Homotopy representations of the unitary groups

TL;DR

This paper develops a criterion for when a complex vector bundle over the classifying space of a compact Lie group contains a subbundle isomorphic to another, using homotopy-theoretic tools and obstruction theory, with applications to maps between unitary group classifying spaces.

Contribution

It introduces a new criterion based on $ ext{Lambda}^*$-functors for subbundle existence, applicable to universal bundles over $BU(n)$, and extends obstruction theory for lifting maps.

Findings

Provides a sufficient condition for subbundle inclusion in classifying spaces.

Applies the criterion to universal bundles over $BU(n)$.

Develops a generalized obstruction theory for lifting maps along fibrations.

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 $\xi|_{BN_p^G}$, 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.