  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.
