"Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K -and L-Theory" 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 L-theory of integral group rings and to the Baum-Connes Co* *njecture on the topological K-theory 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 L-theory, Baum-Connes Conjecture, assembly maps, spa* *ces and spectra over a category AMS-classification 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 G-sets, and morphisms are G-maps. Thi* *s is a useful construct for organizing the study of fixed sets and quotients of G-acti* *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 K-theory Ki(ZH), algebraic L-theory 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 i-th homotopy group* *. We use these functors to give a clean formulation of some of the main conjectures * *of high- ___________________________________ 1Supported by the Alexander von Humboldt-Stiftung and the National Science F* *oundation. James F. Davis wishes to thank the Johannes Gutenberg-Universit"at in Mainz for its hosp* *itality when this work was initiated. 1 dimensional topology: the Isomorphism Conjecture of Farrell-Jones [14] (which i* *mplies the Borel/Novikov Conjecture) and the Baum-Connes 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 L-groups 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 K0-group "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 K-theory. Section 2 is devoted to K-theory, 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 Ktop-ca* *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 C-space X. Th* *e ideas here are well-known 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 C-space X by a free C-CW -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 C-spectrum E.* * We give an Atiyah-Hirzebruch 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% : G-SPACES -! 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 G-CW -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 Farrell-Jones [14] for algebraic K- and L-theory 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 L-theories. When F = FIN , the family of finite subgroups of G* *, and Ktop is the Or(G)-spectra associated with the K-theory of C*-algebras, then the (Kto* *p; FIN ; G)- Isomorphism Conjecture is equivalent to the Baum-Connes 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 well-suited 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 G-homotopy 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* * Weiss-Williams [41] to the equivariant setting. Associated to a homotopy invariant functor E : G-SPACES -! SPECTRA ; we define a new functor E% : G-SPACES -! 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 L-theory and for the topological K-theory of C*-algebras. This g* *ives a fourth point of view on assembly maps. 3 In Section 7 we make explicit the correspondence between G-spaces 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 L-theory 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 L-Theory 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. G-Spaces 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 non-em* *pty set of subgroups of G closed under taking conjugates and subgroups. The orbit categ* *ory Or(G) has as objects homogeneous G-spaces G=H and as morphisms G-maps. 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 G-space 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) C-space 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 C-spaces * *is a natural transformation of such functors. Given C-spaces X and Y , denote by hom C(X; Y * *)the space of mapsQof C-spaces 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 C-set and an RC-module. For a ring R a RC-modul* *e is a functor M from C to the category of R-modules. For two RC-modules 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 RC-modules is an abelian cat* *egory, and thus one can use homological algebra to study RC-modules (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) G-space. Maps of Or(G; 1)-spaces correspond to G-maps. 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 Zp-space 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 G-space 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 C-space* *s a space. They are called the coend and end constructions in category theory [24, pages 2* *19 and 222]. A lot of well-known constructions are special cases of it. Definition 1.4 Let X be a contravariant and Y be a covariant C-space. 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) G-spaces. The balanced product X xG Y of a right G-space X and * *of a left G-space Y can be identified with the tensor product X Or(G;1)Y . The mapping * *space map G(X; Y ) of two left (right) G-spaces 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 C-space, Y be a covariant C-space and Z b* *e a space. Denote by map (Y; Z) the obvious contravariant C-space 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 C-space Y the fun* *ctor - C Y from the category of contravariant C-spaces to the category of spaces and the f* *unctor map (Y; -) from the category of spaces to the category of contravariant C-spac* *es are adjoint. Similarly if N is a covariant RC-module, then there is adjoint to hom RC(N; -),* * namely the tensor product of RC-modules - 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 R-module X, a left R-module Y and an abelian group Z, the tensor product X R Y , the R-m* *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) C-sp* *aces. Let X xY be the obvious covariant (contravariant) C-space. 