"Spaces over a Category and Assembly Maps
in Isomorphism Conjectures in K and
LTheory"
by
1
James F. Davis and Wolfgang L"uck
Introduction
Abstract: We give a unified approach to the Isomorphism Conjecture of Farrell*
* and Jones on
the algebraic K and Ltheory of integral group rings and to the BaumConnes Co*
*njecture on
the topological Ktheory of reduced group C*algebras. The approach is through *
*spectra over the
orbit category of a discrete group G. We give several points of view on the as*
*sembly map for
a family of subgroups and describe such assembly maps by a universal property g*
*eneralizing the
results of Weiss and Williams to the equivariant setting. The main tools are sp*
*aces and spectra
over a category and the study of the associated generalized homology and cohomo*
*logy theories and
homotopy limits.
Key words: Algebraic K and Ltheory, BaumConnes Conjecture, assembly maps, spa*
*ces and
spectra over a category
AMSclassification number: 57
Glen Bredon [5] introduced the orbit category Or(G) of a group G. Object*
*s are ho
mogeneous spaces G=H, considered as left Gsets, and morphisms are Gmaps. Thi*
*s is a
useful construct for organizing the study of fixed sets and quotients of Gacti*
*ons. If G
acts on a set X, there is the contravariant fixed point functor Or(G) ! SETS *
* given by
G=H 7! XH = map G(G=H; X) and the covariant quotient space functor Or(G) ! SE*
*TS
given by G=H 7! X=H = X xG G=H. Bredon used the orbit category to define equiva*
*riant
cohomology theory and to develop equivariant obstruction theory.
Examples of covariant functors from the orbit category of a discrete grou*
*p G to abelian
groups are given by algebraic Ktheory Ki(ZH), algebraic Ltheory Li(ZH), and t*
*he K
theory Ktopi(C*r(H)) of the reduced C*algebra of H. In Section 2, we express e*
*ach of these
as the composite of a functor Or(G) ! SPECTRA with the ith homotopy group*
*. We
use these functors to give a clean formulation of some of the main conjectures *
*of high
___________________________________
1Supported by the Alexander von HumboldtStiftung and the National Science F*
*oundation. James F.
Davis wishes to thank the Johannes GutenbergUniversit"at in Mainz for its hosp*
*itality when this work was
initiated.
1
dimensional topology: the Isomorphism Conjecture of FarrellJones [14] (which i*
*mplies the
Borel/Novikov Conjecture) and the BaumConnes Conjecture in the case of discret*
*e groups.
Our motivation was in part to obtain such a formulation and and in part t*
*o set the
stage for explicit computations based on isomorphism conjectures. We give compu*
*tations of
K and Lgroups of group rings in a separate paper [7]. Our formulation is used*
* by Kimberly
Pearson [27] to show that the Whitehead group Wh(G) and the reduced K0group "K*
*0(ZG)
vanish for two dimensional crystallographic groups. We also hope our formulatio*
*ns will prove
useful in the further study of isomorphism conjectures and in the related study*
* of manifolds
admitting metrics of positive scalar curvature.
Sections 1, 3, 4 and 7 contain foundational background, independent of as*
*sembly maps
and algebraic Ktheory. Section 2 is devoted to Ktheory, and Sections 5 and 6 *
*to assembly
maps. More precisely, in Section 1 we discuss the adjointness of mapping spaces*
* and tensor
(or balanced) products over a category, as well as the notions of spaces and sp*
*ectra over a
category. In Section 2, we define our three main examples of Or(G)spectra: Kal*
*g, L, and
Ktop._They_are all defined by first assigning to an object G=H, the transformat*
*ion groupoid
G=H , whose objects are elements of G=H, and whose morphisms are given by mult*
*iplication
by a group element, and then assigning a spectrum to a groupoid. In the Ktopca*
*se there
is an intermediate step of considering the C*category of a groupoid and a spec*
*trum of a
C*category, derived from Bott periodicity.
In Section 3 we discuss free CW complexes over a category C, the univers*
*al free CW 
complex EC over a category C, and homotopy (co)limits EC CX of a Cspace X. Th*
*e ideas
here are wellknown to the experts (see e.g. [10]), but the approach, relying o*
*n homological
methods and avoiding simplicial methods, may appeal to an algebraist. By approx*
*imating
a Cspace X by a free CCW complex, in Section 4 we define homology HC*(X; E) *
*and
cohomology of a space H*C(X; E) of a space with coefficients in a Cspectrum E.*
* We give an
AtiyahHirzebruch type spectral sequence for these theories.
With regard to the assembly maps arising in the Isomorphism Conjectures, *
*we give
three points of view in Section 5. Let F be a family of subgroups of G, closed*
* under
inclusion and conjugation. Let E : Or(G) ! SPECTRA be a covariant functor.*
* We define
a functor
E% : GSPACES ! SPECTRA
by setting E% (X) = (G=H ! XH )+ Or(G)E. Then ss*(E% (X)) is an equivariant h*
*omology
theory in the sense of Bredon [5]. Let E(G; F) be the classifying space for a *
*family of
subgroups of G, i.e. it is a GCW complex so that E(G; F)H is contractible for*
* subgroups
H in F and is empty for H not in F. The map
ss*E% (E(G; F)) ! ss*E% (G=G)
given by applying E% to the constant map and then taking homotopy groups is cal*
*led the
(E; F; G)assembly map. We say the (E; F; G)isomorphism conjecture holds if th*
*e (E; F; G)
assembly map is an isomorphism. When F = VC, the family of virtual cyclic subgr*
*oups of G,
(i.e. H 2 VC if and only if H has a cyclic subgroup of finite index), the isomo*
*rphism conjec
tures of FarrellJones [14] for algebraic K and Ltheory are equivalent to the*
* (Kalg; VC; G)
2
and (L; VC; G)isomorphism conjectures, where Kalgand L are Or(G)spectra assoc*
*iated to
algebraic K and Ltheories. When F = FIN , the family of finite subgroups of G*
*, and Ktop
is the Or(G)spectra associated with the Ktheory of C*algebras, then the (Kto*
*p; FIN ; G)
Isomorphism Conjecture is equivalent to the BaumConnes Conjecture (see Section*
* 5). When
F = 1, the family consisting only of the trivial subgroup of G, then the (Kalg;*
* 1; G), (L; 1; G),
and (Ktop; 1; G)assembly maps can be identified with maps H*(BG; Kalg(Z)) ! *
*K*(ZG),
H*(BG; L(Z)) ! L*(ZG), and H*(BG; Ktop(C)) ! Ktop*(C*rG):
We give three variant ways of expressing the (E; F; G)assembly map: by a*
*pproximat
ing E by E% as above, in terms of homotopy colimits, and in terms of a generali*
*zed homology
theory over a category. The first definition is the quickest and leads to an ax*
*iomatic char
acterization; the last two are wellsuited for computations.
Let Or(G; F) be the restricted orbit category, where the objects are G=H *
*with H 2 F.
The (E; F; G)assembly map is equivalent to the map
ss*(hocolim E) ! ss*(hocolim E)
Or(G;F) Or(G)
induced by the inclusion of the restricted orbit category in the full orbit cat*
*egory. Since
E(G; F) is only defined up to Ghomotopy type, it is natural for us to define h*
*omotopy
limits and colimits as a homotopy type, rather than a fixed space or spectra; w*
*e take this
approach in Section 3.
Given a family F of subgroups of G, define the Or(G)space {*}F to be the*
* functor
which sends G=H to a point if H is in F and to the empty set otherwise. Let {*}*
* be the
trivial Or(G)space, which sends G=H to a point for all H. The third point of *
*view is to
identify the (E; F; G)assembly map with the map
HOr(G)*({*}F ; E) ! HOr(G)*({*}; E)
induced by the inclusion map of Or(G)spaces, {*}F ! {*}.
Section 6 gives a characterization of assembly maps, generalizing that of*
* WeissWilliams
[41] to the equivariant setting. Associated to a homotopy invariant functor
E : GSPACES ! SPECTRA ;
we define a new functor
E% : GSPACES ! SPECTRA ;
and a natural transformation
A : E%  ! E;
where A(G=H) is a homotopy equivalence for all orbits G=H. Here E% is the "best*
* approx
imation" of E by an excisive functor, in particular ss*(E% (X)) is an equivaria*
*nt homology
theory. When E(X) = Kalg((EG xG X)) where is the fundamental groupoid, then the
map ss*(A(E(G; F))) is equivalent to the (Kalg; F; G)assembly map. An analogou*
*s state
ment holds for Ltheory and for the topological Ktheory of C*algebras. This g*
*ives a fourth
point of view on assembly maps.
3
In Section 7 we make explicit the correspondence between Gspaces and Or(*
*G)spaces
which has been implicit throughout the paper.
We thank Erik Pedersen for warning us about two pitfalls related to the s*
*pectra of
algebraic K and Ltheory and Stephan Stolz for discussions on the material of *
*Section 2.
The paper is organized as follows :
0. Introduction
1. Spaces and Spectra over a Category
2. K and LTheory Spectra over the Orbit Category
3. CW Approximations and Homotopy Limits
4. (Co)Homology Theories Associated to Spectra over a Category
5. Assembly Maps and Isomorphism Conjectures
6. Characterization of Assembly Maps
7. GSpaces and Or(G)spaces
References
Version of June 3, 1996
4
1. Spaces and Spectra over a Category
This section gives basic definitions and examples of spaces and spectra o*
*ver a small
(discrete) category C and discusses the adjointness of the tensor product and m*
*apping space.
Our main example for C is due to Bredon [5]:
Definition 1.1 Let G be a group and F be a family of subgroups, i.e. a nonem*
*pty set
of subgroups of G closed under taking conjugates and subgroups. The orbit categ*
*ory Or(G)
has as objects homogeneous Gspaces G=H and as morphisms Gmaps. The orbit cate*
*gory
Or (G; F) with respect to F is the full subcategory of Or(G) consisting of thos*
*e objects G=H
for which H belongs to F. 
Examples of families are F = {H G  XH 6= ;} for a Gspace X, the finit*
*e subgroups
of G, and the virtually cyclic subgroups of G. Notice that the automorphism gro*
*up of an
object G=H can be identified with the Weyl group W (H) = N(H)=H. Furthermore if*
* H is
finite, then any endomorphism of G=H is invertible, but not in general [23, Lem*
*ma 1.31 on
page 22]. We will always work in the category of compactly generated spaces (se*
*e [37] and
[42, I.4]).
Definition 1.2 A covariant (contravariant) Cspace X over the category C is a *
*covariant
(contravariant) functor
X : C ! SPACES
from C into the category of compactly generated spaces. A map between Cspaces *
*is a natural
transformation of such functors. Given Cspaces X and Y , denote by hom C(X; Y *
*)the space
of mapsQof Cspaces from X to Y with the subspace topology coming from the obvi*
*ous inclusion
into c2Ob(C)map(X(c); Y (c)). 
Likewise we can define a Cset and an RCmodule. For a ring R a RCmodul*
*e is a
functor M from C to the category of Rmodules. For two RCmodules M and N of th*
*e same
variance, hom RC(M; N) is the abelian group of natural transformations from M t*
*o N. We
can form kernels and cokernels, so the category of RCmodules is an abelian cat*
*egory, and
thus one can use homological algebra to study RCmodules (see [23]).
Let G be a group. Let 1 be the family consisting of precisely one elemen*
*t, namely
the trivial group. Then Or (G; 1) is a category with a single object, and the *
*morphisms
can be identified with the group G. A covariant (contravariant) Or(G; 1)space *
*is the same
as a left (right) Gspace. Maps of Or(G; 1)spaces correspond to Gmaps. For *
*a different
example of an orbit category, let Zp be the cyclic group of order p for a prime*
* number p. A
contravariant Or(Zp)space Y is specified by a Zpspace Y (Zp={1}), a space Y (*
*Zp=Zp), and
a map Y (Zp=Zp) ! Y (Zp={1})Zp.
5
Example 1.3 Let Y be a left Gspace and F be a family of subgroups. Define th*
*e associated
contravariant Or(G; F)space map G(; Y ) by
map G(; Y ) : Or(G; F) ! SPACES G=H 7! map G(G=H; Y ) = Y H: *
* 
Next we explain two important constructions which make out of two Cspace*
*s a space.
They are called the coend and end constructions in category theory [24, pages 2*
*19 and 222].
A lot of wellknown constructions are special cases of it.
Definition 1.4 Let X be a contravariant and Y be a covariant Cspace. Define t*
*heir tensor
product to be the space a
X C Y = X(c) x Y (c)= ~
c2Ob(C)
where ~ is the equivalence relation generated by (xOE; y) ~ (x; OEy)for*
* all morphisms
OE : c ! d in C and points x 2 X(d) and y 2 Y (c). Here xOE stands for X(OE)*
*(x) and
OEy for Y (OE)(y). 
Recall that the category of covariant (contravariant) Or(G; 1)spacesis t*
*he category
of left (right) Gspaces. The balanced product X xG Y of a right Gspace X and *
*of a left
Gspace Y can be identified with the tensor product X Or(G;1)Y . The mapping *
*space
map G(X; Y ) of two left (right) Gspaces X and Y can be identified with hom O*
*r(G;1)(X; Y ).
The main property of the tensor product is the following.
Lemma 1.5 Let X be a contravariant Cspace, Y be a covariant Cspace and Z b*
*e a space.
Denote by map (Y; Z) the obvious contravariant Cspace whose value at an object*
* c is the
mapping space map (Y (c); Z). Then there is a homeomorphism natural in X, Y and*
* Z
T = T (X; Y; Z) : map (X C Y; Z) ! homC(X; map(Y; Z))
Proof______: We only indicate the definition of T . Given a map g : X C Y  ! Z*
* , we have to
specify for each object c in C a map T (g)(c) : X(c) ! map (Y (c);.Z)This is t*
*he same as
specifying a map X(c) x Y (c) ! Z which is defined to be the composition of g *
*with the
obvious map from X(c) x Y (c) to X C Y . 
In particular Lemma 1.5 says that for a fixed covariant Cspace Y the fun*
*ctor  C Y
from the category of contravariant Cspaces to the category of spaces and the f*
*unctor
map (Y; ) from the category of spaces to the category of contravariant Cspac*
*es are adjoint.
Similarly if N is a covariant RCmodule, then there is adjoint to hom RC(N; ),*
* namely the
tensor product of RCmodules  RC N (see [9, p. 79], [23, p. 166]). Many proper*
*ties of
these products can be proven via the adjoint property, rather than referring ba*
*ck to the
definition. These products are reminiscent of the analogous situation of a righ*
*t Rmodule
X, a left Rmodule Y and an abelian group Z, the tensor product X R Y , the Rm*
*odule
hom Z(Y; Z). Here there is a natural adjoint isomorphism
hom Z(X R Y; Z) ! hom R(X; homZ(Y; Z)):
6
Lemma 1.6 Let X be a space and let Y and Z be covariant (contravariant) Csp*
*aces. Let
X xY be the obvious covariant (contravariant) Cspace. There is a homeomorphism*
*, natural
in X, Y , and Z
T (X; Y; Z) : hom C(X x Y; Z) ! map(X; homC(Y; Z)): 
Example 1.7 Let be the category of finite ordered sets, i.e. for each nonn*
*egative integer
p we have an object [p] = {0; 1; : :;:p}and morphisms are monotone functions. A*
* simplicial
space X: is by definition a contravariant space and a cosimplicial space is *
*a covariant
space. A simplicial set is a contravariant set. It can be considered a*
*s a simplicial
space by using the discrete topology. Define a covariant space : by assigni*
*ng to [p]
the standard psimplex and to a monotone function the obvious simplicial map. *
*Given
a topological space Y , the associated simplicial set S:Y is given by map (:; *
*Y )d. (The
subscript d indicates that we equip this mapping space with the discrete topolo*
*gy, in contrast
to the usual convention.) The geometric realization X: of a simplicial space *
*X: is the space
X: : . The geometric realization of a simplicial set has has the structure*
* of a CW 
complex where each nondegenerate psimplex corresponds to a pcell.
