“Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K-and L-Theory”

by

James F. Davis^{1} and Wolfgang Lück

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 Conjecture on the topological K-theory of reduced C^{∗}-algebras of groups. The approach is through spectra over the orbit category of a discrete group G. We give several points of view on the assembly map for a family of subgroups and characterize such assembly maps by a universal property generalizing the results of Weiss and Williams to the equivariant setting. The main tools are spaces and spectra over a category and their associated generalized homology and cohomology theories, and homotopy limits.

Key words: Algebraic K and L-theory, Baum-Connes Conjecture, assembly maps, spaces and spectra over a category

AMS-classification number: 57

Introduction

Glen Bredon [5] introduced the orbit category Or(G) of a group G. Objects are homogeneous spaces G/H, considered as left G-sets, and morphisms are G-maps. This is a useful construct for organizing the study of fixed sets and quotients of G-actions. If G acts on a set X, there is the contravariant fixed point functor Or(G) ⟶ SETS given by G/H ↦ X^{H} = map_{G}(G/H, X) and the covariant quotient space functor Or(G) ⟶ SETS given by G/H ↦ X/H = X ×_{G} G/H. Bredon used the orbit category to define equivariant cohomology theory and to develop equivariant obstruction theory.

Examples of covariant functors from the orbit category of a discrete group G to abelian groups are given by algebraic K-theory K_{i}(ℤH), algebraic L-theory L_{i}(ℤH), and the Ktheory K_{i}^{top}(C_{r}^{∗}(H)) of the reduced C^{∗}-algebra of H. In Section 2, we express each 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 highdimensional topology: the Isomorphism Conjecture of Farrell-Jones [15] (which implies the Borel/Novikov Conjecture) and the Baum-Connes Conjecture in the case of discrete groups.

1Supported by the Alexander von Humboldt-Stiftung and the National Science Foundation. James F.

Davis wishes to thank the Johannes Gutenberg-Universität in Mainz for its hospitality when this work was initiated.

Our motivation was in part to obtain such a formulation and and in part to set the stage for explicit computations based on isomorphism conjectures. We give computations of K- and L-groups of group rings in a separate paper [8]. Our formulation is used by Kimberly Pearson [27] to show that the Whitehead group Wh(G) and the reduced K_{0}-group K̃_{0}(ℤG) vanish for two dimensional crystallographic groups. We also hope our formulations 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 assembly 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 spectra over a category. In Section 2, we define our three main examples of Or(G)-spectra: K^{alg}, L, and K^{top}. (These are all non-connective spectra; they have homotopy groups in negative dimensions.) They are all defined by first assigning to an object G/H, the transformation groupoid \overline{G/H}, whose objects are elements of G/H, and whose morphisms are given by multiplication by a group element, and then assigning a spectrum to a groupoid. In the K^{top}-case there is an intermediate step of considering the C^{∗}-category of a groupoid and a spectrum of a C^{∗}-category, derived from Bott periodicity.

In Section 3 we discuss free CW -complexes over a category C, the universal free CW - complex EC over a category C, and homotopy (co)-limits EC ⊗_{C} X of a C-space X. The ideas here are well-known to the experts (see e.g. [11]), but our approach, relying on homological methods and avoiding simplicial methods, may appeal to an algebraist. By approximating a C-space X by a free C-CW -complex, we define in Section 4 homology H_{∗}^{C}(X; E) and cohomology H_{C}^{∗}(X; E) of a space X 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 taking subgroups 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 ⟶ X^{H})_{+}⊗_{Or(G)}E. Then π_{∗}(E_{%}(X)) is an equivariant homology 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

π_{∗}E_{%}(E(G, F)) ⟶ π_{∗}E_{%}(G/G)

given by applying E_{%} to the constant map and then taking homotopy groups is called the (E, F, G)-assembly map. We say the (E, F, G)-isomorphism conjecture holds if the (E, F, G)-assembly map is an isomorphism. When F = VC, the family of virtual cyclic subgroups of G, (i.e. H ∈ VC if and only if H has a cyclic subgroup of finite index), the isomorphism conjectures of Farrell-Jones [15] for algebraic K- and L-theory are equivalent to the (K^{alg}, VC, G)- and (L, VC, G)-isomorphism conjectures, where K^{alg} and L (= L^{⟨−∞⟩}) are Or(G)-spectra associated to algebraic K- and L-theories. When F = FIN, the family of finite subgroups of G, and K^{top} is the Or(G)-spectra associated with the K-theory of C^{∗}- algebras, then the (K^{top}, FIN , G)-Isomorphism Conjecture is equivalent to the Baum-Connes Conjecture for the discrete group G (see Section 5). When F = 1, the family consisting only of the trivial subgroup of G, then the (K^{alg}, 1, G), (L, 1, G), and (K^{top}, 1, G)-assembly maps can be identified with maps H_{∗}(BG; K^{alg}(ℤ)) ⟶ K_{∗}(ℤG), H_{∗}(BG; L(ℤ)) ⟶ L_{∗}(ℤG), and H_{∗}(BG; K^{top}(ℂ)) ⟶ K_{∗}^{top}(C_{r}^{∗}G).

We give three variant ways of expressing the (E, F, G)-assembly map: by approximating E by E_{%} as above, in terms of homotopy colimits, and in terms of a generalized homology theory over a category. The first definition is the quickest and leads to an axiomatic characterization; 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 ∈ F. The (E, F, G)-assembly map is equivalent to the map

π_{∗}(hocolim_{Or(G,F)} E) ⟶ π_{∗}(hocolim_{Or(G)} E)

induced by the inclusion of the restricted orbit category in the full orbit category. Since E(G, F) is only defined up to G-homotopy type, it is natural for us to define homotopy limits and colimits as a homotopy type, rather than a fixed space or spectra; we 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

H_{∗}^{Or(G)}({∗}_{F}; E) ⟶ H_{∗}^{Or(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 [42] 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 approximation” of E by an excisive functor, in particular π_{∗}(E^{%}(X)) is an equivariant homology theory. When E(X) = K^{alg}(Π(EG ×_{G} X)) where Π is the fundamental groupoid, then the map π_{∗}(A(E(G, F))) is equivalent to the (K^{alg}, F, G)-assembly map. (We define K^{alg} of a groupoid in Section 2.) An analogous statement holds for L-theory and for the topological K-theory of C^{∗}-algebras. This gives a fourth point of view on assembly maps.

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 spectra 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

1. Spaces and Spectra over a Category

This section gives basic definitions and examples of spaces and spectra over a small (discrete) category C and discusses the adjointness of the tensor product and mapping 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-empty set of subgroups of G closed under taking conjugates and subgroups. The orbit category Or(G) has as objects homogeneous G-spaces G/H and as morphisms G-maps. The orbit category Or(G, F) with respect to F is the full subcategory of Or(G) consisting of those objects G/H for which H belongs to F.

Examples of families are F = {H ⊂ G | X^{H} ≠ ∅} for a G-space X, the finite subgroups of G, and the virtually cyclic subgroups of G. Notice that the automorphism group 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, Lemma 1.31 on page 22]. We will always work in the category of compactly generated spaces (see [37] and [43, 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 maps of C-spaces from X to Y with the subspace topology coming from the obvious inclusion into ∏_{c∈Ob(C)} map(X(c), Y (c)).

Likewise we can define a C-set and an RC-module. For a ring R a RC-module is a functor M from C to the category of R-modules. For two RC-modules M and N of the same variance, hom_{RC}(M, N) is the abelian group of natural transformations from M to N. We can form kernels and cokernels, so the category of RC-modules is an abelian category, 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 element, namely the trivial group. Then Or(G, 1) is a category with a single object, and G can be identified with the set of morphisms by sending g ∈ G to the automorphism G/1 ⟶ G/1 which maps g′ to g′g^{−1}. 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 ℤ_{p} be the cyclic group of order p for a prime number p. A contravariant Or(ℤ_{p})-space Y is specified by a ℤ_{p}-space Y (ℤ_{p}/{1}), a space Y (ℤ_{p}/ℤ_{p}), and a map Y (ℤ_{p}/ℤ_{p}) ⟶ Y (ℤ_{p}/{1})^{ℤ_{p}}.

Example 1.3 Let Y be a left G-space and F be a family of subgroups. Define the associated contravariant Or(G, F)-space map_{G}(−, Y) by

map_{G}(−, Y) : Or(G, F) ⟶ SPACES G/H ↦ map_{G}(G/H, Y) = Y^{H}.

The tensor product of a contravariant C-space with a covariant C space yields a topological space.

Definition 1.4 Let X be a contravariant and Y be a covariant C-space. Define their tensor product to be the space

X ⊗_{C} Y = ∐_{c∈Ob(C)} X(c) × Y (c)/ ∼

where ∼ is the equivalence relation generated by (xϕ, y) ∼ (x, ϕy) for all morphisms ϕ : c ⟶ d in C and points x ∈ X(d) and y ∈ Y (c). Here xϕ stands for X(ϕ)(x) and ϕy for Y (ϕ)(y).

The tensor product and the hom space are called the coend and end constructions in category theory [24, pages 219 and 222]. A lot of well-known constructions are special cases of it.

Recall that the category of covariant (contravariant) Or(G, 1)-spaces is the category of left (right) G-spaces. The balanced product X ×_{G} 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_{Or(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 be 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) ⟶ hom_{C}(X, map(Y, Z))

\underline{\underline{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 the same as specifying a map X(c) × Y (c) ⟶ Z which is defined to be the composition of g with the obvious map from X(c) × Y (c) to X ⊗_{C} Y.

In particular Lemma 1.5 says that for a fixed covariant C-space Y the functor − ⊗_{C} Y from the category of contravariant C-spaces to the category of spaces and the functor map(Y, −) from the category of spaces to the category of contravariant C-spaces 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 [10, p. 79], [23, p. 166]). Many properties of these products can be proven via the adjoint property, rather than referring back to the definition. These products are reminiscent of the analogous situation of a right R-module X, a left R-module Y and an abelian group Z, the tensor product X ⊗_{R} Y, the R-module hom_{ℤ}(Y, Z). Here there is a natural adjoint isomorphism

hom_{ℤ}(X ⊗_{R} Y, Z) ⟶ hom_{R}(X, hom_{ℤ}(Y, Z)).

Lemma 1.6 Let X be a space and let Y and Z be covariant (contravariant) C-spaces. Let X ×Y be the obvious covariant (contravariant) C-space. There is a homeomorphism, natural in X, Y, and Z

T (X, Y, Z) : hom_{C}(X × Y, Z) ⟶ map(X, hom_{C}(Y, Z)).

Example 1.7 Let Δ be the category of finite ordered sets, i.e. for each non-negative integer p we have an object [p] = {0, 1, …, p} and morphisms are monotone increasing functions. A simplicial space X. is by definition a contravariant Δ-space and a cosimplicial space is a covariant Δ-space. A simplicial set is a contravariant Δ-set. It can be considered as a simplicial space by using the discrete topology. Define a covariant Δ-space Δ. by assigning 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(Δ., Y)_{d}. (The subscript d indicates that we equip this mapping space with the discrete topology, in contrast to the usual convention.) The geometric realization |X.| of a simplicial space X. is the space X. ⊗_{Δ} Δ.. The geometric realization of a simplicial set 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. for a simplicial space X. and a space Y there is a natural homeomorphism

T (X., Y) : map(|X.|, Y) ⟶ hom_{Δ}(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 construction of a CW -approximation of the space Y. For more information about simplicial spaces and sets we refer for instance to [4] [7], [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), σ(n)) | n ∈ ℤ} is a sequence of pointed spaces {E(n) | n ∈ ℤ} together with pointed maps called structure maps σ(n) : E(n) ∧ S^{1} ⟶ E(n + 1). A (strong) map of spectra (sometimes also called function in the literature) f : E ⟶ E′ is a sequence of maps f(n) : E(n) ⟶ E′(n) which are compatible with the structure maps σ(n), i.e. we have f(n + 1) ∘ σ(n) = σ′(n) ∘ (f(n) ∧ id_{S^{1}}) for all n ∈ ℤ. This should not be confused with the notion of map of spectra in the stable category (see [1, III.2.]). Recall that the homotopy groups of a spectrum are defined by

π_{i}(E) = colim_{k→∞} π_{i+k}(E(k))

where the system π_{i+k}(E(k)) is given by the composition

π_{i+k}(E(k)) ⟶^{S} π_{i+k+1}(E(k) ∧ S^{1}) ⟶^{σ(k)_{∗}} π_{i+k+1}(E(k + 1))

of the suspension homomorphism and the homomorphism induced by the structure map. A weak homotopy equivalence of spectra is a map f : E ⟶ F of spectra inducing an isomorphism on all homotopy groups. A spectrum E is called Ω-spectrum if for each structure map, its adjoint E(n) ⟶ ΩE(n + 1) = map(S^{1}, E(n + 1)) is a weak homotopy equivalence of spaces. We denote by Ω-SP ECT RA the corresponding full subcategory of SPECTRA.

A pointed C-space resp. a C-spectrum resp. C-Ω-spectrum is a functor from C to SPACES_{+} resp. SPECTRA resp. Ω-SPECTRA. We have introduced tensor product of C-spaces in Definitions 1.4 and mapping spaces of C-spaces in Definition 1.2. These notions extend to pointed spaces, one simply has to replace disjoint unions ∐ and cartesian products ∏ by wedge products ∨ and smash products ∧ and mapping spaces by pointed mapping 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 which 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 ∈ ℤ} of pointed Cspaces and the structure maps as maps of pointed C-spaces. With this interpretation it is obvious what the tensor product spectrum X ⊗_{C} E of a contravariant pointed C-space and a covariant C- spectrum means. The canonical associativity homeomorphisms

(X ⊗_{C} E(n)) ∧ S^{1} ⟶ X ⊗_{C} (E(n) ∧ S^{1})

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 composition of natural bijections coming from various adjunctions where Z is a pointed space

map ((X ⊗_{C} E(n)) ∧ S^{1}, Z) ⟶ map (X ⊗_{C} E(n), map(S^{1}, Z)) ⟶
hom_{C} (X, map (E(n), map(S^{1}, Z))) ⟶ hom_{C} (X, map(E(n) ∧ S^{1}, Z))
⟶ map(X ⊗_{C} (E(n) ∧ S^{1}), Z).

