NULLIFICATION FUNCTORS AND THE HOMOTOPY TYPE OF THE CLASSIFYING SPACE FOR PROPER BUNDLES RAM'ON J. FLORES Abstract. Let G be a discrete group. In this note we build a bridge betw* *een the homotopy theory of BG and the theory of proper G-actions, by showing that under mild restrictions, the classifying space for proper G-bundles* * has the homotopy type of the W-nullification of BG for some space W. This allows us to use properties of the localization functors to obtain space* *s that are homotopy equivalent to B_G for a wide range of groups, and on the ot* *her hand, we take profit of the existence of well-known geometrical and fini* *te- dimensional models of B_G for some infinite groups for obtaining homotop* *ical information about the p-primary part of their classifying spaces. 1.Introduction Let G be a discrete group. We will say that a CW-complex X is a G-CW- complex if G acts on X by permuting the cells. If the isotropy groups of the ac* *tion are finite, then we will call it a proper G-CW-complex. In 1994, Baum-Connes-Higson introduce in [BCH94 ] the "classifying space for proper actions", that can be described as the unique proper G-CW-complex E_G, up to G-homotopy, that enjoys the following universal property: If X is another proper G-CW-complex, there exists a G-map X -! E_G which is unique up to G-homotopy. The space E_G appears as the principal new feature on the reformulation of the Baum-Connes conjecture stated in the aforementioned article. The conjecture, partially solved, asks if for a locally compact, Hausdorff and second countable* * group G, the C*-algebra K-theory groups Kj(C*r(G)) are isomorphic to the Kasparov K- homology groups KGj(E_G), for j = 0, 1. The great amount of research that have emerged around this subject has led to a growing interest in the theory of prop* *er actions. An important part of the efforts carried out in this direction has been devo* *ted to understand the relationship between the algebraic structure of G and the homoto* *py- theoretic properties of E_G and its quotient space E_G=G, which is currently de* *noted by B_G. Probably the greatest success has been reached interpreting correctly f* *inite- ness group-theoretic conditions over G in order to build models of E_G enjoying various types of finiteness conditions. We will review in section 2.3 the main * *con- tributions in this area. In the same way that happens with classical G-actions, the importance of E_G and in particular of B_G does 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 B_G classi* *fy proper G-bundles (see 2.8 for a definition), and they described how to obtain t* *hem by making pullback on maps X -! B_G, a method of a clear classical flavor. ____________ Partially supported by MCYT grant BFM2001-2035. 1 2 RAM'ON J. FLORES The most important attempt made so far for understanding the homotopy type and properties of B_G is the paper of Leary-Nucinkis [LN01 ]. In it, the author* *s prove that for every CW-complex X there exists a discrete group GX such that B_GX is homeomorphic to X. This Kan-Thurston type result is proved using essentially tools of the theory of graph of groups. As a byproduct, they obtain a precise description of the fundamental group of B_G and a construction of B_G for some subgroups of right-angled Coxeter groups. Although these last results have been very useful for us (particularly the f* *ormula for the fundamental group), our approach to the homotopy type of B_G has been different, and has been carried out with pure homotopy-theoretic tools. Our ide* *a is to find a functor F : Top -! Top that transforms models of BG on spaces that are homotopy equivalent to models of B_G. This functor have enough good properties in order to read information about B_G from BG and vice versa. The appropriate functor turns to be a nullification; a tool that was introdu* *ced by Bousfield in [Bou94 ] in order to study periodic phenomena in unstable homotopy (in fact, he called it "periodization"), and that has been widely used since th* *en. The utility of this functor in this context comes from the fact that it will al* *low us to apply all the machinery of localization developed in the 90's by Bousfield h* *imself ([Bou94 ], [Bou97 ]), Dror-Farjoun ([DF95 ]), Chach'olski ([Cha96 ]) and others. Our main result is the following: Theorem 4.2. Let G be a discreteWgroup such that there exists a finite-dimensio* *nal model for B_G. Denote W1 = BZ=p, where the wedge is extended to all primes. Then we have a homotopy equivalence PW1 B G ' B_G, where PW1 denotes the W1 -nullification functor. Observe that the condition concerning it has to exist a model of B_G with fi* *nite- ness conditions is not too restrictive, in account of the great quantity of gro* *ups that have recently appeared for which these conditions hold (in this sense, see sect* *ion 2.3). Now we describe in more detail the contents of each of the sections of the p* *aper, and in particular we will comment a little bit the consequences of the main the* *orem. In section 2 we make a review of all the needed background and results conce* *rning proper actions, with special emphasis in finiteness conditions for E_G (and hen* *ce for B_G). In section 3 we construct a particular model for the classifying space for f* *amilies EFG that will be very useful in the rest of the work; as a fundamental byproduc* *t, we also obtain an appropriate model for BF G. Section 4 constitutes the bulk of the work, because it is devoted to the pro* *of 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 equiva* *lent to the W -nullification of the nerve of some small category that only depends o* *n G. This nerve turns out to be the model built in the previous section for B_G, and* * we finish by checking that in the conditions of the theorem B_G is W -null. The rest of the paper is devoted to take out some consequences of the main theorem. So, in section 5 we describe the behaviour of the functor B_with respe* *ct to various fundamental constructions in homotopy theory, namely products wedges or colimits. Moreover, we identify in some cases the universal cover of B_G and* * we obtain some conditions about preservation of fibrations under passing to classi* *fying spaces. We begin the following section with a short proof of the fact that if G is a locally finite group which cardinal is smaller than @!, B_G is contractible. La* *ter, NULLIFICATION FUNCTORS AND THE HOMOTOPY TYPE OF B_G 3 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 G in this class admits a finite-dimensional model for B_G* *, then B_G ' BH for some quotient group H that we identify. In particular, in this case B_G is nilpotent as a space if G is nilpotent as a group. We finish this paragr* *aph by studying the BZ=p-nullification, of classifying spaces of supersoluble group* *s. In section 7 we take the opposite point of view, showing that the main the- orem 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 the w* *ell- known geometric properties of them we obtain via B_G a lot of information on the B Z=p-nullification of their classifying spaces. Last section is devoted to the study of a canonical map BG -! BF G (defined previously) that always relates the classical and proper classifying spaces. M* *ore concretely, we prove that the homotopy fiber of that map can be described as a homotopy colimit of classifying spaces of groups of F over a contractible categ* *ory. 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 we* *re 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. 