Multi-dimensional Sphere Problem from 1950s Solved.

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.