Exotic sphere
In differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one (hence the name "exotic").
The first exotic spheres were constructed by John Milnor (1956) in dimension n = 7 as S3-bundles over S4. He showed that there are at least 7 differentiable structures on the 7-sphere. In any dimension Milnor (1959) showed that the diffeomorphism classes of oriented exotic spheres form the non-trivial elements of an abelian monoid under connected sum, which is a finite abelian group if the dimension is not 4. The classification of exotic spheres by Michel Kervaire and John Milnor (1963) showed that the oriented exotic 7-spheres are the non-trivial elements of a cyclic group of order 28 under the operation of connected sum.
The unit n-sphere, Sn, is the set of all (n+1)-tuples (x1, x2, ... xn+1) of real numbers, such that the sum x12 + x22 + ... + xn+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.
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