The action of the Torelli group on the homology of representation spaces is nontrivial
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.
scientific article; zbMATH DE number 1463174