arXiv subjects
Publications and source records attributed to G. Chelnokov.
We prove that any invariant of a 4-quiver, that is piecewise polynomial, moreover, polynomial for fixed signs of entries, is a function of determinant of a quiver.
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.
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.
There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable and four are non-orientable. The aim of this paper is to describe all types of $n$-fold coverings over orientable Euclidean manifolds $\mathcal{G}_{2}$ and $\mathcal{G}_{4}$, and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental groups $π_1(\mathcal{G}_{2})$ and $π_1(\mathcal{G}_{4})$ up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index $n$. The manifolds $\mathcal{G}_{2}$ and $\mathcal{G}_{4}$ are uniquely determined among the others orientable forms by their homology groups $H_1(\mathcal{G}_{2})=\mathbb{Z}_2\times \mathbb{Z}_2 \times \mathbb{Z}$ and $H_1(\mathcal{G}_{4})=\mathbb{Z}_2 \times \mathbb{Z}$.
There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable and four are non-orientable. The aim of this paper is to describe all types of $n$-fold coverings over orientable Euclidean manifolds $\mathcal{G}_{3}$ and $\mathcal{G}_{5}$, and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental groups $π_1(\mathcal{G}_{3})$ and $π_1(\mathcal{G}_{5})$ up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index $n$. The manifolds $\mathcal{G}_{3}$ and $\mathcal{G}_{5}$ are uniquely determined among the others orientable forms by their homology groups $H_1(\mathcal{G}_{3})=\ZZ_3\times \ZZ$ and $H_1(\mathcal{G}_{5})= \ZZ$.
There are just 10 closed flat 3-manifolds, following [1], we call them platycosms. The aim of this paper is to classify types of n-coverings over amphicosms, i.e. some kinds of platycosms, and enumerate the numbers of them. Key words: platycosm, amphicosms, flat 3-manifold, non-equivalent covering over manifold, crystallographic group.