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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ...

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 set of subsets of M (whose union is the whole of M), and a set of open subsets of an n-dimensional vector space: In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

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. In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

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. 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. In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... Zorns lemma, also known as the Kuratowski-Zorn lemma, is a proposition of set theory that states: Every non-empty partially ordered set in which every chain (i. ...

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, see Donaldson's theorem (confer Hilbert's fifth problem). Hassler Whitney (23 March 1907 – 10 May 1989) was an American mathematician who was one of the founders of singularity theory, PhB, Yale University, 1928; MusB, 1929; ScD (Honorary), 1947; PhD, Harvard University, under G.D. Birkhoff, 1932. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, Donaldsons theorem states that a positive definite intersection form of a simply connected smooth manifold of dimension 4 is diagonalisable to the unit matrix. ... In mathematics, a Lie group (IPA ) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...

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. In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...

Differential structures on spheres of dimensions from 1 to 18

The following table lists the numbers of 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. In mathematics, an exotic sphere is a differential manifold M, such that from a topological point of view M is a sphere, but not from the point of view of its differential structure. ...

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. In mathematics, the Poincaré conjecture is a conjecture about the characterisation of the three-dimensional sphere amongst 3-manifolds. ...

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 Randon for dimension 1 and 2, and by Moise in dimension 3. By using Obstruction theory, Kirby and Siebenman were able to show that the number of differential structures for topological manifolds of dimension greater than 4 is finite. Furthermore they proved that 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 n. Obstruction theory is a mathematical theory concerned with when a topological manifold has a piecewise linear structure and when a piecewise linear manifold has a differentiable structure. ...

In case of dimension 4, the situation is much 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 trivial 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 In algebraic topology, the Betti number of a topological space is, in intuitive terms, a way of counting the maximum number of cuts that can be made without dividing the space into two pieces. ...

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)

See also

• Atlas
• Exotic R⁴
• Exotic sphere
• Manifold