ON THE FARRELLJONES CONJECTURE FOR HIGHER
ALGEBRAIC KTHEORY
ARTHUR BARTELS, HOLGER REICH
Abstract.We prove the FarrellJones Conjecture about the algebraic K
theory of a group ring R in the case where the group is the fundament*
*al
group of a closed Riemannian manifold with strictly negative sectional c*
*urva
ture. The coefficient ring R is an arbitrary associative ring with unit *
*and the
result applies to all dimensions.
Contents
1. Introduction 1
1.1. Acknowledgments 5
2. Preliminaries 5
2.1. Conventions and notation 5
2.2. Assembly as a öf rget controlmap" 6
3. Outline of the proof 7
4. Gaining control via the geodesic flow 11
4.1. Geometric preparations 11
4.2. Foliated control with decay speed 12
4.3. Gaining control via the geodesic flow 13
5. The transfer 15
5.1. Setup 16
5.2. Homotopy finite chain complexes 17
5.3. The fiber complex 18
5.4. The transfer functor 19
5.5. An element in the Swan group 21
6. A foliated control theorem for higher Ktheory 23
6.1. Flow cell structures 24
6.2. A family of flow cell structures 26
6.3. Construction of the decay speed S 27
6.4. Statement of the Foliated Control Theorem 29
6.5. First reduction  Delooping in the Bdirection 30
6.6. Second reduction  Discretization 31
6.7. Induction over the skeleta 31
6.8. Comparison to a Euclidean standard situation 34
7. From strong control to continuous control 36
7.1. A space with infinite cyclic isotropy 36
7.2. Strong control maps to continuous control 37
8. Appendix 37
____________
Date: August 5, 2003.
0
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 1
8.1. Homotopy finite chain complexes 38
8.2. The tildeconstruction 38
8.3. Swan group actions on Ktheory 39
8.4. Some fibration sequences 41
References 43
1.Introduction
Conjecturally the algebraic Ktheory groups Kn(Z ), n 2 Z, of the integral
group ring Z of every torsion free group can be expressed in terms of the gr*
*oup
homology of and the algebraic Ktheory of the integers. More precisely there *
*is
the following conjecture, compare [Hsi84, Section VI].
Conjecture 1.1. For a torsion free group the so called assembly map [Lod76]
A: Hn(B ; K1 (Z)) ! Kn(Z )
is an isomorphism for all n 2 Z.
Here B is the classifying space of the group and we denote by K1 (R)
the nonconnective algebraic Ktheory spectrum of the ring R. The homotopy
groups of this spectrum are denoted Kn(R) and coincide with Quillen`s algebraic
Kgroups of R in positive dimensions [Qui73] and with the negative Kgroups
of Bass [Bas68] in negative dimensions. The homotopy groups of the spectrum
X+ ^K1 (Z) are denoted Hn(X; K1 (Z)). They yield a generalized homology
theory and in particular standard computational tools like the AtiyahHirzebruch
spectral sequence apply to the left hand side of the assembly map above.
As a corollary of the main result of this paper we prove Conjecture 1.1 in th*
*e case
where is the fundamental group of a closed Riemannian manifold with strictly
negative sectional curvature. In fact our result is more general and applies to*
* group
rings R , where R is a completely arbitrary coefficient ring.
Note that if one replaces in Conjecture 1.1 the coefficient ring Z by an arbi*
*trary
coefficient ring R the corresponding conjecture would be false already in the s*
*implest
nontrivial case: if = C is the infinite cyclic group the BassHellerSwan fo*
*rmula
[BHS64 ], [Gra76, p.236] for Kn(RC) = Kn(R[t 1]) yields that
Kn(RC) ~=Kn1(R) Kn(R) NK n(R) NK n(R),
where NK n(R) is defined as the cokernel of the split inclusion Kn(R) ! Kn(R[t])
and does not vanish in general. But since S1 is a model for BC one obtains on t*
*he
left hand side of the assembly map only
Hn(BC; K1 (R)) ~=Kn(R) Kn1(R).
In some sense this is all that goes wrong. Combining our main result Theorem *
*1.4
with Proposition 1.8 in [BFJR03 ] we obtain the following generalization of the*
* Bass
HellerSwan formula.
Corollary 1.2. Let be a fundamental group of a closed Riemannian manifold
with strictly negative sectional curvature, then for every associative ring wit*
*h unit
R we have M
Kn(R ) ~=Hn(B ; K1 (R)) NK n(R) NK n(R)
I
2 ARTHUR BARTELS, HOLGER REICH
for all n 2 Z. Here the sum on the right is indexed over the set I of conjugacy
classes of maximal cyclic subgroups of .
Recall that a ring R is called (right) regular if it is (right) Noetherian and
every finitely generated (right) Rmodule admits a finite dimensional projective
resolution. Principal ideal domains are examples of regular rings. It is known
[Bas68, Chapter XII], [Qui73, p.122] that for a regular ring NK n(R) = 0 for all
n 2 Z. Hence for regular coefficient rings the expression in Corollary 1.2 simp*
*lifies
and proves the more general version of Conjecture 1.1 where the coefficient rin*
*g Z
is replaced by an arbitrary regular coefficient ring R.
We proceed to describe the FarrellJones Conjecture about algebraic Ktheory
of group rings [FJ93a] which is the correct conceptional framework for these ki*
*nds
of results and which applies also to groups which contain torsion. Here is some
more notation.
A set of subgroups of a given group is called a family of subgroups if it is
closed under conjugation with elements from and closed under taking subgroups.
We denote by
{1}, Cyc , VCyc and All
the families which consist of the trivial subgroup, all cyclic subgroups, all v*
*irtually
cyclic subgroups, respectively all subgroups of . Recall that a group is call*
*ed
virtually cyclic if it contains a cyclic subgroup of finite index.
For every family F of subgroups of there exists a classifying space for the
family F denoted E (F), compare [tD72], [tD87, I.6], and [FJ93a, Appendix]. It
is characterized by the universal property that for every CWcomplex X whose
isotropy groups are all in the family F there exists an equivariant continuous *
*map
X ! E (F) which is unique up to equivariant homotopy. A CWcomplex E is
a model for the classifying space E (F) if the fixpoint sets EH are contractib*
*le
for H 2 F and empty otherwise. Note that the one point space ptis a model for
E (All) and that E ({1}) is the universal covering of the classifying space B .
In [DL98 ] Davis and Lück construct a generalized equivariant homology the
ory for CWcomplexes X 7! Hn(X; K1R) associated to a jazzedup version of
the nonconnective algebraic Ktheory spectrum functor. If one evaluates this 
homology theory on a homogeneous space =H one obtains Hn( =H; K1R) ~=
Kn(RH). Using this language the FarrellJones Conjecture for the algebraic K
theory of group rings can be formulated as follows.
Conjecture 1.3 (The FarrellJones Conjecture for Kn(R )). Let be a group and
let R be an associative ring with unit. Then the map
AVCyc: Hn(E (VCyc ); K1R) ! Hn(pt; K1R) ~=Kn(R ).
which is induced by the projection E (VCyc ) ! ptis an isomorphism for all n 2 *
*Z.
This conjecture was formulated in [FJ93a] for R = Z and stated in this more
general form in [BFJR03 ]. Above we used the language developed by Davis and
Lück in [DL98 ] to formulate the conjecture. The identification of this formula*
*tion
with the original formulation in [FJ93a] which used [Qui82] is carried out in [*
*HP03 ].
For more information on this and related conjectures the reader should consult *
*[LR ].
Our main result proves Conjecture 1.3 for the class of groups that was already
mentioned above.
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 3
Theorem 1.4. Let R be an associative ring with unit. Let be the fundamental
group of a closed Riemannian manifold with strictly negative sectional curvatur*
*e.
Then for all n 2 Z the assembly map
ACyc:Hn(E (Cyc ); K1R) ! Kn(R )
induced by the projection E (Cyc ) ! ptis an isomorphism.
Note that the groups considered in this theorem are torsion free and that for
a torsion free group the family VCyc of all virtually cyclic subgroups reduces*
* to
the family Cyc of all cyclic subgroups. The result extends the results of [BFJR*
*03 ],
where surjectivity in low dimensions and injectivity was proven. Results which *
*are
strongly related to the result above about the low dimensional Ktheory of the
integral group ring, pseudoisotopy spectra and the structure set in surgery the*
*ory
were proven by Farrell and Jones in [FJ86], [FJ87], [FJ89] and [FJ91]. Apart fr*
*om
[Wal78] the result above seems to be the first integral result of this type whi*
*ch
applies to the higher algebraic Ktheory of group rings.
From the fact that we do not make any assumptions on the coefficient ring R
one can derive a corresponding isomorphism statement for an assembly map for
NK groups.
Corollary 1.5. Let R be an associative ring with unit. Let be the fundamen
tal group of closed Riemannian manifold with strictly negative sectional curvat*
*ure.
Then for all n 2 Z the assembly map for NK groups
ACyc:Hn(E (Cyc ); NK1R) ! Hn(pt; NK1R) ~=NK n(R )
is an isomorphism.
Proof.Since there is a splitting K1R[t]' K1R _ NK1R the isomorphism result f*
*or
two of the assembly maps associated to K1R, K1R[t], respectively NK1R implie*
*s it
for the third, compare [BFJR03 , Proposition 7.4].
In particular one can conclude that the vanishing of NK n(R) for n N implies
the vanishing of NK n(R ) for n N. Note that even if R is regular it is not at
all clear if R is regular. If one uses that R[ 1 x 2] = R[ 1][ 2] and iterate*
*s one
obtains the following corollary.
Corollary 1.6. Suppose = 1x 2x . .x. k, where each i is the fundamental
group of a closed Riemannian manifold with strictly negative sectional curvatur*
*e.
If R is a regular ring, then the assembly map
A: Hn(B ; K1 (R)) ! Kn(R )
is an isomorphism for all n 2 Z.
The proof of Theorem 1.4 relies on the fact (see Subsection 2.2 and [BFJR03 ])
that the assembly map AVCyc can be described as a öf rgetcontrol map" in the
sense of controlled topology, compare [Qui82], [PW89 ]. In order to prove a sur*
*jec
tivity result we hence have to äg in control". More precisely the negatively cu*
*rved
manifold M whose fundamental group we want to treat can be used in order to
construct a geometric model for the map E ! E (Cyc ) given by the universal
property. We will consider suitable additive categories of Rmodules and morphi*
*sms
over E x [1, 1) where the morphisms satisfy control conditions. The assembly
map is obtained by applying Ktheory to an inclusion of additive categories, wh*
*ere
4 ARTHUR BARTELS, HOLGER REICH
the larger category differs from the smaller one by a relaxed control condition*
* on
the morphisms.
The geometric program in order to gain control stems from [FJ86] and consists
mainly of three steps:
(I)Construct a transfer from the manifold M to a suitable subbundle of its
sphere bundle SM in such a way that transferred morphisms are in as
ymptotic position, i.e. they are in a good starting position for the geo*
*desic
flow. Make sure that transferring up and projecting down again does not
change the Ktheory class.
(II)Consider the foliation on the sphere bundle SM given by the flow lines of
the geodesic flow. Use the geodesic flow on SM in order to achieve folia*
*ted
control.
(III)Prove a öF liated Control Theorem" in order to improve from öf liated
control" to ö rdinary control". At least do so away from the "short"
closed geodesics.
Note that the closed geodesics that appear in Step (III) are in bijection with *
*con
jugacy classes of cyclic subgroups. Hence the family Cyc which appears in Theo
rem 1.4 shows up quite naturally in the proof.
We refer to Section 3 for a more detailed outline of the proof. In the follow*
*ing we
only discuss why new techniques were necessary in order to treat higher algebra*
*ic
Ktheory along the lines of the program above.
One main difficulty was to construct a suitable transfer as required in Step *
*(I)
of the program. Looking at the analogous situations for hcobordisms or Atheor*
*y,
where a transfer is given by pullback, it is in principle clear what the algeb*
*raic
analogue in our situation should be. However the obvious naive approaches are n*
*ot
üf nctorial enough" to induce a map in higher Ktheory. Hence one needs to come
up with a suitably refined transfer which takes care of the functoriality probl*
*ems
(e.g. work with singular chain complexes) but at the same time does not destroy
the control requirements. In order to treat the question whether transferring *
*up
and projecting down yields the identity on Ktheory we prove in Proposition 5.9
a formula for the kind of transfers we construct. Transferring up and projecti*
*ng
down yields multiplication by a certain element in the Swan group. The Swan
group element depends on the homology groups of the fiber considered as modules
over the fundamental group of the base.
Another main achievement in this paper is the Foliated Control Theorem 6.17
for higher algebraic Ktheory. Earlier foliated control theorems (see for exam*
*ple
Theorem 1.6 in [FJ86] or Theorem 1.1 in [BFJR ]) were formulated for individual
Ktheory elements. It is however difficult to explicitly describe elements in h*
*igher
Ktheory groups. Hence we had to find a way to formulate and prove a foliated
control theorem in a more üf nctorial" fashion. We would like to emphasize that
the Foliated Control Theorem 6.17 does not rely on a squeezing result or any ki*
*nd of
torus trick. Essentially only the existence of the long exact sequence associat*
*ed to a
Karoubi filtration [CP97 ] and Eilenbergswindles are used as the abstract buil*
*ding
blocks for such an argument. Of course on the geometric side the existence of l*
*ong
andthin cell structures proven in [FJ86] is crucial. In particular our techni*
*que
should prove analogous foliated control theorems in the context of algebraic L
theory or topological Ktheory of C*algebras since the corresponding tools are
available in those setups, compare [HPR97 ].
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 5
The reader who is familiar with the work of FarrellJones (in particular [FJ9*
*1]
and [FJ93b]) may wonder why we cannot weaken the assumption in Theorem 1.4
from strictly negative curvature to nonpositive curvature. The reason is that *
*the
focal transfer which is used in [FJ91] and [FJ93b], in contrast to the asymptot*
*ic
transfer used in this paper, is definitely not functorial and it is hence even *
*more
difficult to describe a corresponding transfer for higher algebraic Ktheory.
It also remains open whether the program can be adapted to prove cases of the
BaumConnes Conjecture [BCH94 ] or to treat algebraic Ltheory with arbitrary
coefficients. In both cases a crucial question is whether a suitable transfer *
*can
be constructed. (A geometric version of an Ltheory transfer is one of the many
ingredients in [FJ89].)
1.1. Acknowledgments. We would like to thank Tom Farrell and Lowell Jones
for many discussions on the subject. We would also like to thank Wolfgang Lück
for help with the Swan group actions. Our research was supported by the SFB 478
 Geometrische Strukturen in der Mathematik  Münster.
2.Preliminaries
2.1. Conventions and notation. In this section we briefly introduce some nota
tion that is used throughout the proof. For more details the reader should cons*
*ult
Section 2 in [BFJR03 ].
2.1.1. The functor K1 . We will denote by K1 the functor which associates to
an additive category its nonconnective Ktheory spectrum, see [PW85 ] or [CP97*
* ].
We assume that the reader is familiar with the standard properties of this func*
*tor,
compare [BFJR03 , Subsection 2.1]. A statement about exact functors between
additive categories is true ä fter applying Ktheoryö r ö n the level of Kthe*
*ory"
will always mean after applying K1 .
2.1.2. Modules and morphisms over a space. As explained below in Subsection 2.2
the assembly map will be described as a öf rgetcontrol map" between suitable
additive categories of (geometric) modules. An Rmodule M over the space X is
a family (Mx)x2X of finitely generated free Rmodules Mx indexed over points of
X, which is locally finite in the sense that x2K Mx is finitely generated for *
*every
compact subset K X. A morphism OE: M ! N is an Rlinear map OE: x2X
Mx ! y2X Ny. Such a map can of course be decomposed and written as a matrix
OE = (OEy,x) indexed over X x X. The additive category of all such modules and
morphisms would be denoted C(X) or C(X; R) and is equivalent to the category of
finitely generated free Rmodules if X is a compact space.
