Timeline for Poincaré duality
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Jan 4, 2023 at 20:27 | history | edited | Victor | CC BY-SA 4.0 |
added 19 characters in body
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| Jan 4, 2023 at 16:49 | answer | added | Connor Malin | timeline score: 5 | |
| Jan 4, 2023 at 16:22 | comment | added | Jim Conant | Yeah, I was just thinking that it wasn't a manifold of the appropriate kind, but should have thought further about it! | |
| Jan 4, 2023 at 8:05 | comment | added | Aleksandar Milivojević | @JimConant I’m getting that it works for spheres minus finitely many points. E.g. the one point compactification of a two-sphere minus two points has the homotopy type of a two-sphere wedge a circle. So we have 1,1,0 on the LHS and 0,1,1 on the RHS. (edit: sorry, @AchimKrause’s comment appeared while I was writing this) | |
| Jan 4, 2023 at 8:01 | comment | added | Achim Krause | For a sphere minus finitely many points it looks ok, doesn't it? The version with homology and cohomology swapped (i.e. $H_q(M) \cong \widetilde{H}^{n-q}(M^+)$) is usual Poincaré duality combined with the observation that, if the extra point in the $1$-point compactification has a contractible neighbourhood basis, cohomology with compact support agrees with this reduced cohomology. And for $M$ finite type, there shouldn't be any problems dualizing this. | |
| Jan 4, 2023 at 5:53 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing
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| Jan 4, 2023 at 3:38 | history | became hot network question | |||
| Jan 4, 2023 at 2:55 | comment | added | Jim Conant | Or even a sphere minus more than one point. | |
| Jan 3, 2023 at 22:51 | comment | added | Ryan Budney | I suppose it fails for essentially the same reason when $M= \mathbb{Z}$. | |
| Jan 3, 2023 at 22:29 | vote | accept | Victor | ||
| Jan 3, 2023 at 22:29 | |||||
| Jan 3, 2023 at 21:39 | history | edited | Victor | CC BY-SA 4.0 |
edited body
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| Jan 3, 2023 at 20:47 | comment | added | Ryan Budney | It fails for $M = S^1 \setminus C$ where $C$ is a Cantor set in $S^1$. | |
| Jan 3, 2023 at 20:11 | review | Close votes | |||
| Jan 8, 2023 at 3:09 | |||||
| Jan 3, 2023 at 19:49 | comment | added | mme | @IgorBelegradek OP specifies that M should be orientable (otherwise probably the LHS should be taken with $w_1$-twisted coefficients). | |
| Jan 3, 2023 at 19:44 | comment | added | Igor Belegradek | If $M$ is an open Moebius band, then $M^+=RP^2$, and your formula fails for $q=1$. | |
| S Jan 3, 2023 at 19:16 | history | suggested | Evgeny Kuznetsov | CC BY-SA 4.0 |
tipo at the end corrected
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| Jan 3, 2023 at 19:08 | review | Suggested edits | |||
| S Jan 3, 2023 at 19:16 | |||||
| Jan 3, 2023 at 19:07 | review | Suggested edits | |||
| Jan 3, 2023 at 19:07 | |||||
| Jan 3, 2023 at 19:04 | history | asked | Victor | CC BY-SA 4.0 |