Degree-k homology vanishing for open manifolds

Determine whether every open Riemannian manifold with Ric_k ≥ 0 and positive asymptotic volume ratio has trivial homology in degree k as well as in all higher degrees.

Background

For complete open manifolds with Ric_k ≥ 0 and positive asymptotic volume ratio, the paper proves that all homology groups in degrees at least k+1 vanish. In the endpoint cases k=1 and k=n−1, the authors obtain the stronger degree-k conclusion.

The intermediate cases remain unresolved: the paper explicitly asks whether the degree-k vanishing conclusion extends to all k.

References

Therefore, similarly as for Theorem~\ref{T:local_top}, it remains open whether the conclusion of Theorem~\ref{T:open_mfs} also holds for degree $k$.

— Local Topology of Riemannian Manifolds with Lower Intermediate Ricci Curvature Bounds  (2609.10268 - Reiser et al., 9 Sep 2026) in Section 1, paragraph following Theorem 1.5 (labeled T:open_mfs)