The algebra of secondary cohomology operations
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.
scientific article; zbMATH DE number 5015524
scientific article