Let
be a compact
connected Lie group and let
be complex vector bundles over the classifying space
. The problem we
consider is whether
contains a subbundle which is isomorphic to
. The necessary condition
is that for every prime ,
the restriction
,
where
is a maximal
–toral subgroup
of , contains a subbundle
isomorphic to
.
We provide a criterion when this condition is sufficient, expressed in terms of
–functors
of Jackowski, McClure & Oliver, and we prove that this criterion applies for bundles
which
are induced by unstable Adams operations, in particular for the universal bundle over
. 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.
Keywords
homotopy representation, classifying space, unitary group