Similarly one defines the mapping space spectrum hom_{C}(X, E) of a pointed C-space X and a C-spectrum E using the canonical map of pointed spaces (which is not a homeomorphism in general)

hom_{C}(X, E(n)) ∧ S^{1} ⟶ hom_{C}(X, E(n) ∧ S^{1}).

This map assigns to ϕ ∧ z the map of C-spaces from X to E(n) ∧ S^{1} which sends x ∈ X(c) to ϕ(c)(x) ∧ z ∈ E(n)(c) ∧ S^{1} for c ∈ Ob(C).

A homotopy of maps of spectra f_{k} : E ⟶ F is a map of spectra h : [0, 1]_{+} ∧ E ⟶ F whose composition with the inclusion i_{k} : E ⟶ [0, 1]_{+} ∧ E e ↦ k ∧ e is f_{k} for k = 0, 1.

Let C and D be two categories. A C-D-space is a covariant C × D^{op}-space where D^{op} is the opposite of D which has the same objects as D and is obtained by reversing the direction of all arrows in D. This is the analogue of a R-S-bimodule for two rings R and S. Let F : C ⟶ D be a covariant functor. We get a D-C-space mor_{D}(F (?),??) where we use the discrete topology on the set of morphisms. Here ? is the variable in C and ?? is the variable in D. Analogously one defines a C-D-space mor_{D}(??, F (?)).

Definition 1.8 Given a covariant (resp. contravariant) C-space X, define the induction of X with F to be the covariant (resp. contravariant) D-space

F_{∗}X = mor_{D}(F (?),??) ⊗_{C} X

respectively

F_{∗}X = X ⊗_{C} mor_{D}(??, F (?))

and the coinduction of X with F to be the covariant (resp. contravariant) D-space

F_{!}X = hom_{C}(mor_{D}(??, F (?)), X)

respectively

F_{!}X = hom_{C}(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 ∘ F.

There are corresponding definitions for C-sets and RC-modules (see [10, p. 80], [23, p. 166] for induction of modules). For example, if M is a covariant RC-module, then F_{∗}M = R mor_{D}(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 are the following adjoint properties.

Lemma 1.9 There are natural adjunction homeomorphisms

hom_{D}(F_{∗}X, Y) ⟶ hom_{C}(X, F^{∗}Y);
hom_{C}(F^{∗}X, Y) ⟶ hom_{D}(X, F_{!}Y);
F_{∗}X ⊗_{D} Y ⟶ X ⊗_{C} F^{∗}Y;
Y ⊗_{D} F_{∗}X ⟶ F^{∗}Y ⊗_{C} X;

for a C-space X and D-space Y of the required variance.

\underline{\underline{Proof}} : Notice for a covariant D-space Y that there are natural homeomorphisms of covariant C-spaces

mor_{D}(??, F (?)) ⊗_{D} Y ⟶ F^{∗}Y ⟶ hom_{D} (mor_{D}(F (?),??), Y)

and analogously for contravariant Y. Now the claim follows from the adjointness of tensor product and hom and the associativity of tensor product.

2. K- and L-Theory Spectra over the Orbit Category

In this section we construct the main examples of spectra over the orbit category

K^{alg} : Or(G) ⟶ Ω- SPECTRA,
L : Or(G) ⟶ Ω- SPECTRA,
K^{top} : Or(G) ⟶ Ω- SPECTRA .

These functors are necessary for the statements of the various Isomorphism Conjectures. First we outline what we would naively like to do, explain why this does not work 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

K^{alg} : RINGS ⟶ Ω- SPECTRA,
L : RINGS^{inv} ⟶ Ω- SPECTRA,
K^{top} : C^{∗}- ALGEBRAS ⟶ Ω- SPECTRA,

where RINGS^{inv} is the category of rings with involution. The classical functors were defined by Gersten [17] and Wagoner [39] for algebraic K-theory, by Quinn-Ranicki [33] for algebraic L-theory, and by using Bott periodicity for C^{∗}-algebras (see [40] for a discussion of Bott periodicity for C^{∗}-algebras and also the end of this section for a functorial approach). The homotopy groups of these spectra give the algebraic K-groups of Quillen-Bass, the surgery obstruction L-groups of Wall, and the topological K-groups of C^{∗}-algebras. These are all non-connective spectra; the homotopy groups in negative dimensions are non-trivial. In L-theory our notation is an abbreviation for L = L^{⟨j⟩} for j ∈ ℤ ⨿ {−∞}, j ≤ 2, where the superscript refers to the K-theory allowed. We would like our functors defined on the orbit category to have the property that the spectra K^{alg}(G/H), L(G/H) and K^{top}(G/H) have the weak homotopy type of the spectra K^{alg}(ℤH), L(ℤH) and K^{top}(C_{r}^{∗}H) respectively, where ℤH is the integral group ring and C_{r}^{∗}H is the 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 multiplication r_{g} : G/H ⟶ G/K, g′H ↦ g′gK provided g ∈ G satisfies g^{−1}Hg ⊂ K. The induced homomorphism c_{g} : H ⟶ K, h ↦ g^{−1}hg gives a map of rings (with involution) from ℤH to ℤK, and, at least if the index of c_{g}(H) in K is finite, a map on reduced C^{∗}-algebras. We would like the functors applied to the morphism r_{g} in the orbit category to match up with the “classical” functors on rings, rings with involution, and C^{∗}-algebras.

The naive approach is define K^{alg}(G/H), L(G/H) and K^{top}(G/H) as the spectra K^{alg}(ℤH), L(ℤH) and K^{top}(C_{r}^{∗}H) respectively. This definition works fine for objects, but fails for morphisms. The problem is that g in c_{g} is not unique, because for any k ∈ K, clearly g and gk define the same morphism in the orbit category. Hence this definition makes sense only if c_{k} : K ⟶ K induces the identity on the various spectra associated 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 and 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 constructing the functor K^{top} : C^{∗}- ALGEBRAS ⟶ Ω- SPECTRA are particularly involved. P. Baum and J. Block, and P. Baum and G. Comezana have approaches to this construction, quite 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 isomorphisms.) Let GROUPOIDS^{inj} be the subcategory consisting of those functors F : G_{0} ⟶ G_{1} which are faithful, i.e. for any two objects x, y in G_{0} the induced map mor_{G_{0}}(x, y) ⟶ mor_{G_{1}}(F (x), F (y)) is injective. 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. A left G-set S defines a groupoid \overline{S} where Ob(\overline{S}) = S and for s, t ∈ S, mor(s, t) = \{g ∈ G | gs = t\}. The composition law is given by group multiplication. Obviously a map of left G-sets defines a functor of the associated groupoids. The category \overline{G/H} is equivalent to the category Or(H, 1) = H and hence \overline{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^{∗}-algebra of a group to the category of groupoids. The composition of this extension with the functor GR above yields covariant functors from the orbit category to the category of spectra. We will see that their value at each object G/H is homotopy equivalent to the corresponding 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 functors we want to construct:

K^{alg} : GROUPOIDS ⟶ Ω- SPECTRA,
L : GROUPOIDS ⟶ Ω- SPECTRA,
K^{top} : GROUPOIDS^{inj} ⟶ Ω- SPECTRA .

For this purpose we must introduce some additional structures on categories. Recall 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 be 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 mor_{C}(x, y) × mor_{C}(y, z) ⟶ mor_{C}(x, z) for all objects x, y and z in C.

Suppose that R comes with an involution of rings R ⟶ R r ↦ \overline{r}. A R-category with involution is a R-category C with a collection of maps

∗_{x,y} : mor_{C}(x, y) ⟶ mor_{C}(y, x) x, y, ∈ Ob(C)

such that the following conditions are satisfied:

1. ∗_{x,y}(λ · f + μ · g) = \overline{λ} · ∗_{x,y}(f) + \overline{μ} · ∗_{x,y}(g) for all λ, μ ∈ R, objects x, y ∈ Ob(C), and morphisms f, g : x ⟶ y;

2. ∗_{x,y} ∘ ∗_{y,x} = id for all objects x, y ∈ Ob(C);

3. ∗_{x,z}(g ∘ f) = ∗_{x,y}(f) ∘ ∗_{y,z}(g) for all x, y, z ∈ Ob(C) and all morphisms f : x ⟶ y and g : y ⟶ z.

In the sequel we abbreviate ∗_{x,y}(f) by f^{∗}. In this notation the conditions above become (λf + μg)^{∗} = \overline{λ}f^{∗} + \overline{μ}g^{∗}, (f^{∗})^{∗} = f and (g ∘ f)^{∗} = f^{∗} ∘ g^{∗}.

We call a R-category (with involution) an additive R-category (with involution) if it possesses a sum ⊕ and the obvious compatibility conditions with the R-module structures (and the involution) on the morphisms are fulfilled.

The notion of a C^{∗}-category was defined by Ghez-Lima-Roberts [18] 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 such that for each two objects x, y ∈ Ob(C) there is a norm ∥ ∥_{x,y} on each complex vector space mor_{C}(x, y) such that the following conditions are satisfied:

1. (mor_{C}(x, y), ∥ ∥_{x,y}) is a Banach space for all objects x, y ∈ Ob(C);

2. ∥ g ∘ f ∥_{x,z} ≤ ∥ g ∥_{y,z} · ∥ f ∥_{x,y} for all x, y, z ∈ Ob(C) and all morphisms f : x ⟶ y and g : y ⟶ z;

3. ∥ f^{∗} ∘ f ∥_{x,x} = ∥ f ∥_{x,y}^{2} for all x, y ∈ Ob(C) and all morphisms f : x ⟶ y.

4. For every f ∈ mor_{C}(x, y), there is a g ∈ mor_{C}(x, x) so that f^{∗} ∘ f = g^{∗} ∘ g.

In the sequel we abbreviate ∥ f ∥_{x,y} by ∥ f ∥ and we will consider a C^{∗}-category 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 structure of a R-category on C gives mor_{C}(x, x) the structure of a central R-algebra with unit id_{x}. The additional structure of an involution is given by a map ∗ : mor_{C}(x, x) ⟶ mor_{C}(x, x) satisfying:

∗(λ · f + μ · g) = \overline{λ} · ∗(f) + \overline{μ} · ∗(g), ∗ ∘ ∗ = id and ∗ (g ∘ f) = ∗(f) ∘ ∗(g).

The structure of a C^{∗}-category on C is the same as the structure of a C^{∗}-algebra on the set mor_{C}(x, x) with id_{x} as unit. The structure of a topological category on C is the structure of a topological space on mor_{C}(x, x) such that composition defines a continuous map.

Next we construct from a category (for example, a groupoid) other categories with the structures described above. Given a category C, the associated R-category RC has 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 mor_{C}(x, y) generated by the set mor_{C}(x, y). The composition is induced by the composition in C in the obvious way. Notice that the functor C ↦ 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. Then RG inherits the structure of a R-category with involution by defining

(∑_{i=1}^{r} λ_{i}f_{i})^{∗} := ∑_{i=1}^{r} \overline{λ}f_{i}^{−1}.

Next we explain how the category with involution ℂG can be completed to a C^{∗}- category C_{r}^{∗}G. It will have the same objects as G. Consider two objects x, y ∈ Ob(G). If mor_{G}(x, y) is empty, put mor_{C_{r}^{∗}G}(x, y) = 0. Suppose that mor_{G}(x, y) is non-empty. Choose some object z ∈ Ob(G) such that mor_{G}(z, x) is non-empty, for instance one could choose z = x. For a set S let l^{2}(S) be the Hilbert space with S as Hilbert basis. Define a ℂ-linear map

i_{x,y;z} : ℂ mor_{G}(x, y) ⟶ B(l^{2}(mor_{G}(z, x)), l^{2}(mor_{G}(z, y)))

by sending f ∈ mor_{G}(x, y) to the bounded operator from l^{2}(mor_{G}(z, x)) to l^{2}(mor_{G}(z, y)) given by composition with f. On the target of i_{x,y;z} we have the operator norm ∥ ∥. Define:

∥ u ∥_{x,y} := ∥ i_{x,y;z}(u) ∥ for u ∈ mor_{ℂG}(x, y) = ℂ mor_{G}(x, y).

One easily checks that this norm ∥ ∥_{x,y} is independent of the choice of z. The Banach space of morphisms in C_{r}^{∗}G from x to y is the completion of mor_{ℂG}(x, y) with respect to the norm ∥ ∥_{x,y}. We will denote the induced norm on the completion mor_{C_{r}^{∗}G}(x, y) again by ∥ ∥_{x,y} and sometimes abbreviate by ∥ ∥. One easily checks that ∗_{x,y} : mor_{ℂG}(x, y) ⟶ mor_{ℂG}(y, x) is an isometry since it is compatible with applying the maps i_{x,y;z} and i_{y,x;z} and taking adjoints of operators. Therefore it induces an isometry denoted in the same way

∗_{x,y} : mor_{C_{r}^{∗}G}(x, y) ⟶ mor_{C_{r}^{∗}G}(y, x).

Composition defines a ℂ-bilinear map mor_{ℂG}(x, y) × mor_{ℂG}(y, z) ⟶ mor_{ℂG}(x, z) which satisfies ∥ g ∘ f ∥_{x,z} ≤ ∥ g ∥_{y,z} · ∥ f ∥_{x,y}. Hence it induces a map on the completions

mor_{C_{r}^{∗}G}(x, y) × mor_{C_{r}^{∗}G}(y, z) ⟶ mor_{C_{r}^{∗}G}(x, z)

with the same inequality for the norms. This is the composition in C_{r}^{∗}G. One easily verifies that C_{r}^{∗}G 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_{r}^{∗}G is just the reduced group C^{∗}-algebra C_{r}^{∗}G under the identifications of Example 2.1.

The assignment of a C^{∗}-category C_{r}^{∗}G 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-condition that a functor F : G_{0} ⟶ G_{1} is faithful is used to guarantee that the map mor_{ℂG_{0}}(x, y) ⟶ mor_{ℂG_{1}}(F (x), F (y)) extends to mor_{C_{r}^{∗}G_{0}}(x, y) ⟶ mor_{C_{r}^{∗}G_{1}}(F (x), F (y)), for all x, y ∈ Ob(G_{0}).

