Whitney embedding theorem

Whitney embedding theorem

In differential topology, the Whitney Embedding Theorem states that any smooth -dimensional manifold can be embedded in Euclidean 2m-space.

Table of contents

1 A little about the proof

1.1 Whitney trick
1.2 Other things comming from Whitney trick

2 History

A little about the proof

Cases and can be done by hand. For a general position argument show that there is an immersion R^2m with transversal self-intersections. Then apply the Whitney trick, i.e. the following procedure which removes self-inersections one by one.

Whitney trick

Suppose R^2m is a point of self-intersection and such that . Connect and by a smooth curve

so that is a simple closed curve in R^2m. Construct an embedding of a 2-disc R^2m with boundary .

By a general position argument it can be constructed with no self-intersections and with no intersections with (here we use that ). Then one can deform in a little neighborhood of so that the self-intersecton disappears. (The last statement is very easy to see once you visualize this picture properly)

Other things comming from Whitney trick

...

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 (which had been implicit in Riemann's work, Lie group theory, and general relativity for many years); building on Hermann Weyl's book The Idea of a Riemann surface.