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Exotic sphere
In 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 in dimension ''n'' = 7 as ''S''3-bundles over ''S''4. 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 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.
==Introduction==
The unit ''n''-sphere, ''S''n, is the set of all ''n''+1-tuples (''x''1, ''x''2, ... ''x''n+1) of real numbers, such that the sum ''x''12 + ''x''22 + ... + ''x''n+12 = 1. (''S''1 is a circle; ''S''2 is the surface of an ordinary ball of radius one in 3 dimensions.) Topologists consider a space, ''X'', to be an ''n''-sphere if every point in ''X'' can be assigned to exactly one point in the unit ''n''-sphere in a continuous way, which means that sufficiently nearby points in ''X'' get assigned to nearby points in ''S''n and vice versa. For example a point ''x'' on an ''n''-sphere of radius ''r'' can be matched with a point on the unit ''n''-sphere by adjusting its distance from the origin by 1/''r''.
In differential topology, a more stringent condition is added, that the functions matching points in ''X'' with points in ''S''n should be smooth, that is they should have derivatives of all orders everywhere. To calculate derivatives, one needs to have local coordinate systems defined consistently in ''X.'' Mathematicians were surprised in 1956 when John Milnor showed that consistent coordinate systems could be set up on the 7-sphere in two different ways that were equivalent in the continuous sense, but not in the differentiable sense. Milnor and others set about trying to discover how many such exotic spheres could exist in each dimension and to understand how they relate to each other. No exotic structures are possible on the 1-, 2-, 3-, 5-, 6- or 12-spheres. Some higher-dimensional spheres have only two possible differentiable structures, others have thousands. Whether exotic 4-spheres exist, and if so how many, is an important unsolved problem in mathematics.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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