**Mathematical Surveys and Monographs**

Volume: 227;
2017;
193 pp;
Hardcover

MSC: Primary 18; 20; 55; 57;
Secondary 37; 53

Print ISBN: 978-1-4704-4146-3

Product Code: SURV/227

List Price: $116.00

AMS Member Price: $92.80

MAA Member Price: $104.40

**Electronic ISBN: 9781-4704-4319-1
Product Code: SURV/227.E**

List Price: $116.00

AMS Member Price: $92.80

MAA Member Price: $104.40

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#### Supplemental Materials

# Bounded Cohomology of Discrete Groups

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*Roberto Frigerio*

The theory of bounded cohomology, introduced by Gromov in the late
1980s, has had powerful applications in geometric group theory and the
geometry and topology of manifolds, and has been the topic of active
research continuing to this day. This monograph provides a unified,
self-contained introduction to the theory and its applications, making
it accessible to a student who has completed a first course in
algebraic topology and manifold theory. The book can be used as a
source for research projects for master's students, as a thorough
introduction to the field for graduate students, and as a valuable
landmark text for researchers, providing both the details of the
theory of bounded cohomology and links of the theory to other closely
related areas.

The first part of the book is devoted to settling the fundamental
definitions of the theory, and to proving some of the (by now
classical) results on low-dimensional bounded cohomology and on
bounded cohomology of topological spaces. The second part describes
applications of the theory to the study of the simplicial volume of
manifolds, to the classification of circle actions, to the analysis of
maximal representations of surface groups, and to the study of flat
vector bundles with a particular emphasis on the possible use of
bounded cohomology in relation with the Chern conjecture. Each chapter
ends with a discussion of further reading that puts the presented
results in a broader context.

#### Readership

Graduate students and researchers interested in geometry and topology.

#### Reviews & Endorsements

The author manages a near perfect equilibrium between necessary technicalities (always well motivated) and geometric intuition, leading the readers from the first simple definition to the most striking applications of the theory in 13 very pleasant chapters. This book can serve as an ideal textbook for a graduate topics course on the subject and become the much-needed standard reference on Gromov's beautiful theory.

-- Michelle Bucher

#### Table of Contents

# Table of Contents

## Bounded Cohomology of Discrete Groups

- Cover Cover11
- Title page iii4
- Contents v6
- Introduction ix10
- Chapter 1. (Bounded) cohomology of groups 118
- Chapter 2. (Bounded) cohomology of groups in low degree 926
- 2.1. (Bounded) group cohomology in degree zero and one 926
- 2.2. Group cohomology in degree two 926
- 2.3. Bounded group cohomology in degree two: quasimorphisms 1229
- 2.4. Homogeneous quasimorphisms 1330
- 2.5. Quasimorphisms on abelian groups 1431
- 2.6. The bounded cohomology of free groups in degree 2 1532
- 2.7. Homogeneous 2-cocycles 1633
- 2.8. The image of the comparison map 1835
- 2.9. Further readings 2037

- Chapter 3. Amenability 2340
- Chapter 4. (Bounded) group cohomology via resolutions 3350
- 4.1. Relative injectivity 3350
- 4.2. Resolutions of Γ-modules 3552
- 4.3. The classical approach to group cohomology via resolutions 3855
- 4.4. The topological interpretation of group cohomology revisited 3956
- 4.5. Bounded cohomology via resolutions 4057
- 4.6. Relatively injective normed Γ-modules 4158
- 4.7. Resolutions of normed Γ-modules 4158
- 4.8. More on amenability 4461
- 4.9. Amenable spaces 4562
- 4.10. Alternating cochains 4865
- 4.11. Further readings 4966

- Chapter 5. Bounded cohomology of topological spaces 5370
- 5.1. Basic properties of bounded cohomology of spaces 5370
- 5.2. Bounded singular cochains as relatively injective modules 5471
- 5.3. The aspherical case 5673
- 5.4. Ivanov’s contracting homotopy 5673
- 5.5. Gromov’s Theorem 5875
- 5.6. Alternating cochains 5976
- 5.7. Relative bounded cohomology 6077
- 5.8. Further readings 6279

- Chapter 6. ℓ¹-homology and duality 6582
- 6.1. Normed chain complexes and their topological duals 6582
- 6.2. ℓ¹-homology of groups and spaces 6683
- 6.3. Duality: first results 6784
- 6.4. Some results by Matsumoto and Morita 6885
- 6.5. Injectivity of the comparison map 7087
- 6.6. The translation principle 7188
- 6.7. Gromov equivalence theorem 7390
- 6.8. Further readings 7592

- Chapter 7. Simplicial volume 7794
- 7.1. The case with non-empty boundary 7794
- 7.2. Elementary properties of the simplicial volume 7895
- 7.3. The simplicial volume of Riemannian manifolds 7996
- 7.4. Simplicial volume of gluings 8097
- 7.5. Simplicial volume and duality 8299
- 7.6. The simplicial volume of products 83100
- 7.7. Fiber bundles with amenable fibers 83100
- 7.8. Further readings 84101

- Chapter 8. The proportionality principle 87104
- 8.1. Continuous cohomology of topological spaces 87104
- 8.2. Continuous cochains as relatively injective modules 88105
- 8.3. Continuous cochains as strong resolutions of \R 90107
- 8.4. Straightening in non-positive curvature 92109
- 8.5. Continuous cohomology versus singular cohomology 92109
- 8.6. The transfer map 93110
- 8.7. Straightening and the volume form 95112
- 8.8. Proof of the proportionality principle 97114
- 8.9. The simplicial volume of hyperbolic manifolds 97114
- 8.10. Hyperbolic straight simplices 98115
- 8.11. The seminorm of the volume form 99116
- 8.12. The case of surfaces 100117
- 8.13. The simplicial volume of negatively curved manifolds 100117
- 8.14. The simplicial volume of flat manifolds 101118
- 8.15. Further readings 101118

- Chapter 9. Additivity of the simplicial volume 105122
- Chapter 10. Group actions on the circle 113130
- 10.1. Homeomorphisms of the circle and the Euler class 113130
- 10.2. The bounded Euler class 114131
- 10.3. The (bounded) Euler class of a representation 115132
- 10.4. The rotation number of a homeomorphism 116133
- 10.5. Increasing degree one map of the circle 119136
- 10.6. Semi-conjugacy 120137
- 10.7. Ghys’ Theorem 122139
- 10.8. The canonical real bounded Euler cocycle 126143
- 10.9. Further readings 129146

- Chapter 11. The Euler class of sphere bundles 131148
- 11.1. Topological, smooth and linear sphere bundles 131148
- 11.2. The Euler class of a sphere bundle 133150
- 11.3. Classical properties of the Euler class 136153
- 11.4. The Euler class of oriented vector bundles 138155
- 11.5. The euler class of circle bundles 140157
- 11.6. Circle bundles over surfaces 142159
- 11.7. Further readings 143160

- Chapter 12. Milnor-Wood inequalities and maximal representations 145162
- Chapter 13. The bounded Euler class in higher dimensions and the Chern conjecture 169186
- Index 181198
- List of Symbols 185202
- Bibliography 187204
- Back Cover Back Cover1213