2.1.3. Support conditions. We are however only interested in subcategories of m*
*od
ules and morphisms satisfying certain support conditions. The support of a modu*
*le
M or a morphism OE are defined as suppM = {x 2 X  Mx 6= 0} X respectively
suppOE = {(x, y)  OEy,x6= 0} X x X. For a morphism control condition E (a set
of subsets of X x X) and an object support condition F (a set of subsets of X) *
*we
denote by
C(X, E, F)
the subcategory of C(X) consisting of modules M, for which there exists an F 2 F
such that suppM F , and morphisms OE between such modules, for which there
6 ARTHUR BARTELS, HOLGER REICH
exists an E 2 E with suppOE E. We will often refer to such morphisms as E
controlled morphisms. The conditions one needs to impose on E and F in order to
assure that this yields in fact an additive category are spelled out in Subsect*
*ion 2.3
of [BFJR03 ]. A basic example of a morphism control condition on a metric space
(X, d) is Ed, consisting of all E XxX for which there is ff > 0 such that (x,*
* y) 2 E
implies d(x, y) < ff. Measuring control via a map p: X ! Y is formalized by pul*
*ling
back E (living on Y ), i.e. forming p1E = {(p x p)1(E)  E 2 E}. In the case,
where p is the inclusion of a subspace we usually omit p and write E instead of
p1E. Similar notational conventions apply to the F's.
2.1.4. Equivariant versions. We usually deal with equivariant versions where X
is assumed to be a free space and modules and morphisms are required to be
invariant under the action. The corresponding category is denoted
C (X, E, F).
Under suitable conditions about the E's and F's a equivariant map f :X ! Y
induces a functor on such categories which sends M (a module over X) to f*M (a
module over Y ) given by f*My = x2f1({y})Mx.
2.1.5. Thickenings. If E is a neighborhood of the diagonal in X x X and A a
subset of X, then we define the Ethickening AE of A in X to be the set of all
points x 2 X for which there exists a point a 2 A such that (a, x) belongs to E.
In the case where E is determined by a constant ffi via a metric or by a pair (*
*ff, ffi)
using the öf liated distance" (compare Subsection 4.2) we write Affi, respectiv*
*ely
Aff,ffifor the corresponding thickenings.
2.1.6. Germs. We will often use Karoubi quotients of the categories C (X, E, F)
introduced above: let F0 be another object support condition which is Ethicken*
*ing
closed, i.e. for every F 2 F0 and E 2 E there exists an F 02 F0 such that F E *
*F 0.
Then C (X, E, F)>F0 is defined as the additive category which has the same obje*
*cts
as C (X, E, F), but where morphisms are identified whenever their difference fa*
*ctors
over a module with support in F0 2 F0. We think of this construction as taking
germs away from F0. If our space is X x [1, 1) and F0 = {X x [1, t]  t 2 [1, 1*
*)}
then we write C1 rather than C>F0 , see Example 8.8. Further formal properties
of these constructions which will be needed throughout the proof are discussed *
*in
Appendix 8.4.
2.2. Assembly as a öf rget controlmap".
2.2.1. Resolutions. A space is called compact if it is the_ orbit of some *
*com
pact subspace. A_resolution_of the space X is a map p: X ! X of CW
complexes, where X is a free space_and every compact set in X is the image
under p of some compact set in X . For every space the projection X x ! X is
a functorial resolution called the standard resolution.
2.2.2. The functor D . In [BFJR03 , Definition 2.7] we defined for a not necess*
*arily
free CW complex X the notion of equivariant continuous control. This is a
morphism support condition denoted_E cc(X) on the_space X x [1, 1). The set
of all compact_subsets_of_X is denoted_F c(X ). The object support condition
p1_XF c(X ), where p__X:Xx [1, 1) ! X denotes the projection, is our standard
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 7
__
object support condition on X x [1, 1). It is shown in [BFJR03 , Section 3 and *
*5]
that up to equivalence
__ __ 1 1 __
D (X ; p) = C (X x [1, 1); (p x id) E cc(X), p__XF c(X ))
does not depend upon the chosen resolution (hence one can always use X x ! X
as a resolution;_in this case we denote the category by D (X)) and that X 7!
K1 D (X ; p) yields an equivariant homology theory on the level of homotopy
groups.
2.2.3. Assembly. The map induced by
D (E (VCyc )) ! D (pt)
is on the level of Ktheory up to an index shift a model for the generalized as*
*sembly
map that was discussed above. Compare [BFJR03 , Corollary 6.3].
3.Outline of the proof
The injectivity part of Theorem 1.4 is proven in [BFJR03 ]. We will prove sur*
*jec
tivity. Our first observation is that it suffices to prove surjectivity of the *
*map which
is induced on the level of Ktheory by D (X(1)) ! D (pt), where X(1) is any
CW complex all whose nontrivial isotropy groups are infinite cyclic. In fact *
*by
the universal property of E (Cyc ) such a map always factorizes over D (E (Cyc *
*)).
It follows from [BFJR03 , Proposition 3.5] that instead of considering the map *
*of
standard resolutions (compare 2.2.1) we can equally well work with any map of
resolutions which covers X(1) ! pt. We proceed to construct a space X(1) and
such a map of resolutions.
Let M~ be the universal covering of a closed Riemannian manifold with strictly
negative sectional curvature. The hyperbolic enlargement of ~Mis the warped pro*
*d
uct (compare [BO69 ])
HM~ = R xcosh(t)~M.
It is the differentiable manifold R x ~M equipped with the Riemannian metric de
termined by dg2HM~= dt2+ cosh(t)2dg2~M. We refer to the Rfactor as the hyperbo*
*lic
enlargement direction or briefly the Hdirection. Let SHM~ denote the unitsphe*
*re
subbundle of the tangent bundle of HM~. For a subset A R we denote by SHA ~M
the restriction of this bundle to the subspace HA ~M HM~ which is defined as
A x ~M R x ~M= HM~. Throughout the paper we also fix the notation
B = [0, 1) and T = [1, 1).
The space SHM~ x B x T will be important in our context and we will generically
use h, fi and t to denote its H, B and Tcoordinate.
In Section 7 we factorize the natural projection SHM~ ! M~via a map called pX
over a certain compact free space_X,_i.e._we_have_a_commutative_diagram____*
*______________________________________________________________@
____________((___________________________________*
*________________________________________
SHM~ _pX_//_X____//~M.
Roughly speaking X is obtained from SHM~ by collapsing SH(1,1]~Malong the
Hdirection to SH{1}M~ and similarly SH[1,1)~Mto SH{1}~M. Note in particular
that SM~ SH{0}~Msits naturally as a subspace in X. For details about the map
8 ARTHUR BARTELS, HOLGER REICH
pX see Subsection 7.1 (resp. [BFJR03 , 14.5]). In Subsection 7.1 we also constr*
*uct
a quotient map
p: X x B ! X(1).
Here X(1) is a space all whose isotropy groups are cyclic. It is obtained as *
*the
infinite mapping telescope of a sequence of maps which collapse more and more
lines in SM~ X. Here the Bdirection is the telescope direction and the lines
correspond to preimages of closed flow lines of the geodesic flow under the cov*
*ering
projection SM~ ! SM. (Although X(1) is not a model for E (Cyc ) it is fairly
close to being one. With more effort one could probably work with an actual mod*
*el
at its place.) The map of resolutions alluded to above is now
X x B ____//_~M
p *
fflffl fflffl
X(1) ____//_pt.
This map of resolutions induces the bottom map in the following diagram of addi*
*tive
categories.
(5)
C ((SHM~ x B x T)", Ew, FB)1oo________C ((SHM~ x T)", Egeo)1
OO PPP OO ____________________*
*________________________________
(7) PPPP (3) _________________*
*__________________________________
 PPPP  _______________*
*________________________________
~ PPPP(6)PP1 ~ 1 _____________*
*__________________
C ((SHM x B x T)", Es, FB) PPPP C (SH{0}M x T, Easy) __(4)_________*
*______________________________
 PPPP  ________________*
*___________________________________
(8) PPPP (2) __________________*
*________________________________
fflffl PPP(( fflfvv__________________*
*_______________fl
C (X x B x T, (p x idT)1E cc(X(1)), FB)1 C (M~ x T, Ed)1
= (1)
fflffl fflffl
D (X x B, p)________________________//_D (M~, *).
Here (SHM~ x T)" is the subspace of SHM~ x T consisting of all points (v, t) wh*
*ere
the absolute value of the Hcoordinate satisfies h(v) t. Similarly the subs*
*pace
(SHM~ x B x T)" consists of all (v, fi, t) with h(v) t + ~nfi. The constant
here is ~n = 10n+3, where n is the dimension of SHM~. (The subscript " is
supposed to remind the reader of the shape of the region it describes.) We will
mostly be interested in these subspaces, but all maps and all object and morphi*
*sm
support conditions can and will be defined on the whole spaces. The correspondi*
*ng
restrictions to the "subspaces will not appear in the notation.
All maps in the diagram except (3) and (8) are induced by the obvious pro
jections, inclusions or identity maps of the underlying spaces. The map (3) is
essentially given by the geodesic flow and discussed in detail in Section 4. Th*
*e map
of spaces underlying (8) is induced by pX :SHM~ ! X.
The essential information in the diagram is however contained in the different
object and morphism support conditions which will be explained in detail below.
Going clockwise around the diagram from the lower left hand corner to the lower
right hand corner should be thought of as forgetting more and more control. Our
task is to step by step gain control going counterclockwise. (The existence of *
*the
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 9
wrong way maps (5) and (8) says that in between we gain more control than we
actually need.)
We have the following statements about the diagram above:
(i)It will be clear from the construction that the diagram without the map
(3) commutes.
(ii)The triangle consisting of the bended arrow (4) and the maps (3) and (2)
induces a triangle in Ktheory which commutes up to homotopy. This will
be proven in Corollary 5.10.
(iii)The map (2) induces a split surjection in Ktheory and hence by (ii) al*
*so
the maps (4) and (6). This is an immediate consequence of Proposition 5.1
(compare the discussion before that Proposition).
(iv)The map (7) induces an equivalence in Ktheory by the Foliated Control
Theorem 6.17.
(v)The map (1) induces an equivalence in Ktheory by the easy Lemma 3.1.
These statements imply that the bottom map induces a split surjection in Ktheo*
*ry
and hence our main Theorem 1.4 follows.
We will now describe the diagram in more detail and explain some aspects of
statements (ii), (iii) and (iv). We proceed counterclockwise.
(1) Equip T with the standard metric. Let d denote any product metric (e.g. t*
*he
maxmetric) on ~MxT and let Ed be the corresponding morphism support condition.
Observe that the continuouscontrol condition *1E ccis a weaker condition. The
resulting forgetcontrol map (1) is shown to induce an equivalence in Lemma 3.1
below.
(2) The base space of the bundle SH{0}~Mis M~. The map (2) is induced by the
bundle projection. The space SHM~ comes equipped with two different foliations,
the asymptotic foliation Fasyand the geodesic foliation Fgeo. These are explain*
*ed
in Section 4. In that section we will also define the notion of foliated contro*
*l with
a prescribed decay speed depending on a foliation F and a set of decay speed
functions S. The important point about the map (2) is that in the source we have
foliated control with a certain carefully chosen decay speed (denoted Sasy) with
respect to the asymptotic foliation Fasy. This kind of control will be denoted *
*Easy.
Section 5 is devoted to proving that the map (2) induces a split surjection in *
*K
theory, see in particular Proposition 5.1. To prove this we will construct a tr*
*ansfer
map going essentially the other way. In fact the target of the transfer map wil*
*l not
be C (SH{0}~Mx T, Easy)1 but a formally enlarged version of that category which
yields the same Ktheory. The construction of the transfer map will depend on
the choice of a sequence d = (ffi0, ffi1, ffi2 . .).of decay speed functions wh*
*ich will be
responsible for the decay speed one can achieve in the target. The geometry of *
*our
situation enters in verifying that a suitable sequence of decay speed functions*
* exists,
compare Lemma 5.8. The map on the Ktheory of C (M~ x T, Ed)1 induced by the
composition of the transfer and the projection is described by an element in a *
*Swan
group in Proposition 5.9. In order to achieve that this element is the identity*
* in
our case, we have to work with the subbundle S+ H{0}~Mof SH{0}~Mwhose fiber is
contractible. This explains the necessity of the hyperbolic enlargement: it all*
*ows
us to pick out this subbundle.
10 ARTHUR BARTELS, HOLGER REICH
(3) In Section 4 we will study the map (3) which is induced by
SHM~ x T ! SHM~ x T
(v, t)7! ( t(v), t),
where denotes the geodesic flow on SHM~. Via this map one gains control in
the directions transverse to the geodesic flow. We prove in Theorem 4.9 that the
map turns foliated control with respect to the asymptotic foliation into foliat*
*ed
control with respect to the geodesic foliation. In fact in the target we will *
*have
foliated control (with respect to the geodesic foliation) with exponential deca*
*y speed
(depending on the upper bound for the sectional curvature) and our choice of de*
*cay
speed Sasy for Easy in (2) is made in such a way that we achieve this. Since
t(SH{0}~M) SH[t,t]~Mall objects in the image will lie in (SHM~ x T)".
(4) The bended arrow (4) is induced by the projection SHM~ ! M~. This col
lapses the noncompact Hdirection and is therefore not a proper map. In genera*
*l,
non proper maps do not induce functors on our categories of modules over a spac*
*e.
(Such a map does not preserve the local finiteness condition, compare 2.1.2.) H*
*ow
ever, the restriction of the projection to (SHM~ x T)" is proper and we obtain a
well defined functor on objects. Since SHM~ ! M~ does not increase distances and
Egeocontrolled morphisms are in particular bounded with respect to the product
metric we obtain a well defined functor. In Corollary 5.10 we show that the tri*
*angle
induced in Ktheory by the maps (2), (3) and (4) commutes up to homotopy. Here
we use a variant of the Lipschitz homotopy argument from [HPR97 ].
(5) The map (5) is induced by the inclusion of SHM~ x {0} x T into SHM~ x
B x T. This inclusion is clearly compatible with the "subspaces. We equip B
with the standard (Euclidean) Riemannian metric and SHM~ x B with the product
Riemannian structure. Also we extend the geodesic foliation (by taking the prod*
*uct
with the trivial 0dimensional foliation of B) to a foliation Fw of SHM~ x B. N*
*ow
Ew is defined similar to Egeo as foliated control with a certain carefully chos*
*en
decay speed S (defined in Subsection 6.3) with respect to the foliation Fw. It
will follow from the construction (see Proposition 6.13 (i)) that the inclusion*
* maps
Egeocontrol to Ewcontrol. The FB object support condition consists of all sub*
*sets
whose projection to B is contained in a compact interval [0, fi0].
(6) The map (6) is induced by the projection SHM~ x B ! M~. As in (4) this
projection is not proper since it collapses the noncompact H and Bdirection.
However the restriction to the "subspace and the FB condition ensure that we
nevertheless have a well defined functor on objects. By construction Ewcontrol
dominates metric control and the projection does not increase distances. Theref*
*ore
(6) is also compatible with the morphism control conditions.
(7) The fact that the map (7) induces an equivalence in Ktheory should be
thought of as a öf liated control theorem". It is proven as Theorem 6.17. Very
roughly, this theorem improves foliated control to metric control (with decay s*
*peed)
on compact subsets (in the SHM~coordinate) that do not meet preimages of "shor*
*t"
closed geodesics in M. The relatively long Section 6 is devoted to the proof of*
* this
theorem. The only difference between source and target of the map (7) are the
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 11
morphism support conditions. As explained above the weak control condition Ew
is essentially, i.e. up to the added Bdirection, foliated control with respect*
* to the
geodesic foliation with a certain carefully chosen decay speed S. The stronger
control condition Es is obtained from Ew, by adding a metric control condition *
*with
decay speed S over a certain subset S SHM~ x B x T. The subset and the precise
control condition will be explained in Subsection 6.4.
(8) The map (8) is induced from the projection pX :SHM~ ! X. In particular
this map collapses the non compact Hdirection and hence the remaining FBobject
support in the target is just the usual compact support. The Es condition is *
*shown
to be strong enough to induce a map in Proposition 7.1.
This finishes the outline of the proof.
