Nullification functors and the homotopy type of the classifying space for proper bundles

ISSN 1472-2739 (on-line) 1472-2747 (printed) 1141 Algebraic & Geometric Topology ATG Volume 5 (2005) 1141–1172 Published: 18 September 2005 Nullification functors and the homotopy type of the classifying space for proper bundles Ram´ on J. Flores Abstract Let Gbe a discrete group for which the classifying space for proper G-actions is finite-dimensional. We find a space Wsuch that for any such G, the classifying space BGfor proper G-bundles has the homotopy type of the W-nullification of BG. We use this to deduce some results concerning BGand in some cases where there is a good model for BGwe obtain information about the BZ/p-nullification of BG. AMS Classification 55P20; 55P80 Keywords (Co)localization, finite groups, Eilenberg-MacLane spaces 1 Introduction Let Gbe a discrete group. We will say that a G-CW-complex Xis proper if the isotropy groups of the action are finite. In 1971, J.P. Serre introduced in [37] the “classifying space for proper actions”, that can be described as the unique proper G-CW-complex EG, up to G- homotopy, that enjoys the following universal property: “If Xis another proper G-CW-complex, there exists a G-map X−→ EG which is unique up to G-homotopy”. The space EGappears as the principal new feature in the reformulation of the Baum-Connes conjecture stated in [3] by Baum-Connes-Higson. The conjecture, partially solved, asks if for a locally compact, Hausdorff and second countable group G, the assembly map from the Kasparov K-homology groups KG j(EG) to the C∗-algebra K-theory groups Kj(C∗ r(G)) is an isomorphism for j= 0,1. The great amount of research that has emerged around this subject has led to a growing interest in the theory of proper actions. An important part of the efforts carried out in this direction has been devoted to understand the relationship between the algebraic structure of Gand the c  Geometry & Topology Publications 1142 Ram´on J. Flores homotopy-theoretic properties of EGand its quotient space EG/G, which is currently denoted by BG. Probably the greatest success has been reached by interpreting correctly finiteness group-theoretic conditions over Gin order to build models of EGenjoying various types of finiteness conditions. See ([28], section 5) for an excellent survey on this topic. In the same way that happens with classical G-actions, the importance of EG and in particular of BGdoes not come only because they reflect geometrically the algebraic properties of the group G, but because the importance of these spaces in the theory of G-bundles. Baum-Connes-Higson already pointed out that BGclassifes proper G-bundles (see [3], definition 8), and they described how to obtain them by making pullback on maps X−→ BG, a method of a clear classical flavor. Moreover, if Gis a group for which Baum-Connes conjecture holds, the knowledge of the homotopical structure of BGplays a rˆole in the computation of the K-theory of C∗(G), and in general gives information about the integral homology of G. See ([28], section 8) for details about these results. The most important attempt made so far to understand the homotopy type and properties of BGis the paper of Leary-Nucinkis [24]. In it, the authors prove that for every CW-complex Xthere exists a discrete group GXsuch that BGXis homeomorphic to X. This “Kan-Thurston-like” result is proved using essentially tools of the theory of graph of groups. As a by-product, they obtain a precise description of the fundamental group of BGand a construction of BG for some subgroups of right-angled Coxeter groups. Although these results have been very useful for us (particularly the formula for the fundamental group), our approach to the homotopy type of BGhas been different, and has been carried out with pure homotopy-theoretic tools. Our idea is to find a functor Fin the topological category that transforms models of BGon spaces that are homotopy equivalent to models of BG. This functor have enough good properties in order to read information about BGfrom BG and vice versa. The appropriate functor turns to be a nullification; a tool that was introduced by Bousfield in [4] in order to study periodic phenomena in unstable homotopy (in fact, he called it “periodization”), and that has been widely used since then. The utility of this functor in this context comes from the fact that it will allow us to apply all the machinery of localization developed in the 90’s by Bousfield himself [4, 5], Dror-Farjoun [13], Chach´olski [8] and others. Our main result is the following: Theorem 3.2 Let Gbe a discrete group such that there exists a finitedimensional model for BG. Let W∞denote WBZ/p, where the wedge is in- Algebraic & Geometric Topology, Volume 5 (2005) Nullification functors and the homotopy type of classifying spaces 1143 dexed by all primes. Then we have a homotopy equivalence PW∞BG≃BG, where PW∞denotes the W∞-nullification functor. Observe that the condition concerning the existence of a model of BGwith finiteness conditions is not too restrictive, in account of the great quantity of groups that have recently appeared for which these conditions hold. Now we describe in more detail the contents of each of the sections of the paper, and in particular we will comment a little bit the consequences of the main theorem. In section 2 we