A universal space for plus-constructions, by A. J. Berrick and Carles Casacuberta

To appear in Topology. We exhibit a two-dimensional, acyclic, Eilenberg-MacLane space W such that, for every space X, the plus-construction X^+ with respect to the largest perfect subgroup of pi _1(X) coincides, up to homotopy, with the W-nullification of X; that is, the natural map from X to X^+ is homotopy initial among maps from X to 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. This leads to a question concerning which spaces W have the property that W-nullification coincides with the plus-construction on spaces of type BGLR; such spaces may be said to define algebraic K-theory.


A. J. Berrick <berrick@math.nus.edu.sg>
Carles Casacuberta <casac@mat.uab.es>