Cerf theory

Summary

Cerf theory studies one-parameter families of smooth real-valued functions on a manifold and the ways in which they fail to be Morse functions. In a generic family, the function is a Morse function with distinct critical values at all but finitely many times. At those times either a pair of critical points is born or dies, or two critical values cross. Jean Cerf developed it to study diffeomorphisms of manifolds, in particular to prove pseudo-isotopy results. Through the correspondence between Morse functions and handle decompositions, it describes how any two handle decompositions of a manifold are related.