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.[break]
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.