We start the proof by the following easy lemma about the map (1).
Lemma 3.1. The map (1) induces an equivalence in Ktheory.
Proof.The map C (M~ x T, Ed) ! C (M~ x T, *1E cc) induces a map between the
two corresponding "germs at infinity"fibrations, see Example 8.8. In the resul*
*ting
ladder the two middle terms allow an Eilenbergswindle towards infinity along T
(compare [BFJR03 , 4.4, 4.5]) and the left hand terms are even equal.
4.Gaining control via the geodesic flow
There are two foliations on SHM~, the geodesic foliation and the asymptotic f*
*oli
ation. In this section we will define the notion of foliated control with a pre*
*scribed
decay speed and we will prove in Theorem 4.9 that the geodesic flow can be used
to turn foliated control with a certain decay speed with respect to the asympto*
*tic
foliation into foliated control with exponential decay speed with respect to th*
*e geo
desic foliation. Finally we show that after forgetting control the map induced *
*by the
geodesic flow is homotopic to the identity (more precisely to a certain inclusi*
*on),
see Theorem 4.11.
4.1. Geometric preparations. Recall from Section 3 that HM~ denotes the hy
perbolic enlargement and SHM~ its sphere bundle. The space SHM~ will be equipped
with two foliations, the geodesic foliation and the asymptotic foliation. Let *
* :
R x SHM~ ! SHM~, (t, v) 7! t(v) denote the geodesic flow. The geodesic foliati*
*on
Fgeois simply the 1dimensional foliation given by the flow lines of . Two poi*
*nts
v and w in SHM~ are called asymptotic if the distance between t(v) and t(w)
stays bounded if t tends to +1. This defines an equivalence relation and the set
of equivalence classes, denoted S(1), can be naturally equipped with a topology
in such a way that the map a : SHM~ ! S(1) given by sending a vector to its
equivalence class restricts to a homeomorphism on each fiber SHM~x of the bundle
SHM~ ! HM~, compare Section 1 in [EO73 ]. The preimages a1(`) for ` 2 S(1)
are the leaves of a foliation Fasywhich we will call the asymptotic foliation. *
*Since
M is compact there are positive constants a and b such that the sectional curva*
*ture
K satisfies
(4.1) b2 K a2.
The same inequalities hold for the hyperbolic enlargement (compare [FJ86]).
12 ARTHUR BARTELS, HOLGER REICH
The homeomorphisms SHM~x ! S(1) are used to define the fiber transport
~= ~=
ry,x: SHM~x ! S(1)  SHM~y
for the bundle SHM~ ! HM~. The fiber transport will play an important role
in Section 5. Since we have curvature bounds the fiber transport is known to be
Höldercontinuous. More precisely a consequence of Proposition 2.1. in [AS85 ] *
*is
the following lemma which will be used in Theorem 4.9 and Lemma 5.8.
Lemma 4.2. For all ff > 0 there is a constant C0(ff) > 0 such that for x, y 2 H*
*M~
with d(x, y) ff and v, w 2 SHM~x we have
d(ry,x(v), ry,x(w)) C0(ff) . d(v, w)a_b.
Later we will have to quantitatively analyze how the flow t : SHM~ ! SHM~
may increase distances. For this purpose we introduce the function Cflw(t) in t*
*he
following lemma.
Lemma 4.3. There exists a monotone increasing function Cflw(t) such that d s
Cflw(t) for all s t. In particular for arbitrary v, w 2 SHM~ we have
d( t(v), t(w)) Cflw(t) . d(v, w).
Proof.This again holds since we have curvature bounds. The differential of the
geodesic flow can be expressed in terms of Jacobi fields, compare [EHS93 , Sec
tion 2.3]. These satisfy a second order differential equation involving the sec*
*tional
curvature as coefficients [CE75 , p.1516] and the result can be deduced from t*
*his
equation using standard arguments about ordinary differential equations, compare
e.g. [Per01, p.79].
4.2. Foliated control with decay speed. We now want to define the notions of
metric respectively foliated control with decay speed S. Let F be a foliation o*
*f a
Riemannian manifold N. For x, y 2 N we will write
dF (x, y) (ff, ffi)
if there is a piecewise C1path of arclength shorter than ff which is entirely *
*con
tained in one leaf of the foliation and whose start respectively endpoint lie*
*s within
distance ffi_2of x respectively y. (Elsewhere we used ffi instead of ffi_2but c*
*ompare Re
mark 4.5.) Suppose we are given a set S of functions from T to [0, 1). (We often
use ffit as the name for the function which sends t to ffit.) Suppose S satisfi*
*es the
following conditions.
(A) For each ffit 2 S and every ff 2 R there exists ffi0tand t0 1 such that
ffit+ff ffi0tfor all t t0 + ff.
(B) Given ffit, ffi0t2 S there exists ffi00t2 S and t0 1 such that ffit+ f*
*fi0t ffi00tfor
all t t0.
Given such a set of functions S we make the following definitions.
Definition 4.4 (Foliated and metric control with decay speed).
(i)Suppose X = (X, d) is a metric space, then we define a morphism support
condition E = E(X, S) on X xT by requiring that a subset E of (X xT)x2
belongs to E if there exists a ffit 2 S and constants ff > 0 and t0 > 1 *
*such
that for all (x, t, x0, t0) 2 E we have
t  t0 ff, d(x, x0) ff
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 13
and if t, t0> t0 then
d(x, x0) ffimin(t,t0).
This condition will be called metric control with decay speed S.
(ii)Suppose N is a Riemannian manifold equipped with a foliation F . We
define E = E(N, F, S) a set of subsets of (N x T)x2 by requiring that a
subset E belongs to E if there exists a function ffit2 S and constants f*
*f > 0
and t0 > 1 such that for all (x, t, x0, t0) 2 E we have
t  t0 ff, d(x, x0) ff
and if t, t0> t0 then
dF (x, x0) (ff, ffimin(t,t0)).
If E defines a morphismcontrol condition then it will be called foliated
control with respect to the foliation F with decay speed S.
Remark 4.5. Observe that for ff = 0 the foliated condition in (ii) specializes *
*to the
metric condition in (i), i.e. if ff = 0 then an (ff, ffit)foliated controlled *
*morphism is
ffitcontrolled in the metric sense.
Remark 4.6. Note that these definitions only depend on the behaviour of functio*
*ns
in S in a neighborhood of 1, i.e what is really important about S is the set of
germs (at infinity) of functions it determines.
Warning 4.7. While conditions (A) and (B) guarantee that in the metric case
E(X, S) is a morphismcontrol condition, in general this may fail in the foliat*
*ed
case for E = E(N, F, S) because it is not clear that E is closed under composit*
*ion.
However in the two cases we are interested in E is closed under composition by
Lemma 4.10 combined with Lemma 4.8 (ii) below.
4.3. Gaining control via the geodesic flow. Now set
Sgeo = {A . exp(at)A > 0},
Sasy = {A . exp(~(Cflw(2t + B) + (t + B)2))A > 0, B 2 R, ~ > 0},
where in the first line a comes from the upper curvature bound in (4.1). In the
second line we take Cflw(2t + B) = Cflw(0) for 2t + B < 0. Clearly conditions (*
*A)
and (B) before Definition 4.4 are satisfied in both cases. Here Sasyis designed*
* in
such a way that the geodesic flow will turn Sasydecay speed into Sgeodecay sp*
*eed,
see Theorem 4.9 which uses property (i) of the following elementary lemma.
Lemma 4.8. Apart from (A) and (B) before Definition 4.4 Sasysatisfies
(i)For ffit2 Sasyand each ff > 0 there exists a ffi0t2 Sgeoand t0 1 such *
*that
Cflw(t + ff) . ffit ffi0tfor all t .t0
(ii)If ffit2 Sasyand ~ > 0 then (ffit)~ 2 Sasy.
We now define Egeo to be foliated control with exponential decay speed Sgeo
with respect to the geodesic foliation Fgeo and Easy to be foliated control with
decay speed Sasywith respect to the asymptotic foliation Fasy, i.e.
Egeo = E(SHM~, Fgeo, Sgeo),
Easy = E(SHM~, Fasy, Sasy).
14 ARTHUR BARTELS, HOLGER REICH
We are now prepared to formulate the main result of this section. Recall that
(SHM~ x T)" is the subspace given by h t, where h denotes the Hcoordinate
and t the Tcoordinate.
Theorem 4.9. The map (v, t) 7! ( t(v), t) on SHM~ x T turns Easycontrol into
Egeocontrol. In particular, it induces a well defined map
C (SH{0}~Mx T, Easy)1 ! C ((SHM~ x T)", Egeo)1 .
Proof.We recall some results which were discussed in [BFJR03 ] in Proposition 1*
*4.2
and Lemma 14.3 and rely on [HIH77 ]. Let a and b be the curvature constants from
(4.1). With the constants C = (1 + b2)1=2, D = 1=a and the function E(ff) =
2_ _b 1_ ~
bsinh(2(ff + a)) we have for any pair v, w 2 SHM of asymptotic vectors with
d(v, w) ff the following inequality
0 at
dFgeo( t(v), t0(w)) C . (ff + t  t  + D), C . 2E(ff).. e
Now consider (v, t, w, t0), where v and w are no longer assumed to be asymptoti*
*c.
Assume t0 > t then we have because of Lemma 4.3 for any monotone decreasing
function ffit that
dFasy(v, w) (ff, ffit) and t0 t ff
implies
at
dFgeo( t(v), t0(w)) C . (2ff + D), C . 2E(ff) . e + Cflw(t.+ ff)ffit
By Lemma 4.8 (i) and property (B) for Sgeo this implies the claim about the
morphism control conditions. Since the flowspeed in the Hdirection is at most*
* 1
we see that (v, t) 7! ( t(v), t) maps SH{0}~Mx T to (SHM~ x T)".
We still have to verify that Egeoand Easy are well defined morphismsupport
conditions, compare Warning 4.7. This is a consequence of Lemma 4.8 and the
following lemma.
Lemma 4.10 (Foliated triangle inequalities). For u, v, w 2 SHM~ we have:
(i)If dFgeo(u, v) (ff, ffi) and dFgeo(v, w) (fi, ffl) then
dFgeo(u, w) (ff + fi, ffi + Cflw(ff)(ffl +)ffi+ ffl).
(ii)If dFasy(u, v) (ff, ffi) and dFasy(v, w) (fi, ffl) then
dFasy(u, w) (Cff + C(ffi_+_ffl2) + fi, ffi + 2C0(ff + ffi_+_ffl2)(C + 1)a_b*
*(ffi_+_ffl2)a_b+ ffl).
Here the constant C0 stems from Lemma 4.2 and C from the proof of Theorem 4.9.
Proof.In both cases there are u0, v0, v1 and w1 within distance ffi respectivel*
*y ffl
from u, v respectively w such that u0 and v0 respectively v1 and w1 can be join*
*ed
by a curve of length ff respectively fi contained in a leaf of the foliation in*
* question.
For the first statement assume that t(u0) = v0 and let ~u0= t(v1). Then ~u0
and w1 are contained in a leaf of Fgeoand can be used to prove the first inequa*
*lity
(using 4.3). For the second statement let x and y denote the foot points of u0 *
*and
v1 and set ~u0= rx,y(v1). Then ~u0and w1 are contained in a leaf of Fasyand can
be used to prove the second inequality (using 4.2 and [BFJR03 , 14.3]).
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 15
We finish this section by comparing the map induced by the flow to the map
induced by the inclusion SH{0}~M ! SHM~. This is only possible after relaxing
the control conditions in the target. We denote by Ed the control condition com*
*ing
from the product metric on SHM~ x T.
Theorem 4.11. The map SH{0}~Mx T ! (SHM~ x T)" defined by (v, t) 7!
( t(v), t) and the inclusion induce homotopic maps
C (SH{0}~Mx T, Easy)1 ! C ((SHM~ x T)", Ed)1 .
on the level of Ktheory.
Proof.Let us abbreviate the two categories from above by C0 and C1. Let ^Z
SH{0}~Mx [1, 1)x T consist of all (v, s, t) with s t. Let p : ^Z! SH{0}~Mx T
denote the obvious projection. We will use ^C= C (Z^, ^E)1 . Here ^E= Ed \ p=1E*
*asy
where Ed denotes metric control with respect to a product metric on Z^. The
arguments used in the proof of Theorem 4.9 can also be used to check that (v, s*
*, t) 7!
( s(v), t) induces a functor H : ^C! C1. Moreover, (v, t) 7! (v, 1, t) and (v, *
*t) 7!
(v, t, t) induce functors I, J : C0 ! ^Cwhile (v, s, t) 7! (v, t) induces P : ^*
*C! C0.
The claim of the theorem is that H O I and H O J induce equivalent maps in K
theory. Clearly, P O I = P O J = idC0. It is now sufficient to show that I indu*
*ces an
isomorphism in Ktheory, since then I O P = id^Cand H O I = H O I O P O J = H O*
* J
in Ktheory. Now I : C0 ! ^Cis equivalent to a Karoubi filtration with quotient
C (Z^, ^E)>Fq, where Fq consists of all sets of the form {(v, s, t)s t and (s
N ort N)} for some N. We claim that this category is flasque. Indeed, the map
(v, s, t) 7! (v, s  1, t) induces an Eilenberg swindle on it. A little care is*
* needed
in producing the swindle from this map, since it not welldefined on (v, s, t) *
*for
s < 2 (because then s  1 62 [1, 1)). However, in the quotient category in ques*
*tion
modules over SH{0}~Mx [1, 2] x T can be ignored. Compare 2.1.6.
Corollary 4.12. The triangle consisting of the maps (3),(4) and (2) in the main
diagram of Section 3 commutes up to homotopy after applying Ktheory.
Proof.Compose the maps in Theorem 4.11 with the map induced by the projection
SHM~ x T ! M~x T.
5.The transfer
Our aim in this section is to prove that the map (2) in our main diagram in
Section 3 induces a split surjective map in Ktheory. We define S+ HM~ to be the
subbundle of the sphere bundle SHM~ T HM~ = T R x T ~M= R x T ~Mconsisting
of all vectors with nonnegative Rcoordinate. Note that the fiber of this subb*
*undle
is a disk and hence contractible. This is important, because we will show below
that the transfer on a bundle whose fiber has interesting topology is in general
not a splitting of the bundle projection, cf. Proposition 5.9. Since the projec*
*tion
S+ H{0}~M! M~ factorizes as
"
S+ H{0}~MØ____//SH{0}~M____//~M
surjectivity of the map (2) is implied by the following proposition.
16 ARTHUR BARTELS, HOLGER REICH
Proposition 5.1. The map
C (S+ H{0}~Mx T, Easy)1 ! C (M~ x T; Ed)1
induced by the bundle projection induces a split surjective map in Ktheory.
In order to prove this proposition we will produce a transfer map in the reve*
*rse
direction. In fact we will construct the following (noncommutative) diagram.
(5.2) C (S+ H{0}~Mx T, Easy)1____//_echhfC (S+ H{0}~Mx T, Easy)1
hhh44
 trdhhhhhhh 
 hhhhhh 
fflfflhhh fflffl
C (M~ x T; Ed)1____________//echhfC (M~ x T; Ed)1 .
The diagonal arrow is the promised transfer. It will depend upon the choice of a
sequence d = (ffi0, ffi1, ffi2, . .).of decay speed function from Sasy. The ho*
*rizontal
arrows induce equivalences in Ktheory and the square without the transfer map
commutes. In Subsection 5.5 we will show that the lower triangle commutes in
Ktheory up to multiplication by a certain element in a Swan group which is de
termined by the homology of the fiber of the bundle S+ H{0}~M. Since the fiber *
*of
this bundle is a disk we know that the triangle induces a commutative triangle,*
* see
Corollary 5.10.
Remark 5.3. In the diagram above and in the proof below we have to deal with
certain Waldhausen categories (categories with cofibrations and weak equivalenc*
*es
[Wal85]) which are categories of chain complexes. There seems to be no good def*
*i
nition of nonconnective Ktheory in the literature which applies in this gener*
*ality.
However in our situation we can always make an adhoc construction of a non
connective Ktheory spectrum as follows. Let X be a free space and E a control
condition on X. Let Ed be the standard Euclidean metric control condition on Rn.
