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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

Let G be a compact connected simple Lie group. Any principal G–bundle over S4 is classified by an integer k π3(G), and we denote the corresponding gauge group by Gk(G). We prove that Gk(PU(3)) G(PU(3)) if and only if (24,k) = (24,), and Gk(PSp(2)) (p)G(PSp(2)) for any prime p if and only if (40,k) = (40,), where (m,n) is the gcd of integers m,n.

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gauge group, Samelson product
Mathematical Subject Classification 2010
Primary: 55P35
Secondary: 55Q15
Received: 23 June 2015
Revised: 29 September 2015
Accepted: 11 December 2015
Published: 1 July 2016
Sho Hasui
Department of Mathematics
Kyoto University
Kyoto 606-8502
Daisuke Kishimoto
Department of Mathematics
Kyoto University
Kyoto 606-8502
Akira Kono
Department of Mathematical Science, Faculty of Science and Engineering
Doshisha University
Kyoto 610-0394
Takashi Sato
Department of Mathematics
Kyoto University
Kyoto 606-8502