Metrics with positive Ricci eigenvalue sums on ${\mathbb{T}}^k\times {\mathbb{S}}^m$

We construct metrics with $k$-positive Ricci curvature on ${\mathbb{T}}^k\times {\mathbb{S}}^m$, for $k,m\ge 2$, arbitrarily close to the standard flat-round product in the smooth topology. The two-torus case disproves Wolfson’s conjecture that every closed manifold with two-positive Ricci curvature has virtually free fundamental group. We also establish a local obstruction to two-positive Ricci curvature near closed backgrounds carrying three orthonormal parallel one-forms.