COEFFICIENTS FOR
THE FARRELL-JONES CONJECTURE
ARTHUR BARTELS AND HOLGER REICH
Abstract.We introduce the Farrell-Jones Conjecture with coefficients in *
*an
additive category with G-action. This is a variant of the Farrell-Jones *
*Conjec-
ture about the algebraic K- or L-Theory of a group ring RG. It allows to*
* treat
twisted group rings and crossed product rings. The conjecture with coeff*
*icients
is stronger than the original conjecture but it has better inheritance p*
*roper-
ties. Since known proofs using controlled algebra carry over to the set-*
*up with
coefficients we obtain new results about the original Farrell-Jones Conj*
*ecture.
The conjecture with coefficients implies the fibered version of the Farr*
*ell-Jones
Conjecture.
1.Introduction
The Farrell-Jones Conjecture predicts that the algebraic K- or L-theory of a
group ring RG can be described in terms of the K- respectively L-theory of group
rings RH, where H ranges over the family of virtually cyclic subgroups of G,
compare [FJ93 ]. More formally the conjecture says that the assembly map
HG*(EF G; KR ) ! K*(RG),
which assembles K*(RG) from the pieces K*(RH), H 2 F, is an isomorphism if F
is the family of virtually cyclic subgroups of G. Here HG*(-; KR ) is a G-equiv*
*ariant
homology theory and EF G denotes the classifying space for the family of subgro*
*ups
F. A group is called virtually cyclic if it contains a cyclic subgroup of finit*
*e index.
For more explanations see [LR05 ]. There is a similar formulation for L-theory.
The goal of this paper is to define, and prove in many cases, a "Farrell-Jon*
*es
Conjecture with coefficients in an additive category with G-action". A precise
formulation is given in Section 3 and makes essential use of Definition 2.1. T*
*he
conjecture with coefficients is a generalization of the original conjecture. Th*
*e special
case where the additive category with G-action is the category of finitely gene*
*rated
free R-modules equipped with the trivial G-action corresponds to the usual Farr*
*ell-
Jones Conjecture.
The reason for considering this sort of generalization of the Farrell-Jones *
*Con-
jecture is twofold. First, the conjecture with coefficients has better inherit*
*ance
properties.
Theorem 1.1. The Farrell-Jones Conjecture with coefficients in an additive cate-
gory with G-action passes to arbitrary subgroups and more generally it "pulls b*
*ack"
under arbitrary group homomorphisms, where of course the family needs to be pul*
*led
back as well. But even more is true: the injectivity and surjectivity part of t*
*he con-
jecture have these inheritance properties independently.
For a precise statement see Conjecture 3.2, Corollary 4.3 and Theorem 4.5.
Second, we obtain as a special case of the conjecture with coefficients the *
*correct
conjecture for the K-theory of twisted group rings and more generally crossed
product rings. For example let G operate through ring homomorphisms on the ring
____________
Date: October 27, 2005.
1
2 ARTHUR BARTELS AND HOLGER REICH
R, i.e. we have a group homomorphism ff: G ! Aut(R), and let RffG denote the
twisted group ring. Then of course one expects that K*(RffG) can be assembled
from the K*(RffH), where H ranges over the virtually cyclic subgroups of G. Our
conjecture makes this precise, compare Conjecture 6.18.
Recall that crossed product rings play an important role in Moody's Induction
Theorem, see [Moo87 ] and Chapter 8 in [Pas89].
Of course all this is only useful if there are techniques which prove the mo*
*re
general conjecture. Many results on the Farrell-Jones Conjecture (without coeff*
*i-
cients) use the concept of controlled algebra and the description of the assemb*
*ly
map as a "forget-control map". In Section 7 we extend these concepts to the case
with coefficients and formulate a "forget-control version" of the Farrell-Jones*
* Con-
jecture with Coefficients. By simply inspecting existing proofs in the literatu*
*re, see
Section 8, one obtains results about the Farrell-Jones Conjecture with Coeffici*
*ents.
Combined with the inheritance properties we obtain the following new results ab*
*out
the original Farrell-Jones Conjecture.
Corollary 1.2. Let G be the fundamental group of a closed Riemannian manifold
of strictly negative sectional curvature and let be a subgroup of G. Then for*
* every
ring R the assembly map
H*(EVCyc ; KR ) ! K*(R )
is an isomorphism.
This is an extension of the main result from [BR05 ]. It follows from Theore*
*m 4.5,
Remark 3.3 and Theorem 8.1.
Corollary 1.3. Let be a subgroup of a hyperbolic group in the sense of Gromov
and let R be a ring. Then the assembly map
H*(EFin ; KR ) ! K*(R )
is split injective.
This generalizes [Ros04] and uses the fact proven in [RS04 ] that hyperbolic*
* groups
in the sense of Gromov satisfy the assumptions of Theorem 8.2.
Corollary 1.4. Let G be group of finite asymptotic dimension that admits a fini*
*te
model for the classifying space BG. Let be a subgroup of G. Then for every ri*
*ng
R the assembly map
H*(E ; KR ) ! K*(R )
is split injective.
This is a generalization of [Bar03] and follows because of the inheritance p*
*rop-
erties immediately from Theorem 8.3.
In [FJ93 ] Farrell and Jones develop the Fibred Isomorphism Conjecture, a di*
*ffer-
ent generalization of the Farrell-Jones Conjecture, which also has better inher*
*itance
properties. The fibred version is however not so well adapted to proofs which u*
*se
controlled algebra as opposed to controlled topology. The precise relationship *
*be-
tween the Fibered Farrell-Jones Conjecture and the Farrell-Jones Conjecture with
Coefficients is discussed in Remark 4.4.
In the context of topological K-theory of C*-algebras there is an analog to *
*the
Farrell-Jones Conjecture with Coefficients 3.2, the Baum-Connes Conjecture with
Coefficients. Of course the development of a Farrell-Jones Conjecture with Coef*
*fi-
cients was motivated by this analogy. The reader should be warned that the Baum-
Connes Conjecture with Coefficients is know to be wrong [HLS02 ]. At present it
is not clear whether the Farrell-Jones Conjecture with Coefficients 3.2 fails f*
*or the
groups considered in [HLS02 ].
COEFFICIENTS FOR THE FARRELL-JONES CONJECTURE 3
We would like to thank David Rosenthal for helpful discussions on his paper
[Ros04].
The paper is organized as follows :
1. Introduction
2. The category A *G T
3. The Farrell-Jones Conjecture with Coefficients
4. Inheritance properties
5. L-theory
6. Crossed Products
7. Controlled Algebra
8. Applications
9. The Swan group action
References
2. The category A *G T
In the following we will consider additive categories A with a right G-actio*
*n, i.e.
to every group element g we assign an additive covariant functor g*: A ! A, such
that 1* = idand composition of functors (denoted O) relates to multiplication in
the group via g* O h* = (hg)*.
Definition 2.1. Let A be an additive category with a right G-action and let T
be a left G-set. We define a new additive category denoted A *G T as follows. An
object A in A *G T is a family
A = (At)t2T
of objects in A where we require that {t 2 T | At6= 0} is a finite set. A morph*
*ism
OE: A ! B is a collection of morphisms
OE = (OEg,t)(g,t)2GxT,
where
OEg,t:At! g* (Bgt)
is a morphism in A. We require that the set of pairs (g, t) 2 Gx T with OEg,t6=*
* 0 is
finite. Addition of morphisms is defined componentwise. Composition of morphisms
is defined as follows. Let OE = (OEg,t): A ! B and OE0= (OE0g,t): B ! C be give*
*n then
the composition _ = OE0O OE: A ! C has components
X
(2.2) _g,t= h*(OE0k,ht) O OEh,t.
h,k2G g=kh
The reader could now pass immediately to Section 3 in order to see how the
Farrell-Jones Conjecture with Coefficients is formulated.
Remark 2.3 (Naturality of A *G T ). The construction A *G T is natural in A,
i.e. if F :A ! A0 is an additive functor which is equivariant with respect to t*
*he
G-action then
(F *G T (A))t= F (At) and (F *G T (OE))g,t= F (OEg,t)
defines a functor F *G T :A *G T ! A0*G T . If the functor F is an equivalence *
*of
categories, then F *G T is an equivalence of categories.
