On maps from \(BS^1\) to classifying spaces of certain gauge groups. II (Q1127112)

The purpose of the paper is to generalize a result of a previous paper of the author [ibid., No. 2, 327-342 (1997; Zbl 0891.55020)]. Let \(\pi :P \to X\) be a principal \(SU(2)\) bundle over a simply connected closed 4-manifold \(X\) and let \(\mathcal G\) be its gauge group. Theorem 1.1. The following conditions are equivalent: 1. There exists a homotopically nontrivial map from \(BS^1\) to \(B\mathcal G\). 2. There exists a nontrivial homomorphism from \(S^1\) to \(\mathcal G\). 3. The structure group of \(P\) reduces to \(S^1\). In his previous paper the author proved such a result assuming that \(X\) is a smooth simply connected spin 4-manifold or \(\mathbf {C}P^2\).