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.