The cohomology of Torelli groups is algebraic

Abstract

The Torelli group of $W_g={\#}^g{S}^n\times S^n$ is the subgroup of the diffeomorphisms of $W_g$ fixing a disc which act trivially on $H_n(W_g;Z)$. The rational cohomology groups of the Torelli group are representations of an arithmetic subgroup of $S{p}_{2g}(Z)$ or $O_{g,g}(Z)$. In this paper we prove that for $2n\ge 6$ and $g\ge 2$, they are in fact algebraic representations. Combined with previous work, this determines the rational cohomology of the Torelli group in a stable range. We further prove that the classifying space of the Torelli group is nilpotent.

49 pages, to appear in Forum of Mathematics, Sigma