Talk:Exotic sphere

Order

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

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

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)

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)

What value of k in Brieskorn's complex manifold equation (the intersection of that manifold and a "small" (10-)sphere around the origin) yeilds a "non-exotic" homotopy 7-sphere?

"As shown by Egbert Brieskorn (1966, 1966b) (see also (Hirzebruch & Mayer 1968)) the intersection of the complex manifold of points in C⁵ satisfying

$a^{2}+b^{2}+c^{2}+d^{3}+e^{6k-1}=0\$

with a small sphere around the origin for k = 1, 2, ..., 28 gives all 28 possible smooth structures on the oriented 7-sphere." First of all, does the radius of the sphere matter? If not, the word 'small' should probably be removed. But my main question is, which of the 28 possible values of k in the equation for the non-spherical complex manifold results in a manifold whose intersection with a 10-sphere around the origin (presumably of any radius) yeilds a smooth structure on the oriented 7-sphere that is diffeomorphic to a 7-sphere? Kevin Lamoreau (talk) 22:29, 16 October 2010 (UTC)

(Corrected to replace "is not" with "is.") Kevin Lamoreau (talk) 19:33, 27 November 2010 (UTC)

Didn't Brieskorn answer that question in his paper, cited in the main article? Rybu (talk) 15:35, 11 April 2011 (UTC)

Probably, but I couldn't find it in the first of the three links cited in that area of the article, and the other two are in German (which I don't speak) and are either only partially shown or largely restricted to subsribers. Kevin Lamoreau (talk) 20:35, 29 May 2011 (UTC)

Making the article accessible

This article is currently impenetrable even to fairly technical folks who aren't math PhDs. A visual example, hint of what these might be useful for in the real world, or step-by-step walkthrough of the basic definition would be very helpful. -- Beland (talk) 06:42, 4 April 2011 (UTC)

One thing I would like to see explained in plain English is whether an exotic sphere is, or may be, simply connected: can any loop on it be shrunk to a point? — Cheers, Steelpillow (Talk) 17:09, 10 April 2011 (UTC)

Steelpillow, your question is answered in the first sentence of the article. All exotic spheres are simply connected since they're homeomorphic to n-dimensional spheres and there are no exotic spheres in dimension 0 or 1. Rybu (talk) 15:22, 11 April 2011 (UTC)

OK, thanks. I guess another question then is, does the lack of a diffeomorphism mean that a sphere and an exotic sphere have different topologies, or just different metrics? And a third is, do you begin to see how long technical words do not bring out the simple principles the way that simple words can? — Cheers, Steelpillow (Talk) 19:05, 11 April 2011 (UTC)

Steelpillow, exotic spheres are all homeomorphic to $S^{n}$ so it's not clear what you're referring to by "different topologies". These are smooth manifolds -- they are of course metrizable but metrics are not relevant to their definitions. Your last sentence is far from clear to me. What technical words and simple principles are you talking about? Rybu (talk) 21:12, 11 April 2011 (UTC)

Well, there you are you see, I'm not entirely sure which technical words I should be chasing down. An exotic sphere is in some way different from an ordinary one, but in what way? You say it's not a difference in topology, and that metrics are not relevant. So I am stumped as to the distinction. The article talks about diffeomorphism but that article is worse than this one, I have not the faintest idea what story it has to tell me. What in plain English is the difference between an ordinary sphere and an exotic one, and which fancy words does this point me at? — Cheers, Steelpillow (Talk) 17:27, 12 April 2011 (UTC)

Beland, it's not clear what "real world" applications you're looking for. The article is on a fairly technical topic and I don't think anyone is making a claim this is an accessible topic to someone not trained in formal mathematics. Rybu (talk) 15:22, 11 April 2011 (UTC)

Well, what real-world situations are modeled using exotic spheres? For example, are they used in quantum mechanics or other branches of physics? Or are people studying them merely as a thought experiment? -- Beland (talk) 18:21, 12 April 2011 (UTC)

