memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: 66b1395664fc31222820eeb52776df924ae950a1
Parent: 9e76324b67a948e1d1b2c2a963e778db205c5aaa
Author: Pablo <pablo-pie@riseup.net>
Date: Sun, 23 Jun 2024 08:58:02 +0000

Fixed some TODOs

Diffstat

3 files changed, 7 insertions, 9 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	1	0	1
Modified	sections/presentation.tex	14	7	7
Modified	sections/representations.tex	1	0	1

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -294,7 +294,6 @@ The symplectic representation already allows us to compute some important
 examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 =
 S_1\) and the once-punctured torus \(S_{1, 1}\).
 
-% TODO: Draw a diagram?
 \begin{example}[Alexander trick]\label{ex:alexander-trick}
   The group \(\Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) of homeomorphisms of the
   unit disk \(\mathbb{D}^2 \subset \mathbb{Z}\) is contractible. In particular,

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -114,18 +114,18 @@ strands.
 In his seminal paper on braid groups, Artin \cite{artin} gave the following
 finite presentation of \(B_n\).
 
-% TODO: Align this a little better?
 \begin{theorem}[Artin]
   \[
+    \arraycolsep=1.2pt
     B_n =
     \left\langle
     \sigma_1, \ldots, \sigma_{n - 1} :
-    \begin{aligned}
-      \sigma_i \sigma_{i+1} \sigma_i & = \sigma_{i+1} \sigma_i \sigma_{i+1}
-      \ \text{for all} \ i, \\
-      \sigma_i \sigma_j & = \sigma_j \sigma_i
-      \ \text{for} \ j \ne i + 1 \ \text{and} \ j \ne i - 1
-    \end{aligned}
+    \begin{array}{rll}
+      \sigma_i \sigma_{i+1} \sigma_i & = \sigma_{i+1} \sigma_i \sigma_{i+1} &
+      \quad \text{for all} \ i, \\
+      \sigma_i \sigma_j & = \sigma_j \sigma_i &
+      \quad \text{for} \ j \ne i + 1 \ \text{and} \ j \ne i - 1
+    \end{array}
     \right\rangle.
   \]
 \end{theorem}

diff --git a/sections/representations.tex b/sections/representations.tex
@@ -286,7 +286,6 @@ representations.
     \label{fig:korkmaz-proof-subsurface}
   \end{figure}
 
-  % TODO: Add more comments on the injectivity of this map?
   We claim that it suffices to find a \(m\)-dimensional
   \(\Mod(R)\)-invariant\footnote{Here we view $\Mod(R)$ as a subgroup of
   $\Mod(S_g^p)$ via the inclusion homomorphism $\Mod(R) \to \Mod(S_g^p)$ from