Homotopy comultiplications on the \(k\)-fold wedge of spheres
type: Item
instance of: scholarly article
title: Homotopy comultiplications on the \(k\)-fold wedge of spheres
zbMATH Open document ID: 1420.55025
author: Dae-Woong Lee
author: Sunyoung Lee
published in: Topology and its Applications
publication date: 2019-01-23T00:00:00Z
review text: The paper under review is the natural continuation of the works by \textit{M. Arkowitz} and \textit{D.-W. Lee} [Sci. China, Math. 54, No. 1, 9--22 (2011; Zbl 1229.55009); Topology Appl. 157, No. 9, 1607--1621 (2010; Zbl 1203.55007)]. The focus is on the cardinality of the set of homotopy comultiplications on the \(k\)-fold wedge of spheres \(Z=\bigvee_{i=1}^k\mathbb{S}^{n_i}\) for arbitrary and sufficiently large values of \(k\), where \(2\leq n_1<n_2<\dots<n_k\). The main ingredients are basic Whitehead products on \(Z\) and the Hopf-Hilton invariants. The authors develop formulas for three cases. The first formula gives the number of comultiplications, the second the number of comultiplications which are associative and the third the number of comultiplications which are commutative. The results are not simple even to state and involve many technical details and notation. We illustrate one of the cases, namely the number of comultiplications. As a consequence of one of the main results, Theorem 2.14, the authors show: Corollary 2.15. The number of comultiplications of \(Z=\mathbb{S}^{n_1}\vee \mathbb{S}^{n_2}\vee\dots \vee\mathbb{S}^{n_k}\) is \[ |\mathcal{C}(Z)|=\Pi_{w_2\in W_ 2^{*}}|\pi_{n_2}(\mathbb{S}^{h_{w_2}})|\times \Pi_{w_3\in W_3^{*}}|\pi_{n_3}(\mathbb{S}^{h_{w_3}})|\times \dots\times \Pi_{w_k\in W_k^{*}}|\pi_{n_k}(\mathbb{S}^{h_{w_k}})|. \] The sets \(W_i^{*}\) are certain basic Whitehead products which involve elements from the homotopy of each of the factors of \(Z\vee Z\) and the spheres \(\mathbb{S}^{h_{w_i}}\) are given by Hilton's formula for the description of the homotopy of \(Z\). Then two similar results concerning the cardinality of associative and commutative comultiplications are given. At the end the authors present some figures to analyse certain regularities and patterns of the commutative comultiplications.
reviewed by: Daciberg Lima Gonçalves
Mathematics Subject Classification ID: 55Q20
Mathematics Subject Classification ID: 55Q25
Mathematics Subject Classification ID: 55Q40
Mathematics Subject Classification ID: 55Q15
Mathematics Subject Classification ID: 55P65
Mathematics Subject Classification ID: 55P45
zbMATH DE Number: 7005612
zbMATH Keywords: comultiplication
zbMATH Keywords: co-\(H\)-space
zbMATH Keywords: Whitehead product
zbMATH Keywords: Hilton's theorem
zbMATH Keywords: Hopf-Hilton invariant
zbMATH Keywords: homotopy associative
zbMATH Keywords: homotopy commutative
MaRDI profile type: Publication
full work available at URL: j.topol.2019.01.001
OpenAlex ID: W2909430168
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Wikidata QID: Q128625567
DOI: 10.1016/J.TOPOL.2019.01.001
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description: scientific article; zbMATH DE number 7005612