A UNIVERSAL SPACE FOR PLUS-CONSTRUCTIONS

Abstract

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 π 1( 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.