We get from Lemma 1.5 that these two functors are adjoint, i.e. there is*
* a natural
homeomorphism for a simplicial space X: and a space Y
T (X:; Y ) : map (X:; Y ) ! hom (X:; S:Y ):
In particular we get for a space Y the natural map given by the adjoint of the *
*identity on
S:Y
t(Y ) : S:Y  ! Y
which is known to be a weak homotopy equivalence. Hence t(Y ) is a functorial c*
*onstruction
of a CW approximation of the space Y . For more information about simplicial s*
*paces and
sets we refer for instance to [4] [6], [22] and [25]. 
Next we introduce spectra over a category C. Let SPACES +be the category*
* of pointed
spaces. Recall that objects are compactly generated spaces X with base points *
*for which
the inclusion of the base point is a cofibration and morphisms are pointed maps*
*. We define
the category SPECTRA of spectra as follows. A spectrum E = {(E(n); oe(n)) *
* n 2 Z}is
a sequence of pointed spaces {E(n)  n 2 Z}together with pointed maps called st*
*ructure
maps oe(n) : E(n) ^ S1 ! E(n + 1). A (strong) map of spectra f : E ! E0is a s*
*equence
of maps f(n) : E(n) ! E0(n)which are compatible with the structure maps oe(n),*
* i.e. we
have f(n + 1) O oe(n) = oe0(n) O (f(n) ^fidS1)or all n 2 Z. This should not *
*be confused
with the notion of map of spectra in the stable category (see [1, III.2.]). Re*
*call that the
homotopy groups of a spectrum are defined by
ssi(E) = colimssi+k(E(k))
k!1
where the system ssi+k(E(k)) is given by the composition
oe(k)*
ssi+k(E(k)) S!ssi+k+1(E(k) ^ S1) ! ssi+k+1(E(k + 1))
7
of the suspension homomorphism and the homomorphism induced by the structure ma*
*p. A
weak homotopy equivalence of spectra is a map f : E ! F of spectra inducing an*
* isomor
phism on all homotopy groups. A spectrum E is called spectrum if for each str*
*ucture
map, its adjoint E(n) ! E(n + 1) = map (S1; E(n + 1)) is a weak homotopy equiv*
*alence
of spaces. We denote by SP ECT RA the corresponding full subcategory of SPECTR*
*A .
A pointed Cspace resp. a Cspectrum resp. Cspectrum is a functor fro*
*m C to
SPACES + resp. SPECTRA resp. SPECTRA . We have introduced tensor pro*
*duct of
Cspaces in Definitions 1.4 and mapping spaces of Cspaces in Definition`1.2. T*
*hese notions
extendQto pointed spaces, one simply has to replace disjoint unions and carte*
*sian products
by wedge products _ and smash products ^ and mapping spaces by pointed mapp*
*ing
spaces. All the`adjunction properties remain true. Any Cspace X determines a*
* pointed
Cspace X+ = X {*} by adjoining a base point. Here {*} denotes a Cspace whic*
*h assigns
to any object a single point. It is called the trivial Cspace.
A Cspectrum E can also be thought of as a sequence {E(n)  n 2 Z}of poin*
*ted C
spaces and the structure maps as maps of pointed Cspaces. With this interpreta*
*tion it is
obvious what the tensor product spectrum X C E of a contravariant pointed Csp*
*ace and a
covariant C spectrum means. The canonical associativity homeomorphisms
(X C E(n)) ^ S1  ! X C (E(n) ^ S1)
are used in order to define the structure maps. It is given on representatives*
* by sending
(x C e) ^ z to x C (e ^ z). More abstractly, it is induced by the following com*
*position of
natural bijections coming from various adjunctions where Z is a pointed space
1 1
map (X C E(n)) ^ S ; Z  ! map X C E(n); map(S ; Z) !
1 1
map X; homC E(n); map(S ; Z) ! map X; homC(E(n) ^ S ; Z)
! map(X C (E(n) ^ S1); Z):
Similarly one defines the mapping space spectrum hom C(X; E) of a pointed Cspa*
*ce X and
a Cspectrum E using the canonical map of pointed spaces (which is not a homeom*
*orphism
in general)
homC(X; E(n)) ^ S1  ! hom C(X; E(n) ^ S1):
This map assigns to OE ^ z the map of Cspaces from X to E(n) ^ S1 which sends *
*x 2 X(c)
to OE(c)(x) ^ z 2 E(n)(c) ^fS1or c 2 Ob (C).
A homotopy of maps of spectra fk : E ! F is a map of spectra h : [0; 1]+*
* ^ E ! F
whose composition with the inclusion ik : E ! [0; 1]+ ^ E e 7! ki^sefk for k*
* = 0; 1.
Let C and D be two categories. A CDspace is a covariant C x Dopspace w*
*here Dop
is the opposite of D which has the same objects as D and is obtained by reversi*
*ng the
direction of all arrows in D. This is the analogue of a RSbimodule for two ri*
*ngs R and S.
Let F : C ! D be a covariant functor. We get a DCspace mor D(F (?); ??) if w*
*e use the
discrete topology on the set of morphisms. Here ? is the variable in C and ?? i*
*s the variable
in D. Analogously one defines a CDspace morD (??; F (?)).
8
Definition 1.8 Given a covariant (contravariant) Cspace X, define the inducti*
*on of X
with F to be the covariant (contravariant) Dspace
F*X = morD(F (?); ??) C X
respectively
F*X = X C morD(??; F (?))
and the coinduction of X with F to be the covariant (contravariant) Dspace
F!X = homC(mor D(??; F (?)); X)
respectively
F!X = homC(mor D(F (?); ??); X):
Given a covariant (contravariant) Dspace Y , define the restriction of Y with *
*F to be the
covariant (contravariant) Cspace F *Y = Y O F . 
There are corresponding definitions for Csets and RCmodules (see [9, p.*
* 80], [23,
p. 166] for induction of modules). For example, if M is a covariant RCmodule*
*, then
F*M = R morD(F (?); ??) RC M, where for a set S the notation RS is the free R*
*module
generated by the set S. The key properties of (co)induction and restriction ar*
*e the following
adjoint properties.
Lemma 1.9 There are natural adjunction homeomorphisms
homD (F*X; Y ) ! hom C(X; F *Y );
homC(F *X; Y ) ! hom D (X; F!Y );
F*X D Y  ! X C F *Y ;
Y D F*X  ! F *Y C X;
F *Y C X  ! Y D F*X;
X C F *Y  ! F*X D Y ;
for a Cspace X and Dspace Y of the required variance.
Proof______: Notice for a covariant Dspace Y that there are natural homeomor*
*phisms of co
variant Cspaces
morD(??; F (?)) D Y ! F *Y ! homD (morD (F (?); ??); Y )
and analogously for contravariant Y . Now the claim follows from the adjointnes*
*s of tensor
product and hom and the associativity of tensor product. 
Version of June 3, 1996
9
2. K and LTheory Spectra over the Orbit Category
In this section we construct the main examples of spectra over the orbit *
*category
Kalg: Or(G) ! SPECTRA ;
L: Or(G) ! SPECTRA ;
Ktop: Or(G) ! SPECTRA :
These functors are necessary for the statements of the various Isomorphism Conj*
*ectures.
First we outline what we would naively like to do, explain why this does not wo*
*rk and then
give the details of the correct construction.
The three functors defined over the orbit category will be related to the*
* more classical
functors
Kalg: RINGS  ! SPECTRA ;
_______
L: RINGS  ! SPECTRA ;
Ktop : C*ALGEBRAS  ! SPECTRA ;
_______
where RINGS is the category of rings with involution. The classical functors w*
*ere defined
by Gersten [16] for algebraic Ktheory, by QuinnRanicki [33] for algebraic Lt*
*heory, and
by using Bott periodicity for C*algebras (see [39] for a discussion of Bott pe*
*riodicity for
C*algebras and also the end of this section for a functorial approach). The h*
*omotopy
groups of these spectra give the algebraic Kgroups of QuillenBass, the surger*
*y obstruc
tion Lgroups of Wall, and the topological Kgroups of C*algebras. We would l*
*ike our
functors defined on the orbit category to have the property that the spectra Ka*
*lg(G=H),
L(G=H) and Ktop(G=H) have the weak homotopy type of the spectra Kalg(ZH), L<*
*j>(ZH)
and Ktop(C*rH) respectively, where ZH is the integral group ring and C*rH is th*
*e reduced
C*algebra of H (see [29] for a definition). We would also like our functor to*
* be correct
on morphisms. Notice that a morphism from G=H to G=K is given by right multipli*
*cation
rg : G=H ! G=K; g0H 7! g0gK provided g 2 G satisfies g1Hg K. The induced*
* ho
momorphism cg : H ! K; h 7! g1hggives a map of rings (with involution) from*
* ZH to
ZK, and, at least if the index of cg(H) in K is finite, a map on reduced C*alg*
*ebras. We
would like the functors applied to the morphism rg in the orbit category to mat*
*ch up with
the "classical" functors on rings, rings with involution, and C*algebras.
The naive approach is define Kalg(G=H), L(G=H) and Ktop(G=H) as the sp*
*ectra
Kalg (ZH), L(ZH) and Ktop(C*rH) respectively. This definition works fine for*
* objects, but
fails for morphisms. The problem is that g in cg is not unique, because for an*
*y k 2 K,
clearly g and gk define the same morphism in the orbit category. Hence this de*
*finition
makes sense only if ck : K ! K induces the identity on the various spectra as*
*sociated to
K. This is actually true on the level of homotopy groups, but not on the level *
*of the spectra
themselves. However, it is important to construct these functors for spectra an*
*d not only for
homotopy groups of spectra in order to deal with assembly maps and the various *
*Isomorphism
Conjectures. Thus we must thicken up the spectra. The problems with construct*
*ing the
10
functor Ktop : C*ALGEBRAS ! SPECTRA are particularly involved. P. Ba*
*um and
J. Block, and P. Baum and G. Comezana have approaches to this construction, qui*
*te different
from ours.
The general strategy for a solution of this problem is the following. Let*
* GROUPOIDS
be the category of (discrete) groupoids with functors of groupoids as morphisms*
*. (A groupoid
is a small category, all of whose morphisms are isomophisms.) Let GROUPOIDS *
*injbe the
full subcategory consisting of those functors F : G0 ! G1 which are faithful, *
*i.e. for any
two objects x; y in G0 the induced map mor G0(x; y) ! morG1(F (x); F (y))is in*
*jective. In
the first step one defines a covariant functor
GR : Or(G) ! GROUPOIDS inj
from the orbit category_to the category_of groupoids as follows. Namely, a lef*
*t Gset S
defines a groupoid S where Ob (S ) = S and for s; t 2 S, mor(s; t) = {g 2 G  g*
*s = t}. The
composition law is given by group multiplication. Obviously_a map of left Gse*
*ts defines
a functor of the associated_groupoids._ The category G=H is equivalent to the*
* category
Or (H; 1) = H and hence G=H can serve as a substitute for the subgroup H.
Next one extends the definition of the algebraic K and Ltheory spectra *
*of the integral
group ring of a group and the topological Ktheory spectrum of the reduced C*a*
*lgebra of
a group to the category of groupoids. The composition of this extension with t*
*he functor
GR above yields covariant functors from the orbit category to the category of s*
*pectra. We
will see that their value at each object G=H is homotopy equivalent to the corr*
*esponding
spectrum associated to H. The main effort is now to construct these extensions*
* to the
category of groupoids, which will be denoted in the same way as the three funct*
*ors we want
to construct:
Kalg : GROUPOIDS  ! SPECTRA ;
L: GROUPOIDS  ! SPECTRA ;
Ktop : GROUPOIDS inj ! SPECTRA :
For this purpose we must introduce some additional structures on categories. R*
*ecall that
a category C is small if the objects in C form a set and for any two objects x *
*and y the
morphisms from x to y form a set. In the sequel all categories are assumed to b*
*e small. We
will recall and introduce additional structures on C.
Let R be a commutative ring with unit. We call C a Rcategory if for any *
*two objects
x and y the set mor C(x; y) of morphisms from x to y carries the structure of a*
* Rmodule
such that composition induces a Rbilinear map morC(x; y) x morC(y; z) ! morC(*
*x; z)for
all objects x, y and z in C. We also require the existence of an object 0 so th*
*at morC(0; 0)
is the zero Rmodule.
Suppose that R comes with an involution of rings R ! R r 7! _r. A Rcat*
*egory with
involution is a Rcategory C with a collection of maps
*x;y: morC(x; y) ! morC(y; x) x; y; 2 Ob (C)
11
such that the following conditions are satisfied:
__ __
1. *x;y( . f + . g) = . *x;y(f) + . *x;y(g)for all ; 2 R , objects x; *
*y 2 Ob (C), and
morphisms f; g : x !;y
2. *x;yO *y;x = idfor all objects x; y 2 Ob (C);
3. *x;z(g O f) = *x;y(f) O *y;z(g)for all x; y; z 2 Ob (C)and all morphisms *
*f : x ! yand
g : y ! z.
In the_sequel we abbreviate *x;y(f)by f*. In this notation the conditions*
* above become
(f + g)* = f* + __g* , (f*)* = fand (g O f)* = f* O.g*
We call a Rcategory (with involution) an additive Rcategory (with invol*
*ution) if it
possesses a sum and the obvious compatibility conditions with the Rmodule str*
*uctures
(and the involution) on the morphisms are fulfilled.
The notion of a C*category was defined by GhezLimaRoberts [17] and we *
*give the
definition below in our language. Equip the complex numbers with the involution*
* of rings
given by complex conjugation. A C*category C is a Ccategory with involution s*
*uch that for
each two objects x; y 2 Ob (C)there is a norm k kx;yon each complex vector spac*
*e morC(x; y)
such that the following conditions are satisfied:
1. (mor C(x; y); k kx;y)is a Banach space for all objects x; y 2 Ob (C);
2. k g O f kx;z k g ky;z. k f kx;yfor all x; y; z 2 Ob (C)and all morphisms *
*f : x ! y
and g : y ! z;
3. k f* O f kx;x= k f k2x;yfor all x; y 2 Ob (C)and all morphisms f : x ! y.
4. For every f 2 morC(x; y), there is a g 2 morC(x; x) so that f* O f = g* O*
* g.
In the sequel we abbreviate k f kx;yby k f kand we will consider a C*cat*
*egory as a
topological category by equipping the set of objects with the discrete topology*
* and the set
mor C(x; y) with the topology which is induced by the norm.
Example 2.1 Let C be a category with precisely one object x. Then the struct*
*ure of a
Rcategory on C gives mor C(x; x) the structure of a central Ralgebra with uni*
*t idx. The
additional structure of an involution is given by a map * : morC(x; x) ! morC(*
*x; x)satis
fying:
__ __
*( . f + . g) = . *(f) + . *(g); * O * = id and * (g O f) = *(f)*
* O *(g):
The structure of a C*category on C is the same as the structure of a C*algebr*
*a on the set
mor C(x; x) with idxas unit. The structure of a topological category on C is th*
*e structure of
a topological space on morC(x; x)such that composition defines a continuous map*
*. 
12
Next we construct from a category (for example, a groupoid) other categor*
*ies with the
structures described above. Given a category C, the associated Rcategory RC ha*
*s the same
objects as C and its morphism set mor RC(x; y) from x to y is given by the free*
* Rmodule
R morC(x; y) generated by the set morC(x; y). The composition is induced by the*
* composition
in C in the obvious way. Notice that the functor C 7! RC is the left adjoint of*
* the forgetful
functor from the category of Rcategories to the category of small categories.
