  We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have
free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and
Gonzalez-Acuna, 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 relate the latter to the systolic
category of M.
