A generalization of moment-angle manifolds with non-contractible orbit spaces
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866910383300673536 |
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| author | Yu, Li |
| author_facet | Yu, Li |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2011_10366 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A generalization of moment-angle manifolds with non-contractible orbit spaces Yu, Li Algebraic Topology Geometric Topology 57S12 (Primary) 57N65, 57S17, 57S25 (Secondary) 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. |
| title | A generalization of moment-angle manifolds with non-contractible orbit spaces |
| topic | Algebraic Topology Geometric Topology 57S12 (Primary) 57N65, 57S17, 57S25 (Secondary) |
| url | https://arxiv.org/abs/2011.10366 |