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A lower bound in the problem of realization of cycles
Journal of Topology ( IF 0.8 ) Pub Date : 2023-11-28 , DOI: 10.1112/topo.12320
Vasilii Rozhdestvenskii 1, 2, 3
Affiliation  

We consider the classical Steenrod problem on realization of integral homology classes by continuous images of smooth oriented manifolds. Let k ( n ) $k(n)$ be the smallest positive integer such that any integral n $n$ -dimensional homology class becomes realizable in the sense of Steenrod after multiplication by  k ( n ) $k(n)$ . The best known upper bound for k ( n ) $k(n)$ was obtained independently by Brumfiel and Buchstaber in 1969. All known lower bounds for k ( n ) $k(n)$ were very far from this upper bound. The main result of this paper is a new lower bound for k ( n ) $k(n)$ that is asymptotically equivalent to the Brumfiel–Buchstaber upper bound (in the logarithmic scale). For n < 24 $n<24$ , we prove that our lower bound is exact. Also, we obtain analogous results for the case of realization of integral homology classes by continuous images of smooth stably complex manifolds.

中文翻译:

循环实现问题的下界

我们考虑通过平滑定向流形的连续图像实现积分同调类的经典 Steenrod 问题。让 k n $k(n)$ 是最小的正整数,使得任何积分 n $n$ 乘以 后,维同调类在 Steenrod 意义上变得可实现  k n $k(n)$ 。最著名的上限为 k n $k(n)$ 是由 Brumfiel 和 Buchstaber 于 1969 年独立获得的。所有已知的下界 k n $k(n)$ 离这个上限还很远。本文的主要结果是一个新的下界 k n $k(n)$ 渐近等于 Brumfiel-Buchstaber 上限(以对数标度)。为了 n < 24 $n<24$ ,我们证明我们的下界是准确的。此外,对于通过平滑稳定复流形的连续图像实现积分同调类的情况,我们获得了类似的结果。
更新日期:2023-11-29
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