2. Background results In this section we give a brief review of the main concepts of the theory of* * proper actions and classifying spaces for families of subgroups, that will be used lat* *er in our work. The main references used here have been [BCH94 ], [LN01 ], and [MV01 ]. * *For more information about classical theory of G-actions, look at [Die87], for exam* *ple. 2.1. Proper actions. Definition 2.1. Let G be a discrete group, X a G-space. The action of G in X is said proper if for every point p 2 X there exists a triple (U, H, æ) such th* *at the following conditions hold: (1)H is a finite subgroup of G. (2)U is an open neighborhood of p such that gu 2 U for each (g, u) 2 G x U. (3)æ is a G-map from U to G=H. It is not hard to see that all the isotropy groups of a proper G-space are f* *inite. In particular, if in addition the action is free, then the canonical projection X * *-! X=G is a trivial principal G-bundle. Now we will define the subcategory of the category of G-spaces where we will usually work. Definition 2.2. A G-CW-complex is a G-space X endowed of a filtration X0 X1 X2 . . .X by G-subspaces such that the following axioms hold: (1)Xn is closed in X for each n. (2)X0Sis a discrete subspace of X. (3) n 0Xn = X. 4 RAM'ON J. FLORES (4)If n 2, there is a discrete G-space n and G-maps f : Sn-1 x n -! Xn-1, g : Bn x n -! Xn such that the following diagram is a pushout f Sn-1 x n _____//Xn-1 | | | | fflffl|g |fflffl Bn x n _______//Xn where the vertical maps are inclusions. (5)A subspace Y X is closed if and only if Y \ Xn is closed for every n * * 0. We use the conventions X-1 = ;, 0 = X0. If x 2 n, the subspaces g(x x intBn) and g(x x Bn) are called respectively the open cells and closed cells of* * X. The concept of G-CW-subcomplex is defined in the obvious way. Definition 2.3. A G-map f : X - ! Y between G-spaces is called a weak G- equivalence if for every finite subgroup H < G the induced map on the fixed poi* *nt spaces fH : XH - ! Y H is a weak homotopy equivalence. It can be seen ([Lüc89], I.2.4) that if X is a G-CW-complex, the notion of G- homotopy equivalence coincides with the concept of weak G-equivalence defined above. Moreover, a G-CW-complex is proper if and only if the isotropy group of every point is a finite subgroup of G. So, we can say that if X is a proper G-CW-complex, G acts essentially by permuting the cells. Definition 2.4. A G-CW-complex is n-dimensional if X = Xn for some n, and finite-dimensional if it is n-dimensional for some n. X is said to be finite i* *f it consists in a finite number of open cells. X is finitely-dominated if there exi* *sts a finite G-CW-complex Y and G-maps r : Y -! X and i : X -! Y such that r O i is G-homotopy equivalent to the identity of X. It is clear that if X is finite,* * then it is finite-dimensional and finitely-dominated. Some of these finiteness conditions will be repeatedly used throughout our w* *ork. 2.2. Classifying spaces for families of subgroups. In this section we describe the objects that will be studied on this article: the classifying spaces for fa* *milies of subgroups and particularly their quotient spaces by the action of G. Definition 2.5. Suppose F is a family of subgroups of a discrete group G that is closed under conjugation and taking subgroups. We will say that a G-CW-complex Y is a model for EFG if the isotropy group of each point belongs to F and for e* *ach H 2 F, the fixed-point space Y H is contractible. It is easy to see that if F is the family that only contains the trivial gro* *up, then E FG = EG, the universal space for principal G-bundles. The object EFG is characterized by the following universal property: Proposition 2.6. If X is a model for EF G, then for each proper G-CW-complex there is a map X - ! EF G which is unique up to G-homotopy. Moreover, two models for EF G are always G-homotopy equivalent. Conversely, if Y is a G-CW- complex for which this universal property holds, Y is a model for EF G. Proof. See [Die87], section 1.6. Because of this property, the space EF G is usually called the classifying s* *pace for the family F. NULLIFICATION FUNCTORS AND THE HOMOTOPY TYPE OF B_G 5 There are various standard constructions for EF G, such as for example the f* *ol- lowing, that resembles the classical construction of Milnor of the classifying * *space por principal G-bundles. Take the disjoint union of all the homogeneous spaces G=H, H 2 F, and call it X. Denote by X(n) the n-join X * . .*.X. If we con- siderSthe obvious inclusions X(n) ,! X(n + 1), we can define the infinite union n2NX(n). Then we have S Proposition 2.7. n2NX(n) is a model for EF G. Proof. See [BCH94 ], appendix 1 or [MV01 ], section 1. The quotient space EF G=G is usually denoted by BF G. As EF G is unique up to G-homotopy equivalence, BF G is unique up to homotopy equivalence. But, on the contrary to what it happens with EF G, this does not means that every space X homotopy equivalent to a model of BF G is a model of BF G. For example, by definition BFG is a point if and only if G is finite, but, as it is known (and * *we shall see in 6.1), there are a lot of infinite groups for which BF G is contractible. Now, if G, G0 are discrete groups, F and F0 families of subgroups of G and G0 that are closed by conjugation and taking subgroups, and f : G -! G0 is a homomorphism such that f(H) 2 F0for each H 2 F, it is not difficult to see that f gives rise to a G-map EF G -! EF0G0, which is unique up to G-homotopy, and to a pointed map BF G -! BF0G0which is unique up to homotopy. If there is no explicit mention against it, we will suppose from here that F* * is the family of finite subgroups of G. In this case, it is standard to denote EFG and* * BFG by E_G and B_G, respectively. In particular, according to the previous paragrap* *h, every group homomorphism G -! G0induces a G-map E_G -! E_G0(respectively a pointed map B_G -! B_G0) which is unique up to G-homotopy (respectively up to homotopy). We know that if F = {e}, BFG coincides with the usual classifying space for * *prin- cipal G-bundles BG. Now, following [BCH94 ], we will see that B_G is a classif* *ying space in a certain interesting sense. Definition 2.8. Let X be a metrizable space. A proper G-bundle over X is a pair (Z, ß), where Z is a proper G-space and ß : Z -! X is a continuous map such that (1)If (g, z) belongs to G x Z, then ß(gz) = ß(z). (2)The induced map G\Z -! X induced by ß is a homeomorphism. Baum-Connes-Higson also define the suitable notions of homotopy equivalence and isomorphism between G-proper bundles, which allows them to prove the fol- lowing important theorem: Theorem 2.9. Let P (G, X) be the set of homotopy classes of proper G-bundles ov* *er X. Then, if F is the family of the finite subgroups of G, there exists a biject* *ion of sets [X, BFG] -'! P (G, X). Proof. The desired map is obtained assigning to every homotopy class of maps OE : X -! BFG the pullback along OE of EFG over BF G. See [BCH94 ], appendix 3. So B_G is really a classifying space. According to [LN01 ], we have adopted * *the terminology "proper G-bundle" in place of the original "proper G-space" because that one makes explicit the parallelism between the role of BG in the theory of classical G-actions and B_G in the theory of proper actions. 6 RAM'ON J. FLORES 2.3. Finiteness conditions for E_G and B_G . We want to finish this preliminary section by reviewing some of the finiteness conditions for E FG that have been obtained in the last years for different classes of discrete groups G, with spe* *cial attention to the case in which F is the family of finite subgroups of G. These conditions have a fundamental importance for our work, because the groups that enjoy them are precisely those for which our main theorem 4.2 holds. Note that * *if X is a model for EF G, X=G is a model for BF G and dim X = dim X=G. Thus, the finiteness conditions over EFG will be also valid over BF G. We begin with an easy result coming from graph theory. Proposition 2.10. If G acts on a tree T by a simplicial and continuous action whose isotropy groups are finite, then T is a model for E_G. Serre shows ([Ser80]) that if G1, G2 and H are finite groups, the pushout G = G1*H G2 acts on a tree T in the previous way. This tree is constructed making t* *he double mapping cylinder of the diagram G=G1 - G=H -! G=G2. So this gives us a big family of groups for which E_G is not only finite-dim* *ensional, but 1-dimensional. Recently Leary-Nucinkis ([LN01 ], prop. 4) and Platten ([Pla02]) have genera* *l- ized the result of Serre to the case of EFG, being F any family of finite subgr* *oups of G. In [Lüc00] we can find the following finiteness result (theorem 0.1), that u* *ses as input Bredon coefficient systems on proper complexes (definition and properties* * of this category can be found in section 2 of that paper or in [Bre72]): Theorem 2.11. Let G be a discrete group, F a family of subgroups of G that is closed under conjugation and taking subgroups, and OF the corresponding orbit category, that is, the category whose objects are the homogeneous spaces G=H, H 2 F, and whose morphisms are G-maps. Then we have, for d 3: (1)There is a d-dimensional G-CW-model for EFG if and