- Commit
- 7057b04bf6c633bf7df7bc5b26d516eb1fff98cb
- Parent
- 4127c23b2acd5c54bccffff057640c690cd118eb
- Author
- Pablo <pablo-pie@riseup.net>
- Date
Hydrated the 3rd chapter
memoire-m2
My M2 Memoire on mapping class groups & their representations
Diffstat
15 files changed, 3248 insertions, 47 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Added | images/braid-group-center.svg | 194 | 194 | 0 |
| Added | images/braid-group-generator.svg | 290 | 290 | 0 |
| Added | images/hyperelliptic-covering.svg | 608 | 608 | 0 |
| Added | images/hyperelliptic-involution.svg | 336 | 336 | 0 |
| Added | images/hyperelliptic-relation.svg | 213 | 213 | 0 |
| Added | images/lantern-relation-disk-filling.svg | 109 | 109 | 0 |
| Added | images/lantern-relation-trivial-abelianization.svg | 275 | 275 | 0 |
| Added | images/lantern-relation.svg | 249 | 249 | 0 |
| Added | images/sphere-rotation.svg | 164 | 164 | 0 |
| Added | images/wajnryb-presentation-curves.svg | 377 | 377 | 0 |
| Modified | preamble.tex | 4 | 3 | 1 |
| Modified | references.bib | 49 | 49 | 0 |
| Modified | sections/introduction.tex | 2 | 1 | 1 |
| Modified | sections/presentation.tex | 395 | 362 | 33 |
| Modified | sections/twists.tex | 30 | 18 | 12 |
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diff --git a/preamble.tex b/preamble.tex @@ -5,7 +5,7 @@ \usepackage{enumitem, xfrac, xcolor, cancel, multicol, tabularx, relsize} \usepackage[normalem]{ulem} \usepackage{hyperref, refcount} -\usepackage{fancyhdr, titling} +\usepackage{fancyhdr, titling, caption} \usepackage[backend=biber, citetracker=true, citestyle=alphabetic, bibstyle=ieee-alphabetic, sorting=nty, sortcites=true]{biblatex} \usepackage{pgfplots, tikz, tikz-cd} \usepackage{graphicx, wrapfig} @@ -154,6 +154,7 @@ % Display long arrows instead of short ones \renewcommand{\to}{\longrightarrow} \renewcommand{\mapsto}{\longmapsto} +\renewcommand{\hookrightarrow}{\lhook\joinrel\longrightarrow} % Isomorphism arrow \newcommand{\isoto}{\xlongrightarrow{\sim}} @@ -171,6 +172,7 @@ % Group of homeomorphisms \DeclareMathOperator{\Homeo}{Homeo} +\DeclareMathOperator{\SHomeo}{SHomeo} % General linear group \DeclareMathOperator{\GL}{GL}
diff --git a/references.bib b/references.bib @@ -61,6 +61,55 @@ year = {2023}, } +@article{korkmaz-mccarthy, + author = {Korkmaz, Mustafa and McCarthy, John D.}, + doi = {10.1017/S0305004199004259}, + journal = {Mathematical Proceedings of the Cambridge Philosophical Society}, + number = 1, + pages = {35--53}, + title = {Surface mapping class groups are ultrahopfian}, + volume = {129}, + year = {2000}, +} + +@article{artin, + author = {Artin, Emil}, + doi = {10.2307/1969218}, + issn = {0003-486X}, + journal = {The Annals of Mathematics}, + month = jan, + number = 1, + pages = {101--126}, + title = {Theory of Braids}, + volume = {48}, + year = {1947}, +} + +@article{birman-hilden, + ISSN = {0003486X}, + author = {Birman, Joan and Hilden, Hugh}, + journal = {Annals of Mathematics}, + number = 3, + pages = {424--439}, + publisher = {Annals of Mathematics}, + title = {On Isotopies of Homeomorphisms of Riemann Surfaces}, + volume = {97}, + year = {1973}, +} + +@article{wajnryb, + author = {Wajnryb, Bronislaw}, + doi = {10.1007/bf02774014}, + issn = {1565-8511}, + journal = {Israel Journal of Mathematics}, + month = jun, + number = {2--3}, + pages = {157--174}, + title = {A simple presentation for the mapping class group of an orientable surface}, + volume = 45, + year = {1983}, +} + @incollection{julien, author = {Marché, Julien}, booktitle = {Topology and geometry. A collection of essays dedicated to Vladimir G. Turaev},
diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -72,7 +72,7 @@ \section{First Computations} % TODO: Explain the Alexander trick -\begin{example}[Alexander trick] +\begin{example}[Alexander trick]\label{ex:alexander-trick} \(\Mod(\mathbb{D}) = 1\) \end{example}
diff --git a/sections/presentation.tex b/sections/presentation.tex @@ -1,35 +1,104 @@ \chapter{Relations Among Twists \& Wajnryb's Presentation} -\begin{example} - Lantern relation +Having acomplished the milestones of Theorem~\ref{thm:lickorish-gens} and +Corollary~\ref{thm:humphreys-gens}, we now find ourselves ready to study some +of the group-theoretic aspects of \(\Mod(S)\). We should note, however, that +our current understanding of its group structure is quite lacking: even though +we know the generators of \(\Mod(S)\), we have already seen in +Example~\ref{ex:braid-relation} that these must satisfy nontrivial relations. + +This poses the question: what relations between Dehn twists are there? The goal +of this chapter is to highlight some of these relations and the geometric +intuition behind them. We start by perhaps the simplest of these, known as +\emph{the lantern relation}. + +\begin{example}\label{ex:lanter-relation} + Let \(S_0^4\) be the surface the of genus \(0\) with \(4\) boundary + components -- i.e. the \emph{lantern-like} surface we get by subtracting + \(4\) disjoint open disks from \(\mathbb{S}^2\). If \(\alpha, \beta, \gamma, + \delta_1, \ldots, \delta_4 \subset S_0^4\) are as in + Figure~\ref{fig:latern-relation} then from the Alexander method we get the so + called \emph{lantern relation} in \(\Mod(S_0^4)\). + \begin{equation}\label{eq:latern-relation} + \tau_\alpha \tau_\beta \tau_\gamma + = \tau_{\delta_1} \tau_{\delta_2} \tau_{\delta_3} \tau_{\delta_4} + \end{equation} \end{example} +\begin{figure}[ht] + \centering + \includegraphics[width=.25\linewidth]{images/lantern-relation.eps} + \caption{The curves from the latern relation of $S_0^4$.} + \label{fig:latern-relation} +\end{figure} + +We may exploit different embedings \(S_0^4 \hookrightarrow S\) and their +corresponding inclusion homomorphisms \(\Mod(S_0^4) \to \Mod(S)\) to obtain +interesting relations between the corresponding Dehn twists in \(\Mod(S)\). For +example\dots + \begin{proposition}\label{thm:trivial-abelianization} - If \(g \ge 3\) then the Abelianization \(\Mod(S_g^b)^\ab = - \mfrac{\Mod(S_g^b)}{[\Mod(S_g), \Mod(S_g)]}\) is trivial. In other words, - \(\Mod(S_g) = [\Mod(S_g), \Mod(S_g)]\) is a perfect group for \(g \ge 3\). + The Abelianization \(\Mod(S_g^b)^\ab = \mfrac{\Mod(S_g^b)}{[\Mod(S_g), + \Mod(S_g)]}\) is cyclic. Moreover, if \(g \ge 3\) then \(\Mod(S_g^b)^\ab = + 1\). In other words, \(\Mod(S_g)\) is a perfect group for \(g \ge 3\). \end{proposition} -% TODO: Explain how to get the groups in the low-genus settings we use explicit -% presentations of Mod(S_g) +\begin{proof} + By Theorem~\ref{thm:lickorish-gens}, \(\Mod(S_g^b)^\ab\) is generated by the + image of the Lickrish generators, which are all conjugate and thus represent + the same class in the Abelianization. In fact, any nonseparating \(\alpha + \subset S_g^b\) is conjugate to the Lickorish generators too, so + \(\Mod(S_g^b)^\ab = \langle [\alpha] \rangle\). + + Now for \(g \ge 3\) we can embed \(S_0^4\) in \(S_g^b\) in such a way that + all the corresponding curves \(\alpha, \beta, \gamma, \delta_1, \ldots, + \delta_4 \subset S_g^b\) are nonseparating, as shown in + Figure~\ref{fig:latern-relation-trivial-abelianization}. The lantern relation + (\ref{eq:latern-relation}) then becomes + \[ + 3 \cdot [\tau_\alpha] + = [\tau_\alpha] + [\tau_\beta] + [\tau_\gamma] + = [\tau_{\delta_1}] + [\tau_{\delta_2}] + + [\tau_{\delta_3}] + [\tau_{\delta_4}] + = 4 \cdot [\tau_\alpha], + \] + in \(\Mod(S_g^b)^\ab\). In other words, \([\tau_\alpha] = 0\) and thus + \(\Mod(S_g^b)^\ab = 0\). +\end{proof} + +\begin{figure}[ht] + \centering + \includegraphics[width=.5\linewidth]{images/lantern-relation-trivial-abelianization.eps} + \caption{The embedding of $S_0^4$ in $S_g^b$ for $g \ge 3$.} + \label{fig:latern-relation-trivial-abelianization} +\end{figure} + +% TODO: Fix the spacing of this table +We should note that in general \(\Mod(S)^\ab\) needs not be trivial. For +example, in \cite[Section~5.1.3]{farb-margalit} Farb-Margalit use different +presentations of \(\Mod(S_g)\) for \(g \le 2\) to show the Abelianization is given by \begin{center} \begin{tabular}{ r|c|l } \(g\) & \(S_g\) & \(\Mod(S_g)^\ab\) \\ \hline \(0\) & \(\mathbb{S}^2\) & \(0\) \\ \(1\) & \(\mathbb{T}\) & \(\mathbb{Z}/12\) \\ - \(2\) & \(S_2\) & \(\mathbb{Z}/10\) + \(2\) & \(S_2\) & \(\mathbb{Z}/10\) \\ \end{tabular} \end{center} +for closed surfaces with small genus. + +In \cite{korkmaz-mccarthy} Korkmaz-McCarthy showed that +eventhough \(\Mod(S_2^b)\) is not perfect, its commutator subgroup is. +In addition, they also show \([\Mod(S_g^b), \Mod(S_g^b)]\) is normaly generated +by a single mapping class. -% TODO: Look for a proof of this? In any case, cite a reference \begin{proposition}\label{thm:commutator-is-perfect} The commutator subgroup \(\Mod(S_2^b)' = [\Mod(S_2^b), \Mod(S_2^b)]\) is perfect -- i.e. \(\Mod(S_2^b)^{(2)} = [\Mod(S_2^b)', \Mod(S_2^b)']\) is the whole of \(\Mod(S_2^b)'\). \end{proposition} -% TODO: Comment on the proof of this? \begin{proposition}\label{thm:commutator-normal-gen} If \(g \ge 2\) and \(\alpha, \beta \subset S_g\) are simple closed curves with \(\#(\alpha \cap \beta) = 1\) then \(\Mod(S_g)'\) is \emph{normally @@ -37,15 +106,54 @@ \tau_\beta^{-1} \in N \normal \Mod(S_g)'\) then \(\Mod(S_g)' \subset N\). \end{proposition} +These past few results combined paint a remarkably clear picture of the +Abelianization \(\Mod(S)^\ab\) and the commutator \(\Mod(S)'\), but this is +still a far cry from a general undertanding of the structure of \(\Mod(S)\) +itself. For that, we need to intruduce some extra relations. To that end, we +study certain branched covers \(S \to \mathbb{D} \setminus \{x_1, \ldots, +x_r\}\) and how they may be used to relate both mapping class groups. This is +what is known as\dots + \section{The Birman-Hilden Theorem} +Let \(S_{0, r}^1 = \mathbb{D} \setminus \{x_1, \ldots, x_r\}\) be the surface +of genus \(0\) with \(r\) punctures and \(1\) boundary component. We begin our +investigation by providing an alternative description of its mapping class +group. Namely, we show that \(\Mod(S_{0, r}^1)\) is the braid group on \(r\) +strands. + \begin{definition} The \emph{braid group on \(n\) strands} \(B_n\) is the fundamental group \(\pi_1(C(\mathbb{D}, n), *)\) of the unordered configuration space - \(C(\mathbb{D}, n)\) of \(n\) distinct points in the disk \(\mathbb{D}\). + \(C(\mathbb{D}, n) = \mfrac{C^{\operatorname{ord}}(\mathbb{D}, + n)}{\mathfrak{S}_n}\) of \(n\) distinct points in the interior of the disk. + The elements of \(B_n\) are referred to as \emph{braids}. \end{definition} -% TODO: Cite the paper by Artin +\begin{example} + Given \(i = 1, \ldots, n-1\), we define \(\sigma_i \in B_n\) as in + Figure~\ref{fig:braid-group-generator}. +\end{example} + +\begin{figure}[ht] + \centering + \includegraphics[width=.25\linewidth]{images/braid-group-generator.eps} + \caption{The braid $\sigma_i$.