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 β¬1 and β¬2 and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental groups of β¬1 and β¬2 up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index n. The manifolds β¬1 and β¬2 are uniquely determined among the other non-orientable forms by their homology groups and .
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, Ξ is abelian iff Ξβ€β¨a2,b,cβ©
2Keep in mind that in the sum the term Ο1(l)l is not the amount of subgroups Ξ, such that l(Ξ) = l. Contrary, it is the amount of subgroups Ξ, such that .
3In Stanley notations: and .
4In other words, Ξ is abelian iff Ξβ€β¨Ξ±2,Ξ²,Ξ³β©