Lecture Notes in Mathematics

Introduction to ℓ²-invariants

Authors: Kammeyer, Holger

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  • An up-to-date and user-friendly introduction to the rapidly developing field of ℓ ²-invariants
  • Proceeds quickly to the research level after thoroughly covering all the basics
  • Contains many motivating examples, illustrations, and exercises
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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.

About the authors

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.

Table of contents (6 chapters)

Table of contents (6 chapters)
  • Introduction

    Pages 1-8

    Kammeyer, Holger

  • Hilbert Modules and von Neumann Dimension

    Pages 9-33

    Kammeyer, Holger

  • ℓ 2-Betti Numbers of CW Complexes

    Pages 35-66

    Kammeyer, Holger

  • ℓ 2-Betti Numbers of Groups

    Pages 67-86

    Kammeyer, Holger

  • Lück’s Approximation Theorem

    Pages 87-126

    Kammeyer, Holger

Buy this book

eBook n/a
  • ISBN 978-3-030-28297-4
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
Softcover n/a
  • ISBN 978-3-030-28296-7
  • Free shipping for individuals worldwide
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Bibliographic Information

Bibliographic Information
Book Title
Introduction to ℓ²-invariants
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
2247
Copyright
2019
Publisher
Springer International Publishing
Copyright Holder
Springer Nature Switzerland AG
eBook ISBN
978-3-030-28297-4
DOI
10.1007/978-3-030-28297-4
Softcover ISBN
978-3-030-28296-7
Series ISSN
0075-8434
Edition Number
1
Number of Pages
VIII, 183
Number of Illustrations
37 b/w illustrations
Topics