Remark 2.3 We make a few remarks on functoriality (or lack thereof) of C^{∗}-algebras, which motivate our use of C^{∗}-categories. First note that the assignment of a C^{∗}-algebra C_{r}^{∗}H to a group H cannot be extended to a functor from the category of groups to the category 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 behaved with respect to functoriality. To a discrete groupoid G, one can associate the complex groupoid ring ℂG, which as a ℂ-vector space has a basis consisting of the morphisms in the groupoid. The product of two basis elements is the composite if defined and is zero otherwise. The completion of ℂG in B(l^{2}(G), l^{2}(G)) in the operator norm is called the reduced C^{∗}-algebra of the groupoid and which we denote C_{r}^{∗}G- alg. If G is connected (any two objects are isomorphic), and H is the automorphism group of an object, then it can be shown (via Morita theory) that the spectra K^{top}(C_{r}^{∗}G- alg) and K^{top}(C_{r}^{∗}H) have the same weak homotopy type. The second naive approach to the construction of a functor

K^{top} : Or(G) ⟶ Ω- SPECTRA

is to define K^{top}(G/H) to be K^{top}(C_{r}^{∗}\overline{G/H}- alg). While this approach is basically 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^{inj} to C^{∗}-ALGEBRAS. Indeed, consider the groupoid G[n] with n objects and precisely one morphism between two 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 but this is impossible for n ≥ 2 as the associated C^{∗}-algebra of G[n] is the simple algebra M_{n}(ℂ). Another counterexample comes from a morphism in the orbit category. Let G be any infinite group and consider the map of groupoids \overline{G/1} ⟶ \overline{G/G} where G acts on G/1 effectively and transitively 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 mistakes 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 K^{top}-functor from C^{∗}- CATEGORIES to SPECTRA. Note that after applying homotopy groups, one gets 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.

Let C be a R-category. We define a new R-category C_{⊕}, called the symmetric monoidal R-category associated to C with an associative and commutative sum ⊕ as follows. The objects in C_{⊕} are n-tuples \underline{x} = (x_{1}, x_{2}, …, x_{n}) consisting of objects x_{i} ∈ Ob(C) for n = 0, 1, 2, . . .. We will think of the empty set as 0-tuple which we denote by 0. The R-module of morphisms from \underline{x} = (x_{1}, …, x_{m}) to \underline{y} = (y_{1}, …, y_{n}) is given by

mor_{C_{⊕}}(\underline{x}, \underline{y}) := ⊕_{1≤i≤m,1≤j≤n} mor_{C}(x_{i}, y_{j})

Given a morphism f : \underline{x} ⟶ \underline{y}, we denote by f_{i,j} : x_{i} ⟶ y_{j} the component which belongs to i ∈ {1, …, m} and j ∈ {1, …, n}. If x or y is the empty tuple, then mor_{C_{⊕}}(x, y) is defined to be the trivial R-module. The composition of f : \underline{x} ⟶ \underline{y} and \underline{g} : \underline{y} ⟶ \underline{z} for objects \underline{x} = (x_{1}, …, x_{m}), \underline{y} = (y_{1}, …, y_{n}) and \underline{z} = (z_{1}, …, z_{p}) is defined by

(g ∘ f)_{i,k} = ∑_{j=1}^{n} g_{j,k} ∘ f_{i,j}.

The sum on C_{⊕} is defined on objects by sticking the tuples together, i.e. for \underline{x} = (x_{1}, …, x_{m}) and \underline{y} = (y_{1}, …, y_{n}) define

\underline{x} ⊕ \underline{y} := (x_{1}, …, x_{m}, y_{1}, …, y_{n}).

The definition of the sum of two morphisms is now obvious. Notice that this sum is (strictly) associative, i.e. (\underline{x} ⊕ \underline{y}) ⊕ \underline{z} and \underline{x} ⊕ (\underline{y} ⊕ \underline{z}) are the same objects and analogously for morphisms. Moreover, there is a natural isomorphism

\underline{x} ⊕ \underline{y} ⟶ \underline{y} ⊕ \underline{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_{⊕}. Notice that the functor C ↦ 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 following category. An object in P(C) is an endomorphism p : x ⟶ x in C which is an idempotent, i.e. p ∘ p = p. A morphism in P(C) from p : x ⟶ x to q : y ⟶ y is a morphism f : x ⟶ y in C satisfying q ∘ f ∘ p = f. The identity on the object p : x ⟶ x in P(C) is given by the morphism p : x ⟶ x in C. If C has the structure of a R-category or of a a symmetric monoidal 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 objects as C, but whose morphisms are the isomorphisms of C. If C is a symmetric monoidal R-category, then so is Iso(C).

Let C be a symmetric monoidal R-category, all of whose morphisms are isomorphisms. Then its group completion is the following symmetric monoidal R-category C . An object in C is a pair (x, y) of objects in C. A morphism in C from (x, y) to (\overline{x}, \overline{y}) is given by equivalence classes of triples (z, f, g) consisting of an object z in C and isomorphisms f : x ⊕ z ⟶ \overline{x} and g : y ⊕ z ⟶ \overline{y}. We call two such triples (z, f, g) and (z′, f′, g′) equivalent if there is an isomorphism h : z ⟶ z′ which satisfies f′ ∘ (id_{x} ⊕h) = f and g′ ∘ (id_{y} ⊕h) = g. The sum on C is given by

(x, y) ⊕ (\overline{x}, \overline{y}) := (x ⊕ \overline{x}, y ⊕ \overline{y}).

If C is a C^{∗}-category, then C_{⊕} and P(C) inherit the structure of a C^{∗}-category where one should modify the definition of P(C) by requiring that each object p : x ⟶ x is a selfadjoint idempotent, i.e. p ∘ p = p and p^{∗} = p. Moreover, C_{⊕}, P(C_{⊕}) and (Iso(P(C_{⊕}))) inherit the structure of topological categories where the set of objects always gets the discrete topology.

Next we can construct the desired functors from GROUPOIDS and GROUPOIDS^{inj} to Ω-SPECTRA. The covariant functor non-connective algebraic K-theory spectrum of a groupoid with coefficients in R

K^{alg} : GROUPOIDS ⟶ Ω- SPECTRA

assigns to a groupoid G the non-connective K-theory spectrum of the small additive category (Iso(P(RG_{⊕}))) . (See [28] for the construction of the non-connective Ktheory spectrum of a small additive category.)

Next we define the covariant functor periodic algebraic L-theory spectrum of a groupoid with coefficients in R

L = L^{h} : GROUPOIDS ⟶ Ω- SPECTRA

where we assume that R is a commutative ring with unit and involution. Then RG and hence RG_{⊕} inherit an involution. We apply the construction of the periodic algebraic Ltheory spectrum in [33, Example 13.6 on page 139]. If one uses the idempotent completion one gets the projective version

L^{p} : GROUPOIDS ⟶ Ω- SPECTRA .

Taking the Whitehead torsion into account yields the simple version

L^{s} : GROUPOIDS ⟶ Ω- SPECTRA .

More generally one obtains for j ∈ ℤ ⨿ {−∞}, j ≤ 2

L^{⟨j⟩} : GROUPOIDS ⟶ Ω- SPECTRA .

where L^{⟨j⟩} is L^{s}, L^{h}, L^{p} for j = 2, 1, 0.

Next we construct the covariant functor non-connective topological K-theory spectrum

K^{top} : GROUPOIDS^{inj} ⟶ Ω- SPECTRA .

We do this by composing the functor

C_{r}^{∗} : GROUPOIDS^{inj} ⟶ C^{∗}- CATEGORIES,

with the functor

K^{top} : C^{∗}- CATEGORIES ⟶ Ω- 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 \underline{1}. We have introduced the category ℂ_{⊕} before. We denote by \underline{n} the n-fold sum of the object \underline{1}. In this notation ℂ_{⊕} has as objects \{\underline{n} | n = 0, 1, 2 … \}, the sum is \underline{m} ⊕ \underline{n} = \underline{m + n} for m, n = 0, 1, 2 … and the Banach space of morphisms from \underline{m} to \underline{n} is just given by the (n, m)-matrices with complex entries. Let C be any ℂ-category. We define a functor

⊗ : ℂ_{⊕} × C_{⊕} ⟶ C_{⊕}

as follows. We assign to an object \underline{n} ∈ ℂ_{⊕} and an object \underline{x} ∈ C_{⊕} the object \underline{n} ⊗ \underline{x} which is the n-fold direct sum ⊕_{i=1}^{n}\underline{x}. Let f : \underline{m} ⟶ \underline{n} be a morphism in ℂ_{⊕} and g : \underline{x} ⟶ \underline{y} be a morphism in C_{⊕}. Define f ⊗ g : \underline{m} ⊗ \underline{x} ⟶ \underline{n} ⊗ \underline{y}, to be the morphism whose component from the i-th copy of \underline{x} in \underline{m} ⊗ \underline{x} to the j-th copy of \underline{y} in \underline{n} ⊗ \underline{y} is f_{i,j} · g, where f_{i,j} ∈ ℂ is the component of f from the i-th coordinate of \underline{m} to the j-th coordinate of \underline{n}. One easily checks that f ⊗ g is a functor. For objects \underline{m} and \underline{n} in ℂ_{⊕} and an object \underline{x} in C_{⊕} we have (\underline{m} ⊕ \underline{n}) ⊗ \underline{x} = (\underline{m} ⊗ \underline{x}) ⊕ (\underline{n} ⊗ \underline{x}). For an object \underline{n} in ℂ_{⊕} and objects \underline{x} and \underline{y} in C_{⊕} we have a natural isomorphism \underline{n} ⊗ (\underline{x} ⊕ \underline{y}) ≅ (\underline{n} ⊗ \underline{x}) ⊕ (\underline{n} ⊗ \underline{y}). Obviously this functor sends the subcategories \{\underline{0}\} × C_{⊕} and ℂ_{⊕} × \{0\} to \{0\} where \{\underline{0}\} and \{0\} denote the obvious subcategories with one object.

Let C be any C^{∗}-category. Then the construction above applies to P(C_{⊕}). It extends to a functor

⊗ : (Iso(ℂ_{⊕})) × (Iso(P(C_{⊕}))) ⟶ (Iso(P(C_{⊕})))

in the obvious way. Notice that (Iso(P(C_{⊕}))) inherits from C the structure of a topological category for which the set of objects is discrete. With respect to these topological structures the functor above is a functor of topological categories. Given a topological category D, let BD be its classifying space [34] (whose construction takes the topology into account). Given topological categories D and D′, the projections induce a homeomorphism

B(D × D′) ⟶ BD × BD′.

Hence the functor above induces a map

B(Iso(ℂ_{⊕})) × B(Iso(P(C_{⊕}))) ⟶ B(Iso(P(C_{⊕})))

for any C^{∗}-category C. Since it sends B(Iso(ℂ_{⊕})) ∨ B(Iso(P(C_{⊕}))) to the base point B{0} ⊂ B(Iso(P(C_{⊕}))) , we obtain a map, natural in C,

μ : B(Iso(ℂ_{⊕})) ∧ B(Iso(P(C_{⊕}))) ⟶ B(Iso(P(C_{⊕}))) .

The category Iso(ℂ_{⊕}) can be identified with the disjoint union ∐_{n≥0} GL(n, ℂ). Let GL(ℂ) = colim_{n→∞} GL(n, ℂ). Let ℤ × GL(ℂ) be the symmetric monoidal category whose objects (and monoidal sum) are given by the integers, and so that mor_{ℤ×GL(ℂ)}(m, n) is empty if m ≠ n and is GL(ℂ) if m = n. There is an obvious functor Iso(ℂ_{⊕}) ⟶ ℤ×GL(ℂ). Using Quillen’s group completion theorem [19, pages 220–221], it follows that B Iso(ℂ_{⊕}) has the homotopy type of ℤ × BGL(ℂ). Let b : S^{2} ⟶ B Iso(ℂ_{⊕}) be a fixed representative of the Bott element in π_{2}(B Iso(ℂ_{⊕}) ) = K^{−2}({pt.}). Then b and μ yield a map, natural in C,

S^{2} ∧ B(Iso(P(C_{⊕}))) ⟶ B(Iso(P(C_{⊕}))) .

Its adjoint is also natural in C and denoted by

β : B(Iso(P(C_{⊕}))) ⟶ Ω^{2}B(Iso(P(C_{⊕}))) .

Define the non-connective topological K-theory spectrum K^{top}(C) of the C^{∗}-category C by the space B(Iso(P(C_{⊕}))) in even dimensions, by the space ΩB(Iso(P(C_{⊕}))) in odd dimensions and by the structure maps which are the identity in odd dimensions and β in even dimensions. (Another construction is suggested by [14, Remark VIII.4.4. on page 186].) We claim that the proof of Bott periodicity for C^{∗}-algebras carries over to C^{∗}-categories. Hence K^{top}(C) is a Ω-spectrum. We will only be interested in the case where C is C_{r}^{∗}G for a connected 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(RG_{⊕}) to the category of finitely generated free RG-modules resp. finitely generated projective RGmodules as defined in [23, section 9]. Notice that these module categories are not small, in contrast to RG_{⊕} and P(RG_{⊕}). A functor F : G_{0} ⟶ G_{1} induces a functor from the category of finitely generated free resp. projective RG_{0}-modules to the corresponding category over G_{1} by induction. However, if we have a second functor G : G_{1} ⟶ G_{2}, then the functor induced on the module categories by G ∘ F and the composition of the functors induced by F and G on the module categories are not the same, they agree only up to natural equivalence. 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 the same as the functor induced by G ∘ F, so that we get honest functors from GROUPOIDS to Ω-SPECTRA.

