L2-Invariants: Theory and Applications to Geometry and K-Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics (44))

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. It is particularly these interactions with different fields, that make L2-invariants very powerful and exciting. The book presents a comprehensive introduction to this area of research, as well as its most recent results and developments. It is written in a way which enables the reader to pick out his favourite topic and to find the result she or he is interested in quickly and without being forced to go through other material. 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. It is particularly these interactions with different fields that make L2-invariants very powerful and exciting. The book presents a comprehensive introduction to this area of research, as well as its most recent results and developments. It is written in a way which enables the reader to pick out a favourite topic and to find the result she or he is interested in quickly and without being forced to go through other material. 1. L[superscript 2]-betti Numbers -- 2. Novikov-shubin Invariants -- 3. L[superscript 2]-torsion -- 4. L[superscript 2]-invariants Of 3-manifolds -- 5. L[superscript 2]-invariants Of Symmetric Spaces -- 6. L[superscript 2]-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. L[superscript 2]-invariants And The Simplicial Volume -- 15. Survey On Other Topics Related To L[superscript 2]-invariants -- 16. Solutions Of The Exercises. Wolfgang Lück. Includes Bibliographical References (p. [559]-581) And Index.