Differential structure

In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes it 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^k differential structure 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 (or any n-dimensional vector space):

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 C⁰-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 C⁰-compatible with a C⁰-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.

Existence and uniqueness theorems

On any manifold with a C^k structure for k>0, there is a unique C^k-compatible C^∞-structure, a theorem due to Whitney. On the other hand, there exist topological manifolds which admit no differential structures, a result proved by Kervaire (1960), and later explained in the context of Donaldson's theorem (compare Hilbert's fifth problem).

When people count differential structures on a manifold, they usually count them modulo orientation-preserving homeomorphisms. There is only one differential structure of any manifold of dimension smaller than 4. For all manifolds of dimension greater than 4 there is a finite number of differential structures on any compact manifold. There is only one differential structure on except when n = 4, in which case there are uncountably many.

Differential structures on spheres of dimensions from 1 to 18

The following table lists the numbers of (smooth) differential structures (modulo orientation-preserving diffeomorphism) on the n-sphere for dimensions 1 up to dimension 18. Spheres with differential structures different from the usual one are known as exotic spheres.

It is not currently known how many differential structures there are on the 4-sphere, beyond 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. Most mathematicians believe that this conjecture is false, i.e. there are more than one differential structure on the 4-sphere. The problem is connected with the existence of more than one differential structure for the open 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. By using Obstruction theory, Robion Kirby and Laurent 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 like 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

See also

• Atlas
• Exotic R⁴
• Exotic sphere

References

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations where appropriate. (May 2009)

• Hirsch, Morris, Differential Topology, Springer (1997), ISBN 0-387-90148-5. 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 0-691-08190-5.
• Asselmeyer-Maluga, T. and Brans, C.H., Exotic Smoothness in Physics. World Scientific Singapore, 2007 (for more information see the web-page http://loyno.edu/~cbta), ISBN 978-981-02-4195-7