Let G be a groupoid and R a commutative ring with unit and involution. T*
*hen RG
inherits the structure of a Rcategory with involution by defining
r ! *
X Xr __
ifi := f1i:
i=1 i=1
Let G be a groupoid. Next we explain how the category with involution CG*
* can be
completed to a C*category C*rG. It will have the same objects as G. Consider t*
*wo objects
x; y 2 Ob (G). If morG(x; y) is empty, put morC*rG(x; y) =.0Suppose that morG(x*
*; y) is non
empty. Choose some object z 2 Ob (G)such that morG (z; x)is nonempty, for inst*
*ance one
could choose z = x. Define a Clinear map
ix;y;z: C morG(x; y) ! B(l2(mor G(z; x)); l2(mor G(z; y)))
by sending f 2 morG(x; y)to the bounded operator from l2(mor G(z; x))to l2(mor *
*G(z; y))
given by composition with f. On the target of ix;y;zwe have the operator norm k*
* k. Define:
k u kx;y:= k ix;y;z(u) k for u 2 morCG(x; y) = C morG(x; y):
One easily checks that this norm k kx;yis independent of the choice of z. The B*
*anach space
of morphisms in C*rG from x to y is the completion of morCG(x; y) with respect *
*to the norm
k kx;y. We will denote the induced norm on the completion morC*rG(x; y)again by*
* k kx;yand
sometimes abbreviate by k k. One easily checks that *x;y: morCG(x; y) ! morCG*
*(y; x)is
an isometry since it is compatible with applying the maps ix;y;zand iy;x;zand t*
*aking adjoints
of operators. Therefore it induces an isometry denoted in the same way
*x;y: morC*rG(x; y) ! morC*rG(y; x):
Composition defines a Cbilinear map morCG(x; y) x morCG(y; z) ! morCG(x; z)wh*
*ich sat
isfies k g O f kx;z k g ky;z. k f kx;y. Hence it induces a map on the completio*
*ns
mor C*rG(x; y) x morC*rG(y; z) ! morC*rG(x; z)
with the same inequality for the norms. This is the composition in C*rG. One ea*
*sily verifies
that C*rG satisfies all the axioms of a C*category.
Example 2.2 Let G be a group. It defines a groupoid G with one object and G *
*as its
automorphism group. Then RG is just the group ring RG and C*rG is just the redu*
*ced group
C*algebra C*rG under the identifications of Example 2.1. 
13
The assignment of a C*category C*rG to a groupoid G gives a functor
C*r: GROUPOIDS inj! C*CATEGORIES ;
where C*CATEGORIES is the category of small C*categories. The injcondit*
*ion that
a functor F : G0 ! G1 is faithful is used to guarantee that the map mor CG0(x*
*; y)  !
mor CG1(F (x); F (y)) extends to morC*rG0(x; y) ! morC*rG1(F (x); F,(y))for al*
*l x; y 2 Ob (G0):
Remark 2.3 We make a few remarks on functoriality (or lack thereof) of C*al*
*gebras, which
motivate our use of C*categories. First note that the assignment of a C*algeb*
*ra C*rH to
a group H cannot be extended to a functor from the category of groups to the ca*
*tegory of
C*algebras. For instance, the reduced C*algebra C*r(Z * Z) of the free group *
*on two letters
is simple [31] and hence admits no C*homomorphism to the reduced C*algebra C *
*of the
trivial group.
There is a notion of the C*algebra of a groupoid, but it is poorly behav*
*ed with respect
to functoriality. To a discrete groupoid G, one can associate the complex grou*
*poid ring
CG, which as a Cvector space has a basis consisting of the morphisms in the gr*
*oupoid.
The product of two basis elements is the composite if defined and is zero other*
*wise. The
completion of CG in B(l2(G); l2(G)) in the operator norm is called the reduced *
*C*algebra
of the groupoid and which we denote C*rGalg. If G is connected (any two objec*
*ts are
isomorphic), and H is the automorphism group of an object, then it can be shown*
* (via Morita
theory) that the spectra Ktop(C*rGalg) and Ktop(C*rH) have the same weak homot*
*opy type.
The second naive approach to the construction of a functor
Ktop : Or(G) ! SPECTRA
_____
is to define Ktop(G=H) to be Ktop(C*rG=H alg). While this approach is basical*
*ly correct
for algebraic K and Ltheory, it fails for C*algebras because the C*algebra *
*of a groupoid
does not define a functor from the category GROUPOIDS injto C*ALGEBRAS .*
* Indeed,
consider the groupoid G[n] with n objects and precisely one morphism between tw*
*o objects.
Notice that the obvious functor from G[n] to G[1] has an obvious right inverse.*
* Hence it
would induce a surjective C*homomorphism between the associated C*algebras bu*
*t this
is impossible for n 2 as the associated C*algebra of G[n] is M(n; n; C). An*
*other coun
terexample comes from a morphism_in_the_orbit_category. Let G be any infinite g*
*roup and
consider the map of groupoids G=1  ! G=G where G acts on G=1 effectively and *
*transi
tively by left multiplication and G acts trivially on G=G. An easy computation*
* with the
operator norm shows that this map of groupoids does not extend to a map of the *
*reduced
C*algebras of the groupoids. We take the trouble to discuss this because mist*
*akes have
been made in the literature on this point and to motivate our definition of the*
* functor
C*r: GROUPOIDS inj! C*CATEGORIES . Below we will define the Ktopfuncto*
*r from
C*CATEGORIES to SPECTRA . Note that after applying homotopy groups, one g*
*ets maps
on the Ktheory of reduced C*algebras of the groupoids, independent of Morita *
*theory and
without maps on the C*algebras themselves. 
We recall some basic constructions we will need later.
14
Let C be a Rcategory. We define a new Rcategory C , called the symmetri*
*c monoidal
Rcategory associated to C with an associative and commutative sum as follows.*
* The
objects in C are ntuples x_= (x1; x2; : :;:xn)consisting of objects xi2 Ob (C*
*)for n = 0, 1,
2, : :.:We will think of the empty set as 0tuple which we denote by 0. The Rm*
*odule of
morphisms from x_= (x1; : :;:xmt)o y_= (y1; : :;:yn)is given by
mor C (x_; y_) := 1im;1jn mor C(xi; yj)
Given a morphism f : x_! y_, we denote by fi;j: xi ! yjthe component which be*
*longs to
i 2 {1; : :;:m}and j 2 {1; : :;:n}. If x or y is the empty tuple, then morC (x;*
* y) is defined
to be the trivial Rmodule. The composition of f : x_! y_and g_: y_! z_for o*
*bjects
x ____= (x1; : :;:xm,)y_= (y1; : :;:yn)and z_= (z1; : :;:zp)is defined by
Xn
(g O f)i;k= gj;kO fi;j:
j=1
The sum on C is defined on objects by sticking the tuples together, i.e. for x*
*_= (x1; : :;:xm )
and y_= (y1; : :;:yn)define
x_ y_:= (x1; : :;:xm ; y1; : :;:yn):
The definition of the sum of two morphisms is now obvious. Notice that this sum*
* is (strictly)
associative, i.e. (x_ y_) z_and x_ (y_ z_) are the same objects and analogousl*
*y for mor
phisms. Moreover, there is a natural isomorphism
x_ y_! y_ x_
and all obvious compatibility conditions hold. The zero object is given by the *
*empty tuple
0. These data define the structure of a symmetric monoidal Rcategory on C . No*
*tice that
the functor C 7! C is the left adjoint of the forgetful functor from symmetric*
* monoidal
Rcategories to Rcategories.
Given a category C, define its idempotent completion P(C) to be the follo*
*wing category.
An object in P(C) is an endomorphism p : x ! xin C which is an idempotent, i.e*
*. p O p = p.
A morphism in P(C) from p : x ! xto q : y ! yis a morphism f : x ! yin C sat*
*isfying
q O f O p = f. The identity on the object p : x ! x in P(C) is given by the m*
*orphism
p : x ! x in C. If C has the structure of a Rcategory or of a a symmetric mo*
*noidal
Rcategory, then P(C) inherits such a structure in the obvious way.
For a category C, let Iso(C) be the subcategory of C with the same object*
*s as C, but
whose morphisms are the isomorphisms of C. If C is a symmetric monoidal Rcateg*
*ory, then
so is Iso(C).
Let C be a symmetric monoidal Rcategory, all of whose morphisms are isom*
*orphisms.
Then its group completion is the following symmetric monoidal Rcategory Cb. An*
* object in C
is a pair (x; y) of objects in C. A morphism in Cb from (x; y) to (x0; y0) is g*
*iven by equivalence
classes of triples (z; f; g)consisting of an object z in C and isomorphisms f :*
* x z ! x0
15
and g : y z ! y0. We call two such triples (z; f; g)and (z0; f0; g0)equivalen*
*t if there is an
isomorphism h : z ! z0which satisfies f0O (idxh) = f and g0O (idyh) = g. The s*
*um on
Cb is given by
(x; y) (x0; y0) := (x x0; y y0):
If C is a C*category, then C and P(C) inherit the structure of a C*cat*
*egory where one
should modify the definition of P(C) by requiring that each object p : x ! xis*
* a selfadjoint
idempotent, i.e. p O p = pand p* = p. Moreover, C , P(C ) and (Iso(P(C )))b inh*
*erit the
structure of topological categories where the set of objects always gets the di*
*screte topology.
Next we can construct the desired functors from GROUPOIDS and GROUPOID*
*S inj
to SPECTRA . The covariant functor nonconnective algebraic Ktheory spectr*
*um of a
groupoid with coefficients in R
Kalg: GROUPOIDS ! SPECTRA
assigns to a groupoid G the nonconnective Ktheory spectrum of a small additiv*
*e category
(see [28]) where the additive category is (Iso(P(RG )))b.
Next we define the covariant functor periodic algebraic Ltheory spectrum*
* of a groupoid
with coefficients in R
L = Lh : GROUPOIDS ! SPECTRA
where we assume that R is a comutative ring with unit and involution. Then RG a*
*nd hence
RG inherit an involution. We apply the construction of the periodic algebraic*
* Ltheory
spectrum in [33, Example 13.6 on page 139]. If one uses the idempotent completi*
*on one gets
the projective version
Lp : GROUPOIDS ! SPECTRA :
Taking the Whitehead torsion into account yields the simple version
Ls : GROUPOIDS ! SPECTRA :
More generally one obtains for j 2 Z q {1}; j 2
L: GROUPOIDS ! SPECTRA :
where Lis Ls, Lh, Lp for j = 2; 1;.0
Next we construct the covariant functor nonconnective topological Ktheo*
*ry spectrum
Ktop: GROUPOIDS inj! SPECTRA :
We do this by composing the functors
GR : Or(G) ! GROUPOIDS inj;
C*r: GROUPOIDS inj! C*CATEGORIES ;
16
with the functor
Ktop : C*CATEGORIES  ! SPECTRA ;
which we are about to construct. Let C denote both the complex numbers and the *
*obvious
C*category with precisely one object denoted by 1_. We have introduced the cat*
*egory C
before. We denote by n_the nfold sum of the object 1_. In this notation C has*
* as objects
{n___ n = 0; 1; 2,:t:}:he sum is m_ n_= m_+_n_ for m; n = 0; 1; 2 :a:n:d the *
*Banach space
of morphisms from m_to n_is just given by the (m; n)matrices with complex entr*
*ies. Let C
be any Ccategory. We define a functor
: C x C ! C
as follows. We assign to an object n_2 C and an object x_2 C the object n_ x*
*_ which
is the nfold direct sum ni=1x_. Let f : m_! n_be a morphism in C and g : x_*
*! y_be
a morphism in C . Define f g : m_ x_! n_ y_ , to be the morphism whose compon*
*ent
from the ith copy of x_in m_ x_to the jth copy of y_in n_ y_is fi;j. g, where*
* fi;j2 C is
the component of f from the ith coordinate of m_ to the jth coordinate of n_.*
* One easily
checks that f g is a functor. For objects m_ and n_in C and an object x_in C *
* we have
(m___ n_) x_= (m_ x_) (n_ x_). For an object n_in C and objects x_and y_in *
*C we
have a natural isomorphism n_ (x_ y_) ~=(n_ x_) (n_ y_). Obviously this functo*
*r sends
the subcategories {0_} x C and C x {0} to {0} where {0_} and {0} denote the ob*
*vious
subcategories with one object.
Let C be any C*category. Then the construction above applies to P(C ). I*
*t extends
to a functor
: (Iso(C ))b x (Iso(P(C )))b ! (Iso(P(C )))b
in the obvious way. Notice that (Iso(P(C )))b inherits from C the structure of *
*a topological
category for which the set of objects is discrete. With respect to these topolo*
*gical structures
the functor above is a functor of topological categories. Given a topological c*
*ategory D, let
BD be it classifying space [34] (whose construction takes the topology into acc*
*ount). Given
topological categories D and D0, the projections induce a homeomorphism
B(D x D0) ! BD x BD0:
Hence the functor above induces a map
B(Iso(C ))b x B(Iso(P(C )))b ! B(Iso(P(C )))b
for any C*category C. Since it sends B(Iso(C ))b _ B(Iso(P(C )))b to the base*
* point
B{0} B(Iso(P(C )))b , we obtain a map, natural in C,
: B(Iso(C ))b ^ B(Iso(P(C )))b ! B(Iso(P(C )))b :
`
The category Iso(C ) can be identified with the disjoint union n0 GL(n;*
* C). Let
GL(C) = colimn!1 GL(n; C): Let Z x GL(C) be the symmetric moniodal category who*
*se
objects (and monoidal sum) are given by the integers, and so that morZxGL(C)(m;*
* n) is empty
17
if m 6= n and is GL(C) if m = n. There is an obvious functor Iso(C ) ! ZxGL(C)*
*. Using
Quillen's group completion theorem [18, pages 220221], it follows that B Iso(C*
* )b has the
homotopy type of Z x BGL(C). Let b : S2 ! B Iso(C )bbe a fixed representative *
*of the
Bott element in ss2(B Iso(C )b ) = K2({pt:}). Then b and yield a map, natural*
* in C,
S2 ^ B(Iso(P(C )))b ! B(Iso(P(C )))b :
Its adjoint is also natural in C and denoted by
fi : B(Iso(P(C )))b ! 2B(Iso(P(C )))b :
Define the nonconnective topological Ktheory spectrum Ktop(C)of the C*catego*
*ry C by the
space B(Iso(P(C )))b in even dimensions, by the space B(Iso(P(C )))b in odd dim*
*ensions
and by the structure maps which are the identity in odd dimensions and fi in ev*
*en dimensions.
Another construction is suggested by [13, Remark VIII.4.4. on page 186]. We cla*
*im that the
proof of Bott periodicity for C*algebras carries over to C*categories. Hence *
*Ktop(C) is a
spectrum. We will only be interested in the case where C is C*rG for a connect*
*ed groupoid
and in this case the claim follows from Bott periodicity for the reduced group *
*C*algebra of
the automorphism group of an object in G and Lemma 2.4.
We make some remarks about the constructions of the spectra of groupoids *
*above and
give some basic properties.
There are obvious equivalences of additive categories from RG resp. P(R*
*G ) to
the category of finitely generated free RGmodules resp finitely generated proj*
*ective RG
modules as defined in [23, section 9]. Notice that these module categories are *
*not small in
contrast to RG and P(RG ). A functor F : G0 ! G1 induces a functor from the c*
*ategory
of finitely generated free resp. projective RG0modules to the corresponding ca*
*tegory over G1
by induction. However, if we have a second functor G : G1 ! G2, then the funct*
*or induced
on the module categories by G O F and the composition of the functors induced b*
*y F and
G on the module categories are not the same, they agree only up to natural equi*
*valence. In
order to avoid this technical problem, we prefer the small category RG and its*
* idempotent
completion since there the composition of the functors induced by F and G is th*
*e same
as the functor induced by G O F , so that we get honest functors from GROUPOIDS*
* to
SPECTRA .
