- Commit
- a06cad908b55623bdb1a5f33821e0341aa9f404c
- Parent
- bdebdaa781646c3fe22916bd41b17fa76adf96e4
- Author
- Pablo <pablo-pie@riseup.net>
- Date
Revised and spellchecked the text
Also renamed an environment
memoire-m2
My M2 Memoire on mapping class groups & their representations
Diffstat
7 files changed, 509 insertions, 471 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | memoir.tex | 4 | 2 | 2 |
| Modified | memoires.cls | 2 | 1 | 1 |
| Modified | preamble.tex | 2 | 1 | 1 |
| Modified | sections/introduction.tex | 278 | 149 | 129 |
| Modified | sections/presentation.tex | 313 | 160 | 153 |
| Modified | sections/representations.tex | 104 | 53 | 51 |
| Modified | sections/twists.tex | 277 | 143 | 134 |
diff --git a/memoir.tex b/memoir.tex @@ -29,8 +29,8 @@ Our main references are the beautiful book by Farb-Margalit \cite{farb-margalit}, as well as the 2023 article by Korkmaz \cite{korkmaz}. We assume the reader is already familiar with basic topology and group - theory. Cruacially, we \emph{do not} assume any familiarity with - representation-theory. + theory. Crucially, we \emph{do not} assume any familiarity with + representation theory. \section*{Acknowledgments}
diff --git a/memoires.cls b/memoires.cls @@ -738,7 +738,7 @@ \begin{titlepage}% \vspace*{1cm} % Some space at the top \begin{center} - This work is licensed under a \textbf{Creative Commons Attributions 4.0 + This work is licensed under the \textbf{Creative Commons Attributions 4.0 International License} \par \vspace{1em}
diff --git a/preamble.tex b/preamble.tex @@ -23,7 +23,7 @@ \newtheorem{fundamental-observation}[theorem]{Fundamental Observation} \newtheorem{fact}[theorem]{Fact} \theoremstyle{remark} -\newtheorem*{note}{Remark} +\newtheorem*{remark}{Remark} % Use \blacksquare for \qed \renewcommand{\qedsymbol}{\ensuremath{\blacksquare}}
diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -1,7 +1,7 @@ \chapter{Introduction}\label{ch:introduction} Ever since Mankind first stepped foot on the surface of Earth, humanity has -been asking what is the shape of the planet we enhabit. More recently, +been asking what is the shape of the planet we inhabit. More recently, mathematicians have spent the past centuries trying to understand the topology of manifolds and, in particular, surfaces. Orientable compact surfaces were first classified in the 1920s by Radò\footnote{The classification of closed @@ -14,103 +14,114 @@ proof. \begin{theorem}[Classification of surfaces]\label{thm:classification-of-surfaces} Any closed connected orientable surface is homeomorphic to the connected sum - \(\Sigma_g\) of \(g \ge 0\) copies of the torus \(\mathbb{T}^2 = - \mfrac{\mathbb{R}^2}{\mathbb{Z}^2}\). Any compact connected orientable - surface \(\Sigma\) is isomorphic to the surface \(\Sigma_g^p\) obtained from \(\Sigma_g\) by - removing \(p \ge 0\) open disks with disjoint closures. + \(\Sigma_g\) of the sphere \(\mathbb{S}^2\) with \(g \ge 0\) copies of the + torus \(\mathbb{T}^2 = \mfrac{\mathbb{R}^2}{\mathbb{Z}^2}\). Any compact + connected orientable surface \(\Sigma\) is isomorphic to the surface + \(\Sigma_g^p\) obtained from \(\Sigma_g\) by removing \(p \ge 0\) open disks + with disjoint closures. \end{theorem} The integer \(g \ge 0\) in Theorem~\ref{thm:classification-of-surfaces} is -called \emph{the genus of \(\Sigma\)}. We also have the noncompact surface \(\Sigma_{g, -r}^p = \Sigma_g^p \setminus \{x_1, \ldots, x_r\}\), where \(x_1, \ldots, x_r\) 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. +called \emph{the genus of \(\Sigma\)}. We also have the noncompact surface +\(\Sigma_{g, r}^p = \Sigma_g^p \setminus \{x_1, \ldots, x_r\}\), where \(x_1, +\ldots, x_r\) lie in the interior of \(\Sigma_g^p\). The points \(x_1, \ldots, +x_r\) are called the \emph{punctures} of \(\Sigma_{g, r}^p\). Throughout these +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 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 +transversely. It is interesting to remark that, aside from the homeomorphism type of a -surface \(\Sigma\), Theorem~\ref{thm:classification-of-surfaces} also informs the -geometry of the curves in \(\Sigma\) 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 \(\alpha, \beta \subset \Sigma\), 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 - \(\phi(\alpha') = \beta'\). + Given oriented nonseparating simple closed curves \(\alpha, \beta \subset + \Sigma\), we can find an orientation-preserving homeomorphism \(\phi : \Sigma + \isoto \Sigma\) 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} +% TODO: Prove the easy case too? \begin{proof} - 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 \(\delta \subset \partial \Sigma_{\alpha \beta}\), 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 \(\Sigma\). By - identifying the pairs of arcs corresponding to the same curve we obtain the - surface \(\mfrac{\Sigma_{\alpha \beta}}{\sim} \cong S\). - - 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 \(\tilde\phi : \Sigma_{\alpha \alpha'} \isoto - \Sigma_{\beta \beta'}\). 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 - \(S \cong \mfrac{\Sigma_{\alpha \alpha'}}{\sim} \cong \mfrac{\Sigma_{\beta - \beta'}}{\sim}\). Moreover, \(\phi\) is such that \(\phi(\alpha) = \alpha'\) - and \(\phi(\beta) = \beta'\), as desired. + 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 \(\delta \subset \partial \Sigma_{\alpha + \alpha'}\), so \(\Sigma_{\alpha \alpha'} \cong \Sigma_{g-1,r}^{p+1}\). The + boundary component \(\delta\) is naturally subdivided into the four arcs in + Figure~\ref{fig:change-of-coordinates}, each 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 + \(\mfrac{\Sigma_{\alpha \alpha'}}{\sim} \cong S\). + + 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 homeomorphism \(\tilde\phi : \Sigma_{\alpha + \alpha'} \isoto \Sigma_{\beta \beta'}\). 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 \(S \cong \mfrac{\Sigma_{\alpha \alpha'}}{\sim} + \cong \mfrac{\Sigma_{\beta \beta'}}{\sim}\). Moreover, \(\phi\) is such that + \(\phi(\alpha) = \alpha'\) and \(\phi(\beta) = \beta'\), as desired. \end{proof} \begin{figure}[ht] \centering \includegraphics[width=.8\linewidth]{images/change-of-coords-cut.eps} \caption{By cutting $\Sigma_{g, r}^p$ across $\alpha$ we obtain $\Sigma_{g-1, - r}^{p+2}$, where $\alpha'$ deterimines a yellow arc joining the two - additional boundary components. Now by cutting $\Sigma_{g-1, r}^{p+2}$ across this - arc we obtain $\Sigma_{g-1,r}^p$, with the added boundary component subdivided - into the four arcs corresponding to $\alpha$ and $\alpha'$.} + r}^{p+2}$, where $\alpha'$ determines a yellow arc joining the two + additional boundary components. Now by cutting $\Sigma_{g-1, r}^{p+2}$ across + this arc we obtain $\Sigma_{g-1,r}^p$, with the added boundary component + subdivided into the four arcs corresponding to $\alpha$ and $\alpha'$.} \label{fig:change-of-coordinates} \end{figure} 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 \(\Sigma\), 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, +unanswered questions about surfaces and their 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 \(\alpha, \beta \subset \Sigma\), can we find \(\phi \in - \Homeo^+(\Sigma, \partial \Sigma)\) 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 \(\Homeo^+(\Sigma, \partial \Sigma)\)? What - about its connected components? + \item What are the conjugacy classes of \(\Homeo^+(\Sigma, \partial + \Sigma)\)? What about its connected components? - \item Does \(\Homeo^+(\Sigma, \partial \Sigma)\) determine \(\Sigma\)? If the answer is - \emph{no}, what about in the closed case? + \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 \(\Homeo^+(\Sigma, \partial)\) 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 \(\phi \simeq \psi : \Sigma \isoto \Sigma\) determine the same automorphism \(\phi_* = \psi_*\) at the levels of -homotopy and homology. This leads us to consider the group of connected -components of \(\Homeo^+(\Sigma, \partial \Sigma)\), also known as \emph{the mapping -class group}. This will be the focus of the dissertation at hand. +Unfortunately, however, the algebraic structure of \(\Homeo^+(\Sigma, \partial +\Sigma)\) is typically too complex to tackle. More importantly, all of this +complexity is arguably unnecessary for most topological applications, in the +sense that usually we are only really interested in considering +\emph{homeomorphisms up to 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 \(\Homeo^+(\Sigma, \partial +\Sigma)\), 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 \(\Mod(\Sigma)\) 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 \(\Sigma\). + 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 \(\Sigma\). \[ \Mod(\Sigma) = \mfrac{\Homeo^+(\Sigma, \partial \Sigma)}{\simeq} \] @@ -119,28 +130,33 @@ class group}. This will be the focus of the dissertation at hand. There are many variations of the Definition~\ref{def:mcg}. For example\dots \begin{example}\label{ex:action-on-punctures} - 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 \(f\). + Any \(\phi \in \Homeo^+(\Sigma, \partial \Sigma)\) extends uniquely 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 independent of the choice of representative + \(\phi\) of \(f\). \end{example} +% TODO: Change this notation? \begin{definition} - 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 \(\Sigma\). + Given an orientable surface \(\Sigma\) and a puncture \(x \in + \widebar\Sigma\) 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 \(\alpha \subset \Sigma\), denote by \(\vec{[\alpha]}\) and \([\alpha]\) the isotopy classes of \(\alpha\) with - 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 + and without orientation, respectively -- i.e \(\vec{[\alpha]} = + \vec{[\beta]}\) if \(\alpha \simeq \beta\) as functions and \([\alpha] = + [\beta]\) if \(\vec{[\alpha]} = \vec{[\beta]}\) or \(\vec{[\alpha]} = + \vec{[\beta^{-1}]}\). 