Farrell–Jones conjecture

In mathematics, the Farrell–Jones conjecture,[1] named after F. Thomas Farrell and Lowell E. Jones, states that certain assembly maps are isomorphisms.

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

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

The K-theoretic Farrell–Jones conjecture for a group G states that the map

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 computation of the algebraic K-groups and the L-groups of a group ring

satisfies the Farrell–Jones conjecture for algebraic K-theory.

for the classifying space for virtually cyclic subgroups: Choose

-pushouts and apply the Mayer-Vietoris sequence to them: This sequence simplifies to: This means that if any group satisfies a certain isomorphism conjecture one can compute its algebraic K-theory (L-theory) only by knowing the algebraic K-Theory (L-Theory) of virtually cyclic groups and by knowing a suitable model for

One might also try to take for example the family of finite subgroups into account.

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.

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.

The class of groups which satisfies the fibered Farrell–Jones conjecture contain the following groups Furthermore the class has the following inheritance properties: Fix an equivariant homology theory

One could say, that a group G satisfies the isomorphism conjecture for a family of subgroups

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 One gets immediately that in this situation

also satisfies the fibered isomorphism conjecture for the family

The transitivity principle is a tool to change the family of subgroups to consider.

satisfies the (fibered) isomorphism conjecture with respect to the family

satisfies the fibered isomorphism conjecture with respect to the family

if and only if it satisfies the (fibered) isomorphism conjecture with respect to the family

and suppose that G"' satisfies the fibered isomorphism conjecture for a family F of subgroups.

Then also H"' satisfies the fibered isomorphism conjecture for the family

agrees with the family of virtually cyclic subgroups of H. For suitable

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

It is known that if one of the following maps is rationally injective, then the Novikov-conjecture holds for

The Kaplansky conjecture predicts that for an integral domain

module by taking the image of the right multiplication with