This question was prompted by my interpretation of a question by cosmologist Berian James.
Background
Some cosmologists have suggested using the cosmological dark matter density, which defines a function $f:M\to \mathbb{R}$ with $M$ the spatial universe, in order to probe the topology of $M$. (edit: Berian comments below that this may not be what this is about! Listen to him… not to me!) The original reference seems to be this paper by Gott et al.. Although the paper does not mention it explicitly, it seems that the natural mathematical framework for this proble is Morse theory.
Berian is interested not just in functions, but also e.g., vector fields or more generally sections of bundles on $M$. Hence the following
Question
Can one extend Morse theory beyond functions to sections of bundles? or perhaps to differentiable maps $f: M \to X$?
Pointers to the literature would be most welcome.
Cheers.
Best Answer
You are asking a very classical question. It seems to me, that general answer is No. But there were some interesting attempts (could not find referencies..). In some cases one can prove that there is no "Morse theory" for a class of maps, since they satisfy h-principle and one can eliminate (almost all) singularities.
I formulate a result (due to Gromov) of a Morse-type estimates of a singular points of smooth mappings to the plane. Consider a generic smooth map of a closed manifold $M$ to the plane. Consider the image of its singular points. It is a curve having cusps and double self-intersections as singular points only. Denote by $C$ a number of its cusps, by $X$ a number of double self-intersections and by $Comp$ a number of components. Then the sum $B(M)$ of Betti numbers of $M$ is restricted by $C+2X+2Comp$.