A generalization of moment-angle manifolds with noncontractible orbit spaces

A generalization of moment-angle manifolds with noncontractible orbit spaces LI YUWe 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 ᐆ Q via a construction called rim-cubicalization of Q. From this, we derive a formula to compute the integral cohomology group of ᐆ 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 ᐆ 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 ᐆ Q .Using this general construction, we can compute the equivariant cohomology ring of ᐆ 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 Rᐆ Q over Q as well.