BOUSFIELD LOCALIZATIONS OF CLASSIFYING SPACES OF NILPOTENT GROUPS

WILLIAM G. DWYER, , EMMANUEL DROR FARJOUN, , AND DOUGLAS C. RAVENEL,

1. Introduction

Let G be a finitely generated nilpotent group. The object of this paper is to identify the Bousfield localization L_{h}BG of the classifying space BG with respect to a multiplicative complex oriented homology theory h_{∗}. We show that L_{h}BG is the same as the localization of BG with respect to the ordinary homology theory determined by the ring h_{0}. This is similar to what happens when one localizes a space X with respect to a connected ring theory E: it follows from results of Bousfield [Bou79, Theorem 3.1] that L_{E}X is the localization of X with respect to ordinary homology with coefficients in the ring E_{0}. The point in this paper is that we do not require the spectrum h to be connected.

Our main result is

Theorem 1. Let G be a finitely generated nilpotent group, and let h_{∗} be a multiplicative complex oriented homology theory. Then L_{h}BG = L_{R}BG, where R is the ring h_{0} and L_{R} is localization with respect to the ordinary homology theory determined by R.

The hypothesis that h_{∗} be multiplicative is not essential to any of our arguments. We include it mainly to avoid cumbersome statements, and because most complex oriented theories of interest, such as Morava K-theory, are multiplicative. By modifying somewhat the results it seems that one could remove the assumption that h_{∗} is multiplicative. However, do not consider this case. Complex orientability is used in an essential way, in the proof of Theorem 3.

Our method of proof is to begin with finite p-groups and proceed by induction on the order of the group. We show that if G has a normal subgroup H such that BH is h_{∗}-local, G/H = Z/p, and B(G/H) is R-local for R = h_{0}, then BG is h_{∗}-local. We do this by studying the fibration

BH ⟶ BG ⟶ B(G/H) = BZ/p

To pass to arbitrary finitely generated nilpotent groups, we use the arithmetic square decomposition of L_{h}X due to Mislin and Bousfield.

Our main task is to study the fibration displayed above. In general it is not true that if the base and fibre of a fibration are local with respect to some homology theory, then the total space is also local. For example in the fibration

S^{2} ⟶ RP^{2} ⟶ BZ/(2)

All three authors were partially supported by the US-Israel Binational Science Foundation, and the first and third authors by the National Science Foundation.

both fiber and base are local with respect to ordinary integral homology, but the total space is not [BK72], [DDK77].

Our technique for dealing with this problem is to use the following lemma. If B is a space and C is a class of fibrations over B, say that C has h_{∗}-accessible fibres if any h_{∗}-equivalence (over B) between fibrations in C induces an h_{∗}-equivalence on fibres.

Lemma 2. Let h_{∗} be an arbitrary homology theory, and consider the diagram

F  E  B F′  L_{h}E  B w_{_{f} u}  ^{π} w_{_{g} u} w  _{^{L_{h}π} w}

in which each row is a fiber sequence. Suppose that B and F are both h_{∗}-local, and that there exists some class C of fibrations over B which has h_{∗}-accessible fibres and contains both π and L_{h}π. Then E is h_{∗}-local.

Proof. Since C has h_{∗}-accessible fibres and h_{∗}(g) is an equivalence, h_{∗}(f) is also an equivalence. The space F is h_{∗}-local by assumption, and F′ is h_{∗}-local since it is the homotopy fiber of a map between h_{∗}-local spaces. Therefore f, being and h_{∗}-equivalence between h_{∗}-local spaces, is an equivalence. It follows that g is also an equivalence and E is h_{∗}-local.

In order to use this lemma, we show in the next section that if h_{∗} is a multiplicative complex oriented homology theory with the property that h_{0} is a vector space over Z/p, then the class of all fibrations over BZ/p has h_{∗}-accessible fibres.

2. Fibrations over BZ/p

In this section we prove the following theorem. The results and arguments are inspired by the work of Kriz [Kri].

Theorem 3. Let h_{∗} be a multiplicative complex oriented homology theory such that the ring h_{0} is a mod p vector space. Suppose that E and E′ are fibrations over BZ/p with fibres F and F′, respectively, and that f : E → E′ is a map over BZ/p which induces an isomorphism h_{∗}E ≅ h_{∗}E′. Then f also induces an isomorphism h_{∗}F ≅ h_{∗}F′.

