DISCRETE MODELS FOR THE p-LOCAL HOMOTOPY THEORY OF COMPACT LIE GROUPS AND p-COMPACT GROUPS CARLES BROTO, RAN LEVI, AND BOB OLIVER Abstract. We define and study a certain class of spaces which includes p-* *completed classifying spaces of compact Lie groups, classifying spaces of p-compact* * groups, and p-completed classifying spaces of certain locally finite discrete gro* *ups. These spaces are determined by fusion and linking systems over "discrete p-tora* *l groups" _ extensions of (Z=p1 )r by finite p-groups _ in the same way that classify* *ing spaces of p-local finite groups as defined in [BLO2 ] are determined by fusion a* *nd linking systems over finite p-groups. We call these structures "p-local compact g* *roups". In our earlier paper [BLO2 ], we defined and studied a certain class of spac* *es which in many ways behave like p-completed classifying spaces of finite groups. These* * spaces occur as "classifying spaces" of certain algebraic objects called p-local finit* *e groups. The purpose of this paper is to generalize the concept of p-local finite groups* * to what we call p-local compact groups. The motivation for introducing this family come* *s from the observation that p-completed classifying spaces of finite and compact Lie g* *roups, as well as classifying spaces of p-compact groups [DW ], share many similar ho* *motopy theoretic properties, but earlier studies of these properties usually required * *different techniques for each case. Moreover, while p-completed classifying spaces of fin* *ite and, more generally, compact Lie groups arise from the algebraic and geometric struc* *ture of the groups in question, p-compact groups are purely homotopy theoretic objec* *ts. Unfortunately, many of the techniques used in the study of p-compact groups fai* *l for p-completed classifying spaces of general compact Lie groups. With the approach* * pre- sented here, we propose a framework general enough to include p-completed class* *ifying spaces of arbitrary compact Lie groups as well as p-compact groups. The new idea here is to replace fusion systems over finite p-groups, as handl* *ed in [BLO2 ], by fusion systems over discrete p-toral groups. A discrete p-toral g* *roup is a group which contains a discrete p-torus (a group of the form (Z=p1 )r for finit* *e r 0) as a normal subgroup of p-power index. A p-local compact group consists of a tr* *iple (S, F, L), where S is a discrete p-toral group, F is a saturated fusion system * *over S (a collection of fusion data between subgroups of S arranged in the form of a c* *ategory and satisfying certain axioms), and L is a centric linking system associated to* * F (a category whose objects are a certain distinguished subcollection of the object * *of F, and of which the corresponding full subcategory of F is a quotient category). * *The linking system L allows us to define the classifying space of this p-local comp* *act group ___________ 1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 57T1* *0. Key words and phrases. Classifying space, p-completion, fusion, compact Lie g* *roups, p-compact groups. C. Broto is partially supported by MEC grant MTM2004-06686. R. Levi is partially supported by EPSRC grant MA022 RGA0756. B. Oliver is partially supported by UMR 7539 of the CNRS. All three of the authors were partially supported by the Mittag-Leffler Insti* *tute in Sweden, and also by the EU grant nr. HPRN-CT-1999-00119. 1 2 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups to be the p-completed nerve |L|^p. If S is a finite p-group, then the theory re* *duces to the case of p-local finite groups as studied in [BLO2 ]. We hope that working with this setup will make it possible to prove results o* *f in- terest in a uniform fashion for the entire family. In this paper (Theorem 7.1),* * we give a combinatorial description of the space of self equivalences of |L|^pin terms * *of auto- morphisms of the category L, and a description of the group Out (|L|^p) of homo* *topy classes of self equivalences in terms of "fusion preserving automorphisms" of S* *. We also show that a p-local compact group (S, F, L) is determined up to isomorphism by * *the homotopy type of its classifying space |L|^p. One future goal is to show that t* *he mod p cohomology of the classifying space |L|^pof a p-local compact group (S, F, L)* * can always be described in terms of the fusion system F, as a ring of "stable eleme* *nts" in the cohomology of S. Other goals are to define connected p-local compact groups* *, and understand their properties and their relation to connected p-compact groups; a* *nd to characterize algebraically (connected) p-compact groups among all (connected) p* *-local compact groups. Finally, a more general question which is still open is whethe* *r the p-completion of the classifying space of every finite loop space is the classif* *ying space of a p-local compact group. As one might expect, passing from a finite to an infinite setup introduces an* * array of problems one must deal with in order to produce a coherent theory. Some of the * *basic properties of fusion systems over discrete p-toral groups are analogous or even* * identical to the finite case, whereas other aspects are more delicate. Once the definiti* *on of a saturated fusion system over a discrete p-toral group is given and their basic * *properties are studied, one defines associated centric linking systems and p-local compact* * groups in a fashion more or less identical to the finite case. However, while in the * *finite case, any finite group G gives rise automatically to a saturated fusion system * *and an associated centric linking system, the corresponding construction for compac* *t Lie groups is less obvious. Similar complications present themselves when dealing w* *ith the fusion system and the centric linking system associated to a p-compact group. * *It is for that reason that the only aims of this paper are to establish the setup, st* *udy some basic properties, and prove that the classifying spaces which are the obvious c* *andidates to give rise to p-local compact groups indeed do so. We proceed by describing the contents of the paper in some detail. In Sectio* *n 1, we define and list some properties of discrete p-toral groups. We show why this* * class of groups is a natural one to consider for our purposes, and study some of its * *useful properties. Then in Section 2, we define saturated fusion systems over discrete* * p-toral groups. The definitions in this section are very similar to those given in [BLO* *2 ] for the finite case, but some modifications are needed due to having given up finitenes* *s. Much of the work on p-local finite groups makes implicit use of the fact that* * the categories one works with are finite. If S is an infinite discrete p-toral grou* *p, then any fusion system over it will have infinitely many objects. In Section 3 we show t* *hat any saturated fusion system F over a discrete p-toral group S contains a full subca* *tegory with finitely many objects, which in the appropriate sense determines F complet* *ely. More precisely, we show that F contains only finitely many objects which are bo* *th centric and radical, and then prove the appropriate analog of Alperin's fusion * *theorem. The latter, roughly speaking, says that in a saturated fusion system, every mor* *phism can be factored into a sequence of morphisms each of which is the restriction o* *f an automorphism of a centric radical subgroup. Carles Broto, Ran Levi, and Bob Oliver * * 3 Linking systems associated to fusion systems over discrete p-toral groups are* * defined in Section 4. In fact, the definition is identical to that used when working ov* *er a finite p-group, and the proof that the nerve |L| of a linking system is p-good is esse* *ntially identical to that in the finite case. The connection between linking systems as* *sociated to a given fusion system F and rigidifications of the homotopy functor P 7! BP * *on the orbit category Oc(F) is then studied. Higher limits over the orbit category of a fusion system are investigated in * *Section 5. We first describe how to reduce the general problem to one of higher limits* * over a finite subcategory, and then show how those can be computed with the help of * *the graded groups *( ; M) introduced in [JMO ]. These general results are then ap* *plied to prove the acyclicity of certain explicit functors whose higher limits appear* * later as obstruction groups. Spaces of maps Map (BQ, |L|^p) are studied in Section 6, when Q is a discrete* * p-toral group and |L|^pis the classifying space of a p-local compact group, and the spa* *ce of self equivalences of |L|^pis handled in Section 7. In both cases, the descriptions w* *e obtain in this new situation (in Theorems 6.3 and 7.1) are the obvious generalizations* * of those obtained in [BLO2 ] for linking systems over finite p-groups. We also prove (T* *heorem 7.4) that a p-local compact group is determined by the homotopy type of its cla* *ssifying space: if (S, F, L) and (S0, F0, L0) are p-local compact groups such that |L|^p* *' |L0|^p, then they are isomorphic as triples of groups and categories. We finish with three sections of examples: certain infinite locally finite g* *roups in Section 8, including linear torsion groups; compact Lie groups in Section 9; an* *d p- compact groups in Section 10. In all cases, we show that the groups in questio* *n fit into our theory: they have saturated fusion systems and associated linking syst* *ems, defined in a unique way (unique up to isomorphism at least), and the classifyin* *g spaces of the resulting p-local compact groups are homotopy equivalent to the p-comple* *ted classifying spaces of the groups in the usual sense. The first and third authors would like to thank the University of Aberdeen fo* *r it's hospitality during several visits; in particular, this work began when we got t* *ogether there in January 2002. All three authors would like to thank the Mittag-Leffler* * Insti- tute, where this work was finished over a period of several months. 1.Discrete p-toral groups When attempting to generalize the theory of p-local finite groups to certain * *infinite groups, the first problem is to decide which groups should replace the finite p* *-groups over which we studied fusion systems in [BLO2 ]. The following is the class of* * groups we have chosen for this purpose. Let Z=p1 ~=Z[1_p]=Z denote the union of the c* *yclic p-groups Z=pn under the obvious inclusions. Definition 1.1. A discrete p-toral group is a group P , with normal subgroup P0* * C P , such that P0 is isomorphic to a finite product of copies of Z=p1 , and P=P0 is * *a finite p-group. The subgroup P0 will be called the identity component of P , and P wi* *ll be called connected if P = P0. Set ss0(P ) def=P=P0: the group of components of P . The identity component P0 of a discrete p-toral group P can be characterized * *as the subset of all infinitely p-divisible elements in P , and also as the minimal su* *bgroup of 4 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups finite index in P . Define rk(P ) = k if P0 ~=(Z=p1 )k, and set |P | def=(rk(P ), |ss0(P )|) = (rk(P ), |P=P0|) . We regard the order of a discrete p-toral group as an element of N2 with the le* *xico- graphical ordering. Thus |P | |P 0| if and only if rk(P ) < rk(P 0), or rk(P * *) = rk(P 0) and |ss0(P )| |ss0(P 0)|. In particular, P 0 P implies |P 0| |P |, with eq* *uality only if P 0= P . The obvious motivation for choosing this class is the role they play as "Sylo* *w p- subgroups" in compact Lie groups and p-compact groups. But in fact, it seems d* *if- ficult to construct fusion systems with interesting properties over any larger * *class of subgroups. The reason for this is that discrete p-toral groups are characteriz* *ed by certain finiteness properties, which are needed in order for fusion systems ove* *r them to be manageable, and for related homotopy theoretic phenomena to be controled * *by p-local information. A group G is locally finite if every finitely generated subgroup of G is fini* *te, and is a locally finite p-group if every finitely generated subgroup of G is a finite p-* *group. The class of locally finite (p-)groups is closed under subgroups and quotient group* *s. It is also closed under group extensions, since finite index subgroups of finitely ge* *nerated groups are again finitely generated. A group G is artinian (satisfies the minimum condition in the terminology of * *[W ]) if every non-empty set of subgroups of G, partially ordered by inclusion, has a mi* *nimal element. Equivalently, G is artinian if its subgroups satisfy the descending ch* *ain con- dition. The class of artinian groups is closed under taking subgroups, quotient* *s, and extensions. Every artinian group is a torsion group (since an infinite cyclic * *group is not artinian). If G is artinian and ' 2 Inj(G, G) is an injective endomorphism * *of G, then ' is an automorphism, since otherwise {'n(G)} would be an infinite descend* *ing chain. This is just one example of why it will be important that the groups we * *work with are artinian; the descending chain condition will be used in other ways la* *ter. It is an open question whether every artinian group is locally finite (see [K* *W , pp. 31-32] for a discussion of this). If one restricts attention to groups all of w* *hose elements have p-power order for some fixed prime p, then artinian groups are known to be* * locally finite if p = 2 [KW , Theorem 1.F.6], but this seems to be unknown for odd pri* *mes. However, any counterexample to these questions would probably be far too wild f* *or our purposes. Hence it is natural to restrict attention to locally finite group* *s, and since we are working with local structure at a prime p, to locally finite p-groups. T* *he next proposition tells us that in fact, this restricts us to the class of discrete p* *-toral groups. It is included only as a way to help motivate this choice of groups to work wit* *h. Proposition 1.2. A group is a discrete p-toral group if and only if it is artin* *ian and a locally finite p-group. Proof.The group Z=p1 is clearly a locally finite p-group and artinian. Since b* *oth of these properties are preserved under extensions of groups, they are satisfied b* *y every discrete p-toral group. Conversely, assume that G is artinian and a locally finite p-group. By [KW ,* * Theorem 5.8], every locally finite artinian group is a ~Cernikov group; in particular, * *it contains a normal abelian subgroup with finite index. By [Fu , Theorems 25.1 & 3.1], every* * abelian artinian group is a finite product of groups of the form Z=qm where q is a prim* *e and Carles Broto, Ran Levi, and Bob Oliver * * 5 m 1. Thus G is an extension of the form 1 ---! A -----! G -----! ss ---! 1, where ss is a finite p-group, and A is a finite product of groups Z=pm for m * *1. The subgroup of A generated by the factors Z=p1 is the subgroup of infinitely p-di* *visible elements, thus a characteristic subgroup of A, and a normal subgroup of G of p-* *power index. It follows that G is a discrete p-toral group. We next note some of the other properties which make discrete p-toral groups * *con- venient to work with. Lemma 1.3. Any subgroup or quotient group of a discrete p-toral group is a disc* *rete p-toral group. Any extension of one discrete p-toral group by another is a dis* *crete p-toral group. Proof.These statements are easily checked directly. They also follow at once f* *rom Proposition 1.2, since the classes of locally finite p-groups and artinian grou* *ps are both closed under these operations. Clearly, the main difficulty when working with infinite discrete p-toral grou* *ps, in- stead of finite p-groups, is that they have infinitely many subgroups and infin* *ite au- tomorphism groups. We next investigate what finiteness properties these groups* * do have. Lemma 1.4. The following hold for each discrete p-toral group P . (a)For each n 0, P contains finitely many conjugacy classes of subgroups of o* *rder pn. (b)P contains finitely many conjugacy classes of elementary abelian p-subgroups. Proof.Clearly, for each n, P0 contains finitely many subgroups of order pn, sin* *ce they are all contained inside the pn-torsion subgroup of P0 which is finite. So to p* *rove (a), it suffices, for each finite subgroup A P0 and each subgroup B = eB=P0 P=P0* *, to show that there are finitely many P -conjugacy classes of subgroups Q P such * *that Q \ P0 = A and QP0 = eB. Let Q be the set of all such subgroups, and assume Q 6* *= ?. Then Q 2 Q if and only if Q=A \ P0=A = 1 and QP0=P0 = B; and this implies that A C QP0 = eBand that Q=A is the image of a splitting of the extension 1 ___! P0=A ___! eB=A ___! B ___! 1. In other words, Q is in one-to-one correspondence with the set of splittings of* * this extension. The set of P0-conjugacy classes of such splittings (if there are an* *y) is in one-to-one correspondence with the elements of H1(B; P0=A) (see [Bw , Propositi* *on IV.2.3]). Since this cohomology group is finite, so is the set of conjugacy cl* *asses of such extensions. This proves point (a). Point (b) follows from (a), together with the observat* *ion that for any elementary abelian subgroup E P , rk(E) rk(P ) + rkp(P=P0). We next check what can be said about finiteness in automorphism groups. Proposition 1.5. Let P be a discrete p-toral group. (a)Any torsion subgroup of Aut (P ) is an extension of an abelian group by a fi* *nite group. 6 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups (b)Any torsion subgroup of Out(P ) is finite. (c)For each Q P , OutP (Q) is a finite p-group. Proof.Assume first that P ~=(Z=p1 )r: a discrete p-torus of rank r 0. Then Aut(P ) ~=GLr(bZp), and it is well known that the subgroup (1 + p2Mr(bZp))x of * *ma- trices which are congruent modulo p2 to the identity is torsion free. This foll* *ows, for example, from the inverse bijections log (1 + p2Mr(bZp))x ------!------p2Mr(bZp) exp defined by the usual power series: while logis not a homomorphism, it does sati* *sfy the relation log(Xr) = r log(X). So if H is a torsion subgroup of Aut(P ) (equivale* *ntly, of GLr(bZp)), then the composite =p2 2 H -----! GLr(bZp) -----! GLr(Z=p ) is injective, and thus H is finite. Now let P be an arbitrary discrete p-toral group with connected component P0 * *and group of components ss = P=P0. There is an exact sequence 0 ----! H1(ss; P0) ----! Aut(P )= AutP0(P ) ----! Aut(P0) x Aut(ss) (cf. [Sz, 2.8.7]), where Aut P0(P ) = {cx 2 Aut(P ) | x 2 P0}. We have just s* *een that every torsion subgroup of Aut(P0) is finite, and H1(ss; P0) and Aut(ss) are cle* *arly finite. Hence every torsion subgroup of Aut(P )= InnP0(P ) is finite. This proves (b); * *and also proves (a) (every torsion subgroup of Aut(P ) is an extension of an abelian gro* *up by a finite group) since AutP0(P ) is abelian. Point (c) follows immediately from (b* *), since P is a torsion group all of whose elements have p-power order. In the next section (in Definition 2.2), we will need some more precise bound* *s on the size of normalizers and centralizers. Lemma 1.6. Let S be any discrete p-toral group, and set N = |ss0(S)|rk(S)+1. T* *hen for all P S, |ss0(CS(P ))| N, |ss0(NS(P )=P )| N, and |ss0(NS(P ))| N.|ss* *0(P )|. Proof.Set T = S0 for short,Qand set Q = P T=T . Let NQ :T ___! T be the norm m* *ap for the Q action: NQ(x) = gT2Qgxg-1. The image of NQ is connected and central* *izes P , and thus Im(NQ) CS(P )0 = CT(P )0. If x 2 CT(P ), then x|Q|= NQ(x) 2 CT(P )0. Thus every element in CT(P )=CT(P )0 has order dividing |Q|, and it follows that |ss0(CT(P ))| = |CT(P )=CT(P )0| |Q|rk(S) |ss0(S)|rk(S). Thus |ss0(CS(P ))| |ss0(CT(P ))|.|S=T | N. If x 2 NT(P ), then Y x|Q|= NQ(x). [x, g] 2 CT(P )0.P NT(P )0.P. gT2Q Thus fi ffi fi fiNT(P ) NT(P )0.(T \ P ) fi |Q|rk(S) |ss0(S)|rk(S), and hence |ss0(NS(P )=P )| N, by the same arguments as those used for ss0(CS(* *P )). The last inequality is now immediate. Carles Broto, Ran Levi, and Bob Oliver * * 7 Note that discrete p-toral groups are all solvable, but (in contrast to finit* *e p-groups) need not be nilpotent. For instance, the infinite dihedral group, a split exte* *nsion of Z=21 by Z=2, is a discrete 2-toral group which is not nilpotent (since the nil* *potency class of D2n is n - 1). The following lemma contains some generalizations of a standard theorem about automorphisms of finite p-groups: if ff 2 Aut (P ) is the identity on Q C P an* *d on P=Q, then it has p-power order. Lemma 1.7. The following hold for any discrete p-toral group P and any automor- phism ff 2 Aut(P ). (a)Assume, for some Q C P , that ff|Q = IdQ and ff Id (mod Q). Then every ff-orbit in P is finite of p-power order. If, in addition, [P : Q] < 1, then* * ff has finite order. (b)ff has finite order if and only if ff|P0 has finite order. (c)Set P(1)= {g 2 P0| gp = 1}. If ff|P(1)= Id and ff Id (mod P0), then each o* *rbit of ff acting on P has p-power order. Proof.