $S_{g-1,r}^p$, with the added boundary component subdivided + 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 subdivided into the four arcs corresponding to $\alpha$ and $\alpha'$.} \label{fig:change-of-coordinates} \end{figure} @@ -83,111 +82,108 @@ geometry of the curves in \(S\) and their intersections. For example\dots More generally, despite the apparent clarity of the picture painted by Theorem~\ref{thm:classification-of-surfaces}, there are still plenty of unawsered questions about surfaces and their -homeomorphisms. Given a surface \(S\), the group \(\Homeo^+(S, \partial S)\) of -orientation-preserving homeomorphism of \(S\) fixing its boundary pointwise is -a topological group\footnote{Here we endow \(\Homeo^+(S, \partial S)\) with the +homeomorphisms. Given a surface \(\Sigma\), the group \(\Homeo^+(\Sigma, \partial \Sigma)\) of +orientation-preserving homeomorphism of \(\Sigma\) fixing its boundary pointwise is +a topological group\footnote{Here we endow \(\Homeo^+(\Sigma, \partial \Sigma)\) with the compact-open topology.} with a rich geometry. It is not hard to come up with interesting questions about such group. For example, \begin{enumerate} - \item Given closed curves \(\alpha, \beta \subset S\), can we find \(\phi \in - \Homeo^+(S, \partial S)\) with \(\phi(\alpha) = \beta\)? + \item Given closed curves \(\alpha, \beta \subset \Sigma\), can we find \(\phi \in + \Homeo^+(\Sigma, \partial \Sigma)\) with \(\phi(\alpha) = \beta\)? - \item What are the conjugacy classes of \(\Homeo^+(S, \partial S)\)? What + \item What are the conjugacy classes of \(\Homeo^+(\Sigma, \partial \Sigma)\)? What about its connected components? - \item Does \(\Homeo^+(S, \partial S)\) determine \(S\)? If the answer is + \item Does \(\Homeo^+(\Sigma, \partial \Sigma)\) determine \(\Sigma\)? If the answer is \emph{no}, what about in the closed case? \end{enumerate} -Unfortunately, however, the algebraic structure \(\Homeo^+(S, \partial)\) is +Unfortunately, however, the algebraic structure \(\Homeo^+(\Sigma, \partial)\) is tipically too complex to tackle. More importantly, all of this complexity is arguably unnecessary for most topological applications, in the sence that usually we are only really interested in considering \emph{homeomorphisms up to -isotopy}. For instance, isotopic homeomorphisms \(\phi \simeq \psi : S \isoto -S\) determine the same automorphism \(\phi_* = \psi_*\) at the levels of +isotopy}. For instance, isotopic homeomorphisms \(\phi \simeq \psi : \Sigma \isoto \Sigma\) determine the same automorphism \(\phi_* = \psi_*\) at the levels of homotopy and homology. This leads us to consider the group of connected -components of \(\Homeo^+(S, \partial S)\), also known as \emph{the mapping +components of \(\Homeo^+(\Sigma, \partial \Sigma)\), also known as \emph{the mapping class group}. This will be the focus of the dissertation at hand. \begin{definition}\label{def:mcg} - The \emph{mapping class group \(\Mod(S)\) of an orientable surface \(S\)} is - the group of isotopy classes of orientation-preserving homeomorphisms \(S - \isoto S\), where both the homeomorphisms and the isotopies are assumed to - fix the points of \(\partial S\) and the punctures of \(S\). + The \emph{mapping class group \(\Mod(\Sigma)\) of an orientable surface \(\Sigma\)} is + the group of isotopy classes of orientation-preserving homeomorphisms \(\Sigma \isoto \Sigma\), where both the homeomorphisms and the isotopies are assumed to + fix the points of \(\partial \Sigma\) and the punctures of \(\Sigma\). \[ - \Mod(S) = \mfrac{\Homeo^+(S, \partial S)}{\simeq} + \Mod(\Sigma) = \mfrac{\Homeo^+(\Sigma, \partial \Sigma)}{\simeq} \] \end{definition} There are many variations of the Definition~\ref{def:mcg}. For example\dots \begin{example}\label{ex:action-on-punctures} - Any \(\phi \in \Homeo^+(S, \partial S)\) extends to a homomorphism - \(\tilde\phi\) of \(\bar{S}\) that permutes the set \(\{x_1, \ldots, x_r\} = - \bar{S} \setminus S\) of punctures of \(S\). We may thus define an action - \(\Mod(S) \leftaction \{x_1, \ldots, x_r\}\) via \(f \cdot x_i = - \tilde\phi(x_i)\) for \(f = [\phi] \in \Mod(S)\) -- which is independant of + Any \(\phi \in \Homeo^+(\Sigma, \partial \Sigma)\) extends to a homomorphism + \(\tilde\phi\) of \(\widebar\Sigma\) that permutes the set \(\{x_1, \ldots, x_r\} = + \widebar\Sigma \setminus \Sigma\) of punctures of \(\Sigma\). We may thus define an action + \(\Mod(\Sigma) \leftaction \{x_1, \ldots, x_r\}\) via \(f \cdot x_i = + \tilde\phi(x_i)\) for \(f = [\phi] \in \Mod(\Sigma)\) -- which is independant of the choice of representative \(\phi\) of \(f\). \end{example} \begin{definition} - Given an orientable surface \(S\) and a puncture \(x \subset \bar{S}\) 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 \(S\). + Given an orientable surface \(\Sigma\) and a puncture \(x \subset \widebar\Sigma\) of + \(\Sigma\), denote by \(\Mod(\Sigma, x) \subset \Mod(\Sigma)\) the subgroup of mapping + classes that fix \(x\). The \emph{pure mapping class group \(\PMod(\Sigma) \subset + \Mod(\Sigma)\) of \(\Sigma\)} is the subgroup of mapping classes that fix every + puncture of \(\Sigma\). \end{definition} \begin{example}\label{ex:action-on-curves} - Given a simple closed curve \(\alpha \subset S\), denote by + Given a simple closed curve \(\alpha \subset \Sigma\), denote by \(\vec{[\alpha]}\) and \([\alpha]\) the isotopy classes of \(\alpha\) with - and without orientation, respectively. There are natural actions \(\Mod(S) - \leftaction \{ \vec{[\alpha]} : \alpha \subset S \}\) and \(\Mod(S) - \leftaction \{ [\alpha] : \alpha \subset S \}\) given by + and without orientation, respectively. There are natural actions \(\Mod(\Sigma) + \leftaction \{ \vec{[\alpha]} : \alpha \subset \Sigma \}\) and \(\Mod(\Sigma) + \leftaction \{ [\alpha] : \alpha \subset \Sigma \}\) given by \begin{align*} f \cdot \vec{[\alpha]} & = \vec{[\phi(\alpha)]} & f \cdot [\alpha] & = [\phi(\alpha)] \end{align*} - for \(f = [\phi] \in \Mod(S)\). + for \(f = [\phi] \in \Mod(\Sigma)\). \end{example} \begin{definition} - Given a simple closed curve \(\alpha \subset S\), we denote by - \(\Mod(S)_{\vec{[\alpha]}} = \{ f \in \Mod(S) : f \cdot \vec{[\alpha]} = - \vec{[\alpha]} \}\) and \(\Mod(S)_{[\alpha]} = \{ f \in \Mod(S) : f \cdot + Given a simple closed curve \(\alpha \subset \Sigma\), we denote by + \(\Mod(\Sigma)_{\vec{[\alpha]}} = \{ f \in \Mod(\Sigma) : f \cdot \vec{[\alpha]} = + \vec{[\alpha]} \}\) and \(\Mod(\Sigma)_{[\alpha]} = \{ f \in \Mod(\Sigma) : f \cdot [\alpha] = [\alpha] \}\) the subgroups of mapping classes that fix the isotopy classes of \(\alpha\). \end{definition} -While trying to understand the mapping class group of \(S\), it is interesting -to consider how the geometric relationship between \(S\) and other surfaces -affects \(\Mod(S)\). Indeed, different embeddings \(R \hookrightarrow S\) +While trying to understand the mapping class group of \(\Sigma\), it is interesting +to consider how the geometric relationship between \(\Sigma\) and other surfaces +affects \(\Mod(\Sigma)\). Indeed, different embeddings \(\Sigma' \hookrightarrow \Sigma\) translate to homomorphisms at the level of mapping class groups. \begin{example}[Inclusion homomorphism]\label{ex:inclusion-morphism} - 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 \(\tilde{\phi}(x) = x\) for \(x \in S\) outside of - \(R\) -- which is well defined since \(\phi\) fixes every point in \(\partial - R\). This contruction yields a group homomorphism + Let \(\Sigma' \subset \Sigma\) be a closed subsurface. Given some \(\phi \in \Homeo^+(\Sigma', + \partial \Sigma')\), we may extend \(\phi\) to \(\tilde{\phi} \in \Homeo^+(\Sigma, + \partial \Sigma)\) by setting \(\tilde{\phi}(x) = x\) for \(x \in S\) outside of + \(\Sigma'\) -- which is well defined since \(\phi\) fixes every point in \(\partial \Sigma'\). This contruction yields a group homomorphism \begin{align*} - \Mod(R) & \to \Mod(S) \\ + \Mod(\Sigma') & \to \Mod(\Sigma) \\ [\phi] & \mapsto [\tilde\phi], \end{align*} known as \emph{the inclusion homomorphism}. \end{example} \begin{example}[Capping exact sequence]\label{ex:capping-seq} - Let \(\delta \subset \partial S\) be an oriented boundary component of \(S\). - We refer to the inclusion homomorphism \(\operatorname{cap} : \Mod(S) \to - \Mod(S \cup_\delta (\mathbb{D}^2 \setminus \{0\}))\) as \emph{the capping + Let \(\delta \subset \partial \Sigma\) be an oriented boundary component of \(\Sigma\). + We refer to the inclusion homomorphism \(\operatorname{cap} : \Mod(\Sigma) \to + \Mod(\Sigma \cup_\delta (\mathbb{D}^2 \setminus \{0\}))\) as \emph{the capping homomorphism}. There is an exact sequence \begin{center} \begin{tikzcd} 1 \rar & \langle \tau_\alpha \rangle \rar & - \Mod(S) \rar{\operatorname{cap}} & - \Mod(S \cup_\delta (\mathbb{D}^2 \setminus \{0\}), 0) \rar & + \Mod(\Sigma) \rar{\operatorname{cap}} & + \Mod(\Sigma \cup_\delta (\mathbb{D}^2 \setminus \{0\}), 0) \rar & 1, \end{tikzcd} \end{center} @@ -196,17 +192,17 @@ translate to homomorphisms at the level of mapping class groups. \end{example} \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 + Given a simple closed curve \(\alpha \subset \Sigma\), denote by + \(\Mod(\Sigma)_{\vec{[\alpha]}} \subset \Mod(\Sigma)\) 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 + \Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\) has a representative \(\phi \in + \Homeo^+(\Sigma_{g+1})\) fixing \(\alpha\) point-wise, so that \(\phi\) restricts + to a homeomorphism of \(\Sigma \setminus \alpha\). 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}], + \operatorname{cut} : \Mod(\Sigma)_{\vec{[\alpha]}} + & \to \Mod(\Sigma\setminus\alpha) \\ + [\phi] & \mapsto [\phi\!