The homotopy types of \(\mathrm{SU}(n)\)-gauge groups over \(S^6\) (Q2291596)

Let \(G\) be a compact connected Lie group and let \(\pi:P\to X\) be a principal \(G\)-bundle over a space \(X\). Then the gauge group of \(P\) is the topological group consisting of \(G\)-equivariant automorphisms of \(P\). When \(X=S^6\) and \(G=\mathrm{SU}(n)\), principal \(\mathrm{SU}(n)\)-bundles over \(S^6\) are classified by their third Chern classes \(k\in\mathbb{Z}\) up to isomorphism. Denote the isomorphism class of principal \(\mathrm{SU}(n)\)-bundles with third Chern class \(k\) by \(P_k\) and its associated gauge group by \(\mathcal{G}(P_k)\). In this article, the authors give a necessary condition for homotopy equivalence \(\mathcal{G}(P_k)\simeq\mathcal{G}(P_l)\). According to \textit{M. F. Atiyah} and \textit{R. Bott} [Philos. Trans. R. Soc. Lond., Ser. A 308, 523--615 (1983; Zbl 0509.14014)] and \textit{D. H. Gottlieb} [Trans. Am. Math. Soc. 171, 23--50 (1972; Zbl 0251.55018)] the classifying space \(B\mathcal{G}(P_k)\) is homotopy equivalent to a connected component of the mapping space \(\text{Map}(S^6, B\mathrm{SU}(n))\). There is a fibration sequence \(\mathrm{SU}(n)\overset{\alpha_k}{\to}\Omega^5_0\mathrm{SU}(n)\to B\mathcal{G}(P_k)\overset{ev}{\to}B\mathrm{SU}(n)\), where \(ev\) is the evaluation map and \(\alpha_k\) is the boundary map. Apply the functor \([\Sigma^{2n-5}\mathbb{CP}^2,-]\) to the sequence and obtain \([\Sigma^{2n-5}\mathbb{CP}^2,B\mathcal{G}(P_k)]\cong\text{Coker}(\alpha_k)_*\). In Section 3 the authors use unstable \(K^1\)-theory to compute \(\text{Coker}(\alpha_k)_*\). Let \(W_n=\mathrm{SU}(\infty)/\mathrm{SU}(n)\). Then, there is an inclusion \(\lambda:[\Sigma^{2n-5}\mathbb{CP}^2,\Omega W_n]\to H^{2n}(\Sigma^{2n-5}\mathbb{CP}^2)\oplus H^{2n+2}(\Sigma^{2n-5}\mathbb{CP}^2)\). Furthermore, \(\alpha_k\) corresponds to the Samelson product \(\langle\epsilon,k\rangle:\Sigma^{5}\mathrm{SU}(n)\to \mathrm{SU}(n)\) which factors through \(\langle\epsilon,k\rangle':\Sigma^5\mathrm{SU}(n)\to\Omega W_n\). Using the formula of \(\lambda\circ\langle\epsilon,k\rangle'\) given in [\textit{H. Hamanaka} and \textit{A. Kono}, J. Math. Kyoto Univ. 43, No. 2, 333--348 (2003; Zbl 1070.55007)], the authors calculate the order of \((\alpha_k)_*\) and hence \([\Sigma^{2n-5}\mathbb{CP}^2,B\mathcal{G}(P_k)]\). In Section 4, they compare \([\Sigma^{2n-5}\mathbb{CP}^2,B\mathcal{G}(P_k)]\) with \([\Sigma^{2n-5}\mathbb{CP}^2,B\mathcal{G}(P_l)]\) and obtain the result.