(a) The proof is identical to the proof for finite p-groups (see [Go , T* *heorem 5.3.2]), and in fact applies whenever all elements of Q have p-power order. Fo* *r any g 2 P , ff(g) = gx for some x 2 Q (since ff Id(mod Q)), and ff(x) = x since f* *f|Q = Id. k k * * i Thus ffn(g) = gxn for all n, and ffp (g) = g if p = |x|. Since the order of {* *ff (g)} depends only on the coset gQ, this also shows that |ff| is finite (and a power * *of p) if P=Q is finite. (b) If ff|P0 has finite order, then there is n 1 such that ffn|P0 = Id and f* *fn Id (mod P0). Then ffn has finite order by (a), so ff also has finite order. (c) For each m 1, let P(m) P0 be the pm -torsion in P0. Fix g 2 P , and s* *et x = g-1ff(g), pk = |x|, and Q = . The P(m)are all ff-invariant, and so* * Q is also ff-invariant since g-1ff(g) 2 P(k). Also, ff acts via the identity on P(1)by as* *sumption, hence on P(i)=P(i-1)for all 1 i k, and also on Q=P(k). So by (a) (and sinc* *e Q is a finite group), ff|Q has p-power order. In particular, the ff-orbit of g ha* *s p-power order. The next lemma is another easy generalization of a standard result about fini* *te p-groups. Lemma 1.8. If P Q are distinct discrete p-toral groups, then P NQ(P ). Proof.When [Q:P ] < 1, this follows by the same proof as for finite p-groups. M* *ore precisely, when Q=P is finite, the action of P on Q=P (defined by x(gP ) = xgP * *for x 2 P and g 2 Q) factors through a finite quotient group P=N of P . Also, P=N i* *s a p-group since P is a p-torsion group. Thus |NQ(P )=P | = |(Q=P )P=N| |Q=P | 0 (mod p), and so NQ(P )=P 6= 1. Now assumenthat [Q:P ] is infinite; i.e., that P0 Q0. For each n, set An = * *{x 2 Q0| xp = 1}. Then An C Q, and in particular is normalized by P . For n large enough, An P , so P P An Q, P NPAn(P ) since [P An:P ] < 1, and thus P NQ(P ). 8 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups We will also need the following well known result about finite subgroups of d* *iscrete p-toral groups. Lemma 1.9. For any discrete p-toral group P , there is a finite subgroup Q P such that P = QP0. There is also anSincreasing sequence Q1 Q2 Q3 . . .of finite subgroups of P such that P = 1n=1Qn. More generally, for any finite su* *bgroup K Aut(P ), the Qi can be chosen to be K-invariant. Proof.Fix any (finite) set X of coset representatives for P0 in P , and set Q =* * . Then Q is K-invariant, Q is finite since P is locally finite, and P * *= QP0 by construction. For each n 1, let Pn P0 be the pn-torsionSsubgroup, and s* *et Qn = QPn. Then the Qn are also finite and K-invariant, and P = 1n=1Qn. To finish the section, we consider maps between the p-completed classifying s* *paces of discrete p-toral groups. This following lemma is implicit in [DW ] and [DW2* * ] (the spaces in question are classifying spaces of p-compact groups). But it does not* * seem to be stated explicitly anywhere there. Lemma 1.10. For any pair P, Q of discrete p-toral groups, B :Rep (P, Q) -------! [BP ^p, BQ^p] is a bijection. In particular, any homotopy equivalence BP ^p-'-!BQ^pis induced* * by an isomorphism P ~= Q. Also, for any homomorphism ae: P --! Q, the homomorphism (incl,ae) CQ(ae(P )) x P ----! Q induces a homotopy equivalence BCQ(ae(P ))^p---'---!Map (BP ^p, BQ^p)Bae. Proof.For any pair G, H of discrete groups, [BG, BH] ~=Rep (G, H) and Map (BG, BH)Bae' BCH (ae(G)) for each ae 2 Hom (G, H). See, for example, [BKi , Proposition 7.1] for a proof. By [DW2 , Proposition 3.1], the homotopy fiber of the map BQ --! BQ^pis a K(* *V, 1) for some bQp-vector space V . Using this, together with standard obstruction th* *eory and the fact that eH*(BQ; Q) = 0, one checks that [BP ^p, BQ^p] ~=[BP, BQ] ~=Rep (P, Q). Now fix some ae 2 Hom (P, Q). By [DW , Propositions 5.1 & 6.22], Map (BP ^p,* * BQ^p)Bae is the classifying space of some p-compact group X, and in particular is p-comp* *lete. Since Map (BP, BQ)Bae' CQ(ae(P )) (P and Q are both discrete), we will be done * *upon showing that the completion map Map (BP, BQ)Bae-----! Map (BP, BQ^p)Bae (1) is a mod p homology equivalence. Fix a sequence of finite subgroups P1 P2 . . .whose union is P . Since Q* * is artinian, CQ(ae(Pn)) = CQ(ae(P )) for n sufficiently large. Also, Map (BP, BQ^p* *)Baeis the homotopy inverse limit of the mapping spaces Map (BPn, BQ^p)Bae. So if (1) is a* * mod p equivalence upon replacing P by Pn for each n, it is also a mod p equivalence* * for P . In other words, it suffices to prove this when P is a finite p-group. Let X be the homotopy fiber of the completion map BQ ---! BQ^p. As noted abov* *e, X is a K(V, 1) where V is a rational vector space. Since the map from Map (BP, * *BQ) Carles Broto, Ran Levi, and Bob Oliver * * 9 to Map (BP, BQ^p) is a bijection on components, the homotopy fiber of the map i* *n (1) is XhP for a proxy action of P on X (in the sense of [DW ]) induced by ae. Consider the fibration sequence prO- XhP -----! Map (BP, XhP){1}-----! Map (BP, BP )Id, where prdenotes the projection of XhP to BP , and the total space is the set of* * all maps f :BP ---! XhP such that prOf ' Id. Since XhP is the total space of a fibratio* *n over BP with fiber X, it is a K(ss, 1) where V C ss and ss=V ~=P . Since P is a fini* *te p-group and V is a rational vector space, this extension splits, and the splitting is u* *nique up to conjugacy by elements of V . It follows that [BP, XhP]-----! [BP, BP ] ~=Rep(P,ss) ~=Rep(P,P) is a bijection. Also, the induced map ss1(Map (BP, XhP){1})-----!ss1(Map (BP, BP )Id) ~=Css(P) ~=Z(P) is surjective, and its kernel V P (where the action of P on V is induced by the* * action on X) is a rational vector space. Thus XhP ' K(V P, 1). It follows that XhP is mod p acyclic, and hence that (1* *) is a mod p equivalence. This finishes the proof. 2.Fusion systems over discrete p-toral groups We now define saturated fusion systems over dicrete p-toral groups and study * *their basic properties. The definitions are almost identical to those in the finite c* *ase ([BLO2 , x1]). Definition 2.1. A fusion system F over a discrete p-toral group S is a category* * whose objects are the subgroups of S, and whose morphism sets Hom F (P, Q) satisfy th* *e fol- lowing conditions: (a)Hom S(P, Q) Hom F(P, Q) Inj(P, Q) for all P, Q S. (b)Every morphism in F factors as an isomorphism in F followed by an inclusion. Two subgroups P, P 0 S are called F-conjugate if IsoF(P, P 0) 6= ?. Definition 2.2. Let F be a fusion system over a discrete p-toral group S. o A subgroup P S is fully centralized in F if |CS(P )| |CS(P 0)| for all P * *0 S which is F-conjugate to P . o A subgroup P S is fully normalized in F if |NS(P )| |NS(P 0)| for all P 0* * S which is F-conjugate to P . o F is a saturated fusion system if the following three conditions hold: (I)For each P S which is fully normalized in F, P is fully centralized in * *F, Out F(P ) is finite, and OutS (P ) 2 Sylp(Out F(P )). 10 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups (II)If P S and ' 2 Hom F(P, S) are such that '(P ) is fully centralized, an* *d if we set N' = {g 2 NS(P ) | 'cg'-1 2 AutS('(P ))}, _ _ then there is ' 2 Hom F(N', S) such that '|P = '. (III)If P1S P2 P3 . . .is an increasing sequence of subgroups of S, with P1 = 1n=1Pn, and if ' 2 Hom (P1 , S) is any homomorphism such that '|Pn 2 Hom F(Pn, S) for all n, then ' 2 Hom F(P1 , S). By Lemma 1.6, there is a global upper bound for |ss0(CS(P ))| and |ss0(NS(P )* *)|, taken over all subgroups P of any given S. In particular, for any given subgroup P * * S, |CS(P 0)| and |NS(P 0)| take on maximal values among all P 0which are F-conjuga* *te to P . This proves that the conjugacy class of P always contains fully centra* *lized subgroups and fully normalized subgroups. It is very convenient, in the above definition, to be working with a class of* * groups where the concept of "order" of subgroups is defined. However, there are other * *ways to define fully normalized and fully centralized subgroups in a fusion system, and* * hence to define saturation; and this property was not a factor in our decision to res* *trict attention to fusion systems over discrete p-toral groups. The crucial propertie* *s of these groups, which seem to be needed frequently when developing the theory, are that* * they are artinian and locally finite. When F is a saturated fusion system over the discrete p-toral subgroup S, the* *n by (I), OutF (P ) = AutF (P )= Inn(P ) is finite for fully normalized P S, and h* *ence for all P S. Since Inn(P ) is discrete p-toral (being a quotient group of P ), AutF (* *P ) inherits many of the properties of discrete p-toral groups. In particular, it is artinia* *n, locally finite, and contains a unique conjugacy class of maximal discrete p-toral subgr* *oups. This condition that OutF (P ) be finite does simplify slightly the definition o* *f a saturated fusion system, but it is in fact unnecessary, as is shown by the following prop* *osition. Proposition 2.3. Let F be a fusion system over the discrete p-toral group S. As* *sume that axiom (II) in Definition 2.2 holds, and that (I) holds for all finite full* *y normalized subgroups of S. Then OutF (P ) is finite for all P S. m Proof.Fix P S. For all m 1, set P(m) = {g 2 P0| gp = 1}. By Proposition 1.5(b), to show that Out F(P ) is finite, it suffices to show that Aut F(P ) is* * a torsion group. Fix ff 2 Aut F(P ). We want to show that ff has finite order; by Lemma 1.7(b* *), it suffices to do this when P = P0 is connected. After replacing ff by ffn for* * some appropriate n 1, we can assume that ff|P(1)= Id. Then by Lemma 1.7(c), ffm d* *ef= ff|P(m)has p-power order for all m. For each m, there is 'm 2 Hom F (P(m), S)* * such that 'm (P(m)) is fully normalized, and by (I), 'm (P(m)) is fully centralized,* * and 'm can_be chosen such that 'm ffm '-1m2 AutS ('m (P(m))). Also, 'm can be extended* * to 'm 2 Hom F(S0, S) by (II), so 'm (P(m)) S0, and hence | AutS('m (P(m)))| |S* *=S0|. Thus (ffm )|S=S0|= IdP(m)for each m, so ff|S=S0|= IdP, and ff has finite order. In fact, one can show that in the definition of a saturated fusion system, it* * suffices to require that (I) holds for all finite fully normalized subgroups P S; it t* *hen follows that (I) holds for all fully normalized subgroups. When F is a (saturated) fusion system over a discrete p-toral group S, we thi* *nk of the identity component S0 as the "maximal torus" of the fusion system, and thin* *k of Carles Broto, Ran Levi, and Bob Oliver * * 11 AutF (S0) as its "Weyl group". The following lemma describes how morphisms betw* *een subgroups of the maximal torus are controlled by the Weyl group. Lemma 2.4. Let F be a saturated fusion system over a discrete p-toral group S w* *ith connected component T = S0. Then the following hold for all P T . (a)For every P 0 S which is F-conjugate to P and fully centralized in F, P 0 * *T , and there exists some w 2 AutF (T ) such that w|P 2 IsoF(P, P 0). (b)Every ' 2 Hom F(P, T ) is the restriction of some w 2 AutF (T ). Proof.We first prove the following statement. (c)For each ' 2 Hom F (P, S) such that P 0def='(P ) is fully centralized in F, * *there exists w 2 AutF (T ) such that w|P = '. _ By assumption, P T CS(P_). By condition_(II) in Definition 2.2, there is '* * 2 Hom F(CS(P ), S) such that '|P = '. Then '(T ) T since T is connected (infini* *tely _ def_ p-divisible), and so '(T ) = T since T is artinian. Thus w = '|T 2 AutF (T ) i* *s such that w|P = '. This proves (c), and also proves (a) since P 0= w(P ) T . Now fix any ' 2 Hom F (P, T ). Let Q be a fully centralized subgroup of S in* * the F-conjugacy class of P and '(P ), and choose _ 2 IsoF('(P ), Q). By (c), there* * are elements u, v 2 AutF (T ) such that u|P = _ O' and v|'(P)= _. So if we set w = * *v-1u, then w|P = '. By Proposition 2.3, Out F(P ) is finite for every subgroup P S. The follo* *wing lemma extends this statement. Lemma 2.5. Let F be a saturated fusion system over a discrete p-toral group S. * *Then for all P, Q S, the set Rep F(P, Q) def=Inn(Q)\ Hom F(P, Q) is finite. Proof.As just noted, OutF (P ) is finite for all P S. Also, if ', '02 Hom F(P* *, Q) and Im(') = Im('0), then '0= ' Off for some ff 2 AutF (P ) by condition (b) in Defi* *nition 2.1. So there is a bijection ~= 0 fi 0 ffi Rep F(P, Q)= OutF(P ) -----! P Q fiP F-conjugate to P (Q-conjugacy),(1) which sends the class of a homomorphism to the conjugacy class of its image. By Lemma 2.4, the F-conjugacy class (P0) of P0 is just its orbit under the ac* *tion of Aut F(S0), and hence a finite set. By Lemma 1.4(a), for any given Q 2 (P0), there are only finitely many NS(Q)=Q-conjugacy classes of subgroups of order |P* *=P0| in NS(Q)=Q. Hence there are only finitely many S-conjugacy classes of subgroups P 0 S which are F-conjugate to P and such that P00= Q. This shows that the tar* *get set in (1) is finite, and hence that RepF (P, Q) is also finite. The definitions of centric and radical subgroups in a fusion system over a di* *screte p-toral group are essentially the same as those in the finite case. Definition 2.6. Let F be a fusion system over a discrete p-toral group S. A sub* *group P S is called F-centric if P and all its F-conjugates contain their S-central* *izers. A subgroup P S is called F-radical if Op(Out F(P )) = 1; i.e., if Out F(P ) con* *tains no nontrivial normal p-subgroup. 12 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups Notice that any F-centric subgroup is fully centralized. Conversely, if P S* * is fully centralized and centric in S; that is, Z(P ) = CS(P ), then it is F-centric. T* *he next proposition says that the set of F-centric subgroups is closed under overgroups. Proposition 2.7. Let F be a saturated fusion system over the discrete p-toral g* *roup S, and let P Q S be such that P is F-centric. Then Q is also F-centric. Proof.Fix any Q0 which is F-conjugate to Q, choose ' 2 IsoF(Q, Q0), and set P 0= '(P ). Then CS(Q0) CS(P 0) P 0 Q0, where the second inequality holds since P is F-centric. So Q is also F-centric. The next proposition gives another important property of F-centric subgroups;* * one which is much less obvious. Proposition 2.8. Let F be a saturated fusion system over the discrete p-toral g* *roup S. Then for each P Q S such that P is F-centric, and each ', '02 Hom F(Q, S) such that '|P = '0|P, there is some g 2 Z(P ) such that ' = '0Ocg. Proof.The hypothesis implies that ' O'0-1|'0(P)= Id'0(P), and we must show that ' O'0-1= Id'0(Q). It thus suffices to prove, for P Q S and ' 2 Hom F(Q, S) * *where P is F-centric, that '|P = IdP implies ' = cg for some g 2 Z(P ). Assume first that P C Q. Then for each x 2 Q, c'(x)|P = cx|P. Thus '(x) x (* *mod CS(P )), and CS(P ) P since P is F-centric. In particular, this shows that '(* *Q) = Q, and thus that ' 2 AutF (Q). It also shows that ' induces the identity on Q=P . * *Since Q=P has finite order, ' has p-power order by Lemma 1.7(a). Without loss of generality, we can replace Q by any other subgroup in its F- conjugacy class. In particular, we can assume that Q is fully normalized, and * *hence that Out S(Q) 2 Sylp(Out F(Q)). So every p-subgroup of Aut F(Q) is conjugate t* *o a subgroup of AutS(Q). Thus there is O 2 AutF (Q) such that O O' OO-1 = cy for so* *me y 2 NS(Q). Since '|P = IdP, cy acts as the identity on '(P ), which is also F-c* *entric, hence y 2 CS('(P )) = '(Z(P )). Set x = O-1(y); then ' = cx. Now assume P is not normal in Q. Let Q be the set of subgroups Q0 Q containi* *ng P such that '|Q0 = cg|Q0 for some g 2 Z(P ). If P Q0 Q and Q0 2 Q, then NQ(Q0) Q0 by Lemma 1.8, and NQ(Q0) 2 Q since the proposition holds for the normal pair Q0C NQ(Q0). Hence if Q contains a maximal element, it must be Q its* *elf. S 1 Let Q1 Q2 . . .be any increasing chain in Q, and set Q1 = n=1Qn. Let gn 2 Z(P ) be such that '|Qn = cgn|Qn. Since P is F-centric, so are the Qn, and* * thus Z(Q1) Z(Q2) . .i.s a decreasing sequence of subgroups. Since S is artinian,* * there is some k such that Z(Qn) = Z(Qk) for all n k. This shows that gn gk (mod Z(Qk)) for all n k, hence that '|Q1 = cgk|Q1 , and hence that Q1 2 Q. Thus by Zorn's lemma, Q contains a maximal element, so Q 2 Q, and this finishes the proof. 3.A finite retract of a saturated fusion system A fusion system F over a discrete p-toral group S generally has infinitely ma* *ny isomorphism classes of objects. In this section, we construct a subcategory Fo* * of F with only finitely many isomorphism classes of objects, together with a retra* *ction Carles Broto, Ran Levi, and Bob Oliver * * 13 functor from F to Fo which is a left adjoint to the inclusion. This means that* * in many cases, it will suffice to work over the "finite" subcategory Fo rather tha* *n the full fusion system F. As a first application, we show that Ob (Fo) contains all F-ce* *ntric F-radical subgroups, and hence that there are only finitely many conjugacy clas* *ses of such subgroups. A second application is Alperin's fusion theorem in this se* *tting: restriction to Fo allows us to repeat the same inductive argument as that used * *for fusion systems over a finite p-group. Following the group theorists' usual notation, whenever is a group of autom* *or- phisms of a group G and H G, we write C (H) = {fl 2 | fl|H = IdH}. The following definitions were motivated by some constructions of Benson [Be ],* * which he in fact used to prove a version of Alperin's fusion theorem for compact Lie * *groups. Definition 3.1. Let F be a saturated fusion system over a discrete p-toral grou* *p S, let T = S0 be the identity component of S, and set W = AutF (T ) = OutF (T ) (the "* *Weyl group"). Set k pm = exp(S=T ) def=min{pk| xp 2 T 8 x 2 S}. (a)For each P T , set fi I(P ) = T CW (P)= t 2 T fiw(t) = t 8 w 2 W such thatw|P = IdP ; and let I(P )0 be the identity component of I(P ). m (b)For each P S, let P [m]= T , and set P o= P .I(P [m])0 def={gt | g 2 P, t 2 I(P [m])0}. (c)Set H(F) = {I(P )0| P T }, and Ho(F) = {P o| P S}; and let Fo F be the full subcategory with object set Ho(F). Thus for P T , I(P ) is the maximal subgroup of T such that for all w 2 W ,* * w|P = Idif and only w|I(P)= Id. In particular, for all v and w in W , v|P = w|P if an* *d only if v|I(P)= w|I(P). Together with Lemma 2.4(b), this implies that every ' 2 Hom F(P* *, T ) extends to a unique I(') 2 Hom F(I(P ), T ), which is obtained by first extendi* *ng ' to T and then restricting to I(P ). In other words, every F-isomorphism ': P ___!* * Q between subgroups of T extends to a unique F-isomorphism I('): I(P ) ! I(Q). For an arbitrary subgroup P S, P [m]is a subgroup of T , and the above argu* *ments apply. Since P [m]C P , any x 2 P normalizes P [m], and hence also normalizes I* *(P [m]). Thus P normalizes I(P [m])0, and this shows that the subset P odef=P .I(P [m])0* * is a group. More generally, for any k m, we could define subgroups P ok P for each P * * S by setting P ok= P .I(P [k]). This can be different from P o, but P 7! P okhas* * all of the same properties which we prove here for P o. However, the only way in which this generalization might be needed would be if we wanted to compare these "bul* *let functors" for two different fusion systems over two different discrete p-toral * *groups, and that will not be needed in this paper. Lemma 3.2. The following hold for every saturated fusion system F over a discre* *te p-toral group S. 14 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups (a)The set H(F) is finite, and the set Ho(F) contains finitely many S-conjugacy classes of subgroups of S. (b)For all P S, (P o)o = P o. (c)If P Q S, then P o Qo. (d)If P S is F-centric, then Z(P o) = Z(P ). Proof.Let T = S0 C S be the identity component, and set W = AutF (T ) and pm = exp(S=T ). Note that for any P Q T , CW (P ) CW (Q), and hence I(P ) I(* *Q). Also, CW (I(P )) = CW (P ) by definition, and hence I(I(P )) = I(P ). (a) By definition, each subgroup in H(F) has the form I(P )0 = (T K)0 for some P T , where K = CW (P ) W . Since the finite group W = Out F(T ) has a finite number of subgroups, this shows that H(F) is finite. Also, for any P * * S, P0 P [m] I(P [m]), and so (P o)0 = I(P [m])0 2 H(F). In particular, there ar* *e only finitely many possibilities for identity components of subgroups in Ho(F). Fix P S, and set K = CW (P [m]). Since P [m]is generated by all pm -powers * *in P (and pm = exp(S=T )), P [m] T and [P :P [m]] = [P :(P \T )].[(P \T ):P [m]] |S=T |.pm.rk(T). Here, the last inequality holds since (P \ T )=P [m]is abelian with exponent at* * most pm and rank at most rk(T ). Also, since P [m].I(P [m])0 = P [m].(T K)0 T K, |ss0(P o)| = |ss0(P .(T K)0)| |ss0(P [m].(T K)0)|.|P=P [m]| |ss0(T K)|.|P=P [m]| |ss0(T K)|.|S=T |.pm.