12/03/17 23:35:58.50
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なるほど・・
URLﾘﾝｸ(en.wikipedia.org)
Homeomorphism and diffeomorphism
It is easy to find a homeomorphism that is not a diffeomorphism, but it is more difficult to find a pair of homeomorphic manifolds that are not diffeomorphic.
In dimensions 1, 2, 3, any pair of homeomorphic smooth manifolds are diffeomorphic.
In dimension 4 or greater, examples of homeomorphic but not diffeomorphic pairs have been found. The first such example was constructed by John Milnor in dimension 7.
He constructed a smooth 7-dimensional manifold (called now Milnor's sphere) that is homeomorphic to the standard 7-sphere but not diffeomorphic to it.
There are in fact 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere (each of them is a total space of the fiber bundle over the 4-sphere with the 3-sphere as the fiber).
Much more extreme phenomena occur for 4-manifolds: in the early 1980s, a combination of results due to Simon Donaldson and Michael Freedman led to the discovery of exotic R4s:
there are uncountably many pairwise non-diffeomorphic open subsets of R4 each of which is homeomorphic to R4,
and also there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to R4 that do not embed smoothly in R4.