Homotopy idempotent functors on classifying spaces

arXiv:1210.7772v1 [math.AT] 29 Oct 2012 Homotopy idempotent functors on classifying spaces Nat` alia Castellana and Ram´ on Flores October 30, 2012 Abstract Fix a prime p. Since their definition in the context of Localization Theory, the homotopy functors PBZ/p and CWBZ/p have shown to be powerful tools to understand and describe the mod pstructure of a space. In this paper, we study the effect of these functors on a wide class of spaces which includes classifying spaces of compact Lie groups and their homotopical analogues. Moreover, we investigate their relationship in this context with other relevant functors in the analysis of the mod phomotopy, such as Bousfield-Kan completion and Bousfield homological localization. 1 Introduction Let Aand Xbe two connected topological spaces. The study of the homotopical properties of Xthat are visiblethrough the mapping space map (A, X)is called the A-homotopy theory of Xand was proposed by E. Dror-Farjoun in [Far96]. In this context, it is particularly important to describe the behaviour of the nullification PΣiAand the cellularization CWΣiA(see definitions in Section 2), which are functors that play in A-homotopy theory the same role as the connected covers and Postnikov pieces play in classical (S0-) homotopy theory. Let pbe a prime. If Xis a space and we are interested in describing the p-primary part of Xthrough its A-homotopy theory for some space A, there are some choices of Athat become apparent. Probably the easiest one are the Moore spaces M(Z/pn,1) and their suspensions; this task was undertaken in the nineties by Rodriguez-Scherer in the case of cellularization [RS01] and Bousfield [Bou97], who did not only described PM(Z/pm,n)X N. Castellana: Departamento de Matem´aticas, Universidad Aut´onoma de Barcelona, 08193 Bellaterra, Spain; e-mail: [email protected]ab.es R. Flores: Departamento de Estad´ıstica, Universidad Carlos III, 28029 Colmenarejo, Spain; e-mail: [email protected] (corresponding author) Mathematics Subject Classification (2010): Primary 55P20; Secondary 55P80 1 2 Nat`alia Castellana and Ram´on Flores for a wide number of spaces, but remarked the close relationship between these functors and the vn-periodic homotopy theory. In this paper we deal with the case A=BZ/p. After Miller’s solution of Sullivan conjecture [Mil84], and subsequentworkof Lannes, Dwyer-Zabrodskyand others, a number of new powerful tools were available to researchers attempting to study the mapping space map (BZ/p, X), and overwhelming success was reached, particularly for nilpotent spaces X. In the framework we are interested in, we should emphasize the work of Neisendorfer [Nei95], where the author proves that the functor PBZ/p can often recover the p-primary homotopy of Xfrom that of its n-connected cover, or [CCS07-2], about the BZ/p-homotopy of H-spaces. The first motivation for our work came from two different sources: the study undertaken by W. Dwyer in [Dwy96] concerning BZ/p-nullification of classifying spaces of compact Lie groups whose group of components is a p-group, and its relationship with the homology decompositions of BG; and Question 11 in Farjoun’s book ([Far96, page 175]), where he asked about the cellularity of the p-completion of BG. This seemed a natural extension of the problems considered by the second author concerning BZ/phomotopy of finite groups (see [Flo07], [FS07], and [FF]), so it was natural to investigate this structure with similar methods. We show that under certain hypothesis, we are able to characterize the effect of the nullification functor PBZ/p by means of a fibration. Theorem 4.1. Let Xbe a connected space with finite fundamental group and suchthat (PBZ/p(Xh1i))∧ p≃ ∗. Then there is a fibration LZ[1 p](Xp)→PBZ/p(X)→B(π1(X)/Tp(π1(X))) where Xpis the covering of Xwhose fundamental group is Tp(π1(X)), and LZ[1 p](Xp) denotes Bousfield homological localization of Xpwith respect to H∗(−;Z[1 p]). In particular, one can compute the homotopy groups of PBZ/p(X)in terms of those of Xif Xis good enough. This result is quite general, and in fact describes in a single statement a phenomenon which was previously known for finite groups, p-compact groups and some compact Lie groups, but not for p-local compact or Kaˇc-Moody groups (Corollary 4.13); so, it can be read then as a common property of a big family of homotopy meaningful spaces. Moreover, finite loop spaces also satisfy this property (Corollary 4.12). Another source of examples is the theory of infinite loop spaces. McGibbon [McG] shows that infinite loop spaces satisfy the hypothesis of Theorem 4.1 (see Corollary 4.14). We also obtain some interesting consequences of these results, including a detailed analysis of the relationship between the BZ/p-nullification and Z[1/p]-localization of Homotopy idempotent functors on classifying spaces 3 these spaces -which is very much in the spirit of [Dwy96, Section 6]- and the commutation of nullity functors on them, a situation that was discussed in [RS00] in a general framework. The second part of the paper deals with the effect of the cellularization functor CWBZ/p on classifying spaces of compact Lie groups. We show a Serre-type dichotomy theorem. Theorem 6.9. Let Gbe a compact connected Lie group. If there exists a non pcohomologically central element of order p, then the BZ/p-cellullarization of BG has infinitely many nonzero homotopy groups. Otherwise, it has the homotopy type of a K(V, 1), where Vis a finite elementary abelian p-group. This statement is in fact a consequence of a more general statement which extends Proposition 2.3 in [FS07]. Theorem 6.1. Let Xbe a connected