Positive curvature, local and global symmetry, and fundamental groups

A π₁-invariant torus-action on a manifold M is a T^k-action on the universal covering which extends to the action of a semi-direct product π₁(M)×_ρT^k. In particular, the T^k-action is the lift of a T^k-action on M if ρ is the identity map. The main result asserts that if a compact manifold Mⁿ of positive sectional curvature admits a π₁-invariant isometric T^k-action, then the fundamental group has a cyclic subgroup of index ≤ w(n). This refines the main result in [Ro1].