only if the constant ZOF-module Z_has a projective resolution of length d. (2)There is a finite G-CW-model for EF G if and only if EF G has a G-CW- model with finite 2-skeleton and the constant ZOF-module Z_has a finite free resolution over OF. (3)There is a G-CW-model with finite 2-skeleton for E_G if and only if there are only finitely many conjugacy classes of finite subgroups H < G and f* *or any finite subgroup H < G the group N(H)=H is finitely presented. In the aforementioned article [Lüc00] some classes of groups with finiteness* * con- ditions are also described: Proposition 2.12. For the following classes of groups there exists a model of f* *inite dimension for E_G and B_G: o The word-hyperbolic groups, with the word-length metric. The model for E_G is the Rips complex. o The discrete cocompact subgroups of a Lie group G with a finite number of connected components. If K is the maximal compact subgroup of G, G=K is the aforementioned model. o The extensions of Zn by finite groups. In this case the model is Rn, and the group acts via isometries. o The virtually-polycyclic groups. Other groups that have recently deserved attention in this context have been* * the hierarchically decomposable groups, that we are going to define right now. NULLIFICATION FUNCTORS AND THE HOMOTOPY TYPE OF B_G 7 Definition 2.13. Let C be a class of groups. We write H1C for the class of grou* *ps G that admit a finite-dimensional contractible G-CW-complex with isotropy groups in C, and HC for the smallest class C0such that C < C0H1C0= C0. If F is the cla* *ss of finite groups, then HF is the class of hierarchically decomposable groups. Mislin and Kropholler give ([KM98 ], theorem A) the following finiteness re* *sult: Theorem 2.14. Let G be an hierarchically decomposable group of type FP1 . Then there is a finite-dimensional model for E_G. Moreover, if (G) is the poset of non-trivial finite subgroups of G and B(G,* * Z) is the G-operator ring of bounded functions from G to Z, they obtain (theorem B* *): Theorem 2.15. If G is an HF-group such that m = proj. dimZGB(G, Z) and d = dim| (G)| are finite, then there is a model of E_G of finite dimension, and* * this dimension can be made smaller than 2d7 - 1, if m 2, or smaller than 2d+1(m + 1) - 1 if m > 2. Mislin ([Mis01], 4.1-4.5) has recently gone a little bit further in this lin* *e of research. Among the results he has obtained, we can quote the following ones, t* *hat are used in this note (see section 6): Proposition 2.16. Let K -! G -! Q be an extension of groups, and assume that Q 2 H1F and the order of its torsion subgroups is bounded. Then if dimK E_K is finite, dimG E_G is finite too. Remark 2.17. In ([Lüc00] 3.1), we can find a refinement of the previous stateme* *nt: if moreover we know that dimQE_Q is finite, we can calculate dimGE_G using as i* *nput dim KE_K and dimQE_Q. Other finiteness conditions of öc homological kind" can be found in this paper ([Lüc00], 4.2, 5.2 and 6.4) Proposition 2.18. If G is a locally finite group such that |G| < @!, then G adm* *its a finite-dimensional E_G. Proposition 2.19. Let G be a soluble group with |G| < @!. Then the following are equivalent: (1)The torsion-free rank of G is finite. (2)dim GE_G is finite. (3)The rational cohomological dimension of G is finite. Finally, we would like to mention a result of Leary-Nucinkis ([LN01 ], prop.* * 8), because its essentially geometric nature, and because it will be very useful for clarifying to what extent the functor B_preserve fibrations. We need before some definitions: Definition 2.20. A right-angled Coxeter group is a group generated by a set of elements of order two that are subject uniquely to commutativity relations. The* *se generators are called the Coxeter generators. A Coxeter presentation of G can be always encoded in a simplicial complex K = K(G) that has one vertex for every Coxeter generator, and one simplex for every subset of the set of generators that commute. This is sometimes called the Coxeter complex of G. It is not hard to prove that every flag complex is the Co* *xeter complex of some right-angled Coxeter group G. Definition 2.21. We define the Davis' complex = (G) as the realization of the poset whose elements are the cosets of finite Coxeter subgroups of G. The action of G on the cosets induces a proper action over , and it can be * *seen ([BLN01 ], section 3, see also [Dav83 ]) that this action turns to a model of* * E_G. Now we can give the promised curious result: 8 RAM'ON J. FLORES Proposition 2.22. Let G be the right-angled Coxeter group corresponding to a fl* *ag complex K, and H the index two subgroup of G whose elements are the words that can be written with an even number of letters. Then =H, which is a model for B_H, is isomorphic to the suspension of the barycentric division of K. The study of these index two subgroups was one of the first motivations for * *our work. 3. A useful model for EF G and B FG Our description of the relationship between BG and BF G is essentially based* * in the construction of the classifying space for proper actions and its orbit spac* *e as nerves of small categories. This is the main goal of this section, but we need* * to recall previously some concepts. Recall from 2.11 the definition of the orbit category associated to a family* * F; it is not hard to see that there is a bijective map Mor (G=K, G=H) = {g 2 G | g-1Kg H}=H given by f -! f(eK), where e is the identity element of G. The key definition we need for building the desired model of EF G is the following (see [Dwy97 ], sec* *tion 2, for details): Definition 3.1. Let D be 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 f is defined as the category whose objects are the pairs (d, x), with x 2 D and x 2 f(d), and where* * a morphism (d, x) ! (d0, x0) is a pair (u, v) where u : d ! d0is a morphism in D * *and v : f(u)(x) ! x0 is a morphism in f(d0). The composition is made in the obvious way. The main feature of this construction, due to Thomason, is the following: Theorem 3.2. Let D be a small category, F : D -! C 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 N denotes the nerve. Proof. See [Tho79 ], 1.2. Now we can describe our model of the universal space EFG. Proposition 3.3. Let G be 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 Grothe* *ndieck construction of f is a model for EF G. Proof. For convenience, we will denote X = |N (Gr (R))|. This space has a natur* *al action of G given by the left action of G in every homogeneus space G=H. We will prove firstly that for every x 2 X the isotropy group Gx of x belongs to F. The action of G over X is simplicial and is induced from the action of G over the homogeneous spaces, so by definition of nerve we will only need to study the NULLIFICATION FUNCTORS AND THE HOMOTOPY TYPE OF B_G 9 action of G over the vertices of X. Hence, let (G=H, a) be the pair associated * *with the vertex x. It is clear that Gx = {g 2 G | gaH = aH} = {g 2 G | 9h 2 H s. t.gah = a} and this amounts to say that g 2 aHa-1. So, Gx = aHa-1, that belongs to F because H does. Now we will see that for every K 2 F, the set of fixed points XK is contrac* *tible. We can reduce again to the case in which x is a vertex of X. A point x 2 X is f* *ixed by the action of K if, given the pair (G=H, aH) associated to x, we have that, * *for every k 2 K, kaH = aH. Thus, we see that [ XK = {(G=H, aH) | a-1Ka H}=H. H2F So, for every element (G=H, aH) x 2 XK there exists one and only one morphism (G=K, eK) - ! (G=H, aH), or in other words, XK can be identified with the nerve of the undercategory (see section 8 for a definition) associated to the e* *lement (G=K, eK) of the category Gr (R). This nerve is contractible, and then XK is contractible too. So we are done. Note that if we consider the action of G over Gr (R) via functors, the objec* *ts 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 BF G. Remark 3.4. Observe that if we take the realizations of the nerves of Gr (R) and OF, we obtain models of EFG and BF G in the category of (topological) spaces. Remark 3.5. The idea of the construction comes from ([AD01 ], section 2), al- though they only describe it in the case of G finite, being F the family of fin* *ite subgroups of G, and with another purpose. In the language of that paper, we have proved that X is the F-approximation to a point. We conclude this section with a modification of the previous models that will be useful later. So, if {H1, . .,.Hn} are subgroups of G that belong to F and s* *uch that for every 0 < i < n there exists a G-equivariant map G=Hi -! G=Hi+1, we define G=H1 -! . .