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Hardcover ISBN:  9781470441463 
Product Code:  SURV/227 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470443191 
Product Code:  SURV/227.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470441463 
eBook ISBN:  9781470443191 
Product Code:  SURV/227.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 227; 2017; 193 ppMSC: Primary 18; 20; 55; 57; Secondary 37; 53
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, selfcontained 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 lowdimensional 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.
ReadershipGraduate students and researchers interested in geometry and topology.

Table of Contents

Chapters

(Bounded) cohomology of groups

(Bounded) cohomology of groups in low degree

Amenability

(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

MilnorWood inequalities and maximal representations

The bounded Euler class in higher dimensions and the Chern conjecture


Additional Material

Reviews

The author very well succeeds to present to the reader an overview of all important applications of bounded cohomology.
Thilo Kuessner, Zentralblatt MATH 
This book provides a careful, uniform treatment of the main results of bounded cohomology of discrete groups and its applications, also reflecting recent developments. Moreover, important techniques are highlighted...The 'further readings' sections form a valuable collection of references to current research that will be helpful both for students and for researchers.
Clara Löh, Mathematical Reviews 
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 muchneeded standard reference on Gromov's beautiful theory.
Michelle Bucher


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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, selfcontained 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 lowdimensional 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.

Chapters

(Bounded) cohomology of groups

(Bounded) cohomology of groups in low degree

Amenability

(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

MilnorWood inequalities and maximal representations

The bounded Euler class in higher dimensions and the Chern conjecture

The author very well succeeds to present to the reader an overview of all important applications of bounded cohomology.
Thilo Kuessner, Zentralblatt MATH 
This book provides a careful, uniform treatment of the main results of bounded cohomology of discrete groups and its applications, also reflecting recent developments. Moreover, important techniques are highlighted...The 'further readings' sections form a valuable collection of references to current research that will be helpful both for students and for researchers.
Clara Löh, Mathematical Reviews 
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 muchneeded standard reference on Gromov's beautiful theory.
Michelle Bucher