Isomorphism Conjectures in K- and L-Theory

This monograph is devoted to the Isomorphism Conjectures formulated by Baum and Connes, and by Farrell and Jones. These conjectures are central to the study of the topological K-theory of reduced group C-algebras and the algebraic K- and L*-theory of group rings. They have far-reaching applications in algebra, geometry, group theory, operator theory, and topology.

The book provides a detailed account of the development of these conjectures, their current status, methods of proof, and their wide-ranging implications. These conjectures are not only powerful tools for concrete computations but also play a crucial role in proving other major conjectures. Among these are the Borel Conjecture on the topological rigidity of aspherical closed manifolds, the (stable) GromovLawsonRosenberg Conjecture on the existence of Riemannian metrics with positive scalar curvature on closed Spin-manifolds, Kaplansky's Idempotent Conjecture and the related Kadison Conjecture, the Novikov Conjecture on the homotopy invariance of higher signatures, and conjectures concerning the vanishing of the reduced projective class group and the Whitehead group of torsionfree groups.

The state of the art in Baum-Connes, Farrell-Jones and other conjectures Highly detailed and unified exposition, linking various parts of mathematics An invaluable reference work by a leading expert

Autorentext

Wolfgang Lück has worked on topology, K-theory, and global analysis. He completed his PhD in 1984 under the supervision of Prof. Tammo tom Dieck at Göttingen, where he also obtained the venia legendi in 1989. He has held permanent positions at the universities at Lexington, Mainz and Münster, and is currently Professor at the University of Bonn. He was awarded the Max Planck Research Award in 2003, the Gottfried Wilhelm Leibniz Award in 2008 and the von Staudt Prize in 2025. His other honors include membership of the Leopoldina (since 2010) and of the Nordrhein-Westfälische Akademie der Wissenschaft und der Künste (since 2013), Fellowship of the American Mathematical Society (since 2013), and he was president of the Deutsche Mathematiker Vereinigung during 2009–2010. In addition, he was a Max Planck Fellow from 2013–2023 and obtained an ERC Advanced Grant in 2014. To date, he has directed the theses of 30 PhD students. In Bonn, he was the director of the Hausdorff Institute from 2011–2017 and the spokesperson of the Cluster of Excellence Hausdorff Center for Mathematics from 2019–2022. He has been married to Sibylle Lück since 1984 and has four children and four grandchildren.

Inhalt

1 Introduction.- Part I: Introduction to K- and L-theory.- 2 The Projective Class Group.- 3 The Whitehead Group.- 4 Negative Algebraic K-Theory.- 5 The Second Algebraic K-Group.- 6 Higher Algebraic K-Theory.- 7 Algebraic K-Theory of Spaces.- 8 Algebraic K-Theory of Higher Categories.- 9 Algebraic L-Theory.- 10 Topological K-Theory.- Part II: The Isomorphism Conjectures.- 11 Classifying Spaces for Families.- 12 Equivariant Homology Theories.- 13 The FarrellJones Conjecture.- 14 The BaumConnes Conjecture.- 15 The (Fibered) Meta- and Other Isomorphism Conjectures.- 16 Status.- 17 Guide for Computations.- 18 Assembly maps.- Part III: Methods of Proofs.- 19 Motivation, Summary, and History of the Proofs of the FarrellJones Conjecture.- 20 Conditions on a Group Implying the FarrellJones Conjecture.- 21 Controlled Topology Methods.- 22 Coverings and Flow Spaces.- 23 Transfer.- 24 Higher Categories as Coefficients.- 25 Analytic Methods.- 26 Solutions to the Exercises.