nilpotent ΣnBZ/p-null space for some n≥0. Then the BZ/p-cellullarization of Xhas the homotopy type of a Postnikov piece with homotopy groupsare concentratedin degrees 1to n, or else ithas infinitely many nonzero homotopy groups. Moreover, if Xis 1-connected of finite type, then the fundamental group π1(CWBZ/p(X)) is a finite elementary abelian p-group. This result opens the way to describe with precision (up to p-completion) the BZ/pcellularization of BG for an ample class of Lie groups which includes p-toral groups and their discrete approximations, the 3-sphere, extensions of elementary abelian groups by groups of order prime to p-which generalize [FS07, Corollary 3.3]-, or BSO(3). In particular, we find examples of both cases of the dichotomy statement. It is interesting to remark here that we use intensively the fact that CWApreserves nilpotent spaces (Lemma 2.5), a fact that was conjectured in [Far96], but which to our knowledge has not been so far exploited in the literature. Notation: Let Rbe a commutative ring, R∞(X)denotes Bousfield-Kan p-completion of a space X([BK72]). When R=Z/p for a prime p,R∞(X)will be used instead of X∧ p. Moreover, LR(X)denotes the HR-localization of Bousfield ([Bou75]). All spaces are assumed to have the homotopy type of a CW-complex. 2 The cellularization and nullification functors Let Abe a connected space. In this section we will define the functors CWAand PA, which are the main tools we use to describe the p-primary structure of the spaces of interest in our work. Only some particular features of these functors, that will be crucial in our further developments, will be described while their relationship with Bousfield-Kan completion will be studied in the next section. A thorough account to these constructions can be found in [Far96]. 4 Nat`alia Castellana and Ram´on Flores Definition 2.1. Let Aand Xbe spaces. Then Xis called A-null if the inclusion of constant maps X ֒→map (A, X)is a weak equivalence. This is equivalent to the condition that map ∗(A, X)is weakly contractible when Xis connected. Dror-Farjoun defines a coaugmented and idempotent functor PA:Spaces → Spaces where PAXis A-null for every X, and such that the coaugmentation X→PAX induces a weak equivalence map (PAX, Y )→map (X, Y )for every A-null space Y. The corresponding definitions in the pointed context are completely analogous. Note that in the language of homotopy localization, PAis the localization with regard to the constant map A→ ∗, and the notation comes from Postnikov sections, which are in fact Snnullifications. Moreover, a space Xsuch that PAX≃ ∗ is called A-acyclic. Now we consider the cellular construction, which is somewhat dual of the previous construction, although not completely (see Theorem 2.3 below). Definition 2.2. Given pointed spaces Aand X,Xis said A-cellular if it can be built from Aby means of pointed homotopy colimits, possibly iterated. Moreover, a map X→Yis said to be an A-equivalence if it induces a weak equivalence map ∗(A, X)→ map ∗(A, Y ). The A-cellularization (or A-cellular approximation) is a canonical way of turning every space into an A-cellular space from the point of view of A-equivalences, which generalizes the classic process of cellular approximation. There exists an augmented endofunctor CW Aof the category of pointed spaces, such that for every space Xthe augmentation CWAX→Xis an A-equivalence, and in initial among all maps Y→Xwhich induce A-equivalence. Unlike PA, this functor only makes sense in the pointed context ([Cha96, 7.4]), and can be characterized in several ways [Far96, 2.E.8]. The remaining of the section is devoted to describe some properties of these functors that we will frequently use later. We begin with a theorem of W. Chach´olski that can be considered the most powerful tool to compute cellularization of spaces in an explicit way. The proof can be found in [Cha96, 20.3]. Theorem 2.3. Let Aand Xbe pointed spaces, and let Cbe the homotopy cofibre of the evaluation W[A,X]∗A→X, where the wedge is taken over all the homotopy classes of maps A→X. Then CWAXhas the homotopy type of the fibre of the map X→PΣAC. Next we will describe two preservation properties, that will be used extensively as we will frequently focus our interest in simply connected spaces and, more generally, nilpotent spaces. Lemma 2.4. [Bou94, 2.9] If Xis 1-connected then PA(X)is also 1-connected. Homotopy idempotent functors on classifying spaces 5 In particular, note that, according to a famous result of Neisendorfer [Nei95, Thm 0.1], there is no analogous result for higher degrees of connectivity. The second preservation property concerns to cellularization and it answers question 7stated by Dror-Farjoun in his book [Far96, p.175]. It is remarkable that the analogous problem in the category of groups was solved in [FGS07]. Lemma 2.5. If Xis a nilpotent space then CWA(X)is also nilpotent. Proof. Apply [BK72, V.5.2] to the fibration CWA(X)→X→PΣACin Theorem 2.3. From the definitions, one can check that if Xis A-null then CWA(X)≃ ∗ since ∗֒→Xis an A-equivalence. In general, the A-cellularization functor also preserves ΣnA- nullity for n≥1. Lemma 2.6. Let Xbe a space which is ΣnA-null for some n≥1then CWA(X)is also ΣnA-null. Proof. Again from Theorem 2.3 we have a fibre sequence CWA(X)→X→PΣA(X). Since the base space is ΣA-null, it is also ΣnA-null for any n≥1. The result follows since the nullification functor PΣApreserves then the fibration [Far96, 3.D.3]. If we specialize now to A=BZ/p, which is the case of interest in this paper, and we turn our attention to Eilenberg-MacLane spaces, it is interesting to observe that given an arbitrary group G, the BZ/p-nullity properties