Invariance properties of Miller-Morita-Mumford characteristic numbers of fibre bundles

Abstract

Characteristic classes of fibre bundles $E^{d+n}\to B^n$ in the category of closed oriented manifolds give rise to characteristic numbers by integrating the classes over the base. Church, Farb and Thibault [CFT] raised the question of which generalised Miller-Morita-Mumford classes have the property that the associated characteristic number is independent of the fibering and depends only on the cobordism class of the total space $E$. Here we determine a complete answer to this question in both the oriented category and the stably almost complex category. An MMM class has this property if and only if it is a fibre integral of a vector bundle characteristic class that satisfies a certain approximate version of the additivity of the Chern character.