\(v_ 1\)-periodicity in the unstable Adams spectral sequence
type: Item
instance of: scholarly article
title: \(v_ 1\)-periodicity in the unstable Adams spectral sequence
zbMATH Open document ID: 0675.55012
DOI: 10.1007/BF02570877
author: Donald M. Davis
author: Mark Mahowald
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183774
review text: In this paper, we show that the unstable Adams spectral sequence (ASS) converging to the 2-primary homotopy groups of the sphere \(S^{2n+1}\) is periodic above a line of slope 1/5, with periodicity operator \(v_ 1\) which moves along lines of slope \(1/2\). We also show that in this region this unstable ASS agrees with the classical ASS for the stable homotopy groups of \(\Sigma^{2n+1}P^{2n}\) with \(P^{2n}\) a real projective space. We give a complete calculation of the \(v_ 1\)-periodic homotopy groups of \(S^{2n+1}\), and we obtain an analogue of Adams' periodicity theorem, which applies to Ext in the category of finite unstable A-modules. Odd-primary analogues of all these results are also obtained.
Mathematics Subject Classification ID: 55T15
Mathematics Subject Classification ID: 55Q40
Mathematics Subject Classification ID: 55S10
zbMATH DE Number: 4105868
zbMATH Keywords: unstable module over the Steenrod algebra
zbMATH Keywords: homotopy groups of sphere
zbMATH Keywords: unstable Adams spectral sequence
zbMATH Keywords: Adams' periodicity theorem
zbMATH Keywords: unstable A-modules
MaRDI profile type: Publication
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OpenAlex ID: W2042370050
description: scientific article; zbMATH DE number 4105868