memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: 73ae3808ad8e75b7f57f9b674a59bafc358095a1
Parent: 63449610f5a478d808f080b2c710b7ce7d7ec22e
Author: Pablo <pablo-pie@riseup.net>
Date: Tue, 25 Jun 2024 15:59:08 +0000

Removed unnecessary whitespace

Diffstat

4 files changed, 12 insertions, 12 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	4	2	2
Modified	sections/presentation.tex	4	2	2
Modified	sections/representations.tex	12	6	6
Modified	sections/twists.tex	4	2	2

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -254,7 +254,7 @@ Here we collect a few fundamental examples of linear representations of
 \end{observation}
 
 Now by choosing \(k = 1\) we obtain the so called \emph{symplectic
-representation.} 
+representation.}
 
 \begin{observation}
   Recall \(H_1(\Sigma_g, \mathbb{Z}) \cong \mathbb{Z}^{2g}\), with standard basis
@@ -431,7 +431,7 @@ Another fundamental class of examples of representations are the so called
     & = \mathcal{F}(\Sigma) \otimes \mathcal{F}(\Sigma') &
     \mathcal{F}([W] \otimes [W'])
     & = \mathcal{F}([W]) \otimes \mathcal{F}([W']),
-  \end{align*} 
+  \end{align*}
   where \(\Vect\) denotes the category of finite-dimensional complex vector
   spaces.
 \end{definition}

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -353,7 +353,7 @@ to obtain other relations. Since \(\iota\) has \(2g + 2\) fixed points in
   If \(g \ge 2\) then we have an exact sequence
   \begin{center}
     \begin{tikzcd}
-      1 \rar 
+      1 \rar
       & \langle [\iota] \rangle \rar
       & C_{\Mod(\Sigma_g)}([\iota]) \rar
       & \Mod(\Sigma_{0, 2g + 2}) \rar
@@ -457,7 +457,7 @@ explained in terms of the geometry of curves in \(\Sigma_g\).
       \end{align*}
 
     \item The hyperelliptic relation \([a_{2g} \cdots a_1 a_1 \cdots a_{2g}, d]
-      = 1\), where \(d = n_g\) for \(n_1 = a_1\), \(n_2 = b_0\) and 
+      = 1\), where \(d = n_g\) for \(n_1 = a_1\), \(n_2 = b_0\) and
       \begin{align*}
         n_{i + 2} & = w_i n_i w_i^{-1} \\
         w_i & = (a_{2i + 4} a_{2i + 3} a_{2i + 2} n_{i + 1})

diff --git a/sections/representations.tex b/sections/representations.tex
@@ -128,7 +128,7 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
   some Abelian subgroup of \(\GL_n(\mathbb{C})\).
 
   \begin{enumerate}[leftmargin=1.9cm]
-    \item[\bfseries\color{highlight}(1) \& (4)] 
+    \item[\bfseries\color{highlight}(1) \& (4)]
       By the change of coordinates principle, both \(L_{\alpha_1}\) and
       \(L_{\beta_1}\) are conjugate to \(L_{\alpha_2} = \lambda\). But the
       only matrix conjugate to \(\lambda\) is \(\lambda\) itself. Hence
@@ -143,7 +143,7 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
       and \(L_{\beta_1}\) lie inside the subgroup of diagonal matrices -- an
       Abelian subgroup of \(\GL_n(\mathbb{C})\).
 
-    \item[\bfseries\color{highlight}(3) \& (6)] 
+    \item[\bfseries\color{highlight}(3) \& (6)]
       As before, it follows from the disjointness relations that \(E_{\alpha_2
       = \lambda} = \ker (L_{\alpha_2} - \lambda)\) and \(\ker (L_{\alpha_2} -
       \lambda)^2\) are invariant under both \(L_{\alpha_1}\) and
@@ -382,14 +382,14 @@ representations.
   individually.
 
   \begin{enumerate}
-    \item[\bfseries\color{highlight}(1)] 
+    \item[\bfseries\color{highlight}(1)]
       Here we use the change of coordinates principle: each \(L_{\alpha_i},
       L_{\beta_i}, L_{\gamma_i},  L_{\eta_i}\) is conjugate to \(L_{\alpha_g} =
       \lambda\), so all Lickorish generators of \(\Mod(\Sigma_g^p)\) act on
       \(\mathbb{C}^n\) as scalar multiplication by \(\lambda\) as well. Hence
       \(\rho(\Mod(\Sigma_g^p)) = \langle \lambda \rangle\) is Abelian.
 
-    \item[\bfseries\color{highlight}(2)] 
+    \item[\bfseries\color{highlight}(2)]
       In this case, \(W = \ker (L_{\alpha_g} - \lambda)^2\) is a
       \(2\)-dimensional \(\Mod(\Sigma)\)-invariant subspace.
   \end{enumerate}
@@ -494,7 +494,7 @@ main lemma}. Namely\dots
   \begin{align*}
     B_3^n & \to \GL_m(\mathbb{C}) \\
     a_i
-    & \mapsto 
+    & \mapsto
     \left(
     \begin{array}{c|c|c}
       1_{2(i-1)} & 0                                            & 0 \\ \hline
@@ -503,7 +503,7 @@ main lemma}. Namely\dots
     \end{array}
     \right) \\
     b_i
-    & \mapsto 
+    & \mapsto
     \left(
     \begin{array}{c|c|c}
       1_{2(i-1)} & 0                                             & 0 \\ \hline

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -23,7 +23,7 @@ classes fixing such arcs and curves. Formally, this translates to\dots
     \item \([\alpha_i] \ne [\alpha_j]\) for \(i \ne j\).
     \item Each pair \((\alpha_i, \alpha_j)\) crosses at most once.
     \item Given distinct \(i, j, k\), at least one of \(\alpha_i \cap \alpha_j,
-      \alpha_i \cap \alpha_k, \alpha_j \cap \alpha_k\) is empty. 
+      \alpha_i \cap \alpha_k, \alpha_j \cap \alpha_k\) is empty.
     \item The surface obtained by cutting \(\Sigma\) across the \(\alpha_i\) is a
       disjoint union of disks and once-punctured disks.
   \end{enumerate}
@@ -209,7 +209,7 @@ too.
   is not hard to check that \(\tau_\beta \tau_\alpha \cdot [\beta] =
   [\alpha]\). 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_\beta\), from which follows the \emph{braid relation}
   \[
     \tau_\alpha \tau_\beta \tau_\alpha = \tau_\beta \tau_\alpha \tau_\beta.
   \]