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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$74 Individual Members: US$44.40
Institutional Members: US\$59.20
Order Code: MEMO/214/1006

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.

• 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