Introduction to bounded cohomology
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Syllabus and Literature
A list of precise references on topics dealt with during the course.
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Week
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Topics
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1 (Sept 23)
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Singular homology and singular cohomology of a topological space, bounded cohomology of a topological space
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2 (Sept 30)
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Module, group ring, cohomology of discrete groups via homogeneous and inhomogeneous cocomplexes, group cohomology in low degree
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3 (Oct 7)
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Bounded cohomology of discrete groups via homogeneous and inhomogeneous cocomplexes, bounded cohomology in low degree, definition of quasimorphisms
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4 (Oct 14)
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Relation between quasimorphisms and second bounded cohomology, second bounded cohomology of free groups, homogeneous quasimorphisms
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5 (Oct 21)
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Example of different behaviour of bounded cohomology with real and with integer coefficients, Eilenberg-Maclane spaces, topological resolution in group cohomology and in bounded cohomology
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6 (Oct 28)
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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
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7 (Nov 4)
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Amenable groups, characterization of amenable groups via bounded cohomology, consequences
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8 (Nov 11)
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Mapping Theorem, relatively injective module, examples of relatively injective module, amenability and relatively injectivity, contracting homotopy, relatively injective strong RG-resolutions
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9 (Nov 18)
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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
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10 (Nov 25)
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Definition of simplicial volume, examples, properties, vanishing results, simplicial volume of hyperbolic manifolds, proportionality principle
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11 (Dec 2)
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Simplicial volume and bounded cohomology, duality principle and several consequences, basic facts about hyperbolic space
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12 (Dec 16)
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Simplicial volume of hyperbolic manifolds, actions by homeomorphisms on the circle
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