For a natural number n and some k which may be a non-negative integer or infinity, an n-dimensional Ck differential structure[1] is defined using a Ck-atlas, which is a set of bijections called charts between subsets of M (whose union is the whole of M) and open subsets of
are Ck-compatible if are open, and the transition maps have continuous partial derivatives of order k. If k = 0, we only require that the transition maps are continuous, consequently a C0-atlas is simply another way to define a topological manifold.
Two atlases are Ck-equivalent if the union of their sets of charts forms a Ck-atlas.
For simplification of language, without any loss of precision, one might just call a maximal Ck−atlas on a given set a Ck−manifold.
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.
A bit loosely, one might express this by saying that the smooth structure is (essentially) unique.
Namely, there exist topological manifolds which admit no C1−structure, a result proved by Kervaire (1960),[2] and later explained in the context of Donaldson's theorem (compare Hilbert's fifth problem).
There is an "essentially unique" smooth structure for any topological manifold of dimension smaller than 4.
The following table lists the number of smooth types of the topological m−sphere Sm for the values of the dimension m from 1 up to 20.
The problem is connected with the existence of more than one smooth type of the topological 4-disk (or 4-ball).
As mentioned above, in dimensions smaller than 4, there is only one differential structure for each topological manifold.
[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.
[4] 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.
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.