COMMUTING HOMOTOPY LIMITS AND SMASH PRODUCTS

WOLFGANG LÜCK, HOLGER REICH, AND MARCO VARISCO

Abstract. In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute. In this paper we investigate under what additional assumptions these two processes do commute. In fact we deal with an equivariant generalization that involves spectra and smash products over the orbit category of a discrete group. Such a situation naturally occurs if one studies the equivariant homology theory associated to topological cyclic homology. The main theorem of this paper will play a role in the generalization of the results obtained by Bökstedt, Hsiang and Madsen about the algebraic K-theory Novikov Conjecture to the assembly map for the family of virtually cyclic subgroups.

to Hyman Bass on his seventieth birthday

1. Introduction

Let I be a small category and let E : I → Ω-SPECTRAbe a contravariant functor, then for any space X there is a natural map

X_{+} ∧ holim_{I} E → holim_{I} X_{+} ∧ E.

In general this map is not a weak equivalence. A special case of our main result says that the map is an equivalence if we assume that:

(i) The spectra E(c) for objects c in I are uniformly bounded below.

(ii) The homology of X is degreewise finitely generated as an abelian group. (iii) The category I admits a finite dimensional contravariant classifying space

(compare Definition 3.4). In fact we will generalize the question in several directions. On the one hand

we will work throughout in an equivariant setting. On the other hand we will investigate general mapping space constructions, of which homotopy limits are just special cases. Moreover we will also try to obtain weaker assumptions that still suffice to conclude that the map induces an isomorphism on rationalized homotopy groups. And finally we will discuss a chain complex analogue of our result.

In order to state our main result we introduce some more notation. Let G be a discrete group. Let Or(G) be the orbit category, whose objects are the homogeneous left G-spaces G/H and whose morphisms are G-maps. Every left G-space X, in particular every G-CW-complex (compare Section 8), gives rise to a contravariant Or(G)-space, i.e. a contravariant functor from the orbit category to spaces, by sending G/H to X^{H} = map_{G}(G/H, X). If we furthermore have a covariant functor E from Or(G) to SPECTRA(from now on called an Or(G)-spectrum) then we can form the smash product over the orbit category

X_{+} ∧_{Or (G)} E

Date: August 21, 2003.

2000 Mathematics Subject Classification. 55P42, 55P91.

Key words and phrases. Homotopy inverse limits of spectra, smash products, orbit category.

to obtain an ordinary spectrum (compare Section 3). This is an important construction since sending X to π_{∗}(X_{+} ∧_{Or (G)} E) yields an equivariant homology theory.

Our basic conventions concerning the categories SPECTRAand Ω-SPECTRAcan be found in Section 2. In particular we will denote by

R : SPECTRA→ Ω-SPECTRA

a fibrant replacement functor that replaces a spectrum by a weakly equivalent Ωspectrum, and r: E → R(E) will denote the natural equivalence from a spectrum to its replacement, compare (2.2).

The homotopy limit of a contravariant functor E : I → SPECTRAis defined as

holim_{I} E = map_{I}(EI_{+}, R ∘ E),

see Definition 3.6. (If E takes already values in Ω-SPECTRA , then one can work with E instead of R ∘ E, but in general the replacement is crucial.) The I-mapping spectrum construction is explained in Section 3 and EI is the contravariant classifying space of the category I (see Definition 3.4). In particular EI is a contravariant I-CW -complex, i.e. a contravariant functor from I to spaces which is built up out of free cells (see Definition 3.3). Even if one is eventually only interested in homotopy limits it is important for the proof of our main result to deal with other I-CW -complexes in place of EI, because the proof will proceed via induction over the skeleta.

We will now state the question we want to investigate in full generality. Let I be a small category and Y a contravariant I-CW -complex. Furthermore let

E: I^{op} × Or(G) → Ω-SPECTRA

be a functor and X a G-CW -complex. There are maps of spectra t′ and t′′, which are natural in X, Y and E. They are given as follows:

t′: X_{+} ∧_{Or (G)} map_{I}(Y_{+}, E) → map_{I}(Y_{+}, X_{+} ∧_{Or (G)} E)

sends x ∧ ϕ to the map from Y to X_{+} ∧ E which is given by y ↦ x ∧ ϕ(y) and

t′′: map_{I}(Y_{+}, X_{+} ∧_{Or (G)} E) → map_{I} (Y_{+}, R (X_{+} ∧_{Or (G)} E))

is induced by the map r (X_{+} ∧_{Or (G)} E), the weak equivalence to the fibrant replacement.

Question 1.1. Under what assumptions is the map

t: X_{+} ∧_{Or (G)} map_{I}(Y_{+}, E) → map_{I} (Y_{+}, R (X_{+} ∧_{Or (G)} E)) ,

defined as the composition t′′ ∘ t′, a weak equivalence?

We still need to explain a mild condition that we will always impose on our functor E. Let GROUPOIDS denote the category of small groupoids and functors between them. For a G-set S let G^{G}(S) denote the transport groupoid, whose set of objects is S and for which the set of morphisms from s_{1} to s_{2} consists of {g ∈ G|gs_{1} = s_{2}}. Composition of morphisms is induced from the group structure on G. We obtain a covariant functor G^{G}: Or(G) → GROUPOIDS by assigning G^{G}(G/H) to G/H. We call a functor F: GROUPOIDS → SPECTRAa homotopy functor if it sends an equivalence of groupoids to a weak equivalence of spectra. It seems that all interesting examples of Or(G)-spectra can be obtained by composing a suitable homotopy functor F : GROUPOIDS → SPECTRAwith the functor G^{G}. For a subgroup H ⊂ G we denote by Z_{G}H = {g ∈ G | gh = hg for all h ∈ H} its centralizer.

Theorem 1.2. Consider the following data:

• A functor E: I^{op} × Or(G) → Ω-SPECTRAwhich factorizes as

E = F ∘ (id ×G^{G}),

where F is such that for each object c in I the functor F(c, −) is a homotopy functor.

• A contravariant I-CW complex Y.

• A G-CW complex X. Suppose that there are numbers d, n and N ∈ ℤ with d ≥ 0 such that the following conditions are satisfied:

(A) Y is d-dimensional.

(B) The spectra E(c, G/H) are uniformly (N −1)-connected, i.e. for all objects c in I and all orbits G/H we have π_{q}(E(c, G/H)) = 0 for q < N. (C) The G-CW complex X has only finite isotropy groups and only finitely

many orbit types. (D) For each finite subgroup H ⊂ G and every p ≤ n + d − N the homology

group

H_{p}(Z_{G}H\X^{H}; ℤ)

is a finitely generated ℤ-module. Then for each p ≤ n the map

π_{p}(t): π_{p} (X_{+} ∧_{Or (G)} map_{I}(Y_{+}, E)) → π_{p} (map_{I} (Y_{+}, R (X_{+} ∧_{Or (G)} E)))

is an isomorphism.

This theorem will be proven in Section 5. In Example 1.9 we discuss that none of the assumptions can be dropped. A chain complex version of Theorem 1.2 will be discussed in Section 6.

If one is only interested in the question whether the map t induces an isomorphism on rational homotopy groups one can obtain results where assumption (D) is replaced by a weaker assumption. A very naive guess here would be to require the rationalized homology groups to be finitely generated as ℚ-vector spaces. But this is too weak, compare Remark 5.6. In order to formulate the weakest possible assumption and strongest conclusion we introduce the notions of almost finitely generated modules and almost isomorphisms, compare Section 4.

A ℤ-module M is almost trivial if there is an element r ∈ ℤ, r ≠ 0 such that rm = 0 holds for all m ∈ M. A ℤ-module M is almost finitely generated if M/ tors(M) is a finitely generated ℤ-module and tors(M) is almost trivial. A ℤ-homomorphism is an almost isomorphism if its kernel and cokernel are almost trivial. An almost isomorphism becomes an isomorphism after rationalization.

Addendum 1.3. If we weaken assumption (D) in Theorem 1.2 by requiring the homology groups to be almost finitely generated ℤ-modules, then we obtain the conclusion that the map π_{p}(t) is an almost isomorphism and in particular a rational isomorphism for all p ≤ n.

As already explained we obtain results about homotopy limits by specializing to classifying spaces. For the convenience of the reader we formulate the corresponding result explicitly.

Corollary 1.4. If I admits a finite dimensional model for its contravariant classifying space EI, then there is a natural equivalence

X_{+} ∧_{Or (G)} holim_{I} E → holim_{I} X_{+} ∧_{Or (G)} E

for a functor E : I^{op} × Ω-SPECTRAas in Theorem 1.2 and a G-CW complex X provided that:

(B) The spectra E(c, G/H) are uniformly bounded below.

(C) The G-CW complex X has only finite isotropy groups and only finitely

many orbit types. (D) For every finite subgroup H ⊂ G and every n the homology group

H_{n}(Z_{G}H\X^{H}; ℤ)

is a finitely generated ℤ-module.

Of course there is again a version with an “almost finitely generated”-assumption and correspondingly an “almost isomorphism”-conclusion for the map induced on homotopy groups. Also there is the analogous refinement specifying the range of dimensions where we have to impose the finiteness conditions in order to obtain the conclusion for a given dimension.

In the case where G is the trivial group assumption (C) is redundant, assumption (D) simply says that the ordinary homology groups of X are degreewise finitely generated ℤ-modules and the conclusion is about the map

X_{+} ∧ holim_{I} E → holim_{I} X_{+} ∧ E.

Example 1.5. An important example is the case where the category is N, the category whose objects are the non-negative integers and for which mor_{N} (i, j) is empty if i > j, and consists of one element if i ≤ j. In that case a contravariant functor E : N → SPECTRAis simply an inverse system of spectra

E_{0} ← E_{1} ← E_{2} ← ….

There exists a 1-dimensional model for the contravariant classifying space EN, compare Section 7.

Example 1.6. Illuminating is also the extreme case where the only morphisms in I are the identities. In that case I can be identified with an index set I and a contravariant functor I → Ω-SPECTRAis just a family of Ω-spectra {E_{i} | i ∈ I} indexed by I. In the case where G is the trivial group the above specializes to results about the map

X_{+} ∧ ∏_{i∈I} E_{i} → ∏_{i∈I} R (X_{+} ∧ E_{i})

and is a variation on [Ada95, Theorem 15.2 on page 326].