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)] @@ -150,22 +166,25 @@ There are many variations of the Definition~\ref{def:mcg}. For example\dots \begin{definition} 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 \(\alpha\). + \(\Mod(\Sigma)_{\vec{[\alpha]}} = \{ f \in \Mod(\Sigma) : f \cdot + \vec{[\alpha]} = \vec{[\alpha]} \}\) 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 \(\Sigma\), 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. +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 \(\Sigma' \subset \Sigma\) 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 + Let \(\Sigma' \subset \Sigma\) be a closed subsurface. Given \(\phi \in + \Homeo^+(\Sigma', \partial \Sigma')\), we may extend \(\phi\) to + \(\tilde{\phi} \in \Homeo^+(\Sigma, \partial \Sigma)\) by setting + \(\tilde{\phi}(x) = x\) for \(x \in \Sigma\) outside of \(\Sigma'\) -- which + is well defined since \(\phi\) fixes every point in \(\partial \Sigma'\). + This construction yields a group homomorphism \begin{align*} \Mod(\Sigma') & \to \Mod(\Sigma) \\ [\phi] & \mapsto [\tilde\phi], @@ -173,15 +192,16 @@ translate to homomorphisms at the level of mapping class groups. known as \emph{the inclusion homomorphism}. \end{example} +% TODOOO: You haven't defined Dehn twists yet!!!!! \begin{example}[Capping exact sequence]\label{ex:capping-seq} - 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 + 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 & + \langle \tau_\delta \rangle \rar & \Mod(\Sigma) \rar{\operatorname{cap}} & \Mod(\Sigma \cup_\delta (\mathbb{D}^2 \setminus \{0\}), 0) \rar & 1, @@ -191,22 +211,21 @@ translate to homomorphisms at the level of mapping class groups. \cite[Proposition~3.19]{farb-margalit} for a proof. \end{example} +% TODOOO: You haven't defined Dehn twists yet!!!!! \begin{example}[Cutting homomorphism]\label{ex:cutting-morphism} - 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 + Given a simple closed curve \(\alpha \subset \Sigma\), any \(f \in \Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\) 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 + \Homeo^+(\Sigma, \partial \Sigma)\) fixing \(\alpha\) point-wise -- so that + \(\phi\) restricts to a homeomorphism of \(\Sigma \setminus \alpha\). There + is a group homomorphism \begin{align*} \operatorname{cut} : \Mod(\Sigma)_{\vec{[\alpha]}} & \to \Mod(\Sigma\setminus\alpha) \\ - [\phi] & \mapsto [\phi\!\restriction_{\Sigma_{g+1} \setminus \alpha}], + [\phi] & \mapsto [\phi\!\restriction_{\Sigma \setminus \alpha}], \end{align*} known as \emph{the cutting homomorphism}. Furthermore, \(\ker \operatorname{cut} = \langle \tau_\alpha \rangle\) -- see - \cite[Propostion~3.20]{farb-margalit} for a proof. + \cite[Proposition~3.20]{farb-margalit} for a proof. \end{example} As goes for most groups, another approach to understanding the mapping class @@ -227,7 +246,7 @@ Here we collect a few fundamental examples of linear representations of 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 + independent of the choice of representative \(\phi\) of \(f\). By the functoriality of homology groups we then get a \(\mathbb{Z}\)-linear action \(\Mod(\Sigma) \leftaction H_k(\Sigma, R)\), given by \(f \cdot \xi = \phi_*(\xi)\) for \(f = [\phi] \in \Mod(\Sigma)\). @@ -273,7 +292,7 @@ representation.} \centering \includegraphics[width=\linewidth]{images/homology-generators.eps} \captionof{figure}{The curves $\alpha_1, \beta_1, \ldots, \alpha_g, \beta_g - \subset \Sigma_g$ that generate its first singular homology group.} + \subset \Sigma_g$ that generate its first homology group.} \label{fig:homology-basis} \end{minipage} \hspace{.5cm} % @@ -304,7 +323,7 @@ examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 = \end{cases} \end{align*} that ``fixes the band \(\{ z \in \mathbb{D}^2 : |z| \ge 1 - t \}\) and does - \(\phi\) inside the subdisk \(\{ z \in \mathbb{D}^2 : |z| \le 1 - t\}\)'' + \(\phi\) inside the sub-disk \(\{ z \in \mathbb{D}^2 : |z| \le 1 - t\}\)'' joins \(\phi = \phi_0\) and \(1 = \phi_1\). \end{example} @@ -333,8 +352,8 @@ examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 = in Figure~\ref{fig:cut-torus-across}. Now by the Alexander trick from Example~\ref{ex:alexander-trick}, \(\tilde\phi\) must be isotopic to the identity. The isotopy \(\tilde\phi \simeq 1 \in \Homeo^+(\mathbb{D}^2, - \mathbb{S}^1)\) then decends to an isotopy \(\phi \simeq 1 \in - \Homeo^+(\mathbb{T}^2)\), so \(f = 1 \in \Mod(\mathbb{T}^2\) as desired. + \mathbb{S}^1)\) then descends to an isotopy \(\phi \simeq 1 \in + \Homeo^+(\mathbb{T}^2)\), so \(f = 1 \in \Mod(\mathbb{T}^2)\) as desired. \end{example} \begin{figure}[ht] @@ -347,22 +366,23 @@ examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 = \end{figure} \begin{example}\label{ex:punctured-torus-mcg} - By the same token, \(\Mod(\Sigma_{1, 1}) \cong \operatorname{SL}_2(\mathbb{Z})\). + 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 % Bigelow-Budney? -\begin{note} +\begin{remark} Despite the fact \(\psi : \Mod(\mathbb{T}^2) \to \operatorname{SL}_2(\mathbb{Z})\) 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 \(\Mod(\Sigma_g)\) - 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 \(\Mod(\Sigma_g)\) in the \(g \ge 3\) case -- if one such - representation exists. -\end{note} + fact, the question of existence of injective linear representations of + \(\Mod(\Sigma_g)\) remains wide-open. Recently, Korkmaz + \cite[Theorem~3]{korkmaz} established the lower bound of \(3 g - 3\) for the + dimension of an injective representation of \(\Mod(\Sigma_g)\) in the \(g \ge + 3\) case -- if one such representation exists. +\end{remark} Another fundamental class of examples of representations are the so called \emph{TQFT representations}. @@ -382,7 +402,7 @@ Another fundamental class of examples of representations are the so called 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 $\varphi(\partial_\pm W) = \partial_\pm W'$ + identifications in the sense 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 \[ @@ -401,7 +421,7 @@ Another fundamental class of examples of representations are the so called \end{definition} \begin{definition}[TQFT]\label{def:tqft} - A \emph{topological quantum field theory} (abreviated by \emph{TQFT}) + A \emph{topological quantum field theory} (abbreviated by \emph{TQFT}) is a functor \(\mathcal{F} : \Cob \to \Vect\) satisfying \begin{align*} \mathcal{F}(\emptyset) & = \mathbb{C} & @@ -420,7 +440,7 @@ Another fundamental class of examples of representations are the so called 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 + independent 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} @@ -445,7 +465,7 @@ 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} +Hence constructing an actual functor typically 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} \(\Mod(\Sigma_g) \times \mathbb{Z}\). We refer the reader to @@ -459,12 +479,12 @@ topology. Besides Example~\ref{ex:symplectic-rep} and Example~\ref{ex:tqft-reps}, not a 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 +representation theory of mapping class groups remains at mystery 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' proof of the triviality of low-dimensional representations and comment on his -classfication of \(2g\)-dimensional representations \cite{korkmaz}. To that -end, in Chapter~\ref{ch:dehn-twists} and Chapter~\ref{ch:relations} we survay +classification of \(2g\)-dimensional representations \cite{korkmaz}. To that +end, in Chapter~\ref{ch:dehn-twists} and Chapter~\ref{ch:relations} we survey the group structure of mapping class groups: its relations and known presentations.