If f :T ! T 0is a G-equivariant map then
(A *G f(A))t0= t2f-1(t0)At and (A *G f(OE))g,t0= t2f-1(t0)OEg,t
defines (almost) a functor A *G f :A *G T ! A *G T 0. The minor problem, that
this definition involves the choice of a direct sum, can be resolved by redefin*
*ing an
object in A *G T to be an object A as before together with a choice of direct s*
*um
s2SAs for every subset S of T . We prefer to ignore this problem.
4 ARTHUR BARTELS AND HOLGER REICH
Example 2.4 (Trivial action). A ring R can be considered as a category with one
object. Let R denote the additive category obtained from R by formally adding
sums, compare [Mac71 , Exercise 5 on p.194]. This is a small model for the cate*
*gory
of finitely generated free R-modules. If we equip R with the trivial right G-a*
*ction
then there is a canonical identification
R *G T = RGG (T ) ,
where the right hand side is constructed in [DL98 , Section 2] and plays an imp*
*ortant
role in the construction of the assembly map by Davis and L"uck. This identific*
*ation
is natural with respect to maps of G-sets T ! T 0. Let us specialize to the ca*
*se
where T = G=H. Then the inclusion of the full subcategory consisting of objects
A = (AgH ), with AgH = 0 for gH 6= eH, induces an equivalence
(2.5) Ff(RH) ' R *G G=H.
Here Ff(RH) denotes the category of finitely generated free RH-modules.
Example 2.6 (Twisted group rings). Suppose the group G acts via ring homomor-
phisms on R, i.e. we are given a group homomorphism ff: G ! Aut(R). Then the
twisted group ring RffG is defined to be RG as an abelian group with the twisted
multiplication determined by gr = ff(r)g for r 2 R and g 2 G. There is a right
G-operation defined on the category of R-modules, where g*M = resff(g)M, i.e.
g*M has the same underlying abelian group but the R-module structure is twisted
by ff, i.e. r .resff(g)Mm = ff(g)(r)m for r 2 R and m 2 M. Let Ff(R) denote a
small model for the category of finitely generated free right R-modules. One can
arrange that Ff(R) is closed under the G-operation. We show in Section 6 that
there is an equivalence of categories
Ff(R) *G pt' Ff(RffG)
and that more generally
Ff(R) *G G=H ' Ff(Rff|HH).
Example 2.7 (Group extensions). Suppose K is a normal subgroup of and let
p: ! =K = G denote the quotient homomorphism. If the group extension
1 ! K ! ! G ! 1 splits we can choose a group homomorphism s: G !
such that p O s = id. If we define ff(fl): RK ! RK as conjugation with s(fl) we
see that the group ring R can be written as a twisted group ring RKffG, compare
Example 2.6. If however the extension is non-split and s is only a set-theoreti*
*cal
section (with s(1) = 1) then the group ring R is a crossed product ring
R = RKff,oG,
and no longer a twisted group ring, compare [Pas89] and Section 6 below. In
particular fl 7! ff(fl) no longer defines an action of G = =K on RK and correc*
*tion
terms, expressible in terms of the cocycle o(fl, fl0) = s(fl)s(fl0)s(flfl0)-1 w*
*ill appear.
The language developed above absorbs these extra difficulties. We will see t*
*hat
we can work with an honest action if we are working in the context of actions on
additive categories. By (2.5)we have
Ff(RK) '-!R * =K.
The category on the right should be thought of as a "fat" version of the catego*
*ry
Ff(RK) of finitely generated free RK-modules, which has the advantage that it
carries an honest naturally defined right =K-action. Now applying - * =K pt
COEFFICIENTS FOR THE FARRELL-JONES CONJECTURE 5
should be thought of as forming a "fattened" twisted group ring, compare Exam-
ple 2.6. In fact we have an additive equivalence
~=
(R * =K) * =K pt -! R * x =K ( =K x pt)
'-! R
* pt= R
by an application of Proposition 2.8(i)respectively (ii)below.
Let K and G be groups. If A is an additive category with right K-action and S
is a K-G biset. Then A *K S can be equipped with a right G-action as follows. If
A = (As)s2S is an object and OE = (OEk,s)(k,s)2KxSis a morphism in A *K S then
for g 2 G we define g*A and g*OE by
(g*A)s = Asg-1 and (g*OE)k,s= OEk,sg-1.
Proposition 2.8. (i)Let K and G be groups. Suppose A is an additive cat-
egory with right K-action, let S be a K-G biset and let T be a left G-s*
*et.
Then there is an additive isomorphism of additive categories
~=
(A *K S) *G T -! A *KxG (S x T ).
Here in order to form the category on the right hand side we let K x G *
*act
from the right on A via the projection to K, and from the left on the s*
*et
S x T by (k, g)(s, t) = (ksg-1, gt) for (k, g) 2 K x G and (s, t) 2 S x*
* T .
(ii)Let N be a normal subgroup of G. Let A be an additive category with a
right G-action such that N acts trivially. Let T be a left G-set such t*
*hat N
operates freely. Then there is an additive functor which is an equivale*
*nce
of categories
A *G T -'!A *G=N (N\T ).
(iii)Let H be a subgroup of G and A be an additive category with right G-act*
*ion.
We denote by resHA the additive category A considered with the H action
obtained by restriction. Then for an H-set T the map T ! Gx HT defined
by t 7! (1G , t) induces an equivalence of additive categories
(resHA) *H T ! A *G (Gx HT ).
Proof.(i)The functor F :(A *K S) *G T ! A *KxG (S x T ) is given by
(F (A))(s,t)= (At)s and (F (OE))(k,g)(s,t)= (OEg,t)k,s.
Note that if OE: A ! B is a morphism in (A *K S) *G T then OEg,t:At ! g*(Bgt)
is a morphism in A *K S and (OEg,t)k,s:(At)s ! k*((g*Bgt)ks) is a morphism in A.
The target of this last morphism is
k*((g*Bgt)ks) = k*((Bgt)ksg-1) = k*(F (B)(ksg-1,gt)) = (k, g)*(F (B)(k,g)(s,*
*t))
by definition of the G-action on A *K S and the K x G-action on A and S x T . In
particular this is a correct target for F (OE)(k,g)(s,t). Clearly, F is an isom*
*orphism of
categories. To verify that F is indeed an additive functor is lengthy but strai*
*ght-
forward.
(ii)Let p: T ! N\T denote the projection and consider A *G p: A *G T !
A*G (N\T ), see Remark 2.3. Next we define an additive functor F :A*G (N\T ) !
A *G=N (N\T ). For both these categories objects are given by sequences (At)t2T
of objects in A indexed by T and we define F as the identity on objects. Let
OE: A ! B be a morphism in A *G (N\T ). For g 2 G, n 2 N and t 2 T OEgn,Ntis a
morphism
ANt ! (gn)*BgnNt = g*BgNt
6 ARTHUR BARTELS AND HOLGER REICH
and we define F on morphisms by
X
(F (OE))gN,Nt= OEgn,Nt.
n2N
Then the composition F O (A *G p) is full and faithful and hence an equivalence*
* of
additive categories.
(iii)It is straight forward to check that this functor is full and faithful.
Let : K ! G be a group homomorphism. For a given additive category A with
right K-action we define ind A, a category with right G-action, as
ind A = A *K res G.
This is a special case of the construction discussed before Proposition 2.8. H*
*ere
res G denotes G considered as K-G biset via . Our main motivation for Propo-
sition 2.8 was the following corollary which will play a key role when we will *
*study
inheritance properties for Isomorphism Conjectures.
Corollary 2.9. Let A be an additive category with a right K-action. For a group
homomorphism : K ! G and a G-set T there is an additive functor which is an
equivalence of categories
(ind A) *G T -'!A *K (res T ).
Proof.According to Proposition 2.8(i)and (ii)we have an isomorphism respectively
an equivalence
~= '
(A *K res G) *G T -! A *KxG (G x T ) -! A *K G xG T = A *K res T.
