Higher commutators in the loop space homology of \(K\)-products
type: Item
instance of: scholarly article
title: Higher commutators in the loop space homology of \(K\)-products
zbMATH Open document ID: 1197.55004
DOI: 10.1134/S0081543809030055
author: N. E. Dobrinskaya
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2010-01-12T00:00:00Z
review text: If \(K\) is an abstract simplicial complex on \(m\)-vertices \([m]=\{1,\ldots, m\}\), and \(X\) a pointed space, one defines the ``\(K\)-product'' or ``polyhedral product'' \(\underline{X}^K\) to be the set of all \((x_1,\dots, x_m)\in X^m\) such that \(\{i\mid x_i\neq *\}\) is in \(K\). For example the cartesian product \(X^m\) corresponds to the full simplex on \([m]\), while the wedge corresponds to \(K\) being a disconnected set of vertices. The purpose of this interesting paper is to discuss structural results on the homology algebra of the loop space \(\Omega \underline{X}^K\), and then to give complete homological calculations for various choices of \(K\) and \(X\). The paper at hand builds on earlier work of the author where she gives an explicit description of the loop homology algebra \(H_*(\Omega \underline{X}^K)\), for simply connected \(X\), as the quotient of \[ \bigoplus_{I\in{\mathbb N}^m}H_K(I)\otimes\widetilde H_*(\Omega \underline{X})^{\otimes I} \] via identifications induced from ``doubling'' operations defined on \(H_K:=\bigoplus_{I\in{\mathbb N}^m}H_K(I)\); a multigraded algebra obtained as the homology of a free algebra generated by the simplexes of \(K\) with explicit differential (Theorem 2.3). Unless \(H_*(\Omega X;{\mathbb Z} )\) is torsion-free, homology is taken with field coefficients. In order to get a hold on this loop homology algebra, which in general is a difficult task, the author introduces ``higher commutators'' or ``Samelson products'' \[ \{,\}: \widetilde H_*(\Omega \underline{X}^{L_1})\times\cdots\times \widetilde H_*(\Omega \underline{X}^{L_s})\longrightarrow \widetilde H_*(\Omega \underline{X}^{K}) \] defined for \((L_1,\dots, L_s)\) any set of subcomplexes satisfying a novel property called the ``\(\partial\)-property'' (Definition 3.2). It turns out that these commutators are closely related to work of \textit{F. D. Williams} [J. Pure Appl. Algebra 2, 249--260 (1972; Zbl 0239.55017)]. An open problem is whether these hig…
reviewed by: Sadok Kallel
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 13D07
zbMATH DE Number: 5655902
zbMATH Keywords: loop space
zbMATH Keywords: polyhedral product
zbMATH Keywords: Samelson product
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5655902