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

or the L-theory of a group ring

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 subgroupss of G.

The Baum–Connes conjecture formulates a similar statement, for the topological K-theory.

Formulation

equivariant homology theories satisfying

denotes the group ring.

The K-theoretic Farrell–Jones conjecture for a group G induces an isomorphism on homology

denotes the classifying space of the group G with respect to the family of virtually cyclic subgroups, i.e. a G-CW-complex whose isotropy groups are virtually cyclic and for any virtually cyclic subgroup of G the fixed point set is contractible.

The L-theoretic Farrell–Jones conjecture is analogous.

Computational aspects

is motivated by obstructions living in those groups (see for example Wall's finiteness obstruction, surgery obstruction, Whitehead torsion for the classifying space for virtually cyclic subgroups:

-pushouts and apply the Mayer-Vietoris sequence to them:

This sequence simplifies to:

.

Why the family of virtually cyclic subgroups?

acts freely by translations. Using the properties of equivariant K-theory we get

The Bass-Heller-Swan decomposition gives

Indeed one checks that the assembly map is given by the canonical inclusion.

is a regular ring. So in this case one can really use the family of finite subgroups. On the other hand this shows that the isomorphism conjecture for algebraic K-Theory and the family of finite subgroups is not true. One has to extend the conjecture to a larger family of subgroups which contains all the counterexamples. Currently no counterexamples for the Farrell–Jones conjecture are known. If there is a counterexample, one has to enlarge the family of subgroups to a larger family which contains that counterexample.

Inheritances of isomorphism conjectures

The class of groups which satisfies the fibered Farrell–Jones conjecture contain the following groups

• virtually cyclic groups (definition)

• hyperbolic groups (see )

• CAT(0) groups (see )

• solvable groups (see )

• mapping class groups (see )

Furthermore the class has the following inheritance properties:

• Closed under finite products of groups.

• Closed under taking subgroups.

Meta-conjecture and fibered isomorphism conjectures

. One could say, that a group G induces an isomorphism on homology:

The group G satisfies the fibered isomorphism conjecture for the family of subgroups F if and only if for any group homomorphism the group H satisfies the isomorphism conjecture for the family

1. .

.

Transitivity principle

.

Isomorphism conjectures and group homomorphisms

and suppose that G"' satisfies the fibered isomorphism conjecture for a family F of subgroups. Then also agrees with the family of virtually cyclic subgroups of H.

one can use the transitivity principle to reduce the family again.

Connections to other conjectures

Novikov conjecture

There are also connections from the Farrell–Jones conjecture to the Novikov conjecture. It is known that if one of the following maps

. See for example,.

Bost conjecture

The Bost conjecture (named for Jean-Benoît Bost) states that the assembly map

. Composing the upper assembly map with this homomorphism one gets exactly the assembly map occurring in the Baum–Connes conjecture.

Kaplansky conjecture

The Kaplansky conjecture predicts that for an integral domain. There are theorems relating the Kaplansky conjecture to the Farrell–Jones conjecture (compare ).

References