On the fundamental group of certain polyhedral products
type: Item
instance of: scholarly article
title: On the fundamental group of certain polyhedral products
zbMATH Open document ID: 1311.55026
author: Mentor Stafa
published in: Journal of Pure and Applied Algebra
publication date: 2015-02-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1310.3504
review text: Let \(K\) be an abstract simplicial complex with \(n\) vertices and \((\underline{X},\underline{A})=\{(X_{i},A_{i})\}_{i=1}^{n}\) be a sequence of pairs of spaces with \(A_{i}\subset X_{i}\) for \(i=1,\ldots,n\). Then, a topological space \(\mathcal{Z}_{K}(\underline{X},\underline{A})\), called a polyhedral product, can be defined. If \(X_{i}=X\) and \(A_{i}=A\) for all \(i=1,\ldots, n\), then \(\mathcal{Z}_{K}(\underline{X},\underline{A})\) is denoted by \(\mathcal{Z}_{K}(X,A)\). A polyhedral product naturally appears in toric topology, e.g. \(\mathcal{Z}_{K}(\mathbb{C}, \mathbb{C}^{*})\) is a coordinate space arrangement, \(\mathcal{Z}_{K}(D^2, S^1)\) is a moment-angle complex, \(\mathcal{Z}_{K}(\mathbb{C}P^{\infty}, *)\) is a Davis-Januszkiewicz space, see \textit{V. M. Buchstaber} and \textit{T. E. Panov} [Torus actions and their applications in topology and combinatorics. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1012.52021)]. So, recently, topological properties of polyhedral products (in particular their homotopy types) have been studied by many topologists. In the paper under review, the author proves the following two results about homotopy types of polyhedral products: 1. for non-trivial discrete groups \(G_{1},\ldots, G_{n}\), the polyhedral product \(\mathcal{Z}_{K}(\underline{BG})\) is an Eilenberg-MacLane space if and only if \(K\) is a flag complex, where \(\underline{BG}=\{(BG_{i}, *_{i})\}_{i=1}^{n}\) and a flag complex is a simplicial complex with no minimal non-faces. Equivalently, \(\mathcal{Z}_{K}(\underline{EG},\underline{G})\) is an Eilenberg-MacLane space if and only if \(K\) is a flag complex. 2. Let \(X_{1},\ldots, X_{n}\) be \(1\)-connected CW complexes. Then the polyhedral product \(\mathcal{Z}_{K}(\underline{C \Omega X}, \underline{\Omega X})\) is \(1\)-connected. Moreover, the author asks the following problem which is motivated by a theorem of Adem, Cohen and Torres-Giese [\textit{A. Adem} et al., Math. Proc. Camb.…
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 55Q05
zbMATH DE Number: 6398761
zbMATH Keywords: polyhedral product
zbMATH Keywords: simplicial complexes
zbMATH Keywords: fundamental group
reviewed by: Shintaro Kuroki
MaRDI profile type: Publication
OpenAlex ID: W2026727644
arXiv ID: 1310.3504
cites work: Commuting elements, simplicial spaces and filtrations of classifying spaces
cites work: The polyhedral product functor: A method of decomposition for moment-angle complexes, arrangements and related spaces
cites work: Q4537714
cites work: Q4266116
cites work: Convex polytopes, Coxeter orbifolds and torus actions
cites work: Cohomology computations for Artin groups, Bestvina-Brady groups, and graph products.
cites work: Moment-angle complexes, monomial ideals and Massey products
cites work: Fundamental group of homotopy colimits.
cites work: Q4819371
cites work: Construction of universal bundles. II
cites work: The geometric realization of a semi-simplicial complex
cites work: Q5789034
DOI: 10.1016/J.JPAA.2014.09.001
description: scientific article