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The book develops the theory of secondary cohomology operations for singular cohomology theory. The author develops the subject in terms of elementary constructions from general homotopy theory. Among many applications considered are the Hopf invariant one theorem (for all primes $p$, including $p = 2$), Browder's theorem on higher Bockstein operations, and cohomology theory of Massey-Peterson fibrations. Numerous examples and exercises help readers to gain a working knowledge of the theory. A summary of more advanced parts of the core material is included in the first chapter. Prerequisite is basic algebraic topology, including the Steenrod operations. The book is written for graduate students and research mathematicians interested in algebraic topology and can be used for self-study or as a textbook for an advanced course on the topic.
The algebra of primary cohomology operations computed by the well-known Steenrod algebra is one of the most powerful tools of algebraic topology. This book computes the algebra of secondary cohomology operations which enriches the structure of the Steenrod algebra in a new and unexpected way. The book solves a long-standing problem on the algebra of secondary cohomology operations by developing a new algebraic theory of such operations. The results have strong impact on the Adams spectral sequence and hence on the computation of homotopy groups of spheres.
Cohomology operations are at the center of a major area of activity in algebraic topology. This treatment explores the single most important variety of operations, the Steenrod squares. It constructs these operations, proves their major properties, and provides numerous applications, including several different techniques of homotopy theory useful for computation. 1968 edition.
Written and revised by D. B. A. Epstein.