Intermediate Ricci Curvatures and Gromov’s Betti number bound

We consider intermediate Ricci curvatures Ricₖ R i c k on a closed Riemannian manifold Mⁿ M n . These interpolate between the Ricci curvature when $$k=n-1$$ k = n - 1 and the sectional curvature when $$k=1$$ k = 1 . By establishing a surgery result for Riemannian metrics with Ricₖ>0 R i c k > 0 , we show that Gromov’s upper Betti number bound for sectional curvature bounded below fails to hold for Ricₖ>0 R i c k > 0 when n/2 +2 ≤ k ≤ n-1. ⌊ n / 2 ⌋ + 2 ≤ k ≤ 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).