Cellular decomposition of quaternionic spherical space forms
type: Item
instance of: scholarly article
title: Cellular decomposition of quaternionic spherical space forms
zbMATH Open document ID: 1262.57025
author: O. Manzoli Neto
author: Thiago de Melo
author: M. Spreafico
published in: Geometriae Dedicata
publication date: 2013-02-01T00:00:00Z
review text: The paper under review has two main purposes: The first one is to give a cellular decomposition of the quaternionic spherical space forms for all free actions of a generalized quaternionic group; the second one is to classify up to homeomorphism the space forms, i.e. the quotient of the sphere by the action. The result about cellular decomposition is given in Corollary 1 and is obtained constructing geometrically the decomposition; the construction is inspired in the case where the sphere is \(S^3\). For the classification, the strategy is to compute the Whitehead torsion of the resolution which was found in the first part from the cellular decomposition. It turns out that the classification is similar to the classical result for lens spaces. The main result is: Theorem 1 Let \(Q(4t; q_1,...,q_n)\) and \(Q(4t; q_1',...,q_n')\) be two quaternionic spherical space forms. Then, \(Q\) and \(Q'\) are homeomorphic if and only if there are integers numbers \(a\) and \(\epsilon_1,....,\epsilon_n,\) with \(a\) prime to \(2t\) and \(\epsilon_l=\pm 1\), and some permutation \(\sigma\) of \(\{1,...,n\}\) such that \[ (q_1',..., q_n')=(\epsilon_1aq_{\sigma(1)},...,\epsilon_naq_{\sigma(n)}) \;\;mod \;2t. \] A good summary of Whitehead torsion is presented which makes it easier to understand the work.
Mathematics Subject Classification ID: 57N60
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 18G35
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 55U15
zbMATH DE Number: 6133883
zbMATH Keywords: quaternionic groups
zbMATH Keywords: lens space
zbMATH Keywords: spherical space forms
zbMATH Keywords: cellular decomposition
zbMATH Keywords: projective resolution
zbMATH Keywords: Whitehead torsion
reviewed by: Daciberg Lima Gonçalves
MaRDI profile type: Publication
full work available at URL: s10711-012-9714-4
OpenAlex ID: W2018287503
cites work: Q3241211
cites work: Q5676841
cites work: Complexes à automorphismes et homéomorphic différentiable
cites work: Über die Torsion einer Überdeckung.
cites work: Whitehead torsion
cites work: Homotopieringe und Linsenräume
cites work: Q5546515
DOI: 10.1007/S10711-012-9714-4
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description: scientific article; zbMATH DE number 6133883