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Maeda, Hiroshi; Masuda, Mikiya; Panov, Taras

Torus graphs and simplicial posets. (English) Zbl 1119.55004

Adv. Math. 212, No. 2, 458-483 (2007).
The authors define the notion of torus graph as a regular \(n\)-valent graph with vector labels on its edges, associated to a manifold acted on by the torus. It allows them to translate the important topological properties of torus manifolds into the language of combinatorics. The notion of equivariant cohomology of a torus graph is introduced and it is shown 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 an application, it is proved that a simplicial poset is Cohen-Macaulay if its face ring is Cohen-Macaulay and thus the algebraic characterisation of Cohen-Macaulay posets is complete. Blow-ups of torus graphs and manifolds from both the algebraic and the topological points of view are also studied.
Reviewer: Gheorghe Pitiş (Braşov)

Cited in 16 Documents

MSC:

55N91 Equivariant homology and cohomology in algebraic topology
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
06A11 Algebraic aspects of posets

Keywords:

torus graph; torus manifold; simplicial poset; equivariant cohomology; Thom class; blow-up
PDFBibTeX XMLCite
\textit{H. Maeda} et al., Adv. Math. 212, No. 2, 458--483 (2007; Zbl 1119.55004)
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