The plus construction, Postnikov towers and universal central module extensions

Abstract

Given a connected spaceX, we consider the effect of Quillen’s plus construction on the homotopy groups ofX in terms of its Postnikov decomposition. Specifically, using universal properties of the fibration sequenceAX→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 II-central fibrations and a theory of universal central extensions of modules, analogous to universal central extensions of perfect groups.