Cohomology of Diffeomorphism Groups

Organizers: Sam Narmian, Katie Mann

Introduction and content

In this seminar, we'll cover both classical theory (Gelfand-Fuks cohomology, work of Segal, McDuff, Thurston, Morita...) and some recent developments in homology and cohomology of diffeomorphism groups and relationships to classifying foliations, realization problems, and more. Topics will depend on the interests of the participants.

A central goal is to include work from many areas (dynamics, geometry, topology, more specifically homotopy theory, etc...) that are all united by the general theme of understanding diffeomorphism groups and their homology.

Here is a *reading list* (somewhat annotated) and organized into topics we might cover. E-mail one of the organizers if you are interested in presenting a paper, have suggestions to add, or would like to hear any of the optional topics.

Schedule

Talks will be held at Stanford (Friday afternoons, 2:15) and occasionally at Berkeley (time TBD), with schedule depending on availability of the participants.
  • Friday, Sept. 26. 2:15-3:15; 380D. Organizational meeting, and a proof that H_1(Diff_c(M)) = 0. (Notes)

    References:
    1. Haller, Rybicki, Teichmann. "Smooth perfectness for the group of diffoemorphisms" (arxiv)
    2. My exposition of a simplified version of H-R-T's argument, available here . (This is the main content of the talk).
    3. Banyaga, "The structure of classical diffeomorphism groups" (book). Chapter 2 gives some of the history of the problem, and a proof following some of Thurston's arguments.

  • Oct. 3. 2:15-3:15; 380D. Segal's proof of Mather's theorem. (Notes)

    Reference:
    Segal, "Classifying spaces related to foliations" (1978).

  • Introduction to the Godbillon-Vey class. (Notes)