In differential topology, a mathematical discipline, 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.
Read more about Exotic Sphere: Introduction, Classification, Explicit Examples of Exotic Spheres, Twisted Spheres, Applications, 4-dimensional Exotic Spheres and Gluck Twists
Famous quotes containing the words exotic and/or sphere:
“I do not approve of anything that tampers with natural ignorance. Ignorance is like a delicate exotic fruit; touch it and the bloom is gone.”
—Oscar Wilde (1854–1900)
“Every person is responsible for all the good within the scope of his abilities, and for no more, and none can tell whose sphere is the largest.”
—Gail Hamilton (1833–1896)