\restriction_{\Sigma_{g+1} \setminus \alpha}], \end{align*} known as \emph{the cutting homomorphism}. Furthermore, \(\ker \operatorname{cut} = \langle \tau_\alpha \rangle\) -- see @@ -214,43 +210,43 @@ translate to homomorphisms at the level of mapping class groups. \end{example} As goes for most groups, another approach to understanding the mapping class -group of a given surface \(S\) is to study its actions. We have already seen +group of a given surface \(\Sigma\) is to study its actions. We have already seen simple example of such actions in Example~\ref{ex:action-on-punctures} and Example~\ref{ex:action-on-curves}. A particularly important class of actions -of \(\Mod(S)\) are its \emph{linear representations} -- i.e. the group -homomorphisms \(\Mod(S) \to \GL_n(\mathbb{C})\). These may be seen as actions -\(\Mod(S) \leftaction \mathbb{C}^n\) where each \(f \in \Mod(S)\) acts via some +of \(\Mod(\Sigma)\) are its \emph{linear representations} -- i.e. the group +homomorphisms \(\Mod(\Sigma) \to \GL_n(\mathbb{C})\). These may be seen as actions +\(\Mod(\Sigma) \leftaction \mathbb{C}^n\) where each \(f \in \Mod(\Sigma)\) acts via some \(\mathbb{C}\)-linear isomorphism \(\mathbb{C}^n \isoto \mathbb{C}^n\). \section{Representations} Here we collect a few fundamental examples of linear representations of -\(\Mod(S)\). +\(\Mod(\Sigma)\). \begin{observation} - Given \(k \ge 0\) and \(f = [\phi] \in \Mod(S)\), we may consider the map - \(\phi_* : H_k(S, \mathbb{Z}) \to H_k(S, \mathbb{Z})\) induced at the level + Given \(k \ge 0\) and \(f = [\phi] \in \Mod(\Sigma)\), we may consider the map + \(\phi_* : H_k(\Sigma, \mathbb{Z}) \to H_k(\Sigma, \mathbb{Z})\) induced at the level of singular homology. By homotopy invariance, the map \(\phi_*\) is independant of the choice of representative \(\phi\) of \(f\). By the functoriality of homology groups we then get a \(\mathbb{Z}\)-linear action - \(\Mod(S) \leftaction H_k(S, R)\), given by \(f \cdot \xi = \phi_*(\xi)\) for - \(f = [\phi] \in \Mod(S)\). + \(\Mod(\Sigma) \leftaction H_k(\Sigma, R)\), given by \(f \cdot \xi = \phi_*(\xi)\) for + \(f = [\phi] \in \Mod(\Sigma)\). \end{observation} Now by choosing \(k = 1\) we obtain the so called \emph{symplectic representation.} \begin{observation} - Recall \(H_1(S_g, \mathbb{Z}) \cong \mathbb{Z}^{2g}\), with standard basis - given by \([\alpha_1], [\beta_1], \ldots, [\alpha_g], [\beta_g] \in H_1(S_g, + Recall \(H_1(\Sigma_g, \mathbb{Z}) \cong \mathbb{Z}^{2g}\), with standard basis + given by \([\alpha_1], [\beta_1], \ldots, [\alpha_g], [\beta_g] \in H_1(\Sigma_g, \mathbb{Z})\) for \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g\) as - in Figure~\ref{fig:homology-basis}. The Abelian group \(H_1(S_g, + in Figure~\ref{fig:homology-basis}. The Abelian group \(H_1(\Sigma_g, \mathbb{Z})\) is endowed with a natural \(\mathbb{Z}\)-bilinear alternating form given by the \emph{algebraic intersection number} \([\alpha] \cdot [\beta] = \sum_{x \in \alpha \cap \beta} \operatorname{ind}\,x\) -- where the index \(\operatorname{ind}\,x = \pm 1\) of an intersection point is given by Figure~\ref{fig:intersection-index}. In terms of the standard basis of - \(H_1(S_g, \mathbb{Z})\), this form is given by + \(H_1(\Sigma_g, \mathbb{Z})\), this form is given by \begin{align}\label{eq:symplectic-form} [\alpha_i] \cdot [\beta_j] & = \delta_{i j} & [\alpha_i] \cdot [\alpha_j] & = 0 & @@ -261,23 +257,23 @@ representation.} \end{observation} \begin{example}[Symplectic representation]\label{ex:symplectic-rep} - Consider the \(\mathbb{Z}\)-linear action \(\Mod(S_g) \leftaction H_1(S_g, + Consider the \(\mathbb{Z}\)-linear action \(\Mod(\Sigma_g) \leftaction H_1(\Sigma_g, \mathbb{Z}) \cong \mathbb{Z}^{2g}\). Since pushforwards by orientation-preserving homeomorphisms preserve the index of intersection points, \((f \cdot [\alpha]) \cdot (f \cdot [\beta]) = [\alpha] \cdot - [\beta]\) for all \(\alpha, \beta \subset S_g\) and \(f \in \Mod(S_g)\). In - light of (\ref{eq:symplectic-form}), this implies \(\Mod(S_g)\) acts on + [\beta]\) for all \(\alpha, \beta \subset \Sigma_g\) and \(f \in \Mod(\Sigma_g)\). In + light of (\ref{eq:symplectic-form}), this implies \(\Mod(\Sigma_g)\) acts on \(\mathbb{Z}^{2g}\) via \(\mathbb{Z}\)-linear symplectomorphisms. We thus - obtain a group homomorphism \(\psi : \Mod(S_g) \to + obtain a group homomorphism \(\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z}) \subset \GL_{2g}(\mathbb{C})\), known as - \emph{the symplectic representation of \(\Mod(S_g)\)}. + \emph{the symplectic representation of \(\Mod(\Sigma_g)\)}. \end{example} \begin{minipage}[b]{.45\linewidth} \centering \includegraphics[width=\linewidth]{images/homology-generators.eps} \captionof{figure}{The curves $\alpha_1, \beta_1, \ldots, \alpha_g, \beta_g - \subset S_g$ that generate its first singular homology group.} + \subset \Sigma_g$ that generate its first singular homology group.} \label{fig:homology-basis} \end{minipage} \hspace{.5cm} % @@ -292,7 +288,7 @@ representation.} The symplectic representation already allows us to compute some important examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 = -S_1\) and the once-punctured torus \(S_{1, 1}\). +\Sigma_1\) and the once-punctured torus \(\Sigma_{1, 1}\). \begin{example}[Alexander trick]\label{ex:alexander-trick} The group \(\Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) of homeomorphisms of the @@ -351,7 +347,7 @@ S_1\) and the once-punctured torus \(S_{1, 1}\). \end{figure} \begin{example}\label{ex:punctured-torus-mcg} - By the same token, \(\Mod(S_{1, 1}) \cong \operatorname{SL}_2(\mathbb{Z})\). + By the same token, \(\Mod(\Sigma_{1, 1}) \cong \operatorname{SL}_2(\mathbb{Z})\). \end{example} % TODO: Add comments on the proof of linearity of Mod(S_2) by Korkmaz and @@ -361,10 +357,10 @@ S_1\) and the once-punctured torus \(S_{1, 1}\). \operatorname{SL}_2(\mathbb{Z})\) is an isomorphism, the symplectic representation is \emph{not} injective for surfaces of genus \(g \ge 2\) -- see \cite[Section~6.5]{farb-margalit} for a description of its kernel. In - fact, the question of existance of injective representations of \(\Mod(S_g)\) + fact, the question of existance of injective representations of \(\Mod(\Sigma_g)\) remains wide-open. Recently, Korkmaz \cite[Theomre~3]{korkmaz} established the lower bound of \(3 g - 3\) for the dimension of an injective - representation of \(\Mod(S_g)\) in the \(g \ge 3\) case -- if one such + representation of \(\Mod(\Sigma_g)\) in the \(g \ge 3\) case -- if one such representation exists. \end{note} @@ -372,31 +368,33 @@ Another fundamental class of examples of representations are the so called \emph{TQFT representations}. \begin{definition} - A \emph{cobordism} between closed oriented surfaces \(R\) and \(S\) is a - triple \((W, \phi_+, \phi_-)\) where \(W\) is a smooth oriented compact - \(3\)-manifold with \(\partial W = \partial_+ W \amalg \partial_- W\), - \(\phi_+ : S \isoto \partial_+ W\) is an orientation preserving - diffeomorphism and \(\phi_- : R \isoto \partial_- W\) is an - orientation-reversing diffeomorphism. + A \emph{cobordism} between closed oriented surfaces \(\Sigma\) and + \(\Sigma'\) is a triple \((W, \phi_+, \phi_-)\) where \(W\) is a smooth + oriented compact \(3\)-manifold with \(\partial W = \partial_+ W \amalg + \partial_- W\), \(\phi_+ : \Sigma \isoto \partial_+ W\) is an orientation + preserving diffeomorphism and \(\phi_- : \Sigma' \isoto \partial_- W\) is an + orientation-reversing diffeomorphism. We may abuse the notation and denote + \(W = (W, \phi_+, \phi_-)\). \end{definition} \begin{definition} We denote by \(\Cob\) the category whose objects are (possibly disconnected) - closed oriented surfaces and whose morphisms \(R \to S\) are diffeomorphism - classes\footnote{Here we only consider orientation-preserving diffeomorphisms - $\varphi : W \isoto W'$ that are compatible with the boundary identifications - in the sence that $\varphi(\partial_\pm W) = \partial_\pm W'$ and $\psi_\pm = - \varphi \circ \phi_\pm$.} of cobordisms between \(R\) and \(S\), with - composition given by + closed oriented surfaces and whose morphisms \(\Sigma \to \Sigma'\) are + diffeomorphism classes\footnote{Here we only consider orientation-preserving + diffeomorphisms $\varphi : W \isoto W'$ that are compatible with the boundary + identifications in the sence that $\varphi(\partial_\pm W) = \partial_\pm W'$ + and $\psi_\pm = \varphi \circ \phi_\pm$.} of cobordisms between \(\Sigma\) + and \(\Sigma'\), with composition given by \[ [W, \phi_-, \phi_+] \circ [W', \psi_-, \psi_+] = [W \cup_{\psi_- \circ \phi_+^{-1}} W', \phi_-, \psi_+] \] - for \([W, \phi_-, \phi_+] : R \to S\) and \([W', \psi_-, \phi_+] : S \to L\). - We endow \(\Cob\) with the monoidal structure given by + for \([W, \phi_-, \phi_+] : \Sigma \to \Sigma'\) and \([W', \psi_-, \phi_+] : + \Sigma' \to \Sigma''\). We endow \(\Cob\) with the monoidal structure given + by \begin{align*} - S \otimes R - & = S \amalg R & + \Sigma \otimes \Sigma' + & = \Sigma \amalg \Sigma' & [W,\phi_+,\phi_-] \otimes [W',\psi_+,\psi_-] & = [W \amalg W', \phi_+ \amalg \psi_+, \phi_- \amalg \psi_-]. \end{align*} @@ -405,37 +403,37 @@ Another fundamental class of examples of representations are the so called \begin{definition}[TQFT]\label{def:tqft} A \emph{topological quantum field theory} (abreviated by \emph{TQFT}) is a functor \(\mathcal{F} : \Cob \to \Vect\) satisfying - \begin{gather*} - \begin{aligned} - \mathcal{F}(\emptyset) & = \mathbb{C} & - \mathcal{F}(S \otimes R) & = \mathcal{F}(S) \otimes \mathcal{F}(R) - \end{aligned} \\ - \mathcal{F}([W,\phi_+,\phi_-] \otimes [W',\psi_+,\psi_-]) - = \mathcal{F}([W,\phi_+,\phi_-]) \otimes \mathcal{F}([W',\psi_+,\psi_-]), - \end{gather*} + \begin{align*} + \mathcal{F}(\emptyset) & = \mathbb{C} & + \mathcal{F}(\Sigma \otimes \Sigma') + & = \mathcal{F}(\Sigma) \otimes \mathcal{F}(\Sigma') & + \mathcal{F}([W] \otimes [W']) + & = \mathcal{F}([W]) \otimes \mathcal{F}([W']), + \end{align*} where \(\Vect\) denotes the category of finite-dimensional complex vector spaces. \end{definition} \begin{observation} - Given \(\phi \in \Homeo^+(S_g)\), we may consider the so called \emph{mapping - cylinder} \(M_\phi = (S_g \times [0, 1], \phi, 1)\), a cobordism between - \(S_g\) and itself -- where \(\partial_+ (S_g \times [0, 1]) = S_g \times 0\) - and \(\partial_- (S_g \times [0, 1]) = S_g \times 1\). The diffeomorphism - class of \(M_\phi\) is independant of the choice of representative of \(f = - [\phi] \in \Mod(S_g)\), so \(M_f = [M_\phi] : S_g \to S_g\) is a well defined - morphism in \(\Cob\). + Given \(\phi \in \Homeo^+(\Sigma_g)\), we may consider the so called + \emph{mapping cylinder} \(M_\phi = (\Sigma_g \times [0, 1], \phi, 1)\), a + cobordism between \(\Sigma_g\) and itself -- where \(\partial_+ (\Sigma_g + \times [0, 1]) = \Sigma_g \times 0\) and \(\partial_- (\Sigma_g \times [0, + 1]) = \Sigma_g \times 1\). The diffeomorphism class of \(M_\phi\) is + independant of the choice of representative of \(f = [\phi] \in + \Mod(\Sigma_g)\), so \(M_f = [M_\phi] : \Sigma_g \to \Sigma_g\) is a well + defined morphism in \(\Cob\). \end{observation} \begin{example}[TQFT representations]\label{ex:tqft-reps} - It is clear that \(M_1\) is the identity morphism \(S_g \to S_g\) in - \(\Cob\). In addition, \(M_{f \cdot g} = M_f \circ M_g\) in \(\Cob\) for all - \(f, g \in \Mod(S_g)\) -- see \cite[Lemma~2.5]{costantino}. Now given a TQFT - \(\mathcal{F} : \Cob \to \Vect\), by functoriality we obtain a linear - representation + It is clear that \(M_1\) is the identity morphism \(\Sigma_g \to \Sigma_g\) + in \(\Cob\). In addition, \(M_{f \cdot g} = M_f \circ M_g\) in \(\Cob\) for + all \(f, g \in \Mod(\Sigma_g)\) -- see \cite[Lemma~2.5]{costantino}. Now + given a TQFT \(\mathcal{F} : \Cob \to \Vect\), by functoriality we obtain a + linear representation \begin{align*} - \rho_{\mathcal{F}} : \Mod(S_g) & \to \GL(\mathcal{F}(S_g)) \\ - f & \mapsto \mathcal{F}(M_f). + \rho_{\mathcal{F}} : \Mod(\Sigma_g) & \to \GL(\mathcal{F}(\Sigma_g)) \\ + f & \mapsto \mathcal{F}(M_f). \end{align*} \end{example} @@ -443,14 +441,14 @@ As simple as the construction in Example~\ref{ex:tqft-reps} is, in practice it is not that easy to come across functors as the ones in Definition~\ref{def:tqft}. This is because, in most interesting examples, we are required to attach some extra data to our surfaces to get a well defined -association \(S_g \mapsto \mathcal{F}(S_g)\). Moreover, the condition +association \(\Sigma_g \mapsto \mathcal{F}(\Sigma_g)\). Moreover, the condition \(\mathcal{F}([W] \circ [W']) = \mathcal{F}([W]) \circ \mathcal{F}([W'])\) may only hold up to multiplication by scalars. Hence constructing an actual functor tipically requires \emph{extending} \(\Cob\) and \emph{tweaking} \(\Vect\). These ``extended TQFTs'' give rise to linear and projective representations of the \emph{extended mapping class -groups} \(\Mod(S_g) \times \mathbb{Z}\). We refer the reader to +groups} \(\Mod(\Sigma_g) \times \mathbb{Z}\). We refer the reader to \cite{costantino, julien} for constructions of one such extended TQFT and its corresponding representations: the so called \emph{\(\operatorname{SU}_2\) TQFT of level \(r\)}, first introduced by Witten and Reshetikhin-Tuarev @@ -460,7 +458,7 @@ topology. % TODOO: Add comments on Costantino's idea to get a faithful representation? Besides Example~\ref{ex:symplectic-rep} and Example~\ref{ex:tqft-reps}, not a -lot of other linear representations of \(\Mod(S_g)\) are known. Indeed, the +lot of other linear representations of \(\Mod(\Sigma_g)\) are known. Indeed, the representation theory of mapping class groups remains at mistery at large. In Chapter~\ref{ch:representations} we provide a brief overview of the field, as well as some recent developments. More specifically, we highlight Korkmaz'
diff --git a/sections/presentation.tex b/sections/presentation.tex @@ -1,22 +1,22 @@ \chapter{Relations Between Twists}\label{ch:relations} -Having found a conveniant set of genetors for \(\Mod(S)\), it is now natural to +Having found a conveniant set of genetors for \(\Mod(\Sigma)\), it is now natural to ask what are the relations between such generators. In this chapter, we highlight some further relations between Dehn twists and the geometric intuition behind them, culminating in the statement of a presentation for -\(\Mod(S_g)\) whose relations can be entirely explained in terms of the -geometry of curves in \(S_g\) -- see Theorem~\ref{thm:wajnryb-presentation}. +\(\Mod(\Sigma_g)\) whose relations can be entirely explained in terms of the +geometry of curves in \(\Sigma_g\) -- see Theorem~\ref{thm:wajnryb-presentation}. We start by the so called \emph{lantern relation}. % TODO: Add a sketch of proof? \begin{fundamental-observation} - Let \(S_0^4\) be the surface the of genus \(0\) with \(4\) boundary + Let \(\Sigma_0^4\) be the surface the of genus \(0\) with \(4\) boundary components -- i.e. the \emph{lantern-like} surface we get by subtracting \(4\) disjoint open disks from \(\mathbb{S}^2\). If \(\alpha, \beta, \gamma, - \delta_1, \ldots, \delta_4 \subset S_0^4\) are as in + \delta_1, \ldots, \delta_4 \subset \Sigma_0^4\) are as in Figure~\ref{fig:latern-relation} then from the Alexander method we get the - \emph{lantern relation} (\label{eq:lantern-relation}) in \(\Mod(S_0^4)\). + \emph{lantern relation} (\label{eq:lantern-relation}) in \(\Mod(\Sigma_0^4)\). \begin{equation}\label{eq:latern-relation} \tau_\alpha \tau_\beta \tau_\gamma = \tau_{\delta_1} \tau_{\delta_2} \tau_{\delta_3} \tau_{\delta_4} @@ -26,31 +26,31 @@ We start by the so called \emph{lantern relation}. \begin{figure}[ht] \centering \includegraphics[width=.25\linewidth]{images/lantern-relation.eps} - \caption{The curves from the latern relation of $S_0^4$.} + \caption{The curves from the latern relation of $\Sigma_0^4$.} \label{fig:latern-relation} \end{figure} -We may exploit different embedings \(S_0^4 \hookrightarrow S\) and their -corresponding inclusion homomorphisms \(\Mod(S_0^4) \to \Mod(S)\) to obtain -interesting relations between the corresponding Dehn twists in \(\Mod(S)\). For +We may exploit different embedings \(\Sigma_0^4 \hookrightarrow \Sigma\) and their +corresponding inclusion homomorphisms \(\Mod(\Sigma_0^4) \to \Mod(\Sigma)\) to obtain +interesting relations between the corresponding Dehn twists in \(\Mod(\Sigma)\). For example\dots \begin{proposition}\label{thm:trivial-abelianization} - The Abelianization \(\Mod(S_g^p)^\ab = \mfrac{\Mod(S_g^p)}{[\Mod(S_g), - \Mod(S_g)]}\) is cyclic. Moreover, if \(g \ge 3\) then \(\Mod(S_g^p)^\ab = - 1\). In other words, \(\Mod(S_g)\) is a perfect group for \(g \ge 3\). + 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 \(\Mod(\Sigma_g^p)^\ab = + 1\). In other words, \(\Mod(\Sigma_g)\) is a perfect group for \(g \ge 3\). \end{proposition} \begin{proof} - By Theorem~\ref{thm:lickorish-gens}, \(\Mod(S_g^p)^\ab\) is generated by the + By Theorem~\ref{thm:lickorish-gens}, \(\Mod(\Sigma_g^p)^\ab\) is generated by the image of the Lickrish generators, which are all conjugate and thus represent the same class in the Abelianization. In fact, any nonseparating \(\alpha - \subset S_g^p\) is conjugate to the Lickorish generators too, so - \(\Mod(S_g^p)^\ab = \langle [\alpha] \rangle\). + \subset \Sigma_g^p\) is conjugate to the Lickorish generators too, so + \(\Mod(\Sigma_g^p)^\ab = \langle [\alpha] \rangle\). - Now for \(g \ge 3\) we can embed \(S_0^4\) in \(S_g^p\) in such a way that + Now for \(g \ge 3\) we can embed \(\Sigma_0^4\) in \(\Sigma_g^p\) in such a way that all the corresponding curves \(\alpha, \beta, \gamma, \delta_1, \ldots, - \delta_4 \subset S_g^p\) are nonseparating, as shown in + \delta_4 \subset \Sigma_g^p\) are nonseparating, as shown in Figure~\ref{fig:latern-relation-trivial-abelianization}. The lantern relation (\ref{eq:latern-relation}) then becomes \[ @@ -60,27 +60,27 @@ example\dots + [\tau_{\delta_3}] + [\tau_{\delta_4}] = 4 \cdot [\tau_\alpha] \] - in \(\Mod(S_g^p)^\ab\). In other words, \([\tau_\alpha] = 0\) and thus - \(\Mod(S_g^p)^\ab = 0\). + in \(\Mod(\Sigma_g^p)^\ab\). In other words, \([\tau_\alpha] = 0\) and thus + \(\Mod(\Sigma_g^p)^\ab = 0\). \end{proof} \begin{figure}[ht] \centering \includegraphics[width=.5\linewidth]{images/lantern-relation-trivial-abelianization.eps} - \caption{The embedding of $S_0^4$ in $S_g^p$ for $g \ge 3$.} + \caption{The embedding of $\Sigma_0^4$ in $\Sigma_g^p$ for $g \ge 3$.} \label{fig:latern-relation-trivial-abelianization} \end{figure} -To get extra relations we need to consider certain branched covers \(S \to +To get extra relations we need to consider certain branched covers \(\Sigma \to \mathbb{D}^2 \setminus \{x_1, \ldots, x_r\}\) and how they may be used to relate both mapping class groups. This is what is known as\dots \section{The Birman-Hilden Theorem} -Let \(S_{0, r}^1 = \mathbb{D}^2 \setminus \{x_1, \ldots, x_r\}\) be the surface +Let \(\Sigma_{0, r}^1 = \mathbb{D}^2 \setminus \{x_1, \ldots, x_r\}\) be the surface of genus \(0\) with \(r\) punctures and \(1\) boundary component. We begin our investigation by providing an alternative description of its mapping class -group. Namely, we show that \(\Mod(S_{0, r}^1)\) is the braid group on \(r\) +group. Namely, we show that \(\Mod(\Sigma_{0, r}^1)\) is the braid group on \(r\) strands. \begin{definition} @@ -131,14 +131,14 @@ finite presentation of \(B_n\). \] \end{theorem} -As promised, we now show that \(B_n\) coincides with \(\Mod(S_{0, n}^1)\). +As promised, we now show that \(B_n\) coincides with \(\Mod(\Sigma_{0, n}^1)\). Recall from Theorem~\ref{thm:birman-exact-seq} that there is an exact sequence \begin{center} \begin{tikzcd} 1 \rar & B_n \rar{\operatorname{push}} - & \Mod(S_{0, n}^1) \rar + & \Mod(\Sigma_{0, n}^1) \rar & \Mod(\mathbb{D}^2) \rar & 1, \end{tikzcd} @@ -148,7 +148,7 @@ Example~\ref{ex:alexander-trick}. But \(\Mod(\mathbb{D}^2) = 1\). Hence we get\dots \begin{proposition} - The map \(\operatorname{push} : B_n \to \Mod(S_{0, n}^1)\) is a group + The map \(\operatorname{push} : B_n \to \Mod(\Sigma_{0, n}^1)\) is a group isomorphism. \end{proposition} @@ -156,10 +156,10 @@ we get\dots \begin{minipage}[b]{.45\linewidth} \begin{observation}\label{ex:braid-group-center} Using the capping exact sequence from Example~\ref{ex:capping-seq} and - the Alexander method, one can check that the center \(Z(\Mod(S_{0, n}^1))\) - of \(\Mod(S_{0, n}^1)\) is freely generated by the Dehn twist \(\tau_\delta\) - about the boundary \(\delta = \partial S_{0, n}^1\). It is also not very hard - to see that \(\operatorname{push} : B_n \to \Mod(S_{0, n}^1)\) takes + the Alexander method, one can check that the center \(Z(\Mod(\Sigma_{0, n}^1))\) + of \(\Mod(\Sigma_{0, n}^1)\) is freely generated by the Dehn twist \(\tau_\delta\) + about the boundary \(\delta = \partial \Sigma_{0, n}^1\). It is also not very hard + to see that \(\operatorname{push} : B_n \to \Mod(\Sigma_{0, n}^1)\) takes \(\sigma_1 \cdots \sigma_{n-1}\) to the rotation by \(\sfrac{2\pi}{n}\) as in Figure~\ref{fig:braid-group-center}, which is an \(n\)-th root of \(\tau_\delta\). Hence the center \(Z(B_n)\) is freely @@ -171,18 +171,18 @@ we get\dots \centering \includegraphics[width=.4\linewidth]{images/braid-group-center.eps} \captionof{figure}{The clockwise rotation by $\sfrac{2\pi}{n}$ about an axis center - around the punctures $x_1, \ldots, x_n$ of $S_{0, n}^1$.} + around the punctures $x_1, \ldots, x_n$ of $\Sigma_{0, n}^1$.} \label{fig:braid-group-center} \end{minipage} \smallskip -To get from \(S_{0, n}^1\) to surfaces of genus \(g > 0\) we may consider the -\emph{hyperelliptic involution} \(\iota : S_g \isoto S_g\) which rotates -\(S_g\) by \(\pi\) around some axis, as shown in +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 shown in Figure~\ref{fig:hyperelliptic-involution}. Given \(\ell < g\) and \(p = 1, 2\), -we can also embed \(S_\ell^p\) in \(S_g\) in such way that \(\iota\) restricts -to an involution\footnote{This involution does not fix $\partial S_\ell^p$ -point-wise.} \(S_\ell^p \isoto S_\ell^p\). +we can also embed \(\Sigma_\ell^p\) in \(\Sigma_g\) in such way that \(\iota\) restricts +to an involution\footnote{This involution does not fix $\partial \Sigma_\ell^p$ +point-wise.} \(\Sigma_\ell^p \isoto \Sigma_\ell^p\). \begin{figure}[ht] \centering @@ -192,18 +192,18 @@ point-wise.