Topological implications of bounded-diameter intermediate-Ricci manifolds

Determine what topological implications follow for closed Riemannian manifolds with a lower bound on the k-th intermediate Ricci curvature and an upper diameter bound, without assuming a lower volume bound.

Background

The paper studies the hierarchy of curvature conditions between sectional and Ricci curvature, focusing on manifolds satisfying Ric_k ≥ K. The authors establish a local topological theorem under the additional assumption of a uniform lower volume bound, showing that sufficiently small metric balls have trivial homology in degrees at least k+1.

The broader question remains unresolved when only the lower intermediate-Ricci curvature bound and diameter bound are imposed. The lower volume assumption is essential for the local theorem proved in the paper, as demonstrated by shrinking-sphere examples.

References

While this question has been studied extensively in recent years , the following basic question is not yet understood. Given $D>0$ and $K\inR$, what topological implications hold for closed Riemannian manifolds of $Ric_k\geq K$ and $\mathrm{diam}\leq D$?

— Local Topology of Riemannian Manifolds with Lower Intermediate Ricci Curvature Bounds  (2609.10268 - Reiser et al., 9 Sep 2026) in Section 1, Introduction and main results, Question 1.1 (labeled Q:Ric_k-top)

Thus, the following question arises. Can the conclusion of Theorem~\ref{T:local_top} be improved to degree $k$? Can it be improved even further when $k\leq\frac{n}{2}$?

— Local Topology of Riemannian Manifolds with Lower Intermediate Ricci Curvature Bounds  (2609.10268 - Reiser et al., 9 Sep 2026) in Section 1, paragraph following the discussion of the homology degree in Theorem A

By interpolating between these two endpoint results, one is led to the following question. Is there a constant $C>0$ such that for any closed, $n$-dimensional Riemannian manifold of $Ric_k\geq K$ and $\mathrm{diam}\leq D$ we have \sum_{i=k}n b_i(M)\leq C, where $b_i(M)$ denotes the $i$-th Betti number of $M$ with coefficients in some ring? In the constructions of and for positive Ricci curvature were generalized to $Ric_k>0$ to show that, in general, there is no bound on the Betti numbers $b_i$ with $i\leq k-1$ when $k\geq\frac{n}{2}+1$. However, Question \ref{Q:betti-numbers} as stated remains open.

— Local Topology of Riemannian Manifolds with Lower Intermediate Ricci Curvature Bounds  (2609.10268 - Reiser et al., 9 Sep 2026) in Section 1, Question 1.3 (labeled Q:betti-numbers)