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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
##### Keywords:
gauge group; mod $$p$$ decomposition
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