Torelli spaces of high-dimensional manifolds

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.

29 pages; v2 accepted for publication in the Journal of Topology

References in corpus (3)

• Homological stability for moduli spaces of high dimensional manifolds. I
• Homological stability for moduli spaces of high dimensional manifolds
• The diffeomorphism group of a K3 surface and Nielsen realization

Cited by in corpus (9)

• Generalised Miller-Morita-Mumford classes for block bundles and topological bundles
• Mapping class groups of highly connected $(4k+2)$-manifolds
• On the cohomology of Torelli groups
• Some finiteness results for groups of automorphisms of manifolds
• Borel's stable range for the cohomology of arithmetic groups
• Variations of rational higher tangential structures
• A note on rational homological stability for automorphisms of manifolds
• Characteristic classes of fiberwise branched surface bundles via arithmetic
• Some rational homology computations for diffeomorphisms of odd-dimensional manifolds