As mentioned earlier, the functors K^{alg}, L^{⟨j⟩}, and K^{top} defined on the orbit category are given by the composition of the groupoid-valued functor GR and the spectra-valued functors defined above. The automorphism group of the object eH in \overline{G/H} for the identity element e ∈ G is just the subgroup H. Hence the next lemma proves what we have already claimed before, namely, that the spectra we assign to \overline{G/H} are homotopy equivalent to the spectra associated to H. In particular we get for all n ∈ ℤ and j ∈ ℤ ⨿ {−∞}, j ≤ 2

π_{n}(K^{alg}(G/H)) ≅ K_{n}^{alg}(ℤH)
π_{n}(L^{⟨j⟩}(G/H)) ≅ L_{n}^{⟨j⟩}(ℤH)
π_{n}(K^{top}(G/H)) ≅ K_{n}(C^{∗}H)

Lemma 2.4 1. If F_{i} : G_{0} ⟶ G_{1} for i = 0, 1 are functors of groupoids and T : F_{0} ⟶ F_{1} is a natural transformation between them, then the induced maps of spectra

K^{alg}(F_{i}) : K^{alg}(G_{0}) ⟶ K^{alg}(G_{1})

are homotopy equivalent and analogously for L^{⟨j⟩} and K^{top};

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 ∈ Ob(G), let G_{x} be the full subgroupoid with precisely one object, namely x. Then the inclusion i_{x} : G_{x} ⟶ G induces a homotopy equivalence

K^{alg}(i_{x}) : K^{alg}(G_{x}) ⟶ K^{alg}(G)

and K^{alg}(G_{x}) is isomorphic to the spectrum K^{alg} associated to the group ring R aut_{G}(x). The analogous statements hold for L^{⟨j⟩} and K^{top}.

\underline{\underline{Proof}} : Obviously 2.) follows from 1.). We indicate the proof of 1.) in the case of K^{top}, the other cases are analogous if one inspects the definitions in [28] and [33]. One easily checks that a natural transformation between F_{0} to F_{1} induces a natural transformation between the induced functors from (Iso(P(C_{r}^{∗}G_{0}_{⊕}))) to (Iso(P(C_{r}^{∗}G_{1}_{⊕}))) . Let [1] be the category having two objects, namely 0 and 1, and three morphisms, namely the identities on 0 and 1 and one morphism from 0 to 1. Then the natural transformation above can be viewed as a functor of topological categories from (Iso(P(C_{r}^{∗}G_{0}_{⊕}))) × [1] to (Iso(P(C_{r}^{∗}G_{1}_{⊕}))) . 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_{r}^{∗}G_{0}_{⊕}))) × [0, 1] ⟶ B(Iso(P(C_{r}^{∗}G_{1}_{⊕}))) .

One easily checks that this induces the desired homotopy of maps of spectra.

3. CW -Approximations and Homotopy Limits

In this section we give the basic definitions and properties of spaces and CW -complexes over a small category C. We show that the Whitehead Theorem and CW -approximations carry over from spaces to C-spaces. We emphasize the parallels between a category and a group, thinking of a group as a category with a single object, all of whose morphisms 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 ×_{G} 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 Ob(C)-spaces is a collection of maps {f(c) : X(c) ⟶ Y (c) | c ∈ Ob(C)}. There is a forgetful 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 ∐_{c∈Ob(C)} mor_{C}(−, c) × X(c). In the covariant case one uses mor_{C}(c, −).

Lemma 3.1 The functor B is the left adjoint of F.

\underline{\underline{Proof}} : This means that there is a natural bijection

T (X, Y) : hom_{C}(B(X), Y) ⟶ hom_{Ob(C)}(X, F (Y))

for all Ob(C)-spaces X and for all C-spaces Y. Actually T (X, Y) will even be a homeomorphism. For f : B(X) = ∐_{c∈Ob(C)} mor_{C}(−, c) × X(c) ⟶ Y (−) define T (X, Y)f by restricting f to X(−) = {id_{−}} × X(−). The inverse T (X, Y)^{−1} assigns to a map g of Ob(C)-spaces the following transformation

B(X) = ∐_{c∈Ob(C)} mor_{C}(−, c) × X(c) ⟶ Y (−), (ϕ, x) ↦ Y (ϕ) ∘ g(c)(x).

Let R be a ring. There is also an adjoint to the forgetful functor from RC- MOD to Ob(C)- SETS. It is defined as B(X(−)) = ⊕_{c∈Ob(C)}R(mor_{C}(−, c) × X(c)). A free RC-module is a module isomorphic to one in the image of B. Notice the analogy between Lemma 3.1 and the adjoint pair consisting of the forgetful functor from R-modules to sets and the functor assigning to a set S the free R-module RS generated by S.

We have already mentioned that the category of Or(G, 1)-spaces is the category of G-spaces and the category Ob(Or(G, 1))-spaces is the category of spaces. Under this identification the forgetful functor F just forgets the G-action and B sends a space Z to the G-space G × Z where G acts in the obvious way.

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 notions in the category SPACES objectwise. For instance, the pushout of a diagram of C-spaces X_{1} ⟵ X_{0} ⟶ X_{2} is defined as the functor X : C ⟶ SPACES whose value at an object c in C is the pushout of the diagram of spaces X_{1}(c) ⟵ X_{0}(c) ⟶ X_{2}(c). We mention that sometimes in the literature the terms direct limit and inverse limit are used instead of colimit and limit. We will always use the names colimit and limit.

Given a C-space X and a space Y, we obtain the C-space X × Y by assigning to an object c the space X(c) × Y. Taking Y = [0, 1], it is now clear what a homotopy of maps of C-spaces means. 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 of aut(c)- spaces for all objects c in C. The proof of this fact is a simple abstract manipulation 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 will 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 generalize 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 patterns.

Definition 3.2 A contravariant free C-CW -complex X is a contravariant C-space X together with a filtration

∅ = X_{−1} ⊂ X_{0} ⊂ X_{1} ⊂ X_{2} ⊂ … ⊂ X_{n} ⊂ … ⊂ X = ⋃_{n≥0} X_{n}

such that X = colim_{n→∞} X_{n} and for any n ≥ 0 the n-skeleton X_{n} is obtained from the (n − 1)-skeleton X_{n−1} by attaching free contravariant C-n-cells, i.e. there exists a pushout of C-spaces of the form

∐_{i∈I_{n}} mor_{C}(−, c_{i}) × S^{n−1} ⟶ X_{n−1}
↓  ↓
∐_{i∈I_{n}} mor_{C}(−, c_{i}) × D^{n} ⟶ X_{n}

where the vertical maps are inclusions, I_{n} is an index set, and the c_{i} are objects of C. We refer to the inclusion functor mor_{C}(−, c_{i}) × int D^{n} ⟶ X as a free C-n-cell based at c_{i}. A free C-CW -complex has dimension ≤ n if X = X_{n}. The definition of a covariant free C-CW -complex is analogous.

Note that the trivial contravariant (covariant) C-space which sends every object to a point is not in general a free C-CW -complex unless C has a final (initial) object.

The more general notion of a C-CW -complex was defined by Dror Farjoun [11, 1.16 and 2.1] (see also [30]). We shall deal almost exclusively with free C-CW -complexes. For a free C-CW -complex X, the cellular chain complex C_{∗}(X)(−), c ↦ 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.

If Y is a G-CW -complex, then map_{G}(−, Y) (which sends G/H ↦ 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 Or(G)-cell of map_{G}(−, Y) based at G/H. Recall that the category of Or(G, 1)-spaces coincides 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.

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 π_{k}(f, x) : π_{k}(X, x) ⟶ π_{k}(Y, f(x)) is 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 equivalence) if for all objects c the map of spaces f(c) : X(c) ⟶ Y (c) is n-connected (a weak homotopy equivalence).

The constant map EG ⟶ {∗} is a weak homotopy equivalence, but not a homotopy equivalence of Or(G, 1)-spaces.

The following result is well-known for ordinary CW -complexes [43, IV. Theorem 7.16 and 7.17 on page 182]. See also [11, 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 map on homotopy classes of maps between C-spaces induced by composition with f is denoted 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 -complex 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.

\underline{\underline{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 figure out the full proof following the classical proof in [43, IV. Theorem 7.16 and 7.17 on page 182].

We begin with the if statement. Suppose that [X, Y]^{C} consists of one element for each free C-CW -complex X. We then choose X = mor_{C}(−, c) × S^{k}, for a fixed c ∈ Ob(C). From Lemma 3.1 we obtain a natural homeomorphism

hom_{C}(mor_{C}(−, c) × S^{k}, Y) ⟶ map(S^{k}, Y (c))

and thus a natural bijection

[mor_{C}(−, c) × S^{k}, Y]^{C} ⟶ [S^{k}, Y (c)].

Hence for all objects c in C any map from S^{k} to Y (c) is nullhomotopic. This implies that f is a weak homotopy equivalence.

Next we prove the only if statement. Suppose that f is a weak homotopy equivalence. 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 is obtained from X by attaching C-cells. Therefore it suffices to show that any map of C-spaces mor_{C}(−, c) × S^{n−1} ⟶ Y can be extended to a map mor_{C}(−, c) × D^{n} ⟶ Y. Such a problem reduces to extending a map from S^{n−1} to Y (c) to D^{n}. 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 homotopy equivalence.

\underline{\underline{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 ∘ g] = [id_{X}]. 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) : (X′, A′) ⟶ (X, A)

consists of a free C-CW -pair (X′, A′) 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 -approximation 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 topological space X (see [43, 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) : (X′, A′) ⟶ (X, A) and (a, b) : (Y′, B′) ⟶ (Y, B), then there exists a map of pairs (f′, g′) : (X′, A′) ⟶ (Y′, B′) so that the diagram

(X′, A′) ⟶^{(u,v)} (X, A)
(f′,g′)↓  ↓^{(f,g)}
(Y′, B′) ⟶^{(a,b)} (Y, B)

commutes up to homotopy. Furthermore the map (f′, g′) is unique up to homotopy.

\underline{\underline{Proof}} : Existence of a C-CW -approximation is an inductive construction done by attaching n-cells to obtain a n-connected map and finally taking a colimit. Uniqueness follows 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 contractible for all objects c.

Since EC is a C-CW -approximation of the trivial C-space, EC exists and is unique up to homotopy type. Note there is a contravariant EC and a covariant EC. They are not closely related, but one can identify the contravariant EC with the covariant EC^{op}. 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 more 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 trivial C-space, which is a single C-0-cell based at the final object. Similarly, if C has an initial object, the trivial C-space is a covariant EC. If G is a crystallographic group, i.e. a discrete subgroup of the isometries of ℝ^{n} so that ℝ^{n}/G is compact, then (G/H ⟼ (ℝ^{n})^{H}) is a contravariant 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 ⟼ E(G, F)^{H}) is a model for E Or(G, F). This example is expanded on in Section 7.

Example 3.9 Let ω be the category whose objects are the non-negative integers and whose morphisms are given by the arrows below, their composites, and the identity maps.

0 ⟶ 1 ⟶ 2 ⟶ 3 ⟶ ⋯

Then we may take a contravariant Eω to be (Eω)(i) = [i, ∞), whose zero skeleton is obtained by intersecting each space with the integers. For each non-negative integer i, there is C-0-cell and a C-1-cell based at i. We may take the covariant Eω to be the trivial 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 Cspace.

The classifying space BC is a CW -complex defined only up to homotopy type. 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-spaces. Then for any contravariant free C-CW -complex X the induced map

id_{X} ⊗_{C}f : X ⊗_{C} Y ⟶ X ⊗_{C} Z

is a weak homotopy equivalence. A similar statement holds for weak homotopy equivalences of contravariant C-spaces.

Let X be a covariant (contravariant) free C-CW -complex and f : Y ⟶ Z be a weak homotopy equivalence of covariant (contravariant) C-spaces. Then the induced map

hom_{C}(id, f) : hom_{C}(X, Y) ⟶ hom_{C}(X, Z)

is a weak homotopy equivalence.

\underline{\underline{Proof}} : We will prove the claim by induction over the skeletons and the cells in X. We only consider the case id_{X} ⊗_{C}f. The functor − ⊗_{C} Y is compatible with colimits, using the standard trick from category theory that a functor with a right adjoint commutes with arbitrary colimits (see [24, Chapter V, section 5]). Hence the pushout specifying how X_{n} is obtained from X_{n−1} by attaching cells remains a pushout after applying − ⊗_{C} Y. Moreover, the left vertical arrow in this pushout is a cofibration and id_{X_{n}} ⊗_{C}f is the pushout of three weak homotopy equivalences. Hence it is itself a weak homotopy equivalence by excision theorem of Blakers-Massey [43, VII.7]. Analogously one argues to show that the colimit of the maps id_{X_{n}} ⊗_{C}f is id_{X} ⊗_{C}f and each inclusion X_{n} ⊗_{C} Y ⟶ X_{n+1} ⊗_{C} Y is a cofibration. This implies that id_{X} ⊗_{C}f is a weak homotopy equivalence. The proof of the assertion for hom is similar.

Next we give some definitions, which are in close analogy with group cohomology 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 groups

colim_{C} M = ℤ ⊗_{ℤC} M  and  lim_{C} M = hom_{ℤC}(ℤ, M).

Define the colimit of X over C and the limit of X over C to be the topological spaces

colim_{C} X = {∗} ⊗_{C} X  and  lim_{C} X = hom_{C}({∗}, X).

Define the colimit of E over C and the limit of E over C to be the spectra

colim_{C} E = {∗} ⊗_{C} E  and  lim_{C} E = hom_{C}({∗}, E).

The above definitions are standard and the universal properties follow from the adjunctions 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 colimits and limits of contravariant functors over C, by applying the above definitions to the functors considered as covariant functors on C^{op}. We next discuss the higher derived functors of the above limits.

Definition 3.13 If M is a covariant ℤC-module, define

H_{i}(C; M) = H_{i}(C_{∗}(EC) ⊗_{ℤC} M)  and  H^{i}(C; M) = H^{i}(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_{C} X = EC ⊗_{C} X  and  holim_{C} X = hom_{C}(EC, X).

If E is a covariant C-spectrum, define the homotopy colimit and the homotopy limit of E over C as

hocolim_{C} E = EC ⊗_{C} E  and  holim_{C} E = hom_{C}(EC, E).

One must be careful about the variances on EC in the above definitions. In the lefthand appearances of EC we are taking the contravariant version, while on the right we want the covariant version. In the definition of the higher limits H^{i} and colimits H_{i}, the ℤC-chain complex C_{∗}(EC) can be replaced by any projective ℤC-resolution of ℤ. As above we define homology, cohomology, hocolimits, and holimits of contravariant functors by considering them as functors defined on the opposite category. For properties of H_{i} and H^{i}, see, for example, [23] and for properties of homotopy limits see for instance [4], [12, §9] and [21]. One obtains functorial definitions if one uses the functorial construction E^{bar}C for EC. Since EC maps to {∗}, there are maps hocolim_{C} X ⟶ colim_{C} X and lim_{C} X ⟶ holim_{C} X. They are not, in general, weak homotopy equivalences, although the first map is if X is a free C-CW -complex. The basic property of homotopy limits is that if X ⟶ Y is a weak homotopy equivalence, then so are the induced maps hocolim_{C} X ⟶ hocolim_{C} Y and holim_{C} Y ⟶ holim_{C} X; this follows from Theorem 3.11.

