论文标题
在封闭的非定向欧几里得歧管的封面上
On the coverings of closed non-orientable Euclidean manifolds $\mathcal{B}_{3}$ and $\mathcal{B}_{4}$
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}本文的目的是描述所有类型的$ n $折叠覆盖物上的不可取向的欧几里得歧管$ \ Mathcal {b} _ {3} $和$ \ MATHCAL {B} _ {4} $，并计算每种类型的非平均封面数量。歧管$ \ MATHCAL {B} _ {3} $和$ \ MATHCAL {B} _ {4} $由其同源组$ H_1（\ Mathcal {B} _ {3} _ {3}）= \ ZZ_2 \ ZZ_2 \ Zz_2 \ ZZ $＆TIMES $ H_1（\ MATHCAL {B} $ H_1（\ Mathcal {B} _ {4}）= \ zz_4 \ times \ zz $。 我们将子组分类为基本组$π_1（\ Mathcal {b} _ {3}）$和$π_1（\ Mathcal {b} _ {4}）$ to isomormormorphism。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生成功能。
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 $π_1(\mathcal{B}_{3})$ and $π_1(\mathcal{B}_{4})$ 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.
