On the coverings of closed non-orientable Euclidean manifolds $\mathcal{B}_{3}$ and $\mathcal{B}_{4}$

TL;DR

This paper classifies all n-fold coverings over two specific non-orientable Euclidean 3-manifolds, describing their types, counting non-equivalent coverings, and analyzing subgroup structures of their fundamental groups.

Contribution

It provides a complete classification and enumeration of coverings over $_{3}$ and $_{4}$, including subgroup analysis and generating functions, which was previously unaddressed.

Findings

Classified all n-fold coverings over $_{3}$ and $_{4}$.

Calculated the number of non-equivalent coverings for each type.

Derived Dirichlet generating functions for the covering counts.

Abstract

There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable $\mathcal{G}_1,\dots,\mathcal{G}_6$ and four are non-orientable $\mathcal{B}_1,\dots,\mathcal{B}_4$. The aim of this paper is to describe all types of $n$-fold coverings over the non-orientable Euclidean manifolds $\mathcal{B}_{3}$ and $\mathcal{B}_{4}$, and calculate the numbers of non-equivalent coverings of each type. The manifolds $\mathcal{B}_{3}$ and $\mathcal{B}_{4}$ are uniquely determined among non-orientable forms by their homology groups $H_1(\mathcal{B}_{3})=\ZZ_2\times \ZZ_2 \times \ZZ$ and $H_1(\mathcal{B}_{4})=\ZZ_4 \times \ZZ$. We classify subgroups in the fundamental groups $\pi_1(\mathcal{B}_{3})$ and $\pi_1(\mathcal{B}_{4})$ up to isomorphism. Given index $n$, we calculate the numbers of subgroups and the numbers of conjugacy classes of subgroups for each isomorphism type and provide the Dirichlet generating functions for the above sequences.