The homology of simplicial complement and the cohomology of the moment-angle complexes

A simplicial complement P is a sequence of subsets of [m] and the simplicial complement P corresponds to a unique simplicial complex K with vertices in [m]. In this paper, we defined the homology of a simplicial complement \(H_{i,\sigma}(\Lambda^{*,*}[P], d)\) over a principle ideal domain k and pro...

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Bibliographic Details
Published in:arXiv.org 2010-11
Main Authors: Wang, Xiangjun, Zheng, Qibing
Format: Article
Language:English
Subjects:
Apexes
Complement
Homology
Ring structures
Online Access:Get full text
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Summary:A simplicial complement P is a sequence of subsets of [m] and the simplicial complement P corresponds to a unique simplicial complex K with vertices in [m]. In this paper, we defined the homology of a simplicial complement \(H_{i,\sigma}(\Lambda^{*,*}[P], d)\) over a principle ideal domain k and proved that \(H_{*,*}(\Lambda[P], d)\) is isomorphic to the Tor of the corresponding face ring k(K) by the Taylor resolutions. As applications, we give methods to compute the ring structure of Tor_{*,*}^{k[x]}(k(K), k)\(, \)link_{K}\sigma\(, \)star_{K}\sigma$ and the cohomology of the generalized moment-angle complexes.
ISSN:2331-8422