- Commit
- 0eb06874f69e5cf392f056f14722be6681ed315b
- Parent
- 7ef18d837c07281c6da1e8dd763c228c9078c8f3
- Author
- Pablo <pablo-pie@riseup.net>
- Date
Hydrated the 1st chapter
memoire-m2
My M2 Memoire on mapping class groups & their representations
Diffstat
12 files changed, 1589 insertions, 125 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Added | images/change-of-coords-cut.svg | 265 | 265 | 0 |
| Modified | images/cutting-homeo.svg | 28 | 10 | 18 |
| Added | images/half-twist-disk.svg | 91 | 91 | 0 |
| Added | images/homology-generators.svg | 240 | 240 | 0 |
| Added | images/intersection-index.svg | 296 | 296 | 0 |
| Added | images/torus-cut.svg | 151 | 151 | 0 |
| Modified | preamble.tex | 5 | 5 | 0 |
| Modified | references.bib | 93 | 83 | 10 |
| Modified | sections/introduction.tex | 529 | 440 | 89 |
| Modified | sections/presentation.tex | 4 | 2 | 2 |
| Modified | sections/representations.tex | 4 | 2 | 2 |
| Modified | sections/twists.tex | 8 | 4 | 4 |
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diff --git a/preamble.tex b/preamble.tex @@ -69,6 +69,7 @@ % General linear group \DeclareMathOperator{\GL}{GL} +\DeclareMathOperator{\PGL}{PGL} % Group Abelianization \newcommand{\ab}{{\operatorname{ab}}} @@ -82,3 +83,7 @@ % A normal subobject in a pointed cathegory \newcommand{\normal}{\triangleleft} + +% Useful category definitions +\newcommand{\Cob}{\mathbf{Cob}_3^+} +\newcommand{\Vect}{\mathbf{Vect}(\mathbb{C})}
diff --git a/references.bib b/references.bib @@ -1,5 +1,5 @@ @book{farb-margalit, - author = {Farb, Benson and Margalit, Dan}, + author = {Farb, Benson and Margalit, Dan}, isbn = {978-0-691-14794-9}, language = {English}, publisher = {Princeton University Press}, @@ -10,7 +10,7 @@ } @article{korkmaz, - author = {Korkmaz, Mustafa}, + author = {Korkmaz, Mustafa}, doi = {10.1112/topo.12305}, issn = {1753-8416}, journal = {Journal of Topology}, @@ -24,7 +24,7 @@ } @article{funar, - author = {Louis Funar}, + author = {Funar, Louis}, doi = {10.1090/S0002-9939-2010-10555-5}, journal = {Proceedings of the American Mathematical Society}, language = {English}, @@ -36,7 +36,7 @@ } @article{franks-handel, - author = {Franks, John and Handel, Michael}, + author = {Franks, John and Handel, Michael}, doi = {10.1090/S0002-9939-2013-11556-X}, issn = {0002-9939}, journal = {Proceedings of the American Mathematical Society}, @@ -62,7 +62,7 @@ } @article{korkmaz-mccarthy, - author = {Korkmaz, Mustafa and McCarthy, John D.}, + author = {Korkmaz, Mustafa and McCarthy, John D.}, doi = {10.1017/S0305004199004259}, journal = {Mathematical Proceedings of the Cambridge Philosophical Society}, number = 1, @@ -87,7 +87,7 @@ @article{birman-hilden, ISSN = {0003486X}, - author = {Birman, Joan and Hilden, Hugh}, + author = {Birman, Joan and Hilden, Hugh}, journal = {Annals of Mathematics}, number = 3, pages = {424--439}, @@ -98,7 +98,7 @@ } @article{wajnryb, - author = {Wajnryb, Bronislaw}, + author = {Wajnryb, Bronislaw}, doi = {10.1007/bf02774014}, issn = {1565-8511}, journal = {Israel Journal of Mathematics}, @@ -111,7 +111,7 @@ } @article{lickorish, - author = {Lickorish, William Bernard Raymond}, + author = {Lickorish, William Bernard Raymond}, doi = {10.1017/s030500410003824x}, issn = {1469-8064}, journal = {Mathematical Proceedings of the Cambridge Philosophical Society}, @@ -124,8 +124,68 @@ year = {1964}, } +@article{witten, + author = {Witten, Edward}, + doi = {10.1007/bf01217730}, + issn = {1432-0916}, + journal = {Communications in Mathematical Physics}, + month = sep, + number = {3}, + pages = {351--399}, + publisher = {Springer Science and Business Media LLC}, + title = {Quantum field theory and the Jones polynomial}, + volume = {121}, + year = {1989}, +} + +@article{reshetikhin-turaev, + title = {Invariants of $3$-manifolds via link polynomials and quantum groups}, + volume = {103}, + issn = {1432-1297}, + doi = {10.1007/bf01239527}, + number = {1}, + journal = {Inventiones Mathematicae}, + publisher = {Springer Science and Business Media LLC}, + author = {Reshetikhin, Nicolai and Turaev, Vladimir Georgievich}, + year = {1991}, + month = dec, + pages = {547--597}, +} + +@article{rado, + author = {Rad{\'o}, Tibor}, + journal = {Acta Litt. Sci. Szeged}, + number = {101-121}, + pages = {10}, + title = {{\"U}ber den begriff der riemannschen fl{\"a}che}, + volume = {2}, + year = {1925}, +} + +@article{thomassen, + author = {Thomassen, Carsten}, + doi = {10.2307/2324180}, + issn = {0002-9890}, + journal = {The American Mathematical Monthly}, + month = feb, + number = {2}, + pages = {116}, + publisher = {JSTOR}, + title = {The Jordan-Schonflies Theorem and the Classification of Surface}, + volume = {99}, + year = {1992}, +} + +@book{kerekjarto, + author = {Kerékjártó, Béla}, + doi = {10.1007/978-3-642-50825-7}, + isbn = {9783642508257}, + publisher = {Springer}, + title = {Vorlesungen \"{u}ber Topologie}, + year = {1923}, +} @incollection{julien, - author = {Marché, Julien}, + author = {Marché, Julien}, booktitle = {Topology and geometry. A collection of essays dedicated to Vladimir G. Turaev}, doi = {10.4171/IRMA/33-1/7}, isbn = {978-3-98547-001-3; 978-3-98547-501-8}, @@ -136,8 +196,21 @@ year = {2021}, } +@inproceedings{costantino, + author = {Costantino, Francesco}, + booktitle = {Winter Braids Lecture Notes}, + doi = {10.5802/wbln.7}, + issn = {2426-0312}, + month = feb, + pages = {1--45}, + publisher = {Cellule MathDoc/CEDRAM}, + title = {Notes on Topological Quantum Field Theories}, + volume = {2}, + year = {2015}, +} + @book{hatcher, - author = {Hatcher, Allen}, + author = {Hatcher, Allen}, edition = {1}, isbn = {9780521795401; 0521795400}, language = {English},
diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -1,9 +1,115 @@ \chapter{Introduction}\label{ch:introduction} -% TODO: Motivation -% TODO: Talk about the classification of surfaces +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, +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 +orientable surfaces admitting a triangulation was already known in the +mid-nineteenth century and is often attributed to Möbius, but at that time it +was not yet known that all closed surfaces admit a triangulation. Radò +\cite{rado} would go on to establish this fact in 1925.}, Kerékjártó and others +\cite{rado, kerekjarto}. We refer the reader to \cite{thomassen} for a complete +proof. -\begin{definition} +\begin{theorem}[Classification of surfaces]\label{thm:classification-of-surfaces} + Any closed connected orientable surface is homeomorphic to the connected sum + \(S_g\) of \(g \ge 0\) copies of the torus \(\mathbb{T}^2 = + \mfrac{\mathbb{R}^2}{\mathbb{Z}^2}\). Any compact connected orientable + surface \(S\) is isomorphic to the surface \(S_g^p\) obtained from \(S_g\) by + removing \(p \ge 0\) open disks with disjoint closures. +\end{theorem} + +The integer \(g \ge 0\) in Theorem~\ref{thm:classification-of-surfaces} is +called \emph{the genus of \(S\)}. We also have the noncompact surface \(S_{g, +r}^p = S_g^p \setminus \{x_1, \ldots, x_r\}\), where \(x_1, \ldots, x_r\) in +the interior of \(S_g^p\). The points \(x_1, \ldots, x_r\) are called the +\emph{puctures} of \(S_{g, r}^p\). Throught these notes, all surfaces +considered will be of the form \(S = S_{g, r}^p\). Any such \(S\) admits a +natural compactification \(\bar S\) obtained by filling the its punctures. We +denote \(S_{g, r} = S_{g, r}^0\). All closed curves \(\alpha, \beta \subset S\) +we consider lie in the interior \(S\degree\) of \(S\) and intersect +transversily. + +It is interesting to remark that, aside from the homeomorphism type of a +surface \(S\), Theorem~\ref{thm:classification-of-surfaces} also informs the +geometry of the curves in \(S\) and their intersections. For example\dots + +\begin{lemma}[Change of coordinates principle]\label{thm:change-of-coordinates} + Given oriented nonseparating simple closed curves \(\alpha, \beta \subset + S\), we can find an orientation-presering homeomorphis \(\phi : S \isoto S\) + fixing \(\partial S\) pointwise such that \(\phi(\alpha) = \beta\) with + orientation. Even more so, if \(\alpha', \beta' \subset S\) are nonseparating + curve such that each pair \((\alpha, \alpha'), (\beta, \beta')\) crosses only + once, then we can choose \(\phi\) with \(\phi(\alpha) = \beta'\) and + \(\phi(\alpha') = \beta'\). +\end{lemma} + +\begin{proof} + Let \(S = S_{g, r}^b\) and consider the surface \(S_{\alpha \alpha'}\) + obtained by cutting \(S\) across \(\alpha\) and \(\alpha'\), as in + Figure~\ref{fig:change-of-coordinates}. Since \(\alpha\) and \(\alpha'\) are + nonseparating, this surface has genus \(g - 1\) and one additional boundary + component \(\delta \subset \partial S_{\alpha \beta}\), so \(S_{\alpha \beta} + \cong S_{g-1,r}^{b+1}\). The boundary component \(\delta\) is naturally + subdived into the four arcs in Figure~\ref{fig:change-of-coordinates}, each + corresponding to one of the curves \(\alpha\) and \(\alpha'\) in \(S\). By + identifying the pairs of arcs corresponding to the same curve we obtain the + surface \(\mfrac{S_{\alpha \beta}}{\sim} \cong S\). + + Similarly, \(S_{\beta \beta'} \cong S_{g-1, r}^{b+1}\) also has an additional + boundary component \(\delta' \subset \partial S_{\beta \beta'}\) subdivided + into four arcs. Now by the classification of surfaces we can find an + orientation-preserving homemorphism \(\tilde\phi : S_{\alpha \alpha'} \isoto + S_{\beta \beta'}\). Even more so, we can choose \(\tilde\phi\) taking each + one of the arcs in \(\delta\) to the corresponding arc in \(\delta'\). Now + \(\tilde\phi\) descends to a self-homeomorphism \(\phi\) the quotient surface + \(S \cong \mfrac{S_{\alpha \alpha'}}{\sim} \cong \mfrac{S_{\beta + \beta'}}{\sim}\). 