Intermediate Ricci Curvatures and Gromov’s Betti number bound

Abstract

We consider intermediate Ricci curvatures Ric_k on a closed Riemannian manifold M^n. These interpolate between the Ricci curvature when k=n-1 and the sectional curvature when k=1. By establishing a surgery result for Riemannian metrics with Ric_k>0, we show that Gromov’s upper Betti number bound for sectional curvature bounded below fails to hold for Ric_k>0 when lfloor n/2 rfloor +2 le k le n-1. This was previously known only in the case of positive Ricci curvature (Sha and Yang in J Differ Geom 29(1):95–103, 1989, J Differ Geom 33:127–138, 1991).