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Refined gauge group decompositions. (English) Zbl 1321.55007
Let \(G\) be a simple, simply connected, compact Lie group and let \(P_{k}\to S^4 \) be the principal \(G\)-bundle whose second Chern class has value \(k\). In the paper under review, the authors give a \(p\)-local homotopy decomposition of the gauge group, \({\mathcal{G}}_{k}(G)\), of \(P_{k}\), when \(G\) is a matrix group and \(p\) is an odd prime. Their main result consists of the following homotopy equivalences, for \(n\geq 2\) and any \(k\in {\mathbb Z}\), \[ {\mathcal{G}}_{k}({\mathrm SU}(n)) \simeq \prod_{i=1,i\neq v-1,v}^{p-1} (B_{i}\times \Omega^4_{0}B_{i+2})\times X_{v-1}\times X_{v}. \] The spaces \(B_{\bullet}\) come from a \(p\)-local homotopy equivalence, \(G\simeq \prod_{i=1}^{p-1} B_{i}\), established by M. Mimura et al. [Publ. Res. Inst. Math. Sci. 13, 627–680 (1977; Zbl 0383.22007)]. Also, there is a homotopy fibration, \(\Omega^4_{0}B_{t+2}\to X_{t}\to B_{t}\), for \(t\in\{v-1,v\}\). A similar decomposition of \({\mathcal{G}}_{k}({\mathrm Sp}(n))\) is also provided.

MSC:
55P35 Loop spaces
54C35 Function spaces in general topology
81T13 Yang-Mills and other gauge theories in quantum field theory
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References:
[1] R. Bott, A note on the Samelson product in the classical groups , Comment. Math. Helv. 34 (1960), 249-256. · Zbl 0094.01503
[2] C. P. Boyer, B. M. Mann, and D. Waggonner, On the homology of \(\operatorname{SU} (n)\) instantons , Trans. Amer. Math. Soc. 323 (1991), no. 2, 529-561. · Zbl 0725.53065
[3] D. H. Gottlieb, Applications of bundle map theory , Trans. Amer. Math. Soc. 171 (1972), 23-50. · Zbl 0251.55018
[4] H. Hamanaka and A. Kono, Unstable \(K^{1}\)-group and homotopy type of certain gauge groups , Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 149-155. · Zbl 1103.55004
[5] B. Harris, On the homotopy groups of the classical groups , Ann. of Math. (2) 74 (1961), 407-413. · Zbl 0118.18501
[6] D. Kishimoto, Generating varieties, Bott periodicity and instantons , Topology Appl. 157 (2010), 657-668. · Zbl 1184.81070
[7] D. Kishimoto and A. Kono, Note on mod-\(p\) decompositions of gauge groups , Proc. Japan Acad. Ser. A. Math. Sci. 86 (2010), 15-17. · Zbl 1190.55007
[8] D. Kishimoto, A. Kono, and M. Tsutaya, Mod-\(p\) decompositions of gauge groups , Algebr. Geom. Topol. 13 (2013), 1757-1778. · Zbl 1276.57036
[9] A. Kono, A note on the homotopy type of certain gauge groups , Proc. Roy. Soc. Edinburgh Sect. A 117 (1991), 295-297. · Zbl 0722.55008
[10] G. E. Lang, The evaluation map and \(EHP\) sequences , Pacific J. Math. 44 (1973), 201-210. · Zbl 0217.20003
[11] M. Mimura, G. Nishida, and H. Toda, Mod-\(p\) decomposition of compact Lie groups , Publ. Res. Inst. Math. Sci. 13 (1977/78), 627-680. · Zbl 0383.22007
[12] S. D. Theriault, Odd primary decompositions of gauge groups , Algebr. Geom. Topol. 10 (2010), 535-564. · Zbl 1196.55009
[13] S.D. Theriault, The homotopy types of \(\operatorname{Sp} (2)\)-gauge groups , Kyoto J. Math. 50 2010, 591-605. · Zbl 1202.55004
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