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 $S_{g, r}^b$ across $\alpha$ we obtain $S_{g-1, + r}^{b+2}$, where $\alpha'$ deterimines a yellow arc joining the two + additional boundary components. Now by cutting $S_{g-1, r}^{b+2}$ across this + arc we obtain $S_{g-1,r}^b$, with the added boundary component subdivided + into the four arcs corresponding to $\alpha$ and $\alpha'$.} + \label{fig:change-of-coordinates} +\end{figure} + +More generally, despite the apparent clarity of the picture painted by +Theorem~\ref{thm:classification-of-surfaces}, there are still plenty of +unawsered questions about surfaces and their +homeomorphisms. Given a surface \(S\), the group \(\Homeo^+(S, \partial S)\) of +orientation-preserving homeomorphism of \(S\) fixing its boundary pointwise is +a topological group\footnote{Here we endow \(\Homeo^+(S, \partial S)\) with the +compact-open topology.} with a rich geometry. It is not hard to come up with +interesting questions about such group. For example, +\begin{enumerate} + \item Given closed curves \(\alpha, \beta \subset S\), can we find \(\phi \in + \Homeo^+(S, \partial S)\) with \(\phi(\alpha) = \beta\)? + + \item What are the conjugacy classes of \(\Homeo^+(S, \partial S)\)? What + about its connected components? + + \item Does \(\Homeo^+(S, \partial S)\) determine \(S\)? If the answer is + \emph{no}, what about in the closed case? +\end{enumerate} + +Unfortunately, however, the algebraic structure \(\Homeo^+(S, \partial)\) is +tipically too complex to tackle. More importantly, all of this complexity is +arguably unnecessary for most topological applications, in the sence that +usually we are only really interested in considering \emph{homeomorphisms up to +isotopy}. For instance, isotopic homeomorphisms \(\phi \simeq \psi : S \isoto +S\) determine the same automorphism \(\phi_* = \psi_*\) at the levels of +homotopy and homology. This leads us to consider the group of connected +components of \(\Homeo^+(S, \partial S)\), also known as \emph{the mapping +class group}. This will be the focus of the dissertation at hand. + +\begin{definition}\label{def:mcg} The \emph{mapping class group \(\Mod(S)\) of an orientable surface \(S\)} is the group of isotopy classes of orientation-preserving homeomorphisms \(S \isoto S\), where both the homeomorphisms and the isotopies are assumed to @@ -13,10 +119,55 @@ \] \end{definition} +There are many variations of the Definition~\ref{def:mcg}. For example\dots + +\begin{example}\label{ex:action-on-punctures} + Any \(\phi \in \Homeo^+(S, \partial S)\) extends to a homomorphism + \(\tilde\phi\) of \(\bar{S}\) that permutes the set \(\{x_1, \ldots, x_r\} = + \bar{S} \setminus S\) of punctures of \(S\). We may thus define an action + \(\Mod(S) \leftaction \{x_1, \ldots, x_r\}\) via \(f \cdot x_i = + \tilde\phi(x_i)\) for \(f = [\phi] \in \Mod(S)\) -- which is independant of + the choice of representative \(\phi\) of \(f\). +\end{example} + +\begin{definition} + Given an orientable surface \(S\) and a puncture \(x \subset \bar{S}\) of + \(S\), denote by \(\Mod(S, x) \subset \Mod(S)\) the subgroup of mapping + classes that fix \(x\). The \emph{pure mapping class group \(\PMod(S) \subset + \Mod(S)\) of \(S\)} is the subgroup of mapping classes that fix every + puncture of \(S\). +\end{definition} + +\begin{example}\label{ex:action-on-curves} + Given a simple closed curve \(\alpha \subset S\), denote by + \(\vec{[\alpha]}\) and \([\alpha]\) the isotopy classes of \(\alpha\) with + and without orientation, respectively. There are natural actions \(\Mod(S) + \leftaction \{ \vec{[\alpha]} : \alpha \subset S \}\) and \(\Mod(S) + \leftaction \{ [\alpha] : \alpha \subset S \}\) given by + \begin{align*} + f \cdot \vec{[\alpha]} & = \vec{[\phi(\alpha)]} & + f \cdot [\alpha] & = [\phi(\alpha)] + \end{align*} + for \(f = [\phi] \in \Mod(S)\). +\end{example} + +\begin{definition} + Given a simple closed curve \(\alpha \subset S\), we denote by + \(\Mod(S)_{\vec{[\alpha]}} = \{ f \in \Mod(S) : f \cdot \vec{[\alpha]} = + \vec{[\alpha]} \}\) and \(\Mod(S)_{[\alpha]} = \{ f \in \Mod(S) : f \cdot + [\alpha] = [\alpha] \}\) the subgroups of mapping classes that fix the + isotopy classes of \(\alpha\). +\end{definition} + +While trying to understand the mapping class group of \(S\), it is interesting +to consider how the geometric relationship between \(S\) and other surfaces +affects \(\Mod(S)\). Indeed, different embeddings \(R \hookrightarrow S\) +translate to homomorphisms at the level of mapping