Bounded Cohomology of Discrete Groups
About this Title
Roberto Frigerio, University of Pisa, Pisa, Italy
Publication: Mathematical Surveys and Monographs
Publication Year: 2017; Volume 227
ISBNs: 978-1-4704-4146-3 (print); 978-1-4704-4319-1 (online)
MathSciNet review: MR3726870
MSC: Primary 57T10; Secondary 20J06, 37C85, 55N10, 57M07
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.
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.
Graduate students and researchers interested in geometry and topology.
Table of Contents
- (Bounded) cohomology of groups
- (Bounded) cohomology of groups in low degree
- (Bounded) group cohomology via resolutions
- Bounded cohomology of topological spaces
- $\ell ^1$-homology and duality
- Simplicial volume
- The proportionality principle
- Additivity of the simplicial volume
- Group actions on the circle
- The Euler class of sphere bundles
- Milnor-Wood inequalities and maximal representations
- The bounded Euler class in higher dimensions and the Chern conjecture