Now we would like to briefly explain the concrete situation that motivated the investigations in this paper. In [LRRV03] we study to what extent the methods used by Bökstedt, Hsiang and Madsen in [BHM93] to prove the algebraic K-theory Novikov Conjecture can be applied to prove rational injectivity results for generalized assembly maps in algebraic K-theory. The main tool in both cases is to use the so called cyclotomic trace to topological cyclic homology in order to detect elements in K-theory. The K-theoretic Novikov Conjecture says that the “classical” assembly map corresponding to the family consisting of the trivial subgroup is rationally injective. Using generalized assembly maps we can detect a much bigger portion in K-theory. The method of comparison via the cyclotomic trace thus naturally leads to study generalized assembly maps for topological cyclic homology:

EG(F)_{+} ∧_{Or (G)} T C(A(?); p) → T C(A(G); p).

Here EG(F) is the classifying space for a family of subgroups F (compare Section 8), A is a fixed coefficient ring(spectrum), and for a fixed prime p the Or(G)- spectrum T C(A(?); p) is defined as a certain homotopy limit

T C(A(?); p) = holim_{RF} T HH(A(?))^{C_{p^{n}}} .

The category RF is described in Section 7 and we verify in Proposition 7.3 that it has a 2-dimensional model for its contravariant classifying space. Moreover the spectra T HH(A(G/H))^{C_{p^{n}}} are uniformly bounded below, so that conditions (A) and (B) are satisfied. In order to study the generalized assembly map it is hence particularly important to understand the conditions (C) and (D) in the case where X is the classifying space of a family of subgroups. The following proposition, which is proven in Section 8, says that for certain families the conditions can be formulated entirely in terms of group homology.

Proposition 1.7. Let F be a family of finite subgroups of G such that the set of conjugacy classes (F) = {(H) | H ∈ F} is finite.

If H_{n}(BZ_{G}H; ℤ) is an almost finitely generated ℤ-module for all n ≥ 0 and all H ∈ F, then the G-CW -complex EG(F) has finite isotropy groups and only finitely many orbit types, and for any finite group H ⊂ G and any n ≥ 0 the homology group H_{n}(Z_{G}H\EG(F)^{H}; ℤ) is an almost finitely generated ℤ-module.

Remark 1.8. The notions “almost finitely generated ℤ-module” and “almost isomorphism” allow to prove simultaneously a result about commuting homotopy limits and smash products as in Addendum 1.3 and the above Proposition 1.7.

If one drops “almost”, then our main Theorem 1.2 works, but the proof of a corresponding version of Proposition 1.7 breaks down, because then EG(F) may have infinitely many equivariant cells. One could get away with the stronger assumption that EG(F) has a model with finite skeleta. But that is a lot more restrictive and often not true in the applications we have in mind.

On the other hand a version of Proposition 1.7 where the assumption and the conclusion is that the rationalized homology groups are finitely generated as ℚmodules does exist. But under such an assumption one cannot prove an analogue of Theorem 1.2, compare Remark 5.6.

We finish this introduction with an example that shows that no proper subset of the conditions appearing in Theorem 1.2 is sufficient.

Example 1.9. Let G be the trivial group and I be the trivial category. Fix two sequences of integers 0 ≤ m_{0} ≤ m_{1} ≤ m_{2} ≤ … and 0 ≤ n_{0} ≤ n_{1} ≤ n_{2} ≤ . . .. Put X_{+} = ⋁_{i=0}^{∞} S^{m_{i}} and Y_{+} = (∐_{j=0}^{∞} S^{n_{j}})_{+}. (Notice that here we consider for simplicity pointed spaces instead of spaces with a disjoint base point added). Let E be an Ω-spectrum. The transformation t appearing in Question 1.1 can be identified with the map

⋁_{i=0}^{∞} S^{m_{i}} ∧ ∏_{j=0}^{∞} map(S^{n_{j}}, E) → ∏_{j=0}^{∞} map (S^{n_{j}}, ⋁_{i=0}^{∞} S^{m_{i}} ∧ E)

→ ∏_{j=0}^{∞} map (S^{n_{j}}; R (⋁_{i=0}^{∞} S^{m_{i}} ∧ E)) .

Recall that for a spectrum F we have π_{p}(S^{n} ∧F) = π_{p−n}(F) and π_{p}(map(S^{n}, F)) = π_{p+n}(F), and for a collection {X_{i} | i ∈ I} of pointed CW -complexes the canonical map

⨁_{i∈I} π_{p} (X_{i} ∧ F) ⟶^{≅} π_{p} (⋁_{i∈I} X_{i} ∧ F)

is bijective. Using Lemma 2.3 and Lemma 3.9 one can identify the map above with the canonical map

⨁_{i=0}^{∞} ∏_{j=0}^{∞} π_{n_{j}−m_{i}+p}(E) → ∏_{j=0}^{∞} ⨁_{i=0}^{∞} π_{n_{j}−m_{i}+p}(E).

This map is always injective. But it is surjective if and only if there exists i_{0} such that for all i ≥ i_{0} and all j we have π_{n_{j}−m_{i}+p}(E) = 0. Note that condition (A) corresponds to the existence of a constant D such that n_{j} ≤ D for all j. Condition (B) says that there exists a constant N such that π_{q}(E) = 0 for q ≤ N. Condition (C) is redundant in the non-equivariant case and condition (D) is satisfied if and only if lim_{i→∞} m_{i} = ∞. It is then easy to find examples showing that we cannot drop any of the assumptions (A), (B) or (D).

2. Basics about spectra

The question we are studying is not even well posed in the stable homotopy category of spectra. We are hence forced to explain what we mean by a spectrum, and in what sense we understand the basic constructions concerning spectra.

Throughout this paper we work in the category of compactly generated spaces (see [Ste67], [Whi78, I.4]). So space means compactly generated space and all constructions like mapping spaces and products are to be understood in this category. We will always assume that the inclusion of the base point into a pointed space is a cofibration and that maps between pointed spaces preserve the base point.

We define the category SPECTRAas follows. A spectrum E = {(E(n), σ(n)) | n = 0, 1, 2, … } is a sequence of pointed spaces {E(n) | n = 0, 1, 2, … } together with pointed maps (called structure maps) σ(n) : E(n) ∧ S^{1} → E(n + 1). A map of spectra (sometimes also called function in the literature) f : E → E′ is a sequence of maps of pointed spaces f(n): E(n) → E′(n) that 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. This should not be confused with the notion of map of spectra in the stable homotopy category (see [Ada95, III.2]). Recall that the homotopy groups of a spectrum are defined as

π_{p}(E) = colim_{k→∞} π_{p+k}(E(k)),

where the maps in this system are given by the composition

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

of the suspension homomorphism and the homomorphism induced by the structure map. A weak equivalence of spectra is a map f : E → F inducing an isomorphism on all homotopy groups. A spectrum E is called Ω-spectrum if the adjoint of each structure map

\overline{σ}(n): E(n) → ΩE(n + 1) = map(S^{1}, E(n + 1))

is a weak homotopy equivalence of spaces. We denote by Ω-SPECTRAthe corresponding full subcategory of SPECTRA .

In order to define the covariant fibrant replacement functor

R: SPECTRA→ Ω-SPECTRA

we first need to recall the definition and a key property of sequential homotopy colimits. Consider a sequence of pointed spaces and maps X_{0} → X_{1} → X_{2} → … This can be thought of equivalently as a covariant functor X from the category N to pointed spaces (see Example 1.5). Its homotopy colimit is defined as the pointed space

hocolim_{n→∞} X_{n} = hocolim_{N} X = EN_{+} ∧_{N} X,

compare Definition 3.6. Using the model for EN described in Example 7.1 one gets the well-known reduced mapping telescope construction. Since any map from a compact space to hocolim_{n→∞} X_{n} factors for some N through the natural map X_{N} → hocolim_{n→∞} X_{n}, we have that for any k ≥ 0

(2.1)  π_{k}(hocolim_{n→∞} X_{n}) ≅ colim_{n→∞} π_{k}(X_{n}).

Now for any spectrum E we define its fibrant replacement as the spectrum RE whose k-th pointed space is hocolim_{n→∞} Ω^{n}E(n + k). Here the homotopy colimit is taken with respect to the system of maps

Ω^{n}\overline{σ}(n + k): Ω^{n}E(n + k) → Ω^{n}ΩE(n + k + 1) = Ω^{n+1}E((n + 1) + k)

where \overline{σ}(k) : E(k) → ΩE(k + 1) is the adjoint of the structure map of E. The adjoints of the structure maps of RE are given by the composition

hocolim_{n→∞} Ω^{n}E(n + k) ⟶^{hocolim_{n→∞} Ω^{n}\overline{σ}(k+n)} hocolim_{n→∞} Ω^{n}ΩE(n + k + 1)

= hocolim_{n→∞} ΩΩ^{n}E(n + (k + 1)) ⟶^{μ} Ω hocolim_{n→∞} Ω^{n}E(n + (k + 1)),

where the first map is the obvious weak equivalence induced by shifting the system, and μ is the canonical map, which is also a weak equivalence as one sees applying (2.1) above. Hence RE is indeed an Ω-spectrum. There is a natural map

r(E): E → RE (2.2)

coming from the canonical maps E(k) → hocolim_{n→∞} Ω^{n}E(n+k). Using (2.1) one computes that r(E) is a weak equivalence.

Many constructions for spaces can be carried over to spectra by applying them levelwise, i.e. to each of the individual spaces E(n). Usually there is then an obvious way to define either the structure maps or their adjoints. For example the product ∏_{i∈I} E_{i} of a collection of spectra {E_{i} | i ∈ I} is the spectrum whose n-th space is ∏_{i∈I} E_{i}(n). For the structure maps observe that Ω commutes with products. For us the main reason to consider Ω-spectra is the following lemma, which will play an important role below and does not hold for arbitrary spectra.