Example 3.14 Let ω be the category from Example 3.9. Let M and N be covariant and contravariant ℤC-modules respectively. Then it is easy to see that H_{i}(ω; M) is colim_{j→∞} M(j) for i = 0 and zero for i > 0, that H^{i}(ω; M) is M(0) for i = 0 and zero for i > 0, that H_{i}(ω; N) is N(0) for i = 0 and zero for i > 0, and that H^{i}(ω; N) is lim_{j→∞} N(j) for i = 0, Milnor’s lim_{j→∞}^{1} N(j) for i = 1, and zero for i > 1.

Let X and Y be covariant and contravariant C-spaces respectively. Then with the Eω’s from Example 3.9 hocolim_{ω} X is the infinite mapping telescope of

X(0) ⟶ X(1) ⟶ X(2) ⟶ X(3) ⟶ · · · .

Clearly holim_{ω} X = X(0) and hocolim_{ω} Y = Y (0). Now holim_{ω} Y is a bit bigger, it is the subspace of

map([0, ∞), Y (0)) × map([1, ∞), Y (1)) × map([2, ∞), Y (2)) × ⋯,

consisting of all tuples (γ_{0}, γ_{1}, γ_{2}, ⋯) so that the composite of [i, ∞) ⟶^{γ_{i}} Y (i) ⟶ Y (i − 1) equals γ_{i−1} restricted to [i, ∞).

Definition 3.15 Let X be a C-space and M a ℤC-module. Let X′ ⟶ X be a C-CW -ap-proximation. If X is contravariant and M is covariant, define

H_{p}^{C}(X; M) = H_{p}(C_{∗}(X′) ⊗_{ℤC} M),

where C_{∗}(X′) is the cellular chain complex of X′. There is a similar definition if X is covariant and M is contravariant. If X and M have the same variance, define

H_{C}^{p}(X; M) = H^{p}(hom_{ℤC}(C_{∗}(X′), M)).

When C = Or(G, 1), H_{p}^{C}(X; M) is Borel equivariant homology H_{p}^{G}(X; M) = H_{p}(EG×_{G} X; M). When C = Or(G) and X is the the fixed point functor G/H ↦ Z^{H} of a G-CW - complex Z, then H_{p}^{C}(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 category was equivariant obstruction theory, and it is clear that all the ingredients 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 reader motivated by specific applications. Local coefficient systems are particularly subtle, see [26].

Next we recall functorial constructions of classifying spaces and C-CW -approximations (see for instance [4], [21], [34]). We will need some of the details later in Section 6. View the ordered set [p] = {0, 1, 2, …, p} as a category, namely, objects are the elements and there is precisely one morphism from i to j if i ≤ j and no morphism otherwise. Continuing with the terminology from Example 1.7, we get a covariant functor

[ ] : Δ ⟶ 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 : Δ ⟶ SETS, [p] ↦ func([p], C).

More explicitly, N_{p}C consists of diagrams in C of the form

c_{0} ⟶^{ϕ_{0}} c_{1} ⟶^{ϕ_{1}} c_{2} ⟶^{ϕ_{2}} … ⟶^{ϕ_{p−1}} c_{p}.

The bar resolution model B^{bar}C 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 using the discrete topology. It has the nice properties (see [34]) that it is functorial, that B^{bar}(C × D) = B^{bar}C × B^{bar}D, that B^{bar}C = B^{bar}(C^{op}), and that a natural transformation from a functor F_{0} to a functor F_{1} induces a homotopy between the maps B^{bar}F_{0} and B^{bar}F_{1} 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 Example 1.7 we get that B^{bar}C comes with a canonical CW -complex structure such that there is a bijective correspondence between the set of sequences of composable morphisms

c_{0} ⟶^{ϕ_{0}} c_{1} ⟶^{ϕ_{1}} c_{2} ⟶^{ϕ_{2}} … ⟶^{ϕ_{p−1}} c_{p}

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 diagram ? ⟶^{α} c ⟶^{β}?? in C. A morphism from ? ⟶^{α} c ⟶^{β}?? to ? ⟶^{α′} c′ ⟶^{β′} ?? is a commutative diagram in C of the shape

? ⟶^{α} c ⟶^{β} ??
id↓  ^{ϕ}↓  ^{id}↓
? ⟶^{α′} c′ ⟶^{β′} ??

Let \overline{mor_{C}(?,??)} be the category whose set of objects is mor_{C}(?,??) and whose only morphisms are the identity morphism of objects. Consider the functor

pr : ?↓C ↓?? ⟶ \overline{mor_{C}(?,??)} (? ⟶^{α} c ⟶^{β}??) ↦ (β ∘ α : ? ⟶??) .

Lemma 3.17 The map of contravariant C × C^{op}-spaces

B^{bar} pr : B^{bar} ?↓C ↓?? ⟶ B^{bar}\overline{mor_{C}(?,??)} = mor_{C}(?,??)

is a C × C^{op}-CW -approximation.

\underline{\underline{Proof}} : First we verify that B^{bar} pr is a weak homotopy equivalence. Fix objects c, c′ of C. Define functors

j : \overline{mor_{C}(c, c′)} ⟶ c↓C ↓c′ (c ⟶^{α} c′) ↦ (c ⟶^{id} c ⟶^{α} c′) .

pr(c, c′) : c↓C ↓c′ ⟶ \overline{mor_{C}(c, c′)} (c ⟶^{α} d ⟶^{β} c′) ↦ (β ∘ α : c ⟶ c′) .

These give homotopy equivalences after applying B^{bar}, since pr(c, c′) ∘ j is the identity and there is a natural transformation S : j ∘ pr(c, c′) ⟶ id defined by assigning to an object

c ⟶^{α} d ⟶^{β} c′ in c↓C ↓c′ the morphism in c↓C ↓c′

c ⟶^{id} c ⟶^{β∘α} c′
id↓  ^{α}↓  ^{id}↓
c ⟶^{α} d ⟶^{β} c′

We next show that B^{bar} ?↓C ↓?? is a free C × C^{op}-CW -complex. The canonical skeletal filtration on the classifying space of a category induces a filtration on B^{bar} ?↓C ↓?? such that

B^{bar} ?↓C ↓?? = colim_{p→∞} B_{p}^{bar} ?↓C ↓?? .

Moreover, there is a pushout of contravariant C × C^{op}-spaces

(n.d.N_{p}?↓C ↓?? ) × S^{p−1} ⟶ B_{p−1}^{bar} ?↓C ↓??
↓  ↓
(n.d.N_{p}?↓C ↓?? ) × D^{p} ⟶ B_{p}^{bar} ?↓C ↓??

where n.d.N_{p}?↓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 mor_{C}(?, c_{0}) × mor_{C}(c_{p},??) where the disjoint union runs over the sequences

c_{0} ⟶^{ϕ_{0}} c_{1} ⟶^{ϕ_{1}} c_{2} ⟶^{ϕ_{2}} … ⟶^{ϕ_{p−1}} c_{p}

where no morphism ϕ_{i} is the identity. Such sequences thus give the indexing set 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 not 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 over the variable ?? yields contravariant C-spaces X ⊗_{C} B^{bar} ?↓C ↓?? and X ⊗_{C} mor_{C}(?,??). Define a map of contravariant C-spaces

p_{X} : X ⊗_{C} B^{bar} ?↓C ↓?? ⟶^{id ⊗_{C}B^{bar} pr} X ⊗_{C} mor_{C}(?,??) ⟶^{≅} X

where the second map is the canonical isomorphism given by x ⊗ ϕ ↦ X(ϕ)(x). Define a map of contravariant C-spaces

a_{X} : |S.X| ⊗_{C} B^{bar} ?↓C ↓?? ⟶^{t⊗_{C}id} X ⊗_{C} B^{bar} ?↓C ↓?? ⟶^{p_{X}} X.

Lemma 3.19 Let X be a contravariant C-space. Then:

1. p_{X} is a weak homotopy equivalence of contravariant C-spaces, i.e. p_{X}(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. for each object c in C there is a CW -structure on X(c) and each morphism f : c ⟶ c′ in C induces a cellular map X(f) : X(c′) ⟶ X(c). Suppose Y is a contravariant free D × C^{op}-CW -complex. Then the contravariant D-space X ⊗_{C} Y inherits the structure of a free D-CW -complex;

3. The map a_{X} : |S.X| ⊗_{C} B^{bar} ?↓C ↓?? ⟶ X is a C-CW -approximation.

\underline{\underline{Proof}} : 1.) Fix an object c in C. Then

B pr(c,??) : B^{bar} c↓C ↓?? ⟶ mor_{C}(c,??)

is a weak homotopy equivalence of free C-CW -complexes, hence is a C-homotopy equivalence. Thus p_{X}(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=p} X_{i} ⊗_{C} Y_{j}. A free D × C^{op}-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 Φ : D^{i} ⟶ X(c) is a characteristic map for a i-cell of X(c) and if Ψ : mor_{D}(?, d) × mor_{C}(c,??) × D^{j} ⟶ Y is a characteristic map for a free D × C^{op}-j-cell of Y based at (d, c), then the characteristic map

mor_{D}(?, d) × D^{i} × D^{j} ⟶ X ⊗_{C} Y

is given by

(f, a, b) ↦ [Φ(a), Ψ(f, id_{c}, b)].

3.) follows from Lemma 3.17, 1.), 2.) above, and Theorem 3.11.

If one takes X = {∗} in the construction above, one obtains the contravariant bar C-CW -approximation of {∗}

E^{bar}C := {∗} ⊗_{C} B^{bar} ?↓C ↓?? .

More explicitly it is given as follows. For an object ? in C let ?↓C be the category of objects under ?. An object in ?↓C is a morphism ϕ : ? ⟶ c in C with ? as source. A morphism in ?↓ C from ϕ_{0} : ? ⟶ c_{0} to ϕ_{1} : ? ⟶ c_{1} is given by a morphism h : c_{0} ⟶ c_{1} in C satisfying ϕ_{1} = h ∘ ϕ_{0}. A morphism ψ : c ⟶ d in C defines a functor ψ ↓ C : d ↓ C ⟶ c ↓ C by composition with ψ from the right. Then

E^{bar}C : C ⟶ SPACES, c ↦ B^{bar} c↓C .

One easily checks that E^{bar}C ⊗_{C} {∗} = B^{bar}C and thereby justifies our notation.

4. (Co-)Homology Associated to Spectra over a Category

In this section we introduce the homology and cohomology theories associated to a spectrum over a category. We then explain a kind of Atiyah-Hirzebruch type spectral sequence.

Definition 4.1 Let (X, A) be a pair of pointed C-spaces. Denote the reduced cone of the pointed space A by cone(A). For a C-spectrum E of the opposite variance as (X, A) define

E_{p}^{C}(X, A) = π_{p}(X ∪_{A} cone(A) ⊗_{C} E).

For a C-spectrum E of the same variance as (X, A), define

E_{C}^{p}(X, A) = π_{−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 morphism, then the homology and cohomology as defined in Definition 4.1 reduces to the classical definition of the reduced homology and cohomology of a pair with coefficients in a spectrum. This is obvious for homology whereas for cohomology one uses the natural bijection induced by the adjunction

π_{p+k}(map(X, E(k))) ⟶ [X ∧ S^{p+k}, E(k)].

Notice that writing homology and cohomology in terms of tensor product and mapping space spectra is analogous to the definition of the homology and cohomology of a chain 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 generalized reduced homology and cohomology theories for pointed C-spaces.

\underline{\underline{Proof}} : The proof is exactly as in the case of spaces, i.e. where C is the trivial category. For instance, let us check the long cohomology sequence of a pair (X, A) of pointed C-spaces. The following diagram is a pushout

A ⟶^{i} X ∪_{A} (A ∧ [0, 1]_{+})
p↓  ↓^{q}
{∗} ⟶^{j} X ∪_{A} cone A

where i is the cofibration given by the inclusion and p and q are the projections. The functor hom_{C}(−, Y) for a fixed pointed covariant C-space Y has a left adjoint, namely −⊗_{C} Y. Hence the following diagram is a pullback and hom_{C}(i, id_{E(n)}) is a fibration for all n ∈ ℤ.

hom_{C}(X ∪_{A} cone(A), E(n)) ⟶^{hom_{C}(q,id_{E(n)})} hom_{C}(X ∪_{A} (A ∧ [0, 1]_{+}), E(n))
hom_{C}(j,id_{E(n)})↓  ↓^{hom_{C}(i,id_{E(n)})}
hom_{C}({∗}, E(n))V  ⟶^{hom_{C}(p,id_{E(n)})}  hom_{C}(A, E(n))

Hence we get for n ∈ ℤ fibrations of pointed spaces

hom_{C}(X ∪_{A} cone(A), E(n)) ⟶^{hom_{C}(q,id_{E(n)})} hom_{C}(X ∪_{A} (A ∧ [0, 1]_{+}), E(n))

⟶^{hom_{C}(i,id_{E(n)})} hom_{C}(A, E(n)).

They are compatible with the structure maps. Now the colimit over their long homotopy sequences yields the desired long cohomology sequence of the pair since the canonical projection from X ∪_{A} (A ∧ [0, 1]_{+}) to X is a homotopy equivalence of pointed C-spaces.

The suspension isomorphism is induced by the following identifications

π_{p+k}(hom_{C}(X ∧ S^{1}, E(k))) = π_{p+k}(map(S^{1}, hom_{C}(X, E(k)))
= π_{p+k}(Ω hom_{C}(X, E(k))) = π_{p+k+1}(hom_{C}(X, E(k))).

