A note on the Samelson products in \(\pi_*(SO(2n))\) and the group \([SO(2n),SO(2n)]\)
type: Item
instance of: scholarly article
title: A note on the Samelson products in \(\pi_*(SO(2n))\) and the group \([SO(2n),SO(2n)]\)
zbMATH Open document ID: 1119.55006
author: Hiroaki Hamanaka
author: Akira Kono
published in: Topology and its Applications
publication date: 2007-02-02T00:00:00Z
review text: Consider the map \(j\, ':BSO(2n)\to BSO(2n+1)\) with fibre \(\alpha ':S^{2n}\to BSO(2n),\) where \(j\, '=Bj,\) \(j:SO(2n)\to SO(2n+1)\) is the inclusion and \(\alpha '\) is the adjoint of an element \(\alpha \in \pi_{2n-1}(SO(2n)).\) For odd \(p\) the authors obtain elements whose Samelson products with \(\alpha \) are non-zero in \(\pi _*(SO(2n))_{(p)}.\) They use this result to show that the group \([SO(2n),SO(2n)]_{(p)}\) is commutative if, and only if, \(p>4n-5\) (for \(n\neq 3,4,\) or \(6\)) or \(p\geq 4n-5\) (for \(n= 3,4,\) or \(6\)).
Mathematics Subject Classification ID: 55Q05
Mathematics Subject Classification ID: 55P60
Mathematics Subject Classification ID: 55Q15
zbMATH DE Number: 5121449
zbMATH Keywords: self-maps
zbMATH Keywords: orthogonal group
zbMATH Keywords: localisation
reviewed by: John W. Rutter
MaRDI profile type: Publication
full work available at URL: j.topol.2006.07.010
OpenAlex ID: W2087862487
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DOI: 10.1016/J.TOPOL.2006.07.010
description: scientific article; zbMATH DE number 5121449