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 Atiyah-Hirzebruch spectral sequence, the structure of \(S^1\)-equivariant \(K\)-theory, and the rational behaviour of cyclotomic spectra and the topological cyclic homology construction.

Readership

Graduate students and research mathematicians working in algebraic topology.

Table of Contents

• General introduction

• Introduction to Part I
• Topological building blocks
• Maps between \(\mathcal F\)-free \(\mathbb T\)-spectra
• Categorical reprocessing
• Assembly and the standard model
• The torsion model

• Introduction to Part II
• Induction, coinduction and geometric fixed points
• Algebraic inflation and deflation
• Inflation, Lewis-May fixed points and quotients

• Introduction to Part III
• Homotopy Mackey functors and related constructions
• Classical miscellany
• Cyclic and Tate cohomology
• Cyclotomic spectra and topological cyclic cohomology

• Introduction
• Torsion functors
• Torsion functors for the semifree standard model
• Wide spheres and representing the semifree torsion functor
• Torsion functors for the full standard model
• Product functors
• The tensor-Hom adjunction
• The derived tensor-Hom adjunction
• Smash products, function spectra and Lewis-May fixed points
• Appendix A. Mackey functors
• Appendix B. Closed model categories
• Appendix C. Conventions
• Appendix D. Indices
• Appendix E. Summary of models
• Bibliography