Realizing cohomology classes as Euler classes

Abstract

Abstract For a space X, let E k(X), E ks(X) and E k○ (X) denote respectively the set of Euler classes of oriented k-plane bundles over X, the set of Euler classes of stably trivial k-plane bundles over X and the spherical classes in H k(X; ℤ). We prove some general facts about the sets E k(X), E ks(X) and E k○ (X). We also compute these sets in the cases where X is a projective space, the Dold manifold P(m, 1) and obtain partial computations in the case that X is a product of spheres.