Let p : Rn x X ! Rn and q : Rn x X ! X be the projections and K() be Wald
hausen's connective Ktheory functor which applies to Waldhausen categories. It
is well known that the spaces
KC (Rn x X; p1Ed \ p1E)
together with structure maps derived from swindles coming from a decomposition
Rn = Rn+[Rnyield a model for the non connective Ktheory spectrum of C (X; E)
(compare the last page in [CP97 ]). The same construction applies to categories*
* like
chfC(X; E) and chhfC(X; E) and all variants which will be used below. Hence in
each case it makes sense to talk of the nonconnective Ktheory. In all constru*
*ctions
and arguments below the Rnfactor will play the role of a dummy variable. In
order to facilitate the exposition we hence formulated all arguments only for t*
*he 0
th spaces, i.e. for connective Ktheory. It is straightforward to make the nece*
*ssary
modifications to obtain the analogous statement for the other spaces of the spe*
*ctrum
and to check compatibility with the structure maps.
5.1. Setup. The construction of the transfer works in the following generality*
*. Let
~Bbe the universal covering of the compact space B. Let denote the fundamental
group which acts on ~Bfrom the left. Let ß : ~B! B denote the covering projecti*
*on.
Suppose p : E ! B is a smooth fiber bundle with compact fiber. We form the
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 17
pullback ~Eand use the following notation.
~E__ß__//E
~p p
fflfflßfflffl
~B____//_B.
Suppose ~Bis equipped with a invariant metric d. Suppose E is a Riemannian
manifold and ~Eis equipped with the pulled back Riemannian structure and a 
invariant foliation F . Furthermore let S be a set of decay speed functions (co*
*mpare
Subsection 4.2) and assume that E = E(E~, F, S) really defines a morphism contr*
*ol
condition, compare Warning 4.7.
We also assume that we are given a fiber transport r, i.e. a homeomorphism
of fibers rb0,b: ~Eb! ~Eb0for each pair of points b and b0 in ~Bwhich fulfills *
*the
following requirements.
Assumption 5.4. The fiber transport has the following properties:
(i)It is functorial, i.e. rb,b= id~Eband rb00,b0O rb0,b= rb00,bfor all b00,*
* b0and
b 2 ~B.
(ii)It is invariant, i.e. for all b, b02 ~Band all g 2 we have
lg O rb0,bO l1g= rgb0,gb.
Here lg : ~Eb! ~Egbis the restriction of the left action of on ~E.
(iii)It is compatible with the foliation in the following strong sense. The*
*re
exists a constant C 1 such that for all b, b0 2 ~Bwith d(b, b0) ff a*
*nd
every e 2 ~Ebwe have
dF (rb0,be, e) (Cff, 0),
i.e. there is a path of length no longer than Cff inside one leaf which
connects rb,b0e and e.
An additional requirement will be formulated in Assumption 5.6 below. All the
assumptions are fulfilled in our situation where E ! B is the bundle S+ H{0}M !
M, the foliation is the asymptotic foliation Fasyand S = Sasy. Assumption (iii)
follows from [BBE85 , 1.1]: if OE(t) is a geodesic in M~ from b to b0, then the*
* path
t 7! rOE(t),be is contained in a leaf of Fasyand no longer than (1 + b2)1=2. ff*
*, where
b2 is a lower bound for the curvature of M.
5.2. Homotopy finite chain complexes. Below we would like to work with sin
gular chain complexes, which have the advantage that they do not depend on
any further choices (like triangulations or CWstructures). On the other hand
in order to define Ktheory we need to impose some finiteness conditions, i.e.
we want to work with homotopy finite chain complexes. We now introduce the
necessary_notation. Given a free space X and a controlcondition E we de
fine C (X; E) completely analogous to C (X; E) but we do not require that the
modules are locally finite. We hence allow objects M = (Mx) whose support
suppM = {x 2 X  Mx 6= 0} is an arbitrary subset of X. Also the free Rmodules
Mx need not be finitely generated. (We should however require that the cardinal*
*ity
of the bases are bounded by some fixed large enough cardinal. This allows us to
choose small models for all the categories that will appear below.) We think of
18 ARTHUR BARTELS, HOLGER REICH
__
the full subcategory C (X; E) inside C (X; E) as the category of "finiteö bjec*
*ts
and define as explained in the Appendix 8.1 the categories of finite, respectiv*
*ely
homotopy finite chain complexes
chfC (X; E) and chhfC (X; E).
__
Both categories are full subcategories of the category chC (X; E) of ä ll" chain
complexes and are naturally equipped with the structure of a Waldhausen categor*
*y,
see Appendix 8.1. The natural inclusions
C (X; E) ! chfC (X; E) ! chhfC (X; E)
induce equivalences on Ktheory, compare Lemma 8.1 and Remark 5.3. Analogous
considerations_apply to C (X x T; E)1 considered as the subcategory of "finite"
objects in C (X x T; E)1 .
__
5.3. The fiber complex. A chain complex C 2 chC (E~xT, E) is called a fiberwise
chain complex if no differential connects different fibers, i.e. if a pair of p*
*oints lies
in the support of a differential, then both points lie in the same fiber of the*
* bundle
~Ex T ! ~Bx T. Given such a fiberwise complex and a point (b, t) in the base we
define the fiber C(b,t)to be the largest subcomplex such that the support of al*
*l its
modules lie in the fiber ~Ex T(b,t)= ~Ebx {t}.
We define the fiberwise complex F and for a given ffi = ffit 2 S the fiberwise
complex F ffiby
F(b,t)= Csing(Eß(b)) and F(ffib,t)= Cffitsing(Eß(b)).
Here Csingdenotes the singular chain complex and Cffitsingdenotes the subcomplex
generated by all singular simplices oe : ! Eß(b)which have the property that
oe( ) has diameter ffit in E. The complexes F(b,t)and F(ffib,t)are complexes *
*over
Eß(b)by gluing each singular simplex to the image of its barycenter. The comple*
*xes
F and F ffibecome invariant complexes over ~Ex T via the maps
Eß(b)ooß~=E~b__~=_//~Ebx {t}inc//_~Ex.T
Observe that for F ffithe condition on the size of the singular simplices assur*
*es
that_each differential is (0, ffit) and hence E = E(F, S)controlled so that F*
* ffi2
chC (E~, E). This is not true for the full singular chain complex F .
Given_an Rmodule M 2 C (B~x T, Ed; R) and a fiberwise Zchain complex
C 2 chC (E~x T, E; Z) we define the fiberwise Rchain complex
__
M C 2 chC (E~x T, E; R)
by requiring that its fibers are given by (M C)(b,t)= M(b,t) ZC(b,t). In parti*
*cular
we consider the fiberwise complex M F ffi.
Using part of Remark 5.5 below and the fact that our fibers admit arbitrarily
fine triangulations one_can show that for ffi 2 S the complex M F ffiis homot*
*opy
equivalent inside chC (E~x T; E) to a locally finite complex and hence
M F ffi2 chhfC (E~x T; E).
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 19
Remark 5.5. Let T be a triangulation of the metric space X = T  such that the
diameter of each simplex is smaller than ffi. Let C(T ) denote the chain compl*
*ex
associated to the triangulation and let Cfflsing(X) denote the subcomplex of th*
*e sin
gular chain complex of X generated by all singular simplices which are smaller *
*than
ffl. Both complexes can be considered as complexes over X using the barycenters.
If 0 < ffi ffi1 ffi2 then there are natural inclusions
C(T ) ! Cffi1(X) ! Cffi2(X).
Both maps are chain homotopy equivalences and one can show that the homotopy
inverse and the homotopies can be chosen to be 10ffi2controlled when considere*
*d as
morphisms over X.
5.4. The transfer functor. The discussion above suffices to define the desired
transfer functor on objects. In order to define it on morphisms we need the fib*
*er
transport. For (b, t) and (b0, t0) 2 ~Bx T the map
rb0,b ~=
Eß(b)o~ß=o~Eb____//_~Eb0ß_//Eß(b0)
induces a chain map
F(b,t)! F(b0,t0)
which we will denote by r(b,t),(b0,t0). Using Assumption 5.4 (i) one checks that
M 7! M F
f = (f(b0,t0),(b,t))7!f r = (f(b0,t0),(b,t) r(b0,t0),(b,t))
defines a functor to homotopy finite chain complexes over ~Ex T if one ignores *
*the
control condition. But since rb0,bcan stretch simplices it does not induce a we*
*ll
defined map on the singular simplices of a fixed restricted size. This means th*
*at
the analogous definition with F ffidoes not work. In order to deal with this pr*
*oblem
we formally enlarge our category.
In Appendix 8.2 we construct for every Waldhausen category W satisfying some
mild conditions (which are satisfied for categories of chain complexes in an ad*
*ditive
category) a Waldhausen category fW. Objects in this category are sequences
C0 __c0_//C1_c1_//C2_c2_//. . .
where the Ciare objects in W and all cn are simultaneously cofibrations and weak
equivalences (trivial cofibrations) in W. A morphism f in fW is represented by a
sequence (fm , fm+1 , fm+2 , . .).of morphisms in W which fit into a commutative
diagram
cm+1 cm+2
Cm ___cm__//Cm+1_______//Cm+2_____//_. . .
fm fm+1 fm+2
fflffldm+~ fflffldm+~+1fflffldm+~+2
Dm+~ _____//Dm+~+1_____//Dm+~+2____//_... .
Here m and ~ are nonnegative integers. If we enlarge m or ~ the resulting di
agram represents the same morphism, i.e. we identify (fm , fm+1 , fm+2 , . .).w*
*ith
the sequence (fm+1 , fm+2 , fm+3 , . .).but also with (dm O fm , dm+1 O fm+1 , *
*dm+2 O
fm+2 , . .).. Sending an object to the constant sequence yields an obvious inc*
*lu
sion W ! fW and according to Proposition 8.2 this inclusion induces an equiva
lence in connective Ktheory. In the case where W = chhfC (E~x T; E) we write
20 ARTHUR BARTELS, HOLGER REICH
echhfC (E~x T; E) for the corresponding enlargement and using Remark 5.3 we con
clude that the inclusion induces an equivalence in nonconnective Ktheory.
Let d = (ffi0, ffi1, ffi2, . .).be a monotone increasing sequence of decay sp*
*eed func
tions ffii 2 S, i.e. for all i 0 and all t 2 T we have ffiit ffii+1t. Then f*
*or a module
M in C (B~x T; Ed)
0 ffi1 ffi2
M F d= (M F ffi! M F ! M F ! . .).
defines an object in echhfC (E~xT; E). Here the maps in the sequence are the na*
*tural
inclusion maps. They are shown to be Econtrolled homotopy equivalences using
again Remark 5.5.
Assumption 5.6. Suppose for each ff 0 there exists an integer ~(ff) such that*
* the
following holds:
If d(b, b0) ff and t  t0 ff then for all e 2 ~Eb0we have
i ffii+~(ff)
rb,b0({e}ffit) rb,b0({e}) t0,
for all sufficiently large t, t0.
Here the thickenings are taken in ~Eband ~Eb0with respect to distance in the am*
*bient
manifold.
Under this assumption we immediately obtain the following proposition.
Proposition 5.7. Suppose d is a sequence of decay speed functions satisfying As
sumption 5.6. Then there exists a functor
trd: C (B~x T; Ed) ! echhfC (E~x T; E)
which sends f : M ! N to the morphism in echhfC (E~x T; E) represented by
M F ffi0___//_M F ffi1__//_M F ffi2_//_. . .
fr fr fr
fflffl fflffl fflffl
N F ffi~___//N F ffi~+1//_N F ffi~+2//_. . .
for suitably chosen ~ depending on the bound of f.
It remains to check, that in our situation where the fiber bundle is S+ H{0}M*
* !
M, r is the asymptotic fiber transport and S = Sasy, we can find a suitable
sequence d of decay speed functions. The proof will use the fact that the fiber
transport is Hölder continuous, compare Lemma 4.2.
Lemma 5.8. There exists a sequence d = (ffi0, ffi1, ffi2, . .).of decay speed f*
*unctions
ffii 2 Sasysatisfying Assumption 5.6 with respect to the asymptotic fiber trans*
*port.
Proof.We abbreviate the constant a_bappearing in Lemma 4.2 by ~. Lemma 4.2
implies that rb,b0(effi) is contained in rb,b0(e)C0(ff).ffi~. Thus our task is*
* to find
ffii 2 Sasysuch that for all ff > 0 there is ~(ff) such that for all sufficient*
*ly large t, t0
with t  t0 ff we have
C0(ff)(ffiit)~ ffiit0.
In the following it will be convenient to extend all functions on T to functions
on R that are constant on (1, 1]. Start with an arbitrary ffi0 2 Sasy. Choose
inductively ffii 2 Sasysuch that
i . (ffi0ti+ . .+.ffii1ti)~ < ffiit
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 21
for all sufficiently large t. This is indeed possible because of Lemma 4.8 (ii)*
* and
(A) and (B) before Definition 4.4. Choose now ~(ff) 2 N larger than ff and C0(f*
*f).
Then
C0(ff) . (ffiit)~ (i + ~(ff)) . (ffiit)~ ffi~(ff)+it+i+~(ff) f*
*fi~(ff)+it0,
for sufficiently large t, t0 with t  t0 ff. (For the last inequality note*
* that all
functions in Sasyare monotone decreasing.)
5.5. An element in the Swan group. It remains to study the (noncommutative)
triangle
echhfC (E~x T, E)1
kk55k
ktrdkkkkkk p
kkkk fflffl
C (B~x T, Ed)1_inc_//echhfC (B~x T, Ed)1 .
We will denote the induced maps in Ktheory by the same symbols. Recall that
the inclusion incinduces an isomorphism. What we would like to understand is the
self map inc1O p O trd. To describe the result we need some preparation. Let us
fix a point b0 2 ~B. Observe that in general the diagram
~Egb0____rb0,gb0____//~E
F y b0
FFßFF ß yyy
~= FF##F __y~=yyy
Eß(b0)
does not commute. In fact we use it to define a left operation on Eß(b0)by le*
*tting
g 2 act via ßOrb0,gb0Oß1. The singular chain complex F0 = Csing(Eß(b0)) hence
becomes a complex of Z modules. As a Zchain complex F0 is homotopy equivalent
to a finite complex of finitely generated Zmodules. The homology groups Hi(F0)
are hence Z modules which are finitely generated as Zmodules. Such a module
defines an element in the Swan ring Sw ( ; Z). Ktheory becomes a module over
the Swan ring via maps
Sw( ; Z) Z Kn(R ) ! Kn(R ).
The Swan ring, its action on Ktheory and certain variants we need below in the
proof are discussed in Appendix 8.3.
Proposition 5.9. Under the identification
Kn(R ) ~=Kn+1(C (B~x T, Ed)1 )
coming from the germs at infinity fibration (compare Example 8.8) the map inc1O
p O trdcorresponds to multiplication with
1i=0(1)i[Hi(F0)]2 Sw( ; Z).
Corollary 5.10. Diagram (5.2)induces a commutative diagram in Ktheory.
Proof.The fiber of S+ HM~ ! M~is contractible and hence the Swan group element
is represented by the trivial Z module Z, which acts as the identity on Ktheo*
*ry.
22 ARTHUR BARTELS, HOLGER REICH
The rest of this subsection is devoted to the proof of Proposition 5.9. Again
we will only discuss the argument for connective Ktheory. This yields Proposi
tion 5.9 for n 1. The general result follows by filling in extra Rnfactors, *
*compare
Remark 5.3.
We first want to get rid of the f construction. Consider
pOtrd
_________________________________________________________*
*______________________________________________________________@
C (B~ x T, Ed)_r_//chhfC (B~x T, Ed)inc//_echhfC (B~x T,.Ed)
Here the functor  r is given by M 7! M F and f 7! f r, where now F is
considered as a complex over ~Bx T and we do not care how it is distributed over
each fiber.
Lemma 5.11. There is a natural transformation between incO ( r) and p O trd
which is objectwise a weak equivalence.
