Homological algebra and poset versions of the Garland method
Combinatorics
2026-05-05 v4 Algebraic Topology
Group Theory
Abstract
Garland introduced a vanishing criterion for a characteristic zero cohomology group of a locally finite and locally connected simplicial complex. The criterion is based on the spectral gaps of the graph Laplacians of the links of faces and has turned out to be effective in a wide range of examples. In this note we extend the approach to include a range of non-simplicial (co)chain complexes associated to combinatorial structures we call Garland posets and elaborate further on the case of cubical complexes.
Keywords
Cite
@article{arxiv.2308.00972,
title = {Homological algebra and poset versions of the Garland method},
author = {Eric Babson and Volkmar Welker},
journal= {arXiv preprint arXiv:2308.00972},
year = {2026}
}
Comments
Corollary 2.3 added, Figures 1 and 5 added, description of simplicial and cubical cases expanded, typos corrected