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, ⁠${\displaystyle {\mathbb{R}}^{2m},}$⁠ 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 ⁠${\displaystyle f:M\to {\mathbb{R}}^{2m}}$⁠ 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 ⁠${\displaystyle {\mathbb{R}}^{2m}.}$⁠ Since ⁠${\displaystyle {\mathbb{R}}^{2m}}$⁠ 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 ⁠${\displaystyle {\mathbb{R}}^{2m}}$⁠ 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 ⁠${\displaystyle {\alpha }_m:{\mathbb{R}}^m\to {\mathbb{R}}^{2m}}$⁠ which are approximately linear outside of the unit ball, but containing a single double point. For m = 1 such an immersion is given by

${\displaystyle \{\begin{array}{l}\alpha :{\mathbb{R}}^1\to {\mathbb{R}}^2\\ \alpha (t)=\left(\frac{1}{1+t^2},t-\frac{2t}{1+t^2}\right)\end{array}}$

Notice that if α is considered as a map to ⁠${\displaystyle {\mathbb{R}}^3}$⁠ like so:

${\displaystyle \alpha (t)=\left(\frac{1}{1+t^2},t-\frac{2t}{1+t^2},0\right)}$

then the double point can be resolved to an embedding:

${\displaystyle \beta (t,a)=\left(\frac{1}{(1+t^2)(1+a^2)},t-\frac{2t}{(1+t^2)(1+a^2)},\frac{ta}{(1+t^2)(1+a^2)}\right).}$

Notice β(t, 0) = α(t) and for a ≠ 0 then as a function of t, β(t, a) is an embedding.

For higher dimensions m, there are α_m that can be similarly resolved in ⁠${\displaystyle {\mathbb{R}}^{2m+1}.}$⁠ For an embedding into ⁠${\displaystyle {\mathbb{R}}^5,}$⁠ for example, define

${\displaystyle {\alpha }_2(t_1,t_2)=\left(\beta (t_1,t_2),t_2\right)=\left(\frac{1}{(1+t_1^2)(1+t_2^2)},t_1-\frac{2t_1}{(1+t_1^2)(1+t_2^2)},\frac{t_1{t}_2}{(1+t_1^2)(1+t_2^2)},t_2\right).}$

This process ultimately leads one to the definition:

${\displaystyle {\alpha }_m(t_1,t_2,\cdots ,t_m)=\left(\frac{1}{u},t_1-\frac{2t_1}{u},\frac{t_1{t}_2}{u},t_2,\frac{t_1{t}_3}{u},t_3,\cdots ,\frac{t_1{t}_m}{u},t_m\right),}$

where

${\displaystyle u=(1+t_1^2)(1+t_2^2)\cdots (1+t_m^2).}$

The key properties of α_m is that it is an embedding except for the double-point α_m(1, 0, ... , 0) = α_m(−1, 0, ... , 0). Moreover, for |(t₁, ... , t_m)| large, it is approximately the linear embedding (0, t₁, 0, t₂, ... , 0, t_m).

Eventual consequences of the Whitney trick

The Whitney trick was used by Stephen 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.