On certain classes of hyperbolic 3-manifolds
type: Item
instance of: scholarly article
title: On certain classes of hyperbolic 3-manifolds
zbMATH Open document ID: 0947.57013
author: Alberto Cavicchioli
author: Luisa Paoluzzi
published in: Manuscripta Mathematica
publication date: 2000-11-01T00:00:00Z
review text: The authors study an interesting class of closed orientable (mostly hyperbolic) 3-manifolds \(M(n,k)\), defined for positive integers \(n\) and \(k\), from various different points of view. These manifolds can be obtained by glueing in pairs the faces of certain 3-dimensional polyhedra, by an identification pattern with a rotational symmetry of order \(n\). Using handle decompositions (Heegaard splittings) and handle cancellations (or alternatively, coloured graphs) it is shown that the manifolds \(M(n,k)\) are exactly the \(n\)-fold cyclic branched coverings of the Whitehead link, with branching orders \(n\) and \(n/d\) associated to its two components where \(d = (n,k)\) is the greatest common divisor (so, if \(d>1\), these coverings are not strongly cyclic, and this is one of the first studies of such a situation). This generalizes constructions in a paper by \textit{H. Helling, A. C. Kim} and \textit{J. L. Mennicke} [Some honey-combs in hyperbolic 3-space, Commun. Algebra 23, No. 14, 5169--5206 (1995; Zbl 1002.57503)] for the case \(d=1\) (where hyperbolic realizations of the polyhedra are constructed). The manifolds \(M(n,k)\) are also \(n/d\)-fold strongly cyclic branched coverings of \((d+1)\)-component links \(L_{d+1}\) in the 3-sphere formed by a chain of \(d\) unknotted circles plus their axis of \(d\)-symmetry (\(L_2\) is just the Whitehead link). Presentations of the fundamental groups and the homology groups of the manifolds are obtained, and the manifolds are classified up to homeomorphism resp. isometry. Combinatorial descriptions of the manifolds by coloured graphs are given. Finally, the manifolds obtained by Dehn surgery on the components of the strongly invertible links \(L_{d+1}\) are studied, obtaining railroad systems and spines associated to presentations of their fundamental groups; using the Montesinos algorithm, they are 2-fold branched coverings of links in the 3-sphere which are explicitly determined.
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 1463440
zbMATH Keywords: 3-manifold
zbMATH Keywords: classification
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: s002290050227
OpenAlex ID: W2070064289
DOI: 10.1007/S002290050227
description: scientific article; zbMATH DE number 1463440