Relative plus construction

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 $X\rightarrow X^{+h}_H$ 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.

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