Functions with minimal number of critical points

Abstract

In 1934 Lusternik and Schnirelmann introduced a new numerical topological invariant (the Lusternik-Schnirelmann category) during research into the global calculus of variation. They showed that this invariant carries important information about both the existence of critical points and the cardinality of the critical set. Further results on Lusternik-Schnirelmann category distract the attention from the original problem: finding crit(M) the minimal number of critical points of a smooth function on a given smooth manifold M. Only Cornea in 1998 analyzed the notion of crit, originally introduced by Takens in 1968. Inspired by Takens' paper and by Morse theory in the Smale setting we have analyzed the crit for products of manifolds. Using concepts from the graph theory we have found upper bounds of crit for products of special manifolds with generalized tori. The proofs of these results are based on the fusing lemma, which establishes sufficient conditions to construct a triad function with at most one critical point from a triad function with two critical points. In order to compute the crit for products of lens spaces with spheres we prove that the ball category is a lower bound for the number of the critical points of a function. The thesis contains also a paragraph about crit of manifolds with boundary.