# Generalized Tate Cohomology

Share this page
*J. P. C. Greenlees; J. P. May*

This book presents a systematic study of a new equivariant cohomology theory \(t(k_G)^*\) constructed from any given equivariant cohomology theory \(k^*_G\), where \(G\) is a compact Lie group. Special cases include Tate-Swan cohomology when \(G\) is finite and a version of cyclic cohomology when \(G = S^1\). The groups \(t(k_G)^*(X)\) are obtained by suitably splicing the \(k\)-homology with the \(k\)-cohomology of the Borel construction \(EG\times _G X\), where \(k^*\) is the nonequivariant cohomology theory that underlies \(k^*_G\). The new theories play a central role in relating equivariant algebraic topology with current areas of interest in nonequivariant algebraic topology. Their study is essential to a full understanding of such “completion theorems” as the Atiyah-Segal completion theorem in \(K\)-theory and the Segal conjecture in cohomotopy. When \(G\) is finite, the Tate theory associated to equivariant \(K\)-theory is calculated completely, and the Tate theory associated to equivariant cohomotopy is shown to encode a mysterious web of connections between the Tate cohomology of finite groups and the stable homotopy groups of spheres.

#### Table of Contents

# Table of Contents

## Generalized Tate Cohomology

- Contents v6 free
- Introduction 110 free
- Part I: General theory 1423 free
- §0. Preamble: definitions, change of universe, and split G-spectra 1423
- §1. Invariance properties of the functors f,c, and t 2029
- §2. Basic properties of the theories represented by f(k[sub(G)]), c(k[sub(G)]), and t(k[sub(G)]) 2332
- §3. Homotopical behavior of the functors f,c, and t 2635
- §4. Completion at the augmentation ideal of the Burnside ring 3039
- §5. Transfer and the fixed point spectra of Tate G-spectra 3746

- Part II: Eilenberg-Maclane G-spectra and the spectral sequences 4352
- §6. Eilenberg-MacLane G-spectra and their associated theories 4352
- §7. Mackey functors and coefficient systems 4857
- §8. Products in the theories associated to Eilenberg-MacLane G-spectra 5261
- §9. Chain level calculation of the coefficient groups 5564
- §10. The f,c, and t Tate Atiyah-Hirzebruch spectral sequences 6170

- Part III: Specializations and calculations 6877
- §11. Tate-Swan cohomology and the spectral sequences for finite groups 6877
- §12. Some remarks on nonequivariant stable homotopy theory 7483
- §13. The Tate K-theory of finite groups and related calculations 7887
- §14. Cyclic cohomology and the spectral sequences for the circle group 8392
- §15. Calculations in homotopy and K-theory for the circle group 9099
- §16. Free G-spheres and periodicity phenomena 94103

- Part IV: The generalization to families 98107
- §17. Families and their f,c, and t G-spectra 98107
- §18. Cohomological and homological completion phenomena 104113
- §19. The generalized Tate G-spectra of periodic K-theory 107116
- §20. Theories associated to Mackey functors and coMackey functors 113122
- §21. Amitsur-Dress-Tate cohomology theories 119128
- §22. The generalized Tate Atiyah-Hirzebruch spectral sequences 124133
- §23. Some calculational methods and examples: groups of order pq 129138
- §24. Equivariant root invariants of stable homotopy groups of spheres 135144
- §25. Proof of the root invariant theorem 142151

- Appendix A: Splittings of rational G-spectra for finite groups G 146155
- Appendix B: Generalized Atiyah-Hirzebruch spectral sequences 154163
- Bibliography 164173
- Index 168177