Timeline for Is there a purely topological definition of $\text{Spin}(p,q)$?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 26 at 18:39 | vote | accept | WillG | ||
| S Feb 25 at 19:00 | history | bounty ended | CommunityBot | ||
| S Feb 25 at 19:00 | history | notice removed | CommunityBot | ||
| Feb 24 at 12:44 | answer | added | Tom Goodwillie | timeline score: 2 | |
| Feb 24 at 0:34 | answer | added | Moishe Kohan | timeline score: 4 | |
| Feb 22 at 15:20 | comment | added | Will Sawin | I think "The unique (up to isomorphism) double cover that, restricted to $\operatorname{SO}(p)$, is nontrivial, and, restricted to $\operatorname{SO}(q)$, is nontrivial" does the trick for $p,q>1$ and you can replace "nontrivial" by "connected" per preference | |
| Feb 22 at 15:18 | comment | added | Will Sawin | @IanAgol I don't follow the logic in your second-to-last sentence: A space with fundamental group $\mathbb Z \times \mathbb Z$ has three connected two-fold covers (since $\operatorname{Hom}(\mathbb Z \times \mathbb Z, S_2)$ has four elements but one is disconnected) and of these three, only one is diagonal. | |
| Feb 17 at 18:02 | comment | added | Ryan Budney | That said, you could make some good arguments that the covering space you'd want to call $Spin(p,q)$ should be a 4-sheeted cover of $SO^+(p,q)$ when $p,q > 1$. I suppose it depends on what you want to get out of the covering space. i.e. are you interested specifically in spacelike / timelike vector bundle constructions or are you really just looking for a direct analogy to the definite case? | |
| Feb 17 at 17:51 | comment | added | Ryan Budney | Qiaochu describes the precise double cover, so I would say the answer is yes -- basically just what Qiaochu said. I wasn't aware of that Wikipedia page, probably the best way to get it fixed would be to put some comments in the "talk" section of the page, and the people that wrote it could fix the oversight. But I'm a bit confused about your question. If you believe you won't be able to recognize a correct answer, why ask the question? Reading the Wikipedia page it looks like they aren't really making mistakes, just overlooking the details when the form isn't definite. | |
| S Feb 17 at 17:33 | history | bounty started | WillG | ||
| S Feb 17 at 17:33 | history | notice added | WillG | Draw attention | |
| Nov 15, 2023 at 6:57 | comment | added | Ryan Budney | Yes, you can make a definition of that form. Connected covering spaces of Lie groups are always Lie groups (you can lift the group structure) so your question boils down to finding $2$-sheeted covering spaces. The number of sheets in the covering space corresponds to the index of the subgroup of the fundamental group of the case. | |
| S Nov 15, 2023 at 5:04 | review | First questions | |||
| Nov 15, 2023 at 5:11 | |||||
| S Nov 15, 2023 at 5:04 | history | asked | WillG | CC BY-SA 4.0 |