As mentioned earlier, the functors Kalg, L, and Ktopdefined on the orb*
*it category are
given by the composition of the groupoidvalued functor GR_and_the spectravalu*
*ed functors
defined above. The automorphism group of the object eH in G=H for the identity*
* element
e 2 G is just the subgroup H. Hence the next lemma_proves what we have already *
*claimed
before, namely, that the spectra we assign to G=H are homotopy equivalent to t*
*he spectra
associated to H. In particular we get for all n 2 Z and j 2 Z q {1}; j 2
ssn(Kalg(G=H)) ~= Kalgn(ZH)
ssn(L(G=H))~= Ln(ZH)
ssn(Ktop(G=H)) ~= Kn(C*H)
18
Lemma 2.4 1. If Fi: G0 ! G1 for i = 0; 1 are functors of groupoids and T *
*: F0 ! F1
is a natural transformation between them, then the induced maps of spectra
Kalg(Fi) : Kalg(G0) ! Kalg(G1)
are homotopy equivalent and analogously for Land Ktop;
2. Let G be a groupoid. Suppose that G is connected, i.e there is a morphism*
* between any
two objects. For an object x 2 Ob (G), let Gx be the full subgroupoid wit*
*h precisely one
object, namely x. Then the inclusion ix : Gx ! Ginduces a homotopy equiv*
*alence
Kalg(ix) : Kalg(Gx) ! Kalg(G)
and Kalg(Gx) is isomorphic to the spectrum Kalgassociated to the group ri*
*ng R autG(x).
The analogous statements hold for Land Ktop.
Proof______: Obviously 2.) follows from 1.). We indicate the proof of 1.) i*
*n the case of
Ktop , the other cases are analogous if one inspects the definitions in [28] an*
*d [33]. One easily
checks that a natural transformation between F0 to F1 induces a natural transfo*
*rmation from
the induced functors from (Iso(P(C*rG0 )))bto (Iso(P(C*rG1 )))b. Let [1] be the*
* category
having two objects, namely 0 and 1 and three morphisms, namely the identities o*
*n 0 and
1 and one morphism from 0 to 1. Then the natural transformation above can be vi*
*ewed as
a functor of topological categories from (Iso(P(C*rG0 )))b x [1]to (Iso(P(C*rG1*
* )))b. Since
the classifying space of a product is the product of the classifying spaces and*
* the classifying
space of [1] is [0; 1], we obtain a map
h : B(Iso(P(C*rG0 )))b x [0; 1] ! B(Iso(P(C*rG1 )))b:
One easily checks that this induces the desired homotopy of maps of spectra. *
* 
Version of June 3, 1996
19
3. CW Approximations and Homotopy Limits
In this section we give the basic definitions and properties of spaces an*
*d CW complexes
over a small category C. We show that the Whitehead Theorem and CW approximati*
*ons
carry over from spaces to Cspaces. We emphasize the parallels between a catego*
*ry and a
group, thinking of a group as a category with a single object, all of whose mor*
*phisms are
invertible. We define EC, the universal free contractible Cspace, and use this*
* to define the
homotopy colimit EC C X, the analogue of the Borel construction EG xG X.
Consider the set Ob (C) as a small category in the trivial way, i.e. the *
*set of objects is
Ob (C) itself and the only morphisms are the identity morphisms. A map of two O*
*b(C)spaces
is a collection of maps {f(c) : X(c) ! Y (c)  c 2 Ob (C)}. There is a forgetf*
*ul functor
F : CSPACES  ! Ob(C)SPACES
Define a functor
B : Ob (C)SPACES  ! CSPACES
`
by sending a contravariant Ob (C)space X() to c2Ob(C)morC(; c) x X(c). In *
*the covari
ant case one uses morC(c; ).
Lemma 3.1 The functor B is the left adjoint of F .
Proof______: This means that there is a natural bijection
T (X; Y ) : hom C(B(X); Y ) ! homOb(C)(X; F (Y ))
for all Ob(C)spaces X`and for all Cspaces Y. Actually T (X; Y ) will even be *
*a homeomor
phism. For f : B(X) = c2Ob(C)morC(; c) x X(c) ! Y ()define T (f)() by res*
*tricting
f to X() = {id} x X() . The inverse T (X; Y )1 assigns to a map g of Ob (C)*
*spacesthe
following transformation
a
B(X) = morC(; c) x X(c) ! Y (); (OE; x) 7! Y (OE) O g(c)(x*
*): 
c2Ob(C)
Let R be a ring. There is also an adjoint to the forgetful functor from R*
*CMOD to
Ob (C)SETS . It is defined as B(X()) = c2Ob(C)R(mor C(; c) x X(c)). A free R*
*Cmodule
is a module isomorphic to one in the image of B. Notice the analogy between Lem*
*ma 3.1
and the case of the forgetful functor from Rmodules to sets and the functor as*
*signing to a
set S the free Rmodule RS generated by S.
We have already mentioned that the category of Or (G; 1)spacesis the cat*
*egory of
Gspaces and the category Ob (Or (G; 1)spaces)is the category of spaces. Under*
* this iden
tification the forgetful functor F just forgets the Gaction and B sends a spac*
*e Z to the
Gspace G x Z where G acts in the obvious way.
20
Notice that the notions of coproduct, product, pushout, pullback, colimit*
*, and limit
exist in the category of Cspaces. They are constructed by applying these noti*
*ons in
the category SPACES objectwise. For instance, the pushout of a diagram of C*
*spaces
X1  X0 ! X2 is defined as the functor X : C ! SPACES whose value at an*
* object
c in C is the pushout of the diagram of spaces X1(c)  X0(c) ! X2(c) . We men*
*tion that
sometimes in the literature the terms direct limit and inverse limit are used i*
*nstead of colimit
and limit. We will always use the names colimit and limit.
A map f : X ! Y of Cspaces is a cofibration (fibration) of Cspaces if*
* it has the
homotopy extension property (homotopy lifting property) for all Cspaces. If f*
* is a (co)
fibration of Cspaces, its evaluation f(c) : X(c) ! Y (c)is a (co)fibration o*
*f aut(c)spaces
for all objects c in C. The proof of this fact is a simple abstract manipulatio*
*n of the homotopy
lifting (extension) property and various adjunctions. Notice that the converse *
*is not true.
Next we extend the notion of a CW complex for spaces to Cspaces. We wi*
*ll see
that the notion of a free CCW complex is very similar to the the notion of an*
* ordinary
CW complex and that standard results and their proofs for CW complexes genera*
*lize in a
straightforward manner to the case of free CCW complexes. This leads to easy*
* proofs of
known and new results whose strategy is very close to classical ideas and patte*
*rns.
Definition 3.2 A contravariant free CCW complex X is a contravariant Cspace*
* X to
gether with a filtration
[
; = X1 X0 X1 X2: : :Xn : : :X = Xn
n0
such that X = colimn!1 Xn and for any n 0 the nskeleton Xn, is obtained from *
*the
(n  1)skeleton Xn1 by attaching free Cncells, i.e. there exists a pushout *
*of Cspaces of
the form
` n1
i2InmorC(;?ci) x S ! Xn1?
? ?
y y
` n
i2InmorC(; ci) x D ! Xn
where the vertical maps are inclusions, In is an index set, and the ci are obje*
*cts of C. We
refer to the inclusion functor mor C(; ci) x intDn  ! X as a free Cncell ba*
*sed at ci.
A free CCW complex has dimension n if X = Xn. The definition of a covariant*
* free
CCW complex is analogous. 
A CCW complex was defined by Dror Farjoun [10, 1.16 and 2.1] (see also *
*[30]). We
shall deal almost exclusively with free CCW complexes. For a free CCW compl*
*ex X, the
cellular chain complex C*(X)(), c 7! C*(X)(c) is a Cchain complex of free ZC*
*modules.
Notice that a free CCW complex X defines a functor from C to CW COMPLEXES *
* , but
not any functor from C to CW COMPLEXES is a free CCW complex.
21
If Y is a GCW complex, then map G(; Y ) (which sends G=H 7! Y H) is an*
* example
of a free Or(G)CW complex. A Gcell of Y of orbit type G=H corresponds to a O*
*r(G)cell
of map G(; Y ) based at G=H. Recall that the category of Or(G; 1)spacescoinc*
*ides with
the category of Gspaces. Under this identification a free Or(G; 1)CW complex*
* is the same
as a free GCW complex.
Given a Cspace X and a space Y , we obtain the Cspace X x Y by assignin*
*g to an
object c the space X(c) x Y . Taking Y = [0; 1], it is now clear what a homotop*
*y of maps
of Cspaces means. Recall that a map f : X ! Y of spaces is nconnected for n*
* 0 if and
only if for all points x in X the induced map ssk(f; x) : ssk(X; x) ! ssk(Y;if*
*(x))s bijective
for all k < n and surjective for k = n. It is a weak homotopy equivalence if it*
* is nconnected
for all n 0.
Definition 3.3 A map f : X ! Y of Cspaces is nconnected (a weak homotopy e*
*quiv
alence) if for all objects c the map of spaces f(c) : X(c) ! Y (c)is nconnect*
*ed (a weak
homotopy equivalence). 
The constant map EG ! {*} is a weak homotopy equivalence, but not a homo*
*topy
equivalence of Or(G; 1)spaces.
The following result is wellknown for ordinary CW complexes [42, IV. Th*
*eorem 7.16
and 7.17 on page 182]. See also [10, Proposition 2.9] and [30, Theorem 3.4].
Theorem 3.4 Let f : Y  ! Z be a map of Cspaces and X be a Cspace. The ma*
*p on
homotopy classes of maps between Cspaces induced by composition with f is deno*
*ted by
f* : [X; Y ]C ! [X; Z]C.
1. Then f is nconnected if and only if f* is bijective for any free CCW c*
*omplex X with
dim(X) < n and surjective for any free CCW complex X with dim(X) n;
2. Then f is a weak homotopy equivalence if and only if f* is bijective for *
*any free C
CW complex X.
Proof______: We only give the proof of the second assertion in the special case*
* where Z is the
trivial Cspace, i.e. Z(c) = {*} for all objects c in C. Then it is easy to fig*
*ure out the full
proof following the classical proof in [42, IV. Theorem 7.16 and 7.17 on page 1*
*82].
We begin with the if statement. Suppose that [X; Y ]C consists of one ele*
*ment for each
free CCW complex X. We then choose X = morC(; c) x Sk, for a fixed c 2 Ob (C*
*). From
Lemma 3.1 we obtain a natural homeomorphism
homC(mor C(; c) x Sk; Y ) ! map(Sk; Y (c))
22
and thus a natural bijection
[mor C(; c) x Sk; Y ]C  ! [Sk; Y (c)]:
Hence for all objects c in C any map from Sk to Y (c) is nullhomotopic. This im*
*plies that f
is a weak homotopy equivalence.
Next we prove the only if statement. Suppose that f is a weak homotopy e*
*quiva
lence. We must show for any free CCW complex X that any map of Cspaces g : X*
* ! Y
is nullhomotopic, or in other words, extends to the cone on X. The cone on X i*
*s ob
tained from X by attaching Ccells. Therefore it suffices to show that any map *
*of Cspaces
mor C(; c) x Sn1 ! Y can be extended to a map mor C(; c) x Dn ! Y. Such a *
*prob
lem reduces to extending a map from Sn1 to Y (c) to Dn. This can be done as Y *
*(c) has the
weak homotopy type of a point by assumption. 
Corollary 3.5 A weak homotopy equivalence between free CCW complexes is a ho*
*motopy
equivalence.
Proof______: Let f : Y  ! X be a weak homotopy equivalence between free CCW *
*complexes.
By Theorem 3.4, there is a g : X ! Y so that f*[g] = [f O g] = [idX]. Thus g*
* is a weak
homotopy equivalence. To show that g is the homotopy inverse of f, we need only*
* show that
g has a right homotopy inverse, but this follows by Theorem 3.4 again. 
Definition 3.6 Let (X; A) be a pair of Cspaces. A CCW approximation
(u; v) : (X0; A0) ! (X; A)
consists of a free CCW pair (X0; A0) together with a map of pairs (u; v) of C*
*spaces such
that both u and v are weak homotopy equivalences of Cspaces. A CCW approxima*
*tion of a
space X is a CCW approximation of the pair (X; ;). 
This is a categorical generalization of the notion of a CW approximation*
* for a topo
logical space X (see [42, V.3]. By taking (f; g) to be the identity in Theorem *
*3.7 below we
see that CCW approximations exist and are unique up to homotopy.
Theorem 3.7 Let (X; A) be a pair of Cspaces.
1. (existence) There exists a CCW approximation of (X; A);
2. (uniqueness) Given a map of pairs (f; g) : (X; A) ! (Y; B) of Cspaces *
*and given
CCW approximations (u; v) : (X0; A0) ! (X; A) and (a; b) : (Y 0; B0) *
* ! (Y; B);
then there exists a map of pairs (f0; g0) : (X0; A0) ! (Y 0; B0) so tha*
*t the diagram
23
(X0; A0)(u;v)!(X; A)
? ?
(f0;g0)?y ?y(f;g)
(a;b)
(Y 0; B0)! (Y; B)
commutes up to homotopy. Furthermore the map (f0; g0) is unique up to hom*
*otopy.
Proof______: Existence of a CCW approximation is an inductive construction do*
*ne by attaching
ncells to obtain a nconnected map and finally taking a colimit. Uniqueness fo*
*llows from
the relative versions of Theorem 3.4 and Corollary 3.5. 
Definition 3.8 Let EC denote any free CCW complex so that EC(c) is contracti*
*ble for all
objects c. 
Since EC is a CCW approximation of the trivial Cspace, EC exists and i*
*s unique up
to homotopy type. Note there is a contravariant EC and a covariant EC. They a*
*re not
closely related, but one can identify the contravariant EC with the covariant E*
*Cop. There
are functorial constructions of CCW approximations and hence for EC, which we*
* describe
at the end of this section. However, often it is useful to have smaller and mo*
*re flexible
models.
If C = Or(G; 1), then EC can be identified with EG, a contractible free G*
*CW complex.
If C has a final object, then we may take the contravariant EC to be the trivia*
*l Cspace,
which is a single C0cell based at the final object. Similarly, if C has an in*
*itial object, the
trivial Cspace is a covariant EC. If G is a crystallographic group, i.e. a dis*
*crete subgroup
of the isometries of Rn so that Rn=G is compact, then (G=H 7 ! (Rn)H ) is a co*
*ntravariant
E Or(G; FIN ), where FIN is the family of finite subgroups. More generally, if*
* E(G; F) is
classifying space for a family of subgroups of a discrete group G, then (G=H 7*
* ! E(G; F)H )
is a model for E Or(G; F). This example is expanded on in Section 7.
Example 3.9 Let ! be the category whose objects are the nonnegative integers*
* and whose
morphisms are given by the arrows below, their composites, and the identity map*
*s.
0 ! 1 ! 2 ! 3 ! . . .
Then we may take the contravariant E! to be (E!)(i) = [i; 1), whose zero skelet*
*on is
obtained by intersecting each space with the integers. For each nonnegative in*
*teger i, there
is C0cell and a C1cell based at i. We may take the covariant E! to be the t*
*rivial Cspace.

Definition 3.10 The classifying space of a category C is the space BC = EC C {*
**}, where
{*} is the trivial Cspace and EC is a contravariant CCW approximation of the*
* trivial C
space. 
24
The classifying space BC is a CW complex defined only up to homotopy typ*
*e. We will
recall its functorial definition later in this section.
Theorem 3.11 Let f : Y  ! Z be a weak homotopy equivalence of covariant Cs*
*paces.
Then for any contravariant free CCW complex X the induced map
idX Cf : X C Y ! X C Z
is a weak homotopy equivalence. A similar statement holds for weak homotopy equ*
*ivalences
of contravariant Cspaces.
Let X be a covariant (contravariant) free CCW complex and f : Y  ! Z b*
*e a weak
homotopy equivalence of covariant (contravariant) Cspaces. Then the induced map
hom C(id; f) : hom C(X; Y ) ! homC(X; Z)
is a weak homotopy equivalence.
Proof______: We will prove the claim by induction over the skeletons and the c*
*ells in X. We
only consider the case idX Cf. The functor  C Y is compatible with colimits*
*, using
the standard trick from category theory that a functor with a right adjoint com*
*mutes with
arbitrary colimits (see [24, Chapter V, section 5]). Hence the pushout specifyi*
*ng how Xn is
obtained from Xn1 by attaching cells remains a pushout after applying  C Y .*
* Moreover,
the left vertical arrow in this pushout is a cofibration and idXnCf is the pu*
*shout of three
weak homotopy equivalences. Hence it is itself a weak homotopy equivalence by *
*excision
theorem of BlakersMassey [42, VII.7]. Analogously one argues to show that the *
*colimit of
the maps idXnCf is idXCf and each inclusion Xn C Y  ! Xn+1 C Y is a cof*
*ibration.
