EULER CLASSES OF VECTOR BUNDLES OVER MANIFOLDS

Let E-k denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle alpha over X, the Euler class e(alpha) = 0. We show that if X is an element of E2n+1 is orientable, then X is a rational homology sphere and pi(1)(X) is perfect. We also show that E-8 = empty set and derive additional cohomlogical restrictions on orientable manifolds in E-k. (C) 2021 Mathematical Institute Slovak Academy of Sciences