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 α over X , the Euler class e ( α ) = 0. We show that if X ∈ 𝓔 2 n +1 is orientable, then X is a rational homology sphere and π 1 ( X ) is perfect. We also show that 𝓔 8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in 𝓔 k .

中文翻译：

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流形上向量束的欧拉类

令𝓔k表示闭连通光滑k流形X的微分类集，其性质为，对于X上的任何定向矢量束α，欧拉类e（α）=0。我们证明，如果X∈𝓔2 n + 1是可定向的，则X是有理同性球，而π1（X）是完美的。我们还表明𝓔8 =∅并推导𝓔k中可定向流形的其他同调限制。