Differential structures | Surgery theory | Spheres | Differential topology

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 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 Michel Kervaire and Milnor 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. (Wikipedia).

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Related pages

Kervaire invariant | Fiber bundle | Homeomorphism | Derivative | J-homomorphism | Poincaré conjecture | Surgery exact sequence | Surgery theory | Continuous function | H-cobordism | Index of a subgroup | Complex manifold | Parallelizable manifold | Quaternion | Piecewise linear manifold | Homotopy sphere | Pseudoisotopy theorem | Bernoulli number | Boundary (topology) | Finite group | Monoid | Connected sum | Differentiable manifold | Diffeomorphism | Exotic R4 | Cyclic group | Critical point (mathematics) | N-sphere | Orientability | Smooth structure | Differential topology | Exotic sphere | Edwin E. Moise | Tuple | Atlas (topology) | Cerf theory | Seven-dimensional space | List of unsolved problems in mathematics | Abelian group | Moise's theorem | Clutching construction