Actions on lens spaces which respect a Heegaard decomposition
type: Item
instance of: scholarly article
title: Actions on lens spaces which respect a Heegaard decomposition
zbMATH Open document ID: 1020.57015
DOI: 10.1016/S0166-8641(02)00196-7
author: John Kalliongis
author: Andy Miller
published in: Topology and its Applications
publication date: 2003-05-27T00:00:00Z
review text: For \(L= L(p,q)\) a 3-dimensional lens space, the authors consider smooth finite group actions on \(L\) which leave a Heegaard torus and each of its complementary components invariants. In this paper they enumerate and classify two type of actions (rotational and dirotational) up to equivalence (two actions are equivalent if their images are conjugate in the group of self-diffeomorphisms of \(L\)). First, they establish the geometrization and rigidity for actions which respect a Heegaard decomposition. Then the results are combined to describe the classification of the equivalence classes of actions of rotational type. They also examine the fixed point data of each equivalence of cyclic actions of rotational type. Finally, in the main result of this paper, going beyond the geometrization and rigidity conjectures they give a complete enumeration of the equivalence classes of group actions on \(L\) which respect a Heegaard decomposition. Using recent progress on Thurston's geometrization conjecture these results give more complete solutions to the classification problem of symmetries on lens spaces.
reviewed by: Corina Mohorianu
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 1917841
zbMATH Keywords: geometric action
zbMATH Keywords: lens space
zbMATH Keywords: finite group action
zbMATH Keywords: equivalence of actions
zbMATH Keywords: equivalence
zbMATH Keywords: symmetries
MaRDI profile type: Publication
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full work available at URL: s0166-8641(02)00196-7
OpenAlex ID: W2063227580
description: scientific article; zbMATH DE number 1917841