3.The Farrell-Jones Conjecture with coefficients
Let K-1 : Add Cat ! Sp be the functor that associates the non-connective
K-theory spectrum to an additive category (using the split exact structure). Th*
*is
functor is constructed in [PW85 ]. See [BFJR04 , Section 2.1 and 2.5] for a b*
*rief
review of this functor and its properties. Let G be a group and OrG be the orbit
category of G, whose objects are transitive G-sets of the form G=H and whose
morphisms are G-equivariant maps. For any OrG-spectrum E, i.e. for any functor
E: OrG ! Sp, Davis and L"uck construct a G-equivariant homology theory for
G-CW -complexes by
HG*(X; E) = ss*(X+ O^rGE),
where X+ ^OrG E denotes the balanced smash product of X+ = map G(?, X+ )
considered as a contravariant OrG-space and the covariant OrG-spectrum E. For
more details see [DL98 , Section 4]. For a group G and a family F of subgroups,*
* i.e.
a collection of subgroups that is closed under subconjugation, there is a G-CW -
complex EF G with the property that for a subgroup H of G the set of fixed poin*
*ts
EF GH is empty if H 2= F and contractible if H 2 F, see for example [L"uc04].
The triple (E, F, G) is said to satisfy the Isomorphism Conjecture if the so ca*
*lled
assembly map
HG*(EF G; E) ! HG*(pt; E) = ss*(E(G=G))
induced by the projection EF G ! ptis an isomorphism, see [DL98 , Definition 5.*
*1].
In this paper we will use the following OrG-spectra.
Definition 3.1. Let A be an additive category with right G-action. The Or G-
spectrum KA is defined by
KA (T ) = K-1 (A *G T ).
COEFFICIENTS FOR THE FARRELL-JONES CONJECTURE 7
Conjecture 3.2 (Algebraic K-Theory Farrell-Jones-Conjecture with Coefficients).
Let G be a group and let VCyc be the family of virtually cyclic subgroups of G.*
* Let
A be an additive category with a right G-action. Then the assembly map
HG*(EVCycG; KA ) ! HG*(pt; KA )
is an isomorphism.
Remark 3.3. If A = R then by Example 2.4 KA can be identified with the func-
tor introduced in [DL98 , Section 2]. In particular Conjecture 3.2 implies the *
*original
conjecture by Farrell and Jones in [FJ93 ]. (The formulation of Davis and L"uck*
* has
been identified with the original formulation of Farrell and Jones in [HP04 ].)
4.Inheritance properties
By definition a family of subgroups of a group G is a collection of subgroups
closed under taking subgroups and conjugation. If : K ! G is group homomor-
phism and F is a family of subgroups of G then we define a family of subgroups *
*of
K by setting
*F = {H G | H is a subgroup ofK and (H) 2 F}.
Remark 4.1. The K-CW -complex res EF G is a model for the classifying space
E *F K, because it satisfies the characterizing property concerning the fixed p*
*oint
sets.
Proposition 4.2. Let : K ! G be a group homomorphism. Let A be an additive
category with right K-action and let F be a family of subgoups of G. Then the
assembly map
HG*(EF G; Kind A) ! HG*(pt; Kind A)
is equivalent to the assembly map
HK*(E *F K; KA ) ! HK*(pt; KA ).
Proof.Because of Corollary 2.9 and since K-1 preserves equivalences we have
equivalences of OrG-spectra Kind A '-!KA O res and therefore for every G-space
X a natural isomorphism
HG*(X; Kind A) ~=HG*(X; KA O resOE)
For a K-space Y define ind Y to be the quotient of G x Y by the right K action
given by (g, y)k = (g (k), k-1y). For every G-space X there is an isomorphism
res X?+ = map K(?, res X)+
~= map G(ind (?), X)+
~= map G(??, X)+ ^ map (ind (?), ??)+
OrG G
= X??+^OrGmapG(ind (?), ??)+
of contravariant pointed OrK-spaces. Here and in the next display ? denotes fun*
*c-
toriality in OrK and ?? denotes functoriality in OrG. For every covariant funct*
*or
F from K-sets to spectra there is an isomorphism of covariant OrG-spectra
map G(ind (?), ??)+ O^rKF(?)~=map K (?, res(??))+ O^rKF(?)
~= F O res(??).
Combining these isomorphisms with associativity of balanced smash products we
obtain an isomorphism of spectra
res X+ O^rKF ~=X+ O^rGF O res
8 ARTHUR BARTELS AND HOLGER REICH
and in particular a natural isomorphism
HK (resOEX; KA ) ~=HG (X; KA O res).
It remains to observe that res pt= ptand that res EF G is a model for E *F K.
Proposition 4.2 has the following immediate consequence.
Corollary 4.3. Let : K ! G be a group homomorphism. Let F be a family
of subgroups of G. Suppose that for every additive category A with G-action the
assembly map
HG*(EF G; KA ) ! HG*(pt; KA )
is injective. Then for every additive category B with K-action the assembly map
HK*(E *F K; KB) ! HK*(pt; KB)
is injective. The same statement holds with injectivity replaced by surjectivi*
*ty in
assumption and conclusion.
Remark 4.4 (With coefficients is stronger than fibered). The fibered version of*
* the
Farrell-Jones Conjecture in algebraic K-theory for a group G (and a ring R), [F*
*J93 ,
Section 1.7] can be formulated as follows: for every group homeomorphism : K !
G the assembly map HK*(E *VCycK; KR ) ! HK*(pt; KR ) is an isomorphism, see
Section 6 and in particular Remark 6.6 in [BL04 ]. Therefore by Corollary 4.3
the Farrell-Jones Conjecture with Coefficients 3.2 implies the Fibered Farrell-*
*Jones
Conjecture.
Corollary 4.3 implies in particular the following theorem about the Farrell-*
*Jones
Conjecture with Coefficients 3.2.
Theorem 4.5. Let H be a subgroup of G. Suppose that for every additive category
A with G-action the assembly map
HG*(EVCycG; KA ) ! HG*(pt; KA )
is injective or surjective respectively. Then for every additive category B wit*
*h right
H-action the assembly map
HH*(EVCycH; KB) ! HH*(pt; KB)
is injective or surjective respectively. In particular, if the Farrell-Jones Co*
*njecture
with Coefficients 3.2 holds for a group G, then it holds for every subgroup of *
*G.
Similar as for rings there exists a suspension category and hence results th*
*at
hold without a condition on the coefficient category can always be shifted down.
More precisely the following holds.
Proposition 4.6. For every additive category A with G-action there is an additi*
*ve
category A with G-action such that for every family of subgroups F and every
n 2 Z the assembly map
HGn(EF G; KA ) ! HGn(pt; KA )
is isomorphic to the assembly map
HGn-1(EF G; K A ) ! HGn-1(pt; K A ).
Proof.We use a construction of A that is similar to the construction from [PW8*
*5 ].
For a given A there is a natural construction of a Karoubi filtration of additi*
*ve
categories A0 A whose quotient we denote by A. Here A0is naturally equiv-
alent to A and there is an Eilenberg swindle on A, see Example 7.2. Therefore
A ! A ! A induces a fibration sequence in (non-connective) K-theory by
[CP95 , Theorem 1.28] and K* A = 0. Because the construction is natural, there
COEFFICIENTS FOR THE FARRELL-JONES CONJECTURE 9
are G-actions on A and A. Both, the Karoubi filtration and the Eilenberg swin-
dle are preserved by the passage from A to A *G T . Therefore we have a fibrati*
*on
sequence of OrG-spectra,
KA ! K A ! K A
that gives long exact sequences of the associated homology groups for every G-
space X. By the Eilenberg swindle on A *G T the groups HG*(X; K A ) vanish and
the boundary map in the long exact sequence yields the desired identification of
assembly maps.
From Proposition 4.6 we obtain the following analog of [BFJR04 , Corollary 7*
*.3].
Corollary 4.7. Let F be a family of subgroups of the group G. If for every addi*
*tive
category A with right G-action the assembly map
HG*(EF G; KA ) ! HG*(pt; KA )
is injective or surjective respectively in a fixed degree * = n, then this asse*
*mbly map
is injective or surjective respectively in all degrees * = j with j n.
