A π1-invariant torus-action on a manifold M is a Tk-action on the universal covering which extends to the action of a semi-direct product π1(M) ⋉ ρ Tk. In particular, the Tk-action is the lift of a Tk-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 Tk-action, then the fundamental group has a cyclic subgroup of index ≤ w(n). This refines the main result in [Ro1].
All Science Journal Classification (ASJC) codes