My M2 Memoire on mapping class groups & their representations
Moved some comments
Fixed the use of a definition before its statement: moved comments on the capping exact sequence and the kernel of the cutting homomorphism to after the definition of Dehn twists
3 files changed, 62 insertions, 47 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/introduction.tex | 21 | 3 | 18 |
| Modified | sections/presentation.tex | 38 | 19 | 19 |
| Modified | sections/twists.tex | 50 | 40 | 10 |
diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -208,26 +208,13 @@ 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} +\begin{example}[Capping exact homomorphism]\label{ex:capping-morphism} 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_\delta \rangle \rar & - \Mod(\Sigma) \rar{\operatorname{cap}} & - \Mod(\Sigma \cup_\delta (\mathbb{D}^2 \setminus \{0\}), 0) \rar & - 1, - \end{tikzcd} - \end{center} - known as \emph{the capping exact sequence} -- see - \cite[Proposition~3.19]{farb-margalit} for a proof. + \emph{the capping homomorphism}. \end{example} -% TODOOO: You haven't defined Dehn twists yet!!!!! \begin{example}[Cutting homomorphism]\label{ex:cutting-morphism} Given a simple closed curve \(\alpha \subset \Sigma\), any \(f \in \Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\) has a representative \(\phi \in @@ -239,9 +226,7 @@ class groups. & \to \Mod(\Sigma\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[Proposition~3.20]{farb-margalit} for a proof. + known as \emph{the cutting homomorphism}. \end{example} As goes for most groups, another approach to understanding the mapping class
diff --git a/sections/presentation.tex b/sections/presentation.tex @@ -26,16 +26,16 @@ Theorem~\ref{thm:wajnryb-presentation}. = \tau_\beta \tau_{\delta_4}^{-1} \in \Mod(\Sigma_{0, 1}^3). \] - Using the capping exact sequence from Example~\ref{ex:capping-seq}, we can - then see \(\tau_{\delta_2} \tau_\alpha^{-1} \tau_{\delta_3} \tau_\gamma^{-1}, - \tau_\beta \tau_{\delta_4}^{-1} \in \Mod(\Sigma_0^4)\) differ by a power of - \(\tau_{\delta_1}\). In fact, it follows from the Alexander method that - \((\tau_{\delta_2} \tau_\alpha^{-1} \tau_{\delta_3} \tau_\gamma^{-1}) - (\tau_\beta \tau_{\delta_4}^{-1})^{-1} = \tau_{\delta_1}^{-1} \in - \Mod(\Sigma_0^4)\). Now the disjointness relations \([\tau_{\delta_i}, - \tau_\alpha] = [\tau_{\delta_i}, \tau_\beta] = [\tau_{\delta_i}, \tau_\gamma] - = 1\) give us the \emph{lantern relation} (\ref{eq:lantern-relation}) in - \(\Mod(\Sigma_0^4)\). + Using the capping exact sequence from Observation~\ref{ex:capping-seq}, we + can then see \(\tau_{\delta_2} \tau_\alpha^{-1} \tau_{\delta_3} + \tau_\gamma^{-1}, \tau_\beta \tau_{\delta_4}^{-1} \in \Mod(\Sigma_0^4)\) + differ by a power of \(\tau_{\delta_1}\). In fact, it follows from the + Alexander method that \((\tau_{\delta_2} \tau_\alpha^{-1} \tau_{\delta_3} + \tau_\gamma^{-1}) (\tau_\beta \tau_{\delta_4}^{-1})^{-1} = + \tau_{\delta_1}^{-1} \in \Mod(\Sigma_0^4)\). Now the disjointness relations + \([\tau_{\delta_i}, \tau_\alpha] = [\tau_{\delta_i}, \tau_\beta] = + [\tau_{\delta_i}, \tau_\gamma] = 1\) give us the \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} @@ -185,15 +185,15 @@ get\dots \noindent \begin{minipage}[b]{.47\linewidth} \begin{observation}\label{ex:braid-group-center} - Using the capping exact sequence from Example~\ref{ex:capping-seq} and - the Alexander method, one can check that the center \(Z(\Mod(\Sigma_{0, n}^1))\) - of \(\Mod(\Sigma_{0, n}^1)\) is freely generated by the Dehn twist \(\tau_\delta\) - about the boundary \(\delta = \partial \Sigma_{0, n}^1\). It is also not very