An overview of rationalization theories of non-simply connected spaces and non-nilpotent groups

TL;DR

This paper reviews five rationalization theories for non-simply connected spaces and groups, extending classical rationalization methods to more complex topological and algebraic structures.

Contribution

It provides a comprehensive overview of various rationalization theories for spaces and groups, highlighting their extensions beyond classical cases.

Findings

Summarizes five space rationalization theories including Bousfield-Kan and Sullivan.

Details three group rationalization theories extending Malcev completion.

Connects space and group rationalizations through their algebraic and topological frameworks.

Abstract

We give an overview of five rationalization theories for spaces (Bousfield-Kan's $\mathbb Q$-completion; Sullivan's rationalization; Bousfield's homology rationalization; Casacuberta-Peschke's $\Omega$-rationalization; G\'{o}mez-Tato-Halperin-Tanr\'{e}'s $\pi_1$-fiberwise rationalization) that extend the classical rationalization of simply connected spaces. We also give an overview of the corresponding rationalization theories for groups ($\mathbb Q$-completion; $H\mathbb Q$-localization; Baumslag rationalization) that extend the classical Malcev completion.