Geodesic
The homotopy types of PU(3)– and PSp(2)–gauge groups
Hasui, Sho1 ; Kishimoto, Daisuke1 ; Kono, Akira2 ; Sato, Takashi1
1Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
2Department of Mathematical Science, Faculty of Science and Engineering, Doshisha University, Kyoto 610-0394, Japan
Algebraic and Geometric Topology, Tome 16 (2016) no. 3, pp. 1813-1825
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Let G be a compact connected simple Lie group. Any principal G–bundle over S4 is classified by an integer k ∈ ℤ≅π3(G), and we denote the corresponding gauge group by Gk(G). We prove that Gk(PU(3)) ≃Gℓ(PU(3)) if and only if (24,k) = (24,ℓ), and Gk(PSp(2)) ≃(p)Gℓ(PSp(2)) for any prime p if and only if (40,k) = (40,ℓ), where (m,n) is the gcd of integers m,n.

DOI : 10.2140/agt.2016.16.1813
Classification : 55P35, 55Q15
Keywords: gauge group, Samelson product

Hasui, Sho  1   ; Kishimoto, Daisuke  1   ; Kono, Akira  2   ; Sato, Takashi  1

1 Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
2 Department of Mathematical Science, Faculty of Science and Engineering, Doshisha University, Kyoto 610-0394, Japan 
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Hasui, Sho; Kishimoto, Daisuke; Kono, Akira; Sato, Takashi. The homotopy types of PU(3)– and PSp(2)–gauge groups. Algebraic and Geometric Topology, Tome 16 (2016) no. 3, pp. 1813-1825. doi: 10.2140/agt.2016.16.1813

[1] J F Adams, Vector fields on spheres, Ann. of Math. (2) 75 (1962) 603

[2] M F Atiyah, R Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983) 523

[3] R Bott, A note on the Samelson product in the classical groups, Comment. Math. Helv. 34 (1960) 249

[4] D H Gottlieb, Applications of bundle map theory, Trans. Amer. Math. Soc. 171 (1972) 23

[5] H Hamanaka, A Kono, Unstable K1–group and homotopy type of certain gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 149

[6] S Kaji, D Kishimoto, Homotopy nilpotency in p–regular loop spaces, Math. Z. 264 (2010) 209

[7] Y Kamiyama, D Kishimoto, A Kono, S Tsukuda, Samelson products of SO(3) and applications, Glasg. Math. J. 49 (2007) 405

[8] D Kishimoto, Homotopy nilpotency in localized SU(n), Homology, Homotopy Appl. 11 (2009) 61

[9] D Kishimoto, A Kono, Mod p decompositions of non-simply connected Lie groups, J. Math. Kyoto Univ. 48 (2008) 1

[10] D Kishimoto, S D Theriault, M Tsutaya, The homotopy type of G2–gauge groups, preprint

[11] A Kono, A note on the homotopy type of certain gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 117 (1991) 295

[12] G E Lang Jr., The evaluation map and EHP sequences, Pacific J. Math. 44 (1973) 201

[13] M Mimura, On the number of multiplications on SU(3) and Sp(2), Trans. Amer. Math. Soc. 146 (1969) 473

[14] M Mimura, G Nishida, H Toda, Localization of CW–complexes and its applications, J. Math. Soc. Japan 23 (1971) 593

[15] M Mimura, H Toda, Homotopy groups of SU(3), SU(4) and Sp(2), J. Math. Kyoto Univ. 3 (1963/1964) 217

[16] E Rees, Embeddings of real projective spaces, Topology 10 (1971) 309

[17] S D Theriault, The homotopy types of Sp(2)–gauge groups, Kyoto J. Math. 50 (2010) 591

[18] G W Whitehead, On products in homotopy groups, Ann. of Math (2) 47 (1946) 460

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