Lemma 2.3. Let {E_{i} | i ∈ I} be a family of Ω-spectra, where I is an arbitrary index set. Then the canonical map induced by the various projections pr_{i}: ∏_{i∈I} E_{i} → E_{i}

π_{n} (∏_{i∈I} E_{i}) ⟶^{≅} ∏_{i∈I} π_{n}(E_{i})

is bijective for all n ∈ ℤ.

Proof. Notice that for an Ω-spectrum F the canonical map

π_{n}(F (0)) → π_{n}(F) = colim_{k→∞} π_{n}(Ω^{k}F (n + k))

is bijective and that ∏_{i∈I} E_{i} is an Ω-spectrum. Hence the claim follows from the corresponding statement for spaces. □

The following example shows that Lemma 2.3 does not hold without the assumption that each E_{i} is an Ω-spectrum.

Example 2.4. Let E be an Ω-spectrum such that A = π_{0}(E) ≅ π_{k}(E(k)) is a nontrivial abelian group. Denote by E_{(i,∞)} the spectrum obtained from E by replacing the spaces E(0), E(1), …, E(i) by the one-point space {pt}. We have

π_{k}(E_{(i,∞)}(k)) = \{ \begin{matrix}0 & if k ≤ i \\ A & if k > i \end{matrix}

and the maps π_{k}(E_{(i,∞)}(k)) → π_{k+1}(E_{(i,∞)}(k + 1)) are isomorphisms for k > i. We see that for all i we have π_{0}(E_{(i,∞)}) = A and

π_{0}(∏_{i∈ℕ} E_{(i,∞)}) = colim_{k} π_{k}(∏_{i∈ℕ} E_{(i,∞)}(k)) = colim_{k} ⨁_{i=1}^{k} A = ⨁_{i=1}^{∞} A.

The natural map

π_{0}(∏_{i∈ℕ} E_{(i,∞)}) → ∏_{i∈ℕ} π_{0}(E_{(i,∞)})

can be identified with the natural inclusion ⨁_{i∈ℕ} A → ∏_{i∈ℕ} A and is not an isomorphism.

Now we fix some notations and conventions about homotopy pullbacks and homotopy pushouts of spectra. Consider a commutative square D_{E} of spectra

\xymatrix{E_{0} \ar[r]^{f_{1}} \ar[d]_{f_{2}} & E_{1} \ar[d]^{g_{1}} \\ E_{2} \ar[r]_{g_{2}} & E_{12}.}

We denote by hopb(E_{2} → E_{12} ← E_{1}) the levelwise homotopy pullback spectrum and by hopo(E_{2} ←E_{0} →E_{1}) the levelwise homotopy pushout spectrum, i.e. the kth spaces are given by the homotopy pullback respectively the homotopy pushout of the corresponding diagrams of pointed spaces. For the structure maps use the fact that homotopy pullbacks commute with Ω and homotopy pushouts commute with S^{1} ∧ − up to natural homeomorphisms. There are canonical maps of spectra

a: E_{0} → hopb(E_{2} →E_{12} ←E_{1}), b: hopo(E_{2} ←E_{0} →E_{1}) → E_{12}.

We call D_{E} homotopy cartesian if the map a is a weak equivalence of spectra; dually we call D_{E} homotopy cocartesian if b is a weak equivalence. We use the analogous terminology for spaces.

Remark 2.5. A homotopy pullback is a special case of a homotopy limit (the indexing category is 2 ← 12 → 1 for contravariant functors). In Definition 3.6 below, we discuss that in order to define a homotopy limit of a diagram of spectra, one should first replace all spectra in sight with Ω-spectra by applying the fibrant replacement functor R, and then take the levelwise homotopy limit. At first glance this seems to be inconsistent with the definition of homotopy pullback given above. But Lemma 2.6 below implies in particular that the canonical map r induces a weak equivalence of spectra

hopb(E_{2} →E_{12} ←E_{1}) → hopb(RE_{2} →RE_{12} ←RE_{1}).

Finally we recall the well-known fact that a commutative square of spectra is homotopy cartesian if and only if it is homotopy cocartesian. We would like to stress that it is important for later proofs to have such a statement for hopb(E_{2} → E_{12} ← E_{1}) as opposed to hopb(RE_{2} → RE_{12} ← RE_{1}). We have not been able to find an explicit reference (in particular none dealing with this extra subtlety) in the literature, therefore we include a proof.

Lemma 2.6. A commutative square D_{E} of spectra is homotopy cocartesian if and only if it is homotopy cartesian. In this case there is a natural long exact Mayer-Vietoris sequence

… ⟶^{∂_{n+1}} π_{n}(E_{0}) ⟶^{π_{n}(f_{1})⊕π_{n}(f_{2})} π_{n}(E_{1}) ⊕ π_{n}(E_{2})

⟶^{π_{n}(g_{1})−π_{n}(g_{2})} π_{n}(E_{12}) ⟶^{∂_{n}} π_{n−1}(E_{0}) ⟶^{π_{n−1}(f_{1})⊕π_{n−1}(f_{2})} …

Example 2.7. A special case of this Lemma yields the following well-known fact: For any pair of spectra E_{1} and E_{2} the natural map E_{1} ∨ E_{2} → E_{1} × E_{2} is a weak equivalence.

Proof of Lemma 2.6. The key step in the proof is to show the following Claim: If a commutative square of spectra is levelwise a homotopy cocartesian

square of spaces, then it is homotopy cartesian. We first explain why this claim implies the lemma. Any levelwise homotopy cartesian square of spectra yields a natural long exact Mayer-Vietoris sequence. Using the natural map a we get a natural Mayer-Vietoris sequence for every homotopy cartesian square. For every commutative square D_{E} we consider the levelwise homotopy cocartesian square D_{po}

\xymatrix{E_{0} \ar[r]^{f_{1}} \ar[d]_{f_{2}} & E_{1} \ar[d]^{_{\overline{g}_{1}}} \\ E_{2} \ar[r]_{_{\overline{g}_{2}}} & hopo(E_{2} ←E_{0} →E_{1}).}

The claim implies that D_{po} is homotopy cartesian, and therefore there is a long exact Mayer-Vietoris sequence associated to D_{po}.

Now assume that D_{E} is homotopy cartesian. In this case both D_{po} and D_{E} induce long exact Mayer-Vietoris sequences, and the map b induces a natural transformation between them. The Five-Lemma then implies that b is a weak equivalence of spectra, i.e. D_{E} is homotopy cocartesian. Conversely, assume that D_{E} is homotopy cocartesian. Using the natural weak equivalence b we get a long exact Mayer-Vietoris sequence for D_{E} from the one for D_{po}. Then a Five-Lemma argument shows that the map a is a weak equivalence, i.e. D_{E} is homotopy cartesian.

It remains to prove the claim. This is based on the following well-known stable range statement about homotopy cocartesian squares of spaces (a version of Blakers-Massey homotopy excision theorem, see for example [Goo92, Section 2, in particular Theorem 2.3]). Assume that the commutative square of spaces

\xymatrix{X_{0} \ar[r]^{f_{1}} \ar[d]_{f_{2}} & X_{1} \ar[d]^{g_{2}} \\ X_{2} \ar[r]_{g_{1}} & X_{12}}

is homotopy cocartesian, and that f_{1} is k_{1}-connected and f_{2} is k_{2}-connected. Then the natural map : a: X_{0} → hopb(X_{2} →X_{12} ←X_{1}) is at least (k_{1}+k_{2}−1)-connected.

Now suppose we are given a levelwise homotopy cocartesian square D_{E} of spectra. Since homotopy pushouts commute with suspensions we have for every n, k ∈ ℕ a homotopy cocartesian square of spaces

\xymatrix{Σ^{k}E_{0}(n) \ar[r]^{Σ^{k}f_{1}(n)} \ar[d]_{Σ^{k}f_{2}(n)} & Σ^{k}E_{1}(n) \ar[d]^{Σ^{k}g_{2}(n)} \\ Σ^{k}E_{2}(n) \ar[r]_{Σ^{k}g_{1}(n)} & Σ^{k}E_{12}(n).}

Now consider the following diagram

[figure: Σ^{k}E_{0}(n); Σ^{k}a(n); Σ^{k} hopb(E_{2}(n)→E_{12}(n)←E_{1}(n)); η; Σ^{k}E_{0}(n); σ; a_{k,n}; hopb(Σ^{k}E_{2}(n)→Σ^{k}E_{12}(n)←Σ^{k}E_{1}(n)); μ; E_{0}(n + k); a(n+k); hopb(E_{2}(n + k)→E_{12}(n + k)←E_{1}(n + k))]

where σ comes from the structure maps of E_{0}, η is the natural map, and μ the map induced by the structure maps of E_{2}, E_{12} and E_{1}. One checks that the diagram commutes and that the composition μ ∘ η is the structure map of the spectrum hopb(E_{2} → E_{12} ← E_{1}) (recall that for the homotopy pullback spectrum one actually defines the adjoints of the structure maps, using the fact that homotopy pullbacks and loops commute up to natural homeomorphism). From the Blakers-Massey connectivity statement recalled above we get that the natural map a_{k,n} in the middle row of the diagram is at least (2k − 3)-connected. Hence we can apply Lemma 2.8 below to conclude that the natural map of spectra a: E_{0} → hopb(E_{2} → E_{12} ←E_{1}) is a weak equivalence, i.e. D_{E} is homotopy cartesian. □

Lemma 2.8. Let f : E → F be a map of spectra and l ∈ ℤ be an integer. Suppose that for all n ∈ ℕ there exist a k = k(l, n) ∈ ℕ and a commutative diagram

\xymatrix{Σ^{k}E(n) \ar[d] \ar[dd] \ar[r]^{Σ^{k}f(n)} & Σ^{k}F (n) \ar[d] \ar[dd] \\ E_{k,n} \ar[d] \ar[r]^{f_{k,n}} & F_{k,n} \ar[d] \\ E(k + n) \ar[r]^{f(k+n)} & F (k + n)}

where f_{k,n} is (l + k + n)-connected, and the outer vertical maps are given by the structure maps of the spectra. Then f : E → F is l-connected.

