Maps between classifying spaces of unitary groups

We give a classification theorem for maps between classifying spaces of unitary groups in certain range of dimensions. We show a construction of families of exceptional (i.e. not induced by homomorphisms of groups and Adams operations) maps and show that for each n these include exceptional representations of U(n) of the smallest possible dimension. In particular we closely investigate and calculate in several cases obstructions needed to lift a map from a colimit to a homotopy colimit. We give a criterion for lifting a map from homotopy colimit by a fibration.