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We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily havefree fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga andGonzalez-Acuna, by generalizing their result in dimension 3, to all higherdimensions. We also obtain some general results on the relations between thefundamental group of a closed manifold M, the dimension of M, and theLusternik-Schnirelmann category of M, and relate the latter to the systoliccategory of M.

Small values of the Lusternik–Schnirelmann category for manifolds