review the needed background and results concerning proper actions, and we construct a particular model for the classifying space for families EFGthat will be very useful in the rest of the work; as a fundamental byproduct, we also obtain an appropriate model for BFG. Section 3 constitutes the bulk of the work, because it is devoted to the proof of the main theorem we stated before. The technique is the following: we apply the functor PW∞to a suitable model of BG, and we obtain that it is homotopy equivalent to the W∞-nullification of the nerve of some small category that only depends on G. This nerve turns out to be the model built in the previous section for BG, and we finish by checking that in the conditions of the theorem BGis W∞-null. The rest of the paper is devoted to take out some consequences of the main theorem. So, in section 4 we describe the behaviour of the functor B with respect to various fundamental constructions in homotopy theory, namely products, wedges or colimits. Moreover, we identify in some cases the universal cover of BGand we obtain some conditions about preservation of fibrations under passing to classifying spaces. We begin the following section with a short new proof of the well-known fact that if Gis a locally finite group which cardinal is smaller than ℵω, BGis contractible (indeed, the statement is true for all locally finite groups, see [11]). Later, we treat the case of groups for which the normalizer condition holds, an ample class of discrete groups that include, for example, all the nilpotent groups. We prove that if a group Gin this class admits a finite-dimensional model for BG, then BG≃BHfor some quotient group Hthat we identify. In particular, in this case BGis nilpotent as a space if Gis nilpotent as a group. We finish this paragraph by studying the BZ/p-nullification of classifying spaces of supersoluble groups. In section 6 we take the opposite point of view, showing that the main theorem can give information in the two directions. More concretely, we focus our attention on groups of isometries of the real plane, and taking profit of well-known Algebraic & Geometric Topology, Volume 5 (2005) 1144 Ram´on J. Flores geometric properties of them we obtain via BGa lot of information on the BZ/p-nullification of their classifying spaces. Last section is devoted to the study of a canonical map BG−→ BFG(defined previously) that always relates the classical and proper classifying spaces. More concretely, we prove that the homotopy fiber of that map can be described as a homotopy colimit of classifying spaces of groups of Fover a contractible category. We finish by proving a technical and interesting statement that appears in the proof and concerns the localization of a comma category. Acknowledgements I wish to thank Carles Broto, for turning my attention to proper actions and the world of Geometric Group Theory, and for all the time we have spent discussing about these topics. The results about crystallographic groups were motivated by a suggestion of Ian Leary, whom I acknowledge his interest in my work. I am also grateful to Emmanuel Dror-Farjoun, who pointed out some very useful observations that enriched the results of last section. Finally, I would like to thank the Institute Galil´ee, Universit´e Paris XIII, for their hospitality in the seven months I spent there, and the referee, for the detailed report and numerous suggestions. The author is partially supported by MCYT grant BFM2001-2035. 2 Useful models for EFGand BFG We will begin by recalling the definition of the classifying space for families. Further information about proper actions and their classifying spaces can be found in [3], [24], [33], or [12]. Definition 1 Suppose Fis a family of subgroups of a discrete group Gthat is closed under conjugation and taking subgroups. We will say that a G-CW- complex Yis a model for EFGif the isotropy group of each point belongs to Fand for each H∈ F , the fixed-point space YHis contractible. The G-space EFGis characterized by the following universal property: Proposition 2.1 If Xis a model for EFG, then for each G-CW-complex Y whose isotropy groups lie in Fthere is a map Y−→ Xwhich is unique up to G- homotopy. Moreover, two models for EFGare always G-homotopy equivalent. Conversely, if Xis a G-CW-complex for which this universal property holds, Xis a model for EFG. Algebraic & Geometric Topology, Volume 5 (2005) Nullification functors and the homotopy type of classifying spaces 1145 The quotient space EFG/G is usually denoted by BFG. As EFGis unique up to G-homotopy equivalence, BFGis unique up to homotopy equivalence; see ([24], section 2) for a review of the main properties of BFG. If there is no explicit mention against it, we will suppose from here that Fis the family of finite subgroups of G. In this case, it is standard to denote EFG and BFGby EGand BG, respectively. In particular, it can be seen that every group homomorphism G−→ G′induces a G-map EG−→ EG′(respectively a map BG−→ BG′) which is unique up to G-homotopy (respectively up to homotopy). A number of geometric constructions for EFGand its quotient space is available in the literature (see section 4 of [27] for a description of some of them), but to describe the relationship between BGand BFGwe will need to build these spaces as nerves of small categories. This is the main goal of