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

摘要

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.