5.L-theory
Everything we did for algebraic K-theory has an analog in L-theory and we wi*
*ll
state the corresponding conjecture and inheritance result here quickly. An add*
*i-
tive category with involution is an additive category A together with an additi*
*ve
contravariant functor # = (-)# :A ! A such that # O # = id. We consider now
additive categories with involution and right G-action, where we require in add*
*ition
that for every g 2 G the covariant functor g* is compatible with the involution*
* #,
i.e. # O g* = g* O #. If T is a G-set then
(A# )t= (At)# and (OE# )g,t= g*((OEg-1,gt)# )
defines an involution on A *G T . There is a functor L-1 : Add Cat Inv ! Sp
that associates the L-theory spectrum to an additive category with involution c*
*on-
structed by Ranicki [Ran92 ]. We consider the OrG-spectrum LA defined by
LA (T ) = L-1 (A *G T ).
Conjecture 5.1 (L-Theory Farrell-Jones-Conjecture with Coefficients). Let G be
a group and let VCyc be the family of virtually cyclic subgroups of G. Let A be*
* an
additive category with involution with a right G-action. Then the assembly map
HG (EVCycG); LA ) ! HG (pt; LA )
is an isomorphism.
The only property of the functor K-1 that was used in the proof of Proposi-
tion 4.2 is that it sends equivalences of categories to equivalences of spectra*
*. Because
this property holds also for the functor L-1 there is also the L-theory versio*
*n of
Proposition 4.2. Therefore there are also L-theory versions of Corollary 4.3, *
*Re-
mark 4.4 and Theorem 4.5. We spell out only the analog of Theorem 4.5.
Theorem 5.2. Let H be a subgroup of G. Suppose that for every additive category
A with involution with G-action the assembly map
HG*(EVCycG; LA ) ! HG*(pt; LA )
is injective or surjective respectively. Then for every additive category B wit*
*h invo-
lution with right H-action the assembly map
HH*(EVCycH; LB) ! HH*(pt; LB)
10 ARTHUR BARTELS AND HOLGER REICH
is injective or surjective respectively. In particular, if the L-Theory Farrell*
*-Jones
Conjecture with Coefficients 5.1 holds for a group G, then it also holds for ev*
*ery
subgroup of G.
6. Crossed products
In this section we show that the Farrell-Jones Conjecture with Coefficients *
*3.2
covers crossed product rings. We first recall the notion of a crossed product r*
*ing,
compare [Pas89].
Let R be a ring, G be a group and ff: G ! Aut(R), g 7! ffg and o :Gx G !
Rx , (g, h) 7! og,hbe maps. Here Rx are the units of R and Aut(R) denotes the
group of ring-automorphisms of R. We require that
(6.1) og,hogh,k = ffg(oh,k)og,hk
(6.2) og,hffgh(r)= (ffg O ffh)(r)og,h
for g, h, k 2 G, r 2 R. We will also assume that ffe = idR, where e denotes the*
* unit
element in G.
The crossed product ring Rff,oG is as an additive group RG, but is equipped
with a twisted multiplication .ff,owhere
(6.3) (rg) .ff,o(sh) = rffg(s)og,hgh
for r, s 2 R and g, h 2 G. The element 1R e is the unit of Rff,oG.
Setting g = e or h = e in (6.2) we conclude that for g 2 G,
(6.4) oe,gand og,elie in the centerRof.
Example 6.5. The notion of a crossed product ring naturally appears in the fol-
lowing situation. Consider an extension of groups
1 ! K ! p-!G ! 1.
Choose a set-theoretical section s of p such that s(1) = 1. Let S be a ring, l*
*et
R = SK and set ffg(r) = s(g)rs(g)-1 and og,h= s(g)s(h)s(gh)-1. Then
Rff,oG ~=S .
Our aim is now to define an additive category Aff,owith a right G-action such
that the category A *G pt is equivalent to the category of finitely generated f*
*ree
Rff,oG-modules.
We start with the category Ff(R) of finitely generated free R-modules. If ' *
*is an
automorphism of R, then we define a functor M 7! res'M where the latter is the *
*R-
module obtained by twisting the R-module structure by ', i.e. r.res'Mv = '(r).M*
* v.
This defines a right action of Aut(R) on Ff(R). (If we want a small category we
can restrict attention to modules of the form res'Rn.)
We digress for a moment and discuss the special case where o 1R . Then
ff is a group homomorphism and we obtain an action of G on Ff(R). This is
the desired category with G-action in this special case. The equivalence to the
category of finitely generated free RffG-modules sends the morphism OE: M ! N
in Ff(R) *G pt with components
OEg: M ! resffgN
to the RffG-linear map
RffG R M ! RffG R N, x v 7! xg-1 OEg(v).
We continue with the explanation of the general case. In general, Log,hdefin*
*es
a natural transformation from resffghto resffhO resffg. (Here we denoted the map
v 7! rv for r 2 R by Lr. The expression rv is formed with respect to the the
original module multiplication on M.)
COEFFICIENTS FOR THE FARRELL-JONES CONJECTURE 11
The category Aff,ois now obtained by rigidifying this operation as follows. *
*Ob-
jects of Aff,oare pairs (M, g) where M is a finitely generated free R-module and
g 2 G. Morphisms from (M, g) to (N, h) are R-linear maps ': resffgM ! resffhN
and composition is composition of linear maps. The right action of fl 2 G is de*
*fined
by (M, g) 7! (M, gfl) on objects and by
(6.6) ' 7! fl*' = L-1oh,flO ' O Log,fl
for a morphism ': (M, g) ! (N, h). The only thing one has to check is that
ffi*(fl*') = (flffi)*' for a morphism ': (M, g) ! (N, h). Recall that such a mo*
*rphism
is given by an additive map ': M ! N for which 'OLffg(r)= Lffh(r)O' for v 2 M,
r 2 R. Using this and (6.1)we can compute
ffi*(fl*')= L-1ohfl,ffiO L-1oh,flO ' O Log,flO Logfl,ffi
= L-1oh,flffiO L-1ffh(ofl,ffi)O ' O Lffg(ofl,ffi)O Log,f*
*lffi
= L-1oh,flffiO L-1ffh(ofl,ffi)O Lffh(ofl,ffi)O ' O Log,f*
*lffi
= L-1oh,flffiO ' O Log,flffi
= (flffi)*'.
Proposition 6.7. The categories Aff,o*G ptand the category Ff(Rff,oG) of finite*
*ly
generated free Rff,oG modules are equivalent as additive categories.
Proof.We start by listing a number of useful consequences of (6.1), (6.2) and
ffe = idR,
(6.8) ffa(ob,c)= oa,boab,co-1a,bc
(6.9) ff-1a(r)= o-1a-1,affa-1(r)oa-1,a
(6.10) (ffa O ffb)(r)=oa,bffab(r)o-1a,b
for a, b, c 2 G and r 2 R. From the definition of the product . = .ff,oin (6.3)*
* we
recall
(6.11) a . r= ffa(r) . a
(6.12) r . a= a . ff-1a(r)
(6.13) a . b= oa,b. ab
for a, b 2 G and r 2 R.
Denote by (Aff,o*G pt)0 the full subcategory of Aff,o*G ptwhose objects are *
*of
the form (M, e). It is easy to check that the inclusion (Aff,o*G pt)0 ! Aff,o*G
pt is an equivalence of additive categories. We define a functor F :(Aff,o*G
pt)0 ! Ff(Rff,oG) as follows. For an object (M, e) in (Aff,o*G pt)0 let F (M, e*
*) =
Rff,oG RM. A morphism OE: (M, e) ! (N, e) in (Aff,o*G pt) is by definition a
sequence (OEfl)fl2Gwhere OEfl:M ! resffflN is an R-linear map. Because we can
add morphisms in additive categories it will suffice to discuss morphisms for w*
*hich
OEfl= 0 for all but one fl 2 G; we write (', g) for the morphism given by OEfl=*
* ' if
fl = g and OEfl= 0 otherwise, in particular ' is an additive map M ! N for which
(6.14) '(rv) = ffg(r)'(v) for allr 2 R, v 2 M.
We define F (', g): F (M, e) ! F (N, e) as the linear map
(6.15) x v 7! x . g-1 . o-1g,g-1'(v).
Note that
xr . g-1 . o-1g,g-1= x . g-1 . ff-1g-1(r)o-1g,g-1by(6.12)
= x . g-1 . o-1g,g-1ffg(r) by (6.9).
12 ARTHUR BARTELS AND HOLGER REICH
Because of (6.14) this means that F (', g) defines indeed a well defined map on*
* the
tensor product. (This explains the appearance of o-1g,g-1in (6.15); without thi*
*s term
the map is ill defined.)
