Under finite regular covering spaces, each preimage class of the base spaces is covered by some preimage classes of the covering spaces. We find an explicit relationship between those preimage classes in terms of covering transformation groups. Thereby we obtain a relationship of the homomorphism indices between them in terms of covering transformation groups. This is a local averaging formula and is the main result of our paper. The local formula gives rise to a global formula for the preimage homomorphism index and a formula for the preimage Reidemeister trace.

中文翻译：

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原像类别的原像同态索引

在有限的规则覆盖空间下，基本空间的每个原像类别都由覆盖空间的一些原像类别覆盖。我们在覆盖转换组方面发现了这些原像类之间的显式关系。由此，我们根据覆盖的变换组获得了它们之间同构性指标的关系。这是一个局部平均公式，是本文的主要结果。局部公式产生了原像同质性指数的整体公式和原像Reidemeister迹线的公式。