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 L 2-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 L 2-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.

A comprehensive introduction to the field of L2-Invariants Presents the most recent results and developments Chapters are kept as independent of one other as possible Each chapter includes exercises; hints to their solution are given Contains an extensive index The book will become a standard reference work in the field of L2-invariants Includes supplementary material: sn.pub/extras

Auteur
Univ.-Prof. (em.) Dr. Dr. h.c. Wolfgang Lück, Wirtschaftsprüfer, Technische Universität München, Lehrstuhl für Betriebswirtschaftslehre, Accounting - Auditing - Consulting, Vorsitzender des Vorstandes des International Accounting and Auditing Research Institute, Wiesbaden. Gründer des Arbeitskreises "Externe und Interne Überwachung der Unternehmung" der Schmalenbachgesellschaft für Betriebswirtschaftslehre e.V. Zahlreiche Veröffentlichungen zur Betriebswirtschaftslehre, Wirtschaftsprüfung, Controlling, Interne Revision, Unternehmensberatung, Internationalisierung, Herausgabe diverser Lexika und Schriftenreihen, Beiträge in Sammelwerken und in verschiedenen Zeitschriften.

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.