A vanishing theorem for tautological classes of aspherical manifolds

Abstract

Tautological classes, or generalised Miller-Morita-Mumford classes, are basic characteristic classes of smooth fibre bundles, and have recently been used to describe the rational cohomology of classifying spaces of diffeomorphism groups for several types of manifolds. We show that rationally tautological classes depend only on the underlying topological block bundle, and use this to prove the vanishing of tautological classes for many bundles with fibre an aspherical manifold.

arXiv commentv4: 38 pages, final version, to appear in G&T

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Cited by in corpus (7)

• On the cohomology of Torelli groups
• A homological approach to pseudoisotopy theory. I
• The family signature theorem
• Tautological classes of definite 4-manifolds
• Bundle Transfer of L-Homology Orientation Classes for Singular Spaces
• Some rational homology computations for diffeomorphisms of odd-dimensional manifolds
• Mapping class groups of $h$-cobordant manifolds