Proof.At M the natural transformation is given by the natural inclusion
M F ffi0__//_M F ffi1_//M F ffi2_//. . .
  
  
fflffl= fflffl= fflffl=
M F ______//_M F_____//_M F_____//_... .
The functors and the natural transformation in Lemma 5.11 are compatible
with the germs at infinity fibrations. The middle terms in these fibrations are
contractible since we work with the product metric. (Compare Example 8.8 and
[BFJR03 , Proposition 4.4, Example 4.5] for such arguments.) Hence there is a
version of Lemma 5.11 for the germs at infinity categories. We have reduced our
question to comparing the two maps
_inc_//
C (B~, Ed)___//chhfC (B~, Ed).
 r
Here  r is the obvious restriction of the functor above with the same name.
Since we assume that ~Bis compact the Edcondition is no extra condition and
we omit it in the following. For the same reason the inclusion of the orbit b0*
* ! ~B
for some fixed b0 2 ~Binduces equivalences on the categories and we are reduced*
* to
comparing the upper horizontal map in the following (non commutative) diagram
to the natural inclusion. (The diagram does commute if one replaces the horizon*
*tal
maps by the natural inclusions.)
(5.12) C ( b0; R)__r__//chhfC ( b0; R)
 
fflffl idF0 fflffl
C(ß(b0); R )____//chhfC(ß(b0); R ).
The vertical functors in this diagram are equivalences given by sending an R
module M = (Mgb0) over b0 to g2 Mgb0considered as an R module, compare
Lemma 2.8 in [BFJR03 ]. The lower horizontal map sends an R module N to the
complex of R modules N Z F0. Here F0 is the singular chain complex of the
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 23
fiber Eß(b0)considered as a Z module as explained towards the beginning of this
subsection and operates diagonally on N ZF0. On morphisms the functor sends
f to f idF0. Diagram (5.12) does not commute but we have the following lemma.
Lemma 5.13. There is a natural transformation between the two ways through
Diagram (5.12) which is objectwise an isomorphism.
Proof.Let M = (Mgb0) be an Rmodule over b0. Both ways through (5.12) send
M to ( g2 Mgb0) F . However, the actions are different. If we go first right
and then down acts only on the first factor, if we go down and then right a*
*cts
diagonally on the tensor product. The natural transformation from right/down to
down/right sends m v 2 Mgb0 F0 to m gv 2 Mgb0 F0.
It remains to compare  idF0to the natural inclusion. In Appendix 8.3 we
explain a variant Sw ch( ; Z) of the Swan group together with its action on K
theory. The complex F0 is a complex of Z modules which is degreewise free as a
Zcomplex and whose homology is finitely generated as an abelian group. Such a
complex defines an element in Swch( ; Z) and inc1O( idF0) describes the action
of this element on Ktheory. Proposition 5.9 now follows from Proposition 8.3 a*
*nd
the discussion following that proposition.
Remark 5.14. In the case we are interested in, where the fiber is a disk, we co*
*uld
avoid the Swan group and proceed differently after Lemma 5.13. In that case
the augmentation ffl: F0 ! Z is a equivariant map which is a nonequivariant
chain homotopy equivalence. In particular it induces for every free R module N
a homology isomorphism N Z F0 ! N Z Z. Each chain module of N Z F0 is
noncanonically isomorphic to the R module with the same underlying abelian
group but where operates only on the left tensor factor. Hence both complexes
are complexes of free R modules and the homology isomorphism is in fact an R 
chain homotopy equivalence. This yields a natural transformation between  F0
and the inclusion which is objectwise a weak equivalence.
6. A foliated control theorem for higher Ktheory
In this section we will show that a certain relax control map induces an equi*
*va
lence in Ktheory. Roughly speaking the map relaxes control from metric control
to foliated control with respect to the geodesic foliation. Unfortunately the p*
*recise
statement is more complicated and to formulate it we need some rather lengthy
preparations. The reader should right away take a look at Subsection 6.4 to get*
* a
first idea about the statement we are aiming at.
Ignoring the technicalities the argument can be summarized as follows. We know
that metric control leads to homological behavior. In particular we have the lo*
*ng
exact sequences associated to pairs of spaces in order to work inductively over
the skeleta of a cell structure. One task is now to formulate and prove analogo*
*us
results for foliated control using the skeleta of a long and thin cell structur*
*e. The
crucial step is a öf liated excision" result that reduces the statement about t*
*he
relax control map to a comparison result for a collection of long and thin cells
(compare Proposition 6.24). Carefully bookkeeping the errorterms one can even
assume that one has a collection of long and thin cells in Euclidean space equi*
*pped
with a "standard" 1dimensional foliation. The comparison result is then easily
established: an Eilenberg swindle is used to reduce the question to transversal
24 ARTHUR BARTELS, HOLGER REICH
cells. On transversal cells metric control and foliated control coincide (comp*
*are
Lemma 6.26).
6.1. Flow cell structures. Let N be an ndimensional Riemannian manifold
which is equipped with a smooth flow . The flow determines a onedimensional
foliation which will be called F . In our application N will always be SHM (or *
*its
universal covering) equipped with the geodesic flow and the corresponding folia*
*tion
Fgeo.
The following definition of a flow cell combines Definition 7.1 and Lemma 8.1*
* in
[FJ86]. Whereas in [FJ86] Lemma 8.1 the information about the length of a cell
is contained in the map ge we require the map to roughly preserve the length and
use instead a long parametrizing interval Ae. Below Rn = R x Rn1 is equipped
with the standard Euclidean metric and foliated by the lines parallel to the fi*
*rst
coordinate axis. We denote this foliation by FRn. Moreover ~n = 10n+3 is the
constant which depends only on the dimension that appears in Proposition 7.2 in
[FJ86]. Recall that for a subset Y Rn we denote by Y (ff,ffi)the set of all x*
* 2 Rn
for which there is a y 2 Y such that dFRn(x, y) (ff, ffl), compare Subsection*
* 4.2.
Definition 6.1 (fiflow cell). Let fi > 0 be given. A cell e N is called a f*
*i
flow cell if there exist cells Ae R, Be Rn1, a number ffle > 0 and a smooth
embedding ge:(Ae x Be)(fi,ffle)! N such that
(i)We have ge(Ae x Be) = e.
(ii)The map ge preserves the foliation, i.e. for each y 2 Rn1 the segment
R x {y} \ (Ae x Be)(fi,ffle)is mapped to a segment of a flow line in N.
(iii)If Ae R is not a 0cell then it is an interval of length exactly fi.
(iv)For all tangent vectors v 2 T ((Ae x Be)(fi,ffle)) which are tangential *
*to the
flow lines we have
v < dge(v) ~n=5 . v.
There are two sorts of flow cells: A flow cell where Ae is a 0cell will be c*
*alled
transversal. If Ae is a 1cell we call the cell e a long cell. Observe that fro*
*m (iii)
and (iv) it follows that such a cell is filong in the sense that for every y 2*
* Be
the segment ge(Ae x {y}) has arclength strictly larger than fi (and shorter than
fi . ~n=5).
Remark 6.2. A simple compactness argument shows that for a flow cell we addi
tionally have the following.
(v)There exists a constant Ce > 1 such that for all tangent vectors v 2
T ((Ae x Be)(fi,ffle)) we have
C1e. v dge(v) Ce . v.
Remark 6.3. Since (iv) and (v) hold over the (fi, ffle)thickening of Ae x Be, *
*foli
ated distances (compare Subsection 4.2) between points in the cell which are sm*
*all
compared to (fi, ffle) can be approximately determined in Euclidean space using*
* the
chart. More precisely: Given z = ge(v) and z0= ge(v0) with z, z02 e and
dF (z, z0) (ff, ffi) (5~1n. fi, C1e. ffle)
we have
dFRn(v, v0) (ff, Ceffi).
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 25
The other way round
dFRn(v, v0) (ff, ffi) implies dF (z, z0) (~n . ff, Ce . ffi).
A cell structure L for a compact subset of N all whose cells are fiflow cell*
*s will
be called a fiflow cell structure. Given a fiflow cell structure we will alwa*
*ys fix
choices of charts ge and constants ffle and Ce as in Definition 6.1. For a give*
*n cell
structure L we denote by L N its underlying topological space. We recall the
main result of Proposition 7.2 and Lemma 8.1 in [FJ86].
Theorem 6.4. Let n = dim N and ~n = 10n+3. Let N ~nfidenote the union
of all leaves which are shorter than ~nfi. For arbitrarily large fi and any com*
*pact
subset K N  N ~nfithere exists a fiflowcell structure L with K L.
Given a cell structure L we denote by L[k]the set of all kcells and by L(k)t*
*he
set of cells of dimension less than or equal to k. The kskeleton is L(k). We*
* define
a filtration
N(1) N(0)__._. .N(n)= N
of N as follows. Set N(1)= N  Land N(k)= N(1)[ L(k). Observe that a
cell e 2 L[k]may already be contained in N(1)and hence does not contribute to
the kth filtration step. We hence define L{k}to be the set of those kcells wh*
*ich
do not lie entirely in N(1). Note that such a cell e can meet N(1)only with i*
*ts
boundary, which we denote @e.
Let ~Ndenote the universal cover of N and let be the fundamental group which
acts via deck transformations on ~N. The lifted cell structure will be denoted *
*~Land
~N(k)denotes the preimage filtration of N~under the covering projection. Also we
will use ~L[k]and ~L{k}to denote the obvious sets of cells of ~L. Note that a f*
*low cell
in N gives rise to a whole orbit of flow cells in ~Nfor which one can simulta*
*neously
use the constants appearing in Definition 6.1 and Remark 6.2.
Roughly speaking the following proposition says that if one removes an (ff, *
*)
thickening of the (k  1)st filtration step from the kth filtration step, the*
*n the
remaining pieces of the interiors of the kcells are at least (ff, ffi)foliate*
*d apart from
one another. This fact will later play a crucial role in Proposition 6.24.
Proposition 6.5. Let L be a fiflow cell structure for a compact subset L N.
There exists an ffl0L> 0 and a function L(ff, ffi) defined for 0 ff fi and*
* 0 < ffi <
ffl0Lsuch that L(ff, ffi) ffi and the following holds:
(i)Suppose e 2 ~L{k}is a kcell and x 2 ~N(k). Whenever L(ff, ffi) is defi*
*ned
and L(ff, ffi) then
dF (e  @e(ff,,)x) (ff, ffi)
implies that x 2 e.
(ii)For fixed ff the function (ff, ffi) tends monotone to zero when ffi do*
*es.
Remark 6.6. Observe that Proposition 6.5 says in particular that for two 0cell*
*s e
and e02 ~L{0}with
dF (e, e0) < (fi, ffl0L)
we have e = e0since @e = ; for e 2 ~L[0].
Proof of Proposition 6.5.For e 2 ~L{k}, > 0 and ff 0 let Y (e, ff, ) N~
consist of all points x = ge(t0 + t, y) where ge(t0, y) 2 e  @e(ff, )and the p*
*ath
ø 7! ge(t0 + ø, y) for 0 ø t (resp. t ø 0) has arclength ff. (If the *
*flow
26 ARTHUR BARTELS, HOLGER REICH
has unit speed then Y (e, ff, ) coincides with [ff,ff](e  @e(ff,)).) If ff *
* fi then
Y (e, ff, ) is disjoint from ~N(k1)and every cell e02 ~L[k]unless e = e0. For*
* a long
cell e this is immediate from the construction (and the fact that we assume >*
* 0).
To see it in the case where e is a transversal cell observe that a cell in ~L{k*
*}can
meet N~(1)only at its boundary and hence points near (but not in) e  @e(ff, )
which lie on a flow line which meets e@e(ff, )must lie in the interior of a lo*
*ng cell
of dimension bigger than k. (Here one uses the fact that by definition the long*
* cells
are strictly longer than fi.) Define X(e) = ~N(k1)[ {e0  e02 L{k}, e06= e}.*
* As a
first approximation to the foliated distance appearing in (i) we discuss the di*
*stance
d(e, ff, ) = d(Y (e, ff, ), X(e)).
Observe that even though X(e) is usually not compact only the intersection of it
with some sufficiently large compact set matters. For small enough we know
that Y (e, ff, ) 6= ; and hence d(e, ff, ) is a positive number, say 4ffl0e. *
*Moreover
0implies d(e, ff, ) d(e, ff, ) and d(e, ff, ) tends to 0 if does un*
*less e
is a 0cell. For cells e which are not 0cells and 0 < ffi < ffl0ewe define e(*
*ff, ffi) as the
minimal for which d(e, ff, ) 2ffi. Since each deck transformation fl 2 a*
*cts
by isometries, preserves the foliation, respects the filtration and permutes th*
*e cells,
we know that d(e, ff, ) = d(fle, ff, ). Since there are only finitely many or*
*bits of
cells we can define ffl0as the minimal ffl0e, where e ranges over all cells whi*
*ch are in
L{k}for some k 0 and (ff, ffi) as the maximal e(ff, ffi), where e ranges ov*
*er all
cells which are in L{k}for some k 1. For 0 ff fi and 0 < ffi < ffl0Land e*
*very
e 2 L{k} we have d(e, ff, ) 2ffi for all (ff, ffi) and (ff, ffi) tends*
* to 0 if ffi
does. It remains to improve the established inequalities slightly. Note first t*
*hat for
ff 0 there is a constant Cflw(ff) such that for x, y 2 ~L with d(x, y) 1 *
*and all
t with ff t ff we have d( t(x), t(y)) < Cflw(ff) . d(x, y) by a compactne*
*ss
argument. (For the geodesic flow on SHM~ this holds even over all of SHM~ by
4.3.) Because of our symmetric definition of foliated distance in Subsection 4.*
*2 we 0
see that L(ff, ffi) = (ff, (Cflw(ff) + 2) . ffi_2) and ffl0L= min(1, (Cflw(fi*
*) + 1)1 . ffl_2)
satisfy our requirements. Compare also Lemma 4.10 (i).
6.2. A family of flow cell structures. In order to prove foliated control resul*
*ts
for SHM~ equipped with the geodesic foliation Fgeowe need longer and longer cell
structures (necessarily missing more and more closed geodesics) but we also want
to cover larger and larger chunks in the non compact Hdirection (because the
flow moves things in that direction). This naturally leads us to choose flowc*
*ell
structures Lfi,iindexed by N0x N which are filong and cover the [i, i]part i*
*n the
Hdirection (each individual cell structure will only cover a compact region). *
*Here
are the details:
Let fi > 0 be given. Let SHM ~nfidenote the subset of SHM that consists
of all closed geodesics of length ~nfi. Here ~n = 10n+3 with n = dim SHM,
cf. Theorem 6.4. Observe that SHM ~nfilies in SH{0}M because all compact flow
lines have an Hcoordinate which is constantly 0.
For a fixed fi 2 N0 we choose a monotone decreasing sequence of tubular neigh
borhoods Tfi,i, i 2 N of SHM ~nfisuch that
"
Tfi,i= SHM ~nfi.
i2N
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 27
We will use the tildenotation, i.e. SHM~ ~nfiand ~Tfi,i, to denote the obvious*
* preim
ages under the universal covering projection SHM~ ! SHM. Throughout the rest
of this section we also fix a choice of a fiflow cell structure Lfi,ifor fi 2 *
*N0 and
i 2 N such that
SH[i,i]M  Tfi,i Lfi,i SHM,
compare Theorem 6.4. Since its cells are shorter than fi . ~n=5 we can also arr*
*ange
that
Lfi,i SH[i~nfi,i+~nfi]M.
To our choice of flow cell structures Lfi,iwe will now associate certain sequ*
*ences
of constants and functions. First recall that for a cell e in a single flow cel*
*l structure
L we have the constants ffle and Ce appearing in Definition 6.1 and Remark 6.2.
Moreover there are the constant ffl0Land the function L(ff, ffi) from Proposit*
*ion 6.5.
We set
(6.7) fflL = min{fflee 2 L} [ {ffl0L} and CL = max{Cee 2 L}.
