TITLE: Is there a Morse theory for sections of bundles or more generally for maps?
QUESTION [9 upvotes]: 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.

REPLY [4 votes]: There is some really cool recent work of Gay and Kirby on "Morse 2-functions", ie generic maps $f$ from an $n$-manifold $X$ to a surface $\Sigma$ (both compact, connected, oriented, smooth) (see http://arxiv.org/abs/1102.0750).
In the $n=4$, $\Sigma = S^2$ case, they have a reconstruction theorem (http://arxiv.org/abs/1202.3487) for recovering $X$ and $f$ from a combinatorial diagram on $S^2$, which in particular records the critical values (the "Cerf graphic").  They even understand what happens in families of Morse 2-functions, ie the analogue of birth-death moves.