Rational S^{1}-Equivariant Stable Homotopy Theory.

Intro -- Contents -- Chapter 0. General Introduction -- 0.1. Motivation -- 0.2. Overview -- Part I. The algebraic model of rational T-spectra -- Chapter 1. Introduction to Part I -- 1.1. Outline of the algebraic models -- 1.2. Reading Guide for Part I -- 1.3. Haeberly's example -- 1.4. McClure's Chern character isomorphism for F-spaces -- Chapter 2. Topological building blocks -- 2.1. Natural cells and basic cells -- 2.2. Separating isotropy types -- 2.3. The single strand spectra E(H) -- 2.4. Operations: self-maps of E(H) -- Chapter 3. Maps between F-free T-spectra -- 3.1. The Adams short exact sequence -- 3.2. The Whitehead and Hurewicz theorems for T-spectra over H -- 3.3. The injective case -- 3.4. Injectives in the category of torsion Q[c[sub(H)]]-modules -- 3.5. Proof of Theorem 3.1.1 -- Chapter 4. Categorical reprocessing -- 4.1. Recollections about derived categories -- 4.2. Split linear triangulated categories -- 4.3. The uniqueness theorem -- 4.4. The algebraicization of the category of T-spectra over H -- 4.5. The algebraicization of the category of F-spectra -- 4.6. Euler classes revisited -- Chapter 5. Assembly and the standard model -- 5.1. Assembly -- 5.2. The ring t[sup(f)][sub(*)] -- 5.3. Global assembly -- 5.4. The standard model category -- 5.5. Homological algebra in the standard model -- 5.6. The algebraicization of rational T-spectra -- 5.7. Maps between injective spectra -- 5.8. Algebraic cells and spheres -- 5.9. Explicit models -- 5.10. Hausdorff modules -- Chapter 6. The torsion model -- 6.1. Practical calculations -- 6.2. The torsion model -- 6.3. Homological algebra in the torsion model -- 6.4. The derived category of the torsion model -- 6.5. Equivalence of derived categories of standard and torsion models -- 6.6. Relationship to topology -- Part II. Change of groups functors in algebra and topology.

Chapter 7. Introduction to Part II -- 7.1. General outline -- 7.2. Modelling functors changing equivariance -- 7.3. Functors between split triangulated categories -- Chapter 8. Induction, coinduction and geometric fixed points -- 8.1. Forgetful, induction and coinduction functors -- 8.2. The Lewis-May T-fixed point functor -- 8.3. An algebraic model for geometric fixed points -- 8.4. Analysis of geometric fixed points -- Chapter 9. Algebraic inflation and deflation -- 9.1. Algebraic inflation and deflation of omitted f-modules -- 9.2. Inflation and its right adjoint on the torsion model category -- Chapter 10. Inflation, Lewis-May fixed points and quotients -- 10.1. The topological inflation and Lewis-May fixed point functors -- 10.2. Inflation on objects -- 10.3. Correspondence of Algebraic and geometric inflation functors -- 10.4. A direct approach to the Lewis-May fixed point functor -- 10.5. The homotopy type of Lewis-May fixed points -- 10.6. Quotient functors -- Part III. Applications -- Chapter 11. Introduction to Part III -- 11.1. General Outline -- 11.2. Prospects and problems -- Chapter 12. Homotopy Mackey functors and related constructions -- 12.1. The homotopy Mackey functor on A -- 12.2. Eilenberg-MacLane spectra -- 12.3. coMackey functors and spectra representing ordinary homology -- 12.4. Brown-Comenetz spectra -- Chapter 13. Classical miscellany -- 13.1. The collapse of the Atiyah-Hirzebruch spectral sequence -- 13.2. Orbit category resolutions -- 13.3. Suspension spectra -- 13.4. K-theory revisited -- 13.5. The geometric equivariant rational Segal conjecture for T -- Chapter 14. Cyclic and Tate cohomology -- 14.1. Cyclic cohomology -- 14.2. Rational Tate spectra -- 14.3. The integral T-equivariant Tate spectrum for complex K-theory -- Chapter 15. Cyclotomic spectra and topological cyclic cohomology -- 15.1. Cyclotomic spectra.

15.2. Free loop spaces and THH -- 15.3. The definition of topological cyclic homology -- 15.4. Topological cyclic homology of rational spectra -- Part IV. Tensor and Horn in algebra and topology -- Chapter 16. Introduction -- 16.1. General outline -- 16.2. Modelling of the smash product and the function spectrum -- 16.3. Torsion Functors -- 16.4. Modelling of the product spectrum -- 16.5. Modelling the Lewis-May fixed point functor -- 16.6. Genera of small objects -- Chapter 17. Torsion functors -- 17.1. Context -- 17.2. Torsion k[c]-modules -- 17.3. F-finite torsion omitted f-modules -- 17.4. The torsion model -- Chapter 18. Torsion functors for the semifree standard model -- 18.1. Maps out of spheres -- 18.2. The torsion functor -- 18.3. Calculations of the torsion functor -- Chapter 19. Wide spheres and representing the semifree torsion functor -- 19.1. Some indecomposable objects -- 19.2. Wide spheres -- 19.3. Corepresenting the torsion functor -- 19.4. Duals of wide spheres -- 19.5. Representing the torsion functor -- Chapter 20. Torsion functors for the full standard model -- 20.1. Maps out of spheres -- 20.2. The torsion functor -- 20.3. Calculations of the torsion functor -- Chapter 21. Product functors -- 21.1. General discussion -- 21.2. Torsion modules -- 21.3. The torsion model -- 21.4. The standard models -- Chapter 22. The tensor-Horn adjunction -- 22.1. General discussion -- 22.2. Calculations in the semifree standard model -- 22.3. Construction of the semifree Horn functor -- 22.4. Flabbiness of the Horn object -- 22.5. Calculations in the standard model -- 22.6. Construction of the standard Horn functor -- Chapter 23. The derived tensor-Horn adjunction -- 23.1. The case of finite flat dimension -- 23.2. Flat dimension of the semifree category -- 23.3. Flat dimension of the standard model -- 23.4. The case without enough flat objects.

23.5. Torsion k[c]-modules -- Chapter 24. Smash products, function spectra and Lewis-May fixed points -- 24.1. Models of smash products -- 24.2. Models of function spectra -- 24.3. Proof of Theorem 24.2.1 -- 24.4. The Lewis-May K-fixed point functor -- Appendix A. Mackey functors -- Appendix B. Closed model categories -- Appendix C. Conventions -- C.1. Conventions for spaces and spectra -- C 2. Standing conventions -- Appendix D. Indices -- D.1. Index of definitions and terminology -- D.2. Index of notation -- Appendix E. Summary of models -- E.1. The standard model -- E.2. The torsion model -- Bibliography.

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