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Concept
Exotic sphere
Summary
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 in dimension as -bundles over . He showed that there are at least 7 differentiable structures on the 7-sphere. In any dimension 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 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.
Specifically, this means that the elements of this group (n ≠ 4) 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.
The unit n-sphere, , is the set of all (n+1)-tuples of real numbers, such that the sum . For instance, is a circle, while is the surface of an ordinary ball of radius one in 3 dimensions. Topologists consider a space, X, to be an n-sphere if there is a homeomorphism between them, i.e. every point in X may be assigned to exactly one point in the unit n-sphere in a bicontinuous (i.e. continuous and invertible with continuous inverse) manner. 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 .
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