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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 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. 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