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Odd primary homotopy decompositions of gauge groups. (English) Zbl 1196.55009
Given a principal \(G\)-bundle \(P \to X\), the gauge group \(G(P)\) is the group of equivariant automorphisms of \(P\) over \(X\). The problem of determining the homotopy types of gauge groups has been of interest for some time. The rational homotopy type was found in [Y. Félix and J. Oprea, Proc. Am. Math. Soc. 137, No. 4, 1519–1527 (2009; Zbl 1168.55010)]. Here, for certain specific bundles, the author generalizes the rational decomposition to a \(p\)-local decomposition. In particular, the author achieves this for \(SU(n)\), \(Sp(n)\) and \(Spin(n)\)-bundles over simply connected \(4\)-manifolds and for \(U(n)\)-bundles over compact orientable Riemann surfaces when the prime \(p\) is restricted in specified ways. A main tool for all results is the spatial decomposition of a (\(p\)-localized) group \(G\) corresponding to the well-known algebra decomposition \(H_*(G) \cong \otimes_{i=1}^{p-1} \Lambda(V_i)\), where \(V_i\) consists of generators constrained by degree in terms of \(p\).

MSC:
55P35 Loop spaces
55R10 Fiber bundles in algebraic topology
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[1] M F Atiyah, R Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983) 523 · Zbl 0509.14014
[2] A Borel, F Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958) 458 · Zbl 0097.36401
[3] R Bott, A note on the Samelson product in the classical groups, Comment. Math. Helv. 34 (1960) 249 · Zbl 0094.01503
[4] F R Cohen, J A Neisendorfer, A construction of \(p\)-local \(H\)-spaces (editors I Madsen, B Oliver), Lecture Notes in Math. 1051, Springer (1984) 351 · Zbl 0582.55010
[5] R L Cohen, R J Milgram, The homotopy type of gauge-theoretic moduli spaces (editors G E Carlsson, R L Cohen, W C Hsiang, J D S Jones), Math. Sci. Res. Inst. Publ. 27, Springer (1994) 15 · Zbl 0799.57019
[6] M C Crabb, Fibrewise homology, Glasg. Math. J. 43 (2001) 199 · Zbl 0987.55016
[7] M C Crabb, W A Sutherland, Counting homotopy types of gauge groups, Proc. London Math. Soc. \((3)\) 81 (2000) 747 · Zbl 1024.55005
[8] S K Donaldson, Connections, cohomology and the intersection forms of \(4\)-manifolds, J. Differential Geom. 24 (1986) 275 · Zbl 0635.57007
[9] Y Félix, J Oprea, Rational homotopy of gauge groups, Proc. Amer. Math. Soc. 137 (2009) 1519 · Zbl 1168.55010
[10] E M Friedlander, Exceptional isogenies and the classifying spaces of simple Lie groups, Ann. Math. \((2)\) 101 (1975) 510 · Zbl 0308.55016
[11] D H Gottlieb, Applications of bundle map theory, Trans. Amer. Math. Soc. 171 (1972) 23 · Zbl 0251.55018
[12] B Gray, Homotopy commutativity and the \(EHP\) sequence (editors M Mahowald, S Priddy), Contemp. Math. 96, Amer. Math. Soc. (1989) 181 · Zbl 0698.55011
[13] J Grbić, S D Theriault, Odd primary self-maps of low rank Lie groups, Canad. J. Math 62 (2010) 284 · Zbl 1200.55014
[14] H Hamanaka, A Kono, Unstable \(K^1\)-group and homotopy type of certain gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 149 · Zbl 1103.55004
[15] B Harris, On the homotopy groups of the classical groups, Ann. of Math. \((2)\) 74 (1961) 407 · Zbl 0118.18501
[16] P Hilton, G Mislin, J Roitberg, Localization of nilpotent groups and spaces, North-Holland Math. Studies 15, North-Holland Publishing Co. (1975) · Zbl 0323.55016
[17] I M James, The topology of Stiefel manifolds, London Math. Soc. Lecture Note Ser. 24, Cambridge Univ. Press (1976) · Zbl 0337.55017
[18] D Kishimoto, A Kono, Note on mod \(p\) decompositions of gauge groups, Proc. Japan Acad. Ser. A 86 (2010) 15 · Zbl 1190.55007
[19] A Kono, A note on the homotopy type of certain gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 117 (1991) 295 · Zbl 0722.55008
[20] A Kono, S Tsukuda, A remark on the homotopy type of certain gauge groups, J. Math. Kyoto Univ. 36 (1996) 115 · Zbl 0865.57018
[21] G E Lang Jr., The evaluation map and \rmEHP sequences, Pacific J. Math. 44 (1973) 201 · Zbl 0248.55016
[22] G Masbaum, On the cohomology of the classifying space of the gauge group over some \(4\)-complexes, Bull. Soc. Math. France 119 (1991) 1 · Zbl 0729.57006
[23] M Mimura, G Nishida, H Toda, \(\mathrm{Mod} p\) decomposition of compact Lie groups, Publ. Res. Inst. Math. Sci. 13 (1977) 627 · Zbl 0383.22007
[24] M Mimura, H Toda, Cohomology operations and homotopy of compact Lie groups. I, Topology 9 (1970) 317 · Zbl 0204.23803
[25] W A Sutherland, Function spaces related to gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 121 (1992) 185 · Zbl 0761.55007
[26] S Terzić, The rational topology of gauge groups and of spaces of connections, Compos. Math. 141 (2005) 262 · Zbl 1068.55013
[27] S D Theriault, The \(H\)-structure of low-rank torsion free \(H\)-spaces, Q. J. Math. 56 (2005) 403 · Zbl 1158.55300
[28] 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 · Zbl 1117.55005
[29] H Toda, A topological proof of theorems of Bott and Borel-Hirzebruch for homotopy groups of unitary groups, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math. 32 (1959) 103 · Zbl 0106.16403
[30] H Toda, On iterated suspensions. I, II, J. Math. Kyoto Univ. 5 (1966) 87, 209 · Zbl 0149.40801
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