} \(S_\ell^p \isoto S_\ell^p\). \end{figure} It is clear from Figure~\ref{fig:hyperelliptic-involution} that the quotients -\(\mfrac{S_\ell^1}{\iota}\) and \(\mfrac{S_\ell^2}{\iota}\) are both disks, -with boundary corresponding to the projection of the boundaries of \(S_\ell^1\) -and \(S_\ell^2\), respectively. Given \(p = 1, 2\), the quotient map \(S_\ell^p -\to \mfrac{S_\ell^p}{\iota} \cong \mathbb{D}^2\) is a double cover with \(2\ell + +\(\mfrac{\Sigma_\ell^1}{\iota}\) 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 \(2\ell + b\) branch points corresponding to the fixed points of \(\iota\). We may thus -regard \(\mfrac{S_\ell^p}{\iota}\) as the disk \(S_{0, 2\ell + b}^1\) with +regard \(\mfrac{\Sigma_\ell^p}{\iota}\) 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 \(\alpha_1, -\ldots, \alpha_{2\ell} \subset S_\ell^p\) of the Humphreys generators of -\(\Mod(S_g)\). Since these curves are invariant under the action of \(\iota\), +\ldots, \alpha_{2\ell} \subset \Sigma_\ell^p\) of the Humphreys generators of +\(\Mod(\Sigma_g)\). Since these curves are invariant under the action of \(\iota\), they descend to arcs \(\bar{\alpha}_1, \ldots, \bar{\alpha}_{2\ell + b} \subset -S_{0, 2\ell + b}^1\) joining the punctures of the quotient surface. +\Sigma_{0, 2\ell + b}^1\) joining the punctures of the quotient surface. \begin{figure}[ht] \centering @@ -213,91 +213,91 @@ S_{0, 2\ell + b}^1\) joining the punctures of the quotient surface. \end{figure} \begin{observation}\label{ex:push-generators-description} - The map \(\operatorname{push} : B_{2\ell + b} \to \Mod(S_{0, 2\ell + b}^1)\) + 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 S_{0, 2\ell + + \(h_{\bar{\alpha}_i}\) about the arc \(\bar{\alpha}_i \subset \Sigma_{0, 2\ell + b}^1\). \end{observation} -We now study the homemorphisms of \(S_\ell^1\) and \(S_\ell^2\) that descend to +We now study the homemorphisms of \(\Sigma_\ell^1\) and \(\Sigma_\ell^2\) that descend to the quotient surfaces and their mapping classes, known as \emph{the symmetric mapping clases}. \begin{definition} Let \(\ell \ge 0\) and \(p = 1, 2\). The \emph{group of symmetric - homeomorphisms of \(S_\ell^p\)} is \(\SHomeo^+(S_\ell^p, \partial S_\ell^p) = - \{\phi \in \Homeo^+(S_\ell^p, \partial S_\ell^p) : [\phi, \iota] = 1\}\). The - \emph{symmetric mapping class group of \(S_\ell^p\)} is the subgroup - \(\SMod(S_\ell^1) = \{ [\phi] \in \Mod(S_\ell^p) : \phi \in - \SHomeo^+(S_\ell^p, \partial S_\ell^p) \}\). + 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 + \(\SMod(\Sigma_\ell^1) = \{ [\phi] \in \Mod(\Sigma_\ell^p) : \phi \in + \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p) \}\). \end{definition} Fix \(p = 1\) or \(2\). It follows from the universal property of quotients -that any \(\phi \in \SHomeo^+(S_\ell^p, \partial S_\ell^p)\) defines a -homeomorphism \(\bar \phi : S_{0, 2\ell+b}^1 \isoto S_{0, 2\ell+b}^1\). This +that any \(\phi \in \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)\) defines a +homeomorphism \(\bar \phi : \Sigma_{0, 2\ell+b}^1 \isoto \Sigma_{0, 2\ell+b}^1\). This yeilds a homomorphism of topological groups \begin{align*} - \SHomeo^+(S_\ell^p, \partial S_\ell^p) - & \to \Homeo^+(S_{0, 2\ell + b}^1, \partial S_{0, 2\ell + b}^1) \\ + \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p) + & \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 \(\psi \in \Homeo^+(S_{0, 2\ell + b}^1, -\partial S_{0, 2\ell + b}^1)\) lifts to \(S_\ell^p\). +which is surjective because any \(\psi \in \Homeo^+(\Sigma_{0, 2\ell + b}^1, +\partial \Sigma_{0, 2\ell + b}^1)\) lifts to \(\Sigma_\ell^p\). -It is also not hard to see \(\SHomeo^+(S_\ell^p, \partial S_\ell^p) \to -\Homeo^+(S_{0, 2\ell + b}^1, \partial S_{0, 2\ell + b}^1)\) is injective: the +It is also not hard to see \(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p) \to +\Homeo^+(\Sigma_{0, 2\ell + b}^1, \partial \Sigma_{0, 2\ell + b}^1)\) is injective: the only cadidates for elements of its kernel are \(1\) and \(\iota\), but -\(\iota\) is not an element of \(\SHomeo^+(S_\ell^p, \partial S_\ell^p)\) since -it does not fix \(\partial S_\ell^p\) point-wise. Now since we have a +\(\iota\) 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 continuous bijective homomorphism we find \[ \begin{split} - \pi_0(\SHomeo^+(S_\ell^p, \partial S_\ell^p)) - & \cong \pi_0(\Homeo^+(S_{0, 2\ell+b}^1, \partial S_{0, 2\ell+b}^1)) \\ - & = \mfrac{\Homeo^+(S_{0,2\ell+b}^1, \partial S_{0, 2\ell+b}^1)}{\simeq} \\ - & = \Mod(S_{0, 2\ell+b}^1) \\ + \pi_0(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)) + & \cong \pi_0(\Homeo^+(\Sigma_{0, 2\ell+b}^1, \partial \Sigma_{0, 2\ell+b}^1)) \\ + & = \mfrac{\Homeo^+(\Sigma_{0,2\ell+b}^1, \partial \Sigma_{0, 2\ell+b}^1)}{\simeq} \\ + & = \Mod(\Sigma_{0, 2\ell+b}^1) \\ & \cong B_{2\ell + b}. \end{split} \] -We would like to say \(\pi_0(\SHomeo^+(S_\ell^p, \partial S_\ell^p)) = -\SMod(S_\ell^p)\), but a priori the story looks a little more complicated: -\(\phi, \psi \in \SHomeo^+(S_\ell^p, \partial S_\ell^p)\) define the same class -in \(\SMod(S_\ell^p)\) if they are isotopic, but they may not lie in same -connected component of \(\SHomeo^+(S_\ell^p, \partial S_\ell^p)\) if they are +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 not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden \cite{birman-hilden} showed that this is never the case. \begin{theorem}[Birman-Hilden] - If \(\phi, \psi \in \SHomeo^+(S_\ell^p, \partial S_\ell^p)\) are isotopic + If \(\phi, \psi \in \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)\) are isotopic then \(\phi\) and \(\psi\) are isotopic through symmetric homeomorphisms. In particular, there is an isomorphism \begin{align*} - \SMod(S_\ell^p) & \isoto \Mod(S_{0, 2\ell + b}) \\ + \SMod(\Sigma_\ell^p) & \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 \(\SMod(S_\ell^p) \isoto \Mod(S_{0, 2g + b})\) + Birman-Hilden isomorphism \(\SMod(\Sigma_\ell^p) \isoto \Mod(\Sigma_{0, 2g + b})\) takes \(\tau_{\alpha_i}\) to the half twist \(h_{\bar{\alpha}_i} \in - \Mod(S_{0, 2g + b})\). This can be checked by looking at + \Mod(\Sigma_{0, 2g + b})\). This can be checked by looking at \(\iota\)-invaratiant anular neighborhoods of the curves \(\alpha_i\) -- \cite[Section~9.4]{farb-margalit}. \end{observation} \begin{fundamental-observation} - The Birman-Hilden isomorphism \(\SMod(S_\ell^1) \isoto \Mod(S_{0, - 2\ell+1}^1)\) takes the twists \(\tau_\delta \in \SMod(S_\ell^1)\) about the - boundary \(\delta = \partial S_\ell^1\) to \(\tau_{\bar\delta}^2 \in - \Mod(S_{0, 2\ell+1}^1)\). Similarly, \(\SMod(S_\ell^2) \isoto \Mod(S_{0, - 2\ell+2})\) takes \(\tau_{\delta_1} \tau_{\delta_2} \in \SMod(S_\ell^2)\) to + The Birman-Hilden isomorphism \(\SMod(\Sigma_\ell^1) \isoto \Mod(\Sigma_{0, + 2\ell+1}^1)\) takes the twists \(\tau_\delta \in \SMod(\Sigma_\ell^1)\) about the + boundary \(\delta = \partial \Sigma_\ell^1\) to \(\tau_{\bar\delta}^2 \in + \Mod(\Sigma_{0, 2\ell+1}^1)\). Similarly, \(\SMod(\Sigma_\ell^2) \isoto \Mod(\Sigma_{0, + 2\ell+2})\) takes \(\tau_{\delta_1} \tau_{\delta_2} \in \SMod(\Sigma_\ell^2)\) 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 \(\SMod(S_\ell^p) \subset \Mod(S_g)\). + \emph{\(k\)-chain relations} in \(\SMod(\Sigma_\ell^p) \subset \Mod(\Sigma_g)\). \[ \arraycolsep=1.4pt \begin{array}{rlcrll} @@ -314,9 +314,9 @@ not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden \] \end{fundamental-observation} -We may also exploit the quotient \(\mfrac{S_g}{\iota} \cong \mathbb{S}^2\) to -obtain other relations. Since \(\iota\) has \(2g + 2\) fixed points in \(S_g\), -we get branched double cover \(S_g \to S_{0, 2g+2}\). +We may also exploit the quotient \(\mfrac{\Sigma_g}{\iota} \cong \mathbb{S}^2\) to +obtain other relations. Since \(\iota\) has \(2g + 2\) fixed points in \(\Sigma_g\), +we get branched double cover \(\Sigma_g \to \Sigma_{0, 2g+2}\). \begin{theorem}[Birman-Hilden without boundary]\label{thm:boundaryless-birman-hilden} If \(g \ge 2\) then we have an exact sequence @@ -324,36 +324,36 @@ we get branched double cover \(S_g \to S_{0, 2g+2}\). \begin{tikzcd} 1 \rar & \langle [\iota] \rangle \rar - & C_{\Mod(S_g)}([\iota]) \rar - & \Mod(S_{0, 2g + 2}) \rar + & C_{\Mod(\Sigma_g)}([\iota]) \rar + & \Mod(\Sigma_{0, 2g + 2}) \rar & 1, \end{tikzcd} \end{center} - where \(C_{\Mod(S_g)}([\iota]) \subset \Mod(S_g)\) is the commutator subgroup - of \([\iota]\) and the right map takes \([\phi] \in C_{\Mod(S_g)}([\iota])\) - to \([\bar \phi] \in \Mod(S_{0, 2g + 2})\). + where \(C_{\Mod(\Sigma_g)}([\iota]) \subset \Mod(\Sigma_g)\) is the commutator subgroup + of \([\iota]\) and the right map takes \([\phi] \in C_{\Mod(\Sigma_g)}([\iota])\) + to \([\bar \phi] \in \Mod(\Sigma_{0, 2g + 2})\). \end{theorem} \begin{fundamental-observation} - Let \(\alpha_1, \ldots, \alpha_{2g}, \delta \subset S_g\) be and \(\delta - \subset S_g\) be as in Figure~\ref{fig:hyperellipitic-relations}. Then + Let \(\alpha_1, \ldots, \alpha_{2g}, \delta \subset \Sigma_g\) be and \(\delta + \subset \Sigma_g\) be as in Figure~\ref{fig:hyperellipitic-relations}. Then \begin{equation}\label{eq:hyperelliptic-eq} [\iota] = \tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta. \end{equation} - Indeed, \(C_{\Mod(S_g)}([\iota]) \to \Mod(S_{0, 2g+2})\) takes \(\tau_\delta + Indeed, \(C_{\Mod(\Sigma_g)}([\iota]) \to \Mod(\Sigma_{0, 2g+2})\) takes \(\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1}\) to the rotation from Figure~\ref{fig:hyperelliptic-relation-rotation}, while \(\tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta\) is taken to its inverse. By Theorem~\ref{thm:boundaryless-birman-hilden}, \(\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots - \tau_{\alpha_{2g}} \tau_\delta \in \ker (C_{\Mod(S_g)}([\iota]) \to - \Mod(S_{0, 2g+2})) = \langle [\iota] \rangle \cong \mathbb{Z}/2\). Given the + \tau_{\alpha_{2g}} \tau_\delta \in \ker (C_{\Mod(\Sigma_g)}([\iota]) \to + \Mod(\Sigma_{0, 2g+2})) = \langle [\iota] \rangle \cong \mathbb{Z}/2\). Given the fact \(\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta\) inverts the orientation of \(\alpha_1\), (\ref{eq:hyperelliptic-eq}) follows. In particular, we obtain - the so called \emph{hyperelliptic relations} of \(\Mod(S_g)\). + the so called \emph{hyperelliptic relations} of \(\Mod(\Sigma_g)\). \begin{align*} (\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta)^2 @@ -368,7 +368,7 @@ we get branched double cover \(S_g \to S_{0, 2g+2}\). \centering \includegraphics[width=.7\linewidth]{images/hyperelliptic-relation.eps} \vspace*{.5cm} - \captionof{figure}{The curves from the Humphreys generators of $\Mod(S_g)$ + \captionof{figure}{The curves from the Humphreys generators of $\Mod(\Sigma_g)$ and the curve $\delta$ from the hyperelliptic relations.} \label{fig:hyperellipitic-relations} \end{minipage} @@ -377,25 +377,25 @@ we get branched double cover \(S_g \to S_{0, 2g+2}\). \centering \includegraphics[width=.33\linewidth]{images/sphere-rotation.eps} \captionof{figure}{The clockwise rotation by $\sfrac{\pi}{g + 1}$ about an - axis centered around the punctures of $S_{0, 2g + 1}$.