Bounded Cohomology of Discrete Groups

Cover......Page 1
Title page......Page 4
Contents......Page 6
Introduction......Page 10
1.1. Cohomology of groups......Page 18
1.3. Bounded cohomology of groups......Page 20
1.5. The bar resolution......Page 22
1.7. Further readings......Page 23
2.2. Group cohomology in degree two......Page 26
2.3. Bounded group cohomology in degree two: quasimorphisms......Page 29
2.4. Homogeneous quasimorphisms......Page 30
2.5. Quasimorphisms on abelian groups......Page 31
2.6. The bounded cohomology of free groups in degree 2......Page 32
2.7. Homogeneous 2-cocycles......Page 33
2.8. The image of the comparison map......Page 35
2.9. Further readings......Page 37
Chapter 3. Amenability......Page 40
3.1. Abelian groups are amenable......Page 42
3.2. Other amenable groups......Page 43
3.3. Amenability and bounded cohomology......Page 44
3.4. Johnson’s characterization of amenability......Page 45
3.5. A characterization of finite groups via bounded cohomology......Page 46
3.6. Further readings......Page 47
4.1. Relative injectivity......Page 50
4.2. Resolutions of Γ-modules......Page 52
4.3. The classical approach to group cohomology via resolutions......Page 55
4.4. The topological interpretation of group cohomology revisited......Page 56
4.5. Bounded cohomology via resolutions......Page 57
4.7. Resolutions of normed Γ-modules......Page 58
4.8. More on amenability......Page 61
4.9. Amenable spaces......Page 62
4.10. Alternating cochains......Page 65
4.11. Further readings......Page 66
5.1. Basic properties of bounded cohomology of spaces......Page 70
5.2. Bounded singular cochains as relatively injective modules......Page 71
5.4. Ivanov’s contracting homotopy......Page 73
5.5. Gromov’s Theorem......Page 75
5.6. Alternating cochains......Page 76
5.7. Relative bounded cohomology......Page 77
5.8. Further readings......Page 79
6.1. Normed chain complexes and their topological duals......Page 82
6.2. ℓ¹-homology of groups and spaces......Page 83
6.3. Duality: first results......Page 84
6.4. Some results by Matsumoto and Morita......Page 85
6.5. Injectivity of the comparison map......Page 87
6.6. The translation principle......Page 88
6.7. Gromov equivalence theorem......Page 90
6.8. Further readings......Page 92
7.1. The case with non-empty boundary......Page 94
7.2. Elementary properties of the simplicial volume......Page 95
7.3. The simplicial volume of Riemannian manifolds......Page 96
7.4. Simplicial volume of gluings......Page 97
7.5. Simplicial volume and duality......Page 99
7.7. Fiber bundles with amenable fibers......Page 100
7.8. Further readings......Page 101
8.1. Continuous cohomology of topological spaces......Page 104
8.2. Continuous cochains as relatively injective modules......Page 105
8.3. Continuous cochains as strong resolutions of \R......Page 107
8.5. Continuous cohomology versus singular cohomology......Page 109
8.6. The transfer map......Page 110
8.7. Straightening and the volume form......Page 112
8.9. The simplicial volume of hyperbolic manifolds......Page 114
8.10. Hyperbolic straight simplices......Page 115
8.11. The seminorm of the volume form......Page 116
8.13. The simplicial volume of negatively curved manifolds......Page 117
8.15. Further readings......Page 118
9.1. A cohomological proof of subadditivity......Page 122
9.2. A cohomological proof of Gromov additivity theorem......Page 124
9.3. Further readings......Page 127
10.1. Homeomorphisms of the circle and the Euler class......Page 130
10.2. The bounded Euler class......Page 131
10.3. The (bounded) Euler class of a representation......Page 132
10.4. The rotation number of a homeomorphism......Page 133
10.5. Increasing degree one map of the circle......Page 136
10.6. Semi-conjugacy......Page 137
10.7. Ghys’ Theorem......Page 139
10.8. The canonical real bounded Euler cocycle......Page 143
10.9. Further readings......Page 146
11.1. Topological, smooth and linear sphere bundles......Page 148
11.2. The Euler class of a sphere bundle......Page 150
11.3. Classical properties of the Euler class......Page 153
11.4. The Euler class of oriented vector bundles......Page 155
11.5. The euler class of circle bundles......Page 157
11.6. Circle bundles over surfaces......Page 159
11.7. Further readings......Page 160
12.1. Flat sphere bundles......Page 162
12.2. The bounded Euler class of a flat circle bundle......Page 166
12.3. Milnor-Wood inequalities......Page 168
12.4. Flat circle bundles on surfaces with boundary......Page 171
12.5. Maximal representations......Page 179
12.6. Further readings......Page 183
13.1. Ivanov-Turaev cocycle......Page 186
13.2. Representing cycles via simplicial cycles......Page 190
13.3. The bounded Euler class of a flat linear sphere bundle......Page 191
13.4. The Chern conjecture......Page 195
13.5. Further readings......Page 196
Index......Page 198
List of Symbols......Page 202
Bibliography......Page 204
Back Cover......Page 213