this section. Recall first the definition of the orbit category associated to a group Gand a family of subgroups F: the objects are the homogeneous spaces G/H , with H∈ F , and the morphisms are the G-maps. It is not hard to see that there is a bijective map Mor(G/K, G/H) = {g∈G|g−1Kg ⊆H}/H given by f−→ f(eK), where eis the identity element of G. The key definition we need for building the desired model of EFGis the following (see [15], section 2, for details): Definition 2 Let Dbe a small category, Cat the category where the objects are the small categories and whose morphisms are functors, and f:D → Cat a functor. The Grothendieck construction Gr(f) associated to fis defined as the category whose objects are the pairs (d, x), with x∈ D and x∈f(d), and where a morphism (d, x)→(d′, x′) is a pair (u, v) where u:d→d′is a morphism in Dand v:f(u)(x)→x′is a morphism in f(d′). The composition is made in the obvious way. The main feature of this construction, due to Thomason, is the following: Theorem 2.2 Let Dbe a small category, F:D −→ Cat a functor, and Gr(F)the Grothendieck construction of F. Then there exists a natural weak homotopy equivalence: N(Gr(F)) ≃hocolim N(F) where Ndenotes the nerve. Algebraic & Geometric Topology, Volume 5 (2005) 1146 Ram´on J. Flores Proof See [39], 1.2. Now we can describe our model of the universal space EFG. Proposition 2.3 Let Gbe a discrete group. Consider the functor R:OF−→ Cat that sends every homogeneous space G/H to the category G/H (whose objects are the elements of G/H and there is only identity morphisms), and the morphisms to the obvious functors. In these conditions, we have that the nerve of the Grothendieck construction of fis a model for EFG. Proof For convenience, we will denote X=|N(Gr(R))|. This space has a natural action of Ggiven by the left action of Gin every homogeneous space G/H . We will prove firstly that for every x∈Xthe isotropy group Gxof x belongs to F. The action of Gover Xis simplicial and is induced from the action of Gover the homogeneous spaces, so by definition of nerve we will only need to study the action of Gover the vertices of X. Hence, let (G/H, a) be the pair associated with the vertex x. It is clear that Gx={g∈G|gaH =aH}={g∈G| ∃h∈Hs. t. gah =a} and this amounts to say that g∈aHa−1. So, Gx=aHa−1, that belongs to F because Hdoes. Now we will see that for every K∈ F , the set of fixed points XKis contractible. Now, a point on a simplex is fixed by a subgroup if and only if all vertices are fixed, because the action of Gis defined over the vertices and then extended to the rest of the simplices by linearity. So, consider a vertex x∈X. This point is fixed by the action of Kif, given the pair (G/H, aH) associated to it, we have that, for every k∈K,kaH =aH . Thus, we see that XK=[ H∈F {(G/H, aH)|a−1Ka ⊆H}/H. So, for every element (G/H, aH)∈XKthere exists one and only one morphism (G/K, eK)−→ (G/H, aH), or in other words, XKcan be identified with the nerve of the undercategory (in the sense of [30], II.6) associated to the element (G/K, eK) of the category Gr(R). This nerve is contractible, and then XKis contractible too. So we are done. Algebraic & Geometric Topology, Volume 5 (2005) Nullification functors and the homotopy type of classifying spaces 1147 Note that if we consider the action of Gover Gr(R) via functors, the objects of Gr(R)/G are the homogeneous spaces G/H ,H < G finite, and the morphisms are the G-maps. So, this quotient category is identified in a natural way with the orbit category OF, and in particular N(OF) is a model for BFG. Remark 2.4 Observe that if we take the realizations of the nerves of Gr (R) and OF, we obtain models of EFGand BFGin the category of (topological) spaces. Remark 2.5 The idea of the construction comes from ([2], section 2), although they only describe it in the case of Gfinite, being Fthe family of finite subgroups of G, and with another purpose. In the language of that paper, we have proved that Xis the F-approximation to a point. We conclude this section with a modification of the previous models that will be useful in the sequel. So, if {H1,...,Hn}are subgroups of Gthat belong to Fand such that for every 0 < i < n there exists a G-equivariant map G/Hi−→ G/Hi+1 , we define G/H1−→ ...−→ G/Hnas the small category whose elements are n-uples (a1H1,...,anHn) such that for every 0 < i < n there exists a G-equivariant map fi:G/Hi−→ G/Hi+1 with fi(aiHi) = ai+1Hi+1 , and whose morphisms are the identity maps. Now, if Γis the poset category of non-degenerate simplices of N(OF), we define a functor S:Γop −→ Cat that takes the simplex represented by the chain of maps {G/H1−→ ...−→ G/Hn}to the category G/H1−→ ...−→ G/Hn, and the face maps to the obvious functors. Then we have the following: Proposition 2.6 In the previous conditions, hocolimΓN(S)is a model for EFG, and N(Γ)is a model for BFG. Proof By theorem 2.2, hocolimΓN(S)≃N(Gr(S)), and on the other hand, the left action of Gover every homogeneous space G/H , (H∈ F ) induces, via functors, another one over the categories G/H1−→ ...−→ G/Hn. Now, observe that N(Gr(S)) is the subdivision (in the sense of [19], III.4) of N(Gr(R)), where Ris the functor defined in proposition 2.3. In fact, if we take a nondegenerate simplex of N(Gr(R)) that is represented by a chain of morphisms (G/H1, a1H1)−→ ... −→ (G/Hn, anHn), its barycenter is the vertex of the nerve N(Gr(S)) represented by the object (G/H1−→ ...