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 non-n* *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 p-simplex 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 non-degenerate p-simplex corresponds to a p-cell. 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 C-space resp. a C-spectrum resp. C--spectrum is a functor fro* *m C to SPACES + resp. SPECTRA resp. -SPECTRA . We have introduced tensor pro* *duct of C-spaces in Definitions 1.4 and mapping spaces of C-spaces 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 C-space X determines a* * pointed C-space X+ = X {*} by adjoining a base point. Here {*} denotes a C-space whic* *h assigns to any object a single point. It is called the trivial C-space. A C-spectrum 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 C-spaces. With this interpreta* *tion it is obvious what the tensor product spectrum X C E of a contravariant pointed C-sp* *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 C-spa* *ce X and a C-spectrum 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 C-spaces 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 C-D-space is a covariant C x Dop-space 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 R-S-bimodule for two ri* *ngs R and S. Let F : C -! D be a covariant functor. We get a D-C-space 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 C-D-space morD (??; F (?)). 8 Definition 1.8 Given a covariant (contravariant) C-space X, define the inducti* *on of X with F to be the covariant (contravariant) D-space 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) D-space F!X = homC(mor D(??; F (?)); X) respectively F!X = homC(mor D(F (?); ??); X): Given a covariant (contravariant) D-space Y , define the restriction of Y with * *F to be the covariant (contravariant) C-space F *Y = Y O F . | There are corresponding definitions for C-sets and RC-modules (see [9, p.* * 80], [23, p. 166] for induction of modules). For example, if M is a covariant RC-module* *, 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 C-space X and D-space Y of the required variance. Proof______: Notice for a covariant D-space Y that there are natural homeomor* *phisms of co- variant C-spaces 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 L-Theory 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 K-theory, by Quinn-Ranicki [33] for algebraic L-t* *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 K-groups of Quillen-Bass, the surger* *y obstruc- tion L-groups of Wall, and the topological K-groups 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 g-1Hg K. The induced* * ho- momorphism cg : H -! K; h 7! g-1hggives 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 G-set 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 G-se* *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 L-theory spectra * *of the integral group ring of a group and the topological K-theory 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 R-category 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* * R-module such that composition induces a R-bilinear 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 R-module. Suppose that R comes with an involution of rings R -! R r 7! _r. A R-cat* *egory with involution is a R-category 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 R-category (with involution) an additive R-category (with invol* *ution) if it possesses a sum and the obvious compatibility conditions with the R-module str* *uctures (and the involution) on the morphisms are fulfilled. The notion of a C*-category was defined by Ghez-Lima-Roberts [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 C-category 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 R-category on C gives mor C(x; x) the structure of a central R-algebra 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 R-category 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* * R-module 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 R-categories 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 R-category with involution by defining r ! * X Xr __ ifi := f-1i: 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 non-empty, for inst* *ance one could choose z = x. Define a C-linear 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 C-bilinear 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 inj-condit* *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 C-vector 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*rG-alg. 