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)
But the chart regarding the number of differentiable structures doesn't make clear whether one is referring to smooth structures or Ck structures, for some unspecified k. —Preceding unsigned comment added by 72.82.227.155 (talk) 03:09, 29 March 2008 (UTC)
Mistake in article regarding differentiable structures on topological manifolds
[edit]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)
- We don't. I've removed it 130.237.198.176 (talk) 12:44, 3 July 2008 (UTC)
- Are we sure we don't? Because it looks to me like we do need it. In particular, having constructed the equivalence class of Ck-compatible atlases, one would first need to equip it with a partial ordering (trivial: otherwise maximality means nothing; we take the partial ordering to be the usual set inclusion), then show that a maximal atlas in that class exists, and finally show that it is unique (trivial: if it is not, there are at least two such maximal atlases, so their union is a member of the class and properly contains both, which contradicts their maximality). The second step (existence), though, can be carried out through Zorn's lemma (since every chain has, indeed, an upper bound: the union of its elements) & Zorn's lemma is equivalent to the Axiom of Choice. Am I making a mistake somewhere? Athenray (talk) 22:56, 3 November 2008 (UTC)
- Zorn's lemma is unnecessary here because we are able to identify the maximal atlas explicitly. That is, we do not need to use Zorn's lemma to explain why the union of all atlases in a given equivalence class is something that exists and is itself an atlas. You have given a valid argument using Zorn's lemma to show the existence of a maximal atlas, which may or may not be the union of all atlases. The point is there is another existence argument that does not involve Zorn's lemma. Compare with the application to ideals in rings in the Zorn's lemma article. In that case, we can't "explicitly" name a maximal ideal for an arbitrary ring. Orthografer (talk) 16:40, 5 November 2008 (UTC)
- Thanks, I don't know why I got so hung up on this - in fact, I got confused by the fact that the maximal atlas itself belongs to the equivalence class over which the union ranges; I mistakenly took this to be AC's "signature" for some reason. The next day I recalled the construction of the smallest algebra containing a given set the construction of which is exactly analogous (w/ the partial ordering inverted so that in the end one takes the intersection instead of the union). Thanks again! Athenray (talk) 18:29, 5 November 2008 (UTC)
- Zorn's lemma is unnecessary here because we are able to identify the maximal atlas explicitly. That is, we do not need to use Zorn's lemma to explain why the union of all atlases in a given equivalence class is something that exists and is itself an atlas. You have given a valid argument using Zorn's lemma to show the existence of a maximal atlas, which may or may not be the union of all atlases. The point is there is another existence argument that does not involve Zorn's lemma. Compare with the application to ideals in rings in the Zorn's lemma article. In that case, we can't "explicitly" name a maximal ideal for an arbitrary ring. Orthografer (talk) 16:40, 5 November 2008 (UTC)
- Are we sure we don't? Because it looks to me like we do need it. In particular, having constructed the equivalence class of Ck-compatible atlases, one would first need to equip it with a partial ordering (trivial: otherwise maximality means nothing; we take the partial ordering to be the usual set inclusion), then show that a maximal atlas in that class exists, and finally show that it is unique (trivial: if it is not, there are at least two such maximal atlases, so their union is a member of the class and properly contains both, which contradicts their maximality). The second step (existence), though, can be carried out through Zorn's lemma (since every chain has, indeed, an upper bound: the union of its elements) & Zorn's lemma is equivalent to the Axiom of Choice. Am I making a mistake somewhere? Athenray (talk) 22:56, 3 November 2008 (UTC)
Smooth Structure
[edit]Smooth structure redirects here, but the term is only really mentioned with prior knowledge taken for granted. LokiClock (talk) 06:29, 8 October 2009 (UTC)
Existence and uniqueness theorems
[edit]This section needs to be clarified. What does Ck-compatible C∞-structure mean? Presumably it just means C∞-structure that's a subset of the given Ck-structure? It's not unique; Hirsch, in his Differential Topology book, says that it's unique up to C∞ diffeomorphism (Thm. 2.9 on p. 51).
What does it mean for Ck-structures to be equivalent? This is never defined in the article; it defines equivalent atlases, but not equivalent structures. I think perhaps "equal" is meant instead of "equivalent"? Given a C∞ maximal atlas, there's a unique Ck maximal atlas containing it, which is just gotten by adding in all charts that are Ck-compatible with all the charts in the C∞ maximal atlas. I presume that this is what is meant?
It's should be made clear that it's easy to consider a C∞ manifold as a Ck manifold, but the converse takes work (that must be Whitney's result, and it's proved in Hirsch's book). Kier07 (talk) 00:12, 23 August 2010 (UTC)
I tried to clarify the terminology, but the content about the number of "smooth types" should also be checked and supplied with references. This is not part of my expertise. SepIHw (talk) 15:54, 8 January 2011 (UTC)
- Are you sure it was Johann Radon who proved uniqueness of differentiable structures in dimension 1 and 2? Wasn't it Tibor Radó? --84.75.61.32 (talk) 09:01, 13 October 2011 (UTC)
Differential structures on spheres of dimensions from 1 to 18
[edit]More precision of speech would be obtained here by systematically utilizing definitions, e.g. as follows: Let m be a natural number and let k be a natural number or ∞. Let M be an m−dimensional Ck−manifold defined as a maximal Ck−atlas on a given set. Call two maximal C∞−atlases on the same set equivalent iff (i.e. if and only if) there is a smooth, i.e. C∞−diffeomorphism between them, and there is a Ck−diffeomorphism between one (hence both) of them and M, and they both define the same topology as M. Note that the requirement of existence of a Ck−diffeomorphism is redundant in the case where k = 0. The equivalence classes are then called smooth types of M. If orientation here plays any role, one should similarly define oriented smooth types. Note that an oriented Ck−manifold is most conveniently defined as a maximal oriented atlas. Here "oriented" means (for k ≠ 0) that the Jacobian determinant of any "chart change" is positive.
Accepting these definitions, one may then say with precision, e.g. that the set of smooth types of the topological manifold R4 is uncountable, whereas Rn has only one smooth type for any natural number n ≠ 4. Instead of saying that "it is not currently known how many differential structures there are on the 4-sphere" one could more precisely say that it is not currently known how many (oriented) smooth types the topological 4-sphere S4 has. SepIHw (talk) 12:38, 9 January 2011 (UTC)
Moving stuff regarding smooth structure to relevant article
[edit]IMHO things would be clearer if section about exotic spheres and parts of other sections would be moved to article about smooth structure, but I dont want to touch this text since I dont understand it. — Preceding unsigned comment added by Alesak23 (talk • contribs) 15:08, 7 December 2011 (UTC)
Edit request
[edit]The sentence FRAGMENT "Thus there is only one class of pairwise smoothly diffeomorphic smooth, i.e. C∞−structures in a Ck−manifold." requires fixing. I'm going to add a comma but it seems possible that the sentence is garbled in some other way and the "i.e. ...Ck-manifold" is the parenthetical i.e. phrase hence the subject (noun or noun phrase) is missing.173.184.17.151 (talk) 13:16, 11 October 2019 (UTC)