−→ G/Hn, a1H1 −→ ... −→ anHn) of Gr(S). Then, by ([21] prop. 12-14), there exists a homotopy equivalence |N(GrR)| ≃ |N(Gr S)|, that is a G-equivalence by construction, and then N(GrS) is a model for EFG. A similar line of reason proves that N(S) is a model for BFG. Algebraic & Geometric Topology, Volume 5 (2005) 1148 Ram´on J. Flores The main advantage of these models is that they reconstruct EFGand BFGas homotopy colimits over a poset category, and in particular they have structure of simplicial complexes. These facts will be very useful in the final section. 3 BGis homotopy equivalent to a nullification Let Gbe a discrete group, Fa family of subgroups of Gclosed under conjugation and taking subgroups. As a first step in our study of the relation between the classical and proper classifying spaces, we will describe a canonical map that always relates BGand BFG. So, consider models of EGand EFG; both of them are G-spaces, and then we can make the Borel construction EG×GEFG. Now, let p1be the projection EG×GEFGp1 −→ EG/G ≃BG. The action of Gover EGis free, so the map p1is a fibration, and its homotopy fiber EFGis contractible. Thus, p1is a homotopy equivalence, and EG×GEFG is a model of BG. Consider now the projection over the second component EG×GEFGp2 −→ EFG/G ≃BFG. We have seen that EG×GEFGand EFG/G are respectively models for BG and BFG, and then p2can be thought as a map BG−→ BFGthat we will call fin the rest of the section. The map fis not a fibration in general, because the action of Gover EFGis not free. In fact, if x∈BFG, we have that f−1(x) has the homotopy type of EG×GG/Hx, being Hxthe isotropy group of x, that belongs to F. Now, EG×GG/Hxis a model for BHx, and hence all the fibers of the map fhave the homotopy type of classifying spaces of groups of F. This fact gave us the intuition that the map we have studied could encode a functorial way of passing from the usual classifying space of Gto the classifying space for proper G-bundles, and what is more important, to obtain valuable information of the latter starting from the homotopy structure of BG, and viceversa. More concretely, we searched for a functor Fsuch that the following conditions hold: (1) F“kills” the homotopy fiber of f. (2) F(f) is a weak equivalence. Algebraic & Geometric Topology, Volume 5 (2005) Nullification functors and the homotopy type of classifying spaces 1149 (3) F(BG)≃BG. The two first conditions give the impression, according to ([13], 1.H.1 and 3.D.3), of Fbeing a localization functor Lin the sense of Dror-Farjoun, and in fact, the functor we have found has been the A-nullification functor with respect to a certain space A. For the main properties of these functors you can look at [4], [8] or [13], although we will recall here the definition. Let Aand Xbe spaces; Xis said A-null if the mapping space map(A, X) is homotopy equivalent to Xvia the inclusion of constant maps X−→ map(A, X). The A-nullification of Xis a functor PA:Spaces −→ Spaces that takes every space Xto an A-null space PAXsuch that there exists a universal map X−→ PAXwhich induces a weak homotopy equivalence map(PAX, Y )≃map(X, Y ) for every A-null space Y. It can be seen that any other A-null space Xfor which the last property holds is indeed homotopy equivalent to PAX. So, we have defined a functor in the category of unpointed spaces (that can also be defined in the category of pointed spaces, although we we will only work here with unpointed spaces), which is always coaugmented and idempotent, and kills the structure of Xthat “depends” on A. In fact, PAXis the localization of X with respect to the constant map A−→ ∗, and the spaces Xfor which PAX is contractible are called A-acyclic. Consider now the set of all prime numbers {p1, p2, p3...}with the usual order, and let Xbe a space. In the remaining of the paper we will denote by Wnthe space BZ/p1∨...∨BZ/pn, and by W∞the wedge WBZ/p extended over of the all prime numbers. The next key lemma is one strong reason that suggests that the W∞-nullification is the functor we need. Theorem 3.1 If Gis a finite group, then PW∞BGis contractible. Proof A point is always null, so we only need to prove that for every W∞-null space Xthere is an equivalence X≃map(PW∞BG, X). But the W∞-null spaces are, in particular, Wn-null for every n; hence, map(PW∞BG, X)≃map(BG, X)≃map(PWnBG, X) for every n. Now, suppose that |G|=pn1 j1pn2 j2. . . pnm jm, with j1< . . . jm. Because of ([17], 3.3), we know that PWkBGis contractible for every k≥jm. This implies map(PW∞BG, X)≃map(PWnBG, X)≃X as we claimed. Algebraic & Geometric Topology, Volume 5 (2005) 1156 Ram´on J. Flores Proposition 5.1 Let Gbe a locally finite group that admits a finite-dimen- sional model for BG. Then BGis contractible. Proof It is known (see for example [31], 9.8) that every locally finite group is the colimit of the directed system of its finite subgroups. Thus, we have a homotopy equivalence BG≃hocolimCBH, where Cis a contractible poset category (because it has an initial object given by the trivial group) and BH represents all the classifying spaces of finite groups Hof G. So, by ([13], 1.D.3), we obtain PW∞BG≃PW∞(hocolimCPW∞BH) = PW∞(|C|) = PW∞(∗) = ∗ and we are done. Remark 5.2 By [11], this result applies to all locally finite groups whose cardinal is smaller than ℵω. Now we can prove the following result, that concerns to the classifying space for proper G-bundles of extensions of locally finite groups. Proposition 5.3 Let {1} −→ K−→ G−→ Q−→ {1} be an extension of groups, Ka locally finite group whose cardinal is smaller than ℵω, and assume there is a bound on the order of finite subgroups of Q. Then if Qadmits a finite model for BQ,Gadmits a finite model for BG, and then BQis homotopy equivalent to BG. Proof If we apply the results 3.2 and 5.1 we obtain the statement is true if there is a finite-dimensional model for BG, and this happens by proposition 4.4 of [32]. 