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Rational \(S^1\)-Equivariant Stable Homotopy Theory
J. P. C. Greenlees, University of Sheffield, England

Memoirs of the American Mathematical Society
1999; 289 pp; softcover
Volume: 138
ISBN-10: 0-8218-1001-4
ISBN-13: 978-0-8218-1001-9
List Price: US$73
Individual Members: US$43.80
Institutional Members: US$58.40
Order Code: MEMO/138/661
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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 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.


Graduate students and research mathematicians working in algebraic topology.

Table of Contents

  • General introduction
Part I. The algebraic model of rational \(\mathbb T\)-spectra
  • 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
Part II. Change of groups functors in algebra and topology
  • Introduction to Part II
  • Induction, coinduction and geometric fixed points
  • Algebraic inflation and deflation
  • Inflation, Lewis-May fixed points and quotients
Part III. Applications
  • Introduction to Part III
  • Homotopy Mackey functors and related constructions
  • Classical miscellany
  • Cyclic and Tate cohomology
  • Cyclotomic spectra and topological cyclic cohomology
Part IV. Tensor and Hom in algebra and topology
  • 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
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