Timeline for The core question of topology
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Mar 2, 2010 at 5:07 | comment | added | Igor Belegradek | Mike, my confusion came from the fact that in the recent edit you talk about Smale's h-cobordism theorem which has nothing to do with Smale's theorem 7.1 that you mention in comments: the latter more or less says that two stable regular neighborhoods are diffeomorphic if and only if they are tangentially homotopy equivalent (cf. paper by Lickorish-Siebenmann titled "Regular neighbourhoods and the stable range"). | |
| Mar 2, 2010 at 1:44 | comment | added | Mike Usher | I don't exactly see how my comment lends itself to that interpretation, but in case someone else is confused, yes, of course, the Theorem 7.1 to which I was referring has different hypotheses than the (much more important) h-cobordism theorem, which has the assumptions on the manifolds that you indicate. | |
| Mar 1, 2010 at 22:41 | comment | added | Igor Belegradek | In your comment you sound like the h-cobordism theorem does NOT apply to manifolds that are closed. In fact it applies to closed simply-connected manifolds of dimension >4: two such manifolds are h-cobordant iff they are diffeomorphic. | |
| Mar 1, 2010 at 17:39 | comment | added | Mike Usher | Sorry--in a hasty effort to find a clean statement in the literature without using cobordism language I overlooked some obviously-rather-important parts of the hypothesis of Theorem 7.1 of Smale's "On the structure of manifolds"...namely that the manifolds need to have vanishing cohomology in degrees above around half the dimension (so obviously they can't be closed, among other serious restrictions). I've edited the error | |
| Mar 1, 2010 at 17:39 | history | edited | Mike Usher | CC BY-SA 2.5 |
Corrected misstatement of theorem of Smale
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| Feb 28, 2010 at 3:42 | comment | added | Igor Belegradek | Smale did not show that "two smooth manifolds are diffeomorphic as soon as there is a homotopy equivalence between them which pulls back the tangent bundle on one to the tangent bundle of the other", because this is not true. I think, exotic 7-spheres should give a counterexample. | |
| Nov 16, 2009 at 19:05 | comment | added | Jason DeVito - on hiatus | Could you provide a reference that homotopy equivalence which pulls back tangent bundles is a diffeomorphism? Does it somehow follow from the H-cobordism theorem? | |
| Oct 18, 2009 at 8:41 | vote | accept | Tejus | ||
| Oct 18, 2009 at 8:41 | vote | accept | Tejus | ||
| Oct 18, 2009 at 8:41 | |||||
| Oct 17, 2009 at 16:51 | history | answered | Mike Usher | CC BY-SA 2.5 |