} + axis centered around the punctures of $\Sigma_{0, 2g + 1}$.} \label{fig:hyperelliptic-relation-rotation} \end{minipage} \medskip \section{Presentations of Mapping Class Groups} -There are numerous known presentations of \(\Mod(S_{g, r}^p)\), such as the +There are numerous known presentations of \(\Mod(\Sigma_{g, r}^p)\), such as the ones due to Birman-Hilden \cite{birman-hilden} and Gervais \cite{gervais}. -Wajnryb \cite{wajnryb} derived a presentation of \(\Mod(S_g)\) using the +Wajnryb \cite{wajnryb} derived a presentation of \(\Mod(\Sigma_g)\) using the relations discussed in Chapter~\ref{ch:dehn-twists} and Chapter~\ref{ch:relations}. This is a particularly satisfactory presentation, since all of its relations can be explained in terms of the geometry of curves -in \(S_g\). +in \(\Sigma_g\). \begin{theorem}[Wajnryb]\label{thm:wajnryb-presentation} Suppose \(g \ge 3\). 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 \(\Mod(S_g)\) + Figure~\ref{fig:humphreys-gens} and \(a_i = \tau_{\alpha_i} \in \Mod(\Sigma_g)\) + are the Humphreys generators, then there is a presentation of \(\Mod(\Sigma_g)\) with generators \(a_0, \ldots a_{2g}\) subject to the following relations. \begin{enumerate} \item The disjointness relations \([a_i, a_j] = 1\) for \(\alpha_i\) and @@ -434,7 +434,7 @@ in \(S_g\). \begin{note} The mapping classes \(b_0, \ldots, b_3, d\) in the statement of Theorem~\ref{thm:wajnryb-presentation} correspond to the Dehn twists about - the curves \(\beta_0, \ldots, \beta_3, \delta \subset S_g\) highlighted in + the curves \(\beta_0, \ldots, \beta_3, \delta \subset \Sigma_g\) highlighted in Figure~\ref{fig:wajnryb-presentation-curves}, so Wajnryb's presentation is not as intractible as it might look at first glance. \end{note} @@ -447,42 +447,42 @@ in \(S_g\). \end{figure} Different presentations can be used to compute the Abelianization of -\(\Mod(S_g)\) for \(g \le 2\). Indeed, if \(G = \langle g_1, \ldots, g_n : R +\(\Mod(\Sigma_g)\) for \(g \le 2\). Indeed, if \(G = \langle g_1, \ldots, g_n : R \rangle\) is a finitely-presented group, then \(G^\ab = \langle g_1, \ldots, g_n : R, [g_i, g_j] \text{ for all } i, j \rangle\). Using this approach, Farb-Margalit \cite[Section~5.1.3]{farb-margalit} show the Abelianization is given by \begin{center} \begin{tabular}{r|c|l} - \(g\) & \(S_g\) & \(\Mod(S_g)^\ab\) \\[1pt] + \(g\) & \(\Sigma_g\) & \(\Mod(\Sigma_g)^\ab\) \\[1pt] \hline & & \\[-10pt] \(0\) & \(\mathbb{S}^2\) & \(0\) \\ \(1\) & \(\mathbb{T}^2\) & \(\mathbb{Z}/12\) \\ - \(2\) & \(S_2\) & \(\mathbb{Z}/10\) \\ + \(2\) & \(\Sigma_2\) & \(\mathbb{Z}/10\) \\ \end{tabular} \end{center} for closed surfaces of small genus. In \cite{korkmaz-mccarthy} Korkmaz-McCarthy -showed that eventhough \(\Mod(S_2^p)\) is not perfect, its commutator subgroup -is. In addition, they also show \([\Mod(S_g^p), \Mod(S_g^p)]\) is normaly +showed that eventhough \(\Mod(\Sigma_2^p)\) is not perfect, its commutator subgroup +is. In addition, they also show \([\Mod(\Sigma_g^p), \Mod(\Sigma_g^p)]\) is normaly generated by a single mapping class. \begin{proposition}\label{thm:commutator-is-perfect} - The commutator subgroup \(\Mod(S_2^p)' = [\Mod(S_2^p), \Mod(S_2^p)]\) is - perfect -- i.e. \(\Mod(S_2^p)^{(2)} = [\Mod(S_2^p)', \Mod(S_2^p)']\) is the - whole of \(\Mod(S_2^p)'\). + 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 \(\Mod(\Sigma_2^p)'\). \end{proposition} \begin{proposition}\label{thm:commutator-normal-gen} - 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 + If \(g \ge 2\) and \(\alpha, \beta \subset \Sigma_g\) are simple closed crossing + only once, then \(\Mod(\Sigma_g)'\) 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 \(\Mod(S_g)' \subset N\). + \Mod(\Sigma_g)'\) then \(\Mod(\Sigma_g)' \subset N\). \end{proposition} -The different presentations of \(\Mod(S_g)\) may also be used to study its +The different presentations of \(\Mod(\Sigma_g)\) may also be used to study its representations. Indeed, in light of Theorem~\ref{thm:wajnryb-presentation}, a -representation \(\rho : \Mod(S_g) \to \GL_n(\mathbb{C})\) is nothing other than +representation \(\rho : \Mod(\Sigma_g) \to \GL_n(\mathbb{C})\) is nothing other than a choice of \(2g + 1\) matrices \(\rho(\tau_{\alpha_1}), \ldots, \rho(\tau_{\alpha_{2g}}) \in \GL_n(\mathbb{C})\) satisfying the relations \strong{(i)} to \strong{(v)} as above. In the next chapter, we will discuss how
diff --git a/sections/representations.tex b/sections/representations.tex @@ -2,7 +2,7 @@ Having built a solid understanding of the combinatorics of Dehn twists, we are now ready to attack the problem of classifying the representations of -\(\Mod(S_g)\) of sufficiently small dimension. As promised, our strategy is to +\(\Mod(\Sigma_g)\) of sufficiently small dimension. As promised, our strategy is to make us of the \emph{geometrically-motivated} relations derrived in Chapter~\ref{ch:relations}. @@ -14,8 +14,8 @@ goal of this chapter is providing a concise account of Korkmaz' results, starting by\dots \begin{theorem}[Korkmaz]\label{thm:low-dim-reps-are-trivial} - Let \(S_g^p\) be the surface of genus \(g \ge 1\) and \(b\) boundary - components and \(\rho : \Mod(S_g^p) \to \GL_n(\mathbb{C})\) be a linear + Let \(\Sigma_g^p\) be the surface of genus \(g \ge 1\) and \(b\) boundary + components and \(\rho : \Mod(\Sigma_g^p) \to \GL_n(\mathbb{C})\) 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 \(\rho : \Mod(S_2^p) \to \GL_n(\mathbb{C})\) with \(n \le 3\), the + Given \(\rho : \Mod(\Sigma_2^p) \to \GL_n(\mathbb{C})\) with \(n \le 3\), the image of \(\rho\) is Abelian. \end{proposition} \begin{proof}[Sketch of proof] - Given \(\alpha \subset S_2^p\), let \(L_\alpha = \rho(\tau_\alpha)\) and + Given \(\alpha \subset \Sigma_2^p\), 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 \(\alpha_1, \alpha_2, \beta_1, \beta_2, - \gamma, \eta_1, \ldots, \eta_{p-1} \subset S_2^p\) be the curves of the + \gamma, \eta_1, \ldots, \eta_{p-1} \subset \Sigma_2^p\) 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(S_2^p)) \subset \GL_1(\mathbb{C}) = + If \(n = 1\) then \(\rho(\Mod(\Sigma_2^p)) \subset \GL_1(\mathbb{C}) = \mathbb{C}^\times\) is Abelian. Now if \(n = 2\) or \(3\), by Propositon~\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 \(\Mod(S_2^p)' \subset \ker \rho\) -- i.e. - \(\rho(\Mod(S_2^p))\) is Abelian. Given the braid relation + \rho\) and thus \(\Mod(\Sigma_2^p)' \subset \ker \rho\) -- i.e. + \(\rho(\Mod(\Sigma_2^p))\) 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}, @@ -164,8 +164,8 @@ 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 \(\Mod(S_2^p)\)-invariant. Indeed, by change of - coordinates we can always find \(f, g, h_i \in \Mod(S_2^p)\) with + \(E_{\alpha_2 = \lambda}\) is \(\Mod(\Sigma_2^p)\)-invariant. Indeed, by change of + coordinates we can always find \(f, g, h_i \in \Mod(\Sigma_2^p)\) with \begin{align*} f \cdot [\alpha_2] & = [\alpha_1] & @@ -213,13 +213,13 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g E_{\eta_1 = \lambda} = \cdots = E_{\eta_{p-1} = \lambda}\) is invariant under the action of all Lickorish generators. - Hence \(\rho\) restricts to a subrepresentation \(\bar \rho : \Mod(S_2^p) \to + Hence \(\rho\) restricts to a subrepresentation \(\bar \rho : \Mod(\Sigma_2^p) \to \GL(E_{\alpha_2 = \lambda}) = \GL_2(\mathbb{C})\) -- recall \(E_{\alpha_2 = \lambda} = \mathbb{C} e_1 \oplus \mathbb{C} e_2\). By case (2), \(\bar - \rho(f) = 1\) for all \(f \in \Mod(S_2^p)'\), given that \(\bar - \rho(\Mod(S_2^p))\) is Abelian. Thus + \rho(f) = 1\) for all \(f \in \Mod(\Sigma_2^p)'\), given that \(\bar + \rho(\Mod(\Sigma_2^p))\) is Abelian. Thus \[ - \rho(\Mod(S_2^p)') \subset + \rho(\Mod(\Sigma_2^p)') \subset \begin{pmatrix} 1 & 0 & * \\ 0 & 1 & * \\ @@ -228,7 +228,7 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g \] lies inside the group of upper triangular matrices, a solvalbe subgroup of \(\GL_3(\mathbb{C})\). Now by Proposition~\ref{thm:commutator-is-perfect} we - get \(\rho(\Mod(S_2^p)') = 1\): any homomorphism from a perfect group to a + get \(\rho(\Mod(\Sigma_2^p)') = 1\): any homomorphism from a perfect group to a solvable group is trivial. Finally, if \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\) and @@ -261,46 +261,47 @@ 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(S_g^p) \to + Let \(g \ge 1\), \(p \ge 0\) and fix \(\rho : \Mod(\Sigma_g^p) \to \GL_n(\mathbb{C})\) 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 \(S_{g - + 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 Abelian image. Let \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g, \gamma_1, \ldots, - \gamma_{g - 1}, \eta_1, \ldots, \eta_{p-1} \subset S_g^p\) be the curves - from the Lickorish generators of \(\Mod(S_g^p)\), as in + \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 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 \(R \cong S_{g - - 1}^1\) be the closed subsurface highlighted in + \(L_\alpha\) associated to \(\lambda \in \mathbb{C}\). Let \(\Sigma \cong + \Sigma_{g - 1}^1\) be the closed subsurface highlighted in Figure~\ref{fig:korkmaz-proof-subsurface}. \begin{figure}[ht] \centering \includegraphics[width=.35\linewidth]{images/lickorish-gens-korkmaz-proof.eps} - \caption{The subsurface $R \subset S_g^p$.} + \caption{The subsurface $\Sigma \subset \Sigma_g^p$.