  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.
