On the coverings of closed orientable Euclidean manifolds ${\mathcal{G}}_2$ and ${\mathcal{G}}_4$

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 ${\mathcal{G}}_2$ and ${\mathcal{G}}_4$, and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental groups ${\pi }_1({\mathcal{G}}_2)$ and ${\pi }_1({\mathcal{G}}_4)$ up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index $n$. The manifolds ${\mathcal{G}}_2$ and ${\mathcal{G}}_4$ are uniquely determined among the others orientable forms by their homology groups $H_1({\mathcal{G}}_2)={\mathbb{Z}}_2\times {\mathbb{Z}}_2\times \mathbb{Z}$ and $H_1({\mathcal{G}}_4)={\mathbb{Z}}_2\times \mathbb{Z}$.