Timeline for Non-zero homotopy/homology in diffeomorphism groups
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 14, 2018 at 12:55 | vote | accept | Thomas Rot | ||
| Feb 13, 2018 at 9:06 | comment | added | Thomas Rot | I should know better than to comment during my commute. I just accepted that the time-1 map would be a section. Anyway, the corrected answer is still very nice. It gives me an easy class of examples where I can play with. I'll leave the question open longer to see if there are other ideas. | |
| Feb 13, 2018 at 2:05 | comment | added | Ryan Budney | To extend John's comment, the map $Diff(M) \to M$ extends to maps $Diff(M) \to C_n(M)$ and $Diff(M) \to Emb(X,M)$. In the first case you restrict the diffeomorphisms to a finite subset of $M$, in the second case you restrict the diffeomorphisms to a submanifold. Often these maps are informative. | |
| Feb 13, 2018 at 0:37 | comment | added | John Klein | Duh...I was a trigger happy when I wrote the above. A flow doesn't suffice. However, when $M$ is a Lie group there is a section. | |
| Feb 13, 2018 at 0:32 | history | edited | John Klein | CC BY-SA 3.0 |
added 77 characters in body
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| Feb 12, 2018 at 23:51 | comment | added | Michael Albanese | @DylanWilson: Ah, I missed that assumption. My bad. | |
| Feb 12, 2018 at 22:45 | comment | added | Dylan Wilson | (@MichaelAlbanese that's not a counterexample to his claim because $S^2$ has no nonvanishing vector field... also, his claim holds whenever $M$ is itself a group, and for $S^7$, so the first sphere which could give a counterexample is $S^9$. That said- I still don't understand why the claim should be true.) | |
| Feb 12, 2018 at 19:59 | comment | added | Michael Albanese | I don't think your claim that the cohomology of $M$ injects into the cohomology of $\operatorname{Diff}(M)$ is true. Smale showed thay $\operatorname{Diff}(S^2)$ is homotopy equivalent to $O(3)$, and $H^2(S^2; \mathbb{Z}) \cong \mathbb{Z}$ while $H^2(O(3); \mathbb{Z}) \cong H^2(\mathbb{RP}^3\sqcup\mathbb{RP}^3; \mathbb{Z}) = \mathbb{Z}_2\oplus\mathbb{Z}_2$, so the cohomology does not inject. | |
| Feb 12, 2018 at 19:11 | comment | added | Oscar Randal-Williams | I don't believe this. To what diffeomorphism does your section send $x \in M$? | |
| Feb 12, 2018 at 19:11 | comment | added | Dylan Wilson | I’m probably missing something silly, but how does a flow define a section? I can see how to define a map from R to Diff(M) or a map from M to Map(R,M), but I don’t see a map from M to Diff(M). | |
| Feb 12, 2018 at 17:55 | comment | added | Thomas Rot | Excellent, this is exactly the type of answer that I was hoping for! I'll leave the question open for a bit longer, but I am already very happy! | |
| Feb 12, 2018 at 17:29 | history | answered | John Klein | CC BY-SA 3.0 |