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On the Algebraic Foundations of Bounded Cohomology
Theo Bühler, ETH Zurich, Switzerland
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Memoirs of the American Mathematical Society
2011; 97 pp; softcover
Volume: 214
ISBN-10: 0-8218-5311-2
ISBN-13: 978-0-8218-5311-5
List Price: US$70
Individual Members: US$42
Institutional Members: US$56
Order Code: MEMO/214/1006
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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

  • Introduction and main results
Part 1. Triangulated categories
  • Triangulated categories
  • The derived category of an exact category
  • Abstract truncation: \(t\)-structures and hearts
Part 2. Homological algebra for bounded cohomology
  • Categories of Banach spaces
  • Derived categories of Banach \(G\)-modules
Part 3. Appendices
  • Appendix A. Mapping cones, homotopy push-outs, mapping cylinders
  • Appendix B. Pull-back of exact structures
  • Appendix C. Model categories
  • Appendix D. Standard Borel \(G\)-spaces are regular
  • Appendix E. The existence of Bruhat functions
  • Bibliography
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