TY - JOUR

T1 - On hyperelliptic Euclidean 3-manifolds

AU - Mednykh, A. D.

AU - Vuong, B.

N1 - The authors are grateful to professor Louis H. Kauffman for helpful comments on the preliminary results of the paper and professor N. A. Abrosimov whose remarks and suggestions assisted greatly in preparation of the text. Also, the authors are very thankful to an anonymous referee for valuable remarks and suggestions. The work has been supported by the Russian Science Foundation (Project 19-41-02005). Publisher Copyright: © 2021 World Scientific Publishing Company.

PY - 2021/9/1

Y1 - 2021/9/1

N2 - In this paper, we study closed orientable Euclidean manifolds which are also known as flat three-dimensional manifolds or just Euclidean 3-forms. Up to homeomorphism, there are six of them. The first one is the three-dimensional torus. In 1972, Fox showed that the 3-torus is not a double branched covering of the 3-sphere. So, it is not a hyperelliptic manifold. In this paper, we show that all the remaining Euclidean 3-forms are hyperelliptic manifolds.

AB - In this paper, we study closed orientable Euclidean manifolds which are also known as flat three-dimensional manifolds or just Euclidean 3-forms. Up to homeomorphism, there are six of them. The first one is the three-dimensional torus. In 1972, Fox showed that the 3-torus is not a double branched covering of the 3-sphere. So, it is not a hyperelliptic manifold. In this paper, we show that all the remaining Euclidean 3-forms are hyperelliptic manifolds.

KW - branched covering

KW - Euclidean form

KW - fundamental group

KW - homology group

KW - hyperelliptic manifold

KW - π-orbifold

UR - http://www.scopus.com/inward/record.url?scp=85121243393&partnerID=8YFLogxK

U2 - 10.1142/S0218216521400010

DO - 10.1142/S0218216521400010

M3 - Article

AN - SCOPUS:85121243393

VL - 30

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 10

M1 - 21400015

ER -