This implies that idXCY is a weak homotopy equivalence. The proof of the asse*
*rtion for
hom is similar. 
Next we give some definitions, which are in close analogy with group coho*
*mology and
homological algebra.
Definition 3.12 Let M be a covariant ZCmodule, X a covariant Cspace, and E a*
* covariant
Cspectrum. Define the colimit and the limit of M over C to be the abelian grou*
*ps
colimM = Z ZC M and lim M = homZC(Z; M):
C C
Define the colimit of X over C and the limit of X over C to be the topological *
*spaces
colimX = {*} C X and lim X = homC({*}; X):
C C
Define the colimit of E over C and the limit of E over C to be the spectra
colimE = {*} C E and limE = homC({*}; E): 
C C
25
The above definitions are standard and the universal properties follow fr*
*om the ad
junctions in Lemma 1.5 and Lemma 1.6. Here Z represents the trivial ZCmodule,*
* with
Z(c) Z and {*} is the trivial Cspace. It is also convenient to define colimit*
*s and limits of
contravariant functors over C, by applying the above definitions to the functor*
*s considered
as covariant functors on Cop. We next discuss the higher derived functors of th*
*e above limits.
Definition 3.13 If M is a covariant ZCmodule, define
Hi(C; M) = Hi(C*(EC) ZC M) and Hi(C; M) = Hi(Hom ZC(C*(EC); M)):
If X is a covariant Cspace, define the homotopy colimit and the homotopy limit*
* of X over
C as
hocolim X = EC C X and holim X = homC(EC; X):
C C
If E is a covariant Cspectrum, define the homotopy colimit and the homotopy li*
*mit of E
over C as
hocolimE = EC C E and holimE = homC(EC; E): 
C C
One must be careful about the variances on EC in the above definitions. I*
*n the left
hand appearances of EC we are taking the contravariant version, while on the ri*
*ght we want
the covariant version. In the definition of the higher limits Hi and colimits H*
*i, the ZCchain
complex C*(EC) can be replaced by any projective ZCresolution of Z. As above w*
*e define
homology, cohomology, hocolimits, and holimits of contravariant functors by con*
*sidering
them as functors defined on the opposite category. For properties of Hi and Hi*
*, see, for
example, [23] and for properties of homotopy limits see for instance [4], [11, *
*x9] and [21].
One obtains the functorial definitions if one uses the functorial construction *
*EbarC for EC.
Since EC maps to {*}, there are maps hocolimCX ! colimCX and limCX ! holimCX.
They are not, in general, weak homotopy equivalences, unless X is a free CCW *
*complex.
The basic property of homotopy limits is that if X ! Y is a weak homotopy equi*
*valence,
then so are the induced maps hocolimCX ! hocolimCY and holimCY  ! holimCX; th*
*is
follows from Theorem 3.11.
Example 3.14 Let ! be the category from Example 3.9. Let M and N be covarian*
*t and con
travariant ZCmodules respectively. Then it is easy to see that Hi(!; M) is col*
*imj!1 M(j)
for i = 0 and zero for i > 0, that Hi(!; M) is M(0) for i = 0 and zero for i > *
*0, that
Hi(!; N) is N(0) for i = 0 and zero for i > 0, and that Hi(!; N) is limj!1 N(j)*
* for i = 0,
Milnor's lim1j!1N(j) for i = 1, and zero for i > 1.
Let X and Y be covariant and contravariant Cspaces respectively. Then wi*
*th the E!'s
from Example 3.9 hocolim!X is the infinite mapping telescope of
X(0) ! X(1) ! X(2) ! X(3) ! . .:.
26
Clearly holim!X = X(0) and hocolim!Y = Y (0). Now holim!Y is a bit bigger, it i*
*s the
subspace of
map ([0; 1); Y (0)) x map ([1; 1); Y (1)) x map ([2; 1); Y (2)) x map (I; *
*Y (3)) x . .;.
f*
*li
consisting of all tuples (fl0; fl1; fl2; . .).so that the composite of [i; 1) *
*! Y (i) ! Y (i  1)
equals fli1restricted to [i; 1). 
Definition 3.15 Let X be a Cspace and M a ZCmodule. Let X0 ! X be a CCW *
*ap
proximation. If X is contravariant and M is covariant, define
HCp(X; M) = Hp(C*(X0) ZC M);
where C*(X0) is the cellular chain complex of X0. There is a similar definitio*
*n if X is
covariant and M is contravariant. If X and M have the same variance, define
HpC(X; M) = Hp(hom ZC(C*(X0); M)): 
When C = Or(G; 1), HCp(X; M) is Borel equivariant homology HGp(X; M) = Hp*
*(EGxG
X; M). When C = Or(G) and X is the the fixed point functor G=H 7! ZH of a GC*
*W 
complex Z, then HCp(X; M) is Bredon equivariant homology of Z with coefficients*
* in M.
Remark 3.16 One of the original motivations for Bredon's introduction of the*
* orbit cate
gory was equivariant obstruction theory, and it is clear that all the ingredien*
*ts are in place for
the development of obstruction theory for the study of Cmaps between a free C*
*CW space
and a Cspace, but we leave the task of finding the precise formulation to a re*
*ader motivated
by specific applications. Local coefficient systems are particularly subtle, se*
*e [26]. 
Next we recall functorial constructions of classifying spaces and CCW a*
*pproximations
(see for instance [4], [21], [34]). We will need some of the details later in S*
*ection 6. View the
ordered set [p] = {0; 1; 2; : :;:p}as a category, namely, objects are the eleme*
*nts and there is
precisely one morphism from i to j if i j and no morphism otherwise. Continuin*
*g with
the terminology from Example 1.7, we get a covariant functor
[ ] : ! CATEGORIES
from the category of finite ordered sets into the category of small categories.*
* The nerve of a
category C is the simplicial set
N:C : N ! SETS ; [p] 7! func([p]; C):
More explicitly, NpC consists of diagrams in C of the form
OE0 OE1 OE2 OEp1
c0  ! c1  ! c2  ! : : :! cp:
27
The bar resolution model BbarC for the classifying space of a category C is the*
* geometric
realization N:C of its nerve where we regard a simplicial set as a simplicial*
* space by us
ing the discrete topology. It has the nice properties (see [34]) that it is fu*
*nctorial, that
Bbar(C x D) = BbarC x BbarD , that BbarC = Bbar(Cop), and that a natural transf*
*ormation
from a functor F0 to a functor F1 induces a homotopy between the maps BbarF0 an*
*d BbarF1
on the bar resolution models. In particular an equivalence of categories gives *
*a homotopy
equivalence on the bar resolution models of the classifying spaces. From Examp*
*le 1.7 we
get that BbarC comes with a canonical CW complex structure such that there is *
*a bijective
correspondence between the set of sequences of composable morphisms
OE0 OE1 OE2 OEp1
c0  ! c1  ! c2  ! : : :! cp
where no morphism is the identity and the set of pcells. Any functor induces a*
* cellular map.
We will justify the term "model of the classifying space" shortly.
Given two objects ? and ?? in C, define the category ?#C #?? as follows. *
*An object is a
fi ff fi ff0 0 fi0
diagram ? ff!c ! ??in C. A morphism from ? ! c ! ??to ? ! c ! ??is a comm*
*utative
diagram in C of the shape
fi
?  ff!c ! ??
? ? ?
id?y OE?y id?y
0 fi0
?  ff!c0! ??
__________
Let mor C(?; ??)be the category whose set of objects is morC(?; ??) and whose o*
*nly morphisms
are the identity morphism of objects. Consider the functor
__________ i ff fi j
pr: ?#C #?? ! mor C(?; ??) ? ! c !?? 7! (fi O ff : ? !??):
Lemma 3.17 The map of contravariant C x Copspaces
__________
Bbarpr: Bbar?#C #?? ! Bbarmor C(?; ??)= morC(?; ??)
is a C x CopCW approximation.
Proof______: First we verify that Bbarpr is a weak homotopy equivalence. Fix ob*
*jects c; c0of C.
Define functors
__________ i ff j i id ff j
j : mor C(c; c0)! c#C #c0 c ! c0 7! c ! c ! c0:
__________ i ff fi j
pr(c; c0) : c#C #c0 ! mor C(c; c0) c ! d ! c07! (fi O ff : c !:*
*c0)
These give homotopy equivalences after applying Bbar, since pr(c; c0) O j is th*
*e identity and
there is a natural transformation S : j O pr(c; c0) !didefined by assigning to*
* an object
fi 0 0 0
c ff!d ! cin c#C #c the morphism in c#C #c
28
c id!c fiOff!c0
? ? ?
id?y ff?y id?y
fi 0
c ff!d ! c
We next show that Bbar?#C #?? is a free C x CopCW complex. The canonica*
*l skeletal
filtration on the classifying space of a category induces a filtration on Bbar?*
*#C #?? such that
Bbar?#C #?? = colimBbarp?#C #?? :
p!1
Moreover, there is a pushout of contravariant C x Copspaces
(n:d:Np?#C #?? ) x Sp1! Bbarp1?#C #??
? ?
? ?
y y
(n:d:Np?#C #?? ) x Dp ! Bbarp?#C #??
where n:d:Np?#C #?? is the set of nondegenerate psimplices of the nerve of ?#*
*C #?? . This
set can be identified with the disjoint union of the CCsets morC(?; c0) x mor*
*C(cp; ??)where
the disjoint union runs over the sequences
OE0 OE1 OE2 OEp1
c0  ! c1  ! c2  ! : : :! cp
where no morphism OEi is the identity. Such sequences thus give the indexing s*
*et for the
pcells. 
From Example 1.7 we get that for any Cspace X, there is a weak homotopy *
*equivalence
of Cspaces
t : S:X ! X:
such that S:X is functor from C to CW COMPLEXES . Notice that this does n*
*ot mean
that S:X itself is a free CCW complex.
Definition 3.18 Let X be a contravariant Cspace. The tensor product taking o*
*ver the
variable ?? yields contravariant Cspaces X C Bbar?#C #?? and X C morC(?; ??)*
*. Define
a map of contravariant Cspaces
idCBbarpr ~=
pX : X C Bbar?#C #?? ! X C morC(?; ??) ! X
where the second map is the canonical isomorphism given by x OE 7! X(OE)(x). *
*Define a
map of contravariant Cspaces
tCid bar pX
aX : S:X C Bbar?#C #?? ! X C B ?#C #?? ! X: 
29
Lemma 3.19 Let X be a contravariant Cspace. Then:
1. pX is a weak homotopy equivalence of contravariant Cspaces, i.e. pX (c*
*) is a weak
equivalence of spaces for all objects c in C;
2. Suppose that X is a contravariant functor from C to CW COMPLEXES , i.*
*e. there is
a CW structure on X(c) for each object c in C such that each morphism f *
*: c ! c0
in C induces a cellular map X(f) : X(c0) ! X(c). Suppose Y is a contrava*
*riant free
D x CopCW complex. Then the contravariant Dspace X C Y inherits the *
*structure
of a free DCW complex;
3. The map aX : S:X C Bbar?#C #?? ! X is a CCW approximation.
Proof______: 1.) Fix an object c in C. Then
B pr(c; ??) : Bbarc#C #?? ! morC(c; ??)
is a weak homotopy equivalence of free CCW complexes, hence is a Chomotopy e*
*quivalence.
Thus pX (c) is a homotopy equivalence.
2.) We will only indicate what the skeleta and cells are. The pskeleton of X*
* C Y is
[i+j=pXiCYj. A free D x Copjcell of Y based at (d; c) together with a icell *
*of X(c) gives
rise to a free Di + jcell based at d. More precisely, if : Di ! X(c) is a c*
*haracteristic
map for a icell of X(c) and if : morD (?; d) x morC(c; ??) x Dj ! Y is a cha*
*racteristic
map for a free D x Copjcell of Y based at (d; c), then the characteristic map
mor D(?; d) x Dix Dj ! X C Y
is given by
(f; a; b) 7! [(a); (f; idc; b)]:
3.) follows from Lemma 3.17, 1.), 2.), and Theorem 3.11. 
If one takes X = {*} in the construction above, one obtains the contravar*
*iant bar
CCW approximation of {*}
EbarC := {*} C Bbar?#C #?? :
More explicitly it is given as follows. For an object ? in C let ?#C be the *
*category of
objects under C. An object in ?#C is a morphism OE : ? ! cin C with ? as sou*
*rce. A
morphism in ?# Cfrom OE0 : ? ! c0to OE1 : ? ! c1is given by a morphism h : c0*
* ! c1in C
satisfying OE1 = h O OE0. A morphism : c ! din C defines a functor # C : d*
* # C ! c # C
by composition with from the right. Then
EbarC : C ! SPACES ; c 7! Bbarc#C :
One easily checks that EbarC C {*} = BbarC and thereby justifies our notation.
Version of June 3, 1996
30
4. (Co)Homology Associated to Spectra over a
Category
In this section we introduce the homology and cohomology theories associa*
*ted to a
spectrum over a category. We then explain a kind of AtiyahHirzebruch type spe*
*ctral se
quence.
Definition 4.1 Let (X; A) be a pair of pointed Cspaces. Denote the reduced c*
*one of the
pointed space A by cone(A). For a Cspectrum E of the opposite variance as (X; *
*A) define
ECp(X; A) = ssp(X [A cone(A) C E):
Given a Cspectrum E of the same variance as (X; A), define
EpC(X; A) = ssp(hom C(X [A cone(A); E)):
If A is just a point, we omit A from the notation. 
If C is the trivial category consisting of precisely one object and one m*
*orphism, then
the homology and cohomology as defined in Definition 4.1 reduces to the classic*
*al definition
of the reduced homology and cohomology of a pair with coefficients in a spectru*
*m. This is
obvious for homology whereas for cohomology one uses the natural bijection indu*
*ced by the
adjunction
ssp+k(map (X; E(k))) ! [X ^ Sp+k; E(k)]:
Notice that writing homology and cohomology in terms of tensor product and mapp*
*ing space
spectra is analogous to the definition of the homology and cohomology of a chai*
*n complex
C* with coefficients in a module M as the homology of C* M respectively Hom (C*
**; M).
Lemma 4.2 The homology and cohomology groups defined in Definition 4.1 are g*
*eneralized
reduced homology and cohomology theories for pointed Cspaces.
Proof______: The proof is exactly as in the case of spaces, i.e. where C is th*
*e trivial category.
For instance, let us check the long cohomology sequence of a pair (X; A) of poi*
*nted Cspaces.
The following diagram is a pushout
A  i! X [A (A ^ [0; 1]+)
? ?
p?y ?yq
j
{*}  ! X [A coneA
where i is the cofibration given by the inclusion and p and q are the projectio*
*ns. The functor
hom C(; Y ) for a fixed pointed covariant Cspace Y has a left adjoint, namel*
*y CY . Hence
the following diagram is a pullback and hom C(i; idE(n)) is a fibration for all*
* n 2 Z.
31
homC(q;idE(n))
hom C(X [A cone(A); E(n))! hom C(X [A (A ^ [0; 1]+); E(n))
? ?
homC(j;idE(n))?y ?yhomC(i;idE(n))
homC(p;idE(n))
homC({*}; E(n)) ! homC(A; E(n))
Hence we get for n 2 Z fibrations of pointed spaces
homC(q;idE(n))
hom C(X [A cone(A); E(n)) ! homC(X [A (A ^ [0; 1]+); E(n))
homC(i;idE(n))
! hom C(A; E(n)):
They are compatible with the structure maps. Now the colimit over their long h*
*omotopy
sequences yields the desired long cohomology sequence of the pair since the can*
*onical pro
jection from X [A (A ^ [0; 1]+) to X is a homotopy equivalence of pointed Cspa*
*ces.
