Research output: Contribution to journal › Article › peer-review
On the coverings of Hantzsche-Wendt manifold. / Chelnokov, Grigory; Mednykh, Alexander.
In: Tohoku Mathematical Journal, Vol. 74, No. 2, 2022, p. 313-327.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - On the coverings of Hantzsche-Wendt manifold
AU - Chelnokov, Grigory
AU - Mednykh, Alexander
N1 - Funding Information: 2010 Mathematics Subject Classification. Primary 20H15; Secondary 57M10, 05A15, 55R10. Key words and phrases. Euclidean form, platycosm, flat 3-manifold, non-equivalent coverings, crystallographic group, Dirichlet generating series, number of subgroups, number of conjugacy classes of subgroups. ∗The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no. 0314-2019-0007). Publisher Copyright: © 2022 Tohoku University, Mathematical Institute. All rights reserved.
PY - 2022
Y1 - 2022
N2 - There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable G1, . . . , G6 and four are non-orientable B1, . . . , B4. In the present paper we investigate the manifold G6, also known as Hantzsche-Wendt manifold; this is the unique Euclidean 3-form with finite first homology group H1(G6) = Z24 . The aim of this paper is to describe all types of n-fold coverings over G6 and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental group π1(G6) 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 series for the above sequences.
AB - There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable G1, . . . , G6 and four are non-orientable B1, . . . , B4. In the present paper we investigate the manifold G6, also known as Hantzsche-Wendt manifold; this is the unique Euclidean 3-form with finite first homology group H1(G6) = Z24 . The aim of this paper is to describe all types of n-fold coverings over G6 and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental group π1(G6) 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 series for the above sequences.
KW - crystallographic
KW - Euclidean form
KW - flat 3-manifold
KW - non-equivalent coverings
KW - platycosm
UR - http://www.scopus.com/inward/record.url?scp=85135184687&partnerID=8YFLogxK
U2 - 10.2748/tmj.20210308
DO - 10.2748/tmj.20210308
M3 - Article
AN - SCOPUS:85135184687
VL - 74
SP - 313
EP - 327
JO - Tohoku Mathematical Journal
JF - Tohoku Mathematical Journal
SN - 0040-8735
IS - 2
ER -
ID: 36743928