The Borel Conjecture states that any homotopy equivalence between compact aspherical manifolds which is a homeomorphism on the boundaries is isotopic, rel boundary, to a homeomorphism. The Borel Conjecture holds for 4-manifolds with good fundamental group by topological surgery and the Farrell-Jones Conjecture.

This talk will report on joint work with Jonathan Hillman undertaking the classification (enumeration) of such manifolds, their fundamental groups, and their boundary components. The simply connected case, due to Freedman, is a model: Any closed homology 3-sphere bounds a compact, contractible 4-manifold, unique up to homeomorphism.