Contents

Chapter 1. Introduction 1

Chapter 2. Basic properties of coarse cohomology 7

2.1. The uniformly homologous categories 7

2.2. Definition of coarse theory 9

2.3. Examples 12

2.4. Product structure on coarse theory 17

Chapter 3. Computation of coarse cohomology 21

3.1. Review of Cech theory 21

3.2. The main theorem 24

3.3. Alternative definitions of coarse theory 30

3.4. When is c an isomorphism? 31

3.5. Bornotopy 33

3.6. Examples 34

Chapter 4. Cyclic cohomology and index theory 39

4.1. Operator algebras 39

4.2. The Connes character map 42

4.3. Index theory 44

4.4. The computation for R

2 m

49

4.5. The general index theorem (even case) 51

4.6. The odd case 52

Chapter 5. Coarse cohomology and coinpactification 57

5.1. Higson's corona space 57

5.2. K+{BH) and K+{yM) 60

5.3. The transgression map 63

5.4. Relation to index theory 66

Chapter 6. Examples and Applications 69

6.1. Index theorem for partitioned manifolds 69

6.2. Ultraspherical manifolds 70