# A vanishing theorem for tautological classes of aspherical manifolds

## TL;DR

This paper proves that tautological classes for certain fibre bundles over aspherical manifolds vanish, showing they depend only on the topological structure and not on smooth details.

## Contribution

It demonstrates that rational tautological classes depend solely on the topological block bundle and establishes their vanishing for many bundles with aspherical fibres.

## Key findings

- Rational tautological classes depend only on the underlying topological block bundle.
- Tautological classes vanish for many bundles with aspherical manifold fibres.
- Provides new insights into the cohomology of diffeomorphism classifying spaces.

## 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.