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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
##### Keywords:
principal bundle; gauge group; unitary group
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