On The Algebraic Foundations Of Bounded Cohomology
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On The Algebraic Foundations Of Bounded Cohomology
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Author : Theo Bühler
language : en
Publisher: American Mathematical Soc.
Release Date : 2011
On The Algebraic Foundations Of Bounded Cohomology written by Theo Bühler and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Mathematics categories.
It is a widespread opinion among experts that (continuous) bounded cohomology cannot be interpreted as a derived functor and that triangulated methods break down. The author proves that this is wrong. He uses the formalism of exact categories and their derived categories in order to construct a classical derived functor on the category of Banach $G$-modules with values in Waelbroeck's abelian category. This gives us an axiomatic characterization of this theory for free, and it is a simple matter to reconstruct the classical semi-normed cohomology spaces out of Waelbroeck's category. The author proves that the derived categories of right bounded and of left bounded complexes of Banach $G$-modules are equivalent to the derived category of two abelian categories (one for each boundedness condition), a consequence of the theory of abstract truncation and hearts of $t$-structures. Moreover, he proves that the derived categories of Banach $G$-modules can be constructed as the homotopy categories of model structures on the categories of chain complexes of Banach $G$-modules, thus proving that the theory fits into yet another standard framework of homological and homotopical algebra.
On The Algebraic Foundation Of Bounded Cohomology
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Author : Theo Bühler
language : en
Publisher:
Release Date : 2008
On The Algebraic Foundation Of Bounded Cohomology written by Theo Bühler and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with categories.
Bounded Cohomology And Simplicial Volume
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Author : Caterina Campagnolo
language : en
Publisher: Cambridge University Press
Release Date : 2022-11-17
Bounded Cohomology And Simplicial Volume written by Caterina Campagnolo and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-11-17 with Mathematics categories.
Since their introduction by Gromov in the 1980s, the study of bounded cohomology and simplicial volume has developed into an active field connected to geometry and group theory. This monograph, arising from a learning seminar for young researchers working in the area, provides a collection of different perspectives on the subject, both classical and recent. The book's introduction presents the main definitions of the theories of bounded cohomology and simplicial volume, outlines their history, and explains their principal motivations and applications. Individual chapters then present different aspects of the theory, with a focus on examples. Detailed references to foundational papers and the latest research are given for readers wishing to dig deeper. The prerequisites are only basic knowledge of classical algebraic topology and of group theory, and the presentations are gentle and informal in order to be accessible to beginning graduate students wanting to enter this lively and topical field.
Bounded Cohomology Of Discrete Groups
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Author : Roberto Frigerio
language : en
Publisher: American Mathematical Soc.
Release Date : 2017-11-21
Bounded Cohomology Of Discrete Groups written by Roberto Frigerio and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-21 with Algebra, Homological categories.
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.
Cohomology In Banach Algebras
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Author : Barry Edward Johnson
language : en
Publisher: American Mathematical Soc.
Release Date : 1972
Cohomology In Banach Algebras written by Barry Edward Johnson and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1972 with Mathematics categories.
Let [Fraktur capital]A be a Banach algebra, [Fraktur capital]X a two-sided Banach [Fraktur capital] A-module, and define the cohomology groups [italic]H[italic superscript]n([Fraktur capital]A, [Fraktur capital]X) from the complex of bounded cochains in exact analogy to the classical (algebraic) case. This article gives an introduction to several aspects of the resulting theory.
Continuous Bounded Cohomology Of Locally Compact Groups
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Author : Nicolas Monod
language : en
Publisher: Springer
Release Date : 2003-07-01
Continuous Bounded Cohomology Of Locally Compact Groups written by Nicolas Monod and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-07-01 with Mathematics categories.
Recent research has repeatedly led to connections between important rigidity questions and bounded cohomology. However, the latter has remained by and large intractable. This monograph introduces the functorial study of the continuous bounded cohomology for topological groups, with coefficients in Banach modules. The powerful techniques of this more general theory have successfully solved a number of the original problems in bounded cohomology. As applications, one obtains, in particular, rigidity results for actions on the circle, for representations on complex hyperbolic spaces and on Teichmüller spaces. A special effort has been made to provide detailed proofs or references in quite some generality.
Foundations Of Algebraic Topology
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Author : Samuel Eilenberg
language : en
Publisher:
Release Date : 1952
Foundations Of Algebraic Topology written by Samuel Eilenberg and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1952 with Mathematics categories.
"The principal contribution of this book is an axiomatic approach to that part of algebraic topology called Homology theory. It is the oldest and most extensively developed portion of algebraic topology and may be regarded as the main body of the subject" - preface.
Twenty Four Hours Of Local Cohomology
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Author : Srikanth B. Iyengar
language : en
Publisher: American Mathematical Society
Release Date : 2022-07-19
Twenty Four Hours Of Local Cohomology written by Srikanth B. Iyengar and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-07-19 with Mathematics categories.
This book is aimed to provide an introduction to local cohomology which takes cognizance of the breadth of its interactions with other areas of mathematics. It covers topics such as the number of defining equations of algebraic sets, connectedness properties of algebraic sets, connections to sheaf cohomology and to de Rham cohomology, Gröbner bases in the commutative setting as well as for $D$-modules, the Frobenius morphism and characteristic $p$ methods, finiteness properties of local cohomology modules, semigroup rings and polyhedral geometry, and hypergeometric systems arising from semigroups. The book begins with basic notions in geometry, sheaf theory, and homological algebra leading to the definition and basic properties of local cohomology. Then it develops the theory in a number of different directions, and draws connections with topology, geometry, combinatorics, and algorithmic aspects of the subject.
Gromov S Theory Of Multicomplexes With Applications To Bounded Cohomology And Simplicial Volume
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Author : Roberto Frigerio
language : en
Publisher: American Mathematical Society
Release Date : 2023-03-09
Gromov S Theory Of Multicomplexes With Applications To Bounded Cohomology And Simplicial Volume written by Roberto Frigerio and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-03-09 with Mathematics categories.
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Author :
language : en
Publisher: World Scientific
Release Date :
written by and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on with categories.