Timeline for Poincare duality spaces vs. manifolds via lifting maps, the obstruction theory and the role of simply connectedness
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Jul 23, 2018 at 8:26 | comment | added | truebaran | I've edited my post accordingly, indeed I'm interest in the compact case. | |
| Jul 23, 2018 at 8:25 | history | edited | truebaran | CC BY-SA 4.0 |
added 42 characters in body
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| Jul 23, 2018 at 7:24 | comment | added | Francesco Polizzi | "$X$ must satisfy Poincare duality, namely there must be a class in the top homology of $X$ such that the cap product with this class induces isomorphism between homology and cohomology". So you are at least assuming that $X$ is compact. In fact, there are plenty of open manifolds that do not satisfy Poincaré duality. | |
| Jul 23, 2018 at 0:17 | answer | added | Tom Goodwillie | timeline score: 14 | |
| Jul 22, 2018 at 22:43 | comment | added | Igor Belegradek | The existence of a manifold structure on a Poincare complex is described by the surgery exact sequence, see en.wikipedia.org/wiki/Surgery_exact_sequence which works regardless of the fundamental group. In the simply-connected case the L-groups are known and quite simple, but this is also true for other classes of fundamental groups. The sequence doesn't use local coefficients in the sense they are used in the obstruction theory. Incidentally, the obstruction theory works reasonably well with local coefficients - no problem here. | |
| Jul 22, 2018 at 21:42 | history | asked | truebaran | CC BY-SA 4.0 |