The Spaces That Define Algebraic K-theory

We characterize spaces W such that the W -nullification functor PW , applied to any BGL(R), gives BGL(R) . Let X be a pointed space and X the Quillen plus-construction on X with respect to the maximal perfect subgroup of π1X. If W is a pointed CW-complex, let PW (X) denote the W -nullification of X. (Recall that PWX is up to homotopy the initial space Y under X which is W -null in the sense that the pointed mapping space Map ∗ (W,Y ) is weakly contractible.) For some time it has been known that there exist universal spaces W such that for any X, PW (X) is equivalent to X . We will say that such a space W defines the plus-construction. Given that the plus-construction was originally applied to spaces of the form BGL(R) in order to construct the higher algebraicK-groups of a ring R, it seems natural to consider spaces W with the property that for any discrete, associative ring R with unit, PW BGL(R) is equivalent to BGL(R). We will say that such a W defines algebraic K-theory. For then one can use W to define the algebraic K-groups of a ring R as