Exotic sphere

In an area of mathematics called 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 ${\displaystyle n=7}$ as ${\displaystyle S^{3}}$-bundles over ${\displaystyle S^{4}}$. 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 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.

More generally, in any dimension n ≠ 4, there is a finite Abelian group whose elements are the equivalence classes of smooth structures on Sⁿ, where two structures are considered equivalent if there is an orientation preserving diffeomorphism carrying one structure onto the other. The group operation is defined by [x] + [y] = [x + y], where x and y are arbitrary representatives of their equivalence classes, and x + y denotes the smooth structure on the smooth Sⁿ that is the connected sum of x and y. It is necessary to show that such a definition does not depend on the choices made; indeed this can be shown.