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*rG-alg) 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 L-theory, 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 Ktop-functo* *r from C*-CATEGORIES to SPECTRA . Note that after applying homotopy groups, one g* *ets maps on the K-theory 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 R-category. We define a new R-category C , called the symmetri* *c monoidal R-category associated to C with an associative and commutative sum as follows.* * The objects in C are n-tuples x_= (x1; x2; : :;:xn)consisting of objects xi2 Ob (C* *)for n = 0, 1, 2, : :.:We will think of the empty set as 0-tuple which we denote by 0. The R-m* *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 R-module. 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 R-category on C . No* *tice that the functor C 7! C is the left adjoint of the forgetful functor from symmetric* * monoidal R-categories to R-categories. 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 R-category or of a a symmetric mo* *noidal R-category, 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 R-categ* *ory, then so is Iso(C). Let C be a symmetric monoidal R-category, all of whose morphisms are isom* *orphisms. Then its group completion is the following symmetric monoidal R-category 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 non-connective algebraic K-theory spectr* *um of a groupoid with coefficients in R Kalg: GROUPOIDS -! -SPECTRA assigns to a groupoid G the non-connective K-theory 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 L-theory 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* * L-theory 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 non-connective topological K-theo* *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 n-fold 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 C-category. 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 n-fold 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 i-th copy of x_in m_ x_to the j-th copy of y_in n_ y_is fi;j. g, where* * fi;j2 C is the component of f from the i-th coordinate of m_ to the j-th 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 220-221], 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 non-connective topological K-theory 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 RG-modules 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 RG0-modules 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 groupoid-valued functor GR_and_the spectra-valu* *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 C-spaces. 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 C-space, 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 : C-SPACES - ! Ob(C)-SPACES Define a functor B : Ob (C)-SPACES - ! C-SPACES ` 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 C-spaces 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* *C-MOD to Ob (C)-SETS . It is defined as B(X(-)) = c2Ob(C)R(mor C(-; c) x X(c)). A free R* *C-module 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 R-modules to sets and the functor as* *signing to a set S the free R-module RS generated by S. We have already mentioned that the category of Or (G; 1)-spacesis the cat* *egory of G-spaces and the category Ob (Or (G; 1)-spaces)is the category of spaces. Under* * this iden- tification the forgetful functor F just forgets the G-action and B sends a spac* *e Z to the G-space 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 C-spaces. 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 C-spaces is a cofibration (fibration) of C-spaces if* * it has the homotopy extension property (homotopy lifting property) for all C-spaces. If f* * is a (co)- fibration of C-spaces, 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 C-spaces. We wi* *ll see that the notion of a free C-CW -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 C-CW -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 C-CW -complex X is a contravariant C-space* * X to- gether with a filtration [ ; = X-1 X0 X1 X2: : :Xn : : :X = Xn n0 such that X = colimn!1 Xn and for any n 0 the n-skeleton Xn, is obtained from * *the (n - 1)-skeleton Xn-1 by attaching free C-n-cells, i.e. there exists a pushout * *of C-spaces of the form ` n-1 i2InmorC(-;?ci) x S ---! Xn-1? ? ? 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 C-n-cell ba* *sed at ci. A free C-CW -complex has dimension n if X = Xn. The definition of a covariant* * free C-CW -complex is analogous. | A C-CW -complex was defined by Dror Farjoun [10, 1.16 and 2.1] (see also * *[30]). We shall deal almost exclusively with free C-CW -complexes. For a free C-CW -compl* *ex X, the cellular chain complex C*(X)(-), c 7! C*(X)(c) is a C-chain complex of free ZC-* *modules. Notice that a free C-CW -complex X defines a functor from C to CW -COMPLEXES * * , but not any functor from C to CW -COMPLEXES is a free C-CW -complex. 