Whitney embedding theorem

In differential topology, the Whitney embedding theorem states that

Any smooth second countable $m$-dimensional manifold can be embedded in Euclidean 2m-space.

The result is sharp, in particular the projective m-space can not be embeded into Euclidean (2m − 1)-space

A little about the proof

Cases $m=1$ and $2$ can be done by hand. For $m\geq 3$ a general position argument show that there is an immersion $f:M\to$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 $p\in$R^2m is a point of self-intersection and $x,y\in M$ such that $f(x)=f(y)=p$. Connect $x$ and $y$ by a smooth curve

$c:[0,1]\to M$

so that $f\circ c$ is a simple closed curve in R^2m. Construct an embedding of a 2-disc $h:D^{2}\to$R^2m with boundary $f\circ c$.

By a general position argument it can be constructed with no self-intersections and with no intersections with $f(M)$ (here we use that $m\geq 3$). Then one can deform $f$ in a little neighborhood of $h(D^{2})$ so that the self-intersecton disappears. (The last statement is very easy to see once you visualize this picture properly)

Other things coming from Whitney trick

Whitney trick is used to prove h-cobordism theorem, it also shows that two oriented submanifolds of complimentary dimensions in a simply connected manifold of dimension $\geq 5$ are isotopic to submanifolds such that all points of intesections have the same sign.

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.