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On the comparison of stable and unstable \(p\)-completion. (English) Zbl 1410.55005
Completion is a fundamental analytic idea that has long found its way also into the toolbox of algebraists and topologists. The note under review aims at completing the literature on this topic by providing an account of the comparison between unstable and stable \(p\)-completion. The authors show that a \(p\)-complete nilpotent space has a \(p\)-complete suspension spectrum if and only if its homotopy groups are bounded \(p\)-torsion (Theorem 4.7).
The Introduction is followed by a section on \(p\)-completion and one on generalized Serre theory. In Section 4, the authors study the relation between \(p\)-completion for spectra and for spaces under the infinite loop space functor, and then they prove their main theorem. The final Section 5 exhibits rational classes in the stable homotopy groups of certain Eilenberg-Mac Lane spaces.
MSC:
55P60 Localization and completion in homotopy theory
55P42 Stable homotopy theory, spectra
Keywords:
completion
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