Morse Functions and Nondepraved Critical Points

Abstract

Classical Morse theory is concerned with the critical points of a class of smooth proper functions f from a manifold Z to the real numbers, called Morse functions. For our generalization, we will let Z be a closed Whitney stratified space in some ambient smooth manifold M. We will need analogues for the notions of smooth function, critical point, and Morse function for this setting. A smooth function on Z will be a function which extends to a smooth function on M. A critical point of a smooth function f will be a critical point of the restriction of f to any stratum S of Z. A proper function f is called Morse if (1) its restriction to each stratum has only nondegenerate critical points, (2) its critical values are distinct, and (3) the differential of f at a critical point p in S does not annihilate any limit of tangent spaces to a stratum other than S. This third condition is a sort of nondegeneracy condition normal to the stratum. If Z is subanalytic (which includes the real and complex analytic cases), then the set of Morse functions forms an open dense subset of the space of smooth functions, and Morse functions are structurally stable, just as in the classical case [P1].