} + \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 Example~\ref{ex:braid-relation}. +\end{note} + +In his seminal paper on braid groups, Artin \cite{artin} gave the following +finite presentation of \(B_n\). + \begin{theorem}[Artin] \[ B_n = @@ -55,40 +163,201 @@ \sigma_i \sigma_{i+1} \sigma_i & = \sigma_{i+1} \sigma_i \sigma_{i+1} \ \text{for all} \ i, \\ \sigma_i \sigma_j & = \sigma_j \sigma_i - \ \text{for} \ j \ne i + 1 \ \text{and} \ j \ne i - 1 + \ \text{for} \ j \ne i + 1 \ \text{and} \ j \ne i - 1 \end{aligned} \right\rangle \] \end{theorem} -\begin{corollary} +As promised, we now show that \(B_n\) coincides with \(\Mod(S_{0, n}^1)\). +Recall from Theorem~\ref{thm:generalized-birman-seq} that there is an exact +sequence +\begin{center} + \begin{tikzcd} + 1 \rar + & B_n \rar{\operatorname{push}} + & \Mod(S_{0, n}^1) \rar + & \Mod(\mathbb{D}) \rar + & 1, + \end{tikzcd} +\end{center} +given that \(\Homeo^+(\mathbb{D}, \mathbb{S}^1)\) is contractible -- see +Example~\ref{ex:alexander-trick}. But \(\Mod(\mathbb{D}) = 1\). Hence +we get\dots + +\begin{proposition} The map \(\operatorname{push} : B_n \to \Mod(S_{0, n}^1)\) is a group isomorphism. -\end{corollary} +\end{proposition} -\begin{example} - The center \(Z(B_n)\) is freely generated by \(z = (\sigma_1 \cdots \sigma_{n - - 1})^n\). + +\begin{minipage}[b]{.45\linewidth} +\begin{example}\label{ex:braid-group-center} + Using the capping exact sequence from Proposition~\ref{ex:capping-seq} and + the Alexander method, one can check that the center \(Z(\Mod(S_{0, n}^1))\) + of \(\Mod(S_{0, n}^1)\) is freely generated by the Dehn twist \(\tau_\delta\) + about the boundary \(\delta = \partial S_{0, n}^1\). It is also not very hard + to see that \(\operatorname{push} : B_n \to \Mod(S_{0, n}^1)\) takes + \(\sigma_1 \cdots \sigma_{n-1}\) to the rotation by + \(\sfrac{2\pi}{n}\) as in Figure~\ref{fig:braid-group-center}, which is an + \(n\)-th root of \(\tau_\delta\). Hence the center \(Z(B_n)\) is freely + generated by \(z = (\sigma_1 \cdots \sigma_{n - 1})^n\). +\end{example} +\end{minipage} +\hspace{.5cm} % +\begin{minipage}[b]{.45\textwidth} + \centering + \includegraphics[width=.18\linewidth]{images/braid-group-center.eps} + \captionof{figure}{The clockwise rotation by $\sfrac{2\pi}{n}$ about an axis center + around the punctures $x_1, \ldots, x_n$ of $S_{0, n}^1$.} + \label{fig:braid-group-center} +\end{minipage} + +To get from \(S_{0, n}^1\) to surfaces of genus \(g > 0\) we may consider the +\emph{hyperelliptic involution} \(\iota : S_g \isoto S_g\) which rotates +\(S_g\) by \(\pi\) around some axis, as shown in +Figure~\ref{fig:hyperelliptic-involution}. Given \(\ell < g\) and \(b = 1, 2\), +we can also embed \(S_\ell^b\) in \(S_g\) in such way that \(\iota\) restricts +to an involution\footnote{This involution does not fix $\partial S_\ell^b$ +point-wise.} \(S_\ell^b \isoto S_\ell^b\). + +\begin{figure}[ht] + \centering + \includegraphics[width=\linewidth]{images/hyperelliptic-involution.eps} + \caption{The hyperellipctic involution $\iota$.