Farrell–Jones conjecture
Encyclopedia
In mathematics, the Farrell–Jones conjecture, named after F. Thomas Farrell (now at SUNY Binghamton) and Lowell Edwin Jones (now at SUNY Stony Brook) states that certain assembly map
s are isomorphism
s. These maps are given as certain homomorphism
s.
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 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 
.
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.
is motivated by obstructions living in those groups (see for example Wall's finiteness obstruction, surgery obstruction, Whitehead torsion
). 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 (L-theory) only by knowing the algebraic K-Theory (L-Theory) of virtually cyclic groups and by knowing a suitable model for
.
. 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.
Furthermore the class has the following inheritance properties:
. 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 
.
of subgroups of 
. Suppose every group 
satisfies the (fibered) 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 (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 
. 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.
. It is known that if one of the following maps
is rationally injective then the Novikov-conjecture holds for
. See for example,.
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.
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 (compare ).
Assembly map
In mathematics, assembly maps are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left...
s are isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
s. These maps are given as certain homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
s.
The motivation is the interest in the target of the assembly maps; this may be, for instance, the algebraic K-theory
Algebraic K-theory
In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....
of a group ring
Group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
or the L-theory
L-theory
Algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall,with L being used as the letter after K. Algebraic L-theory, also known as 'hermitian K-theory',is important in surgery theory.-Definition:...
of a group ring
,where G is some group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
.
The sources of the assembly maps are equivariant homology theory evaluated on the classifying space
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
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
or .The Baum-Connes conjecture formulates a similar statement, for the topological K-theory
Topological K-theory
In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on general topological spaces, by means of ideas now recognised as K-theory that were introduced by Alexander Grothendieck...
of reduced group
-algebras .Formulation
One can find for any ring equivariant homology theories satisfying-
respectively 
Here
denotes the group ringGroup ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
.
The K-theoretic Farrell–Jones conjecture for a group G states that the map
induces an isomorphism on homologyHere
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 (see for example Wall's finiteness obstruction, surgery obstruction, Whitehead torsionWhitehead torsion
In geometric topology, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau, which is an element in the Whitehead group Wh. These are named after the mathematician J. H. C...
). 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
-pushoutsand 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
.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 ringRegular ring
In commutative algebra, a regular ring is a commutative noetherian ring, such that the localization at every prime ideal is a regular local ring: that is, every such localization has the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension.Jean-Pierre...
. 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)
- CAT(0)-groups (see )
- hyperbolic 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
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 (fibered) 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 (fibered) isomorphism conjecture with respect to the family .Isomorphism conjectures and group homomorphisms
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.Novikov conjecture
There are also connections from the Farrell–Jones conjecture to the Novikov conjectureNovikov conjecture
The Novikov conjecture is one of the most important unsolved problems in topology. It is named for Sergei Novikov who originally posed the conjecture in 1965....
. 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 mapis 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.Kaplansky 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 conjectureKaplansky conjecture
The mathematician Irving Kaplansky is notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras...
to the Farrell–Jones conjecture (compare ).