memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
Log
Files
Refs
README
LICENSE

Commit: b14978121a73375c6426854aa0e772cf3af1344a
Parent: f81481919a75ac63f8b9e50ed8d0e551b2f5bc59
Author: Pablo <pablo-pie@riseup.net>
Date: Wed, 26 Jun 2024 12:20:54 +0000

Condensed one example

Condensed the example of the symplectic representation

In the case of surfaces, the only action Mod(Σ) ↺ H_k(Σ, ℤ) of interest is k = 1: the action on the other homology groups is trivial

Diffstat

1 file changed, 25 insertions, 31 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	56	25	31

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -243,27 +243,15 @@ Here we collect a few fundamental examples of linear representations of
 \(\Mod(\Sigma)\).
 
 \begin{observation}
-  Given \(k \ge 0\) and \(f = [\phi] \in \Mod(\Sigma)\), we may consider the map
-  \(\phi_* : H_k(\Sigma, \mathbb{Z}) \to H_k(\Sigma, \mathbb{Z})\) induced at the level
-  of singular homology. By homotopy invariance, the map \(\phi_*\) is
-  independent of the choice of representative \(\phi\) of \(f\). By the
-  functoriality of homology groups we then get a \(\mathbb{Z}\)-linear action
-  \(\Mod(\Sigma) \leftaction H_k(\Sigma, R)\), given by \(f \cdot \xi = \phi_*(\xi)\) for
-  \(f = [\phi] \in \Mod(\Sigma)\).
-\end{observation}
-
-Now by choosing \(k = 1\) we obtain the so called \emph{symplectic
-representation.}
-
-\begin{observation}
-  Recall \(H_1(\Sigma_g, \mathbb{Z}) \cong \mathbb{Z}^{2g}\), with standard basis
-  given by \([\alpha_1], [\beta_1], \ldots, [\alpha_g], [\beta_g] \in H_1(\Sigma_g,
-  \mathbb{Z})\) for \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g\) as
-  in Figure~\ref{fig:homology-basis}. The Abelian group \(H_1(\Sigma_g,
-  \mathbb{Z})\) is endowed with a natural \(\mathbb{Z}\)-bilinear alternating
-  form given by the \emph{algebraic intersection number} \([\alpha] \cdot
-  [\beta] = \sum_{x \in \alpha \cap \beta} \operatorname{ind}\,x\) -- where the
-  index \(\operatorname{ind}\,x = \pm 1\) of an intersection point is given by
+  Recall \(H_1(\Sigma_g, \mathbb{Z}) \cong \mathbb{Z}^{2g}\), with standard
+  basis given by \([\alpha_1], [\beta_1], \ldots, [\alpha_g], [\beta_g] \in
+  H_1(\Sigma_g, \mathbb{Z})\) for \(\alpha_1, \ldots, \alpha_g, \beta_1,
+  \ldots, \beta_g\) as in Figure~\ref{fig:homology-basis}. The Abelian group
+  \(H_1(\Sigma_g, \mathbb{Z})\) is endowed with a natural
+  \(\mathbb{Z}\)-bilinear alternating form given by the \emph{algebraic
+  intersection number} \([\alpha] \cdot [\beta] = \sum_{x \in \alpha \cap
+  \beta} \operatorname{ind}\,x\) -- where the index \(\operatorname{ind}\,x =
+  \pm 1\) of an intersection point is given by
   Figure~\ref{fig:intersection-index}. In terms of the standard basis of
   \(H_1(\Sigma_g, \mathbb{Z})\), this form is given by
   \begin{align}\label{eq:symplectic-form}
@@ -276,16 +264,22 @@ representation.}
 \end{observation}
 
 \begin{example}[Symplectic representation]\label{ex:symplectic-rep}
-  Consider the \(\mathbb{Z}\)-linear action \(\Mod(\Sigma_g) \leftaction H_1(\Sigma_g,
-  \mathbb{Z}) \cong \mathbb{Z}^{2g}\). Since pushforwards by
-  orientation-preserving homeomorphisms preserve the index of intersection
-  points, \((f \cdot [\alpha]) \cdot (f \cdot [\beta]) = [\alpha] \cdot
-  [\beta]\) for all \(\alpha, \beta \subset \Sigma_g\) and \(f \in \Mod(\Sigma_g)\). In
-  light of (\ref{eq:symplectic-form}), this implies \(\Mod(\Sigma_g)\) acts on
-  \(\mathbb{Z}^{2g}\) via \(\mathbb{Z}\)-linear symplectomorphisms. We thus
-  obtain a group homomorphism \(\psi : \Mod(\Sigma_g) \to
-  \operatorname{Sp}_{2g}(\mathbb{Z}) \subset \GL_{2g}(\mathbb{C})\), known as
-  \emph{the symplectic representation of \(\Mod(\Sigma_g)\)}.
+  Given \(f = [\phi] \in \Mod(\Sigma_g)\), we may consider the map \(\phi_* :
+  H_1(\Sigma_g, \mathbb{Z}) \to H_1(\Sigma_g, \mathbb{Z})\) induced at the
+  level of singular homology. By homotopy invariance, the map \(\phi_*\) is
+  independent of the choice of representative \(\phi\) of \(f\). By the
+  functoriality of homology groups we then get a \(\mathbb{Z}\)-linear action
+  \(\Mod(\Sigma_g) \leftaction H_1(\Sigma_g, \mathbb{Z}) \cong
+  \mathbb{Z}^{2g}\), given by \(f \cdot [\alpha] = \phi_*([\alpha]) =
+  [\phi(\alpha)]\). Since pushforwards by orientation-preserving homeomorphisms
+  preserve the index of intersection points, \((f \cdot [\alpha]) \cdot (f
+  \cdot [\beta]) = [\alpha] \cdot [\beta]\) for all \(\alpha, \beta \subset
+  \Sigma_g\) and \(f \in \Mod(\Sigma_g)\). In light of
+  (\ref{eq:symplectic-form}), this implies \(\Mod(\Sigma_g)\) acts on
+  \(\mathbb{Z}^{2g}\) via symplectomorphisms. We thus obtain a group
+  homomorphism \(\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})
+  \subset \GL_{2g}(\mathbb{C})\), known as \emph{the symplectic representation
+  of \(\Mod(\Sigma_g)\)}.
 \end{example}
 
 \noindent