Дифференциальные структуры и гладкость многообразий.

Mathematical structure
In mathematics, an n dimensional differential structure (or differentiable structure) on a set M makes M into an n dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold. If M is already a topological manifold, it is required that the new topology be identical to the existing one.

For any integer k > 0 and any n−dimensional Ck−manifold, the maximal atlas contains a C∞−atlas on the same underlying set by a theorem due to Hassler Whitney. It has also been shown that any maximal Ck−atlas contains some number of distinct maximal C∞−atlases whenever n > 0, although for any pair of these distinct C∞−atlases there exists a C∞−diffeomorphism identifying the two. It follows that there is only one class of smooth structures (modulo pairwise smooth diffeomorphism) over any topological manifold which admits a differentiable structure, i. e. The C∞−, structures in a Ck−manifold. A bit loosely, one might express this by saying that the smooth structure is (essentially) unique. The case for k = 0 is different. Namely, there exist topological manifolds which admit no C1−structure, a result proved by , and later explained in the context of Donaldson's theorem (compare Hilbert's fifth problem). Smooth structures on an orientable manifold are usually counted modulo orientation preserving smooth homeomorphisms. There then arises the question whether orientation reversing diffeomorphisms exist. There is an "essentially unique" smooth structure for any topological manifold of dimension smaller than 4. For compact manifolds of dimension greater than 4, there is a finite number of "smooth types", i. e. equivalence classes of pairwise smoothly diffeomorphic smooth structures. In the case of Rn with n ≠ 4, the number of these types is one, whereas for n = 4, there are uncountably many such types. One refers to these by exotic R4.

As mentioned above, in dimensions smaller than 4, there is only one differential structure for each topological manifold. That was proved by Tibor Radó for dimension 1 and 2, and by Edwin E. Moise in dimension 3. By using obstruction theory, Robion Kirby and Laurent C. Siebenmann were able to show that the number of PL structures for compact topological manifolds of dimension greater than 4 is finite. John Milnor, Michel Kervaire, and Morris Hirsch proved that the number of smooth structures on a compact PL manifold is finite and agrees with the number of differential structures on the sphere for the same dimension (see the book Asselmeyer Maluga, Brans chapter 7) By combining these results, the number of smooth structures on a compact topological manifold of dimension not equal to 4 is finite. Dimension 4 is more complicated. For compact manifolds, results depend on the complexity of the manifold as measured by the second Betti number b2. For large Betti numbers b2 > 18 in a simply connected 4 manifold, one can use a surgery along a knot or link to produce a new differential structure. With the help of this procedure one can produce countably infinite many differential structures. But even for simple spaces such as one doesn't know the construction of other differential structures. For non compact 4 manifolds there are many examples like having uncountably many differential structures.