-.! G=Hn as 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 is the poset category of non-degenerate simplices of N* *(OF), we define a functor S : -! 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 3.6. In the previous conditions, hocolim N(S) is a model for EF G, and N( ) is a model for BF G. Proof. By the result 3.2, hocolim N(S) ' N(Gr (S)), and on the other hand, the left action of G over every homogeneous space G=H, (H 2 F) induces, via func- tors, another one over the categories G=H1 -! . .-.! G=Hn . Now, observe that N (Gr (S)) is the subdivision (in the sense of [GJ99 ], III.4) of N (Gr (R)), w* *here R is the functor defined in 3.3. In fact, if we take a non-degenerate simplex of N (Gr (R)) represented by a chain of morphisms (G=H1, a1H1) - ! . . .-! (G=Hn, anHn), its barycenter is the vertex of N(Gr (S)) represented by the obje* *ct (G=H1 -! . .-.! G=Hn , a1H1 -! . .-.! anHn) of Gr (S). Then, by ([Jar02] prop. 12-14), there exists a homotopy equivalence |N (Gr R)| ' |N (Gr S)|, tha* *t is 10 RAM'ON J. FLORES a G-equivalence by construction, and then N(Gr S) is a model for EFG. A similar line of reason proves that N(S) is a model for BF G. The main advantage of this models is that they reconstruct E FG and B FG as homotopy colimits over a poset category, and in particular these models have structure of simplicial complexes. These facts will be very useful in last sect* *ion. 4. B_G is homotopy equivalent to a nullification Let G be a discrete group, F a family of subgroups of G closed under conjuga* *tion 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 BG and BF G. So, consider models of EG and EF G; both of them are G-spaces, and then we can make the Borel construction EG xG EF G. Now, let p1 be the projection EG xG EF G -p1!EG=G ' BG. The action of G over EG is free, so the map p1 is a fibration, and its homot* *opy fiber EFG is contractible. Thus, p1 is a homotopy equivalence, and EG xG EF G is a model of BG. Consider now the projection over the second component E G xG EF G -p2!EFG=G ' BFG. We have seen that EG xG EF G and EFG=G ' BG are respectively models for BG and BF G, and then p2 can be thought as a map BG -! BF G that we will call f in the rest of the section. The map f is not a fibration in general, because * *the action of G over EFG is not free. In fact, if x 2 BFG, we have that f-1 (x) has* * the homotopy type of EG xG G=Hx, being Hx the isotropy group of x, that belongs to F. Now, EG xG G=Hx is a model for BHx, and hence all the fibers of the map f have 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 G to the classify* *ing space for proper G-bundles, and what is more important, to obtain valuable info* *r- mation of the latter starting from the homotopy structure of BG, and viceversa. More concretely, we searched for a functor F such that the following conditions hold: (1)F "kills" the homotopy fiber of f. (2)F (f) is a weak equivalence. (3)F (B_G) ' B_G. The two first conditions give the impression, according to ([DF95 ], 1.H.1 a* *nd 3.D.3), of F being a localization functor L in the sense of Dror-Farjoun, and in fact, the functor we have found has been the A-nullification functor with respe* *ct a certain space A. For the main properties of these functors you can look at [Bou* *94 ], [Cha96 ] or [DF95 ], although we will recall here the definition. Let A and X be spaces. Recall that X is said A-null if the pointed mapping space map *(A, X) is contractible. The A-nullification of X is a functorial way* * of turning every space into an A-null space, and can be defined as the unique space PA X up to homotopy that is A-null and such that there exists a universal map X -! PA X which induces a weak homotopy equivalence map *(PA X, Y ) ' map *(X, Y ) for every A-null space Y . In this way there is defined a functor in the catego* *ry of pointed spaces (although it can be defined similarly over unpointed spaces) whi* *ch is NULLIFICATION FUNCTORS AND THE HOMOTOPY TYPE OF B_G 11 always coaugmented and idempotent, and kills the structure of X that "dependsö * * n A. This is, in fact, the localization of X with respect to the constant map A -* *! *. Moreover, the spaces X for which PA X is contractible are called A-acyclic. Consider now the set of all prime numbers {p1, p2, p3. .}.with the usual ord* *er, and let X be a space.WIn the rest we will denote by Wn the space BZ=p1_. ._.BZ=* *pn, and by W1 the wedge BZ=p extended over of the all prime numbers. The next key lemma is one strong reason that suggests that the W1 -nullification is the func* *tor we need. Theorem 4.1. If G is a finite group, then PW1 B G is contractible. Proof. A point is always null, so we only need to prove that map *(PW1 B G, X) * *is contractible for every W1 -null space X. But the W1 -null spaces are, in partic* *ular, Wn-null for every n; hence, map *(PW1 B G, X) ' map *(B G, X) ' map *(PWn BG, X) for every n. Now, suppose that |G| = pn1j1pn2j2.p.n.mjm, with j1 < . .j.m. Beca* *use of ([Flo03], 3.3), we know that PWk BG is contractible for every k jm . This imp* *lies map*(PW1 B G, X) ' map *(PWn BG, X) ' * as we claimed. Now we suppose that G is a discrete group such that there exists a finite- dimensional model for B_G. We are now in position of stating our main theorem: Theorem 4.2. With the previous notation, we have that B_G is homotopy equivalent to PW1 B G. Proof. Consider the model of E_G given in section 3 as the realization of the n* *erve of the Grothendieck construction of a certain functor R : OF -! Cat described there. We have seen that EG xG E_G is a model of BG, so using 3.2 we obtain B G ' EG xG E_G ' EG xG (hocolimOFR) ' hocolimOF(E G xG R(-)) where the previous equivalence is a simple application of ([DH00 ], 6.5). Now,* * if we apply the nullification functor PW1 to the previous string of equivalences,* * we obtain a weak equivalence PW1 B G ' PW1 hocolimOF (E GxG R(-)), and the latter is equivalent, by ([DF95 ] 1.H.1), to PW1 hocolimOF PW1 (E G xG R(-)). Observe that the spaces that appear in the target of the functor EG xG N(R(-)) : OF -! Spaces have the homotopy type of classifying spaces of finite subgroups of G. Hence, i* *f we apply the previous proposition, we have that PW1 O (E G xG R(-)) is equivalent* * to the constant functor, and then PW1 B G ' PW1 hocolimOF * ' PW1 (N (OF)) ' PW1 (B_G). Then, by the solution of Miller to the Sullivan conjecture ([Mil84]), we know t* *hat the space map *(W1 , B_G) is contractible, and hence B_G is W1 -null. This means that PW1 B_G is homotopy equivalent to B_G, and we are done. The following generalization, that will be widely applied in section 7, is a* *n im- mediate consequence of the proof of the previous theorem: 12 RAM'ON J. FLORES Corollary 4.3. If F is a family of finite subgroups of G closed under conjugati* *on and taking subgroups, and PA is a nullification functor such that PA BH ' * for every H 2 F, then the map f : BG -! BFG is an equivalence after A-nullification. If we particularize for the family of all the finite groups, we obtain the f* *ollowing: Corollary 4.4. If G is a discrete group, the classifying spaces BG and BF G are always equivalent after W1 -nullification; moreover, the map f that was describ* *ed at the beginning of this section is, in fact, equivalent to the W1 -nullificati* *on map if G admits a finite-dimensional model for BF G. Proof. The fact that BG and BF G are PW1 -equivalent is a particular case of the previos corollary. For the second statement, if we have the fibration Fibf -! BG -! BFG (where Fibf stand for the homotopy fibre of f), the base is W1 -null, and then * *by ([DF95 ], 3.D.3) the fibration is preserved under W1 -nullification. Now the re* *sult is an easy consequence of ([Cha96 ], 14.2). Remark 4.5. It is worth to point out that the finiteness conditions under which* * the main theorem holds is that there exists a model of B_G for which map *(W1 , B_G* *) is weakly contractible). This is weaker than having a finite-dimensional model for* * B_G, but the condition that will always hold for the groups that appear in the rest * *of this note will be the latter, because it is the usual one that