Introduction to ℓ²-invariants, 1st ed. 2019
Lecture Notes in Mathematics Series, Vol. 2247

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Language: English

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This book introduces the reader to the most important concepts and problems in the field of ?²-invariants. After some foundational material on group von Neumann algebras, ?²-Betti numbers are defined and their use is illustrated by several examples. The text continues with Atiyah's question on possible values of ?²-Betti numbers and the relation to Kaplansky's zero divisor conjecture. The general definition of ?²-Betti numbers allows for applications in group theory. A whole chapter is dedicated to Lück's approximation theorem and its generalizations. The final chapter deals with ?²-torsion, twisted variants and the conjectures relating them to torsion growth in homology.
 
The text provides a self-contained treatment that constructs the required specialized concepts from scratch. It comes with numerous exercises and examples, so that both graduate students and researchers will find it useful for self-study or as a basis for an advanced lecture course.
- Introduction. - Hilbert Modules and von Neumann Dimension. - l2-Betti Numbers of CW Complexes. - l2-Betti Numbers of Groups. - Lück’s Approximation Theorem. - Torsion Invariants.

Holger Kammeyer studied Mathematics at Göttingen and Berkeley. After a postdoc position in Bonn he is now based at Karlsruhe Institute of Technology. His research interests range around algebraic topology and group theory. The application of ℓ ²-invariants forms a recurrent theme in his work. He has given introductory courses on the matter on various occasions.

An up-to-date and user-friendly introduction to the rapidly developing field of ℓ²-invariants

Proceeds quickly to the research level after thoroughly covering all the basics

Contains many motivating examples, illustrations, and exercises