Properties of comultiplications on a wedge of spheres
type: Item
instance of: scholarly article
title: Properties of comultiplications on a wedge of spheres
zbMATH Open document ID: 1203.55007
author: Martin Arkowitz
author: Dae-Woong Lee
published in: Topology and its Applications
publication date: 2010-05-19T00:00:00Z
review text: A comultiplication \(\varphi\) on a space \(X\) is a map \(\varphi : X \to X \vee X\) satisfying \(j \circ \varphi \simeq \Delta\) where \(j : X \vee X \to X \times X\) is the inclusion and \(\Delta : X \to X \times X \) is the diagonal. The authors study the set of comultiplications when \(X\) is a wedge of \(t\) spheres for \(t > 2\) building on their previous work for \(t = 2\) [\textit{M. Arkowitz} and \textit{D.-W. Lee}, ``Comultiplications on a wedge of two spheres'', to appear]. A comultiplication \(\varphi\) in this case is characterized by a perturbation \(P = (P_1, \dots, P_t)\) where \(P_j\) satisfies \(\varphi \circ k_j \simeq \iota_1 \circ k_j + \iota_2 \circ k_j + P_j\). Here \(k_j\) is the inclusion of the \(j\)th sphere in the wedge \(X\) and the \(\iota_1, \iota_2 : X \to X \vee X\) are the inclusions. By Hilton's Theorem, the \(P_j\) may be expressed in terms of Whitehead products. The authors consider the cases where each \(P_j\) involves only length \(\leq 2\) or length \(\leq 3\) Whitehead products and prove extensive results on homotopy associativity, commutativity, inversivity etc. in these cases. The techniques used are classical with Hopf-Hilton invariants playing a central role.
Mathematics Subject Classification ID: 55Q25
zbMATH DE Number: 5710098
zbMATH Keywords: co-H-spaces
zbMATH Keywords: comultiplications
zbMATH Keywords: homotopy associativity
zbMATH Keywords: homotopy commutativity
zbMATH Keywords: Whitehead products
zbMATH Keywords: Hopf-Hilton invariants
zbMATH Keywords: wedges of spheres
zbMATH Keywords: algebraic loops
zbMATH Keywords: inversivity
zbMATH Keywords: power-associativity
zbMATH Keywords: the Moufang property
reviewed by: Samuel Bruce Smith
MaRDI profile type: Publication
full work available at URL: j.topol.2010.03.001
OpenAlex ID: W2059305909
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DOI: 10.1016/J.TOPOL.2010.03.001
description: scientific article; zbMATH DE number 5710098