Whitney Embedding Theorem - A Little About The Proof

A Little About The Proof

The general outline of the proof is to start with an immersion 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 has boundary, one can remove the self-intersections simply by isotoping into itself (the isotopy being in the domain of ), to a submanifold of that does not contain the double-points. Thus, we are quickly led to the case where 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 . Since is simply-connected, one can assume this path bounds a disc, and provided one can further assume (by the weak Whitney embedding theorem) that the disc is embedded in such that it intersects the image of only in its boundary. Whitney then uses the disc to create a 1-parameter family of immersions, in effect pushing 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 of into which are approximately linear outside of the unit ball, but containing a single double point. For such an immersion is defined as with . Notice that if is considered as a map to i.e.: then the double point can be resolved to an embedding: . Notice and for then as a function of, is an embedding. Define . can similarly be resolved in, this process ultimately leads one to the definition: with for all . The key properties of is that it is an embedding except for the double-point . Moreover, for large, it is approximately the linear embedding .