Next we check that F is compatible with composition, a somewhat tedious cal-
culation. Let (_, h): (N, e) ! (L, e) be a second morphism in (Aff,o*G pt)0, in
particular _ is an additive map N ! L for which
(6.16) _(rv) = ffh(r)_(v) for allr 2 R, v 2 N.
Then F (_, h) O F (', g) maps x v to
x . g-1 . o-1g,g-1. h-1 . o-1h,h-1_('(v))
= x . g-1 . h-1 . ff-1h-1(o-1g,g-1)o-1h,h-1_('(v)) by (6.12)
= x . og-1,h-1. (hg)-1 . ff-1h-1(o-1g,g-1)o-1h,h-1_('(v))by(6.13)
= x . (hg)-1 . ff-1(hg)-1(og-1,h-1)ff-1h-1(o-1g,g-1)o-1h,h-1_('(v))by(6.12)
= x . (hg)-1 . A _('(v))
where
A = ff-1(hg)-1(og-1,h-1)ff-1h-1(o-1g,g-1)o-1h,h-1
i j
= o-1hg,(hg)-1ffhg(og-1,h-1)ohg,(hg)-1
i j
o-1h,h-1ffh(o-1g,g-1)oh,h-1o-1h,h-1 by (6.9)
i j
= o-1hg,(hg)-1o-1h,gffh(ffg(og-1,h-1))oh,g
ohg,(hg)-1o-1h,h-1ffh(o-1g,g-1) by (6.10)
i j
= o-1hg,(hg)-1o-1h,gffh(og,g-1oe,h-1o-1g,(hg)-1)
oh,gohg,(hg)-1o-1h,h-1ffh(o-1g,g-1) by (6.8)
-1
= o-1hg,(hg)-1o-1h,gffh(og,g-1)ffh(oe,h-1)ffh(og,(hg)-1)
oh,gohg,(hg)-1o-1h,h-1ffh(og,g-1)-1
i i j j
= o-1hg,(hg)-1o-1h,goh,gohg,g-1o-1h,eoh,eoh,h-1o-1h,h-1
i -1j i -1 j
oh,gohg,(hg)-1o-1h,h-1oh,gohg,(hg)-1o-1h,h-1oh,gohg,g-1o-1h,eby(6.8)
= o-1hg,(hg)-1ohg,g-1oh,eo-1hg,g-1o-1h,g
= o-1hg,(hg)-1o-1h,goh,e by (6.4).
Compute the composition in (Aff,o*G pt)0 as follows
(_, h) O (',=g)(g*_ O ', hg) by (2.2)
= (L-1oh,gO _ O Loe,gO ', hg) by (6.6).
Therefore F ((_, h) O (', g)) maps x v to
(x . (hg)-1 . o-1hg,(hg)-1o-1h,g_(oe,g'(v))
= x . (hg)-1 . o-1hg,(hg)-1o-1h,gffh(oe,g)_('(v)) by (6.16)
= x . (hg)-1 B_('(v))
COEFFICIENTS FOR THE FARRELL-JONES CONJECTURE 13
where
B = o-1hg,(hg)-1o-1h,gffh(oe,g)
= o-1hg,(hg)-1o-1h,goh,eoh,go-1h,g by (6.8)
= o-1hg,(hg)-1o-1h,goh,e.
Thus A = B and this shows F (_, h) O F (', g) = F ((_, h) O (', g)). Thus F is
indeed a functor. It is straight forward to check that F is full and faithful, *
*i.e. an
equivalence of categories.
The following sharpening of Proposition 6.7 is obtained by formal arguments.
Corollary 6.17. Suppose we are given a crossed product situation
R, ff: G ! Aut(R), o :G x G ! Rx .
Then there exists an additive category Aff,owith a right G-action, such that for
every orbit G=H the category
Aff,o*G G=H and the category Ff(Rff|,o|H)
of finitely generated Rff|,o|H-modules are equivalent. Here ff| and o| denote *
*the
restriction of ff and o to H respectively H x H. In particular there is for ev*
*ery
G=H and every n 2 Z an isomorphism
Kn(Aff,o*G G=H) ~=Kn(Rff|,o|H).
Proof.We have a chain of equivalences
Aff,o*G G=H ' (resHAff,o) *H pt
' Aff|,o|*H pt
' Rff|,o|H .
Here the first equivalence is a special case of Proposition 2.8 (iii)and the la*
*st
follows immediately from the previous Proposition 6.7. The second equivalence is
induced from the H-equivariant inclusion Aff|,o|! resHAff,owhich sends (M, h)
to the same element considered as an object of Aff,o. This inclusion is clearly*
* full
and faithful and every object (M, g) in the target is isomorphic to (resffgM, e*
*).
One then checks that in general an H-equivariant equivalence A ! B induces an
equivalence A *H T ! BH * T for every G-set T .
Observe that in particular the G-equivariant homology theory HG*(-; KAff,o)
evaluated on an orbit G=H is isomorphic to K*(Rff|,o|H). The following special
case of Conjecture 3.2 hence makes precise the idea that K*(Rff,oG) should be
assembled from the pieces K*(Rff|,o|H), where H ranges over the virtually cyclic
subgroups of G.
Conjecture 6.18. Suppose Rff,oG is a crossed product ring, then the assembly
map
HG*(EVCycG; KAff,o) ! HG*(pt; KAff,o) ~=K*(Rff,oG)
induced from EVCycG ! ptis an isomorphism.
7. Controlled algebra
Many results on the Farrell-Jones conjecture (without coefficients) use the *
*con-
cept of controlled algebra. In this section we briefly indicate how the fundame*
*ntal
concepts of controlled algebra extend from rings to additive categories with gr*
*oup
actions.
The following generalizes the definitions in [BFJR04 , Section 2].
14 ARTHUR BARTELS AND HOLGER REICH
Definition 7.1. Let A be an additive category with a right G-action and let X be
a free G-space. Define the additive category with right G-action
C(X; A)
as follows. Objects are families A = (Ax)x2X of objects in A such that suppA =
{x 2 X | Ax 6= 0} is locally finite. A morphism OE: A ! B is a family (OEy,x)(y*
*,x)2XxX,
where OEy,x:Ax ! By is a morphism in A and for fixed x the set of y with OEy,x6*
*= 0 is
finite and for fixed y the set of x with OEy,x6= 0 is finite. The composition _*
* = OE0OOE
is defined to be X
_z,x= OE0z,yO OEy,x.
y2X
The element g 2 G acts via the covariant additive functor g* which is given by
(g*A)x = g*(Agx) and (g*OE)y,x= g*(OEgy,gx).
It now makes sense to consider the fixed category C(X; A)G . An object A and*
* a
morphism OE in the fixed category satisfy
Ax = g*(Agx) and OEy,x= g*(OEgy,gx).
Observe that in the case where the category A is R for some ring R equipped
with the trivial G-action we obtain the category which was denoted CG (X; R) in
[BFJR04 , Section 2], compare also Example 2.4.
7.1. Support conditions. As usual we define the support of an object A, respec-
tively a morphism OE as
supp A = {x 2 X | Ax 6= 0} and suppOE = {(x, y) 2 X x X | OEy,x6= 0}.
If a set E of subsets of X x X and a set F of subsets of X satisfies the condit*
*ions
(i)-(iv) listed in [BFJR04 , Subsection 2.3] then we speak of E respectively F *
*as
morphism and object support conditions and define
C(X; E, F; A)
to be the subcategory of C(X; A) consisting of objects A for which there exists*
* an
F 2 F such that suppA F and morphisms OE for which there exists an E 2 E
such that suppOE E. Observe that for g 2 G we have
suppg*A = g-1(supp A) and suppg*OE = g-1(supp OE).
We say that E is G-invariant if for every g 2 G, E 2 E we have g(E) 2 E, where G
acts diagonally on X x X. We say that F is G-invariant if for every g 2 G, F 2 F
we have g(F ) 2 F. For G-invariant object and morphism support conditions E and
F there is a G-action on C(X; E, F; A) and we can consider the corresponding fi*
*xed
category, which we denote
CG (X; E, F; A).
The following example was used in Proposition 4.6.