5.2 Groups with the normalizer condition We study now the groups for which the normalizer condition holds. It is greatly remarkable that this class contains all the nilpotent groups. Recall that a group Gis said to satisfy the normalizer condition if every proper subgroup of Gis distinct from its normalizer. In this case the following holds (see [22], page 215): Algebraic & Geometric Topology, Volume 5 (2005) Nullification functors and the homotopy type of classifying spaces 1157 (1) For every prime p, there exists a normal p-group Tpsuch that if x∈G and the order of xis a power of p, then x∈Tp. (2) The elements of finite order of Gform a normal subgroup of Gwhich is isomorphic to Qp prime Tp. Throughout this section we will impose to the groups for which the normalizer condition holds that the torsion p-subgroups Tpthat we have just defined are locally finite. We need this condition because it is not known if the BZ/pnullification of the classifying space of a general p-group is contractible if the group is not locally finite. Among the few examples that have been described of non-locally finite p-groups we can remark the Burnside groups B(n, e) for n > 1 and e > 664 or the “monsters” of Tarski-Olshanskii. See [1] and [34] for more information about these families of groups. All these facts have the following interesting consequence: Proposition 5.4 If Gis a discrete group for which the normalizer condition holds, and p1. . . pnis a collection of primes, we have a homotopy equivalence PBZ/p1∨...∨BZ/pnBG≃B(G/Tp1×...×Tpn).In particular, there is an equivalence PW∞BG≃B(G/ Qp prime Tp). Proof For simplicity, we will only prove the case of one prime p(the generalization to a family is immediate). It is clear that BTpis BZ/p-acyclic, and B(G/Tp) is BZ/p-null, so if we BZ/p-nullify the fibration BTp−→ BG−→ B(G/Tp) we obtain the desired homotopy equivalence. If we suppose that Gis such that exists a finite-dimensional model for BGwe have: Corollary 5.5 BG≃B(G/ Qp prime Tp). So we have a complete description of the homotopy type of BG. Other case that can be solved with the same tools is the following: Proposition 5.6 Let Gbe a discrete group, Ha normal subgroup of Gfor which the normalizer conditions holds, and such that G/H does not have ptorsion. If Tpis the p-torsion subgroup of H, then the BZ/p-nullification of BGfits into the following fibration sequence: B(H/Tp)−→ PBZ/pBG−→ B(G/H) and hence it is an Eilenberg-MacLane space. Algebraic & Geometric Topology, Volume 5 (2005) 1158 Ram´on J. Flores Proof The base of the fibration BH−→ BG−→ B(G/H) is BZ/p-null, so by ([13], 3.D.3) the fibre is preserved under BZ/p-nullification. The result now follows from proposition 5.4. Taking into account the main theorem 3.2, the following corollary is immediate: Corollary 5.7 In the hypotheses of the previous proposition, if G/H is torsion-free, Tis the torsion subgroup of Hand there exists a finite-dimensional model for BG, then the fibration B(H/T)−→ BG−→ B(G/H) defines the classifying space for proper G-bundles, which is again an Eilenberg- MacLane space. We conclude this paragraph by focusing on nilpotent groups, that is a distinguished class of discrete groups for which the normalizer condition holds. The following result proves that the BZ/p-nullification preserves nilpotency when it is applied on classifying spaces of nilpotent groups, and in fact, the functor Bsends nilpotent groups (for which the finiteness conditions hold) to nilpotent spaces. Corollary 5.8 If Gis a nilpotent group, the nullification PBZ/p1∨...∨BZ/pnBG is, for every set of primes {p1,...,pn}, the classifying space of a nilpotent group. If moreover Gadmits a finite-dimensional model for BG, we obtain that BGis again the classifying space of a nilpotent group, and hence nilpotent as a space. Proof Using the previous results, it is enough recalling that the quotient of a nilpotent group is always nilpotent, and that according to ([34], 2.7.1), every nilpotent p-group is locally finite. In particular, using ([32], 4.5) we have that the part of the previous corollary that alludes to BGis always true if Gis a nilpotent group whose cardinal is smaller than ℵωand whose torsion-free rank is finite. Algebraic & Geometric Topology, Volume 5 (2005) Nullification functors and the homotopy type of classifying spaces 1159 5.3 Supersoluble groups In this paragraph we will compute, for podd, the BZ/p-nullification of classifying spaces of supersoluble groups. In this case we obtain no result about the homotopy type of BG(for reasons that will be explained at the end) but we include here this computation because of its intrinsical interest, and because the way it has been worked out generalizes in some sense the methods we have used to compute the