Parametrized cobordism categories and the Dwyer-Weiss-Williams index theorem

Abstract

We define parametrized cobordism categories and study their formal properties as bivariant theories. Bivariant transformations to a strongly excisive bivariant theory give rise to characteristic classes of smooth bundles with strong additivity properties. In the case of cobordisms between manifolds with boundary, we prove that such a bivariant transformation is uniquely determined by its value at the universal disk bundle. This description of bivariant transformations yields a short proof of the Dwyer-Weiss-Williams family index theorem for the parametrized A-theory Euler characteristic of a smooth bundle.

22 pages; minor changes; to appear in the Journal of Topology

References in corpus (1)

• On the infinite loop space structure of the cobordism category

Cited by in corpus (6)

• A cobordism model for Waldhausen $K$-theory
• Diffeomorphisms of odd-dimensional discs, glued into a manifold
• Parametrized geometric cobordism and smooth Thom stacks
• Parametrised moduli spaces of surfaces as infinite loop spaces
• A K-theory spectrum for cobordism cut and paste groups
• Topological manifold bundles and the $A$-theory assembly map