* *rk(T). We have already seen that (T K)0 is the identity component of P o, and we have * *just shown that the number of components of P ois bounded by an integer which depends only on K (and on S). Since NS((T K)0)=(T K)0 has only finitely many conjugacy classes of finite subgroups of any given order (Lemma 1.4(a)), this shows that * *there are only finitely many conjugacy classes of subgroups in Ho(F) corresponding to any* * given K W ; and thus (since W is finite) only finitely many conjugacy classes of su* *bgroups in Ho(F). (b) Fix P S. Since P normalizes I(P [m])0, for any g 2 P and any x 2 I(P [m]* *)0, m pm [m] [m] [m] (gx)p 2 g .I(P )0 P .I(P )0. This proves the second inequality on the following line: P [m] (P o)[m] P [m].I(P [m])0 I(P [m]), and the others are clear. Since I(-) is idempotent and preserves order, this sh* *ows that I((P o)[m]) = I(P [m]). Hence (P o)o = P o.I(P [m])0 = P o. (c) If P Q, then P [m] Q[m], so I(P [m]) I(Q[m]), and hence P o Qo. (d) For any P S, P P o. Thus if P is F-centric, then so is P o, and Z(P o) * * Z(P ). To see that this is an equality, it suffices to show that every element in Z(P * *) commutes with I(P [m]). For all x 2 Z(P ), cx (as an element of W = AutF (T )) lies in C* *W (P [m]), [m]) hence commutes with all elements of I(P [m]) = T CW (P , and in particular with* * all elements of I(P [m])0. We are now ready to prove the main, crucial, property of these subgroups P o. Carles Broto, Ran Levi, and Bob Oliver * * 15 Proposition 3.3. Let F be a saturated fusion system over a discrete p-toral gro* *up S. Fix P, Q S and ' 2 Hom F(P, Q). Then ' extends to a unique homomorphism 'o 2 Hom F(P o, Qo); and this makes P 7! P ointo a functor from F to itself. Proof.The functoriality of P 7! P oand ' 7! 'o (i.e., the fact that (IdP)o = Id* *Poand (_ O')o = _o O'o) follows immediately from the existence and uniqueness of these extensions. So this is what we need to prove. As usual, we set T = S0 and W = AutF (T ). For all Q T , CW (Q) = CW (I(Q))* * by definition of I(-). This will be used frequently throughout the proof. We first check that there is at most one morphism 'o which extends '. Assume * *that _, _02 Hom F(P o, Qo) are two such extensions. By Lemma 2.4(b), there are eleme* *nts w, w02 W such that _|P[m].I(P[m])0= w|P[m].I(P[m])0and _0|P[m].I(P[m])0= w0|P[m* *].I(P[m])0. Since w|P[m]= w0|P[m], w-1w0 2 CW (P [m]) = CW (I(P [m])), so w|I(P[m])= w0|I(P* *[m])as well. It follows that _ = _0, since they take the same values on P and on I(P [* *m])0. It remains to prove the existence of 'o. By Lemma 3.2(c), it suffices to prov* *e this when ' 2 IsoF(P, Q). Recall that P o= P .I(P [m])0. Fix u 2 W = AutF (T ) such * *that u|P[m]= '|P[m]. Define 'o by setting, for all g 2 P and all x 2 I(P [m])0, 'o(gx) = '(g)u(x). After two preliminary steps, we show in Step 3 that 'o is well defined and a ho* *momor- phism, and in Step 4 that it is a morphism in F. Step 1: Fix A, A0 T , and w 2 W such that w(A) = A0. We show here that A B I(A), _ 2 Hom F(B, T ), _|A = w|A =) _ = w|B ; (1) and also that A B A.I(A)0, _ 2 Hom F(B, S), _|A = w|A =) _(B) T and _ = w|B . (2) If _(B) T , then _ = w0|B for some w0 2 W by Lemma 2.4(b), w-1w0 2 CW (A) = CW (I(A)), and thus _ = w0|B = w|B . This proves (1). Now assume B A.I(A)0. By Lemma 2.4(a), there is w0 2 W such that w0(B) is fully centralized in F. It thus suffices to prove (2) when B is fully centrali* *zed. Set B0= _(B) for short. Now, B0 A0and B0 is abelian. So for all x 2 B0, if we regard cx as an elemen* *t of W = AutF (T ), then cx 2 CW (A0) = CW (I(A0)). Thus I(A0) = w(I(A)) CS(B0). By axiom (II) (and since B = _-1(B0) is fully centralized), _-1 extends to an F-mo* *rphism defined on B0.CS(B0), and in particular to a morphism fi 2 Hom F(B0.I(A0), S). * *Since fi|A0= w-1|A0 and fi(I(A0)0) T , fi|A0.I(A0)0= w-1|A0.I(A0)0by (1). Thus for all x 2 B0, fi(x) = _-1(x) 2 B A.I(A)0 = fi(A0.I(A0)0). Since fi * *is injective, this shows that x 2 A0.I(A0)0 T . So B0 T , and (2) now follows f* *rom (1). Step 2: We next show that for all x 2 I(P [m]) and all g 2 P , the following i* *dentity holds: u(gxg-1) = '(g)u(x)'(g)-1; (3) or equivalently that c-1'(g)OuOcg(x) = u(x). Set w = c-1'(g)OuOcg 2 W for short* *. Then (3) holds for x 2 P [m]since '|P[m]= u|P[m], and thus w|P[m]= u|P[m]. So w|I(P[m])=* * u|I(P[m]) by (1), and this proves (3) for all x 2 I(P [m]). 16 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups Step 3: Recall that we defined 'o(gx) = '(g)u(x) for all g 2 P and x 2 I(P [m* *])0. By assumption, '|P[m]= u|P[m]. Hence the restrictions of ' and u to P [m].(P \ * *I(P0[m])) are equal by (2), and this shows that 'o is well defined. For all g, g02 P and all x, x02 I(P [m])0, 0 -1 0 0 'o((gx)(g0x0))= '(gg0).u(g0-1xg0x0) = '(gg0). '(g ) u(x)'(g ) .u(x ) = '(g)u(x)'(g0)u(x0) = 'o(gx).'o(g0x0), where the second equality follows from Step 2. Thus 'o is a homomorphism. Step 4: It remains to show that 'o 2 IsoF(P o, Qo); i.e., that 'o is a morphi* *sm in the category F. By condition (III) in Definition 2.2, together with Zorn's l* *emma, there is a maximal subgroup P 0 P ocontaining P such that 'o|P0 2 Hom F(P 0, Q* *o). Assume P 0 P o; and set '0 = 'o|P0 and P 00= NPo(P 0) P 0. By condition (II* *) in Definition 2.2, '0 extends to some morphism _ 2 Hom F (P 00, S) (the existence * *of the homomorphism 'o shows that N'0 P 00). By (2) again, the restrictions of _, u, * *and 'o to P 00\ (P [m].I(P [m])0) are equal. Since P 00= P .(P 00\ I(P [m])0), this* * shows that _ = 'o|P00. This contradicts the maximality assumption about P 0; so P 0= P o, * *and we are done. (-)o o Note in particular that by Lemma 3.2(c), the functor F ---! F of Proposition* * 3.3 sends inclusions of subgroups to inclusions. Corollary 3.4. The functor (-)o is a left adjoint to the inclusion of Fo as a f* *ull subcategory of F. Proof.Fix any P in F and any Q in Fo. Since Q = Qo by Lemma 3.2(b), every ' 2 Hom F (P, Q) extends to a unique 'o 2 Hom F (P o, Q) by Proposition 3.3. T* *he restriction map Hom F(P o, Q) --Res----!HomF(P, Q) is thus a bijection, and this proves adjointness. Corollary 3.4 will later be extended to orbit and linking categories associat* *ed to F and Fo. Corollary 3.5. Let F be a saturated fusion system over a discrete p-toral group* * S. Then all F-centric F-radical subgroups of S are in Ho(F), and in particular the* *re are only finitely many conjugacy classes of such subgroups. Proof.Assume P is F-centric and F-radical. We claim that I(P [m])0 P , and th* *us that P = P o2 Ho(F). Assume otherwise. Then P o P , and hence NPo(P ) P by Lemma 1.8. Thus NPo(P )=P 6= 1, and since P is F-centric, this group can be identified with a p* *-subgroup of Out F(P ). By Proposition 3.3, any ff 2 Aut F(P ) extends to an automorphis* *m of P o, and in particular to an automorphism of NPo(P ). This shows that NPo(P )=P* * C OutF (P ), which contradicts the assumption that P is F-radical. The last statement now follows since Ho(F) contains only finitely many conjug* *acy classes by Lemma 3.2(a). As a third consequence of Proposition 3.3, we now prove Alperin's fusion theo* *rem in our context. This theorem was originally formulated for finite groups in [Al* *], and then for saturated fusion systems over finite p-groups by Puig [Pu ] (see also * *[BLO2 , Carles Broto, Ran Levi, and Bob Oliver * * 17 Theorem A.10]). Our approach here (and our definition of P o) is modelled on Be* *nson's proof of the theorem for fusion in compact Lie groups [Be ]. Theorem 3.6 (Alperin's fusion theorem). Let F be a saturated fusion system over a discrete p-toral group S. Then for each ' 2 IsoF(P, P 0), there exist sequen* *ces of subgroups of S P = P0, P1, . .,.Pk = P 0 and Q1, Q2, . .,.Qk, and elements 'i2 AutF (Qi), such that (a)Qi is fully normalized in F, F-radical, and F-centric for each i; (b)Pi-1, Pi Qi and 'i(Pi-1) = Pi for each i; and (c)' = 'k O'k-1O . .O.'1. Proof.For each P S, let (P ) be the number of F-conjugacy classes of subgrou* *ps in Ho(F) which contain P . We prove the theorem by induction on (P ). Using Proposition 3.3, we can assume that P, P 02 Ho(F). The claim is clear when (P * *) = 1 (i.e., P = S). Assume P S. Let P 00 S be any subgroup which is F-conjugate to P and fully normalized in F, and fix _ 2 IsoF(P, P 00). The theorem holds for ' 2 IsoF(P, P* * 0) if it holds for _ and for _ O'-1 2 IsoF(P 0, P 00). So we are reduced to proving the * *theorem when the target group P 0is fully normalized in F. Since P 0is fully normalized, the p-subgroup 'OAut S(P )O'-1 of AutF(P 0) is * *conjugate to a subgroup of AutS(P 0). Let O 2 AutF (P 0) be such that_(OO')OAut S(P )O(OO* *')-1 AutS(P_0). By condition (II) in Definition 2.2, there is ' 2 Hom F(NS(P ), S) s* *uch that '|P = O O'. Since NS(P_) P (since P S) and P 2 Ho(F), (NS(P )) < (P ), and the theorem holds for ' (as an isomorphism to its image) by the induction hypot* *hesis. So it holds for ' if and only if it holds for O. Hence it now remains only to p* *rove it when P = P 0is fully normalized in F, P 2 Ho(F), and ' 2 AutF (P ). In particular, P is fully centralized in F. So if P_is not F-centric, then by* * condition (II) in Definition 2.2, ' extends to an automorphism ' 2 Aut F(CS(P ).P ). Si* *nce (CS(P ).P ) < (P ), the theorem holds for ' by the induction hypothesis. Now assume that P is not F-radical. Let K Aut F(P ) be the subgroup such that K= Inn(P ) = Op(Out F(P )) 6= 1. Since P is fully normalized in F, Out S(* *P ) 2 Sylp(Out F(P )), and so K AutS(P ). In particular, fi NKS(P ) def=g 2 NS(P ) ficg|P 2 K P since K Inn(P ). Also, for each g 2 NKS(P ), 'cg'-1 2 K (since K C Aut F(P )* *), and hence 'cg'-1 = ch for some h 2 NKS(P ). So by condition (II) in Definition * *2.2, ' extends to an automorphism of NKS(P ) P , and the theorem again holds for '* * by the induction hypothesis. Finally, if ' 2 AutF (P ) and P 2 Ho(F) is a fully normalized F-centric F-rad* *ical subgroup of S, then the theorem holds for trivial reasons. 18 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups 4. Linking systems over discrete p-toral groups We are now ready to define linking systems associated to a fusion system over* * a dis- crete p-toral group, and to study the relationship between linking systems and * *certain finite full subcategories. Definition 4.1. Let F be a fusion system over the discrete p-toral group S. A c* *entric linking system associated to F is a category L whose objects are the F-centric * *subgroups of S, together with a functor ss :L ------! Fc, ffiP and "distinguished" monomorphisms P --! AutL (P ) for each F-centric subgroup P S, which satisfy the following conditions. (A) ss is the identity on objects and surjective on morphisms. More precisely, * *for each pair of objects P, Q 2 L, Z(P ) acts freely on Mor L(P, Q) by composition (u* *pon identifying Z(P ) with ffiP(Z(P )) AutL(P )), and ss induces a bijection ~= MorL (P, Q)=Z(P ) ------! Hom F(P, Q). (B) For each F-centric subgroup P S and each g 2 P , ss sends ffiP(g) 2 AutL(* *P ) to cg 2 AutF (P ). (C) For each f 2 Mor L(P, Q) and each g 2 P , the following square commutes in * *L: f P ______! Q | | ffiP(g)| |ffiQ(ss(f)(g)) # # f P ______! Q . More generally, if F0 Fc is any subcategory, then a linking system associated* * to F0 is a category L0, together with a functor L0 -ss0-!F0 and distinguished monomor* *phisms ffiP P --! AutL0(P ) for P 2 Ob (F0) = Ob (L0), which satisfy conditions (A), (B), * *and (C) above. It is now clear, by analogy with the finite case, how to define p-local compa* *ct groups. Definition 4.2. A p-local compact group is a triple (S, F, L), where S is a dis* *crete p- toral group, F is a saturated fusion system over S, and L is a linking system a* *ssociated to F. The classifying space of such a triple (S, F, L) is the p-completed nerve* * |L|^p. The following very basic lemma about linking systems extends [BLO2 , Lemma 1* *.10] to this situation. Lemma 4.3. Fix a p-local compact group (S, F, L), and let ss :L --! Fc be the p* *ro- jection. Fix F-centric subgroups P, Q, R in S. Then the following hold. ' _ c -1 (a)Fix any sequence P --! Q --! R of morphisms in F , and let _e 2 ssQ,R(_) and f_' 2 ss-1P,R(_') be arbitrary liftings. Then there is a unique morphis* *m e'2 Mor L(P, Q) such that e_Oe'= f_'; (1) and furthermore ssP,Q('e) = '. Carles Broto, Ran Levi, and Bob Oliver * * 19 (b)If e', e'02 Mor L(P, Q) are such that the homomorphisms ' def=ssP,Q('e) and * *'0 def= ssP,Q('e0) are conjugate (differ by an element of Inn(Q)), then there is a u* *nique element g 2 Q such that e'0= ffiQ(g) Oe'in Mor L(P, Q). Proof.Part (a) is an easy application of axiom (A) for a linking system. Part (* *b) is first reduced to the case where ' = '0using axiom (B), and this case then follo* *ws from (A) and (C). For more detail, see the proof of [BLO2 , Lemma 1.10]. We next show that the nerve of a linking system is p-good, and hence that the classifying space of a p-local compact group is p-complete. Proposition 4.4. Let (S, F, L) be any p-local compact group at the prime p. The* *n |L| is p-good. Also, the composite ss1(`) ^ S --------! ss1(|L|) ----! ss1(|L|p), induced by the inclusion BS --`-!|L|, factors through a surjection ss0(S) -i ss* *1(|L|^p). Proof.For each F-centric subgroup P S, fix a morphism 'P 2 Mor L(P, S) which lifts the inclusion (and set 'S = IdS). By Lemma 4.3(a), for each P Q S, th* *ere is a unique morphism 'QP2 Mor L(P, Q) such that 'Q O'QP= 'P. Regard the vertex S as the basepoint of |L|. Define ! :Mor (L) -----! ss1(|L|) by sending each ' 2 Mor L(P, Q) to the loop formed by the edges 'P, ', and 'Q (* *in that order). Clearly, !(_ O') = !(_).!(') whenever _ and ' are composable, and !('QP) = !('P) = 1 for all P Q S. Also, ss1(|L|) is generated by Im(!) sinc* *e any loop in |L| can be split up as a composite of loops of the above form. By Theorem 3.6 (Alperin's fusion theorem), each morphism in F, and hence each morphism in L, is (up to inclusions) a composite of automorphisms of fully norm* *alized F-centric subgroups. Thus ss1(|L|) is generated by the subgroups !(Aut L(P )) f* *or all fully normalized F-centric P S. Let K C ss1(|L|) be the subgroup generated by all infinitely p-divisible elem* *ents. For each fully normalized F-centric P S, AutL (P ) is generated by its Sylow subg* *roup NS(P ) together with elements of order prime to p. Hence ss1(|L|) is generated * *by K together with the subgroups !(NS(P )); and !(NS(P )) !(S) for each P . This s* *hows that ! sends S surjectively onto ss1(|L|)=K, and hence (since the identity comp* *onent of S is infinitely divisible) factors through a surjection of ss0(S) onto ss1(|* *L|)=K. In particular, this quotient group is a finite p-group. Set ss = ss1(|L|)=K for short. Since K is generated by infinitely p-divisible* * elements, the same is true of its abelianization, and hence H1(K; Fp) = 0. Thus, K is p-p* *erfect. Let X be the cover of |L| with fundamental group K. Then X is p-good and X^pis simply connected since ss1(X) is p-perfect [BK , VII.3.2]. Also, since ss is a * *finite p-group, it acts nilpotently on Hi(X; Fp) for all i. Hence X^p--! |L|^p--! Bss is a fi* *bration sequence and |L|^pis p-complete by [BK , II.5.1]. So |L| is p-good, and ss1(|L|* *^p) ~=ss is a quotient group of ss0(S). Recall, from Section 3, that for any saturated fusion system F, we defined a * *finite subcategory Fo such that the inclusion Fo F has a left adjoint (-)o. We next * *show that we can do the same on the level of linking systems. 20 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups Proposition 4.5. Let F be a saturated fusion system over a discrete p-toral gro* *up S, and let Fco Fc be the full subcategory whose objects are the F-centric sub* *groups contained in Ho(F). (a)Let L be a centric linking system associated to F, and let Lo L be the full subcategory with Ob (Lo) = Ob (Fco). Then the inclusion Lo ,! L has a left a* *djoint, which sends P to P ofor each F-centric P S. In particular, the inclusion |* *Lo| |L| is a homotopy equivalence. (b)Let Lo be a linking system associated to Fco. Let L be the category whose ob* *jects are the F-centric subgroups of S, and where o o fi o MorL (P, Q) = ' 2 Mor Lo(P , Q ) fiss (')(P ) Q}; and let ffiP :P ---! AutL(P ) be the restriction of Po. In other words, L* * is the pullback category in the following square: L ______!Lo | | |ss sso| # # (-)o co Fc _____! F . Then L is a centric linking system associated to F. Proof.(a) For each F-centric subgroup P S, fix a morphism 'P 2 Mor L(P, S) such that ss('P) is the inclusion (and such that 'S = IdS). For any pair of F-c* *entric subgroups P Q S, the same group Z(P ) acts freely and transitively on the s* *ets of morphisms in L covering the inclusions P Q and P S, and hence there is a un* *ique morphism 'QP2 Mor L(P, Q) such that 'Q O'QP= 'P. Now let ' 2 Hom F(P, Q) be any morphism in Fc. By Proposition 3.3, ' has a un* *ique extension to 'o 2 Hom F(P o, Qo). Also, by Lemma 3.2(d), Z(P o) = Z(P ). Hence * *by condition (A) in the definition of a linking system, restriction sends the morp* *hisms in ss-1('o) bijectively to the morphisms in ss-1('). Thus for any _ 2 Mor L(P, Q)* * such that ss(_) = ', there is a unique "extension" _o 2 Mor L(P o, Qo) of _; i.e., a* * unique o Qo morphism such that _o O'PP = 'Q O_. Thus, if we define ` :L --! Lo by setting `(P ) = P oand `(_) = _o, then ` is* * well defined. This also shows that Mor L(P, Q) = Mor L(P o, Q) when Q = Qo, and thus* * that ` is a left adjoint functor to the inclusion. Since the inclusion has a left a* *djoint, it follows that it induces a homotopy equivalence |Lo| ' |L|. (b) Since Z(P ) = Z(P o) for all F-centric P S (Lemma 3.2(d) again), axiom (* *A) for L follows from the same axiom applied to Lo. Axioms (B) and (C) for L foll* *ow immediately from axioms (B) and (C) for Lo by restriction. We finish the section with a description of the relation between linking syst* *ems associated to a given fusion system F0, and rigidifications of the homotopy fun* *ctor B :Oc(F0) ---! hoTop defined by setting B (P ) = BP . Each linking system L0 i* *n- duces a rigidification of B, which in turn defines a decomposition of |L0| as a* * homotopy colimit. More precisely, by a "rigidification of the homotopy functor B" in the* * following proposition is meant a functor eB:O(F0) ---! Top together with a natural homoto* *py equivalence of functors (in hoTop ) from B to ho OeB; i.e., a natural transform* *ation of functors to hoTop which defines a homotopy equivalence BP ---! eB(P ) for each* * P . A Carles Broto, Ran Levi, and Bob Oliver * * 21 natural homotopy equivalence of rigidifications from eBto eB0is a natural trans* *formation Be---~--! eB0 of functors to Top such that ho(~) commutes with the functors from B. Two rigid* *ifica- tions eB1and eB2are equivalent if there is a third rigidification eB0and natura* *l homotopy equivalences eB1---! Be0--- eB2; this is seen to be an equivalence relation b* *y taking pushouts. By a linking system L0 in the following proposition is always meant the categ* *ory L0 together with the projection to the associated fusion system and the disting* *uished monomorphisms. Hence an isomorphism of linking systems means an isomorphism of the categories which is natural with respect to these other structures. Proposition 4.6. Fix a saturated fusion system F over a discrete p-toral group * *S, and let F0 Fc be any full subcategory. Then there are mutually inverse biject* *ions 8 9 8 9 < linking systems = ke < rigidifications O(F0) ___! Top= associated to F0 -------!------- of the homotopy functor B . : up to isomorphism; ls : up to natural homotopy equivalence; More precisely, the following hold for any linking system L0 associated to F0 a* *nd any rigidification eBof the homotopy functor B on O(F0). (a)The left homotopy Kan extension ke(L0) of the constant functor L0 -*-!Top al* *ong the projection ess0:L0 --! O(F0) is a rigidification of B , and there is a h* *omotopy equivalence |L0| ' hocolim-----!(ke(L0)) . (1) O(F0) (b)There is a linking system ls(Be) associated to F0, and a natural homotopy eq* *uiva- lence of functors ke(ls(Be)) ---'--!Be. Furthermore, if Be0is another rigidification of B , any natural homotopy equ* *iva- lence of rigidifications ~: eB--! eB0induces an isomorphim ~]:ls(Be) --! ls* *(Be0) of linking systems. (c)There is an isomorphism L0 ~=ls(ke(L0)) of linking systems associated to F0. We define ke([L0]) = [ke(L0)] for each L0, and ls([Be]) = [ls(Be)] for each eB. Proof.The left homotopy Kan extension is natural with respect to isomorphisms L0 --! L00of linking systems. Thus ke sends isomorphic systems to natural homot* *opy equivalent functors O(F0) --! Top , these are rigidifications of B by (a), and * *hence ke is well defined. Point (b) implies that lsis well defined, and it also implies * *that lsOke is the identity. Finally, (c) implies that ke Ols is the identity. Hence the Pr* *oposition follows once we prove (a), (b), and (c). (a) Fix L0, and set Be = ke(L0) for short. Recall that we write Rep F(P, Q)* * = Mor O(F)(P, Q). By definition, for each P in F0, eB(P ) is the nerve (homotopy * *colimit of the point functor) of the overcategory ess0#P , whose objects are pairs (Q, * *ff) for Q in L0 and ff 2 RepF (Q, P ), and where fi Mor ess0#P(Q, ff), (R, fi) = ' 2 Mor L(Q, R) fiff = fi Oess0(')(.2) Since |L0| ~=hocolim-----!L0(*), (1) holds by [HV , Theorem 5.5]. 