Four-manifolds with positive curvature operator

A compact surface with positive mean scalar curvature must be diffeomorphic to the sphere S 2 or the real projective space RP 2 . A compact three-manifold with positive Ricci curvature must be diffeomorphic to the sphere S 3 or a quotient of it by a finite group of fixed point free isometries in the standard metric, such as the real projective space RP 3 or a lens space L 3 p q . This was proven in 1. Our main result is the following generalization to four dimensions.