Let G be a finite group. It is an unsolved problem to classify closed connected manifolds with fundamental group G. This thesis represents a first approximation to solving this problem. We consider the universal covers of such manifolds, and require that these covers be connected up to, but not including, the middle dimension, and that they satisfy a specific formulation of Poincaré Duality originally set out by Lefschetz. Using results from homological algebra, in particular the work of Johnson and Remez in constructing diagonal resolutions for metacyclic groups, we are able to construct purely algebraic chain complexes and invariants which act as a first approximation to these universal covers for the cases G cyclic and metacyclic.