} \label{fig:korkmaz-proof-subsurface} \end{figure} We claim that it suffices to find a \(m\)-dimensional - \(\Mod(R)\)-invariant\footnote{Here we view $\Mod(R)$ as a subgroup of - $\Mod(S_g^p)$ via the inclusion homomorphism $\Mod(R) \to \Mod(S_g^p)$ 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 representations + \(\Mod(\Sigma)\)-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 + 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 + representations \begin{align*} - \rho_1 : \Mod(R) & \to \GL(W) \cong \GL_m(\mathbb{C}) + \rho_1 : \Mod(\Sigma) & \to \GL(W) \cong \GL_m(\mathbb{C}) & - \rho_2 : \Mod(R) & \to \GL(\mfrac{\mathbb{C}^n}{W}) + \rho_2 : \Mod(\Sigma) & \to \GL(\mfrac{\mathbb{C}^n}{W}) \cong \GL_{n - m}(\mathbb{C}) \end{align*} - fall into the induction hypotesis -- i.e. \(\rho_i(\Mod(R))\) is - Abelian. In particular, \(\rho_i(\Mod(R)') = 1\) and we can find some + fall into the induction hypotesis -- i.e. \(\rho_i(\Mod(\Sigma))\) is + Abelian. In particular, \(\rho_i(\Mod(\Sigma)') = 1\) and we can find some basis for \(\mathbb{C}^n\) with respect to which \[ \rho(f) = @@ -311,26 +312,27 @@ representations. \end{array} \right) \] - for all \(f \in \Mod(R)'\) -- where \(1_k\) denotes the \(k \times k\) + for all \(f \in \Mod(\Sigma)'\) -- where \(1_k\) denotes the \(k \times k\) identity matrix. Since the group of upper triangular matrices is solvable, it follows from Proposition~\ref{thm:commutator-is-perfect} that \(\rho\) - annihilates all of \(\Mod(R)'\) and, in particular, \(\tau_{\alpha_1} + 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 \(\Mod(S_g^p)'\) is normally - generated by \(\tau_{\alpha_1} \tau_{\beta_1}^{-1}\), from which we conclude - \(\rho(\Mod(S_g^p)') = 1\), as desired. + Proposition~\ref{thm:commutator-normal-gen} that \(\Mod(\Sigma_g^p)'\) is + normally generated by \(\tau_{\alpha_1} \tau_{\beta_1}^{-1}\), from which we + conclude \(\rho(\Mod(\Sigma_g^p)') = 1\), as desired. As before, we exhaustively analyse all possible Jordan forms of \(L_{\alpha_g}\). First, consider the case where we can find eigenvalues \(\lambda_1, \ldots, \lambda_k\) of \(L_{\alpha_g}\) such that the sum \(W = \bigoplus_i E_{\alpha_g = \lambda_i}\) of the corresponding eigenspaces has dimension \(m\) with \(2 \le m \le n - 2\). In this case, it suffices to - observe that since \(\alpha_g\) lies outside of \(R\), each \(E_{\alpha_g = - \lambda_i}\) is \(\Mod(R)\)-invariant: the Lickorish generators - \(\tau_{\alpha_1}, \ldots, \tau_{\alpha_{g - 1}}, \tau_{\beta_1}, \ldots, - \tau_{\beta_{g - 1}}\), \(\tau_{\gamma_1}, \ldots, \tau_{\gamma_{g - 2}}\) of - \(R \cong S_{g - 1}^1\) all commute with \(\tau_{\alpha_g}\) and thus preserve - the eigenspaces of its action on \(\mathbb{C}^n\). + observe that since \(\alpha_g\) lies outside of \(\Sigma\), each + \(E_{\alpha_g = \lambda_i}\) is \(\Mod(\Sigma)\)-invariant: the Lickorish + generators \(\tau_{\alpha_1}, \ldots, \tau_{\alpha_{g - 1}}, \tau_{\beta_1}, + \ldots, \tau_{\beta_{g - 1}}\), \(\tau_{\gamma_1}, \ldots, \tau_{\gamma_{g - + 2}}\) of \(\Sigma \cong \Sigma_{g - 1}^1\) all commute with + \(\tau_{\alpha_g}\) and thus preserve the eigenspaces of its action on + \(\mathbb{C}^n\). If no sum of the form \(\bigoplus_i E_{\alpha_g = \lambda_i}\) has dimension lying between \(2\) and \(n - 2\) there must be at most \(2\) distinct @@ -381,29 +383,29 @@ 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 \(\Mod(S_g^p)\) act on + \lambda\), so all Lickorish generators of \(\Mod(\Sigma_g^p)\) act on \(\mathbb{C}^n\) as scalar multiplication by \(\lambda\) as well. Hence - \(\rho(\Mod(S_g^p)) = \langle \lambda \rangle\) is Abelian. + \(\rho(\Mod(\Sigma_g^p)) = \langle \lambda \rangle\) is Abelian. \item[\bfseries\color{highlight}(2)] In this case, \(W = \ker (L_{\alpha_g} - \lambda)^2\) is a - \(2\)-dimensional \(\Mod(R)\)-invariant subspace. + \(2\)-dimensional \(\Mod(\Sigma)\)-invariant subspace. \end{enumerate} For cases (3) and (4), we consider two situations: \(E_{\alpha_g = \lambda} \ne E_{\beta_g = \lambda}\) or \(E_{\alpha_g = \lambda} = E_{\beta_g = \lambda}\). If \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\), then \(W = E_{\alpha_g = \lambda} \cap E_{\beta_g = \lambda}\) is a \((n - - 2)\)-dimensional \(\Mod(R)\)-invariant subspace: since \(L_{\alpha_g}\) and - \(L_{\beta_g}\) are conjugate and \(\beta_g\) lies outside of \(R\), both - \(E_{\alpha_g = \lambda}\) and \(E_{\beta_g = \lambda}\) are - \(\Mod(R)\)-invariant \((m - 1)\)-dimensional subspaces. + 2)\)-dimensional \(\Mod(\Sigma)\)-invariant subspace: since \(L_{\alpha_g}\) + and \(L_{\beta_g}\) are conjugate and \(\beta_g\) lies outside of \(\Sigma\), + both \(E_{\alpha_g = \lambda}\) and \(E_{\beta_g = \lambda}\) are + \(\Mod(\Sigma)\)-invariant \((m - 1)\)-dimensional subspaces. Finally, we consider the case where \(E_{\alpha_g = \lambda} = E_{\beta_g = \lambda}\). 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 \(f_i, g_i, h_i \in - \Mod(S_g^p)\) with + \Mod(\Sigma_g^p)\) with \begin{align*} f_i \tau_{\alpha_g} f_i^{-1} & = \tau_{\alpha_i} & @@ -437,34 +439,34 @@ representations. \end{pmatrix}. \] Since the group of upper triangular matrices is solvable and - \(\Mod(S_g^p)\) is perfect, it follows that \(\rho(\Mod(S_g^p))\) is - trivial. This concludes the proof \(\rho(\Mod(S_g^p))\) is Abelian. - - To see that \(\rho(\Mod(S_g^p)) = 1\) for \(g \ge 3\) we note that, since - \(\rho(\Mod(S_g^p))\) is Abelian, \(\rho\) factors though the Abelinization - map \(\Mod(S_g^p) \to \Mod(S_g^p)^\ab = \mfrac{\Mod(S_g^p)}{[\Mod(S_g^p), - \Mod(S_g^p)]}\). Now recall from Proposition~\ref{thm:trivial-abelianization} - that \(\Mod(S_g^p)^\ab = 0\) for \(g \ge 3\). In other words, \(\rho\) + \(\Mod(\Sigma_g^p)\) 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 Abelinization + 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, \(\rho\) factors though the homomorphism \(1 \to \GL_n(\mathbb{C})\). We are done. \end{proof} Having established the triviality of the low-dimensional representations \(\rho -: \Mod(S_g^p) \to \GL_n(\mathbb{C})\), all that remains for us is to understand -the \(2g\)-dimensional reprensentations of \(\Mod(S_g^p)\). We certainly know a +: \Mod(\Sigma_g^p) \to \GL_n(\mathbb{C})\), all that remains for us is to understand +the \(2g\)-dimensional reprensentations of \(\Mod(\Sigma_g^p)\). We certainly know a nontrivial example of such, namely the symplectic representation \(\psi : -\Mod(S_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) from +\Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) from Example~\ref{ex:symplectic-rep}. Surprinsgly, this turns out to be \emph{essentially} the only example of a nontrivial \(2g\)-dimensional representation in the compact case. More precisely, \begin{theorem}[Korkmaz]\label{thm:reps-of-dim-2g-are-symplectic} - Let \(g \ge 3\) and \(\rho : \Mod(S_g^p) \to \GL_{2g}(\mathbb{C})\). Then + Let \(g \ge 3\) and \(\rho : \Mod(\Sigma_g^p) \to \GL_{2g}(\mathbb{C})\). Then \(\rho\) is either trivial or conjugate to the symplectic - representation\footnote{Here the map $\Mod(S_g^p) \to + representation\footnote{Here the map $\Mod(\Sigma_g^p) \to \operatorname{Sp}_{2g}(\mathbb{Z})$ is given by the composition of the - inclusion morphism $\Mod(S_g^p) \to \Mod(S_g)$ with the usual symplect - representation $\psi : \Mod(S_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})$.} - \(\Mod(S_g^p) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(S_g^p)\). + inclusion morphism $\Mod(\Sigma_g^p) \to \Mod(\Sigma_g)$ with the usual symplect + representation $\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})$.} + \(\Mod(\Sigma_g^p) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(\Sigma_g^p)\). \end{theorem} Unfortunately, the limited scope of these master thesis does not allow us to @@ -514,10 +516,10 @@ to as \emph{the 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 Lickrish generators \(\tau_{\alpha_1}, \ldots, -\tau_{\alpha_g}, \tau_{\beta_1}, \ldots, \tau_{\beta_g} \in \Mod(S_g^p)\) on -\(H_1(S_g, \mathbb{C}) \cong \mathbb{C}^{2g}\) -- represented in the standard +\tau_{\alpha_g}, \tau_{\beta_1}, \ldots, \tau_{\beta_g} \in \Mod(\Sigma_g^p)\) 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 -\(H_1(S_g, \mathbb{C})\). +\(H_1(\Sigma_g, \mathbb{C})\). \begin{align*} (\tau_{\alpha_i})_* & = \left( @@ -537,28 +539,28 @@ basis \([\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]\) for \right) \end{align*} -Hence by embeding \(B_3^g\) in \(\Mod(S_g^p)\) via +Hence by embeding \(B_3^g\) in \(\Mod(\Sigma_g^p)\) via \begin{align*} - B_3^g & \to \Mod(S_g^p) \\ + B_3^g & \to \Mod(\Sigma_g^p) \\ a_i & \mapsto \tau_{\alpha_i} \\ b_i & \mapsto \tau_{\beta_i} \end{align*} -we can see that any \(\rho : \Mod(S_g^p) \to \GL_{2g}(\mathbb{C})\) in a +we can see that any \(\rho : \Mod(\Sigma_g^p) \to \GL_{2g}(\mathbb{C})\) in a certain class of representation satisfying some technical conditions must be -conjugate to the symplectic representation \(\Mod(S_g^p) \to +conjugate to the symplectic representation \(\Mod(\Sigma_g^p) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) when restricted to \(B_3^g\). Korkmaz then goes on to show that such technical conditions are met for any -nontrivial \(\rho : \Mod(S_g^p) \to \GL_{2g}(\mathbb{C})\). Furthermore, +nontrivial \(\rho : \Mod(\Sigma_g^p) \to \GL_{2g}(\mathbb{C})\). 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 \(\Mod(S_g^p)\) on -\(H_1(S_g, \mathbb{C})\), concluding the classification of \(2g\)-dimensional +\rho(\tau_{\eta_{p-1}})\) also agrees with the action of \(\Mod(\Sigma_g^p)\) 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 \(\Mod(S_g^p)\) for \(g \ge 7\) in terms of certain twisted +representations of \(\Mod(\Sigma_g^p)\) for \(g \ge 7\) in terms of certain twisted \(1\)-cohomology groups.
diff --git a/sections/twists.tex b/sections/twists.tex @@ -2,11 +2,11 @@ With the goal of studying the linear representations of mapping class groups in mind, in this chapter we start investigating the group structure of -\(\Mod(S)\). We begin by computing some fundamental examples. We then explore +\(\Mod(\Sigma)\). We begin by computing some fundamental examples. We then explore how we can use these examples to understand the structure of the mapping class groups of other surfaces. Namely, we compute \(\Mod(\mathbb{S}^1 \times [0, 1]) \cong \mathbb{Z}\), and discuss how its generators gives rise to a conveniant -generating set for \(\Mod(S)\), known as the set of \emph{Dehn twists}. +generating set for \(\Mod(\Sigma)\), known as the set of \emph{Dehn twists}. The idea here is to reproduce the proof of injectivity in Example~\ref{ex:torus-mcg}: by cutting across curves and arcs, we can always @@ -16,18 +16,18 @@ Example~\ref{ex:mdg-once-punctured-disk} then imply the triviality of mapping classes fixing such arcs and curves. Formally, this translates to\dots \begin{proposition}[Alexander method]\label{thm:alexander-method} - Let \(\alpha_1, \ldots, \alpha_n \subset S\) be essencial simple closed + Let \(\alpha_1, \ldots, \alpha_n \subset \Sigma\) be essencial simple closed curves or proper arcs satisfying the following conditions. \begin{enumerate} \item \([\alpha_i] \ne [\alpha_j]\) for \(i \ne j\). \item Each pair \((\alpha_i, \alpha_j)\) crosses at most once. \item Given distinct \(i, j, k\), at least one of \(\alpha_i \cap \alpha_j, \alpha_i \cap \alpha_k, \alpha_j \cap \alpha_k\) is empty. - \item The surface obtained by cutting \(S\) across the \(\alpha_i\) is a + \item The surface obtained by cutting \(\Sigma\) across the \(\alpha_i\) is a disjoint union of disks and once-punctured disks. \end{enumerate} - Suppose \(f \in \Mod(S)\) is such that \(f \cdot \vec{[\alpha_i]} = - \vec{[\alpha_i]}\) for all \(i\). Then \(f = 1 \in \Mod(S)\). + Suppose \(f \in \Mod(\Sigma)\) is such that \(f \cdot \vec{[\alpha_i]} = + \vec{[\alpha_i]}\) for all \(i\). Then \(f = 1 \in \Mod(\Sigma)\). \end{proposition} See \cite[Proposition~2.8]{farb-margalit} for a proof of @@ -77,20 +77,20 @@ applications of the Alexander method. \label{fig:hald-twist-disk} \end{minipage} -Let \(S\) be an orientable surface, possibly with punctures and non-empty -boundary. Given some simple closed curve \(\alpha \subset S\), we may envision -doing