On the topology of manifolds with positive intermediate curvature

TL;DR

This work links the topology of a manifold's universal cover to the existence of metrics with positive $m$-intermediate curvature, proving the conjecture for dimensions $3\le n\le5$ and most cases when $n=6$. The authors blend Schoen–Yau style minimal-surface arguments with the $m$-intermediate curvature framework of Brendle–Hirsch–Johne, employing $\mu$-bubbles, weighted variational techniques, and generalized Bonnet–Myers diameter estimates to derive contradictions from assumed positive curvature. A central technical advance is a Frankel-type theorem for positive conformal Ricci curvature, which ensures connected boundaries in the sliced, diced configurations that appear in higher-codimension arguments. The paper also extends the results to a mapping version and provides a classification-style corollary under degree-raising maps, offering evidence toward the $K(\pi,1)$ conjecture in dimension six. Overall, the results illuminate how universal-cover topology constrains intermediate-curvature metrics and reinforce the interplay between global topology and geometric analysis in moderate dimensions.

Abstract

We formulate a conjecture relating the topology of a manifold's universal cover with the existence of metrics with positive $m$-intermediate curvature. We prove the result for manifolds of dimension $n\in\{3,4,5\}$ and for most choices of $m$ when $n=6$. As a corollary, we show that a closed, aspherical 6-manifold cannot admit a metric with positive $4$-intermediate curvature.

Theorems & Definitions (68)

• Theorem 1
• Theorem 2: Brendle-Hirsch-Johne brendle2024generalization
• Conjecture 3
• Theorem 4: Main Theorem
• Corollary 5
• Theorem 6
• Theorem 7
• Definition 8
• Theorem 9
• Remark 10