A remark on the homotopy type of the classifying space of certain gauge groups
Let \(X\) be a closed simply connected 4-manifold and let \(P_k\) be the principal SU(2)-bundle over \(X\) with \(c_2(P_k)=k\) whose gauge group is denoted by \({\mathfrak g}_k\). In this paper, by examining the \(k\)-invariant of the localization of \(\text{Map}_k(X, HP^\infty)\) at prime \(p\), the author proves that a homotopy equivalence \(B{\mathfrak g}_k\simeq B{\mathfrak g}_m\) between classifying spaces implies \(gcd(k,p)=gcd(m,p)\) for any prime \(p\).
scientific article; zbMATH DE number 950583