Divisible groups in the \(K\)-theory completion of \(SU(n)\)
type: Item
instance of: scholarly article
title: Divisible groups in the \(K\)-theory completion of \(SU(n)\)
zbMATH Open document ID: 1361.55017
author: Peter L. Gregory
published in: Topology and its Applications
publication date: 2015-11-16T00:00:00Z
review text: The article under review studies the unstable homotopy groups of the special unitary group SU\((n)\). The main tool is the unstable Adams spectral sequence based on the generalized cohomology theory \(E(1)\), see [\textit{M. Bendersky} and \textit{R. D. Thompson}, Am. J. Math. 122, No. 3, 599--635 (2000; Zbl 0963.55010)]. Here \(E(1)\) is the usual summand of complex \(K\)-theory \(KU\). This unstable Adams spectral sequence converges to the homotopy groups of the \(K\)-theory completion of SU\((n)\). Another key tool is the fibration \[ \mathrm{SU}(n-1) \rightarrow \mathrm{SU}(n) \rightarrow S^{2n-1}, \] which allows for arguments by induction on \(n\). The article states the results of explicit calculations in an easy-to-use format. These results rely on previous related calculations, including results from [\textit{M. Bendersky}, Topology 31, No. 1, 47--64 (1992; Zbl 0751.55011)] and [\textit{D. M. Davis}, J. Lond. Math. Soc., II. Ser. 43, No. 3, 529--544 (1991; Zbl 0760.55013)].
reviewed by: Daniel C. Isaksen
Mathematics Subject Classification ID: 55Q40
Mathematics Subject Classification ID: 55Q52
Mathematics Subject Classification ID: 55P60
Mathematics Subject Classification ID: 55T15
zbMATH DE Number: 6509434
zbMATH Keywords: algebraic topology
zbMATH Keywords: unstable homotopy
zbMATH Keywords: Bousfield-Kan spectral sequence
zbMATH Keywords: unstable Adams spectral sequence
zbMATH Keywords: \(K\)-theory completion
zbMATH Keywords: special unitary group
zbMATH Keywords: divisible groups
zbMATH Keywords: Lie group
MaRDI profile type: Publication
full work available at URL: j.topol.2015.09.005
OpenAlex ID: W2103542274
cites work: The unstable Adams spectral sequence for generalized homology
cites work: The Bousfield-Kan spectral sequence for periodic homology theories
cites work: Homotopy limits, completions and localizations
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cites work: The ν 1 -Periodic Homotopy Groups of SU(n ) at Odd Primes
cites work: Tractable formulas for v 1periodic homotopy groups of SU (n)when n≤p 2 -p + 1
cites work: Some calculations in the unstable Adams-Novikov spectral sequence
DOI: 10.1016/J.TOPOL.2015.09.005
description: scientific article; zbMATH DE number 6509434