On the coverings of Euclidean manifolds ℬ₁ and ℬ₂

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 non-orientable Euclidean manifolds ℬ₁ and ℬ₂ and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental groups of ℬ₁ and ℬ₂ up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index n. The manifolds ℬ₁ and ℬ₂ are uniquely determined among the other non-orientable forms by their homology groups Z₂ X Z² and H₁B₂ = Z².