diff --git a/sections/presentation.tex b/sections/presentation.tex @@ -1,11 +1,12 @@ \chapter{Relations Between Twists}\label{ch:relations} -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 +Having found a convenient set of generators for \(\Mod(\Sigma)\), it is now +natural to ask what are the relations between such generators. In this chapter, +we highlight some additional relations between Dehn twists and the geometric intuition behind them, culminating in the statement of a presentation for \(\Mod(\Sigma_g)\) whose relations can be entirely explained in terms of the -geometry of curves in \(\Sigma_g\) -- see Theorem~\ref{thm:wajnryb-presentation}. +geometry of curves in \(\Sigma_g\) -- see +Theorem~\ref{thm:wajnryb-presentation}. We start by the so called \emph{lantern relation}. @@ -16,8 +17,8 @@ We start by the so called \emph{lantern relation}. \(4\) disjoint open disks from \(\mathbb{S}^2\). If \(\alpha, \beta, \gamma, \delta_1, \ldots, \delta_4 \subset \Sigma_0^4\) are as in Figure~\ref{fig:latern-relation} then from the Alexander method we get the - \emph{lantern relation} (\label{eq:lantern-relation}) in \(\Mod(\Sigma_0^4)\). - \begin{equation}\label{eq:latern-relation} + \emph{lantern relation} (\ref{eq:lantern-relation}) in \(\Mod(\Sigma_0^4)\). + \begin{equation}\label{eq:lantern-relation} \tau_\alpha \tau_\beta \tau_\gamma = \tau_{\delta_1} \tau_{\delta_2} \tau_{\delta_3} \tau_{\delta_4} \end{equation} @@ -26,33 +27,34 @@ 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 $\Sigma_0^4$.} + \caption{The curves from the lantern relation of $\Sigma_0^4$.} \label{fig:latern-relation} \end{figure} -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 +We may exploit different embeddings \(\Sigma_0^4 \hookrightarrow \Sigma\) 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 \(\Mod(\Sigma_g^p)^\ab = \mfrac{\Mod(\Sigma_g^p)}{[\Mod(\Sigma_g), - \Mod(\Sigma_g)]}\) 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 \(g \ge 3\). + The Abelianization \(\Mod(\Sigma_g^p)^\ab = + \mfrac{\Mod(\Sigma_g^p)}{[\Mod(\Sigma_g), \Mod(\Sigma_g)]}\) is cyclic. + Moreover, if \(g \ge 3\) then \(\Mod(\Sigma_g^p)^\ab = 0\). In other words, + \(\Mod(\Sigma_g)\) is a perfect group for \(g \ge 3\). \end{proposition} \begin{proof} - By Theorem~\ref{thm:lickorish-gens}, \(\Mod(\Sigma_g^p)^\ab\) is generated by the - image of the Lickrish generators, which are all conjugate and thus represent - the same class in the Abelianization. In fact, any nonseparating \(\alpha - \subset \Sigma_g^p\) is conjugate to the Lickorish generators too, so - \(\Mod(\Sigma_g^p)^\ab = \langle [\alpha] \rangle\). - - Now for \(g \ge 3\) we can embed \(\Sigma_0^4\) in \(\Sigma_g^p\) in such a way that - all the corresponding curves \(\alpha, \beta, \gamma, \delta_1, \ldots, - \delta_4 \subset \Sigma_g^p\) are nonseparating, as shown in + By Theorem~\ref{thm:lickorish-gens}, \(\Mod(\Sigma_g^p)^\ab\) is generated by + the image of the Lickorish generators, which are all conjugate and thus + represent the same class in the Abelianization. In fact, any nonseparating + \(\alpha \subset \Sigma_g^p\) is conjugate to the Lickorish generators too, + so \(\Mod(\Sigma_g^p)^\ab = \langle [\alpha] \rangle\). + + Now for \(g \ge 3\) we can embed \(\Sigma_0^4\) in \(\Sigma_g^p\) in such a + way that all the corresponding curves \(\alpha, \beta, \gamma, \delta_1, + \ldots, \delta_4 \subset \Sigma_g^p\) are nonseparating, as shown in Figure~\ref{fig:latern-relation-trivial-abelianization}. The lantern relation - (\ref{eq:latern-relation}) then becomes + (\ref{eq:lantern-relation}) then becomes \[ 3 \cdot [\tau_\alpha] = [\tau_\alpha] + [\tau_\beta] + [\tau_\gamma] @@ -71,24 +73,25 @@ example\dots \label{fig:latern-relation-trivial-abelianization} \end{figure} -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 +To get extra relations we need to investigate certain branched covers \(\Sigma +\to \mathbb{D}^2 \setminus \{x_1, \ldots, x_r\}\) and relationship between +\(\Mod(\Sigma)\) and \(\Mod(\mathbb{D}^2 \setminus \{x_1, \ldots, x_r\})\). +This is what is known as\dots -\section{The Birman-Hilden Theorem} +\section{The Birman-Hilden Theorem}\label{birman-hilden} -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 \(\Mod(\Sigma_{0, r}^1)\) is the braid group on \(r\) -strands. +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 \(\Mod(\Sigma_{0, r}^1)\) is the braid group +on \(r\) strands. \begin{definition} The \emph{braid group on \(n\) strands} \(B_n\) is the fundamental group - \(\pi_1(C(\mathbb{D}^2, n), *)\) of the unordered configuration space - \(C(\mathbb{D}^2, n) = \mfrac{C^{\operatorname{ord}}(\mathbb{D}^2, - n)}{S_n}\) of \(n\) distinct points in the interior of the disk. - The elements of \(B_n\) are referred to as \emph{braids}. + \(\pi_1(C(\mathbb{D}^2, n), *)\) of the configuration space \(C(\mathbb{D}^2, + n) = \mfrac{C^{\operatorname{ord}}(\mathbb{D}^2, n)}{S_n}\) of \(n\) points + in the interior of the disk. The elements of \(B_n\) are referred to as + \emph{braids}. \end{definition} \begin{example} @@ -103,17 +106,14 @@ strands. \label{fig:braid-group-generator} \end{figure} -\begin{note} - We would like to point out that to the third Reidemeister move translates to - the so called \emph{braid relations} - \[ - \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_1 \sigma_i, - \] - in \(B_n\), which motivates the name used in Observation~\ref{ex:braid-relation}. -\end{note} - -In his seminal paper on braid groups, Artin \cite{artin} gave the following -finite presentation of \(B_n\). +The third Reidemeister move translates to the so called \emph{braid +relations} +\[ + \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_1 \sigma_i +\] +in \(B_n\), which motivates the name used in +Observation~\ref{ex:braid-relation}. In his seminal paper on braid groups, +Artin \cite{artin} gave the following finite presentation of \(B_n\). \begin{theorem}[Artin] \[ @@ -143,9 +143,9 @@ sequence & 1, \end{tikzcd} \end{center} -given that \(\Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) is contractible -- see -Example~\ref{ex:alexander-trick}. But \(\Mod(\mathbb{D}^2) = 1\). Hence -we get\dots +given that \(\Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) is contractible by +Example~\ref{ex:alexander-trick}. But \(\Mod(\mathbb{D}^2) = 1\). Hence we +get\dots \begin{proposition} The map \(\operatorname{push} : B_n \to \Mod(\Sigma_{0, n}^1)\) is a group @@ -170,40 +170,41 @@ we get\dots \begin{minipage}[b]{.45\textwidth} \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 $\Sigma_{0, n}^1$.} + \captionof{figure}{The clockwise rotation by $\sfrac{2\pi}{n}$ about an axis + center around the punctures $x_1, \ldots, x_n$ of $\Sigma_{0, n}^1$.} \label{fig:braid-group-center} \end{minipage} \smallskip -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 +To get from \(\Sigma_{0, n}^1\) to surfaces of genus \(g > 0\) we may consider +the \emph{hyperelliptic involution} \(\iota : \Sigma_g \isoto \Sigma_g\), which +rotates \(\Sigma_g\) by \(\pi\) around some axis as in Figure~\ref{fig:hyperelliptic-involution}. Given \(\ell < g\) and \(p = 1, 2\), -we can also embed \(\Sigma_\ell^p\) in \(\Sigma_g\) in such way that \(\iota\) restricts -to an involution\footnote{This involution does not fix $\partial \Sigma_\ell^p$ -point-wise.} \(\Sigma_\ell^p \isoto \Sigma_\ell^p\). +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 \includegraphics[width=\linewidth]{images/hyperelliptic-involution.eps} - \caption{The hyperellipctic involution $\iota$.} + \caption{The hyperelliptic involution $\iota$.