Exotic Sphere - Introduction

Introduction

The unit n-sphere, Sn, is the set of all n+1-tuples (x₁, x₂, ... x_n+1) of real numbers, such that the sum x₁2 + x₂2 + ... + x_n+12 = 1. (S1 is a circle; S2 is the surface of an ordinary ball of radius one in 3 dimensions.) Topologists consider a space, X, to be an n-sphere if every point in X can be assigned to exactly one point in the unit n-sphere in a continuous way, which means that sufficiently nearby points in X get assigned to nearby points in Sn and vice-versa. For example a point x on an n-sphere of radius r can be matched with a point on the unit n-sphere by adjusting its distance from the origin by 1/r.

In differential topology, a more stringent condition is added, that the functions matching points in X with points in Sn should be smooth, that is they should have derivatives of all orders everywhere. To calculate derivatives, one needs to have local coordinate systems defined consistently in X. Mathematicians were surprised in 1956 when John Milnor showed that consistent coordinate systems could be set up on the 7-sphere in two different ways that were equivalent in the continuous sense, but not in the differentiable sense. Milnor and others set about trying to discover how many such exotic spheres could exist in each dimension and to understand how they relate to each other. No exotic structures are possible on the 1-, 2-, 3-, 5-, 6- or 12-spheres. Some higher dimensional spheres have only two possible differentiable structures, others have thousands. Whether exotic 4-spheres exist, and if so how many, is an important unsolved problem in mathematics.