The enumeration of coverings of closed orientable Euclidean manifolds $G_3$ and $G_5$
Abstract
There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable and four are nonorientable. The aim of this paper is to describe all types of $n$fold coverings over orientable Euclidean manifolds $\mathcal{G}_{3}$ and $\mathcal{G}_{5}$, and calculate the numbers of nonequivalent coverings of each type. We classify subgroups in the fundamental groups $\pi_1(\mathcal{G}_{3})$ and $\pi_1(\mathcal{G}_{5})$ up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index $n$. The manifolds $\mathcal{G}_{3}$ and $\mathcal{G}_{5}$ are uniquely determined among the others orientable forms by their homology groups $H_1(\mathcal{G}_{3})=\ZZ_3\times \ZZ$ and $H_1(\mathcal{G}_{5})= \ZZ$.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 DOI:
 10.48550/arXiv.1905.13558
 arXiv:
 arXiv:1905.13558
 Bibcode:
 2019arXiv190513558C
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Group Theory
 EPrint:
 arXiv admin note: text overlap with arXiv:1805.08146