something similar to Example~\ref{ex:mcg-annulus} in \(S\) by looking at +Let \(\Sigma\) be an orientable surface, possibly with punctures and non-empty +boundary. Given some simple closed curve \(\alpha \subset \Sigma\), we may envision +doing something similar to Example~\ref{ex:mcg-annulus} in \(\Sigma\) by looking at anular neighborhoods of \(\alpha\). These are the precisely the \emph{Dehn twists}, illustrated in Figure~\ref{fig:dehn-twist-bitorus} in the case of the -bitorus \(S_2\). +bitorus \(\Sigma_2\). \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 + Given a simple closed curve \(\alpha \subset \Sigma\), fix a closed annular + neighborhood \(A \subset \Sigma\) of \(\alpha\) -- i.e. \(A \cong \mathbb{S}^1 \times [0, 1]\). Let \(f \in \Mod(A) \cong \Mod(\mathbb{S}^1 \times [0, 1]) \cong \mathbb{Z}\) be the generator from Example~\ref{ex:mcg-annulus}. The - \emph{Dehn twist \(\tau_\alpha \in \Mod(S)\) about \(\alpha\)} is defined as - the image of \(f\) under the inclusion homomorphism \(\Mod(A) \to \Mod(S)\). + \emph{Dehn twist \(\tau_\alpha \in \Mod(\Sigma)\) about \(\alpha\)} is defined as + the image of \(f\) under the inclusion homomorphism \(\Mod(A) \to \Mod(\Sigma)\). \end{definition} \begin{figure}[ht] @@ -105,16 +105,16 @@ 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 \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\})\) gives rise the so called \emph{half-twists}. These are examples of mapping -classes that permute the punctures of \(S\). +classes that permute the punctures of \(\Sigma\). \begin{definition} - 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 + Given an arc \(\alpha \subset \Sigma\) joining two punctures in the interior of + \(\Sigma\), fix a closed neighborhood \(D \subset \Sigma\) of \(\alpha\) with \(D \cong \mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}\). Let \(f \in \Mod(\mathbb{S}^1 \times [0, 1]) \cong \Mod(D) \cong \mathbb{Z}\) be the generator from Example~\ref{ex:mcg-twice-punctured-disk}. The - \emph{half-twist \(h_\alpha \in \Mod(S)\) about \(\alpha\)} is defined as the - image of \(f\) under the inclusion homomorphism \(\Mod(D) \to \Mod(S)\). + \emph{half-twist \(h_\alpha \in \Mod(\Sigma)\) about \(\alpha\)} is defined as the + image of \(f\) under the inclusion homomorphism \(\Mod(D) \to \Mod(\Sigma)\). \end{definition} It is interesting to study how the geometry of two curves affects the @@ -132,15 +132,15 @@ 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) + Let \(\alpha \subset \Sigma\) be a simple closed curve and \(T_\alpha\) be a + representative of \(\tau_\alpha \in \Mod(\Sigma)\). Then \(\# (T_\alpha^k(\beta) \cap \beta) = |k| \cdot \#(\alpha \cap \beta)^2\) for any \(k \in \mathbb{Z}\). In particular, if \(\alpha\) is nontrivial then \(\tau_\alpha\) has infinite order. \end{proposition} \begin{observation} - Given \(\alpha, \beta \subset S\), \(\tau_\alpha = \tau_\beta \iff [\alpha] = + Given \(\alpha, \beta \subset \Sigma\), \(\tau_\alpha = \tau_\beta \iff [\alpha] = [\beta]\). 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} @@ -152,25 +152,25 @@ Many other relations between Dehn twists can derrived be in a geometric fashion too. \begin{observation}\label{ex:conjugate-twists} - Given \(f = [\phi] \in \Mod(S)\), \(\tau_{\phi(\alpha)} = f \tau_\alpha + Given \(f = [\phi] \in \Mod(\Sigma)\), \(\tau_{\phi(\alpha)} = f \tau_\alpha f^{-1}\). \end{observation} \begin{observation} - Given \(f \in \Mod(S)\), \([f, \tau_\alpha] = 1 \iff f \cdot [\alpha] = + Given \(f \in \Mod(\Sigma)\), \([f, \tau_\alpha] = 1 \iff f \cdot [\alpha] = [\alpha]\). In particular, \([\tau_\alpha, \tau_\beta] = 1\) for \(\alpha\) and \(\beta\) disjoint. \end{observation} \begin{observation} - 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 \(f \cdot [\alpha] = [\beta]\) + If \(\alpha, \beta \subset \Sigma\) are both nonseparing then \(\tau_\alpha, + \tau_\beta \in \Mod(\Sigma)\) are conjugate. Indeed, by the change of coordinates + principle we can find \(f \in \Mod(\Sigma)\) with \(f \cdot [\alpha] = [\beta]\) and then apply Observation~\ref{ex:conjugate-twists}. \end{observation} \begin{fundamental-observation}\label{ex:braid-relation} - Given \(\alpha, \beta \subset S\) with \(\#(\alpha \cap \beta) = 1\), it is + Given \(\alpha, \beta \subset \Sigma\) with \(\#(\alpha \cap \beta) = 1\), it is not hard to check that \(\tau_\beta \tau_\alpha \cdot [\beta] = [\alpha]\). From Observation~\ref{ex:conjugate-twists} we then get \((\tau_\alpha \tau_\beta) \tau_\alpha (\tau_\alpha \tau_\beta)^{-1} = \tau_\beta\), from which follows @@ -183,9 +183,9 @@ too. A perhaps less obvious fact about Dehn twists is\dots \begin{theorem}\label{thm:mcg-is-fg} - Let \(S_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with \(r\) + Let \(\Sigma_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with \(r\) punctures and \(b\) boundary components. Then the pure mapping class group - \(\PMod(S_{g, r}^p)\) is generated by finitely many Dehn twists about + \(\PMod(\Sigma_{g, r}^p)\) is generated by finitely many Dehn twists about nonseparating curves or boundary components. \end{theorem} @@ -198,72 +198,72 @@ 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 \(S_{g, r}^p\) and -that of \(S_{g, r+1}^p = S_{g, r}^p \setminus \{ x \}\) for some \(x\) in the -interior \((S_{g, r}^p)\degree\) of \(S_{g, r}^p\). Indeed, this will later +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 allow us to establish the induction on the number of punctures \(r\). -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 +Given an orientable surface \(\Sigma\) and \(x_1, \ldots, x_n \in \Sigma\degree\), +denote by \(\Mod(\Sigma \setminus \{x_1, \ldots, x_n\})_{\{x_1, \ldots, +x_n\}} \subset \Mod(\Sigma \setminus \{x_1, \ldots, x_n\})\) 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, +surjective homomorphism \(\operatorname{forget} : \Mod(\Sigma \setminus \{x_1, +\ldots, x_n\})_{\{x_1, \ldots, x_n\}} \to \Mod(\Sigma)\) which ``\emph{forgets} the +additional punctures \(x_1, \ldots, x_n\) of \(\Sigma \setminus \{x_1, \ldots, x_n\}\),'' 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 +To answer this question, we consider the configuration space \(C(\Sigma, n) = +\mfrac{C^{\operatorname{ord}}(\Sigma, n)}{\mathfrak{S}_n}\) of \(n\) (unordered) +points in the interior of \(\Sigma\) -- where \(C^{\operatorname{ord}}(\Sigma, n) = \{ +(x_1, \ldots, x_n) \in (\Sigma\degree)^n : x_i \ne x_j \ \text{for}\ i \ne j \}\). +Denote \(\Homeo^+(\Sigma, \partial \Sigma)_{x_1, \ldots, x_n} = \{\phi \in \Homeo^+(\Sigma, +\partial \Sigma) : \phi(x_i) = x_i \}\). From the fibration\footnote{See \cite[Chapter~4]{hatcher} for a reference.} \[ \arraycolsep=1.4pt \begin{array}{ccrcl} - \Homeo^+(S, \partial S)_{x_1, \ldots, x_n} - & \hookrightarrow & \Homeo^+(S, \partial S) - & \relbar\joinrel\twoheadrightarrow & C(S, n) \\ + \Homeo^+(\Sigma, \partial \Sigma)_{x_1, \ldots, x_n} + & \hookrightarrow & \Homeo^+(\Sigma, \partial \Sigma) + & \relbar\joinrel\twoheadrightarrow & C(\Sigma, n) \\ & & \phi & \mapsto & [\phi(x_1), \ldots, \phi(x_n)] \end{array} \] and its long exact sequence in homotopy we then get\dots \begin{theorem}[Birman exact sequence]\label{thm:birman-exact-seq} - Suppose \(\pi_1(\Homeo^+(S, \partial S), 1) = 1\). Then there is an exact + Suppose \(\pi_1(\Homeo^+(\Sigma, \partial \Sigma), 1) = 1\). Then there is an exact sequence \begin{center} \begin{tikzcd}[cramped] 1 \rar - & \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\}} + & \pi_1(C(\Sigma, n), [x_1, \ldots, x_n]) \rar{\operatorname{push}} + & \Mod(\Sigma \setminus \{x_1, \ldots, x_n\})_{\{x_1, \ldots, x_n\}} \rar{\operatorname{forget}} - & \Mod(S) \rar + & \Mod(\Sigma) \rar & 1. \end{tikzcd} \end{center} \end{theorem} \begin{note} - Notice that \(C(S, 1) = S\degree \simeq S\). Hence for \(n = 1\) + Notice that \(C(\Sigma, 1) = \Sigma\degree \simeq S\). Hence for \(n = 1\) Theorem~\ref{thm:birman-exact-seq} gives us a sequence \begin{center} \begin{tikzcd} 1 \rar - & \pi_1(S, x) \rar{\operatorname{push}} - & \Mod(S \setminus \{x\}, x) \rar{\operatorname{forget}} - & \Mod(S) \rar + & \pi_1(\Sigma, x) \rar{\operatorname{push}} + & \Mod(\Sigma \setminus \{x\}, x) \rar{\operatorname{forget}} + & \Mod(\Sigma) \rar & 1. \end{tikzcd} \end{center} \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 +We may regard a simple loop \(\alpha \subset C(\Sigma, n)\) based at \([x_1, \ldots, +x_n]\) as \(n\) disjoint curves \(\alpha_1, \ldots, \alpha_n \subset \Sigma\) 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 +\mathfrak{S}_n\). The element \(\operatorname{push}([\alpha]) \in \Mod(\Sigma)\) 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 @@ -272,14 +272,14 @@ show\dots \begin{fundamental-observation}\label{ex:push-simple-loop} Using the notation of Figure~\ref{fig:push-map}, \(\operatorname{push}([\alpha]) = \tau_{\delta_1} \tau_{\delta_2}^{-1} \in - \Mod(S)\). + \Mod(\Sigma)\). \end{fundamental-observation} \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 + \caption{The inclusion $\operatorname{push} : \pi_1(\Sigma, x) \to \Mod(\Sigma)$ maps + a simple loop $\alpha \subset \Sigma$ 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.