On the 2-primary \(v_{1}\)-periodic homotopy groups of spaces
type: Item
instance of: scholarly article
title: On the 2-primary \(v_{1}\)-periodic homotopy groups of spaces
zbMATH Open document ID: 1098.55009
author: A. K. Bousfield
published in: Topology
publication date: 2005-03-08T00:00:00Z
review text: The \(p\)-primary, \(v_1\)-periodic homotopy groups \(v^{-1}_1\pi_*(X)\) are defined in terms of localization of a portion of homotopy groups, see \textit{D. M. Davis} and \textit{M. Mahowald} [Adams Memorial Symposium, London Math Society Lecture Notes, vol. 176, 55--72 (1992; Zbl 0753.55004)]. \textit{A. K. Bousfield} [Topology 38, 1239--1264 (1999; Zbl 0933.57034)] showed that these groups are naturally isomorphic to \(\pi_*\tau_p(\Phi_1(X))\), with \(\tau_p\) being torsion and \(\Phi_1\) a \(v_1\)-stabilization, when \(p\) is an odd prime. The most modest description of this paper might be to call it a extension of those earlier results to the case \(p=2\). But in fact, it is a very impressive tour de force involving Atiyah's real \(K\)-theory, Brown-Comenetz duality, Bott sequences, stable Adams operations, etc. Of the many applications, the author recovers earlier results of Mahowald and Davis. The results are also used by Davis in his important paper [\textit{D. M. Davis}, Homology Homotopy Appl. 5, 297--324 (2003; Zbl 1031.55008)].
reviewed by: Donald W. Kahn
Mathematics Subject Classification ID: 55Q51
Mathematics Subject Classification ID: 55P60
Mathematics Subject Classification ID: 55S25
zbMATH DE Number: 2142203
zbMATH Keywords: \(v_1\)-Stabilizations
zbMATH Keywords: \(p\)-Adic \(K\)-theory
zbMATH Keywords: United \(K\)-theory
zbMATH Keywords: Representation theory
MaRDI profile type: Publication
full work available at URL: j.top.2004.10.003
OpenAlex ID: W2046199623
DOI: 10.1016/J.TOP.2004.10.003
description: scientific article; zbMATH DE number 2142203