Relative plus constructions

TL;DR

This paper introduces a functorial relative plus construction in algebraic topology, generalizing Quillen's plus construction using homology theories and Bousfield localization, with applications to fundamental groups and cell attachment.

Contribution

It constructs a functorial relative plus construction for spaces using homology theories, extending previous plus constructions with new functorial properties and clarifying conditions on groups involved.

Findings

Defines a functorial relative plus construction as a Bousfield localization.

Provides a functorial and well-defined counterpart to cell attachment plus construction.

Clarifies the role of strongly R-perfect groups in characteristic zero.

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