{"id":"hexagon/2610.00218","pdf":"https://hexagonmath.org/pdf/2610.00218v2","url":"https://hexagonmath.org/abs/2610.00218v2","tags":["math.AT","msc:22A22","Ample Groupoids","HK-Conjecture"],"score":0,"title":"A Note on Laminar Presentations and Finite Chain Models for Ample Groupoid Homology","domain":"math.KT","source":"hexagon","authors":["Luciano Melodia"],"reviews":0,"subject":"math.KT","summary":"A finite free chain model for the homology of a Hausdorff ample groupoid \\(G\\) is cut out of the Matui complex by a finite clopen partition of each nerve space. We ask what such a model must supply and isolate three requirements: a single model that is a quasi-isomorphism, a chain of models that is a complete system with ranks and transition maps read off the presenting data, and a bound on the number of atoms in terms of the presenting cover. An \\(n\\)-Helly axiom on the presenting covers supplies none of them, being satisfied by every finite atomization for every \\(n\\), and a \\(1\\)-Helly…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-14","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}