The suspension isomorphism is induced by the following identifications
ssp+1+k(hom C(X ^ S1; E(k))) = ssp+1+k(map (S1; homC(X; E(k)))
= ssp+1+k( homC(X; E(k))) = ssp+k(hom C(X; E(k))): 
Recall that a weak homotopy equivalence of Cspaces is a Cmap X ! Y s*
*o that
X(c) ! Y (c) is a weak homotopy equivalence for all objects c 2 Ob (C). The W*
*HEaxiom
says that a weak homotopy equivalence f : X ! Y of pointed spaces induces isom*
*orphisms
on homology resp. cohomology. This is not necessarily satisfied for ECpand Ep*
*Cas the
following example shows. Let G be a group and C = Or(G; 1). Recall that a contr*
*avariant
pointed Or(G; 1)space is a space with a base point preserving right Gaction. *
*Let E be the
ordinary EilenbergMacLane spectrum with ss0(E) = Z, considered as a covariant *
*Or(G; 1)
spectrum by the trivial Gaction. The projection p : EG+ ! {*}+ is a weak ho*
*motopy
equivalence of pointed Or(G; 1)spaces. We get
EOr(G;1)q(EG+) = Hq(BG) and EOr(G;1)q({*}+) = Hq({*}):
where H* is ordinary homology. Obviously these two groups do not coincide in ge*
*neral.
Our goal is to get unreduced homology and cohomology theories for (unpoin*
*ted) C
spaces which satisfy both the disjoint union axiom and the WHEaxiom. To be mor*
*e precise,
a homology theory means that homotopic maps of pairs of Cspaces induce the sam*
*e maps on
the homology groups, that there are long exact sequences of pairs (X; A), and f*
*or a pushout
of Cspaces
X0 i1!X1
? ?
i2?y ?yj1
j2
X2 ! X
32
the map (j2; i1) : (X2; X0) ! (X; X1)induces an isomorphism on homology provid*
*ed that
i2 : X0 ! X2 is a cofibration of Cspaces. If the homology theory satisfies th*
*e WHEaxiom,
it suffices to require that for each object c the map i2(c) : X0(c) ! X2(c)is *
*a cofibration of
spaces. The disjoint union axiom says that for an arbitrary disjoint union the *
*obvious map
from the direct sum of the homology groups of the various summands to the homol*
*ogy of
the disjoint union is an isomorphism. (For cohomology the direct sum has to be *
*substituted
by the direct product and the map goes the other way round.) For this purpose *
*we need
CCW approximations (Definition 3.6) in order to generalize the usual procedur*
*e for spaces
to Cspaces (cf [38, 7.68]).
Definition 4.3 Let (X; A) be a pair of Cspaces. Let (u; v) : (X0; A0) ! (X*
*; A) be a
CCW approximation. For a Cspectrum E of the opposite variance as (X; A), de*
*fine the
homology of (X; A) with coefficients in E by
HCp(X; A; E) = ECp(X0+; A0+):
and
HCp(X; E) = HCp(X; ;; E):
Given a Cspectrum E of the same variance as (X; A), define the cohomology of (*
*X; A) with
coefficients in E by
HpC(X; A; E) = EpC(X0+; A0+)
and
HpC(X; E) = HpC(X; ;; E): 
The above homology and cohomology are welldefined by the existence and u*
*niqueness
of CCW approximations. Furthermore, by Theorem 3.4, given a map of pairs of C*
*spaces
(X; A) ! (Y; B), there is an induced map of their CCW approximations which i*
*s uniquely
up to homotopy determined by the property that the following diagram commutes u*
*p to
homotopy
(X0; A0)! (X; A)
? ?
? ?
y y
(Y 0; B0)! (Y; B)
Thus for a map of Cpairs, there are corresponding maps of homology and cohomol*
*ogy
groups. We always have natural maps
HCp(X; A; E) ! ECp(X; A)
and
EpC(X; A) ! HpC(X; A; E):
They are isomorphisms if (X; A) is a free CCW pair, but not in general.
33
Lemma 4.4 HCp(X; A; E) and HpC(X; A; E) are unreduced homology and cohomolog*
*y theories
on pairs of Cspaces which satisfy the WHEaxiom. The homology theory satisfies*
* the disjoint
union axiom. The cohomology theory satisfies the disjoint union axiom provided *
*that E is a
Cspectrum.
Proof______: The first claim follows from Lemma 4.2 and Theorem 3.4.
The homology theory satisfies the disjoint union axiom for finite disjoin*
*t unions. We get
the disjoint union axiom for arbitrary coproducts, if we show for that the homo*
*logy theory
commutes with arbitrary colimits. This follows from the fact that the functor *
* C E(k)
has a right adjoint and commutes therefore with arbitrary colimits and that two*
* colimits of
systems of abelian groups commute.
To check the`disjoint union axiom for the cohomology theory, it suffices *
*to do this for
a disjoint union i2IXi of free CCW complexes. We conclude from Theorem 3.11*
* for any
free CCW complex Y that hom C(Y; E) is a spectrum since E is a Cspectrum a*
*nd hence
ssp(hom C(Y; E)) = ssp+k(hom C(X; E(k)));
provided p + k 0. Now the claim follows from the adjunction homeomorphism
0 ! 1
a ~=Y
hom C@ Xi ; E(k)A ! hom C((Xi)+; E(k)): 
i2I + i2I
Notice that without the condition that E is a Cspectrum the associated *
*cohomology
theory does not have to satisfy the disjoint union axiom.
Lemma 4.5 Let X be a Cspace with a filtration
; = X1 X0 X1 X2 : : :X
such that X = colimn!1 Xn . Let E be a Cspectrum with the opposite respectivel*
*y the same
variance as X.
1. The natural map
colim HCp(Xn; E) ! HCp(X; E)
n!1
is an isomorphism for p 2 Z;
2. Let E be a Cspectrum. There is a natural exact sequence
{0} ! lim1Hp1C(Xn; E) ! HpC(X; E) ! lim HpC(X; E) ! {0}
n!1 n!1
for all p 2 Z.
34
Proof_____: The proof is exactly as in the case where C is the trivial category*
* which is due to
________
Milnor and can be found for instance in [38, 7.53,7.66,7.73] or [42, Theorem XI*
*II.1.1 on page
604 and Theorem XIII.1.3 on page 605]. 
Lemma 4.4 and Lemma 4.5 imply
Lemma 4.6 Let E and F be Cspectra and f : E ! F be a (strong) map of Cspe*
*ctra. It
induces a natural transformation
f* : HC*(X; E) ! HC*(X; F):
If f is a weak equivalence, then f* is an isomorphism . The analogous statement*
* holds for
cohomology provided that E and F are Cspectra. 
Any cohomology theory on the category of CW complexes satisfying the dis*
*joint union
axiom can be represented by a spectrum. This is a consequence of Brown's repre*
*sentation
theorem and proven for instance in [38, chapter 9]. The proof goes through wit*
*h some
obvious modifications also in the case of free CCW complexes. This does not c*
*ontradict the
remark in [10, 5.8] since in our setting we allow for free CCW complexes only*
* cells of the
type mor(; c) and the objects of C form a set by assumption whereas in [10] al*
*l homotopy
types of orbits can occur and these homotopy types do not form a set.
Finally, we remark that a filtration of X gives a spectral sequence.
Theorem 4.7 Let X be a contravariant Cspace with a filtration
; = X1 X0 X1 X2 : : :X
such that X = colimn!1 Xn .
1. Let E be a covariant Cspectrum E. Then there is a spectral (homology) s*
*equence
Erp;q; drp;q: Erp;q! Erpr;q+r1whose E1term is given by
E1p;q= HCp+q(Xp; Xp1; E)
and the first differential is the composition
d1p;q: E1p;q= HCp+q(Xp; Xp1; E) ! HCp+q1(Xp1; E)
! HCp+q1(Xp1; Xp2; E) = E1p1;q
where the first map is the boundary operator of the pair (Xp; Xp1) and t*
*he second
induced by the inclusion. The E1 term is given by
E1p;q= colimErp;q:
r!1
This spectral sequence converges to HCp+q(X; E), i.e. there is an ascend*
*ing filtration
Fp;mpHCm(X; E) of HCm(X; E) such that
Fp;qHCp+q(X; E)=Fp1;q+1HCp+q(X; E) ~=E1p;q;
35
2. Let E be a contravariant Cspectrum. Then there is a spectral (cohomolog*
*y) sequence
Ep;qr; dp;qr: Erp;q! Erp+r;qr+1whose E1term is given by
Ep;q1= Hp+qC(Xp; Xp1; E)
and the first differential is the composition
d1p;q: E1p;q= Hp+qC(Xp; Xp1; E)  ! Hp+qC(Xp; E)  ! Hp+q+1C(Xp+1; Xp; E*
*) = E1p+1;q
where the first map is induced by the inclusion and the second is the bou*
*ndary operator
of the pair (Xp+1; Xp). The E1 term is given by
E1p;q= lim Erp;q:
r!1
There is a descending filtration F p;mplimn!1HmC(Xn; E) of limn!1 HmC(Xn*
*; E) such
that there is an exact sequence
0 ! F p;qlimHp+qC(Xn; E)=F p+1;q1limHp+qC(Xn; E) ! Ep;q1
n!1 n!1
! lim1Hp+qC(Xp+m ; Xp; E) ! lim1Hp+qC(Xp+m ; Xp1; E):
m!1 m!1
If one of the following conditions is satisfied
(a)The filtration is finite, i.e. there is n 1 such that X = Xn;
(b)The inclusion of Xp into Xp+1 is pconnected for p 2 Z and there is m*
* 2 Z such
that ssq(E(C)) vanishes for all objects c 2 Ob (C) and q > m;
then the spectral sequence converges to Hp+qC(X; E), i.e. there is a desc*
*ending filtration
F p;mpHmC(X; E)of HmC(X; E) such that
F p;qHp+qC(X; E)=F p+1;q1Hp+qC(X; E) ~=Ep;q1:
Proof______: Again this is a variation of the case where C is the trivial categ*
*ory (see [38, 7.75,15.6
and Remark 3 on page 352]) or [42, Theorem XIII.3.2. on page 614 and Theorem XI*
*II.3.6.
on page 616]. 
Suppose in Theorem 4.7 that X is a free CCW complex and Xn its nskelet*
*on. Then
the E2term respectively E2term of the spectral sequence in Theorem 4.7 can be*
* identified
with
E2p;q= HCp(X; HCq({*}; E)) = HCp(X; ssq(E))
respectively
Ep;q2= HpC(X; HqC({*}; E)) = HpC(X; ssq(E)):
One gets the same spectral sequence as in Theorem 4.7 if one takes a dual*
* point of
view. Namely, one does not filter X by its skeleta, but uses a Postnikov decomp*
*osition of
E. The AtiyahHirzebruch spectral sequence [38, 15.7]) is a special case of Th*
*eorem 4.7.
36
Quinn's spectral sequence [32, Theorem 8.7] coincides with Theorem 4.7 when the*
* stratified
system of fibrations is given by a group action.
Taking X = EC, filtering by skeleta, and identifying the E2 and E1 terms*
*, one gets
the homotopy colimit spectral sequence
Hp(C; ssq(E)) =) ssp+q(hocolim E)
C
and the homotopy limit spectral sequence
Hp(C; ssq(E)) =) sspq(holim E)
C
analogous to those of BousfieldKan [4] [XII,5.7 on page 339 and XI,7.1 on page*
* 309].
Version of June 3, 1996
37
5. Assembly Maps and Isomorphism Conjectures
In this section we give three equivalent definitions of assembly maps, ea*
*ch of which
corresponds to a certain point of view. Then we explain the Isomorphism Conject*
*ures for
the three Or(G)spectra introduced in Section 2. We will define assembly maps *
*given the
following data: a (discrete) group G, a nonempty family of subgroups F, close*
*d under
inclusion and conjugation, and a covariant Or(G)spectrum E.
1. Assembly by Extension from Homogeneous Spaces to GSpaces
Let E be a covariant Or(G)spectrum. We define an extension of E to the c*
*ategory of
Gspaces by
E% : GSPACES  ! SPECTRA X 7! mapG (; X)+ Or(G)E
Recall that a
map G(; X)+ Or(G)E = XH+^ E(G=H)= ~
H2F
where ~ is the equivalence relation generated by (xOE; y) ~ (x; OEy)for x 2 XK+*
*, y 2 E(G=H)
and OE : G=H ! G=K . This construction is functorial in E, i.e. a map of Or(*
*G)spectra
T : E ! F induces a map of GSPACES spectra T% : E% ! F% .
Let E(G; F) be a classifying space of G with respect to a family F (see [*
*5] or [9]),
i.e a GCW complex such that the Hfixed point set is contractible if H 2 F an*
*d empty
otherwise. Such classifying spaces were introduced by tom Dieck [8], [9] and ar*
*e unique up
to Ghomotopy type. We will give another point of view on these spaces in Secti*
*on 7. The
projection induces a map
E% (pr) : E% (E(G; F)) ! E% (G=G) = E(G=G)
which is called assembly map. The map ss*(E% (pr)) is the (E; F; G)assembly ma*
*p referred
to in the introduction. 
2. Assembly as Homotopy Colimit
We first discuss the behavior of homotopy limits under change of category*
*. Consider
a covariant functor F : C ! D . We introduced F*X in Definition 1.8. Since EC *
*is a free
CCW complex, we can apply Theorem 3.4 to the weak homotopy equivalence of Cs*
*paces
F *ED ! {*} , and get a Cmap EC ! F *ED , which is unique up to homotopy. *
*It induces
a map of Dspaces f : F*EC ! ED by Lemma 1.9. Let X be a covariant Dspace. T*
*hen
the assembly map
F* : hocolimF *X ! hocolimX
C D
38
is given by the composition
g fDid
EC C F *X ! F*EC D X ! ED D X
where the map g is the homeomorphism from Lemma 1.9. This assembly map is uniq*
*ue
up to homotopy. There is also an assembly map if the covariant Dspace X is rep*
*laced by
a covariant Dspectrum E. If one uses the functorial models EbarC and EbarD, t*
*here is a
functorial construction of the map EbarC ! F *EbarD and hence of the assembly *
*map.
Let
I : Or(G; F) ! Or (G)
be the inclusion functor. Define the assembly map
I* : hocolimI*E ! hocolimE = E(G=G):
Or(G;F) Or(G)
where the homotopy colimit over the orbit category of E is E(G=G) because the o*
*rbit cate
gory has the terminal object G=G. This assembly map can be identified with the *
*assembly
map defined earlier by taking E Or(G) = {*} and E Or(G; F) = map G(; E(G; F)).*
* The
(E; F; G)assembly map is obtained by applying homotopy groups. 
3. Assembly from the Homological Point of View
Let {*}F be the Or(G)space defined by setting {*}F(G=H) to be a point if*
* H 2 F
and to be empty otherwise. Let inc: {*}F ! {*} be the inclusion map of Or(G)s*
*paces.
It follows from definitions that the (E; F; G)assembly map can be identified w*
*ith the map
HOr(G)i(inc) : HOr(G)i({*}F ; E) ! HOr(G)i({*}; E) = ssi(E(G=G)): *
* 
Definition 5.1 The (E; F; G)Isomorphism Conjecture for a discrete group G, a *
*family of
subgroups F, and a covariant Or(G)spectrum E is that the (E; F; G)assembly ma*
*p is an
isomorphism. For an integer i, the (E; F; G; i)Isomorphism Conjecture is that *
*the (E; F; G)
assembly map is an isomorphism in dimension i.
Of course for an arbitrary (E; F; G), the Isomorphism Conjecture need not*
* be valid.
However, the Isomorphism Conjecture is always true (and therefore pointless!) w*
*hen F is
the family of all subgroups. The main problem is given G and E to find a small *
*family F
for which the Isomorphism Conjecture is true. The proper F to choose for the fu*
*nctors K,
L, and Ktop will be discuss later in this section.