of K(G, n)for small values of nimply the BZ/p-nullity for every value of n, as well as some group-theoretic features of G. Lemma 2.7. Let Gbe an abelian discrete group. K(G, 2) is BZ/p-null if and only if pis invertible in Gand K(G, n)is BZ/p-null for all n≥1. Proof. We only need to show that if K(G, 2) is BZ/p-null then pis invertible in G and K(G, n)is BZ/p-null for all n≥1. Since K(G, 1) ≃ΩK(G, 2) is BZ/p-null, Hom(Z/p, G) = [BZ/p, BG]∗= 0. Therefore Ghas no elements of order p. Then, multiplication by pgives rise to a short exact sequence 0→Gp →G→G/pG →0. Now consider the induced fibration K(G, 1) →K(G/pG, 1) →K(G, 2). Since both K(G, 1) and K(G, 2) are BZ/p-null, by [Far96, 3.D.3], we see that B(G/pG)is also BZ/p-null. Therefore G/pG has no elements of order p, so it must be trivial. That is Gp →Gis an isomorphism and pis invertible in G. A standard transferargument (see e.g. [Bro82, Prop III.10.1]) showsthat ˜ H∗(BZ/p;G) is trivial. In particular, map ∗(BZ/p, K(G, n)) is weakly contractible for all n≥1. We finish this preliminary section by describing a context in which we can obtain information about the homology and homotopy groups of the cellularization. 6 Nat`alia Castellana and Ram´on Flores Lemma 2.8. If Ris a ring of coefficients and ˜ H∗(A;R) = 0, then ˜ H∗(CWA(X); R) = 0. If Xis nilpotent and R⊂Qthen πi(CWA(X)) ⊗R= 0 for i > 0. Proof. Under the hypothesis of the theorem, K(R, n)is A-local for n > 0, then the space map ∗(CWA(X), K(R, n)) is weakly contractible. By Lemma 2.5, we can apply [BK72, V.3.1]. 3BZ/p-homotopy and p-completion We devote this section to the description of the behaviour of the functors CWAand PA with respect to the p-completion functor of Bousfield and Kan. In particular, if η:X→ X∧ pis the p-completion, we want to characterize when the maps CWA(η)and PA(η)are mod pequivalences. This will be fundamental in our approach to the BZ/p-nullification and BZ/p-cellularization of classifying spaces, which will be undertaken in the last two sections and is the main goal of our note. A first approximation to these kind of questions appears in the work of Miller in the solution of the Sullivan Conjecture, which implies immediately a statement about BZ/pnullity. Theorem 3.1. [Mil84, Thm 1.5] Let Wbe a connected space with ˜ H∗(W;Z[1 p]) = 0 and let Xbe a nilpotent space. Then η:X→X∧ pis a W-equivalence. Corollary 3.2. If Xis a nilpotent space, the p-completion η:X→X∧ pis a BZ/pequivalence. Observe that if Xis 1-connected, we can p-complete our target space, if necessary, before computing CWBZ/pX. This statement, and the fact that the BZ/p-cellularization is constructed using copies of BZ/p as pieces, may lead to think that CWBZ/pXis always ap-complete space. Next lemma shows that this is true in certain cases but, as we will see in Example 3.4, not always. Lemma 3.3. If Xis a nilpotent space, then CWBZ/p(X)is p-complete if and only if ˜ H∗(CWBZ/p(X)∧ p;Q) = 0. Proof. Since BZ/p is both Q-acyclic and Fq-acyclic for q6=p,CWBZ/p(X)is so ([Far96, D.2.5] or Lemma 2.8), and then the rationalization and q-completions of CWBZ/p(X)are trivial. By Lemma 2.5, CWBZ/p(X)is also a nilpotent space, so it admits a Sullivan arithmetic square decomposition. The result follows. Example 3.4. Consider the space X=K(Z/p∞,2).Xis BZ/p-cellular by [CCS07-2, Lemma 3.3], but it is not p-complete since X∧ p≃K(Z∧ p,3) and ˜ H∗(X∧ p;Q)6= 0. In fact, the p-completion η:K(Z/p∞,2) →K(Z∧ p,3) induces a BZ/p-cellular equivalence, Homotopy idempotent functors on classifying spaces 7 then CWBZ/p(X∧ p)≃X. On the other hand, taking for example p= 2 and X=BΣ3, the classifying space of the symmetric group in three letters, it is not nilpotent, and the cellularization is not complete. See [FS07, Example 2.6] for details. Weproceednowto a systematicstudyofthe inducedmap CWBZ/p(η): CWBZ/p(X)→ CWBZ/p(X∧ p). We want to show under which conditions it becomes a mod pequivalence. The first step is a reduction concerning the fundamental group, for which we need the following definition. Definition 3.5. We say that an element x∈π1(X)lifts to Xif there exists a homotopy lift X  B(hxi) 88 q q q q q q q q q q qihxi//B(π1(X)). Proposition 3.6. Let X be a connected space. There is a fibration CWBZ/p(X)→X→Z with π1(Z)∼ =π1(X)/S, where Sis the normal subgroup generated by the elements of order pwhich lift to X. Proof. The fibration in the proposition isthe one constructed by Chach´olski (see Theorem 2.3) where Z=PΣBZ/p(CX). The subgroup Sis constructed in [CCS07-2, Prop. 2.1] in a way that E→Xis a BZ/p-cellular equivalence, where Eis the homotopy pullback E//  X  BS Bi //B(π1(X)). By construction π1(E)∼ =Sis generated by elements of order pwhich lift to E. Then the Chach´olski’s cofibre CE(see Theorem 2.3) is 1-connected and PΣBZ/p(CE)is too by Lemma 2.4. Since E→Xis a BZ/p-equivalence, from the following diagram of fibrations CWBZ/pE≃//  CWBZ/pX  E//  X  PΣBZ/p(CE)//PΣBZ/p(C). where Cis Chach´olski’s cofibre for X, we see that the fundamental group of PΣBZ/p(C) is π1(X)/S. 8 Nat`alia Castellana and Ram´on Flores Corollary 3.7. Let Xbe a connected space such that π1(X)is generated by elements of order pwhich lift to X. There is a bijection [BZ/p, CWBZ/p(X)] ∼ =[BZ/p, X]between unpointed homotopy classes of maps. Proof. Since CWBZ/p(X)→Xis a BZ/p-homotopy equivalence, there is a bijection [BZ/p, CWBZ/p(X)]∗∼ =[BZ/p, X]∗between pointed homotopy classes of maps. The following diagram [BZ/p, CWBZ/p(X)]∗//  [BZ/p, X]∗  [BZ/p, CWBZ/p(X)] //[BZ/p, X] shows that the quotient map is also a bijection since the induced morphism on fundamental groups π1(CWBZ/p(X)) →π1(X)is an epimorphism by Proposition 3.6. We can get information about the