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)
“In the new science of the twenty-first century, not physical force but spiritual force will lead the way. Mental and spiritual gifts will be more in demand than gifts of a physical nature. Extrasensory perception will take precedence over sensory perception. And in this sphere woman will again predominate.”
—Elizabeth Gould Davis (b. 1910)