Timeline for Characteristic classes of non-linear sphere bundles
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 29, 2022 at 14:33 | comment | added | Connor Malin | @IgorBelegradek The map is nontrivial because $O(n+1) \rightarrow \operatorname{Diff}(S^n) \rightarrow \operatorname{Top}(n+1)$ is. | |
| Jun 29, 2022 at 14:23 | comment | added | Igor Belegradek | @ConnorMalin: I am asking about the map, not the right hand side. Say, why is the map rationally nontrivial? You seem to claim it is. | |
| Jun 29, 2022 at 14:15 | comment | added | Jason DeVito - on hiatus | @ConnorMalin: Thanks to you, too. By the way, the Weiss article your linked is the same Weiss article Igor linked a few comments above ;-). | |
| Jun 29, 2022 at 14:14 | comment | added | Jason DeVito - on hiatus | @IgorBelegradek: Thanks for the references! | |
| Jun 29, 2022 at 14:14 | vote | accept | Jason DeVito - on hiatus | ||
| Jun 29, 2022 at 13:53 | comment | added | Connor Malin | @IgorBelegradek Yes, just extend via cone. The right hand side is studied by Weiss here. | |
| Jun 29, 2022 at 13:42 | comment | added | Igor Belegradek | @ConnorMalin: what can be said rationally about the map $Diff(S^n)\to Top(n+1)$? Could you point me to a reference dealing with the issue? I assume the map takes a diffeomorphism of $S^n$ to an origin-preserving homeomorphism of $\mathbb R^{n+1}$ via an open cone; if not, how is it defined? | |
| Jun 29, 2022 at 13:18 | comment | added | Igor Belegradek | Some references: M. Weiss'es arxiv.org/pdf/1507.00153.pdf and Igusa's "Pontrjagin classes and higher torsion of sphere bundles", researchgate.net/publication/…. | |
| Jun 29, 2022 at 12:36 | history | became hot network question | |||
| Jun 29, 2022 at 12:15 | answer | added | archipelago | timeline score: 11 | |
| Jun 29, 2022 at 4:16 | comment | added | Connor Malin | Rationally, stable characteristic classes exist and satisfy the same formulas. This is because we have maps $B\operatorname{Diff(S^n)} \rightarrow B\operatorname{Top}(n+1)\rightarrow B\operatorname{Top} $ and the latter is rationally equivalent to $BO$ because $\operatorname{Top}/O$ has only rational homotopy groups. In the torsion case I have no clue. I would ask Watanabe if he has ever calculated the characteristic classes of the bundles he constructed in dimension 4. | |
| Jun 29, 2022 at 2:04 | history | asked | Jason DeVito - on hiatus | CC BY-SA 4.0 |