# The cohomology of Torelli groups is algebraic

## TL;DR

This paper proves that the rational cohomology groups of Torelli groups for high-dimensional manifolds are algebraic representations, and establishes their stable cohomology and nilpotency of the classifying space.

## Contribution

It demonstrates that for dimensions at least 6 and genus at least 2, the cohomology of Torelli groups are algebraic, advancing understanding of their structure.

## Key findings

- Cohomology groups are algebraic representations for g ≥ 2 and 2n ≥ 6.
- Determines the stable rational cohomology of Torelli groups.
- Classifying space of Torelli group is shown to be nilpotent.

## 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 $Sp_{2g}(Z)$ or $O_{g,g}(Z)$. In this paper 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.