Higher commutators in the loop space homology of \(K\)-products
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 higher commutators on elements of \(H_*(\Omega X)\), and their iterations, generate \(H_*(\Omega \underline{X}^{K})\). The answer is positive for a certain class of complexes as addressed in the last section of this paper. In that section, \(H_*(\Omega \underline{X}^{K})\) is explicitly worked out for: a flag simplicial complex \(K\) and field coefficients (Corollary 4.1); \(K\) the Alexander dual of a shellable complex (Theorem 4.12); \(X\) the Davis-Januszkiewicz space \(DJ_K = ({\mathbb C}P^{\infty})^K\), \(K\) as above and integer coefficients (Corollary 4.2 and Corollary 4.13).
scientific article; zbMATH DE number 5655902