} @@ -289,7 +289,7 @@ show\dots \section{The Modified Graph of Curves} Having established Theorem~\ref{thm:birman-exact-seq}, we now need to adress -the induction step in the genus \(g\) of \(S_{g, r}^p\). Our strategy is to +the induction step in the genus \(g\) of \(\Sigma_{g, r}^p\). Our strategy is to apply the following lemma from geomtric group theory. \begin{lemma}\label{thm:ggt-lemma} @@ -301,15 +301,15 @@ apply the following lemma from geomtric group theory. \(G\) is generated by \(G_v\) and \(g\). \end{lemma} -We are interested, of course, in the group \(G = \PMod(S_{g, r}^p)\). As for +We are interested, of course, in the group \(G = \PMod(\Sigma_{g, r}^p)\). As for the graph \(\Gamma\), we consider\dots \begin{definition} - The \emph{modified graph 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 + The \emph{modified graph of nonseparating curves \(\hat{\mathcal{N}}(\Sigma)\) + of a surface \(\Sigma\)} is the graph whose vertices are (un-oriented) isotopy + classes of nonseparating simple closed curves in \(\Sigma\) and \[ - \text{\([\alpha]\) --- \([\beta]\) in \(\hat{\mathcal{N}}(S)\)} + \text{\([\alpha]\) --- \([\beta]\) in \(\hat{\mathcal{N}}(\Sigma)\)} \iff \#(\alpha \cap \beta) = 1, \] where \(\#(\alpha \cap \beta)\) is the geometric intersection number of @@ -317,43 +317,43 @@ the graph \(\Gamma\), we consider\dots \end{definition} It is clear from Lemma~\ref{thm:change-of-coordinates} that the actions of -\(\Mod(S_{g, r}^p)\) on \(V(\hat{\mathcal{N}}(S_{g, r}^p))\) and \(\{([\alpha], -[\beta]) \in V(\hat{\mathcal{N}}(S_{g, r}^p))^2 : \#(\alpha \cap \beta) = 1 -\}\) are both transitive. But why should \(\hat{\mathcal{N}}(S_{g, r}^p)\) be +\(\Mod(\Sigma_{g, r}^p)\) on \(V(\hat{\mathcal{N}}(\Sigma_{g, r}^p))\) and \(\{([\alpha], +[\beta]) \in V(\hat{\mathcal{N}}(\Sigma_{g, r}^p))^2 : \#(\alpha \cap \beta) = 1 +\}\) are both transitive. But why should \(\hat{\mathcal{N}}(\Sigma_{g, r}^p)\) be connected? Historically, the modified graph 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{graph of curves \(\mathcal{C}(S)\) of - \(S\)} is the graph whose vertices are (un-oriented) isotopy classes of - essential simple closed curves in \(S\) and + Given a surface \(\Sigma\), the \emph{graph of curves \(\mathcal{C}(\Sigma)\) of + \(\Sigma\)} is the graph whose vertices are (un-oriented) isotopy classes of + essential simple closed curves in \(\Sigma\) and \[ - \text{\([\alpha]\) --- \([\beta]\) in \(\mathcal{C}(S)\)} + \text{\([\alpha]\) --- \([\beta]\) in \(\mathcal{C}(\Sigma)\)} \iff \#(\alpha \cap \beta) = 0. \] - The \emph{graph of nonseparating curves \(\mathcal{N}(S)\)} is the subgraph - of \(\mathcal{C}(S)\) whose vertices consist of nonseparating curves. + The \emph{graph of nonseparating curves \(\mathcal{N}(\Sigma)\)} is the subgraph + of \(\mathcal{C}(\Sigma)\) 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. +cases, \(\mathcal{C}(\Sigma_{g, r})\) is connected. \begin{theorem}[Lickorish] - If \(S_{g, r}\) is not one \(S_0 = \mathbb{S}^2, S_{0, 1}, \ldots, S_{0, 4}, - S_1 = \mathbb{T}^2\) and \(S_{1, 1}\) then \(\mathcal{C}(S_{g, r})\) is + If \(\Sigma_{g, r}\) is not one \(\Sigma_0 = \mathbb{S}^2, \Sigma_{0, 1}, \ldots, \Sigma_{0, 4}, + \Sigma_1 = \mathbb{T}^2\) and \(\Sigma_{1, 1}\) then \(\mathcal{C}(\Sigma_{g, r})\) is connected. \end{theorem} -In other words, given \([\alpha], [\beta] \in \mathcal{C}(S_{g, r})\), we can +In other words, given \([\alpha], [\beta] \in \mathcal{C}(\Sigma_{g, r})\), 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 +[\beta]\) in \(\mathcal{C}(\Sigma_{g, r})\). Now if \(\alpha\) and \(\beta\) are nonseparating, by inductively adjusting this path we then get\dots \begin{corollary}\label{thm:mofied-graph-is-connected} - If \(g \ge 2\) then both \(\mathcal{N}(S_{g, r})\) and - \(\hat{\mathcal{N}}(S_{g, r})\) are connected. + If \(g \ge 2\) then both \(\mathcal{N}(\Sigma_{g, r})\) and + \(\hat{\mathcal{N}}(\Sigma_{g, r})\) are connected. \end{corollary} See \cite[Section~4.1]{farb-margalit} for a proof of @@ -361,36 +361,36 @@ 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 \(S_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with \(r\) + Let \(\Sigma_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with \(r\) punctures and \(b\) boundary components. We want to establish that - \(\PMod(S_{g, r}^p)\) is genetery by a finite number of Dehn twists about + \(\PMod(\Sigma_{g, r}^p)\) is genetery by a finite number of Dehn twists about nonseparating simple closed curves or boundary components. - First, observe that if \(p \ge 1\) and \(\partial S_{g, r}^p = \delta_1 \cup + First, observe that if \(p \ge 1\) and \(\partial \Sigma_{g, r}^p = \delta_1 \cup \cdots \cup \delta_p\) then, by recursively applying the capping exact sequence \begin{center} \begin{tikzcd} 1 \rar & \langle \tau_{\delta_1} \rangle \rar & - \PMod(S_{g, r}^p) \rar{\operatorname{cap}} & - \PMod(S_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\})) \rar & + \PMod(\Sigma_{g, r}^p) \rar{\operatorname{cap}} & + \PMod(\Sigma_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\})) \rar & 1 \end{tikzcd} \end{center} - from Example~\ref{ex:capping-seq}, it suffices to show that \(S_{g, n}\) + from Example~\ref{ex:capping-seq}, it suffices to show that \(\Sigma_{g, n}\) is finitely generated by twists about nonseparating simple closed curves. - Indeed, if \(\PMod(S_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\) + Indeed, if \(\PMod(\Sigma_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\) is finitely generated by twists about nonseparing curves or boundary - components, then we may lift the generators of \(\PMod(S_{g, r}^p + components, then we may lift the generators of \(\PMod(\Sigma_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\) to Dehn twists about the - corresponding curves in \(S_{g, r}^p\) and add \(\tau_{\delta_1}\) to the + corresponding curves in \(\Sigma_{g, r}^p\) and add \(\tau_{\delta_1}\) to the generating set. - It thus suffices to consider the boudaryless case \(S_{g, r}\). As promised, + It thus suffices to consider the boudaryless case \(\Sigma_{g, r}\). 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}^2) \cong \Mod(S_{1, 1}) \cong + \(\Mod(\mathbb{T}^2) \cong \Mod(\Sigma_{1, 1}) \cong \operatorname{SL}_2(\mathbb{Z})\) are generated by the Dehn twists about the curves \(\alpha\) and \(\beta\) from Figure~\ref{fig:torus-mcg-generators}, each corresponding to one of the @@ -412,68 +412,68 @@ Theorem~\ref{thm:mcg-is-fg}. \centering \includegraphics[width=.55\linewidth]{images/torus-mcg-generators.eps} \caption{The curves $\alpha$ and $\beta$ whose Dehn twists generate - $\Mod(\mathbb{T}^2)$ and $\Mod(S_{1, 1})$.} + $\Mod(\mathbb{T}^2)$ and $\Mod(\Sigma_{1, 1})$.} \label{fig:torus-mcg-generators} \end{figure} - Now suppose \(\PMod(S_{g, r})\) is finitely-generated by twists about + Now suppose \(\PMod(\Sigma_{g, r})\) 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, + case, \(\chi(\Sigma_{g, r}) = 2 - 2g - r < 0\) and thus \(\pi_1(\Homeo^+(\Sigma_{g, r})) = 1\) -- 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, r}, x) \rar{\operatorname{push}} - & \PMod(S_{g, r + 1}) \rar{\operatorname{forget}} - & \PMod(S_{g, r}) \rar + & \pi_1(\Sigma_{g, r}, x) \rar{\operatorname{push}} + & \PMod(\Sigma_{g, r + 1}) \rar{\operatorname{forget}} + & \PMod(\Sigma_{g, r}) \rar & 1, \end{tikzcd} \end{center} - 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. + where \(\Sigma_{g, r + 1} = \Sigma_{g, r} \setminus \{x\}\). Since \(g \ge 1\), + \(\pi_1(\Sigma_{g, r}, x)\) is generated by finitely many nonseparating loops. We have seen in Observation~\ref{ex:push-simple-loop} that \(\operatorname{push} - : \pi_1(S_{g, r}, x) \to \Mod(S_{g, r+1}, x)\) takes nonseparation simple + : \pi_1(\Sigma_{g, r}, x) \to \Mod(\Sigma_{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 + we may once again lift the generators of \(\PMod(\Sigma_{g, r})\) to Dehn twists + about nonseparating simple curves in \(\Sigma_{g, r + 1}\). This goes to show that + \(\PMod(\Sigma_{g, r + 1})\) 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 + each \(r \ge 0\), \(\PMod(\Sigma_{g, r})\) 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 + for \(\Mod(\Sigma_{g + 1})\). To that end, we consider the action \(\Mod(\Sigma_{g + + 1}) \leftaction \hat{\mathcal{N}}(\Sigma_{g + 1})\). Since \(g + 1 \ge 2\), + \(\hat{\mathcal{N}}(\Sigma_{g + 1})\) is connected and the conditions of Lemma~\ref{thm:ggt-lemma} are met. Now recall from Observation~\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 \(\Mod(S_{g + 1})_{[\alpha]} = \{ f \in \Mod(S_{g + 1}) : f + \subset \Sigma_{g + 1}\) crossing once, \(\tau_\beta \tau_\alpha \cdot [\beta] = + [\alpha]\). Hence by Lemma~\ref{thm:ggt-lemma} \(\Mod(\Sigma_{g + 1})\) is + generated by \(\Mod(\Sigma_{g + 1})_{[\alpha]} = \{ f \in \Mod(\Sigma_{g + 1}) : f \cdot [\alpha] = [\alpha]\}\) 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 + In turn, \(\Mod(\Sigma_{g + 1})_{[\alpha]}\) has an index \(2\) subgroup + \(\Mod(\Sigma_{g + 1})_{\vec{[\alpha]}} = \{ f \in \Mod(\Sigma_{g + 1}) : f \cdot \vec{[\alpha]} = \vec{[\alpha]}\}\). One can check that \(\tau_\beta - \tau_\alpha^2 \tau_\beta \in \Mod(S_{g + 1})_{[\alpha]}\) inverts the + \tau_\alpha^2 \tau_\beta \in \Mod(\Sigma_{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 + \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\)-coset in \(\Mod(\Sigma_{g+1})_{[\alpha]}\). In + particular, \(\Mod(\Sigma_{g+1})\) is generated by + \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\), \(\tau_\beta \tau_\alpha\) 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. First observe that \(S_{g+1} - \setminus \alpha \cong S_{g,2}\), as shown in + Finally, we claim \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\) is generated by finitely + many twists about nonseparating curves. First observe that \(\Sigma_{g+1} + \setminus \alpha \cong \Sigma_{g,2}\), 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{\operatorname{cut}} & - \PMod(S_{g,2}) \rar & + \Mod(\Sigma_{g+1})_{\vec{[\alpha]}} \rar{\operatorname{cut}} & + \PMod(\Sigma_{g,2}) \rar & 1. \end{tikzcd} \end{equation} @@ -481,40 +481,40 @@ Theorem~\ref{thm:mcg-is-fg}. \begin{figure}[ht] \centering \includegraphics[width=.75\linewidth]{images/cutting-homeo.eps} - \caption{The homeomorphism $S_{g + 1} \setminus \alpha \cong S_{g, 2}$: + \caption{The homeomorphism $\Sigma_{g + 1} \setminus \alpha \cong \Sigma_{g, 2}$: 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 $S_{g, 2}$.} + disks, which gives us $\Sigma_{g, 2}$.} \label{fig:cut-along-nonseparating-adds-two-punctures} \end{figure} - Recall that, by the induction hypothesis, \(\PMod(S_{g, 2})\) is + Recall that, by the induction hypothesis, \(\PMod(\Sigma_{g, 2})\) 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 + \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\). Now by (\ref{eq:cutting-seq}) we get that + \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\) 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 -generators of \(\Mod(S_g^p)\) by addapting the induction steps in the proof of +generators of \(\Mod(\Sigma_g^p)\) 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 \(\Mod(S_g^p)\) is generated by the Dehn twists about the + If \(g \ge 1\) then \(\Mod(\Sigma_g^p)\) 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 Figure~\ref{fig:lickorish-gens} \end{theorem} In the boundaryless case \(p = 0\), we can write \(\tau_{\mu_3}, \ldots, -\tau_{\mu_g} \in \Mod(S_g)\) as products of the twists about the remaining +\tau_{\mu_g} \in \Mod(\Sigma_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 Dehn twists aboud the + If \(g \ge 2\) then \(\Mod(\Sigma_g)\) is generated by the Dehn twists aboud the curves \(\alpha_0, \ldots, \alpha_{2g}\) as in Figure~\ref{fig:humphreys-gens}. \end{corollary} @@ -522,13 +522,13 @@ curves, from which we get the so called \emph{Humphreys generators}. \begin{minipage}[b]{.45\linewidth} \centering \includegraphics[width=\linewidth]{images/lickorish-gens.eps} - \captionof{figure}{The curves from Lickorish generators of $\Mod(S_g^p)$.} + \captionof{figure}{The curves from Lickorish generators of $\Mod(\Sigma_g^p)$.} \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 $\Mod(S_g)$.} + \captionof{figure}{The curves from Humphreys generators of $\Mod(\Sigma_g)$.} \label{fig:humphreys-gens} \end{minipage}