On splitting topological R4-bundles

AbstractLet ξ4 denote any topological R4-bundle over a connected Poincaré complex X of formal dimension ⩽4. Let α be any bundle of lines or planes based on X. We determine necessary and sufficient conditions in terms of characteristic classes of ξ4 and α, whenever possible, for ξ to split off a subbundle isomorphic to α. That is, the existence of a Whitney sum decomposition, ξ4≈α⊤β where both α and β have positive fiber dimension, is stablished in terms of characteristic classes. In particular, our results extend the work of Hirzebruch and Hopf [3] on oriented vector bundles of dimension 4 to arbitrary R4-bundles