In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:
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, R 2 m , {\displaystyle \mathbb {R} ^{2m},} 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 Whitney immersion theorem.
About the proof
Weak embedding theorem The weak Whitney embedding is proved through a projection argument. When the manifold is compact, one can first use a covering by finitely many local charts and then reduce the dimension with suitable projections.
Strong embedding theorem The general outline of the proof is to start with an immersion f : M → R 2 m {\displaystyle f:M\to \mathbb {R} ^{2m}} 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 R 2 m . {\displaystyle \mathbb {R} ^{2m}.} Since R 2 m {\displaystyle \mathbb {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 2 m {\displaystyle \mathbb {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 (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 immersions α m : R m → R 2 m {\displaystyle \alpha _{m}:\mathbb {R} ^{m}\to \mathbb {R} ^{2m}} which are approximately linear outside of the unit ball, but containing a single double point. For m = 1 such an immersion is given by
{ α : R 1 → R 2 α ( t ) = ( 1 1 + t 2 , t − 2 t 1 + t 2 ) {\displaystyle {\begin{cases}\alpha :\mathbb {R} ^{1}\to \mathbb {R} ^{2}\\\alpha (t)=\left({\frac {1}{1+t^{2}}},\ t-{\frac {2t}{1+t^{2}}}\right)\end{cases}}}
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