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Mod \(p\) decompositions of gauge groups. (English) Zbl 1276.57036
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 M. Mimura, G. Nishida and 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.

MSC:
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
55R70 Fibrewise topology
54C35 Function spaces in general topology
55P15 Classification of homotopy type
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