Borel Conjecture for compact aspherical 4-manifolds
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# For a fixed boundary and $\pi$, prove the conjecture via surgery and the Farrell-Jones Conjecture | # 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. | + | 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 $T^4$. 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]]. | This problem was posed by Jim Davis, following discussions with Jonathan Hillman, Monday January 14th at [[:Category:MATRIX 2019 Interactions|MATRIX]]. |
Revision as of 12:18, 14 January 2019
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.