Farrell–Jones conjecture

In mathematics, the Farrell–Jones conjecture,[1] 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

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

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 ${\displaystyle K_{n}^{top}(C_{*}^{r}(G))}$.

Formulation

One can find for any ring ${\displaystyle R}$ equivariant homology theories ${\displaystyle KR_{*}^{?},LR_{*}^{?}}$ satisfying

${\displaystyle KR_{n}^{G}(\{\cdot \})\cong K_{n}(R[G])}$ respectively ${\displaystyle LR_{n}^{G}(\{\cdot \})\cong L_{n}(R[G]).}$

Here ${\displaystyle R[G]}$ denotes the group ring.

The K-theoretic Farrell–Jones conjecture for a group G states that the map ${\displaystyle p:E_{VCYC}(G)\rightarrow \{\cdot \}}$ induces an isomorphism on homology

${\displaystyle KR_{*}^{G}(p):KR_{*}^{G}(E_{VCYC}(G))\rightarrow KR_{*}^{G}(\{\cdot \})\cong K_{*}(R[G]).}$

Here ${\displaystyle E_{VCYC}(G)}$ 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 ${\displaystyle R[G]}$ is motivated by obstructions living in those groups (see for example Wall's finiteness obstruction, surgery obstruction, Whitehead torsion). So suppose a group ${\displaystyle G}$ satisfies the Farrell–Jones conjecture for algebraic K-theory. Suppose furthermore we have already found a model ${\displaystyle X}$ for the classifying space for virtually cyclic subgroups:

${\displaystyle \emptyset =X^{-1}\subset X^{0}\subset X^{1}\subset \ldots \subset X}$

Choose ${\displaystyle G}$-pushouts and apply the Mayer-Vietoris sequence to them:

${\displaystyle KR_{n}^{G}(\coprod _{j\in I_{i}}G/H_{j}\times S^{i-1})\rightarrow KR_{n}^{G}(\coprod _{j\in I_{i}}G/H_{j}\times D^{i})\oplus KR_{n}^{G}(X^{i-1})\rightarrow KR_{n}^{G}(X^{i})}$${\displaystyle \rightarrow KR_{n-1}^{G}(\coprod _{j\in I_{i}}G/H_{j}\times S^{i-1})\rightarrow KR_{n-1}^{G}(\coprod _{j\in I_{i}}G/H_{j}\times D^{i})\oplus KR_{n-1}^{G}(X^{i-1})}$

This sequence simplifies to:

${\displaystyle \bigoplus _{j\in I_{i}}K_{n}(R[H_{j}])\oplus \bigoplus _{j\in I_{i}}K_{n-1}(RH_{j})\rightarrow \bigoplus _{j\in I_{i}}K_{n}(RH_{j})\oplus KR_{n}^{G}(X^{i-1})\rightarrow KR_{n}^{G}(X^{i})}$${\displaystyle \rightarrow \bigoplus _{j\in I_{i}}K_{n-1}(RH_{j})\oplus \bigoplus _{j\in I_{i}}K_{n-2}(RH_{j})\rightarrow \bigoplus _{j\in I_{i}}K_{n-1}(RH_{j})\oplus KR_{n-1}^{G}(X^{i-1})}$

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 ${\displaystyle E_{VCYC}(G)}$.

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 ${\displaystyle \mathbb {Z} }$. A model for ${\displaystyle E_{FIN}(\mathbb {Z} )}$ is given by the real line ${\displaystyle \mathbb {R} }$, on which ${\displaystyle \mathbb {Z} }$ acts freely by translations. Using the properties of equivariant K-theory we get

${\displaystyle K_{n}^{\mathbb {Z} }(\mathbb {R} )=K_{n}(S^{1})=K_{n}(pt)\oplus K_{n-1}(pt)=K_{n}(R)\oplus K_{n-1}(R).}$

The Bass-Heller-Swan decomposition gives

${\displaystyle K_{n}^{\mathbb {Z} }(pt)=K_{n}(R[\mathbb {Z} ])\cong K_{n}(R)\oplus K_{n-1}(R)\oplus NK_{n}(R)\oplus NK_{n}(R).}$

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

${\displaystyle K_{n}(R)\oplus K_{n-1}(R)\hookrightarrow K_{n}(R)\oplus K_{n-1}(R)\oplus NK_{n}(R)\oplus NK_{n}(R)}$

So it is an isomorphism if and only if ${\displaystyle NK_{n}(R)=0}$, which is the case if ${\displaystyle R}$ 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 [2])
• CAT(0)-groups (see [3])
• solvable groups (see [4])
• mapping class groups (see [5])

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 ${\displaystyle H_{*}^{?}}$. One could say, that a group G satisfies the isomorphism conjecture for a family of subgroups${\displaystyle F}$, if and only if the map induced by the projection ${\displaystyle E_{F}(G)\rightarrow \{\cdot \}}$ induces an isomorphism on homology:

${\displaystyle H_{*}^{G}(E_{F}(G))\rightarrow H_{*}^{G}(\{\cdot \})}$

The group G satisfies the fibered isomorphism conjecture for the family of subgroups F if and only if for any group homomorphism ${\displaystyle \alpha :H\rightarrow G}$ the group H satisfies the isomorphism conjecture for the family

${\displaystyle \alpha ^{*}F:=\{H'\leq H|\alpha (H)\in F\}}$.

