In mathematics, particularly in differential topology, the strong Whitney embedding theorem states that any connected smooth m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in Euclidean 2m-space. This is the best linear bound on the smallest-dimensional Euclidean space that all m-dimensional manifolds embed in, as the real projective spaces of even dimension m cannot be embedded into Euclidean (2m − 1)-space (as can be seen from a characteristic class argument, also due to Whitney). Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... This article or section is in need of attention from an expert on the subject. ... 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. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology, a second-countable space is a topological space satisfying the second axiom of countability. Specifically, a space is said to be second countable if its topology has a countable base. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. ...

The weak Whitney embedding theorem states that any continuous function from an n-dimensional manifold to an m-dimensional manifold may be approximated by a smooth embedding provided m>2n. Whitney similarly proved that such a map could be approximated by an immersion provided m>2n-1. This last result is sometimes called the weak Whitney immersion theorem.

A little about the proof

The general outline of the proof is to start with an immersion with transversal self-intersections. These are known to exist from Whitney's earlier work on the weak immersion theorem. Transversality of the double points follows from a general-position argument. The idea is to then somehow remove all the self-intersections. If M has boundary, one can remove the self-intersections simply by isotoping M into itself (the isotopy being in the domain of f), to a submanifold of M that does not contain the double-points. Thus, we are quickly led to the case where M has no boundary. Sometimes it is impossible to remove the double-points via an isotopy -- consider for example the figure-8 immersion of the circle in the plane. In this case, one needs to introduce a local double point. Introducing opposite double point Once one has two opposite double points, one constructs a closed loop connecting the two, giving a closed path in R^2m. Since R^2m is simply-connected, one can assume this path bounds a disc, and provided 2m > 4 one can further assume (by the weak Whitney embedding theorem) that the disc is embedded in R^2m such that it intersects the image of M only in its boundary. Whitney then uses the disc to create a 1-parameter family of immersions, in effect pushing M across the disc, removing the two double points in the process. In the case of the figure-8 immersion with its introduced double-point, the push across move is quite simple: Cancelling double points using Whitney disc This process of eliminating opposite sign double-points by pushing the manifold along a disc is called the Whitney Trick. This is a glossary of terms specific to differential geometry and differential topology. ...

Eventual Consequences of the Whitney trick

The Whitney trick was used by Steve Smale to prove the h-cobordism theorem; from which follows the Poincare conjecture in dimensions , and the classification of smooth structures on discs (also in dimensions 5 and up). A cobordism W between M and N is an h-cobordism if the inclusion maps M → W and N → W are homotopy equivalences. ...

Given two oriented submanifolds of complementary dimensions in a simply connected manifold of dimension , one can apply an isotopy to one of the submanifolds so that all the points of intersection have the same sign.

History

The occasion of the proof by Hassler Whitney of the embedding theorem for smooth manifolds is said (rather surprisingly) to have been the first complete exposition of the manifold concept precisely because it brought together and unified the differing concepts of manifolds at the time: no longer was there any confusion as to if abstract manifolds (defined via charts) were any more or less general than submanifolds of Euclidean space. See Manifold. 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. ... 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. ...

Sharper results

Although every n-manifold embeds in R²ⁿ, one can frequently do better. Let e(n) denote the smallest integer so that all compact connected n-manifolds embed in R^e(n), then Whitney's strong embedding theorem states that . For n = 1,2 this inequality is the best possible, as the circle and the Klein bottle show. C.T.C. Wall improved on Whitney's result by showing that e(3) = 5. At present the function e(n) is not known in closed-form for all integers (compare to the Whitney immersion theorem). In differential topology, the Whitney embedding theorem states that Any smooth second-countable -dimensional manifold can be embedded in Euclidean -space. ...

See also

• Whitney immersion theorem
• Nash embedding theorem