Homotopy type of gauge groups of quaternionic line bundles over spheres
type: Item
instance of: scholarly article
title: Homotopy type of gauge groups of quaternionic line bundles over spheres
zbMATH Open document ID: 1163.55004
author: M. H. Andrade Claudio
author: M. Spreafico
published in: Topology and its Applications
publication date: 2009-03-02T00:00:00Z
review text: The authors study the homotopy types of groups of self-equivalences (gauge groups) \(\mathcal{G}_{f}\) of principal \(S^3\)-bundles \(P_{f}\) over \(S^{n}\) by methods introduced in [\textit{A. Kono}, Proc. R. Soc. Edinb., Sect. A 117, No. 3--4, 295--297 (1991; Zbl 0722.55008)]. Those bundles are classified by elements \([f]\in\pi_{n-1}(S^{3})\) and the authors use detailed knowledge on the low-dimensional homotopy groups of \(S^{3}\), mostly covered by \textit{H. Toda} [Composition methods in homotopy groups of spheres. Annals of Mathematics Studies. 49. Princeton, N. J.: Princeton University Press (1962; Zbl 0101.40703)]. In particular, they derive a formula for the connecting map in the homotopy sequence of the evaluation fibration \(\mathcal{G}_f\to S^{3}\). From this formula they draw their descriptions of the homotopy types of \(\mathcal{G}_{f}\) in the range \(5 \leq n \leq 25\) (the case \(n = 21\) the authors are not able to solve, they rather formulate a conjecture). The typical statement they give is like ``On \(S^n\) there are \dots principal \(S^{3}\)-bundles. The gauge groups of \(P_f\) and \(P_{f'}\) have the same homotopy type if and only if \([f ]\) and \([f']\) satisfy the relation\dots'' (where the dots have to filled out according to \(n\)).
reviewed by: Christoph Wockel
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 55R10
zbMATH DE Number: 5522082
zbMATH Keywords: gauge groups
zbMATH Keywords: homotopy type
MaRDI profile type: Publication
full work available at URL: j.topol.2008.08.016
OpenAlex ID: W1979566072
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DOI: 10.1016/J.TOPOL.2008.08.016
description: scientific article; zbMATH DE number 5522082