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 $H_1(\mathcal{G}_{5})= \ZZ$.