## 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, would like to hear any of the optional topics, or just want to be added to our e-mail list for announcements.

### 2015 Schedule

**Monday, April 27.**

1.

**Cohomolgogy of Thompsons group in Homeo(S^1)**

[time and location TBA]

Speaker: Katie Mann

Abstract: This talk will present ideas from Ghys and Sergiescu's beautiful paper "Sur un groupe remarquable de diffeomorphisms du cercle." The "remarkable group" in question is Thompson's group F. Although F often comes up as a counterexample, Ghys-Sergiescu argue that F is not really pathological at all by

a) Computing its cohomology using McDuff--Segal style arguments

b) Finding a (remarkable for many reasons!) doppleganger of the Godbillon-Vey class,

and

c) showing that, though F is defined as a group of (non-differentiable) homeomorphisms, it is actually conjugate to a group of diffeomorphisms.

This talk will cover a) and b).

Reference: Ghys and Sergiescu's paper is available here

**Monday, March 2.**Two talks!

1.

**Characteristic classes of surface bundles and bounded cohomology**

2:30 - 3:30 pm in 384I

Speaker: Sam Nariman

Abstract: We will discuss Morita's theorem on vanishing of MMM-classes for surface bundles with amenable holonomy groups.

2.

**Automatic continuity for homeomorphism groups**

4:00 - 5:00 pm in 380 F.

Speaker: Katie Mann

Abstract: In this talk, I'll demonstrate a remarkable property of the group of homeomorphisms of a manifold: If M is a compact manifold, then any abstract homomorphism from Homeo(M) to any other separable topological group is necessarily continuous. The proof uses a deep result of Edwards and Kirby on homeomorphism groups, but other than that is very hands-on. Time permitting, I'll also discuss some nice applications.

**Monday, February 9.**

1.

**Kister's theorem.**(Notes)

2:30-3:30 pm in 384-I.

Speaker: Sander Kupers

Abstract: We will explain why the embeddings R^n -> R^n are homotopy equivalent to the homeomorphisms R^n -> R^n, a statement whose proof is non-trivial, unlike in the smooth case. This implies that every microbundle with fiber R^n contains a bundle, eg. making tangent bundles to topological manifolds behave a bit more like you expect them to.

2.

**Bott-Haefliger's conjecture.**

4-5 pm in 381-U.

Speaker: Sam Nariman

Abstract: We will sketch the proof of a conjecture due to Bott and Haefliger. They conjectured that the continuous lie algebra cohomology of vector fields of a compact manifold or an open manifold which is the interior of a manifold with boundary (so called Gelfand-Fuks cohomology) is the same as the cohomology of some section space. Haefliger, Trauber, Bott and Segal independently could resolve this conjecture. In this, talk we will sketch cosimplicial techniques due to Segal to explain the idea of the proof.

### Fall 2014 Schedule

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.

Reference:

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

Reference:

Morita "Geometry of characteristic classes", chapter 3.

References:

1. Morita "Geometry of characteristic classes", chapter 3.

2. Ghys L'invariant de Godbillon-Vey

3. Reinhart-Wood A metric formula for the Godbillon-Vey invariant for foliations

3. This picture?

We covered only pages 1-8 of these notes in depth, the rest will be covered in the next lecture.

References:

1. Bott On characteristic classes in the framework of Gelfand-Fuks cohomology

2. Bott Lectures on characteristic classes and foliations.

3. Morita's book

References:

1. Same as the previous lecture, plus

2. The proof of Proposition 2.3 of The geometry of infinite-dimensional groups by Khesin and Wendt.

This proposition shows that (the Thurston cocycle formula for) the Godbillon-vey class generates H^2_{GF}(S^1).

3. Bott "On some formulas for the characteristic classes of group actions"

4. Here are links to some of the original articles:

Bott-Segal The cohomology of the vector fields on a manifold

Gelfand-Fuchs (original paper with spectral sequence and computations) Cohomologies of the Lie algebra of tangential vector fields of a smooth manifold

Instead of our usual meeting, Christian Zickert (University of Maryland) will give a topology seminar talk.

Abstract

(Notes)

This will be independent from the other talks (but closely related, as the proof relies on cohomology of diffeomorphism groups!)

Title:

Point-pushing and Nielsen realization

Abstract:

Let M be a manifold with mapping class group Mod(M). Any subgroup G < Mod(M) can be represented by a collection of diffeomorphisms that form a group up to isotopy. The Nielsen realization problem asks whether or not G can be represented as an honest subgroup of diffeomorphisms. We will discuss a special case of this problem when M is a locally symmetric manifold and G \simeq pi_1(M) is the "point-pushing" subgroup. This generalizes work of Bestvina-Church-Souto.

Reference: Tshishiku's paper, Cohomological obstructions to Nielsen realization