We omit the elementary proof, which just uses the definition of homotopy groups of spectra, basic properties of sequential colimits, and diagram chases.

3. Basics about spaces, spectra and modules over a category

In this section we recall some basic notions about spaces, spectra and modules over a small category I. More information for spaces and spectra can be found in [DL98] and for modules in [Lüc89, II.9].

A covariant (contravariant) pointed I-space X is a covariant (contravariant) functor from I to the category of pointed (compactly generated) spaces. A map between I-spaces is a natural transformation of such functors. For pointed Ispaces X and Y of the same variance let map_{I}(X, Y) be the space of pointed maps from X to Y with the subspace topology coming from the obvious inclusion into ∏_{c∈obj(I)} map(X(c), Y (c)). Let X be a contravariant and Y a covariant I-space. Define their smash product to be the space

X ∧_{I} Y = ⋁_{c∈obj(I)} X(c) ∧ Y (c)/ ∼, (3.1)

where ∼ is the equivalence relation generated by (xϕ, y) ∼ (x, ϕy) for all morphisms ϕ: c → d in I and all points x ∈ X(d) and y ∈ Y (c). Here xϕ stands for X(ϕ)(x) and ϕy for Y (ϕ)(y). If I is the trivial category with only one object and only one morphism, then X and Y are just pointed spaces and X ∧_{I} Y is the ordinary smash product X ∧ Y. For an ordinary pointed space Z let map(Y, Z) denote the obvious contravariant pointed I-space whose value at an object c is the mapping space map(Y (c), Z) of base point preserving maps. One easily checks the following lemma.

Lemma 3.2. Let X be a contravariant pointed I-space, Y be a covariant pointed I-space and Z be a pointed space. Then there is a homeomorphism

T: map(X ∧_{I} Y, Z) ⟶^{≅} map_{I}(X, map(Y, Z))

which is natural in X, Y and Z.

A covariant (contravariant) I-spectrum is a covariant (contravariant) functor E: I → SPECTRA . A covariant (contravariant) I-Ω-spectrum is a covariant (contravariant) functor E: I → Ω-SPECTRA . Given a contravariant pointed I-space X, a covariant I-spectrum E and a contravariant I-spectrum F we obtain spectra

X ∧_{I} E and map_{I}(X, F)

in the obvious way by applying the constructions discussed above levelwise. Notice that for an I-Ω-spectrum E the spectrum X ∧_{I} E is not necessarily an Ω-spectrum.

Definition 3.3. A contravariant I-CW -complex X is a contravariant unpointed I-space X: I → SPACEStogether 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 I-n-cells, i.e. there exists a pushout of I-spaces of the form

\xymatrix{∐_{i∈I_{n}} mor_{I}(−, c_{i}) × S^{n−1} \ar[d] \ar[r] & X_{n−1} \ar[d] \\ ∐_{i∈I_{n}} mor_{I}(−, c_{i}) × D^{n} \ar[r] & X_{n},}

where the vertical maps are inclusions, I_{n} is an index set, and the c_{i} are objects of I. (In [DL98, Definition 3.2 on page 221] this is called a free I-CW -complex, but we will drop the word free here.)

Definition 3.4. A contravariant I-CW -complex EI whose evaluation at each object of I is a contractible space is called a contravariant classifying space for I.

Remark 3.5. Any two models for EI are I-homotopy equivalent. A bar-construc-tion model for EI, which is functorial in I, is given in [DL98, page 230].

Definition 3.6. If X is a covariant pointed I-space and Y a contravariant pointed I-space the homotopy colimit respectively the homotopy limit are defined as the pointed spaces given by

hocolim_{I} X = EI_{+} ∧_{I} X respectively holim_{I} Y = map_{I}(EI_{+}, Y),

compare [DL98, Definition 3.13 on page 225]. For a covariant I-spectrum E and a contravariant I-spectrum F we define similarly

hocolim_{I} E = EI_{+} ∧_{I} E and holim_{I} F = map_{I}(EI_{+}, R ∘ F).

But note the replacement functor that appears in the homotopy limit and the fact that even if E is an I-Ω-spectrum the homotopy colimit need not be an Ω-spectrum.

Remark 3.7. The homotopy type of the homotopy colimit or homotopy limit does not depend on the choice of the model for EI. If the bar-construction model for EI is used, then our definition agrees with the obvious modification of the one given in [BK72, Chapters XI and XII] for diagrams of simplicial sets. Some properties of homotopy colimits and homotopy limits become obvious with our definition. For instance Ω commutes with homotopy limits of spaces because of the adjunctions

map(S^{1}, map_{I}(EI_{+}, X)) ≅ map_{I}(S^{1} ∧_{I} EI_{+}, X) ≅ map_{I}(EI_{+}, map(S^{1}, X))

(see [DL98, Lemma 1.6 on page 206]). This fact is used to define the structure maps of the homotopy limit of spectra.

We collect some results that will be needed below (compare [DL98, Theorem 3.11 on page 224 and Lemma 4.6 on page 234]).

Lemma 3.8. Let Y be a contravariant I-CW -complex.

(i) Let f: Z_{1} → Z_{2} be a map of covariant or contravariant respectively Ispaces such that f(c) is a weak equivalence for all objects c in I. Then the induced maps

Y ∧_{I} Z_{1} → Y ∧_{I} Z_{2}

and

map_{I}(Y, Z_{1}) → map_{I}(Y, Z_{2})

respectively are weak equivalences. (ii) Let f: E → F be a map of covariant I-spectra such that f(c) is a weak

equivalence for all objects c in I. Then the induced map

Y ∧_{I} E → Y ∧_{I} F

is a weak equivalence. (iii) Let f: E → F be a map of contravariant I-Ω-spectra such that f(c) is a

weak equivalence for all objects c in I. Then the induced map

map_{I}(Y, E) → map_{I}(Y, F)

is a weak equivalence.

Lemma 3.9. Let E: I → Ω-SPECTRAbe a contravariant functor and let Y be a contravariant I-space. Then map_{I}(Y_{+}, E) is an Ω-spectrum.

Proof. The adjoint of the k-th structure map of map_{I}(Y_{+}, E) is the composition

map_{I}(Y_{+}, E(k)) ⟶^{map_{I}(id,\overline{σ}(k))} map_{I}(Y_{+}, ΩE(k + 1)) ⟶^{≅} Ω map_{I}(Y_{+}, E(k + 1)),

where the first map is a weak homotopy equivalence by Lemma 3.8 (i) and the second one is the canonical homeomorphism (compare Remark 3.7). □

The constructions discussed above for spaces and spectra all have analogues in the world of modules and chain complexes. We will usually refer to this analogy as the “linear case”. We collect some basics. Fix a commutative associative ring R with unit. A covariant (contravariant) RI-module is a covariant (contravariant) functor from I to the category of R-modules. Morphisms are natural transformations.

Given a contravariant RI-module M and a covariant RI-module N, their tensor product is the R-module

M ⊗_{RI} N = ⨁_{c∈obj(I)} M(c) ⊗_{R} N(c)/ ∼,

where ∼ is defined similar as in (3.1). Given two RI-modules M and N of the same variance, define hom_{RI}(M, N) to be the R-module of natural transformations from M to N. The obvious analogue of Lemma 3.2 is true for ⊗_{RI} and hom_{RI}.

The category RI-MODULES of covariant (contravariant) RI-modules inherits the structure of an abelian category from the category of R-modules. In particular the notion of a projective RI-module makes sense. There are enough injectives and projectives in RI-MODULES (see [Lüc89, 9.16 on page 167], [Wei94, Exercise 2.3.5 on page 42 and Example 2.3.13 on page 43]). Hence standard homological algebra applies to RI-MODULES. For instance, one can define the R-module Tor_{p}^{RI}(M, N) for a contravariant RI-module M and a covariant RI-module N.

An RI-chain complex is a chain complex in RI-MODULES, or, equivalently, a covariant (contravariant) functor from I to the category of R-chain complexes. We denote by ch(RI-MODULES) the category of RI-chain complexes.

We now specialize to the case where I is the orbit category Or(G), whose objects are the homogeneous G-spaces G/H and whose morphisms are G-maps. A (left) G-space X defines a contravariant Or(G)-space

X: Or(G) → SPACES, G/H ↦ map_{G}(G/H, X) = X^{H}.

If X is a G-CW -complex, then X considered as a functor is a contravariant Or(G)- CW -complex (see [DL98, Theorem 7.4 on page 250]). Every Or(G)-CW -complex defines a functor from Or(G) to the category of CW -complexes, which we can compose with the functor cellular chain complex. Thus we associate to an Or(G)- CW -complex X its cellular ℤOr(G)-chain complex C_{∗}^{ℤOr (G)}(X). This is always a free ℤOr(G)-chain complex in the sense of [Lüc89, 9.16 on page 167] and in particular projective.

Furthermore we will also need the category Sub(G), which is a quotient category of Or(G). Its objects are the subgroups H of G. For two subgroups H and K of G denote by conhom_{G}(H, K) the set of group homomorphisms f: H → K, for which there exists an element g ∈ G with gHg^{−1} ⊂ K such that f is given by conjugation with g, i.e. f = c(g): H → K, h ↦ ghg^{−1}. Notice that c(g) = c(g′) holds for two elements g, g′ ∈ G with gHg^{−1} ⊂ K and g′H(g′)^{−1} ⊂ K if and only if g^{−1}g′ lies in the centralizer Z_{G}H = {g ∈ G | gh = hg for all h ∈ H} of H in G. The group of inner automorphisms of K acts on conhom_{G}(H, K) from the left by composition. Define the set of morphisms

mor_{Sub (G)}(H, K) = Inn(K)\ conhom_{G}(H, K).

There is a natural projection

(3.10)  pr: Or(G) → Sub(G)

which sends a homogeneous space G/H to H. Given a G-map f: G/H → G/K, we can choose an element g ∈ G with gHg^{−1} ⊂ K and f(g′H) = g′g^{−1}K. Then pr(f) is represented by c(g): H → K. Notice that mor_{Sub (G)}(H, K) can be identified with the quotient mor_{Or (G)}(G/H, G/K)/Z_{G}H, where g ∈ Z_{G}H acts on mor_{Or (G)}(G/H, G/K) by composition with R_{g^{−1}} : G/H → G/H, g′H ↦ g′g^{−1}H.