appears in the litera* *ture of Geometric Group Theory. It would be interesting to find cohomological conditions to be a Miller space (that is, spaces X for which map *(B Z=p), X) ' *) for all* * the primes at the same time, because to these spaces we could also apply the previo* *us theorem. Recall that there already exist well-known cohomological conditions of this kind for isolated primes, such as for example the Lannes-Schwartz Theorem ([LS89 ]). We finish this section by showing that the discreteness of G is necessary in theorem 4.2. Example 4.6. Let us consider the classifying space of S1. As the circle is com- pact, the classifying space for proper S1-bundles is a point by definition. On * *the other hand, consider the rationalization map B S1 -! K(Q, 2). By the homo- topy long exact sequence, the homotopy fibre of this map has the homotopy type * *of K( Zp1 , 1), where the direct sum runs over all primes. As every Prüfer group Z* *p1 is a colimit of a telescope of injections between p-groups, the results ([DF95 * *] 1.D.3) and ([Flo03], 3.3) imply that PW1 K( Zp1 , 1) is contractible. Then, by ([DF95 ] 1.H.1) the rationalization is preserved by W1 -nullification, and as K(Q, 2) is* * clearly W1 -null, we have that PW1 B S1 ' K(Q, 2), which is clearly non-contractible. In fact, it seems plausible to conjecture that the W1 -nullification of the classi* *fying space of a compact Lie group is homotopy equivalent to its rationalization. 5. The homotopy type of B_G In this paragraph we are going to prove some interesting consequences that t* *he main theorem 4.2 has over the homotopy type B_G. Essentially, the idea is to use properties of the nullification functors for describing the classifying spa* *ce for G-proper bundles. Remark 5.1. From now on, we particularize for the case F being the family of all the finite subgroups of G, although a great part of the results we obtain in the next sections remains valid for any subfamily of F that is subgroup-closed * *and NULLIFICATION FUNCTORS AND THE HOMOTOPY TYPE OF B_G 13 conjugation-closed. We will also suppose that the finiteness conditions of 4.2 hold for all the groups that appear in this section. We begin by analyzing the behaviour of the functor B_under products. Proposition 5.2. Let G1 and G2 discrete groups. Then the following holds: o A model for B_(G1 x G2) is given by B_G1 x B_G2. o The wedge B_G1 _ B_G2 is a model for B_(G1*G2). Proof. It is known that B(G1x G2) ' BG1x BG2. Using that B_G1x B_G2 is W1 - null (because the finiteness) and the preservation property ([DF95 ] 1.A.8, pro* *p. 4), we obtain that B_(G1 x G2) ' PW1 (B_(G1 x G2)) ' PW1 (B_G1) x PW1 (B_G2) ' B_G1 x B_G2. The proof of the second statement is similar, using that B(G1*G2) ' BG1_B G2 and recalling that we can apply ([DF95 ], 1.D.5) because a wedge of a special c* *ase of pointed homotopy colimit. It is worth to point out that these results are known, although this way of * *proving them is probably new. On the other hand, the second of them can be generalized to other colimits, like some telescopes and pushouts. Proposition 5.3. Let {Gi}i2N be a family of discrete groups. Then we have: o If we have the pushout of groups G1 _ff_//_G2 fi|| || fflffl| fflffl| G3 _____//_G and the homomorphisms ff and fi are injective, then the pushout of the induced diagram of classifying spaces for proper G-bundles is a model for B_G. o If G1 -! G2 -! G3 -! . .i.s a telescope of groups where the maps are injective and we denote by G the colimit of the telescope, we have that * *the colimit of the telescope induced by B_is a model for B_G. Proof. To prove the first statement, recall that by Whitehead's theorem ([Bro82* *], II.7.3) the pushout of the classical classifying spaces is the classifying spac* *e of the pushout. As the inclusions BG1 ,! BG2 and BG1 ,! BG3 are cofibrations, BG has the homotopy type of the homotopy pushout. If we apply now the functor PW1 to the diagram, the result is deduced from 4.2, ([DF95 ], 1.D.3) and the fact that* * there exists (as we imposed at the beginning of the section) a finite-dimensional mod* *el for the homotopy pushout of the induced diagram B_G2 - B_G1 -! B_G3. The second statement can be proved in an analogous way using again the relation- ship between localization and colimits given in ([DF95 ], 1.D.3) and the fact t* *hat the strict colimit of a telescope of cofibrations has always the homotopy type * *of the homotopy colimit. As Whitehead's theorem is not true if the maps that appear in the diagram are not injective, it should not be expected that the functor B_preserve colimits i* *n full generality. 14 RAM'ON J. FLORES Recall now that in ([LN01 ], prop. 3) it is identified for any discrete gro* *up G the fundamental group of B_G, that is the quotient of G by the (normal) subgroup generated by the torsion elements. Using the main theorem 4.2, we can identify * *in some cases the universal cover of B_G. Proposition 5.4. Let G be a discrete group, and let T < G be the subgroup gener- ated by the torsion elements. If the quotient G=T is torsion-free, then the uni* *versal cover of B_G has the homotopy type of the W1 -nullification of BT . Proof. We know that T is normal in G, so we have a fibration BT -! BG -! B(G=T ). As G=T is torsion-free, its classifying space is W1 -null. Thus the previous fi* *bration is preserved by W1 -nullification, and we obtain another one: PW1 B T -! B_G -! B(G=T ). Note that, as T is a subgroup of G, every model for E_G is also a model for E_T* * . Hence, B_T is a model for PW1 B T , and in particular ß1(PW1 B T ) = ß1(B_T ) =* * {1}. This implies that PW1 B T is simply-connected, and we are done. The last important consequence of the main theorem that we are going to prove here has to do with fibrations, and will have great importance in the remaining* * of this note. It is a well-known fact of basic algebraic topology that is we have a group * *ex- tension, then the sequence induced at the level of classifying spaces is a fibr* *ation sequence. Using the description of 4.2, we find sufficient conditions that guar* *antee that the analogous result for B_G holds, and we show by means of an easy example that the statement does not need to be true if those hypotheses fail to be fulf* *illed. So, suppose we have a short exact sequence of groups {1} -! G1 -! G -! G2 -! {1}. Then the following result is true: Proposition 5.5. If G2 is torsion-free or G1 admits a contractible model for B_* *G1, the homotopy fiber of the induced map B_G -! B_G2 is homotopy equivalent to B_G* *1. Proof. It is enough to combine the results ([DF95 ], 1.H.1 and 3.D.3) of Dror-F* *arjoun with our description of B_G as a nullification (4.2). The next example will show that the above conditions are necessary. Example 5.6. Consider the group D1 x D1 , and H the index two subgroup whose elements are the words that can be written with an even number of letters. We have an extension H -! D1 x D1 - ! Z=2 that induces a sequence of maps B_H -! B_(D1 x D1 ) -! B_Z=2. It is not hard to see that a model for E_(D1 x D1 ) is given by R2, and the quo* *tient by the action of (D1 x D1 ) is a square, which is contractible. By 2.22, B_H is homotopy equivalent to the 2-sphere, and on the other hand, Z=2 is finite and so B_Z=2 is contractible. This means that the aforementioned sequence cannot be a fibration sequence. Of course, neither of the conditions of the proposition hol* *d in this case. NULLIFICATION FUNCTORS AND THE HOMOTOPY TYPE OF B_G 15 6. Homotopy models of B_G for some families of discrete groups In this section we will use the theorem 4.2 for describing the homotopy type* * of B_G for a wide range of groups. As a byproduct, we will obtain that for every G nilpotent such that it admits a finite-dimensional model for B_G, B_G is nilpot* *ent as a space, and we also determine the BZ=p-nullification of the classifying spa* *ces of wom3 supersoluble groups. Let us start by considering the class of locally f* *inite groups. 