Timeline for Two different spin structures of the real projective space $\Bbb RP^3$
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Feb 22, 2023 at 20:17 | answer | added | Marco Golla | timeline score: 5 | |
| Feb 22, 2023 at 18:01 | comment | added | Ryan Budney | @DaveBenson: it depends on your conventions. Traditionally spin structures are defined in the context where your manifold has a fixed orientation, so orientation-reversing automorphisms would not be allowed. But there are many definitions of spin structures where you could think of the spin structure defining an orientation, i.e. you could think of an extended structure that carries both the orientation and the spin structure. In that case, orientation-reversing automorphisms generally are fine. But you are talking about more than just traditional spin structures in this setting. | |
| Feb 22, 2023 at 16:57 | comment | added | Dave Benson | In the paragraph before this claim, they give a rather convincing way to distinguish between the two spin structures, so I think the last paragraph of IV.B.(i) must be wrong, in which case I should edit my answer. | |
| Feb 22, 2023 at 16:50 | review | Close votes | |||
| Feb 24, 2023 at 10:39 | |||||
| Feb 22, 2023 at 16:46 | comment | added | Dave Benson | Indeed, I see that the discussion in Dabrowski and Trautman is confused on this point. At the end of IV.B.(i) of that paper, it is claimed that there are two spin structures on ${\mathbb R}P^{4n-1}$, and they are interchanged by an orientation reversing automorphism. If you allow orientation reversal as they do, are there four, interchanged in pairs, or are there only two? I'm now confused. | |
| Feb 22, 2023 at 16:02 | comment | added | Ryan Budney | You might want to clarify exactly which definition of spin structure you are using. Because if you accept Benson's argument, then $S^3$ has two spin structures yet $H^1(S^3, \mathbb Z_2)$ is trivial, i.e. the theory of spin structures you are using does not correspond to the "spin structures" you are using. | |
| Feb 22, 2023 at 16:02 | comment | added | mme | In fact an oriented diffeomorphism of RP3 is isotopic to the identity, so there is no orientation-preserving map $f: \Bbb{RP}^3 \to \Bbb{RP}^3$ with $f^* s_1 = s_2$. | |
| Feb 22, 2023 at 14:45 | vote | accept | user302934 | ||
| Feb 22, 2023 at 9:50 | answer | added | Dave Benson | timeline score: 5 | |
| Feb 22, 2023 at 6:15 | history | undeleted | user302934 | ||
| Feb 22, 2023 at 6:15 | history | deleted | user302934 | via Vote | |
| Feb 22, 2023 at 6:10 | history | asked | user302934 | CC BY-SA 4.0 |