BZ/p-nullification in the previous sections. Recall that a group Gis called supersoluble if it has cyclic normal series of finite length. It is known that every finitely generated nilpotent group is supersoluble, and that every supersoluble group is polycyclic. Our key result for computing PBZ/pBGis the following ([35], page 67): Proposition 5.9 If Gis a supersoluble group, there exists a characteristic series 1ELEMEG, in such a way that Lis finite with odd order, M/L is a finitely-generated torsion-free nilpotent group and G/M is a finite 2-group. In the sequel we will use the notation of this proposition. Let pbe an odd prime, and consider the fibration: BL−→ BM−→ B(M/L). As M/L is torsion-free, its classifying space is automatically BZ/p-null, and then by ([13], 3.D.3) we have the nullified fibration: PBZ/pBL−→ PBZ/pBM−→ B(M/L). Using ([17], 3.3), the fundamental group of PBZ/pBMis identified by an extension L/TZ/pL−→ π1PBZ/pBM−→ M/L where TZ/pLis the minimal normal subgroup of Lthat contains all the ptorsion (the Z/p-radical), and the universal cover of PBZ/pBMis homotopy equivalent to Z[1/p]∞(BTZ/pL), where Z[1/p]∞denotes Bousfield-Kan Z[1/p]- completion (see [6] for a definition). Now, we have the fibration that involves Mand G: BM−→ BG−→ B(G/M). This fibration is again preserved under BZ/p-nullification, because G/M is a 2-group and pis odd. The long exact sequence of the nullified fibration proves that the fundamental group of PBZ/pBGfits into the following exact sequence: π1PBZ/pBM−→ π1PBZ/pBG−→ G/M Algebraic & Geometric Topology, Volume 5 (2005) 1160 Ram´on J. Flores where the kernel has already been described. On the other hand, the universal cover of PBZ/pBGis the same as the universal cover of PBZ/pBMwhich is Z[1/p]∞BTZ/pL, as we said before. Thus, we have described the desired nullification by means of a covering fibration. On the other hand, the fact that the classifying space of the quotient G/M is not BZ/2-null makes these methods useless for computing the BZ/2-nullification of BG. As an easy consequence of this, B(G/M) is not W∞-null (in the notation of theorem 3.2) and then we cannot get any homotopical description of BGin this way. 6 Nullifying classifying spaces of groups of isometries via proper actions So far we have applied theorem 3.2 for obtaining results about BGusing properties of the nullification functors. In this section we will go the other way round, using geometric characteristics of the group Gfor describing topological features of the classifying space. Our analysis has been focused in some of the crystallographic groups of the plane, also known as wallpaper groups. Recall that these are groups of isometries of R2that fix a pattern that is invariant under translations in the directions of two lineally independent vectors. It is known that they are exactly seventeen of these groups, and they are always finite extensions of Z⊕Zby a finite group. The main references available about the structure of these groups are [36] (that has been specially interesting for us because the big amount of pictures fundamental domains, mirror lines, rotation centers, generating regions, etc. that can be found on it), [26], [25], [10], [14] and [9]; we refer the reader to them for the details of the structure of the groups that in the sequel will stand without any explicit proof. The general idea is to describe, for a prime pand a discrete group Gthat has p-torsion, the homotopy type of the BZ/p-nullification of the classifying space of Gusing the main theorem 3.2. We have chosen wallpaper groups essentially for two reasons: the first of them is the following structure result, that is a particular case of proposition ([27], 5.2): Lemma 6.1 Let Gbe one of the seventeen wallpaper groups. Then R2, endowed with the natural action of G, is a model for EG. The second feature of the wallpaper groups that we are going to use is that all of them possess a well-known model for the orbit space R2/G, which in fact Algebraic & Geometric Topology, Volume 5 (2005) Nullification functors and the homotopy type of classifying spaces 1161 is always described as a finite-dimensional orbifold. A list of these standard models can be found in [26]. According to the previous lemma, these spaces can be also interpreted as models for BG, and using this we will apply theorem 3.2 for obtaining the value of the BZ/p-nullification of the classifying space of G. Now, if Gis a wallpaper group that only has torsion in a family of primes P={p1. . . pr}, it is an easy consequence of corollary 3.4 that every model for BGis a model for BFG, being Fthe family of finite subgroups of G whose order is divided only for primes of the family P. In particular, if G has torsion only in Pand admits a finite-dimensional model for BG, we have BG= BFG≃PBZ/p1∨...