Isomorphism Conjectures in K- and L-Theory / by Wolfgang Lück.

Format:

Book

Author/Creator:

Lück, Wolfgang.

Contributor:

Winges, Christoph.

Series:

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 2197-5655 ; 78

Language:

English

Subjects (All):

K-theory.

Algebraic topology.

Manifolds (Mathematics).

Algebra, Homological.

Global analysis (Mathematics).

Topological groups.

Lie groups.

K-Theory.

Algebraic Topology.

Manifolds and Cell Complexes.

Category Theory, Homological Algebra.

Global Analysis and Analysis on Manifolds.

Topological Groups and Lie Groups.

Local Subjects:

K-Theory.

Algebraic Topology.

Manifolds and Cell Complexes.

Category Theory, Homological Algebra.

Global Analysis and Analysis on Manifolds.

Topological Groups and Lie Groups.

Physical Description:

1 online resource (881 pages)

Edition:

1st ed. 2025.

Place of Publication:

Cham : Springer Nature Switzerland : Imprint: Springer, 2025.

Summary:

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) Gromov–Lawson–Rosenberg 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.

Contents:

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 Farrell–Jones Conjecture

14 The Baum–Connes 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 Farrell–Jones Conjecture

20 Conditions on a Group Implying the Farrell–Jones 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.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

3-031-98976-7

9783031989766

OCLC:

1543123295