The homotopy types of \(PSp(n)\)-gauge groups over \(S^{2m}\) (Q2222125)

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: \textbf{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 [\textit{M. F. Atiyah} and \textit{R. Bott}, Philos. Trans. R. Soc. Lond., Ser. A 308, 523--615 (1983; Zbl 0509.14014); \textit{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 [\textit{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 [\textit{T. Cutler}, Topology Appl. 236, 44--58 (2018; Zbl 1383.55005)] to obtain the last two statements of the theorem.