Stratification of smooth maps from R^n to R? - MathOverflow

Stratification of smooth maps from R^n to R?

2011-02-10T21:25:22Z

I'm interested in stratifications of smooth maps $\mathbb{R}^n\to\mathbb{R}$ (or more generally of any $n$-manifold $M^n\to\mathbb{R}$). The codimension 0 stratum should be Morse functions, and the codimension 1 stratum should be Morse cancellations, e.g. the $t=0$ value of the following 1-parameter family of maps $$ (x_1,\ldots,x_n) \mapsto tx_1 + x_1^3 \pm x_2^2 \pm\cdots\pm x_n^2 . $$ Is there a good reference for the general codimension $k$ case?

Another way of phrasing the question: given a $k$-parameter family of smooth maps $F: P^k\times \mathbb{R}^n\to\mathbb{R}$, is there a known list of specific singularities such that we may assume that $F(p, \cdot)$ has only these singularities after a small perturbation? I suppose the way to start is to make $F$ Morse as a map from an $(n+k)$-manifold to $\mathbb{R}$, then look at the ways the coordinate axes of $P\times \mathbb{R}$ line up with gradients and the eigenspaces of the hessian of the Morse singularities of $F$. But I would rather cite the details than work them out for myself.

If the general case is messy (instability, cross-ratios, etc.), I would also be interested in an answer for $n=2$.

Answer by Ryan Budney for Stratification of smooth maps from R^n to R?

2011-02-10T21:43:05Z

A standard reference is:

F. Sergeraert "Un theoreme de fonctions implicites sur certains espaces de Frechet et quelques applications," Ann. Sci. Ecole Norm. Sup. (4) 5 (1972), 599-660.

This isn't a stratification of the space of maps $M \to \mathbb R$ but it is a stratification of an infinite co-dimension subspace of the space of all smooth maps $M \to \mathbb R$. It's a relatively popular stratification to use among geometric topologists, in that it produces Cerf theory. Rubinstein, Hong and McCullough use it in their work on the homotopy-type of $\operatorname{Diff}(L_{p,q})$. (which is how I learned of it)

http://front.math.ucdavis.edu/0411.5016

Is this roughly what you're looking for?

Answer by John Klein for Stratification of smooth maps from R^n to R?

2011-02-10T22:54:51Z

It looks to me that what you are really interested in is the Thom-Boardman stratification of the function space. For that I would recommend the well-written, Stable Mappings and Their Singularities by Guillemin and Golubitsky (in the Springer GTM series).