Recall that the homotopy coequalizer of a pair of maps f, g : X → Y is obtained by taking the cylinder X × [0, 1] and gluing one end to Y by f and the other end to Y by g. This construction is sometimes also called the double mapping cylinder of f and g. Given maps of pairs f, g : (X, A) → (Y, B), the homotopy coequalizer of f and g is the pair (Z, C), where Z is the homotopy coequalizer of the two maps X → Y and C is the homotopy coequalizer of the two maps B → C. A diagram of pairs equivalent to

(X, A)  (Y, B)  (Z, C)^{f, g} w  w

is said to be a homotopy coequalizer diagram. The following lemma is elementary.

Lemma 4. Let

(X, A)  (Y, B)  (Z, C)^{f, g} w  w

2 WILLIAM G. DWYER, UNIVERSITY OF NOTRE DAME, NOTRE DAME, IN 46556 USA, EMMANUEL DROR FARJOUN, HEBREW UNIVERSITY, JERUSALEM, ISRAEL, AND DOUGLAS C. RAVENEL, UNIVERSITY OF ROCHESTER, ROCHESTER, NY 14627 USA

be a homotopy coequalizer diagram. Suppose that h_{∗} is a homology theory. Then there is a natural long exact sequence

⋯ ⟶ h_{i}(X, A) ⟶^{f_{∗}−g_{∗}} h_{i}(Y, B) ⟶ h_{i}(Z, C) ⟶ h_{i−1}(X, A) ⟶ ⋯

Lemma 5. Let G be a group of order p with generator g, and V a mod p vector space with an action of G. Then the endomorphism (1 − g) of V is nilpotent (in the sense that for some integer k, (1 − g)^{k} = 0). In particular, the kernel of (1 − g) : V → V is nontrivial.

Proof. It is possible to choose k = p, since, in view of the fact that we are working mod p, (1 − g)^{p} = 1 − g^{p} = 0.

Lemma 6. Suppose that h_{∗} is a multiplicative complex oriented homology theory. Consider a homotopy fibre square

Ẽ  Ẽ′ E  E′^{g} w_{_{q} u}  _{_{q′} u}_{^{f} w}

in which q and q′ are principal S^{1}-bundles. If f induces on isomorphism on h_{∗}, then so does g.

Proof. Let ξ′ be the complex line bundle over E′ associated to q′ and ξ the complex line bundle over E associated to q. Denote the Thom spaces of these bundles by M(ξ) and M(ξ′) respectively. There is a map of cofibration sequences

Ẽ  E  M(ξ) Ẽ′  E′  M(ξ′)^{q} w_{_{g} u}  w_{_{f} u}  _{_{M(f)} u}_{^{q′} w}  w

The map h_{∗}(f) is an isomorphism by hypothesis. By the Thom isomorphism for h_{∗}, the map h_{∗}(M(f)) can be identified with h_{∗}(f) and so it too is an isomorphism. The fact that h_{∗}(g) is an isomorphism follows from looking at long exact homology sequences and using the five lemma.

Proof of Theorem 3. Let G denote the group Z/p, and g ∈ G some chosen generator. We can assume that F and F′ are G-spaces, and that f is obtained up to homotopy by taking the Borel construction β(f) on a G-map F → F′. (One way to obtain a suitable G-space equivalent to F, for instance, is to take the pullback over E → BG of the universal cover of BG.)

Suppose that X is a G-space (in our case either F or F′). Note that the homotopy coequalizer of the G-maps 1, g : G → G is the circle S^{1} with the usual rotation action of G. More generally, the homotopy coequalizer of 1, g : X×G → X×G is the product G-space X × S^{1} Taking Borel constructions gives a homotopy coequalizer diagram

β(X × G)  β(X × G)  β(X × S^{1})^{u,v} w  w

where u and v are the appropriate induced Borel construction maps. It is clear that β(X × S^{1}) is the total space of a principal S^{1}-bundle over β(X), in fact, the total space of the pullback along the map β(X) → BG of the usual principal S^{1}- bundle β(S^{1}) → BG. The Borel construction β(X × G), on the other hand, can be identified up to homotopy with X itself in such a way that u and v can be identified with the original maps 1 and g.

Let Ẽ denote the Borel construction B(G, F × S^{1}) and Ẽ′ the Borel construction B(G, F′ × S^{1}). According to the above considerations we have a homotopy coequalizer diagram

(F′, F)  (F′, F)  (Ẽ′, Ẽ)^{1, g} w  w

Since h_{∗}(E′, E) vanishes by assumption, h_{∗}(Ẽ′, Ẽ) vanishes by Lemma 6. Lemma 4 then implies that the endomorphism (1 − g_{∗}) of h_{∗}(F′, F) is an automorphism, but by Lemma 5 this can happen only if h_{∗}(F′, F) = 0, in other words, only if the map F → F′ induces an h_{∗}-isomorphism.