} \label{fig:hyperelliptic-involution} \end{figure} It is clear from Figure~\ref{fig:hyperelliptic-involution} that the quotients -\(\mfrac{\Sigma_\ell^1}{\iota}\) and \(\mfrac{\Sigma_\ell^2}{\iota}\) are both disks, -with boundary corresponding to the projection of the boundaries of \(\Sigma_\ell^1\) -and \(\Sigma_\ell^2\), respectively. Given \(p = 1, 2\), the quotient map \(\Sigma_\ell^p -\to \mfrac{\Sigma_\ell^p}{\iota} \cong \mathbb{D}^2\) is a double cover with \(2\ell + -b\) branch points corresponding to the fixed points of \(\iota\). We may thus -regard \(\mfrac{\Sigma_\ell^p}{\iota}\) as the disk \(\Sigma_{0, 2\ell + b}^1\) with -\(2\ell + b\) punctures in its interior, as shown in +\(\mfrac{\Sigma_\ell^1}{\iota}\) and \(\mfrac{\Sigma_\ell^2}{\iota}\) are both +disks, with boundary corresponding to the projection of the boundaries of +\(\Sigma_\ell^1\) and \(\Sigma_\ell^2\), respectively. Given \(p = 1, 2\), the +quotient map \(\Sigma_\ell^p \to \mfrac{\Sigma_\ell^p}{\iota} \cong +\mathbb{D}^2\) is a double cover with \(2\ell + p\) branch points corresponding +to the fixed points of \(\iota\). We may thus regard +\(\mfrac{\Sigma_\ell^p}{\iota}\) as the disk \(\Sigma_{0, 2\ell + p}^1\) with +\(2\ell + p\) punctures in its interior, as shown in Figure~\ref{fig:hyperelliptic-covering}. We also draw the curves \(\alpha_1, \ldots, \alpha_{2\ell} \subset \Sigma_\ell^p\) of the Humphreys generators of -\(\Mod(\Sigma_g)\). Since these curves are invariant under the action of \(\iota\), -they descend to arcs \(\bar{\alpha}_1, \ldots, \bar{\alpha}_{2\ell + b} \subset -\Sigma_{0, 2\ell + b}^1\) joining the punctures of the quotient surface. +\(\Mod(\Sigma_g)\). Since these curves are invariant under the action of +\(\iota\), they descend to arcs \(\bar{\alpha}_1, \ldots, \bar{\alpha}_{2\ell + p} \subset \Sigma_{0, 2\ell + p}^1\) joining the punctures of the quotient +surface. \begin{figure}[ht] \centering @@ -213,15 +214,14 @@ they descend to arcs \(\bar{\alpha}_1, \ldots, \bar{\alpha}_{2\ell + b} \subset \end{figure} \begin{observation}\label{ex:push-generators-description} - The map \(\operatorname{push} : B_{2\ell + b} \to \Mod(\Sigma_{0, 2\ell + b}^1)\) - takes \(\sigma_i\) to the half-twist - \(h_{\bar{\alpha}_i}\) about the arc \(\bar{\alpha}_i \subset \Sigma_{0, 2\ell + - b}^1\). + The map \(\operatorname{push} : B_{2\ell + p} \to \Mod(\Sigma_{0, 2\ell + + p}^1)\) takes \(\sigma_i\) to the half-twist \(h_{\bar{\alpha}_i}\) about + the arc \(\bar{\alpha}_i \subset \Sigma_{0, 2\ell + p}^1\). \end{observation} -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}. +We now study the homeomorphisms of \(\Sigma_\ell^1\) and \(\Sigma_\ell^2\) that +descend to the quotient surfaces and their mapping classes, known as \emph{the +symmetric mapping classes}. \begin{definition} Let \(\ell \ge 0\) and \(p = 1, 2\). The \emph{group of symmetric @@ -234,30 +234,30 @@ mapping clases}. Fix \(p = 1\) or \(2\). It follows from the universal property of quotients that any \(\phi \in \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)\) defines a -homeomorphism \(\bar \phi : \Sigma_{0, 2\ell+b}^1 \isoto \Sigma_{0, 2\ell+b}^1\). This -yeilds a homomorphism of topological groups +homeomorphism \(\bar \phi : \Sigma_{0, 2\ell+p}^1 \isoto \Sigma_{0, 2\ell+p}^1\). This +yields a homomorphism of topological groups \begin{align*} \SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p) - & \to \Homeo^+(\Sigma_{0, 2\ell + b}^1, \partial \Sigma_{0, 2\ell + b}^1) \\ + & \to \Homeo^+(\Sigma_{0, 2\ell + p}^1, \partial \Sigma_{0, 2\ell + p}^1) \\ \phi & \mapsto \bar \phi, \end{align*} -which is surjective because any \(\psi \in \Homeo^+(\Sigma_{0, 2\ell + b}^1, -\partial \Sigma_{0, 2\ell + b}^1)\) lifts to \(\Sigma_\ell^p\). +which is surjective because any \(\psi \in \Homeo^+(\Sigma_{0, 2\ell + p}^1, +\partial \Sigma_{0, 2\ell + p}^1)\) lifts to \(\Sigma_\ell^p\). 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 +\Homeo^+(\Sigma_{0, 2\ell + p}^1, \partial \Sigma_{0, 2\ell + p}^1)\) is injective: the +only candidates for elements of its kernel are \(1\) and \(\iota\), but \(\iota\) is not an element of \(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)\) since it does not fix \(\partial \Sigma_\ell^p\) point-wise. Now since we have a continuous bijective homomorphism we find \[ \begin{split} \pi_0(\SHomeo^+(\Sigma_\ell^p, \partial \Sigma_\ell^p)) - & \cong \pi_0(\Homeo^+(\Sigma_{0, 2\ell+b}^1, \partial \Sigma_{0, 2\ell+b}^1)) \\ - & = \mfrac{\Homeo^+(\Sigma_{0,2\ell+b}^1, \partial \Sigma_{0, 2\ell+b}^1)}{\simeq} \\ - & = \Mod(\Sigma_{0, 2\ell+b}^1) \\ - & \cong B_{2\ell + b}. + & \cong \pi_0(\Homeo^+(\Sigma_{0, 2\ell+p}^1, \partial \Sigma_{0, 2\ell+p}^1)) \\ + & = \mfrac{\Homeo^+(\Sigma_{0,2\ell+p}^1, \partial \Sigma_{0, 2\ell+p}^1)}{\simeq} \\ + & = \Mod(\Sigma_{0, 2\ell+p}^1) \\ + & \cong B_{2\ell + p}. \end{split} \] @@ -274,30 +274,31 @@ not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden then \(\phi\) and \(\psi\) are isotopic through symmetric homeomorphisms. In particular, there is an isomorphism \begin{align*} - \SMod(\Sigma_\ell^p) & \isoto \Mod(\Sigma_{0, 2\ell + b}) \\ + \SMod(\Sigma_\ell^p) & \isoto \Mod(\Sigma_{0, 2\ell + p}) \\ [\phi] & \mapsto [\bar \phi]. \end{align*} \end{theorem} \begin{observation} Using the notation of Figure~\ref{fig:hyperelliptic-covering}, the - Birman-Hilden isomorphism \(\SMod(\Sigma_\ell^p) \isoto \Mod(\Sigma_{0, 2g + b})\) - takes \(\tau_{\alpha_i}\) to the half twist \(h_{\bar{\alpha}_i} \in - \Mod(\Sigma_{0, 2g + b})\). This can be checked by looking at - \(\iota\)-invaratiant anular neighborhoods of the curves \(\alpha_i\) -- + Birman-Hilden isomorphism \(\SMod(\Sigma_\ell^p) \isoto \Mod(\Sigma_{0, 2g + + p})\) takes \(\tau_{\alpha_i}\) to the half twist \(h_{\bar{\alpha}_i} \in + \Mod(\Sigma_{0, 2g + p})\). This can be checked by looking at + \(\iota\)-invariant annular neighborhoods of the curves \(\alpha_i\) -- \cite[Section~9.4]{farb-margalit}. \end{observation} \begin{fundamental-observation} 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}, + 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 \(\SMod(\Sigma_\ell^p) \subset \Mod(\Sigma_g)\). + \emph{\(k\)-chain relations} in \(\SMod(\Sigma_\ell^p) \subset + \Mod(\Sigma_g)\). \[ \arraycolsep=1.4pt \begin{array}{rlcrll} @@ -314,9 +315,9 @@ not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden \] \end{fundamental-observation} -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 \(\Sigma_g \to \Sigma_{0, 2g+2}\). +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 @@ -329,47 +330,51 @@ we get branched double cover \(\Sigma_g \to \Sigma_{0, 2g+2}\). & 1, \end{tikzcd} \end{center} - 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 \([\bar \phi] \in \Mod(\Sigma_{0, 2g + 2})\). + 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 \(\alpha_1, \ldots, \alpha_{2g}, \delta \subset \Sigma_g\) be and \(\delta - \subset \Sigma_g\) be as in Figure~\ref{fig:hyperellipitic-relations}. Then + Let \(\alpha_1, \ldots, \alpha_{2g}, \delta \subset \Sigma_g\) 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, \(C_{\Mod(\Sigma_g)}([\iota]) \to \Mod(\Sigma_{0, 2g+2})\) 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 + 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}, \(\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta \in \ker (C_{\Mod(\Sigma_g)}([\iota]) \to - \Mod(\Sigma_{0, 2g+2})) = \langle [\iota] \rangle \cong \mathbb{Z}/2\). 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 \(\Mod(\Sigma_g)\). - \begin{align*} + \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} + (\ref{eq:hyperelliptic-rel-1}) and (\ref{eq:hyperelliptic-rel-2}) in + \(\Mod(\Sigma_g)\). + \begin{align}\label{eq:hyperelliptic-rel-1} (\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta)^2 & = 1 \\ + \label{eq:hyperelliptic-rel-2} [\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta, \tau_\delta] & = 1 - \end{align*} + \end{align} \end{fundamental-observation} \begin{minipage}[b]{.45\textwidth} \centering \includegraphics[width=.7\linewidth]{images/hyperelliptic-relation.eps} \vspace*{.5cm} - \captionof{figure}{The curves from the Humphreys generators of $\Mod(\Sigma_g)$ - and the curve $\delta$ from the hyperelliptic relations.