On maps from \(BS^1\) to classifying spaces of certain gauge groups (Q1380142)

Let \(G\) be a compact connected Lie group, \(\pi: P\to X\) a principal \(G\)-bundle over a compact connected manifold \(X\) and \({\mathcal G}\) its gauge group. The author establishes some equivalent conditions for the existence of a homotopically nontrivial map from \(BS^1\) to \(B{\mathcal G}\) in the following cases: \(X=S^n\), \(G=SU(m)\) or \(Sp(m)\) and \(X\) is a simply connected spin 4-manifold of \(CP^2\), \(G=SU(2)\). The cohomology of \(\text{Map} (CP^2, BSU(2))\) in low degree is also determined.