Polyhedral products over finite posets

Abstract

Polyhedral products were defined by Bahri, Bendersky, Cohen, and Gitler, to be spaces obtained as unions of certain product spaces indexed by the simplices of an abstract simplicial complex. In this paper, we give a very general homotopy theoretic construction of polyhedral products over arbitrary pointed posets. We show that under certain restrictions on the poset $\mathcal{P}$, which include all known cases, the cohomology of the resulting spaces can be computed as an inverse limit over $\mathcal{P}$ of the cohomology of the building blocks. This motivates the definition of an analogous algebraic construction—the polyhedral tensor product. We show that for a large family of posets, the cohomology of the polyhedral product is given by the polyhedral tensor product. We then restrict attention to polyhedral posets, a family of posets that includes face posets of simplicial complexes, and simplicial posets, as well as many others. We define the Stanley–Reisner ring of a polyhedral poset and show that, as in the classical cases, these rings occur as the cohomology of certain polyhedral products over the poset in question. For any pointed poset $\mathcal{P}$, we construct a simplicial poset $s(\mathcal{P})$, and show that if $\mathcal{P}$ is a polyhedral poset, then polyhedral products over $\mathcal{P}$ coincide up to homotopy with the corresponding polyhedral products over $s(\mathcal{P})$.