Moment-angle complexes from simplicial posets

Details Moment-angle complexes from simplicial posets Moment-angle complexes from simplicial posets

2009-12-11

arxiv.org/abs/0912.2219v3

Cent. Eur. J. Math. 9 (2011), no.4, 715-730

Mathematics

Algebraic Topology

17 pages, 2 pictures; v2 revised, v3 minor corrections

Scientific paper

10.2478/s11533-011-0041-z

We extend the construction of moment-angle complexes to simplicial posets by associating a certain T^m-space Z_S to an arbitrary simplicial poset S on m vertices. Face rings Z[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen--Macaulay rings. Our primary motivation is to study the face rings Z[S] by topological methods. The space Z_S has many important topological properties of the original moment-angle complex Z_K associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Z_S is isomorphic to the Tor-algebra of the face ring Z[S]. This leads directly to a generalisation of Hochster's theorem, expressing the algebraic Betti numbers of the ring Z[S] in terms of the homology of full subposets in S. Finally, we estimate the total amount of homology of Z_S from below by proving the toral rank conjecture for the moment-angle complexes Z_S.

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