Timeline for Can you prove equivalence without being able to calculate it?
Current License: CC BY-SA 2.5
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| Sep 11, 2021 at 19:39 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Jan 7, 2010 at 23:53 | comment | added | Ryan Budney | For some exotic spheres there are very explicit homeomorphisms with the standard sphere. Although I don't know of any that are written up well! The group of exotic $n+1$-spheres is isomorphic to $\pi_0 Diff(D^n)$ for $n$ large enough. I believe Allen Hatcher and Kiyoshi Igusa have some very explicit descriptions of some of these non-trivial elements of $\pi_0 Diff(D^n)$. In particular you get a homeomorphism to the standard sphere using the Alexander trick of "crushing" the diffeomorphism (a rescaling/pinching argument). | |
| Jan 7, 2010 at 22:59 | comment | added | Daniel Moskovich | This is an example I had in mind when writing the question! But Ryan pointed out (elsewhere) that, to find diffeomorphisms between such objects, one would not follow the Morse theory proof, but would rather use a direct construction following from a choice of triangulation (handle decomposition), at least in dimension 3. So I started to doubt it... maybe there actually is a (combinatorial) procedure to find explicit diffeomorphisms between two given examples... | |
| Jan 7, 2010 at 17:13 | history | answered | Lennart Meier | CC BY-SA 2.5 |