The Borel Conjecture for closed manifolds implies that two closed aspherical manifolds with isomorphic fundamental group are homeomorphic. The Borel conjecture for compact aspherical manifolds with boundary states that a homotopy equivalence which is homeomorphism on the boundary is homotopic to a homeomorphism.

Jonathan Hillman and I classify and prove the Borel Conjecture for all compact aspherical four manifolds with boundary with good (= elementary amenable) fundamental group. We classify all possible fundamental groups and all possible 3-manifold boundaries.