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A generalization of moment-angle manifolds with noncontractible orbit spaces

Li Yu

Algebraic & Geometric Topology 24 (2024) 449–492

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 𝒵Q via a construction called rim-cubicalization of Q. From this, we derive a formula to compute the integral cohomology group of 𝒵Q 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 𝒵Q 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 𝒵Q. Using this general construction, we can compute the equivariant cohomology ring of 𝒵Q 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 𝒵Q over Q as well.

moment-angle manifold, topological face ring, manifold with corners, equivariant cohomology, Davis–Januszkiewicz space
Mathematical Subject Classification
Primary: 57S12
Secondary: 57N65, 57S17, 57S25
Received: 16 October 2021
Revised: 11 July 2022
Accepted: 27 September 2022
Published: 18 March 2024
Li Yu
Department of Mathematics
Nanjing University

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