Cohomology operations are at the center of a major area of activity in algebraic topology. This technique for supplementing and enriching the algebraic structure of the cohomology ring has been instrumental to important progress in general homotopy theory and in specific geometric applications. For both theoretical and practical reasons, the formal properties of families of operations have received extensive analysis.
This text focuses on the single most important sort 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. In the later chapters, the authors place special emphasis on calculations in the stable range. The text provides an introduction to methods of Serre, Toda, and Adams, and carries out some detailed computations. Prerequisites include a solid background in cohomology theory and some acquaintance with homotopy groups.

Contenuti

Preface 1. Introduction to cohomology operations 2. Construction of the Steenrod squares 3. Properties of the squares 4. Application: the Hopf invariant 5. Application: vector fields on spheres 6. The Steenrod algebra 7. Exact couples and spectral sequences 8. Fibre spaces 9. Cohomology of K(pi, n) 10. Classes of Abelian groups 11. More about fiber spaces 12. Applications: some homotopy groups of spheres 13. n-Type and Postnikov systems 14. Mapping sequences and homotopy classification 15. Properties of the stable range 16. Higher cohomology operations 17. Compositions in the stable homotopy of spheres 18. The Adams spectral sequence Bibliography Index

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