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:
“Anthropology has always struggled with an intense, fascinated repulsion towards its subject.... [The anthropologist] submits himself to the exotic to confirm his own inner alienation as an urban intellectual.”
—Susan Sontag (b. 1933)
“If today there is a proper American “sphere of influence” it is this fragile sphere called earth upon which all men live and share a common fate—a sphere where our influence must be for peace and justice.”
—Hubert H. Humphrey (1911–1978)