Editing Talk:Exotic sphere (section)

==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. -- [[User:Beland|Beland]] ([[User talk: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? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk: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. [[User:Rybu|Rybu]] ([[User talk: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? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 19:05, 11 April 2011 (UTC)

: Steelpillow, exotic spheres are all homeomorphic to <math>S^n</math> 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?  [[User:Rybu|Rybu]] ([[User talk: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? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk: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. [[User:Rybu|Rybu]] ([[User talk: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? -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 18:21, 12 April 2011 (UTC)
:::I found [http://www.math.niu.edu/~rusin/known-math/99/exotic 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. [[User:Charles Matthews|Charles Matthews]] ([[User talk: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. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk: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.  [[User:Rybu|Rybu]] ([[User talk: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. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk: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 [[chart (mathematics)|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. [[User:Charles Matthews|Charles Matthews]] ([[User talk: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. [[User:Rybu|Rybu]] ([[User talk: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.--[[User:ArnoldReinhold|agr]] ([[User talk:ArnoldReinhold|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! &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 16:07, 31 May 2011 (UTC)