memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: 823af40175aabbc4d09137582736306de69d4d6e
Parent: 838b421afaf9aa157d877619768ebcdd903fb812
Author: Pablo <pablo-pie@riseup.net>
Date: Tue, 25 Jun 2024 13:35:49 +0000

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

Diffstat

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 &