The homotopy types of \(SU(n)\)-gauge groups over \(S^{2m}\) (Q2670951)

Given positive integers \(m,n\) such that \(m\leq n\), let \(P_{n,k}\) be the principal \(SU(n)\)-bundle over \(S^{2m}\) with Chern class \(c_m(P_{n,k})=k\). Then \(P_{n,k}\) is the pullback of the universal bundle \(ESU(n)\) by \(k\epsilon:S^{2m}\to BSU(n)\) where \(\epsilon\) represents a generator of \(\pi_{2m}(BSU(n))\cong\mathbb{Z}\). The gauge group \(\mathcal{G}_{k,m}(SU(n))\) of \(P_{n,k}\) is the topological group consisting of equivariant automorphisms of \(P_{n,k}\) that fix \(S^{2m}\). In this paper the author studies the homotopy types of gauge groups \(\mathcal{G}_{k,m}(SU(n))\). According to Atiyah-Bott and Gottlieb, \(B\mathcal{G}_k(S^n,G)\) is homotopy equivalent to \(\text{Map}_k(S^{2m},BSU(n))\), the connected component of \(\text{Map}(S^{2m}, BSU(n))\) that contains \(k\epsilon\). The evaluation map \(ev:\text{Map}_k(S^{2m},BSU(n))\to BSU(n)\) gives the Puppe fibration sequence \[ SU(n)\overset{\alpha_k}{\longrightarrow}\Omega^{2m-1}_0SU(n)\longrightarrow B\text{Map}_k(S^{2m},BSU(n))\overset{ev}{\longrightarrow}BSU(n), \] and the adjoint of the connecting map \(\alpha_k\) is the Samelson product \(k\langle\epsilon,id\rangle:S^{2m-1}\wedge SU(n)\to SU(n)\). The order of \(\langle{\epsilon,id}\rangle\) is the minimum positive integer \(r\) such that \(r\langle{\epsilon,id_G}\rangle\) is null homotopic. It is known that the greatest common divisor \(\text{gcd}(k,r)\) essentially determines the homotopy type of \(\mathcal{G}_{k,m}(SU(n))\). In Sections 3 and 4 the author applies unstable \(K\)-theory to calculate the order of \(\langle{\epsilon,\jmath}\rangle\), where \(\jmath:\Sigma\mathbb{CP}^{n-1}\to SU(n)\) is the inclusion that is a rational homotopy equivalence, and hence proves a necessary condition for \(\mathcal{G}_{k,m}(SU(n))\simeq\mathcal{G}_{\ell,m}(SU(n))\) in Theorem 1.1. In Sections 5 and 6 the author calculates the order of \(\langle{\epsilon,\jmath}\rangle\) when \((n,m)=(5,2)\) and \((n,m)=(6,2)\), and hence obtains Theorem 1.2.