On the unstable homotopy spectral sequences
type: Item
instance of: scholarly article
title: On the unstable homotopy spectral sequences
zbMATH Open document ID: 0607.55010
DOI: 10.1007/BF01171031
author: Nobuyuki Oda
author: Yoshimi Shitanda
published in: Manuscripta Mathematica
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155155
review text: The authors study the spectral sequence that arises from the homotopy classes of maps \([S^ kY,X]\), either by a filtration coming from the skeleta of the domain, or that coming from the Postnikov terms of the range. Such spectral sequences have been studied by several authors. \textit{C. R. F. Maunder} [Proc. Camb. Philos. Soc. 59, 567-574 (1963; Zbl 0116.146)] treated the case where the range is an \(\Omega\)-spectrum. The reviewer [Am. J. Math. 94, 1131-1154 (1972; Zbl 0254.55021)] studied the case of a finite complex, but with the additional module structure over the stable homotopy ring of spheres. Working with connected CW-complexes and simple target spaces, the present authors prove the isomorphism of the spectral sequences, coming from the different filtrations, for \(r>2\). They then give applications, beginning with a theorem about the image of differentials lying in torsion subgroups (related to the finite-order of certain k-invariants), and ending with an application of the famous result of \textit{H. Miller} [Ann. Math., II. Ser. 120, 39-87 (1984; Zbl 0552.55014)].
Mathematics Subject Classification ID: 55T99
Mathematics Subject Classification ID: 55Q05
Mathematics Subject Classification ID: 55S45
zbMATH DE Number: 3982067
zbMATH Keywords: spectral sequence
zbMATH Keywords: homotopy classes of maps
zbMATH Keywords: differentials
zbMATH Keywords: k-invariants
reviewed by: Donald W. Kahn
MaRDI profile type: Publication
cites work: Rationale Homotopietypen
cites work: Homotopy limits, completions and localizations
cites work: Q5725255
cites work: The Sullivan conjecture on maps from classifying spaces
cites work: On rationalized H- and co-H-spaces. With an appendix on decomposable H- and co-H-spaces
cites work: Counting elements in homotopy sets
cites work: On phantom maps and a theorem of H. Miller
description: scientific article; zbMATH DE number 3982067