Cellular decomposition and free resolution for split metacyclic spherical space forms
Let \(G\) be a finite group which acts freely on an odd sphere \(S^{2n-1}\). An interesting problem is to compute a \(G\)-equivariant cell decomposition of the sphere. This result has many good consequences, like for example to find a free resolution for the group \(G\) and to compute the Reidemeister torsion. Many years ago this question was solved for cyclic groups. Only recently, some of the authors and others started to look at other groups which have periodic cohomology of period 4. The goal of the paper is to study this problem for \(G\) the metacyclic groups of period \(4\). A key ingredient on the solution of the problem is the use of a decomposition of \(S^n\) as a join of circles, inspired by the case \(n=3\). Besides the use of that decomposition, a lot more work needs to be done to obtain the result. The method has already been used successfully in \textit{O. Manzoli Neto, T. de Melo} and \textit{M. Spreafico} [Geom. Dedicata 162, 9--24 (2013; Zbl 1262.57025)]. The geometric description of the fundamental domain which is obtained is too complicated and long to be described here, as well as the chain complex. They are given in Proposition 3.2, Lemma 4.1 and Theorem 5.1. To give a flavor of the results, in the case of the sphere \(S^3\), the fundamental domain has a \(G\)-equivariant decomposition with one \(0\)-cell, three \(1\)-cells, three \(2\)-cells and one \(4\)-cell. Several comments about the minimality of the rank of the resolution are made at the end.
scientific article; zbMATH DE number 6169096