A gap theorem for Lusternik–Schnirelmann π1-category

The Lusternik–Schnirelmann π1-category, catπ1X, of a topological space X is the least integer n such that X can be covered by n+1 open subsets U0,…,Un, every loop in each of which is contractible in X. In this paper we will prove a gap theorem that catπ1Mn≠n−1 for any closed connected n-dimensional manifold Mn. With the fact that the fundamental group of a compact Kähler manifold is not a nontrivial free group, we see as a corollary that the π1-category of a compact Kähler surface is even.