The plus construction, Postnikov towers and universal central module extensions

Given a connected space X, we consider the effect of Quillen’s plus construction on the homotopy groups of X in terms of its Postnikov decomposition. Specifically, using universal properties of the fibration sequence AX → X → X+, we explain the contribution of πnX to πnX +, πn+1X + and πnAX, πn+1AX explicitly in terms of the low dimensional homology of πnX regarded as a module over π1X. Key ingredients developed here for this purpose are universal Π-central fibrations and a theory of universal central extensions of modules, analogous to universal central extensions of perfect groups.