Intermediate curvatures and highly connected manifolds

We show that after forming a connected sum with homotopy sphere, all (2j-1)-connected 2j-parallelisable manifolds in dimension 4j+1, j > 0, can be equipped Riemannian metrics of 2-positive Ricci curvature. The condition curvature is defined to mean the two smallest eigenvalues tensor positive at every point. This result counterpart previous authors concerning existence on highly dimensions 4j-1 for 1, and 4j+1 0 torsion-free cohomology.