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 τ 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 τ M -fibrations, Baut(τ M ), and its cohomology ring, i.e., the ring of characteristic classes of τ M -fibrations.For a bundle ξ over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal orientable ξ-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(ξ) 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 τ M -fibration comes from a manifold bundle. Contents 24 4.1. Even spheres 24 4.2. Odd spheres 25 4.3. Complex projective spaces 27 References 41

Algebraic models for classifying spaces of fibrations

“…(Note that the middle vertical arrow does land inside (P n−1 Λ) n+1 ⊂ (P n Λ) n+1 since d has no linear part and Λ has no elements of degree 1.) The snake lemma applied to (9) shows that the kernel of the right-hand vertical arrow in ( 9) is the cokernel of the inclusion…”

“…is known to induce an isomorphism on π k (−) for k > 0. To see this, one can apply[9, Proposition 3.7] to the Sullivan model Λ = C * (L). Let us add that this is essentially equivalent to the statement, going back to Tanré [67, VII.4.…”