My M2 Memoire on mapping class groups & their representations
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
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