The enumeration of coverings of closed orientable Euclidean manifolds G3 and G5

Grigory Chelnokov, Alexander Mednykh

Research output: Contribution to journalArticlepeer-review

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 the orientable Euclidean manifolds G3 and G5, and calculate the numbers of non-equivalent coverings of each type. The manifolds G3 and G5 are uniquely determined among other forms by their homology groups H1(G3)=Z3×Z and H1(G5)=Z. We classify subgroups in the fundamental groups π1(G3) and π1(G5) 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.

Original languageEnglish
Pages (from-to)48-66
Number of pages19
JournalJournal of Algebra
Volume560
DOIs
Publication statusPublished - 15 Oct 2020

Keywords

  • Crystallographic group
  • Euclidean form
  • Flat 3-manifold
  • Non-equivalent coverings
  • Platycosm
  • NONEQUIVALENT COVERINGS
  • REPRESENTATIONS
  • SUBGROUPS
  • SURFACES

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