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. © 1999 Elsevier Science Ltd. All rights reserved.
Berrick, A. J., & Casacuberta, C. (1999). A universal space for plus-constructions. Topology, 38(3), 467–477. https://doi.org/10.1016/S0040-9383(97)00073-6