hard - to see that \(\operatorname{push} : B_n \to \Mod(\Sigma_{0, n}^1)\) takes - \(\sigma_1 \cdots \sigma_{n-1}\) to the rotation by - \(\sfrac{2\pi}{n}\) as in Figure~\ref{fig:braid-group-center}, which is an - \(n\)-th root of \(\tau_\delta\). Hence the center \(Z(B_n)\) is freely - generated by \(z = (\sigma_1 \cdots \sigma_{n - 1})^n\). + Using the capping exact sequence from Observation~\ref{ex:capping-seq} and + the Alexander method, one can check that the center \(Z(\Mod(\Sigma_{0, + n}^1))\) of \(\Mod(\Sigma_{0, n}^1)\) is freely generated by the Dehn twist + \(\tau_\delta\) about the boundary \(\delta = \partial \Sigma_{0, n}^1\). It + is also not very hard to see that \(\operatorname{push} : B_n \to + \Mod(\Sigma_{0, n}^1)\) takes \(\sigma_1 \cdots \sigma_{n-1}\) to the + rotation by \(\sfrac{2\pi}{n}\) as in Figure~\ref{fig:braid-group-center}, + which is an \(n\)-th root of \(\tau_\delta\). Hence the center \(Z(B_n)\) is + freely generated by \(z = (\sigma_1 \cdots \sigma_{n - 1})^n\). \end{observation} \end{minipage} \hspace{.6cm} %
diff --git a/sections/twists.tex b/sections/twists.tex @@ -121,7 +121,37 @@ classes that permute the punctures of \(\Sigma\). the inclusion homomorphism \(\Mod(D) \to \Mod(\Sigma)\). \end{definition} -It is interesting to study how the geometry of two curves affects the +We can use the Alexander method to describe the kernel of capping and cutting +morphisms in terms of Dehn twists. + +\begin{observation}[Capping exact sequence]\label{ex:capping-seq} + Let \(\delta \subset \partial \Sigma\) be a boundary component of \(\Sigma\) + and \(\operatorname{cap} : \Mod(\Sigma) \to \Mod(\Sigma \cup_\delta + (\mathbb{D}^2 \setminus \{0\}))\) be the corresponding the capping + homomorphism from Example~\ref{ex:capping-morphism}. There is an exact + sequence + \begin{center} + \begin{tikzcd} + 1 \rar & + \langle \tau_\delta \rangle \rar & + \Mod(\Sigma) \rar{\operatorname{cap}} & + \Mod(\Sigma \cup_\delta (\mathbb{D}^2 \setminus \{0\}), 0) \rar & + 1, + \end{tikzcd} + \end{center} + known as \emph{the capping exact sequence} -- see + \cite[Proposition~3.19]{farb-margalit} for a proof. +\end{observation} + +\begin{observation}\label{ex:cutting-morphism-kernel} + Let \(\alpha \subset \Sigma\) be a simple closed curve and + \(\operatorname{cut} : \Mod(\Sigma)_{\vec{[\alpha]}} \to \Mod(\Sigma + \setminus \alpha)\) be the cutting homomorphism from + Example~\ref{ex:cutting-morphism}. Then \(\ker \operatorname{cut} = \langle + \tau_\alpha \rangle \cong \mathbb{Z}\). +\end{observation} + +It is also interesting to study how the geometry of two curves affects the relationship between their corresponding Dehn twists. For instance, by investigating the geometric intersection number \[ @@ -385,14 +415,14 @@ Theorem~\ref{thm:mcg-is-fg}. 1 \end{tikzcd} \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 \(\PMod(\Sigma_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus - \{0\})) \cong \PMod(\Sigma_{g, r+1}^{p-1})\) 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. + from Observation~\ref{ex:capping-seq}, it suffices to show that \(\Sigma_{g, + n}\) is finitely generated by twists about nonseparating simple closed + curves. Indeed, if \(\PMod(\Sigma_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 + \setminus \{0\})) \cong \PMod(\Sigma_{g, r+1}^{p-1})\) 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 @@ -476,7 +506,7 @@ Theorem~\ref{thm:mcg-is-fg}. 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 + Observation~\ref{ex:cutting-morphism-kernel} then gives us an exact sequence \begin{equation}\label{eq:cutting-seq} \begin{tikzcd} 1 \rar &