Whitney embedding theorem
- The strong Whitney embedding theorem states that any smooth real m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real 2m-space (R2m), if m > 0. 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 dimension m cannot be embedded into real (2m − 1)-space if m is a power of two (as can be seen from a characteristic class argument, also due to Whitney).
- 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 f : M → R2m with transverse 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.
Once one has two opposite double points, one constructs a closed loop connecting the two, giving a closed path in R2m. Since R2m 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 R2m 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 (pictured).
This process of eliminating opposite sign double-points by pushing the manifold along a disc is called the Whitney Trick.
To introduce a local double point, Whitney created a family of immersions αm : Rm → R2m which are approximately linear outside of the unit ball, but containing a single double point. For m = 1 such an immersion is defined as
Notice that if α1 is considered as a map to R3 i.e.:
then the double point can be resolved to an embedding:
Notice β1(t1, 0) = α1(t1) and for a ≠ 0 then as a function of t1, β1(t1, a) is an embedding. Define
α2 can similarly be resolved in R5, this process ultimately leads one to the definition:
The key properties of αm is that it is an embedding except for the double-point αm(1, 0, … , 0) = αm(−1, 0, … , 0). Moreover, for |(t1, … , tm)| large, it is approximately the linear embedding (0, t1, 0, t2, … , 0, tm).
Eventual consequences of the Whitney trick 
The Whitney trick was used by Steve Smale to prove the h-cobordism theorem; from which follows the Poincaré conjecture in dimensions m ≥ 5, and the classification of smooth structures on discs (also in dimensions 5 and up). This provides the foundation for surgery theory, which classifies manifolds in dimension 5 and above.
Given two oriented submanifolds of complementary dimensions in a simply connected manifold of dimension ≥ 5, one can apply an isotopy to one of the submanifolds so that all the points of intersection have the same sign.
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 whether abstract manifolds, intrinsically defined via charts, were any more or less general than manifold extrinsically defined as submanifolds of Euclidean space. See also the history of manifolds and varieties for context.
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 Re(n). Whitney's strong embedding theorem states that e(n) ≤ 2n. For n = 1, 2 we have e(n) = 2n, as the circle and the Klein bottle show. More generally, for n = 2k we have e(n) = 2n, as the 2k-dimensional real projective space show. Whitney's result can be improved to e(n) ≤ 2n − 1 unless n is a power of 2. This is a result of Haefliger–Hirsch (n > 4) and C.T.C. Wall (n = 3); these authors used important preliminary results and particular cases proved by M. Hirsch, W. Massey, S. Novikov and V. Rokhlin. At present the function e is not known in closed-form for all integers (compare to the Whitney immersion theorem, where the analogous number is known).
Restrictions on manifolds 
One can strengthen the results by putting additional restrictions on the manifold. For example, the n-sphere always embeds in Rn + 1 – which is the best possible (closed n-manifolds cannot embed in Rn). Any compact orientable surface and any compact surface with non-empty boundary embeds in R3, though any closed non-orientable surface needs R4.
If N is a compact orientable n-dimensional manifold, then N embeds in R2n − 1 (for n not a power of 2 the orientability condition is superfluous). For n a power of 2 this is a result of A. Haefliger–M. Hirsch (n > 4) and F. Fang (n = 4); these authors used important preliminary results proved by J. Bo'echat-A. Haefliger, S. Donaldson, M. Hirsch and W. Massey. Haefliger proved that if N is a compact n-dimensional k-connected manifold, then N embeds in R2n − k provided 2k + 3 ≤ n.
Isotopy versions 
A relatively 'easy' result is to prove that any two embeddings of a 1-manifold into R4 are isotopic. This is proved using general position, which also allows to show that any two embeddings of an n-manifold into R2n + 2 are isotopic. This result is an isotopy version of the weak Whitney embedding theorem.
Wu proved that for n ≥ 2, any two embeddings of an n-manifold into R2n + 1 are isotopic. This result is an isotopy version of the strong Whitney embedding theorem.
As an isotopy version of his embedding result, Haefliger proved that if N is a compact n-dimensional k-connected manifold, then any two embeddings of N into R2n − k + 1 are isotopic provided 2k + 2 ≤ n. The dimension restriction 2k + 2 ≤ n is sharp: Haefliger went on to give examples of non-trivially embedded 3-spheres in R6 (and, more generally, (2d − 1)-spheres in R3d). See further generalizations.
See also 
- See section 2 of Skopenkov (2008)
- Hassler Whitney; collected papers. Hassler Whitney, James Eells, Domingo Toledo. Nelson Thornes, 1992
- Lectures on the h-cobordism theorem. John Milnor. Princeton University Press. 1965
- Embeddings and immersions, by Masahisa Adachi, translated by Kiki Hudson
- Skopenkov, A. (2008), "Embedding and knotting of manifolds in Euclidean spaces", In: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes. 347 (2): 248–342, arXiv:math/0604045