Whitney immersion theorem

Remarks

The weak version gives an immersion of any smooth $n$-manifold, for every $n \ge 1$, into $\R^{2n}$, and shows that any smooth map $M \to \R^{2n}$ can be approximated by immersions. It follows from a general position argument: the maps whose differential drops rank form a set of codimension $2n - n + 1 = n + 1 > n$ in the space of $1$-jets, so a generic map avoids them. The hypothesis $n \ge 2$ is needed: the circle $S^1$ admits no immersion into $\R^1$. Such an immersion would be a local diffeomorphism, so its image would be open and compact in $\R$, which is impossible. Example: the real projective plane $\R P^2$ immerses in $\R^3$, as Boy's surface shows. For compact $M$, Ralph Cohen proved the sharper bound $\R^{2n - \alpha(n)}$, where $\alpha(n)$ is the number of ones in the binary expansion of $n$.

Going deeper

For a student

Compare with the Whitney embedding theorem, which needs dimension $2n$ to avoid self-crossings. The easy bound $2n$ comes from transversality of the $1$-jet of a map to the set of rank-deficient jets. The step down to $2n - 1$ needs a finer study of the singularities of a generic map into $\R^{2n-1}$. The Smale–Hirsch theorem later reduced the existence of immersions to a question about vector bundles.