Topological manifold bundles and the 𝐴-theory assembly map

Abstract

We give a new proof of an index theorem for fiber bundles of compact topological manifolds due to Dwyer, Weiss, and Williams, which asserts that the parametrized A A -theory characteristic of such a fiber bundle factors canonically through the assembly map of A A -theory. Furthermore our main result shows a refinement of this statement by providing such a factorization for an extended A A -theory characteristic, defined on the parametrized topological cobordism category. The proof uses a convenient framework for bivariant theories and recent results of Gomez-Lopez and Kupers on the homotopy type of the topological cobordism category. We conjecture that this lift of the extended A A -theory characteristic becomes highly connected as the manifold dimension increases.