INTERMEDIATE CURVATURES AND HIGHLY CONNECTED MANIFOLDS

We show that after forming a connected sum with a homotopy sphere, all (2j-1)-connected 2j-parallelisable manifolds in dimension 4j +1, j = 2, can be equipped with Riemannian metrics of 2-positive Ricci curvature. The condition of 2-positive Ricci curvature is defined to mean that the sum of the two smallest eigenvalues of the Ricci tensor is positive at every point. This result is a counterpart to a previous result of the authors concerning the existence of positive Ricci curvature on highly connected manifolds in dimensions 4j-1 for j = 2, and in dimensions 4j +1 for j = 1 with torsion-free cohomology. A key technical innovation involves performing surgery on links of spheres within 2-positive Ricci curvature.