We determine the fundamental group of a closed n-manifold of positive sectional curvature on which a torus T k (k large) acts effectively and isometrically. Our results are: (A) If k > (n−3)/4 and n 17, then the fundamental group π1(M) is isomorphic to the fundamental group of a spherical 3-space form. (B) If k (n/6)+1 and n =11,15,23, then any abelian… (More)