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

TL;DR

This paper characterizes which Miller-Morita-Mumford characteristic numbers of fibre bundles depend solely on the cobordism class of the total space, revealing precise conditions for their invariance across different fibrations.

Contribution

It provides a complete classification of invariant Miller-Morita-Mumford classes in both oriented and stably almost complex categories based on approximate additivity conditions.

Findings

Invariant MMM classes are fiber integrals of characteristic classes satisfying approximate additivity.

The classification applies to both oriented and stably almost complex fibre bundles.

The results answer a question posed by Church, Farb, and Thibault about invariance properties.

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.