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

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.