Programma del corso ``Coomologia limitata''





  • 29/2: A short introduction to the course. (Bounded) singular cohomology of topological spaces: basic definitions.\vspace{.1cm}\\ The comparison map. The comparison map is not surjective in general. (Bounded) cohomology of groups with coefficients: basic definitions. A geometric interpretation of group cohomology. Eilenberg-Steenrod's results on the relationship between group cohomology and singular cohomology (proof only sketched). Gromov's results on the coincidence of the bounded cohomology of a space with the bounded cohomology of its fundamental group (without proof).
  • 1/3: The bar resolution. Geometric intepretation of the differential in the bar resolution. (Bounded) cohomology of groups in degree 0 and 1. Exact bounded cohomology. Exact bounded cohomology in degree 2: quasimorphisms. Homogeneous quasimorphisms. The case of abelian groups.
  • 7/3: Non-trivial quasimorphisms on non-abelian free groups. Counting quasimorphisms and Cayley graphs. Some equivalent definitions of amenable groups. Abelian groups are amenable.
  • 8/3: Constructions of amenable groups via extensions and by taking subgroups. Elementary amenable groups. Amenability and Folner sequences. Groups of intermediate growth are amenable. The bounded cohomology of an amenable group with values in a dual Banach module vanishes in positive degree.
  • 14/3: Groups containing non-abelian free groups are not amenable. Von Neumann's conjecture. Bounded cohomology via resolutions: relative injectivity and strong resolutions. The standard resolution is strong and relatively injective. Bounded cochains on the universal covering provide a relatively injective module.
  • 15/3: In the aspherical case, bounded cochains on the universal covering provide a relatively injective strong resolution. Resolutions that compute the norm in bounded cohomology. If X is aspherical, its bounded cohomology is canonically isomorphic to the bounded cohomology of its fundamental group. The non-aspherical case: Ivanov's contracting homotopy. Simply connected spaces have vanishing bounded cohomology with real coefficients.
  • 21/3: The bounded cohomology of a space is isometrically isomorphic to the bounded cohomology of its fundamental group. Amenable actions (on discrete sets). Bounded cochains on amenable sets isometrically compute the bounded cohomology of groups. Gromov's mapping Theorem. Relative bounded cohomology of topological spaces.
  • 22/3: Isometric isomorphism between the relative bounded cohomology and the absolute one for pair of spaces with amenable subspaces. Simplicial volume: definition and first examples. The simplicial volume is multiplicative with respect to finite coverings. The case of maps of degree d. Duality. Closed manifolds with an amenable fundamental group have vanishing simplicial volume.
  • 5/4: A characterization of the vanishing of the simplicial volume in terms of the comparison map. Gromov's proportionality principle (statement in the general case). Nonpositively curved manifolds. Straightening. Continuous (bounded) cohomology. Continuous (bounded) cohomology is isometrically isomorphic to singular (bounded) cohomology for nonpositively curved spaces.
  • 11/4: The volume form. Proof of the proportionality principle for nonpositively curved manifolds. The simplicial volume of hyperbolic manifolds.
  • 12/4: Simplicial volume of negatively curved and of flat manifolds. Stable integral simplicial volume. Gromov's conjecture about the relationship between simplicial volume and Euler characteristic for aspherical manifolds. Simplicial volume of products. Subadditivity of simplicial volume with respect to gluings along amenable boundary components.
  • 18/4: Additivity of the simplicial volume with respect to gluings along pi_1-injective amenable boundary components. Central extensions and cohomology in degree 2.
  • 19/4: Group actions on the circle. The euler class and the bounded euler class. The rotation number of a homeomorphism.
  • 26/4: Semi-conjugacy. Minimal actions. Actions sharing the same bounded euler class are semiconjugate.
  • 2/5: A characterization of actions having a fixed point. Ghys' Theorem: two actions are semi-conjugate if and only if they share the same bounded Euler class. Euler class of circle bundles. A circle bundle over a cellular complex admits a section if and only if its Euler class vanishes.
  • 3/5: A characterization of actions having a fixed point. Ghys' Theorem: two actions are semi-conjugate if and only if they share the same bounded Euler class. Euler class of circle bundles. A circle bundle over a cellular complex admits a section if and only if its Euler class vanishes.
  • 9/5: Every integer is the Euler number of a circle bundle over a surface. Some equivalent definitions of flatness for circle bundles. Relationship between the Euler class of a flat bundle and the Euler class of the associated representation. Milnor-Wood inequalities.