Characteristic classes for families of bundles

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.