fundamental group of the cellularization since being BZ/p-cellular imposes some restrictions on the fundamental group of the space. Lemma 3.8. If Xis a BZ/p-cellular space, its fundamental group is generated by elements of order pwhich lift to X. Moreover, if Xis a finite type 1-connected space,then π1(CWBZ/p(X)) is a finitely generated abelian generated by elements of order pwhich lift to CWBZ/p(X). Proof. Let Sbe the normal subgroup of π1(X)generated by elements of order pwhich lift to X. Consider the pullback diagram E  //X  BS Bi//Bπ1(X). By [CCS07-2, Prop 2.1], the map E→Xis a BZ/p-cellular equivalence. Since X is BZ/p-cellular, there exists a map f:E→CWBZ/p(E)such that i◦f≃id where i:CWBZ/p(E)→E. In fact, this implies that p:E→PΣBZ/p(CE)is nullhomotopic, p≃p◦i◦f≃ ∗◦f≃ ∗, therefore CWBZ/p(E)≃E×PBZ/p(ΩCE). But this implies that Eis BZ/p-cellular since CWBZ/p(E)is BZ/p-acyclic, and then E≃X. In particular, π1(X) = S. To prove the second statement, it remains to prove that π1(CWBZ/p(X)) is a finitely generated abelian group. Since Xis 1-connected, then the Chach´olski’s cofibre CXis 1- connected and PΣBZ/p(CX)is too by Lemma 2.4. Then, we see that π2(PΣBZ/p(CX)) ∼ = H2(PΣBZ/p(CX); Z)is a quotient of H2(CX;Z), which in turn is a quotient of the finitely generated group H2(X;Z). Homotopy idempotent functors on classifying spaces 9 Next we need a technical lemma which describes the somewhat intrincate relationship between completion and nullification and that is a key result to understand under which conditions PA(η): PA(X)→PA(X∧ p)is a mod pequivalence (Corollary 3.11). Lemma 3.9. Let Abe a connected space, and let Xsuch that PA(X∧ p)and PA(X) are p-good spaces. Assume that PA(X)∧ pand PA(X∧ p)∧ pare A-null spaces. Then the pcompletion map ηX:X→X∧ pinduces a mod pequivalence PA(η): PA(X)→PA(X∧ p). Proof. Let ǫ:PA(X∧ p)→(PA(X))∧ pbe the unique map up to homotopy such that the right square of the following diagram commutes: XηX// ιX  X∧ p id // ιX∧ p  X∧ p (ιX)∧ p  PA(X)PA(ηX) //PA(X∧ p)ǫ//PA(X)∧ p. Note that ǫexists because PA(X)∧ pis A-null by hypothesis. The left square commutes by naturality, so (ιX)∧ p◦ηX≃ǫ◦PA(ηX)◦ιX. But also, (ιX)∧ p◦ηX≃ηPA(X)◦ιX by naturality of the completion. Because of the universal property of the nullification functor, ǫ◦PA(ηX)≃ηPA(X). Since PA(X)is p-good, η∗ PA(X)is an isomorphism in mod pcohomology. In particular, ǫ∗is a monomorphism and PA(ηX)∗is an epimorphism. Now consider the following commutative diagram: X∧ p id // ιX∧ p  X∧ p (ηX)∧ p// (ιX)∧ p  X∧ p (ιX∧ p)∧ p  PA(X∧ p)ǫ//PA(X)∧ p PA(ηX)∧ p //PA(X∧ p)∧ p. That is (ιX∧ p)∧ p◦(ηX)∧ p≃PA(η)∧ p◦ǫ◦ιX∧ p. But we also have (ιX∧ p)∧ p◦(ηX)∧ p≃(ηPA(X∧ p))◦ ιX∧ p. By hypothesis PA(X∧ p)∧ pis A-null, then the universal property of the nullification functor implies that PA(ηX)∧ p◦ǫ≃ηPA(X∧ p). Since PA(X∧ p)is p-good, (ηPA(X∧ p))∗is an isomorphism and hence (PA(ηX)∧ p)∗is a monomorphism. Therefore PA(ηX)∗is so, and we are done. Remark 3.10. If Xhas finite fundamental group, then both PA(X∧ p)and PA(X)are p-good spaces since they also have finite fundamental groups. Corollary 3.11. If Xis a 1-connected space and Ais such that ˜ H∗(A;Z[1 p]) = 0 then PA(η): PA(X)→PA(X∧ p)is a mod pequivalence. Proof. If Xis 1-connected then X∧ pis also 1-connected and both spaces are p-good. Moreover the BZ/p-nullification of a 1-connected space is also 1-connected. Miller’s theorem [Mil84, Thm 1.5] implies that the spaces PA(X)∧ pand PA(X∧ p)∧ pare A-null. The hypothesis of Lemma 3.9 are then satisfied. 16 Nat`alia Castellana and Ram´on Flores are trivial since the homotopy groups πi(PBZ/p(Xh1i)) ∼ =πi(LZ[1 p](Xh1i)) are Z[1 p]- modules. That is, ¯ Xis a connected space with finite fundamental group generated by p-torsion whose generators lift to ¯ X. In order to apply Theorem 4.2, it remains to check that map ∗(¯ Xh1i, Z)≃ ∗ for any connected BZ/p-null p-complete space Z. Recall that ¯ Xh1i ≃ PBZ/p(Xh1i). Then map ∗(PBZ/p(Xh1i), Z)≃map ∗(Xh1i, Z)≃ ∗ where the last equivalence follows by hypothesis. Remark 4.7. The proof of Theorem 4.1 also holds if we replace the analysis at one prime pfor a set of primes Sand imposing that the hypothesis on pointed mapping spaces are satisfied for any prime pin the set S. In that case we have to replace LZ[1 p]by LZ[S−1], and PBZ/p by PWwhere W=∨p∈SBZ/p. 4.1 Examples We want to explore the implications of these results on classifying spaces of Lie groups, which was the original motivation for our work. For this sake we need the following Lemma, which was proved by Dwyer [Dwy96, Theorem 1.2] using an induction. We include here a shorter proof, based on the homology decomposition of BG via p-toral subgroups. The key point here is that this decomposition is indexed over an mod pacyclic category, and this opens the way for computing PBZ/p for a more general class of p-good spaces (see Corollary 4.11). Lemma 4.8. Let Zbe a connected BZ/p-null and p-complete space. Let F:C→Top be a functor such that for each object c∈C,F(c)is connected and PBZ/p(F(c)∧ p)is mod pacyclic. If |C|∧ p≃ ∗, then map ∗(hocolimCF(c), Z)≃ ∗. Proof. The statement follows from a sequence of equivalences: map (hocolim CF(c), Z)≃holim Cmap (F(c), Z)≃holim Cmap (PBZ/p(F(c)∧ p), Z). Under the hypothesis of the lemma, this last mapping space is homotopy equivalent to map (|C|, Z)≃Zif |C|∧ p≃ ∗. Corollary 4.9. Let pbe a prime. If Gis a compact Lie group and Xis a connected p-complete BZ/p-null space, then map ∗(BG∧ p, X)is weakly contractible. Proof. The proof is divided into two steps. In the first one we assume that Gis a ptoral group, and then we use the existence of mod phomology decompositions of BG with respect to certain families of p-toral