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arXiv · 2208.13438

Intermediate Ricci curvatures and Gromov's Betti number bound

Philipp Reiser·David J. Wraith
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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.

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Philipp Reiser, David J. Wraith. 2022-08-29. Intermediate Ricci curvatures and Gromov's Betti number bound. https://arxiv.org/abs/2208.13438

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