L2-Invariants: Theory and Applications to Geometry and K-Theory

In algebraic topology some classical invariants - such as Betti numbers and Reidemeister torsion - are defined for compact spaces and finite group actions. They can be generalized using von Neumann algebras and their traces, and applied also to non-compact spaces and infinite groups. These new L2-invariants contain very interesting and novel information and can be applied to problems arising in topology, K-Theory, differential geometry, non-commutative geometry and spectral theory. The book, written in an accessible manner, presents a comprehensive introduction to this area of research, as well as its most recent results and developments.

Contenu

0. Introduction.- 1. L2-Betti Numbers.- 2. Novikov-Shubin Invariants.- 3. L2-Torsion.- 4. L2-Invariants of 3-Manifolds.- 5. L2-Invariants of Symmetric Spaces.- 6. L2-Invariants for General Spaces with Group Action.- 7. Applications to Groups.- 8. The Algebra of Affiliated Operators.- 9. Middle Algebraic K-Theory and L-Theory of von Neumann Algebras.- 10. The Atiyah Conjecture.- 11. The Singer Conjecture.- 12. The Zero-in-the-Spectrum Conjecture.- 13. The Approximation Conjecture and the Determinant Conjecture.- 14. L2-Invariants and the Simplicial Volume.- 15. Survey on Other Topics Related to L2-Invariants.- 16. Solutions of the Exercises.- References.- Notation.