# On the Algebraic Foundations of Bounded Cohomology

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*Theo Bühler*

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

# Table of Contents

## On the Algebraic Foundations of Bounded Cohomology

- Introduction and Main Results vii8 free
- Part 1. Triangulated Categories 124 free
- Part 2. Homological Algebra for Bounded Cohomology 4568
- Part 3. Appendices 7396