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On \(p\)-local homotopy types of gauge groups. (English) Zbl 1305.55005
The gauge group, \(\mathcal G P\), of a principal \(G\)-bundle, \(P\), is the topological group of \(G\)-equivariant self-maps of \(P\) covering the identity on the base space, \(K\). This paper is concerned with the determination of the \(p\)-local homotopy types of \(\mathcal G P\) as \(P\) ranges over all principal \(G\)-bundles with base \(K\) for a Lie group \(G\). In a previous paper, [Algebr. Geom. Topol. 13, No. 3, 1757–1778 (2013; Zbl 1276.57036)], the authors gave such a classification for principal \(\mathrm{SU}(n)\)-bundles over \(S^4\), with \(p\) an odd prime number such that \((p-1)(p-2)\geq n-1\).
In the paper under review, they generalize this situation to any simply connected, simple compact Lie group, \(G\), and any sphere \(S^{2d}\) as base space. Let \(\varepsilon\in\pi_{2d-1}(G)\), \(k\in {\mathbb Z}\) and \(P_{k}\) be the principal \(G\)-bundle classified by \(k\varepsilon\). The authors classify the \(p\)-local homotopy type of \(\mathcal G P\), determined by the divisibility of the classifying map by \(p\). One may observe that, when \(\varepsilon\) is of infinite order, “infinitely many principal bundles are divided into finite classes.” Finally, in the cases \(n\leq 2p-1\) and \(2\leq d\leq p-1\), the authors give a concrete classification of gauge groups for \(\mathrm{SU}(n)\)-bundles over \(S^{2d}\).

MSC:
55P15 Classification of homotopy type
55P60 Localization and completion in homotopy theory
81T13 Yang-Mills and other gauge theories in quantum field theory
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
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