The enumeration of coverings of closed orientable Euclidean manifolds $G_3$ and $G_5$

Abstract

There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable and four are non-orientable. The aim of this paper is to describe all types of $n$-fold coverings over orientable Euclidean manifolds ${\mathcal{G}}_3$ and ${\mathcal{G}}_5$, and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental groups ${\pi }_1({\mathcal{G}}_3)$ and ${\pi }_1({\mathcal{G}}_5)$ up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index $n$. The manifolds ${\mathcal{G}}_3$ and ${\mathcal{G}}_5$ are uniquely determined among the others orientable forms by their homology groups $H_1(\mathcal{G}_{3})=\ZZ_3\times \ZZ$ and $H_1(\mathcal{G}_{5})= \ZZ$.

arXiv admin note: text overlap with arXiv:1805.08146