Hom Complexes of Set Systems

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 Engstrom of Lovasz's Hom complex construction, the author associates a cell complex $\mathrm{Hom}(\mathcal{S},\mathcal{T})$ 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.

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