Contents

Chapter 1. Introduction 1

1.1. Motivation 1

1.2. Forms of K-theory 2

1.3. The complex case 4

1.4. Highlights of Chapter 2 6

1.5. Highlights of Chapter 3 7

1.6. Highlights of Chapter 4 7

1.7. Highlights of Chapter 5 8

1.8. Highlights of Chapter 6 9

1.9. Highlights of Chapter 7 10

1.10. Highlights of Chapter 8 11

1.11. Highlights for elementary abelian groups 11

1.12. Conclusions 13

1.13. History and comparisons with other methods 13

1.14. Comparisons with other theories 15

1.15. Prerequisites 16

1.16. Reading this book 17

1.17. Thanks 17

Chapter 2. K-Theory with Reality 19

2.1. Representation theory 19

2.2. The periodic case 21

2.3. Clifford modules 24

2.4. The connective case 30

2.5. The local cohomology and completion theorems 32

2.6. Calculations 34

2.7. The local cohomology theorem and positive scalar curvature 35

2.8. Periodic K-theory and maps into classical fibrations 36

2.9. Near the edge of periodicity 41

Chapter 3. Descent, Twisting and Periodicity 49

3.1. Fixed points, homotopy fixed points and geometric fixed points 49

3.2. Descent 50

3.3. Statement of periodicity for equivariant connective real K-theory 52

3.4. Periodicity for connective real K-theory 54

3.5. Twistings of kR 56

3.6. Splittings 60

3.7. The RO(Q)-graded homotopy of kR 61

3.8. Six truncations of the periodic Tate spectral sequence 63

3.9. Three twists of the homotopy fixed point spectral sequence 66

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