Characteristic classes for families of bundles

TL;DR

This paper develops a framework for understanding characteristic classes of families of bundles via the classifying space for $ au_M$-fibrations, enabling computations of their cohomology and applications to manifold theory.

Contribution

It constructs a Sullivan model for the classifying space of $ au_M$-fibrations and provides explicit tools for computing their rational cohomology rings.

Findings

Computed cohomology rings for spheres and complex projective spaces.

Provided explicit cocycle representatives for characteristic classes.

Discussed applications to tautological rings and manifold bundle recognition.

Abstract

The generalized Miller-Morita-Mumford classes of a manifold bundle with fiber $M$ depend only on the underlying $\tau_M$-fibration, meaning the family of vector bundles formed by the tangent bundles of the fibers. This motivates a closer study of the classifying space for $\tau_M$-fibrations, $Baut(\tau_M)$, and its cohomology ring, i.e., the ring of characteristic classes of $\tau_M$-fibrations. For a bundle $\xi$ over a simply connected Poincar\'e duality space, we construct a relative Sullivan model for the universal orientable $\xi$-fibration together with explicit cocycle representatives for the characteristic classes of the canonical bundle over its total space. This yields tools for computing the rational cohomology ring of $Baut(\xi)$ as well as the subring generated by the generalized Miller-Morita-Mumford classes. To illustrate, we carry out sample computations for spheres and complex projective spaces. We discuss applications to tautological rings of simply connected manifolds and to the problem of deciding whether a given $\tau_M$-fibration comes from a manifold bundle.