Relative plus constructions

Abstract

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 $h$-perfect normal subgroup $H$ of the fundamental group ${\pi }_1(X)$ an $h$-acyclic map $X\to X_H^{+h}$ inducing the quotient by $H$ on the fundamental group. 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.

15 pages; in this revised version the universal property of this relative plus construction is stated