On the coverings of Euclidian manifolds $\mathcal{B}_1$ and $\mathcal{B}_2$

TL;DR

This paper classifies and counts different types of n-coverings over certain flat 3-manifolds called amphicosms, expanding understanding of their topological and geometric structures.

Contribution

It provides a classification scheme and enumeration of n-coverings over amphicosms, a specific class of flat 3-manifolds, which was not previously detailed.

Findings

Classification of n-coverings over amphicosms

Enumeration formulas for coverings

Insights into the structure of flat 3-manifolds

Abstract

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.