Characterisation of polyhedral products with finite generalised Postnikov decomposition (Q2199336)

For a pair \((X,A)\) of spaces and an abstract simplicial complex \(K\), let \(\mathcal{Z}_K(X,A)\) denote the polyhedral product of \((X,A)\) with respect to \(K\), and recall that a generalized Postnikov tower for \(X\) is a tower of principal fibrations with fibers of generalized Eilenberg-MacLane spaces, whose inverse limit is weakly homotopy equivalent to \(X\). For a simple graph \(\Gamma\) with vertex set \([m]=\{1,2,\dots ,m\}\) and a collection \(\mathrm{G}=\{G_i\}_{i=1}^m\) of groups, let \(\mathrm{G}^{\Gamma}\) denote the graph product given by \(\mathrm{G}^{\Gamma}=\langle G_1,\dots ,G_m: [G_i,G_j]=1\text{ for any edge }\{i,j\} \text{ in }\Gamma\rangle,\) and \(K_{\Gamma}(\mathrm{G})\) denotes the kernel of the canonical map \(\mathrm{G}^{\Gamma}\to \prod_{i=1}^mG_i\). In this paper, the authors give a characterization of spaces \(\mathcal{Z}_K(X,A)\) whose universal cover either admits a generalized Postnikov tower of finite length, or is a homotopy retract of a space which admits such a tower. They also obtain the \(p\)-local and rational versions of the above result. Moreover, as an application they consider the problem of group products. Let \(\mathrm{G}=\{G_i\}_{i=1}^m\) and \(\mathrm{H}=\{H_i\}_{i=1}^m\) be collections of groups. Then they show that there is an isomorphism \(K_{\Gamma}(\mathrm{G})\cong K_{\Gamma}(\mathrm{H})\) if \(G_i\) and \(H_i\) have the same cardinality for each \(1\leq i\leq m\). The proof of the final result is purely homotopy theoretic and it seems interesting.