Timeline for $ \mathbb{R}P^n $ bundles over the circle
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Jan 23, 2022 at 19:09 | comment | added | Tom Goodwillie | Well, then there are the mixtures such as classifying smooth bundles up to fiber homotopy equivalence. Maybe that really is what you meant. | |
| Jan 23, 2022 at 18:11 | comment | added | Tom Goodwillie | Three different questions are possible: classification of smooth fiber bundles up to fiber-preserving diffeomorphism, classification of topological fiber bundles up to fiber-preserving homeomorphism, classification of fibrations up to fiber homotopy equivalence. I addressed the first, Vaintrob addressed the third. | |
| Jan 23, 2022 at 15:24 | history | edited | Dmitry Vaintrob | CC BY-SA 4.0 |
edited typos
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| Jan 23, 2022 at 15:18 | comment | added | Dmitry Vaintrob | @IanGershonTeixeira That is correct. Here it's important that by "homotopic" we mean as a map, not as a diffeomorphism (i.e., the intermediate maps are allowed to not be invertible). Note that once you have a homotopy, you can always perturb it to be smooth. | |
| Jan 23, 2022 at 15:11 | comment | added | Ian Gershon Teixeira | OK, just confirming that your answer here is essentially an exact answer to the second part of my question from the original MSE post: "Is every diffeomorphism of even dimensional real projective space homotopic to the identity and every diffeomorphism of odd dimensional projective space is homotopic to either the identity or to an orientation reversing isometry?" And that furthermore this answer is equivalent to classifying bundles up to homotopy. | |
| Jan 23, 2022 at 14:42 | comment | added | Igor Belegradek | As I say it is Corollary 6 in "Coverings of fibrations" by Becker and Gottlieb. | |
| Jan 23, 2022 at 14:41 | comment | added | Dmitry Vaintrob | @IgorBelegradek Hmm, I couldn't find it there. Is it in one of the linked papers? | |
| Jan 23, 2022 at 14:33 | history | edited | Dmitry Vaintrob | CC BY-SA 4.0 |
replaced "topological" by "homotopy"
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| Jan 23, 2022 at 14:30 | comment | added | Igor Belegradek | A reference for the above computation of $\pi_0(Aut(RP^n))$ can be found in mathoverflow.net/questions/407388/…. | |
| Jan 23, 2022 at 14:16 | vote | accept | Ian Gershon Teixeira | ||
| Jan 23, 2022 at 13:58 | comment | added | Tom Goodwillie | And the MSE question had "diffeomorphism" in the title. | |
| Jan 23, 2022 at 13:44 | comment | added | Dmitry Vaintrob | Right. Tom Goodwillie's answer came while I was writing mine, and I added a disclaimer. | |
| Jan 23, 2022 at 13:43 | history | edited | Dmitry Vaintrob | CC BY-SA 4.0 |
added: this works for Serre bundles
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| Jan 23, 2022 at 13:43 | comment | added | Neil Strickland | You are apparently classifying up to fibre homotopy equivalence. As Tom Goodwillie points out, the classification up to diffeomorphism is different. | |
| Jan 23, 2022 at 13:41 | history | answered | Dmitry Vaintrob | CC BY-SA 4.0 |