On the coverings of Euclidean manifolds ℬ₁ and ℬ₂

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 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 $H_1\left({\mathcal{ℬ}}_1\right)={\mathbb{Z}}_2\times {\mathbb{Z}}^2$ and $H_1\left({\mathcal{ℬ}}_2\right)={\mathbb{Z}}^2$.

KEYWORDS:

• Amphicosms
• crystallographic group
• Euclidean form
• flat 3-manifold
• nonequivalent coverings
• platycosm

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20H15
• 57M10
• 55R10

Acknowledgment

The authors are grateful to V. A. Liskovets, R. Nedela, M. Shmatkov for helpful discussions during the work on this paper. We also thank the unknown referee for his/her valuable comments and suggestions.

Notes

¹In other words, Δ is abelian iff Δ≤⟨a²,b,c⟩

²Keep in mind that in the sum ${\sum }_{2l|n}{\sigma }_1\left(l\right)l$ the term σ₁(l)l is not the amount of subgroups Δ, such that l(Δ) = l. Contrary, it is the amount of subgroups Δ, such that $l\left(\Delta \right)=\frac{n}{2l}$.

³In Stanley notations: $j_G\left(n\right)=c_G\left(n\right)$ and $u_G\left(n\right)=s_G\left(n\right)$.

⁴In other words, Δ is abelian iff Δ≤⟨α²,β,γ⟩