Configuration spaces and polyhedral products
A ring \(R=\mathbf{k}[v_1,\cdots ,v_n]/I\) is called \textit{Golod} if all products and higher Massey products in \(\mathrm{Tor}^+_{\mathbf{k}[v_1,\cdots ,v_n]}(R,\mathbf{k})\) vanish, where \(\mathbf{k}\) is a commutative ring and \(I\) is a homogenous ideal in \(\mathbf{k}[v_1,\cdots ,v_n]\). Similarly, a simplicial complex \(K\) on the vertex set \([n]=\{1,2,\cdots ,n\}\) is said to be \textit{Golod over} \(\mathbf{k}\) if the Stein-Reisner ring \(\mathbf{k}[K]\) is Golod, and it is called \textit{Golod} if it is Golod over \(\mathbf{k}\) for all fields \(\mathbf{k}\) or \(\mathbf{k}=\mathbb{Z}\). Now we recall the conjecture that a moment-angle complex \((D^2,S^1)^K\) is a co-\(H\)-space if \(K\) is Golod. It follows from the result due to Iriye-Kishimoto that a moment-angle complex \((D^2,S1)^K\) is a co-\(H\)-space iff there is a homotopy equivalence \((D^2,S^1)^K\simeq \bigvee_{I\subset [n]}\Sigma^{| I| +2} | K_I|\). So it seems interesting and useful to study the most general combinatorial conditions under which a moment-angle complex \((D^2,S^1)^K\) is a co-\(H\)-space. In this paper, the authors consider this problem and they prove that localized at any sufficiently large prime, \((D^2,S^1)^K\) is a co-\(H\)-space iff \(K\) is Golod over \(\mathbb{Z}/p\). Moreover, they show that if \(K\) is coherently homotopy Golod, then \((D^2,S^1)^K\) is a co-\(H\)-space. Their proof is based on a certain generalization of the scanning maps defined on labelled configuration spaces.
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