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The homotopy types of \(Sp(3)\)-gauge groups. (English) Zbl 1383.55005
Let \(G\) be a simple, simply connected compact Lie group and let \(BG\) denote its classifying space. Let \(P_k\) denote the principal \(G\)-bundle over \(S^4\) with the second Chern number \(k\in \mathbb{Z}\cong [S^4,BG]\) and let \(\mathcal{G}_k\) denote the gauge group of the principal \(G\)-bundle \(P_k\). It is known that the gauge groups \(\{\mathcal{G}_k\}_{k\in \mathbb{Z}}\) have only finitely many distinct homotopy types.
In this paper, the author studies the homotopy type classification problem of the gauge groups \(\{\mathcal{G}_k\}_{k\in \mathbb{Z}}\) when \(G=SP(3)\). More precisely, he proves that if there is a homotopy equivalence \(\mathcal{G}_k\simeq \mathcal{G}_{l}\) then \((84,k)=(84,l)\), where \((a,b)\) denotes the greatest common divisor of \(| a|\) and \(| b|\) for integers \(a,b\in \mathbb{Z}\). Moreover, he shows that if \((336,k)=(336,l)\), then there is a \(p\)-local homotopy equivalence \(\mathcal{G}_k\simeq_p \mathcal{G}_{l}\) for any prime \(p\) or for \(p=0\).
As an application he also obtains that for any odd prime \(p\geq 3\), there is a \(p\)-local homotopy equivalence \(\mathcal{G}_k\simeq_p\mathcal{G}_l\) iff \((21,k)=(21,l)\). The proof is based on the calculation of the order of the Samelson product \(Sp(1)\wedge Sp(3)\mathop{\longrightarrow}\limits^{\langle \iota,1\rangle}Sp(3)\) and the fact that there is a homotopy equivalence \(B\mathcal{G}_k\simeq \mathrm{Map}_k(S^4,BG)\).

MSC:
55P15 Classification of homotopy type
55P10 Homotopy equivalences in algebraic topology
55P35 Loop spaces
55Q15 Whitehead products and generalizations
54C35 Function spaces in general topology
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
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[1] Bonora, L.; Schnabl, M.; Sheikh-Jabbari, M. M.; Tomasiello, A., Noncommutative SO(n) and sp(n) gauge theories, Nucl. Phys. B, 589, 461, (2000) · Zbl 1060.81607
[2] Bott, R., A note on the Samelson product in the classical groups, Comment. Math. Helv., 34, 245-256, (1960) · Zbl 0094.01503
[3] Cooke, G.; Smith, L., Mod p decompositions of co-H-spaces and applications, Math. Z., 157, 155-177, (1977) · Zbl 0386.55013
[4] Crabb, M.; Sutherland, W., Counting homotopy types of gauge groups, Proc. Lond. Math. Soc., 83, 747-768, (2000) · Zbl 1024.55005
[5] Gottlieb, D., On fibre spaces and the evaluation map, Ann. Math., 87, 42-55, (1968) · Zbl 0173.25901
[6] Hamanaka, H.; Kono, A., Unstable \(K^1\)-group \([\operatorname{\Sigma}^{2 n - 2} \mathbb{C} P^2, U(n)]\) and homotopy type of certain gauge groups, Proc. R. Soc. Edinb. Sect. A, 136, 149-155, (2006)
[7] Husemoller, D., Fibre bundles, (1994), Springer-Verlag New York
[8] James, I., On Lie groups and their homotopy groups, Proc. Camb. Philol. Soc., 55, 244-247, (1959) · Zbl 0093.03304
[9] James, I.; Whitehead, J., The homotopy theory of sphere-bundles over spheres I, Proc. Lond. Math. Soc., 4, 196-218, (1954) · Zbl 0056.16703
[10] Kono, A., A note on the homotopy type of certain gauge groups, Proc. R. Soc. Edinb. Sect. A, 117, 295-297, (1991) · Zbl 0722.55008
[11] Kono, A.; Tsukuda, S., A remark on the homotopy type of certain gauge groups, J. Math. Kyoto Univ., 36, 115-121, (1996) · Zbl 0865.57018
[12] Lang, G., The evaluation map and EHP sequences, Pac. J. Math., 44, 201-210, (1973) · Zbl 0217.20003
[13] Lin, J., H-spaces with finiteness conditions, (1995), North Holland · Zbl 0941.55002
[14] Miller, H., Stable splittings of Stiefel manifolds, Topology, 24, 411-419, (1985) · Zbl 0581.55006
[15] Mimura, M., On the number of multiplications on SU(3) and sp(2), Transl. Am. Math. Soc., 146, 473-492, (1969) · Zbl 0198.56204
[16] Mimura, M.; Nishida, G.; Toda, H., Mod p decompositions of compact Lie groups, Publ. Res. Inst. Math. Sci., Kyoto Univ., 13, 627-680, (1977) · Zbl 0383.22007
[17] Mimura, M.; Toda, H., Homotopy groups of SU(3), SU(4) and sp(2), J. Math. Kyoto Univ., 3, 217-250, (1963)
[18] Mimura, M.; Toda, H., Homotopy groups of symplectic groups, J. Math. Kyoto Univ., 3, 251-273, (1963) · Zbl 0129.15405
[19] Mimura, M.; Toda, H., A representation and the quasi p-regularity of the compact Lie group, Jpn. J. Math. (N.S.), 1, 101-109, (1975) · Zbl 0317.57020
[20] Oguchi, K., Generators of 2-primary components of homotopy groups of spheres, unitary groups and symplectic groups, J. Fac. Sci., Univ. Tokyo, 11, 65-111, (1964) · Zbl 0129.38902
[21] Sutherland, D., Function spaces related to gauge groups, Proc. R. Soc. Edinb. Sect. A, 121, 185-190, (1992) · Zbl 0761.55007
[22] Theriault, S., The homotopy types of sp(2)-gauge groups, Kyoto J. Math., 50, 591-605, (2010) · Zbl 1202.55004
[23] Theriault, S., The homotopy types of SU(5)-gauge groups, Osaka J. Math., 52, 15-31, (2015) · Zbl 1315.55005
[24] Toda, H., Composition methods in homotopy groups of spheres, Ann. Math. Stud., vol. 49, (1962), Princeton University Press Princeton N.J. · Zbl 0101.40703
[25] Toda, H., On iterated suspensions, I, J. Math. Kyoto Univ., 5, 87-142, (1966) · Zbl 0149.40801
[26] Witten, E., Current algebra, baryons, and quark confinement, Nucl. Phys. B, 223, 2, 433-444, (1983)
[27] Yamaguchi, A., On the p-regularity of Stiefel manifolds, Publ. Res. Inst. Math. Sci., Kyoto Univ., 25, 355-380, (1989) · Zbl 0696.55015
[28] Yokota, I., On the cells of symplectic groups, Proc. Jpn. Acad., 32, 399-400, (1956) · Zbl 0070.25804
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