Note on mod \(p\) decompositions of gauge groups. (English) Zbl 1190.55007

Summary: We give fibrewise mod \(p\) decompositions of the adjoint bundle of a principal \(G\)-bundle \(P\) when the topological group \(G\) has mod \(p\) decompositions by automorphisms as in [B. Harris, Ann. Math. (2) 74, 407–413 (1961; Zbl 0118.18501)], which imply mod \(p\) decompositions of the gauge group of \(P\).


55R70 Fibrewise topology
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
55R10 Fiber bundles in algebraic topology


Zbl 0118.18501
Full Text: DOI


[1] M. F. Atiyah and R. Bott, The Yangmhy Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523-615. · Zbl 0509.14014
[2] A. Borel and J.-P. Serre, Groupes de Lie et puissances réduites de Steenrod, Amer. J. Math. 75 (1953), 409-448. · Zbl 0050.39603
[3] M. Crabb and I. James, Fibrewise homotopy theory , Springer, London, 1998. · Zbl 0905.55001
[4] J. A. Wolf and A. Gray, Homogeneous spaces defined by Lie group automorphisms. I, J. Differential Geometry 2 (1968), 77-114. · Zbl 0169.24103
[5] B. Harris, On the homotopy groups of the classical groups, Ann. of Math. (2) 74 (1961), 407-413. · Zbl 0118.18501
[6] P. Hilton, G. Mislin and J. Roitberg, Localization of nilpotent groups and spaces , North-Holland, Amsterdam, 1975. · Zbl 0323.55016
[7] D. Kishimoto and A. Kono, Splitting of gauge groups, Trans. Amer. Math. Soc. (to appear). · Zbl 1210.57034
[8] A. Kono and S. Tsukuda, Note on the triviality of adjoint bundles, Contemp. Math. (to appear). · Zbl 1209.55004
[9] J. P. May, Fibrewise localization and completion , Trans. Amer. Math. Soc. 258 (1980), no. 1, 127-146. · Zbl 0429.55004
[10] J. M. M\oller, Nilpotent spaces of sections , Trans. Amer. Math. Soc. 303 (1987), no. 2, 733-741. · Zbl 0628.55007
[11] S.D. Theriault, Odd primary homotopy decompositions of gauge groups. (Preprint). · Zbl 1196.55009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.