Simplicial volume and essentiality of manifolds fibered over spheres

Pub Date : 2023-02-06 DOI:10.1112/topo.12286

Thorben Kastenholz, Jens Reinhold

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Abstract

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 $2 n + 1 ⩾ 7$ 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.

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球体上纤维流形的简单体积和本质

我们研究了一个在球体上纤维的流形何时可以是合理的本质，或者具有正的单纯形体积的问题。更具体地说，我们证明了维数为2n+1⩾7$2n+1\geqslant 7$的流形（其基群可以是相当任意的）与非零单体的映射tori是非常常见的。这与维度为d⩾2$d\geqslant 2$的球面上的纤维束的情况形成了对比：我们证明了如果d \10878.; 3$d\getqslant 3$，它们的总空间是合理的不重要的，并且总是具有单纯体积0。利用Dranishnikov的结果，我们还推导了宏观维数的一个令人惊讶的性质，并分别给出了正标量曲率和特征类的两个应用。

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