Hom complexes of set systems (Q1953374)

Summary: A set system is a pair \(\mathcal{S}\) = (\(V(\mathcal{S}),\Delta(\mathcal{S})\)), where \(\Delta(\mathcal{S})\) is a family of subsets of the set \(V(\mathcal{S})\). We refer to the members of \(\Delta(\mathcal{S})\) as the stable sets of \(\mathcal{S}\). A homomorphism between two set systems \(\mathcal{S}\) and \(\mathcal{T}\) is a map \(f : V(\mathcal{S}) \rightarrow V(\mathcal{T})\) such that the preimage under \(f\) of every stable set of \(\mathcal{T}\) is a stable set of \(\mathcal{S}\). Inspired by a recent generalization due to Engström of Lovász's Hom complex construction, the author associates a cell complex \(\operatorname{Hom}(\mathcal{S},\mathcal{T})\) to any two finite set systems \(\mathcal{S}\) and \(\mathcal{T}\). The main goal of the paper is to examine basic topological and homological properties of this cell complex for various pairs of set systems.