I found this quickly, by Googling for "exotic sphere"+"string theory". This is from the attitude of mathematical physics. Google is good for certain types of queries. What this Wikipedia article aims to do is to be precise about what exactly is meant by the concept in question. It could certainly do with better exposition: it is rather compressed. But, to make an important point, the concept of smooth structure is one of the most subtle in mathematics; and indeed the discovery of exotic spheres in 1957 was one of the most unexpected turns in the century. It is unreasonable to ask for plain English as well as precision in this sort of case. An exotic smooth structure is something like a tablecloth you can't get spread out perfectly on a table - always rucked up somewhere. Charles Matthews (talk) 21:33, 12 April 2011 (UTC)

Many thanks. Diffeomorphisms do seem to be the root of the issue, but until I can gain a clearer understanding of whether and how one can do differential calculus in a zone where metrics are irrelevant, as Ryubu put it, I guess I'm on hold for this one. — Cheers, Steelpillow (Talk) 19:46, 14 April 2011 (UTC)

Steelpillow, the diffeomorphism article is the wrong place to start when trying to understand what an exotic smooth sphere is. It's relevant, but not the first thing you need to absorb. The thing you need to absorb first is the difference between a topological manifold and a smooth manifold. Not every topological manifold can be given a smooth structure (and it's not easy to provide examples of this) but every smooth manifold is a topological manifold (just like every integer is a rational number -- they're not exactly the same thing, but they're closely related in that integers can be thought of as specializations of rational numbers). IMO it's not quite fair to expect every Wikipedia page will be totally accessible to some global basic education standard -- some topics are technical and require additional background to understand. Rybu (talk) 01:58, 30 May 2011 (UTC)

Thanks. However this just moves my problem to understanding how a manifold can be understood as differentiable (which any smooth manifold apparently is) when it has no metric. FWIW I do not expect advanced articles to be totally accessible, but I live in hope that any accessible aspects of the topic can be made so. — Cheers, Steelpillow (Talk) 13:53, 30 May 2011 (UTC)

Whitney's embedding theorem can be considered to have cleared up that matter - it is foundational. We might as well be talking about submanifolds of Euclidean space throughout. What is being said is that there are such submanifolds that are, from the point of view of topology, perfectly good spheres. What they are not are "round spheres" - roughly speaking no way is available to establish the expected smooth "latitude and longitude" consistently on them. But that is not quite accurate, given the way latitude and longitude break down at the poles. There is no consistent way of "charting" them all over that is compatible with the "normal" way on a higher-dimensional sphere. There is a system of charts all right, because locally that isn't a problem in Euclidean space. When you try to specify a complete correspondence with the sphere of the right number of dimensions, there is an inevitable "glitch" somewhere. This is a global phenomenon (at least outside dimension 4): smaller patches can be fixed in a matching way, but not the whole manifold. Charles Matthews (talk) 12:59, 31 May 2011 (UTC)

I suppose I'm confused by your confusion. There's no metric in the definition of differentiable manifold, so why would you expect a metric to be relevant? Your comment "which any smooth manifold apparently is" might get at the heart of your confusion -- smooth manifolds admit metrics (in that it is possible to find a metric compatible with the topology), but a metric is not part of their definition. Similarly, wood may be painted, but just because something is made of wood does not mean it is necessarily painted. Rybu (talk) 22:34, 30 May 2011 (UTC)

Perhaps a better way to answer Steelpillow is to explain that the definition of a differentiable manifold effectively creates local metrics in a region about each point on the manifold by the chart maps. Different chart maps create different metrics, in general, but in a region of overlap, the transformation from one metric to another is required to be smooth by the definition. Local metrics are all one needs to define derivatives and the overlap condition insures that everything is consistent.--agr (talk) 15:26, 31 May 2011 (UTC)

My thanks to all for your explanations. It will take a fair bit of plodding to get my head round it all - I'll go away and put it on my reading list, please don't wait up on my account! — Cheers, Steelpillow (Talk) 16:07, 31 May 2011 (UTC)

