Three-dimensional hyperelliptic manifolds and Hamiltonian graphs
A three-dimensional manifold \(M^3\) is called hyperelliptic if it admits an involution \(\tau\) such that \(M^3/\langle\tau\rangle\) is homeomorphic to the sphere \(S^3\). Existence of hyperelliptic manifolds in each of the eight three-dimensional Thurston geometries was proven by \textit{A.~D. Mednykh} [Ann. Global Anal. Geom. 8, No. 1, 13-19 (1990; Zbl 0712.57007)]. The key role in the proof was played by right-angled polyhedra which have Hamiltonian cycles. In the article under review, a constructive method for obtaining hyperelliptic manifolds is suggested. The method is based on coloring of a polyhedron which admits a Hamiltonian cycle. It is also generalized to polyhedra admitting \(\theta\)-Hamiltonian or \(K_4\)-Hamiltonian cycles. Along with proofs of the constructive theorems, the article contains several illustrative examples. In particular, constructions are given of a hyperelliptic manifold of the Löbell type obtained from a Hamiltonian polyhedron and of a hyperelliptic manifold obtained from the Grinberg polyhedron. The Grinberg polyhedron was known as an example of a non-Hamiltonian polyhedron, but the authors show that it is \(\theta\)-Hamiltonian, and so the methods proposed may be applied to it.
scientific article; zbMATH DE number 1396472
scientific article