Positive curvature, local and global symmetry, and fundamental groups

A π 1 -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 π 1 ( M )× ρ T 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 M n of positive sectional curvature admits a π 1 -invariant isometric T k -action, then the fundamental group has a cyclic subgroup of index ≤ w ( n ). This refines the main result in [Ro1].