memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: aa7daec951b8a73a3122359011066f72fba4fa26
Parent: f48da189d8d8cfb546f57b29b62391d766c0a7c9
Author: Pablo <pablo-pie@riseup.net>
Date: Sat, 22 Jun 2024 15:01:21 +0000

Created different example categories to help highlight the more important ones

Diffstat

4 files changed, 44 insertions, 39 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	preamble.tex	3	3	0
Modified	sections/introduction.tex	12	6	6
Modified	sections/presentation.tex	32	16	16
Modified	sections/twists.tex	36	19	17

diff --git a/preamble.tex b/preamble.tex
@@ -18,6 +18,9 @@
 \theoremstyle{definition}
 \newboxedtheorem{definition}{Definition}
 \newtheorem{example}[theorem]{Example}
+\newtheorem{fundamental-example}[theorem]{Fundamental Example}
+\newtheorem{observation}[theorem]{Observation}
+\newtheorem{fundamental-observation}[theorem]{Fundamental Observation}
 \newtheorem{fact}[theorem]{Fact}
 \theoremstyle{remark}
 \newtheorem*{note}{Remark}

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -227,7 +227,7 @@ homomorphisms \(\Mod(S) \to \GL_n(\mathbb{C})\). These may be seen as actions
 Here we collect a few fundamental examples of linear representations of
 \(\Mod(S)\).
 
-\begin{example}
+\begin{observation}
   Given \(k \ge 0\) and \(f = [\phi] \in \Mod(S)\), we may consider the map
   \(\phi_* : H_k(S, \mathbb{Z}) \to H_k(S, \mathbb{Z})\) induced at the level
   of singular homology. By homotopy invariance, the map \(\phi_*\) is
@@ -235,12 +235,12 @@ Here we collect a few fundamental examples of linear representations of
   functoriality of homology groups we then get a \(\mathbb{Z}\)-linear action
   \(\Mod(S) \leftaction H_k(S, R)\), given by \(f \cdot \xi = \phi_*(\xi)\) for
   \(f = [\phi] \in \Mod(S)\).
-\end{example}
+\end{observation}
 
 Now by choosing \(k = 1\) we obtain the so called \emph{symplectic
 representation.} 
 
-\begin{example}
+\begin{observation}
   Recall \(H_1(S_g, \mathbb{Z}) \cong \mathbb{Z}^{2g}\), with standard basis
   given by \([\alpha_1], [\beta_1], \ldots, [\alpha_g], [\beta_g] \in H_1(S_g,
   \mathbb{Z})\) for \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g\) as
@@ -258,7 +258,7 @@ representation.}
   \end{align}
   and thus coincides with the pullback of the standard \(\mathbb{Z}\)-bilinear
   symplectic form in \(\mathbb{Z}^{2g}\).
-\end{example}
+\end{observation}
 
 \begin{example}[Symplectic representation]\label{ex:symplectic-rep}
   Consider the \(\mathbb{Z}\)-linear action \(\Mod(S_g) \leftaction H_1(S_g,
@@ -416,7 +416,7 @@ Another fundamental class of examples of representations are the so called
   spaces.
 \end{definition}
 
-\begin{example}
+\begin{observation}
   Given \(\phi \in \Homeo^+(S_g)\), we may consider the so called \emph{mapping
   cylinder} \(M_\phi = (S_g \times [0, 1], \phi, 1)\), a cobordism between
   \(S_g\) and itself -- where \(\partial_+ (S_g \times [0, 1]) = S_g \times 0\)
@@ -424,7 +424,7 @@ Another fundamental class of examples of representations are the so called
   class of \(M_\phi\) is independant of the choice of representative of \(f =
   [\phi] \in \Mod(S_g)\), so \(M_f = [M_\phi] : S_g \to S_g\) is a well defined
   morphism in \(\Cob\).
-\end{example}
+\end{observation}
 
 \begin{example}[TQFT representations]\label{ex:tqft-reps}
   It is clear that \(M_1\) is the identity morphism \(S_g \to S_g\) in

