Homotopy sets of multiplications on spheres and their algebraic structure
type: Item
instance of: scholarly article
title: Homotopy sets of multiplications on spheres and their algebraic structure
zbMATH Open document ID: 0847.55009
DOI: 10.1007/BF01265299
author: Norbert Endres
published in: Geometriae Dedicata
publication date: 1996-10-15T00:00:00Z
review text: In an earlier paper [ibid. 48, 267-294 (1993; Zbl 0813.55005)] the author introduced an algebraic structure on sets of homotopy classes \([X,S^n]\), given by the reflecting product \(\bullet\) on spheres \(S^n\) \((x \bullet y : = - y + 2 \langle x; y \rangle x\) where \(x,y \in S^n)\), which makes these sets into symmetric groupoids. In this article the algebraic structure of the symmetric groupoids \(([S^n \times S^n; S^n], \bullet)\) for \(n \neq 1, 3, 7\) is determined. Here a symmetric groupoid is a set \(R\) with a multiplication \(\bullet : R \times R \to R\) satisfying \(x \bullet x = x\); \(x \bullet (x \bullet y) = y\); \(x \bullet (y \bullet z) = (x \bullet y) \bullet (x \bullet z)\). The paper is very technical. (revised by eds.)
Mathematics Subject Classification ID: 55Q05
zbMATH DE Number: 806199
zbMATH Keywords: homotopy sets
zbMATH Keywords: multiplication on spheres
zbMATH Keywords: reflecting products on spheres
reviewed by: Werner End
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01265299
OpenAlex ID: W2091806759
description: scientific article; zbMATH DE number 806199