In mathematics, the Farrell–Jones conjecture, named after F. Thomas Farrell and Lowell Edwin 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 Gwith 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 or. The Baum–Connes conjecture formulates a similar statement, for the topological K-theory of reduced group -algebras.
Formulation
One can find for any ring equivariant homology theories satisfying Here denotes the group ring. The K-theoretic Farrell–Jones conjecture for a group G states that the map induces an isomorphism on homology Here 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
The computation of the algebraic K-groups and the L-groups of a group ring is motivated by obstructions living in those groups. So suppose a group satisfies the Farrell–Jones conjecture for algebraic K-theory. Suppose furthermore we have already found a model 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 only by knowing the algebraic K-Theory of virtually cyclic groups and by knowing a suitable model for.
Why the family of virtually cyclic subgroups ?
One might also try to take for example the family of finite subgroups into account. This family is much easier to handle. Consider the infinite cyclic group. A model for is given by the real line, on which 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. So it is an isomorphism if and only if, which is the case if 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
Meta-conjecture and fibered isomorphism conjectures
Fix an equivariant homology theory. One could say, that a group G satisfies the isomorphism conjecture for a family of subgroups, if and only if the map induced by the projection 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.
Transitivity principle
The transitivity principle is a tool to change the family of subgroups to consider. Given two families of subgroups of. Suppose every group satisfies the isomorphism conjecture with respect to the family. Then the group satisfies the fibered isomorphism conjecture with respect to the family if and only if it satisfies the isomorphism conjecture with respect to the family.
Given any group homomorphism 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. For example if has finite kernel 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.
There are also connections from the Farrell–Jones conjecture to the Novikov conjecture. It is known that if one of the following maps is rationally injective then the Novikov-conjecture holds for. See for example,.
Bost conjecture
The Bost conjecture states that the assembly map is an isomorphism. The ring homomorphism induces maps in K-theory. Composing the upper assembly map with this homomorphism one gets exactly the assembly map occurring in the Baum-Connes conjecture.
The Kaplansky conjecture predicts that for an integral domain and a torsionfree group the only idempotents in are. Each such idempotent gives a projective module by taking the image of the right multiplication with. Hence there seems to be a connection between the Kaplansky conjecture and the vanishing of. There are theorems relating the Kaplansky conjecture to the Farrell–Jones conjecture.