6.1. Locally finite groups. It is known that the classifying space for proper G- bundles of a group G is contractible if the group G is locally finite. We begin* * this section by presenting an easy proof of this fact in an ample range of cases. Proposition 6.1. Let G be a locally finite group that admits a finite-dimension* *al model for B_G. Then B_G is contractible. Proof. It is known (see for example [Mil84], 9.8) that every locally finite gro* *up is the colimit of the directed system of its finite subgroups. Thus, we have a hom* *otopy equivalence BG ' hocolimCBH, where C is a contractible poset category (because * *it has an initial object given by the trivial group) and BH represents all the cla* *ssifying spaces of finite groups H of G. So, by ([DF95 ], 1.D.3), we obtain PW BG ' PW (hocolimCPW BH) = PW (|C|) = PW (*) = * and we are done. Remark 6.2. By (2.18), this result applies, in particular, to all the locally f* *inite groups whose cardinal is smaller than @!. Now we can prove the following result, that concerns to the classifying spac* *e for proper G-bundles of extensions of locally finite groups. Proposition 6.3. Let {1} -! K -! G -! Q -! {1} be an extension of groups, K a locally finite group whose cardinal is smaller t* *han @!, and assume there is a bound in the order of finite subgroups of Q. Then if Q admits a finite model for B_Q, G admits a finite model for B_G, and then B_Q * *is homotopy equivalent to B_G. Proof. If we apply the results 4.2 and 6.1 we obtain the statement is true if t* *here is a finite-dimensional model for B_G, and this happens by 2.16. 6.2. Groups with the normalizer condition. We study now the groups for which the normalizer condition holds. It is greatly remarkable that this class * *con- tains all the nilpotent groups. Recall that a group G is said to satisfy the normalizer condition if every p* *roper subgroup of G is distinct from its normalizer. In this case the following holds* * (see [Kur60 ], pag. 215): (1)For every prime p, there exists a normal p-group Tp such that if x 2 G a* *nd the order of x is a power of p, then x 2 Tp. (2)The elements ofQfinite order of G form a normal subgroup of G which is isomorphic to p primeTp. Throughout this section we will impose to the groups for which the normalizer condition holds that the torsion p-subgroups Tp that we have just defined are l* *ocally finite. We need this condition because it is not known if the BZ=p-nullificatio* *n of the classifying space of a general p-group is contractible if the group is not * *locally 16 RAM'ON J. FLORES 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 and the öm nstersö f Tarski-Olshanskii. See [Ady75 ] and [Ols89] for more informat* *ion about these families of groups. All these facts have the following interesting consequence: Proposition 6.4. If G is a discrete group for which the normalizer condition holds, and p1. .p.nis a collection of primes, we have a homotopy equivalence PBZ=p1_..._BZ=pnBGQ' B(G=Tp1x . .x.Tpn). In particular, there is an equivalence PW1 B G ' B(G= p primeTp). Proof. For simplicity, we will only prove the case of one prime p (the generali* *zation to a family is immediate). It is clear that BTp is BZ=p-acyclic, and B(G=Tp) is B Z=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 G is such that exists a finite-dimensional model for B_G * *we have: Q Corollary 6.5. B_G ' B(G= p primeTp). So we have a complete description of the homotopy type of B_G. Other case that can be solved with the same tools is the following: Proposition 6.6. Let G be a discrete group, H a normal subgroup of G for which the normalizer conditions holds, and such that G=H does not have p-torsion. If Tp is the p-torsion subgroup of H, then the BZ=p-nullification of BG fits into * *the following fibration sequence: B(H=Tp) -! PBZ=pBG -! B(G=H) and hence it is an Eilenberg-MacLane space. Proof. The base of the fibration BH -! BG -! B(G=H) is BZ=p-null, so by ([DF95 ], 3.D.3) the fibre is preserved under BZ=p-nullific* *ation. The result now follows from 6.4. Taking into account the main theorem 4.2, the following corollary is immedia* *te: Corollary 6.7. In the hypothesis of the previous proposition, if G=H is torsion* *-free and T is the torsion subgroup of H, then the fibration B(H=T ) -! B_G -! 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 distin- guished class of discrete groups for which the normalizer condition holds. The following result proves that the BZ=p-nullification preserves nilpotency when i* *t is applied on classifying spaces of nilpotent groups, and in fact, the functor B_s* *ends nilpotent groups (for which the finiteness conditions hold) to nilpotent spaces. NULLIFICATION FUNCTORS AND THE HOMOTOPY TYPE OF B_G 17 Corollary 6.8. If G is a nilpotent group, the nullification PBZ=p1_..._BZ=pnBG is, for every set of primes p1, p2. .,.the classifying space of a nilpotent gro* *up. If moreover G admits a finite-dimensional model for B_G, we obtain that B_G is aga* *in 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 ([Ols89], 2.7.1), ev* *ery nilpotent p-group is locally finite. In particular, using 2.19 we have that the part of the previous corollary th* *at alludes B_G is always true if G is a nilpotent group whose cardinal is smaller * *than @! and whose torsion-free rank is finite. 6.3. Supersoluble groups. In this paragraph we will compute, for p odd, the B Z=p-nullification of classifying spaces of supersoluble groups. In this case* * we obtain no result about the homotopy type of B_G (for reasons that will be expla* *ined at the end) but we include here this computation because its intrinsical intere* *st, 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 G is called supersoluble if it has cyclic normal series * *of finite length. It is known that every finitely generated nilpotent group is supersolub* *le, and that every supersoluble group is polycyclic. Our key result for computing PBZ=pBG is the following ([Rob72 ], page 67): Proposition 6.9. If G is a supersoluble group, there exists a characteristic se* *ries 1 E L E M E G, in such a way that L is 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 p be 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 t* *hen by ([DF95 ], 3.D.3) we have the nullified fibration: PBZ=pBL -! PBZ=pBM -! B(M=L). Using ([Flo03], 3.3), the fundamental group of PBZ=pBM is identified by an exte* *n- sion L=TZ=pL -! ß1PBZ=pBM -! M=L where TZ=pL is the minimal normal subgroup of L that contains all the p-torsion (the Z=p-radical), and the universal cover of PBZ=pBM is homotopy equivalent to Z[1=p]1 (B TZ=pL), where Z[1=p]1 denotes Bousfield-Kan Z[1=p]-completion (s* *ee [BK72 ] for a definition). Now, we have the fibration that involves M and G: B M -! BG -! B(G=M). This fibration is again preserved under B Z=p-nullification, because G=M is a 2- group and p is odd. The long exact sequence of the nullified fibration proves t* *hat the fundamental group of PBZ=pBG fits into the following exact sequence: ß1PBZ=pBM -! ß1PBZ=pBG -! G=M 18 RAM'ON J. FLORES where the kernel has already been described. On the other hand, the univer- sal cover of PBZ=pBG is the same as the universal cover of PBZ=pBM which is Z[1=p]1 BTZ=pL, as we said before. Thus, we have described the desired nullifi- cation by means of a Postnikov fibration. On the other hand, the fact that the classifying space of the quotient G=M is not B Z=2-null makes these methods use- less for computing the BZ=2-nullification of BG. As an easy consequence of this, B (G=M) is not W1 -null (in the notation of 4.2) and then we cannot get any ho- motopical description of B_G in this way. 7.Nullifying classifying spaces of groups of isometries via proper actions So far we have applied theorem 4.2 for obtaining results about B_G using pro* *per- ties of the nullification functors. In this section we will walk the inverse pa* *th, using geometric characteristics of the group G for 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 they are groups of isometries of R2 that 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 Z by a finite group.* * The main references available about the structure of these groups are [Sch78] (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 f* *ound on it), [Lev02], [Lee02], [CM65 ], [DDS99 ] and [Con90 ]; we refer the reader t* *o them for the details of the structure of the groups that in the sequel will stand wi* *thout any explicit proof. The general idea is to describe, for a prime p and a discrete group G that h* *ave p-torsion, the homotopy type of the BZ=p-nullification of the classifying space* * of G using the main theorem 4.2. We have chosen wallpaper groups essentially for two reasons: the first of them is the following structure result, that is a particu* *lar case of 2.12: Lemma 7.1. Let G be one of the seventeen wallpaper groups. Then R2, endowed with the natural action of G, is a model for E_G. 