Now back to the family (Lfi,i)(fi,i)2N0xNwe have chosen above. We set
(6.8) ffli= min {fflLfi,i0 fi i},
(6.9) Ci = max {CLfi,i0 fi i},
(6.10) i(ffi)= max { Lfi,i(ff, ffi)ff 2 N0 and 0 ff fi i} forffi < ffli.
Note that i(ffi) ffi and for i fixed i(ffi) tends to 0 with ffi, compare Pr*
*oposi
tion 6.5 (ii). For ffi fixed i(ffi) is monotone increasing with i. Making the *
*fflismaller
and the Ci bigger if necessary we will assume that ffli tends monotone to 0 and*
* the
Ci form an increasing sequence of numbers > 1.
Later on, we will be in a situation where we can ignore all Lfi,iwith fi > i.
(Compare Proposition 6.18 and the definition of FT(fi)before that proposition.)
With the above definitions the constants and functions labeled with i have the
desired properties simultaneously for all cell structures Lfi,iwith fi i.
More precisely we have the following lemma.
Lemma 6.11.
(i)For fixed i Remark 6.3 with ffli and Ci instead of ffle and Ce applies s*
*imul
taneously to all cells in all the cell structures Lfi,iwith fi i.
(ii)Similarly for a fixed ff 2 N0 Proposition 6.5 (i) applies with ffli ins*
*tead of
ffl0Lto all cell structures Lfi,iwith ff fi i.
6.3. Construction of the decay speed S. Let t = (t1, t2, . .).be a sequence of
numbers with t1 = 1 and ti< ti+1. Given a sequence (ffii) we define the associa*
*ted
stepfunction stept((ffii)) to be the function on T whose value on the interval*
* [ti, ti+1)
is ffii. This defines a map from the space of sequences to the space of functio*
*ns.
Our aim is now to construct a certain set of sequences T = {(ffii)} which wil*
*l then
(after a choice of a suitable sequence t) lead to the set of functions S = step*
*t(T )
used to describe the decay speed in the Ewcontrol condition. In fact because *
*of
Remark 4.6 we are really only interested in the germs at infinity of such seque*
*nces.
Lemma 6.12. There exists a nonempty set T = {(ffii)i2N} of sequences of positive
numbers (each of which tends to zero) satisfying:
(i)Each sequence (ffii) 2 T is eventually smaller than the sequence (ffli) *
*defined
in equation (6.8), i.e. there exists an i0 2 N such that ffii< fflifor a*
*ll i i0.
28 ARTHUR BARTELS, HOLGER REICH
(ii)For every (ffii) 2 T the sequence ( i(ffii)), eventually defined by (i*
*), lies
again in T .
(iii)For every (ffii) 2 T the sequence (Ci. ffii) lies again in T .
Moreover we have the following more elementary properties corresponding to (A)
and (B) before Definition 4.4.
(A) For (ffii) 2 T and k 2 Z we have (ffii+k)i2N 2 T . (Here we set ffii+k *
*= ffi1
for i + k 0.)
(B) Given (ffii), (ffi0i) 2 T there exists (ffi00i) 2 T such that ffii+ ff*
*i0i ffi00ifor all
i 2 N.
Proof.It will be convenient to define 0i(ffi) = max {2ffi, i(ffi), Ci. ffi} f*
*or ffi ffli
and 1 otherwise. For fixed i this tends monotone to 0 with ffi, for fixed ffi *
*it
is monotone increasing with i. The space T consisting of all (ffii) satisfyin*
*g the
following condition
8k, l 2 Z, j 2 N0 9i0 8i i0 ( 0i+l)Oj(ffii+k) < ffli
is nonempty and satisfies (i), (ii), (iii), (A) and (B): Properties (i) and (A)*
* are clear
from this construction. To check property (ii) observe that
æ 0 Oj+1
( 0i+l)Oj( i+k(ffii+k)) (( i+l)0Oj+(ffii+k)ifi1+ l i + k.
i+k) (ffii+k)ifi + l i + k
Property (iii) follows by replacing 0i+kby Ci+kin this inequality. To check (B*
*) first
observe that we have ( 0i)Oj(2ffi) ( 0i)Oj+1(ffi). Thus, (ffii) 2 T implies (*
*2ffii) 2 T .
Moreover (ffii) 2 T and ffi0i ffii implies ffi0i2 T . Finally, by construction*
*, (ffii), (ffi0i) 2
T implies max(ffii, ffi0i) 2 T . So we get (B) since ffii+ ffi0i 2 max(ffii, f*
*fi0i). Since 0i(ffi)
tends to 0 with ffi we can find ffii such that ( 02i)O2i(ffii) < ffl2i. Then
( 0i+l)Oj(ffii+k) ( 0i+l)Oi+l(ffii+k) ( 02(i+k))O2(i+k)(ffii+k) e2(i+*
*k)< ffli,
for sufficiently large i, i.e. if i + l > j, 1 i + l 2(i + k) and i 2(i +*
* k). Thus
T contains (ffii) and is indeed not empty.
Now choose an increasing sequence t = (t1, t2, . .).with t1 = 1 such that ti+*
*1
ti i (this will be important in Proposition 6.20) and (eati) 2 T . Here a is*
* the
curvature bound from (4.1). The next statement is immediate from Lemma 6.12.
Proposition 6.13. The set S = steptT satisfies the standard properties (A) and
(B) of a class of decay speed functions introduced before Definition 4.4, each *
*ffit2 S
tends to zero for t ! 1 and moreover:
(i)For each A . eat 2 Sgeothere exists a ffit 2 S with A . eat ffit, i.*
*e. we
can "relax control" from Sgeoto S.
(ii)For a given ffit= stept(ffii) 2 S = steptT take i0 2 N as in Lemma 6.12*
*(i)
and define for t ti0the function t = stept( i(ffii)). This function l*
*ies
again in S.
(iii)For a given ffit= stept(ffii) 2 S the function Ct.ffit= stept(Ci.ffii) *
*lies again
in S.
Condition (i) is important to obtain the map (5) in our main diagram in Secti*
*on 3,
i.e. to connect up the following constructions with the kind of control we obta*
*ined
via the geodesic flow. The second and third condition will play an important ro*
*le
in Proposition 6.24 respectively in Lemma 6.25 below.
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 29
6.4. Statement of the Foliated Control Theorem. Now we are prepared to
define the control structures Ew and Es on SHM~ x B x T which were already
mentioned in the outline of the proof in Section 3. Let us recall what happened*
* so
far. For all natural numbers fi 0 and i 1 we chose a tubular neighborhood T*
*fi,i
of SHM~ ~nfi, and fiflow cell structures Lfi,isuch that SH[i,i]M  Tfi,i Lf*
*i,i.
The associated constants CLfi,i, the fflLfi,i(see (6.7), Remarks 6.2 and 6.3) a*
*nd the
functions Lfi,i(ff, ffi) given by Proposition 6.5 were used to define the sequ*
*ences ffli,
Ciand i(ffi). In Lemma 6.12 we produced a space of sequences T out of this dat*
*a.
Before Proposition 6.13 we then chose a sequence t = (t1, t2, . .).and defined *
*the
set of decay speed functions S = stept(T ).
Now set
[
(6.14) S = Lfi,i x [fi, fi + 1] x [ti, ti+1] SHM~ x B x T.
fi,i
The Foliated Control Theorem 6.17 will improve control precisely over S. Equip B
with the Euclidean metric and SHM~ x B with some product metric. Equip B with
the 0dimensional foliation by points and let Fw denote the product foliation w*
*ith
the foliation Fgeoon SHM~ given by the geodesic flow.
Definition 6.15 (Weak and strong control).
(i)The "weak" morphism control condition Ew on SHM~ x B x T is defined as
foliated control with respect to the foliation Fw on SHM~ x B with decay
speed S = stept(T ), i.e. Ew = E(SHM~ x B, Fw, S), cf. Definition 4.4.
(ii)The "very strong" morphism control condition Evson SHM~ x B x T is de
fined as metric control with respect to the product metric on SHM~xB with
decay speed S = stept(T ), i.e. Evs= E(SHM~xB, S), cf. Definition 4.4. We
let E0vsdenote the metric Evscontrol over the subset S SHM~ x B x T.
(This is a control condition over SHM~ x B x T as explained in Defini
tion 8.12.)
(iii)The "strong" morphism control condition Es is defined as Ew \ E0vs, i.e.
foliated control everywhere and metric control over S.
Observe that strong and weak control differ only over the subset S, where Es
requires the stronger metric control instead of only foliated control.
Remark 6.16. For the foliation Fw there is a foliated triangle inequality analo*
*gous
to Lemma 4.10(i). Together with (A) and (B) (compare Proposition 6.13) it hence
follows that Ew is closed under composition and indeed defines a morphism contr*
*ol
condition, compare Warning 4.7.
Recall that in Section 3 we introduced the object support condition
FB = { {(v, fi, t)  fi fi0}  fi0 2 B}
on SHM~ x B x T and the subspace (SHM~ x B x T)" consisting of all (v, fi, t) w*
*ith
h(v) t + ~nfi. Here h, fi and t denote the H, B respectively Tcoordinate*
* of a
point (v, fi, t) 2 SHM~ x B x T and ~n = 10n+3 with n = dimSHM~. After all these
preparations we can finally formulate the main result of this Section.
Theorem 6.17 (Foliated Control Theorem). The forget control map
C ((SHM~ x B x T)", Es, FB)1 ! C ((SHM~ x B x T)", Ew, FB)1
30 ARTHUR BARTELS, HOLGER REICH
given by relaxing the Escontrol condition to the Ewcondition induces an equiv*
*alence
in Ktheory.
After two preliminary reduction steps this result will be proven by induction
over the skeleta of a relative cell structure. The proof will occupy the rest o*
*f this
section.
6.5. First reduction  Delooping in the Bdirection. Suppose we are given
an (ff, ffit)controlled morphism. Then in a region of the space SHM~ x B x T w*
*here
fi is larger than ff and t is very large the morphism is quite well adapted to *
*the
flow cells we will find there. Conversely we would like to ignore a certain reg*
*ion
where we have no hope to prove a comparison result between foliated and metric
control. To capture this idea we introduce further object support conditions. We
define analogous to FB the following object support conditions on SHM~ x B x T
FT = { {(v, fi, t)  t t0}  t0 2 T}
FT(fi) = { {(v, fi, t)  t t0(fi)}  t0: B ! T a continuous function}
F = FB [ FT(fi).
Observe that germs away from FT are the usual germs at infinity. Since the def
inition of F only involves the B and the Tcoordinate we will later use the sa*
*me
notation for other (subsets of) spaces of the form X x B x T. We can reformulate
the Foliated Control Theorem as follows.
Proposition 6.18. The map in Theorem 6.17 induces an equivalence in Ktheory
if and only if the map
C ((SHM~ x B x T)", Es)>F ! C ((SHM~ x B x T)", Ew)>F
induces an equivalence in Ktheory.
Observe that there is no longer a compactness condition in the Bdirection but
instead of germs at infinity we now have germs away from F.
Proof.For the purpose of this proof we introduce the following abbreviation. For
object support conditions F0 and F00and a morphism support condition E on
SHM~ x B x T set
00 0 >F00
(E, F0)>F = C ((SHM~ x B x T)", E, F ) .
We have the following commutative diagram
(Es, FB)>FT(fi)__//_(Es)>FT(fi)_//_(Es)>F
  
  
fflffl fflffl fflffl
(Ew, FB)>FT(fi)__//_(Ew)>FT(fi)_//(Ew)>F .
Here the vertical map on the left is the map in the Foliated Control Theorem 6.*
*17
because under the presence of the FBobject support condition there is no diffe*
*rence
between germs away from FT (alias germs at infinity) and germs away from FT(fi).
According to Lemma 8.7 (iii) both rows yield fibration sequences in Ktheory and
it hence suffices to show that both categories in the middle admit an Eilenberg
swindle and are therefore contractible. Pick ffit 2 S. Then a swindle is induce*
*d in
both cases by the map (x, fi, t) 7! (x, fi +ffit, t) on SHM~ xBxT, compare [BFJ*
*R03 ,
Proposition 4.4].
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 31
6.6. Second reduction  Discretization. We would like to use the cell structures
on the chunks Li,fi x [fi, fi + 1] x [ti, ti+1]. But we do not know how aroun*
*d the
boundary of the squares [fi, fi +1]x[ti, ti+1] the different cell structures fi*
*t together.
To avoid this problem we use a MayerVietoris argument.
We define the following subsets of B respectively T.
[ [
Be = [fi, fi + 1] Bo = [fi, fi + 1]
fi2N0 fi2N0
fi even fi odd
[ [
Te = [ti, ti+1] To = [ti, ti+1].
ii2Neven i2N
i odd
Moreover we set
Be\o= Be \ Bo = N B
and
Te\o= Te \ To = {t1, t2, . .}. T.
For a subspace Y SHM~ x B x T we denote the intersection of Y with (SHM~ x
B x T)" by Y".
The condition that there are no nontrivial morphisms between different path
components of Y will be denoted by Eß0(Y )or briefly Eß0. Formally this can be
defined as the pullback of the morphism control condition consisting only of the
diagonal on ß0(Y ) via the natural projection Y ! ß0(Y ).
Warning 6.19. If X is a subset of Y one should not confuse Eß0(Y )restricted to
X, which is again denoted Eß0(Y,)with Eß0(X).
Proposition 6.20. If for all 9 spaces SHM~ x Bp x Tq with p, q 2 {e, o, e \ o} *
*the
maps
C ((SHM~ x Bp x Tq)", Es \ Eß0)>F ! C ((SHM~ x Bp x Tq)", Ew \ Eß0)>F
induce equivalences in Ktheory then so does the map in Proposition 6.18.
Proof.If one drops the extra Eß0condition on both sides, this follows easily by
applying a MayerVietoris argument (compare Remark 8.10) in the T and then
again in the Bdirection. However, dropping the Eß0condition does not change
the categories: if the support of a morphism in C ((SHM~ x Bp x Tq)", Es) or
C ((SHM~ x Bp x Tq)", Ew) violates this condition, it does so only on a set in F
and this can be ignored since we take germs away from F. For the Tdirection
this follows from the fact that in the definition of foliated and metric contro*
*l we
always require a bound in the Tdirection and the fact that the distance between
the ti increases with i. For the Bdirection it follows from the ffitcontrol *
*in this
direction.
From now on we will restrict our attention to the space SHM~ x Bex Te, all the
other cases are completely analogous.
6.7. Induction over the skeleta. We next define a filtration for Y = SHM~ x
Be x Te. Recall that
[
S = Lfi,i x [fi, fi + 1] x [ti, ti+1] SHM~ x B x T
fi,i
32 ARTHUR BARTELS, HOLGER REICH
and set ___________________
Y (1)= (SHM~ x Be x Te)  S
[ (k)
Y (k)= Y (1)[ Lfi,i x [fi, fi + 1] x [ti, ti+1].
fi,i even
For a subspace X Y and a morphism support condition E on Y we define the
object support condition
XE = {XE E 2 E}.
(Recall that XE = {y 2 Y  there exists anx 2 X with(x, y) 2 E} denotes the
Ethickening of X in Y .)
Proposition 6.21 (Induction Step). For k = 0, 1, . .,.n = dim SHM~ the relax
control map
(k1)Es\Eß0(Y )
C (Y"(k), Es \ Eß0(Y))>F[(Y )


fflffl
(k1)Ew\Eß0(Y )
C (Y"(k), Ew \ Eß0(Y))>F[(Y )
induces an equivalence in Ktheory.
This proposition is proven by combining Proposition 6.24, Lemma 6.25 and
Lemma 6.26. Before we proceed we note that Proposition 6.21 implies the Foli
ated Control Theorem 6.17.
Corollary 6.22. The map
C (Y", Es \ Eß0(Y))>F! C (Y", Ew \ Eß0(Y))>F
induces an equivalence in Ktheory. By Propositions 6.20 and 6.18 this implies *
*the
Foliated Control Theorem 6.17.
