Talk:Differential structure

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Mathematics rating:	Start Class	Low Priority	Field: Geometry
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Geometry guy 22:04, 3 June 2007 (UTC)

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 C^k 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)

But the chart regarding the number of differentiable structures doesn't make clear whether one is referring to smooth structures or C^k structures, for some unspecified k. —Preceding unsigned comment added by 72.82.227.155 (talk) 03:09, 29 March 2008 (UTC)

[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)

Why do we need Zorn's lemma in: "For each distinct differential structure the existence of a single maximal atlas can be shown using Zorn's lemma. It is the union of all of the atlases in the equivalence class." —Preceding unsigned comment added by 128.125.38.103 (talk) 01:02, 9 September 2007 (UTC)