Homotopy decompositions of stable gauge groups (Q481016)

Let \(G\) be a topological group and let \(P\to B\) be a principal \(G\)-bundle over a base space \(B\). Let \(\mathcal{G}(P)\) denote the group consisting of all \(G\)-equivariant automorphisms of \(P\) which fix \(B\). When \(B\) is a simply connected four dimensional manifold with \(d\) two-cells and \(P\) is a principal \(SU(n)\)-bundle over \(M\) with the second Chern class \(k\), we denote the corresponding gauge group by \(\mathcal{G}_{k,n}(M)\). Similarly, if \(B\) is a compact oriented surface \(\Sigma_g\) of genus \(g\) and \(P\) is a principal \(U(n)\)-bundle with the first Chern class \(k\), we denote by \(\mathcal{G}_{k,n}(\Sigma_g)\) the corresponding gauge group. In this paper, the author studies the homotopy decomposition of the groups \(\mathcal{G}_{k,n}(M)\) and \(\mathcal{G}_{k,n}(\Sigma_g)\) for \(n=\infty\). From now on, we write \(\mathcal{G}_{k}(B)=\mathcal{G}_{k,\infty}(B)\) for \(B=M\) or \(\Sigma_g\). Then the author shows that there are homotopy equivalences of \(H\)-spaces \[ \begin{cases} \mathcal{G}_k(M)&\;\simeq SU(\infty)\times \Omega^4_0SU(\infty)\times (\Omega^2SU(\infty))^d \\ \mathcal{G}_k(\Sigma_g)&\;\simeq U(\infty)\times \Omega^4_0U(\infty)\times (\Omega^2U(\infty))^g \end{cases} \]