The main point of the validity of the (E; F; G)Isomorphism Conjecture is*
* that it
allows the computation of ss*(E(G=G)) from ss*(E(G=H)) for H 2 F and the struct*
*ure
of the restricted orbit category Or(G; F). Here are two examples which were hi*
*storically
important in algebraic Ktheory.
39
Example 5.2 Let G be an amalgamated free product of H1 and H2 along a subgrou*
*p K.
Let F be the smallest family (closed under subgroups and conjugation) containin*
*g H1 and
H2. The E(G; F) can be taken to be a tree, where the isotropy group of an edge *
*is conjugate
to K and the isotropy group of a vertex is conjugate to H1 or H2. The (E; F; G)*
*Isomorphism
Conjecture and the material in Section 4, give a long exact MayerVietoris exac*
*t sequence
. ..! ssi(E(G=H1)) ssi(E(G=H2)) ! ssi(E(G=K)) ! ssi(E(G=G)) ! . . .

Example 5.3 Let G be a semidirect product given by the action of an infinite *
*cyclic group
on a group K. Let F be the family of all subgroups of K. Then E(G; F) can be ta*
*ken to
be a R, with the isotropy group K at every point. The (E; F; G)Isomorphism Con*
*jecture
and the material in Section 4, give a long exact Wang exact sequence
. ..! ssi(E(G=K)) ! ssi(E(G=G)) ! ssi1(E(G=K)) ! . . .

The following observation both motivates Isomorphism Conjectures and can *
*be helpful
in computation of H*(BG) for a generalized homology theory H and a discrete gro*
*up G.
Lemma 5.4 Let S be a fixed spectrum and G be a discrete group. Define an Or(*
*G)spectrum
E by E(G=H) = (EG xG G=H)+ ^ S. For any family F of subgroups of G, the (E; F; *
*G)
Isomorphism Conjecture is valid.
Proof______: Let r : Or(G) ! SPACES be the covariant functor r(G=H) = G=H: *
*Note that
the Or(G)space r has a left Gaction defined by left multiplication of an elem*
*ent g on G=H.
We have
E% (E(G; F)) = E(G; F)H Or(G)(EG xG G=H)+ ^ S)
= (EG xG (E(G; F)H Or(G)r))+ ^ S
= (EG xG E(G; F))+ ^ S
A! (EG x
G G=G)+ ^ S
= E% (G=G):
The first, second, and fourth equalities are clear. The third equality holds s*
*ince one can
identify any left Gspace X with the left Gspace XH Or(G)r by by Theorem 7.4 (*
*a).
The map A is the assembly map E% (pr). Since {e} 2 F, we see E(G; F) = E(G; F){*
*e}is
40
contractible, and hence EG xG E(G; F) ! EG xG G=G is a homotopy equivalence. *
*The
AtiyahHirzebruch spectral sequence then shows A is a weak homotopy equivalence*
*. 
Given a contravariant functor E : Or(G) ! SPECTRA , there is a dual *
*assembly
map obtained by reversing arrows and replacing Or(G)by hom Or(G), hocolimits by*
* holimits,
and homology by cohomology. The analogue of the last lemma remains valid.
Now we consider the covariant Or(G)spectra of Section 2. When E equals t*
*he algebraic
Ktheory spectra Kalg or the algebraic Ltheory spectra L<1> of Section 2 and *
*F is the
family VC of virtually cyclic subgroups of G, then the Isomorphism Conjecture i*
*s the one
of FarrellJones [14]. An element of VC is a subgroup of G which in turn has *
*a cyclic
subgroup of finite index. Farrell and Jones use Quinn's version of the assembly*
* map which
can be identified with the one presented here by the characterization given in *
*Section 6 and
the fact that the source of Quinn's assembly map is a homology theory on the ca*
*tegory
of GVC CW complexes [32, Proposition 8.4 on page 421]. The Isomorphism Conj*
*ecture
computes the algebraic K resp. L<1>groups of the integral group ring of G in*
* terms of the
corresponding groups for all virtually cyclic subgroups of G. The Isomorphism C*
*onjecture
for Kalghas been proven rationally for discrete cocompact subgroups of virtuall*
*y connected
Lie groups by Farrell and Jones [14]. The (Kalg; VC; G; i)Isomorphism Conjectu*
*re for such
groups with i < 2 also follows from [14]. The Isomorphism Conjecture for Lh*
*as been
proven for crystallographic groups if one inverts 2 by Yamasaki [43]. Notice t*
*hat after
inverting 2 the spectrum Lis independent of j. The Isomorphism Conjecture f*
*or Kalg
and L<1> together imply the Novikov Conjecture and (for dimensions greater tha*
*n 4) the
Borel Conjecture. The Borel Conjecture says that two aspherical closed manifol*
*ds with
isomorphic fundamental groups are homeomorphic and any homotopy equivalence bet*
*ween
them is homotopic to a homeomorphism. A survey on these conjectures is given i*
*n [15].
Related issues are discuss in [40, Chapter 14].
When E equals the topological Ktheory spectrum Ktop defined in Section 2*
* and F
is the family FIN of finite subgroups of G, then the Isomorphism Conjecture is*
* the Baum
Connes Conjecture [3, Conjecture 3.15 on page 254]. The identification is not o*
*bvious. It
follows from the material in Section 6 if one reformulates the BaumConnes Conj*
*ecture in
terms of spectra. Such a reformulation has been constructed very recently by Hi*
*gson, Roe
and Stolz [20]. Namely, they construct a functor
KG : GCW COMPLEXES ! SPECTRA
with the following properties:
1. For a GCW complex X the homotopy groups of KG(X) can be identified with*
* the
equivariant Khomology groups in the sense of Kasparov, provided that X i*
*s proper
and cocompact, and hence with the source of the BaumConnes map, provided*
* that
X is proper;
2. Under this identification the map KG(E(G; FIN )) ! KG(G=G) coming fro*
*m the
projection induces the BaumConnes map on homotopy groups;
41
3. KG is weakly excisive for the family F of all groups in the sense of Sect*
*ion 6. (For our
purposes it suffices to know that it is FIN excisive);
4. There is a weak equivalence of Or(G)spectra from KGOr(G)to Ktop.
For more information on the BaumConnes map we refer to [19].
Example 5.5 Let E be a contravariant Or(G)spectrum and F = 1 the trivial fam*
*ily. The
domain of the (E; 1; G)assembly map is E% (E(G; 1)) = EG+ ^G E(G=1). Now_supp*
*ose_
there is a functor J : GROUPOIDS_ __inj!_ SPECTRA so that E(G=H) = J(G=H )*
*: Then
the morphism of groupoids G=1 ! 1=1 gives a map of spectra E(G=1) ! E(1=1) *
*which is
Gequivariant, where E(G=1) is given the G = autOr(G)(G=1)action and E(1=1) is*
* given the
trivial Gaction. Now suppose J has the additional property that given functors*
* of groupoids
Fi : G0 ! G1 for i = 0; 1 and a natural transformation T : F0 ! F1, then th*
*e maps of
spectra J(F0) and J(F1) are homotopic. (See Lemma_2.4_to_see that these hypothe*
*ses are
valid where E is Kalg, L, or Ktop.) Since G=1 ! 1=1 is a natural equivalen*
*ce of groupoids,
the map E(G=1) ! E(1=1) is a homotopy equivalence, which is in addition a Gm*
*ap. It
follows that
E% (E(G=1)) = EG+ ^G E(G=1) ! BG+ ^ E(1=1)
is a weak homotopy equivalence.
Thus the (E; 1; G)assembly map for the three Or(G)spectra of Section 2 *
*can be iden
tified with the "classical" assembly maps
A : Hi(BG; Kalg(Z)) ! Ki(ZG);
A : Hi(BG; L<1>(Z)) ! L<1>i(ZG);
A : Hi(BG; Ktop(C)) ! Ktop(C):
The last map has an interpretation in terms of taking the index of elliptic ope*
*rators. The
Novikov Conjecture is equivalent to the conjecture that the bottom two maps are*
* rationally
injective.
It is easy to check that there are finite groups G for which none of the *
*three assembly
maps above is an isomorphism. However, it is conjectured that when G is torsion*
*free, that all
three maps are isomorphisms. Indeed, the (Kalg; VC; G), (L<1> ; VC; G), and (K*
*top; FIN ; G)
Isomorphism Conjectures applied to a torsion free group G are equivalent to the*
* conjectures
that the maps labeled A are isomorphisms. This is obvious in the (Ktop; FIN ; G*
*)case, and
is shown by FarrellJones [14, 1.6.1 and Remark A.11] is the other two cases. *
* 
Version of June 3, 1996
42
6. Characterization of Assembly Maps
In this section we characterize assembly maps by a universal property. Th*
*is is useful
for identifying different constructions of assembly maps and generalizes work o*
*f Weiss and
Williams [41] from the case of a trivial group to the case of a general discret*
*e group G.
We associate to a covariant Or(G; F)spectrum E an extension
EF%: GSPACES  ! SPECTRA X 7! mapG (; X)+ Or(G;F)E:
Notice that this construction depends on F. If E is a Or(G)spectrum, we have i*
*ntroduced
E% already in Section 5. There is a natural transformation S : (E Or(G;F))F%*
*! E%of
GSPACES spectra. A GFspace (GFCW complex) is a Gspace (GCW complex) *
*such
that the isotropy group Gx of each point x 2 X is contained in the family F. Th*
*e map S(X)
is an isomorphism if X is a GFCW complex but not in general. For instance fo*
*r X = G=G
and F the trivial family 1 we get (E Or(G;F))F%(G=G) = E(G=1)=Gand E% (G=G) = *
*E(G=G) .
We will omit the superscript F in EF%when it is clear from the context. Notice*
* that this
construction is functorial in E, i.e. a map of Or(G; F)spectra T : E ! F ind*
*uces a map
of G  SPACES spectra T% : E% ! F% . Recall that a map (isomorphism) of sp*
*ectra
f : E ! F is a collection of maps (homeomorphisms) f(n) : E(n) ! F(n)which a*
*re com
patible with the structure maps. An isomorphism of Cspectra is a map of Cspec*
*tra whose
evaluation at each object is an isomorphism of spectra.
Lemma 6.1 Let E be a covariant Or(G; F)spectrum. Then:
1. The canonical map E% (X) [E%(f)E% (Y ) ! E% (X [f Y )is an isomorphism, *
*where
f : A ! Y is a Gmap and A is a closed, Ginvariant subset of X;
2. The canonical map colimn!1 E% (Xn) ! E% (colimn!1 Xn) is an isomorphism,*
* where
X0 ! X1 ! X2 ! : : :is a sequence of Gcofibrations;
3. The canonical map Z+ ^ E% (X) ! E% (Z x X) is an isomorphism, where Z i*
*s a
space and X is a Gspace;
4. The canonical map E% (G=H) ! E(G=H) is an isomorphism for all H 2 F.
Proof______: It can be checked directly that the Hfixed point set functor map*
* G(G=H; )
commutes with attaching a Gspace to a Gspace along a Gmap and with colimits *
*of G
cofibrations indexed by the nonnegative integers. Parts 1. and 2. follow from *
*the fact that
 Or(G;F)E commutes with colimits, since it has an right adjoint by Lemma 1.5*
*. Parts 3.
and 4. follow from the definition of E% . 
Lemma 6.2 Let E is a covariant Or (G; F)spectrum. Then the extension E 7! *
*E% is
uniquely determined on the category of GFCW complexes up to isomorphism of G*
*F
CW COMPLEXES spectra by the properties of Lemma 6.1
43
Proof_____: Let E 7! E be another such extension. There is a (a priori not n*
*ecessarily
________ $
continuous) settheoretic natural transformation
T(X) : E% (X) = X+ Or(G;F)E ! E$(X)
which sends an element represented by (x : G=H ! X; e) 2 map G(G=H; X) x E(G=H*
*) to
E$(x)(e). Since any GFCW complex is constructed from orbits G=H with H 2 F *
*via
products with disks, attaching a Gspace to a Gspace along a Gmap, and colimi*
*ts over the
nonnegative integers, T(X) is continuous and is an isomorphism for all GFCW *
*complexes
X. 
Lemma 6.2 is a characterization of E 7! E% up to isomorphism. Next we g*
*ive a
homotopy theoretic characterization.
A covariant functor E : GFCW COMPLEXES ! SPECTRA is called (w*
*eakly) F
homotopy invariant if it sends Ghomotopy equivalences to (weak) homotopy equiv*
*alences
of spectra. The functor E is (weakly) Fexcisive if it has the following four *
*properties.
First, it is (weakly) Fhomotopy invariant. Second, E(;) is contractible. Third*
*, it respects
homotopy pushouts up to (weak) homotopy equivalence, i.e. if the GFCW comple*
*x X is
the union of GCW subcomplexes X1 and X2 with intersection X0, then the canoni*
*cal map
from the homotopy pushout of E(X2) ! E(X0)  E(X2) , which is obtained by g*
*luing
the mapping cylinders together along E(X0), to E(X) is a (weak) homotopy equiva*
*lence
of spectra. Finally, E respects`countable disjoint unions up to (weak) homotop*
*y, i.e. the
natural map _i2IE(Xi) ! E( i2IXi) is a (weak) homotopy equivalence for all co*
*untable
index sets I. The last condition implies that the natural map from the homotop*
*y colimit
of the system E(Xn) coming from the skeletal filtration of a GFCW complex X,*
* i.e. the
infinite mapping telescope, to E(X) is a (weak) homotopy equivalence of spectra*
*. Notice that
E is weakly Fexcisive if and only if ssq(E(X)) defines a homology theory on th*
*e category of
GFCW complexes, satisfying the disjoint union axiom for countable disjoint u*
*nions.
Theorem 6.3 1. Suppose E : Or (G; F) ! SPECTRA is a covariant functor*
*. Then
E% is Fexcisive;
2. Let T : E ! F be a transformation of (weakly) Fexcisive functors E and*
* F from
GFCW COMPLEXES to SPECTRA so that T(G=H) is a (weak) homotopy eq*
*uiva
lence of spectra for all H 2 F. Then T(X) is a (weak) homotopy equivalenc*
*e of spectra
for all GFCW complexes X;
3. For any (weakly) Fhomotopy invariant functor E from GFCW COMPLEXES *
* to
SPECTRA , there is a (weakly) Fexcisive functor E% from GFCW COMPLE*
*XES
to SPECTRA and there are natural transformations
AE : E%  ! E;
BE : E%  ! (E Or(G;F))% ;
which induce (weak) homotopy equivalences of spectra AE(G=H) for all H 2 *
*F and
(weak) homotopy equivalences of spectra BE(X) for all GFCW complexes X*
*. E is
44
(weakly) Fexcisive if and only if AE(X) is a (weak) homotopy equivalence*
* of spectra
for all GFCW complexes X.
Proof______: 1.) follows from Lemma 6.1.
2.) Use the fact that a (weak) homotopy colimit of homotopy equivalences of spe*
*ctra is again
a (weak) homotopy equivalence of spectra.
3.) Define E% (X) by the spectrum
map G( x :; X)d Or(G;F)x Bbar?#Or (G; F) x #?? Or(G;F)x E( x :)
where  resp. : runs over Or(G) resp. , the subscript d in map G( x :; X)d i*
*ndicates
that we equip this mapping space in contrast to the usual convention with the d*
*iscrete
topology and Bbar? # Or(G; F) x #?? was introduced at the end of Section 3. D*
*efine the
transformation AE(X) : E% (X) ! E(X) by the following diagram
E(X)
x
c1??
map G( x :; X)d Or(G;F)x E( x :)
x
pmapG(x:;X)did??
map G( x :; X)d Or(G;F)x Bbar?#Or (G; F) x #?? Or(G;F)x E( x :)
where pmapG(x:;X)d was introduced in Definition 3.18 and here and in the next *
*diagram
ck refers to the canonical map whose definition is obvious from the context. D*
*efine the
45
transformation BE(X) : E% (X) ! (E Or(G;F))% (X)by the following diagram
map G( x :; X)d Or(G;F)x Bbar?#Or (G; F) x #?? Or(G;F)x E( x :)
?
ididE(pr)?y
mapG ( x :; X)d Or(G;F)x Bbar?#Or (G; F) x #?? Or(G;F)x E()
?
idc2id?y~=
map G ( x :; X)d Or(G;F)x Bbar?#Or (G; F)#?? x Bbar?# #?? Or(G;F)x E()
?
c3?y~=
map G( x :; X)d Bbar?# #?? {*} Or(G;F)Bbar?#Or (G; F)#?? Or(G;F)E()
?
