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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.

A stratified space is a filtered space with manifolds as its strata. Connolly and Vajiac proved an end theorem for stratified spaces, generalizing earlier results of Siebenmann and Quinn. Their main result states that there is a single $K$-theoretical obstruction to completing a tame-ended stratified space. A necessary condition to completeness is to find an exhaustion of the stratified space, i.e. an increasing sequence of stratified spaces with bicollared boundaries, whose union is the original space. In this paper we give an example of a stratified space that is not exhaustible. We also prove that the Connolly-Vajiac end obstructions can be realized.

LetG be a finite group acting on a Riemannian manifoldM by isometries. We introduce analytic torsion ρan(M,M1;V ) ∈ R⊗Z RepR(G) PL-torsion ρpl(M,M1;V ) ∈ K1(RG) Poincare torsion ρpd(M,M1;V ) ∈ K1(RG) and Euler characteristic χ(M,M1;V ) ∈ RepR(G) for ∂M the disjoint union of M1 and M2 and V an equivariant coefficient system. The analytic torsionis defined in terms of the spectrum of the Laplace operator, the PL-torsionis based on the cellular chain complex and Poincare torsionmeasures the failure of equivariant Poincare duality in the PL-setting, which does hold in the analytic context. Denote by RepR(G) the subgroup of RepR(G) generated by the irreducible representations of real or complex type. We define an isomorphism