Small values of the Lusternik–Schnirelmann category for manifolds

Abstract

We prove that manifolds of Lusternik–Schnirelmann category $2$ necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larrañaga and Gonzalez-Acuña by generalizing their result in dimension $3$ to all higher dimensions. We also obtain some general results on the relations between the fundamental group of a closed manifold $M$, the dimension of $M$ and the Lusternik–Schnirelmann category of $M$, and we relate the latter to the systolic category of $M$.