Torus graphs and simplicial posets |
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Authors: | Hiroshi Maeda |
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Institution: | a Hakuryo Co., Ltd., Japan b Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan c Department of Geometry and Topology, Faculty of Mathematics and Mechanics, Moscow State University, Leninskiye Gory, Moscow 119992, Russia d Institute for Theoretical and Experimental Physics, Moscow 117259, Russia |
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Abstract: | For several important classes of manifolds acted on by the torus, the information about the action can be encoded combinatorially by a regular n-valent graph with vector labels on its edges, which we refer to as the torus graph. By analogy with the GKM-graphs, we introduce the notion of equivariant cohomology of a torus graph, and show that it is isomorphic to the face ring of the associated simplicial poset. This extends a series of previous results on the equivariant cohomology of torus manifolds. As a primary combinatorial application, we show that a simplicial poset is Cohen-Macaulay if its face ring is Cohen-Macaulay. This completes the algebraic characterisation of Cohen-Macaulay posets initiated by Stanley. We also study blow-ups of torus graphs and manifolds from both the algebraic and the topological points of view. |
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Keywords: | Torus graphs Simplicial posets Cohen-Macaulay posets Torus manifolds GKM-graphs Equivariant cohomology Blow-ups |
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