Talk:Differential structure
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This page had some errors in it, which I have attempted to fix, but it's still far from perfect. In particular, the heading "differential structure" is probably not very good. People more commonly speak of "smooth structures" (or perhaps "differentiable structures") on topological manifolds. So, I urge that this page be retitled "smooth structures". John Baez
- I believe the reason "differential" is preferable to "smooth" here is because we are considering Ck structures for possibly finite k. Orthografer 00:36, 15 April 2006 (UTC)
- It may be of some relevance that according to the crude metric of google searches for exact phrases, all three variations are fairly common, with "differential structure" the most commonly used, and "smooth structure" a close second, but "smooth" to me means
Elroch 01:19, 15 April 2006 (UTC)
- It may be of some relevance that according to the crude metric of google searches for exact phrases, all three variations are fairly common, with "differential structure" the most commonly used, and "smooth structure" a close second, but "smooth" to me means
[edit] Mistake in article regarding differentiable structures on topological manifolds
The following sentence is wrong:
"Kirby and Siebenman were able to show that the number of differential structures for topological manifolds of dimension greater than 4 are the same numbers as for the exotic spheres in the table above."
Every topological sphere has at least one differentiable structure (up to equivalence), but some piecewise linear (PL) manifolds (and hence some topological ones) have no differentiable structure at all; the first example of this phenomenon was found by Michel Kervaire in about 1958. There even exist manifolds homeomorphic to a simplicial complex that are not PL, and manifolds that are not even homeomorphic to any simplicial complex; either of these two bizarre conditions is sufficient to conclude the manifold has no differentiable structure at all.Daqu 19:13, 26 September 2006 (UTC)
- It'd be really cool if you could provide these references in the proper section - then at least this article would have some references! Besides - at this point, it's just your word against that of the editor who put in the sentence (which doesn't sound to me like something that would just be concocted out of nothing). Orthografer 01:46, 27 September 2006 (UTC)
