The almost cyclicity of the fundamental groups of positively curved manifolds

Recall that a pure F-structure is a kind of generalized torus action. The main result asserts that if a compact positively curved manifold M(n) admits an invariant pure F-structure such that each orbit has positive dimension, then the fundamental group has a finite cyclic subgroup with index less than w(n), a constant depending only on n. As an application, ave conclude that for all 0 < delta less than or equal to 1, the fundamental group of a delta-pinched n-manifold either has a cyclic subgroup with index less than w(n) or has order less than w(n,delta), a constant depending only on n and delta. In particular, this substantially improves the main result in [Ro1].