On the Adams-Novikov spectral sequence and products of \(\beta\)-elements
type: Item
instance of: scholarly article
title: On the Adams-Novikov spectral sequence and products of \(\beta\)-elements
zbMATH Open document ID: 0605.55013
author: Katsumi Shimomura
published in: Hiroshima Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: Let \(p\geq 5\) be a prime, BP the Brown-Peterson spectrum at p, and \((A,\Gamma)=(BP_*,BP_*BP)\) the Hopf-algebroid corresponding to BP. With common notation, \(A={\mathbb{Z}}_{(p)}[v_ 1,...,v_ n,..]\) and \(\Gamma =A[t_ 1,...,t_ n,...]\). \textit{H. R. Miller}, \textit{D. C. Ravenel} and \textit{W. S. Wilson} [Ann. Math., II. Ser. 106, 469-516 (1977; Zbl 0374.55022)] studied the Adams-Novikov spectral sequence \(Ext^*_{\Gamma}(A,A) \Rightarrow \pi_*(S^ 0)\) by means of a chromatic spectral sequence \(Ext^*_{\Gamma}(A,M^*_ n) \Rightarrow Ext^*_{\Gamma}(A,N^ 0_ n)\); here \(N^ 0_ n=A/(p,v_ 1,...,v_{n-1})\) and, inductively, \(M^ s_ n=v^{-1}_{n+s} N^ s_ n\) with short exact sequences \(N^ s_ n\to M^ s_ n\to N_ n^{s+1}.\) In the paper under review the groups \(Ext^ s_{\Gamma}(A,M^ 1_ 1)\) are computed completely, using a change of rings argument and Ext- calculations over \(B={\mathbb{Z}}_{(p)}[v_ 1,v_ 2,v_ 2^{-1}]\). It turns out that the Adams-Novikov spectral sequence starting with \(E_ 2=Ext^*_{\Gamma}(A,M^ 1_ 1)\) must be trivial. As an application the author obtains new results about the vanishing or nonvanishing of products \(\beta_ s\cdot \beta_{tp/j}\) and \(\beta_{sp/i}\cdot \beta_{tp/j}\) in \(\pi_*(S^ 0)\).
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 55T15
Mathematics Subject Classification ID: 55T25
zbMATH DE Number: 3977930
zbMATH Keywords: stable homotopy groups of spheres
zbMATH Keywords: \(\beta \)-elements
zbMATH Keywords: Brown-Peterson spectrum
zbMATH Keywords: Adams-Novikov spectral sequence
reviewed by: Erich Ossa
MaRDI profile type: Publication
Recommended article: On the generalized Novikov first Ext group modulo a prime
Recommended article: Q4534414
Recommended article: On the chromatic Ext\(^0(M^1_{n-1})\) on \(\Gamma(m+1)\) for an odd prime
Recommended article: On the Adams \(E_2\)-term for elliptic cohomology
Recommended article: Q4288649
Recommended article: An exact sequence related to Adams-Novikov \(E_ 2\)-terms of a cofibering
Recommended article: On a conjecture of Mahowald on the cohomology of finite sub-Hopf algebras of the Steenrod algebra
Recommended article: Q3810549
Recommended article: Inverting the Hopf map
Recommended article: The microstable Adams-Novikov spectral sequence
description: scientific article; zbMATH DE number 3977930