On products in a real moment-angle manifold
  In this paper we give a necessary and sufficient condition for a (real) moment-angle complex to be a topological manifold. The cup and cap products in a real moment-angle manifold are studied: the Poincar\'{e} duality via cap products is equivalent to the Alexander duality of the defining complex $K$. Consequently, the cohomology ring (with coefficients integers) of a polyhedral product by pairs of disks and their bounding spheres is isomorphic to that of a differential graded algebra associated to $K$, and the dimensions of the disks. 
Li Cai
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