The secondary differentials on the third line of the Adams spectral sequence
type: Item
instance of: scholarly article
title: The secondary differentials on the third line of the Adams spectral sequence
zbMATH Open document ID: 1168.55014
published in: Topology and its Applications
publication date: 2009-03-02T00:00:00Z
review text: As the title indicates, the author computes the Adams differential \(d_2\) on the \(E_2\)-term Ex\(^3=\) Ext\(_{A_*}^3({\mathbb Z}/p,{\mathbb Z}/p)\) of the Adams spectral sequence converging to the stable homotopy groups of spheres at a prime greater than three. He decomposes the \(E_2\)-term into three summands \(T\), \(C\) and \(N\), and shows that \(T\) is the image of the Thom map \(\Phi: H^3BP_*\to\) Ex\(^3\) from the \(E_2\)-term of the Adams-Novikov spectral sequence to that of the Adams spectral sequence, and the elements of \(C\) correspond to elements in \(H^sBP_*\) for \(s<3\). This implies that \(d_2\) acts trivially on \(T\oplus C\). On every generator of the other summand \(N\), the differential \(d_2\) is computed to be non-zero. The computation is based on the relation \(d_2(h_{i+1})=a_0b_i\) and the relations on the matrix Massey products.
reviewed by: Katsumi Shimomura
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 55T15
zbMATH DE Number: 5522066
zbMATH Keywords: stable homotopy
zbMATH Keywords: Adams spectral sequence
zbMATH Keywords: Adams-Novikov spectral sequence
zbMATH Keywords: Thom map
author: Xiangjun Wang
MaRDI profile type: Publication
full work available at URL: j.topol.2007.09.014
OpenAlex ID: W2058454118
cites work: Q4067782
cites work: 3-dimensional cohomology of the mod $p$ Steenrod algebra.
cites work: Odd primary infinite families in stable homotopy theory
cites work: A Chain Functor For Bordism
cites work: The factorization of cyclic reduced powers by secondary cohomology operations
cites work: Matric Massey products
cites work: Periodic phenomena in the Adams-Novikov spectral sequence
cites work: The chromatic \(E_1\)-term \(H^0 M_1^2\) for \(p>3\)
cites work: THE METHODS OF ALGEBRAIC TOPOLOGY FROM THE VIEWPOINT OF COBORDISM THEORY
cites work: Q3747338
cites work: Q3844096
cites work: The homotopy groups \(\pi_ * (L_ 2 S^ 0)\)
cites work: The convergence of \(\tilde\alpha_s^{(n)}h_0h_k\)
DOI: 10.1016/J.TOPOL.2007.09.014
description: scientific article; zbMATH DE number 5522066