22 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups It remains to show that Be is a rigidification of the homotopy functor B . F* *ix a section eoe:Mor(O(F0)) --! Mor(L0) of ess0which sends identity morphisms to id* *entity morphisms. For each P , let B(P ) be the category with one object oP and morphi* *sm group P (so |B(P )| ~=BP ), and define functors `P P B(P ) ------! ess0#P ------! B(P ) as follows. Let `P(oP) = (P, Id), and `P(g) = ffiP(g) (as a morphism in ess0#P* * us- ing (2)) for all g 2 P . Set P(Q, ff) = oP; and let P send each morphism ' 2 Mor ess0#P((Q, ff), (R, fi)) to the unique element g 2 P (unique by Lemma 4.3(b* *)) such that the following square commutes: ' Q _____! R | | eoe(ff)| eoe(fi)| # # ffiP(g) P _____! P . Clearly, P O`P = IdB(P). As for the other composite, define f :Id --! `P O P* * by sending each object (Q, ff) to the morphism eoe(ff) 2 Mor L(Q, P ). This is cl* *early a natural transformation of functors, and thus eB(P ) = |ess0#P | ' |B(P )| ' BP. To finish the proof that eB is a rigidification of the homotopy functor B , w* *e must show, for any ' 2 Hom F (P, Q), that the following square commutes up to natural transformation: `P B(P )_______! ess0#P | |_ |B' |'O- # # `Q B(Q) _______! ess0#Q . Here, ['] 2 Rep F(P, Q) denotes the class of '. This means constructing a natu* *ral transformation F1 ---! F2 of functors B(P ) --! ess0#Q, where F1 = (['] O-) O`P* * and F2 = `Q OB' are given by the formulas F1(oP) = (P, [']), F1(g) = ffiP(g), and F2(oP) = (Q, Id), F2(g) = ff* *iQ('(g)). Let e'2 Mor L(P, Q) be any lifting of '. Then by condition (C), can be define* *d by sending the object oP to the morphism e'2 Mor ess0#P(P, [']), (Q, Id) . (b) We first fix some notation. For any space X and any x, x02 X, ss1(X; x, x0)* * denotes the set of homotopy classes of paths in X (relative endpoints) from x to x0. Fo* *r any u 2 ss1(X; x, x0), u* denotes the induced isomorphism from ss1(X, x) to ss1(X, * *x0). Also, for any map of spaces f : X ! Y , f* denotes the induced map from ss1(X; x, x0)* * to ss1(Y ; f(x), f(x0)). Now fix a rigidification Be:O(F0) ____! Top; we want to define a linking sys* *tem L0 = ls(Be) associated to F. Since eB is a rigidification of the homotopy func* *tor B , fflP we are given homotopy equivalences BP ---! eB(P ) such that the following squa* *re commutes up to homotopy for each ' 2 Hom F(P, Q): fflP BP _______! eB(P ) | B' | |eB([']) # ffl # BQ _______Q! eB(Q) . Carles Broto, Ran Levi, and Bob Oliver * * 23 Here, ['] 2 Rep F(P, Q) denotes the class of ' (mod Inn(Q)). For each P in F0,* * let *P 2 eB(P ) be the image under fflP of the base point of BP , and let ~= flP :P ------! ss1(Be(P ), *P) be the isomorphism induced by fflP on fundamental groups. Let L0 = ls(Be) be the category with Ob (L0) = Ob (F0), and with fi Mor L0(P, Q) = (', u) fi' 2 RepF (P, Q), u 2 ss1(Be(Q); eB'(*P), *Q) . Composition is defined by setting (_, v) O(', u) = (_', v . eB_*(u)), where paths are composed from right to left. Let ss0: L0 --! F0 be the functor * *which is the identity on objects, and which sends (', u) 2 Mor L0(P, Q) to the compos* *ite flP Be'* u* fl-* *1Q P ----!~ ss1(Be(P ), *P) ----! ss1(Be(Q), eB'(*P)) ----! ss1(Be(Q), *Q) ----!* * Q. = ~= Also, for each P , define ffiP :P ----! AutL0(P ) by setting ffiP(g) = (IdP, flP(g)). Axioms (A), (B), and (C) for a centric linking system are easily seen to hold f* *or L0. For example, (C) follows as an immediate consequence of the definition of ss0. Now set B1 = ke(L0) = ke(ls(Be)): the left homotopy Kan extension along the projection ess0:L0 --! O(F0) of the constant point functor on L0. Thus B1(P ) = |B1(P )| for each P , where B1(P ) is the category with objects the pairs (Q, f* *f) for ff 2 RepF (Q, P ), and with morphism sets fi Mor B1(P)(Q, ff), (R, fi) = b'2 Mor L0(Q, R) fiff = fi Oess0('b) fi = (', u) fi' 2 RepF (Q, R), ff = fi O', u 2 ss1(Be(R); eB'(*Q), * **R) . We define a natural homotopy equivalence of functors : B1 ---! eBas follows. F* *or all P , maps P :B1(P ) ___! eB(P ) are defined inductively, one skeleton at a* * time, (and simultaneously for all P ) as follows. o Each vertex (Q, ff) in B1(P ) = |B1(P )| is sent to eB(ff)(*Q) 2 eB(P ). (',u) o For each edge oe = (Q, ') ---! (P, Id) in B1(P ), where ' 2 RepF (Q, P ) and u 2 ss1(Be(P ); eB'(*Q), *P), P|oe= ^ufor some path ^uin the homotopy class of u. (',u) o For each edge oe = (Q, ') ---! (R, fi) in B1(P ), where fi 6= IdP, write o* *e0 = (',u) (Q, ') ---! (R, Id) (an edge in B1(R)), and set P|oe= eB(fi) O( R|oe0). o Consider a simplex of dimension m 2 in B1(P ) of the form i j oe = (Q0, ff0) ---! (Q1, ff1) ---! . .-.--!(Qm , ffm ) . If (Qm , ffm ) = (P, Id), then let P|oebe any singular simplex in eB(P ) wh* *ose bound- ary is as already defined. Otherwise, let oe0 be the unique simplex in B1(Qm* * ) rep- resenting a chain ending in (Qm , Id) such that oe = B1(ffm )(oe0), and set * * P|oe= Be(ffm ) O( Qm |oe0). 24 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups Since B1(P ) ' eB(P ) ' BP (where P is given the discrete topology), the above * *con- struction is always possible, and defines a homotopy equivalence. It induces th* *e identity on fundamental groups, under their given identifications with P . By constructi* *on, the P form a natural morphism of functors from B1 to eB. Let Be0, {ffl0P} be another rigidification of B , and let ~: eB---! Be0be a* * natural homotopy equivalence of rigidifications. We have already chosen our basepoint ** *P = fflP(*), where * 2 BP is a fixed basepoint, and we now set *0P= ffl0P(*). Fix, * *for each P , a homotopy HP between ~P OfflP and ffl0P. The restriction of HP to the base* * point of BP provides a canonical path in eB0(P ) from ~P(*P) to *0P, whose homotopy clas* *s we denote wP 2 ss1(Pe; ~P(*P), *0P). We now define ~]:L0 -----! L00 to be the identity on objects, and for (', u) 2 Mor L0(P, Q), ~](', u) = (', wP . ~Q*(u) . eB0'*(wP)-1) . (1) It is straightforward to show that ~ is a well defined isomorphim of linking sy* *stems; i.e., an isomorphism of categories which is natural with respect to the project* *ions to F0 and the distinguished monomorphisms. (c) Now assume that L0 is given; it remains to construct an isomorphism L0 ~= ls(ke(L0)) of linking systems associated to F0. Set eB= ke(L0) and L1 = ls(Be)* * for short. By definition, L0 and L1 have the same objects, and a morphism in L1 fr* *om P to Q is a pair (', u), where ' 2 RepF (P, Q) and u 2 ss1(Be(Q); eB'(*P), *Q).* * Also, eB(P ) = |ess0#P | where ess0is the projection of L0 onto O(F0); in particular,* * we choose *P to be the vertex of (P, Id). Define : L0 ___! L1 by sending each object to* * itself, and by sending ff 2 Mor L0(P, Q) to (ess0(ff), [ff]), where [ff] is the homotop* *y class of ff, regarded as an edge in |ess0#Q| from (P, ess0(ff)) = ess0ff(*P) to (Q, Id) = *Q* *. This is easily checked to be an isomorphism of categories, and to commute with the distinguish* *ed monomorphisms and the projections to F0. 5. Higher limits over orbit categories If F is any fusion system over a discrete p-toral group S, then O(F) will den* *ote its orbit category: the category whose objects are the subgroups of S, and where Mor O(F)(P, Q) = RepF (P, Q) def=Inn(Q)\ Hom F(P, Q). Also, we write Oc(F) = O(Fc) to denote the full subcategory of O(F) whose objec* *ts are the F-centric subgroups of S; and more generally write O(F0) to denote the * *full subcategory of O(F) corresponding to any full subcategory F0 of F. By Lemma 2.5, the morphism sets in the orbit category are all finite. There * *is a canonical projection functor F ! O(F) which is the identity on objects and the natural projection Hom F(P, Q) ! RepF (P, Q) on morphisms. Throughout this section, when C is a category, we frequently write C-mod to * *denote the category of functors Cop ---! Ab . This notation will not be used in the st* *atements of results here, but it is used in several of the proofs. Carles Broto, Ran Levi, and Bob Oliver * * 25 Lemma 5.1. Let F be a saturated fusion system over a discrete p-toral group S, * *and let F0 F be any full subcategory such that P 2 Ob (F0) implies P o2 Ob (F0). * *Set Fo0= F0 \ Fo. Then there are well defined functors (-)o Oc(F) ------!------O(Fco) , incl where (-)o sends P to P oand ['] to ['o]. Also, (-)o is a left adjoint to the i* *nclusion. Proof.This follows from Corollary 3.4. The only thing to check is that (-)o is* * well defined on morphisms in the orbit category. If '1, '2 2 Hom F (P, Q) represent* * the same morphism in the orbit category, then '1 = cgO '2 for some g 2 Q, so 'o1= c* *gO 'o2 by functoriality, and hence ['o1] = ['o2] in RepF (P o, Qo). The following proposition shows that the problem of describing higher limits * *over the orbit categories we are considering can always be reduced to one over a fin* *ite subcategory. Proposition 5.2. Let F be a saturated fusion system over a discrete p-toral gro* *up S. Let F0 F be any full subcategory such that P 2 Ob (F0) implies P o2 Ob (F0), * *and set Fo0= F0\ Fo. Then for any functor F :O(F0)op--! Z(p)-mod, restriction to F* *o0 induces an isomorphism lim-*(F~)=lim-*(F |O(Fo0)). O(F0) O(Fo0) Proof.Consider the functors R O(F0)-mod ------!------O(Fo0)-mod , T where R is given by restriction and T by composition with the functor (-)o. The* *n T is a left adjoint to R, since (-)o is a left adjoint to the inclusion by Lemma * *5.1. Also, T and R are both exact functors, and R sends injectives to injectives since it * *is right adjoint to an exact functor. Let Z_be the constant functor on O(Fo0) which sends all objects to Z. Then T * *(Z_) is the constant functor on O(F0), and hence for any functor F on O(F0), lim-(F )= Hom O(F0)-mod(T (Z_), F ) ~=Hom O(Fo0)-mod(Z_, R(F )) = lim-(R(F* * )). O(F0) O(Fo0) Since R is exact and sends injectives to injectives, it sends injective resolut* *ions to in- jective resolutions, and thus induces an isomorphism between higher limits over* * O(F0) and over O(Fo0). We next want to show that the techniques which we have already developped for handling higher limits over orbit categories in the finite case [BLO2 , x3] al* *so apply in this new situation. The proof of this is similar to the proof in [BLO2 ] of th* *e analogous result for fusion systems over finite p-groups, and is in fact a special case o* *f a very general result which we prove here. For any group (not necessarily finite), and any set H of subgroups of , we* * define OH ( ) to be the corresponding orbit category of : the category with Ob(OH ( )* *) = H, and with morphism sets Mor OH( )(H, H0) = H0\N (H, H0) ~=Map ( =H, =H0). 26 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups Here, N (H, H0) is the transporter set: N (H, H0) = {g 2 | gHg-1 H0}. If 1 2 H, then for any Z[ ]-module M, we define *H( ; M) = lim-*(FM ), OH( ) where FM :OH ( )op---! Ab is the functor FM (H) = 0 if H 6= 1 and FM (1) = M. It is important to distinguish between the orbit category of a group and the * *orbit category of a fusion system. When G is a finite group and S 2 Sylp(G), the orb* *it category of the fusion system FS(G) is not the same as the orbit category OS(G)* * (the orbit category of G with objects the subgroups of S). Proposition 5.3. Fix a category C, a group , a set H of subgroups of such th* *at 1 2 H, and a functor ff: OH ( ) ------! C. Set c0 = ff(1). For each object d in C, we regard the set Mor C(c0, d) as a -s* *et via ff and composition. Assume that the following conditions hold: (a)ff sends = AutOH( )(1) bijectively to End C(c0). (b)For each d 2 Ob (C) such that d 6~= c0, all isotropy subgroups of the -acti* *on on Mor C(c0, d) are nontrivial and conjugate to subgroups in H. (c)For each , 2 Mor (OH ( )), ff(,) is an epimorphism in the categorical sense:* * ' O ff(,) = _ Off(,) implies ' = _. (d)For any H 2 H, any d 2 Ob (C), and any ' 2 Mor C(c0, d) which is H-invariant, _ _ H there is some ' 2 Mor C(ff(H), d) such that ' = ' Off(incl1). Let : Cop ------! Ab be any functor which vanishes except on the isomorphism class of c0. Then the n* *atural map * * * lim-*( )---ff---!~lim-( Off)= H ( ; (c0)) C = OH( ) is an isomorphism. Proof.Consider the functors ff* OH ( )-mod ------------!C-mod , Rff where ff* is composition with ffop, and Rffis the right Kan extension of ffop. * *Specif- ically, for d 2 Ob (C), let ff#d be the overcategory whose objects are pairs (H* *, ') for ' 2 Mor C(ff(H), d), and where a morphism from (H, ') to (K, _) is a morphi* *sm O 2 Mor OH( )(H, K) such that _ Off(O) = '. Let ~d: ff#d ---! OH ( ) be the for* *getful functor. Then (ff#d)op = d#ffop (the undercategory), and for F :OH ( )op---! * *Ab , Rff(F ) is defined by setting Rff(F )(d) = lim-(F O~dop). (ff#d)op Carles Broto, Ran Levi, and Bob Oliver * * 27 On morphisms, Rff(F ) sends f 2 Mor C(d, d0) to the morphism induced by the fun* *ctor fO- ff#d ----! ff#d0 . By [McL , xX.3, Theorem 1], Rffis right adjoint to ff*. In * *particular, since ff* preserves exact sequences, Rffsends injectives to injectives. Fix H 2 H and d 2 Ob (C). Consider the map ~: Mor C(ff(H), d) ------! Mor C(c0, d) defined by composition with the "inclusion" morphism ff(inclH1). This map is in* *jective by (c), and Im (~) Mor C(c0, d)H by (d). Also, Im (~) is contained in Mor C* *(c0, d)H since inclH1Ox = inclH1for all x 2 H. Thus ~ induces a bijection ~0 H MorC(ff(H), d) ------!~ Mor C(c0, d) . (1) = Fix representatives {'di}i2Idfor the -orbits in Mor C(c0, d), and let di * * be the stabilizer subgroup of 'di. By (b), we can choose the 'disuch that di2 H for a* *ll i. By (1), each 'dihas a unique "extension" to _di2 Mor C(ff( di), d); i.e., there is* * a unique d _disuch that 'di= _diOff(incl1i). Also, for any (H, O) in ff#d, there is a uniq* *ue i 2 Id and a unique morphism O0 2 Mor OH( )(H, di) such that O = _diOO0. So each obje* *ct ( di, _di) is a final object in its connected component of the overcategory ff#* *d. Thus for any F in OH ( )-mod , Y Rff(F )(d) ~= F ( di). (2) i2Id In particular, Rffis an exact functor. Let Z_denote the constant functor on Cop which sends each object to Z and each morphism to the identity. Then ff*Z_is the constant functor on OH ( )op. If F* * : Cop ___! Ab is any functor, then lim-(F )~=HomC-mod(Z_, F ); C and similarly for functors in OH ( )-mod . Assume H 2 H is such that ff(H) ~=ff(1) = c0. Since all endomorphisms of c0 a* *re automorphisms (by (a)), Mor C(c0, ff(H)) contains only isomorphisms, and in par* *ticular ff(inclH1) is an isomorphism. Also, inclH1Ox = inclH1for all x 2 H, so ff(x) = * *Idc0for all x 2 H. By (a) again, this implies that H = 1. The functor ff* = Offop : OH ( )op ___! Z(p)-mod thus sends the object 1 * *to (c0) (with the given action of ), and sends all other objects to 0. Then Rffs* *ends an injective resolution I* of ff* to an injective resolution Rff(I*) of Rff(ff* )* *. It follows that * *H( ; (c0)) def=lim-*(ff*~)=H* Mor OH( )-mod(ff Z_, I*) OH( ) ~=H* Mor C-mod(Z_, Rff(I*)) ~=lim*(R (ff* )). -C ff It remains only to show that Rff(ff* ) ~= . For each d 2 Ob (C), if d 6~= c0* *, then Mor C(c0, d) is a disjoint union of orbits = di, where 1 6= di2 H by (b). So * *by (2), Y Rff(ff* )(d) = Rff( Off)(d) ~= (ff(Hi)) = 0; i where the last equality holds since we already showed that H 6= 1 implies ff(H)* * 6~= c0. If d ~= c0, then Mor C(c0, d) consists of one free orbit of (by (a)), an* *d hence Rff(ff* )(d) ~= (ff(1)) ~= (c0). This finishes the proof that Rff(ff* ) ~= . 28 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups Our first application of Proposition 5.3 is to the case where C is the orbit * *category of a saturated fusion system over a discrete p-toral group. As in [JMO ] and [* *BLO2 ], when is finite and H is the set of p-subgroups of (or the set of subgroups * *of a given Sylow p-subgroup), we write *( ; M) = *H( ; M) (and the prime p is understood* *). Proposition 5.4. Let F be a saturated fusion system over S. Let : Oc(F)op------! Z(p)-mod be any functor which vanishes except on the isomorphism class of some fixed F-c* *entric subgroup Q S. Then lim-*( )~= *(Out F(Q); (Q)). Oc(F) Proof.It suffices to do this when Q is fully normalized. Set = Out F(Q) and * * = OutS(Q) 2 Sylp( ), and let H be the set of subgroups of . Since ~= NS(Q)=Q, each subgroup of has the form OutP (Q) for some unique P NS(Q) containing Q. Define ff: O ( ) -----! Oc(F) on objects by setting ff(Out P(Q)) = P for Q P NS(Q). If ' 2 AutF (Q) is su* *ch that ['] 2 N (Out P(Q), OutP0(Q)) (the set_of elements which conjugate OutP (Q)* * into OutP0(Q)),_then ' can be extended to some ' 2 Hom F(P, P 0) by axiom (II), the * *class of ' in the orbit category is uniquely_determined by ' by Proposition 2.8, and * *ff sends the class of ['] to the class of '. We apply Proposition 5.3 to this functor ff. Condition (a) is clear, (c) hold* *s for Oc(F) by Proposition 2.8, and (d) holds by axiom (II) of a saturated fusion system. A* *s for (b), since every morphism in F is the composite of an isomorphism followed by an inc* *lusion, it suffices to prove that the stabilizer in of an inclusion inclPQ2 Hom F(Q, * *P ), where Q P , is a nontrivial p-subgroup. But the stabilizer is OutP (Q) ~=NP(Q)=Q, w* *hich is nontrivial by Lemma 1.8. All of the hypotheses of Proposition 5.3 thus hold* *, and the result follows. Using the terminology of [BLO2 ], we say that a category C has bounded limit* *s at p if there is k > 0 such that for any functor : Cop ---! Z(p)-mod, lim-i( ) = 0 * *for all i > k. The following is a first corollary of Proposition 5.4. Corollary 5.5. Let F be a saturated fusion system over a discrete p-toral group* * S, and let F0 Fc be a full subcategory such that P 2 Ob (F0) implies P o2 Ob (F0* *). Then the orbit category O(F0) has bounded limits at p. Proof.By Proposition 5.2, it suffices to prove this when F0 Fo; in particular* *, when F0 has only finitely many isomorphism classes. By [JMO2 , Proposition 4.11], f* *or each finite group , there is some k such that i( ; M) = 0 for all Z(p)[ ]-modules* * M and all i > k . Let k be the maximum of the kOutF(P)for all P 2 Ob (F0). Then by Proposition 5.4, for each functor : O(F0)op---! Z(p)-mod which vanishes exce* *pt on one orbit type, lim-i( ) = 0 for i > k. The same result for an arbitrary p-loca* *l functor on O(F0) now follows from the exact sequences of higher limits associated to * *short exact sequences of functors. In practice, when computing higher limits over orbit categories Oc(F), it is * *useful to combine Propositions 5.2 and 5.4, as illustrated by the following corollary. Carles Broto, Ran Levi, and Bob Oliver * * 29 Corollary 5.6. Let F be a saturated fusion system over a discrete p-toral group* * S. Let F :Oc(F)op ___! Z(p)-mod be a functor with the property that for each F-ce* *ntric subgroup P 2 Ho(F), *(Out F(P ); F (P )) = 0. Then lim-*(F=)0. Proof.Let F0: Oc(Fo)op ___! Z(p)-mod be the restriction of F . By Proposition * *5.2, lim-*(F~)= lim-*(F0). Oc(F) Oc(Fo) Assume first that F0 vanishes except on the conjugacy class of one subgroup P 2 Ho(F). Let F 0be the functor on Oc(F) which takes the same value on the conjuga* *cy class of P and vanishes on all other subgroups. Then lim-*(F0)~=lim-*(F 0)~= *(Out F(P ); F (P )) Oc(Fo) Oc(F) by Propositions 5.2 and 5.4, and this is zero by assumption. By Lemma 3.2(a), the category Oc(Fo) contains only finitely many isomorphism classes. Hence there is a sequence 0 = 0 1 . . . k = F0 of subfunctors defined on Oc(Fo), with the property that for each i, i= i-1van* *ishes except on the conjugacy class of one subgroup P , and ( i= i-1)(P ) ~=F (P ). W* *e have just seen that lim-*( i= i-1)= 0 for all i; and hence lim-*(F0)= 0 by the relat* *ive long exact sequences of higher limits. The following lemma will be useful in showing that certain functors on the or* *bit category are acyclic. As usual, when F is a fusion system over S, a subgroup P * * S will be called weakly closed in F if it is the only subgroup in its F-conjugacy* * class. Lemma 5.7. Let F be any saturated fusion system over a discrete p-toral group S* *, and let Q C S be any F-centric subgroup which is weakly closed in F. Set = OutF (* *Q), and let F Q Fc be the full subcategory whose objects are the subgroups which * *contain Q. Define the functor : O(F Q )op------! Op( ) by sending an object P to OutP (Q) , and by sending a morphism ' 2 RepF (P, * *P 0) to the class of '|Q 2 N ( (P ), (P 0)). Then for any pair of functors F :Oc(F)op------! Z(p)-mod and : Op( )op------! Z(p)-mod such that O ~= F |O(F Q), and such that Out Q(P ) ~= NPQ (P )=P acts trivial* *ly on F (P ) for all P S, lim-*(F~)=lim-*( ). Oc(F) Op( ) Proof.Define a functor F 0:Oc(F)op------! Z(p)-mod by setting F 0(P ) = F (P ) if P Q and F 0(P ) = 0 otherwise. Regard F 0as a * *quotient functor of F , and set F 00= Ker[F -i F 0]. If P S is F-centric and P Q, then Out Q(P ) ~= NPQ (P )=P 6= 1, and by assumption this group acts trivially on F (P ) ~=F 00(P ). Hence the kernel of * *the action of Out F(P ) on F 00(P ) has order a multiple of p, and so *(Out F(P ); F 00(P* * )) = 0 by [JMO , Proposition 5.5]. Thus lim-*(F 00)= 0 by Corollary 5.6, and hence lim-*(F )~=lim-*(F.0) Oc(F) Oc(F) 