memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: 5865e61fbc16a4115c9645b8e0f73c53e5dd0927
Parent: 9eb5957567e45263d37fb9dd8ebc31adc9aa6fa9
Author: Pablo <pablo-pie@riseup.net>
Date: Tue, 25 Jun 2024 11:27:08 +0000

Added some further details on the lantern relation

Diffstat

3 files changed, 551 insertions, 226 deletions

Status	File Name	N° Changes	Insertions	Deletions
Added	images/lantern-relation-capped.svg	173	173	0
Modified	images/lantern-relation.svg	544	332	212
Modified	sections/presentation.tex	60	46	14

diff --git a/images/lantern-relation-capped.svg b/images/lantern-relation-capped.svg
@@ -0,0 +1,173 @@
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diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -1,6 +1,6 @@
 \chapter{Relations Between Twists}\label{ch:relations}
 
-Having found a convenient set of generators for \(\Mod(\Sigma)\), it is now
+Having found a convenient set of generators for \(\Mod(\Sigma_g)\), 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
@@ -10,26 +10,58 @@ Theorem~\ref{thm:wajnryb-presentation}.
 
 We start by the so called \emph{lantern relation}.
 
-% TODO: Add a sketch of proof?
 \begin{fundamental-observation}
   Let \(\Sigma_0^4\) 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 \(\alpha, \beta, \gamma,
-  \delta_1, \ldots, \delta_4 \subset \Sigma_0^4\) are as in
-  Figure~\ref{fig:latern-relation} then from the Alexander method we get the
-  \emph{lantern relation} (\ref{eq:lantern-relation}) in \(\Mod(\Sigma_0^4)\).
+  components and \(\alpha, \beta, \gamma, \delta_1, \ldots, \delta_4 \subset
+  \Sigma_0^4\) be as in Figure~\ref{fig:latern-relation}. Consider the
+  surfaces \(\Sigma_0^3 = \Sigma_0^4 \cup_{\delta_1} \mathbb{D}^2\) and
+  \(\Sigma_{0,1}^3 = \Sigma_0^4 \cup_{\delta_1} (\mathbb{D}^2 \setminus \{ 0
+  \})\), as well as the map \(\operatorname{push} : \pi_1(\Sigma_0^3, 0) \to
+  \Mod(\Sigma_{0,1}^3)\). Let \(\eta_1, \eta_2, \eta_3 \subset \Sigma_0^3\) be
+  the loops from Figure~\ref{fig:lantern-relation-capped}, so that \([\eta_1]
+  \cdot [\eta_2] = [\eta_3]\). From Observation~\ref{ex:push-simple-loop} we
+  obtain
+  \[
+    (\tau_{\delta_2} \tau_\alpha^{-1}) (\tau_{\delta_3} \tau_\gamma^{-1})
+    = \operatorname{push}([\eta_1]) \cdot \operatorname{push}([\eta_2])
+    = \operatorname{push}([\eta_3])
+    = \tau_\beta \tau_{\delta_4}^{-1}
+    \in \Mod(\Sigma_{0, 1}^3).
+  \]
+  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
+  \(\Mod(\Sigma_0^4)\).
   \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}
 \end{fundamental-observation}
 
-\begin{figure}[ht]
+\noindent
+\begin{minipage}[t]{.5\linewidth}
   \centering
-  \includegraphics[width=.25\linewidth]{images/lantern-relation.eps}
-  \caption{The curves from the lantern relation of $\Sigma_0^4$.}
+  \includegraphics[width=\linewidth]{images/lantern-relation.eps}
+  \captionof{figure}{Two views of $\Sigma_0^4$: on the left-hand side we see
+  the \emph{lantern-like} surface we get by subtracting \(4\) disjoint open
+  disks from \(\mathbb{S}^2\), and on the right-hand side we see the disk with
+  three open disks subtracted from its interior.}
   \label{fig:latern-relation}
-\end{figure}
+\end{minipage}
+\hspace{.6cm} %
+\begin{minipage}[t]{.44\linewidth}
+  \centering
+  \includegraphics[width=.45\linewidth]{images/lantern-relation-capped.eps}
+  \captionof{figure}{The curves $\eta_1, \eta_2, \eta_3 \subset \Sigma_0^3$
+  from the proof of the lantern relation.}
+  \label{fig:lantern-relation-capped}
+\end{minipage}
 
 We may exploit different embeddings \(\Sigma_0^4 \hookrightarrow \Sigma\) and
 their corresponding inclusion homomorphisms \(\Mod(\Sigma_0^4) \to
@@ -74,9 +106,9 @@ twists in \(\Mod(\Sigma)\). For example\dots
 \end{figure}
 
 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
+\to \mathbb{D}^2 \setminus \{x_1, \ldots, x_r\}\), as well as the 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}\label{birman-hilden}