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A panel structure on a topological space is just a locally finite family ofclosed subspaces. A space together with a panel structure is called a spacewith faces. In this paper, we define the notion of polyhedral product over aspace with faces. This notion provides a unifying viewpoint on theconstructions of polyhedral products and generalized moment-angle complexes invarious settings. We compute the stable decomposition of these spaces and useit to study their cohomology ring structures by the partial diagonal maps.Besides, we can compute the equivariant cohomology ring of the moment-anglecomplex over a space with faces with respect to the canonical torus action. Thecalculation leads to a notion of topological face ring of a space with faces,which generalizes the classical notion of face ring of a simplicial complex. Wewill see that many known results in the study of polyhedral products andmoment-angle complexes can be reinterpreted from our general theorems on thepolyhedral product over a space with faces. Moreover, we can derive some newresults via our approach in some settings.

A unifying view toward polyhedral products through panel structures