The Algebra of Secondary Cohomology Operations

Hans-Joachim Baues (auth.)9783764374488, 3-7643-7448-9, 3764374497

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.

Table of contents :
Front Matter….Pages 1-2
Primary Cohomology Operations….Pages 3-24
Track Theories and Secondary Cohomology Operations….Pages 25-53
Calculus of Tracks….Pages 55-80
Stable Linearity Tracks….Pages 81-118
The Algebra of Secondary Cohomology Operations….Pages 119-148
Front Matter….Pages 149-150
The Algebra Structure of Secondary Cohomology….Pages 151-175
The Borel Construction and Comparison Maps….Pages 177-190
Power Maps and Power Tracks….Pages 191-220
Secondary Relations for Power Maps….Pages 221-258
Künneth Tracks and Künneth-Steenrod Operations….Pages 259-288
The Algebra of Δ-tracks….Pages 289-317
Secondary Hopf Algebras….Pages 319-334
The Action of β on Secondary Cohomology….Pages 335-348
Interchange and the Left Action….Pages 349-374
The Uniqueness of the Secondary Hopf Algebra β ….Pages 375-397
Computation of the Secondary Hopf Algebra β ….Pages 399-427