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

Abstract

We give an overview of four rationalization theories for spaces (Bousfield-Kan's Q-completion; Bousfield's homology rationalization; Casacuberta-Peschke's Ω-rationalization; Gómez-Tato-Halperin-Tanré's π 1 -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.

Formality and finiteness in rational homotopy theory

“…In work of Malcev [93], Lazard [84], and Hilton [71] (see also [22], [72], [74]), this construction was extended to arbitrary nilpotent groups. The Malcev completion functor is left adjoint to the embedding of the category of rational nilpotent groups into the category of nilpotent groups.…”

“…To every space , Sullivan [135], [137], [138] associated in a functorial way its rationalization, denoted Q ; we refer to [21], [53], [51], [120], and [74] for more details on this construction. The rationalization of may be viewed as a geometric realization of the Sullivan minimal model, M ( ), for the PL ( ).…”