论文标题
在汉兹·温德特歧管的封面上
On the coverings of Hantzsche-Wendt manifold
论文作者
论文摘要
只有10种欧几里得形式，它们是平坦的三个维歧管的平坦形式：6个是可定向的$ \ Mathcal {g} _1，\ dots，\ Mathcal {g} _6 $，而四个是不可定向的$ \ Mathcal {B}在本文中，我们研究了歧管$ \ MATHCAL {G} _6 $，也称为Hantzsche-Wendt歧管；这是带有有限的第一同源组$ H_1（\ Mathcal {G} _6）= \ MathBb {Z}^2_4 $的独特欧几里得$ 3 $ -FORM。 本文的目的是描述所有类型的$ n $折叠覆盖物，覆盖$ \ Mathcal {g} _ {6} $，并计算每种类型的非当量覆盖物的数量。我们将子组在基本组$π_1（\ Mathcal {g} _ {6}）中分类为同构。给定的索引$ n $，我们计算了每种同构类型的子组的亚组数量和子组的共轭类别的数量，并为上述序列提供了Dirichlet生成序列。
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 $π_1(\mathcal{G}_{6})$ 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.
