Cerf theory

From Wikipedia, the free encyclopedia

In mathematics, Cerf theory in the subfield of differential topology is the study of families of smooth functions

where M is a smooth manifold.

Contents 1 Motivation 2 An analogy to stratified spaces 3 Origins 4 Applications 5 Generalization 6 References

[edit] Motivation

Marston Morse proved that any continuous function

could be approximated by a Morse function. So for many purposes, one can replace arbitrary functions on M by Morse functions. As a next step, one could ask, 'if you have a 1-parameter family of functions which start and end at Morse functions, can you assume the whole family is Morse?' In general the answer is no: for example the function

considered as a 1-parameter family of functions with starts at

with no critical points, but at

it has two critical points

Cerf showed that a 1-parameter family of functions between two Morse functions could be approximated by one that is Morse at all but finitely many degenerate times. The degeneracies involve a birth/death transition of critical points, as in the above example when t = 0 an index 0 and index 1 critical point are created (as t increases).

[edit] An analogy to stratified spaces

Let denote the space of Morse functions

and the space of smooth functions

Morse proved that

is an open and dense subset in the topology. For the purposes of intuition, here is an analogy. Think of the Morse functions as the top-dimensional open stratum in a stratification of (we make no claim that such a stratification exists, but suppose one does). Notice that in stratified spaces, the co-dimension 0 open stratum is open and dense. For notational purposes, reverse the conventions for indexing the stratifications in a stratified space, and index the open strata not by their dimension, but by their co-dimension. This is convenient since is infinite-dimensional if M is not a finite set. By assumption, the open co-dimension 0 stratum of is , ie: . In a stratified space X, frequently X⁰ is disconnected. An essential property of the co-dimension 1 stratum X¹ is that any path in X which starts and ends in X⁰ can be approximated by a path that intersects X¹ transversely in finitely many points, and does not intersect Xⁱ for all i > 1.

Thus Cerf theory is the study of the positive co-dimensional strata of , i.e.: for i > 0. In the case of

only for t = 0 is the function not Morse, and

has a cubic degenerate critical point corresponding to the birth/death transition.

[edit] Origins

Marston Morse gave an alternative proof of the h-cobordism theorem where the idea was to simplify the Morse function on a manifold rather than eliminating the handles in a handle-presentation of an h-cobordism. In the process, he created 1-parameter families of functions that were Morse at all but finitely many degenerate times.^[1]

[edit] Applications

Cerf theory was pioneered by Jean Cerf^[2], who used it to prove that every diffeomorphism of the 3-sphere is isotopic to an isometry of the 3-sphere (i.e.: an element of the orthogonal group O₄).^[3] He also used it in his proof of the pseudoisotopy theorem.^[4] Robion Kirby used it as a key step in justifying the Kirby calculus.

[edit] Generalization

Nowadays, Cerf theory is considered a particularly nice case of catastrophe theory, or singularity theory.

[edit] References

1. ^ R.Bott, Marston Morse and his mathematical works. Bull. Amer. Math. Soc. 3 (1980), 907--950.
2. ^ French mathematician, born 1928.[1]
3. ^ J.Cerf, Sur les difféomorphismes de la sphère de dimension trois (Γ₄=0), Lecture Notes in Mathematics, No. 53. Springer-Verlag, Berlin-New York 1968.
4. ^ J.Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. No. 39 (1970) 5--173.