Cellular decomposition of quaternionic spherical space forms
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.
scientific article; zbMATH DE number 6133883
scientific article