On the coverings of closed orientable Euclidean manifolds G(2) and G(4)

Grigory Chelnokov, Alexander Mednykh

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1 Citation (Scopus)

Abstract

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 G(2) and G(4) and calculate the numbers of nonequivalent coverings of each type. We classify subgroups in the fundamental groups pi(1)(G(2)) and pi(1)(G(4)) up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index n. The manifolds and are uniquely determined among the others orientable forms by their homology groups H-1(G(2)) = Z(2) x Z(2) x Z and H-1(G4) = Z(2) x Z.

Original languageEnglish
Pages (from-to)2725-2739
Number of pages15
JournalCommunications in Algebra
Volume48
Issue number7
DOIs
Publication statusPublished - 2 Jul 2020

Keywords

  • Crystallographic group
  • Euclidean form
  • flat 3-manifold
  • nonequivalent coverings
  • platycosm
  • SUBGROUPS
  • ENUMERATING REPRESENTATIONS

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