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

2002
ISBN: 978-3-540-43566-2
Verlag: Springer Berlin Heidelberg

Buch, Englisch, Band 44, 595 Seiten, Format (B × H): 160 mm x 241 mm, Gewicht: 1080 g

Reihe: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics

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.

Zielgruppe

Research

Autoren/Hrsg.

Lück, Wolfgang

Fachgebiete

• Mathematik | Informatik Mathematik Topologie Mengentheoretische Topologie
• Mathematik | Informatik Mathematik Geometrie Elementare Geometrie: Allgemeines
• Mathematik | Informatik Mathematik Algebra Homologische Algebra
• Mathematik | Informatik Mathematik Topologie Algebraische Topologie

Inhaltsverzeichnis

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.