Recall that a weak homotopy equivalence of C-spaces is a C-map X ⟶ Y so that X(c) ⟶ Y (c) is a weak homotopy equivalence for all objects c ∈ Ob(C). The WHE-axiom says that a weak homotopy equivalence f : X ⟶ Y of pointed spaces induces isomorphisms on homology resp. cohomology. This is not necessarily satisfied for E_{p}^{C} and E_{C}^{p} as the following example shows. Let G be a group and C = Or(G, 1). Recall that a contravariant 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 π_{0}(E) = ℤ, considered as a covariant Or(G, 1)- spectrum by the trivial G-action. The projection p : EG_{+} ⟶ {∗}_{+} is a weak homotopy equivalence of pointed Or(G, 1)-spaces. We get

E_{q}^{Or(G,1)}(EG_{+}) = H_{q}(BG)  and  E_{q}^{Or(G,1)}({∗}_{+}) = H_{q}({∗}).

where H_{∗} is ordinary homology. Obviously these two groups do not coincide in general.

Our goal is to get unreduced homology and cohomology theories for (unpointed) Cspaces which satisfy both the disjoint union axiom and the WHE-axiom. To be more precise, a homology theory means that homotopic maps of pairs of C-spaces induce the same maps on the homology groups, that there are long exact sequences of pairs (X, A), and for a pushout of C-spaces

X_{0} ⟶^{i_{1}} X_{1}
i_{2}↓  ↓^{j_{1}}
X_{2} ⟶^{j_{2}} X

the map (j_{2}, i_{1}) : (X_{2}, X_{0}) ⟶ (X, X_{1}) induces an isomorphism on homology provided that i_{2} : X_{0} ⟶ X_{2} is a cofibration of C-spaces. If the homology theory satisfies the WHE-axiom, it suffices to require that for each object c the map i_{2}(c) : X_{0}(c) ⟶ X_{2}(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 homology 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 procedure for spaces to C-spaces (cf [38, 7.68]).

Definition 4.3 Let (X, A) be a pair of C-spaces. Let (u, v) : (X′, A′) ⟶ (X, A) be a C-CW -approximation. For a C-spectrum E of the opposite variance as (X, A), define the homology of (X, A) with coefficients in E by

H_{p}^{C}(X, A; E) = E_{p}^{C}(X′_{+}, A′_{+}).

and

H_{p}^{C}(X; E) = H_{p}^{C}(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

H_{C}^{p}(X, A; E) = E_{C}^{p}(X′_{+}, A′_{+})

and

H_{C}^{p}(X; E) = H_{C}^{p}(X, ∅; E).

The above homology and cohomology are well-defined by the existence and uniqueness 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 is uniquely up to homotopy determined by the property that the following diagram commutes up to homotopy

(X′, A′) ⟶ (X, A)
↓  ↓
(Y′, B′) ⟶ (Y, B)

Thus for a map of C-pairs, there are corresponding maps of homology and cohomology groups. We always have natural maps

H_{p}^{C}(X, A; E) ⟶ E_{p}^{C}(X, A)

and

E_{C}^{p}(X, A) ⟶ H_{C}^{p}(X, A; E).

They are isomorphisms if (X, A) is a free C-CW -pair, but not in general.

Lemma 4.4 H_{p}^{C}(X, A; E) and H_{C}^{p}(X, A; E) are unreduced homology and cohomology 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-Ω-spectrum.

\underline{\underline{Proof}} : The first claim follows from Lemma 4.2 and Theorem 3.4.

The homology theory satisfies the disjoint union axiom for finite disjoint unions. We get the disjoint union axiom for arbitrary coproducts, if we show that the homology 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 ∐_{i∈I} X_{i} 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 Ω-spectrum since E is a C-Ω-spectrum and hence

π_{p}(hom_{C}(Y, E)) = π_{p+k}(hom_{C}(Y, E(k))),

provided p + k ≥ 0. Now the claim follows from the adjunction homeomorphism

hom_{C} ((∐_{i∈I} X_{i})_{+} , E(k)) ⟶^{≅} ∏_{i∈I} hom_{C} ((X_{i})_{+}, E(k)) .

Notice that without the condition that E is a C-Ω-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} ⊂ X_{0} ⊂ X_{1} ⊂ X_{2} ⊂ … ⊂ X

such that X = colim_{n→∞} X_{n}. Let E be a C-spectrum with the opposite respectively the same variance as X.

1. The natural map

colim_{n→∞} H_{p}^{C}(X_{n}; E) ⟶ H_{p}^{C}(X; E)

is an isomorphism for p ∈ ℤ;

2. Let E be a C-Ω-spectrum. There is a natural exact sequence

0 → lim^{1}_{n→∞}H_{C}^{p−1}(X_{n}; E) → H_{C}^{p}(X; E) → lim_{n→∞} H_{C}^{p}(X_{n}; E) → 0

for all p ∈ ℤ.

\underline{\underline{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 [43, Theorem XIII.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-spectra. It induces a natural transformation

f_{∗} : H_{∗}^{C}(X; E) ⟶ H_{∗}^{C}(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-Ω-spectra.

Any cohomology theory on the category of CW -complexes satisfying the disjoint union axiom can be represented by a Ω-spectrum. This is a consequence of Brown’s representation theorem and proven for instance in [38, chapter 9]. The proof goes through with some obvious modifications also in the case of free C-CW -complexes. This does not contradict the remark in [11, 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 [11] all 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} ⊂ X_{0} ⊂ X_{1} ⊂ X_{2} ⊂ … ⊂ X

such that X = colim_{n→∞} X_{n}.

1. Let E be a covariant C-spectrum E. Then there is a spectral (homology) sequence E_{p,q}^{r}, d_{p,q}^{r} : E_{p,q}^{r} ⟶ E_{p−r,q+r−1}^{r} whose E_{1}-term is given by

E_{p,q}^{1} = H_{p+q}^{C}(X_{p}, X_{p−1}; E)

and the first differential is the composition

d_{p,q}^{1} : E_{p,q}^{1} = H_{p+q}^{C}(X_{p}, X_{p−1}, E) ⟶ H_{p+q−1}^{C}(X_{p−1}, E)
⟶ H_{p+q−1}^{C}(X_{p−1}, X_{p−2}; E) = E_{p−1,q}^{1}

where the first map is the boundary operator of the pair (X_{p}, X_{p−1}) and the second induced by the inclusion. The E^{∞}-term is given by

E_{p,q}^{∞} = colim_{r→∞} E_{p,q}^{r}.

This spectral sequence converges to H_{p+q}^{C}(X; E), i.e. there is an ascending filtration F_{p,m−p}H_{m}^{C}(X, E) of H_{m}^{C}(X, E) such that

F_{p,q}H_{p+q}^{C}(X, E)/F_{p−1,q+1}H_{p+q}^{C}(X, E) ≅ E_{p,q}^{∞};

2. Let E be a contravariant C-Ω-spectrum. Then there is a spectral (cohomology) sequence E_{r}^{p,q}, d_{r}^{p,q} : E_{p,q}^{r} ⟶ E_{p+r,q−r+1}^{r} whose E^{1}-term is given by

E_{1}^{p,q} = H_{C}^{p+q}(X_{p}, X_{p−1}; E)

and the first differential is the composition

d_{p,q}^{1} : E_{p,q}^{1} = H_{C}^{p+q}(X_{p}, X_{p−1}, E) ⟶ H_{C}^{p+q}(X_{p}, E) ⟶ H_{C}^{p+q+1}(X_{p+1}, X_{p}; E) = E_{p+1,q}^{1}

where the first map is induced by the inclusion and the second is the boundary operator of the pair (X_{p+1}, X_{p}). The E^{∞}-term is given by

E_{p,q}^{∞} = lim_{r→∞} E_{p,q}^{r}.

There is a descending filtration F^{p,m−p} lim_{n→∞} H_{C}^{m}(X_{n}; E) of lim_{n→∞} H_{C}^{m}(X_{n}; E) such that there is an exact sequence

0 ⟶ F^{p,q} lim_{n→∞} H_{C}^{p+q}(X_{n}; E)/F^{p+1,q−1} lim_{n→∞} H_{C}^{p+q}(X_{n}; E) ⟶ E_{∞}^{p,q}

⟶ lim^{1}_{m→∞} H_{C}^{p+q}(X_{p+m}, X_{p}; E) ⟶ lim^{1}_{m→∞} H_{C}^{p+q}(X_{p+m}, X_{p−1}; E).

If one of the following conditions is satisfied

(a) The filtration is finite, i.e. there is n ≥ −1 such that X = X_{n};

(b) The inclusion of X_{p} into X_{p+1} is p-connected for p ∈ ℤ and there is m ∈ ℤ such that π_{q}(E(C)) vanishes for all objects c ∈ Ob(C) and q > m;

then the spectral sequence converges to H_{C}^{p+q}(X; E), i.e. there is a descending filtration F^{p,m−p}H_{C}^{m}(X, E) of H_{C}^{m}(X, E) such that

F^{p,q}H_{C}^{p+q}(X; E)/F^{p+1,q−1}H_{C}^{p+q}(X; E) ≅ E_{∞}^{p,q}.

\underline{\underline{Proof}} : Again this is a variation of the case where C is the trivial category (see [38, 7.75,15.6 and Remark 3 on page 352]) or [43, Theorem XIII.3.2. on page 614 and Theorem XIII.3.6. on page 616].

Suppose in Theorem 4.7 that X is a free C-CW -complex and X_{n} its n-skeleton. Then the E^{2}-term respectively E_{2}-term of the spectral sequence in Theorem 4.7 can be identified with

E_{p,q}^{2} = H_{p}^{C}(X; H_{q}^{C}({∗}; E)) = H_{p}^{C}(X; π_{q}(E))

respectively

E_{2}^{p,q} = H_{C}^{p}(X; H_{C}^{q}({∗}; E)) = H_{C}^{p}(X; π_{−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 decomposition of E. The Atiyah-Hirzebruch spectral sequence [38, 15.7]) is a special case of Theorem 4.7.

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 E^{2} and E^{∞}-terms, one gets the homotopy colimit spectral sequence

H_{p}(C; π_{q}(E)) ⟹ π_{p+q}(hocolim_{C} E)

and the homotopy limit spectral sequence

H^{p}(C; π_{−q}(E)) ⟹ π_{−p−q}(holim_{C} E)

analogous to those of Bousfield-Kan [4] [XII,5.7 on page 339 and XI,7.1 on page 309].

5. Assembly Maps and Isomorphism Conjectures

In this section we give three equivalent definitions of assembly maps, each of which corresponds to a certain point of view. Then we explain the Isomorphism Conjectures 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, closed 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 category of G-spaces by

E_{%} : G- SPACES ⟶ SPECTRA X ↦ map_{G}(−, X)_{+} ⊗_{Or(G)} E

Recall that

map_{G}(−, X)_{+} ⊗_{Or(G)} E = ∐_{H⊂G} X_{+}^{H} ∧ E(G/H)/ ∼

where ∼ is the equivalence relation generated by (xϕ, y) ∼ (x, ϕy) for x ∈ X_{+}^{K}, y ∈ E(G/H) and ϕ : 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 [10]), i.e a G-CW -complex such that the H-fixed point set is contractible if H ∈ F and empty otherwise. Such classifying spaces were introduced by tom Dieck [9], [10] and are unique up to G-homotopy type. We will give another point of view on these spaces in Section 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 π_{∗}(E_{%}(pr)) is the (E, F, G)-assembly map 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-spaces 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. Then the assembly map

F_{∗} : hocolim_{C} F^{∗}X ⟶ hocolim_{D} X

is given by the composition

EC ⊗_{C} F^{∗}X ⟶^{g}F_{∗}EC ⊗_{D} X ⟶^{f⊗_{D}id} ED ⊗_{D} X

where the map g is the homeomorphism from Lemma 1.9. This assembly map is unique up to homotopy. There is also an assembly map if the covariant D-space X is replaced by a covariant D-spectrum E. If one uses the functorial models E^{bar}C and E^{bar}D, there is a functorial construction of the map E^{bar}C ⟶ F^{∗}E^{bar}D and hence of the assembly map.

Let

I : Or(G, F) ⟶ Or(G)

be the inclusion functor. Define the assembly map

I_{∗} : hocolim_{Or(G,F)} I^{∗}E ⟶ hocolim_{Or(G)} E = E(G/G).

where the homotopy colimit over the orbit category of E is E(G/G) because the orbit category 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 ∈ F and to be empty otherwise. Let inc : {∗}_{F} ⟶ {∗} be the inclusion map of Or(G)-spaces. It follows from definitions that the (E, F, G)-assembly map can be identified with the map

H_{i}^{Or(G)}(inc) : H_{i}^{Or(G)}({∗}_{F}; E) ⟶ H_{i}^{Or(G)}({∗}; E) = π_{i}(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 map 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!) when 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 functors K, L^{⟨j⟩}, and K^{top} will be discussed later in this section.

The main point of the validity of the (E, F, G)-Isomorphism Conjecture is that it allows the computation of π_{∗}(E(G/G)) from π_{∗}(E(G/H)) for H ∈ F and the structure of the restricted orbit category Or(G, F). Here are two examples which were historically important in algebraic K-theory.

Example 5.2 Let G be an amalgamated free product of H_{1} and H_{2} along a subgroup K. Let F be the smallest family (closed under subgroups and conjugation) containing H_{1} and H_{2}. 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 H_{1} or H_{2}. The (E, F, G)-Isomorphism Conjecture and the material in Section 4, give a long exact Mayer-Vietoris exact sequence

⋯ ⟶ π_{i}(E(G/K)) ⟶ π_{i}(E(G/H_{1})) ⊕ π_{i}(E(G/H_{2})) ⟶ π_{i}(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 taken to be a ℝ, with the isotropy group K at every point. The (E, F, G)-Isomorphism Conjecture and the material in Section 4, give a long exact Wang exact sequence

⋯ ⟶ π_{i}(E(G/K)) ⟶ π_{i}(E(G/G)) ⟶ π_{i−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 group 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 ×_{G} G/H)_{+} ∧ S. For any family F of subgroups of G, the (E, F, G)- Isomorphism Conjecture is valid.

\underline{\underline{Proof}} : Let ∇ : Or(G) ⟶ SPACES be the covariant functor ∇(G/H) = G/H. Note that the Or(G)-space ∇ has a left G-action defined by left multiplication of an element g on G/H. We have

E_{%}(E(G, F)) = E(G, F)^{H} ⊗_{Or(G)} ((EG ×_{G} G/H)_{+} ∧ S)
= (EG ×_{G} (E(G, F)^{H} ⊗_{Or(G)} ∇))_{+} ∧ S
= (EG ×_{G} E(G, F))_{+} ∧ S
⟶^{A} (EG ×_{G} G/G)_{+} ∧ S
= E_{%}(G/G).

