Maps of Degree One, Lusternik Schnirelmann Category, and Critical Points

Abstract

Let $Crit M$ denote the minimal number of critical points (not necessarily non-degenerate) on a closed smooth manifold $M$. We are interested in the evaluation of $Crit$. It is worth noting that we do not know yet whether $Crit M$ is a homotopy invariant of $M$. This makes the research of $Crit$ a challenging problem. In particular, we pose the following question: given a map $f: M \to N$ of degree 1 of closed manifolds, is it true that $Crit M \geq Crit N$? We prove that this holds in dimension 3 or less. Some high dimension examples are considered. Note also that an affirmative answer to the question implies the homotopy invariance of $Crit$; this simple observation is a good motivation for the research.