On the coverings of Hantzsche-Wendt manifold

TL;DR

This paper classifies all n-fold coverings of the Hantzsche-Wendt manifold, describing subgroup structures, counting non-equivalent coverings, and providing generating series, thus advancing understanding of its topological coverings.

Contribution

It provides a complete classification of n-fold coverings of the Hantzsche-Wendt manifold and computes related subgroup counts and generating functions.

Findings

Classified subgroups of the fundamental group up to isomorphism

Calculated the number of non-equivalent n-fold coverings

Derived Dirichlet generating series for subgroup 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 $\mathcal{B}_1,\dots,\mathcal{B}_4$. In the present paper we investigate the manifold $\mathcal{G}_6$, also known as Hantzsche-Wendt manifold; this is the unique Euclidean $3$-form with finite first homology group $H_1(\mathcal{G}_6) = \mathbb{Z}^2_4$. The aim of this paper is to describe all types of $n$-fold coverings over $\mathcal{G}_{6}$ and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental group $\pi_1(\mathcal{G}_{6})$ 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 series for the above sequences.