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Tautological classes, or generalised Miller-Morita-Mumford classes, are basiccharacteristic classes of smooth fibre bundles, and have recently been used todescribe the rational cohomology of classifying spaces of diffeomorphism groupsfor several types of manifolds. We show that rationally tautological classesdepend only on the underlying topological block bundle, and use this to provethe vanishing of tautological classes for many bundles with fibre an asphericalmanifold.

A vanishing theorem for tautological classes of aspherical manifolds