The mod-$p$ homology of the classifying spaces of certain gauge groups

Abstract: Let $G$ be a simply-connected, simple compact Lie group of type ${n_{1},\ldots,n_{\ell}}$, where $n_{1}\le\cdots \le n_{\ell}$. Let $\mathcal{G}k$ be the gauge group of the principal $G$-bundle (namedright{P}{}{S^{4}}) whose isomorphism class is determined by the the second Chern class having value $k\in\mathbb{Z}$. We calculate the mod-$p$ homology of the classifying space $B\mathcal{G}_k$ provided that $n{\ell}<p-1$ and $p$ does not divide $k$.