Whitney embedding theorem

Whitney embedding theorem states that any smooth $m$-dimensional manifold can be ebeded in Euclidean 2m-space.

Little about Proof

Cases $m=1$ and $2$ can be done by hands. For $m\geq 3$ general position argument show that there is an immersion $f:M\to$R^2m with transversal self-intersections. Then apply the Whitney trick (The Whitney trick gives a way to deform f to remove self-inersections one by one).

Whitney trick

If $p\in$R^2m be a point of self-inetersection 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. Constract an embedding of a 2-disc $h:D^{2}\to$R^2m with boundary $f\circ c$. By general position argument it can be constructed with no self-inersections 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 itersecton disappears.

Other things comming from Whitney trick

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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.