3. Localization of classifying spaces

We begin by recalling a result of Bousfield [Bou82] about localizations of BZ/p. (In that paper he actually determines L_{h}K(A, n) for any homology theory h_{∗} and any abelian group A.) We say that a space is h_{∗}-acyclic if the reduced homology h̃_{∗}(X) vanishes, or equivalently if L_{h}(X) is contractible.

Lemma 7. If h_{∗} is a multiplicative homology theory, then the space BZ/p is h_{∗}- acyclic if p is invertible in h_{0} and h_{∗}-local otherwise. Equivalently, L_{h}BZ/p = L_{R}BZ/p, where R = h_{0} and L_{R} denotes localization with respect to H_{∗}(−; R).

With Theorem 3 in hand we can prove the following.

Theorem 8. Suppose that h_{∗} is a multiplicative complex oriented homology theory, and that G is a finite p-group. The space BG is h_{∗} acyclic if p is invertible in h_{0} and h_{∗}-local otherwise.

Proof. If p is invertible is h_{0} it is obvious from the Atiyah-Hirzebruch spectral sequence that BG is h_{∗}-acyclic, so we assume that p is not invertible in h_{0} and prove that BG is h_{∗}-local. Suppose first that h_{0} is a Z/p-vector space. We argue by induction on the order of G. Let H ⊂ G be a normal subgroup of index p. The space BZ/p is h_{∗}-local by Lemma 7, and so there is a diagram of fibration sequences:

BH  BG  BZ/p F′  L_{h}BG  BZ/p. w_{_{f} u}  _{_{g} u}  w w  w

It thus follows immediately from Lemma 2 and Theorem 3 that BG is h_{∗}-local.

Now consider a general h of the specified type. For a prime q, let h/q denote the smash product of the spectrum h representing h_{∗} with a mod q Moore spectrum

denoted here by M. Clearly h/q is still complex orientable: This is true since a complex orientation for a spectrum E is a class x ∈ E^{2}(CP^{∞}) with certain properties. One has a map E → E ∧ M induced by the unit in M, and one can use the image of x under this map as a complex orientation for E ∧ M.

Alternatively, E is complex orientable iff it is an MU-module spectrum. If E is an MU-module spectrum, so is E ∧ M.

If X is a space, let X_{Q} denote the localization of X with respect to rational homology. Since (BG)_{Q} = ∗, it follows from Proposition 7.2 of [Bou82] that we have a fibration sequence

L_{h}BG ⟶ _{∏q}L_{h/q}BG ⟶ (∏_{q} L_{h/q}BG)_{Q} = pt.

where q runs though the primes not invertible in h_{0}. We know from above that L_{h/p}BG = BG and that L_{h/q}BG = pt. for q ≠ p. It follows that L_{h}BG = BG as claimed.

Slightly more generally we have

Theorem 9. Suppose that G be a finite nilpotent group and that h is a multiplicative complex oriented homology theory. Then L_{h}BG = L_{R}BG, where L_{R} is as in Lemma 7. In particular, if no prime dividing the order of G is invertible in h_{0}, then BG is h_{∗}-local.

Proof. The group G is the direct product of its Sylow p-subgroups G_{p}, so we have BG ≃ ∏_{p} BG_{p} and L_{h}BG ≃ ∏_{p} L_{h}BG_{p}. The factors in this second product can be identified with the help of Theorem 8. There is a similar product formula for L_{R}BG.

We now turn to the proof of the main theorem.

Proof of Theorem 1. It is shown by Bousfield in [Bou82] that for any space X, L_{h}X ≃ L_{h}L_{R}X where L_{R} is localization with respect to H_{∗}(−; R). It is easy to check that a map of spaces is an isomorphism on H_{∗}(−; R) if and only if it is an isomorphism on ⊕_{p}H_{∗}(−; Z/p ⊗ R) as well as an isomorphism on H_{∗}(−; Q ⊗ R). Let P be the set of all primes which are not invertible in R. It follows that a map of spaces is an isomorphism on H_{∗}(−; R) if and only if it is an isomorphism on H_{∗}(−; ⊕_{p∈P}Z/p), as well as, if Q ⊗ R ≠ 0, an isomorphism on H_{∗}(−; Q). Since BG is a nilpotent space, the results of [DDK77] imply that if Q ⊗ R = 0 there is an equivalence

L_{R}BG ≃ ∏_{p∈P} L_{Z/p}BG

while if Q ⊗ R ≠ 0 there is a homotopy fibre square

L_{R}BG  _{∏p∈P}L_{Z/p}BG (BG)_{Q}  (_{∏p∈P}L_{Z/p}BG)_{Q}. w u  u w

We will carry out the proof by showing that L_{R}BG is h_{∗}-local, so that L_{h}BG ≃ L_{h}L_{R}BG ≃ L_{R}BG. To do this we will show that all of the constituents in the above formulas for L_{R}BG are h_{∗}-local, and then appeal to the fact that the class of h_{∗}-local spaces is closed under homotopy inverse limit constructions.

