TY - JOUR

T1 - On the coverings of Euclidean manifolds ℬ1 and ℬ2

AU - Chelnokov, G.

AU - Deryagina, M.

AU - Mednykh, A.

PY - 2017/4/3

Y1 - 2017/4/3

N2 - 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 non-orientable Euclidean manifolds ℬ1 and ℬ2 and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental groups of ℬ1 and ℬ2 up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index n. The manifolds ℬ1 and ℬ2 are uniquely determined among the other non-orientable forms by their homology groups Z2 X Z2 and H1B2 = Z2.

AB - 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 non-orientable Euclidean manifolds ℬ1 and ℬ2 and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental groups of ℬ1 and ℬ2 up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index n. The manifolds ℬ1 and ℬ2 are uniquely determined among the other non-orientable forms by their homology groups Z2 X Z2 and H1B2 = Z2.

KW - Amphicosms

KW - crystallographic group

KW - Euclidean form

KW - flat 3-manifold

KW - nonequivalent coverings

KW - platycosm

KW - NUMBER

KW - SURFACE

KW - SUBGROUPS

UR - http://www.scopus.com/inward/record.url?scp=84998694344&partnerID=8YFLogxK

U2 - 10.1080/00927872.2016.1222396

DO - 10.1080/00927872.2016.1222396

M3 - Article

AN - SCOPUS:84998694344

VL - 45

SP - 1558

EP - 1576

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 4

ER -