Borel Conjecture for compact aspherical 4-manifolds

1 Problem


To apply topological surgery to attack this conjecture, assume that the fundamental group $\pi = \pi_1(M_0) \cong \pi_1(M_1)$$\pi = \pi_1(M_0) \cong \pi_1(M_1)$ is good. One now proceeds to the following problems:

1. Decide which good $\pi$$\pi$ are the fundamental groups of compact $4$$4$-manifolds.
2. Determine the homeomorphism type of the boundaries which can occur for each group in Part 1.
3. For a fixed boundary and $\pi$$\pi$, 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 $T^4$$T^4$. 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.