On the first and second homotopy groups of manifolds with positive curvature

Abstract

We show that an odd dimensional closed manifold with positive curvature cannot contain an incompressible real projective plane in the sense that there is no map of the projective plane into the manifold which is nontrivial on both first and second homotopy groups. Another way to say this is that no element of the fundamental group reverses the orientation of a class in the second homotopy group. Further we show that for a manifold of dimension divisible by four with positive curvature the fundamental group acts trivially on the second homotopy group. The methods involve a careful study of the stability of minimal two spheres in manifolds of positive curvature.