Hopkins and his colleagues think that they have found a way to remove that ambiguity. In their proof, which involves an intricate hierarchy of algebraic systems called homology groups, they show that the factor of two did not exist in any of those dimensions except possibly in the case 126, which, for technical reasons, their proof strategy did not address. (There is actually still another major exception: the 4-D case. Although there are no exotic 1-, 2- or 3-spheres, no one has any clue whether exotic 4-spheres exist or not.)
Although the researchers have not yet published their proof, Hopkins says, “I’m as confident as I possibly could be” without peer review that the proof is correct. Gunnar Carlsson, a topologist at Stanford University, says he has only heard “the most cursory outline of the proposed proof” from Hopkins but is “optimistic that the ingredients may very well be there for a resolution of this problem.” And not a moment too soon, if you’ve stayed up worrying about weird spheres.
Note: This article was originally printed with the title, "Hypersphere Exotica."
*Erratum (10/1/09): This sentence has been edited since posting to correct a numerical error.
This article was originally published with the title Hypersphere Exotica.
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8 Comments
Add Comment2^k-2 would be 1022 not 1026.
Reply | Report Abuse | Link to thisThe earth is not a 2-dimensional object, as to define any place on it you need latitude, longitude AND its radius.
Reply | Report Abuse | Link to thisPerhaps I should also mention that the sphere possesses the coordinates of its central point, its tilt parameters, its trajectory etc. This immediately pumps up the number of its 'dimensions'.
Reply | Report Abuse | Link to thisThere may be dimensions that are Euclidean which we think of as exotic or non-Euclidean. It has a lot to do with how we treat transforms as continous or finite in the quadratic plane. We need a wider term for the idea of dimensions and will find in any base, binary and beyond, there is a place where even and odd calculations vanish in the needed distinction. Our physics tries to put things into quantities which are dimensioned (or dimensionless) but on a deeper level the problems encountered with such physics is this exotic concern that strives to bring things closer to number theory.
Reply | Report Abuse | Link to thisIt is great to see an article on this result in Sci. Am. But the comment by Carlsson gives the misleading impression that details about the proof are not yet available. Though the official paper has not yet been posted, Ravenel gave a series of detailed lectures on the proof in Lisbon in early May, and the extensive notes, plus much other material, is available on his Kervaire Invariant webpage
Reply | Report Abuse | Link to thishttp://www.math.rochester.edu/u/faculty/doug/kervaire.html
So many experts already have a very good idea about how the proof goes.
Great to see this article, but the dimensions of the spheres you mention are off by one. The ``Kervaire spheres" in the first paragraph are in dimensions 2^j - 3: 253, 509, ... . It is manifolds with Arf-Kervaire invariant 1 which live, or don't, in dimensions 2^j - 2. When there are manifolds of Arf-Kervaire invariant 1 in dimension 2^j-2, then the Kervaire spheres in dimension 2^j-3 are standard.
Reply | Report Abuse | Link to thisSee http://en.wikipedia.org/wiki/Kervaire_invariant
In fact, I've just seen that dimension mistake persists to the bottom paragraph of the first page. The remaining ambiguity was for exotic spheres of odd dimension 4n+1, not for even dimensional exotic spheres.
Reply | Report Abuse | Link to thisIt is also worth pointing out that Kervaire and Milnor's result for exotic spheres of dimension 4n+3 also had an ambigiuty of a factor of two which arose from another very important problem in stable homotopy theory which is now called the Adams Conjecture, due to Frank Adams. This Conjecture was settled independently in the 70s by Daniel Quillen, Dennis Sullivan and possibly others.
See http://en.wikipedia.org/wiki/Adams_conjecture
Your article reminds me with a profound application of Milnors sphere in high energy physics which you have indeed alluded to in your article. In this respect I draw the attention of your readers to the following article which I found to be quite surprising because I never thought that such an abstract mathematics should have a bearing on high energy physics. The paper by G.L. Nashed published five years ago is entitled On Milnor seven dimensional sphere, El Nashies E-infinity theory and energy of a Bianchi universe in Chaos, Solitons & Fractals, 19, 2004. Following Alan Connes proposal which he expressed in his classical book On noncommutative geometry, El Naschie introduced a fuzzy Milnor 7 sphere. This sphere has a Hausdorff dimension different from its inductive topological dimension namely 7 + the golden mean to the power of 3. In his book Connes emphasizes the role of the golden number as a topological invariant for the space which he calls x space. This space is noncommutative in the quantum mechanical sense and fuzzy in the more ordinary meaning of the word. Alan Connes took the Penrose universe as an example. The only way to deal with such x space is to replace ordinary calculus with a quantum calculus. Thus integration is replaced by Dix Mier trace and differentiation with something similar to Poissons brackets. I was intrigued to see that this ingenious mathematical treatment of Connes was simplified in a physical way by El Naschie in his E-infinity theory using Weyl scaling. This is of course far more restrictive than Connes treatment but it is the relevant thing for physics of high energy. Weyl scaling is the original idea of gauge theory which failed because it works only in non-smooth spacetime. Such spacetime was not known at the time of Weyl and Einstein. However string theory, loop quantum gravity and E-infinity fractal spacetime clearly shows that non-smooth spacetime is far nearer to what our real spacetime on the quantum scale is. Experimental evidence for the granule nature of spacetime are coming in continuously. You may read about that in a recent article by Anil Ananthaswamy in the New Scientist, 15 August.
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