Example 7.2. Let X = [0, 1). Let E = {Eff| ff > 0}, where Eff= {(x, y) 2
[0, 1)x2 | |x - y| < ff} and F = {[0, r] | r 2 R}. Then there is an Eilenberg
swindle on A = C([0, 1); E; A) induced by the map t 7! t + 1 on [0, 1) and
A0= C([0, 1); E, F; A) A = C([0, 1); E; A) is a Karoubi filtration, see [CP9*
*5 ,
Definition 1.27].
COEFFICIENTS FOR THE FARRELL-JONES CONJECTURE 15
7.2. Assembly as Forget-Control. From this point on it is clear that every
construction and every proof in [BFJR04 ] which treats the category of finitely
generated R-modules in a formal way does have an analog in our context. In
particular there is a category
DG (X; A)
defined analogously to the category DG (X) from Subsection 3.2 in [BFJR04 ] and
this construction is functorial in the G-space X. The functor
X 7! K-1 DG (X).
is a G-equivariant homology theory on the category of G-CW complexes, compare
[BFJR04 , Section 4].
We will now identify this controlled version of a G-equivariant homology the*
*ory
with the G-equivariant homology theory that we defined in Section 3 via the OrG-
spectrum KA from Definition 3.1.
Theorem 7.3. There is an isomorphism between the functors X 7! HG*(X; KA )
and X 7! ss*+1(K-1 D(X; A)G ) from G-CW -complexes to graded abelian groups.
In particular, the map
K*+1(D(EF G, A)G ) ! K*+1(D(pt, A)G )
is a model for the assembly map
HG*(EF G, KA ) ! HG*(pt, KA ).
Proof.Without twisted coefficients this was done in [BFJR04 , Section 6]. The
proof in the case with twisted coefficients is essentially the same. The only *
*step
in the proof where the argument needs to be rethought is Step (ii) in the proof*
* of
[BFJR04 , Proposition 6.2]. This step is redone in lemma 7.4 below.
Lemma 7.4. Let T be a G-set. Let FGc the object support condition on T xG that
contains exactly the G-compact subsets. Let E be the morphism control condition
on T that contains only the diagonal of T . Let p: T xG ! T denote the projecti*
*on.
There is an additive functor
F :A *G T ! C(T x G, p-1E , FGc; A)G
which yields an equivalence of categories. This functor is natural in T .
Proof.It is straightforward to check that
ae * 0
F (A)(t,g-1)= g*(Agt), F (OE)(t0,k-1),(t,g-1)= g (OEkg-1,gt)ift0=itft0*
*6= t
defines an additive functor. Here, in order to check that F (A) satisfies the o*
*bject
support condition observe that G(t, g) 7! g-1t is a bijection between the orbit*
*s of
the left G-set T x G and the set T . Because of the object support condition an
object in the target category can be written as a direct sum of objects support*
*ed
on a single orbit of T x G. Because of the G-invariance an object C = (C(t,g))
supported on a single orbit G(t, g) is determined by its value at one point of *
*the
orbit together with the G-action on the category A. Now the object A 2 A *G T
supported on the single point {t} which is given by At = C(t,e)maps to C under
the functor F . Since the functor is additive we conclude that every object in *
*the
target category is isomorphic to an object in the image of the functor F . The
functor is easily seen to be faithful. It remains to prove that it is full, i.e*
*. surjective
on morphism sets. If f = (f(t0,k-1),(t,g-1)) is a morphism in the target categ*
*ory
then because of the p-1E -condition f(t0,k-1), (t, g-1) is non-trivial only if*
* t0 = t.
If one defines a morphism OE in A *G T by OEk,t= f(t,k-1)(t,e)then one can use *
*the
G-invariance of f in order to check that F (OE) = f.
16 ARTHUR BARTELS AND HOLGER REICH
8. Applications
As already mentioned many arguments in controlled algebra treat the category
A as a formal variable. Consequently existing proofs for results about the Farr*
*ell-
Jones Conjecture without coefficients can often be carried over to the context *
*with
coefficients. We will state three results obtained in this way.
The following is a generalization of the main Theorem in [BR05 ].
Theorem 8.1. Let G be the fundamental group of a closed Riemannian manifold
of strictly negative sectional curvature. Then the algebraic K-theory Farrell-J*
*ones
Conjecture with Coefficients 3.2 holds for G.
Proof.Even though the original proof (injectivity in [BFJR04 ] and surjectivity*
* in
[BR05 ]) is quite lengthy, one can quickly check that the only places in the pr*
*oof
where the arguments need to be rethought are the following.
(i)The functor ind appearing in Proposition 8.3 in [BFJR04 ] needs to be
promoted to a functor ind:DH (X; resHA) ! DG (G xH X; A), where X
is an H-space, H is a subgroup of G and A is an additive category with
G-action. The new formulas for the functor indare
(indM)[g,x]= (g-1)*Mx and (indOE)[g0,x],[g,x]= (g-1)*(OEg-1g0x0,x)
if g-1g02 H and 0 otherwise.
(ii)The proof of injectivity uses the injectivity result for the assembly w*
*ith
respect to the trivial family from [CP95 ], compare (iii) in Subsection*
* 10.3
in [BFJR04 ]. We hence need the version with coefficients of that resul*
*t. It
is a special case of Theorem 8.2 below.
(iii)In order to define the "Nil"-spectra denoted Ni in Subsection 10.2 in
[BFJR04 ] one needs that the assembly map for the infinite cyclic group
with respect to the trivial family is split injective. This is the spec*
*ial case
of Theorem 8.2 below, where G is the infinite cyclic group.
(iv)The construction of the transfer functor and the proof of its properties
in Section 5 of [BR05 ] need to be adapted to the set-up with coefficie*
*nts.
This will occupy the rest of this proof.
It will be convenient to restrict this discussion to connective K-theory becaus*
*e we
use Waldhausen categories. This suffices by Corollary 4.7. (On the other hand,
this discussion can be extended to non-connective K-theory, by giving an adhoc
definition of non-connective K-theory for the Waldhausen categories we encounter
in the following, compare [BR05 , Remark 5.3].)
First we need a replacement for the category of homotopy finite chain comple*
*xes,
defined in Subsection 5.2 and 8.1 in [BR05 ]. Given a category A with a right G-
action and an infinite cardinal number ~ (chosen large enough) we_construct bel*
*ow
in Lemma 9.2 a category_with right G-action A~. Analogously to CG(X; E) from
[BR05 ] we define CG (X; E; A~) by allowing objects M = (Mx)x2X , where Mx is
an object_in A~ and the support of M is an arbitrary subset of X. The cate- *
* __
gory CG(X; E; A~) plays the role of the category that is (unfortunately) called*
* A
in Subsection 8.1 in [BR05 ]. Hence the category chhfCG (X; E; A) is defined to*
* be
__G
the "homotopy closure" of the category chfC(X; E; A) inside chC (X; E; A~). The
fibre complex_F_and its variants from Subsection 5.3 in [BR05 ] can_be_consider*
*ed as
objects in chCG (E"x T; E; F~(Z)), which is defined analogous to chCG (X; E; A~*
*).
Here F~(Z) denotes a small model for the category of those free Z-modules which
admit a basis of cardinality less than or equal to ~. The category F~(Z) car-
ries the trivial G-action. The "tensor product" - -: A~ x F~(Z) ! A~ from
Lemma 9.2 (iii)now allows to construct the the transfer functor M 7! M F ,
COEFFICIENTS FOR THE FARRELL-JONES CONJECTURE 17
OE 7! OE r as before. The proof carries over without change until the end of *
*Sub-
section 5.4 in [BR05 ]. In Proposition 5.9 the action of the Swan group Sw(G; Z*
*) on
Kn(RG) needs to be replaced with the action of the Swan group on Kn(A *G pt)
that we describe below in Section 9. The proof of Proposition 5.9 in [BR05 ] re*
*mains
unchanged until one reaches diagram (5.12). That diagram now gets replaced with
the following diagram
CG (Gb0;OA)-_r_//chhfCGO(Gb0;OA)O
F || chhfF||
| - F0 |
A *G pt_______//chhf(A *G pt).
