eBook ISBN:  9781470402501 
Product Code:  MEMO/138/661.E 
List Price:  $73.00 
MAA Member Price:  $65.70 
AMS Member Price:  $43.80 
eBook ISBN:  9781470402501 
Product Code:  MEMO/138/661.E 
List Price:  $73.00 
MAA Member Price:  $65.70 
AMS Member Price:  $43.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 138; 1999; 289 ppMSC: Primary 55; Secondary 18; 19; 20
The memoir presents a systematic study of rational \(S^1\)equivariant cohomology theories, and a complete algebraic model for them. It provides a classification of such cohomology theories in simple algebraic terms and a practical means of calculation. The power of the model is illustrated by analysis of the Segal conjecture, the behaviour of the AtiyahHirzebruch spectral sequence, the structure of \(S^1\)equivariant \(K\)theory, and the rational behaviour of cyclotomic spectra and the topological cyclic homology construction.
ReadershipGraduate students and research mathematicians working in algebraic topology.

Table of Contents

Chapters

0. General introduction

I. The algebraic model of $\mathbb {T}$spectra

1. Introduction to Part I

2. Topological building blocks

3. Maps between $\mathcal {F}$free $\mathbb {T}$spectra

4. Categorical reprocessing

5. Assembly and the standard model

6. The torsion model

II. Change of groups functors in algebra and topology

7. Introduction to Part II

8. Induction, coinduction and geometric fixed points

9. Algebraic inflation and deflation

10. Inflation, LewisMay fixed points and quotients

III. Applications

11. Introduction to Part III

12. Homotopy Mackey functors and related constructions

13. Classical miscellany

14. Cyclic and Tate cohomology

15. Cyclotomic spectra and topological cyclic cohomology

IV. Tensor and Hom in algebra and topology

16. Introduction

17. Torsion functors

18. Torsion functors for the semifree standard model

19. Wide spheres and representing the semifree torsion functor

20. Torsion functors for the full standard model

21. Product functors

22. The tensorHorn adjunction

23. The derived tensorHorn adjunction

24. Smash products, function spectra and LewisMay fixed points


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The memoir presents a systematic study of rational \(S^1\)equivariant cohomology theories, and a complete algebraic model for them. It provides a classification of such cohomology theories in simple algebraic terms and a practical means of calculation. The power of the model is illustrated by analysis of the Segal conjecture, the behaviour of the AtiyahHirzebruch spectral sequence, the structure of \(S^1\)equivariant \(K\)theory, and the rational behaviour of cyclotomic spectra and the topological cyclic homology construction.
Graduate students and research mathematicians working in algebraic topology.

Chapters

0. General introduction

I. The algebraic model of $\mathbb {T}$spectra

1. Introduction to Part I

2. Topological building blocks

3. Maps between $\mathcal {F}$free $\mathbb {T}$spectra

4. Categorical reprocessing

5. Assembly and the standard model

6. The torsion model

II. Change of groups functors in algebra and topology

7. Introduction to Part II

8. Induction, coinduction and geometric fixed points

9. Algebraic inflation and deflation

10. Inflation, LewisMay fixed points and quotients

III. Applications

11. Introduction to Part III

12. Homotopy Mackey functors and related constructions

13. Classical miscellany

14. Cyclic and Tate cohomology

15. Cyclotomic spectra and topological cyclic cohomology

IV. Tensor and Hom in algebra and topology

16. Introduction

17. Torsion functors

18. Torsion functors for the semifree standard model

19. Wide spheres and representing the semifree torsion functor

20. Torsion functors for the full standard model

21. Product functors

22. The tensorHorn adjunction

23. The derived tensorHorn adjunction

24. Smash products, function spectra and LewisMay fixed points