Homotopy representations of the unitary groups

Abstract

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 $\xi |B{N}_p^G$, where $N_p^G$ is a maximal $p$–toral subgroup of $G$, contains a subbundle isomorphic to $\nu |B{N}_p^G$. We provide a criterion when this condition is sufficient, expressed in terms of ${\Lambda }^{\ast }$–functors of Jackowski, McClure & Oliver, and we prove that this criterion applies for bundles $\nu$ which are induced by unstable Adams operations, in particular for the universal bundle over $BU\left(n\right)$. Our result makes it possible 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.