Kyoto Journal of Mathematics

The homotopy types of Sp(2)-gauge groups

Stephen D. Theriault

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There are countably many equivalence classes of principal Sp(2)-bundles over S4, classified by the integer value second Chern class. We show that the corresponding gauge groups Gk have the property that if there is a homotopy equivalence GkGk, then (40,k)=(40,k), and we prove a partial converse by showing that if (40,k)=(40,k), then Gk and Gk are homotopy equivalent when localized rationally or at any prime.

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Kyoto J. Math., Volume 50, Number 3 (2010), 591-605.

First available in Project Euclid: 11 August 2010

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Zentralblatt MATH identifier

Primary: 54C35: Function spaces [See also 46Exx, 58D15] 55P15: Classification of homotopy type


Theriault, Stephen D. The homotopy types of $\operatorname{Sp}(2)$ -gauge groups. Kyoto J. Math. 50 (2010), no. 3, 591--605. doi:10.1215/0023608X-2010-005.

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  • [AB] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523–615.
  • [B] R. Bott, A note on the Samelson product in the classical groups, Comment. Math. Helv. 34 (1960), 245–256.
  • [CHM] Y. Choi, Y. Hirato, and M. Mimura, Composition methods and homotopy types of the gauge groups of Sp(2) and SU(3), Bull. Belg. Math. Soc. Simon Stevin 15 (2008), 409–417.
  • [CS] M. C. Crabb and W. A. Sutherland, Counting homotopy types of gauge groups, Proc. London Math. Soc. 83 (2000), 747–768.
  • [HKK] H. Hamanaka, S. Kaji, and A. Kono, Samelson products in Sp(2), Topology Appl. 155 (2008), 1207–1212.
  • [HK] H. Hamanaka and A. Kono, Unstable K1-group and homotopy type of certain gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 149–155.
  • [H] J. R. Harper, Secondary Cohomology Operations, Grad. Studies in Math. 49, Amer. Math. Soc., Providence, 2002.
  • [KKKT] Y. Kamiyama, D. Kishimoto, A. Kono, and S. Tsukuda, Samelson products of SO(3) and applications, Glasg. Math. J. 49 (2007), 405–409.
  • [K] A. Kono, A note on the homotopy type of certain gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 117 (1991), 295–297.
  • [KT] A. Kono and S. Tsukuda, A remark on the homotopy type of certain gauge groups, J. Math. Kyoto Univ. 36 (1996), 115–121.
  • [L] G. E. Lang, The evaluation map and EHP sequences, Pacific J. Math. 44 (1973), 201–210.
  • [Mc] C. A. McGibbon, Homotopy commutativity in localized groups, Amer. J. Math. 106 (1984), 665–687.
  • [M] M. Mimura, On the number of multiplications on SU(3) and Sp(2), Trans. Amer. Math. Soc. 146 (1969), 473–492.
  • [MT] M. Mimura and H. Toda, Homotopy groups of SU(3), SU(4), and Sp(2), J. Math. Kyoto Univ. 3 (1963/1964), 217–250.
  • [S] W. A. Sutherland, Function spaces related to gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 121 (1992), 185–190.
  • [T] S. D. Theriault, Odd primary homotopy decompositions of gauge groups, Algebr. Geom. Topol. 10 (2010), 535–564.