One gets immediately that in this situation ${\displaystyle H}$ also satisfies the fibered isomorphism conjecture for the family ${\displaystyle \alpha ^{*}F}$.

Transitivity principle

The transitivity principle is a tool to change the family of subgroups to consider. Given two families ${\displaystyle F\subset F'}$ of subgroups of ${\displaystyle G}$. Suppose every group ${\displaystyle H\in F'}$ satisfies the (fibered) isomorphism conjecture with respect to the family ${\displaystyle F|_{H}:=\{H'\in F|H'\subset H\}}$. Then the group ${\displaystyle G}$ satisfies the fibered isomorphism conjecture with respect to the family ${\displaystyle F}$ if and only if it satisfies the (fibered) isomorphism conjecture with respect to the family ${\displaystyle F'}$.

Isomorphism conjectures and group homomorphisms

Given any group homomorphism ${\displaystyle \alpha \colon H\rightarrow G}$ 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 ${\displaystyle \alpha ^{*}F}$. For example if ${\displaystyle \alpha }$ has finite kernel the family ${\displaystyle \alpha ^{*}VCYC}$ agrees with the family of virtually cyclic subgroups of H.

For suitable ${\displaystyle \alpha }$ 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

${\displaystyle H_{*}^{G}(E_{VCYC}(G),L_{R}^{\langle -\infty \rangle })\rightarrow H_{*}^{G}(\{\cdot \},L_{R}^{\langle -\infty \rangle })=L_{*}^{\langle -\infty \rangle }(RG)}$

${\displaystyle H_{*}^{G}(E_{FIN}(G),K^{top})\rightarrow H_{*}^{G}(\{\cdot \},K^{top})=K_{n}(C_{r}^{*}(G))}$

is rationally injective then the Novikov-conjecture holds for ${\displaystyle G}$. See for example,.[6][7]

Bost conjecture

The Bost conjecture states that the assembly map

${\displaystyle H_{*}^{G}(E_{FIN}(G),K_{l^{1}}^{top})\rightarrow H_{*}^{G}(\{\cdot \},K_{l^{1}}^{top})=K_{*}(l^{1}(G))}$

is an isomorphism. The ring homomorphism ${\displaystyle l^{1}(G)\rightarrow C_{r}(G)}$ induces maps in K-theory ${\displaystyle K_{*}(l^{1}(G))\rightarrow K_{*}(C_{r}(G))}$. Composing the upper assembly map with this homomorphism one gets exactly the assembly map occurring in the Baum-Connes conjecture.

${\displaystyle H_{*}^{G}(E_{FIN}(G),K_{l^{1}}^{top})=H_{*}^{G}(E_{FIN}(G),K^{top})\rightarrow H_{*}^{G}(\{\cdot \},K^{top})=K_{*}(C_{r}(G))}$

Kaplansky conjecture

The Kaplansky conjecture predicts that for an integral domain ${\displaystyle R}$ and a torsionfree group ${\displaystyle G}$ the only idempotents in ${\displaystyle R[G]}$ are ${\displaystyle 0,1}$. Each such idempotent ${\displaystyle p}$ gives a projective ${\displaystyle R[G]}$ module by taking the image of the right multiplication with ${\displaystyle p}$. Hence there seems to be a connection between the Kaplansky conjecture and the vanishing of ${\displaystyle K_{0}(R[G])}$. There are theorems relating the Kaplansky conjecture to the Farrell–Jones conjecture (compare [8]).

References

1. Farrell, F. Thomas, Jones, Lowell E., Isomorphism conjectures in algebraic K-theory, Journal of the American Mathematical Society, v. 6, pp. 249–297, 1993
2. Bartels, Arthur; Lück, Wolfgang; Reich, Holger (2006), "The K-theoretic Farrell-Jones Conjecture for hyperbolic groups", arXiv:math/0609685
3. Bartels, Arthur; Lück, Wolfgang; Reich, Holger (2009), The Borel Conjecture for hyperbolic and CAT(0)-groups, arXiv:0901.0442
4. Wegner, Christian (2013), The Farrell-Jones Conjecture for virtually solvable groups, arXiv:1308.2432, Bibcode:2013arXiv1308.2432W
5. Bartels, Arthur; Bestvina, Mladen (2016), "The Farrell-Jones Conjecture for mapping class groups", arXiv:1606.02844 [math.GT]
6. Ranicki, Andrew A. "On the Novikov conjecture". Novikov conjectures, index theorems and rigidity, Vol. 1, (Oberwolfach 2003). Cambridge, UK: Cambridge University Press. pp. 272–337.
7. Lück, Wolfgang; Reich, Holger (2005). "The Baum-Connes and the Farrell-Jones conjectures in K- and L-theory". Handbook of K-theory. Vol. 1,2. Berlin: Springer. pp. 703–842.
8. Bartels, Arthur; Lück, Wolfgang; Reich, Holger (2008), "On the Farrell-Jones Conjecture and its applications", Journal of Topology, 1 (1): 57–86, arXiv:math/0703548, doi:10.1112/jtopol/jtm008