} + \label{fig:hyperelliptic-involution} +\end{figure} + +It is clear from Figure~\ref{fig:hyperelliptic-involution} that the quotients +\(\mfrac{S_\ell^1}{\iota}\) and \(\mfrac{S_\ell^2}{\iota}\) are both disks, +with boundary corresponding to the projection of the boundaries of \(S_\ell^1\) +and \(S_\ell^2\), respectively. Given \(b = 1, 2\), the quotient map \(S_\ell^b +\to \mfrac{S_\ell^b}{\iota} \cong \mathbb{D}\) is a double cover with \(2\ell + +b\) branch points corresponding to the fixed points of \(\iota\). We may thus +regard \(\mfrac{S_\ell^b}{\iota}\) as the disk \(S_{0, 2\ell + b}^1\) with +\(2\ell + b\) punctures in its interior, as shown in +Figure~\ref{fig:hyperelliptic-covering}. We also draw the curves \(\alpha_1, +\ldots, \alpha_{2\ell} \subset S_\ell^b\) of the Humphreys generators of +\(\Mod(S_g)\). Since these curves are invariant under the action of \(\iota\), +they descend to arcs \(\bar{\alpha}_1, \ldots, \bar{\alpha}_{2\ell + b} \subset +S_{0, 2\ell + b}^1\) joining the punctures of the quotient surface. + +\begin{figure}[ht] + \centering + \includegraphics[width=.77\linewidth]{images/hyperelliptic-covering.eps} + \caption{The double branched covers given by $\iota$.} + \label{fig:hyperelliptic-covering} +\end{figure} + +\begin{example}\label{ex:push-generators-description} + The map \(\operatorname{push} : B_{2\ell + b} \to \Mod(S_{0, 2\ell + b}^1)\) + takes \(\sigma_i\) to the half-twist + \(h_{\bar{\alpha}_i}\) about the arc \(\bar{\alpha}_i \subset S_{0, 2\ell + + b}^1\). \end{example} +We now study the homemorphisms of \(S_\ell^1\) and \(S_\ell^2\) that descend to +the quotient surfaces and their mapping classes, known as \emph{the symmetric +mapping clases}. + \begin{definition} - Let \(\ell \ge 0\) and \(b = 1, 2\). The \emph{symmetric mapping class group - of \(S_\ell^1\)} is the subgroup \(\SMod(S_\ell^1) = \{ [\phi] \in - \Mod(S_\ell^1) : [\phi, \iota] = 1 \}\). + Let \(\ell \ge 0\) and \(b = 1, 2\). The \emph{group of symmetric + homeomorphisms of \(S_\ell^b\)} is \(\SHomeo^+(S_\ell^b, \partial S_\ell^b) = + \{\phi \in \Homeo^+(S_\ell^b, \partial S_\ell^b) : [\phi, \iota] = 1\}\). The + \emph{symmetric mapping class group of \(S_\ell^b\)} is the subgroup + \(\SMod(S_\ell^1) = \{ [\phi] \in \Mod(S_\ell^b) : \phi \in + \SHomeo^+(S_\ell^b, \partial S_\ell^b) \}\). \end{definition} +Fix \(b = 1\) or \(2\). It follows from the universal property of quotients +that any \(\phi \in \SHomeo^+(S_\ell^b, \partial S_\ell^b)\) defines a +homeomorphism \(\bar \phi : S_{0, 2\ell+b}^1 \isoto S_{0, 2\ell+b}^1\). This +yeilds a homomorphism of topological groups +\begin{align*} + \SHomeo^+(S_\ell^b, \partial S_\ell^b) + & \to \Homeo^+(S_{0, 2\ell + b}^1, \partial S_{0, 2\ell + b}^1) \\ + \phi + & \mapsto \bar \phi, +\end{align*} +which is surjective because any \(\psi \in \Homeo^+(S_{0, 2\ell + b}^1, +\partial S_{0, 2\ell + b}^1)\) lifts to \(S_\ell^b\). + +It is also not hard to see \(\SHomeo^+(S_\ell^b, \partial S_\ell^b) \to +\Homeo^+(S_{0, 2\ell + b}^1, \partial S_{0, 2\ell + b}^1)\) is injective: the +only cadidates for elements of its kernel are \(1\) and \(\iota\), but +\(\iota\) is not an element of \(\SHomeo^+(S_\ell^b, \partial S_\ell^b)\) since +it does not fix \(\partial S_\ell^b\) point-wise. Now since we have a +continuous bijective homomorphism we find +\[ + \begin{split} + \pi_0(\SHomeo^+(S_\ell^b, \partial S_\ell^b)) + & \cong \pi_0(\Homeo^+(S_{0, 2\ell+b}^1, \partial S_{0, 2\ell+b}^1)) \\ + & = \mfrac{\Homeo^+(S_{0,2\ell+b}^1, \partial S_{0, 2\ell+b}^1)}{\simeq} \\ + & = \Mod(S_{0, 2\ell+b}^1) \\ + & \cong B_{2\ell + b} + \end{split} +\] + +We would like to say \(\pi_0(\SHomeo^+(S_\ell^b, \partial S_\ell^b)) = +\SMod(S_\ell^b)\), but a priori the story looks a little more complicated: +\(\phi, \psi \in \SHomeo^+(S_\ell^b, \partial S_\ell^b)\) define the same class +in \(\SMod(S_\ell^b)\) if they are isotopic, but they may not lie in same +connected component of \(\SHomeo^+(S_\ell^b, \partial S_\ell^b)\) if they are +not isotopic \emph{through homeomorphisms symmetric homeomorphisms}. +Birman-Hilden \cite{birman-hilden} showed that this is never the case. + \begin{theorem}[Birman-Hilden] - If \(\phi \simeq \psi \in \Homeo^+(S_\ell^1, \partial S_\ell^1)\) both - commute with \(\iota : S_\ell^1 \isoto S_\ell^1\) then \(\phi\) and \(\psi\) - are isotopic through homeomorphisms that commute with \(\iota\). In - particular, there is an isomorphism + If \(\phi, \psi \in \SHomeo^+(S_\ell^b, \partial S_\ell^b)\) are isotopic + then \(\phi\) and \(\psi\) are isotopic through symmetric homeomorphisms. In + particular, there are isomorphism \begin{align*} - \SMod(S_\ell^1) & \isoto B_{2 g + b} \\ + \SMod(S_\ell^b) & \isoto \Mod(S_{0, 2\ell + b}) & + \SMod(S_\ell^b) & \isoto B_{2 g + b} \\ + [\phi] & \mapsto [\bar \phi] & [\phi] & \mapsto \operatorname{push}^{-1}([\bar \phi]). \end{align*} \end{theorem} -\begin{theorem}[Birman-Hilden without boundary] +\begin{example} + Using the notation of Figure~\ref{fig:hyperelliptic-covering}, the + Birman-Hilden isomorphism \(\SMod(S_\ell^b) \isoto \Mod(S_{0, 2g + b})\) + takes \(\tau_{\alpha_i}\) to the half twist \(h_{\bar{\alpha}_i} \in + \Mod(S_{0, 2g + b})\). This can be checked by looking at + \(\iota\)-invaratiant anular neighborhoods of the curves \(\alpha_i\) -- + \cite[Section~9.4]{farb-margalit}. +\end{example} + +\begin{example} + The Birman-Hilden isomorphism \(\SMod(S_\ell^1) \isoto \Mod(S_{0, + 2\ell+1}^1)\) takes the twists \(\tau_\delta \in \SMod(S_\ell^1)\) about the + boundary \(\delta = \partial S_\ell^1\) to \(\tau_{\bar\delta}^2 \in + \Mod(S_{0, 2\ell+1}^1)\). Similarly, \(\SMod(S_\ell^2) \isoto \Mod(S_{0, + 2\ell+2})\) takes \(\tau_{\delta_1} \tau_{\delta_2} \in \SMod(S_\ell^2)\) to + \(\tau_{\bar\delta_1} = \tau_{\bar\delta_2}\). In light of + Example~\ref{ex:push-generators-description}, + Example~\ref{ex:braid-group-center} gives us the so called \emph{\(k\)-chain + relations} in \(\SMod(S_\ell^b) \subset \Mod(S_g)\). + \[ + \begin{array}{rcll} + (\sigma_1 \cdots \sigma_k)^{2k+2} = z^2 \in B_{k + 1} & + \rightsquigarrow & + (\tau_{\alpha_1} \cdots \tau_{\alpha_k})^{2k + 2} + = \tau_\delta & + \text{for } k = 2 \ell \text{ even} \\ + (\sigma_1 \cdots \sigma_k)^{k+1} = z \in B_{k + 1} & + \rightsquigarrow & + (\tau_{\alpha_1} \cdots \tau_{\alpha_k})^{k + 1} + = \tau_{\delta_1} \tau_{\delta_2} & + \text{for } k = 2 \ell + 1 \text{ odd} + \end{array} + \] +\end{example} + +We may also exploit the quotient \(\mfrac{S_g}{\iota} \cong \mathbb{S}^2\) to +obtain other relations. Since \(\iota\) has \(2g + 2\) fixed points in \(S_g\), +we get branched double cover \(S_g \to S_{0, 2g+2}\). + +\begin{theorem}[Birman-Hilden without boundary]\label{thm:boundaryless-birman-hilden} If \(g \ge 2\) then we have an exact sequence \begin{center} \begin{tikzcd} @@ -105,14 +374,56 @@ \end{theorem} \begin{example} - The \(k\)-chain relation. + Let \(\alpha_1, \ldots, \alpha_{2g}, \delta \subset S_g\) be and \(\delta + \subset S_g\) be as in Figure~\ref{fig:hyperellipitic-relations}. Then + \begin{equation}\label{eq:hyperelliptic-eq} + [\iota] + = \tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} + \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta. + \end{equation} + Indeed, \(C_{\Mod(S_g)}([\iota]) \to \Mod(S_{0, 2g+2})\) takes \(\tau_\delta + \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1}\) to the rotation from + Figure~\ref{fig:hyperelliptic-relation-rotation}, while \(\tau_{\alpha_1} + \cdots \tau_{\alpha_{2g}} \tau_\delta\) is taken to its inverse. By + Theorem~\ref{thm:boundaryless-birman-hilden}, \(\tau_\delta + \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} \cdots + \tau_{\alpha_{2g}} \tau_\delta \in \ker (C_{\Mod(S_g)}([\iota]) \to + \Mod(S_{0, 2g+2})) = \langle [\iota] \rangle \cong \mathbb{Z}/2\). Given the + fact \(\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} \tau_{\alpha_1} + \cdots \tau_{\alpha_{2g}} \tau_\delta\) inverts the orientation of + \(\alpha_1\), (\ref{eq:hyperelliptic-eq}) follows. In particular, we obtain + the so called \emph{hyperelliptic relations} of \(\Mod(S_g)\). + \begin{align*} + (\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} + \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta)^2 + & = 1 \\ + [\tau_\delta \tau_{\alpha_{2g}} \cdots \tau_{\alpha_1} + \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta, \tau_\delta] + & = 1 + \end{align*} \end{example} -\begin{example} - The hyperelliptic relations. -\end{example} +\begin{minipage}[b]{.45\textwidth} + \centering + \includegraphics[width=.7\linewidth]{images/hyperelliptic-relation.eps} + \captionof{figure}{The curves from the Humphreys generators of $\Mod(S_g)$ + and the curve $\delta$ from the hyperelliptic relations.} + \label{fig:hyperellipitic-relations} +\end{minipage} +\hspace{.5cm} % +\begin{minipage}[b]{.45\textwidth} + \centering + \includegraphics[width=.33\linewidth]{images/sphere-rotation.eps} + \captionof{figure}{The clockwise rotation by $\sfrac{\pi}{g + 1}$ about an + axis centered around the punctures of $S_{0, 2g + 1}$.} + \label{fig:hyperelliptic-relation-rotation} +\end{minipage} + +Wajnryb used the \(k\)-chain relations and the hyperelliptic relations to +derive a presentation of the mapping class group of a closed surface, +which is widely considered to the the standard presentation of +\(\Mod(S_g)\) \cite{wajnryb}. -% TODO: Cite the original paper? \begin{theorem}[Wajnryb]\label{thm:wajnryb-presentation} If \(\alpha_0, \ldots, \alpha_g\) are as in % TODO: Reference the drawing of the curves somewhere @@ -153,4 +464,22 @@ \end{enumerate} \end{theorem} -% TODO: Add the drawings for the what each one of these curves mean +\begin{note} + The mapping classes \(b_0, \ldots, b_3, d\) in the statement of + Theorem~\ref{thm:wajnryb-presentation} correspond to the Dehn twists about + the curves \(\beta_0, \ldots, \beta_3, \delta \subset S_g\) highlighted in + Figure~\ref{fig:wajnryb-presentation-curves}, so Wajnryb's presentation is + not as intractible as it might look at first glance. +\end{note} + +\begin{figure}[ht] + \centering + \includegraphics[width=.7\linewidth]{images/wajnryb-presentation-curves.eps} + \caption{The curves from Wajnryb's presentation.} + \label{fig:wajnryb-presentation-curves} +\end{figure} + +This presentation paints a much clearer picture of the structure of +\(\Mod(S_g)\). Nevertheless, its linear representations and many other of if +its group-theoretic aspects remain a mistery. This will be the focus of the +next chapter.
