Borel Conjecture for compact aspherical 4-manifolds

Revision as of 09:49, 14 January 2019

1 Problem

Let $== Problem == <wikitex>; Let $M_0$ and $M_1$ be a compact aspherical $-manifolds with boundary. The Borel Conjecture in this setting states that a homotopy equivalence of pairs $f \colon (M_0, \partial M_0) \to (M_1, \partial M_1)$ 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 $\pi = \pi_1(M_0) \cong \pi_1(M_1)$ is good. One now proceeds to the following problems: # Decide which good $\pi$ 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 $\pi$, prove the conjecture via surgery and the Farrell-Jones Conjecture One can of course formulate the above in the smooth category. Smooth counter examples exist: references are needed. </wikitex> == References == {{#RefList:}} [[Category:Questions]] [[Category:Research questions]]M_0$ and $M_1$ be a compact aspherical $4$-manifolds with boundary. The Borel Conjecture in this setting states that a homotopy equivalence of pairs $f \colon (M_0, \partial M_0) \to (M_1, \partial M_1)$ 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 $\pi = \pi_1(M_0) \cong \pi_1(M_1)$ is good. One now proceeds to the following problems:

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

One can of course formulate the above in the smooth category. Smooth counter examples exist: references are needed.

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

2 References