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 is always described as a finite-dimensional orbifold. A list of these standard mod* *els can be found in [Lev02]. According to the previous lemma, these spaces can be also interpreted as models for B_G, and using this we will apply 4.2 for obtain* *ing the value of the BZ=p-nullification of the classifying space of G. Now, if G is a wallpaper group that only has torsion in a family of primes P = {p1. .p.r}, it is an easy consequence of 4.4 that every model for B_G is a * *model for BF G, being F the family of finite subgroups of G whose order is divided on* *ly for primes of the family P . In particular, if G has torsion only in P and admi* *ts a finite-dimensional model for B_G, we have B_G = BF G ' PBZ=p1_..._BZ=prBG. We will constantly use of 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 B Z=p-nullification of the classifying space has homological dimension zero, po* *sitive and infinite, respectively. NULLIFICATION FUNCTORS AND THE HOMOTOPY TYPE OF B_G 19 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 w* *hose axes are perpendicular too. A presentation of the group is given by pmm = {x, y, z, t | xyx-1y-1 = 1, z2 = t2 = 1, zxzx-1 = 1, (tx)2 = (zy)2 =* * 1, tyty-1 = 1}. This group contains reflections and rotations, and as it can be seen in the * *tables of [Sch78], it tessellates the plane with rectangles. The orbit space of the p* *lane by the action of this group is also a rectangle, and hence the classifying spac* *e 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 pr* *ime two, and thus B_pmm is a model for BF2pmm . Now, according to the main theorem 4.2, we conclude that PBZ=2Bpmm ' *. 2. The tritrope group p3. This group is generated by two translations whose directions form an angle of ß=3 and a rotation of angle 2ß=3. A presentation wi* *th these generators is given by: p3= {x, y, z; xyx-1y-1 = 1, z3 = 1, zxz-1y-1x = 1, zyz-1x = 1}. The only distinguished isometries of this group are the 3-rotations, so we h* *ave 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 R2 whose images under the action of p3 cover the plane) is a rhombus, and the action gives rise to a tessellation of R2 by hexagons; in fact, this is the simplest wallpaper gr* *oup 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 poin* *ts (see [Lee02] for details), and in particular it has the homotopy type of a 2-sp* *here. 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 an* *gle of ß=3), a rotation of angle 2ß=3, and a reflection whose axe is the bisectrix * *of the vectors that determine the generating traslations; in particular, the reflectio* *n gives torsion in the prime two and the rotation gives it in the prime three. A presen* *tation with this system of generators is the following: p3m1 = {x, y, z, t; xyx-1y-1 = 1, z3 = t2 = 1, (tz)2 = 1, zxz-1y-1x = 1, zy* *z-1x = 1, (tx)2 = 1, tyty-1x = 1}. The fundamental region is in this case an equilateral triangle, and the latt* *ice is hexagonal, as in the previous example. As one can see in [Sch78], the orbit spa* *ce by the action of p3m1 on R2 is a triangle, and as this is a model for the clas* *sifying space for proper bundles, we have that B_p3m1 is contractible. Applying one more time 4.2, 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 somewhat different strategy. The tryscope group can be seen as an extensi* *on of Z Z by the symmetric group S3, and a consequence of this is that p3 is an index two subgroup of p3m1 . In particular, this gives rise to a fibration: 20 RAM'ON J. FLORES Bp3 -! Bp3m1 -! BZ=2. The base space is B Z=3-null, so according to ([DF95 ], 3.D.3) and our previ- ous description of PBZ=3Bp3 , the BZ=3-nullification of Bp3m1 is identified b* *y a covering fibration: S2 -! PBZ=3Bp3m1 - ! BZ=2. Now observe that the map BZ=3 -! *is a F2-homology equivalence, and hence Hn(B p3m1 ; 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 B Z=3-nullification does, too. So, using universal coefficients theorem, we ob* *tain that PBZ=3Bp3m1 has infinite cohomological dimension. In fact, it is possible to know a little bit more about the homology of this* * space, because the previous fibration is orientable (in the sense of [Swi75], page 344* *). By reductio ab absurdum, suppose that the action of Z=2 over the integer homology of S2 is nontrivial. Then, a computation with the Serre spectral sequence with twisted coefficients shows that the integer cohomology groups Hn(PBZ=3Bp3m1 ) are finite if n > 0. On the other hand, as B_p3m1 ' PBZ=2_BZ=3Bp3m1 and the map BZ=2 _ BZ=3 -! *is a rational homology equivalence, the results of ([LS00 ], section 4) imply that the rational homology of PBZ=3Bp3m1 is nontrivial, which leads to a contradiction. So, the fibration is orientable, and moreover it has a section, because the * *original extension p3 -! p31m -! Z=2 has. Now, an easy calculation with the Gysin sequence shows that the cohomology algebra (over the integers) of PBZ=3Bp3m1 is a tensor product (x) F2(y), where x and y have degree 2. To conclude, note that PBZ=3Bp3m1 cannot be homotopy equivalent to S2 x B Z=2, because in this case the action of Z=2 over p3 in the previous extension, which is induced by the action of Z=2 over the universal cover of PBZ=3Bp3m1 , would be trivial, and this is not true. We think that the line of research followed in this section can give a lot o* *f infor- mation about BZ=p-nullification of classifying spaces of groups of symmetries, * *and we plan to undertake in subsequent work its description for all the crystallogr* *aphic (wallpaper and hyperbolic) groups, and also other groups of symmetries as roset* *te or frieze groups. 8.The homotopy fiber of the natural map B G -! BF G We conclude this note by describing to what extent the homotopy fiber of the map f : BG -! BF G defined at the beginning of section 4 can be built using as pieces classifying spaces of subgroups of G that belong to the family F. To make this decomposition, the main tools that we are going to use are the left homoto* *py Kan extension of a functor and the Gabriel-Zisman localization. Now we will rec* *all briefly these definitions. In the sequel C and D will be small topological categories. Let F : C ! D be* * a functor. If d is an object of D, then we define the overcategory F # d as the c* *ategory whose objects are pairs (c, OE) such that c is an object of C and OE : F (c) ! * *d is a morphism in D. A morphism between two pairs (c, OE) and (c0, OE0) is given by a map _ : c ! c0 in C such that OE(F (c)) = OE0O F (_)(c0). In the same way, t* *he undercategory d # F is defined as the category whose objects are pairs (c, OE) * *with c 2 C and OE : c ! F (d) a morphism in D. A morphism between _ : (c, OE) ! (c0,* * OE0) is a morphism _0: c ! c0such that F (_0)OOE = OE0. When F # d (respectively d #* * F ) NULLIFICATION FUNCTORS AND THE HOMOTOPY TYPE OF B_G 21 is contractible for every object d in D we say that F is left cofinal (respecti* *vely right cofinal). Note 8.1. The overcategory and undercategory are particular cases of öc mma categories". For a complete study of the comma categories in the general context of category theory see ([Mac71 ], II.6). 