Abstract

A new lower bound is obtained relating the rational cup-length of the base and that of the total space of branched coverings of orientable manifolds for the case in which the branched covering is a projection onto the quotient space by the action of commuting involutions on the total space. This bound is much stronger than the classical Berstein–Edmonds 1978 bound which holds for arbitrary branched coverings of orientable manifolds.

In the framework of the theory of branched coverings, results are obtained that are motivated by the problems concerning \(n\)-valued topological groups. We explicitly construct \(m-1\) commuting involutions acting as automorphisms on the torus \(T^m\) with the orbit space \(\mathbb{R}P^m\) for any odd \(m\ge 3\). By the construction thus obtained, the manifold \(\mathbb{R}P^m\) carries the structure of an \(2^{m-1}\)-valued Abelian topological group for all odd \(m\ge 3\).

中文翻译：

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流形上通勤卷合的非自由作用

摘要

获得了一个新的下界，将基的有理杯长与可定向流形的分支覆盖的总空间相关联，其中分支覆盖是通过对合对合作用在商空间上的投影。整体空间。这个界限比经典的 Berstein-Edmonds 1978 界限强得多，后者适用于可定向流形的任意分支覆盖。

在分支覆盖理论的框架内，得到了由涉及\(n\)值拓扑群问题引发的结果。对于任何奇数\( m\ge 3 )，我们明确构造\(m-1\)通勤对合，作为环面\(T^m\)上的自同构，轨道空间\(\mathbb{R}P^m\) \)。通过由此获得的构造，流形\(\mathbb{R}P^m\)对于所有奇数\(m\ge 3 )具有\(2^{m-1}\)值阿贝尔拓扑群的结构\)。