A generalization of moment-angle manifolds with noncontractible orbit spaces

• DOI: 10.2140/agt.2024.24.449
• Publication Date: 2024-03-25T17:09:08Z
• Authors: Li Yu
• Keywords: Mathematics - Geometric Topology; 57S12 (Primary) 57N65, 57S17, 57S25 (Secondary); FOS: Mathematics; Algebraic Topology (math.AT); Geometric Topology (math.GT); Mathematics - Algebraic Topology; 0101 mathematics; 01 natural sciences

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.<br/>46 pages, 2 figures. A new reference is added and some typos are fixed. Final version before publication<br/>