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.
Readership
Graduate students and researchers interested in
geometry and topology.