  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.