diff --git a/sections/twists.tex b/sections/twists.tex @@ -46,6 +46,11 @@ \(\tau_\alpha, \tau_\beta \in \Mod(S)\) are conjugate. \end{fact} + +\begin{example}\label{ex:braid-relation} + Braid relation +\end{example} + \begin{theorem}\label{thm:mcg-is-fg} Let \(S\) be an orientable surface of genus \(g \ge 1\), potentially with punctures and boundary components. Then the pure mapping class group @@ -61,8 +66,8 @@ \begin{center} \begin{tikzcd} 1 \rar - & \pi_1(S, p_0) \rar{\operatorname{push}} - & \Mod(S \setminus \{p_0\}, p_0) \rar{\operatorname{forget}} + & \pi_1(S, x_0) \rar{\operatorname{push}} + & \Mod(S \setminus \{x_0\}, x_0) \rar{\operatorname{forget}} & \Mod(S) \rar & 1. \end{tikzcd} @@ -72,20 +77,21 @@ % TODO: Explan configuration spaces \begin{align*} C^{\operatorname{ord}}(S, n) - & = \{ (p_1, \ldots, p_n) \in S^n : p_i \ne p_j \forall i \ne j \} \\ + & = \{ (x_1, \ldots, x_n) \in \operatorname{int}(S)^n : + x_i \ne x_j \ \text{for}\ i \ne j \} \\ C(S, n) & = \mfrac{C^{\operatorname{ord}}(S, n)}{\mathfrak{S}_n} \end{align*} % TODO: Comment on the proof -\begin{theorem}[Generalized Birman exact sequence] +\begin{theorem}[Generalized Birman exact sequence]\label{thm:generalized-birman-seq} Suppose \(\pi_1(\Homeo^+(S, \partial S), 1) = 1\). Then there is an exact sequence \begin{center} \begin{tikzcd} 1 \rar - & \pi_1(C(S, n), [p_1, \ldots, p_n]) \rar{\operatorname{push}} - & \Mod(S \setminus \{p_1, \ldots, p_n\}) \rar{\operatorname{forget}} + & \pi_1(C(S, n), [x_1, \ldots, x_n]) \rar{\operatorname{push}} + & \Mod(S \setminus \{x_1, \ldots, x_n\}) \rar{\operatorname{forget}} & \Mod(S) \rar & 1. \end{tikzcd} @@ -196,16 +202,16 @@ \begin{center} \begin{tikzcd} 1 \rar - & \pi_1(S_{g, n}, p_0) \rar{\operatorname{push}} + & \pi_1(S_{g, n}, x_0) \rar{\operatorname{push}} & \PMod(S_{g, n + 1}) \rar{\operatorname{forget}} & \PMod(S_{g, n}) \rar & 1, \end{tikzcd} \end{center} - where \(S_{g, n + 1} = S_{g, n} \setminus \{p_0\}\). Since \(g \ge 1\), - \(\pi_1(S_{g, n}, p_0)\) is generated by finitely many nonseparating loops. - We have seen that \(\operatorname{push} : \pi_1(S_{g, n}, p_0) \to - \Mod(S_{g, n+1}, p_0)\) takes simple loops to products of twists about + where \(S_{g, n + 1} = S_{g, n} \setminus \{x_0\}\). Since \(g \ge 1\), + \(\pi_1(S_{g, n}, x_0)\) is generated by finitely many nonseparating loops. + We have seen that \(\operatorname{push} : \pi_1(S_{g, n}, x_0) \to + \Mod(S_{g, n+1}, x_0)\) takes simple loops to products of twists about nonseparating simple curves. Furthermore, we may once again lift the generators of \(\PMod(S_{g, n})\) to Dehn twists about nonseparating simple curves in \(S_{g, n + 1}\). This goes to show that \(\PMod(S_{g, n + 1})\) is @@ -291,7 +297,7 @@ \end{center} \end{theorem} -\begin{corollary}[Humphreys generators] +\begin{corollary}[Humphreys generators]\label{thm:humphreys-gens} If \(g \ge 2\) then \(\Mod(S_g)\) is generated by the twists \(\tau_{\alpha_0}, \ldots, \tau_{\alpha_{2g}} \in \Mod(S_g)\). \end{corollary}