Proof of the Corollary.Let us abbreviate
C(k)x= C (Y"(k), Ex \ Eß0(Y))>F
(k1)Ex\Eß0(Y )
C(k,k1)x= C (Y"(k), Ex \ Eß0(Y))>F[(Y )
for x = w, s. By definition of Es and Ew we have C(1)w= C(1)s. By Lemma 8.7 (*
*iii)
and Remark 8.9 the sequence
C(k1)x____//C(k)x__//C(k,k1)x
induces for x = w or x = s a fibration sequence in Ktheory and relaxing control
from s to w yields a map of fibration sequences. The result follows by induction
using 6.21.
Observe that in Proposition 6.21 taking germs away from Y (k1)means in par
ticular that everything that is relevant happens over S, i.e. the region covere*
*d by
the cell structures (compare (6.14)). Hence we can ignore the difference between
strong and very strong control. (Formally this is an application of Lemma 8.11
from the Appendix.) Moreover we can drop the "subscript because
Y (k) Y"(k) Y (1) Y (k1).
We obtain the following lemma.
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 33
Lemma 6.23. The map in Proposition 6.21 is equal to the map
Evs\Eß0(Y )
C (Y (k), Evs\ Eß0(Y))>F[(Y (k1))


fflffl E
w\Eß0(Y )
C (Y (k), Ew \ Eß0(Y))>F[(Y (k1)) .
Recall that L{k}fi,idenotes those kcells in Lfi,iwhich do not lie in the (1*
*)st
filtration step, i.e. in Y (1). Set
a
Z(k) = (Ae x Be) x [fi, fi + 1] x [ti, ti+1]
e2L{k}fi,i
fi,i even
a
@Z(k) = @(Ae x Be) x [fi, fi + 1] x [ti, ti+1].
e2L{k}fi,i
fi,i even
There is a natural map
g :Z(k)! Y (k)
induced by the charts ge. The map induces a homeomorphism (Z(k) @Z(k)) !
(Y (k) Y (k1)). Recall that the object support condition F is defined for eve*
*ry
space with B and Tcoordinate, compare the beginning of Subsection 6.5. In
particular g1F will again be denoted F. The following proposition is the cruci*
*al
step in the proof of the Foliated Control Theorem and should be thought of as a
öf liated excision" result. It allows to separate cells. (Note the Eß0(Z(k))co*
*ndition
in the source.)
Proposition 6.24 (Excision of the (k1)skeleton). Let E denote either Evsor Ew.
In both cases the natural map g :Z(k)! Y (k)induces an equivalence of categories
(k)g1E\Eß0(Z(k))
C (Z(k), g1E \ Eß0(Z(k)))>F[(@Z )


fflffl
(k1)E\Eß0(Y )
C (Y (k), E \ Eß0(Y))>F[(Y )
for k = 0, 1, . .,.n.
Proof.Observe that the map factorizes over
(k)g1E\g1Eß0(Y )
C (Z(k), g1E \ g1Eß0(Y))>F[(@Z ) .
Since g :Z(k)! Y (k)induces a homeomorphism Z(k) @Z(k)! Y (k) Y (k1)and
all conditions are simply pulled back along g the second map in this factorizat*
*ion
clearly induces an equivalence. We see that the crucial point is whether the fo*
*rget
control map from g1E \ Eß0(Z(k))control to g1E \ g1Eß0(Y)control induces an
equivalence. Note that Eß0(Z(k))does not allow morphisms between different cells
and is hence a lot stronger than g1Eß0(Y )which only separates the different [*
*fi, fi +
1] x [ti, ti+1]blocks. Formally the result will be a consequence of Lemma 8.11*
* in
the Appendix. We will apply that lemma to the case X = Z(k), A = @Z(k),
34 ARTHUR BARTELS, HOLGER REICH
E0 = g1E \ Eß0(Z(k)), E00= g1E \ g1Eß0(Y,)F0 = ; and F = F. We only
formulate the argument for E = Ew. The Evscase is easier and can be obtained by
setting ff = 0, compare Remark 4.5.
Let ff 2 N0 and ffit2 S be given. Write Eff,ffitfor the subset E 2 E determin*
*ed as
in Definition 4.4 (ii) by ff and ffit (we suppress the t0). As in Proposition 6*
*.13 (ii)
define for the given ffit = stept((ffii)) 2 S the function t = stept(( i(ffii)*
*)) for
t ti0. In particular, ffii < ffli for all i i0. By Proposition 6.13 (ii) *
*we have
Eff, t2 E. Using our usual coordinates fi 2 B, t 2 T we define F as the union of
Ft ti0 = {(v, fi, t)  t ti0},
[
Ffi i = SHM~ x [fi, fi + 1] x [ti, ti+1] and
fi i
Ffi ff = {(v, fi, t)  fi ff}.
Observe that Ft ti0[ Ffi i2 FT(fi), Ffi ff2 FB and hence F 2 F. Let E00
g1Eff,ffitresp. E0 g1Eff, tdenote the subset of all pairs of points that sa*
*tisfy
in addition the g1Eß0(Y)resp. Eß0(Z(k))condition. We need to check that with
this notation the condition in Lemma 8.11 is satisfied.0 Since Eff,ffit Eff, t*
*it
suffices to show that if (x, x0) 2 E00and x =2(@Z(k))E [ F then (x, x0) satisfi*
*es the
Eß0(Z(k))condition: let (y, fl, t) = g(x) and (y0, fl0, t0) = g(x0) (here fl i*
*s the B and
t the Tcoordinate). There are i, fi such that ti t, t0 ti+1and fi fl, fl0 *
* fi + 1,
since (x, x0) satisfy the Eß0(Y)condition. We know i fi (since x =2Ffi i), f*
*i > ff
(since x =2Ffi ff) and i > i0 (since x =2Ft ti0). In particular, ffii< ffli ff*
*lLfi,i. There
is e 2 L{k}fi,isuch that y 2 e  @e(ff, )where = i(ffii) Lfi,i(ff, ffii),*
* since x =2
(@Z(k))E0. Finally, dFgeo(y, y0) (ff, ffii), since (x, x0) satisfies the Eff,*
*ffitcondition.
All this allows the application of Lemma 6.5 (i) (compare also Lemma 6.11 (ii))*
* to
conclude that y02 e. Thus (x, x0) does indeed satisfy the Eß0(Z(k))condition.
6.8. Comparison to a Euclidean standard situation. According to the last
proposition we can assume that all the cells which are new in the kth step do *
*not
talk to each other through morphisms (this is formalized in the Eß0(Z(k))condi*
*tion).
In the next step we will use the fact that each flow cell also has a security z*
*one
around it on which we have a very precise control over the foliation and the me*
*tric
to prove that the situation is equivalent to a Euclidean standard situation. Mo*
*re
precisely define
a
W (k) = R x Rn1 x [fi, fi + 1] x [ti, ti+1],
e2L{k}fi,i
fi,i even
We equip each (R x Rn1) x [fi, fi + 1] with the standard Euclidean metric and *
*with
the foliation FRn (i.e. the foliation by lines parallel to the Rfactor). We de*
*fine in
the obvious way the foliated and the metric control structure on W (k). Namely *
*in
both cases we impose Eß0(W(k))control, i.e. different components are infinitel*
*y far
apart. We then let Emet denote metric control with decay speed S together with
Eß0(W(k))and Efolthe foliated control with decay speed S together with Eß0(W(k)*
*).
We consider Z(k)as a subset of W (k)and denote this inclusion by i: Z(k)! W (k).
Note that this map induces an equivalence on ß0 and hence i1Eß0(W(k))= Eß0(Z(k*
*)).
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 35
Now consider the following situation.
g i
(Y (k), Evs)oo_Z(k)_____//(W (k), Emet)
  
 = 
fflfflg fflffli fflffl
(Y (k), Ew)oo_Z(k)_____//(W (k), Efol).
The following proposition reduces our problem to the Euclidean standard situati*
*on
on the right of the diagram above.
Lemma 6.25 (Comparison to a Euclidean situation). For k = 0, 1, . .,.n the cat
egories
(k))E
C (Z(k), E)>F[(@Z
with E = g1Evs\ Eß0(Z(k))resp. E = i1Emet are equivalent. The same holds for
the pair of control conditions E = g1Ew \ Eß0(Z(k))and E = i1Efol.
Proof.We need to show that the four horizontal relax control maps in the follow*
*ing
diagram are equivalences.
g1Evs\ Eß0(Z(k))oo__g1Evs\ Eß0(Z(k))\ i1Emet___//i1Emet
  
  
fflffl fflffl fflffl
g1Ew \ Eß0(Z(k))oo__g1Ew_\ Eß0(Z(k))\ i1Efol___//_i1Efol.
We only treat the lower left hand horizontal map. The other cases are analogous.
Formally the argument is an application of Lemma 8.11.
Let ff and ffit2 S be given. Let E00be determined by g1Eff,ffitand the Eß0(Z*
*(k))
condition. Define E0by i1Eff,Ctffit\E00, where Ctstems from Proposition 6.13 (*
*iii)
which also tells us that Ctffit 2 S. Choose i0 such that for all i i0 we have
Ciffii ffli, see Lemma 6.12 (i) and (iii). Define Ft ti0, Ffi iand Ffi (~n=5)*
*.ffas in
the proof of Proposition 6.24 (but note the constant ~n=5) and let F be the uni*
*on of
these three sets. The condition in Lemma 8.11 is implied if one can show that t*
*wo
points in the same cell of a cell structure Lfi,iwhich are less than (ff, ffiti*
*)foliated
apart, when measured inside the manifold are (ff, Ctiffiti)controlled when mea*
*sured
in Euclidean space using the charts. At least this should be true away from the*
* set
F . But this is the content of Lemma 6.11 (i) which says that Remark 6.3 applies
with Ci instead of Ce and with ffli instead of ffle if fi i. Note that we can*
* assume
ff 5~1nfi and t ti0 which translates into ffii C1iffli. These are just*
* the
assumptions in Remark 6.3.
It remains to prove the comparison result for the Euclidean standard situatio*
*n.
Lemma 6.26. For k = 0, 1, . .,.n the map
1Emet
C (Z(k), i1Emet)>F[(@Z(k))i


fflffl
i1Efol
C (Z(k), i1Efol)>F[(@Z(k))
induces an equivalence in Ktheory.
36 ARTHUR BARTELS, HOLGER REICH
Proof.Write Z(k)= Z(k)t[ Z(k)lwhere the first subspace uses only transversal ce*
*lls
and the second only long cells. It suffices to check the claim separately for Z*
*(k)tand
Z(k)l. On transversal cells i1Emet and i1Efoland the relevant thickenings of *
*the
boundaries agree and we are done for Z(k)t. If Ae = [a, b] let @e= {a} x Be and
write @(Ae x Be) = @e[ @e+where @e+\ @e= {a} x @Be. Set
a
@ Z(k)t= @e x [fi, fi + 1] x [ti, ti+1].
e long cellLin{k}fi,i
fi,i even
Let E denote either i1Emet or i1Efol, then according to Lemma 8.7 (iii) the
Ktheory of the categories we are interested in is the cofiber of the map induc*
*ed by
__(k)E//_(k) (k)E
C (Z(k)l, E, (@ Z(k)l)E)>F[(@+Zl )C (Zl , E)>F[(@+Zl ) .
For both choices of E the map (v, fl, t) 7! (v + (ffit, 0), fl, t) with some fi*
*xed ffit 2 S
induces an Eilenberg swindle for the category on the right. (Here v = (a, b) re*
*fers
to the coordinates in Ae x Be.) By Remark 8.9 the category on the left can be
identified with
(k))E
C (@ Z(k)l, E)>F[(@+Zl .
Note that @ Z(k)lconsists only of transversal pieces and we can repeat the arg*
*ument
from the beginning.
This finishes the proof of Proposition 6.21 and hence by Corollary 6.22 the p*
*roof
of the Foliated Control Theorem 6.17.
7.From strong control to continuous control
In this section we explain the map (8) in the main diagram in Section 3 which
connects up the "strong" control (Escontrol) we obtained so far with the equiv*
*ariant
continuous control on the space X(1) x T (denoted E cc(X(1))control).
7.1. A space with infinite cyclic isotropy. We recall the construction of the
metric space X and the map pX : SHM~ ! X from [BFJR03 , Section 14]. One
can collapse HM~ to the compact space H[1,1]~Mby projecting H(1,1]~Min
the obvious way to H{1}M~ and likewise H[1,1)~Mto H{1}~M. Similarly X is
obtained from SHM~, where additionally the fibers of the bundle SHM~ ! HM~
over H{1}M~ and H{1}~M are collapsed to points. This collapsing map is the
map pX : SHM~ ! X. More details of the construction can be found before
Proposition 14.5 in [BFJR03 ]. We can also project all the way down to M~ and
we hence obtain a factorization of the natural projection SHM~ ! M~ over X.
Restricted to SH[0.5,0.5]~Mthe map pX is essentially the identity and we can h*
*ence
consider SH[0.5,0.5]~Mand for every fi 0 also SHM~ ~nfias a subset of X. (Re*
*call
that SHM~ ~nfidenotes the union of all leaves which are shorter than ~nfi.) Lat*
*er
the metric properties of pX will be important: pX does not increase distances a*
*nd
for i large the map pX contracts SHM~  SH[i,i]~Mrather strongly. The precise
statement is [BFJR03 , 14.5].
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 37
We define X(fi) via the following pushout diagram whose horizontal arrows are
cofibrations
SHM~ ~nfi________//X
 p
 fi
fflffl fflffl
ß0(SHM~ ~nfi)_____//X(fi).
Since SHM~ ~nfi SHM~ ~n(fi+1)we obtain natural maps cfi:X(fi) ! X(fi + 1)
and we define X(1) as the mapping telescope model for hocolimfi 0X(fi), i.e. as
the coequalizer of
` _____//`
fi 0X(fi)____//fi 0X(fi) x [fi, fi + 1],
where the maps are given by sending x 2 X(fi) to (x, fi+1) respectively to (cfi*
*(x), fi).
Note that X(1) is a space all whose isotropy groups are trivial or infinite c*
*yclic.
We obtain natural maps
SHM~ x B q!X x B p!X(1),
where q = pX x idBand p is induced from the maps pfi.
7.2. Strong control maps to continuous control. We will now check that q
does in fact define the arrow labeled (8) in the main diagram in Section 3.
Proposition 7.1. The map q induces a functor
C ((SHM~ x B x T)", Es, FB)1 ! C (X x B x T, (p x idT)1E cc(X(1)), FB)1 .
Proof.Note that the restriction of q x idTto (SHM~ x B x T)" is a proper map.
Compare the discussion of the map (4) in the outline of the proof in Section 3.
We need to check that q maps Escontrol to p1E cccontrol. This has roughly the
following reasons: Firstly, since q does not increase distances ([BFJR03 , 14.5*
*(i)])
it maps very strong control (Evscontrol, compare Definition 6.15) to continuous
control. Since Escontrol implies Evs over the set S (which was defined at the
beginning of Subsection 6.4), Escontrol implies continuous control over S. Sec*
*ondly,
we need to deal with points on a geodesic g that is collapsed to a point in X(1*
*).
Here foliated control (Ewcontrol) already implies continuous control. Thirdly,*
* we
are left with points that are not in S because they have a large Hcoordinate. *
*Here
Escontrol implies only bounded control, but q contracts this part very strongly
([BFJR03 , 14.5(ii)]) and produces continuous control. For a careful argument o*
*ne
needs to construct suitable neighborhoods of g that are invariant under the sta*
*bilizer
of g (compare [BFJR03 , 15.2]) and to use the fact thatTwe did choose the tubul*
*ar
neighborhoods Tfi,imonotone decreasing and such that i2NTfi,i= SHM ~nfi.
8.Appendix
In this Appendix we collect a couple of facts which are more easily treated
independently from the context in which they were used in the main text.
38 ARTHUR BARTELS, HOLGER REICH
__
8.1._Homotopy_finite chain complexes. Let A be an additive category and
A _A_a full additive subcategory. We think of A as the category of "finiteö_ b*
*jects
in A, compare Subsection 5.2. Given such_a situation we denote by chA the categ*
*ory
of bounded below chain complexes in A. The notion of chain homotopy leads to a
notion of weak equivalence and we define cofibrations to be those chain_maps wh*
*ich
are degreewise the inclusion of a direct summand. The category chA becomes a
Waldhausen category (a category with cofibrations and weak equivalences_in the
sense of [Wal85]). We define chfA as the the full subcategory of chA whose obje*
*cts
are bounded below and above complexes where the object in each degree of the
chain complex lies in A._Furthermore_we define chhfA to be the full subcategory
of chain complexes in chA which are homotopy equivalent to a complex in_chfA._
The categories A, chfA and chhfA inherit a Waldhausen structure from chA.
