Timeline for Homotopy type of set of self homotopy-equivalences of a surface
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Mar 19, 2010 at 22:02 | comment | added | Allen Hatcher | @Andy: The topological contractions probably do not preserve a finite group action, but I would expect one could deduce an equivariant contraction by looking at diffeomorphisms of the quotient orbifold. | |
| Mar 19, 2010 at 22:01 | comment | added | Allen Hatcher | Gramain's argument may use Smale's theorem, but there's a simple proof of Smale's theorem (without Poincaré-Bendixson) in the appendix to Cerf's "yellow book", SLN v.53, the $\Gamma_4=0$ theorem. Besides, isn't the Poincaré-Bendixson theorem a fairly simple topological argument: If a nonsingular flow in a square is vertical near the boundary, then every trajectory has to go from the bottom edge to the top edge, otherwise there would be an infinite trajectory spiraling close to itself somewhere, and this could be perturbed to a closed orbit which would have to enclose a singularity. | |
| Mar 19, 2010 at 17:16 | comment | added | Ryan Budney | If I understand the Gramain construction Hatcher is referring to you also have to use Smale's theorem that Diff(S^2) has the homotopy-type of O(3). The main analytical ingredient is the Poincare-Bendixson theorem. | |
| Mar 19, 2010 at 14:33 | comment | added | Andy Putman | Compared to the other proofs, there isn't that much analysis! The two proofs I mentioned both use the measurable Riemann mapping theorem (ie the solution to the Beltrami equation with measurable inputs). Also, the proof of Earle-Eells uses the contractibility of Teichmuller space, while the proof of Earle-McMullen uses the Douady-Earle extension theorem. Compared to all that, Sard's theorem is a triviality! | |
| Mar 19, 2010 at 14:30 | vote | accept | Andy Putman | ||
| Mar 19, 2010 at 8:09 | comment | added | Ryan Budney | IMO "no analysis" is a tad misleading. You use Sard's theorem in a very significant way. | |
| Mar 19, 2010 at 4:05 | comment | added | Andy Putman | Thanks Allen! One beautiful property of the Earle-McMullen contraction of Diff_0 is that it "preserves symmetries" of the surface in the following sense. Let F_t:Diff_0-->Diff_0 be the E-M contraction, so F_0=id and F_1 is the constant map to the identity. Let f in Diff_0 commute with a finite subgroup G of Diff. Then F_t(f) will commute with G for all t. Do any of the other approaches to contacting Diff_0 have similar properties? | |
| Mar 19, 2010 at 3:51 | history | answered | Allen Hatcher | CC BY-SA 2.5 |