Talk:Cerf theory

Critical points

I haven't looked too closely at the math here yet, but I think the example $f_{t}(x)=x^{3}-tx$ could use a bit more explanation. (Maybe even its own section called "Example".) In particular, one would like to see that it is at t = 0 that the family passes through a function that fails to have nondegenerate critical points. And even though the article Morse theory explains degenerate versus nondegenerate critical points, we may want to say a bit more about that here so it's clear what is going wrong as we pass through a family of Morse functions. VectorPosse 20:30, 4 April 2007 (UTC)

yeah, I have in mind a bunch of pictures to include, especially with the stratified space analogy since this kind of thing is so pervasive in mathematics nowadays. Rybu (talk)

Picture request, might as well formalize it:

Attribution

Is the theorem that you attribute to Cerf, actually due to Thom and Mather, as explained in the book by Martinet? My understanding is that what is sometimes called "Cerf theory" (any two handle decompositions are related by critical point cancellation, order changes, and handle slides) is due to Thom-Mather (any generic family of functions is generically Morse and has only birth-death singularities at a finite set of times), Cerf (in a generic family of Morse functions, all critical values are generically distinct but at a finite set of times critical values may cross) and Kirby (in a generic family of Morse functions, all functions are Morse-Smale except for a finite set giving rise to the handle slides). One should point out that only the first Thom-Mather contribution is really non-trivial. Woodwardc 13:46, 30 October 2007 (UTC)

I have not read the Martinet book so I can't comment on its specific contents. But your statement that two handle decompositions are related by critical point cancellations, variations on that can be attributed to Smale and Morse as well. Who deserves the original credit, I'm not certain but Morse was a major proponent of that idea for a long time. Anyhow, that is not the full content of Cerf's "theory". Cerf theory says that any path in the function space that starts and ends at a Morse function can be approximated by one which fails to be Morse at only finitely many points, and those points correspond to birth/death points in the family. So the theorem you quote is the special case of Cerf theory once you realize the space of all functions from M to the real numbers is contractible. The theorem that Martinet talks about is all the Cerf theory you need to know for most low-dimensional topological arguments: kirby calculus, Cerf's proof that orientation preserving diffeomorphisms of S^3 are isotopic to the identity, etc. But it's weaker than what the whole theory is about. And it does not suffice for Rubinstein's proof of the generalized Smale conjecture for lens spaces, for example. Sorting out attributions would be a good thing to do. I should be able to sort the attribution mess soon. I'm not in direct contact with Cerf but I do know his office-mate. Rybu 20:26, 1 November 2007 (UTC)

An update -- I'm in the process of acquiring all the Cerf-theory related references I can find. Thanks for the information about the Martinet book. Rybu (talk) 06:33, 14 November 2009 (UTC)

Recent updates

I recently received a letter from someone who knows the Cerf theory history very well. So I've updated the article taking into account his suggestions. Several references are improved, there's much more detail on the origins, and the generalizations section has much more scope. Still, there's quite a bit to do but the article is much more informative now. Rybu (talk) 09:50, 1 August 2010 (UTC)