Relative plus constructions

Let h be a connective homology theory. We construct a functorial relative plus construction as a Bousfield localization functor in the category of maps of spaces. It allows us to associate to a pair (X,H), consisting of a connected space X and an hperfect normal subgroup H of the fundamental group π1(X), an h-acyclic map X → XH+ h inducing the quotient by H on the fundamental group. We show that this map is terminal among the h-acyclic maps that kill a subgroup of H. When h is an ordinary homology theory with coefficients in a commutative ring with unit R, this provides a functorial and well-defined counterpart to a construction by cell attachment introduced by Broto, Levi, and Oliver in the spirit of Quillen’s plus construction. We also clarify the necessity to use a strongly R-perfect group H in characteristic zero.