# Borel Conjecture for compact aspherical 4-manifolds

## [edit] 1 Problem

Let and be a compact aspherical -manifolds with boundary. The Borel Conjecture in this setting states that a homotopy equivalence of pairs which is a homeomorphism on the boundary is homotopic, relative to the boundary, to a homeomorphism.

To apply topological surgery to attack this conjecture, assume that the fundamental group is good. One now proceeds to the following problems:

- Decide which good are the fundamental groups of compact -manifolds.
- Determine the homeomorphism type of the boundaries which can occur for each group in Part 1.
- For a fixed boundary and , prove the conjecture via surgery and the Farrell-Jones Conjecture

One can of course formulate the above in the smooth category. There are no known smooth counterexamples for closed manifolds; in particular, there is no known exotic smooth structure on . There are smooth counterexamples for manifolds with boundary; in particular, see Akbulut, "A fake compact contractible 4-manifold."

This problem was posed by Jim Davis, following discussions with Jonathan Hillman, Monday January 14th at MATRIX.