} + \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} \hspace{.5cm} % @@ -384,25 +389,26 @@ we get branched double cover \(\Sigma_g \to \Sigma_{0, 2g+2}\). \section{Presentations of Mapping Class Groups} -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 \(\Mod(\Sigma_g)\) 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 \(\Sigma_g\). +There are numerous known presentations of \(\Mod(\Sigma_{g, r}^p)\), such as +the ones due to Birman-Hilden \cite{birman-hilden}, Gervais \cite{gervais} and +many others. Wajnryb \cite{wajnryb} derived a presentation of +\(\Mod(\Sigma_g)\) using the relations discussed in +Chapter~\ref{ch:dehn-twists} and Section~\ref{birman-hilden}. This is a +particularly satisfactory presentation, since all of its relations can be +explained in terms of the geometry of curves 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 \(a_i = \tau_{\alpha_i} \in \Mod(\Sigma_g)\) - 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. + 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 \(\alpha_j\) disjoint. \item The braid relations \(a_i a_j a_i = a_j a_i a_j\) for \(\alpha_i\) - \(\alpha_j\) crossing once. + and \(\alpha_j\) crossing once. \item The \(3\)-chain relation \((a_1 a_2 a_3)^4 = a_0 b_0\), where \[ @@ -431,13 +437,13 @@ in \(\Sigma_g\). \end{enumerate} \end{theorem} -\begin{note} +\begin{remark} 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 \(\beta_0, \ldots, \beta_3, \delta \subset \Sigma_g\) 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} + 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 intractable as it might look at first glance. +\end{remark} \begin{figure}[ht] \centering @@ -447,8 +453,8 @@ in \(\Sigma_g\). \end{figure} Different presentations can be used to compute the Abelianization of -\(\Mod(\Sigma_g)\) 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, +\(\Mod(\Sigma_g)\) 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 @@ -463,28 +469,29 @@ given by \end{tabular} \end{center} for closed surfaces of small genus. In \cite{korkmaz-mccarthy} Korkmaz-McCarthy -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. +showed that even though \(\Mod(\Sigma_2^p)\) is not perfect, its commutator +subgroup is. In addition, they also show \([\Mod(\Sigma_g^p), +\Mod(\Sigma_g^p)]\) is normally generated by a single mapping class. \begin{proposition}\label{thm:commutator-is-perfect} - The commutator subgroup \(\Mod(\Sigma_2^p)' = [\Mod(\Sigma_2^p), \Mod(\Sigma_2^p)]\) is - perfect -- i.e. \(\Mod(\Sigma_2^p)^{(2)} = [\Mod(\Sigma_2^p)', \Mod(\Sigma_2^p)']\) is the - whole of \(\Mod(\Sigma_2^p)'\). + The commutator subgroup \(\Mod(\Sigma_2^p)' = [\Mod(\Sigma_2^p), + \Mod(\Sigma_2^p)]\) 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 \(\alpha, \beta \subset \Sigma_g\) are simple closed crossing - only once, then \(\Mod(\Sigma_g)'\) is \emph{normally generated} by \(\tau_\alpha - \tau_\beta^{-1}\) -- i.e. if \(\tau_\alpha \tau_\beta^{-1} \in N \normal - \Mod(\Sigma_g)'\) then \(\Mod(\Sigma_g)' \subset N\). + 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 \(\tau_\alpha \tau_\beta^{-1} \in + N \normal \Mod(\Sigma_g)'\) then \(\Mod(\Sigma_g)' \subset N\). \end{proposition} 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 \(\rho : \Mod(\Sigma_g) \to \GL_n(\mathbb{C})\) is nothing other than -a choice of \(2g + 1\) matrices \(\rho(\tau_{\alpha_1}), \ldots, +representation \(\rho : \Mod(\Sigma_g) \to \GL_n(\mathbb{C})\) is nothing other +than a choice of \(2g + 1\) matrices \(\rho(\tau_{\alpha_0}), \ldots, \rho(\tau_{\alpha_{2g}}) \in \GL_n(\mathbb{C})\) satisfying the relations \strong{(i)} to \strong{(v)} as above. In the next chapter, we will discuss how -these relations may be used to derrive obstructions to the existance of +these relations may be used to derive obstructions to the existence of nontrivial representations of certain dimensions.
diff --git a/sections/representations.tex b/sections/representations.tex @@ -2,22 +2,22 @@ Having built a solid understanding of the combinatorics of Dehn twists, we are now ready to attack the problem of classifying the representations of -\(\Mod(\Sigma_g)\) of sufficiently small dimension. As promised, our strategy is to -make us of the \emph{geometrically-motivated} relations derrived in +\(\Mod(\Sigma_g)\) of sufficiently small dimension. As promised, our strategy +is to make use of the \emph{geometrically-motivated} relations derived in Chapter~\ref{ch:relations}. Historically, these relations have been exploited by Funar \cite{funar}, Franks-Handel \cite{franks-handel} and others to establish the triviality of -low-dimensional representions, culminating Korkmaz' recent classification of -representations of dimension \(n \le 2 g\) for \(g \ge 3\) \cite{korkmaz}. The -goal of this chapter is providing a concise account of Korkmaz' results, +low-dimensional representations, culminating Korkmaz' \cite{korkmaz} recent +classification of representations of dimension \(n \le 2 g\) for \(g \ge 3\). +The goal of this chapter is to provide a concise account of Korkmaz' results, starting by\dots \begin{theorem}[Korkmaz]\label{thm:low-dim-reps-are-trivial} - Let \(\Sigma_g^p\) be the 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. + Let \(\Sigma_g^p\) be the compact surface of genus \(g \ge 1\) with \(p\) + boundary components and \(\rho : \Mod(\Sigma_g^p) \to \GL_n(\mathbb{C})\) be + 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} Like so many of the results we have encountered so far, the proof of @@ -46,7 +46,7 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g 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} + Proposition~\ref{thm:commutator-normal-gen} it suffices to show \(L_{\alpha_1} = L_{\beta_1}\), so that \(\tau_{\alpha_1} \tau_{\beta_1}^{-1} \in \ker \rho\) and thus \(\Mod(\Sigma_2^p)' \subset \ker \rho\) -- i.e. \(\rho(\Mod(\Sigma_2^p))\) is Abelian. Given the braid relation @@ -56,7 +56,7 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g \end{equation} this amounts to showing \(L_{\alpha_1}\) and \(L_{\beta_1}\) commute. - To that end, we exhausively analyse all of the possible Jordan forms + To that end, we exhaustively analyze all of the possible Jordan forms \begin{align*} \begin{pmatrix} \lambda & 0 \\ @@ -125,7 +125,7 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g For cases (1) to (6), we use the change of coordinates principle and different relations to show \(L_{\alpha_1}\) and \(L_{\beta_1}\) lie inside - some Abelian subgroup of \(\GL_n(\mathbb{C})\) and thus commute. + some Abelian subgroup of \(\GL_n(\mathbb{C})\). \begin{enumerate}[leftmargin=1.9cm] \item[\bfseries\color{highlight}(1) \& (4)] @@ -140,7 +140,8 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g \tau_{\alpha_2}] = [\tau_{\beta_1}, \tau_{\alpha_2}] = 1\) that \(L_{\alpha_1}\) and \(L_{\beta_1}\) preserve the eigenspaces of \(L_{\alpha_2}\), which are all \(1\)-dimensional. Hence \(L_{\alpha_1}\) - and \(L_{\beta_1}\) lie inside the subgroup of diagonal matrices. + and \(L_{\beta_1}\) lie inside the subgroup of diagonal matrices -- an + Abelian subgroup of \(\GL_n(\mathbb{C})\). \item[\bfseries\color{highlight}(3) \& (6)] As before, it follows from the disjointness relations that \(E_{\alpha_2 @@ -164,8 +165,9 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g case. We claim that if \(E_{\alpha_2 = \lambda} = E_{\beta_2 = \lambda}\) then - \(E_{\alpha_2 = \lambda}\) is \(\Mod(\Sigma_2^p)\)-invariant. Indeed, by change of - coordinates we can always find \(f, g, h_i \in \Mod(\Sigma_2^p)\) 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] & @@ -213,11 +215,11 @@ 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 \(\bar \rho : \Mod(\Sigma_2^p) \to - \GL(E_{\alpha_2 = \lambda}) = \GL_2(\mathbb{C})\) -- recall \(E_{\alpha_2 = - \lambda} = \mathbb{C} e_1 \oplus \mathbb{C} e_2\). By case (2), \(\bar - \rho(f) = 1\) for all \(f \in \Mod(\Sigma_2^p)'\), given that \(\bar - \rho(\Mod(\Sigma_2^p))\) is Abelian. Thus + 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 \(f \in \Mod(\Sigma_2^p)'\), given + that \(\bar \rho(\Mod(\Sigma_2^p))\) is Abelian. Thus \[ \rho(\Mod(\Sigma_2^p)') \subset \begin{pmatrix} @@ -226,10 +228,10 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g 0 & 0 & * \end{pmatrix} \] - lies inside the group of upper triangular matrices, a solvalbe subgroup of + lies inside the group of upper triangular matrices, a solvable subgroup of \(\GL_3(\mathbb{C})\). Now by Proposition~\ref{thm:commutator-is-perfect} we - get \(\rho(\Mod(\Sigma_2^p)') = 1\): any homomorphism from a perfect group to a - solvable group is trivial. + 