The cohomology of Torelli groups is algebraic

Abstract

Abstract The Torelli group of $W_g = \#^g S^n \times S^n$ is the group of diffeomorphisms of $W_g$ fixing a disc that act trivially on $H_n(W_g;\mathbb{Z} )$ . The rational cohomology groups of the Torelli group are representations of an arithmetic subgroup of $\text{Sp}_{2g}(\mathbb{Z} )$ or $\text{O}_{g,g}(\mathbb{Z} )$ . In this article we prove that for $2n \geq 6$ and $g \geq 2$ , 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.