An Overview of Rationalization Theories of Non-simply Connected Spaces and Non-nilpotent Groups

We give an overview of five rationalization theories for spaces (Bousfield-Kan’s ℚ-completion; Sullivan’s rationalization; Bousfield’s homology rationalization; Casacuberta-Peschke’s Ω-rationalization; Gómez-Tato-Halperin-Tanré’s π1-fiberwise rationalization) that extend the classical rationalization of simply connected spaces. We also give an overview of the corresponding rationalization theories for groups (ℚ-completion; Hℚ-localization; Baumslag rationalization) that extend the classical Malcev completion.