# The enumeration of coverings of closed orientable Euclidean manifolds $G_3$ and $G_5$

## TL;DR

This paper classifies all n-fold coverings over two specific orientable Euclidean 3-manifolds, describing subgroup structures and counting non-equivalent coverings, thereby advancing understanding of their topological coverings.

## Contribution

It provides a complete classification and enumeration of n-fold coverings over the orientable Euclidean manifolds G_3 and G_5, including subgroup analysis.

## Key findings

- Classified subgroups of fundamental groups up to isomorphism
- Calculated the number of conjugacy classes of subgroups for each index n
- Enumerated all types of n-fold coverings over 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 non-orientable. 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 non-equivalent 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 $H_1(\mathcal{G}_{5})= \ZZ$.