subgroups of G, see [JMO90], to undertake the general case. Homotopy idempotent functors on classifying spaces 17 Consider first when G=T= (S1)n.In thiscase, BT ∧ p≃K(Z∧ p,2)n≃(B(Z/p∞)n)∧ p. As Xis a p-complete space, we have the weak homotopy equivalence map ∗(BT∧ p, X)≃ map ∗(B(Z/p∞)n, X). This mapping space is contractible because Z/p∞∼ =lim Z/prdefined by inclusions Z/pn⊂Z/pn+1, and since Z/pris a p-group, BZ/pris BZ/p-acyclic and map ∗(BZ/pr, X)≃ ∗, and we can apply Lemma 4.8. Now, if G=Pis a p-toral group given by a group extension Tn֒→P→→π, Dwyer and Wilkerson show that BG admits a p-discrete approximation [DW94, Prop 6.9]. There is a sequence of finite p-groups P0⊂P1⊂... such that BP ≃hocolim BPn. Again, by Lemma 4.8, we obtain that map ∗(BP, X)≃ ∗. Let us go now through the general case. Our goal will be to prove that the inclusion of constant maps induces an equivalence X≃map (BG∧ p, X). By work of Jackowski- McClure-Oliver ([JMO90, Thm 4]), the space BG is mod pequivalent to hocolimOpGF, where OpGis the orbit category of stubborn p-toral subgroups of Gand Fis a functor whose values have the homotopy type of classifying spaces of stubborn p-toral subgroups of G. Since the statement holds for p-toral groups, by Lemma 4.8 it is enough to observe that OpGis Fp-acyclic, see [JMO90, Prop 6.1], and we are done. Now we are ready to prove the desired result, which was previously known for finite groups ([Flo07, Theorem 3.5]). Theorem 4.10. Let Gbe a compact Lie group and πits group of components. Let Gpbe the subgroup of Gwhose group of components is Tp(π). Then the BZ/p-nullification of BG fits in the following covering fibration: LZ[1/p]BGp−→ PBZ/pBG −→ B(π/Tpπ) Proof. We have to check that the assumptions on Theorem 4.1 are satisfied when X= BG. Since the universal cover of BG is BG0is again the classifying space of a compact Lie group, by Corollary 4.9 the hypothesis of Theorem 4.2 are satisfied. The proof of Corollary 4.9 applies to other type of spaces which admits mod phomology decompositions, for example p-compact groups (see [CLN07]). The theory of p-local compact groups introduced by Broto, Levi and Oliver in [BLO07] includes both the theory of p-compact groups [DW94] and p-local finite groups [BLO03]. Roughly speaking, ap-local compact is a triple (S, F,L)where Sis a discrete p-toral group and Fand L are categories which model conjugacy relations among subgroups of S. The classifying space of a p-local compact group is |L|∧ p, and one of the main features of p-local compact groups is that this space admits mod p-homology decompositions in terms of classifying spaces of p-compact toral subgroups over mod pacyclic orbit categories (see [BLO07, Proposition 4.6] and [BLO07, Corollary 5.6]). 18 Nat`alia Castellana and Ram´on Flores Proposition 4.11. Let pbe a prime, and (S, F,L)ap-local compact group. Then there is an equivalence LZ[1/p](|L|∧ p)≃PBZ/p(|L|∧ p). Proof. First of all, π1(|L|∧ p)is a finite p-group by [BLO07, Proposition 4.4], therefore Tp(π1(|L|∧ p)) = π1(|L|∧ p). Then, we only need to check the hypothesis in Theorem 4.2. That is, map ∗(Xh1i, Z)≃ ∗ for any connected BZ/p-null p-complete space Z. The same argument used in the proof of Corollary 4.9 using mod phomology decompositions can be applied and it shows that map ∗(|L|∧ p, Z)≃ ∗ for any connected BZ/pnull p-complete space Z. But it is not known in general if the universal cover |L|∧ ph1iis the classifying space of a p-local compact group. Instead, we will check that the proof of Corollary 4.9 applies by showing that |L|∧ ph1iadmits a description, up to p-completion, as a homotopy colimit of BZ/p-acyclic spaces over a mod p-acyclic category. Let P≤Sbe an object in O(F0)(see [BLO07, Proposition 4.6]) and let EP≤Sbe the pullback of |L|∧ ph1i → |L|∧ palong ˜ BP →BS. Then, by naturality there is a map hocolimO(F0)EP≤S→ |L|∧ ph1iwhich fits in a diagram of fibrations by Puppe’s theorem (e.g. [Far96, Appendix]), |L|∧ ph1i//|L|∧ p//Bπ1(|L|∧ p) hocolimO(F0)EP≤S// OO hocolimO(F0)˜ B(P)// OO Bπ1(|L|∧ p). id OO Since the middle vertical arrow is a mod p-equivalence, it follows that the left vertical arrow is also a mod p-equivalence. Moreover, (˜ BP)∧ pis the classifying space of a pcompact toral group, so it follows from the fibration EP≤S→˜ BP →Bπ1(|L|∧ p)that (EP≤S)∧ pis also the classifying space of a p-compact toral group, and therefore BZ/pacyclic. Then the proof of Corollary 4.9 applies. A finite loop space is a triple (X, BX, e)where e:X→ΩBX is a weak equivalence andH∗(X;Z)is finite. Note that if Xis a finite loop space, then BXh1i∧ pis the classifying space of connected p-compact group since H∗(ΩBXh1i;Fp)is finite dimensional (see [DW94]). Corollary 4.12. Let Xbe a finite loop space. Then there is a fibration LZ[1/p]BXp−→ PBZ/pBX −→ B(π/Tpπ) where π=π0(X)and BXpis the covering of Xwhose fundamental group is Tp(π0(X)). Proof. Since BXh1i∧ pis the classifying space of a p-compact group, Proposition 4.11 shows that PBZ/p(BXh1i)∧ p≃ ∗. Therefore we can apply Theorem 4.1. We finish with a somewhat two different examples. Homotopy idempotent functors on classifying spaces 19 Corollary 4.13. Let pbe a prime. Let Kbe a Kaˇ c-Moody group with a finite group of components. Then there is a fibration LZ[1/p]BKp−→ PBZ/pBK −→ B(π1(BK)/Tp(π1(BK)) where Kpis the subgroup of Kwhose group of components is Tp(π1(BK) Proof. First of all, the universal cover of BK is BK0where K0is the connected component of the unit in K. It is shown in Nitu Kitchloo thesis (see [BK02]) that BK is homotopy equivalent to a colimit over a contractible category of classifying spaces of compact Lie groups. Then Corollary 4.9 and its proof apply to show that map ∗(BK, Z)≃ ∗ for any connected BZ/p-null p-complete space Z. A different kind of example arises from the theory of infinite loop spaces and it is a consequence of Theorem 2in [McG]. Corollary 4.14. Let Xbe a connected infinite loop space with finite fundamental group. Then there is a fibration LZ[1 p](Xp)→PBZ/p(X)→B(π1(X)/Tp(π1(X))) where Xpis the covering of Xwhose fundamental group is Tp(π1(X)). 