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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.

Let G be a group and W a G-set. In this work we prove a result that describes geometrically, for a Poincaré duality pair .G; W /, the set of representatives for the G-orbits in W and the family of isotropy subgroups. We also prove, through a cohomological invariant, a necessary condition for a pair .G; W / to be a Poincaré duality pair when W is infinite.

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No abstract