30 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups Recall that = OutF (Q). Since Q is fully normalized in F (it is the unique * *subgroup in its F-conjugacy class), (S) = Out S(Q) 2 Sylp( ). Also, defines a biject* *ion between subgroups of (S) ~= S=Q and subgroups of S which contain Q. For all Q P, P 0 S, (-)|Q 0 RepF(P, P 0) -------! Mor Op( )( (P ), (P )) is injective by Proposition 2.8. If g 2 N ( (P ), (P 0)) is any element in th* *e trans- porter, and g = ['] for ' 2 Aut F(Q), then for all x 2 P there is y 2 P 0such t* *hat 'cx'-1 = cy as automorphisms_of Q. Hence by condition (II)_in Definition 2.2, ' extends to a homomorphism ' 2 Hom F(P, P 0), and sends [' ] 2 RepF (P, P 0) t* *o the class of g. This proves that induces bijections on all morphism sets, and thus is an eq* *uivalence of categories. Hence if is such that O ~=F |O(F Q), then lim-*( )~= lim-*(F |O(F Q))~=lim-*(F~0)=lim-*(F.) Op( ) O(F Q) Oc(F) Oc(F) This can now be applied to prove the acyclicity of certain explicit functors. Proposition 5.8. Let F be any saturated fusion system over a discrete p-toral g* *roup S. Define F1, F2: Oc(F)op------! Z(p)-mod on objects by setting F1(P ) = Z(P )0 and F2(P ) = ss2(B(Z(P ))^p). On morphism* *s, each Fi sends the class of ' 2 Hom F(P, P 0) to the homomorphism induced by the incl* *usion of Z(P 0) into Z('(P )) followed by '-1|Z('(P)). Then F1 and F2 are both acycli* *c. Proof.Set T = S0 (the "maximal torus" in F), Q = CS(T ) C S, and = OutF (Q). Then Q is F-centric, and is weakly closed in F since T is. Let : O(F Q ) ------! Op( ) be the functor of Lemma 5.7. For each p-subgroup , regarded as a group of automorphismsQof Q, let N be the norm map for the action of on T ; i.e., N (* *t) = fl2fl(t) for t 2 T . Define 1( ) = N (T ) and 2( ) = Hom (Z=p1 , T ) . These define functors i:Op( )op ___! Z(p)-mod. For each P S which contains Q, NP=Q(T ) is connected (i.e., infinitely p-di* *visible), and has finite index in Z(P ) since Z(P ) \ T = T Pand T P=NP=Q(T ) has exponen* *t at most |P=Q|. Hence NP=Q(T ) is equal to the identity component Z(P )0, and we ha* *ve F1(P ) = Z(P )0 = NP=Q(T ) = 1( (P )). In general, for any discrete p-toral group P , ss2(BP ^p) = [S2, BP ^p] ~=[BS1, BP ^p] ~=Hom (Z=p1 , P ). Here, the last equivalence follows from Lemma 1.10, while the middle one follow* *s by obstruction theory (since ssi(BP ^p) = 0 for i > 2). Hence for any P S which * *contains Q, F2(P ) = ss2(BZ(P )^p) ~=Hom (Z=p1 , Z(P )) ~=Hom (Z=p1 , T )P=Q = 2( (P )). Carles Broto, Ran Levi, and Bob Oliver * * 31 Thus iO ~=Fi|O(F Q) (for i = 1, 2). Also, for each P S, OutQ (P ) acts t* *rivially on Fi(P ) for i = 1, 2 since Q centralizes Z(P )0 T . So by Lemma 5.7, lim-*(Fi)~=lim-*( i). Oc(F) Op( ) The functors 1 and 2 are both Mackey functors on Op( ) (see [JM , Proposition* * 5.14] or [JMO , Proposition 5.2]), and hence are acyclic. As in Section 4, when F is a saturated fusion system over S, we let B denote * *the homotopy functor B (P ) = BP , and by extension let B ^pdenote the functor B ^p* *(P ) = BP ^p. The following proposition is a first application of Proposition 5.8. It * *shows that there is a bijective correspondence between rigidifications of these two functo* *rs. Proposition 5.9. Let F be a saturated fusion system over a discrete p-toral gro* *up S, and let F0 Fc be any full subcategory which contains Fco. Let bB:O(F0) ---! T* *op be any rigidification of the homotopy functor B ^p. Then there is a f* *unctor eB:O(F0) ---! Top such that Be(P ) ' BP for all P , together with a natural tr* *ans- formation of functors eB___! bBwhich is a homotopy equivalence after p-complet* *ion. Moreover, there is a bijection between equivalence classes of rigidifications o* *f B and equivalence classes of rigidifications of B ^p. Proof.Let O: B ---! B ^pbe the natural transformation of homotopy functors which sends BP to BP ^pby the canonical map. We want to apply Theorem A.3, which is a relative version of the Dwyer-Kan theorem [DK ] for rigidifying centric homot* *opy diagrams. We first check that O is relatively centric in the sense of Theorem A* *.3. This means showing, for each ' 2 Mor O(F0)(P, Q), that the square B'O- Map (BP, BP )Id_______! Map (BP, BQ)B' | | O(P)O-| O(Q)O-| # # B'O- ^ Map (BP, BP ^p)O(P)__! Map (BP, BQp)O(Q)OB' is a homotopy pullback. By a classical result, the top row is a homotopy equiva* *lence, and both mapping spaces have the homotopy type of BZ(P ) (cf. [BKi , Propositi* *on 7.1]). By Lemma 1.10, the second row is also a homotopy equivalence, and both mapping spaces have the homotopy type of BZ(P )^p. So the square is a homotopy pullback. For each i 1, let fii:O(F0)op---! Ab be the functor defined in Theorem A.* *3, where for each P , i O(P)O- j fii(P ) = ssi hofiberMap (BP, BP )Id------! Map (BP, BP ^p)O(P) . BZ(P) BZ(P)^p By [DW2 , Proposition 3.1], this homotopy fiber is a K(V, 1) for some bQp-vect* *or space V . In particular, the fiber is connected, fi1(P ) is abelian for all P , and f* *ii = 0 for all i 2. Also, by the homotopy exact sequence for the fibration, there is a short* * exact sequence of functors 0 ---! F2 -----! fi1 -----! F1 ---! 0, where F1 and F2 are the functors of Proposition 5.8. By Proposition 5.2, for al* *l i 1 and j = 1, 2, lim-i(Fj) ~= lim-i(Fj) ~= lim-i(Fj), O(F0) O(Fo) Oc(F) 32 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups where the last group vanishes by Proposition 5.8. Thus lim-i(fi1) = 0 for all i* * 1. The proposition now follows directly from Theorem A.3. In Section 8, we will also need to work with higher limits over orbit categor* *ies of certain infinite groups. For any (discrete) group G, let Odpt(G) denote the* * orbit category of G whose objects are the discrete p-toral subgroups of G; and define* * (for any Z[G]-module M), ( M if P = 1 *dpt(G; M) = lim-(FM ) where FM (P ) = Odpt(G) 0 if P 6= 1. We are now ready to give a second application of Proposition 5.3. Lemma 5.10. Fix a group G, a discrete p-toral subgroup Q G, and a functor : Odpt(G)op ___! Ab with the property that (P ) = 0 except when P is G-conju* *gate to Q. Let 0:Odpt(NG(Q)=Q)op ___! Ab be the functor 0(P=Q) = (P ). Then lim-*( ) ~= lim-* ( 0) ~= *dpt(NG(Q)=Q; (Q)) . (1) Odpt(G) Odpt(NG(Q)=Q) Proof.We apply Proposition 5.3, where C = Odpt(G), = NG(Q)=Q, and H is the set of discrete p-toral subgroups of . A functor ff: Odpt( ) -----! Odpt(G) is defined by setting ff(P=Q) = P , and by sending the set (P 0=Q)\N (P=Q, P 0=* *Q) to P 0\NG(P, P 0) in the obvious way. The hypotheses of Proposition 5.3 follow easily from the definition of the or* *bit cate- gories, and so the isomorphisms between higher limits follow from the propositi* *on. The following very general lemma will help in certain cases to reduce computa* *tions of higher limits to those taken over finite subcategories. Lemma 5.11. Let C be a (small) category, and let C1 C2 . . .be an increasing sequence of subcategories of C whose union is C. Let F :Cop ---! Ab be a functo* *r such that for each k, i j lim-1lim-k(F |Ci) = 0. i Ci Then the homomorphism ~= i k j lim-k(F ) ------! lim- lim-(F |Ci) C i Ci induced by the restrictions is an isomorphism for all k. Proof.For any category D and any functor : Dop ___! Ab , lim-*( ) is the homo* *logy of the chain complex (C*(D; ), d), defined by setting Y Cn(D; ) = (c0), c0!...!cn where the product is taken over composable n-tuples of morphisms in D, and where n+1X d(,)(c0 ff-!c1 ! . .!.cn+1) = ff*,(c1 ! . .!.cn+1) + ,(c0 ! . .b.ci.!.c.n+1* *). i=1 Carles Broto, Ran Levi, and Bob Oliver * * 33 See, for example, [GZ , Appendix II, Proposition 3.3] or [Ol, Lemma 2]. If D0 * * D is a subcategory, then the restriction homomorphism from lim-*( ) to lim-*( |D0) is * *induced D D0 by the obvious surjections C*(D; ) --i C*(D0; ). In the above situation, the chain complex (C*(C; F ), d) is the limit of an i* *nverse system of chain complexes (C*(Ci; F |Ci), d) with surjections, where the invers* *e system of homology groups of these chain complexes has vanishing lim-1(-). Since lim-* *1(-) vanishes for a (countable directed) inverse system with surjections, we conclud* *e that the cohomology of (C*(C; F ), d) is isomorphic to the inverse limit of the coho* *mology of the complexes (C*(Ci; F |Ci), d). The next lemma describes how, in some cases, the computation of *dpt(G; M) c* *an be reduced to the case where G is finite. When G is a finite group and M is a Z* *[G]- module, we let *(G; M) denote the -functor taken with respect to p-subgroups * *of G. Lemma 5.12. Let G be a locally finite group. Assume there is a discrete p-toral subgroup S G such that every discrete p-toral subgroup of G is conjugate to a* * subgroup of S. Fix a Z[G]-module M, and assume that for some finite subgroup H0 G, *(H; M) = 0 for all finite subgroups H G which contain H0. Then *dpt(G; M) * *= 0. In particular, *dpt(G; M) = 0 if M is a Z(p)[G]-module and the kernel of the a* *ction of G on M contains an element of order p. Proof.By [JMO , Proposition 5.5], for any finite group H and any Z(p)[H]-modul* *e M such that the kernel of the H-action on M has order a multiple of p, *(H; M) =* * 0. Hence the last statement follows as a special case of the first. Fix a Sylow p-subgroup S 2 Sylp(G), and let OS(G) Odpt(G) be the full subca* *te- gory whose objects are the subgroups of S. Since each discrete p-toral subgroup* *s of G is G-conjugate to a subgroup of S, these categories are equivalent, and so we c* *an work over OS(G) instead. Define ( M if P = 1 FM :Cop ---! Ab by setting FM (P ) = 0 if P 6= 1. By definition, *dpt(G; M) = lim-*(FM ), and we must show that this vanishes in* * all degrees. Step 1: To simplify the notation, we write C = OS(G), and let C0 C be the f* *ull subcategory whose objects are the finite subgroups of S. For each subgroup Q * * S and each abelian group A, let IAQin C-mod be the functor Y IAQ(P ) = Map (Mor C(Q, P ), A) ~= A. MorC(Q,P) For any F in C-mod , Hom C-mod(F, IAQ) ~=Hom Z(F (Q), A). Hence IAQis injective* * if A is injective as an abelian group, and each functor on C injects into a product * *of such injectives. Also, when Q is finite, lim-(IAQ|C0) ~=lim-(IAQ) ~=A C0 C (where the second isomorphism holds for arbitrary Q S). Choose a sequence of functors 0 ---! FM ---d0-!I0 --d1--!I1 --d2--!. .,. (1) 34 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups where each Ik is a product of injective functors IAQfor finite subgroups Q S * *and injective abelian groups A, and where (1) is exact after restriction to C0. We* * claim that this is an injective resolution of FM . In other words, the sequence 0 ---! FM (P ) ----! I0(P ) ----! I1(P ) ----! . .,. (2) is exact for all finite P S, and we want to show it is exact for all P S. * *Fix an infiniteSsubgroup P S, and choose finite subgroups P1 P2 . . .such that P = 1j=1Pj (Lemma 1.9). Then FM (P ) = 0 = lim-FM (Pj). For all finite Q S * *and j all A, IAQ(P ) = Map (Mor C(Q, P ), A) = lim-Map (Mor C(Q, Pj), A) j since Mor C(Q, P ) is the union of the Mor C(Q, Pj); and furthermore this is an* * inverse system of surjections. Hence (2) is the inverse limit of the corresponding exa* *ct se- quences for the Pj, all restriction maps Ik(Pj+1) ---! Ik(Pj) are surjective, a* *nd so (2) is also exact. Thus *dpt(G; M) = lim-*(FM ) ~=H*(lim-(Ik), dk) ~=H*(lim-(Ik|C0), dk) ~=lim-*(FM(* *|C0).3) C C C0 C0 StepS2: Fix a sequence S1 S2 S3 . . .of finite subgroups of S such that S = 1j=1Sj (Lemma 1.9). We first construct inductively a sequence of finite s* *ubgroups H1 H2 . . .of G containing H0 such that for each j 1, Hj Sj, and Op(Hj) contains the full subcategory with object set the p-subgroups of Hj-1. Fix j * * 1, and assume that Hj-1 has been constructed. Let Cj be the full subcategory of Of* *inp(G) whose objects are the p-subgroups of (a finite group since G is local* *ly finite). Choose a finite set of morphisms in Cj which generate it, let Xj G be a finit* *e set of elements which induce those morphisms, and set Hj = . Since G is locally fi* *nite, Hj is a finite subgroup. By construction, Op(Hj) Cj; and hence contains both * *O(Sj) and the full subcategory with the same objects as Op(Hj-1). S 1 Set C0def= j=1Op(Hj). This is a full subcategory of Ofinp(G) which contains a* *ll finite subgroups of S as objects. In particular, C0 is equivalent to C0, and hence lim* *-*(FM ) ~= C0 lim-*(FM |C0). Since lim-*(FM |Op(Hj)) = 0 for all j, lim-*(FM |C0) = 0 by Lem* *ma 5.11. C0 Op(Hj) C0 6.Mapping spaces We now look at the spaces of maps from BQ to |L|^p, when Q is a discrete p-to* *ral group and L is a linking system. In general, for any p-local compact group (S, * *F, L) and any discrete p-toral group Q, we define Rep(Q, L) = Hom (Q, S)=~ , where ~ is the equivalence relation defined by setting ae ~ ae0 if there is som* *e O 2 Hom F(ae(Q), ae0(Q)) such that ae0= OOae. We want to show that [BQ, |L|^p] ~=Re* *p (Q, L). The following lemma will be needed to reduce this to the case where Q is fini* *te. The functor (-)o of Section 3 plays an important role when doing this. Carles Broto, Ran Levi, and Bob Oliver * * 35 Lemma 6.1. Fix a discrete p-toral groupSQ, and let Q1 Q2 . . .Q be a sequen* *ce of finite subgroups such that Q = 1n=1Qn. Let (S, F, L) be a p-local compact * *group. Then the following hold. (a)The natural map ~= R: Rep (Q, L) ------! lim-Rep(Qn, L) , n induced by restriction, is a bijection. (b)Assume Q S. Then for n large enough, Qon= Qo Q, and hence restriction induces a bijection Hom F(Q, P ) ~=Hom F (Qn, P ) for all P 2 Ob (Fo). Proof.In general, for any homomorphism ' 2 Hom (H, K), we let ['] denote its cl* *ass in Rep(H, K). (a) Assume first that ', _ 2 Hom (Q, S) are such that R([']) = R([_]). Thus '|* *Qn and _|Qn are F-conjugate for each n; i.e., _|Qn = ffn O'|Qn for some unique ffn* * 2 IsoF('(Qn), _(Qn)). In particular, Ker(')\Qn = Ker(_)\Qn for each n, so Ker(') = Ker(_), and _ = ff O' for some unique ff 2 Iso('(Q), _(Q)). Then ff|Qn = ffn is* * in F for each n, so ff 2 IsoF('(Q), _(Q)) by axiom (III), and [_] = ['] 2 Rep(Q, F). This proves the injectivity of R, and it remains to prove surjectivity. Fix s* *ome {['n]}n 1 2 lim-Rep(Qn, L). n Thus for each n, 'n 2 Hom (Qn, S), and 'n+1|Qn is F-conjugate to 'n. By Lemma 3.2(a), the set {'n(Qn)o| n 1} contains finitely many conjugacy classes. Sin* *ce for all n, 'n(Qn) is F-conjugate to a subgroup of 'n+1(Qn+1), 'n(Qn)o is F-conjugat* *e to a subgroup of 'n+1(Qn+1)o by Lemma 3.2(b) and Proposition 3.3. Hence for some m, 'n(Qn)o is F-conjugate to 'm (Qm )o for all n m. We now construct inductively homomorphisms '0n2 Hom (Qn, S) for all n > m such that ['0n] = ['n] in Rep (Qn, L), and '0n|Qn-1 = '0n-1. Assume '0n-1has been c* *on- structed, and set ffn = 'n O'0n-1-12 Hom F('0n-1(Qn-1), 'n(Qn)). By Proposition* * 3.3 again, this extends to a unique morphism ffon2 Hom F ('0n-1(Qn-1)o, 'n(Qn)o), w* *hich must be an isomorphism since it is injective and the two groups are abstractly * *isomor- phic and artinian. Set '0n= (ffon)-1 O'n; then '0n|Qn-1 = '0n-1. Let ' 2 Hom (Q* *, S) be the union of the '0n; then ['] 2 R-1({['n]}), and this proves the surjectivity * *of R. (b) Now assume Q S. By Lemma 3.2(a,b), for all n, Qon Qon+1 Qo, and the s* *et {Qon| n 1} is finite. Hence Qon Q for n sufficiently large, and this implies* * Qon= Qo. If P = P o S, then every ' 2 Hom F(Qn, P ) extends to a unique 'o 2 Hom F(Qon,* * P ) by Proposition 3.3, and thus Hom F(Q, P ) ~=Hom F (Qn, P ) whenever Qon= Qo. For any linking system L and any discrete p-toral group Q, we let LQ be the c* *ategory whose objects are the pairs (P, ff) for P 2 Ob (L) and ff 2 Hom (Q, P ), and wh* *ere 0 0 0 fi 0 * * 0 MorLQ (P, ff), (P , ff ) = ' 2 Mor L(P, P ) fiff = ss(') Off 2 Hom (Q, * *P ) . We next show that Map (BQ, |L|^p) ' |LQ|^pin this situation. Proposition 6.2. Fix a p-local compact group (S, F, L) and a discrete p-toral g* *roup Q. Let F0 Fc be any full subcategory which contains all F-centric F-radical subg* *roups of S, and such that P 2 Ob (F0) implies P o2 Ob (F0). Let L0 L and LQ0 LQ 36 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups be the full subcategories where Ob (L0) = Ob (F0), and Ob (LQ0) is the set of p* *airs (P, ff) 2 Ob (LQ) such that P 2 Ob (L0). Then there is a bijection ~= ss0(|LQ0|) ------! Rep (Q, L) (1) which sends a vertex (P, ff) to the class of ff as a homomorphism to S. If, fur* *thermore, we define : LQ0x B(Q) ---! L0 by setting i ' 0 0 j (P, ff), oQ = P and (P, ff) --! (P , ff ) , x = ' OffiP(ff(x)* *) , then the map | |0:|LQ0|^p-----! Map (BQ, |L0|^p) (2) adjoint to | | is a homotopy equivalence. Proof.Every vertex (P, ff) in |LQ0| is connected by an edge to the vertex (S, i* *nclSPOff). Furthermore, by the assumption that F0 contains all F-centric F-radical subgrou* *ps, together with Alperin's fusion theorem (Theorem 3.6), two vertices (S, ff) and * *(S, ff0) in |LQ0| are in the same connected component if and only if ff and ff0represent* * the same element of Rep(Q, L). This proves (1). Since (', x) = ' OffiP(ff(x)) = ffiP0(ff0(x)) O' by condition (C), is a we* *ll defined functor. It remains to prove the homotopy equivalence (2). Step 1, where we han* *dle the case Q is finite, is essentially the same as the corresponding proof in [BL* *O2 ]. In Step 2, we extend this to the general case. By assumption, for each P 2 Ob (L0), P o2 Ob (L0). So the functor (-)o of Pro* *po- sition 4.5 restricts to a functor from L0 to Lo0, and also induces a functor fr* *om LQ0to LoQ0. All of these are left adjoint to the inclusion functors, and hence induce* * homotopy equivalences between their geometric realizations. Thus, without loss of genera* *lity, we can assume that L0 = Lo0; i.e., that P = P ofor all P in L0. This assumption wi* *ll be needed at the end of each of Steps 1 and 2 below. Step 1: Assume that Q is a finite p-group. Let O(F0) Oc(F) be the full subca* *te- gory with Ob(O(F0)) = Ob (F0) = Ob (L0), and let ess:L0 --! O(F0) be the projec* *tion functor. Let essQ:LQ0--! O(F0) be the functor essQ(P, ff) = P and essQ(') = es* *s('). Let BeQ, eB:O(F0) -----! Top be the left homotopy Kan extensions over essQand ess, respectively, of the cons* *tant functors *. Then |L0| ' hocolim-----!(Be) and |LQ0| ' hocolim-----!(BeQ)(3) O(F0) O(F0) (cf. [HV , Theorem 5.5]). For each P in O(F0), eB(P ) is the nerve of the overcategory ess#P , whose ob* *jects are the pairs (R, O) for R 2 Ob (L0) = Ob (O(F0)) and O 2 RepF (R, P ), and where 0 0 0 0 Moress#P(R, O), (R , O ) = ' 2 Mor L0(R, R ) | O = O Oess(') . Let B0(P ) be the full subcategory of ess#P with the unique object (P, Id), and* * with morphisms the group of all ffiP(g) for g 2 P . Similarly, BeQ(P ) is the nerve of the category essQ#P , whose objects are th* *e triples (R, ff, O) for R 2 Ob (L0) = Ob (O(F0)), ff 2 Hom (Q, R), and O 2 Rep F(R, P );* * and where 0 0 0 0 0 * * 0 Mor essQ#P(R, ff, O), (R , ff , O ) = ' 2 Mor L0(R, R ) | ff = ss(') Off, O * *= O Oess(') . Carles Broto, Ran Levi, and Bob Oliver * * 37 Let B0Q(P ) be the full subcategory of essQ#P with objects the triples (P, ff, * *Id) for ff 2 Hom (Q, P ). Fix a section eoe:Mor(O(F0)) --! Mor(L0) which sends identity morphisms to i* *den- tity morphisms. Retractions Q 0 ess#P ---! B0(P ) and essQ#P ---! BQ(P ) are defined by setting (R, O) = (P, Id) and Q(R, ff, O) = (P, sseoe(O) Off, Id); and by sending ' in Mor ess#P((R, O), (R0, O0)) or Mor essQ#P((R, ff, O), (R0, * *ff0, O0)) to the automorphism ffiP(g) 2 AutL0(P ), where g 2 P is the unique element such that e* *oe(O0) O ' = ffiP(g) Oeoe(O) in Mor L0(R, P ) (Lemma 4.3(b)). There are natural transfor* *mations Idess#P-----!inclO and IdessQ#P-----!inclO Q of functors which send an object (R, O) to O 2 Mor ess#P((R, O), (P, Id)) and s* *imilarly for an object (R, ff, O). This shows that |B0(P )| |ess#P | and |B0Q(P )| * *|essQ#P | are deformation retracts. We have now shown that for all P 2 Ob (L0), Be(P ) ' |B0(P )| ' BP and eBQ(P ) ' |B0Q(P )| . (4) All morphisms in B0Q(P ) are isomorphisms, two objects (P, ff, Id) and (P, ff0,* * Id) are isomorphic if and only if ff and ff0are conjugate in P , and the automorphism g* *roup of (P, ff, Id) is isomorphic to CP(ffQ). Thus a eBQ(P ) ' BCP(ffQ). (5) ff2Rep(Q,P) Let eB^pand eBQ^pbe the p-completions of eBand eBQ; i.e., (Be^p)(P ) = (Be(P * *))^pand (BeQ^p)(P ) = (BeQ(P ))^p. By (3), and since the spaces eB(P ) and eBQ(P ) are * *all p-good by (4) and (5), i j i * * j |L0|^p' hocolim-----!(Be^p) ^p and |LQ0|^p' hocolim-----!(BeQ^* *p) ^p. O(F0) O(F0) Consider the commutative triangle ______________________// LQ0x B(Q) rL0 PPP rrrr PPPP rrr essQOpr1((PPPPPyyressrrr O(F0) . The left homotopy Kan extension over essQOpr1of the constant functor * is the f* *unctor eBQx BQ, and so the triangle induces a natural transformation of functors 0:BeQx BQ ------! eB. The map e :BeQ--! Map (BQ, eB) adjoint to 0 is also a natural transformation * *of functors from O(F0) to Top, and induces a commutative diagram i hjocolim(e)i j! hocolim-----!(BeQ)^p_^p_!hocolim-----!Map(BQ, eB^p)_^p!MapBQ, hocolim-----!