Topological manifold bundles and the $A$-theory assembly map

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-theory characteristic of such a fiber bundle factors canonically through the assembly map of A-theory. Furthermore our main result shows a refinement of this statement by providing such a factorization for an extended 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-theory characteristic becomes highly connected as the manifold dimension increases