Whitney embedding theorem

When the manifold is compact, one can first use a covering by finitely many local charts and then reduce the dimension with suitable projections.[1]: Ch.

The general outline of the proof is to start with an immersion ⁠

These are known to exist from Whitney's earlier work on the weak immersion theorem.

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.

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.

⁠ 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 ⁠

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 ⁠

⁠ which are approximately linear outside of the unit ball, but containing a single double point.

For m = 1 such an immersion is given by Notice that if α is considered as a map to ⁠

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.

Let e(n) denote the smallest integer so that all compact connected n-manifolds embed in ⁠

⁠ Whitney's strong embedding theorem states that e(n) ≤ 2n.

This is a result of André Haefliger and Morris Hirsch (for n > 4) and C. T. C. Wall (for n = 3); these authors used important preliminary results and particular cases proved by Hirsch, William S. Massey, Sergey Novikov and Vladimir Rokhlin.

[4] 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).

One can strengthen the results by putting additional restrictions on the manifold.

For n a power of 2 this is a result of André Haefliger and Morris Hirsch (for n > 4), and Fuquan Fang (for n = 4); these authors used important preliminary results proved by Jacques Boéchat and Haefliger, Simon Donaldson, Hirsch and William S.

[4] Haefliger proved that if N is a compact n-dimensional k-connected manifold, then N embeds in ⁠

This is proved using general position, which also allows to show that any two embeddings of an n-manifold into ⁠

This result is an isotopy version of the weak Whitney embedding theorem.

This result is an isotopy version of the strong Whitney embedding theorem.

⁠ 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 ⁠