(B* *e)^p O(F0) O(F0) O(F0) ' #| ' | # | |0 ^ |LQ0|^p________________________________________! Map (BQ, |L0|p) . 38 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups For each P S and Q0 Q, Lemma 1.10 (together with (4)) implies that each component of Map (BQ0, B(P )^p) has the form BCP(ae(Q0))^pfor some ae 2 Hom (Q0* *, P ). So all such mapping spaces are p-complete and have finite mod p cohomology in e* *ach degree. Also, O(F0) is a finite category (it has finitely many isomorphism cla* *sses of objects by Lemma 3.2(a) and has finite morphism sets by Lemma 2.5), and it has bounded limits at p by Corollary 5.5. Hence ! is a homotopy equivalence by [BLO* *2 , Proposition 4.2]. It remains only to show that e(P ) is a homotopy equivalence for each P 2 Ob * *(L0). By (4), this means showing that e(P ) restricts to a homotopy equivalence e 0(P ): |B0Q(P )| ------! Map (BQ, |B0(P )|) . Since |B0(P )| ~=BP , and since e 0(P ) is induced by the homomorphisms (incl.f* *f) from CP(ff(Q)) x Q to P , this follows from (5). Step 2: Now let Q be an arbitrary p-toral group. Let Q1 Q2 . .Q.be an increasing sequence of finite subgroups whose union is Q (Lemma 1.9). Then ss0(|LQ0|) ~=lim-ss0(|LQn0|) ~=lim-[BQn, |L0|^p] : (6) n n the first bijection holds by Lemma 6.1 and (1), and the second by Step 1. Fix ' 2 Hom (Q, S), and set 'n = '|Qn. Let Map (BQ, |L0|^p)b'be the space of * *maps f :BQ ---! |L0|^psuch that f|BQn ' B'n for each n. (This contains the connected component of B', but could, a priori, contain other components.) Let (LQ0)' * *LQ0 and (LQn0)' LQn0be the full subcategories with objects those (P, ff) such tha* *t ff is F-conjugate to ' or to 'n, respectively. Thus |(LQ0)'| is the connected compone* *nt of |LQ0| which contains (S, '), and |(LQn0)'| is the connected component which con* *tains (S, 'n). Consider the following commutative diagram, for all n 1: |(LQ0)'|^p_! Map (BQ, |L0|^p)b' | || (7) # # ' ^ |(LQn0)'|^p_!Map(BQn, |L0|p)'n . We want to show that the top row is a homotopy equivalence; the proposition then follows by taking the union of such maps as ' runs through representatives of a* *ll elements of Rep (Q, L). The bottom row is a homotopy equivalence by Step 1. So * *we will be done if we can show that the vertical maps are homotopy equivalences fo* *r n large enough. By Lemma 6.1(b), there is some m such that for all n m, '(Qn)o = '(Q)o, and restriction induces a bijection Rep F('(Q), P ) ~=Rep F('(Qn), P ) for all P 2 * *Ob (L0). (Recall that we are assuming L0 = Lo0.) This implies that |(LQ0)'| ~=|(LQn0)'| * *for all n m. Hence the components Map (BQn, |L0|^p)B'n are all homotopy equivalent for n m by Step 1, so Map (BQ, |L|^p)b'' Map (BQn, |L|^p)B'n for n m, and this * *proves that the vertical maps in (7) are equivalences. The following theorem gives a more explicit description of the set [BQ, |L|^p* *] of homotopy classes of maps, as well as of the individual components in certain ca* *ses. Carles Broto, Ran Levi, and Bob Oliver * * 39 Theorem 6.3. Let (S, F, L) be a p-local compact group, and let ` :BS --! |L|^pbe the natural inclusion followed by completion. Then the following hold, for any * *discrete p-toral group Q. (a)The natural map ~= ^ Rep(Q, L) ------! [BQ, |L|p] is a bijection. Thus each map BQ --! |L|^pis homotopic to ` OBae for some ae* * 2 Hom (Q, S). If ae, ae02 Hom (Q, S) are such that ` OBae ' ` OBae0as maps fro* *m BQ to |L|^p, then there is O 2 Hom F(ae(Q), ae0(Q)) such that ae0= O Oae. (b)For each ae 2 Hom (Q, S) such that ae(Q) is F-centric, the composite incl.Bae ` ^ BZ(ae(Q)) x BQ ------! BS ------! |L|p induces a homotopy equivalence BZ(ae(Q))^p---'--! Map (BQ, |L|^p)`OBae. (c)The evaluation map induces a homotopy equivalence Map (BQ, |L|^p)triv' |L|^p. Proof.We refer to the category LQ, and to the homotopy equivalence | |0:|LQ|^p---'---!Map (BQ, |L|^p) of Proposition 6.2. Point (a) is an immediate consequence of point (1) in the p* *ropo- sition, and (c) holds since the component of LQ which contains the objects (P, * *1) is equivalent to L. If ae 2 Hom (Q, S) is such that ae(Q) is F-centric, then the connected compon* *ent of |LQ| which contains the vertex (ae(Q), ae) contains as deformation retract the * *nerve of the full subcategory with that as its only object. Since Aut LQ(ae(Q), ae) ~= * *Z(ae(Q)), this component has the homotopy type of BZ(ae(Q)), which proves point (b). 7. Equivalences of classifying spaces We next describe the monoid Aut(|L|^p) of self homotopy equivalences of |L|^p* *(The- orem 7.1); and also show that p-local compact groups which have homotopy equiva* *lent classifying spaces are themselves isomorphic (Theorem 7.4). There is some over* *lap between the proofs in this section and those of the corresponding results for p* *-local finite groups in [BLO2 , Sections 8 & 7]; but they differ in some key respects* *, mostly due to the fact that we do not have a way to recover the category L from the sp* *ace |L|^pvia a functor from spaces to categories. We first recall some notation used in [BLO1 ] and [BLO2 ]. For any space X,* * Aut(X) denotes the monoid of self homotopy equivalences of X, and Out(X) = ss0(Aut (X)* *) is the group of homotopy classes of self equivalences. For any discrete category C* *, Aut(C) is the category whose objects are the self equivalences of C and whose morphism* *s are the natural isomorphisms between self equivalences, and Out(C) = ss0(|Aut(C)|) * *is the group of isomorphism classes of self equivalences. We consider Aut(C) as a dis* *crete strict monoidal category, in the sense that composition defines a strictly asso* *ciative functor Aut(-) x Aut(-) ------! Aut(-) 40 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups with strict identity. The nerve of Aut(C) is thus a simplicial monoid, and its * *realization |Aut(C)| is a topological monoid. Consider the evaluation functor ev: Aut(C) x C ------! C which sends a pair of objects ( , c) to (c) 2 Ob (C), and which is defined on * *morphisms by setting i O ' j i 0(')OO(c) j ev ---! 0, c ---! d = (c) --------! 0(d) . =O(d)O (') Upon taking geometric realizations, this defines a map of spaces from |Aut(C)| * *x |C| to |C|, which is adjoint to a homomorphism of topological groups C: |Aut(C)| ---! Aut(|C|). Recall that part of the structure of a centric linking system L associated to* * a fusion ffiP system is a homomorphism P ---! AutL(P ) for each P in L. We write Pffi= Im(ff* *iP), which we think of as a "distinguished subgroup" of AutL (P ) which can be ident* *ified with P . For the purposes of this paper, an equivalence of categories L --! L w* *ill be called isotypical if for each P , P,Psends the subgroup Pffi AutL(P ) to the * *subgroup (P )ffi AutL( (P )). Let Auttyp(L) be the full subcategory of Aut(L) whose ob* *jects are the isotypical equivalences, and set Outtyp(L) = ss0(|Auttyp(L)|). By [BLO2 , Lemma 8.2], when L is a linking system over a finite p-group, an * *equiva- lence : L ---! L is isotypical if and only if the triangle involving and the* * forgetful functor from L to groups commutes up to natural isomorphism. The same proof ap- plies for linking systems over discrete p-toral groups, although we won't be us* *ing that here. Clearly, any equivalence which is naturally isomorphic to an isotypical equiv* *alence is itself isotypical, and any inverse to an isotypical equivalence (inverse up * *to natural isomorphism of functors) is also isotypical. The subcategory Auttyp(L) is thus * *a union of connected components of Aut(L), and Outtyp(L) is a subgroup of Out(L). The main result of this section is the following theorem: Theorem 7.1. Fix a p-local compact group (S, F, L), and set = L. Then the composite (-)^p ^ ^p:|Auttyp(L)| -----! Aut (|L|) -----! Aut (|L|p) induces a homotopy equivalence of topological monoids from |Auttyp(L)|^pto Aut(* *|L|^p). In particular, if we let ssi(BZ^p) denote the functor Oc(F)op---! Ab which se* *nds P to ssi(BZ(P )^p) (each i 1), then Out(|L|^p) ~=Out typ(L) , ssi(Aut (|L|^p)) ~= lim-0(ssi(BZ^p))for i = * *1,,2 Oc(F) and ssi(Aut (|L|^p)) = 0 for i 3. Proof.We prove the isomorphism between groups of components in Step 2, and the homotopy equivalence between the individual components in Step 3. In Step 1, we outline the general procedure for describing the mapping space Aut(|L|^p). Assume we have fixed inclusion morphisms 'P 2 Mor L(P, S) for each P . If * *is an isotypical self equivalence of L, then clearly (S) = S, and hence S,S is an Carles Broto, Ran Levi, and Bob Oliver * * 41 automorphism of AutL(S) which sends Sffi(= Im(ffiS)) to itself. Set _ = ffi-1SO S,S|SffiOffiS 2 Aut(S). For each P 2 Ob (L), axiom (C) and the functoriality of imply that the follow* *ing diagram commutes for all g 2 P : ('P) (P )_____________! S | | (ffiP(g))2|(P)ffi | (ffiS(g))=ffiS(_(g)) # # ('P) (P )_____________! S . Hence ss( ('P))( (P )) = _(P ) (by axiom (C) again). So ('P) = '_(P)O ffP for* * a unique ffP 2 IsoL( (P ), _(P )) by Lemma 4.3(a). Thus is naturally isomorphic* * to an automorphism 0of L such that 0S,S= S,S, and 0(P ) = _(P ) and 0P,S('P) = '* *_(P) for each P . This shows that every object in Auttyp(L) is isomorphic to an isot* *ypical automorphism of L which sends inclusions to inclusions, and from now on we rest* *rict attention to such automorphisms. Step 1: Consider the decomposition pr : hocolim-----!(Be) ---'---!|L| Oc(F) of Proposition 4.6(a), where Be:Oc(F) --! Top is a rigidification of the homot* *opy functor P 7! BP . In the following constructions, we regard hocolim-----!(Be) a* *s the union of skeleta: ian a j. hocolim-----!(n)(Be) = Be(P0) x Di ~ Oc(F) i=0P0!...!Pn where we divide out by the usual face and degeneracy relations. Define functors Z, Z0: Oc(F)op---! Ab and BZ^p:Oc(F)op---! Top by setting Z(P ) = Z(P ), Z0(P ) = Z(P )0, and BZ^p(P ) = BZ(P )^p, and by sending ['] 2 Mor Oc(F)(P, Q) to '-1|Z(Q)or B '-1|Z(Q) ^p. For any eleme* *nt ^ f= fP P2Oc(F)2 lim-[B-, |L|p], Oc(F) let Map (|L|^p, |L|^p)fbe the union of the components of the mapping space whic* *h restrict to f. By [Wo ], the obstructions to this space being nonempty lie in the groups ^ i+1 ^ lim-i+1ssi(Map (B-, |L|p)f-) ~= lim- (ssi(BZp)) Oc(F) Oc(F) for i 1; the functor vanishes for i > 2, and the higher limits vanish for i =* * 2 by Proposition 5.8. Also, if Map (|L|, |L|^p)f6= ?, then the filtration of the map* *ping space i j Map (|L|^p, |L|^p) ' Map hocolim-----!(Be), |L|^p Oc(F) by the skeleta of the homotopy colimit defines a spectral sequence with E2-term ^ E2-i,j= lim-issj(Map (B-, |L|p)f-) , Oc(F) which converges to ssj-i Map (|L|, |L|^p)f . 42 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups By Theorem 6.3(b), 8 >: 0 if j 3. Since ss2(BZ^p) is acyclic by Proposition 5.8, the only obstruction to Map (|L|* *^p, |L|^p)f being nonempty lies in lim-2(Z=Z0); while the spectral sequence takes the form 8 >:- 0 otherwise. Step 2: Let Autfus(S) be the group of fusion preserving automorphisms of S; i.* *e., the group of those ff 2 Aut(S) which induce an automorphism of the fusion system F * *by sending P to ff(P ) and ' 2 Hom F(P, Q) to (ff|Q) O' O(ff|P)-1 2 Hom F(ff(P ), * *ff(Q)). The proof that Out(|L|^p) ~=Out typ(L) is based on the following diagram: ~0 ~0 !0 2 1 _______! lim-1(Z)____! Outtyp(L)____! Out fus(S)_____! lim-(Z) | | | | !1~=#| ss0( ^p)| !2|~= !3|~= (1) # # # ~ ^ ~ ! 2 1 ______! lim-1(Z=Z0)__! Out(|L|p)__! lim-IRep(-, F)_! lim-(Z=Z0) . Here, IRep(P, F) Rep(P, F) denotes the set of classes of injective homomorphi* *sms. All limits are taken over Oc(F), and !1 and !3 are induced by the natural surje* *c- tion of functors from Z onto Z=Z0. They are isomorphisms since lim-i(Z0) = 0 f* *or all i 1 (Proposition 5.8). Also, !2 is induced by the inclusion of Out fus(S* *) = Autfus(S)= AutF(S) into IRep(S, F) = Aut(S)= AutF(S), and Im(!2) = lim-IRep(-, * *F) (thus !2 is a bijection) by definition of fusion preserving. It remains to defi* *ne the two rows, and prove that they are exact and the diagram commutes. It will then fol- low immediately that ss0( ^p) is an isomorphism. Note that this does not requi* *re us to know that lim-IRep(-, F) is a group or that !0 is a homomorphism; only that Im(~) = !-1(0), Im(~0) = !0-1(0), and the inverse image under ~ of each element* * in the target is a coset of Im(~). We first consider the top row, where ~0is defined by restricting an isotypica* *l equiva- lence of L to the image of ffiS. Any_fusion preserving automorphism ff 2 Autfus* *(S) de- fines an isotypical automorphism_ffof F, and !0(ff) is the obstruction of [BLO2* * , Propo- sition 3.1] to lifting ffto an automorphism of L. (The proof in [BLO2 ] applie* *s without change to the case of a linking system over a discrete p-toral group.) Finally* *, the description of Ker(~0) is identical to that shown in [BLO1 , Theorem 6.2]. Mor* *e specifi- cally, a reduced 1-cocycle " 2 Z1(Oc(F); Z) sends each morphism ['] 2 Mor Oc(F)* *(P, Q) to "(') 2 Z(P ) (where "(IdP) = 1), and ~0(["]) is represented by the automorph* *ism A" 2 Aut(L) defined by setting A"(P ) = P for all P , and A"(_) = _ OffiP("([ss* *(_)]))-1 for all _ 2 Mor L(P, Q). This proves the exactness of the top row. As for the bottom row in (1), let ~ be the homomorphism defined by restrictio* *n: ~: Out (|L|^p) --Res---![BS, |L|^p] ~=IRep(S, F). We want to compare i j Map (|L|^p, |L|^p) ' Map hocolim-----!(Be), |L|^p Oc(F) Carles Broto, Ran Levi, and Bob Oliver * * 43 with lim-[B(-), |L|^p] ~= lim-IRep(-, F). Oc(F) Oc(F) By Step 1, the only obstruction to extending any given ff in this last set to a* *n automor- phism of |L|^plies in lim-2(Z=Z0), while if there are liftings, then the set of* * homotopy classes is in bijective correspondence with lim-1(Z=Z0). This proves the exact* *ness of the bottom row in the sense explained above. The second square in (1) clearly commutes. To prove that the first square com* *mutes, fix some " 2 Z1(Oc(F); Z). Then ~0(["]) = [A"] where A" 2 Aut(L) is the automor- phism defined above; and |A"| 2 Aut(|L|) sends each BP B(Aut L(P )) |L| to * *|L| by the inclusion. For each ' 2 Hom Fc(P, Q), let C' L be the subcategory with* * two objects P and Q, whose morphisms are those morphisms in L which get sent to [Id* *P], [IdQ], or ['] in Oc(F). Then |C'| |L| is homeomorphic to the mapping cylinder* * of B': BP ---! BQ; and |A"| sends |C'| to itself by a map which differs from the * *iden- tity via a loop in Map (BP, BQ)B' ' BZ(P ) which represents "([']) 2 Z(P ). Af* *ter taking the p-completion, this shows that [|A"|^p] = ~([~"]), where ~"2 Z1(Oc(F)* *; Z=Z0) is the class of " modulo Z0. This proves that the first square in (1) commutes. Fix ff 2 Outfus(S), and let bffbe the automorphism of the fusion system F ind* *uced by ff. Choose maps ff*P,Q MorL(P, Q) -----! Mor L(ff(P ), ff(Q)) which lift those defined by bff; then !0(ff) is the class of the 2-cocycle fi 2* * Z2(Oc(F); Z) which measures the deviation of the ff*P,Qfrom defining a functor. These same l* *iftings ff*P,Qallow us to define a map of spaces ff*: hocolim-----!(1)(Be) -----! |L|, Oc(F) and the obstruction to extending this to hocolim-----!(2)(Be) is precisely the * *class of the same 2-cocycle fi: but regarded as a 2-cocycle with coefficients in Z=Z0 ~=ss1(Map (B-, |L|^p)ff). This proves that the third square commutes, and finishes the proof that ss0( ^p* *) is an isomorphism. Step 3: Set Z(F) = lim-(Z), regarded as a subgroup of S. Let ~: B(Z(F)) x L -----! L ' be the functor which sends (x, P ---! Q) to ' OffiP(x). This is adjoint to a * *functor from B(Z(F)) to Aut(L), which in turn induces a map j0:BZ(F) ---! |Aut(L)|Id upon taking geometric realizations. On the other hand, if we first take geomet* *ric realizations, then p-complete, and then take the adjoint, we get a map j from B* *Z(F)^p to Aut(|L|^p)Id. These maps now fit together in the following commutative squar* *e: j0 BZ(F) _____'!|Aut(L)|Id | | (-)^p| | (2) # # j ^ BZ(F)^p ____'! Aut(|L|p)Id. Since we are restricting attention to automorphisms of L (as opposed to worki* *ng with all equivalences), Aut(L) is a groupoid, and so ss1(|Aut(L)|) is the group* * of 44 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups natural isomorphisms of functors from IdL to itself. A natural equivalence ff * *sends each object P to an element ff(P ) 2 Aut L(P ), such that for each ' 2 Mor L(P,* * Q), ' Off(P ) = ff(Q) O'. In particular, upon restricting to the case P = Q and ' 2* * ffiP(P ), we see that ss(ff(P )) = IdP for each P , and thus ff(P ) 2 ffiP(Z(P )) ~= Z(P * *). The other relations are equivalent to requiring that ff 2 lim-0(Z) = Z(F). This pro* *ves that ss1(|Aut(L)|) ~=Z(F); and since |Aut(L)|Idis aspherical, shows that j0 is a hom* *otopy equivalence. The E2-term of the spectral sequence for maps defined on a homotopy colimit w* *as described in Step 1: it vanishes except for the row coming from lim-*(Z=Z0), an* *d the po- sition E20,2~=lim-0(ss2(BZ^p)). Hence from the spectral sequence, one sees imme* *diately that for i 1, ssi(Aut (|L|^p)) ~= lim-(ssi(BZ^p)) ~=ssi(BZ(F)^p). Oc(F) By naturality, these isomorphisms are induced by j, and thus j is a homotopy eq* *uiva- lence. It now follows from (2) and from Step 1 that ^pinduces a homotopy equivalence Aut(|L|^p) ' |Auttyp(L)|^p. We also note here the following result, which was shown while proving Theorem* * 7.1. Proposition 7.2. For any p-local compact group (S, F, L), there is an exact seq* *uence 0 ---! lim-1(Z=Z0) -----! Out (|L|^p) -----! Out fus(S) -----! lim-2(Z=Z0), Oc(F) Oc(F) where Z0 Z :Oc(F)op---! Ab are the functors Z(P ) = Z(P ) and Z0(P ) = Z(P )* *0. In Section 9, we will show that for any compact Lie group G, there is a p-loc* *al compact group (S, F, L) = (S, FS(G), LcS(G)) such that |L|^p' BG^p. Hence when G is connected, the exact sequence of Proposition 7.2 gives a new way to descri* *be Out(BG^p), which is different from but closely related to the descriptions in [* *JMO ] and [JMO3 ]. We now turn our attention to maps between p-completed nerves of different lin* *king systems. We first look at the case where the linking systems in question are as* *sociated to the same fusion system. As usual, when we talk about an isomorphism of linki* *ng systems, we mean an isomorphism of categories which is natural with respect to * *the projections to the fusion system and with respect to the distinguished monomorp* *hisms. Lemma 7.3. Let F be a saturated fusion system over a discrete p-toral group S, * *and let F0 Fc be any full subcategory which contains Fco. Let L0 and L00be two li* *nking systems associated to F0. Assume that there is a map f :|L0|^p---! |L00|^psuch * *that the triangle BS ` @ `0 @@R f 0 ^ |L0|^p________! |L0|p is homotopy commutative. Here, ` and `0 are the maps induced by the inclusion * *of B(S) into L0 or L00. Then L0 and L00are isomorphic linking systems associated * *to ~= F0. Furthermore, we can choose an isomorphism L0 ---! L00of linking systems that induces f on p-completed nerves. Carles Broto, Ran Levi, and Bob Oliver * * 45 Proof.Let ke(L0) and ke(L00) be the left homotopy Kan extensions of the constant point functors along the projections "ss0:L0 -! O(F0) and "ss00:L00-! O(F0) res* *pec- tively. Let ~P :ke(L0)(P ) -! |L0|^pbe induced by the forgetful functor from es* *s0#P to L0, and similarly for ~0P:ke(L0)(P ) -! |L00|^p. Then ` and `0 factor through ~* *S and ~0S, and we have a homotopy commutative diagram ~S ^ ke(L0)(P9)______//ke(L0)(S)_____//|L0|p97733hhh ssss oooo hhhhhhhhh | 'sss 'oooo hhhhh | sss oooohhhhhh` | ss ooohhhhhhhh | BP _________//_KBSOVVVV f|| KKKK OOOOOVVVVVVVV0 | KKK OOO VVV`VVV | ' KKK ' OOO VVVVVV | K%% O''O VVVVVV++Vfflffl|~0 ke(L00)(P )_____//ke(L00)(S)____//|L00|^p. S Hence the maps fP :ke(L0)(P ) -! |L00|^pand f0P:ke(L00)(P ) -! |L00|^p, defined* * as the obvious composites shown in the above diagram satisfy: fP 0 ^ ' 0 f0P 0 ^ (a)The composites BP --'! ke(L0)(P ) --! |L0|p and BP --! ke(L0)(P ) --! |L0|p are homotopic, and are centric after p-completion by Theorem 6.3(b). (b)fQ Oke (L0)(') ' fP and f0QOke (L00)(') ' f0Pfor each morphism ': P -! Q of O(F0). Thus ke(L0)^pand ke(L00)^pare equivalent rigidifications of B ^pby Corollary A.* *5; and so ke(L0) and ke(L00) are equivalent rigidifications of B by Proposition 5.9. H* *ence by Proposition 4.6, L0 and L00are isomorphic linking systems associated to F0. More precisely, there is a third rigidification eBof B , and a commutative di* *agram of natural transformations between functors O(F0) ---! Top of the following form: _ _0 0 ke(L0) _____________________! Be _______ ke(L0) | | | | | | # # # f 0 ^ f1 f2 0 ^ |L0|^p______! |L0|p________'!X ________'|L0|p . Here, _(P ) and _0(P ) are homotopy equivalences for each P ; X is some space h* *omotopy equivalent to |L00|^p; all functors in the bottom row of the diagram are consta* *nt functors on O(F0) (sending all objects to the given space and all morphisms to the ident* *ity); and f1 and f2 are homotopy equivalences. Upon taking homotopy colimits of the funct* *ors in the top row, we get the homotopy commutative diagram: ' ' 0 hocolim-----!ke(L0)________________! hocolim-----!