Torelli spaces of high-dimensional manifolds

TL;DR

This paper computes the rational cohomology invariants of the Torelli group for certain high-dimensional, highly connected manifolds, revealing the nontriviality of most Miller--Morita--Mumford classes.

Contribution

It provides the first calculation of the invariant part of the cohomology of Torelli groups for high-dimensional manifolds, extending previous work on diffeomorphism groups.

Findings

Most Miller--Morita--Mumford classes are nontrivial in the cohomology.

The vanishing of classes associated with the Hirzebruch class is explained by the family index theorem.

The work connects surgery theory, pseudoisotopy theory, and arithmetic group results.

Abstract

The Torelli group of a manifold is the group of all diffeomorphisms which act as the identity on the homology of the manifold. In this paper, we calculate the invariant part (invariant under the action of the automorphisms of the homology) of the cohomology of the classifying space of the Torelli group of certain high-dimensional, highly connected manifolds, with rational coefficients and in a certain range of degrees. This is based on Galatius--Randal-Williams' work on the diffeomorphism groups of these manifolds, Borel's classical results on arithmetic groups, and methods from surgery theory and pseudoisotopy theory. As a corollary, we find that all Miller--Morita--Mumford characteristic classes are nontrivial in the cohomology of the classifying space of the Torelli group, except for those associated with the Hirzebruch class, whose vanishing is forced by the family index theorem.