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On the Algebraic Foundations of Bounded Cohomology

Theo Bühler ETH Zurich, Zurich, Switzerland
Available Formats:
Electronic ISBN: 978-1-4704-0623-3
Product Code: MEMO/214/1006.E
List Price: $74.00 MAA Member Price:$66.60
AMS Member Price: $44.40 Click above image for expanded view On the Algebraic Foundations of Bounded Cohomology Theo Bühler ETH Zurich, Zurich, Switzerland Available Formats:  Electronic ISBN: 978-1-4704-0623-3 Product Code: MEMO/214/1006.E  List Price:$74.00 MAA Member Price: $66.60 AMS Member Price:$44.40
• Book Details

Memoirs of the American Mathematical Society
Volume: 2142011; 97 pp
MSC: Primary 18; Secondary 57; 20; 46;

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.

• Chapters
• Introduction and Main Results
• 1. Triangulated Categories
• 1. Triangulated Categories
• 2. The Derived Category of an Exact Category
• 3. Abstract Truncation: $t$-Structures and Hearts
• 2. Homological Algebra for Bounded Cohomology
• 4. Categories of Banach Spaces
• 5. Derived Categories of Banach $G$-Modules
• A. Appendices
• A. Mapping Cones, Homotopy Push-Outs, Mapping Cylinders
• B. Pull-Back of Exact Structures
• C. Model Categories
• D. Standard Borel $G$-Spaces are Regular
• E. The Existence of Bruhat Functions
• Requests

Review Copy – for reviewers who would like to review an AMS book
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Accessibility – to request an alternate format of an AMS title
Volume: 2142011; 97 pp
MSC: Primary 18; Secondary 57; 20; 46;

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.

• Chapters
• Introduction and Main Results
• 1. Triangulated Categories
• 1. Triangulated Categories
• 2. The Derived Category of an Exact Category
• 3. Abstract Truncation: $t$-Structures and Hearts
• 2. Homological Algebra for Bounded Cohomology
• 4. Categories of Banach Spaces
• 5. Derived Categories of Banach $G$-Modules
• A. Appendices
• A. Mapping Cones, Homotopy Push-Outs, Mapping Cylinders
• B. Pull-Back of Exact Structures
• C. Model Categories
• D. Standard Borel $G$-Spaces are Regular
• E. The Existence of Bruhat Functions
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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