On the action of \(\beta_1\) in the stable homotopy of spheres at the prime 3
type: Item
instance of: scholarly article
title: On the action of \(\beta_1\) in the stable homotopy of spheres at the prime 3
zbMATH Open document ID: 0997.55017
author: Katsumi Shimomura
published in: Hiroshima Mathematical Journal
publication date: 2002-11-13T00:00:00Z
review text: This paper investigates multiplicative relations for various \(\beta\)-elements in the stable homotopy groups of spheres \(\pi_*S^0\) localized at the prime~\(3\). In particular, for the family \(\beta_{9t+i}\) with \(i=0,1,2,5,6\) and \(t\geq 0\) and Ravenel's element \(\beta_{6/3}\), it shows that \[ \begin{aligned} \beta_{9t+i}\beta_1^i\neq 0&\Longleftrightarrow i<5, \\ \beta_{9t+i}\beta_2\beta_1^j\neq 0&\Longleftrightarrow j<2,\\ \beta_{9t+i}\beta_{6/3}\beta_1^k\neq 0&\Longleftrightarrow k<4. \end{aligned} \] The methods used are calculational and include the traditional ones involving the Smith-Toda complex \(V(1)\), a self map \(B:\Sigma^{144}V(1)\rightarrow V(1)\) inducing multiplication by \(v_2^9\) in \(BP\)-homology, and the chromatic approach to understanding the \(\text{E}_2\)-term of the Adams-Novikov spectral sequence.
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 55Q51
Mathematics Subject Classification ID: 55Q10
zbMATH DE Number: 1539268
zbMATH Keywords: homotopy groups of spheres
zbMATH Keywords: \(L_2\)-localization
zbMATH Keywords: \(\beta\)-elements
zbMATH Keywords: Adams-Novikov spectral sequence
reviewed by: Andrew Baker
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206124690
OpenAlex ID: W2160958739
DOI: 10.32917/HMJ/1206124690
description: scientific article; zbMATH DE number 1539268