21 If Y is a G-CW -complex, then map G(-; Y ) (which sends G=H 7! Y H) is an* * example of a free Or(G)-CW -complex. A G-cell 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 G-spaces. Under this identification a free Or(G; 1)-CW -complex* * is the same as a free G-CW -complex. Given a C-space X and a space Y , we obtain the C-space 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 C-spaces means. Recall that a map f : X -! Y of spaces is n-connected 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 n-connected for all n 0. Definition 3.3 A map f : X -! Y of C-spaces is n-connected (a weak homotopy e* *quiv- alence) if for all objects c the map of spaces f(c) : X(c) -! Y (c)is n-connect* *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 well-known 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 C-spaces and X be a C-space. The ma* *p on homotopy classes of maps between C-spaces induced by composition with f is deno* *ted by f* : [X; Y ]C -! [X; Z]C. 1. Then f is n-connected if and only if f* is bijective for any free C-CW -c* *omplex X with dim(X) < n and surjective for any free C-CW -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 C-space, 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 C-CW -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 C-CW -complex X that any map of C-spaces 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 C-cells. Therefore it suffices to show that any map * *of C-spaces mor C(-; c) x Sn-1 -! Y can be extended to a map mor C(-; c) x Dn -! Y. Such a * *prob- lem reduces to extending a map from Sn-1 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 C-CW -complexes is a ho* *motopy equivalence. Proof______: Let f : Y - ! X be a weak homotopy equivalence between free C-CW -* *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 C-spaces. A C-CW -approximation (u; v) : (X0; A0) -! (X; A) consists of a free C-CW -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 C-spaces. A C-CW -approxima* *tion of a space X is a C-CW -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 C-CW -approximations exist and are unique up to homotopy. Theorem 3.7 Let (X; A) be a pair of C-spaces. 1. (existence) There exists a C-CW -approximation of (X; A); 2. (uniqueness) Given a map of pairs (f; g) : (X; A) -! (Y; B) of C-spaces * *and given C-CW -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 C-CW -approximation is an inductive construction do* *ne by attaching n-cells to obtain a n-connected 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 C-CW -complex so that EC(c) is contracti* *ble for all objects c. | Since EC is a C-CW -approximation of the trivial C-space, 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 C-CW -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 C-space, which is a single C-0-cell based at the final object. Similarly, if C has an in* *itial object, the trivial C-space 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 non-negative 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 non-negative in* *teger i, there is C-0-cell and a C-1-cell based at i. We may take the covariant E! to be the t* *rivial C-space. ||| Definition 3.10 The classifying space of a category C is the space BC = EC C {* **}, where {*} is the trivial C-space and EC is a contravariant C-CW -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 C-s* *paces. Then for any contravariant free C-CW -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 C-spaces. Let X be a covariant (contravariant) free C-CW -complex and f : Y - ! Z b* *e a weak homotopy equivalence of covariant (contravariant) C-spaces. 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 Xn-1 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 Blakers-Massey [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 ZC-module, X a covariant C-space, and E a* * covariant C-spectrum. 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 ZC-module,* * with Z(c) Z and {*} is the trivial C-space. 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 ZC-module, define Hi(C; M) = Hi(C*(EC) ZC M) and Hi(C; M) = Hi(Hom ZC(C*(EC); M)): If X is a covariant C-space, 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 C-spectrum, 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 ZC-chain complex C*(EC) can be replaced by any projective ZC-resolution 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 C-CW -* *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 ZC-modules 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 C-spaces 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 fli-1restricted to [i; 1). | Definition 3.15 Let X be a C-space and M a ZC-module. Let X0 -! X be a C-CW -* *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 G-C* *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 C-maps between a free C-* *CW -space and a C-space, 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 C-CW -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 OEp-1 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 OEp-1 c0 - ! c1 - ! c2 - ! : : :-! cp where no morphism is the identity and the set of p-cells. 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 Cop-spaces __________ Bbarpr: Bbar?#C #?? -! Bbarmor C(?; ??)= morC(?; ??) is a C x Cop-CW -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 Cop-CW -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 Cop-spaces (n:d:Np?#C #?? ) x Sp-1---! Bbarp-1?#C #?? ? ? ? ? y y (n:d:Np?#C #?? ) x Dp ---! Bbarp?#C #?? where n:d:Np?#C #?? is the set of non-degenerate p-simplices of the nerve of ?