Intermediate Ricci curvatures and Gromov's Betti number bound

TL;DR

This paper investigates intermediate Ricci curvatures on closed Riemannian manifolds, showing that Gromov's Betti number bound for sectional curvature does not extend to certain intermediate Ricci curvatures.

Contribution

It establishes a surgery result for metrics with positive intermediate Ricci curvature, demonstrating the failure of Gromov's Betti number bound in this setting.

Findings

Gromov's Betti number bound fails for certain intermediate Ricci curvatures

A surgery technique for metrics with Ric_k>0 is developed

The result extends known cases from positive Ricci curvature to intermediate cases

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.