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We consider intermediate Ricci curvatures $Ric_k$ on a closed Riemannianmanifold $M^n$. These interpolate between the Ricci curvature when $k=n-1$ andthe sectional curvature when $k=1$. By establishing a surgery result forRiemannian metrics with $Ric_k>0$, we show that Gromov's upper Betti numberbound 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 thecase of positive Ricci curvature.

Intermediate Ricci curvatures and Gromov's Betti number bound