Monotonicity of geometric simplicial gnarledness

Establish whether the geometric gnarledness of finite simplicial complexes is monotone under simplicial maps that induce rational homology isomorphisms.

Background

The paper defines an algebraic version of gnarledness for marked chain complexes and proves that it satisfies a monotonicity property under suitable chain maps. It compares this invariant with the geometric gnarledness of simplicial complexes defined using spanning trees.

Although algebraic gnarledness is monotone, the authors explicitly leave unresolved whether the corresponding geometric invariant enjoys an analogous property for simplicial maps inducing rational homology isomorphisms.

References

Moreover, $G_kQ(C_*\Delta(X))$ satisfies monotonicity (Proposition~\ref{prop:gnarlmono}); this applies, in particular, to simplicial maps that induce $Q$-homology isomorphisms. In contrast, for the gnarledness of simplicial complexes, it is not clear whether such a monotonicity property holds.

— Quantitative chain extensions and integral cohomological fillings  (2610.02987 - Hofmann et al., 2 Oct 2026) in Remark following Proposition on comparison with geometric gnarledness, Section 5.3