Recall that a functor F: GROUPOIDS → SPECTRAis a homotopy functor if it sends an equivalence of groupoids to a weak equivalence of spectra. This is equivalent to requiring that naturally isomorphic functors between groupoids induce the same map after applying π_{∗}(F(−)). Recall also the transport groupoid functor G^{G} defined before Theorem 1.2.

Lemma 3.11. Let F: GROUPOIDS→ SPECTRAbe a homotopy functor, then the functor π_{n}(F ∘ G^{G}(−)) factorizes over the projection pr: Or(G) → Sub(G).

Proof. It suffices to show that the identity and G^{G}(R_{d^{−1}}): G^{G}(G/H) → G^{G}(G/H) are naturally isomorphic for d ∈ Z_{G}H. A natural transformation is given at the object gH by gd^{−1}g^{−1} : gH → gd^{−1}H. □

4. Almost trivial and almost finitely generated modules

In this section we discuss some elementary properties of the notions “almost finitely generated modules” and “almost isomorphisms” that were used in Addendum 1.3. The main aim is to verify that we have the fundamental properties that are necessary to do spectral sequence comparison arguments.

Let R be an integral domain, i.e. an associative commutative ring with unit such that R has no non-trivial zero-divisor. An R-module M is called faithful if its annihilator ideal Ann_{R}(M) = {r ∈ R | rm = 0 for all m ∈ M} is the trivial ideal {0}. We call M almost trivial or non-faithful if M is not faithful, i.e. there exists r ∈ R, r ≠ 0 such that rm = 0 holds for all m ∈ M. Recall for an Rmodule M that tors(M) is the submodule consisting of elements m ∈ M for which there exists some r ∈ R, r ≠ 0 (depending on m) with rm = 0. We call M almost finitely generated if tors(M) is almost trivial and M/ tors(M) is a finitely generated R-module. We will often abbreviate M/ tors(M) by M/ tors.

Recall that a Serre subcategory of an abelian category is an abelian subcategory closed under subobjects, quotients, and extensions (see for example [Wei94, Exercise 10.3.2 on page 384]).

Lemma 4.1. (i) The full subcategory of R-MODULES whose objects are the

almost trivial modules is a Serre subcategory. (ii) If R is Noetherian then the full subcategory of R-MODULES whose objects

are the almost finitely generated modules is a Serre subcategory.

Proof. (i) Obviously a quotient module and a submodule of an almost trivial module are almost trivial again. Let M_{0} → M_{1} → M_{2} be an exact sequence of R-modules such M_{0} and M_{2} are almost trivial. Choose r_{i} in Ann(M_{i}) with r_{i} ≠ 0 for i = 0, 2. One easily checks that the product r_{0}r_{2} lies in Ann(M_{1}) and is different from zero. Hence M_{1} is almost trivial.

(ii) Consider an exact sequence of R-modules 0 → M_{0} → M_{1} → M_{2} → 0. It suffices to prove that both M_{0} and M_{2} are almost finitely generated if and only if M_{1} is almost finitely generated. Define C to be the cokernel of the induced map tors(M_{1}) → tors(M_{2}) and K to be the kernel of the induced map M_{1}/ tors → M_{2}/ tors. Then we get exact sequences

0 → tors(M_{0}) → tors(M_{1}) → tors(M_{2}) → C → 0,
0 → M_{0}/ tors → K → C → 0, and
0 → K → M_{1}/ tors → M_{2}/ tors → 0.

Suppose that M_{1} is almost finitely generated, i.e. tors(M_{1}) is almost trivial and M_{1}/ tors is finitely generated. As R is Noetherian, K is finitely generated. Therefore C is finitely generated and C = tors(C), hence C is almost trivial. We conclude from assertion (i) that tors(M_{0}) and tors(M_{2}) are almost trivial. Since R is Noetherian and M_{1}/ tors is finitely generated, both M_{0}/ tors and M_{2}/ tors are finitely generated. This shows that both M_{0} and M_{2} are almost finitely generated.

Suppose that both M_{0} and M_{2} are almost finitely generated. We conclude from assertion (i) that C and tors(M_{1}) are almost trivial. We obtain an exact sequence

hom_{R}(K, M_{0}/ tors) ⟶ hom_{R}(M_{0}/ tors, M_{0}/ tors) ⟶ Ext_{R}^{1}(C, M_{0}/ tors).

Since C is almost trivial, we can find r ∈ R, r ≠ 0 which annihilates C and hence also Ext_{R}^{1}(C, M_{0}/ tors). Therefore r · id: M_{0}/ tors → M_{0}/ tors is sent to the zero element in Ext_{R}^{1}(C, M_{0}/ tors). Hence we can find a R-map f: K → M_{0}/ tors whose restriction to M_{0}/ tors is r · id: M_{0}/ tors → M_{0}/ tors. Consider x ∈ ker(f). Since r annihilates C, the element r · x lies in M_{0}/ tors and belongs to the kernel of r · id: M_{0}/ tors → M_{0}/ tors. This implies r^{2} · x = 0. Since K is a submodule of M_{1}/ tors and hence is torsionfree, we conclude x = 0. This shows that f: K → M_{0}/ tors is an injection. Since M_{0}/ tors is finitely generated and R is Noetherian, K is finitely generated. Since M_{2}/ tors is finitely generated and R is Noetherian, M_{1}/ tors is finitely generated. Hence M_{1} is almost finitely generated. □

Definition 4.2. A map f: M → N of R-modules is called an almost isomorphism if its kernel and cokernel are almost trivial R-modules.

We conclude from Lemma 4.1 (i) that for two composable R-maps f: L → M and g: M → N all three maps f, g and g ∘ f are almost isomorphisms if two of them are. Moreover there is a Five-Lemma for almost isomorphisms. For an almost R-isomorphism f the F -map F ⊗_{R} f is an F -isomorphism, where F denotes the quotient field of R. An R-module M is almost trivial or almost finitely generated respectively if and only if it is almost isomorphic to the zero module or respectively to a finitely generated R-module.

5. Commuting homotopy limits and smash products

In this section we will prove our main result Theorem 1.2 and the Addendum 1.3. The proof will proceed by induction over the skeleta of Y and hence we start with the following lemma.

Lemma 5.1. Theorem 1.2 and Addendum 1.3 are true if Y is a zero-dimensional I-CW -complex.

Proof. For a large part we will treat the proofs of the theorem and of the addendum simultaneously.

By assumption we can find a family of objects {c_{i} | i ∈ I} such that Y = ∐_{i∈I} mor_{I}(?, c_{i}). Notice that there is a canonical isomorphism

map_{I}(mor_{I}(?, c)_{+}, F) ⟶^{≅} F(c)

for any contravariant I-spectrum F and any object c in I. Hence it suffices to show that the map

t: X_{+} ∧_{Or (G)} ∏_{i∈I} E(c_{i}) → ∏_{i∈I} R (X_{+} ∧_{Or (G)} E(c_{i}))

which is induced by the various projections pr_{i}: ∏_{i∈I} E(c_{i}) → E(c_{i}) followed by the replacement maps r(X_{+} ∧_{Or (G)} E(c_{i})): X_{+} ∧_{Or (G)} E(c_{i}) → R (X_{+} ∧_{Or (G)} E(c_{i})) of (2.2), induces an (almost) isomorphism on homotopy groups π_{p} for all p ≤ n.

For all p there is a commutative diagram

(5.2)  \xymatrix{π_{p} (X_{+} ∧_{Or (G)} ∏_{i∈I} E(c_{i})) \ar[r]^{π_{p}(t)} \ar[d] & π_{p} (∏_{i∈I} R (X_{+} ∧_{Or (G)} E(c_{i}))) \ar[d]^{≅} \\ ∏_{i∈I} π_{p} (X_{+} ∧_{Or (G)} E(c_{i})) \ar[r]^{≅} & ∏_{i∈I} π_{p} (R (X_{+} ∧_{Or (G)} E(c_{i}))) .}

The vertical maps are induced by the obvious projections. The right hand vertical map is an isomorphism by Lemma 2.3. The lower horizontal map is the product of the isomorphisms induced by the fibrant replacement equivalences r(X_{+} ∧_{Or (G)} E(c_{i})): X_{+} ∧_{Or (G)} E(c_{i}) → R (X_{+} ∧_{Or (G)} E(c_{i})) and hence an isomorphism, compare (2.2). We see that it suffices to show that the left hand vertical map is an (almost) isomorphism for p ≤ n. This is what we will prove next.

For any covariant Or(G)-spectrum F and any G-CW -complex Z we get a homological spectral sequence (E_{∗,∗}^{∗}, d_{∗,∗}^{∗}) which converges to π_{p+q} (Z_{+} ∧_{Or (G)} F) and whose E^{2}-term is given by the Bredon homology

E_{p,q}^{2} = H_{p}^{ℤOr (G)}(Z; π_{q}(F))

This follows from [DL98, Theorem 4.7 on page 234, Theorem 7.4 (3) on page 250]. Recall that the Bredon homology of a G-CW -complex Z with coefficients in a covariant ℤOr(G)-module M is defined by

H_{p}^{ℤOr (G)}(Z; M) = H_{p} (C_{∗}^{ℤOr (G)}(Z) ⊗_{ℤOr (G)} M) .

