Euler classes of vector bundles over manifolds
Let Ek denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle α over X, the Euler class e(α) = 0. We show that if X ∈ E2n+1 is orientable, then X is a rational homology sphere and π1(X) is perfect. We also show that E8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in Ek
Naolekar, Aniruddha C., "Euler classes of vector bundles over manifolds" (2021). Journal Articles. 2097.