A reformulation of the Arf invariant one mod p problem and applications to atomic spaces
type: Item
instance of: scholarly article
title: A reformulation of the Arf invariant one mod p problem and applications to atomic spaces
zbMATH Open document ID: 0563.55012
published in: Pacific Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: The author gives detailed proofs of some equivalent formulations of the strong form of the Arf invariant. Namely, he discusses equivalent conditions for when the Whitehead square \([\iota_ n,\iota_ n]\) in \(\pi_{2n-1}S^ n\) is divisible by 2. Odd primary analogues are also given. Let W(n) denote the homotopy theoretic fibre of the double suspension \(E^ 2: S^{2n-1}\to \Omega^{2n+1}\). The author proves that if p is an odd prime with all spaces X localized at p, then a self-map f of X with \(X=\Omega^ 3S^{2n+1}\), \(n>1\), or \(X=W(n)\) is a p-local equivalence if and only if f induces an isomorphism on the first non- vanishing homotopy group of X. Similar results hold when X is the fibre of the pth power map on \(\Omega S^{2n+1}\) if \(p\geq 5\) and \(n\neq 1\) or p. Analogous results in case \(p=2\) are given by the author, \textit{E. Campbell}, \textit{F. Peterson}, and the reviewer in ''The space of maps of Moore spaces to spheres'' (to appear).
Mathematics Subject Classification ID: 55Q40
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55Q45
zbMATH DE Number: 3897829
zbMATH Keywords: Arf invariant
zbMATH Keywords: double suspension
zbMATH Keywords: pth power map
author: Paul Selick
MaRDI profile type: Publication
full work available at URL: pjm.1983.108.431
OpenAlex ID: W2043352375
DOI: 10.2140/PJM.1983.108.431
description: scientific article; zbMATH DE number 3897829