#* *C #?? . This set can be identified with the disjoint union of the C-C-sets morC(?; c0) x mor* *C(cp; ??)where the disjoint union runs over the sequences OE0 OE1 OE2 OEp-1 c0 - ! c1 - ! c2 - ! : : :-! cp where no morphism OEi is the identity. Such sequences thus give the indexing s* *et for the p-cells. | From Example 1.7 we get that for any C-space X, there is a weak homotopy * *equivalence of C-spaces 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 C-CW -complex. Definition 3.18 Let X be a contravariant C-space. The tensor product taking o* *ver the variable ?? yields contravariant C-spaces X C Bbar?#C #?? and X C morC(?; ??)* *. Define a map of contravariant C-spaces 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 C-spaces tCid bar pX aX : |S:X| C Bbar?#C #?? ---! X C B ?#C #?? -! X: | 29 Lemma 3.19 Let X be a contravariant C-space. Then: 1. pX is a weak homotopy equivalence of contravariant C-spaces, 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 Cop-CW -complex. Then the contravariant D-space X C Y inherits the * *structure of a free D-CW -complex; 3. The map aX : |S:X| C Bbar?#C #?? -! X is a C-CW -approximation. Proof______: 1.) Fix an object c in C. Then B pr(c; ??) : Bbarc#C #?? -! morC(c; ??) is a weak homotopy equivalence of free C-CW -complexes, hence is a C-homotopy e* *quivalence. Thus pX (c) is a homotopy equivalence. 2.) We will only indicate what the skeleta and cells are. The p-skeleton of X* * C Y is [i+j=pXiCYj. A free D x Cop-j-cell of Y based at (d; c) together with a i-cell * *of X(c) gives rise to a free D-i + j-cell based at d. More precisely, if : Di -! X(c) is a c* *haracteristic map for a i-cell of X(c) and if : morD (?; d) x morC(c; ??) x Dj -! Y is a cha* *racteristic map for a free D x Cop-j-cell 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 C-CW -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 Atiyah-Hirzebruch type spe* *ctral se- quence. Definition 4.1 Let (X; A) be a pair of pointed C-spaces. Denote the reduced c* *one of the pointed space A by cone(A). For a C-spectrum E of the opposite variance as (X; * *A) define ECp(X; A) = ssp(X [A cone(A) C E): Given a C-spectrum E of the same variance as (X; A), define EpC(X; A) = ss-p(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 C-spaces. 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 C-spaces. 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 C-space 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 C-spa* *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 C-spaces is a C-map X -! Y s* *o that X(c) -! Y (c) is a weak homotopy equivalence for all objects c 2 Ob (C). The W* *HE-axiom 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 G-action. * *Let E be the ordinary Eilenberg-MacLane spectrum with ss0(E) = Z, considered as a covariant * *Or(G; 1)- spectrum by the trivial G-action. 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 WHE-axiom. To be mor* *e precise, a homology theory means that homotopic maps of pairs of C-spaces 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 C-spaces 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 C-spaces. If the homology theory satisfies th* *e WHE-axiom, 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 C-CW -approximations (Definition 3.6) in order to generalize the usual procedur* *e for spaces to C-spaces (cf [38, 7.68]). Definition 4.3 Let (X; A) be a pair of C-spaces. Let (u; v) : (X0; A0) -! (X* *; A) be a C-CW -approximation. For a C-spectrum 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 C-spectrum 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 well-defined by the existence and u* *niqueness of C-CW -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 C-CW -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 C-pairs, 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 C-CW -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 C-spaces which satisfy the WHE-axiom. The homology theory satisfies* * the disjoint union axiom. The cohomology theory satisfies the disjoint union axiom provided * *that E is a C--spectrum. 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 C-CW -complexes. We conclude from Theorem 3.11* * for any free C-CW -complex Y that hom C(Y; E) is a -spectrum since E is a C--spectrum 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 C--spectrum the associated * *cohomology theory does not have to satisfy the disjoint union axiom. Lemma 4.5 Let X be a C-space with a filtration ; = X-1 X0 X1 X2 : : :X such that X = colimn!1 Xn . Let E be a C-spectrum 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 C--spectrum. There is a natural exact sequence {0} ! lim1Hp-1C(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 C-spectra and f : E -! F be a (strong) map of C-spe* *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 C--spectra. | 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 C-CW -complexes. This does not c* *ontradict the remark in [10, 5.8] since in our setting we allow for free C-CW -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 C-space with a filtration ; = X-1 X0 X1 X2 : : :X such that X = colimn!1 Xn . 