Intermediate Ricci curvatures and Gromov's Betti number bound

We consider intermediate Ricci curvatures $Ri{c}_k$ 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 $Ri{c}_k>0$, we show that Gromov’s upper Betti number bound for sectional curvature bounded below fails to hold for $Ri{c}_k>0$ when $\lfloor n\mathrm{/}2\rfloor +2\le k\le n-1$. This was previously known only in the case of positive Ricci curvature.