This spectral sequence reduces to the classical Atiyah-Hirzebruch spectral sequence if G is the trivial group. A map of covariant Or(G)-spectra F → F′ induces a map of the associated spectral sequences. Let (E_{∗,∗}^{∗}, d_{∗,∗}^{∗}) be the spectral sequence associated to F = ∏_{i∈I} E(c_{i}) and (E[i]_{∗,∗}^{∗}, d[i]_{∗,∗}^{∗}) be the spectral sequence associated to F = E(c_{i}). Our assumptions imply that all these spectral sequences live in the first quadrant (possibly shifted down by N). This implies that all convergence questions are trivial, i.e. E_{p,q}^{∞} = E_{p,q}^{r} for r > p + q − N + 1 and the filtration of π_{p+q}(X_{+} ∧_{Or (G)} E(c_{i})), whose quotients are given by the E^{∞}-term, consists of at most p + q − N + 1 stages. Notice that ∏_{i∈I} preserves exact sequences. Hence we can take the product ∏_{i∈I}(E[i]_{∗,∗}^{∗}, d[i]_{∗,∗}^{∗}) and obtain a spectral sequence which converges to ∏_{i∈I} π_{p+q} (X_{+} ∧_{Or (G)} E(c_{i})) and whose E^{2}-term is ∏_{i∈I} H_{p}^{ℤOr (G)}(X; π_{q}(F)). The usual spectral sequence comparison argument shows that the left hand vertical map in (5.2) which is induced by the various projections pr_{i}: ∏_{i∈I} E(c_{i}) → E(c_{i}) is an (almost) isomorphism for each p ≤ n, if the maps induced by the projections pr_{i}: ∏_{i∈I} E(c_{i}) → E(c_{i})

H_{p}^{ℤOr (G)} (X_{+}; π_{q} (∏_{i∈I} E(c_{i}))) → ∏_{i∈I} H_{p}^{ℤOr (G)} (X_{+}; π_{q}(E(c_{i})))

are (almost) isomorphisms for all p, q ∈ ℤ with p + q ≤ n.

The canonical map

π_{q} (∏_{i∈I} E(c_{i})) ⟶^{≅} ∏_{i∈I} π_{q}(E(c_{i}))

is bijective by Lemma 2.3 since we assume that the E(c_{i}) are Ω-spectra. The assumptions about E imply that Lemma 3.11 is applicable and hence the ℤOr(G)- modules π_{q}(E(c_{i})) and ∏_{i∈I} π_{q}(E(c_{i})) factorize through the canonical projection pr: Or(G) → Sub(G). Let F_{X} = {H ⊂ G | X^{H} ≠ ∅} be the family of isotropy groups of X. Let Sub(G; F_{X}) be the full subcategory Sub(G) whose objects are in F_{X}. Let C_{∗}^{ℤSub (G;F_{X})}(X) be the contravariant ℤSub(G; F_{X})-chain complex which is the composition of the “cellular chain complex”-functor with the contravariant functor from Sub(G; F_{X}) to the category of CW -complexes which sends an object H to the CW -complex Z_{G}H\X^{H}. Given a covariant ℤSub(G; F_{X})- module N, define

H_{p}^{ℤSub (G;F_{X})}(X; N) = H_{p} (C_{∗}^{ℤSub (G;F_{X})}(X) ⊗_{ℤSub (G;F_{X})} N) .

If M is a covariant ℤSub(G)-module, then we obtain a natural isomorphism

H_{p}^{ℤOr (G)}(X; pr^{∗} M) ⟶^{≅} H_{p}^{ℤSub (G;F_{X})}(X; M|_{Sub (G;F_{X})}),

where pr^{∗} M is the ℤOr(G)-module M ∘ pr and M|_{Sub (G;F_{X})} is the ℤSub(G; F_{X})- module given by the restriction of M to Sub(G; F_{X}). This follows from the adjunction of induction and restriction [Lüc89, 9.21 on page 169], since the induction of C_{∗}^{ℤSub (G;F_{X})}(X) with respect to the inclusion Sub(G; F_{X}) → Sub(G) and the induction of C_{∗}^{ℤOr (G)}(X) with respect to the projection pr: Or(G) → Sub(G) agree. In the sequel we abbreviate Sub = Sub(G; F_{X}). Hence it remains to prove that for any family {M_{i} | i ∈ I} of covariant ℤSub-modules the canonical map induced by the various projections ∏_{i∈I} M_{i} → M_{i}

H_{p}^{ℤSub} (X; ∏_{i∈I} M_{i}) → ∏_{i∈I} H_{p}^{ℤSub} (X; M_{i})

is an (almost) isomorphism for all p ∈ ℤ with p ≤ n − N. We will prove that for a contravariant projective ℤSub-chain complex C_{∗} (such as C_{∗}^{ℤSub (G;F_{X})}(X)) and a family of covariant ℤSub-modules M_{i} the canonical map

(5.3)  H_{p} (C_{∗} ⊗_{ℤSub} ∏_{i∈I} M_{i}) → ∏_{i∈I} H_{p} (C_{∗} ⊗_{ℤSub} M_{i})

induced by the various projections ∏_{i∈I} M_{i} → M_{i} is an (almost) ℤ-isomorphism for all p ≤ n−N, provided that the ℤ-module H_{p}(C_{∗}(H)) is (almost) finitely generated for any object H in Sub and p ≤ n − N.

We will first treat the case of the addendum, where we assume that the homology modules are almost finitely generated and want to conclude an almost isomorphism.

There are spectral sequences converging to

H_{p+q} (C_{∗} ⊗_{ℤSub} ∏_{i∈I} M_{i}) respectively H_{p+q} (C_{∗} ⊗_{ℤSub} M_{i})

whose E_{p,q}^{2}-term are given by

Tor_{p}^{ℤSub} (H_{q}(C_{∗}), ∏_{i∈I} M_{i}) respectively Tor_{p}^{ℤSub} (H_{q}(C_{∗}), M_{i}).

Taking the product over i ∈ I of the second spectral sequences yields a spectral sequence which converges to ∏_{i∈I} H_{p+q} (C_{∗} ⊗_{ℤSub} M_{i}) and whose E_{p,q}^{2}-term is ∏_{i∈I} Tor_{p}^{ℤSub} (H_{q}(C_{∗}), M_{i}). By the standard spectral sequence comparison argument it suffices to prove that the canonical map

Tor_{p}^{ℤSub} (H_{q}(C_{∗}), ∏_{i∈I} M_{i}) ⟶ ∏_{i∈I} Tor_{p}^{ℤSub} (H_{q}(C_{∗}), M_{i}) (5.4)

is an almost isomorphism for each p, q ∈ ℤ with p + q ≤ n − N. The following diagram commutes

\xymatrix{Tor_{p}^{ℤSub} (H_{q}(C_{∗}), ∏_{i∈I} M_{i}) \ar[d] \ar[r] & ∏_{i∈I} Tor_{p}^{ℤSub} (H_{q}(C_{∗}), M_{i}) \ar[d] \\ _{Tor{ℤSub \atop p} (H_{q}(C_{∗})/ tors, ∏_{i∈I} M_{i})} \ar[r] & ∏_{i∈I} Tor_{p}^{ℤSub} (H_{q}(C_{∗})/ tors, M_{i}).}

As Sub has only finitely many isomorphism classes of objects and tors(H_{q}(C_{∗}(H))) is almost trivial for each object H, there is an integer k ≠ 0 which annihilates tors(H_{q}(C_{∗}(H))) for each object H. Hence Tor_{p}^{ℤSub} (tors(H_{q}(C_{∗})), N) is annihilated by k for each covariant ℤSub-module N. By the long exact Tor-sequence associated to the exact sequence of contravariant ℤSub-modules

0 → tors(H_{q}(C_{∗})) → H_{q}(C_{∗}) → H_{q}(C_{∗})/ tors(H_{q}(C_{∗})) → 0

we conclude that both the kernel and the cokernel of the canonical map

Tor_{p}^{ℤSub} (H_{q}(C_{∗}), N) → Tor_{p}^{ℤSub} (H_{q}(C_{∗})/ tors, N)

are annihilated by k for any covariant ℤSub-module N. This implies that in the commutative square above, the vertical arrows are almost isomorphisms. Hence it suffices to prove that the lower horizontal arrow is an almost ℤ-isomorphism. By assumption the ℤ-module H_{q}(C_{∗}(H))/ tors is finitely generated for all q ≤ n − N.

Since we assume that Sub has only finitely many isomorphism classes of objects and aut_{Sub} (H) = W_{G}H = N_{G}H/H ·Z_{G}H is a finite group for any object H ∈ Sub, the category Sub is finite in the sense of [Lüc89, Definition 16.1 on page 325]. (Notice that Or(G; F_{X}) does not have this property and therefore it is crucial to pass to Sub.) Hence the category of ℤSub-modules is noetherian, in the sense that each submodule of a finitely generated ℤSub-module is itself finitely generated. Moreover a covariant ℤSub-module L is finitely generated if and only if L(H) is a finitely generated ℤ-module for each object H ∈ Sub (see [Lüc89, Lemma 16.10 on page 327]). By assumption H_{q}(C_{∗}(H))/ tors is a finitely generated ℤ-module for each object H in Sub. This implies that there is a finitely generated free (not necessarily finite-dimensional) ℤSub-resolution F_{∗} for H_{q}(C_{∗})/ tors for q ≤ n − N (see [Lüc89, Lemma 17.1 on page 339]). Thus we get for q ≤ n − N identifications

Tor_{p}^{ℤSub} (H_{q}(C_{∗})/ tors, ∏_{i∈I} M_{i}) = H_{p}(F_{∗} ⊗_{ℤSub} ∏_{i∈I} M_{i})

and

∏_{i∈I} Tor_{p}^{ℤSub} (H_{q}(C_{∗})/ tors, M_{i}) = ∏_{i∈I} H_{p}(F_{∗} ⊗_{ℤSub} M_{i})

= H_{p} (∏_{i∈I} F_{∗} ⊗_{ℤSub} M_{i}) ,

under which the map

Tor_{p}^{ℤSub} (H_{q}(C_{∗}), ∏_{i∈I} M_{i}) → ∏_{i∈I} Tor_{p}^{ℤSub} (H_{q}(C_{∗}), M_{i})

becomes the map induced on homology by the canonical ℤ-chain map

(5.5)  F_{∗} ⊗_{ℤSub} ∏_{i∈I} M_{i} → ∏_{i∈I} F_{∗} ⊗_{ℤSub} M_{i}.

But this chain map is a chain isomorphism since each chain module F_{i} is a finitely generated free ℤSub-module. Here one uses the fact that the processes of taking sums over a finite index set and products over an arbitrary index set commute.