∨BZ/prBG. We will frequently use this fact in the sequel. We are going to study here three of the wallpaper groups, namely pmm,p3 and p3m1. The main reason of our choice is that they give examples in which the BZ/p-nullification of the classifying space has homological dimension zero, positive and infinite, respectively. (1) The discope group pmm As a group of symmetries of the real plane, this group is generated by two perpendicular translations and two reflections whose axes are also perpendicular, and in fact it is isomorphic to D∞×D∞. Recall that this group already appeared in the construction of example 4.6. The group pmm contains reflections and rotations, and as it can be seen in the tables of [36], it tessellates the plane with rectangles. The orbit space of the plane by the action of this group is also a rectangle, and hence the classifying space for pmm-proper bundles is contractible. On the other hand, as the rotations that appear in the group are of order two, the group only contains torsion at the prime two, and thus Bpmm is a model for BF2pmm. Now, according to the main theorem 3.2, we conclude that PBZ/2Bpmm ≃ ∗. (2) The tritrope group p3 This group is generated by two translations whose directions form an angle of π/3 and a rotation of angle 2π/3. A presentation with these generators is given by: p3 ={x, y, z;xyx−1y−1= 1, z3= 1, zxz−1y−1x= 1, zyz−1x= 1}. The distinguished isometries of this group are the 3-rotations, so we have no reflections nor glide-reflections and the torsion is concentrated in the prime three. The fundamental region of p3 (that is, the smallest region of R2whose images under the action of p3 cover the plane) is a rhombus, and the action gives rise to a tessellation of R2by hexagons; in fact, this is the simplest wallpaper Algebraic & Geometric Topology, Volume 5 (2005) 1162 Ram´on J. Flores group such that the induced tessellation is not by quadrilaterals. The quotient R2/p3 has then the shape of a non-slit turnover with three corners and no mirror points (see [25] for details), and in particular it has the homotopy type of a 2-sphere. Hence, using an analogous argument to that of the previous case, we obtain that the BZ/3-nullification of Bp3 is homotopy equivalent to S2. (3) The tryscope group p3m1 A convenient system of generators for this group is given by the two usual translations (whose directions form again an angle of π/3), a rotation of angle 2π/3, and a reflection whose axis is the bisectrix of the vectors that determine the generating translations; in particular, the reflection gives torsion in the prime two and the rotation gives it in the prime three. On the other hand, maybe the best way to understand this group is as the group of reflections in the sides of an equilateral triangle. Hence, this is the fundamental region, and the lattice is hexagonal, as in the previous example. As one can see in [36], the orbit space by the action of p3m1 on R2is a triangle, and as this is a model for the classifying space for proper bundles, we have that Bp3m1 is contractible. Applying one more time theorem 3.2, we obtain that the BZ/2∨BZ/3-nullification of the classifying space of p3m1 is a point. Now we are also interested in the BZ/3-nullification of Bp3m1, and we need to use a different strategy. The tryscope group can be seen as an extension of Z⊕Zby the symmetric group Σ3, and a consequence of this is that p3 is an index two subgroup of p3m1. In particular, this gives rise to a fibration: Bp3 −→ Bp3m1 −→ BZ/2. The base space is BZ/3-null, so according to ([13], 3.D.3) and our previous description of PBZ/3Bp3, the BZ/3-nullification of Bp3m1 is identified by a covering fibration: S2−→ PBZ/3Bp3m1 −→ BZ/2. Now observe that the map BZ/3−→ ∗ is a F2-homology equivalence, and hence Hn(Bp3m1;F2) is isomorphic to Hn(PBZ/3Bp3m1;F2). But as p3m1 has 2-torsion, it has nontrivial F2-cohomology in arbitrarily high degrees, and then its BZ/3-nullification does, too. So, using universal coefficients theorem, we obtain that PBZ/3Bp3m1 has infinite cohomological dimension. On the other hand, as Bp3m1 is contractible and the rational homology of BGis that of BGfor any group G, the fact that BZ/3−→ ∗ is a Q-homology equivalence implies that the rational homology of PBZ/3Bp3m1 is trivial. In particular, PBZ/3Bp3m1 is not homotopy equivalent to the product S2×BZ/2. Algebraic & Geometric Topology, Volume 5 (2005) Nullification functors and the homotopy type of classifying spaces 1163 To conclude, note that in this context S2should be seen as two copies of the equilateral triangle glued along their edges, with the action of Z/2 swapping them. Hence, the previous fibration is not orientable (in the sense of [38], page 344). However, PBZ/3Bp3m1 cannot be homotopy equivalent to the projective plane, because this nullification has infinite cohomological dimension. We think that the ideas developed in this section can give a lot of information about BZ/p-nullification of classifying spaces of groups of symmetries, and we plan to undertake in subsequent work its description for all the crystallographic (wallpaper and hyperbolic) groups, and also other groups of symmetries as rosette or frieze groups. 