On the fundamental group of certain polyhedral products
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. Philos. Soc. 152, No. 1, 91--114 (2012; Zbl 1250.57003)], called the extension problem. Let \(\underline{G}=\{G_{1},\ldots, G_{n}\}\) be a family of non-trivial subgroups of a finite discrete group \(G\) and \(K\) be a simplicial complex with \(n\) vertices \(K_{0}\). Then, there is an induced map \(B\varphi:\mathcal{Z}_{K_{0}}(\underline{BG})=BG_{1}\vee \cdots \vee BG_{n}\to BG\) from the natural map \(\varphi: G_{1}*\cdots *G_{n}\to G\). On the other hand there is the natural inclusion \(\mathcal{Z}_{K_{0}}(\underline{BG})\to \mathcal{Z}_{K}(\underline{BG})\). The question is for what abstract simplicial complexes \(K\) does the map \(B\varphi\) extend to \(\mathcal{Z}_{K}(\underline{BG})\)? The author also gives a partial answer to this question.
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