Borel Conjecture for compact aspherical 4-manifolds

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1 Problem

Let $M_0$$== Problem == ; Let M_0 and M_1 be a compact aspherical -manifolds with boundary. The Borel Conjecture in this setting states that a homotopy equivalence of pairs f \colon (M_0, \partial M_0) \to (M_1, \partial M_1) which is a homeomorphism on the boundary is homotopic, relative to the boundary, to a homeomorphism. To apply topological surgery to attack this conjecture, assume that the fundamental group \pi = \pi_1(M_0) \cong \pi_1(M_1) is good. One now proceeds to the following problems: # Decide which good \pi are the fundamental groups of compact -manifolds. # Determine the homeomorphism type of the boundaries which can occur for each group in Part 1. # For a fixed boundary and \pi, prove the conjecture via surgery and the Farrell-Jones Conjecture One can of course formulate the above in the smooth category. There are no known smooth counterexamples for closed manifolds; in particular, there is no known exotic smooth structure on T^4. There are smooth counterexamples for manifolds with boundary; in particular, see Akbulut, "A fake compact contractible 4-manifold." This problem was posed by Jim Davis, following discussions with Jonathan Hillman, Monday January 14th at [[:Category:MATRIX 2019 Interactions|MATRIX]]. == References == {{#RefList:}} [[Category:Problems]] [[Category:Questions]] [[Category:Research questions]]M_0$ and $M_1$$M_1$ be a compact aspherical $4$$4$-manifolds with boundary. The Borel Conjecture in this setting states that a homotopy equivalence of pairs $f \colon (M_0, \partial M_0) \to (M_1, \partial M_1)$$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)$$\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.