Simplicial volume and essentiality of manifolds fibered over spheres

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Topology 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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来源期刊

Journal of Topology 数学-数学

CiteScore

2.00

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Topology publishes papers of high quality and significance in topology, geometry and adjacent areas of mathematics. Interesting, important and often unexpected links connect topology and geometry with many other parts of mathematics, and the editors welcome submissions on exciting new advances concerning such links, as well as those in the core subject areas of the journal. The Journal of Topology was founded in 2008. It is published quarterly with articles published individually online prior to appearing in a printed issue.