Is there a Morse theory for sections of bundles or more generally for maps?

Question

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.

Answer 1

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 $C+2X+2Comp$.

Answer 2

There is a construction for Morse theory for circle-valued functions. I think it is due to Novikov (that's also where the Novikov ring originates), see e.g.
http://www.maths.ed.ac.uk/~aar/papers/ranicki3.pdf

Answer 3

Extending Petya's answer, if you want something like indices, handle attachments and things of that flavour the answer is no. You can prove there's no such thing in that level of generality. But that's a pretty "close read" of Morse theory that does not translate to other contexts. If you take a "broad" reading of Morse theory: classify the local behavior of "generic" maps between manifolds, and study the properties that are conserved via generic homotopies, etc, then what you have is singularity theory, the work of people like Thom and Mather applies. This kind of thing gets applied in many ways -- a nice recent example that I like would be the paper of D.Thurston and Costantino where they use the properties of generic maps from 3-manifolds into $\mathbb R^2$ to construct efficiently-triangulated 4-manifolds bounding triangulated 3-manifolds. But that's only one of many examples.

A question in return: how would you expect a Morse theory of sections of bundles? You don't have "level sets" so perhaps you're interested in properties of generic maps? Or some relations between local properties of the section and a global property of the bundle -- things like Euler characteristics?

edit: I read the "letters to nature" blog you linked to. The author says his motivation is to "classify functions"? It would be helpful if the author could specify what equivalence relation on functions he's considering. Presumably an object like the $C^1$ norm is too restrictive?

Answer 4

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.

Answer 5

Ryan: my use of the phrasing 'classify functions' is both inaccurate and imprecise. But we're in danger of talking past one another, because the classification I am pursuing is on the basis of different physical processes. This is a physics goal, not a mathematical one.

The functions observational cosmologists consider are invariably smooth. The cosmological dark matter density is an example and the calculation of the Euler characteristic for level sets of that function is a nice application of Morse theory to physics. The velocity of the dark matter fluid, which a physicist would call a vector-valued function, is an example of an object to which we would like to apply the same apparatus, but are unsure how to proceed.

I think that you indicate that the concept of a level set is not well-formed for an object of that kind? Could you comment on the generality of that statement bearing in mind the relative triviality of the objects in question?

I would also like to clarify Jose's phrasing. Studies of this kind aim very much to study the properties of $f$, rather than $M$. The study of the topology of the spatial Universe is an excellent and interesting problem as well! But the aim here is to use techniques from Morse theory as a way of probing how physical effects alter the form of $f$.

Thanks for all the help so far! I hope there's more to discuss.

Answer 6

The best extend of Morse Theory to maps is the theory of Lefschetz foliation. More than this would be a theory to deal with such singular foliations.