Groups, Geometry, and Dynamics

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Volume 7, Issue 2, 2013, pp. 293–321
DOI: 10.4171/GGD/183

Published online: 2013-05-07

Isometric group actions and the cohomology of flat fiber bundles

Markus Banagl[1]

(1) Universit├Ąt Heidelberg, Germany

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.

Keywords: Serre spectral sequence, cohomology of fiber bundles, flat bundles, isometric group actions, equivariant cohomology, aspherical manifolds, discrete torsion-free transformation groups, Euler class

Banagl Markus: Isometric group actions and the cohomology of flat fiber bundles. Groups Geom. Dyn. 7 (2013), 293-321. doi: 10.4171/GGD/183