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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 us to do [[differential calculus]] on the manifold. If ''M'' is already a topological manifold, we require that the new topology be identical to the existing one.

==Definition==
For a natural number ''n'' and some ''k'' which may be a non-negative integer or infinity, an '''n-dimensional ''C<sup>k</sup>'' differential structure''' <ref>Hirsch, Morris, ''Differential Topology'', Springer (1997), ISBN 0-387-90148-5. for a general mathematical account of differential structures</ref> is defined using a '''C<sup>k</sup>-atlas''', which is a set of [[bijections]] called '''charts''' between a collection of subsets of ''M'' (whose union is the whole of ''M''), and a set of open subsets of <math>\mathbb{R}^{n}</math>:

:<math>\varphi_{i}:M\supset W_{i}\rightarrow U_{i}\subset\mathbb{R}^{n}</math>

which are ''' ''C<sup>k</sup>''-compatible''' (in the sense defined below):

Each such map provides a way in which certain subsets of the manifold may be viewed as being like open subsets of <math>\mathbb{R}^{n}</math> but the usefulness of this notion depends on to what extent these notions agree when the domains of two such maps overlap.

Consider two charts:

:<math>\varphi_{i}:W_{i}\rightarrow U_{i},\,</math>
:<math>\varphi_{j}:W_{j}\rightarrow U_{j}.\,</math>

The intersection of the domains of these two functions is:

:<math>W_{ij}=W_{i}\cap W_{j}\;</math>

and its map by the two chart maps to the two images: 

:<math>U_{ij}=\varphi_{i}\left(W_{ij}\right),\,</math>
:<math>U_{ji}=\varphi_{j}\left(W_{ij}\right)</math>

The [[transition map]] between the two charts is the map between the two images of this intersection under the two chart maps.

:<math>\varphi_{ij}:U_{ij}\rightarrow U_{ji}</math>

:<math>\varphi_{ij}(x)=\varphi_{j}\left(\varphi_{i}^{-1}\left(x\right)\right).</math>

Two charts <math>\varphi_{i},\,\varphi_{j}</math> are '''C<sup>k</sup>-compatible''' if

:<math>U_{ij},\, U_{ji}</math>

are open, and the transition maps

:<math>\varphi_{ij},\,\varphi_{ji}</math>

have continuous derivatives of order ''k''. If ''k&nbsp;=&nbsp;0'', we only require that the transition maps are continuous, consequently a ''C<sup>0</sup>''-atlas is simply another way to define a topological manifold. If ''k''&nbsp;=&nbsp;∞, derivatives of all orders must be continuous. A family of ''C<sup>k</sup>''-compatible  charts covering the whole manifold is a ''C<sup>k</sup>''-atlas defining a ''C<sup>k</sup>'' differential manifold. Two atlases are ''' ''C<sup>k</sup>''-equivalent''' if the union of their sets of charts forms a ''C<sup>k</sup>''-atlas. In particular, a ''C<sup>k</sup>''-atlas that is ''C<sup>0</sup>''-compatible with a ''C<sup>0</sup>''-atlas that defines a topological manifold is said to determine a ''C<sup>k</sup>'' differential structure on the topological manifold. The ''C<sup>k</sup>'' [[equivalence classes]] of such atlases are the '''distinct C<sup>k</sup> differential structures''' of the [[manifold]]. Each distinct differential structure is determined by a unique maximal atlas, which is simply the union of all atlases in the equivalence class.

For simplification of language, without any loss of precision, one might just call a maximal ''C''<sup>''k''</sup>−atlas on a given set a ''C''<sup>''k''</sup>−manifold. This maximal atlas then uniquely determines both the topology and the underlying set, the latter being the union of the domains of all charts, and the former having the set of all these domains as a basis.

==Existence and uniqueness theorems==

For 0 < ''k'' < ∞ 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|Whitney]]. However, a given maximal ''C''<sup>''k''</sup>−atlas contains ''distinct'' maximal ''C''<sup>∞</sup>−atlases whenever ''n'' > 0 but there is a ''C''<sup>∞</sup>−diffeomorphism between any two of these distinct ''C''<sup>∞</sup>−atlases. Thus there is only one class of pairwise smoothly diffeomorphic smooth, i.e. ''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>{{citation|last=Kervaire|title=A manifold which does not admit any differentiable structure|journal=Coment. Math. Helv.|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>]].

==Differential structures on spheres of dimension from 1 to 20==

The following table lists the number of smooth types of the topological ''m''−sphere '''S'''<sup>''m''</sup> for the values of the dimension ''m'' from 1 up to 20. Spheres with a smooth, i.e. ''C''<sup>∞</sup>−differential structure not smoothly diffeomorphic to the usual one are known as [[exotic sphere]]s.

{| border="1" cellpadding="2"
|-
! Dimension !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12 !! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 19 !! 20
|-
!Smooth types || 1 || 1 || 1 || ? || 1 || 1 || 28 || 2 || 8 || 6 || 992 || 1 || 3 || 2 || 16256 || 2 || 16 || 16 || 523264 || 24
|}

It is not currently known how many smooth types the topological 4-sphere '''S'''<sup>4</sup> has, except that there is at least one.  There may be one, a finite number, or an infinite number.  The claim that there is just one is known as the ''smooth'' [[Poincaré conjecture]] (see [[generalized Poincaré conjecture]]). Most mathematicians believe that this conjecture is false, i.e. that '''S'''<sup>4</sup> has more than one smooth type. The problem is connected with the existence of more than one smooth type of the topological 4-disk (or 4-ball).

==Differential structures on topological manifolds==

As mentioned above, in dimensions smaller than 4, there is only one differential structure for each topological manifold. That was proved by [[Johann Radon]] for dimension 1 and 2, and by [[Edwin E. Moise]] in dimension 3.<ref>Moise, Edwin E., ''Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung''. Annals of Mathematics. Second Series, Vol. 56 pg 96-114 (1952)</ref> By using [[obstruction theory]], [[Robion Kirby]] and Laurent Siebenmann <ref>Kirby, Robion C. and Siebenmann, Laurence C., ''Foundational Essays on Topological Manifolds. Smoothings, and Triangulations''.  Princeton, New Jersey: Princeton University Press (1977), ISBN 0-691-08190-5.</ref> were able to show that the number of [[PL structure]]s 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) <!-- This needs checking: this number agrees with the number of differential structures on the sphere of the same dimension. Thus the table above lists also the number of differential structures for any (metrizable) topological manifold of dimension <math>n</math>.--> By combining these results, the number of smooth structures on a compact topological manifold of dimension not equal to 4 is finite. 

[[4-manifold|Dimension 4]] is  more complicated. For compact manifolds, results depend on the complexity of the manifold as measured by the second [[Betti number]] <math>b_2</math>. For large Betti numbers <math>b_2>18</math> 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 like <math>S^4, {\mathbb C}P^2,...</math> one doesn't know the construction of other differential structures. For non-compact 4-manifolds there are many examples like <math>{\mathbb R}^4,S^3\times {\mathbb R},M^4\setminus\{*\},...</math> having uncountably many differential structures.

==See also==
*[[atlas (topology)|Atlas]]
*[[Exotic R4|Exotic R<sup>4</sup>]]
*[[Exotic sphere]]

== References ==
<references/>

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