On the coverings of Hantzsche-Wendt manifold

G. Chelnokov, A. Mednykh
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引用次数: 4

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.
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只有10种欧几里得形式,即平坦封闭的三维流形:6种是可定向的$\mathcal{G}_1,\dots,\mathcal{G}_6$ 4种是不可定向的$\mathcal{B}_1,\dots,\mathcal{B}_4$。在本文中,我们研究了流形$\mathcal{G}_6$,也称为Hantzsche-Wendt流形;这是唯一的具有有限第一同调群$H_1(\mathcal{G}_6) = \mathbb{Z}^2_4$的欧几里得$3$ -形式。本文的目的是描述$\mathcal{G}_{6}$上所有类型的$n$ -fold覆盖,并计算每种类型的非等效覆盖的数量。我们将基本群$\pi_1(\mathcal{G}_{6})$中的子群划分到同构。给定索引$n$,我们计算了每个同构类型的子群的个数和子群的共轭类的个数,并给出了上述序列的Dirichlet生成级数。
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