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 O p, where p is the projection functor p : F # d ! C. The functor LF (X) value is ca* *lled the homotopy left Kan extension of X along F . The importance of that construction comes mainly from the next result: Theorem 8.2 (Homotopy pushdown theorem). If F : C ! D and X ! Spaces are functors, then there is a homotopy equivalence hocolimD LF (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 ([HV92 ],5.5). Now we will recall the classical definition of localization of a category. Theorem 8.3 (Gabriel-Zisman). Let C be a category. There exists another category L(C) and a functor C -! L(C) such that the following conditions hold: o L inverts the morphisms of C. o If F : C -! D is another functor making the morphisms of C invertible, there exists one and only one functor F 0: L(C) -! D such that F 0OL = F. L(C) is called the category of fractions of C or simply the localization of * *C. Proof. See ([GZ67 ], chapter 1). Recall that if X is a simplicial complex, the simplex category X is the cat* *egory whose objects are the simplexes of X, and whose maps are the face maps (there a* *re no nontrivial degeneracies). We will assume in the rest of the section that we * *will work with the model of BF G constructed in 3.6. In the problem we are intereste* *d, BFGop will play the role of C, and D will be the localization of BFGop. We have developed now all the ingredients we need, and we can give the decom- position, that is based in the concept of öh motopy average", proposed by Dror- Farjoun (see [DF95 ], chapter 9)._So, consider the map BG ! BFG; if S is the fu* *nc- tor defined in section 3, call S the composition of N(S) with the Borel_constru* *ction E G xG (-). According to 3.6 and ([DH00 ], 6.5), we have hocolim BFGopS ' BG. Now, if L is the localization_functor previously defined, we can consider the l* *eft homotopy Kan extension LL (S ). The homotopy pushdown theorem 8.2 implies that we have a homotopy equivalence __ __ hocolimL( BFGop)LL (S ) ' hocolim BFGopS. So joining all this data we obtain a string of maps __ __ hocolimL( BFGop)LL (S ) ' hocolim BFGopS ' BG ! BFG. But now, if oe is a n-simplex_in BF G, it is clear that its inverse image by* * that string of maps is precisely LL S(oe), so we can establish the following 22 RAM'ON J. FLORES Theorem 8.4. If f : BG ! BFG is the map previously defined, then __ Fibf ' LL S(oe) for any simplex oe of BF G. Here Fibf stands for the homotopy fiber of f. __ Proof. We need only check that the homotopy type of LL S(oe) does not depend on the simplex_oe of BF G. We_know,_by the construction of the Kan extension, that LL (S )(oe) = hocolimL#oe(S O p), where p is the projection functor p : L # oe * *! C. So, if oe and oe0 are two distinct simplices of B FG, it is enough to see that * *the overcategories L # oe and L # oe0are equivalent. In order to check this, let g * *: oe ! oe0 be a morphism in L( BFGop), that always exist because BF G is connected. In these conditions, g induces a natural transformation Tg : L # oe -! L # oe0 that sends every object (ø, OE) of L # oe to (ø, g O OE) 2 L # oe0 and the morp* *hisms to the obvious ones. But the morphism g is invertible (because is a morphism in the localized category), and clearly the natural transformations Tg and Tg-1 are inverses one of each other. In other words, the two overcategories are equivale* *nt, and the corresponding homotopy colimits have the same homotopy type. So we are done. The following corollary is immediate: __ Corollary 8.5. The homotopy fiber Fibf has the homotopy type of hocolimL#oe(S O p), and in particular it is a homotopy colimit of classifying spaces of groups * *of F over a contractible category. We conclude this section by proving that the nerves of the two overcategories that appear in the proof of the previous theorem are contractible. We think that this question can have independent interest, and we would like to point out tha* *t, although the result seems to be known (see [DF95 ], 9.E.3), we have found no pr* *oof in the literature, so we give this one. Proposition 8.6. Let X be a simplicial complex, and let L : X -! L( X ) be the Gabriel-Zisman localization functor, where X is the simplex category of X. Then for every simplex oe 2 X the overcategory L # oe is contractible. Proof. The idea of the proof is to build, for every simplex oe 2 X a homotopy between the identity map Id|N(L#oe)|and a constant map. In order to do this, we will prove firstly the existence of a sequence of endofunctors {Fn} : L # oe -! L # oe for every n 0 such that F0 = Idand for every (ø, OE) 2 L # oe there exists a * *natural number n(ø,OE)in such a way that Fm ((ø, OE)) = (oe, Id) for every m n(ø,OE). In the sequel the maps in X and their images in L( X ) will be denoted ind* *is- tinctly by iff, where ff will be an appropriate subindex. The inverse of iffin * *the localized category will be called jff. It is plain from the definition of the localization functor that every eleme* *nt (ø, OE) of L # oe admits a unique expression of the form (ø, jn O in-1 O . .O.j2O i1), * *where we allow that jn or i1 can be the identity (but no one of the other maps that appe* *ar), jt-1 6= i-1t6= jt+1 for every t. So, we begin with F0 = Id. Let us define the functor F1 : L # oe -! L # oe. If (ø, jn Oin-1 O. .O.j2Oi1) is an element of the overcategory, then we say F1(* *(ø, jn O in-1 O . .O.j2 O i1)) = (i1(ø), jn O in-1 O . .O.j2), and the map induced by a * *face NULLIFICATION FUNCTORS AND THE HOMOTOPY TYPE OF B_G 23 map will be sent to the identity map between the images. This can be easily seen a well-defined functor. Now, F2 : L # oe -! L # oe will be defined as F2((ø, jn O in-1 O . .O.j2 O i* *1)) = (j-12O i1(ø), jn O in-1 O . .O.i3). Observe that this is well-defined because * *the localization functor is, in this case, bijective over the objects. Again, the i* *mage of every morphism by F2 will be the identity. It is clear again that this is a fun* *ctor. In an analogous way, we can define, for m odd, Fm ((ø, jn O in-1 O . .O.j2O * *i1)) = (im Ojm-1 Oi1(ø), jnOin-1O. .j.m+1), and for m even, Fm ((ø, jnOin-1O. .O.j2Oi1* *)) = (jm O im-1 O i1(ø), jn O in-1 O . .i.m+1), and the image sends every morphism t* *o the identity map. This is the sequence we were looking for. Our next goal will be to relate all these functors by natural transformation* *s, in order to obtain the desired homotopy. Let m 0 be again a natural number. First we will define the transformation T2m : F2m -! F2m+1 . If (ø, jn O in-1 O . .O.j2O i1) is an object of the overca* *tegory, we define the map F2m ((ø, jnOin-1O. .O.j2Oi1)) -! F2m+1 ((ø, jnOin-1O. .O.j2Oi* *1)) as the obvious map induced by i2m : j2m-1 O . .O.i1(ø) -! i2m O j2m-1 O . .O.i1(ø). On the other hand, we define, for every m 1, the natural transformation T2m-1 : F2m -! F2m-1 in the following way: F2m ((ø, jn O in-1 O . .O.j2 O i1)) * *-! F2m-1 ((ø, jn O in-1 O . .O.j2 O i1)) is the map induced by i2m-1 : j2m-1 O . .O.i1(ø) -! i2m-2 O j2m-3 O . .O.i1(ø). Recall the fact that, by definition of the j's, j-1 represents a morphism in X* * . By the previous arguments we have defined a string of natural transformations Id= F0 T0-!F1 T1-F2 T2-!F3 T3-. . . Before we continue, we shall do a couple of remarks. o It is known ([DH00 ], I.5) the functors Fn define simplicial maps from n* *erves N(Fn) : N(L # oe) -! N(L # oe) which, the same way, define maps |fn| from the realization of the nerve * *to itself. The fact that Fn is always related to Fn+1 by a natural transfor* *ma- tions means that fn is simplicially homotopic to fn+1, and, in addition,* * |fn| is homotopic to |fn+1|. The crucial point here is the homotopies between the realization of the maps are first defined over the vertices of the n* *erve of L # oe x I (with the usual simplicial structure of the product) and t* *hen extended by linearity to all the complex. We will use this fact later. o Let (ø, jn O. .O.i1) be an object of the overcategory. From the definiti* *ons of the functors Fiwe can deduce that FnO. .O.F1((ø, jnO. .O.i1)) = (oe, Id)* *. So, as the chain of maps jn O . .O.i1 is always finite, we can say that for * *every (ø, OE) 2 (L # oe) there exists a minimal natural number n(ø,OE)such that Fnø,OEO . .O.F1((ø, OE)) = (oe, Id). At the level of nerves, we are sayi* *ng that for every vertex v 2 N(L # oe) there exists nv such that fnv O . .O.f1(v* *) = N (oe, Id). For n even, let us call Hn the simplicial homotopy induced by the transforma* *tion Tn. If n is odd, we call H0n-1the homotopy induced by Tn between fn and fn-1, and put Hn-1(x, t) = H0n-1(x, 1 - t), the homotopy that begins in fn-1 and ends at fn. Now we are prepared to define the homotopy between the identity and the con- stant map from the realization to itself with value |N (oe, Id)| (in the rest w* *e will 24 RAM'ON J. FLORES call this element *). So, consider a vertex v 2 N (L # oe). We define a map H : |N (L #)oe| x I -! |N (L #)oe| by 8 1_ >>>|H0|(v, nvt) ift 2 [0, nv] < |H1|(v, nvt - 1) ift 2 [_1_n, 2_n] H(v, t) = > . . v v >>:.. .. |Hn-1|(v, nvt - (n - 1))ift 2 [nv-1_nv, 1] The map H defined in this way lineally extends to all of |N (L # oe)|. Let u* *s see H is the desired map. (1)If v is a vertex of N (L # oe), H(v, 0) = H0(v, 0) = v. In the same way, H(v, 1) = Hnv(v, 1) = *. 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