1. Let E be a covariant C-spectrum E. Then there is a spectral (homology) s* *equence Erp;q; drp;q: Erp;q-! Erp-r;q+r-1whose E1-term is given by E1p;q= HCp+q(Xp; Xp-1; E) and the first differential is the composition d1p;q: E1p;q= HCp+q(Xp; Xp-1; E) -! HCp+q-1(Xp-1; E) -! HCp+q-1(Xp-1; Xp-2; E) = E1p-1;q where the first map is the boundary operator of the pair (Xp; Xp-1) 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;m-pHCm(X; E) of HCm(X; E) such that Fp;qHCp+q(X; E)=Fp-1;q+1HCp+q(X; E) ~=E1p;q; 35 2. Let E be a contravariant C--spectrum. Then there is a spectral (cohomolog* *y) sequence Ep;qr; dp;qr: Erp;q-! Erp+r;q-r+1whose E1-term is given by Ep;q1= Hp+qC(Xp; Xp-1; E) and the first differential is the composition d1p;q: E1p;q= Hp+qC(Xp; Xp-1; 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;m-plimn!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;q-1limHp+qC(Xn; E) -! Ep;q1 n!1 n!1 -! lim1Hp+qC(Xp+m ; Xp; E) -! lim1Hp+qC(Xp+m ; Xp-1; 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 p-connected 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;m-pHmC(X; E)of HmC(X; E) such that F p;qHp+qC(X; E)=F p+1;q-1Hp+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 C-CW -complex and Xn its n-skelet* *on. Then the E2-term respectively E2-term 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; ss-q(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 Atiyah-Hirzebruch 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; ss-q(E)) =) ss-p-q(holim E) C analogous to those of Bousfield-Kan [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 non-empty 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 G-Spaces Let E be a covariant Or(G)-spectrum. We define an extension of E to the c* *ategory of G-spaces by E% : G-SPACES - ! 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 G-SPACES -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 G-CW -complex such that the H-fixed 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 G-homotopy 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 C-CW -complex, we can apply Theorem 3.4 to the weak homotopy equivalence of C-s* *paces F *ED -! {*} , and get a C-map EC -! F *ED , which is unique up to homotopy. * *It induces a map of D-spaces f : F*EC -! ED by Lemma 1.9. Let X be a covariant D-space. 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 D-space X is rep* *laced by a covariant D-spectrum 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 K-theory. 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 Mayer-Vietoris 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)) -! ssi-1(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 G-action 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 G-space X with the left G-space 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 Atiyah-Hirzebruch 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 K-theory spectra Kalg or the algebraic L-theory 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 Farrell-Jones [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 G-VC -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 K-theory 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 Baum-Connes 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 : G-CW -COMPLEXES -! SPECTRA with the following properties: 1. For a G-CW -complex X the homotopy groups of KG(X) can be identified with* * the equivariant K-homology groups in the sense of Kasparov, provided that X i* *s proper and cocompact, and hence with the source of the Baum-Connes 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 Baum-Connes 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 KG|Or(G)to Ktop. For more information on the Baum-Connes 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 G-equivariant, where E(G=1) is given the G = autOr(G)(G=1)-action and E(1=1) is* * given the trivial G-action. 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 G-m* *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 Farrell-Jones [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%: G-SPACES - ! 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 G-SPACES -spectra. A G-F-space (G-F-CW -complex) is a G-space (G-CW -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 G-F-CW -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 C-spectra is a map of C-spec* *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 G-map and A is a closed, G-invariant 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 G-cofibrations; 3. The canonical map Z+ ^ E% (X) -! E% (Z x X) is an isomorphism, where Z i* *s a space and X is a G-space; 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 H-fixed point set functor map* * G(G=H; -) commutes with attaching a G-space to a G-space along a G-map and with colimits * *of G- cofibrations indexed by the non-negative 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 G-F-CW -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) set-theoretic 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 G-F-CW -complex is constructed from orbits G=H with H 2 F * *via products with disks, attaching a G-space to a G-space along a G-map, and colimi* *ts over the non-negative