Homological algebra and poset versions of the Garland method

TL;DR

This paper extends Garland's vanishing criterion to new combinatorial structures called Garland posets, including cubical complexes, broadening its applicability in homological algebra.

Contribution

It introduces a generalized approach to Garland's method for non-simplicial complexes and elaborates on the case of cubical complexes.

Findings

Extended Garland's criterion to Garland posets.

Applied the approach to cubical complexes.

Demonstrated effectiveness in new combinatorial settings.

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.