Complex and real vector bundle monomorphisms

Given two complex vector bundles over a closed smooth manifold, we compare results concerning the existence and the (homotopy) classification of complex vector bundle monomorphisms on one hand and of real ones on the other hand. The two theories are related by transition homomorphisms which turn out to fit into an exact Gysin sequence of normal bordism groups. A detailed study reveals astonishing phenomena, e.g., situations where no complex but infinitely many real monomorphisms exist. Also all possible combinations of finiteness/infiniteness for the following two numbers occur already over products of spheres: 1. e number of complex monomorphisms which become homotopic as real monomorphisms, and he number of real monomorphisms which are not homotopic to complex ones.