Timeline for $S^n \to S^m \to B$ bundle: possible?
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
|
|
| Nov 3, 2010 at 15:31 | answer | added | Igor Belegradek | timeline score: 8 | |
| Nov 3, 2010 at 15:12 | comment | added | Ian Agol | There are also exotic $RP^4$'s constructed by Cappell-Shaneson (although the first examples where the double cover was proved standard were by Fintushel and Stern I think) ams.org/mathscinet-getitem?mr=607896 ams.org/mathscinet-getitem?mr=418125 ams.org/mathscinet-getitem?mr=1081936 | |
| Nov 3, 2010 at 12:28 | comment | added | Spiro Karigiannis | Oops. I didn't read closely enough. Of course, if the base need not be a sphere either, there are the more general Hopf fibrations that Neil describes very well in his answer. | |
| Nov 3, 2010 at 8:51 | answer | added | Tilman | timeline score: 3 | |
| Nov 3, 2010 at 8:26 | answer | added | Neil Strickland | timeline score: 23 | |
| Nov 3, 2010 at 4:28 | comment | added | Tim Perutz | The answer to (1b) is that the category does matter. Milnor found seven smooth 7-manifolds, all different in DIFF but all PL homeomorphic to $S^7$, which are $S^3$-bundles over $S^4$. [Jose: is your quotation of Adams precisely correct? Isn't there an infinite sequence of $S^3$-bundles over $S^4$ with Euler class $\pm 1$ but varying $p_1$, and a subsequence for which the total space is $S^7$ even in DIFF?] | |
| Nov 3, 2010 at 2:46 | comment | added | Romeo | Here we're allowing the base to be arbitrary. | |
| Nov 3, 2010 at 2:29 | comment | added | Spiro Karigiannis | José's comment came at the same time as my answer. I believe him that they're the only examples. | |
| Nov 3, 2010 at 2:25 | answer | added | Spiro Karigiannis | timeline score: 4 | |
| Nov 3, 2010 at 2:21 | comment | added | José Figueroa-O'Farrill | How about the Hopf bundles? Those are the only bundles where the fibre, total space and base are all spheres -- a result, I believe, due to Frank Adams? Experts here will correct me if I'm wrong. | |
| Nov 3, 2010 at 2:17 | history | asked | Romeo | CC BY-SA 2.5 |