The Fundamental Group of Closed Manifolds with Two-positive Ricci Curvature

Let M be a Riemannian n-manifold with n greater than or equal to 3. For k between 1 and n, we say M has k-positive Ricci curvature if at every point of M the sum of any k eigenvalues of the Ricci curvature is strictly positive. In particular, one positive Ricci curvature is equivalent to positive Ricci curvature and n-positive Ricci curvature is equivalent to positive scalar curvature. Let G be the fundamental group of the closed manifold M. We say that G is virtually free if G contains a free subgroup of finite index, or equivalently, if some finite cover of M has a fundamental group that is a free group. In this paper we will prove: Let M be a closed Riemannian n-manifold, with n greater than or equal to 3, such that (n-1)-eigenvalues of the Ricci curvature are strictly positive. Then the fundamental group of M is virtually free. As an immediate consequence we have: Let M be a closed Riemannian n-manifold, with n greater than or equal to 3, with 2-positive Ricci curvature. Then the fundamental group of M is virtually free.