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

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.

• 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