5 Relation of nullification functors with other idempotent functors In this section we compare the effect of nullification PBZ/p on spaces which satisfy the hypothesis of Theorem 4.1 with the effect of some completions or localizations on it. We analyse both functors that are supposed to kill the p-torsion, like LZ[1/p]or Z[1/p]∞, and functors that usually preserve it, as LZ[1/q]and p-completion do. Lemma 5.1. Let Xbe a connected space with finite fundamental group, pand qdifferent primes. Then (X∧ p)∧ qis contractible. Proof. If Xis 1-connected the case of Fq-completion is described in [BK72, VI.5.1]. If Xis not simply-connected, consider the fibration X∧ ph1i → X∧ p→Bπ1(X∧ p)and its fibrewise q-completion, (X∧ ph1i)∧ q→Y→Bπ1(X∧ p). Since the fibre is a 1-connected p-complete space completed at qit is contractible, then Y≃Bπ1(X∧ p). But then (X∧ p)∧ q≃Y∧ q≃Bπ1(X∧ p)∧ qwhich is contractible since Bπ1(X∧ p)is the classifying space of a finite p-group. 20 Nat`alia Castellana and Ram´on Flores We start by showing some direct direct consequences of Theorem 4.1. Proposition 5.2. Let Xbe a space which satisfies the hypothesis of Theorem 4.1. Then π1(PBZ/p(X)) = π1(X)/Tp(π1(X)) and (PBZ/p(X))∧ p≃ ∗. Moreover PBZ/p(X∧ p)≃ LZ[1 p](X∧ p)≃(X∧ p)Qis 1-connected. Proof. Since LZ[1 p](Xp)is 1-connected by Lemma 4.3, it is clear from the fibration in Theorem 4.1 that π1(PBZ/p(X)) = π1(X)/Tp(π1(X)). The space LZ[1 p](Xp)is mod p acyclic by [Dwy96, Lemma 6.2] and the order of π1(X)/Tp(π1(X)) is prime to p, it follows that (PBZ/p(X))∧ pis weakly contractible. The second statement follows from applying Theorem 4.1 to X∧ p. Observe that if X satisfies the hypothesis of the theorem, then X∧ palso does. Moreover, π1(X∧ p)is a finite p-group, then PBZ/p(X∧ p)≃LZ[1 p](X∧ p). It remains to prove that they are equivalent to (X∧ p)Q. Since they are 1-connected we can apply Sullivan’s arithmetic square. We have proved that (PBZ/p(X∧ p))∧ pis weakly contractible. Moreover, if q6=pthen (PBZ/p(X∧ p))∧ q≃(X∧ p)∧ qwhich is weakly contractible by Lemma 5.1. Then PBZ/p(X∧ p)≃ PBZ/p(X∧ p)Q≃(X∧ p)Q. We start by showing that BZ/p-nullification and p-completion behave like opposite functors in this context. Remark 5.3. If we complete in one prime qand BZ/p-nullify with regard to a different prime p, then X∧ qis BZ/p-null and the coaugmentation X→PBZ/pXis an equivalence after q-completion if Xsatisfies the hypothesis of Lemma 3.9. Remark 5.4. Note that in general a connected space Xcould be Z[1/p]-bad if Xis not 1-connected, and then it is not possible in general to replace completion by localization in the previous results. If we know in advance that Xis Z[1/p]-good (this happens, for example, if its fundamental group is Z[1/p]-perfect) then we can do the replacement, and moreover Z[1/p]∞X≃LZ[1/p]X. See for example ([Far96, 1.E]) for more information about the relation between R-localization and R-completion. Proposition 5.5. Let Xbe a space which satisfies the hypothesis of Theorem 4.1 and such that π1(X)∼ =Tq(π1(X)). Then there are homotopy equivalences PBZ/pLZ[1/q]X≃LZ[1/p,1/q]X≃LZ[1/p]PBZ/qX. Proof. Since π1(X)∼ =Tq(π1(X)),LZ[1 q](X)≃PBZ/q(X)is 1-connected. Then, by Theorem 4.1, we have PBZ/p(PBZ/q(X)) ≃PBZ/p(LZ[1 q](X)) ≃LZ[1 p](LZ[1 q](X)) ≃ LZ[1 p](PBZ/q(X)). We finish by establishing the commutativity of the functors PBZ/p and PBZ/q. The problem of commutation of localization functors was extensively studied in [RS00]. Homotopy idempotent functors on classifying spaces 21 Proposition 5.6. Let Xbea connectedspace,pand qtwo differentprimes. Assumethat X satisfies the hypothesis of Theorem 4.1 for both primes pand q. Then there are homotopy equivalences PBZ/pPBZ/qX≃PBZ/p∨BZ/qX≃PBZ/qPBZ/pX. Proof. It is enough to show the first equivalence since the other one will follow by symmetry. Consider the set of primes S={p, q}. By pulling back the universal fibration, there is a fibration XS→X→B(π1(X)/(TS(π1(X))), where π1(XS) = TS(π1(X)). Since the order of π1(X)/TS(π1(X)) is prime to both pand q, the space B(π1(X)/(TS(π1(X))) is both BZ/p-null and BZ/q-null (in particular it is also BZ/p ∨BZ/q-null) the composite of functors PBZ/p ◦PBZ/q and PBZ/p∨BZ/q preserve the fibration [Far96, 3.D.3], and there is a diagram of fibrations PBZ/p(PBZ/q(XS)) //  PBZ/p(PBZ/q(X)) //  B(π1(X)/(TS(π1(X))) id  PBZ/p∨BZ/q(XS)//PBZ/p∨BZ/q(X)//B(π1(X)/(TS(π1(X))). where the first two vertical maps exist because if a space Yis BZ/p ∨BZ/q-null, and then it is also BZ/p-null and BZ/q-null. Then we can assume that the group π1(X) = TS(π1(X)), which we simply denote by πin the sequel, is generated by pand qtorsion. By [RS00, Prop 1.1], we need to show that PBZ/p(PBZ/q(X)) is BZ/q-null and conversely PBZ/q(PBZ/p(X)) is BZ/p-null. In our situation, by symmetry, it is enough to check one of the two conditions. Let us see first that PBZ/p(PBZ/q(X)) is 1-connected. We can apply the fibrewise BZ/p-nullification to the fibration in Theorem 4.1, LZ[1 q](Xq)//  PBZ/q(Xq)//  B(π/Tq(π)) id  PBZ/p(LZ[1 q](Xq)) //¯ P//B(π/Tq(π)), where LZ[1 q](Xq)is 1-connected and PBZ/p(¯ P)≃PBZ/p(PBZ/q(Xq));ˇ then ¯ Phas fundamental group π/Tq(π)which is generated by ptorsion. If ¯ Psatisfies the hypothesis of Theorem 4.1 for the prime pthen PBZ/p(¯ P)≃LZ[1 p](¯ P)is 1-connected. We need to check that for