Course: Bounded cohomology and simplicial volume

  • Program of the course .

    • Lecture class: Tu 10-12 M101.

    Exercise class: Mo 8.30-10 PHY 9.1.10 (Physics). The exercise class is given by Johannes Witzig.

    Bibliography: Frigerio's book that you may find in a preliminary version here and Hatcher's book (online version)

  • Lecture 1: Introduction of the course; definition of singular homology and singular cohomology; Eilenberg-Steenrod axioms for singular (co)homology.

  • Exercise sheet 1: Exercise sheet No. 1

  • Exercise sheet 2: Exercise sheet No. 2 (Deadline: Monday, 28/10/2019)

  • Lecture 2: Introduction to manifolds; local orientation via homology; orientation of a manifold; fundamental class; top dimensional homology group and orientability; Poincaré duality theorem.

  • Exercise sheet 3: Exercise sheet No. 3 (Deadline: Monday, 04/11/2019)

  • Lecture 3: Universal coefficient theorem for (co)homology; Euler characteristic and odd dimensional manifolds; bounded cohomology of topological spaces; comparison map; Eilenberg-Steenrod axioms for bounded cohomology: lack of excision.

  • Exercise sheet 4: Exercise sheet No. 4 (Deadline: Monday, 11/11/2019)

  • Lecture 4: Introduction to (bounded) cohomology of groups with coefficients in (normed) G-modules; homogeneous resolution; comparison map; canonical seminorm; topological interpretation.

  • Exercise sheet 5: Exercise sheet No. 5 (Deadline: Monday, 18/11/2019)

  • Lecture 5: Topological interpretation of cohomology of groups pt.2; aspherical spaces and higher homotopy groups; bar resolution.

  • Exercise sheet 6: Exercise sheet No. 6 (Deadline: Monday, 25/11/2019)

  • Lecture 6: Cohomology of groups in degree 2 vs. central extensions; quasimorphisms; homogeneous quasimorphisms; bounded cohomology in degree 2 for Abelian groups; bounded cohomology in degree 2 for the free non-Abalian group of rank 2; Rolli quasimorphisms.

  • Exercise sheet 7: Exercise sheet No. 7 (Deadline: Monday, 02/12/2019)

  • Lecture 7: Epimorphisms into the free group and bounded cohomology; introduction to amenable groups; examples; equivalent definition of amenable groups in terms of probability measure.

  • Exercise sheet 8: Exercise sheet No. 8 (Deadline: Monday, 09/12/2019)

  • Lecture 8: Properties of amenable groups; elementary amenable groups; groups of intermediate growth; vanishing of the bounded cohomology in positive degree for amenable groups with coefficients in a dual normed module; applications to quasimorphisms; free groups of rank at least 2 are not amenable; Von Neumann's conjecture.

  • Exercise sheet 9: Exercise sheet No. 9 (Deadline: Monday, 16/12/2019)

  • Lecture 9: Bounded cohomology of amenable groups pt.2; Johnson's Theorem; characterization of finite groups via bounded cohomology (without proof).

  • Lecture 10: Resolutions; strong resolutions; relatively injective resolutions; the standard resolution for groups is strong and relatively injective; if the space is aspherical, then the standard resolution of its universal covering is strong and relatively injective.

  • Lecture 11: The bounded cohomology group of a group with coefficients in a module can be computed via strong and relatively injective resolution: the isomorphism is biLipschitz; Special strong and relatively injective resolutions compute isometrically bounded cohomology with the canonical seminorm.

  • Exercise sheet 10: Exercise sheet No. 10 (Deadline: Monday, 13/01/2020)

  • Lecture 12: Bounded cohomology of aspherical spaces: it is isometrically isomorphic to the one of the fundamental group; Ivanov's contracting homotopy: simply connected countable CW-complexes have vanishing bounded cohomology in positive degree (sketch of the argument involving Dold-Thom construction and principal bundles); bounded cohomology of spaces only depends on the fundamental group; Gromov's Mapping Theorem (algebraic and topological version).

  • Exercise sheet 11: Exercise sheet No. 11 (Deadline: Monday, 20/01/2020)

  • Lecture 13: Definition of simplicial volume; examples; simplicial volumes vs. mapping degree; vanishing simplicial volume of spheres and tori; simplicial volume is a homotopy invariant; simplicial volume vs. coverings; duality between simplicial volume and bounded cohomology (no proof yet); simplicial volume of amenable manifolds is zero..

  • Exercise sheet 12: Exercise sheet No. 12 (Deadline: Tuesday, 28/01/2020)

  • Lecture 14: Hahn-Banach theorem (without proof); proof of the duality principle between bounded cohomology and simplicial volume; introduction to the relation between simplicial volume and Riemannian volume; commensurable Riemannian manifolds satisfy the proportionality principle.

  • Exercise sheet 13: Exercise sheet No. 13 (Deadline: Monday, 03/02/2020)

  • Lecture 15: Gromov's proportioanlity principle for non-positively curved manifolds (sketch of the proof); continuous bounded cohomology; straightening; restriction and trasfer maps; volume cocycle; applications to hyperbolic geometry; negatively curved manifolds have positive simplicial volume.
    • Updated 04 Feb 2020
    by Marco Moraschini.