The minimal number of critical points of a smooth function on a closed manifold and the ball category

Abstract

In this dissertation, we discuss the Lusternik-Schnirelmann category and relevant results. We will then introduce variants for a closed manifold M of the Lusternik-Schnirelmann category using Singhof-Takens fillings as well as a variant for the minimum number of critical points for any smooth function on M. We will show that the category of a Singhof-Takens filling by topological balls with corners is related to the minimum number of critical points for any smooth function on M such that the critical points admit a gradient convex ball neighborhood. We will also show that the category of a Singhof-Takens filling by smooth balls with corners is related to the minimum number of critical points for any smooth function on M.