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Introduction to ℓ²-invariants

  • Book
  • © 2019

Overview

Authors:
  1. Holger Kammeyer

    1. Institute for Algebra and Geometry, Karlsruhe Institute of Technology, Karlsruhe, Germany

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  • 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

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2247)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-viii
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  2. Introduction

    • Holger Kammeyer
    Pages 1-8

  3. Hilbert Modules and von Neumann Dimension

    • Holger Kammeyer
    Pages 9-33

  4. ℓ 2-Betti Numbers of CW Complexes

    • Holger Kammeyer
    Pages 35-66

  5. ℓ 2-Betti Numbers of Groups

    • Holger Kammeyer
    Pages 67-86

  6. Lück’s Approximation Theorem

    • Holger Kammeyer
    Pages 87-126

  7. Torsion Invariants

    • Holger Kammeyer
    Pages 127-163

  8. Back Matter

    Pages 165-183
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Keywords

About this book

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.

Reviews

“This is an excellent introductory book, to be recommended to readers looking for an introduction to the field, as well as those that want to have an overview of recent developments.” (Joan Porti, Mathematical Reviews, September, 2020)

Authors and Affiliations

  • Institute for Algebra and Geometry, Karlsruhe Institute of Technology, Karlsruhe, Germany

    Holger Kammeyer

About the author

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.

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