Talk:Exotic sphere

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Interesting example. What is its order in the group of structures?

Charles Matthews 07:40, 27 October 2005 (UTC)

[edit] Group of smooth structures on the n-sphere is not clearly defined

The article makes several references to the group of smooth structures on Sⁿ via connected sum, but never defines the elements of this group clearly or explains why the connected sum operation gives a group (in particular, why an element of this group must have an inverse). I propose adding a clear explanation of the group, and also stating in a table not merely the order of this group but also its group structure.Daqu 08:07, 3 June 2007 (UTC)

I agree. Even more basic — what is the identity in this group? The article starts by excluding the standard sphere from consideration (which is OK, that's not an exotic sphere), but then says: "The monoid of exotic n-spheres is the collection of oriented smooth n-manifolds which are homeomorphic to the n-sphere". Well, does the standard sphere belong to this monoid, or not? If yes, that's self-contradictory; if not, well, then the monoid does not have a unit (in fact, it may be empty!), and thus cannot be a group. I think the way out of this impasse is to either (1) specify that the standard sphere belongs to the "monoid of exotic spheres", or (2) just forget about this notion, and simply talk about the group of homotopy spheres, Θ_n, as Kervaire and Milnor do, in the very title of their (landmark) paper. Any other ideas? Turgidson 16:15, 7 November 2007 (UTC)

It is well known that cobordisms form a group. For an example refer to Milnor and Stasheff, "Characteristic Classes". But it is certainly not clear to the initiate why they should, and certainly, it helps to know what a cobordism is first! So I agree that an explanation would be useful to add here. I also wish to make an additional comment. It does not seem at all clear to me why there is a one to one correspondence between homotopy n-spheres and differentiable n-spheres, as seems to be implied by the first paragraph in the offending section. One would think that homotopy n-spheres represent at most a subset of a possibly larger class of differentiable structures. However, I am aware of the no OR policy of wikipedia. Perhaps this is discussed somewhere in the literature, however? RogueTeddy (talk) 11:47, 6 December 2007 (UTC)

I should clarify my point. Via Smales paper "On the Generalised Poincare Conjecture in dimensions greater than four" from the Annals of Math, it is known that homotopy n spheres for n > 4 are topological n spheres. Since each of these has a different differentiable structure, this establishes some sort of classification. However this does not necessarily mean that all topological n spheres are homotopy n spheres, does it? Or am I just missing something obvious here? For instance as an example where this sort of reasoning does not work R^{4} has one homotopy class (I think) but infinitely many admissible differentiable structures via a result of Donaldson in the early 80s. Pardon a student for a silly question. RogueTeddy (talk) 11:54, 6 December 2007 (UTC)

Have you read the article recently? Your criticisms seem a little off the mark. That topological spheres are homotopy spheres requires no results at all -- it follows tautologically from the definitions. Specifically it follows from the result that the "homotopy equivalence" relation is coarser than the "homeomorphism" relation among topological spaces. Rybu (talk) 17:58, 6 December 2007 (UTC)

Ah, thank you. I see that it was a very silly question. Cheers. RogueTeddy (talk) 20:50, 6 December 2007 (UTC)

With regard to credits for proving the Poincare conjecture in dimension 3, I think it should be given to both Hamilton and Perlman. Hamilton set up the Ricci flow program and solved many of the key steps before Perlman completed his program. I would like to propose a change from 'Perlman' to 'Hamilton-Perlman' in the article on who it is that sovled the Poincare conjecture in dimension 3. —Preceding unsigned comment added by 137.78.73.75 (talk) 16:46, 28 May 2008 (UTC)

[edit] Perelman or Poincare?

The article, as of just before my edit, said that the triviality of omega3 required perelman's proof of the poincare conjecture. Does it depend on the Poincare Conjecture itself (as implied by the statement at the end of the same section that the triviality <=>PC), which happened to have been proved by Perelman, or, as suggested by the pre-edit article, does the statement depend on some method or other result proved by Perelman in his mass of Ricci flow work (which is broader than PC)? I write from total ignorance, but perhaps whoever added that statement can clarify. If it really does depend on Perelman's work rather than merely depending on the bare statement of the Poincare Conj, it would be excellent to have a sentence or reference clarifying this relation. —Preceding unsigned comment added by David Farris (talk • contribs) 21:04, 17 October 2007 (UTC)

The previous definition of the groups theta_n was incorrect -- it was talking about cobordism classes of homotopy n-spheres. In dimension 3, all manifolds are cobordant so theta_3 is trivial by definition. The actual definition of theta_n (due to Rene Thom) is that it is h-cobordism classes of homotopy n-spheres, with the connect-sum operation as the monoid structure. 3-manifolds have an essentially unique connect-sum decomposition and there are no inverses (Kneser and Milnor proved this). So I reformulated the section: first define the monoid of exotic spheres, then define the monoid of homotopy spheres. Then mention when n>4, they are the same. Exotic spheres are homeomorphic to the standard n-sphere. Homotopy spheres are homotopy-equivalent to the standard n-sphere. I think the original author was taking the line of reasoning that if you use Perelman's result (the proof of the Poincare conjecture) then there is only one homotopy 3-sphere (the standard sphere). I doubt the previous author was referring to anything more than truth of the Poincare conjecture. Rybu —Preceding comment was added at 21:27, 17 October 2007 (UTC)