On a conjecture of Ōshima
type: Item
instance of: scholarly article
title: On a conjecture of Ōshima
zbMATH Open document ID: 1185.55017
author: Daisuke Kishimoto
author: Akira Kono
published in: Topology and its Applications
publication date: 2009-08-20T00:00:00Z
review text: It is a classical result of G.W. Whitehead that the self homotopy group \([G,G]\) for a compact connected Lie group \(G\) has the nilpotency class bounded from above by \(\roman{cat} G\), the reduced Lusternik-Schnirelmann category of \(G\). \textit{H. Ōshima} [J. Math. Kyoto Univ. 40, No.~1, 177--184 (2000; Zbl 1009.55007)] stated a conjecture about a lower bound for the nilpotency class of \([G,G]\): if \(G\) is simple, then \([G,G]\) is nilpotent of class \(\geq \roman{rank}\,G\). The authors show that the conjecture is true for \(\text{PU}(p)\), the \(p\)-dimensional projective unitary group, for any odd prime \(p\). This is the first example satisfying the conjecture for a group of rank greater than 3.
reviewed by: Younggi Choi
Mathematics Subject Classification ID: 55Q05
Mathematics Subject Classification ID: 55M30
zbMATH DE Number: 5597559
zbMATH Keywords: self homotopy group
zbMATH Keywords: Lie group
zbMATH Keywords: nilpotency class
Wikidata QID: Q122879368
MaRDI profile type: Publication
full work available at URL: j.topol.2009.04.049
OpenAlex ID: W2055370175
cites work: Noncommutativity of the group of self homotopy classes of Lie groups
cites work: A note on the Samelson product in the classical groups
cites work: Homotopy limits, completions and localizations
cites work: Self homotopy groups with large nilpotency classes
cites work: Regularity of finite H-spaces
cites work: Mod \(p\) decompositions of non-simply connected Lie groups
cites work: COMMUTATIVITY OF THE GROUP OF SELF-HOMOTOPY CLASSES OF LIE GROUPS
cites work: Localization of CW-complexes and its applications
cites work: Self homotopy group of the exceptional Lie group \(G_2\)
cites work: On mappings into group-like spaces
cites work: Groupes d'homotopie et classes de groupes abéliens
DOI: 10.1016/J.TOPOL.2009.04.049
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description: scientific article; zbMATH DE number 5597559