Understanding maps from M to R^n, for n>dim M

Question

I am interested in "approximating" smooth maps from a compact smooth manifold $M$ of dim $m$ into $\mathbb R^n,$ for $n>m,$ by "nice" maps, with properties similar to those of Morse functions. Of course, the question is interesting only for $n\leq 2m,$ since every map can be approximated by an immersion for higher $n$. (The case $dim M<n$ is also interesting but less relevant to my needs.)

What work has been done in this area (beyond Sard's Thm)?

Consider for example topologically stable maps. Are the sets of their critical values always finite? What is known about the structure of these maps near their critical values?

Perhaps there is some other dense subset of $C^\infty(M,\mathbb R^n)$ whose elements are functions with finite numbers of critical values and these critical values well understood?

Answer 1

For example if $m=2$ and $n=3$ then generic maps will have finite number of Whitney umbrella points. In this case generic maps are stable, i.e. stable maps form an open dense subset in $C^\infty(M,R^n).$

Similar statements can be formulated for higher dimensions until we are in the so called nice dimensions of Mather, (roughly n > 7/6 m$.) This is what singularity theory is about. (See for example the book by Arnold, Gusein-Zade, Varchenko: Singularities of smooth maps, volume 1.)

Is this really what you wanted to know?

But your expectations are a bit misled by Morse theory. For example for $m=3, n=4$ the generic maps still will have only Whitney umbrella singularities, but this time they form a one dimensional submanifold in $M$. So this set is not finite.