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The homotopy types of $$Sp(3)$$-gauge groups. (English) Zbl 1383.55005
Let $$G$$ be a simple, simply connected compact Lie group and let $$BG$$ denote its classifying space. Let $$P_k$$ denote the principal $$G$$-bundle over $$S^4$$ with the second Chern number $$k\in \mathbb{Z}\cong [S^4,BG]$$ and let $$\mathcal{G}_k$$ denote the gauge group of the principal $$G$$-bundle $$P_k$$. It is known that the gauge groups $$\{\mathcal{G}_k\}_{k\in \mathbb{Z}}$$ have only finitely many distinct homotopy types.
In this paper, the author studies the homotopy type classification problem of the gauge groups $$\{\mathcal{G}_k\}_{k\in \mathbb{Z}}$$ when $$G=SP(3)$$. More precisely, he proves that if there is a homotopy equivalence $$\mathcal{G}_k\simeq \mathcal{G}_{l}$$ then $$(84,k)=(84,l)$$, where $$(a,b)$$ denotes the greatest common divisor of $$| a|$$ and $$| b|$$ for integers $$a,b\in \mathbb{Z}$$. Moreover, he shows that if $$(336,k)=(336,l)$$, then there is a $$p$$-local homotopy equivalence $$\mathcal{G}_k\simeq_p \mathcal{G}_{l}$$ for any prime $$p$$ or for $$p=0$$.
As an application he also obtains that for any odd prime $$p\geq 3$$, there is a $$p$$-local homotopy equivalence $$\mathcal{G}_k\simeq_p\mathcal{G}_l$$ iff $$(21,k)=(21,l)$$. The proof is based on the calculation of the order of the Samelson product $$Sp(1)\wedge Sp(3)\mathop{\longrightarrow}\limits^{\langle \iota,1\rangle}Sp(3)$$ and the fact that there is a homotopy equivalence $$B\mathcal{G}_k\simeq \mathrm{Map}_k(S^4,BG)$$.

##### MSC:
 55P15 Classification of homotopy type 55P10 Homotopy equivalences in algebraic topology 55P35 Loop spaces 55Q15 Whitehead products and generalizations 54C35 Function spaces in general topology 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology
##### Keywords:
gauge group; homotopy type; Samelson product; principal bundle
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