7 The homotopy fiber of the natural map BG−→ BFG We conclude this note by describing to what extent the homotopy fiber of the map f: BG−→ BFGdefined at the beginning of section 3 can be built using as pieces classifying spaces of subgroups of Gthat belong to the family F. To make this decomposition, the main tools that we are going to use are the left homotopy Kan extension of a functor and the Gabriel-Zisman localization. Now we will recall briefly these definitions. In the sequel Cand Dwill be small topological categories. Let F:C → D be a functor. If dis an object of D, then we define the overcategory F↓das the category whose objects are pairs (c, φ) such that cis an object of Cand φ:F(c)→dis a morphism in D. A morphism between two pairs (c, φ) and (c′, φ′) is given by a map ψ:c→c′in Csuch that φ(F(c)) = φ′◦F(ψ)(c′). In the same way, the undercategory d↓Fis defined as the category whose objects are pairs (c, φ) with c∈ C and φ:c→F(d) a morphism in D. A morphism between ψ: (c, φ)→(c′, φ′) is a morphism ψ′:c→c′such that F(ψ′)◦φ=φ′. When F↓d(respectively d↓F) is contractible for every object din Dwe say that Fis left cofinal (respectively right cofinal). Remark 7.1 The overcategory and undercategory are particular cases of “comma categories”. For a complete study of the comma categories in the general context of category theory see ([30], II.6). From now on, and unless explicit mention against it, we will work in the category Spaces of simplicial spaces (although most of the spaces that will appear will have a simplicial complex structure) Algebraic & Geometric Topology, Volume 5 (2005) 1164 Ram´on J. Flores Let F:C → D be a functor. Segal defined another functor, induced by F, LF:Fun(C,Spaces)−→ Fun(D,Spaces), whose value on every X:C → Spaces is given by LF(X)(d) = hocolimF↓dX◦p, where pis the projection functor p:F↓d→ C. The functor LF(X) is called the homotopy left Kan extension of Xalong F. The importance of that construction comes mainly from the next result: Theorem 7.2 (Homotopy pushdown theorem) If F:C → D and X:C → Spaces are functors, then there is a homotopy equivalence hocolimDLF(X)≃hocolimCX. Proof The proof is done making use of the description of the homotopy left Kan extension as the classifying space of a category. See ([20], 5.5). Now we will recall the classical definition of localization of a category. Theorem 7.3 (Gabriel-Zisman) Let Cbe a category. There exists another category L(C)and a functor C −→ L(C)such that the following conditions hold: • L inverts the morphisms of C. •If F:C −→ D is another functor making the morphisms of Cinvertible, there exists one and only one functor F′:L(C)−→ D such that F′◦ L = F. L(C)is called the category of fractions of Cor simply the localization of C. Proof See ([18], chapter 1). Recall that if Xis a simplicial complex, the simplex category ΓX is the category whose objects are the simplices of X, and whose maps are the face maps (there are no nontrivial degeneracies). We will assume in the rest of the section that we will work with the model of BFGconstructed in proposition 2.6. In the problem we are interested, ΓBFGop will play the role of C, and Dwill be the localization of ΓBFGop . From now on, we will use the model of BFGgiven in Proposition 2.6, and it is not hard to see that in this case, the ΓBFGis exactly the category Γdefined there. Algebraic & Geometric Topology, Volume 5 (2005) Nullification functors and the homotopy type of classifying spaces 1165 We have developed now all the ingredients we need, and we can give the decomposition, that is based in the concept of “homotopy average”, proposed by Dror-Farjoun (see [13], chapter 9). So, consider the map BG→BFG; if Sis the functor defined in section 2, call Sthe composition of N(S) with the Borel construction EG×G(−). According to proposition 2.6 and ([16], 6.5), we have that hocolimΓop S≃BG. Now, if Lis the localization functor previously defined, we can consider the left homotopy Kan extension LL(S). The homotopy pushdown theorem 7.2 implies that we have a homotopy equivalence hocolimΓop S≃hocolimL(Γop)LL(S). So joining all these data we obtain a string of maps hocolimL(Γop)LL(S)≃hocolimΓop S≃BG→BFG. (1) that, up to homotopy equivalence, is the map BG−→ BFGthat we are analyzing. So, we need to describe the inverse image of a simplex σof BFGin hocolimL(Γop)LL(S). Consider first a simplex σ∈Γ; it can be identified with a chain G/H1→...→ G/Hn, and then N(S)(σ) = G/H1(as a discrete G-set). In a similar way, if {G/Hi1→...→G/Hik} −→ {G/H1→...→G/Hn} represents a morphism in Γ, its image by N(S) is represented by the corresponding map G/H1→G/Hi1. Now, the explicit definition over a simplex of N(S) is immediate. Thus, if we identify now hocolimΓop Swith the Grothendieck construction of the nerve of S, every simplex of hocolimΓop Sis represented ([16], 6.5) by a triple (τ, σ1≤... ≤σn, aH1) where τis a simplex of EG,σiis a simplex of Γ,g∈Gand σn=G/H1→... →G/Hn. Observe that σ1≤... ≤σn represents a simplex in N(Γ), which is a model for BFG, as we know. Now the map BG−→ BFGthat we are studying can be seen as the natural map hocolimΓop S−→ hocolimΓop ∗ induced by the natural transformation S→ ∗. Hence the image of the previous triple under this map is the simplex σ, and according to the definition of S we can identify the inverse image of σwith the pair (σ, S(σn)). Observe that, because of the definition of S, the latter is the same as (σ, S(σ)), which is in fact a model for the classifying space of H1. It remains to identify the image of the pair (σ, S(σn)) under the map hocolimΓop S≃hocolimL(Γop)LL(S) Algebraic & Geometric Topology, Volume 5 (2005) 1172 Ram´on J. Flores [38] R M Switzer,Algebraic topology—homotopy and homology, Grundlehren series 212, Springer-Verlag, New York (1975) [39] R W Thomason,Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91–109 Departamento de Matem´aticas, Universidad Aut´onoma de Barcelona E–08193 Bellaterra, Spain Email: [email protected] URL: http://mat.uab.es/∼ ramonj Received: 25 November 2004 Revised: 27 August 2005 Algebraic & Geometric Topology, Volume 5 (2005)