Introduction to (2)-invariants

Holger Kammeyer - Collection Yellow Sale 2023

183 pages, parution le 30/10/2019

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Résumé

The text continues with Atiyah's question on possible values of (2)-Betti numbers and the relation to Kaplansky's zero divisor conjecture.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 Luck'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. - Luck's Approximation Theorem. - Torsion Invariants.

Holger Kammeyer studied Mathematics at Goettingen 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 (2)-invariants forms a recurrent theme in his work. He has given introductory courses on the matter on various occasions.