Differential structure

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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.

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[edit] Definition

For a natural number n and some k which may be a non-negative integer or infinity, an n-dimensional Ck differential structure is defined using a Ck-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 an n-dimensional vector space:

\varphi_{i}: M\supset W_{i}\rightarrow U_{i}\subset\mathbb{R}^{n}.

which are Ck-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 \mathbb{R}^{n} 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:

\varphi_{i}:W_{i}\rightarrow U_{i},\,
\varphi_{j}:W_{j}\rightarrow U_{j}.\,

The intersection of the domains of these two functions is:

W_{ij}=W_{i}\cap W_{j}\;

and is mapped to two images

U_{ij}=\varphi_{i}\left(W_{ij}\right),\,
U_{ji}=\varphi_{j}\left(W_{ij}\right)

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.

\varphi_{ij}:U_{ij}\rightarrow U_{ji}
\varphi_{ij}(x)=\varphi_{j}\left(\varphi_{i}^{-1}\left(x\right)\right).

Two charts \varphi_{i},\,\varphi_{j} are Ck-compatible if

U_{ij},\, U_{ji}

are open, and the transition maps

\varphi_{ij},\,\varphi_{ji}

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 Ck-compatible charts covering the whole manifold is a Ck-atlas defining a Ck differential manifold. Two atlases are Ck-equivalent if the union of their sets of charts forms a Ck-atlas. In particular, a Ck-atlas that is C0-compatible with a C0-atlas that defines a topological manifold is said to determine a Ck differential structure on the topological manifold. The Ck equivalence classes of such atlases are the distinct Ck differential structures of the manifold. For each distinct differential structure the existence of a single maximal atlas can be shown using Zorn's lemma. It is the union of all of the atlases in the equivalence class.

[edit] Existence and uniqueness theorems

On any manifold with a Ck structure for k>0, there is a unique Ck-compatible C∞-structure, a theorem due to Whitney. On the other hand, there exist topological manifolds which admit no differential structures, see Donaldson's theorem (confer 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 \mathbb{R}^{n} except when n = 4, in which case there are uncountably many.

[edit] Differential structures on spheres of dimensions from 1 to 18

The following table lists the numbers of (smooth) differential structures (modulo orientation-preserving homeomorphism) on the n-sphere for dimensions n up to dimension 18. Spheres with differential structures different from 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
Structures 1 1 1  ? 1 1 28 2 8 6 992 1 3 2 16256 2 16 16

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 4-disk D4.

[edit] 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 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 S^4, S^2\times S^2, {\mathbb C}P^2,... one doesn't know the construction of other differential structures. For non-compact 4-manifolds there are many examples like {\mathbb R}^4,S^3\times {\mathbb R},M^3\setminus\{*\},... having uncountably many differential structures

[edit] References

  • Hirsch, Morris, Differential Topology, Springer (1997), ISBN 0-387-90148-5. for a general mathematical account of differential structures
  • 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 informations see the web-page http://loyno.edu/~cbta), ISBN 978-981-02-4195-7

[edit] See also

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