The case where we assume the homology groups to be finitely generated ℤmodules is similar but easier. Using the fact that the category of ℤSub-modules is noetherian one can replace the complex C_{∗} which appears in (5.3) by a homotopy equivalent complex F_{∗} of projective ℤSub-modules which is of finite type. For such a complex a map like the one in (5.5) is an isomorphism and hence so is the map (5.3). □

Remark 5.6. Before we finish the proof of Theorem 1.2 and its addendum we want to discuss where the proof above breaks down if we ask for a rational isomorphisms and weaken assumption (D) to the requirement that H_{n}(Z_{G}H\X^{H}; ℚ) is a finitely generated ℚ-module for all n.

We would have to show that the map appearing in (5.4) is a rational isomorphism. The following example shows that this is not possible. We claim that for a fixed prime p the canonical map

Tor_{1}^{ℤ} (⨁_{m≥2} ℤ/p^{m}, ∏_{n≥2} ℤ/p^{n}) → ∏_{n≥2} Tor_{1}^{ℤ} (⨁_{m≥2} ℤ/p^{m}, ℤ/p^{n})

cannot be a rational isomorphism although (⨁_{m≥2} ℤ/p^{m}) ⊗_{ℤ} ℚ = 0. Namely, the source is ⨁_{m≥2} ∏_{n≥2} ℤ/p^{min(m,n)}, which vanishes after applying − ⊗_{ℤ} ℚ. The target is ∏_{n≥2} ⨁_{m≥2} ℤ/p^{min(m,n)}, which contains ∏_{n≥2} ℤ/p^{n} as a submodule and hence does not vanish after applying − ⊗_{ℤ} ℚ.

Now we finish the proof of Theorem 1.2 and Addendum 1.3.

Proof of Theorem 1.2 and Addendum 1.3. We use induction over the dimension d of Y. If d = 0, the claim has already been proved in Lemma 5.1. The induction step from d − 1 to d ≥ 1 is done as follows. We can write Y as a pushout

\xymatrix{Y_{0} \ar[d]^{i_{2}} \ar[r]^{i_{1}} & Y_{1} \ar[d]^{j_{1}} \\ Y_{2} \ar[r]^{i_{2}} & Y}

where Y_{1} is a (d − 1)-dimensional I-CW -complex, Y_{0} = ∐_{i∈I} mor_{I}(?, c_{i}) × S^{d−1}, Y_{2} = ∐_{i∈I} mor_{I}(?, c_{i}) × D^{d} and i_{2} is the inclusion. In particular Y_{1} is a (d − 1)- dimensional CW -complex and Y_{2} is I-homotopy equivalent to the 0-dimensional CW -complex ∐_{i∈I} mor_{I}(?, c_{i}). Hence the induction hypothesis applies to Y_{0}, Y_{1} and Y_{2}.

The canonical map from the homotopy pushout of Y_{2} ← Y_{0} → Y_{1} to Y is a map of I-CW -complexes whose evaluation at each object c is a homotopy equivalence since i_{2}(c) is a cofibration. Hence it is a I-homotopy equivalence (see [DL98, Corollary 3.5 on page 222]). Since for any covariant I-space Z the functor map_{I}(−, Z) sends a homotopy cocartesian square of spaces to a homotopy cartesian one, the following diagram is levelwise homotopy cartesian for each object G/H ∈ Or(G)

\xymatrix{map_{I}(Y_{+}, E(G/H)) \ar[d] \ar[r] & map_{I}((Y_{1})_{+}, E(G/H)) \ar[d] \\ map_{I}((Y_{2})_{+}, E(G/H)) \ar[r] & map_{I}((Y_{0})_{+}, E(G/H)).}

We conclude with Lemma 2.6 that the square is homotopy cocartesian. By the same line of argument the diagram

\xymatrix{map_{I}(Y_{+}, R(X_{+} ∧_{Or (G)} E(G/?))) \ar[d] \ar[r] & map_{I}((Y_{1})_{+}, R(X_{+} ∧_{Or (G)} E(G/?))) \ar[d] \\ map_{I}((Y_{2})_{+}, R(X_{+} ∧_{Or (G)} E(G/?))) \ar[r] & map_{I}((Y_{0})_{+}, R(X_{+} ∧_{Or (G)} E(G/?)))}

is homotopy cocartesian. Since the functor X_{+} ∧_{Or (G)} − respects homotopy cocartesian squares because of the associativity of ∧, we conclude from Lemma 3.8 (ii) that the following diagram is homotopy cartesian

\xymatrix{X_{+} ∧_{Or (G)} map_{I}(Y_{+}, E) \ar[d] \ar[r] & X_{+} ∧_{Or (G)} map_{I}((Y_{1})_{+}, E) \ar[d] \\ X_{+} ∧_{Or (G)} map_{I}((Y_{2})_{+}, E) \ar[r] & X_{+} ∧_{Or (G)} map_{I}((Y_{0})_{+}, E).}

The various transformations t(X, Y_{i}, E) and t(X, Y, E) yield a map from this homotopy cartesian diagram to the one before. Hence they yield a transformation from the long exact homotopy sequences associated in Lemma 2.6 to these homotopy pullbacks. Now Theorem 1.2 follows from the Five-Lemma for (almost) isomorphisms. □

6. The linear version

In this section we state the linear version of our main Theorem 1.2. This will be needed in [LMR03]. The proof in the linear setting is completely analogous to the one for spaces and spectra presented in Section 5, and therefore we omit it.

Theorem 6.1. Consider the following data:

• A functor E_{∗}: I^{op} × Or(G) → ch(ℤ-MODULES) which factorizes as

E_{∗} = F_{∗} ∘ (id ×G^{G}),

where F_{∗}: I^{op} × GROUPOIDS→ ch(ℤ-MODULES) has the property that for each object c in I the functor F_{∗}(c, −) sends equivalences of groupoids to homotopy equivalences.

• A contravariant chain complex D_{∗} of projective ℤI-modules.

• A G-CW complex X. Suppose that there are numbers d, n and N ∈ ℤ with d ≥ 0 such that the following conditions are satisfied:

(A) D_{∗} is d-dimensional, i.e. D_{k}(c) = 0 for all objects c in I and for all k ∈ ℤ

such that k < 0 or k > d. (B) The chain complexes E_{∗}(c, G/H) are uniformly homologically (N − 1)- bounded below, i.e. for all objects c in I and all orbits G/H we have H_{q}(E_{∗}(c, G/H)) = 0 for q < N.

(C) The G-CW complex X has only finite isotropy groups and only finitely

many orbit types. (D) For each finite subgroup H ⊂ G and every p ≤ n + d − N the homology

group

H_{p}(Z_{G}H\X^{H}; ℤ)

is an (almost) finitely generated ℤ-module. Then for each p ≤ n the map

H_{p} (C_{∗}^{ℤOr (G)}(X) ⊗_{ℤOr (G)} hom_{I}(D_{∗}, E_{∗})) ⟶^{H_{p}(t_{∗})} H_{p} (hom_{I} (D_{∗}, C_{∗}^{ℤOr (G)}(X) ⊗_{ℤOr (G)} E_{∗}))

is an (almost) isomorphism.

Here of course

t_{∗}: C_{∗}^{ℤOr (G)}(X) ⊗_{ℤOr (G)} hom_{I}(D_{∗}, E_{∗}) → hom_{I} (D_{∗}, C_{∗}^{ℤOr (G)}(X) ⊗_{ℤOr (G)} E_{∗})

is the chain map that sends x ⊗ ϕ to the map D_{∗} → C_{∗}^{ℤOr (G)}(X) ⊗_{ℤOr (G)} E_{∗} given by y ↦ x ⊗ ϕ(y). There are also linear versions of Examples 1.6 and 1.9.

We mention that Theorem 6.1 above remains true in a more general setting. Namely we can replace the ring of integers ℤ by any commutative Noetherian ring R, and C_{∗}^{ℤOr (G)}(X) by any contravariant ROr(G)-chain complex satisfying the following conditions: C_{∗} is a complex of projectives concentrated in non-negative degrees; the set S(C_{∗}) := {(H) | S_{G/H}C_{p} ≠ 0 for some p ∈ ℤ} is finite; (H) ∈ S(C_{∗}) implies that H is finite; and for any finite group H ⊂ G and any p ≤ n+d−N the homology group H_{p}((pr_{∗} C_{∗})(H)) is an (almost) finitely generated ℤ-module.

Here (H) denotes the conjugacy class of a subgroup H ⊆ G and S_{G/H} is the splitting functor defined in [Lüc89, 9.26 on page 170]; pr_{∗}C_{∗} is the RSub(G)-chain complex obtained from C_{∗} by induction (see [Lüc89, 9.15 on page 166])) with the projection pr: Or(G) → Sub(G) defined in (3.10).

The situation above is a special case because C_{∗}^{ℤOr (G)}(X) is a free (in the sense of [Lüc89, 9.16 on page 167 and page 356]) and hence a projective ℤOr(G)-chain complex, the set S(C_{∗}^{ℤOr (G)}(X)) is just the set of conjugacy classes of isotropy groups of X, and H_{p}(pr_{∗} C_{∗}^{ℤOr (G)}(X)(H)) ≅ H_{p}(Z_{G}H\X^{H}; ℤ).

7. Examples of categories with finite-dimensional classifying spaces

In this section we show two examples of categories that have a finite-dimensional contravariant classifying space, in the sense of Definition 3.4. The second example is relevant to the construction of topological cyclic homology and will play a role in the applications discussed at the end of the introduction. In both examples we will denote by ℕ the set of non-negative integers {0, 1, 2, …}.

Example 7.1. Let N be the category associated to the partially ordered set ℕ, i.e. the category whose objects are the non-negative integers ℕ and for which mor_{N} (i, j) is empty if i > j, or consists of precisely one element if i ≤ j. There is a 1-dimensional model for the contravariant classifying space EN. In fact, define EN (i) to be the interval [i, ∞). The zero-skeleton is given by the intersection of [i, ∞) with ℤ; the 1-cells are then obvious. (Compare also [DL98, Example 3.9 on page 224].)