Here chhf(A *G pt) is the homotopy closure of chf(A *G pt) in ch(A~ *G pt). The
functor F is a special case of the equivalence from Lemma 7.4 and is given by
F (A)g-1b0= g*(A) and F (OE)k-1b0,g-1b0= g*(OEkg-1). The functor - r is given
by (Mgb0) r = (Mgb0 F0) and (OEgb0,hb0) r = (OEgb0,hb0 rgb0,hb0). We
can equip F0 with a G-action in such way that rgb0,hb0:F0 ! F0 corresponds to
lgh-1, i.e. to left multiplication with gh-1. With this notation the functor - *
* F0
is defined as follows. The object A maps to A F0 the morphism OE = (OEg)
maps to (OEg lg). As opposed to the original diagram in [BR05 ] the diagram n*
*ow
commutes. Let inc:A *G pt! chhf(A *G pt) denote the inclusion. It follows from
the discussion of the Swan group action in Section 9 below thatPon the level of*
* K-
theory inc-1O(- F0) corresponds to multiplication with [F0] = i(-1)i[Hi(F0)] 2
Sw ch(G; Z) ~=Sw(G; Z).
The following is a generalization of a result of Rosenthal [Ros04].
Theorem 8.2. Let G be a group. Suppose that there is a model for EFinG that is
a finite G-CW -complex and admits a compactification X such that
(i)the G-action extends to X;
(ii)X is metrizable;
(iii)XF is contractible for every F 2 Fin;
(iv)(EFinG)F is dense in X for every F 2 Fin;
(v) compact subsets of EFinG become small near Y = X - EFinG. That is,
for every compact subset of EFinG and for every neighborhood U X
of y 2 Y , there exists a neighborhood V X of y such that g 2 G and
gK \ V 6= ; implies gK U.
Let A be an additive category with right G-action. Then the assembly map
HG*(EFinG; KA ) ! HG*(pt; KA )
is split injective.
The following is a generalization of the main result from [Bar03].
Theorem 8.3. Let G be group of finite asymptotic dimension that admits a finite
model for the classifying space BG. Let A be an additive category with right G-
action. Then the assembly map
HG*(EG; KA ) ! HG*(pt; KA )
is split injective.
Both, the proof of Theorem 8.2 and of Theorem 8.3 are trivial modifications *
*of
the original proofs in [Ros04] respectively [Bar03]. Everywhere in these proofs*
* the
category of R-modules is treated as a formal variable and can simply be replaced
by the additive category with G-action A.
18 ARTHUR BARTELS AND HOLGER REICH
9.The Swan group action
In the proof of Theorem 8.1 we used some facts about Swan group actions which
we are going to prove now. The Swan group Sw (G; Z) is the K0-group of the
category of ZG-modules that are finitely generated as Z-modules. Recall from Su*
*b-
section 8.2 of [BR05 ] that there are version Swfr(G; Z) and Swch(G; Z) defined*
* using
ZG-modules that are finitely generated free as Z-modules, respectively bounded *
*be-
low chain complexes of ZG-modules that are free as Z-modules and whose homology
is finitely generated as a ZG-module. We have shown in Proposition 8.3 in [BR05*
* ]
that the natural maps j :Sw fr(G; Z) ! Swch(G; Z) and i: Sw fr(G; Z) ! Sw(G; Z)
are isomorphisms.
Lemma 9.1. Let A be an additive category with right G-action. There exists for
n 1 a commutative diagram
Kn(A *G pt) Z Swfr(G; Z)______//_Kn(A *G pt)
id|j| |inc|
fflffl| fflffl|
Kn(A *G pt) Z Swch(G; Z)____//Kn(chhf(A *G pt)),
where both vertical arrows are isomorphisms. In this way Kn(A *G pt) becomes a
module over the Swan ring Swfr(G; Z) ~=Swch(G; Z) ~=Sw(G; Z).
Using the suspension category A from Proposition 4.6 it is possible to form*
*ulate
a version of the above Lemma that applies to all n. However, for our purposes t*
*he
above formulation suffices.
Proof.Choose a small model Ff(Z) for the category of finitely generated free Z-
modules such that the underlying Z-module of every ZG-module that is used in the
construction of Swfr(G; Z) is contained in Ff(Z).
We replace A by the equivalent category Af from Lemma 9.2 but refer to it
as A in the following. This is justified because - *G pt respects equivalences,
compare Remark 2.3. We can hence assume that there exists a tensor product
- -: A x Ff(Z) ! A with the expected properties. Then define for a Swan
module M the additive functor - M :A *G pt! A *G pt by
A M = A UM, and (OE M)g = OEg lg.
Here UM denotes the underlying Z-module of M and lg denotes left multipli-
cation by g. Note that the ZG-module structure of M enters only in the mor-
phisms. A morphism f :M ! N of Swan modules induces a natural transforma-
tion o(f, A): A M ! A N that is given by o(f, A)g = 0 if g 6= e and o(f, A)*
*e =
idA f. A short exact sequence L ! M ! N of Swan-modules leads to a short
exact sequence of functors, because the underlying sequence UL ! UM ! UN
always splits and being a short exact sequence of functors is checked objectwis*
*e.
One uses [Wal85 , 1.3.2 (4)] in order to check that one obtains the Swfr(G; Z)-*
*action.
For the Sw ch(G; Z)-action arrange that the underlying Z-modules of all Swan
modules that appear in the chain complexes lie in a small model F~(Z) of the
category of finitely generated free modules that admit a basis B of cardinality
card(B) ~. (Strictly speaking we have to have a cardinality assumption when we
define the category of chain complexes that leads to Sw ch(G; Z).) By Lemma 9.2
there exists an inclusion A ! A~ and the "tensor product" we used so far extends
to a tensor product - -: A~ F~(Z) ! A~.
Define chhf(A *G pt) to be the category of chain complexes in ch(A~ *G pt)
that are chain homotopy equivalent to a bounded below and above chain complex
in ch(A *G pt). Similarly let chhfFf(Z) denote the category of chain complexes
COEFFICIENTS FOR THE FARRELL-JONES CONJECTURE 19
in chF~(Z) which are chain homotopy equivalent to a finite complex in chFf(Z),
compare [BR05 , Section 8.1]. The right hand vertical arrow induced by the incl*
*usion
A *G pt! chhf(A *G pt) is an equivalence by [BR05 , Lemma 8.1]. Now for a chain
complex Co which represents an element in Swch(G; Z) one defines a functor
- Co: A *G pt! chhf(A *G pt)
analogously to - M above by A Co = A UCo. This is well defined because
UCo is by definition of Swch(G; Z) a bounded below complex of finitely generated
free Z-modules whose homology is concentrated in finitely many degrees and each
of its homology groups is a finitely generated Z-module. Such a complex is ho-
motopy equivalent to a complex C0oof finitely generated free Z-modules which is
concentrated in finitely many degrees. (In order to prove this assume that Co *
*is
concentrated in non-negative degrees and use induction over the largest number
m such that Hm (Co) 6= 0. In the case m = 0 the complex UCo is a resolution
of the finitely generated Z-module H0(Co) and is hence homotopy equivalent to
a finite resolution. For m 1 choose a finite resolution Do of Hm (Co) and co*
*n-
struct a Hm -isomorphism f :Do ! UCo. Factorize f over its mapping cylinder
cyl(f) ' UCo and study the sequence Do ! cyl(f) ! cone(f).) Consequently
A UCo is homotopy equivalent to A C0oand hence lies in chhfA *G pt. Again a
short exact sequence of chain complexes leads to a short exact sequence of func*
*tors,
because the objects depend only on the underlying Z-chain complexes. A homology
equivalence Co ! Do can be considered as a homotopy equivalence UCo ! UDo
and hence induces a homotopy equivalence A UCo ! A UDo.
In the proof above and in the proof of Theorem 8.1 we used the following lem*
*ma.
Lemma 9.2. Let A be a small additive category with a G-action by additive func-
tors. Let ~ be a fixed infinite cardinal. Denote by F~(Z) some small model for
the category of all free Z-modules which admit a basis B with card(B) ~. Equip
F~(Z) with a tensorproduct functor - Z -. (This of course involves choices.)
Let Ff(Z) be the full subcategory of F~(Z) that consists of finitely generated *
*free
Z-modules.
There exist additive categories Af and A~ with G-action and G-equivariant ad-
ditive inclusion functors
A ! Af ! A~
such that the following conditions hold.
