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 π_nX to π_n X⁺, π_n+1 X⁺ and π_nAX, π_n+1 AX explicitly in terms of the low dimensional homology of π_nX regarded as a module over π₁X. Key ingredients developed here for this purpose are universal II-central fibrations and a theory of universal central extensions of modules, analogous to universal central extensions of perfect groups.