any connected space Zwhich is p-complete and BZ/p-null, map ∗(¯ Ph1i, Z)is weakly contractible. Note that ¯ Ph1i ≃ PBZ/p(LZ[1 q](Xq)), and then map∗(PBZ/p(LZ[1 q](Xq)), Z)≃map∗(LZ[1 q](Xq), Z)≃map∗(PBZ/q(Xq), Z)≃ ≃map∗(Xq, Z), 22 Nat`alia Castellana and Ram´on Flores where the last equivalence follows because (PBZ/q(Xq))∧ p≃(Xq)∧ pand Zis p-complete. Finally Lemma 4.6 tells us that this last mapping space is weakly contractible. We denote by Ythe space PBZ/p(PBZ/q(X)), and we finally should check that it is BZ/q-null. Since it is a 1-connected space, we can use Sullivan’s arithmetic square and check that the mapping spaces map ∗(BZ/q, YQ)and map ∗(BZ/q, Y ∧ r)are weakly contractible for any prime r. If r6=q,map ∗(BZ/q, Y ∧ r)≃ ∗ because (BZ/q)∧ r≃ ∗. Also, since (BZ/q)Q≃ ∗, map ∗(BZ/q, YQ)≃ ∗. We are left to the case r=q, and Y∧ q= (PBZ/p(PBZ/q(X)))∧ q≃ (PBZ/q(X))∧ q≃ ∗ by Proposition 5.2. So we are done. For example, given a compact Lie group, BG satisfies the hypothesis of Theorem 4.1 for any prime p. Remark 5.7. The same proof remains valid if we apply in succession over Xa finite number of BZ/p-nullification functors for different primes assuming Xsatisifes the hypothesis of Theorem 4.1 for each prime. On the other hand, it is likely that that the nullification of Xwith regard to the wedge of the classifying spaces of all primes is homotopy equivalent to the rational localization of X. See [Flo07, Section 3.2] for details. 6BZ/p-cellularization of classifying spaces In this section we will give a Serre-type general dichotomy theorem (Theorem 6.1), which is very much in the spirit of [FS07]. Then, we will use this statement to describe several examples concerning the BZ/p-cellularization of some families of classifying spaces of remarkable groups, such as p-toral groups, finite groups with a p-subgroup of p′-index, BS3or BSO(3) (at the prime 2). Our considerations are also based in the results of the previous sections relating cellularization, nullification and completion. 6.1 The dichotomy theorem The main result of this section is: Theorem 6.1. Let Xbe a connected nilpotent ΣnBZ/p-null space for some n≥0. Then the BZ/p-cellullarization of Xhas the homotopy type of a Postnikov piece with homotopy groupsare concentratedin degrees 1to n, or else it has infinitely many nonzero homotopy groups. Moreover, if Xis 1-connected of finite type, then the fundamental group π1(CWBZ/p(X)) is a finite elementary abelian p-group. The rest of the subsection is devoted to the proof of Theorem 6.1. Even if the statement is similar to the one in [FS07, Proposition 2.3], the authors deal with the situation in which the space is torsion, and this is not the case for BG where Gis a compact connected Homotopy idempotent functors on classifying spaces 23 Lie group. The strategy used in [FS07] for classifying spaces of finite groups can be summarized as follows. Proposition 6.2. Let Xbe a torsion Postnikov piece whose fundamental group is generated by elements of order pwhich lift to X. Assume there exists a prime q6=psuch that X∧ qis torsion and it has infinitely many non-trivial homotopy groups. Then CWBZ/p(X) also has infinitely many non-trivial homotopy groups. Proof. Consider the fibration CWBZ/p(X)→X→PΣBZ/p(C)from Theorem 2.3. Note that PΣBZ/p(C)is 1-connected since Cis so. To prove the statement, we will show that PΣBZ/p(C)has infinitely many non-trivial homotopy groups. We apply Sullivan’s arithmetic square to PΣBZ/p(C)to obtain a pullback diagram PΣBZ/p(C)//  (Qr6=pX∧ r)×(PΣBZ/p(C))∧ p  *//(Qr6=q,p X∧ r)Q×((PΣBZ/p(C))∧ p)Q. which allow us to construct a map s:X∧ q→PΣBZ/p(C)such that η◦s≃id. That is sis a section of the q-completion. Then for n≥2we have that πn(X∧ q)is a direct summand of πn(PΣBZ/p(C)). For example, by Levi’s work in [Lev95], the previous theorem applies when Xis the classifying space of a finite group. Now we need to state some general results concerning to the cellularization of ΣBZ/pnull spaces, that deal with the consequences of imposing that CWBZ/p(X)is a Postnikov piece for a certain space X. Note that this is the “forbidden” case in Theorem 6.1. Lemma 6.3. Let P[n]be a connected ΣBZ/p-null Postnikov piece with n≥3then pis invertible in πn(P[n]). Proof. Note that if P[n]is ΣBZ/p-null then ΩP[n]is BZ/p-null, and also Ωn−1P[n]is so. Since the connected component of the constant in Ωn−1P[n]is an Eilenberg-MacLane space K(πn(P[n]),1), we see that K(πn(P[n]),1) is also BZ/p-null. Similarly, the connected component Eof Ωn−2P[n]is BZ/p-null. There is a fibration K(πn(P[n]),2) →E→K(πn−1(P[n]),1). Since the pointed mapping spaces map ∗(BZ/p, K(πn−1(P[n]),1))cand map ∗(BZ/p, E)are weakly contractible, we obtain from the previous fibration that map (BZ/p, K(πn(P[n]),2)) is also weakly contractible. That is K(πn(P[n]),2) is BZ/p-null. The conclusion now follows from Lemma 2.7. Proposition 6.4. Let Xbea connectednilpotentΣBZ/p-null space suchthat Z[1 p]∞(X)≃ ∗. Then Xhas the homotopy type of a K(G, 1) or it has infinitely many nonzero homotopy groups. 