## Stratification of smooth maps from R^n to R?

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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 $n=2$.

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## 2 Answers

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).

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Guillemin and Golubitsky don't quite set it up in this setting -- they develop the general Mather machine but they don't apply it to real-valued function spaces, at least, not in any real detail. – Ryan Budney Feb 10 2011 at 23:03
Granted. I guess, the other possible reference might be Cerf, at least when the codimension is small. – John Klein Feb 10 2011 at 23:29

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Yup. Cerf starts from the Thom-Mather machine and works out the details of the singularities in the real-valued function case. Guillemin and Golubitsky is probably the most readable account of Thom-Mather theory out there, as far as I know. – Ryan Budney Feb 11 2011 at 1:25
Thanks John, I'll have a look at G&G. – Kevin Walker Feb 11 2011 at 4:15

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A good supplement to G&G which provides an alternative understanding of the Thom-Boardman stratification is Porteous' paper springerlink.com/content/ln6331341j213515 – Sergey Melikhov Feb 11 2011 at 15:58

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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?

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Thanks -- I'll have a look at those references. The paper by J.W. Bruce in the bibliography of Rubinstein et al also looks interesting. – Kevin Walker Feb 10 2011 at 22:00

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Ryan: you misspelled Sergeraert's name. – Thierry Zell Feb 11 2011 at 0:09
Ah, thanks for catching that. I made the same mistake on the Cerf Theory Wikipedia page (twice!). – Ryan Budney Feb 11 2011 at 1:05

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My knowledge of French is pretty minimal, but I didn't see anything like an explicit list in chapters 8 or 9 of Sergeraert. Is the information necessary to produce an explicit list in there, or is the description of the stratification more abstract than that? – Kevin Walker Feb 11 2011 at 4:14
I think it's fairly explicit at some point though I haven't looked at it in much detail in over a year. I think the best thing to do would be to see how Rubinstein & McCullough cite the work, because they're looking for something in the ballpark of what you want. It's one of those moments in the semester where I rarely get my head out of paperwork so I won't have much more to say for a few weeks. – Ryan Budney Feb 14 2011 at 23:54
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