×

The homotopy types of \(PSp(n)\)-gauge groups over \(S^{2m}\). (English) Zbl 1465.55001

Given a topological group \(G\), the isomorphism classes of principal \(G\)-bundles over \(S^n\) are classified by elements in \(\pi_{n-1}(G)\). In particular, when (1) \(G=PSp(2)\) and \(n=8\) or (2) \(G=PSp(3)\) and \(n=4\), \(\pi_{n-1}(G)\cong\mathbb{Z}\). Therefore any principal \(G\)-bundle \(P\) is classified by an integer \(k\) in these two cases. The gauge group of \(P\) is the topological group consisting of \(G\)-equivariant automorphisms of \(P\) that fix \(S^n\), and is denoted by \(\mathcal{G}_k(S^n,G)\). In this paper the author shows the following theorem:
Theorem: Let \((k,l)\) be the greatest common divisor of the integers \(k\) and \(l\).
\(\bullet\) If \(\mathcal{G}_k(PSp(2),S^8)\simeq\mathcal{G}_l(PSp(2),S^8)\), then \((140,k)=(140,l)\);
\(\bullet\) if \((140,k)=(140,l)\), then \(\Omega\mathcal{G}_k(PSp(2),S^8)\simeq\Omega\mathcal{G}_l(PSp(2),S^8)\);
\(\bullet\) if \(\mathcal{G}_k(PSp(3),S^4)\simeq\mathcal{G}_l(PSp(3),S^4)\), then \((84,k)=(84,l)\);
\(\bullet\) if \((672,k)=(672,l)\), then \(\Omega\mathcal{G}_k(PSp(3),S^4)\simeq\Omega\mathcal{G}_l(PSp(3),S^4)\) after localization at any prime.
In Section 2 the author gives the background of his method. According to [M. F. Atiyah and R. Bott, Philos. Trans. R. Soc. Lond., Ser. A 308, 523–615 (1983; Zbl 0509.14014); D. H. Gottlieb, Trans. Am. Math. Soc. 171, 23–50 (1972; Zbl 0251.55018)], the classifying space of the gauge group \(B\mathcal{G}_k(S^n,G)\) is homotopy equivalent to the connected component of \(\text{Map}(S^n, BG)\) that contains \(k\epsilon\). Consider the homotopy fibration sequence \[ G\overset{\alpha_k}{\longrightarrow}\Omega^{n-1}_0G\longrightarrow B\mathcal{G}(P_k)\overset{ev}{\longrightarrow}BG, \] where \(ev\) is the evaluation map at the base point, \(\Omega^{n-1}_0G\) is the connected component of \(\Omega^{n-1}G\) that contains the identity and \(\alpha_k\) is a connecting map. The adjoint of \(\alpha_k\) is the Samelson product \(k\langle{\epsilon,id_G}\rangle:S^{n-1}\wedge G\to G\), where \(\epsilon:S^n\to BG\) is a map representing the generator of \(\pi_{n-1}(BG)\cong\mathbb{Z}\) and \(id_G\) is the identity map on \(G\). We define the order of \(\langle{\epsilon,id}\rangle\) to be the minimum positive integer \(m\) such that \(m\langle{\epsilon,id_G}\rangle\) is null homotopic. It is known that the greatest common divisor \((k,m)\) can essentially determine the homotopy type of \(\mathcal{G}_k(S^n,G)\). In Section 3 the author calculates the order \(m\) of \(\langle{\epsilon,id_G}\rangle\) for \(\mathcal{G}_k(PSp(2),S^8)\) and in Section 4 calculates \(m\) for \(\mathcal{G}_k(PSp(3),S^4)\).
In the \(PSp(2)\) case, let \(\epsilon:S^7\to PSp(2)\) and \(\overline{\epsilon}:S^7\to Sp(2)\) be maps representing the generators of \(\pi_7(PSp(3))\) and \(\pi_7(Sp(3))\). Using the fibration sequence \[ Sp(2)\to PSp(2)\to B\mathbb{Z}/2\mathbb{Z} \] the author shows that the orders of \(\langle\overline{\epsilon},id_{Sp(2)}\rangle\) and \(\langle\epsilon,id_{PSp(2)}\rangle\) are the same. By [H. Hamanaka et al., Topology Appl. 155, No. 11, 1207–1212 (2008; Zbl 1144.55014)] the order of \(\langle\overline{\epsilon},id_{Sp(2)}\rangle\) is 140, so the author obtains the first two statements of his theorem. In the \(PSp(3)\) case, the author uses the same method as [T. Cutler, Topology Appl. 236, 44–58 (2018; Zbl 1383.55005)] to obtain the last two statements of the theorem.

MSC:

55P15 Classification of homotopy type
54C35 Function spaces in general topology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Atiyah, M. F.; Bott, R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. R. Soc. Lond. Ser. A, 308, 523-615 (1983) · Zbl 0509.14014
[2] Cutler, T., The homotopy type of \(S p(3)\)-gauge groups, Topol. Appl., 236, 44-58 (2018) · Zbl 1383.55005
[3] Crabb, M. C.; Sutherland, W. A., Counting homotopy types of gauge groups, Proc. Lond. Math. Soc., 83, 747-768 (2000) · Zbl 1024.55005
[4] Hamanaka, H.; Kono, A., Unstable \(K^1\)-group and homotopy type of certain gauge groups, Proc. R. Soc. Edinb., Sect. A, 136, 149-155 (2006) · Zbl 1103.55004
[5] Hamanaka, H.; Kono, A., Homotopy type of gauge groups of \(S U(3)\)-bundles over \(S^6\), Topol. Appl., 154, 1377-1380 (2007) · Zbl 1120.55006
[6] Hamanaka, H.; Kaji, S.; Kono, A., Samelson products in \(S p(2)\), Topol. Appl., 155, 1207-1212 (2008) · Zbl 1144.55014
[7] Hasui, S.; Kishimoto, D.; Kono, A.; Sato, T., The homotopy types of \(P U(3)\)- and \(P S p(2)\)-gauge groups, Algebraic Geom. Topol., 16, 1813-1825 (2016) · Zbl 1352.55005
[8] Lang, G. E., The evaluation map and EHP sequences, Pac. J. Math., 44, 201-210 (1973) · Zbl 0217.20003
[9] Mohammadi, S.; Asadi-Golmankhaneh, M. A., The homotopy types of \(S U(4)\)-gauge groups over \(S^8\), Topol. Appl., 266, Article 106845 pp. (2019) · Zbl 1429.55007
[10] Mohammadi, S.; Asadi-Golmankhaneh, M. A., The homotopy types of \(S U(n)\)-gauge groups over \(S^6\), Topol. Appl., 270, Article 106952 pp. (2020) · Zbl 1433.55004
[11] Mimura, M.; Toda, H., Homotopy groups of \(S U(3), S U(4)\), and \(S p(2)\), J. Math. Kyoto Univ., 3, 217-250 (1963/1964) · Zbl 0129.15404
[12] Rea, S., Homotopy types of gauge groups of \(P U(p)\)-bundles over spheres
[13] Sutherland, W., Function spaces related to gauge groups, Proc. R. Soc. Edinb., Sect. A, 121, 185-190 (1992) · Zbl 0761.55007
[14] Theriault, S. D., The homotopy types of \(S U(5)\)-gauge groups, Osaka J. Math., 52, 15-31 (2015)
[15] Theriault, S. D., The homotopy types of \(S p(2)\)-gauge groups, Kyoto J. Math., 50, 591-605 (2010) · Zbl 1202.55004
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.