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# Homology, Homotopy and Applications

## Volume 15 (2013)

### Number 1

### Cellular decomposition and free resolution for split metacyclic spherical space forms

Pages: 253 – 278

DOI: http://dx.doi.org/10.4310/HHA.2013.v15.n1.a13

#### Authors

#### Abstract

Given a free isometric action of a split metacyclic group on odd dimensional sphere, we obtain an explicit finite cellular decomposition of the sphere equivariant with respect to the group action. A cell decomposition of the factor space and an explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group follow. In particular, the construction provides a simple explicit 4-periodic free resolution for the split metacyclic groups.

#### Keywords

metacyclic group, fundamental domain, spherical space form

#### 2010 Mathematics Subject Classification

16E05, 18G10, 20J05, 20J06, 57M07, 57M10, 57Q10