Mod \(p\) decompositions of gauge groups
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.
scientific article; zbMATH DE number 6177484
scientific article