Maps of degree 1, 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→N of degree 1 of closed manifolds, is it true that CritM≥CritN? 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.