The Whitehead squares \([l_{2^ i-1}, l_{2^ i-1}]\in \pi_{2^{i+1}-3} (S^{2^ i-1})\) are projective
type: Item
instance of: scholarly article
title: The Whitehead squares \([l_{2^ i-1}, l_{2^ i-1}]\in \pi_{2^{i+1}-3} (S^{2^ i-1})\) are projective
zbMATH Open document ID: 0846.55011
DOI: 10.1016/0040-9383(94)00026-H
author: Wen-Hsiung Lin
published in: Topology
publication date: 1996-09-25T00:00:00Z
review text: A homotopy element in \(\pi_n (X)\) is said to be projective if it can be represented by a map \(f\) that factors through the real projective space \(P^n\). This means \(f = \overline f \sigma : S^n @>\sigma>> P^n @> \overline f >> X\) for some \(\overline f\), where \(\sigma\) is the double covering map. In this paper the following result is proved: Theorem. The Whitehead product \([\iota_n, \iota_n] \in \pi_{2n - 1} (S^n)\) is projective for \(n = 2^i - 1\) and \(i \geq 4\). Discussing the literature the author shows that by this result all \(n\) with projective \([\iota_n, \iota_n]\) are now known. The further values are \(n = 2, 4, 8\), and by trivial reasons \(n = 1, 3, 7\), because in these last 3 cases \([\iota_n, \iota_n] = 0\). The proof of the Theorem uses the \(k\)th 2-primary Brown-Gitler spectrum \(B(k)\) and the Adams spectral sequences for \(\pi^S_* (B(k) \wedge Y)\).
Mathematics Subject Classification ID: 55Q15
zbMATH DE Number: 751256
zbMATH Keywords: Brown-Gitler spectrum
zbMATH Keywords: Adams spectral sequence
zbMATH Keywords: Whitehead product
reviewed by: Werner End
MaRDI profile type: Publication
full work available at URL: 0040-9383(94)00026-h
OpenAlex ID: W2086844326
description: scientific article; zbMATH DE number 751256