A generalization of moment-angle manifolds with noncontractible orbit spaces

We generalize the notion of moment -angle manifold over a simple convex polytope to an arbitrary nice manifold with corners. For a nice manifold with corners Q, we first compute the stable decomposition of the moment -angle manifold degrees Q via a construction called rim-cubicalization of Q. From this, we derive a formula to compute the integral cohomology group of degrees Q via the strata of Q. This generalizes the Hochster's formula for the moment -angle manifold over a simple convex polytope. Moreover, we obtain a description of the integral cohomology ring of degrees Q using the idea of partial diagonal maps. In addition, we define the notion of polyhedral product of a sequence of based CW-complexes over Q and obtain similar results for these spaces as we do for degrees Q. Using this general construction, we can compute the equivariant cohomology ring of degrees Q with respect to its canonical torus action from the Davis-Januszkiewicz space of Q. The result leads to the definition of a new notion called the topological face ring of Q, which generalizes the notion of face ring of a simple polytope. Moreover, the topological face ring of Q computes the equivariant cohomology of all locally standard torus actions with Q as the orbit space when the corresponding principal torus bundle over Q is trivial. Meanwhile, we obtain some parallel results for the real moment -angle manifold R degrees Q over Q as well.