Borel Conjecture for compact aspherical 4-manifolds
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; 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.