Configuration spaces and polyhedral products
type: Item
instance of: scholarly article
title: Configuration spaces and polyhedral products
zbMATH Open document ID: 1369.55006
author: Piotr Beben
author: Jelena Grbić
published in: Advances in Mathematics
publication date: 2017-06-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1409.4462
full work available at URL: FinalConfigPolyhedral.pdf
review text: 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.
reviewed by: Kohhei Yamaguchi
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55R80
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 13F55
zbMATH DE Number: 6730471
zbMATH Keywords: polyhedral product
zbMATH Keywords: moment-angle complex
zbMATH Keywords: toric topology
zbMATH Keywords: Stanley-Reisner ring
zbMATH Keywords: Golod ring
zbMATH Keywords: configuration space
zbMATH Keywords: co-\(H\)-space
zbMATH Keywords: scanning map
MaRDI profile type: Publication
OpenAlex ID: W2963208475
arXiv ID: 1409.4462
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DOI: 10.1016/J.AIM.2017.05.001
description: scientific article