Üzeyir S. - The bounded Euler class

Abstract

Talk

The Euler class associates a homology to a homeomorphism

1. Group cohomology - classifying space model

There are many different ways of defining group cohomology. We’ll be using classifying spaces.

We construct this space as follows: Let the group elements be the vertices. We fill in higher simplices by relations. The result is a space whose first fundamental group is the group.

We next define chains as the set of maps $f:{\Gamma }^{n+1}\to A$ which fulfill $f(g_0,...,g_n)=f(g{g}_0,...,g{g}_n)$. We define a coboundary map which sends $C^n(\Gamma ,\mathbb{R})$ to $C^{n+1}(\Gamma ,\mathbb{R})$. The quotient of kernel and image gives of the $n$-th cohomology. The bar-cohomology removes one dimension in exchange for a more complicated coboundary map. The coboundary maps in the first cohomology however looks like a Quasimorphism.

Example

The first cohomology $H^1(\Gamma ,\mathbb{R})$ is equal to the Homeomorphisms $Hom(\Gamma ,\mathbb{R})$

Let $A$ an abelian group like $\mathbb{R}$ or $\mathbb{Z}$.

Example

The second cohomology parametrises the extensions of the group. We eloborate: A central extension is a short exact sequence $0\to A\stackrel{i}{\to }\tilde{\Gamma }\stackrel{p}{\to }\Gamma \to 1$ where $i(A)$ lies in the Zentrum (Gruppe) of $\tilde{\Gamma }$. If there is a homormophism $s:\Gamma \to \tilde{\Gamma }$ with $o\circ s=id$, then the sequence splits and $\tilde{\Gamma }$ is a direct product of $\Gamma$ and $A$.

We care about all of this, because we will do an extension of a circle group later on.

Example

We show that the second cohomology $H^2(F_n,A)=0$ by showing that there is not central extension.

Where did you learn this style of talk?

2. Euler-class of a group acting on a circle

\newcommand{\H}{\text{Homeo}_{+}(S^{1})} There is a central extension $0\to \mathbb{Z}\to \text{tH}\to \stackrel{˝}{\to }1$ You can now take some subgroup embedded. The embedding defines something called the euler cohomology.

We go over to more fundamental group stuff. Let ${\Sigma }_g$ be a closed, orientable surface of genus $g$. Then its fundamental group has a presentation given by ${\Gamma }_g=⟨a_1,b_1,...,a_g,b_g|{\prod }_{i=1}^g{a}_i,b_i=1⟩$

Theorem

The euler class (defined earlier but I didn’t get how) is a quasimorphism.

Theorem (Wilnor-wood inequality)

For any $\varphi :{\Gamma }_g\to |H$ the bounded Euler class is bounded by $2g-2$

Welche Vorlesung war am wichtigsten, um mit dem ganzen mitzikommen?