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Exotic sphere
Smooth manifold that is homeomorphic but not diffeomorphic to a sphere
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 n = 7 {\displaystyle n=7} as S 3 {\displaystyle S^{3}} -bundles over S 4 {\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...
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