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

Abstract

We examine the relation between the gauge groups of \(\mathrm {SU}(n)\)- and \(\mathrm {PU}(n)\)-bundles over \(S^{2i}\), with \(2\le i\le n\), particularly when n is a prime. As special cases, for \(\mathrm {PU}(5)\)-bundles over \(S^4\), we show that there is a rational or p-local equivalence \(\mathcal {G}_{2,k}\simeq _{(p)}\mathcal {G}_{2,l}\) for any prime p if, and only if, \((120,k)=(120,l)\), while for \(\mathrm {PU}(3)\)-bundles over \(S^6\) there is an integral equivalence \(\mathcal {G}_{3,k}\simeq \mathcal {G}_{3,l}\) if, and only if, \((120,k)=(120,l)\).