Editing Differential structure (section)

==Existence and uniqueness theorems==
For any integer ''k'' > 0 and any ''n''−dimensional ''C''<sup>''k''</sup>−manifold, the maximal atlas contains a ''C''<sup>∞</sup>−atlas on the same underlying set by a theorem due to [[Hassler Whitney]]. It has also been shown that any maximal ''C''<sup>''k''</sup>−atlas contains some number of ''distinct'' maximal ''C''<sup>∞</sup>−atlases whenever ''n'' > 0, although for any pair of these ''distinct'' ''C''<sup>∞</sup>−atlases there exists a ''C''<sup>∞</sup>−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''<sup>∞</sup>−, structures in a ''C''<sup>''k''</sup>−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 manifold]]s which admit no ''C''<sup>1</sup>−structure, a result proved by {{harvtxt|Kervaire|1960}},<ref>{{cite journal|last=Kervaire|first=Michel|authorlink=Michel Kervaire|title=A manifold which does not admit any differentiable structure|journal=[[Commentarii Mathematici Helvetici]]|volume=34|pages=257&ndash;270|year=1960|doi=10.1007/BF02565940}}</ref> 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 [[homeomorphism]]s. 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 '''R'''<sup>''n''</sup> 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|exotic '''R'''<sup>4</sup>]].