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

## TL;DR

This paper provides a new proof of an index theorem relating the parametrized A-theory characteristic of fiber bundles of topological manifolds to the assembly map, and extends this to a refined characteristic on the cobordism category.

## Contribution

It introduces a new proof of the index theorem and extends the A-theory characteristic to the topological cobordism category, using a bivariant theory framework.

## Key findings

- Factorization of the parametrized A-theory characteristic through the assembly map.
- Extension of the A-theory characteristic to the topological cobordism category.
- Conjecture on high connectivity of the lift as manifold dimension increases.

## 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 $A$-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.