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

Abstract

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 $H_{i,\sigma }({\Lambda }^{\ast ,\ast }[P],d)$ over a principle ideal domain k and proved that $H_{\ast ,\ast }(\Lambda [P],d)$ 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}σ$,$star_{K}σ$ and the cohomology of the generalized moment-angle complexes.

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References in corpus (1)

• The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces

Cited by in corpus (2)

• Homology groups of simplicial complements: A new proof of Hochster theorem
• Moment-angle manifolds and connected sums of sphere products