A UNIVERSAL SPACE FOR PLUS-CONSTRUCTIONS

We exhibit a two-dimensional, acyclic, Eilenberg–Mac Lane space W such that, for every space X, the plus-construction X+ with respect to the largest perfect subgroup of π1(X) coincides, up to homotopy, with the W-nullification of X; that is, the natural map X→X+ is homotopy initial among maps X→Y where the based mapping space map∗(W, Y) is weakly contractible. Furthermore, we describe the effect of W-nullification for any acyclic W, and show that some of its properties imply, in their turn, the acyclicity of W.