Composition methods and homotopy types of the gauge groups of \(Sp(2)\) and \(SU(3)\) (Q953975)

For a compact connected simple Lie group \(G\), let \(P_k\) denote the principal \(G\)-bundle over \(S^4\) with the topological quantum number \(k\in\mathbb Z=\pi_4(BG)\) and let \({\mathcal G}_k\) be the corresponding gauge group. In this paper the authors study the homotopy types of the gauge group \({\mathcal G}\) for \(G=Sp(2)\) or \(SU(3)\), and compute the low dimensional homotopy sets related to the group \({\mathcal G}\) by using Toda's composition method. As an application, they obtain several sufficient conditions for integers \((k,l)\) under which two gauge groups \({\mathcal G}_k\) and \({\mathcal G}_l\) are homotopy equivalent. Moreover, they give another proof of a theorem concerning the homotopy types of \({\mathcal G}_k\) for \(G=SU(3)\) obtained by Kono-Hamanaka.