A generalization of moment-angle manifolds with non-contractible orbit spaces

TL;DR

This paper extends the concept of moment-angle manifolds from simple convex polytopes to general nice manifolds with corners, providing formulas for their cohomology and introducing new topological invariants.

Contribution

It generalizes the cohomology computation and the face ring concept for moment-angle manifolds over arbitrary manifolds with corners, beyond simple polytopes.

Findings

Derived a stable decomposition of Z_Q via rim-cubicalization.

Provided a formula for the integral cohomology of Z_Q.

Introduced the topological face ring of Q.

Abstract

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 Z_Q via a construction called rim-cubicalization of Q. From this, we derive a formula to compute the integral cohomology group of Z_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 Z_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 Z_Q. Using this general construction, we can compute the equivariant cohomology ring of Z_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. Meanwhile, we obtain some parallel results for the real moment-angle manifold RZ_Q over Q.