class groups. + \begin{example}[Inclusion homomorphism] Let \(R \subset S\) be a closed subsurface. Given some \(\phi \in \Homeo^+(R, \partial R)\), we may extend \(\phi\) to \(\tilde{\phi} \in \Homeo^+(S, - \partial S)\) by setting \(\tilde{\phi}(p) = p\) for \(p \in S\) outside of + \partial S)\) by setting \(\tilde{\phi}(x) = x\) for \(x \in S\) outside of \(R\) -- which is well defined since \(\phi\) fixes every point in \(\partial R\). This contruction yields a group homomorphism \begin{align*} @@ -26,26 +177,23 @@ known as \emph{the inclusion homomorphism}. \end{example} -\begin{example}[Capping homomorphism] - Let \(\alpha \subset \partial S\) be a boundary component of \(S\) and fix - some orientation of \(\alpha\). We refer to the inclusion homomorphism - \(\operatorname{cap} : \Mod(S) \to \Mod(S \cup_\alpha (\mathbb{D}^2 \setminus - \{0\}))\) as \emph{the capping homomorphism}. -\end{example} - -\begin{proposition}[Capping exact sequence]\label{ex:capping-seq} - Given some oriented boundary component \(\alpha \subset \partial S\) of - \(S\), there is an exact sequence +\begin{example}[Capping exact sequence]\label{ex:capping-seq} + Let \(\delta \subset \partial S\) be an oriented boundary component of \(S\). + We refer to the inclusion homomorphism \(\operatorname{cap} : \Mod(S) \to + \Mod(S \cup_\delta (\mathbb{D}^2 \setminus \{0\}))\) as \emph{the capping + homomorphism}. There is an exact sequence \begin{center} \begin{tikzcd} 1 \rar & \langle \tau_\alpha \rangle \rar & \Mod(S) \rar{\operatorname{cap}} & - \Mod(S \cup_\alpha (\mathbb{D}^2 \setminus \{0\})) \rar & - 1 + \Mod(S \cup_\delta (\mathbb{D}^2 \setminus \{0\}), 0) \rar & + 1, \end{tikzcd} \end{center} -\end{proposition} + known as \emph{the capping exact sequence} -- see + \cite[Proposition~3.19]{farb-margalit} for a proof. +\end{example} \begin{example}[Cutting homomorphism]\label{ex:cutting-morphism} Given a simple closed curve \(\alpha \subset S\), denote by @@ -61,102 +209,298 @@ [\phi] & \mapsto [\phi\!\restriction_{S_{g+1} \setminus \alpha}], \end{align*} known as \emph{the cutting homomorphism}. Furthermore, \(\ker - \operatorname{cut} = \langle \tau_\alpha \rangle\). + \operatorname{cut} = \langle \tau_\alpha \rangle\) -- see + \cite[Propostion~3.20]{farb-margalit} for a proof. \end{example} +As goes for most groups, another approach to understanding the mapping class +group of a given surface \(S\) is to study its actions. We have already seen +simple example of such actions in Example~\ref{ex:action-on-punctures} and +Example~\ref{ex:action-on-curves}. A particularly important class of actions +of \(\Mod(S)\) are its \emph{linear representations} -- i.e. the group +homomorphisms \(\Mod(S) \to \GL_n(\mathbb{C})\). These may be seen as actions +\(\Mod(S) \leftaction \mathbb{C}^n\) where each \(f \in \Mod(S)\) acts via some +\(\mathbb{C}\)-linear isomorphism \(\mathbb{C}^n \isoto \mathbb{C}^n\). + +\section{Representations} + +Here we collect a few fundamental examples of linear representations of +\(\Mod(S)\). + +\begin{example} + Given \(k \ge 0\) and \(f = [\phi] \in \Mod(S)\), we may consider the map + \(\phi_* : H_k(S, \mathbb{Z}) \to H_k(S, \mathbb{Z})\) induced at the level + of singular homology. By homotopy invariance, the map \(\phi_*\) is + independant of the choice of representative \(\phi\) of \(f\). By the + functoriality of homology groups we then get a \(\mathbb{Z}\)-linear action + \(\Mod(S) \leftaction H_k(S, R)\), given by \(f \cdot \xi = \phi_*(\xi)\) for + \(f = [\phi] \in \Mod(S)\). +\end{example} + +Now by choosing \(k = 1\) we obtain the so called \emph{symplectic +representation.} + \begin{example} - Let \(S_{g, r}^b = S_g^b \setminus \{ x_1, \ldots, x_r \}\) be the surface of - genus \(g\) with \(r\) punctures and \(b\) boudary components. Given \(\phi - \in \Homeo^+(S_{g, r}^b, \partial S_{g, r}^b)\), there is some \(\sigma_\phi - \in \mathfrak{S}_r\) with \(\phi(x) \to x_{\sigma(i)}\) as \(x \to x_i\) in - \(S_{g, r}\). It is clear that \(\sigma_\phi = \sigma_\psi\) for \(\phi - \simeq \psi\). Hence \(\Mod(S_{g,r}^b) \leftaction \{ x_1, \ldots, x_r \}\) - via \(f \cdot x_i = x_{\sigma_\phi(i)}\) for \(f = [\phi]\). + Recall \(H_1(S_g, \mathbb{Z}) \cong \mathbb{Z}^{2g}\), with standard basis + given by \([\alpha_1], [\beta_1], \ldots, [\alpha_g], [\beta_g] \in H_1(S_g, + \mathbb{Z})\) for \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g\) as + in Figure~\ref{fig:homology-basis}. The Abelian group \(H_1(S_g, + \mathbb{Z})\) is endowed with a natural \(\mathbb{Z}\)-bilinear alternating + form given by the \emph{algebraic intersection number} \([\alpha] \cdot + [\beta] = \sum_{x \in \alpha \cap \beta} \operatorname{ind}\,x\) -- where the + index \(\operatorname{ind}\,x = \pm 1\) of an intersection point is given by + Figure~\ref{fig:intersection-index}. In terms of the standard basis of + \(H_1(S_g, \mathbb{Z})\), this form is given by + \begin{align}\label{eq:symplectic-form} + [\alpha_i] \cdot [\beta_j] & = \delta_{i j} & + [\alpha_i] \cdot [\alpha_j] & = 0 & + [\beta_i] \cdot [\beta_j] & = 0 + \end{align} + and thus coincides with the pullback of the standard \(\mathbb{Z}\)-bilinear + symplectic form in \(\mathbb{Z}^{2g}\). \end{example} +\begin{example}[Symplectic representation]\label{ex:symplectic-rep} + Consider the \(\mathbb{Z}\)-linear action \(\Mod(S_g) \leftaction H_1(S_g, + \mathbb{Z}) \cong \mathbb{Z}^{2g}\). Since pushforwards by + orientation-preserving homeomorphisms preserve the index of intersection + points, \((f \cdot [\alpha]) \cdot (f \cdot [\beta]) = [\alpha] \cdot + [\beta]\) for all \(\alpha, \beta \subset S_g\) and \(f \in \Mod(S_g)\). In + light of (\ref{eq:symplectic-form}), this implies \(\Mod(S_g)\) acts on + \(\mathbb{Z}^{2g}\) via \(\mathbb{Z}\)-linear symplectomorphisms. We thus + obtain a group homomorphism \(\psi : \Mod(S_g) \to + \operatorname{Sp}_{2g}(\mathbb{Z}) \subset \GL_{2g}(\mathbb{C})\), known as + \emph{the symplectic representation of \(\Mod(S_g)\)}. +\end{example} + +\begin{minipage}[b]{.45\linewidth} + \centering + \includegraphics[width=\linewidth]{images/homology-generators.eps} + \captionof{figure}{The curves $\alpha_1, \beta_1, \ldots, \alpha_g, \beta_g + \subset S_g$ that generate its first singular homology group.} + \label{fig:homology-basis} +\end{minipage} +\hspace{.5cm} % +\begin{minipage}[b]{.45\linewidth} + \centering + \includegraphics[width=\linewidth]{images/intersection-index.eps} + \vspace*{.4cm} + \captionof{figure}{The index of an intersection point $x \in \alpha \cap + \beta$.} + \label{fig:intersection-index} +\end{minipage} + +Another fundamental class of examples of representations are the so called +\emph{TQFT representations}. + \begin{definition} - Given an orientable surface \(S\) and a puncture \(x\) of \(S\), denote by - \(\Mod(S, x) \subset \Mod(S)\) the subgroup of mapping classes that fix - \(x\). The \emph{pure mapping class group \(\PMod(S) \subset \Mod(S)\) of - \(S\)} is the subgroup of mapping classes that fix every puncture of \(S\). + A \emph{cobordism} between closed oriented surfaces \(R\) and \(S\) is a + triple \((W, \phi_+, \phi_-)\) where \(W\) is a smooth oriented compact + \(3\)-manifold with \(\partial W = \partial_+ W \amalg \partial_- W\), + \(\phi_+ : S \isoto \partial_+ W\) is an orientation preserving + diffeomorphism and \(\phi_- : R \isoto \partial_- W\) is an + orientation-reversing diffeomorphism. \end{definition} -\begin{example}\label{ex:inclusion-morphism} - \(\Mod(S) \leftaction \{ \vec{[\alpha]} : \alpha \subset S \}\) and \(\Mod(S) - \leftaction \{ [\alpha] : \alpha \subset S \}\) via +\begin{definition} + We denote by \(\Cob\) the category whose objects are (possibly disconnected) + closed oriented surfaces and whose morphisms \(R \to S\) are diffeomorphism + classes\footnote{Here we only consider orientation-preserving diffeomorphisms + $\varphi : W \isoto W'$ that are compatible with the boundary identifications + in the sence that $\varphi(\partial_\pm W) = \partial_\pm W'$ and $\psi_\pm = + \varphi \circ \phi_\pm$.} of cobordisms between \(R\) and \(S\), with + composition given by + \[ + [W, \phi_-, \phi_+] \circ [W', \psi_-, \psi_+] + = [W \cup_{\psi_- \circ \phi_+^{-1}} W', \phi_-, \psi_+] + \] + for \([W, \phi_-, \phi_+] : R \to S\) and \([W', \psi_-, \phi_+] : S \to L\). + We endow \(\Cob\) with the monoidal structure given by \begin{align*} - f \cdot [\alpha] & = [\phi(\alpha)] & - f \cdot \vec{[\alpha]} & = \vec{[\phi(\alpha)]} + S \otimes R + & = S \amalg R & + [W,\phi_+,\phi_-] \otimes [W',\psi_+,\psi_-] + & = [W \amalg W', \phi_+ \amalg \psi_+, \phi_- \amalg \psi_-]. \end{align*} - for \(f = [\phi] \in \Mod(S)\). -\end{example} +\end{definition} -\begin{example}[Change of coordinates principle]\label{ex:change-of-coordinates} - % Also highlight the fact that Mod(S) acts transitively on the pairs of - % curves crossing once. -\end{example} +\begin{definition}[TQFT]\label{def:tqft} + A \emph{topological quantum field theory} (abreviated by \emph{TQFT}) + is a functor \(\mathcal{F} : \Cob \to \Vect\) satisfying + \begin{gather*} + \begin{aligned} + \mathcal{F}(\emptyset) & = \mathbb{C} & + \mathcal{F}(S \otimes R) & = \mathcal{F}(S) \otimes \mathcal{F}(R) + \end{aligned} \\ + \mathcal{F}([W,\phi_+,\phi_-] \otimes [W',\psi_+,\psi_-]) + = \mathcal{F}([W,\phi_+,\phi_-]) \otimes \mathcal{F}([W',\psi_+,\psi_-]), + \end{gather*} + where \(\Vect\) denotes the category of finite-dimensional complex vector + spaces. +\end{definition} \begin{example} - \(\Mod(S) \leftaction H_k(S, \mathbb{R})\) + Given \(\phi \in \Homeo^+(S_g)\), we may consider the so called \emph{mapping + cylinder} \(M_\phi = (S_g \times [0, 1], \phi, 1)\), a cobordism between + \(S_g\) and itself -- where \(\partial_+ (S_g \times [0, 1]) = S_g \times 0\) + and \(\partial_- (S_g \times [0, 1]) = S_g \times 1\). The diffeomorphism + class of \(M_\phi\) is independant of the choice of representative of \(f = + [\phi] \in \Mod(S_g)\), so \(M_f = [M_\phi] : S_g \to S_g\) is a well defined + morphism in \(\Cob\). \end{example} -\begin{example}\label{ex:symplectic-rep} - The symplectic representation. +\begin{example}[TQFT representations]\label{ex:tqft-reps} + It is clear that \(M_1\) is the identity morphism \(S_g \to S_g\) in + \(\Cob\). In addition, \(M_{f \cdot g} = M_f \circ M_g\) in \(\Cob\) for all + \(f, g \in \Mod(S_g)\) -- see \cite[Lemma~2.5]{costantino}. Now given a TQFT + \(\mathcal{F} : \Cob \to \Vect\), by functoriality we obtain a linear + representation + \begin{align*} + \rho_{\mathcal{F}} : \Mod(S_g) & \to \GL(\mathcal{F}(S_g)) \\ + f & \mapsto \mathcal{F}(M_f). + \end{align*} \end{example} -\begin{example} - TQFT representations. -\end{example} +As simple as the construction in Example~\ref{ex:tqft-reps} is, in practice it +is not that easy to come across functors as the ones in +Definition~\ref{def:tqft}. This is because, in most interesting examples, we +are required to attach some extra data to our surfaces to get a well defined +association \(S_g \mapsto \mathcal{F}(S_g)\). Moreover, the condition +\(\mathcal{F}([W] \circ [W']) = \mathcal{F}([W]) \circ \mathcal{F}([W'])\) may +only hold up to multiplication by scalars. + +Hence constructing an actual functor tipically requires \emph{extending} +\(\Cob\) and \emph{tweaking} \(\Vect\). These ``extended TQFTs'' give rise to +linear and projective representations of the \emph{extended mapping class +groups} \(\Mod(S_g) \times \mathbb{Z}\). We refer the reader to +\cite{costantino, julien} for constructions of one such extended TQFT and its +corresponding representations: the so called \emph{\(\operatorname{SU}_2\) TQFT +of level \(r\)}, first introduced by Witten and Reshetikhin-Tuarev +\cite{witten, reshetikhin-turaev} in their foundational papers on quantum +topology. + +Besides Example~\ref{ex:symplectic-rep} and Example~\ref{ex:tqft-reps}, not a +lot of other linear representations of \(\Mod(S_g)\) are known. Indeed, the +representation theory of mapping class groups remains at mistery at large. In +Chapter~\ref{ch:representations} we provide a brief overview of the field, as +well as some recent developments. More specifically, we highlight Korkmaz' +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 +the group structure of mapping class groups: its relations and known +presentations. +% TODOO: Move this to the next chapter \section{First Computations} -% TODO: Explain the Alexander trick +We begin by the simplest, yet perhaps the most fundamental, of computations in +this entire thesis. + +% TODO: Draw a diagram? \begin{example}[Alexander trick]\label{ex:alexander-trick} - \(\Mod(\mathbb{D}^2) = 1\) + The group \(\Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) of homeomorphisms of the + unit disk \(\mathbb{D}^2 \subset \mathbb{Z}\) is contractible. In particular, + \(\Mod(\mathbb{D}^2) = 1\). Indeed, for any \(\phi \in + \Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) the isotopy + \begin{align*} + \phi_t : \mathbb{D}^2 & \to \mathbb{D}^2 \\ + (z, t) & \mapsto + \begin{cases} + (1 - t) \phi(\sfrac{z}{1 - t}) & \text{if } 0 \le |z| \le 1 - t \\ + z & \text{otherwise} + \end{cases} + \end{align*} + that ``fixes the band \(\{ z \in \mathbb{D}^2 : |z| \ge 1 - t \}\) and does + \(\phi\) inside the subdisk \(\{ z \in \mathbb{D}^2 : |z| \le 1 - t\}\)'' + joins \(\phi = \phi_0\) and \(1 = \phi_1\). \end{example} -\begin{example} +\begin{example}\label{ex:mdg-once-punctured-disk} By the same token, \(\Mod(\mathbb{D}^2 \setminus \{0\}) = 1\). \end{example} -% TODO: Explain this \begin{example}\label{ex:torus-mcg} - The symplectic representation \(\psi : \Mod(\mathbb{T}^2) \to - \operatorname{Sp}_2(\mathbb{Z}) = \operatorname{SL}_2(\mathbb{Z})\) is a - group isomorphism. In particular, \(\Mod(\mathbb{T}^2) \cong - \operatorname{SL}_2(\mathbb{Z})\). + Let \(\mathbb{T}^2 = S_1\) be the torus. The symplectic representation \(\psi + : \Mod(\mathbb{T}^2) \to \operatorname{Sp}_2(\mathbb{Z}) = + \operatorname{SL}_2(\mathbb{Z})\) is a group isomorphism. In particular, + \(\Mod(\mathbb{T}^2) \cong \operatorname{SL}_2(\mathbb{Z})\). To see \(\psi\) + is surjective, first observe \(\mathbb{Z}^2 \subset \mathbb{R}^2\) is + \(\operatorname{SL}_2(\mathbb{Z})\)-invariant. Hence any matrix \(g \in + \operatorname{SL}_2(\mathbb{Z})\) descends to an orientation-preserving + homeomorphism \(\phi_g\) of the quotient \(\mathbb{T}^2 = + \mfrac{\mathbb{R}^2}{\mathbb{Z}^2}\), which satisfies \(\psi([\phi_g]) = g\). + To see \(\psi\) is injective we consider the curves \(\alpha_1\) and + \(\beta_1\) from Figure~\ref{fig:homology-basis}. Given \(f = [\phi] \in + \Mod(\mathbb{T}^2)\) with \(\psi(f) = 1\), \(f \cdot \vec{[\alpha_1]} = + \vec{[\alpha_1]}\) and \(f \cdot \vec{[\beta_1]} = \vec{[\beta_1]}\), so we + may choose a representative \(\phi\) of \(f\) fixing \(\alpha_1 \cup + \beta_1\) pointwise. Such \(\phi\) determines a homeomorphism \(\tilde \phi\) + of the surface \(\mathbb{T}_{\alpha_1 \beta_1}^2 \cong \mathbb{D}^2\) + obtained by cutting \(\mathbb{T}^2\) across \(\alpha_1\) and \(\beta_1\), as + 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. \end{example} +\begin{figure}[ht] + \centering + \includegraphics[width=.55\linewidth]{images/torus-cut.eps} + \caption{By cutting $\mathbb{T}^2$ across $\alpha_1$ we obtain a cylinder, + where $\beta_1$ determines a yellow arc joining the two boundary components. + Now by cutting across this yellow arc we obtain a disk.} + \label{fig:cut-torus-across} +\end{figure} + \begin{example}\label{ex:punctured-torus-mcg} By the same token, \(\Mod(S_{1, 1}) \cong \operatorname{SL}_2(\mathbb{Z})\). \end{example} -\begin{proposition}[Alexander method] +% TODOO: Add comments on Costantino's idea? +\begin{note} + 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 \(\Mod(S_g)\) + remains wide-open. Recently, Korkmaz \cite[Theomre~3]{korkmaz} established a + lower bound for the dimension of an injective representation of \(\Mod(S_g)\) + in the \(g \ge 3\) case -- if one such representation exists. +\end{note} + +By cutting across curves and arcs as in the proof of injectivity in +Example~\ref{ex:torus-mcg}, we can always decompose a surface into copies of +\(\mathbb{D}^2\) and \(\mathbb{D}^2 \setminus \{0\}\). +Example~\ref{ex:alexander-trick} and Example~\ref{ex:mdg-once-punctured-disk} +then imply\dots + +\begin{proposition}[Alexander method]\label{thm:alexander-method} Let \(\alpha_1, \ldots, \alpha_n \subset S\) be essencial simple closed - curves or proper arcs pair-wise in minimal position. Suppose \([\alpha_i] \ne - [\alpha_j]\) for \(i \ne j\) and that there are no triple intersections among - the \(\alpha_i\) -- i.e. given distinct \(i, j, k\), at least one of - \(\alpha_i \cap \alpha_j, \alpha_i \cap \alpha_k, \alpha_j \cap - \alpha_k\) is empty. Fix \(\phi \in \Homeo^+(S, \partial S)\) and \(f = - [\phi] \in \Mod(S)\). Then + curves or proper arcs satisfying the following conditions. \begin{enumerate} - \item If there is \(\sigma \in \mathfrak{S}_n\) such that \(f \cdot - [\alpha_i] = [\alpha_{\sigma(i)}]\) for all \(i\) then there is \(\psi - \in \Homeo^+(S, \partial S)\) isotopic to \(1\) taking \(\cup_i - \alpha_i\) to \(\cup_i \phi(\alpha_i)\). - - % TODO: Explain the meaning of "fill" - \item Suppose \(\alpha_1, \ldots, \alpha_n\) fill \(S\) and \(\#(\alpha_i - \cap \alpha_j) \le 1\) for all \(i, j\). If there is \(\sigma \in - \mathfrak{S}_n\) such that \(f \cdot \vec{[\alpha_i]} = - \vec{[\alpha_{\sigma(i)}]}\) then \(f^{\operatorname{ord}(\sigma)} = 1 - \in \Mod(S)\). In particular, if \(\sigma = 1\) then \(f = 1 \in - \Mod(S)\). + \item \([\alpha_i] \ne [\alpha_j]\) for \(i \ne j\). + \item Each pair \((\alpha_i, \alpha_j)\) crosses at most once. + \item Given distinct \(i, j, k\), at least one of \(\alpha_i \cap \alpha_j, + \alpha_i \cap \alpha_k, \alpha_j \cap \alpha_k\) is empty. + \item The surface obtained by cutting \(S\) across the \(\alpha_i\) is a + disjoint union of disks and once-punctured disks. \end{enumerate} + Let \(f \in \Mod(S)\). If there is \(\sigma \in \mathfrak{S}_n\) such that + \(f \cdot \vec{[\alpha_i]} = \vec{[\alpha_{\sigma(i)}]}\) for all \(i\), then + \(f^{\operatorname{ord}(\sigma)} = 1 \in \Mod(S)\). In particular, if + \(\sigma = 1\) then \(f = 1 \in \Mod(S)\). \end{proposition} -% TODO: Add comments on the proof -% TODO: Check the sign in the formula for ϕ: is it t + s or t - s +See \cite[Proposition~2.8]{farb-margalit} for a proof of +Proposition~\ref{thm:alexander-method}. The Alexander method can also be used +to show two mapping classes \(f, g \in \Mod(S)\) are the same: if we decompose +\(S\) into disks and once-punctured disks with \(\alpha_1, \ldots, \alpha_n\) +such that \(f \cdot [\alpha_i] = g \cdot [\alpha_i]\), then \(f g^{-1} \cdot +[\alpha_i] = [\alpha_i]\) and so we may apply +Proposition~\ref{thm:alexander-method} to the class \(f g^{-1}\). We now +collect some important applications of the method. + \begin{example}\label{ex:mcg-annulus} \(\Mod(\mathbb{S}^1 \times [0, 1]) \cong \mathbb{Z}\) is freely generated by \(f = [\phi]\), where @@ -168,16 +512,6 @@ particular, \(\Mod(\mathbb{S}^1 \times [0, 1]) \cong \mathbb{Z}\). \end{example} -\begin{figure}[ht] - \centering - \includegraphics[width=.3\linewidth]{images/dehn-twist-cylinder.eps} - \caption{The generator $f$ of $\Mod(\mathbb{S}^1 \times [0, 1]) \cong - \mathbb{Z}$ takes the yellow arc in the left-hand side to the arc on the - right-hand side that winds about the curve $\alpha$.