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

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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$k（n）$是最小的正整数，使得任何积分$n$乘以 后，维同调类在 Steenrod 意义上变得可实现 $k（n）$。最著名的上限为$k（n）$是由 Brumfiel 和 Buchstaber 于 1969 年独立获得的。所有已知的下界$k（n）$离这个上限还很远。本文的主要结果是一个新的下界$k（n）$渐近等于 Brumfiel-Buchstaber 上限（以对数标度）。为了，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。