Determine the degree-dependent positive-curvature threshold for higher GLMY homology

Determine whether, for each homological degree p≥2, vanishing of the p-th GLMY path homology follows from a positive Lin–Lu–Yau curvature lower bound depending on p.

Background

The Cartesian products T_r=C_5{\square r} have strictly positive Lin–Lu–Yau curvature 1/(2r) and nonzero GLMY path homology in degrees up to r. Consequently, strict positivity alone cannot force vanishing in any fixed higher degree when the curvature lower bound is allowed to decrease with the number of factors.

The examples leave open whether a stronger, degree-sensitive statement holds: namely, whether there is a positive curvature threshold depending on the homological degree p such that every finite connected simple graph with p-th GLMY path homology nonzero must have minimum curvature at most that threshold, or equivalently whether a sufficiently strong positive lower bound depending on p forces vanishing.

References

This construction does not address whether vanishing follows from a positive curvature lower bound that depends on the homological degree.

— A Sharp Curvature Threshold for GLMY Path Homology  (2608.23187 - Bai et al., 24 Aug 2026) in Section 4.3, immediately before Section 4.3.1, subsection “Positive curvature and higher-dimensional GLMY path homology”