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Let   k   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  ∈  2 n +1  is orientable, then  X  is a rational homology sphere and  π  1 ( X ) is perfect. We also show that  8  = ∅ and derive additional cohomlogical restrictions on orientable manifolds in   k  .

Euler classes of vector bundles over manifolds