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# Hypersphere Exotica: Kervaire Invariant Problem Has a Solution!

A 45-year-old problem on higher-dimensional spheres is solved--probably

Relax. Until recently, lurking in the dark recesses of mathematical existence, there might have been a really weird sphere of 254 dimensions, or 510, or 1,022.* In fact, for all you knew, you might have had to worry about weird spheres when visiting any space with numbers of dimensions of the type 2k - 2.

Not anymore. "We can all sleep a bit better tonight," joked mathematical physicist John Baez of the University of California, Riverside, in his blog. Baez was referring to the announcement made by mathematicians Michael Hopkins of Harvard University, Michael Hill of the University of Virginia and Douglas Ravenel of the University of Rochester that they had cracked a 45-year-old question known as the Kervaire invariant problem. If confirmed, their result puts the finishing touch to a glorious piece of 1960s mathematics: the classification of "exotic," higher-dimensional spheres. The Kervaire problem was a major stumbling block in understanding multidimensional spaces, and its solution could have implications in equally exotic fields of physics such as string theory.

When mathematicians talk about higher-dimensional spaces, they are referring to the number of variables, or dimensions, needed to locate a point in such a space. The surface of the earth is two-dimensional because two coordinates—latitude and longitude—are needed to specify any point on it. In more formal terms, the standard two-dimensional sphere is the set of points equidistant from a point in 2 + 1 = 3 dimensions. More generally, the standard n-dimensional sphere, or n-sphere for short, is the set of points that are at the same distance from a center point in a space of n + 1 dimensions. Spheres are among the most basic spaces in topology, the branch of mathematics that studies which properties are unchanged when an object is deformed without crushing or ripping it. Topology comes up in many studies, including those trying to determine the shape of our universe.

In recent years mathematicians have completed the classification of 3-D spaces that are “compact,” meaning that they are finite and with no edges [see “The Shapes of Space,” by Graham P. Collins; Scientific American, July 2004]. (A sphere is compact, but an infinite plane is not.) Thus, they have figured out the topologies of all possible universes, as long as those universes are compact and three-dimensional. In more than three dimensions, however, the complete classification has turned out to be intractable and even logically impossible. Topologists had hoped at least that spaces as simple as spheres would be easy enough.

John Milnor, now at Stony Brook University, complicated matters somewhat in the 1950s, when he discovered the first “exotic” 7-sphere. An exotic n-sphere is a sphere from the point of view of topology. But it is not equivalent to a standard n-sphere from the point of view of differential calculus, the language in which physics theories are formulated. The discrepancy has consequences for equations such as those that describe the motion of particles or the propagation of waves. It means that solutions to such equations (or even their formulation) on one space cannot be mapped onto the other without developing kinks, or “singularities.” Physically, the two spheres are different, incompatible worlds.

In 1963 Milnor and his colleague Michel Kervaire calculated the number of exotic 7-spheres and found that there were exactly 27 different ones. In fact, they calculated the number of n-spheres for any n from five up. Their counts, however, had an ambiguity—a possible factor of two—when n is an even number. William Browder of Princeton University later removed that ambiguity, except for dimensions of the type n = 2k - 2, starting with k = 7—specifically, 126, 254, 510, and so on. In other words, mathematicians could only guess the number of exotic spheres in these dimensions to within a factor of two, known as the Kervaire invariant because of its relation to an earlier concept invented by Kervaire.

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1. 1. Ralf123 12:31 PM 7/25/09

2^k-2 would be 1022 not 1026.

2. 2. eco-steve 05:07 PM 7/27/09

The earth is not a 2-dimensional object, as to define any place on it you need latitude, longitude AND its radius.

3. 3. eco-steve 05:15 PM 7/27/09

Perhaps 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'.

4. 4. PeSla 05:45 PM 7/27/09

There 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.

5. 5. Nicholas Kuhn 10:48 AM 8/10/09

It 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

http://www.math.rochester.edu/u/faculty/doug/kervaire.html

So many experts already have a very good idea about how the proof goes.

6. 6. Diarmuid Crowley 06:39 AM 8/11/09

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.

See http://en.wikipedia.org/wiki/Kervaire_invariant

7. 7. Diarmuid Crowley 06:49 AM 8/11/09

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.

It 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.

8. 8. Bjorn 03:35 PM 8/14/09

Your article reminds me with a profound application of Milnors 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 Nashies 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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