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -5,14 +5,14 @@ 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.
+Observation~\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}
+\begin{fundamental-observation}
   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,
@@ -23,7 +23,7 @@ intuition behind them. We start by perhaps the simplest of these, known as
     \tau_\alpha \tau_\beta \tau_\gamma
     = \tau_{\delta_1} \tau_{\delta_2} \tau_{\delta_3} \tau_{\delta_4}
   \end{equation}
-\end{example}
+\end{fundamental-observation}
 
 \begin{figure}[ht]
   \centering
@@ -148,7 +148,7 @@ strands.
   \[
     \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_1 \sigma_i,
   \]
-  in \(B_n\), which motivates the name used in Example~\ref{ex:braid-relation}.
+  in \(B_n\), which motivates the name used in Observation~\ref{ex:braid-relation}.
 \end{note}
 
 In his seminal paper on braid groups, Artin \cite{artin} gave the following
@@ -193,7 +193,7 @@ we get\dots
 
 
 \begin{minipage}[b]{.45\linewidth}
-\begin{example}\label{ex:braid-group-center}
+\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 \(Z(\Mod(S_{0, n}^1))\)
   of \(\Mod(S_{0, n}^1)\) is freely generated by the Dehn twist \(\tau_\delta\)
@@ -203,7 +203,7 @@ we get\dots
   \(\sfrac{2\pi}{n}\) 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{observation}
 \end{minipage}
 \hspace{.5cm} %
 \begin{minipage}[b]{.45\textwidth}
@@ -251,12 +251,12 @@ S_{0, 2\ell + b}^1\) joining the punctures of the quotient surface.
   \label{fig:hyperelliptic-covering}
 \end{figure}
 
-\begin{example}\label{ex:push-generators-description}
+\begin{observation}\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}
+\end{observation}
 
 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
@@ -318,24 +318,24 @@ not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden
   \end{align*}
 \end{theorem}
 
-\begin{example}
+\begin{observation}
   Using the notation of Figure~\ref{fig:hyperelliptic-covering}, the
   Birman-Hilden isomorphism \(\SMod(S_\ell^p) \isoto \Mod(S_{0, 2g + b})\)
   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}
+\end{observation}
 
-\begin{example}
+\begin{fundamental-observation}
   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
+  Observation~\ref{ex:push-generators-descriptions},
+  Observation~\ref{ex:braid-group-center} gives us the so called \emph{\(k\)-chain
   relations} in \(\SMod(S_\ell^p) \subset \Mod(S_g)\).
   \[
     \arraycolsep=1.4pt
@@ -351,7 +351,7 @@ not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden
       \; \text{for } k = 2 \ell + 1 \text{ odd}
     \end{array}
   \]
-\end{example}
+\end{fundamental-observation}
 
 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 \(S_g\),
@@ -373,7 +373,7 @@ we get branched double cover \(S_g \to S_{0, 2g+2}\).
   to \([\bar \phi] \in \Mod(S_{0, 2g + 2})\).
 \end{theorem}
 
-\begin{example}
+\begin{fundamental-observation}
   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}
@@ -401,7 +401,7 @@ we get branched double cover \(S_g \to S_{0, 2g+2}\).
     \tau_{\alpha_1} \cdots \tau_{\alpha_{2g}} \tau_\delta, \tau_\delta]
     & = 1
   \end{align*}
-\end{example}
+\end{fundamental-observation}
 
 \begin{minipage}[b]{.45\textwidth}
   \centering

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -139,44 +139,46 @@ we can distinguish between powers of Dehn twists
   has infinite order.
 \end{proposition}
 
-\begin{example}\label{thm:dehn-twist-is-uniq}
+\begin{observation}
   Given \(\alpha, \beta \subset S\), \(\tau_\alpha = \tau_\beta \iff [\alpha] =
   [\beta]\). Indeed, if \(\alpha\) and \(\beta\) are non-isotopic, we can find
   \(\gamma\) with \(\#(\gamma \cap \alpha) > 0\) and \(\#(\gamma \cap \beta) =
   0\). It thus follows from Proposition~\ref{thm:twist-intersection-number}
   that \(\#(T_\alpha(\gamma) \cap \gamma) > \#(T_\beta(\gamma) \cap \gamma)\),
   so \(\tau_\alpha \ne \tau_\beta\).
-\end{example}
+\end{observation}
 
 Many other relations between Dehn twists can derrived be in a geometric fashion
 too.
 