(i)The inclusion A ! Af is an equivalence of categories.
(ii)In A~ there exist categorical sums over indexing setsLJ with card(J) *
*~.
Hence if Aj, j 2 J is a family of objects in A~ then j2J Aj exists.
Moreover one can make a choice for these sum objects such that for every
g 2 G we have an equality
M M
g*( Aj) = g*(Aj).
j2J j2J
(iii)There exists a bilinear bifunctor
- -: A~ x F~(Z) ! A~,
which restricts to
- -: Af x Ff(Z) ! Af.
The functor is compatible with direct sums in the sense that for a fami*
*ly
of objects Aj, j 2 J in A~ with card(J) ~ and a Z-module F 2 F~(Z)
20 ARTHUR BARTELS AND HOLGER REICH
there exists a natural isomorphism
0 1
M M
@ AjA F ~= (Aj F )
j2J j2J
(The analogous statement for finite direct sums holds in both variables
because of the bilinearity.) The bifunctor is also compatible with the*
* G-
action in the sense that for every g 2 G and all morphisms OE: A ! B in
A~ and f :F ! F 0in F~(Z) we have equalities
g*(A F ) = (g*A) F and g*(f OE) = g*(f) OE.
(iv)For objects A in A~ and F , F 0in F~(Z) we have a natural isomorphism
(A F ) F 0~=A (F F 0).
Proof.The construction of the categories Af and A~ makes use of the following t*
*wo
elementary constructions. First construction: let B and C be two Ab -categorie*
*s,
i.e. categories enriched over abelian groups, compare [Mac71 , I.8]. Then we de*
*fine
the Ab -category B C as follows. Objects are pairs of objects which we denote
B C, where B is an object in B and C an object in C. We set
mor B C(B C, B0 C0) = morB(B, B0) Z morC(C, C0).
Composition and identities are defined in the obvious way. The construction is
functorial with respect to additive functors in B and C and preserves additive *
*equiv-
alences.
Second construction: given an Ab-category D and a set I we define the catego*
*ry
D(I) to be the category whose objects are families D = (D(i))i2Iof objects in D
and where a morphism f :D ! D0 is a family of morphisms f(j, i): D(i) ! D(j),
i, j 2 I subject to the condition that for a fixed i 2 I there are only finitel*
*y many
j 2 I such that f(j, i) 6= 0. Composition is the usual matrix multiplication, w*
*here
the components of f0O f are given by
X
(f0O f)(k, i) = f0(k, j) O f(j, i).
j
Now set Af = A Ff(Z). The inclusion A ! Af is given by A 7! A Z1,
where Z1 is some 1-dimensional free Z-module in Ff(Z). It is not difficult to c*
*heck
that this is an equivalence of categories. We define a "tensor product" - -: Afx
Ff(Z) ! Af by (A F ) F 0= A (F F 0). The G-action on Af is defined by
g*(A F ) = (g*A) F .
Next we choose a set I of cardinality ~ and set
f ~
A~ = A F (Z) (I).
The inclusion functor Af ! A~ sends A F to the object which at some fixed
index i0 is given by A F and is zero everywhere else. The G-action extends
via g*((A(i) F (i))i2I) = ((g*A(i)) F (i))i2I. The "tensor product" extends*
* by
((A(i) F (i))i2I) F = (A(i) (F (i) F ))i2I, where of course the A(i) are objects
in A and the F (i) and F are objects in F~(Z). The existence of the required di*
*rect
sums is a consequence of the fact that card(I x I) = card(I) for an infinite se*
*t I,
see for example [Lan02 , Appendix 2, x 3, Theorem 3.6].
References
[Bar03] Arthur C. Bartels. Squeezing and higher algebraic K-theory. K-Theory, 2*
*8(1):19-37,
2003.
[BFJR04]A. Bartels, T. Farrell, L. Jones, and H. Reich. On the isomorphism conj*
*ecture in alge-
braic K-theory. Topology, 43(1):157-213, 2004.
COEFFICIENTS FOR THE FARRELL-JONES CONJECTURE 21
[BL04] A. Bartels and W. L"uck. Isomorphism conjecture for homotopy K-theory a*
*nd groups
acting on trees. Preprintreihe SFB 478 _ Geometrische Strukturen in der*
* Mathematik,
Heft 342, M"unster, arXiv:math.KT/0407489, to appear in J. Pure Appl. A*
*lgebra, 2004.
[BR05] A. Bartels and H. Reich. On the Farrell-Jones conjecture for higher alg*
*ebraic K-theory.
J. Amer. Math. Soc., 18(3):501-545 (electronic), 2005.
[CP95] G. Carlsson and E. K. Pedersen. Controlled algebra and the Novikov conj*
*ectures for
K- and L-theory. Topology, 34(3):731-758, 1995.
[DL98] J. F. Davis and W. L"uck. Spaces over a category and assembly maps in i*
*somorphism
conjectures in K- and L-theory. K-Theory, 15(3):201-252, 1998.
[FJ93] F. T. Farrell and L. E. Jones. Isomorphism conjectures in algebraic K-t*
*heory. J. Amer.
Math. Soc., 6(2):249-297, 1993.
[HLS02] N. Higson, V. Lafforgue, and G. Skandalis. Counterexamples to the Baum-*
*Connes
conjecture. Geom. Funct. Anal., 12(2):330-354, 2002.
[HP04] I. Hambleton and E. K. Pedersen. Identifying assembly maps in K- and L-*
*theory. Math.
Ann., 328(1-2):27-57, 2004.
[Lan02] Serge Lang. Algebra, volume 211 of Graduate Texts in Mathematics. Sprin*
*ger-Verlag,
New York, third edition, 2002.
[LR05] W. L"uck and H. Reich. The Baum-Connes and the Farrell-Jones conjecture*
* in K- and
L-theory. In Handbook of K-Theory, pages 703-842. Springer, Berlin, 200*
*5.
[L"uc04]W. L"uck. Survey on classifying spaces for families of subgroups. Prepr*
*intreihe
SFB 478 _ Geometrische Strukturen in der Mathematik, Heft 308, M"unst*
*er,
arXiv:math.GT/0312378 v1, 2004.
[Mac71] S. MacLane. Categories for the working mathematician. Springer-Verlag, *
*New York,
1971. Graduate Texts in Mathematics, Vol. 5.
[Moo87] John A. Moody. Induction theorems for infinite groups. Bull. Amer. Math*
*. Soc. (N.S.),
17(1):113-116, 1987.
[Pas89] Donald S. Passman. Infinite crossed products, volume 135 of Pure and Ap*
*plied Math-
ematics. Academic Press Inc., Boston, MA, 1989.
[PW85] E. K. Pedersen and C. A. Weibel. A nonconnective delooping of algebraic*
* K-theory. In
Algebraic and geometric topology (New Brunswick, N.J., 1983), volume 11*
*26 of Lecture
Notes in Math., pages 166-181. Springer, Berlin, 1985.
[Ran92] A. A. Ranicki. Algebraic L-theory and topological manifolds, volume 102*
* of Cambridge
Tracts in Mathematics. Cambridge University Press, Cambridge, 1992.
[Ros04] D. Rosenthal. Splitting with continuous control in algebraic K-theory. *
*K-Theory,
32(2):139-166, 2004.
[RS04] D. Rosenthal and D. Sch"utz. On the algebraic K- and L-theory of word h*
*yperbolic
groups. Preprintreihe SFB 478 _ Geometrische Strukturen in der Mathemat*
*ik, Heft
343, M"unster, 2004.
[Wal85] Friedhelm Waldhausen. Algebraic K-theory of spaces. In Algebraic and ge*
*ometric topol-
ogy (New Brunswick, N.J., 1983), volume 1126 of Lecture Notes in Math.,*
* pages 318-
419. Springer, Berlin, 1985.
Westf"alische Wilhelms-Universit"at M"unster, Mathematisches Institut, Einst*
*einstr. 62,
D-48149 M"unster, Germany
E-mail address: bartelsa@math.uni-muenster.de
URL: http://www.math.uni-muenster.de/u/bartelsa/bartels
E-mail address: reichh@math.uni-muenster.de
URL: http://www.math.uni-muenster.de/u/reichh