#### Volume 16, issue 3 (2016)

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The homotopy types of $\mathrm{PU}(3)$– and $\mathrm{PSp}(2)$–gauge groups

### Sho Hasui, Daisuke Kishimoto, Akira Kono and Takashi Sato

Algebraic & Geometric Topology 16 (2016) 1813–1825
##### Abstract

Let $G$ be a compact connected simple Lie group. Any principal $G$–bundle over ${S}^{4}$ is classified by an integer $k\in ℤ\cong {\pi }_{3}\left(G\right)$, and we denote the corresponding gauge group by ${\mathsc{G}}_{k}\left(G\right)$. We prove that ${\mathsc{G}}_{k}\left(PU\left(3\right)\right)\simeq {\mathsc{G}}_{\ell }\left(PU\left(3\right)\right)$ if and only if $\left(24,k\right)=\left(24,\ell \right)$, and ${\mathsc{G}}_{k}\left(PSp\left(2\right)\right){\simeq }_{\left(p\right)}{\mathsc{G}}_{\ell }\left(PSp\left(2\right)\right)$ for any prime $p$ if and only if $\left(40,k\right)=\left(40,\ell \right)$, where $\left(m,n\right)$ is the gcd of integers $m,n$.

##### Keywords
gauge group, Samelson product
Primary: 55P35
Secondary: 55Q15