Timeline for Complex projective manifolds are homeomorphic if homotopy equivalent
Current License: CC BY-SA 4.0
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| Sep 5, 2020 at 13:49 | comment | added | Ian Agol | @JoeT Freedman’s theory carries over to 4-manifolds with sub exponential growth. But one would have to consult an expert for the classification of such manifolds. The only known finitely presented groups with sub exponential growth are polynomial growth, so virtually nilpotent. There may be interesting surfaces with nilpotent fundamental group, and one could try to see how the theory applies in that case. | |
| Sep 5, 2020 at 13:46 | comment | added | Ian Agol | @JoeT the only thing that I’m aware of is that for aspherical manifolds, the Borel conjecture predicts that they should be homotopy rigid (homotopy implies homeomorphism). This is known for products of curves and ball quotients (complex projective spaces), but is open in general. For manifolds with nontrivial π_1, π_2 or π_3, I’m not sure what to expect. Part of the difficulty is that surgery theory is not completely understood in 4D - this has to do with Freedman’s AB-slice conjecture. | |
| Sep 5, 2020 at 13:34 | comment | added | user164740 | what about the surfaces with non-trivial $\pi_1$? | |
| Sep 5, 2020 at 11:21 | vote | accept | CommunityBot | ||
| Sep 5, 2020 at 1:03 | comment | added | Michael Albanese | A nice summary of what is known about the classification of complete intersections can be found here. | |
| Sep 4, 2020 at 3:50 | history | edited | Ian Agol | CC BY-SA 4.0 |
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| Sep 4, 2020 at 3:35 | history | answered | Ian Agol | CC BY-SA 4.0 |