(idc4)idid?y~=
map G( x :; X)d Bbar?# Or(G;F)Bbar?#Or (G; F)#?? Or(G;F)E()
?
(c5id)idid?y~=
map (:; mapG (; X))d Bbar?# Or(G;F)Bbar?#Or (G; F)#?? Or(G;F)E()
?
(idq)idid ?y
(map(:; mapG (; X))d :) Or(G;F)Bbar?#Or (G; F)#?? Or(G;F)E()
?
amapG(;X)id?y
mapG (; X) Or(G;F)E()
where the canonical map q : Bbar?#  ! ? is defined in [4, Example XI.2.6 on *
*page 293]
and amapG(;X)was introduced in Definition 3.18.
Next we show that BE(X) is a (weak) homotopy equivalence provided that X *
*is a GF
CW complex. Since E is (weakly) Fexcisive, the map E(pr) : E(G=H x n) ! E(G=*
*H)
is a (weak) homotopy equivalence for all H 2 F. Hence the first map in the diag*
*ram above
id idE(pr) is a weak homotopy equivalence because of Theorem 3.11. The next*
* four
maps are all isomorphisms. The map
idq : map(:; mapG (; X))d Bbar?#  ! map (:; mapG (; X))d :
is a weak homotopy equivalence of Or (G; F)spaces [4, XII.3.4 on page 331]. B*
*ecause of
Theorem 3.11 the map
bar bar
(idq) id: map (:; mapG (; X))d B ?# Or(G;F)B ?#Or (G; F)#??
 ! (map(:; mapG (; X))d ) Or(G;F)Bbar?#Or (G; F)#??
is a weak Or(G; F)homotopy equivalence of Or(G; F)spaces. Since the domain an*
*d target
are free Or(G; F)CW complexes by Lemma 3.19, it is a homotopy equivalence of *
*Or(G)
spaces by Corollary 3.5. Hence the map (idq) id id in the diagram above is a*
* homotopy
equivalence.
46
As we assume that X is a GFCW complex map G(; X) is a Or(G; F)CW co*
*mplex.
Since amapG(;X)is a Or (G; F)CW approximation by Lemma 3.19 Corollary 3.5 im*
*plies
that it is a homotopy equivalence of Or(G; F)CW complexes. Hence the last map*
* in the
diagram above amapG(;X) id is a homotopy equivalence. This shows that BE(X) i*
*s a
(weak) homotopy equivalence.
In the case X = G=H for H 2 F the composition of the (weak) homotopy equi*
*va
lence BE(G=H) with the canonical isomorphism map G(; G=H) Or(G;F)E() ! E(G=H)
agrees with AE(G=H). Hence AE(G=H) is a (weak) homotopy equivalence for all G=H*
* with
H 2 F . This finishes the proof of Theorem 6.3. 
The map AE is called an assembly map for E.
Example 6.4 For a topological space X, the fundamental groupoid (X) is the ca*
*tegory
whose objects are points in X and whose morphism set mor(X) (x; y) is given by *
*equivalence
classes of paths from x to y, where the equivalence relation is homotopy rel {0*
*; 1}. A map
of spaces gives a map of fundamental groupoids. A homotopy equivalence of space*
*s gives a
natural equivalence of fundamental groupoids. If X is pathconnected and x0 2 X*
*, then the
inclusion of the fundamental group ss1(X; x0) ! (X) is a natural equivalence *
*of groupoids.
Let Kalg : GROUPOIDS ! SPECTRA be the functor from Section 2. By*
* Lemma
2.4, Kalghas the property that a natural equivalence of groupoids gives a homot*
*opy equiv
alence of spectra.
One can define a homotopy invariant functor E : CW COMPLEXES  ! SP*
*ECTRA
by E(X) = Kalg((X)). We apply Theorem 6.3 in the case where G is the trivial gr*
*oup (note
that for G = 1, Theorem 6.3 is due to WeissWilliams [41]). The map BE gives a *
*homotopy
equivalence from E% (X) to X+ ^ Kalg(Z); where Kalg(Z) is the algebraic Kspect*
*rum of the
ring Z. After one applies the nth homotopy group to the assembly map
AE : E% (X) ! E(X)
one obtains the algebraic Ktheory assembly map
A : Hn(X; Kalg(Z)) ! Kalgn(Zss1X):
Next consider a discrete group G and a family of subgroups F. One can the*
*n define
an Fhomotopy invariant functor
E : GCW COMPLEXES ! SPECTRA
by setting E(X) = Kalg((EG xG X)). If X is simplyconnected, there is a natural*
* equiva
lence of groupoids
G = Or(G; 1) ! (EG xG X):
Using this identification, we have a fourth point of view on the (Kalg; F; G)a*
*ssembly map,
namely it is
ss*(AE(E(G; F))) : ss*(E% (E(G; F))) ! ss*(E(E(G; F))):
47
The case of algebraic Ltheory is analogous. For a map of spaces X ! Y *
*, the map
of groupoids (X) ! (Y ) need not be a morphism in GROUPOIDS inj. However,*
* all
relevant maps in the definition of AE and BE have this property, so that the an*
*alogous
statement holds also for the topological Ktheory of C*algebras. 
Next we explain why Theorem 6.3 characterizes the assembly map in the sen*
*se that
AE : E% ! E is the universal approximation from the left by a (weakly) Fexc*
*isive functor
of a (weakly) Fhomotopy invariant functor E from GFCW COMPLEXES to SPEC*
*TRA .
The argument is the same as in [41, page 336]. Namely, let T : F ! E be a tran*
*sformation
of functors from GFCW COMPLEXES to SPECTRA such that F is (weakly) Fe*
*xcisive
and T(G=H) is a (weak) homotopy equivalence for all H 2 F. Then for any GFCW*
* 
complex X the following diagram commutes
AF(X)
F% (X) ! F(X)
? ?
T%(X)?y ?yT(X)
AE(X)
E% (X) ! E(X)
and AF(X) and T% (X) are (weak) homotopy equivalences. Hence one may say that T*
*(X)
factorizes over AE(X).
One may be tempted to define a natural transformation S : E% ! E as ind*
*icated in
the proof of Lemma 6.2. Then S(X) is a welldefined bijection of sets but is no*
*t necessarily
continuous because we do not want to assume that E is continuous, i.e. that the*
* induced
map from hom C(X; Y )to hom C(E(X); E(Y ))is continuous for all GFCW complex*
*es X
and Y . The construction above uses the (weak) Fhomotopy invariance of E inste*
*ad.
Finally we give for a covariant Or (G)spectrum E an equivalent definitio*
*n of E%
which is closer to the construction in [41]. Let simpG (X) be the category hav*
*ing as mor
phisms pairs (G=H x [n]; oe)which consists of an object G=H x [n] in Or(G; F) x*
* and
a Gmap oe : G=H x n ! X . A morphism from (G=H x [n]; oe)to (G=K x [m]; o) *
*is a
morphism f x u : G=H x [n] ! G=K x [m] in Or(G; F) x such that the induced *
*map
G=H x n ! G=K x m composed with o is oe. This is the equivariant versi*
*on of the
construction in [35, Appendix A] applied to the simplicial set S:X associated t*
*o a space
X. Obviously we obtain a covariant functor E( x :) from simpG (X) to SPECTRA*
* by
(G=K x [m]; oe) 7! E(G=K x m ) We briefly indicate how one can identify
E% (X) = hocolimE( x :):
simpG(X)
Let P : simpG(X) ! Or(G) x be the obvious forgetful functor. It suffices t*
*o construct
a natural isomorphism of Or(G) x spaces
Bbar?#simp G(X) simpG(X)morOr(G)x (??; P (?)) !
48
mapG ( x :; X) Or(G)x Bbar??#Or (G) x #  x : :
It will be implemented by the following natural bijection of simplicial sets fo*
*r a given object
G=K x [m] in Or(G) x where p runs over 0; 1; 2; : : :
Np?#simp G(X) simpG(X)morOr(G)x (G=K x [m]; P (?)) !
map G( x :; X) Or(G)x NpG=K x [m]#Or (G) x #  x : :
An element in the source is represented for ? = (G=H x [n]; oe)by the pair
((G=H x [n]; oe) ! (G=H0 x [n0]; oe0) ! . ..! (G=Hp x [np]; oep))
x (G=K x [m] ! G=H x [n]) :
It is sent to the element in the target represented by
oep : G=Hp x np ! X
x (G=K x [m] ! G=H x [n] ! G=H0 x [n0] ! . ..! G=Hp x [np]) :
This is indeed a bijection since G=H0 x [n0] ! . ..! G=Hp x [np] and oep dete*
*rmine oe0,
: : :, oep1.
Version of June 3, 1996
49
7. GSpaces and Or(G)spaces
In this section we discuss the orbit category in more detail, and give a *
*correspondence
between Gspaces with isotropy in F and Or(G; F)spaces. This in turn will give*
* a corre
spondence between classifying spaces of G with respect to F and models of E Or(*
*G; F) and
will thereby give a source of natural examples. As usual, let G be a discrete g*
*roup and F
a nonempty family of subgroups closed under conjugation and inclusion. A Gspa*
*ce X is a
GFspace if the isotropy subgroup of each point in X is contained in F. Let Or*
*(G; F) be
the restricted orbit category whose objects are G=H for H 2 F and whose morphis*
*ms are
Gmaps.
Next we explain how one gets from GFspaces to Or(G; F)spaces and vice *
*versa. We
will get a correspondence up to homeomorphism, not only up to homotopy (cf. [10*
*, Theorem
3.11], [12], [30]).
Definition 7.1 Given a left Gspace Y , define the associated contravariant Or*
*(G; F)space
map G(; Y ) by
Or(G; F) ! SPACES G=H 7! map G(G=H; Y ) = Y H:
Let r be the covariant Or(G; F)space given by sending G=H to itself. Given a c*
*ontravariant
Or (G; F)space X define the associated left GFspace bX by
Xb = X Or(G;F)r:
The left action of an element g 2 G is given by idOr(G;F)Lg where Lg : G=H ! G*
*=H is
the map of covariant Or(G; F)spaces given by left multiplication with g. *
* 
The notation for the functor r is intended to be reminiscent of the cosim*
*plicial space
: from Example 1.7.
Lemma 7.2 The functors in Definition 7.1 are adjoint, i.e. for a contravaria*
*nt Or(G; F)
space X and a left Gspace Y there is a natural homeomorphism
T (X; Y ) : map G(Xb; Y ) ! homOr(G;F)(X; mapG (; Y )):
Proof______: If we neglect the Gaction on Y , we get from Lemma 1.6 a natural *
*homeomorphism
map (Xb; Y ) ! hom Or(G;F)(X; map(; Y )):
Using the transformations Lg and the Gaction on Y one defines appropriate Gac*
*tions on
the source and target of this map and checks that this map is Gequivariant. He*
*nce it induces
a homeomorphism on the Gfixed point set which is just T (X; Y ). Of course one*
* can define
for instance T (X; Y )1 explicitly. Given f : X ! map G(; Y )we define T (X;*
* Y )1(f) by
specifying for each G=H a map X(G=H) x G=H ! Y . It sends (x; gH) to the v*
*alue of
f(G=H)(x) at gH. 
50
Lemma 7.3 The map
f : X(G=1) ! bX x 7! [x; 1]
is a Ghomeomorphism.
Proof______: The inverse f1 : bX ! X(G=1) assigns to an element represented b*
*y (x; gH) the
element X(qgH)(x) where qgH : G=1 ! G=H sends g0to g0gH. 
Let X be a contravariant Or(G; F)space. Obviously the projection pr: G=1*
* ! G=H
induces a map X(pr) : X(G=H) ! X(G=1)H . Now one easily checks using Lemma 7.3
above.
Theorem 7.4 1. Given a left GFspace Y , the adjoint of the identity on m*
*ap G(; Y )
under the adjunction of Lemma 7.2 is a natural Ghomeomorphism
T (Y ) : map"G(; Y )! Y:
It is induced by the map
a
map (G=H; Y )G x G=H ! Y; (OE; gH) 7! OE(gH);
H2F
2. Given a contravariant Or(G; F)space X, the adjoint of the identity on Xb*
* under the
adjunction of Lemma 7.2 is a natural map of Or(G; F)spaces
S(X) : X ! mapG (; bX):
Given H 2 F, the map S(X)(G=H) maps the element x 2 X(G=H) to the eleme*
*nt
H
in map G(G=H; bX) = X Or(G;F)r represented by (x; eH) 2 X(G=H) x G=H*
* . It
is an isomorphism of Or (G; F)spaces if and only if for each H 2 F the p*
*rojection
pr: G=1 ! G=H induces a homeomorphism X(pr) : X(G=H) ! X(G=1)H . T*
*his
condition is satisfied if X is a free Or(G; F)CW complex,
3. If Y is left GFCW complex, then map G(; Y ) is a free Or (G; F)CW *
*complex.
There is a bijective correspondence between the Gcells in Y of type G=H*
* and the
Or(G; F)cells in Or(Y; F) based at the object G=H. The analogous statem*
*ent holds
for a free Or(G; F)CW complex X and bX.
The bar resolution is a natural construction, however, it is a "very big"*
* model. Models
with a fewer number of cells can be very convenient for concrete calculations a*
*nd arise often
as follows.
Definition 7.5 Let G be a group and F be a family of subgroups. A classifying*
* space
E(G; F) of G with respect to F is a left GCW complex such that E(G; F)H is co*
*ntractible
for H 2 F and empty otherwise. 
51
The existence of E(G; F) and proofs that for any GFCW complex X there *
*is precisely
one Gmap up to Ghomotopy from X to E(G; F) and thus that two such classifying*
* spaces
are Ghomotopy equivalent, is given in [8],[9, I.6]. Another construction and *
*proof of the
results above come from Theorem 3.4 and the following result which is a direct *
*consequence
of Theorem 7.4.
Lemma 7.6 Let G be a group and F be a family of subgroups.
1. If E(G; F) is a classifying space of G with respect to F, then the associ*
*ated contravari
ant Or(G; F)space
map G(; E(G; F))
is a model for E Or(G; F);
2. Given a model E Or(G; F), then the Gspace E O"r(G; F)is a classifying sp*
*ace of G
with respect to F. 
Example 7.7 Sometimes geometry yields small examples of classifying spaces an*
*d resolu
tions. We have already mentioned this in the case where G is a crystallographi*
*c group.
Generalizing this, let G be a discrete subgroup of a Lie group L with a finite *
*number of com
ponents. If K is a maximal compact subgroup of L, then L=K is homeomorphic to R*
*n and
L=K can be taken as a model for E(G; FIN ), where FIN is the family of finite *
*subgroups.
Generalizing further, let G be a group of finite virtual cohomological dimensio*
*n. Then there
is finitedimensional classifying space E(G; FIN ) (see [36, Proposition 12]) a*
*nd hence a finite
dimensional model for E Or(G; FIN ). Many examples of such groups are discussed*
* by Serre
in [36]. More examples of nice geometric models for E(G; FIN ) can be found in *
*[3, section
2]. 
Version of June 3, 1996
52
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Addresses
James F. Davis Wolfgang L*
*"uck
Department of Mathematics Fachbereich Mathema*
*tik
Indiana University Johannes GutenbergUniver*
*sit"at
Bloomington, IN 47405 55099 Ma*
*inz
U.S.A. Bundesrepublik Deutschl*
*and
email: jfdavis@indiana.edu lueck@topologie.mathematik.unima*
*inz.de
FAX: 8128550046 06131 393*
*867
Version of June 3, 1996
55