The first, second, and fourth equalities are clear. The third equality holds since one can identify any left G-space X with the left G-space X^{H} ⊗_{Or(G)} ∇ by by Theorem 7.4 (1.). The map A is the assembly map E_{%}(pr). Since {e} ∈ F, we see E(G, F) = E(G, F)^{{e}} is contractible, and hence EG ×_{G} E(G, F) ⟶ EG ×_{G} 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) ⟶ Ω- 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 the algebraic K-theory spectra K^{alg} or the algebraic L-theory spectra L^{⟨−∞⟩} of Section 2 and F is the family VC of virtually cyclic subgroups of G, then the Isomorphism Conjecture is the one of Farrell-Jones [15]. 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 category of G-VC-CW -complexes [32, Proposition 8.4 on page 421]. The Isomorphism Conjecture computes the algebraic K- resp. L^{⟨−∞⟩}-groups of the integral group ring of G in terms of the corresponding groups for all virtually cyclic subgroups of G. Here L^{⟨−∞⟩} = lim_{j→−∞} L^{⟨j⟩}. For the integers ℤ, L^{⟨j⟩}(ℤ) is independent of j, and conjecturely L^{⟨j⟩}(ℤG) is independent of j for any torsion-free group G. The Isomorphism Conjecture for K^{alg} has been proven rationally for discrete cocompact subgroups of virtually connected Lie groups by Farrell and Jones [15]. The (K^{alg}, VC, G, i)-Isomorphism Conjecture for such groups with i < 2 also follows from [15]. The Isomorphism Conjecture for L^{⟨j⟩} has been proven for crystallographic groups if one inverts 2 by Yamasaki [44]. Notice that after inverting 2 the spectrum L^{⟨j⟩} is independent of j. The Isomorphism Conjecture for K^{alg} and L^{⟨−∞⟩} together imply the Novikov Conjecture and (for dimensions greater than 4) the Borel Conjecture. The Borel Conjecture says that two aspherical closed manifolds with isomorphic fundamental groups are homeomorphic and any homotopy equivalence between them is homotopic to a homeomorphism. A survey on these conjectures is given in [16]. Related issues are discussed in [41, Chapter 14].

When E equals the topological K-theory spectrum K^{top} 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 obvious and will explained at the end of Section 6. For information about the Baum-Connes map we refer to [3] or [20].

Example 5.5 Let E be a covariant Or(G)-spectrum and F = 1 the trivial family. The domain of the (E, 1, G)-assembly map is E_{%}(E(G, 1)) = EG_{+} ∧_{G} E(G/1). Now suppose there is a functor J : GROUPOIDS^{inj} ⟶ SPECTRA so that E(G/H) = J(\overline{G/H}). Then the morphism of groupoids \overline{G/1} ⟶ \overline{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 = aut_{Or(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 F_{i} : G_{0} ⟶ G_{1} for i = 0, 1 and a natural transformation T : F_{0} ⟶ F_{1}, then the maps of spectra J(F_{0}) and J(F_{1}) are homotopic. (See Lemma 2.4 to see that these hypotheses are valid where E is K^{alg}, L^{⟨j⟩}, or K^{top}.) Since \overline{G/1} ⟶ \overline{1/1} is a natural equivalence of groupoids, the map E(G/1) ⟶ E(1/1) is a homotopy equivalence, which is in addition a G-map. 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 identified with the “classical” assembly maps

A : H_{i}(BG; K^{alg}(ℤ)) ⟶ K_{i}(ℤG),
A : H_{i}(BG; L^{⟨−∞⟩}(ℤ)) ⟶ L_{i}^{⟨−∞⟩}(ℤG),
A : H_{i}(BG; K^{top}(ℂ)) ⟶ K_{i}^{top}(C_{r}^{∗}(G)).

The last map has an interpretation in terms of taking the index of elliptic operators. The Novikov Conjecture is equivalent to the conjecture that the middle map is rationally injective and is implied by the conjecture that the last map is rationally injective, which is in turn implied by the Baum-Connes conjecture.

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 (K^{alg}, VC, G), (L^{⟨−∞⟩}, 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 (K^{top}, FIN , G)-case, and is shown by Farrell-Jones [15, 1.6.1 and Remark A.11] in the other two cases.

6. Characterization of Assembly Maps

In this section we characterize assembly maps by a universal property. This is useful for identifying different constructions of assembly maps (for example, assembly maps arising controlled topology, or from geometric techniques) and generalizes work of Weiss and Williams [42] from the case of a trivial group to the case of a general discrete group G.

We first review the non-equivariant version. Let E be a spectrum. Then one can define a functor

E_{%} : SPACES ⟶ SPECTRA X ↦ X_{+} ∧ E.

When this functor is restricted to the category of CW -complexes, it is excisive, in particular π_{∗}(E_{%}(−)) is a generalized homology theory.

Now suppose F : SPACES ⟶ SPECTRA

is a (weakly) homotopy invariant functor, i.e. it takes (weak) homotopy equivalences to (weak) homotopy equivalences^{2}. Then Weiss-Williams [42] construct a functor

F^{%} : CW -COMPLEXES ⟶ SPECTRA

and natural transformations

A_{F} : F^{%} ⟶ F;
B_{F} : F^{%} ⟶ (F |_{{∗}})_{%};

which induce a (weak) homotopy equivalence of spectra A_{F}({∗}) and (weak) homotopy equivalences of spectra B_{F}(X) for all CW -complexes X. Thus F^{%} is a (weakly) excisive approximation for F. The map A_{F}(X) should be thought of as an assembly map, and when X = BG and F = K^{alg}(Π(X), applying homotopy groups gives the classical K-theory assembly map

H_{∗}(X; K^{alg}(ℤ)) ⟶ K_{∗}^{alg}(ℤπ_{1}X).

We now proceed to give the equivariant version of the above. We associate to a covariant Or(G, F)-spectrum E an extension

E_{%}^{F} : G- SPACES ⟶ SPECTRA X ↦ map_{G}(−, X)_{+} ⊗_{Or(G,F)} E.

Notice that this construction depends on F. If E is a Or(G)-spectrum, we have introduced 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 G_{x} of each point x ∈ X is contained in the family F. The map S(X) is an isomorphism if X is a G-F-CW -complex but not in general. For instance for X = G/G and F the trivial family 1 we get (E |_{Or(G,F)})_{%}^{F}(G/G) = E(G/1)/G and E_{%}(G/G) = E(G/G). We will omit the superscript F in E_{%}^{F} 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 induces a map of G − SPACES-spectra T_{%} : E_{%} ⟶ F_{%}. Recall that a map (isomorphism) of spectra f : E ⟶ F is a collection of maps (homeomorphisms) f(n) : E(n) ⟶ F(n) which are compatible with the structure maps. An isomorphism of C-spectra is a map of C-spectra whose evaluation at each object is an isomorphism of spectra.

2The example to be kept in mind is F(X) = K^{alg}(Π(X)), the algebraic K-spectrum of the fundamental groupoid. This functor is homotopy invariant, but is neither excisive as a functor of X, nor continuous as a functor of topological categories.

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 colim_{n→∞} E_{%}(X_{n}) ⟶ E_{%}(colim_{n→∞} X_{n}) is an isomorphism, where X_{0} ⟶ X_{1} ⟶ X_{2} ⟶ … is a sequence of G-cofibrations;

3. The canonical map Z_{+} ∧ E_{%}(X) ⟶ E_{%}(Z × X) is an isomorphism, where Z is a space and X is a G-space;

4. The canonical map E_{%}(G/H) ⟶ E(G/H) is an isomorphism for all H ∈ F.

\underline{\underline{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 Gcofibrations 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 ↦ 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.

\underline{\underline{Proof}} : Let E ↦ E_{$} be another such extension. There is a (a priori not necessarily 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) ∈ map_{G}(G/H, X) × E(G/H) to E_{$}(x)(e). Since any G-F-CW -complex is constructed from orbits G/H with H ∈ F via products with disks, attaching a G-space to a G-space along a G-map, and colimits 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 ↦ E_{%} up to isomorphism. Next we give a homotopy theoretic characterization.

A covariant functor E : G-F-CW -COMPLEXES ⟶ SPECTRA is called (weakly) Fhomotopy invariant if it sends G-homotopy equivalences to (weak) homotopy equivalences 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 -complex X is the union of G-CW -subcomplexes X_{1} and X_{2} with intersection X_{0}, then the canonical map from the homotopy pushout of E(X_{2}) ⟶ E(X_{0}) ⟵ E(X_{2}), which is obtained by gluing the mapping cylinders together along E(X_{0}), to E(X) is a (weak) homotopy equivalence of spectra. Finally, E respects countable disjoint unions up to (weak) homotopy, i.e. the natural map ∨_{i∈I}E(X_{i}) ⟶ E(∐_{i∈I} X_{i}) is a (weak) homotopy equivalence for all countable index sets I. The last condition implies that the natural map from the homotopy colimit of the system E(X_{n}) 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 π_{q}(E(X)) defines a homology theory on the category of G-F-CW -complexes, satisfying the disjoint union axiom for countable disjoint unions.

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 equivalence of spectra for all H ∈ F. Then T(X) is a (weak) homotopy equivalence 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 -COMPLEXES to SPECTRA and there are natural transformations

A_{E} : E^{%} ⟶ E;
B_{E} : E^{%} ⟶ (E |_{Or(G,F)})_{%};

which induce (weak) homotopy equivalences of spectra A_{E}(G/H) for all H ∈ F and (weak) homotopy equivalences of spectra B_{E}(X) for all G-F-CW -complexes X. Given a family F′ ⊂ F, E is (weakly) F′-excisive if and only if A_{E}(X) is a (weak) homotopy equivalence of spectra for all G-F′-CW -complexes X.

\underline{\underline{Proof}} : 1.) follows from Lemma 6.1.

2.) Use the fact that a (weak) homotopy colimit of homotopy equivalences of spectra is again a (weak) homotopy equivalence of spectra.

3.) Define E^{%}(X) by the spectrum

map_{G}(− × Δ., X)_{d} ⊗_{Or(G,F)×Δ} B^{bar} ?↓Or(G, F) × Δ↓?? ⊗_{Or(G,F)×Δ} E(− × Δ.)

where − resp. . runs over Or(G) resp. Δ, the subscript d in map_{G}(− × Δ., X)_{d} indicates that we equip this mapping space in contrast to the usual convention with the discrete topology and B^{bar} ?↓Or(G, F) × Δ↓?? was introduced at the end of Section 3. Define the transformation A_{E}(X) : E^{%}(X) ⟶ E(X) by the following diagram

E(X)
c_{1}↑
map_{G}(− × Δ., X)_{d} ⊗_{Or(G,F)×Δ} E(− × Δ.)
p_{map_{G}(−×Δ.,X)_{d}}⊗id↑
map_{G}(− × Δ., X)_{d} ⊗_{Or(G,F)×Δ} B^{bar} ?↓Or(G, F) × Δ↓?? ⊗_{Or(G,F)×Δ} E(− × Δ.)

where p_{map_{G}(−×Δ.,X)_{d}} was introduced in Definition 3.18 and here and in the next diagram c_{k} refers to the canonical map whose definition is obvious from the context. Define the transformation B_{E}(X) : E^{%}(X) ⟶ (E |_{Or(G,F)})_{%}(X) by the following diagram

map_{G}(− × Δ., X)_{d} ⊗_{Or(G,F)×Δ} B^{bar} ?↓Or(G, F) × Δ↓?? ⊗_{Or(G,F)×Δ} E(− × Δ.)

id ⊗ id ⊗E(pr)↓
map_{G}(− × Δ., X)_{d} ⊗_{Or(G,F)×Δ} B^{bar} ?↓Or(G, F) × Δ↓?? ⊗_{Or(G,F)×Δ} E(−)
id ⊗c_{2}⊗id↓≅

map_{G}(− × Δ., X)_{d} ⊗_{Or(G,F)×Δ} B^{bar} ?↓Or(G, F)↓?? × B^{bar} ?↓Δ↓?? ⊗_{Or(G,F)×Δ} E(−)

c_{3}↓≅

(map_{G}(− × Δ., X)_{d} ⊗_{Δ} B^{bar} ?↓Δ↓?? ⊗_{Δ} {∗}) ⊗_{Or(G,F)} B^{bar} ?↓Or(G, F)↓?? ⊗_{Or(G,F)} E(−)

(id ⊗c_{4})⊗id ⊗ id↓≅

(map_{G}(− × Δ., X)_{d} ⊗_{Δ} B^{bar} ?↓Δ ) ⊗_{Or(G,F)} B^{bar} ?↓Or(G, F)↓?? ⊗_{Or(G,F)} E(−)

(c_{5}⊗id)⊗id ⊗ id↓≅

(map(Δ., map_{G}(−, X))_{d} ⊗_{Δ} B^{bar} ?↓Δ ) ⊗_{Or(G,F)} B^{bar} ?↓Or(G, F)↓?? ⊗_{Or(G,F)} E(−)

(id ⊗q)⊗id ⊗ id↓
(map(Δ., map_{G}(−, X))_{d} ⊗_{Δ} Δ.) ⊗_{Or(G,F)} B^{bar} ?↓Or(G, F)↓?? ⊗_{Or(G,F)} E(−)
a_{map_{G}(−,X)}⊗id↓
map_{G}(−, X) ⊗_{Or(G,F)} E(−)

where the canonical map q : B^{bar} ?↓Δ ⟶ Δ_{?} is defined in [4, Example XI.2.6 on page 293] and a_{map_{G}(−,X)} was introduced in Definition 3.18.

Next we show that B_{E}(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 × Δ_{n}) ⟶ E(G/H) is a (weak) homotopy equivalence for all H ∈ F. Hence the first map in the diagram above id ⊗ id ⊗E(pr) is a weak homotopy equivalence because of Theorem 3.11. The next four maps are all isomorphisms. The map

id ⊗q : map(Δ., map_{G}(−, X))_{d} ⊗_{Δ} B^{bar} ?↓Δ ⟶ map(Δ., map_{G}(−, X))_{d} ⊗_{Δ} Δ.

is a weak homotopy equivalence of Or(G, F)-spaces [4, XII.3.4 on page 331]. Because of Theorem 3.11 the map

(id ⊗q) ⊗ id : (map(Δ., map_{G}(−, X))_{d} ⊗_{Δ} B^{bar} ?↓Δ ) ⊗_{Or(G,F)} B^{bar} ?↓Or(G, F)↓??

