The algebra of secondary cohomology operations
type: Item
instance of: scholarly article
title: The algebra of secondary cohomology operations
zbMATH Open document ID: 1091.55001
author: Hans-Joachim Baues
published in: Progress in Mathematics
publication date: 2006-03-28T00:00:00Z
review text: This book develops the algebraic theory of secondary cohomology operations for singular cohomology theory and makes application to the Adams spectral sequence. The book ends with explicit and computer aided calculations. A secondary cohomology operation is a natural transformation which is associated to a sequence with homotopy. See the excellent book written by \textit{J. R. Harper} [Secondary cohomology operations, Graduate Studies in Mathematics 49, Providence, RI: AMS (2002; Zbl 1003.55001)] for the classical definition and the development of this specialized topic in terms of elementary concepts from general homotopy theory. The purpose of this book is the study of the global algebraic structure of secondary operations. For this aim the author introduces the new object \({\mathcal B}\), termed as the algebra of secondary cohomology operations. Secondary operations can be derived from \({\mathcal B}\) in a similar way as Massey products are derived from the structure of differential graded algebra of the cochain complex. This new elaborated theory of secondary classes allows to exhibit the first nontrivial Massey product of the form \(\langle\alpha,\beta, \gamma\rangle\) with \(\alpha, \beta,\gamma\) elements of the Steenrood algebra as well as to prove that the \(E_3\)-term in the Adams spectral sequence is algebraically determined by \({\mathcal B}\). This yields a new upper bound on the homotopy of spheres. Many results of this book rely on the stable track theory of Eilenberg-MacLane spaces. Definitions and more important results are stated in the introduction. The topological construction of both the multiplication and the diagonal in \({\mathcal B}\) and the proof of the uniqueness theorem constitute a substantial amount of work. Essentially all the material of this book is required in the proof of the main result. The book is written for reseach mathematicians interested in secondary cohomology operations and can be used for self-study.
Mathematics Subject Classification ID: 55-02
Mathematics Subject Classification ID: 18-02
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 55T15
Mathematics Subject Classification ID: 55S20
Mathematics Subject Classification ID: 55S30
zbMATH DE Number: 5015524
zbMATH Keywords: Steenrod algebra
zbMATH Keywords: Adams spectral sequence
reviewed by: Jean-Claude Thomas
MaRDI profile type: Publication
Recommended article: Q5860009
Recommended article: Q4552273
Recommended article: Q3027916
Recommended article: The algebra of secondary homotopy operations in ring spectra
Recommended article: Operations on the secondary Hochschild cohomology.
Recommended article: Cohomology operations and algebraic geometry
Recommended article: Secondary Hochschild cohomology and derivations
Recommended article: ON THE ALGEBRA OF OPERATIONS FOR HOPF COHOMOLOGY
Recommended article: Higher-order and secondary Hochschild cohomology
Recommended article: Secondary cohomology and \(k\)-invariants
description: scientific article; zbMATH DE number 5015524