Exotic dihedral actions on homology 3-spheres
type: Item
instance of: scholarly article
title: Exotic dihedral actions on homology 3-spheres
zbMATH Open document ID: 0729.57004
DOI: 10.1016/0166-8641(91)90015-E
author: Charles Livingston
published in: Topology and its Applications
publication date: 1991-01-01T00:00:00Z
review text: The dihedral group \(D_{2n}\), with 2n elements, has linear actions on the odd-dimensional spheres \(S^{4k+3}\) such that the normal cyclic subgroup \({\mathbb{Z}}_ n\subset D_{2n}\) acts freely. In these linear actions, the fixed point sets of the elements of order 2 in \(D_{2n}\) are disjoint k-spheres, any two of which have linking numbers \(\pm 1.\) In a paper of \textit{J. F. Davis} and \textit{T. tom Dieck} [Indiana Univ. Math. J. 37, No.2, 431-450 (1988; Zbl 0634.57021)] non-linear actions of \(D_{2n}\) were constructed on homotopy spheres \(\Sigma^{8\ell +3}\) \((\ell >0)\) and on certain homology 3-spheres, where the fixed sets of the involutions have linking numbers \(\neq \pm 1\). The purpose of the present paper is to construct explicit examples of this type on dimension 3. Theorem. Let \(n>1\) be an odd integer. For every integer x, there is a 2n- fold regular dihedral covering of \(S^ 3\) branched over the Pretzel Knot K(p,q,r) with \(p=1+2xn,\) \(q=-2-2xn,\) and \(r=-n-(4x^ 2n^ 2+6xn+2).\) That cover is a homology 3-sphere, and the linking number of any two components of the branch set is \(-4x(xn+1)-1\).
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 4205406
zbMATH Keywords: dihedral group
zbMATH Keywords: linear actions
zbMATH Keywords: odd-dimensional spheres
zbMATH Keywords: linking numbers
zbMATH Keywords: non-linear actions
zbMATH Keywords: homotopy spheres
zbMATH Keywords: homology 3-spheres
zbMATH Keywords: regular dihedral covering
zbMATH Keywords: Pretzel Knot
reviewed by: Ian Hambleton
MaRDI profile type: Publication
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cites work: Three dimensional manifolds, Kleinian groups and hyperbolic geometry
description: scientific article; zbMATH DE number 4205406