(Be)_hocolim-----!ke(L0) O(F0) O(F0) O(F0) ' | | ' | # # # f 0 ^ f1 f2 0 ^ |L0|^p__________! |L0|p____________'!X ____________'|L0|p . Here, the left and right vertical maps are homotopy equivalences by Proposition* * 4.6(a). This proves that f is a homotopy equivalence. The last statement (an isomorphi* *sm L0 ~=L00can be chosen to induce f) now follows since by Theorem 7.1, every homo* *topy equivalence from |L00|^pto itself is induced by some self equivalence of L00. 46 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups An isomorphism (S, F, L) --! (S0, F0, L0) of p-local compact groups consists * *of a triple (ff, ffF , ffL), where ffF 0 ffL 0 S --ff--!S0, F ----! F , and L ----! L are isomorphisms of groups and categories which satisfy the following compatibi* *lity conditions: (a)ffF (P ) = ffL(P ) = ff(P ) for all P S; (b)ffF and ffL commute with the projections ss :L --! F and ss0:L0---! F0; and (c)ffL commutes with the distinguished monomorphisms ffiP :P --! AutL (P ) and ffi0P:P --! AutL0(P ). We are now ready to show that the isomorphism class of a p-local compact group * *is determined by the homotopy type of its classifying space. This was shown for p-* *local finite groups in [BLO2 , Theorem 7.4]. Theorem 7.4. If (S, F, L) and (S0, F0, L0) are two p-local compact groups such * *that |L|^p' |L0|^p, then (S, F, L) and (S0, F0, L0) are isomorphic as p-local compac* *t groups. f 0 ^ Proof.If |L|^p ---! |L |p is a homotopy equivalence, then by Theorem 6.3(a), t* *here ' are homomorphisms ff 2 Hom (S, S0) and ff02 Hom (S0, S) such that the squares Bff 0 Bff0 BS ________!BS ________! BS | | | `| `0| `| # # # f 0 ^ f0 ^ |L|^p______! |L |p_______! |L|p commute up to homotopy, where f0 is any homotopy inverse to f. The composites ff0Off and ff Off0are F-conjugate to IdSand IdS0by Theorem 6.3(a) again, and th* *us ff is an isomorphism. By yet another application of Theorem 6.3(a), for any P, Q S, fi Hom F(P, Q) = ' 2 Inj(P, Q) fi`|BQ OB' ' `|BP . From this, and the corresponding result for Hom F0(ff(P ), ff(Q)), we see that * *ff induces an isomorphism of categories from F to F0. Upon replacing S0and F0by S and F, we can now assume that L and L0are two lin* *k- f* * 0 ^ ing systems associated to F, for which there is a homotopy equivalence |L|^p---* *! |L |p such that f O` ' `0. Then L ~= L0 (as linking systems associated to F) by Lemma 7.3. 8. Fusion and linking systems of infinite groups We now want to find some general conditions on an infinite group G which guar* *an- tee that we can associate to G a p-local compact group (S, FS(G), LcS(G)) such * *that |LcS(G)|^p' BG^p. This will be done in as much generality as possible. For exam* *ple, we prove the saturation of the fusion system FS(G) in sufficient generality so * *that the result also applies to the case where G is a compact Lie group. Carles Broto, Ran Levi, and Bob Oliver * * 47 At the end of the section, to show that the theory we have built up does cont* *ain some interesting examples, we show that it applies in particular to all linear * *torsion groups in characteristic different from p. We say that a group G "has Sylow p-subgroups" if there is a discrete p-toral * *subgroup S G which contains all discrete p-toral subgroups of G up to conjugacy. For * *any such G, we let Sylp(G) be the set of such maximal discrete p-toral subgroups. Lemma 8.1. Fix a group G, a normal discrete p-toral subgroup Q C G, and a subgr* *oup K G such that G = QK. Assume that K has Sylow p-subgroups. Then G has Sylow p-subgroups, and Sylp(G) = {QS | S 2 Sylp(K)}. Proof.Let Syl0p(G) = {QS | S 2 Sylp(K)}. All subgroups in Syl0p(G) are G-conjug* *ate since all subgroups in Sylp(K) are K-conjugate. If P G is an arbitrary discr* *ete p-toral subgroup, then QP is also discrete p-toral (since Q and QP=Q are discre* *te p-toral), and QP = QK \ QP = Q.(K \ QP ). Thus P QP QS 2 Syl0p(G) for any S 2 Sylp(K) which contains K \ QP . This shows that G has Sylow p-subgroups, and that they are precisely the subgroups in Syl0p(G). We first establish some general conditions on an infinite group G with Sylow * *p- subgroups, which imply that FS(G) is a saturated fusion system for S 2 Sylp(G).* * The following technical lemma will be needed when doing this. Lemma 8.2. Fix a group G, and normal subgroups N, Q C G, with the following properties: (a)Q is a discrete p-toral group. (b)G=QN is a finite group. (c)For each H G such that H N and H=N is finite, H has Sylow p-subgroups. (d)each coset gN 2 G=N contains at least one element of finite order. Then G has Sylow p-subgroups. For any discrete p-toral subgroup P G, P 2 Sylp* *(G) if and only if P Q, P \ N 2 Sylp(N), and P N=QN 2 Sylp(G=QN). Proof.Fix any G0 G such that G0 QN and G0=QN 2 Sylp(G=QN). For every discrete p-toral subgroup P G, P QN=QN is conjugate to a subgroup of G0=QN, hence P is G-conjugate to a subgroup of G0. Hence G has Sylow p-subgroups if G0 does, and in that case, Sylp(G0) is the set of subgroups of G0which are in Sylp* *(G). It thus suffices to prove the lemma when G = G0; i.e., when G=QN is a finite p-gro* *up; and we assume this from now on. Step 1: Assume first that Q = 1, and thus that |G=N| is a finite p-group. Then* * G has Sylow p-subgroups by (c). Throughout this step, we fix some S 2 Sylp(G). We* * first prove that NS = G (hence NS=N 2 Sylp(G=N)) and S \ N 2 Sylp(N). Afterwards, we prove the converse: P \ N 2 Sylp(N) and NP = G imply P is G-conjugate to S, and hence P 2 Sylp(G). If NS G, then NS=N G=N, where the latter is a finite p-group. Since every proper subgroup of a p-group is contained in a proper normal subgroup, there is* * a 48 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups proper normal subgroup Nb C G which contains NS. By (d), there is an element g 2 Gr bN of finite order. Write |g| = mpk where p - m, and set g0 = gm . Then g0 2 Gr bNsince Nb has p-power index, and |g0| = pk. This means that is a f* *inite p-subgroup of G which is not conjugate to a subgroup of S, which contradicts the assumption that S 2 Sylp(G). Thus NS = G. For all S02 Sylp(N), there are elements x 2 G and y 2 N such that xS0x-1 S * *\ N and y(S \ N)y-1 S0. Thus (yx)S0(yx)-1 S0, and this must be an equality sinc* *e S0 is artinian. It follows that S \ N = xS0x-1 2 Sylp(N). Now let P G be any subgroup such that P \N 2 Sylp(N) and P N = G. Fix x 2 G such that xP x-1 S. Then (xP x-1)N = xP Nx-1 = G, xP x-1\N = x(P \N)x-1 S \ N, and this last must be an equality since P \ N 2 Sylp(N). It follows that |G=N| = |xP x-1.N=N| = |xP x-1=(xP x-1 \ N)| |S=(S \ N)| = |SN=N| |G=N|; so these are all equalities, and P = x-1Sx 2 Sylp(G). Step 2: Now consider the general case. By assumption, G=N is an extension of the discrete p-toral group QN=N by the finite p-group G=QN, and hence is dis- crete p-toral. So by Lemma 1.9, there is a finite p-subgroup G0=N G=N such th* *at (G0=N).(QN=N) = G=N, and thus QG0 = G (since G0 N). Then G0 has Sylow p-subgroups by (c). Hence G has Sylow p-subgroups by Lemma 8.1. Also, by Step 1 applied to the pair N C G0 (recall G0=N is a p-group), P 2 Sylp(G0) () P \ N 2 Sylp(N) and P N = G0. Let P G be any discrete p-toral subgroup which contains Q, and set P0 = P \* * G0. In general, for any A, B G and C C G with C A, C.(A \ B) = A \ CB. Thus QP0 = Q.(P \ G0) = P \ QG0 = P \ G = P (1) P0N = (P \ G0).N = P N \ G0. Also, by Lemma 8.1 again, Sylp(G) = {QS | S 2 Sylp(G0)}. Hence P 2 Sylp(G) () P0 = P \ G0 2 Sylp(G0) () P0 \ N 2 Sylp(N) and P0N = G0 (Step 1) () P \ N 2 Sylp(N) and P N = G; where the last equivalence holds by (1) and since P0\ N = P \ (G0\ N) = P \ N. Let G be any group which has Sylow p-subgroups. For any S 2 Sylp(G), we let FS(G) be the fusion system over S with objects the subgroups of S and morphisms Hom FS(G)(P, Q) = Hom G(P, Q) . Proposition 8.3. Let G be a group for which the following conditions hold: (a)For each discrete p-toral subgroup P G, each element of AutG (P ) is conju* *gation by some x 2 NG(P ) of finite order. (b)For each discrete p-toral subgroup P G, and each finite subgroup H=CG(P ) NG(P )=CG(P ), H has Sylow p-subgroups. (c)For each increasing sequence P1 P2 P3 . . .of discrete p-toral subgrou* *ps of G, there is some k such that CG(Pn) = CG(Pk) for all n k. Carles Broto, Ran Levi, and Bob Oliver * * 49 Then for each S 2 Sylp(G), FS(G) is a saturated fusion system. Furthermore, the following hold for each subgroup P S. (1)CG(P ) has Sylow p-subgroups, and P is fully centralized in FS(G) if and onl* *y if CS(P ) 2 Sylp(CG(P )). (2)NG(P ) has Sylow p-subgroups, and P is fully normalized in FS(G) if and only* * if NS(P ) 2 Sylp(NG(P )). Proof.Note first that G has Sylow p-subgroups by (b), applied with P = 1. Fix S 2 Sylp(G), and let P S be any subgroup. By (a), Aut G(P ) is a torsi* *on group, so Out G(P ) is a torsion group, and hence is finite by Proposition 1.5(* *b). We first claim that P .CG(P ) NG(P ) =) has Sylow p-subgroups; (3) and that if we set S0 = S \ , then S0.CG(P ) S0 2 Sylp( ) () CS(P ) 2 Sylp(CG(P )) and _________2 Sylp( =P .CG(P )).(4) P .CG(P ) Points (3) and (4) follow from Lemma 8.2, applied with N = CG(P ) and Q = P . Conditions (c) and (d) of Lemma 8.2 follow from conditions (b) and (a) above. N* *ote that =QN is finite since OutG (P ) ~=NG(P )=QN is finite. We next prove (1) and (2). For all P S, (3) (applied with = NG(P )) impli* *es that there is Q 2 Sylp(NG(P )) such that NS(P ) Q NG(P ). Choose g 2 G such that gQg-1 S; then gQg-1 is a Sylow p-subgroup of gNG(P )g-1 = NG(gP g-1). Since gQg-1 S, gQg-1 = S \ NG(gP g-1) = NS(gP g-1). Hence NS(gP g-1) is a Sylow p- subgroup of NG(gP g-1). If P is fully normalized, then |NS(P )| |NS(gP g-1)| * *= |Q|. Since NS(P ) Q, this implies that NS(P ) = Q 2 Sylp(NG(P )). Conversely, suppose that NS(P ) 2 Sylp(NG(P )). Choose g 2 G such that gP g-1* * S and is fully normalized in FS(G). Then NS(gP g-1) 2 Sylp(NG(gP g-1)), so NS(P )* * ~= NS(gP g-1) since NG(P ) ~=NG(gP g-1), and P is also fully normalized. This proves (2). The proof of (1) (the condition for P to be fully centraliz* *ed) is similar, except that CG(P ) has Sylow p-subgroups by (b). We now prove that FS(G) is saturated. (I) Assume that P S is fully normalized in FS(G). We have already seen that OutG (P ) is finite (since it is a torsion group by (a)). Also, NS(P ) 2 Sylp(* *NG(P )) by (2). So by (4), applied with = NG(P ), CS(P ) 2 Sylp(CG(P )) (hence P is f* *ully centralized by (1)), and OutS(P ) 2 Sylp(Out G(P )). (II) Let P S be an arbitrary subgroup, and let g 2 G be such that P 0def=gP* * g-1 S is fully centralized. Set = NS(P 0).CG(P 0), and define fi -1 0 fi -1 N = x 2 NS(P ) ficg Ocx Ocg 2 AutS(P ) = x 2 NS(P ) figxg 2 . Then CS(P 0) 2 Sylp(CG(P 0)) by (1), and so by (4), NS(P 0) is a Sylow p-subgro* *up of = NS(P 0).CG(P 0) (S0.CG(P ) = in the notation of (4)). Since gNg-1 is a di* *screte p- toral subgroup of , it is -conjugate to a subgroup of NS(P 0). Thus there are* * elements x 2 NS(P 0) and y 2 CG(P 0) such that (xyg)N(xyg)-1 NS(P 0). Then (yg)N(yg)-1 NS(P 0) S, and cyg2 Hom G(N, S) is an extension of cg 2 Hom G(P, S). 50 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups S (III) Let P1 P2 P3 . . .be a sequence of subgroups of S, and set P1 = 1 n=1Pn. Assume ' 2 Hom (P1 , S) is a monomorphism such that '|Pn 2 Hom G(Pn, S) for each n. Fix elements gn 2 G for each n such that '(x) = gnxg-1n, for x 2 Pn* *. Then for all 1 k < n, g-1ngk 2 CG(Pk). By (c), there is k such that CG(Pk) = CG(Pn) = CG(P1 ) for all n k. Hence f* *or all n k and all x 2 Pn, '(x) = gnxg-1n= gkxg-1k, and thus ' = cgk2 Hom G(P1 , S). In general, for any group G, we define a p-centric subgroup of G to be a disc* *rete p-toral subgroup P G such that Z(P ) is the unique Sylow p-subgroup of CG(P ) (i.e., every discrete p-toral subgroup of CG(P ) is contained in Z(P )). Equiva* *lently, P is p-centric if and only if CG(P )=Z(P ) has no elements of order p. Proposition 8.4. Let G be any group which has Sylow p-subgroups, and fix S 2 Sylp(G). Then a subgroup P S is FS(G)-centric if and only if P is p-centric * *in G. Proof.Assume that P is p-centric in G; i.e., that Z(P ) 2 Sylp(CG(P )). For ev* *ery g 2 G such that gP g-1 S, CS(gP g-1) is a discrete p-toral subgroup of CG(gP * *g-1) = gCG(P )g-1, and Z(gP g-1) = gZ(P )g-1 is a Sylow p-subgroup (hence the unique o* *ne) of gCG(P )g-1. It follows that Z(gP g-1) = CS(gP g-1) for all such g, and so P* * is FS(G)-centric. Conversely, suppose that P S is FS(G)-centric. Let Q be any discrete p-tora* *l sub- group of CG(P ). Then QP is a discrete p-toral subgroup, and hence there is an * *element g 2 G such that g(QP )g-1 S. Thus gP g-1 S, and gQg-1 S \ CG(gP g-1) = CS(gP g-1). Since P is FS(G)-centric, this shows that gQg-1 Z(gP g-1), and th* *us that Q Z(P ). In other words, every discrete p-toral subgroup of CG(P ) is co* *ntained in Z(P ), and so P is p-centric in G. We now restrict attention to locally finite groups. For any such group G, fo* *r the purposes of this section, we define Op(G) C G be the subgroup generated by all * *ele- ments of order prime to p. This clearly generalizes the usual definition of Op(* *G) for finite G (although it is not the only generalization). Proposition 8.5. If G is locally finite, then a discrete p-toral subgroup P * *G is p-centric if and only if CG(P ) = Z(P ) x Op(CG(P )) and all elements of Op(CG(* *P )) have order prime to p. Proof.By the above definition, a discrete p-toral subgroup P G is p-centric i* *f and only if CG(P )=Z(P ) has no elements of order p. So if P is not p-centric, then* * either Op(CG(P )) has p-torsion, or CG(P ) is not generated by Z(P ) and Op(CG(P )). Conversely, assume that P is p-centric, and thus that CG(P )=Z(P ) has no p-t* *orsion. Consider the universal coefficient exact sequence 2 0 ---! Ext H1(CG(P )=Z(P )), Z(P ) ----! H (CG(P )=Z(P ); Z(P )) ----! Hom H2(CG(P )=Z(P )), Z(P ) ---! 0. By assumption, all elements of CG(P )=Z(P ) have order prime to p, all elements* * of Z(P ) have p-power order, and both groups are locally finite. Hence for i = 1,* * 2, Hi(CG(P )=Z(P )) is a direct limit of finite abelian groups of order prime to p* *, and thus a torsion group all of whose elements have order prime to p. This shows that al* *l terms in the above sequence vanish. Hence the central extension 1 --! Z(P ) ----! CG(P ) ----! CG(P )=Z(P ) --! 1 Carles Broto, Ran Levi, and Bob Oliver * * 51 splits, so CG(P ) ~=Z(P ) x (CG(P )=Z(P )), and all elements of the group Op(CG* *(P )) ~= CG(P )=Z(P ) have order prime to p. When working with fusion systems over discrete p-toral groups and their orbit* * cate- gories, we are able to reduce certain problems to ones involving finite categor* *ies using the functor (-)o constructed in Section 3. This is not a functor on the orbit c* *ategory of a group, and so we need a different way to make such reductions. For any gro* *up G with Sylow p-subgroups, we let X = X(G) denote the set of all subgroups of G wh* *ich are intersections of (nonempty) subsets of Sylp(G). Since discrete p-toral gro* *ups are artinian, it makes no difference whether we require finite intersections or all* *ow infinite intersections. Lemma 8.6. Let G be a group such that for each discrete p-toral subgroup P G, NG(P ) has Sylow p-subgroups. Assume, for every increasing sequence P(1) P(2) P(3) . . .of discrete p-toral subgroups of G, that the union of the P(i)is aga* *in a discrete p-toral group, and that there is some k such that CG(P(n)) = CG(P(k)) * *for all n k. Then the set X(G) contains finitely many G-conjugacy classes. Proof.For each discrete p-toral subgroup P G, we let P O P denote the inters* *ection of all Sylow p-subgroups of G which contain P . We first prove (1)For each discrete p-toral subgroup P G, there is a finite subgroup P 0 P * *such that P 0O= P O. To see this, set pn = exp(ss0(S)) for S 2 Sylp(G). The discrete p-torus S0 is * *the union of an increasing sequence of finite p-subgroups, and since centralizers s* *tabilize by assumption, therenis a finite subgroup Q P0 such that CG(Q) = CG(P0). Set Q0 = {x 2 P0| xp 2 Q}: also a finite p-subgroup. By Lemma 1.9, there is a fin* *ite subgroup P 0 P such that P 0 Q0and P 0P0 = P . Fix S 2 Sylp(G) which contains P 0. Then S0 Q (since S Q0), and hence S0 CG(Q) = CG(P0). Since S0 is a maximal discrete p-torus in G and S0.P0 is a* *lso a discrete p-torus, this implies that S0 P0. Hence S P 0P0 = P . Since this h* *olds for all S 2 Sylp(G) which contains P 0, we have shown that P 0O= P O; and this fini* *shes the proof of (1). __ Let X (G) be the set of G-conjugacy classes of subgroups_in X(G). We let (P ) denote the conjugacy class of the subgroup P , and_make X(G) into a poset by se* *tting (P ) (Q) if P xQx-1 for some x 2 G. Let P X (G) be the set of all classes (P_) which are contained in infinitely many other classes. We claim that P = ?.* * Since X(G) contains a smallest element which is contained in all the others_(the clas* *s of the intersection of all Sylow p-subgroups of G), P = ? implies that X(G) is finite,* * which is what we want to prove. Assume otherwise: assume P 6= ?. We claim that P has a maximal element. For any totally ordered subset P0 of P, upon restricting to those subgroups of maxi* *mal rank, we obtain a sequence of subgroups P(1) P(2) P(3) . . .whose conjugacy classes are cofinal in P0. If thisSsequence is finite, then P0 clearly has a m* *aximal element. Otherwise, set P(1) = 1i=1P(i), and let P 0 P(1) be a finite subgro* *up such that P 0O= P(1) (apply (1)). Then P 0 P(k)for some k, and so (P(k)) = (P(1)) i* *s a maximal element in P0. Thus by Zorn's lemma, P contains a maximal element (Q), and clearly Q =2Sylp(* *G). Since NG(Q) has Sylow p-subgroups, there is some S 2 Sylp(G) such that every p-* *toral 52 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups subgroup of G containing Q with index p is G-conjugate to a subgroup of NS(Q); * *and hence by Lemma 1.4, there are finitely many G-conjugacy classes_of such subgrou* *ps. Hence since (Q) is contained in infinitely many classes in X(G), the same holds* * for (Q0) for some Q0 G such that Q C Q0with index p. Then (Q0O) 2 P, which contradicts the maximality assumption about Q. So P contains no maximal element, hence must be empty, and so X(G) has finitely many G-conjugacy classes. Now, for any discrete group G which has Sylow p-subgroups, let Lcp(G) be the * *cate- gory whose objects are the p-centric subgroups of G, and where Mor Lcp(G)(P, Q) = NG(P, Q)=Op(CG(P )). For any S 2 Sylp(G), let LcS(G) Lcp(G) be the equivalent full subcategory who* *se objects are the subgroups of S which are p-centric in G. It will be convenient, throughout the rest of this section, to use the term "* *p-group" to mean any group each of whose elements has p-power order. It is not hard to s* *how that if G is locally finite, and has Sylow p-subgroups in the sense described a* *bove, then every p-subgroup of G is a discrete p-toral subgroup. Hence there is no l* *oss of generality to assume this in the following theorem. Theorem 8.7. Let G be any group which satisfies the following conditions: (a)G is locally finite. (b)Each p-subgroup of G is a discrete p-toral group. (c)For any increasing sequence A(1) A(2) A(3) . .o.f finite abelian p-subgro* *ups of G, there is some k such that CG(A(n)) = CG(A(k)) for all n k. Then G has a unique conjugacy class Sylp(G) of maximal discrete p-toral subgrou* *ps. For any S 2 Sylp(G), (S, FS(G), LcS(G)) is a p-local compact group, with classi* *fying space |LcS(G)|^p' BG^p. Proof.We first apply Proposition 8.3 to show that FS(G) is a saturated fusion s* *ystem over S. Once this has been checked, then it easily follows that LcS(G) is a cen* *tric linking system associated to FS(G): condition (A) in Definition 4.1 holds by Propositio* *ns 8.5 and 8.4, and conditions (B) and (C) are immediate. It then will remain only to * *show that |LcS(G)|^p' BG^p. By [KW , Theorem 3.4], conditions (a) and (c) above imply that all maximal p- subgroups of G are conjugate, and hence (by (b)) that G has Sylow p-subgroups. * *Since these three conditions are carried over to subgroups of G, this also shows that* * each subgroup of G has Sylow p-subgroups. This proves condition (b) in Proposition 8* *.3, and condition (a) holds since G is locally finite. It remains to prove condition (c) in Proposition 8.3, which we state here as: (d)For any increasing sequence P(1) P(2) P(3) . .o.f discrete p-toral subgro* *ups of G, there is some k such that CG(P(n)) = CG(P(k)) for all n k. To see this, fix any such sequence, and let P(1) be its union. Let A = (P(1))0 * *be the identity component, and set A(i)= A \ P(i)for all i. Let r be such that P(i)sur* *jects onto ss0(P(1)) for all i r; equivalently, P(i).A = P(1) for all i r. ForSe* *ach i, let A0i A(i)be the finite subgroup of elements of order at most pi. Then A = 1i=* *1A0i; and so by (c) there is r such that CG(A) = CG(A0r). Hence CG(A(i)) = CG(A(r)) f* *or Carles Broto, Ran Levi, and Bob Oliver * * 53 all i r (since A0r A(r) A(i) A). We can assume that r is chosen large enou* *gh so that P(r)surjects onto P(1)=A; i.e., such that P(r).A = P(1). Then for all * *i r, P(i)= A(i).P(r), CG(P(i)) = CG(A(i)) \ CG(P(r)) = CG(A(r)) \ CG(P(r)) = CG(P(r)), and this finishes the proof of (d). We have now shown that the hypotheses of Proposition 8.3 hold, and thus that FS(G) is a saturated fusion system over S. We have already seen that LcS(G) is* * a linking system associated to FS(G), and it remains only to show that |LcS(G)|^p* *' BG^p. As in Section 5, for any discrete group G, we let Op(G) be the category whose* * objects are the discrete p-toral subgroups P G, and where MorOp(G)(P, Q) = Q\NG(P, Q) ~=Map G(G=P, G=Q). Let OX(G) Op(G) be the full subcategory with object set X = X(G): the set of * *all intersections of subgroups in Sylp(G). For each discrete p-toral subgroup P G* *, we let P O2 X denote the intersection of all subgroups in Sylp(G) which contain P . Cl* *early, for any P and Q, NG(P, Q) NG(P O, QO), and so this defines a functor (-)O from Op(G) to OX(G). Since NG(P O, Q) = NG(P, Q) when Q 2 X(G), the two functors incl OX(G) -------!