Example 7.2. Let RF be the following category. The set of objects is ℕ = {0, 1, 2, …}. The set of morphisms from m to n is {(i, j) ∈ ℕ × ℕ | i + j = n − m}. In particular there are no morphisms from m to n if m > n. The identity morphism of n is given by (0, 0). The composition of (i_{0}, j_{0}): n_{0} → n_{1} and (i_{1}, j_{1}): n_{1} → n_{2} is (i_{0} + i_{1}, j_{0} + j_{1}): n_{0} → n_{2}.

Notice that a contravariant functor E from RF to the category of spectra is the same as a sequence of spectra E_{0}, E_{1}, E_{2}, … together with a system of maps

\xymatrix{… \ar[r]^{r_{2}} \ar[r]_{f_{2}} & E_{2} \ar[r]^{r_{1}} \ar[r]_{f_{1}} & E_{1} \ar[r]^{r_{0}} \ar[r]_{f_{0}} & E_{0}}

such that r_{i} ∘ f_{i+1} = f_{i} ∘ r_{i+1} holds for all i ∈ ℕ. Namely, put E_{i} = E(i), r_{i} = E((0, 1): i → (i+1)) and f_{i} = E((1, 0): i → (i+1)). Topological cyclic homology is defined as the homotopy limit of such a system of spectra, where the E_{n}’s are given by the fixed points of cyclic p-groups acting on topological Hochschild homology, and the structure maps are the so-called Restriction and Frobenius maps (see [BHM93]).

Proposition 7.3. There is a two-dimensional model for the contravariant classifying space of RF.

Proof. Define a contravariant RF-space E by sending an object n to the space {(x, y) ∈ ℝ × ℝ | x, y ≥ 0} and a morphism (i, j): m → n to the map (x, y) ↦ (x + i, y + j). Obviously each space E(n) is contractible. It remains to construct the RF-CW -structure.

Define E_{0}(n) = ℕ × ℕ, E_{1}(n) = {(x, y) ∈ ℝ × ℝ | x, y ≥ 0, x or y ∈ ℕ} and E_{2}(n) = E. This yields a filtration E_{0} ⊂ E_{1} ⊂ E_{2} = E of contravariant RFspaces. We claim that this is an RF-CW -structure. We have to construct the relevant pushouts of contravariant RF-spaces. Define for each n ∈ ℕ embeddings of contravariant RF-spaces

Q_{0}(n): mor_{RF}(?, n) → E  (i, j) ↦ (i, j);
Q_{1}^{h}(n): mor_{RF}(?, n) × [0, 1] → E  ((i, j), s) ↦ (i + s, j);
Q_{1}^{v}(n): mor_{RF}(?, n) × [0, 1] → E  ((i, j), s) ↦ (i, j + s);
Q_{2}(n): mor_{RF}(?, n) × [0, 1] × [0, 1] → E  ((i, j), s, t) ↦ (i + s, j + t).

We claim that ∐_{n∈ℕ} im(Q_{0}) = E_{0} and that we get pushouts of contravariant RF-CW -complexes

\xymatrix{∐_{n∈ℕ}(mor_{RF}(?, n) × \{0, 1\} ∐ mor_{RF}(?, n) × \{0, 1\}) \ar[r]^{^{∐^{n∈ℕ}(Q_{1}^{h}| ∐ Q_{1}^{v}|)}} \ar[d] & E_{0} \ar[d] \\ ∐_{n∈ℕ}(mor_{RF}(?, n) × [0, 1] ∐ mor_{RF}(?, n) × [0, 1]) \ar[r]^{_{∐_{n∈ℕ}(Q_{1}^{h} ∐ Q_{1}^{v})}} & E_{1}}

and

\xymatrix{∐_{n∈ℕ} mor_{RF}(?, n) × ∂([0, 1] × [0, 1]) \ar[r]^{∐_{n∈ℕ} Q_{2}|} \ar[d] & E_{1} \ar[d] \\ ∐_{n∈ℕ} mor_{RF}(?, n) × [0, 1] × [0, 1] \ar[r]^{∐_{n∈ℕ} Q_{2}} & E}

where Q_{1}^{h}|, Q_{1}^{v}| and Q_{2}| denote the obvious restrictions. This finishes the proof of Proposition 7.3. □

8. The homology of classifying spaces for families

This section is devoted to the proof of Proposition 1.7. We first recall some definitions.

A G-CW -complex is a CW -complex with G-action such that for an open cell e and g ∈ G with g · e ∩ e ≠ ∅ multiplication with g induces the identity on e. For information about G-CW -complexes we refer for instance to [Lüc89, Sections 1 and 2] and [tD87, Chapter II, Section 1]. A G-CW complex is proper if all isotropy groups are finite. Recall that a family F of subgroups of G is a set of subgroups closed under conjugation and taking subgroups. A G-CW -complex EG(F) all whose isotropy groups belong to F and whose H-fixed point sets are contractible whenever H ∈ F is called a classifying space for the family F. This notion is due to tom Dieck (see [tD87, Chapter I, Section 6]). Any G-CW -complex whose isotropy groups belong to F admits up to G-homotopy precisely one G-map to EG(F). In particular any two models for EG(F) are G-homotopy equivalent. A functorial bar-type construction for EG(F) is given in [DL98, Lemma 7.6].

Lemma 8.1. Let X be a proper G-CW -complex. Suppose that there exists for each p ≥ 0 a positive integer d(p) such that for any isotropy group H of X multiplication with d(p) annihilates H̃_{p}(BH; ℤ).

Then the canonical projection pr_{X}: EG ×_{G} X → G\X induces for all p ≥ 0 a ℤ-almost isomorphism

H_{p}(pr_{X}; ℤ): H_{p}(EG ×_{G} X; ℤ) → H_{p}(G\X; ℤ).

Proof. Let X_{n} be the n-skeleton of X. Then both maps EG×_{G}X_{n} → EG×_{G}X and G\X_{n} → G\X are n-connected and induce isomorphisms on H_{p}(−; ℤ) for p < n. Hence it suffices to prove the claim for each n-skeleton X_{n}. This will be done by induction over n. The induction beginning n = −1 is trivial since X_{−1} = ∅. The induction step from n to n + 1 is done as follows.

We can write X_{n+1} as a G-pushout

\xymatrix{∐_{i∈I} G/H_{i} × S^{n} \ar[d] \ar[r]^{∐_{i∈I} q_{i}} & X_{n} \ar[d] \\ ∐_{i∈I} G/H_{i} × D^{n+1} \ar[r]^{∐_{i∈I} Q_{i}} & X_{n+1}.}

Applying EG ×_{G} − and G\− one obtains two pushout squares whose left vertical arrows are again cofibrations. The projections pr_{Z} where Z runs through the four corners of the G-pushout square yield a map between the two new squares. By a Five-Lemma argument, the Künneth formula and the fact that H_{p}(−; ℤ) satisfies the disjoint union axiom for arbitrary index sets, the induction step follows from the assumption that the map H_{p}(pr_{G/H}; ℤ): H_{p}(EG ×_{G} G/H; ℤ) → H_{p}({pt}; ℤ) is surjective and its kernel is annihilated by d(p) for each isotropy group H of X. □

Lemma 8.2. Let F be a family of finite subgroups of G. Fix an element H ∈ F. Suppose that there exists for each p ≥ 0 a positive integer d(p) such that for any K ∈ F with H ⊂ K the annihilator ideal of H̃_{p}(B(Z_{G}H ∩ K); ℤ) contains d(p). Then H_{p}(Z_{G}H\EG(F)^{H}; ℤ) is an almost finitely generated ℤ-module for all p ≥ 1.

Proof. Fix H ∈ F. The projection EZ_{G}H ×_{Z_{G}H} EG(F)^{H} → BZ_{G}H is a homotopy equivalence since EG(F)^{H} is contractible. Now apply Lemma 8.1 to the proper Z_{G}H-CW -complex EG(F)^{H} and the projection EZ_{G}H ×_{Z_{G}H} EG(F)^{H} → Z_{G}H\EG(F)^{H}. □

We are now prepared to prove Proposition 1.7.

Proof of Proposition 1.7. Obviously EG(F) has finite isotropy groups and only finitely many orbit types. There is a number d such that the order of any element H ∈ F divides d. Hence H̃_{p}(BK; ℤ) is annihilated by d for each K ∈ F and p [Bro94, Section III.10]. Now apply Lemma 8.2. □

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[BHM93] Marcel Bökstedt, Wu-Chung Hsiang, and Ib Madsen. The cyclotomic trace and algebraic K-theory of spaces. Invent. Math., 111(3):465–539, 1993.

[BK72] Aldridge K. Bousfield and Daniel M. Kan. Homotopy limits, completions and localizations, volume 304 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1972.

[Bro94] Kenneth S. Brown. Cohomology of groups, volume 87 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original.

[DL98] James F. Davis and Wolfgang Lück. Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory. K-Theory, 15(3):201–252, 1998.

[Goo92] Thomas G. Goodwillie. Calculus. II. Analytic functors. K-Theory, 5(4):295–332, 1991/92.

[LMR03] Wolfgang Lück, Michel Matthey, and Holger Reich. In preparation, 2003.

[LRRV03] Wolfgang Lück, Holger Reich, John Rognes, and Marco Varisco. In preparation, 2003. [Lüc89] Wolfgang Lück. Transformation groups and algebraic K-theory, volume 1408 of Lecture

Notes in Mathematics. Springer-Verlag, Berlin, 1989. Mathematica Gottingensis.

[Ste67] Norman E. Steenrod. A convenient category of topological spaces. Michigan Math. J., 14:133–152, 1967.

[tD87] Tammo tom Dieck. Transformation groups, volume 8 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1987.

[Wei94] Charles A. Weibel. An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994.

[Whi78] George W. Whitehead. Elements of homotopy theory, volume 61 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1978.

Westfälische Wilhelms-Universität Münster, Mathematisches Institut. Einsteinstr. 62, D-48149 Münster, Germany

   E-mail address: lueck@math.uni-muenster.de
   URL: http://www.math.uni-muenster.de/u/lueck
   E-mail address: reichh@math.uni-muenster.de
   URL: http://www.math.uni-muenster.de/u/reichh
   E-mail address: varisco@math.uni-muenster.de
   URL: http://www.math.uni-muenster.de/u/varisco