We study the question when a manifold that fibers over a sphere can be rationally essential, or have positive simplicial volume. More concretely, we show that mapping tori of manifolds (whose fundamental groups can be quite arbitrary) of dimension $\text{◂⩾▸}2n+1⩾7$ with non-zero simplicial volume are very common. This contrasts the case of fiber bundles over a sphere of dimension $d⩾2$: we prove that their total spaces are rationally inessential if $d⩾3$, and always have simplicial volume 0. Using a result by Dranishnikov, we also deduce a surprising property of macroscopic dimension, and we give two applications to positive scalar curvature and characteristic classes, respectively.

中文翻译：

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球体上纤维化的流形的单纯体积和本质

我们研究的问题是，球体上的纤维流形可以是理性本质的，或者具有正单纯形体积。更具体地说，我们展示了维数流形（其基本群可以是相当任意的）的映射托里$\text{◂⩾▸}\mathrm{2个}n+\mathrm{1个}⩾7$具有非零单纯体积的非常普遍。这与维度球体上的纤维束的情况形成对比$d⩾\mathrm{2个}$：我们证明如果$d⩾\mathrm{3个}$，并且总是有单纯体积 0。使用 Dranishnikov 的结果，我们还推导出宏观维度的一个令人惊讶的性质，我们分别给出了正标量曲率和特征类的两个应用。