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 the Jordan form of \(L_{\alpha_2}\) is given by (8) then @@ -240,14 +242,14 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g \subsetneq V \] is a flag of subspaces invariant under \(L_{\alpha_1}\) and \(L_{\beta_1}\), - for \(\alpha_2\) \(\beta_2\) are disjoint from \(\alpha_1 \cup \beta_1\) and - thus \([\tau_{\alpha_2}, \tau_{\alpha_1}] = [\tau_{\alpha_2}, \tau_{\beta_1}] - = [\tau_{\beta_2}, \tau_{\alpha_1}] = [\tau_{\beta_2}, \tau_{\beta_1}] = 1\). - In this case we can find a basis for \(\mathbb{C}^3\) with respect to wich - the matrices of \(L_{\alpha_1}\) and \(L_{\beta_1}\) are both upper - triangular with \(\lambda\) along the diagonal: take \(v_1, v_2, v_3 \in - \mathbb{C}^3\) with \(v_1 \in E_{\alpha_2 = \lambda} \cap E_{\beta_2 = - \lambda}\), \(v_2 \in V_{L_{\alpha_2}}\) and adjust \(v_3\) to get the + for \(\alpha_2\) and \(\beta_2\) are disjoint from \(\alpha_1 \cup \beta_1\) + and thus \([\tau_{\alpha_2}, \tau_{\alpha_1}] = [\tau_{\alpha_2}, + \tau_{\beta_1}] = [\tau_{\beta_2}, \tau_{\alpha_1}] = [\tau_{\beta_2}, + \tau_{\beta_1}] = 1\). In this case we can find a basis for \(\mathbb{C}^3\) + with respect to which the matrices of \(L_{\alpha_1}\) and \(L_{\beta_1}\) are + both upper triangular with \(\lambda\) along the diagonal: take \(v_1, v_2, + v_3 \in \mathbb{C}^3\) with \(v_1 \in E_{\alpha_2 = \lambda} \cap E_{\beta_2 + = \lambda}\), \(v_2 \in V_{L_{\alpha_2}}\) and adjust \(v_3\) to get the desired diagonal entry. Any such pair of matrices satisfying the braid relation (\ref{eq:braid-rel-induction-basis}) commute. @@ -266,8 +268,8 @@ representations. \(g\). The base case \(g = 1\) is again clear from the fact \(n = 1\) and \(\GL_1(\mathbb{C}) = \mathbb{C}^\times\). The case \(g = 2\) was also established in Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}. Now - suppose \(g \ge 3\) and every \(m\)-dimensional representation of \(\Sigma_{g - - 1}^q\) has Abelian image for \(m < 2(g - 1)\). Let us show \(\rho\) has + 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, @@ -300,7 +302,7 @@ representations. \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. \(\rho_i(\Mod(\Sigma))\) is + fall into the induction hypothesis -- 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 \[ @@ -321,7 +323,7 @@ representations. 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 + As before, we exhaustively analyze 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 @@ -335,7 +337,7 @@ representations. \(\mathbb{C}^n\). 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 + lying between \(2\) and \(n - 2\), then there must be at most \(2\) distinct eigenvalues and \(\dim E_{\alpha_g = \lambda} = 1, n - 1, n\) for all eigenvalues \(\lambda\) of \(L_{\alpha_g}\). Hence the Jordan form of \(L_{\alpha_g}\) has to be one of @@ -399,11 +401,11 @@ representations. 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. + \(\Mod(\Sigma)\)-invariant \((n - 1)\)-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 + Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}, it follows from the change of coordinates principle that there are \(f_i, g_i, h_i \in \Mod(\Sigma_g^p)\) with \begin{align*} @@ -417,7 +419,7 @@ representations. & g_i \tau_{\beta_g} g_i^{-1} & = \tau_{\gamma_i} & - h_i \tau_{\beta_g} h_i^{-1} & = \tau_{\eta_i}. + h_i \tau_{\beta_g} h_i^{-1} & = \tau_{\eta_i} \end{align*} and thus \[ @@ -443,7 +445,7 @@ representations. 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 + \(\rho(\Mod(\Sigma_g^p))\) is Abelian, \(\rho\) factors though the Abelianization map \(\Mod(\Sigma_g^p) \to \Mod(\Sigma_g^p)^\ab = \mfrac{\Mod(\Sigma_g^p)}{[\Mod(\Sigma_g^p), \Mod(\Sigma_g^p)]}\). Now recall from Proposition~\ref{thm:trivial-abelianization} that \(\Mod(\Sigma_g^p)^\ab = 0\) for \(g \ge 3\). In other words, \(\rho\) @@ -452,10 +454,10 @@ representations. Having established the triviality of the low-dimensional representations \(\rho : \Mod(\Sigma_g^p) \to \GL_n(\mathbb{C})\), all that remains for us is to understand -the \(2g\)-dimensional reprensentations of \(\Mod(\Sigma_g^p)\). We certainly know a +the \(2g\)-dimensional representations of \(\Mod(\Sigma_g^p)\). We certainly know a nontrivial example of such, namely the symplectic representation \(\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) from -Example~\ref{ex:symplectic-rep}. Surprinsgly, this turns out to be +Example~\ref{ex:symplectic-rep}. Surprisingly, this turns out to be \emph{essentially} the only example of a nontrivial \(2g\)-dimensional representation in the compact case. More precisely, @@ -464,18 +466,18 @@ representation in the compact case. More precisely, \(\rho\) is either trivial or conjugate to the symplectic representation\footnote{Here the map $\Mod(\Sigma_g^p) \to \operatorname{Sp}_{2g}(\mathbb{Z})$ is given by the composition of the - inclusion morphism $\Mod(\Sigma_g^p) \to \Mod(\Sigma_g)$ with the usual symplect + inclusion morphism $\Mod(\Sigma_g^p) \to \Mod(\Sigma_g)$ with the usual symplectic representation $\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})$.} \(\Mod(\Sigma_g^p) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(\Sigma_g^p)\). \end{theorem} Unfortunately, the limited scope of these master thesis does not allow us to dive into the proof of Theorem~\ref{thm:reps-of-dim-2g-are-symplectic}. The -heart of this proof lies in a somewhat technical result about representations -of the product \(B_3^n = B_3 \times \cdots \times B_3\), which Korkmaz refers -to as \emph{the main lemma}. Namely\dots +heart of this proof lies in a result about representations of the product +\(B_3^n = B_3 \times \cdots \times B_3\), which Korkmaz refers to as \emph{the +main lemma}. Namely\dots -\begin{lemma}[Korkmaz' Main Lemma]\label{thm:main-lemma} +\begin{lemma}[Korkmaz' main lemma]\label{thm:main-lemma} Given \(i = 1, \ldots, n\), denote by \newline \(a_i = (1, \ldots, 1, \sigma_1, 1, \ldots 1)\) and \(b_i = (1, \ldots, 1, \sigma_2, 1, \ldots, 1)\) the \(n\)-tuples in \(B_3^n\) whose \(i\)-th coordinates are \(\sigma_1\) and @@ -483,8 +485,8 @@ to as \emph{the main lemma}. Namely\dots Let \(m \ge 2n\) and \(\rho : B_3^n \to \GL_m(\mathbb{C})\) be a representation satisfying: \begin{enumerate} - \item The only eigenvalue of \(\rho(a_i)\) is \(1\) and it's eigenspace is - \((2g - 1)\)-dimensional. + \item The only eigenvalue of \(\rho(a_i)\) is \(1\) and its eigenspace is + \((m - 1)\)-dimensional. \item The eigenspaces of \(\rho(a_i)\) and \(\rho(b_i)\) associated to the eigenvalue \(1\) do not coincide. \end{enumerate} @@ -515,7 +517,7 @@ 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, +coincide with the action of the Lickorish generators \(\tau_{\alpha_1}, \ldots, \tau_{\alpha_g}, \tau_{\beta_1}, \ldots, \tau_{\beta_g} \in \Mod(\Sigma_g^p)\) on \(H_1(\Sigma_g, \mathbb{C}) \cong \mathbb{C}^{2g}\) -- represented in the standard basis \([\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]\) for @@ -539,7 +541,7 @@ basis \([\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]\) for \right) \end{align*} -Hence by embeding \(B_3^g\) in \(\Mod(\Sigma_g^p)\) via +Hence by embedding \(B_3^g\) in \(\Mod(\Sigma_g^p)\) via \begin{align*} B_3^g & \to \Mod(\Sigma_g^p) \\ a_i & \mapsto \tau_{\alpha_i} \\
