The action of the Torelli group on the homology of representation spaces is nontrivial
type: Item
instance of: scholarly article
title: The action of the Torelli group on the homology of representation spaces is nontrivial
zbMATH Open document ID: 0971.57024
DOI: 10.1016/S0040-9383(99)00044-0
published in: Topology
publication date: 2001-11-01T00:00:00Z
review text: The mapping class group and its subgroup, the Torelli group, of the Riemannian surface \(M\) have a natural action on the space \(R_{\text{SU}(2)}(M)\) of \(\text{SU}(2)\)-representations of the fundamental group of \(M\) and its subspace \(R_{\text{SU}(2)}(M)_{\text{irred}}\) of irreducibles. In this paper the authors compute the cohomologies \(H^*(R_{\text{SU}(2)}(M))\), \(H^*(R_{\text{SU}(2)}(M)_{\text{irred}})\) of both of these spaces and show that the induced action of the Torelli group is non-trivial. They develop an obstruction theory to study a unipotent group action on a short exact sequence where the action is trivial at the two end terms. Applying this theory, the triviality or non-triviality of the action by an element \(\gamma\) of \(I_{(g,1)}\) on \(H^d_{\text{SU}(2)}(R_{\text{SU}(2)}(M)^\#\), \(R_{\text{SU}(2)}(M)^\#_{\text{red}})\) is equivalent to a certain obstruction map \(\Psi_d(\gamma): \ker i^*_d\to \text{coker }i^*_{d- 1}\) being zero or not. The authors combine the methods of Atiyah and Bott with those of Johnson to place the problem in the general framework of computing characteristic classes.
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 57T10
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 20F34
zbMATH DE Number: 1463174
zbMATH Keywords: representation theory
zbMATH Keywords: moduli space
zbMATH Keywords: mapping class group
author: Sylvain Cappell
author: R. Lee
author: E. Y. Miller
reviewed by: Marian Ioan Munteanu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1463174