Algebraic & Geometric Topology

Mod $p$ decompositions of gauge groups

Abstract

We give mod $p$ decompositions of homotopy types of the gauge groups of principal bundles over spheres, which are compatible with mod $p$ decompositions of Lie groups given by Mimura, Nishida and Toda. As an application, we also give some computations on the homotopy types of gauge groups. In particular, we show the $p$–local converse of the result of Sutherland on the classifications of the gauge groups of principal $SU ( n )$–bundles.

Article information

Source
Algebr. Geom. Topol., Volume 13, Number 3 (2013), 1757-1778.

Dates
Revised: 18 September 2012
Accepted: 1 February 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715599

Digital Object Identifier
doi:10.2140/agt.2013.13.1757

Mathematical Reviews number (MathSciNet)
MR3071142

Zentralblatt MATH identifier
1276.57036

Keywords
gauge group mod $p$ decomposition

Citation

Kishimoto, Daisuke; Kono, Akira; Tsutaya, Mitsunobu. Mod $p$ decompositions of gauge groups. Algebr. Geom. Topol. 13 (2013), no. 3, 1757--1778. doi:10.2140/agt.2013.13.1757. https://projecteuclid.org/euclid.agt/1513715599

References

• M F Atiyah, R Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983) 523–615
• M C Crabb, W A Sutherland, Counting homotopy types of gauge groups, Proc. London Math. Soc. 81 (2000) 747–768
• E D Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics 1622, Springer, Berlin (1996)
• E M Friedlander, Exceptional isogenies and the classifying spaces of simple Lie groups, Ann. Math. 101 (1975) 510–520
• H Hamanaka, On Samelson products in $p$-localized unitary groups, Topology Appl. 154 (2007) 573–583
• H Hamanaka, S Kaji, A Kono, Samelson products in ${\rm Sp}(2)$, Topology Appl. 155 (2008) 1207–1212
• H Hamanaka, A Kono, Unstable $K\sp 1$-group and homotopy type of certain gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 149–155
• H Hamanaka, A Kono, A note on Samelson products and mod $p$ cohomology of classifying spaces of the exceptional Lie groups, Topology Appl. 157 (2010) 393–400
• B Harris, On the homotopy groups of the classical groups, Ann. of Math. 74 (1961) 407–413
• Y Kamiyama, D Kishimoto, A Kono, S Tsukuda, Samelson products of $\rm SO(3)$ and applications, Glasg. Math. J. 49 (2007) 405–409
• D Kishimoto, A Kono, Note on mod $p$ decompositions of gauge groups, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010) 15–17
• D Kishimoto, A Kono, On the cohomology of free and twisted loop spaces, J. Pure Appl. Algebra 214 (2010) 646–653
• D Kishimoto, A Kono, Splitting of gauge groups, Trans. Amer. Math. Soc. 362 (2010) 6715–6731
• D Kishimoto, T Nagao, Commutativity in special unitary groups at odd primes, Topology Appl. 157 (2010) 1949–1954
• A Kono, A note on the homotopy type of certain gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 117 (1991) 295–297
• C A McGibbon, Homotopy commutativity in localized groups, Amer. J. Math. 106 (1984) 665–687
• M Mimura, G Nishida, H Toda, ${\rm Mod}\ p$ decomposition of compact Lie groups, Publ. Res. Inst. Math. Sci. 13 (1977/78) 627–680
• W A Sutherland, Function spaces related to gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 121 (1992) 185–190
• S D Theriault, The odd primary $H$-structure of low rank Lie groups and its application to exponents, Trans. Amer. Math. Soc. 359 (2007) 4511–4535
• S D Theriault, The homotopy types of $\rm Sp(2)$-gauge groups, Kyoto J. Math. 50 (2010) 591–605
• S D Theriault, Odd primary homotopy decompositions of gauge groups, Algebr. Geom. Topol. 10 (2010) 535–564
• H Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies 49, Princeton Univ. Press (1962)
• G W Whitehead, On products in homotopy groups, Ann. of Math 47 (1946) 460–475
• C Wilkerson, Self-maps of classifying spaces, from: “Localization in group theory and homotopy theory, and related topics”, (P Hilton, editor), Lecture Notes in Math. 418, Springer, Berlin (1974) 150–157
• C Wilkerson, Genus and cancellation, Topology 14 (1975) 29–36