Homotopy representations of the unitary groups

Let $G$ be a compact connected Lie group and let $\xi,\nu$ be complex vector bundles over the classifying space $BG$. The problem we consider is whether $\xi$ contains a subbundle which is isomorphic to $\nu$. The necessary condition is that for every prime $p$ the restriction $\xi|_{BN_p^G}$, where $N_p^G$ is a maximal $p$-toral subgroup of $G$, contains a subbundle isomorphic to $\nu|_{BN_p^G}$. We provide a criterion when this condition is sufficient, expressed in terms of $\Lambda^*$-functors of Jackowski, McClure \& Oliver and we prove that this criterion applies if $\nu$ is a universal bundle over $BU(n)$. Our result allows to construct new examples of maps between classifying spaces of unitary groups. While proving the main result, we develop the obstruction theory for lifting maps from homotopy colimits along fibrations, which generalizes the result of Wojtkowiak.