Betti numbers, L²multiplicities and an equivariant approximation theorem

Date:
19.12.2017

Speaker:
Steffen Kionke

Time:
13:30 Uhr

Abstract: In recent years several interesting results in group theory and topology were obtained using socalled L²invariants. In the theory of L²invariants, methods from functional analysis are used to generalize classical homological invariants of finite simplicial complexes to infinite simplicial complexes with a cocompact proper action of a group. For example, the Betti numbers of a finite simplicial complex are the vector space dimensions of the rational homology groups of the complex. In a similar vein, L²Betti numbers are defined as von Neumann dimensions of the socalled L²homology groups of the complex. We will introduce the underlying tools and discuss the definition of L²Betti numbers. In fact, we will see how a modification yields a notion of dimension (and Betti numbers) for every character of the underlying group. Some of these new Betti numbers, the L²multiplicities, turn out to be themselves natural generalizations of classical finite dimensional invariants. In the end, we discuss an approximation theorem which relates the L²multiplicities to their finite dimensional analogues and mention some applications.

Place:
1.067 (20.30)