-\begin{example}\label{thm:conjugate-twists}
+\begin{observation}\label{ex:conjugate-twists}
   Given \(f = [\phi] \in \Mod(S)\), \(\tau_{\phi(\alpha)} = f \tau_\alpha
   f^{-1}\).
-\end{example}
+\end{observation}
 
-\begin{example}
-  \([f, \tau_\alpha] = 1 \iff f \cdot [\alpha] = [\alpha]\).
-\end{example}
+\begin{observation}
+  Given \(f \in \Mod(S)\), \([f, \tau_\alpha] = 1 \iff f \cdot [\alpha] =
+  [\alpha]\). In particular, \([\tau_\alpha, \tau_\beta] = 1\) for \(\alpha\)
+  and \(\beta\) disjoint.
+\end{observation}
 
-\begin{example}
+\begin{observation}
   If \(\alpha, \beta \subset S\) are both nonseparing then \(\tau_\alpha,
   \tau_\beta \in \Mod(S)\) are conjugate. Indeed, by the change of coordinates
   principle we can find \(f \in \Mod(S)\) with \(f \cdot [\alpha] = [\beta]\)
-  and then apply Fact~\ref{thm:conjugate-twists}.
-\end{example}
+  and then apply Observation~\ref{ex:conjugate-twists}.
+\end{observation}
 
-\begin{example}\label{ex:braid-relation}
+\begin{fundamental-observation}\label{ex:braid-relation}
   Given \(\alpha, \beta \subset S\) with \(\#(\alpha \cap \beta) = 1\), it is
   not hard to check that \(\tau_\beta \tau_\alpha \cdot [\beta] = [\alpha]\).
-  From Fact~\ref{thm:conjugate-twists} we then get \((\tau_\alpha \tau_\beta)
+  From Observation~\ref{ex:conjugate-twists} we then get \((\tau_\alpha \tau_\beta)
   \tau_\alpha (\tau_\alpha \tau_\beta)^{-1} = \tau_\beta\), from which follows
   the \emph{braid relation} 
   \[
     \tau_\alpha \tau_\beta \tau_\alpha = \tau_\beta \tau_\alpha \tau_\alpha.
   \]
-\end{example}
+\end{fundamental-observation}
 
 A perhaps less obvious fact about Dehn twists is\dots
 
@@ -267,11 +269,11 @@ then be seen as the mapping class that ``\emph{pushes} a neighborhood of
 in Figure~\ref{fig:push-map} for the case \(n = 1\). Indeed, this goes to
 show\dots
 
-\begin{example}\label{ex:push-simple-loop}
+\begin{fundamental-observation}\label{ex:push-simple-loop}
   Using the notation of Figure~\ref{fig:push-map},
   \(\operatorname{push}([\alpha]) = \tau_{\delta_1} \tau_{\delta_2}^{-1} \in
   \Mod(S)\).
-\end{example}
+\end{fundamental-observation}
 
 \begin{figure}[ht]
   \centering
@@ -430,7 +432,7 @@ Theorem~\ref{thm:mcg-is-fg}.
   \end{center}
   where \(S_{g, r + 1} = S_{g, r} \setminus \{x\}\). Since \(g \ge 1\),
   \(\pi_1(S_{g, r}, x)\) is generated by finitely many nonseparating loops.
-  We have seen in Example~\ref{ex:push-simple-loop} that \(\operatorname{push}
+  We have seen in Observation~\ref{ex:push-simple-loop} that \(\operatorname{push}
   : \pi_1(S_{g, r}, x) \to \Mod(S_{g, r+1}, x)\) takes nonseparation simple
   loops to products of twists about nonseparating simple curves. Furthermore,
   we may once again lift the generators of \(\PMod(S_{g, r})\) to Dehn twists
@@ -445,7 +447,7 @@ Theorem~\ref{thm:mcg-is-fg}.
   1}) \leftaction \hat{\mathcal{N}}(S_{g + 1})\). Since \(g + 1 \ge 2\),
   \(\hat{\mathcal{N}}(S_{g + 1})\) is connected and the conditions of
   Lemma~\ref{thm:ggt-lemma} are met. Now recall from
-  Example~\ref{ex:braid-relation} that, given nonseparating \(\alpha, \beta
+  Observation~\ref{ex:braid-relation} that, given nonseparating \(\alpha, \beta
   \subset S_{g + 1}\) crossing once, \(\tau_\beta \tau_\alpha \cdot [\beta] =
   [\alpha]\). Hence by Lemma~\ref{thm:ggt-lemma} \(\Mod(S_{g + 1})\) is
   generated by \(\Mod(S_{g + 1})_{[\alpha]} = \{ f \in \Mod(S_{g + 1}) : f