⟶ (map(Δ., map_{G}(−, X))_{d} ⊗_{Δ} Δ) ⊗_{Or(G,F)} B^{bar} ?↓Or(G, F)↓??

is a weak Or(G, F)-homotopy equivalence of Or(G, F)-spaces. Since the domain and 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 (id ⊗q) ⊗ id ⊗ id in the diagram above is a homotopy equivalence.

As we assume that X is a G-F-CW -complex map_{G}(−, X) is a Or(G, F)-CW -complex. Since a_{map_{G}(−,X)} is a Or(G, F)-CW -approximation by Lemma 3.19 Corollary 3.5 implies that it is a homotopy equivalence of Or(G, F)-CW -complexes. Hence the last map in the diagram above a_{map_{G}(−,X)} ⊗ id is a homotopy equivalence. This shows that B_{E}(X) is a (weak) homotopy equivalence.

In the case X = G/H for H ∈ F the composition of the (weak) homotopy equivalence B_{E}(G/H) with the canonical isomorphism map_{G}(−, G/H) ⊗_{Or(G,F)} E(−) ⟶ E(G/H) agrees with A_{E}(G/H). Hence A_{E}(G/H) is a (weak) homotopy equivalence for all G/H with H ∈ F. This finishes the proof of Theorem 6.3.

The map A_{E} is called an assembly map for E.

Example 6.4 For a topological space X, the fundamental groupoid Π(X) is the category whose objects are points in X and whose morphism set mor_{Π(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 spaces gives a natural equivalence of fundamental groupoids. If X is path-connected and x_{0} ∈ X, then the inclusion of the fundamental group π_{1}(X, x_{0}) ⟶ Π(X) is a natural equivalence of groupoids.

Let K^{alg} : GROUPOIDS ⟶ SPECTRA be the functor from Section 2. By Lemma 2.4, K^{alg} has the property that a natural equivalence of groupoids gives a homotopy equivalence of spectra.

One can define a homotopy invariant functor E : CW -COMPLEXES ⟶ SPECTRA by E(X) = K^{alg}(Π(X)). We apply Theorem 6.3 in the case where G is the trivial group (note that for G = 1, Theorem 6.3 is due to Weiss-Williams [42]). The map B_{E} gives a homotopy equivalence from E^{%}(X) to X_{+} ∧ K^{alg}(ℤ), where K^{alg}(ℤ) is the algebraic K-spectrum of the ring ℤ. After one applies the n-th homotopy group to the assembly map

A_{E} : E^{%}(X) ⟶ E(X)

one obtains the algebraic K-theory assembly map

A : H_{n}(X; K^{alg}(ℤ)) ⟶ K_{n}^{alg}(ℤπ_{1}X).

Next consider a discrete group G and a family of subgroups F. One can then define an F-homotopy invariant functor

E : G-CW - COMPLEXES ⟶ SPECTRA

by setting E(X) = K^{alg}(Π(EG ×_{G} X)). If X is simply-connected, there is a natural equivalence of groupoids

G = Or(G, 1) ⟶ Π(EG ×_{G} X).

Using this identification, we have a fourth point of view on the (K^{alg}, F, G)-assembly map, namely it is

π_{∗}(A_{E}(E(G, F))) : π_{∗}(E^{%}(E(G, F))) ⟶ π_{∗}(E(E(G, F))).

The case of algebraic L-theory is analogous. For a map of spaces X ⟶ Y, the map of groupoids Π(X) ⟶ Π(Y) need not be a morphism in GROUPOIDS^{inj}. However, all relevant maps in the definition of A_{E} and B_{E} have this property, so that the analogous point of view holds also for the topological K-theory of C^{∗}-algebras.

Next we give for a covariant Or(G)-spectrum E an equivalent definition of E^{%} which is closer to the construction in [42]. Let simp_{G}(X) be the category having as morphisms pairs (G/H × [n], σ) which consists of an object G/H × [n] in Or(G, F) × Δ and a Gmap σ : G/H × Δ_{n} ⟶ X. A morphism from (G/H × [n], σ) to (G/K × [m], τ) is a morphism f × u : G/H × [n] ⟶ G/K × [m] in Or(G, F) × Δ such that the composition of the induced map G/H × Δ_{n} ⟶ G/K × Δ_{m} with τ is σ. This is the equivariant version of the construction in [35, Appendix A] applied to the simplicial set S.X associated to a space X. We get a covariant functor E(− × Δ.) from simp_{G}(X) to SPECTRA by (G/K × [m], σ) ↦ E(G/K × Δ_{m}) We briefly indicate how one can identify

E^{%}(X) = hocolim_{simp_{G}(X)} E(− × Δ.).

Let P : simp_{G}(X) ⟶ Or(G) × Δ be the obvious forgetful functor. It suffices to construct a natural isomorphism of Or(G) × Δ-spaces

B^{bar} ?↓simp_{G}(X) ⊗_{simp_{G}(X)} mor_{Or(G)×Δ}(??, P (?)) ⟶
map_{G}(− × Δ., X) ⊗_{Or(G)×Δ} B^{bar} ??↓Or(G) × Δ↓ − × . .

It will be implemented by the following natural bijection of simplicial sets for a given object G/K × [m] in Or(G) × Δ where p runs over 0, 1, 2, …

N_{p}?↓simp_{G}(X) ⊗_{simp_{G}(X)} mor_{Or(G)×Δ}(G/K × [m], P (?)) ⟶

map_{G}(− × Δ., X) ⊗_{Or(G)×Δ} N_{p} G/K × [m]↓Or(G) × Δ↓ − × . .

An element in the source is represented for ? = (G/H × [n], σ) by the pair

((G/H × [n], σ) ⟶ (G/H_{0} × [n_{0}], σ_{0}) ⟶ ⋯ ⟶ (G/H_{p} × [n_{p}], σ_{p}))
× (G/K × [m] ⟶ G/H × [n]) .

It is sent to the element in the target represented by

(σ_{p} : G/H_{p} × Δ_{n_{p}} ⟶ X)
× (G/K × [m] ⟶ G/H × [n] ⟶ G/H_{0} × [n_{0}] ⟶ ⋯ ⟶ G/H_{p} × [n_{p}]) .

This is indeed a bijection since G/H_{0} × [n_{0}] ⟶ ⋯ ⟶ G/H_{p} × [n_{p}] and σ_{p} determine σ_{0}, …, σ_{p−1}.

Next we explain why Theorem 6.3 characterizes the assembly map in the sense that A_{E} : E^{%} ⟶ E is the universal approximation from the left by a (weakly) F-excisive functor of a (weakly) F-homotopy invariant functor E from G-F-CW -COMPLEXES to SPECTRA. The argument is the same as in [42, page 336]. Namely, let T : F ⟶ E be a transformation of functors from G-F-CW -COMPLEXES to SPECTRA such that F is (weakly) F-excisive and T(G/H) is a (weak) homotopy equivalence for all H ∈ F. Then for any G-F-CW - complex X the following diagram commutes

F^{%}(X) ⟶^{A_{F}(X)} F(X)
T^{%}(X)↓  ↓^{T(X)}
E^{%}(X) ⟶^{A_{E}(X)} E(X)

and A_{F}(X) and T^{%}(X) are (weak) homotopy equivalences. Hence one may say that T(X) factorizes over A_{E}(X).

One may be tempted to define a natural transformation S : E_{%} ⟶ E as indicated in the proof of Lemma 6.2. Then S(X) is a well-defined bijection of sets but is not 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 -complexes X and Y. The construction above uses the (weak) F-homotopy invariance of E instead.

Finally we say how one can identify the Baum-Connes-map of [3] with the map induced on homotopy groups by the assembly map

(K^{top})_{%}(E(G, FIN)) ⟶ (K^{top})_{%}(G/G).

This problem has been considered by many people including Baum, Bloch, Carlsson, Comezana, Higson, Pedersen, Roe, and Stolz. Our characterization of the assembly map allows the proof of this identification for all possible models of K^{top} if it has been done for one model. We shall use the following construction due to Carlsson, Pedersen, and Roe [6].

There is a functor P from the category of G-spaces to the category of spectra with G-action and strong maps as morphisms such that the functor

Q : G-CW - COMPLEXES ⟶ SPECTRA

obtained from P by taking the G-fixed point set of G-spectra has the following properties.

1. There are identifications of the homotopy groups of Q(E(G, FIN)) with the source of the Baum-Connes map and of Q(G/G) with the target of the Baum-Connes map such that the Baum-Connes map itself agrees with the map induced by the projection Q(E(G, FIN)) ⟶ Q(G/G);

2. There is a weak equivalence of Or(G)-spectra

U : K^{top} ⟶ Q|_{Or(G)}.

3. Q is weakly FIN -excisive.

Then we obtain from Theorem 6.3 for F the family of all subgroups and F′ = FIN a commutative diagram of spectra whose vertical maps are all weak homotopy equivalences.

(K^{top})_{%} (E(G, FIN)) ⟶ (K^{top})_{%} (G/G)
U_{%}(E(G,FIN))↓  ^{U_{%}(G/G)}↓
(Q|_{Or(G)})_{%} (E(G, FIN)) ⟶ (Q|_{Or(G)})_{%} (G/G)
B_{Q}(E(G,FIN))↑  ^{B_{Q}(G/G)}↑
Q^{%}(E(G, FIN))  ⟶  Q^{%}(G/G)
A_{Q}(E(G,FIN))↓  ^{A_{Q}(G/G)}↓
Q(E(G, FIN))  ⟶  Q(G/G)

Hence the map induced on homotopy groups by the top horizontal arrow is the Baum-Connes map.

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 correspondence 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 group and F a non-empty family of subgroups closed under conjugation and inclusion. A G-space 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 ∈ F and whose morphisms 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. [11, Theorem 3.11], [13], [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 ↦ map_{G}(G/H, Y) = Y^{H}.

Let ∇ be the covariant Or(G, F)-space given by sending G/H to itself. Given a contravariant Or(G, F)-space X define the associated left G-F-space X̂ by

X̂ = X ⊗_{Or(G,F)} ∇.

The left action of an element g ∈ G is given by id ⊗_{Or(G,F)}L_{g} where L_{g} : G/H ⟶ G/H is the map of covariant Or(G, F)-spaces given by left multiplication with g.

The notation for the functor ∇ is intended to be reminiscent of the cosimplicial space Δ. from Example 1.7.

Lemma 7.2 The functors in Definition 7.1 are adjoint, i.e. for a contravariant Or(G, F)- space X and a left G-space Y there is a natural homeomorphism

T (X, Y) : map_{G}(X̂, Y) ⟶ hom_{Or(G,F)}(X, map_{G}(−, Y)).

\underline{\underline{Proof}} : If we neglect the G-action on Y, we get from Lemma 1.5 a natural homeomorphism

map(X̂, Y) ⟶ hom_{Or(G,F)}(X, map(−, Y)).

Using the transformations L_{g} and the G-action on Y one defines appropriate G-actions on the source and target of this map and checks that this map is G-equivariant. Hence 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) × G/H ⟶ Y. It sends (x, gH) to the value of f(G/H)(x) at gH.

Lemma 7.3 The map

f : X(G/1) ⟶ X̂ x ↦ [x, 1]

is a G-homeomorphism.

\underline{\underline{Proof}} : The inverse f^{−1} : X̂ ⟶ X(G/1) assigns to an element represented by (x, gH) the element X(q_{gH})(x) where q_{gH} : G/1 ⟶ G/H sends g′ to g′gH.

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 the following

Theorem 7.4 1. Given a left G-F-space Y, the adjoint of the identity on map_{G}(−, Y) under the adjunction of Lemma 7.2 is a natural G-homeomorphism

T (Y) : { \atop map_{G}(−, Y)} ⟶ Y.

It is induced by the map

∐_{H∈F} map_{G}(G/H, Y) × G/H ⟶ Y, (ϕ, gH) ↦ ϕ(gH),

2. Given a contravariant Or(G, F)-space X, the adjoint of the identity on X̂ under the adjunction of Lemma 7.2 is a natural map of Or(G, F)-spaces

S(X) : X ⟶ map_{G}(−, X̂).

Given H ∈ F, the map S(X)(G/H) maps the element x ∈ X(G/H) to the element in map_{G}(G/H, X̂) = (X ⊗_{Or(G,F)} ∇)^{H} represented by (x, eH) ∈ X(G/H) × G/H. It is an isomorphism of Or(G, F)-spaces if and only if for each H ∈ F the projection pr : G/1 ⟶ G/H induces a homeomorphism X(pr) : X(G/H) ⟶ X(G/1)^{H}. This 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 map_{G}(−, Y) based at the object G/H. The analogous statement holds for a free Or(G, F)-CW -complex X and X̂.

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 and 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 contractible for H ∈ F and empty otherwise.

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 [9],[10, 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 associated 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 { \atop E Or(G, F)} is a classifying space of G with respect to F.

Example 7.7 Sometimes geometry yields small examples of classifying spaces and resolutions. We have already mentioned this in the case where G is a crystallographic group. Generalizing this, let G be a discrete subgroup of a Lie group L with a finite number of components. If K is a maximal compact subgroup of L, then L/K is homeomorphic to ℝ^{n} and L/K can be taken as a model for E(G, FIN), where FIN is the family of finite subgroups. Generalizing further, let G be a group of finite virtual cohomological dimension. Then there is finite-dimensional classifying space E(G, FIN) (see [36, Proposition 12]) and hence a finite dimensional model for E Or(G, FIN). Many examples of such groups are discussed by Serre in [36]. More examples of nice geometric models for E(G, FIN) can be found in [3, section 2].

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Addresses

James F. Davis Wolfgang Lück Department of Mathematics Institut für Mathematik Indiana University Westfaelische Wilhelms-Universtitaet Bloomington, IN 47405 48149 Muenster U.S.A. Bundesrepublik Deutschland email: jfdavis@indiana.edu lueck@math.uni-muenster.de internet: http://wwwmath.uni-muenster.de/math/u/lueck/ FAX: 812-855-0046 0251 8338370