Characterisation of polyhedral products with finite generalised Postnikov decomposition

A generalised Postnikov tower for a space X is a tower of principal fibrations with fibres generalised Eilenberg-MacLane spaces, whose inverse limit is weakly homotopy equivalent to X. In this paper we give a characterisation of a polyhedral product ZK(X, A) whose universal cover either admits a generalised Postnikov tower of finite length, or is a homotopy retract of a space admitting such a tower. We also include p-local and rational versions of the theorem. We end with a group theoretic application.