Farrell–Jones conjecture

In mathematics, the Farrell–Jones conjecture, named after F. Thomas Farrell and Lowell E. Jones, states that certain assembly maps are isomorphisms. These maps are given as certain homomorphisms.

The motivation is the interest in the target of the assembly maps; this may be, for instance, the algebraic K-theory of a group ring

${\displaystyle K_{n}(RG)}$

or the L-theory of a group ring

${\displaystyle L_{n}(RG)}$,

where G is some group.

The sources of the assembly maps are equivariant homology theory evaluated on the classifying space of G with respect to the family of virtually cyclic subgroups of G. So assuming the Farrell–Jones conjecture is true, it is possible to restrict computations to virtually cyclic subgroups to get information on complicated objects such as ${\displaystyle K_{n}(RG)}$ or ${\displaystyle L_{n}(RG)}$.

The Baum–Connes conjecture formulates a similar statement, for the topological K-theory of reduced group ${\displaystyle C^{*}}$-algebras ${\displaystyle K_{n}^{top}(C_{*}^{r}(G))}$.