diff --git a/sections/twists.tex b/sections/twists.tex @@ -2,11 +2,12 @@ With the goal of studying the linear representations of mapping class groups in mind, in this chapter we start investigating the group structure of -\(\Mod(\Sigma)\). 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 \(\Mod(\Sigma)\), known as the set of \emph{Dehn twists}. +\(\Mod(\Sigma)\). 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 generator gives rise to +a convenient 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 @@ -16,7 +17,7 @@ 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 \(\alpha_1, \ldots, \alpha_n \subset \Sigma\) be essencial simple closed + Let \(\alpha_1, \ldots, \alpha_n \subset \Sigma\) be essential simple closed curves or proper arcs satisfying the following conditions. \begin{enumerate} \item \([\alpha_i] \ne [\alpha_j]\) for \(i \ne j\). @@ -35,8 +36,8 @@ Proposition~\ref{thm:alexander-method}. We now state some \emph{fundamental} applications of the Alexander method. \begin{example}\label{ex:mcg-annulus} - \(\Mod(\mathbb{S}^1 \times [0, 1]) \cong \mathbb{Z}\) is freely generated by - \(f = [\phi]\), where + The mapping class group \(\Mod(\mathbb{S}^1 \times [0, 1])\) is freely + generated by \(f = [\phi]\), where \begin{align*} \phi : \mathbb{S}^1 \times [0, 1] & \isoto \mathbb{S}^1 \times [0, 1] \\ (e^{2 \pi i t}, s) & \mapsto (e^{2 \pi i (t - s)}, s) @@ -73,30 +74,31 @@ applications of the Alexander method. \includegraphics[width=.4\linewidth]{images/half-twist-disk.eps} \captionof{figure}{The generator $f$ of $\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}) \cong \mathbb{Z}$ corresponds to the - cclockwise rotation by $\pi$ about the origen.} + clockwise rotation by $\pi$ about the origin.} \label{fig:hald-twist-disk} \end{minipage} 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 \(\Sigma_2\). +boundary. Given some closed \(\alpha \subset \Sigma\), we may envision doing +something similar to Example~\ref{ex:mcg-annulus} in \(\Sigma\) by looking at +annular neighborhoods of \(\alpha\). These are precisely the \emph{Dehn twists}, +illustrated in Figure~\ref{fig:dehn-twist-bitorus} in the case of the bitorus +\(\Sigma_2\). \begin{definition} 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 \(\tau_\alpha \in \Mod(\Sigma)\) about \(\alpha\)} is defined as - the image of \(f\) under the inclusion homomorphism \(\Mod(A) \to \Mod(\Sigma)\). + 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 \(\tau_\alpha \in + \Mod(\Sigma)\) about \(\alpha\)} is defined as the image of \(f\) under the + inclusion homomorphism \(\Mod(A) \to \Mod(\Sigma)\). \end{definition} \begin{figure}[ht] \centering \includegraphics[width=.6\linewidth]{images/dehn-twist-bitorus.eps} - \caption{The Dehn twist about the curve $\alpha$ takes the figure-eight curve + \caption{The Dehn twist about the curve $\alpha$ takes the peanut-shaped curve on the left-hand side to the yellow curve in the right-hand side.} \label{fig:dehn-twist-bitorus} \end{figure} @@ -108,13 +110,14 @@ gives rise the so called \emph{half-twists}. These are examples of mapping classes that permute the punctures of \(\Sigma\). \begin{definition} - 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 \(h_\alpha \in \Mod(\Sigma)\) about \(\alpha\)} is defined as the - image of \(f\) under the inclusion homomorphism \(\Mod(D) \to \Mod(\Sigma)\). + 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 + \(f \in \Mod(D) \cong \Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, + \sfrac{1}{2}\}) \cong \mathbb{Z}\) be the generator from + Example~\ref{ex:mcg-twice-punctured-disk}. The \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 @@ -133,22 +136,23 @@ we can distinguish between powers of Dehn twists \begin{proposition}\label{thm:twist-intersection-number} 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. + 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 \(\alpha, \beta \subset \Sigma\), \(\tau_\alpha = \tau_\beta \iff [\alpha] = - [\beta]\). 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} - that \(\#(T_\alpha(\gamma) \cap \gamma) > \#(T_\beta(\gamma) \cap \gamma)\), - so \(\tau_\alpha \ne \tau_\beta\). + 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} that \(\#(T_\alpha(\gamma) + \cap \gamma) > \#(T_\beta(\gamma) \cap \gamma)\), so \(\tau_\alpha \ne + \tau_\beta\). \end{observation} -Many other relations between Dehn twists can derrived be in a geometric fashion +Many other relations between Dehn twists can derived be in a geometric fashion too. \begin{observation}\label{ex:conjugate-twists} @@ -163,35 +167,35 @@ too. \end{observation} \begin{observation} - 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}. + If \(\alpha, \beta \subset \Sigma\) are both nonseparating 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 \(\alpha, \beta \subset \Sigma\) 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 - the \emph{braid relation} + 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 the \emph{braid relation} \[ - \tau_\alpha \tau_\beta \tau_\alpha = \tau_\beta \tau_\alpha \tau_\alpha. + \tau_\alpha \tau_\beta \tau_\alpha = \tau_\beta \tau_\alpha \tau_\beta. \] \end{fundamental-observation} A perhaps less obvious fact about Dehn twists is\dots \begin{theorem}\label{thm:mcg-is-fg} - Let \(\Sigma_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with \(r\) - punctures and \(b\) boundary components. Then the pure mapping class group - \(\PMod(\Sigma_{g, r}^p)\) is generated by finitely many Dehn twists about - nonseparating curves or boundary components. + Let \(\Sigma_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with + \(r\) punctures and \(p\) boundary components. Then the pure mapping class + group \(\PMod(\Sigma_{g, r}^p)\) is generated by finitely many Dehn twists + about nonseparating curves or boundary components. \end{theorem} The proof of Theorem~\ref{thm:mcg-is-fg} is simple in nature: we proceed by -indution in \(g\) and \(r\). On the other hand, the indutction steps are -somewhat involved and require two ingrediantes we have not encountered so far, +induction in \(g\) and \(r\). On the other hand, the induction steps are +somewhat involved and require two ingredients we have not encountered so far, namely the \emph{Birman exact sequence} and the \emph{modified graph of curves}. @@ -246,7 +250,7 @@ and its long exact sequence in homotopy we then get\dots \end{center} \end{theorem} -\begin{note} +\begin{remark} 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} @@ -258,7 +262,7 @@ and its long exact sequence in homotopy we then get\dots & 1. \end{tikzcd} \end{center} -\end{note} +\end{remark} 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 @@ -279,7 +283,7 @@ show\dots \centering \includegraphics[width=.35\linewidth]{images/push-map.eps} \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 + a simple loop $\alpha \subset \Sigma$ to the mapping class supported at an annular 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.} @@ -288,14 +292,14 @@ show\dots \section{The Modified Graph of Curves} -Having established Theorem~\ref{thm:birman-exact-seq}, we now need to adress +Having established Theorem~\ref{thm:birman-exact-seq}, we now need to address the induction step in the genus \(g\) of \(\Sigma_{g, r}^p\). Our strategy is to -apply the following lemma from geomtric group theory. +apply the following lemma from geometric group theory. \begin{lemma}\label{thm:ggt-lemma} Let \(G\) be a group and \(\Gamma\) be a \emph{connected} graph with \(G \leftaction \Gamma\) via graph automorphisms. Suppose that \(G\) acts - transitively both in \(V(\Gamma)\) and \(\{(v, w) \in V(\Gamma)^2 : + transitively on both \(V(\Gamma)\) and \(\{(v, w) \in V(\Gamma)^2 : v \text{ --- } w \text{ in } \Gamma \}\). If \(v, w \in V(\Gamma)\) are connected by an edge and \(g \in G\) is such that \(g \cdot w = v\) then \(G\) is generated by \(G_v\) and \(g\). @@ -306,7 +310,7 @@ the graph \(\Gamma\), we consider\dots \begin{definition} The \emph{modified graph of nonseparating curves \(\hat{\mathcal{N}}(\Sigma)\) - of a surface \(\Sigma\)} is the graph whose vertices are (un-oriented) isotopy + of a surface \(\Sigma\)} is the graph whose vertices are (unoriented) isotopy classes of nonseparating simple closed curves in \(\Sigma\) and \[ \text{\([\alpha]\) --- \([\beta]\) in \(\hat{\mathcal{N}}(\Sigma)\)} @@ -317,39 +321,41 @@ the graph \(\Gamma\), we consider\dots \end{definition} It is clear from Lemma~\ref{thm:change-of-coordinates} that the actions of -\(\Mod(\Sigma_{g, r}^p)\) on \(V(\hat{\mathcal{N}}(\Sigma_{g, r}^p))\) and \(\{([\alpha], -[\beta]) \in V(\hat{\mathcal{N}}(\Sigma_{g, r}^p))^2 : \#(\alpha \cap \beta) = 1 -\}\) are both transitive. But why should \(\hat{\mathcal{N}}(\Sigma_{g, r}^p)\) be -connected? +\(\Mod(\Sigma_{g, r}^p)\) 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 +Historically, the modified graph of nonseparating curves first arose as a +\emph{modified} version of another graph, known as\dots \begin{definition} - 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 + Given a surface \(\Sigma\), the \emph{graph of curves \(\mathcal{C}(\Sigma)\) + of \(\Sigma\)} is the graph whose vertices are (unoriented) isotopy classes + of essential simple closed curves in \(\Sigma\) and \[ \text{\([\alpha]\) --- \([\beta]\) in \(\mathcal{C}(\Sigma)\)} \iff \#(\alpha \cap \beta) = 0. \] - The \emph{graph of nonseparating curves \(\mathcal{N}(\Sigma)\)} is the subgraph - of \(\mathcal{C}(\Sigma)\) 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, \(\mathcal{C}(\Sigma_{g, r})\) is connected. +Lickorish \cite{lickorish} essentially showed that, apart from a small number +of sporadic cases, \(\mathcal{C}(\Sigma_{g, r})\) is connected. \begin{theorem}[Lickorish] - 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. + 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 \([\alpha], [\beta] \in \mathcal{C}(\Sigma_{g, r})\), we can -find a path \([\alpha] = [\alpha_1] \text{---} \cdots \text{---} [\alpha_n] = -[\beta]\) in \(\mathcal{C}(\Sigma_{g, r})\). Now if \(\alpha\) and \(\beta\) are -nonseparating, by inductively adjusting this path we then get\dots +In other words, given simple closed curves \(\alpha, \beta \subset \Sigma_{g, +r}\), we can find closed \(\alpha = \alpha_1, \alpha_2, \ldots, \alpha_n = +\beta\) in \(\Sigma_{g, r}\) with \(\alpha_i\) and \(\alpha_{i+1}\) disjoint. +Now if \(\alpha\) and \(\beta\) are nonseparating, by inductively adjusting +this sequence of curves we then get\dots \begin{corollary}\label{thm:mofied-graph-is-connected} If \(g \ge 2\) then both \(\mathcal{N}(\Sigma_{g, r})\) and @@ -361,13 +367,13 @@ Corollary~\ref{thm:mofied-graph-is-connected}. We are now ready to show Theorem~\ref{thm:mcg-is-fg}. \begin{proof}[Proof of Theorem~\ref{thm:mcg-is-fg}] - Let \(\Sigma_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with \(r\) - punctures and \(b\) boundary components. We want to establish that - \(\PMod(\Sigma_{g, r}^p)\) is genetery by a finite number of Dehn twists about - nonseparating simple closed curves or boundary components. + Let \(\Sigma_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with + \(r\) punctures and \(p\) boundary components. We want to establish that + \(\PMod(\Sigma_{g, r}^p)\) is generated by a finite number of Dehn twists + about nonseparating simple closed curves or boundary components. - 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 + 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} @@ -380,14 +386,14 @@ Theorem~\ref{thm:mcg-is-fg}. \end{center} 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 \(\PMod(\Sigma_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\) - is finitely generated by twists about nonseparing curves or boundary - components, then we may lift the generators of \(\PMod(\Sigma_{g, r}^p - \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\) to Dehn twists about the - corresponding curves in \(\Sigma_{g, r}^p\) and add \(\tau_{\delta_1}\) to the - generating set. - - It thus suffices to consider the boudaryless case \(\Sigma_{g, r}\). As promised, + Indeed, if \(\PMod(\Sigma_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus + \{0\})) \cong \PMod(\Sigma_{g, r+1}^{p-1})\) is finitely generated by twists + about nonseparating curves or boundary components, then we may lift the + generators of \(\PMod(\Sigma_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus + \{0\}))\) to Dehn twists about the corresponding curves in \(\Sigma_{g, + r}^p\) and add \(\tau_{\delta_1}\) to the generating set. + + It thus suffices to consider the boundaryless case \(\Sigma_{g, r}\). As promised, we proceed by double induction on \(r\) and \(g\). For the base case, it is clear from Example~\ref{ex:torus-mcg} and Example~\ref{ex:torus-mcg} that \(\Mod(\mathbb{T}^2) \cong \Mod(\Sigma_{1, 1}) \cong @@ -433,7 +439,7 @@ Theorem~\ref{thm:mcg-is-fg}. 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 \(\operatorname{push} - : \pi_1(\Sigma_{g, r}, x) \to \Mod(\Sigma_{g, r+1}, x)\) takes nonseparation simple + : \pi_1(\Sigma_{g, r}, x) \to \Mod(\Sigma_{g, r+1}, x)\) takes nonseparating simple loops to products of twists about nonseparating simple curves. Furthermore, 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 @@ -441,31 +447,33 @@ Theorem~\ref{thm:mcg-is-fg}. simple curves, concluding the induction step on \(r\). As for the induction step on \(g\), fix \(g \ge 1\) and suppose that, for - each \(r \ge 0\), \(\PMod(\Sigma_{g, r})\) is finitely generated by twists about - nonseparing curves or boundary components. Let us show that the same holds - 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 + each \(r \ge 0\), \(\PMod(\Sigma_{g, r})\) is finitely generated by twists + about nonseparating curves or boundary components. Let us show that the same + holds 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 \(\alpha, \beta - \subset \Sigma_{g + 1}\) 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, \(\Mod(\Sigma_{g + 1})_{[\alpha]}\) 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 \(\tau_\beta + \subset \Sigma_{g + 1}\) crossing once, \(\tau_\beta \tau_\alpha \cdot + [\beta] = [\alpha]\). It thus follows from Lemma~\ref{thm:ggt-lemma} that + \(\Mod(\Sigma_{g + 1})\) 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, \(\Mod(\Sigma_{g + 1})_{[\alpha]}\) has its index \(2\) subgroup + \(\Mod(\Sigma_{g + 1})_{\vec{[\alpha]}} = \{ f \in \Mod(\Sigma_{g + 1}) : f + \cdot \vec{[\alpha]} = \vec{[\alpha]}\}\) of mapping classes fixing the + orientation of \(\alpha\). One can check that \(\tau_\beta \tau_\alpha^2 \tau_\beta \in \Mod(\Sigma_{g + 1})_{[\alpha]}\) inverts the orientation of \(\alpha\) and is thus a representative of the nontrivial - \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\)-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 \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\) is generated by finitely - many twists about nonseparating curves. First observe that \(\Sigma_{g+1} - \setminus \alpha \cong \Sigma_{g,2}\), as shown in + \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\)-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 \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\) 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} @@ -481,40 +489,41 @@ Theorem~\ref{thm:mcg-is-fg}. \begin{figure}[ht] \centering \includegraphics[width=.75\linewidth]{images/cutting-homeo.eps} - \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 $\Sigma_{g, 2}$.} + \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 $\Sigma_{g, 2}$.} \label{fig:cut-along-nonseparating-adds-two-punctures} \end{figure} 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 - \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\). 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 \(g\). + \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\). 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 \(\Mod(\Sigma_g^p)\) by addapting the induction steps in the proof of -Theorem~\ref{thm:mcg-is-fg}. These are known as the \emph{Lickorish +generators of \(\Mod(\Sigma_g^p)\) by adapting the induction steps in the +proof of Theorem~\ref{thm:mcg-is-fg}. These are known as the \emph{Lickorish generators}. \begin{theorem}[Lickorish generators]\label{thm:lickorish-gens} - If \(g \ge 1\) then \(\Mod(\Sigma_g^p)\) is generated by the Dehn twists 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 + 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 \(\tau_{\mu_3}, \ldots, -\tau_{\mu_g} \in \Mod(\Sigma_g)\) as products of the twists about the remaining -curves, from which we get the so called \emph{Humphreys generators}. +In the boundaryless case \(p = 0\), we can write \(\tau_{\alpha_3}, \ldots, +\tau_{\alpha_g} \in \Mod(\Sigma_g)\) as products of the twists about the +remaining curves, from which we get the so called \emph{Humphreys generators}. \begin{corollary}[Humphreys generators]\label{thm:humphreys-gens} - If \(g \ge 2\) then \(\Mod(\Sigma_g)\) is generated by the Dehn twists aboud the + If \(g \ge 2\) then \(\Mod(\Sigma_g)\) is generated by the Dehn twists about the curves \(\alpha_0, \ldots, \alpha_{2g}\) as in Figure~\ref{fig:humphreys-gens}. \end{corollary}