Refined gauge group decompositions (Q2258615)

Let \(G\) be a simple, simply connected, compact Lie group and let \(P_{k}\to S^4 \) be the principal \(G\)-bundle whose second Chern class has value \(k\). In the paper under review, the authors give a \(p\)-local homotopy decomposition of the gauge group, \({\mathcal{G}}_{k}(G)\), of \(P_{k}\), when \(G\) is a matrix group and \(p\) is an odd prime. Their main result consists of the following homotopy equivalences, for \(n\geq 2\) and any \(k\in {\mathbb Z}\), \[ {\mathcal{G}}_{k}({\mathrm SU}(n)) \simeq \prod_{i=1,i\neq v-1,v}^{p-1} (B_{i}\times \Omega^4_{0}B_{i+2})\times X_{v-1}\times X_{v}. \] The spaces \(B_{\bullet}\) come from a \(p\)-local homotopy equivalence, \(G\simeq \prod_{i=1}^{p-1} B_{i}\), established by \textit{M. Mimura} et al. [Publ. Res. Inst. Math. Sci. 13, 627--680 (1977; Zbl 0383.22007)]. Also, there is a homotopy fibration, \(\Omega^4_{0}B_{t+2}\to X_{t}\to B_{t}\), for \(t\in\{v-1,v\}\). A similar decomposition of \({\mathcal{G}}_{k}({\mathrm Sp}(n))\) is also provided.