Now according to [BK72, VI 2.6, 2.2 and IV §2] the space L_{Z/p}BG ≃ (Z/p)_{∞}BG can be identified as B(G_{p̂}), where G_{p̂} = lim G/Γ_{s}^{p}G is the p-lower-central-series completion of G. In particular L_{Z/p}BG is equivalent to the homotopy inverse limit of the tower {B(G/Γ_{s}^{p}G)}_{s}. If p ∈ P then each space in this tower is h_{∗}-local (Theorem 8), and so L_{Z/p}BG is h_{∗}-local by homotopy inverse limit closure. By the same principle, ∏_{p∈P} L_{Z/p}BG is h_{∗}-local.

We can complete the proof by showing that if Q ⊗ R ≠ 0 then any space W local with respect to rational homology is also local with respect to h_{∗}. Given the definition of what it means for a space to be h_{∗}-local, we have to show that any h_{∗}-equivalence f : X → Y induces a bijection f^{#} : [Y, W] → [X, W] (where the brackets indicate homotopy classes of maps). However, by [Bou82, 3.3], such an f is a rational equivalence, so the fact that f^{#} is a bijection follows from the fact that W is local with respect to rational homology.

4. possible extensions and related problems

It was shown above that the Bousfield localization with respect to certain homology theories of the classifying space BG of a finitely generated nilpotent group G is the same as the localization with respect to a classical homology theory with appropriate coefficients. The question remains open for other (non finitely generated) nilpotent groups and other localization functors. Using the fact that K(F, 2), where F is any free abelian group, is local with respect to complex K- theory it is not hard to see that so is K(G, 1) for any abelian group G and in fact one can show that theorem 1 holds for any abelian group.

To go beyond Eilenberg-MacLane spaces the following is a natural possible extension of the main results above.

Let N be a nilpotent space whose homotopy groups vanish above certain dimension n. Is it true that any Bousfield homological localization of N is equivalent to its localization with respect to a well chosen classical homology theory?

Similar question arise beyond the realm of Bousfield homological localization. Namely, one may ask for analogues of the above questions for an arbitrary homotopical localization L_{f} with respect to an arbitrary map f. In that case it is not true that the localization will be the same as the localization with respect to a well chosen classical homology. This is because the map BZ/p^{2} → BZ/p, induced by the quotient group map, is in fact a homotopy localization map, but it is not an homological localization map. But one does expect that an arbitrary localization L_{f}N of a nilpotent space N as above will also be a nilpotent space with vanishing homotopy groups above a certain dimension that depends only on n.

References

[BK72] A. K. Bousfield and D. M. Kan. Homotopy Limits, Completions and Localizations, volume 304 of Lecture Notes in Mathematics. Springer-Verlag, 1972.

[Bou79] A. K. Bousfield. The localization of spectra with respect to homology. Topology, 18:257– 281, 1979.

[Bou82] A. K. Bousfield. On homology equivalences and homological localizations of spaces. American Journal of Mathematics, 104:1025–1042, 1982.

[DDK77] W. G. Dwyer, E. Dror, and D. M. Kan. An arithmetic square for virtually nilpotent spaces. Illinois Journal of Mathematics, 21:242–254, 1977.

[Kri] I. Kriz. Morava K-theory of classifying spaces: some calculations. To appear in Topology.

[Kur56] A. G. Kurosh. The Theory of Groups, Volume Two. Chelsea Publishing Company, New York, 1956.

6 WILLIAM G. DWYER, UNIVERSITY OF NOTRE DAME, NOTRE DAME, IN 46556 USA, EMMANUEL DROR FARJOUN, HEBREW UNIVERSITY, JERUSALEM, ISRAEL, AND DOUGLAS C. RAVENEL, UNIVERSITY OF ROCHESTER, ROCHESTER, NY 14627 USA

UNIVERSITY OF NOTRE DAME, NOTRE DAME, IN 46556 USA HEBREW UNIVERSITY, JERUSALEM, ISRAEL UNIVERSITY OF ROCHESTER, ROCHESTER, NY 14627 USA