Cellular decomposition and free resolution for split metacyclic spherical space forms
type: Item
instance of: scholarly article
title: Cellular decomposition and free resolution for split metacyclic spherical space forms
zbMATH Open document ID: 1276.57037
author: A. P. T. Galves
author: O. Manzoli Neto
author: Lígia Laís Fêmina
author: M. Spreafico
published in: Homology, Homotopy and Applications
publication date: 2013-05-31T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 16E05
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 6169096
zbMATH Keywords: group action
zbMATH Keywords: space form
zbMATH Keywords: finite periodic groups
zbMATH Keywords: finite metacyclic groups
zbMATH Keywords: fundamental domain
zbMATH Keywords: resolution
zbMATH Keywords: equivariant complex
reviewed by: Daciberg Lima Gonçalves
MaRDI profile type: Publication
DOI: 10.4310/HHA.2013.V15.N1.A13
Recommended article: Cellularization for exceptional spherical space forms and the flag manifold of \(\mathsf{SL}_3 (\mathbb{R})\)
Recommended article: Fundamental Domain and Cellular Decomposition of Tetrahedral Spherical Space Forms
Recommended article: Space forms and group resolutions: the tetrahedral family
Recommended article: Free resolutions for certain metacyclic groups
Recommended article: On minimal resolutions for metacyclic groups with periodic cohomology
Recommended article: Diagonal resolutions for metacyclic groups
Recommended article: The mod-p cohomology rings of metacyclic groups
Recommended article: Local surgery obstructions and space forms
Recommended article: Q3981790
Recommended article: Q3334647
description: scientific article; zbMATH DE number 6169096