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On the Algebraic Foundations of Bounded Cohomology
 
Theo Bühler ETH Zurich, Zurich, Switzerland
Front Cover for On the Algebraic Foundations of Bounded Cohomology
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
Front Cover for On the Algebraic Foundations of Bounded Cohomology
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  • Front Cover for On the Algebraic Foundations of Bounded Cohomology
  • Back Cover for On the Algebraic Foundations of Bounded Cohomology
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.

  • Table of Contents
     
     
    • 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
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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
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