Introduction to ℓ²-invariants

• Main Author: Kammeyer, Holger (Author, http://id.loc.gov/vocabulary/relators/aut)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Cham : Springer International Publishing : Imprint: Springer, 2019.
• Edition: 1st ed. 2019.
• Series: Lecture Notes in Mathematics, 2247
• Subjects: Algebraic topology.; Manifolds (Mathematics).; Complex manifolds.; Functional analysis.; Group theory.; Algebraic Topology.; Manifolds and Cell Complexes (incl. Diff.Topology).; Functional Analysis.; Group Theory and Generalizations.
• Physical Description: VIII, 183 p. 37 illus. online resource.
• ISBN: 9783030282974
• ISSN: 0075-8434 ;
• DOI: 10.1007/978-3-030-28297-4

This book introduces the reader to the most important concepts and problems in the field of ℓ²-invariants. After some foundational material on group von Neumann algebras, ℓ²-Betti numbers are defined and their use is illustrated by several examples. The text continues with Atiyah's question on possible values of ℓ²-Betti numbers and the relation to Kaplansky's zero divisor conjecture. The general definition of ℓ²-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 ℓ²-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.