Isometric Group Actions and the Cohomology of Flat Fiber Bundles

TL;DR

This paper proves that the Leray-Serre spectral sequence collapses for flat fiber bundles with Riemannian fibers under isometric structure group actions, using topological methods related to intersection space theory.

Contribution

It introduces a topological approach to show spectral sequence collapse in flat bundles with isometric actions, and applies this to compute equivariant cohomology and analyze flat sphere bundles.

Findings

Spectral sequence collapses under specified conditions.

Equivariant cohomology can be computed as a direct sum over group cohomology.

Results have implications for the Euler class of flat sphere bundles.

Abstract

Using methods originating in the theory of intersection spaces, specifically a de Rham type description of the real cohomology of these spaces by a complex of global differential forms, we show that the Leray-Serre spectral sequence with real coefficients of a flat fiber bundle of smooth manifolds collapses if the fiber is Riemannian and the structure group acts isometrically. The proof is largely topological and does not need a metric on the base or total space. We use this result to show further that if the fundamental group of a smooth aspherical manifold acts isometrically on a Riemannian manifold, then the equivariant real cohomology of the Riemannian manifold can be computed as a direct sum over the cohomology of the group with coefficients in the (generally twisted) cohomology modules of the manifold. Our results have consequences for the Euler class of flat sphere bundles. Several examples are discussed in detail, for instance an action of a free abelian group on a flag manifold.