Introduction to bounded cohomology
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Syllabus and Literature
A list of precise references on topics dealt with during the course.
Week

Topics

1 (Sept 23)

Singular homology and singular cohomology of a topological space, bounded cohomology of a topological space

2 (Sept 30)

Module, group ring, cohomology of discrete groups via homogeneous and inhomogeneous cocomplexes, group cohomology in low degree

3 (Oct 7)

Bounded cohomology of discrete groups via homogeneous and inhomogeneous cocomplexes, bounded cohomology in low degree, definition of quasimorphisms

4 (Oct 14)

Relation between quasimorphisms and second bounded cohomology, second bounded cohomology of free groups, homogeneous quasimorphisms

5 (Oct 21)

Example of different behaviour of bounded cohomology with real and with integer coefficients, EilenbergMaclane spaces, topological resolution in group cohomology and in bounded cohomology

6 (Oct 28)

Proof of Theorem: for every K(G,1) space the (bounded) cohomology of the space is (isometrically) isomorphic to the (bounded) cohomology of the group G, definition of amenable groups and first examples

7 (Nov 4)

Amenable groups, characterization of amenable groups via bounded cohomology, consequences

8 (Nov 11)

Mapping Theorem, relatively injective module, examples of relatively injective module, amenability and relatively injectivity, contracting homotopy, relatively injective strong RGresolutions

9 (Nov 18)

Definition of bounded cohomology by means of relatively injective strong resolutions, comparison between the seminorm induced in bounded cohomology by a relatively injective strong resolution and the usual supremum seminorm, Ivanov's criterion about seminorms, idea of the proof of Gromov's Theorem

10 (Nov 25)

Definition of simplicial volume, examples, properties, vanishing results, simplicial volume of hyperbolic manifolds, proportionality principle

11 (Dec 2)

Simplicial volume and bounded cohomology, duality principle and several consequences, basic facts about hyperbolic space

12 (Dec 16)

Simplicial volume of hyperbolic manifolds, actions by homeomorphisms on the circle
