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 ${\pi }_1\left(X\right)$, an $h$-acyclic map $X\to X_H^{+h}$ 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.

让$H$是一个连接同调理论。我们构建了一个函子相对加构造作为空间映射类别中的 Bousfield 定位函子。它允许我们联系到一对$(X,H)$, 由一个相连的空间组成$X$和$H$-完全正态子群$H$基本组的${\pi }_{\mathrm{1个}}\left(X\right)$， 一个$H$-无环图$X\to X_H^{+H}$归纳商数$H$关于基本组。我们表明这张地图在$H$-杀死一个子组的非循环映射$H$. 什么时候$H$是具有单位的交换环中的系数的普通同调理论$R$，这为 Broto、Levi 和 Oliver 本着 Quillen 的 plus 构造的精神引入的单元连接构造提供了一个定义明确的函子对应物。我们还阐明了使用强烈$R$-完美组$H$在特征零。