Relative plus constructions

Abstract

Let 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 , consisting of a connected space and an -perfect normal subgroup of the fundamental group , an -acyclic map inducing the quotient by on the fundamental group. We show that this map is terminal among the -acyclic maps that kill a subgroup of . When is an ordinary homology theory with coefficients in a commutative ring with unit , 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 -perfect group in characteristic zero.