Small values of the Lusternik-Schnirelmann category for manifolds

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.