} - \label{fig:dehn-twist-cylinder} -\end{figure} - -% TODO: Can we prove this without using braid groups? \begin{example}\label{ex:mcg-twice-punctured-disk} The mapping class group \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\})\) of the twice punctured unit disk in \(\mathbb{C}\) is @@ -185,10 +519,27 @@ \begin{align*} \phi : \mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\} & \isoto \mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\} \\ - z & \mapsto -z. + z & \mapsto -z \end{align*} - In particular, \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}) - \cong \mathbb{Z}\). + is the map from Figure~\ref{fig:hald-twist-disk}. In particular, + \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}) \cong + \mathbb{Z}\). \end{example} -% TODO: Add a picture explaining that ϕ is counter-clockwise rotation by π +\begin{figure}[ht] + \centering + \includegraphics[width=.3\linewidth]{images/dehn-twist-cylinder.eps} + \caption{The generator $f$ of $\Mod(\mathbb{S}^1 \times [0, 1]) \cong + \mathbb{Z}$ takes the yellow arc in the left-hand side to the arc on the + right-hand side that winds about the curve $\alpha$.} + \label{fig:dehn-twist-cylinder} +\end{figure} + +\begin{figure}[ht] + \centering + \includegraphics[width=.18\linewidth]{images/half-twist-disk.eps} + \caption{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.} + \label{fig:hald-twist-disk} +\end{figure}
diff --git a/sections/presentation.tex b/sections/presentation.tex @@ -1,4 +1,4 @@ -\chapter{Relations Between Twists} +\chapter{Relations Between Twists}\label{ch:relations} Having acomplished the milestones of Theorem~\ref{thm:lickorish-gens} and Corollary~\ref{thm:humphreys-gens}, we now find ourselves ready to study some @@ -193,7 +193,7 @@ we get\dots \begin{minipage}[b]{.45\linewidth} \begin{example}\label{ex:braid-group-center} - Using the capping exact sequence from Proposition~\ref{ex:capping-seq} and + Using the capping exact sequence from Example~\ref{ex:capping-seq} and the Alexander method, one can check that the center \(Z(\Mod(S_{0, n}^1))\) of \(\Mod(S_{0, n}^1)\) is freely generated by the Dehn twist \(\tau_\delta\) about the boundary \(\delta = \partial S_{0, n}^1\). It is also not very hard
diff --git a/sections/representations.tex b/sections/representations.tex @@ -1,4 +1,4 @@ -\chapter{Low-Dimensional Representations} +\chapter{Low-Dimensional Representations}\label{ch:representations} Having built a solid understanding of the combinatorics of Dehn twists, we are now ready to attack the problem of classifying the representations of @@ -264,7 +264,7 @@ representations. We claim that it suffices to find a \(m\)-dimensional \(\Mod(R)\)-invariant\footnote{Here we view $\Mod(R)$ as a subgroup of $\Mod(S_g^b)$ via the inclusion homomorphism $\Mod(R) \to \Mod(S_g^b)$ from - Example~\ref{ex:inclusion-morphism}, which can be shown to be injective in + Lemma~\ref{thm:inclusion-morphism}, which can be shown to be injective in this particular case.} subspace \(W \subset \mathbb{C}^n\) with \(2 \le m \le n - 2\). Indeed, in this case \(m < 2(g - 1)\) and \(\dim \mfrac{\mathbb{C}^n}{W} = n - m < 2(g - 1)\). Thus both representations
diff --git a/sections/twists.tex b/sections/twists.tex @@ -1,4 +1,4 @@ -\chapter{Dehn Twists} +\chapter{Dehn Twists}\label{ch:dehn-twists} We have now seen some concrete examples of mapping class groups. In this chapter, we will investigate how we can use the anulus \(\mathbb{S}^1 \times @@ -302,12 +302,12 @@ Theorem~\ref{thm:mcg-is-fg}. \begin{tikzcd} 1 \rar & \langle \tau_{\delta_1} \rangle \rar & - \Mod(S_{g, r}^b) \rar{\operatorname{cap}} & - \Mod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\})) \rar & + \PMod(S_{g, r}^b) \rar{\operatorname{cap}} & + \PMod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\})) \rar & 1 \end{tikzcd} \end{center} - from Proposition~\ref{ex:capping-seq}, it suffices to show that \(S_{g, n}\) + from Example~\ref{ex:capping-seq}, it suffices to show that \(S_{g, n}\) is finitely generated by twists about nonseparating simple closed curves. Indeed, if \(\PMod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\) is finitely generated by twists about nonseparing curves or boundary