- Commit
- e221d5fd6496104788fee9af632808f7690cfdd9
- Parent
- 8894c953609e975dc716d9cf7ac375986198093e
- Author
- Pablo <pablo-pie@riseup.net>
- Date
Tweak in notation
Replace 𝔻 with 𝔻²
Replaced 𝕋 with 𝕋²
Also updated a drawing
memoire-m2
My M2 Memoire on mapping class groups & their representations
Diffstat
4 files changed, 65 insertions, 55 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | images/torus-mcg-generators.svg | 64 | 37 | 27 |
| Modified | sections/introduction.tex | 20 | 10 | 10 |
| Modified | sections/presentation.tex | 18 | 9 | 9 |
| Modified | sections/twists.tex | 18 | 9 | 9 |
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diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -26,7 +26,7 @@ \begin{example} 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} \setminus + \(\operatorname{cap} : \Mod(S) \to \Mod(S \cup_\alpha (\mathbb{D}^2 \setminus \{0\}))\) as \emph{the capping homomorphism}. \end{example} @@ -38,7 +38,7 @@ 1 \rar & \langle \tau_\alpha \rangle \rar & \Mod(S) \rar{\operatorname{cap}} & - \Mod(S \cup_\alpha (\mathbb{D} \setminus \{0\})) \rar & + \Mod(S \cup_\alpha (\mathbb{D}^2 \setminus \{0\})) \rar & 1 \end{tikzcd} \end{center} @@ -70,18 +70,18 @@ % TODO: Explain the Alexander trick \begin{example}[Alexander trick]\label{ex:alexander-trick} - \(\Mod(\mathbb{D}) = 1\) + \(\Mod(\mathbb{D}^2) = 1\) \end{example} \begin{example} - By the same token, \(\Mod(\mathbb{D} \setminus \{0\}) = 1\). + 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}) \to + 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}) \cong + group isomorphism. In particular, \(\Mod(\mathbb{T}^2) \cong \operatorname{SL}_2(\mathbb{Z})\). \end{example} @@ -137,15 +137,15 @@ % TODO: Can we prove this without using braid groups? \begin{example}\label{ex:mcg-twice-punctured-disk} - The mapping class group \(\Mod(\mathbb{D} \setminus \{-\sfrac{1}{2}, + 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 freely generated by \(f = [\phi]\), where \begin{align*} - \phi : \mathbb{D} \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\} - & \isoto \mathbb{D} \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\} \\ + \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. \end{align*} - In particular, \(\Mod(\mathbb{D} \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}) + In particular, \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}) \cong \mathbb{Z}\). \end{example}
diff --git a/sections/presentation.tex b/sections/presentation.tex @@ -82,7 +82,7 @@ presentations of \(\Mod(S_g)\) for \(g \le 2\) to show the Abelianization is giv \hline & & \\[-10pt] \(0\) & \(\mathbb{S}^2\) & \(0\) \\ - \(1\) & \(\mathbb{T}\) & \(\mathbb{Z}/12\) \\ + \(1\) & \(\mathbb{T}^2\) & \(\mathbb{Z}/12\) \\ \(2\) & \(S_2\) & \(\mathbb{Z}/10\) \\ \end{tabular} \end{center} @@ -110,13 +110,13 @@ 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, +study certain branched covers \(S \to \mathbb{D}^2 \setminus \{x_1, \ldots, x_r\}\) and how they may be used to relate both mapping class groups. This is what is known as\dots \section{The Birman-Hilden Theorem} -Let \(S_{0, r}^1 = \mathbb{D} \setminus \{x_1, \ldots, x_r\}\) be the surface +Let \(S_{0, r}^1 = \mathbb{D}^2 \setminus \{x_1, \ldots, x_r\}\) be the surface of genus \(0\) with \(r\) punctures and \(1\) boundary component. We begin our investigation by providing an alternative description of its mapping class group. Namely, we show that \(\Mod(S_{0, r}^1)\) is the braid group on \(r\) @@ -124,8 +124,8 @@ 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) = \mfrac{C^{\operatorname{ord}}(\mathbb{D}, + \(\pi_1(C(\mathbb{D}^2, n), *)\) of the unordered configuration space + \(C(\mathbb{D}^2, n) = \mfrac{C^{\operatorname{ord}}(\mathbb{D}^2, n)}{\mathfrak{S}_n}\) of \(n\) distinct points in the interior of the disk. The elements of \(B_n\) are referred to as \emph{braids}. \end{definition} @@ -177,12 +177,12 @@ sequence 1 \rar & B_n \rar{\operatorname{push}} & \Mod(S_{0, n}^1) \rar - & \Mod(\mathbb{D}) \rar + & \Mod(\mathbb{D}^2) \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 +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 \begin{proposition} @@ -233,7 +233,7 @@ 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 \(2\ell + +\to \mfrac{S_\ell^b}{\iota} \cong \mathbb{D}^2\) is a double cover with \(2\ell + b\) branch points corresponding to the fixed points of \(\iota\). We may thus regard \(\mfrac{S_\ell^b}{\iota}\) as the disk \(S_{0, 2\ell + b}^1\) with \(2\ell + b\) punctures in its interior, as shown in
diff --git a/sections/twists.tex b/sections/twists.tex @@ -34,14 +34,14 @@ Figure~\ref{fig:dehn-twist-bitorus} in the case of the bitorus \(S_2\). Similarly, using the description of the mapping class group of the twice-puncture disk derived in Example~\ref{ex:mcg-twice-punctured-disk}, the -generator of \(\Mod(\mathbb{D} \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\})\) +generator of \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\})\) gives rise the so called \emph{half-twists}. These are examples of mapping classes that permute the punctures of \(S\). \begin{definition} Given an arc \(\alpha \subset S\) joining two punctures in the interior of \(S\), fix a closed neighborhood \(D \subset S\) of \(\alpha\) with \(D \cong - \mathbb{D} \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}\). Let \(f \in + \mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}\). Let \(f \in \Mod(\mathbb{S}^1 \times [0, 1]) \cong \Mod(D)\) be as in Example~\ref{ex:mcg-twice-punctured-disk}. The \emph{half-twist \(h_\alpha \in \Mod(S)\) about \(\alpha\)} is defined as the image of the generator \(f @@ -279,7 +279,7 @@ cases, \(\mathcal{C}(S_{g, r})\) is connected. \begin{theorem}[Lickorish] If \(S_{g, r}\) is not one \(S_0 = \mathbb{S}^2, S_{0, 1}, \ldots, S_{0, 4}, - S_1 = \mathbb{T}\) and \(S_{1, 1}\) then \(\mathcal{C}(S_{g, r})\) is + S_1 = \mathbb{T}^2\) and \(S_{1, 1}\) then \(\mathcal{C}(S_{g, r})\) is connected. \end{theorem} @@ -312,23 +312,23 @@ Theorem~\ref{thm:mcg-is-fg}. 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} \setminus \{0\})) \rar & + \Mod(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 \(S_{g, n}\) is finitely generated by twists about nonseparating simple closed curves. - Indeed, if \(\PMod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D} \setminus \{0\}))\) + 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 components, then we may lift the generators of \(\PMod(S_{g, r}^b - \cup_{\delta_1} (\mathbb{D} \setminus \{0\}))\) to Dehn twists about the + \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\) to Dehn twists about the corresponding curves in \(S_{g, r}^b\) and add \(\tau_{\delta_1}\) to the generating set. It thus suffices to consider the boudaryless case \(S_{g, r}\). 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}) \cong \Mod(S_{1, 1}) \cong + \(\Mod(\mathbb{T}^2) \cong \Mod(S_{1, 1}) \cong \operatorname{SL}_2(\mathbb{Z})\) are generated by the Dehn twists about the curves \(\alpha\) and \(\beta\) from Figure~\ref{fig:torus-mcg-generators}, each corresponding to one of the @@ -348,9 +348,9 @@ Theorem~\ref{thm:mcg-is-fg}. \begin{figure}[ht] \centering - \includegraphics[width=.33\linewidth]{images/torus-mcg-generators.eps} + \includegraphics[width=.55\linewidth]{images/torus-mcg-generators.eps} \caption{The curves $\alpha$ and $\beta$ whose Dehn twists generate - $\Mod(\mathbb{T})$ and $\Mod(S_{1, 1})$.} + $\Mod(\mathbb{T}^2)$ and $\Mod(S_{1, 1})$.} \label{fig:torus-mcg-generators} \end{figure}