Whitney embedding theorem

From Wikipedia, the free encyclopedia

In mathematics, particularly in differential topology,there are two Whitney embedding theorems:

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 $2 m$ -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 ( $2 m - 1$ )-space (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.

1 A little about the proof
- 1.1 Eventual Consequences of the Whitney trick
2 History
3 Sharper results
4 See also

[edit] A little about the proof

The general outline of the proof is to start with an immersion $f:M\to\mathbb R^{2m}$ 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 double-point.

Once one has two opposite double points, one constructs a closed loop connecting the two, giving a closed path in $R 2 m$ . Since $R 2 m$ is simply-connected, one can assume this path bounds a disc, and provided $2 m > 4$ one can further assume (by the weak Whitney embedding theorem) that the disc is embedded in $R 2 m$ 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).

Cancelling opposite double-points.

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 $α m$ of $R m$ into $R 2 m$ which are approximately linear outside of the unit ball, but containing a single double point. For $m = 1$ such an immersion is defined as $\alpha_1 : R^1 \to R^2$ with $\alpha_1(t_1)=\left(\frac{1}{1+t_1^2}, t_1 - \frac{2t_1}{1+t_1^2}\right)$ . Notice that if $α 1$ is considered as a map to $R 3$ ie: $\alpha_1(t_1) = \left( \frac{1}{1+t_1^2},t_1 - \frac{2t_1}{1+t_1^2},0\right)$ then the double point can be resolved to an embedding: $\beta_1(t_1,a) = \left(\frac{1}{(1+t_1^2)(1+a^2)},t_1 - \frac{2t_1}{(1+t_1^2)(1+a^2)},\frac{t_1a}{(1+t_1^2)(1+a^2)}\right)$ . Notice $β 1 (t 1,0) = α 1 (t 1)$ and for $a \neq 0$ then as a function of $t 1$ , $β 1 (t 1, a)$ is an embedding. Define $\alpha_2(t_1,t_2) = \left(\beta_1(t_1,t_2),t_2\right) = \left(\frac{1}{(1+t_1^2)(1+t_2^2)},t_1 - \frac{2t_1}{(1+t_1^2)(1+t_2^2)},\frac{t_1t_2}{(1+t_1^2)(1+t_2^2)}, t_2 \right)$ . $α 2$ can similarly be resolved in $R 5$ , this process ultimately lead one to the definition: $\alpha_m(t_1,t_2,\cdots,t_m) = \left(\frac{1}{u},t_1 - \frac{2t_1}{u}, \frac{t_1t_2}{u}, t_2, \frac{t_1t_3}{u}, t_3, \cdots, \frac{t_1t_m}{u}, t_m \right)$ with $u=(1+t_1^2)(1+t_2^2)\cdots(1+t_m^2)$ for all $m \geq 1$ . The key properties of $α m$ is that it is an embedding except for the double-point $\alpha_m(1,0,\cdots,0)=\alpha_m(-1,0,\cdots,0)$ . Moreoever, for $|(t_1,\cdots,t_m)|$ large, it is approximately the linear embedding $(0,t_1,0,t_2,\cdots,0,t_m)$ .

[edit] 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 $m \geq 5$ , and the classification of smooth structures on discs (also in dimensions 5 and up).

Given two oriented submanifolds of complementary dimensions in a simply connected manifold of dimension $\geq 5$ , one can apply an isotopy to one of the submanifolds so that all the points of intersection have the same sign.

[edit] 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.

[edit] Sharper results

Although every $n$ -manifold embeds in $R 2 n$ , one can frequently do better. Let $e (n)$ denote the smallest integer so that all compact connected $n$ -manifolds embed in $R e (n)$ . Whitney's strong embedding theorem states that $e(n) \leq 2n$ . 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, where the analogous number is known).

Wu proved that for $n \geq 2$ , any two embeddings of an $n$ -manifold into $R 2 n + 1$ are isotopic. A relatively `easy' result to prove is that any two embeddings of a 1-manifold into $R 4$ are isotopic.

Haefliger proved that if $N$ is a compact $n$ -dimensional $k$ -connected manifold, then $N$ embeds in $R 2 n - k$ provided $2k+3 \leq n$ . Moreover, any two embeddings of $N$ into $R 2 n - k + 1$ are isotopic provided $2k+2 \leq n$ . Haefliger went on to give examples of non-trivially embedded 3-spheres in $R 6$ .