Relative plus constructions

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