On the result of Hill, Hopkins, and Ravenel

The current version of the article says that the Kervaire problem has been announced by "Michael Hopkins (jointly with Hill and Ravenel)." There is no justification for breaking the standard convention of crediting coauthors in alphabetical order, and the way the sentence is written inevitably relegates Mike Hill and Doug Ravenel to an unjustified secondary status. While I am aware that the talk in which the result was announced was given by Michael Hopkins, the sentence should still be rewritten to give all three authors equal credit (and listing Mike Hill first). It is also not clear why the date of the talk is important, and it would seem much more useful to give a reference to the paper (maybe an arxiv listing; I do not know wikipedia's policies on the matter). — Preceding unsigned comment added by 140.112.4.181 (talk) 05:31, 5 July 2011 (UTC)

I agree, so I re-ordered the list of names and removed the date reference. Rybu (talk) 21:15, 11 August 2011 (UTC)

Why did you remove the date reference? At least the year would seem appropriate. --agr (talk) 21:41, 11 August 2011 (UTC)

As was mentioned in the paragraph I was responding to, the date of the talk isn't relevant to the article. Rybu (talk) 03:11, 14 August 2011 (UTC)

homotopy, homotopy equivalent vs isotopy homeomorphic

"It is homotopy equivalent to the standard n-sphere because the gluing map is homotopic to the identity (being an orientation-preserving diffeomorphism, hence degree 1)"

I think that the gluing map is actually ISOTOPIC to the identity, hence a more precise sentence should be

"It is TOPOLOGICALLY EQUIVALENT to the standard n-sphere because the gluing map is ISOTOPIC to the identity (being an orientation-preserving diffeomorphism (or HOMEOMORPHISM), hence degree 1)"

of course the same result follows by poincare... but it is not necessary here --131.114.73.45 (talk) 16:07, 18 April 2013 (UTC)

constructed

According to http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf Milnor did not construct the spheres but rather proved their existence. Briscorn constructed them explicitly. Crasshopper (talk) 21:17, 3 January 2014 (UTC)

7-D result by Goette, Kerin, and Shankar - 2020

Mathematicians Resolve Complex Geometry Question First Asked 60 Years Ago 2020 has link to their online article “Highly connected 7-manifolds and non-negative sectional curvature”. Not sure how/where to update this article. - Rod57 (talk) 08:06, 3 June 2020 (UTC)

4 dimensions

[1] Manolescu and Piccirillo found 5 topologically slice knots with a computer search and proved that if any of them are slice, then there is an exotic 4-sphere. I don't have any clue about whether this is worth mentioning in the article, so am leaving it here for the experts. I didn't even know that the 4D case was unknown. 2602:24A:DE47:BB20:50DE:F402:42A6:A17D (talk) 22:08, 25 April 2021 (UTC)

Truly bad writing

In the section "Introduction" the second paragraph begins as follows:

"In differential topology, the relevant notion of sameness is witnessed by a diffeomorphism, which is a homeomorphism with the additional condition that it is smooth, that is, it should have derivatives of all orders everywhere."

But nobody has the slightest idea what the words "relevant" or "sameness" or "witnessed" mean here.

This is as incomprehensible to most readers as any writing I have ever seen in Wikipedia.

I hope someone knowledgeable about this subject and also capable of clear writing can fix this.

ALSO: The claim that a diffeomorphism "is a homeomorphism with the additional condition that it is smooth, that is, it should have derivatives of all orders everywhere" is simply wrong.

In order to qualify as a smooth diffeomorphism, a homeomorphism between two smooth manifolds must not only be smooth but its inverse function must also be smooth.

Why this is wrong: The real line ℝ is a smooth manifold. Consider the map f : ℝ → ℝ defined via f(x) = x³.

The function f is a homeomorphism with the additional condition that it has derivatives of all orders.

But it is not a diffeomorphism, because its inverse function is defined by f^-1(x) = x^1/3, which is not differentiable at x = 0.)