Editing Exotic sphere (section)

{{Short description|Smooth manifold that is homeomorphic but not diffeomorphic to a sphere}}
{{Use dmy dates|date=September 2020}}
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|''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 {{harvs|authorlink=John Milnor|first=John|last=Milnor|year=1956|txt=yes}} in  dimension <math>n = 7</math> as <math>S^3</math>-[[Fiber bundle|bundles]] over <math>S^4</math>. He showed that there are at least 7 differentiable structures on the 7-sphere. In any dimension {{harvtxt|Milnor|1959}} showed that the [[diffeomorphism class]]es of oriented exotic spheres form the non-trivial elements of an [[abelian monoid]] under [[connected sum]], which is a [[Finite group|finite]] [[abelian group]] if the dimension is not 4. The classification of exotic spheres by {{harvs |authorlink1=Michel Kervaire |first1=Michel |last1=Kervaire |last2=Milnor |year=1963 |txt=yes}} showed that the [[orientability|oriented]] exotic 7-spheres are the non-trivial elements of a [[cyclic group]] of order 28 under the operation of [[connected sum]]. These groups are known as [[Kervaire–Milnor group|Kervaire–Milnor groups]].

More generally, in any dimension ''n ≠ 4'', there is a finite Abelian group whose elements  are the equivalence classes of smooth structures on ''S''<sup>n</sup>, where two structures are considered equivalent if there is an orientation preserving diffeomorphism carrying one structure onto the other. The group operation is defined by [x] + [y]  =  [x + y], where x and y are arbitrary representatives of their equivalence classes, and ''x + y'' denotes the smooth structure on the smooth ''S''<sup>n</sup> that is the connected sum of x and y. It is necessary to show that such a definition does not depend on the choices made; indeed this can be shown.