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.
|Number of pages||19|
|Journal||Communications in Algebra|
|Publication status||Published - 3 Apr 2017|
- crystallographic group
- Euclidean form
- flat 3-manifold
- nonequivalent coverings