Abstract

We give a survey on L2-invariants such as L2-Betti numbers and L2-torsion taking an algebraic point of view. We discuss their basic definitions, properties and applications to problems arising in topology, geometry, group theory and K-theory.

Key words: dimensions theory over finite von Neumann algebras, L2-Betti numbers, Novikov Shubin invariants, L2-torsion, Atiyah Conjecture, Singer Conjecture, algebraic K-theory, geometric group theory, measure theory.

MSC 2000: 57S99, 46L99, 18G15, 19A99, 19B99, 20C07, 20F25.

Introduction

The purpose of this survey article is to present an algebraic approach to L2-invariants such as L2-Betti numbers and L2-torsion. Originally these were defined analytically in terms of heat kernels. Since it was discovered that they have simplicial and homological algebraic counterparts, there have been many applications to various problems in topology, geometry, group theory and algebraic K-theory, which on the first glance do not involve any L2-notions. Therefore it seems to be useful to give a quick and friendly introduction to these notions in particular for mathematicians who have a more algebraic than analytic background. This does not mean at all that the analytic aspects are less important, but for certain applications it is not necessary to know the analytic approach and it is possible and easier to focus on the algebraic aspects. Moreover, questions about L2-invariants of heat kernels such as the Atiyah Conjecture or the zero-in-the-spectrum-Conjecture turn out to be strongly related to algebraic questions about modules over group rings.