 Differential structure

In mathematics, an ndimensional differential structure (or differentiable structure) on a set M makes M into an ndimensional 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.
Contents
Definition
For a natural number n and some k which may be a nonnegative integer or infinity, an ndimensional C^{k} differential structure ^{[1]} is defined using a C^{k}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 :
which are C^{k}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 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:
The intersection of the domains of these two functions is:
and is mapped to two images
by the two chart maps.
The transition map between the two charts is the map between the two images of this intersection under the two chart maps.
Two charts are C^{k}compatible if
are open, and the transition maps
have continuous derivatives of order k. If k = 0, we only require that the transition maps are continuous, consequently a C^{0}atlas is simply another way to define a topological manifold. If k = ∞, derivatives of all orders must be continuous. A family of C^{k}compatible charts covering the whole manifold is a C^{k}atlas defining a C^{k} differential manifold. Two atlases are C^{k}equivalent if the union of their sets of charts forms a C^{k}atlas. In particular, a C^{k}atlas that is C^{0}compatible with a C^{0}atlas that defines a topological manifold is said to determine a C^{k} differential structure on the topological manifold. The C^{k} equivalence classes of such atlases are the distinct C^{k} 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^{k}−atlas on a given set a C^{k}−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^{k}−manifold, the maximal atlas contains a C^{∞}−atlas on the same underlying set by a theorem due to Whitney. However, a given maximal C^{k}−atlas contains distinct maximal C^{∞}−atlases whenever n > 0. Anyway, there is a C^{∞}−diffeomorphism between any two of these distinct C^{∞}−atlases. Thus there is only one class of pairwise smoothly diffeomorphic smooth, i.e. C^{∞}−structures in a C^{k}−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 C^{1}−structure, a result proved by Kervaire (1960),^{[2]} and later explained in the context of Donaldson's theorem (compare Hilbert's fifth problem).
Smooth structures on a orientable manifold are usually counted modulo orientationpreserving smooth homeomorphisms. There then arises the question whether orientationreversing 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^{n} with n ≠ 4, the number of these types in one, whereas for n = 4, there are uncountably many such types. One refers to these by exotic R^{4}.
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^{m} for the values of the dimension m from 1 up to 20. Spheres with a smooth, i.e. C^{∞}−differential structure not smoothly diffeomorphic to the usual one are known as exotic spheres.
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 4sphere S^{4} 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^{4} has more than one smooth type. The problem is connected with the existence of more than one smooth type of the topological 4disk (or 4ball).
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.^{[citation needed]} By using Obstruction theory, Robion Kirby and Laurent Siebenmann ^{[3]} 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 AsselmeyerMaluga, 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 b_{2}. For large Betti numbers b_{2} > 18 in a simply connected 4manifold, 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 one doesn't know the construction of other differential structures. For noncompact 4manifolds there are many examples like having uncountably many differential structures.
In 2010, Akhmedov and Park constructed infinitely many nondiffeomorphic smooth structures on .^{[4]}
See also
References
 ^ Hirsch, Morris, Differential Topology, Springer (1997), ISBN 0387901485. for a general mathematical account of differential structures
 ^ Kervaire (1960), "A manifold which does not admit any differentiable structure", Coment. Math. Helv. 34: 257–270
 ^ Kirby, Robion C. and Siebenmann, Laurence C., Foundational Essays on Topological Manifolds. Smoothings, and Triangulations. Princeton, New Jersey: Princeton University Press (1977), ISBN 0691081905.
 ^ Amar Akhmedov, B. Doug Park, Exotic Smooth Structures On S^2 x S^2, http://arxiv.org/abs/1005.3346
Categories: Differential structures
Wikimedia Foundation. 2010.