Mod \(p\) decompositions of gauge groups
type: Item
instance of: scholarly article
title: Mod \(p\) decompositions of gauge groups
zbMATH Open document ID: 1276.57036
author: Daisuke Kishimoto
author: Akira Kono
author: Mitsunobu Tsutaya
published in: Algebraic \& Geometric Topology
publication date: 2013-06-20T00:00:00Z
review text: Let \(G\) be a topological group and let \(P\) be a principal \(G\)-bundle over a space \(K\). The gauge group \({\mathcal G}(P)\) of \(P\) is the group of all automorphisms of \(P\) equipped with the compact-open topology, where automorphisms of \(P\) are \(G\)-equivariant self-maps of \(P\) covering the identity map of \(K\). As the authors remark, one naively expects that \({\mathcal G}(P)\) inherits properties of \(G\), and they mention cases where this is true. But what is the situation if one localizes at an odd prime \(p\)? If \(G\) is a compact, simply connected, simple Lie group with integral homology \(p\)-torsion free, a theorem of \textit{M. Mimura, G. Nishida} and \textit{H. Toda} [Publ. Res. Inst. Math. Sci., Kyoto Univ. 13, 627--680 (1977; Zbl 0383.22007)] shows that the \(p\)-localization of \(G\) is homotopy equivalent to a product of smaller spaces -- called a mod \(p\) decomposition of \(G\). In the present paper, the authors first give a proof of the theorem af Mimura, Nishida and Toda, in a form suitable for the proof of their own main theorem, which gives a mod \(p\) decomposition of the \(p\)-localization of the gauge group \({\mathcal G}(P)\) for a principle \(G\)-bundle over a sphere, with dimension restrictions on the sphere and for a restricted class of Lie groups.
reviewed by: Vagn Lundsgaard Hansen
Mathematics Subject Classification ID: 57S05
Mathematics Subject Classification ID: 55R70
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 55P15
zbMATH DE Number: 6177484
zbMATH Keywords: gauge group
zbMATH Keywords: mod \(p\) decomposition
MaRDI profile type: Publication
full work available at URL: agt.2013.13.1757
OpenAlex ID: W2052345001
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DOI: 10.2140/AGT.2013.13.1757
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description: scientific article; zbMATH DE number 6177484