Introduction to ℓ²-invariants [electronic resource] / by Holger Kammeyer.
Material type:
TextSeries: Lecture Notes in Mathematics ; 2247Publisher: Cham : Springer International Publishing : Imprint: Springer, 2019Edition: 1st ed. 2019Description: VIII, 183 p. 37 illus. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9783030282974Subject(s): Algebraic topology | Manifolds (Mathematics) | Complex manifolds | Functional analysis | Group theory | Algebraic Topology | Manifolds and Cell Complexes (incl. Diff.Topology) | Functional Analysis | Group Theory and GeneralizationsAdditional physical formats: Printed edition:: No title; Printed edition:: No titleDDC classification: 514.2 LOC classification: QA612-612.8Online resources: Click here to access online
In:
Springer eBooksSummary: 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.
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.