Topological rigidity and low dimensional topology

An aspherical manifold is a manifold whose universal cover is contractible, i.e. a $K(\pi,1)$-manifold. Borel conjectured that any two closed aspherical manifolds with isomorphic fundamental groups are homeomorphic, in fact, that the structure set of an aspherical manifold is trivial. Wall asked if a $K(\pi,1)$-space which satisfies Poincare duality is homotopy equivalent to a closed manifold. There are corresponding conjectures for compact aspherical manifolds with boundary.

 

We will survey these topological rigidity conjectures, their relationship with low-dimensional topology, and eventually focus on the questions: What closed 3-manifolds are the boundary of some compact aspherical 4-manifold? What are the fundamental groups of compact aspherical 4-manifolds?