On a function on the mapping class group of a surface of genus 2
type: Item
instance of: scholarly article
title: On a function on the mapping class group of a surface of genus 2
zbMATH Open document ID: 0946.57019
DOI: 10.1016/S0166-8641(98)00157-6
author: Ryoji Kasagawa
published in: Topology and its Applications
publication date: 2000-10-19T00:00:00Z
review text: The mapping class group \({\mathcal M}_g\) of a surface \(\Sigma_g\) of genus \(g \geq 2\) admits a natural action on \(H^1(\Sigma_g, Z)\). Such an action preserves the intersection pairing on the surface which is a standard symplectic form, and induces the classical representation \(\rho: {\mathcal M}_g \to \Aut H^1(\Sigma, Z)\). By fixing a symplectic basis of \(H^1(\Sigma_g, Z)\), we can identify \(\Aut H^1(\Sigma, Z)\) with the Siegel modular group \(\operatorname {Sp}(2g, Z)\). Hence \(\rho: {\mathcal M}_g \to \text{Sp}(2g, Z)\) is a non-trivially surjective homomorphism with the Torelli group \({\mathcal T}_g\) defined to be the kernel of \(\rho\). The author extends the above representation to a twist-version as the following. Any nonzero class \(w \in H^1(\Sigma_g, Z_2) = \Hom(\pi_1(\Sigma, \star), Z_2)\) defines a nontrivial flat \(O(2)\)-bundle \(P\) over \(\Sigma_g\) (for \(w = 0\), it defines a trivial \(O(2)\)-bundle over \(\Sigma_g\) and everything is back to the classical situation). Let \(G\) be the automorphisms of \(P\) which preserve an orientation of the fiber at the base point \(\star\), and \(M_P\) be the moduli space of flat \(O(2)\)-connections modulo the gauge group \(G\). Let \({\mathcal M}_{g^{\star}}^w\) be the diffeomorphisms of \(\Sigma_g\) preserving the base point, the orientation of \(\Sigma_g\) and the cohomology class \(w\). For \(f \in {\mathcal M}_{g^{\star}}^w\), there is a lift \(f_{\star}\in G\) which provides an action on \(M_P\). This action is independent of the choice of a lift and a representation of the isotopy class of \(f\). The moduli space \(M_P\) can be identified with \(T^{2(g-1)}\) and the action of \({\mathcal M}_{g^{\star}}^w\) gives \[ \rho_{g^{\star}}^w: {\mathcal M}_{g^{\star}}^w \to \Aut H^1(\pi_1(\Sigma_g, \star), Z_w) = \operatorname {Sp}(2(g-1), Z) \] with twisted Torelli group \({\mathcal T}_{g^{\star}}^w\) as the kernel of \(\rho_{g^{\star}}^w\). The main result of the paper is to prove that \(\rho_{g^{…
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 57M20
zbMATH DE Number: 1421596
zbMATH Keywords: Torelli group
zbMATH Keywords: cohomology with twisted coefficients
zbMATH Keywords: representation
zbMATH Keywords: Meyer function
zbMATH Keywords: signature of surface bundle
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1421596