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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.
Content Level »Research
Keywords »Algebra - Algebraic topology - Cohomology theory - Homotopy - Homotopy group - cohomology - homology
Secondary Cohomology and Track Calculus.- Primary Cohomology Operations.- Track Theories and Secondary Cohomology Operations.- Calculus of Tracks.- Stable Linearity Tracks.- The Algebra of Secondary Cohomology Operations.- Products and Power Maps in Secondary Cohomology.- The Algebra Structure of Secondary Cohomology.- The Borel Construction and Comparison Maps.- Power Maps and Power Tracks.- Secondary Relations for Power Maps.- Künneth Tracks and Künneth-Steenrod Operations.- The Algebra of ?-tracks.- Secondary Hopf Algebras.- The Action of ? on Secondary Cohomology.- Interchange and the Left Action.- The Uniqueness of the Secondary Hopf Algebra ?.- Computation of the Secondary Hopf Algebra ?.