24 Nat`alia Castellana and Ram´on Flores Proof. Assume that X≃P[n]is a finite Postnikov piece. First we show that n≤2. Since Z[1 p]∞(X)is weakly contractible (see Lemma 2.8), then πi(P[n])⊗Z[1 p] = 0 for all i > 0 ([BK72, V.4.1]). But if n≥3,pis invertible in πn(P[n]) by the previous Proposition 6.3, then πn(P[n]) = 0. We can apply this argument as long as n≥3. From now on we assume n= 2. Next we prove that π2(P[2]) has no p-torsion. There is a fibration K(π2(P[2]),2) →P[2] →K(π1(P[2]),1), which induces a covering K(π1(P[2]),0) →K(π2(P[2]),2) →P[2]. If π2(P[2]) had p-torsion, then a nontrivial homomorphism Z/p →π2(P[2]) would induce a nontrivial map f: ΣBZ/p → K(π2(P[2]),2) which is nullhomotopic when composed with K(π2(P[2]),2) →P[2] since P[2] is ΣBZ/p-null (see Lemma 2.6). Then we obtain a contradiction since fmust be nullhomotopic. So π2(P[2]) has no p-torsion, and this is again a contradiction since π2(P[2]) ⊗Z[1 p] = 0. We now state our dichotomy theorem for nilpotent ΣBZ/p-null spaces. Theorem 6.5. Let Xbea connectednilpotentΣBZ/p-null space.Then the BZ/p-cellulla- rization of Xhas the homotopy type of a K(G, 1) or it has infinitely many nontrivial homotopy groups. Moreover, if Xis 1-connected of finite type, then π1(CWBZ/p(X)) is a finite elementary abelian p-group. Proof. By Lemma 2.5, CWBZ/p(X)is also nilpotent. Moreover, by Lemma 2.8 we have an equivalence Z[1 p]∞(CWBZ/p(X)) ≃ ∗ , and then we can apply Proposition 6.4 to CWBZ/p(X). Proof of Theorem 6.1. Assume that n≥2, we can assume that Xis BZ/p-cellular and ΣnBZ/p-null. By [Bou94, Theorem 7.2], there is a principal fibration K(P, n)→X→PΣn−1(X) where Pis a p-torsion grup. Since K(P, n)is BZ/p-cellular ([CCS07-2, Lemma 3.3]) and Xtoo by assumption, then PΣn−1(X)is also BZ/p-cellular by [Cha96, Theorem 4.7]. By induction we reduce to the case in which n= 1, which is proved in Theorem 6.5. The next question we need to refer concerning Theorem 6.1 is when the cellularization of a classifying space is again a classifying space, not necessarily of a discrete group. This is important to understand the first part of the previous dichotomy. Proposition 6.6. Let Xbe a space. If CWBZ/p(X)≃BH for some compact Lie group H, then it must be a finite p-group generated by order pelements. Homotopy idempotent functors on classifying spaces 25 Proof. Since the pointed homotopy colimit of acyclics is acyclic for any cohomology theory ([Far96, 2.D.2.5]), it is clear that ˜ H∗(BH;Q) = ˜ H∗(CWBZ/pX;Q) = 0. On the other hand, it is well-known that the rational cohomology of BH are the invariants of the rational cohomology of the classifying space of the maximal torus Tunder the action of the Weyl group W. In fact ˜ H∗(BH;Q) = 0 if and only if His a finite group. Finally, the functor CW is idempotent, so BH must be BZ/p-cellular. Thus, we can apply [Flo07, Prop 4.14 ] to finish the proof. Remark 6.7. The arguments in Proposition 6.6 also work if CWBZ/p(X)≃(BH)∧ p where His a compact Lie group. It is clear then that ˜ H∗(BH;Z∧ p)⊗Q= 0. But again this is only possible if His discrete. If His in particular finite, conditions are known (see [FS07, Corollary 3.3]) under which (BH)∧ pis BZ/p-cellular. See Example 6.14 below. Remark 6.8. When Xis an H-space satisfying the hypothesis of Theorem 6.1, Castellana, Crespo and Scherer proved in [CCS07-2] that the BZ/p-cellularization of Xis always a Postnikov piece. Examples of such spaces are given by H-spaces whose mod p cohomology isfinitely generated as an algebra over the Steenrod algebra (see [CCS07-1]). 6.2 Examples In this subsection we concentrate in the description of the BZ/p-cellularization of classifying spaces of compact Lie groups, generalizing to the continuous case work of the second author in the finite case ([Flo07] and [FS07]). In the study of the homotopy type of classifying spaces of Lie groups, a very useful strategy is to isolate the information at every prime. Theorem 6.1 implies automatically the following dichotomy theorem for classifying spaces of compact Lie groups. We say that an element g∈Gis p-cohomologically central if the map induced by the inclusion BCG(x)→BG is a mod phomology isomorphism. Mislin in [Mis92] shows that there is a natural bijection between the set of conjugacy classes of p-cohomologically central elements of order pin Gwith pZ(G/Op′(G)) where Op′(G)is the largest normal p′-subgroup of Gand pZ(G)are the elements of order pin Z(G). Theorem 6.9. Let Gbe a compact connected Lie group. If there exists a non p-cohomo- logically central element of order p, then the BZ/p-cellullarization of BG has infinitely many nonzero homotopy groups. Otherwise, it has the homotopy type of a K(V, 1), where Vis a finite elementary abelian p-group. Proof. Since Gis assumed to be connected, BG is simply connected. Moreover it is of finite type, and ΣBZ/p-null because of Miller’s solution of the Sullivan conjecture because ΩBG ≃Gis a finite complex. Now, we apply Theorem 6.1. If CWBZ/p(BG) 32 Nat`alia Castellana and Ram´on Flores Let Cbe the cofibre of k. We know that Cis mod 2acyclic and 1-connected. Now if qis an odd prime, since Yis BZ/2-cellular, Yis mod qacyclic and C∧ q≃(BSO(3)∧ 2)∧ q is contractible. Finally CQ≃(BSO(3)∧ 2)Q. Then, by a Sullivan arithmetic square argument, C≃(BSO(3)∧ 2)Qwhich is, in turn, BZ/2-null. In particular Cis ΣBZ/2-null. Therefore, the fibration of the theorem follows from Chach´olski’s fibration describing the cellularization (Theorem 2.3). Remark 6.18. Note that if pis an odd prime, then BSO(3)∧ p≃BN(T)∧ p, where N(T) is the normalizer of the maximal torus, and we analyzed this case in Example 6.15. It seems natural to ask if the problem of computing CWBZ/p(BG)for any compact Lie group Gis accessible at this point. A strategy was developed for finite groups in [FF], based in the description of the strongly closed subgroups of G, which are classified. Recent research has remarked the role of the strongly closed subgroups of discrete p-toral groups in the homotopy theory of compact Lie groups and, more generally, p-local compact groups [Gon10], but to our knowledge there is no available classification of these objects. On the other hand, the nontrivial rational homotopy of BG seems an important obstacle to generalize the arithmetic square arguments of the strategy. 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