zbMATH — the first resource for mathematics

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.
55P60 Localization and completion in homotopy theory
55P42 Stable homotopy theory, spectra
Full Text: DOI arXiv
[1] Baer, Reinhold, The subgroup of the elements of finite order of an abelian group, Ann. of Math. (2), 37, 4, 766-781, (1936) · JFM 62.1093.05
[2] Bousfield, A. K., The localization of spaces with respect to homology, Topology, 14, 133-150, (1975) · Zbl 0309.55013
[3] Bousfield, A. K., The localization of spectra with respect to homology, Topology, 18, 4, 257-281, (1979) · Zbl 0417.55007
[4] Bousfield, A. K.; Kan, D. M., Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, v+348 pp., (1972), Springer-Verlag, Berlin-New York · Zbl 0259.55004
[5] S\'eminaire Henri Cartan de l’Ecole Normale Sup\'erieure, 1954/1955. Alg\`ebres d’Eilenberg-MacLane et homotopie, i+234 pp., (1955), Secr\'etariat math\'ematique, 11 rue Pierre Curie, Paris
[6] Eilenberg, Samuel; Mac Lane, Saunders, On the groups \(H(Π,n)\). II. Methods of computation, Ann. of Math. (2), 60, 49-139, (1954) · Zbl 0055.41704
[7] Goodwillie, Thomas G., Calculus. III. Taylor series, Geom. Topol., 7, 645-711, (2003) · Zbl 1067.55006
[8] Hovey, Mark; Strickland, Neil P., Morava \(K\)-theories and localisation, Mem. Amer. Math. Soc., 139, 666, viii+100 pp., (1999) · Zbl 0929.55010
[9] Kuhn, Nicholas; McCarty, Jason, The mod 2 homology of infinite loopspaces, Algebr. Geom. Topol., 13, 2, 687-745, (2013) · Zbl 1333.55008
[10] May, J. P.; Ponto, K., More concise algebraic topology, Chicago Lectures in Mathematics, xxviii+514 pp., (2012), University of Chicago Press, Chicago, IL · Zbl 1249.55001
[11] prst_pcompletion S. Precht Reeh, T. M. Schlank, and N. Stapleton, \emph A formula for \(p\)-completion by way of the Segal conjecture, ArXiv e-prints (2017).
[12] Richter, Birgit, Divided power structures and chain complexes. Alpine perspectives on algebraic topology, Contemp. Math. 504, 237-254, (2009), Amer. Math. Soc., Providence, RI · Zbl 1187.18010
[13] Sullivan, Dennis, Genetics of homotopy theory and the Adams conjecture, Ann. of Math. (2), 100, 1-79, (1974) · Zbl 0355.57007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.