Timeline for Vector fields on $(4n+1)$-spheres
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 3, 2013 at 14:56 | vote | accept | Chris Gerig | ||
| May 1, 2013 at 1:19 | answer | added | Chris Gerig | timeline score: 2 | |
| Apr 30, 2013 at 13:00 | comment | added | Misha | Chris: You should also specify what "down-to-earth" means: I am nearly sure that every proof (of linear dependence) will involve homotopy groups at some point and in some form. | |
| Apr 30, 2013 at 7:30 | comment | added | Russell | You might want to check out Lawson & Michelson's ``Spin Geometry''. It contains a discussion about how Clifford algebras can be used to study the number of linearly independent vector fields on spheres. I can't recall off the top of my head if this involves the Hopf fibration in an essential way, but you figure at least in these low dimensions the algebras are simple enough -- in the non-technical sense of the expression -- to understand how things play out concretely. At worst, the book contains many references to classic papers on the problem. | |
| Apr 30, 2013 at 5:33 | answer | added | Peter Michor | timeline score: 4 | |
| Apr 30, 2013 at 5:26 | comment | added | Misha | The result for spheres of dimension $4n+1$ was first proven in 1942 independently by Eckmann and Whitehead. Eckmann's paper is in German (and I will not comment on it), but Whitehead's 1942 Annals paper "Homotopy Properties of Real Orthogonal Groups" is readable and is low-tech by comparison to Adams' paper. It is still much harder than Poincare-Hopf, and the proof is homotopy-theoretic. If you are really motivated, you may want to run Whitehead's arguments for $n=1$ and see if you find them sufficiently geometric for your taste. | |
| Apr 30, 2013 at 4:13 | history | edited | Chris Gerig | CC BY-SA 3.0 |
added 688 characters in body; deleted 1 characters in body
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| Apr 30, 2013 at 2:17 | history | asked | Chris Gerig | CC BY-SA 3.0 |