Homotopy types of gauge groups of \(\mathrm{PU}(p)\)-bundles over spheres (Q2023786)

Let us assume that \(G=\mathrm{SU}(n)\) or \(G=\mathrm{PU}(n)\). Note that an isomorphism class of a principal \(G\)-bundle \(P\) over \(S^{2i}\) is classified by its characteristic element in \(\pi_{2i-1}(G)\cong \mathbb{Z}\). We denote by \(P_k\) the principal \(G\) bundle over \(S^{2i}\) with characteristic element \(k\in \mathbb{Z}\), and let \(\mathcal{G}_{i,k}(G)\) denote the gauge group of the principal \(G\)-bundle \(P_k.\) In this paper, the author investigates the relation between the gauge groups \(\mathcal{G}_{i,k}(\mathrm{SU}(n))\) and \(\mathcal{G}_{i,k}(\mathrm{PU}(n))\) when \(2\leq i\leq n\) and \(n\) is a prime. In particular, he proves that there is a \(p\)-local equivalence \(\mathcal{G}_{2,k}(\mathrm{PU}(5))\simeq_{(p)} \mathcal{G}_{2,l}(\mathrm{PU}(5))\) iff \((120,k)=(120,l)\) when \(p=0\) or \(p\) is any prime. Moreover, he also shows that there is a homotopy equivalence \(\mathcal{G}_{3,k}(\mathrm{PU}(3))\simeq \mathcal{G}_{3,l}(\mathrm{PU}(3))\) iff \((120,k)=(120,l)\). The proof is based on the analysis of the Samelson products. Indeed, when \(p\) is a prime and \(2\leq i\leq p\), let \(\epsilon_i\) and \(\delta_i\) denote generators of \(\pi_{2i-1}(\mathrm{PU}(p))\) and \(\pi_{2i-1}(\mathrm{SU}(p))\). Then he shows that the orders of the Samelson products \(\langle \epsilon_i,1\rangle :S^{2i-1}\wedge \mathrm{PU}(p)\to \mathrm{PU}(p)\) and \(\langle \delta_i,1\rangle :S^{2i-1}\wedge \mathrm{SU}(p)\to \mathrm{SU}(p)\) coincide. By using this result with a careful analysis of several Samelson products, the author obtains the desired results.