The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres
type: Item
instance of: scholarly article
title: The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres
zbMATH Open document ID: 0997.55016
published in: Inventiones Mathematicae
publication date: 2002-11-13T00:00:00Z
review text: The authors investigate the Goodwillie tower \(\cdots \rightarrow P_{3}(X) \rightarrow P_{2}(X) \rightarrow P_{1}(X) \simeq Q(X)\) in the case where \(X\) is a sphere localized at a prime \(p\). They show that, if \(X\) is an odd-dimensional sphere localized at \(p\), then the map \(X \rightarrow P_{p^{k}}(X)\) is a \(v_{j}\)-periodic equivalence for all \(k \geq 0\) and all \(0 \leq j \leq k\). Along the way, they show that there are equivalences \(D_{n}(X) \simeq \ast\) for the homotopy fibres \(D_{n}(X)\) of the maps \(P_{n}(X) \rightarrow P_{n-1}(X)\) if \(n\) is not a power of a prime, and that \(D_{p^{k}}(X)\) has only \(p\)-primary torsion. The main result is a substantial generalization of a result of \textit{M. Mahowald} and \textit{R. D. Thompson}, which asserts that \(X \rightarrow P_{2}(X)\) induces an isomorphism in \(v_{1}\)-periodic homotopy if \(X\) is a odd sphere localized at 2 [Topology 31, No. 1, 133-141 (1992; Zbl 0759.55010)]. This paper is beautifully written, and the reviewer will not attempt to paraphrase anything beyond a statement of the main results. The introduction of the paper contains an illuminating account of the genesis and scope of the theory.
Mathematics Subject Classification ID: 55Q40
Mathematics Subject Classification ID: 55P47
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 55S12
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55R12
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 55Q10
Mathematics Subject Classification ID: 55Q51
zbMATH DE Number: 1337386
zbMATH Keywords: homotopy groups of spheres
author: Gregory Arone
author: Mark Mahowald
MaRDI profile type: Publication
full work available at URL: s002220050300
OpenAlex ID: W1997081520
DOI: 10.1007/S002220050300
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description: scientific article; zbMATH DE number 1337386