On the Farrell-Jones and related Conjectures

These extended notes are based on a series of six lectures presented at the summer school ``Cohomology of groups and algebraic $K$-theory’’ which took place in Hangzhou, China from July 1 until July 12 in 2007. They give an introduction to the Farrell-Jones and the Baum-Connes Conjecture.

💡 Research Summary

These lecture notes, compiled from a six‑lecture series given at the 2007 Hangzhou summer school, provide an extensive introduction to the Farrell‑Jones and Baum‑Connes conjectures, two central statements linking algebraic K‑ and L‑theory of group rings with the topological K‑theory of reduced group C(^)-algebras. The authors begin by recalling why direct computation of (K_n(RG)), (L_n(RG)) or (K_n(C_r^(G))) is notoriously difficult, and they motivate the use of assembly maps that translate these groups into equivariant homology groups of suitable classifying spaces, which are far more tractable.

The first part of the notes develops the basic algebraic K‑theory needed for the conjectures. Projective class groups (K_0(R)) are defined as the Grothendieck group of finitely generated projective (R)-modules, and the reduced group (\widetilde K_0(R)) is introduced by quotienting out the classes of free modules. The authors emphasize that (\widetilde K_0(R)) measures the failure of projectives to be stably free, and they illustrate the concept with principal ideal domains, Dedekind domains, and group rings (\mathbb ZG). Swan’s theorem that (\widetilde K_0(\mathbb ZG)) is finite for finite (G) is presented, leading to Wall’s finiteness obstruction (o(X)\in\widetilde K_0(\mathbb Z