  A combinatorial construction is used to analyze the properties of polyhedral
products and generalized moment-angle complexes with respect to certain
operations on CW pairs including exponentiation. This allows for the
construction of infinite families of toric manifolds, associated to a given
one, in a way which simplifies the combinatorial input and consequently, the
presentation of the cohomology rings. The new input is the interaction of a
purely combinatorial construction with natural associated geometric
constructions related to polyhedral products and toric manifolds. Applications
of the methods and results developed here have appeared in literature.
