On the coverings of closed orientable Euclidean manifolds $\mathcal{G}_{2}$ and $\mathcal{G}_{4}$

TL;DR

This paper classifies all n-fold coverings over two specific orientable Euclidean 3-manifolds, $ ext{G}_2$ and $ ext{G}_4$, by analyzing their fundamental groups and subgroup structures.

Contribution

It provides a complete classification of n-fold coverings over $ ext{G}_2$ and $ ext{G}_4$, including enumeration of non-equivalent coverings and subgroup conjugacy classes.

Findings

Classified subgroups of $ ext{G}_2$ and $ ext{G}_4$ fundamental groups

Calculated the number of non-equivalent n-fold coverings

Identified unique homology group characteristics of $ ext{G}_2$ and $ ext{G}_4$

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 $\mathcal{G}_{2}$ and $\mathcal{G}_{4}$, and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental groups $\pi_1(\mathcal{G}_{2})$ and $\pi_1(\mathcal{G}_{4})$ up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index $n$. The manifolds $\mathcal{G}_{2}$ and $\mathcal{G}_{4}$ are uniquely determined among the others orientable forms by their homology groups $H_1(\mathcal{G}_{2})=\mathbb{Z}_2\times \mathbb{Z}_2 \times \mathbb{Z}$ and $H_1(\mathcal{G}_{4})=\mathbb{Z}_2 \times \mathbb{Z}$.