integers, T(X) is continuous and is an isomorphism for all G-F-CW * *-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 : G-F-CW -COMPLEXES -! SPECTRA is called (w* *eakly) F- homotopy invariant if it sends G-homotopy equivalences to (weak) homotopy equiv* *alences of spectra. The functor E is (weakly) F-excisive if it has the following four * *properties. First, it is (weakly) F-homotopy invariant. Second, E(;) is contractible. Third* *, it respects homotopy pushouts up to (weak) homotopy equivalence, i.e. if the G-F-CW -comple* *x X is the union of G-CW -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 G-F-CW -complex X,* * i.e. the infinite mapping telescope, to E(X) is a (weak) homotopy equivalence of spectra* *. Notice that E is weakly F-excisive if and only if ssq(E(X)) defines a homology theory on th* *e category of G-F-CW -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 F-excisive; 2. Let T : E -! F be a transformation of (weakly) F-excisive functors E and* * F from G-F-CW -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 G-F-CW -complexes X; 3. For any (weakly) F-homotopy invariant functor E from G-F-CW -COMPLEXES * * to SPECTRA , there is a (weakly) F-excisive functor E% from G-F-CW -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 G-F-CW -complexes X* *. E is 44 (weakly) F-excisive if and only if AE(X) is a (weak) homotopy equivalence* * of spectra for all G-F-CW -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 G-F- CW -complex. Since E is (weakly) F-excisive, 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 G-F-CW -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 path-connected 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 Weiss-Williams [41]). The map BE gives a * *homotopy equivalence from E% (X) to X+ ^ Kalg(Z); where Kalg(Z) is the algebraic K-spect* *rum of the ring Z. After one applies the n-th homotopy group to the assembly map AE : E% (X) -! E(X) one obtains the algebraic K-theory 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 F-homotopy invariant functor E : G-CW -COMPLEXES -! SPECTRA by setting E(X) = Kalg((EG xG X)). If X is simply-connected, 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 L-theory 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 K-theory 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) F-exc* *isive functor of a (weakly) F-homotopy invariant functor E from G-F-CW -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 G-F-CW -COMPLEXES to SPECTRA such that F is (weakly) F-e* *xcisive and T(G=H) is a (weak) homotopy equivalence for all H 2 F. Then for any G-F-CW* * - 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 well-defined 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 G-F-CW -complex* *es X and Y . The construction above uses the (weak) F-homotopy 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 G-map 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, : : :, oep-1. Version of June 3, 1996 49 7. G-Spaces and Or(G)-spaces In this section we discuss the orbit category in more detail, and give a * *correspondence between G-spaces 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 non-empty family of subgroups closed under conjugation and inclusion. A G-spa* *ce X is a G-F-space 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 G-maps. Next we explain how one gets from G-F-spaces 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 G-space 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 G-F-space 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 G-space Y there is a natural homeomorphism T (X; Y ) : map G(Xb; Y ) -! homOr(G;F)(X; mapG (-; Y )): Proof______: If we neglect the G-action 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 G-action on Y one defines appropriate G-ac* *tions on the source and target of this map and checks that this map is G-equivariant. He* *nce it induces a homeomorphism on the G-fixed 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 G-homeomorphism. Proof______: The inverse f-1 : 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 G-F-space Y , the adjoint of the identity on m* *ap G(-; Y ) under the adjunction of Lemma 7.2 is a natural G-homeomorphism 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 G-F-CW -complex, then map G(-; Y ) is a free Or (G; F)-CW -* *complex. There is a bijective correspondence between the G-cells 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 G-CW -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 G-F-CW -complex X there * *is precisely one G-map up to G-homotopy from X to E(G; F) and thus that two such classifying* * spaces are G-homotopy 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 G-space 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. 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Davis Wolfgang L* *"uck Department of Mathematics Fachbereich Mathema* *tik Indiana University Johannes Gutenberg-Univer* *sit"at Bloomington, IN 47405 55099 Ma* *inz U.S.A. Bundesrepublik Deutschl* *and email: jfdavis@indiana.edu lueck@topologie.mathematik.uni-ma* *inz.de FAX: 812-855-0046 06131 393* *867 Version of June 3, 1996 55