In mathematics, the Farrell–Jones conjecture,[1] 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
Kn(RG)
or the L-theory of a group ring
Ln(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
Kn(RG)
Ln(RG)
The Baum–Connes conjecture formulates a similar statement, for the topological K-theory of reduced group
C*
r | |
K | |
*(G)) |
One can find for any ring
R
? | |
KR | |
* |
G(\{ ⋅ \})\cong | |
KR | |
n |
Kn(R[G])
G(\{ ⋅ \})\cong | |
LR | |
n |
Ln(R[G]).
Here
R[G]
The K-theoretic Farrell–Jones conjecture for a group G states that the map
p:EVCYC(G) → \{ ⋅ \}
G(E | |
KR | |
VCYC |
(G)) →
G(\{ ⋅ \})\cong | |
KR | |
* |
K*(R[G]).
Here
EVCYC(G)
The L-theoretic Farrell–Jones conjecture is analogous.
The computation of the algebraic K-groups and the L-groups of a group ring
R[G]
G
X
\emptyset=X-1\subsetX0\subsetX1\subset\ldots\subsetX
Choose
G
G(\coprod | |
KR | |
j\inIi |
G/Hj x Si-1) →
G(\coprod | |
KR | |
j\inIi |
G/Hj x Di) ⊕
G(X | |
KR | |
n |
i-1) →
G(X | |
KR | |
n |
i)
→
G(\coprod | |
KR | |
j\inIi |
G/Hj x Si-1) →
G(\coprod | |
KR | |
j\inIi |
G/Hj x Di) ⊕
G(X | |
KR | |
n-1 |
i-1)
This sequence simplifies to:
oplus | |
j\inIi |
Kn(R[Hj]) ⊕
oplus | |
j\inIi |
Kn-1(RHj) →
oplus | |
j\inIi |
Kn(RHj) ⊕
G(X | |
KR | |
n |
i-1) →
G(X | |
KR | |
n |
i)
→
oplus | |
j\inIi |
Kn-1(RHj) ⊕ oplus
j\inIi |
Kn-2(RHj) →
oplus | |
j\inIi |
Kn-1(RHj) ⊕
G | |
KR | |
n-1 |
(Xi-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
EVCYC(G)
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
\Z
EFIN(\Z)
\R
\Z
1)=K | |
K | |
n(pt) ⊕ |
Kn-1(pt)=Kn(R) ⊕ Kn-1(R).
The Bass-Heller-Swan decomposition gives
\Z(pt)=K | |
K | |
n(R[\Z])\cong |
Kn(R) ⊕ Kn-1(R) ⊕ NKn(R) ⊕ NKn(R).
Indeed one checks that the assembly map is given by the canonical inclusion.
Kn(R) ⊕ Kn-1(R)\hookrightarrowKn(R) ⊕ Kn-1(R) ⊕ NKn(R) ⊕ NKn(R)
So it is an isomorphism if and only if
NKn(R)=0
R
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
? | |
H | |
* |
F
EF(G) → \{ ⋅ \}
G(E | |
H | |
F(G)) → |
G(\{ ⋅ \}) | |
H | |
* |
The group G satisfies the fibered isomorphism conjecture for the family of subgroups F if and only if for any group homomorphism
\alpha:H → G
\alpha*F:=\{H'\leH|\alpha(H)\inF\}
One gets immediately that in this situation
H
\alpha*F
The transitivity principle is a tool to change the family of subgroups to consider. Given two families
F\subsetF'
G
H\inF'
F|H:=\{H'\inF|H'\subsetH\}
G
F
F'
Given any group homomorphism
\alpha\colonH → G
\alpha*F
\alpha
\alpha*VCYC
For suitable
\alpha
There are also connections from the Farrell–Jones conjecture to the Novikov conjecture. It is known that if one of the following maps
G | |
H | |
*(E |
VCYC
\langle-infty\rangle | |
(G),L | |
R) → |
\langle-infty\rangle | |
H | |
R)= |
\langle-infty\rangle | |
L | |
*(RG) |
G | |
H | |
*(E |
FIN(G),Ktop) →
top | |
H | |
*(\{ ⋅ \},K |
)=
* | |
K | |
r(G)) |
is rationally injective, then the Novikov-conjecture holds for
G
The Bost conjecture (named for Jean-Benoît Bost) states that the assembly map
G | |
H | |
*(E |
FIN
top | |
(G),K | |
l1 |
) →
top | |
H | |
l1 |
1(G)) | |
)=K | |
*(l |
is an isomorphism. The ring homomorphism
l1(G) → Cr(G)
1(G)) → | |
K | |
*(l |
K*(Cr(G))
G | |
H | |
*(E |
FIN
top | |
(G),K | |
l1 |
G | |
)=H | |
*(E |
FIN(G),Ktop) →
top | |
H | |
*(\{ ⋅ \},K |
)=K*(Cr(G))
The Kaplansky conjecture predicts that for an integral domain
R
G
R[G]
0,1
p
R[G]
p
K0(R[G])
. Andrew Ranicki. On the Novikov conjecture. Novikov conjectures, index theorems and rigidity, Vol. 1, (Oberwolfach 2003). 272–337. Cambridge University Press. Cambridge, UK.