Lemma 8.1. The natural inclusions
A ! chfA ! chhfA
induce equivalences in connective Ktheory.
Proof.For the first map see [Bri79] and [TT90 ] or [CP97 ]. The second inclusio*
*n in
duces an equivalence by a standard application of the Approximation Theorem 1.6*
*.7
in [Wal85]. (Mimic the mapping cylinder argument on page 380 in [Wal85] for cha*
*in
complexes.)
8.2. The tildeconstruction. In Section 5 we were forced to artificially enlarge
the Waldhausen category chhfC (E~x T; E) to a category with the same Ktheory
in order to define a transfer. In fact this enlargement is most easily treated *
*in the
generality of Waldhausen categories. In this subsection we briefly describe how
such an enlargement is formally defined and we explain that the natural inclusi*
*on
defines an equivalence in Ktheory under mild conditions.
Let W be a Waldhausen category. We additionally assume:
(M) Cofibrations in W are monomorphisms.
(H) There is a functor C ! D to some other category such that precisely the
weak equivalences are mapped to isomorphism in D.
(Z)The category admits a cylinder functor and satisfies the cylinder axioms,
compare page 348 in [Wal85].
Note that (H) implies the saturation axiom, i.e. the "two out of threeä xiom f*
*or
weak equivalences (see [Wal85] page 327).
Analogously to the Waldhausen categories Fm W defined on page 324 in [Wal85]
we define the Waldhausen category F1 W whose objects
"j0 Ø "j1 Ø "j2
C = (C0 Ø____//C1____//C2____//.). .
are infinite sequences of cofibrations in W. We denote the full Waldhausen subc*
*at
egory on those objects where all the ji are additionally weak equivalences by c*
*W.
The shift functor sh : cW ! cW sends C0 ! C1 ! . .t.o C1 ! C2 ! . .,.i.e. it
simply forgets C0. There is an obvious natural transformation ø from the identi*
*ty
functor to the shift functor.
We define fWto be the category whose objects are the same as the objects in cW
and where the set of morphisms between C and D is given by the colimit over the
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 39
following commutative diagram (which is indexed over the lattice points in a 3_*
*8th
plane).
.O.O. .O.O. .O.O.
sh sh sh
 ø  ø  ø
morcW(shC, D)__*_//_morcW(shC,OshD)*__//morcW(shC,Osh2D)*_//_. . .
OO
sh sh
 ø  ø
morcW(C, D)____*___//morcW(C, shD)__*_//_. . .
Assumption (M) implies that all maps in this diagram are inclusions. We define
the cofibrations in fW to be those morphisms which up to an isomorphism can be
represented by a cofibration in cW in the above colimit. Similarily C ! D is a
weak equivalence if it can be represented by a weak equivalence shnC ! shmD in
cW for some n and m. A lengthy but straightforward argument shows that these
structures indeed define the structure of a Waldhausen category on fW. Assumpti*
*on
(H) is used to verify that isomorphisms are weak equivalences.
Proposition 8.2. Suppose W satisfies (M), (H) and (Z) then the natural inclusion
W ! fW
which sends C to C =! C =! C =! . .i.nduces an equivalence on connective
Ktheory.
Proof.This is an application of Waldhausen's Approximation Theorem 1.6.7 in
[Wal85].
8.3. Swan group actions on Ktheory. In this subsection we will briefly de
scribe several versions of the Swan group and how it acts on Ktheory. We will *
*use
the notation
Sw ( ; Z) and Sw fr( ; Z)
for the K0group of Z modules which are finitely generated as Zmodules respec
tively finitely generated free as Zmodules. In both cases the relations are t*
*he
additivity relation given for all (not necessarily Z or Z split) exact sequen*
*ces.
Furthermore we will need the "chain complex version"
Swch( ; Z),
which is defined as the free abelian group on isomorphism classes of all bounded
below complexes Co of Z modules satisfying
(i)The homology H*(Co) is finitely generated as an abelian group (and in
particular concentrated in finitely many degrees).
(ii)The modules in each degree are free as Zmodules.
modulo the relations
(i)A short exact sequence 0 ! Co ! Do ! Eo ! 0 of Z chain complexes
yields [Do] = [Co] + [Eo].
(ii)If Co ! Do is a Z chain map which induces an isomorphism on homology
then [Co] = [Do].
40 ARTHUR BARTELS, HOLGER REICH
There is a natural map i : Swfr( ; Z) ! Sw( ; Z) and considering a module as a
chain complex concentrated in degree 0 defines a map j : Swfr( ; Z) ! Swch( ; Z*
*).
More remarkable is the map
Ø : Swch( ; Z)! Sw( ; Z)
X
Co 7! (1)i[Hi(Co)].
Proposition 8.3. All three maps in the commutative diagram
j ch Ø
Sw fr( ; Z)___//_____________33________________________________*
*_Sw(/;/Z)_Sw( ; Z)
___________________________________________________*
*______________________________________________________________@
i
are isomorphisms.
Proof.For a Z module M we denote by T M the Ztorsion submodule. For [M] 2
Sw( ; Z) there exists a Z resolution Fo(T M) = (F1 ! F0) of T M by modules
which are finitely generated free as Zmodules. The map : [M] 7! [F1]  [F0] +
[M=T M] is a well defined inverse for i by Lemma 2.2 in [PT78 ]. The claim now
follows if we can prove that j O O Ø is the identity. Let [Co] 2 Sw ch( ; Z)*
* be
given. Without loss of generality we assume that the complex is concentrated in
nonnegative degrees. There is an m 0 such that the degrees of all nonvanish*
*ing
homology groups lie in {0, 1, . .,.m}. We argue by induction over m. Let m = 0 *
*and
put M = H0(Co), i.e. Co is a resolution of M. Choose projective Z resolutions
Po0! T M and Po00! M=T M. Then Po = Po0 Po00is a resolution of M. Standard
arguments produce chain maps Po0! Fo(T M), Po00! M=T M and Po ! Co which
are homology isomorphisms. We have
j O O Ø([Co]) = j O ([M]) = j([F0]  [F1] + [M=T M]) =
[Fo(T M)] + [M=T M] = [Po0] + [Po00] = [Po] = [Co].
Now let m 1. Choose a projective Z resolution of Lo ! Hm (Co) and construct
a Hm isomorphism f : Lo ! Co. The map f factorizes over its mapping cylinder
and we have a short exact sequence
0 ! Lo ! cyl(f) ! cone(f) ! 0.
The claim holds by the induction hypothesis for [cone(f)] and by the argument f*
*or
m = 0 for [Lo] hence also for [Co] = [cyl(f)].
Let C(pt; R ) denote the category of finitely generated free R modules. Each
Z module M which is finitely generated free as a Zmodule yields an exact func*
*tor
 Z M : C(pt; R ) ! C(pt; R ). To check that this is well defined one uses
that for a free R module F there is a non canonical isomorphism of R modules
between F Z M with the diagonal respectively with the left action. Using
Proposition 1.3.1 and 1.3.2 (4) in [Wal85] it is straightforward to check that *
*the
construction leads to maps
(8.4) Sw fr( ; Z) Z Kn(R ) ! Kn(R )
for n 0. Replacing the onepoint space pt by Rn (compare Remark 5.3) one
obtains the corresponding construction for all n 2 Z. We use the isomorphism
i : Swfr( ; Z) ! Sw( ; Z) to define maps
Sw( ; Z) Z Kn(R ) ! Kn(R ).
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 41
Remark 8.5. The tensor product over Z yields a ring structure on Sw fr( ; Z) and
hence on Sw ( ; Z) and Sw ch( ; Z). The map (8.4) gives Kn(R ) the structure of
an Swfr( ; Z)module.
Let chhfC(pt; R ) denote the category of those bounded below chain complexes
which are Z homotopy equivalent to a bounded complex of finitely generated free
Z modules, compare Subsection 8.1. Let Co be a complex which represents an
element in Swch( ; Z).
Lemma 8.6. The functor  Z Co : C(pt; R ) ! chhfC(pt; R ) is well defined.
Proof.Let Q be a finitely generated free Z module. Since each Cn is free as a
Zmodule there is a noncanonical isomorphism between Q Z Cn equipped with
the diagonal action and the same module with the right action. We see that
Q Z Co is a complex of free Z modules. The crucial point is now to verify that
this complex is homotopy equivalent to a bounded complex of finitely generated *
*free
Z modules. We argue by induction over the length of the interval in which the
homology of Co is concentrated. We did this already in the proof of Proposition*
* 8.3
and will use the notation established there. Let m = 0 and let Co be a Z 
resolution of the module M = H0(Co). Since Q is free as a Zmodule the sequence
0 ! Q Z T M ! Q Z M ! Q Z M=T M ! 0 is exact. The diagonalaction
versusrightaction argument used above shows that Q ZFo(T M) and Q ZM=T M
are finitely generated free Z complexes. Using the horseshoelemma we obtain a
resolution Po = (Q Z F1 ! Q Z F0 Q Z M=T M) of M by a complex of finitely
generated free Z modules. Using standard arguments we can construct a Z 
chain map Po ! Q Z Co which is a homology isomorphism and hence a homotopy
equivalence since both complexes are complexes of projective Z modules. For the
induction step one constructs a map f : Lo ! Co and a cylindercone sequence as*
* in
the proof of Proposition 8.3. One uses the induction hypothesis and the horsesh*
*oe
lemma to construct a Z homotopy equivalence Po ! Co with Po bounded and
degreewise finitely generated Z free.
Since the inclusion C(pt; R ) ! chhfC(pt; R ) induces an isomorphism in K
theory (compare Lemma 8.1) it is not difficult to check that we obtain maps
Swch( ; Z) Z Kn(R ) ! Kn(R ).
Via the isomorphism j from Proposition 8.3 this action coincides with the ac
tionPof Sw fr( ; Z) and Sw ( ; Z). In particular  Z Co induces multiplication*
* by
(1)i[Hi(Co)].
8.4. Some fibration sequences. Let E be a morphism support condition on the
space X and let F, F0, F0 and F1 be object support conditions. Slightly abusi*
*ng
settheoretical notation we define F0 \ F1 = {F0 \ F1  F0 2 F0, F1 2 F1} and
similarly F0 [ F1. Also we write F0 F1 if for every F0 2 F0 there is F1 2 F1
such that F0 F1. If F0 F1 and F1 F0 we write F0 ' F1. We also use the
corresponding notation for morphism support conditions.
An object support condition F is called Ethickening closed if for every F 2 F
and E 2 E there exists an F 02 F such that F E F 0. Compare 2.1.5 for notation.
A typical example of such an F is AE = {AE  E 2 E} for a subset A X. If
F and F0 are Ethickening closed object support conditions, then C (X, E, F0\ F*
*0)
is a Karoubi filtration [Kar70] of C (X, E, F) and we define C (X, E, F)>F as *
*the
42 ARTHUR BARTELS, HOLGER REICH
Karoubi quotient. For the definition of Karoubi filtrations and quotients we re*
*fer
to [CP97 ].
Lemma 8.7. For the purpose of this lemma we abbreviate
(F) = C (X, E, F)
0 >F0
(F)>F = C (X, E, F)
and we assume that all object support conditions which occur are Ethickening
closed.
(i)The sequence (F \ F0) ! (F) ! (F)>F0 induces a fibration sequence in
Ktheory.
(ii)The square
(F0 \ F1)______//(F1)
 
 
fflffl fflffl
(F0)_______//(F0 [ F1)
induces a homotopy pushout square of spectra after applying Ktheory.
(iii)The sequence (F \ F0)>F1 ! (F)>F1 ! (F)>F0[F1 induces a fibration
sequence in Ktheory.
Proof.The first statement is just the fact that a Karoubi filtration leads to a
fibration sequence in Ktheory, compare [CP97 ]. One can check that (F1)>F0\F1 !
(F0 [ F1)>F0 is an equivalence of categories. This yields (ii). The square
(F \ F0 \ F1)_______//(F \ F1)
 
 
fflffl fflffl
(F \ F0)_______//(F \ (F0 [ F1))
is a homotopy pushout square by (ii). This square maps to the homotopy push
out square whose vertical maps are identities and both whose horizontal maps are
(F \F0) ! F. The induced square of cofibers is again a homotopy pushout square
and its lower left hand corner is (F \ F0)>F\F0 and hence contractible. Using (*
*i)
this yields the desired fibration sequence in (iii).
Example 8.8. Suppose that E and F are defined on X x [1, 1) = X x T and let
FT = {X x[1, t0]  t0 1}. Then we write C (X xT, E, F)1 for C (X xT, E, F)>FT
because we think of this category as being obtained by taking "germs at infinit*
*y".
Let us assume that the next remark applies to the inclusion X = X x{1} ! X xT.
Then the fibration sequence from Lemma 8.7 (i) becomes
C (X, E, F) ! C (X x T, E, F) ! C (X x T, E, F)1
and we refer to it as the "germs at infinity"fibration.
Remark 8.9. Let iA : A ! X be the inclusion of a invariant subset. Suppose th*
*at
the morphism support condition E satisfies the following properness condition: *
*for
a compact K X and E 2 E the closure of KE is again compact. Then
1F E >F
C (A, i1AE, i1AF0)>iA ! C (X, E, A \ F0)
is an equivalence of categories. (Here F0 and F are again assumed to be E
thickening closed object support conditions.) Most morphism support conditions
ON THE FARRELLJONES CONJECTURE FOR HIGHER ALGEBRAIC KTHEORY 43
used in this paper are defined on a locally compact metric space and contain a
global metric condition and are therefore proper in the above sense. (In [BFJR0*
*3 ]
a weaker properness condition was used, but this will not concern us here.)
Remark 8.10 (MayerVietoris type results). Let A and B be invariant subsets of
X with A [ B = X. Apply Lemma 8.7 (ii) with F0 = AE and F1 = BE. Suppose
that E is proper in the sense of Remark 8.9 and that one can additionally show
AE \ BE ' (A \ B)E.
Then one obtains a MayerVietoris result, i.e. a homotopy pushout square invol*
*ving
the Ktheories of the categories C (Y, i1E), with Y = A \ B, A, B resp. X. (He*
*re
i denotes in each case the relevant inclusion.) If F0 and F are Ethickening cl*
*osed
object support conditions, thenthere1is0a similar homotopy pushout square for*
* the
categories C (Y, i1E, i1F)>i F .
Lemma 8.11. Let X be a space and A X a subspace. Let F and F0 be object
support conditions and E0 E00be morphism support conditions on X. The map
E0 00 >F[AE00
C(X, E0, F0)>F[A ! C(X, E , F0)
induced by relaxing control from E0 to E00is an equivalence of categories provi*
*ded
the following condition is satisfied.
For all E002 E00there exist E02 E0 and F 2 F [ AE0 such that
E00 E0 F x F.
Proof.Note that the0condition0implies F [0AE0=0F [ AE00. But it also implies th*
*at
C(X, E0, F0)>F[AE ! C(X, E00, F0)>F[AE is surjective on objects and bijecti*
*ve
on morphism sets.
The following concept was used to define strong control in Definition 6.15.
Definition 8.12 (Control over a subset). Let E be a control structure on the sp*
*ace
X. Let A X be a subspace. We define Econtrol over A (a morphismcontrol
condition on X) as follows. A morphism OE is Econtrolled over A if there exist*
*s an
E 2 E such that if (x, y) lies in the support of OE and x or y 2 AE then (x, y*
*) 2 E.
Here AE = ((Ac)E )c where Ac denotes the complement of A in X and AE is
the Ethickening of A in X.
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Fachbereich Mathematik, Universität Münster, Einsteinstr. 62, 48149 Münster, *
*Ger
many
Email address: bartelsa@math.unimuenster.de
Email address: reichh@math.unimuenster.de