On the coverings of Euclidean manifolds $\mathbb{G}_2$ and $\mathbb{G}_4$

Communications in Algebra. 2020. Vol. 48. No. 7. P. 2725–2739.

Chelnokov G. R., Mednykh A. D.

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 $H_1(\mathcal{G}_{2})=\ZZ_2\times \ZZ_2 \times \ZZ$ and
$H_1(\mathcal{G}_{4})=\ZZ_2 \times \ZZ$.
 

Keywords: Amphicosms, crystallographic group, Euclidean form, flat 3-manifold, nonequivalent coverings, platycosm