Generalized K(pi,1) conjecture for positive m-intermediate curvature (3 ≤ n ≤ 7)

Prove that if M^n is a closed manifold of dimension 3 ≤ n ≤ 7 whose universal cover 2 satisfies H_n(2;Z) = H_{n-1}(2;Z) = ... = H_{n-m+1}(2;Z) = 0, then M admits no Riemannian metric with positive m-intermediate curvature.

Background

The paper introduces an interpolation of curvature conditions between Ricci curvature and scalar curvature via the m-intermediate curvature. The proposed conjecture relates vanishing homology of the universal cover in top degrees to the nonexistence of metrics with positive m-intermediate curvature.

This conjecture recovers as special cases both the classical K(pi,1) conjecture for positive scalar curvature (when m = n − 1) and the generalized Geroch-type nonexistence results for N^{n−m} × T^m (where the relevant homology vanishing holds). The authors prove the conjecture in dimensions n = 3, 4, 5 for all m, and for most m when n = 6, providing strong evidence but leaving the conjecture open in full generality within 3 ≤ n ≤ 7.

References

In this paper, we propose the following conjecture which relates the topology of a manifold's universal cover with the existence of metrics with positive $m$-intermediate curvature.

Let $M^n$ be a closed manifold of dimension $3\leq n\leq 7$. Assume that the universal cover $\overline M$ of $M$ satisfies

H_n(\overline M,Z) = H_{n-1}(\overline M,Z) = \hdots = H_{n-m+1}(\overline M,Z) = 0.

Then $M$ does not admit a metric with positive $m$-intermediate curvature.

— On the topology of manifolds with positive intermediate curvature (2503.13815 - Mazurowski et al., 18 Mar 2025) in Introduction, Conjecture (label generalized-Kpi1)