-------Op(G) (1) (-)O are adjoint. Step 1: Let I and be the following functors from Op(G) to (G-)spaces: I(P ) = G=P and (P ) = EG xG I(P ) ~=EG=P. Then for any full subcategory C Op(G), ia1 a j. hocolim-----!(I) = G=P0 x n ~ C n=0G=P0!...!G=Pn is the nerve of the category whose objects are the cosets gP for all P 2 Ob (C)* *, and with a unique morphism gP ! hQ exactly when gP g-1 hQh-1. When C = OX(G), this category has as initial object the intersection of all Sylow p-subgroups o* *f G, and hence hocolim-----!OX(G)(I) is contractible. Since the Borel construction comm* *utes with homotopy colimits in this situation (being itself a special case of a homotopy * *colimit), i j hocolim-----!( ) ~=EG xG hocolim-----!(I) ' BG. (2) OX(G) OX(G) Step 2: Fix some Q 2 X which is not p-centric. For each i 0, consider the fu* *nctor ( Hi(BP ; Fp) if P is G-conjugate t* *o Q Fi[Q]:Op(G)op ___! Ab where Fi[Q](P ) = 0 otherwise. The subgroup CG(Q).Q=Q ~=CG(Q)=Z(Q) of AutOp(G)(Q) = N(Q)=Q acts trivially on Fi[Q](Q), and contains an element of order p since Q is not p-centric. Hence by* * Lemmas 5.10 and 5.12, [Q] lim-*(Fi[Q]) ~= lim-*(Fi[Q]) ~= * NG(Q)=Q; Fi (P ) = 0 for all,i OX(G) Op(G) where the first isomorphism follows from the adjoint functors (1). 54 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups Step 3: Now let OcX(G) OX(G) be the full subcategory with objects the p-cent* *ric subgroups which lie in X. Let P1, P2, . .,.Pm S be representatives for thos* *e G- conjugacy classes in X(G) which are not p-centric (a finite set by Lemma 8.6). * * We assume these are ordered such that |Pi| |Pi+1| for each i. For each i 0, consider the functor Fi:Op(G)op ___! Ab , defined by setting* * Fi(P ) = Hi(BP ; Fp) for all P . For all k = 0, . .,.m, define functors ( 0 if P ~ Pj, some j k Fi,k:OX(G)op ___! Ab by setting Fi,k(P ) = G Fi(P ) otherwise. Here, "~ " means "G-conjugate", and these are all defined to be quotient functo* *rs of G Fi|OX(G). In particular, Fi,0= Fi|OX(G) and Fi,m= Fi|OcX(G). Also, for all k, [P ] Ker Fi,k---i Fi,k+1~= Fi k|OX(G), and the higher limits of this last functor vanish by Step 2. So there are isomo* *rphisms lim-*(Fi) = lim-*(Fi,0) ~= lim-*(Fi,1) ~=. .~.=lim-*(Fi,m) ~= lim-*(F* *i); OX(G) OX(G) OX(G) OX(G) OcX(G) whose composite is induced by restriction from OX(G) to OcX(G). The spectral sequence for the cohomology of a homotopy colimit now implies th* *at the inclusion of OcX(G) into OX(G) induces a mod p homology isomorphism of homotopy colimits of , and hence a homotopy equivalence ^ ' ^ hocolim-----!( ) p ------! hocolim-----!( ) p . (3) OcX(G) OX(G) Also, the adjoint functors in (1) restrict to adjoint functors between OcX(G) a* *nd Ocp(G), and hence induce a homotopy equivalence ^ ^ hocolim-----!( ) p ' hocolim-----!( ) p . (4) OcX(G) Ocp(G) Step 4: Let Tpc(G) be the centric transporter category for G: the category wh* *ose objects are the p-centric subgroups of G, and where the set of morphisms from P* * to Q is the transporter NG(P, Q). By exactly the same argument as in [BLO1 , Lemma * *1.2], hocolim-----!( ) ' |Tpc(G)|. (5) Ocp(G) The canonical projection functor Tpc(G) ___! Lcp(G) satisfies all of the hypot* *heses of the functor in [BLO1 , Lemma 1.3], except that we only know that p K(P ) def=KerAutTpc(G)(P ) --i AutLcp(G)(P ) = O (CG(P )) is a locally finite group all of whose elements have order prime to p (not nece* *ssarily a finite group). But this suffices to ensure that coinvariants preserve exact s* *equences of Z(p)[KP]-modules, which is the only way this property of KP is used in the p* *roof of [BLO1 , Lemma 1.3]. Hence the induced map |Tpc(G)| ------! |Lcp(G)|. is a mod p homology equivalence. Together with (2), (3), (4), and (5), this sho* *ws that |LcS(G)|^p' |Lcp(G)|^p' |Tpc(G)|^p' BG^p. Carles Broto, Ran Levi, and Bob Oliver * * 55 We now finish the section by exhibiting a more concrete class of groups which* * satisfy the hypotheses of Theorem 8.7. A linear torsion group is a torsion subgroup of * *GLn(k), for any positive integer n and any (commutative) field k. These are also refer* *red to as "periodic linear groups", since their elements are all periodic transform* *ations (automorphisms of finite order) of a finite dimensional vector space. The following facts about linear torsion groups are the starting point of our* * work here. Proposition 8.8. The following hold for every field k, and every linear torsion* * group G GLn(k). (a)G is locally finite. (b)For p 6= char(k), every p-subgroup of G is a discrete p-toral group. Proof.Point (a) is a theorem of Schur, and is shown in [W , Corollary 4.9]. By * *[W , 2.6], every locally finite p-subgroup of GLn(k) is artinian (when p 6= char(k)), and * *hence is discrete p-toral by Proposition 1.2. In order to apply Theorem 8.7, it remains only to check that centralizers of * *discrete p-toral subgroups of linear torsion groups stabilize in the sense of Theorem 8.* *7. Proposition 8.9. Let A1 A2 A3 . .b.e an increasing sequence of finite abe* *lian p-subgroups of a linear torsion group G GLn(k), where char(k) 6= p. Then ther* *e is r such that CG(Ai) = CG(Ar) for all i r. Proof.Upon replacing k by its algebraic closure if necessary, we can assume tha* *t k is algebraically closed. Hence any representation over k of a finite abelian p-* *group A splits as a sum of 1-dimensional irreducible representations. Moreover, if A * *GLn(k), and kn = U1 . . .Um is the unique decomposition with the property that each Ui* *is a sum of irreducible modules with the same character and different Ui correspon* *d to different characters of A, then Ym Ym CGLn(k)(A) ~= Autk(Ui) ~= GLdi(k). (di= dim(Ui)) i=1 i=1 From this observation, it is clear that for any increasing sequence of such sub* *groups Ai, the centralizers CGLn(k)(Ai) stabilize for i sufficiently large, and hence the * *stabilizers CG(Ai) also stabilize. Propositions 8.8 and 8.9 show that linear torsion groups satisfy all of the h* *ypotheses of Theorem 8.7. So as an immediate consequence, we get: Theorem 8.10. Fix a linear torsion group G, a prime p different than the defini* *ng characteristic of G, and a Sylow subgroup S 2 Sylp(G). Then (S, FS(G), LcS(G)) * *is a p-local compact group, with classifying space |LcS(G)|^p' BG^p. 9. Compact Lie groups Throughout this section, we fix a compact Lie group G and a prime p. Our main result is to show that G defines a p-local compact group whose classifying spac* *e has the homotopy type of BG^p. 56 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups A compact Lie group P is called p-toral if its identity_component is a torus * *and if its group of components is a p-group._ The closure P of a discrete p-toral sub* *group P G is a p-toral_group, since P0 is abelian and connected, hence a torus, and* * has p-power index in P. We will generally denote p-toral groups (including tori) by* * P , Q , T , etc., to distinguish them from discrete p-toral groups P , Q, T , etc. Our * *first task is to identify the maximal (discrete) p-toral subgroups of G. Definition 9.1. (a)For any p-toral group P , Sylp(P ) denotes the set of discre* *te p- toral subgroups P P such that P .P0= P and P contains all p-power torsion * *in P 0. __ (b)A discrete p-toral subgroup P G is snugly embedded if P 2 Sylp(P ). ___ (c)Sylp(G) denotes the set of all p-toral subgroups S G such that the identit* *y com- ponent S0 is a maximal torus of G and S=S 02 Sylp(N(S 0)=S 0). __ (d)Sylp(G) denotes the set of all discrete p-toral subgroups P G such that P* * 2 ___ __ Sylp(G) and P 2 Sylp(P ). __ For any discrete_p-toral_subgroup_P_ G, P0 is_a torus, as noted above, and h* *as finite index in P . Hence P0 = (P )0, and ss0(P ) ~=P=P0. So P is snugly embedd* *ed_in G if and only if P0 is snugly embedded, and this holds exactly when rk(P ) = rk* *(P ). As an example of a subgroup which is not snugly embedded, one can construct a r* *ank one subgroup P ~=Z=p1 which is densely embedded in a torus (S1)r for r > 1. __ __ Clearly, when rk(P ) < rk(P ), we cannot expect BP ^pand BP ^pto have the same homotopy type. But we do get a homotopy equivalence when P is snugly embedded. __ Proposition 9.2. If P G is snugly_embedded, then the inclusion of P in P indu* *ces a homotopy equivalence BP ^p' BP ^p. __ Proof.This means showing that the inclusion of BP into BP induces an isomorphi* *sm on mod p cohomology. See, for example, [Fe, Proposition 2.3]. ___ The following proposition is well known. It says that Sylp(G) is the set of m* *aximal p- toral subgroups of G, that Sylp(G) is the set of maximal discrete p-toral subgr* *oups of G, and that each of these sets contains exactly one G-conjugacy class. Note in par* *ticular the case where G = P is p-toral: there is a unique conjugacy class of discrete * *p-toral subgroups snugly embedded in P , and every discrete p-toral subgroup of P is co* *ntained in a snugly embedded subgroup. Proposition 9.3. The following hold for any compact Lie group G and any p-toral group P . ___ (a)Any two subgroups in Sylp(G) are G-conjugate,_and each p-toral subgroup P G is contained in some subgroup S 2 Sylp(G). (b)Any two subgroups in Sylp(G) are G-conjugate, and each discrete p-toral subg* *roup P G is contained in some subgroup S 2 Sylp(G). ___ Proof.(a) The subgroups in Sylp(G) are clearly all_conjugate_to each other, si* *nce all maximal tori in G are conjugate. For any S 2 Sylp(G) with identity compone* *nt the maximal torus T = S 0, O(G=N(T )) = 1 (see [Br, Proposition 0.6.3]), and h* *ence Carles Broto, Ran Levi, and Bob Oliver * * 57 O(G=S ) is prime to p. If Q is an arbitrary p-toral subgroup of G, then_O((G=S* * )Q ) is congruent mod p to O(G=S ), so (G=S )Q 6= ?, and hence Q gS g-1 2 Sylp(G) for some g 2 G. (b) Assume first that G = P is p-toral. Set T = P 0, and let T T be the subg* *roup of elements of p-power torsion. By definition, Sylp(P ) is the set of all subgr* *oups P T such that P=T is the image of a splitting of the extension 1 --! T =T ---! P=T ---! P=T --! 1. (1) The cohomology groups Hi(P =T ; T =T ) vanish for all i > 0, since P =T is a p* *-group and T =T is uniquely p-divisible. Hence the extension (1) is split, and any two* * splittings are conjugate by an element of T =T . Thus Sylp(P ) 6= ?, and its elements are * *conjugate to each other by elements of T . Now let Q P be an arbitrary discrete p-toral subgroup. Then QT is also a di* *screte p-toral subgroup (since T C P ), and QT=T is the image of a splitting of the ex* *tension of T =T by QT =T . We have seen that any two such splittings are conjugate by elem* *ents of T =T , and hence they all extend to splittings of the extension by P =T . I* *n other words, there is a subgroup P 2 Sylp(P ) which contains QT , and hence contains * *Q. _ Now let G be an arbitrary compact Lie group. For any S, S0 2 Sylp(G), S is G- __ _ ______ _ conjugate to S0 by (a), so S = gS0g-1 for some g 2 G, and S, gS0g-1 2 Sylp(S ).* * We _ have just shown that all subgroups in Sylp(S ) are conjugate, and hence S and S* *0 are * * __ conjugate. If P G is an arbitrary discrete p-toral subgroup, then_its_closure* * P is a p-toral subgroup, and hence contained in some maximal subgroup S 2 Sylp(G) by (* *a) again. So there is some S 2 Sylp(S ) Sylp(G) which contains P . We next need some information about the outer automorphisms of (discrete) p-t* *oral subgroups of G. Lemma 9.4. The following hold for all discrete p-toral subgroups P, Q G. (a)If P Q, then Out _Q(P ) is a finite p-group, and Out _Q(P ) = Out Q(P ) if* * Q is snugly embedded in G. In particular, OutQ_(Q) = 1 if Q is snugly embedded in* * G. __ (b)Out G(P ) and OutG (P ) are both finite. (c)If Q is snugly embedded, then the natural map ~= __ __ Rep G(P, Q) ------! Rep G(P , Q) is a bijection. __ __ __ Proof.(a) Choose Q0 Q such that Q0 = Q and Q0 is snugly embedded. Then OutQ0(P ) is a finite p-group by Proposition 1.5(c). The first statement thus * *follows from the second. Now assume Q is snugly embedded. We must show that Out _Q(P ) = Out Q(P ); or equivalently that AutQ_(P ) = AutQ (P ). Fix x 2 N_Q(P ), and set |P | = pk. __ __ __ Let Q =Q be the set of left cosets gQ for_g 2 Q ,_and let (Q =Q)P be the fixe* *d point set of the left P -action._Then for g 2 Q , gQ 2 (Q =Q)P if and_only_if_g-1P g * * Q. In particular, xQ 2 (Q =Q)P since x normalizes P and P Q. Since Q =Q = Q0=Q0 and 58 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups * * __ the latter group is uniquely p-divisible (since Q is snugly embedded), there is* * y 2 Q 0 k __ P such that yp 2 xQ and yQ 2 (Q =Q) . Set Y k Y k by= (aya-1) = yp . ((y-1ay).a-1) 2 yp Q = xQ; a2P a2P where the inclusion holds since P Q and y-1P y Q. Then by2 C_Q(P ). Since* * x was arbitrary, this proves that N_Q(P ) = C_Q(P ).NQ(P ), and finishes the proo* *f that AutQ_(P ) = AutQ (P ). In the case P = Q, this shows that OutQ_(Q) = OutQ (Q) =* * 1. (b) The kernel of the homomorphism __ OutG(P ) ------! Out G(P ) (1) is OutP_(P ). By (a), this is always finite, and is trivial if P is snugly embe* *dded. If P __ __ __ * * __ is snugly embedded, i.e., if P 2 Sylp(P ), then NG(P ) = P.NG(P ) (any subgroup* * of P __ which is G-conjugate to P is also P -conjugate to_P_), and_hence_the_map_in (1)* * is also surjective. Thus in this case, Out G(P ) ~=Out G(P ) ~=NG(P )=P .CG(P ) is a co* *mpact_ Lie group, all torsion subgroups of which are finite by Theorem 1.5. Thus Out * *G(P ) is finite (since otherwise it would contain S1). If P is an arbitrary discrete* * p-toral subgroup of G, then the kernel and the image of the map in (1) are finite, and * *hence OutG (P ) is also finite in this case. (c) Assume P, Q S, where_Q_is snugly embedded. We must show that the map from RepG (P, Q) to RepG (P , Q) which sends_a_homomorphism_to_its unique conti* *nuous extension is_a_bijection. For any_' 2 Hom G(P , Q), '(P ) is Q -conjugate to a * *subgroup of Q 2 Sylp(Q ), and hence ' is Q-conjugate to a homomorphism which sends P int* *o Q. This proves surjectivity._ To_prove injectivity, fix '1, '2 2 Hom G (P, Q) whic* *h induce the same class in Rep G(P , Q), and set Pi = Im ('i). Then '2 = O O'1 for some O 2 Iso_Q(P1, P2). We must show that O 2 IsoQ(P1, P2), and it suffices to do th* *is when _ O 2 Iso_Q0(P1, P2). In this case, P1Q0 = P2Q0, so O extends to O 2 Aut _Q0(P1Q* *0). Also, P1Q0 is snugly embedded since Q is, so OutQ_0(P1Q0) = 1 by (b), and hence* * O is __ conjugation by an element of Q 0\ P1Q0 = Q0. The fusion system of a compact Lie group is defined exactly as in Section 8. * *For any S 2 Sylp(G), FS(G) is the fusion system over S where for P, Q S, Mor FS(G)(P, Q) = Hom G(P, Q) ~=NG(P, Q)=CG(P ) is the set of homomorphisms from P to Q induced by conjugation by elements of G. Here, as usual, NG(P, Q) = {x 2 G | xP x-1 Q} denotes the transporter set. Lemma 9.5. For each maximal discrete p-toral subgroup S 2 Sylp(G), FS(G) is a saturated fusion system over S. Also, a subgroup P S is fully centralized in * *FS(G) if and only if CS(P ) 2 Sylp(CG(P )). Proof.We must show that conditions (a), (b), and (c) of Proposition 8.3 all hol* *d. For each discrete p-toral subgroup P G, Out G(P ) is finite by Lemma 9.4(b),* * so AutG (P ) ~=NG(P )=CG(P ) is a torsion group. Hence for each g 2 NG(P ), .CG* *(P ) is a finite extension of CG(P ), thus a closed subgroup, and so the coset gCG(P ) * *contains Carles Broto, Ran Levi, and Bob Oliver * * 59 elements of finite order. Also, for each finite subgroup H=CG(P ) in NG(P )=CG(* *P ), H is a closed subgroup of G, and hence has Sylow p-subgroups in the sense of Sect* *ion 8. If P1 P2 P3 . . .is an increasing sequence of discrete p-toral subgroups * *of G, then the centralizers CG(Pi) form a decreasing sequence of closed subgroups of * *G, and hence is constant for i sufficiently large. Thus Proposition 8.3 applies: for any S 2 Sylp(G), FS(G) is a saturated fusi* *on system over S, and a subgroup P S is fully centralized in FS(G) if and only * *if CS(P ) 2 Sylp(CG(P )). Recall from Section 8 that a discrete p-toral subgroup P G is called p-cent* *ric in G if Z(P ) 2 Sylp(CG(P )). By analogy with this definition, a p-toral subgr* *oup ___ P G is called p-centric if Z(P ) 2 Sylp(CG(P )). We next note some conditio* *ns which characterize p-toral and discrete p-toral subgroups of G which are p-cent* *ric. Lemma 9.6. The following hold for any discrete p-toral subgroup P G. __ __ __ __ __ (a)If P is p-centric in G, then NG(P )=P is finite, and CG(P )=Z(P ) is finite* * of order prime to p. __ (b)If P is p-centric in G, then P is p-centric in G. __ (c)If P is p-centric in G and P is snugly embedded, then P is p-centric in G. Proof.(a) Assume P is p-centric in G, and consider the groups CG(P )=Z(P ) ~=P .CG(P )=P and OutG (P ) ~=NG(P )=(P .CG(P )). The first group is finite of order prime to p since Z(P ) is a maximal p-toral * *subgroup of CG(P ) which is also central. The second group is finite by Lemma 9.4(b). He* *nce NG(P )=P is also finite. _____ __ (b) If P is_p-centric_in G,_then Z(P ) Z(P ) is a maximal p-toral subgroup in CG(P ) = CG(P ), and hence P is also p-centric in G. (c) Assume P 2 Sylp(P ). If x 2 Z(P ) has p-power order, then since [x, P0] = * *1, the only elements of p-power order in xP 0 are those in xP0. Since some element of * *xP 0 lies in P and has p-power order, this shows that x 2 P , and hence that x 2 Z(P* * ). In other words, Z(P ) 2 Sylp(Z(P )). So if P is p-centric in G, then Z(P ) is a m* *aximal discrete p-toral subgroup of CG(P ) = CG(P ), and hence P is also p-centric in * *G. We want to apply Proposition 4.6, to construct a centric linking system LcS(G) associated to FS(G), and to show that |LcS(G)|^p' BG^p. This means constructing a rigidification of the homotopy functor B :P 7! BP ; which by Proposition 5.9* * is equivalent to constructing a rigidification of the homotopy functor B ^p:P 7! B* *P ^p. This last is closely related to the homotopy decomposition of BG constructed in* * [JMO ]. ___ For any S 2 Sylp(G), we let OS(G) denote the category whose objects are the p* *-toral subgroups of S , and where Mor OS(G)(P , Q ) = Q \NG(P , Q ). Define B :OS(G) ---! Top by setting Qx .x-1 B (P ) = EG=P and B(P ---! Q ) = (EG=P ---! EG=Q ). 60 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups Let : hocolim-----!(B ) ------! EG=G = BG OS(G) be the map induced by the obvious surjections from B (P ) = EG=P onto BG = EG=* *G. Lemma 9.7. Fix a maximal p-toral subgroup S 2 Sylp(G). Let OcS(G) OS(G) be the full subcategory whose objects are those p-toral subgroups of S which are p* *-centric in G, and let Bc:OcS(G) -----! Top and c: hocolim-----!(B c) -----! BG OcS(G) be the restrictions of B and , respectively. Then c is a mod p homology equiv* *alence. Proof.Define fi X = P S p-toralfi|NG(P )=P | < 1, Op(NG(P )=P ) = 1 or P is p-centric. Let OXS(G) OS(G) be the full subcategory with object set X, and let B X :OXS(G) -----! Top and X :hocolim-----!(B X ) -----! BG OXS(G) be the restrictions of B and . By [JMO , Theorem 1.4], X is a mod p homology equivalence. So to prove the proposition, we must show that the inclusion of hocolim-----!(B c) in hocolim--* *---!(B X ) is a mod p homology equivalence. Set F = H*(B X (-); Fp), regarded as a functor on OXS(G* *)op. Let F0 F be the subfunctor defined by setting F0(P ) = 0 if P is p-centric in* * G and F0(P ) = F (P ) otherwise. We claim that lim-*(F0)= 0. Assuming this, we see th* *at lim-*(H*(B X (-); Fp))~=lim-*(F=F0)~=lim-*(H*(B c(-); Fp)); OXS(G) OXS(G) OcS(G) the last step since there are no morphisms from any object of the subcategory to any object not in the subcategory. This shows that the spectral sequences for * *the cohomology of hocolim-----!(B X ) and hocolim-----!(B c) have isomorphic E2-ter* *ms, and hence that the inclusion is a mod p homology equivalence. It remains to prove that lim-*(F0)= 0. By [JMO , Proposition 1.6], X0 conta* *ins finitely many G-conjugacy classes. Hence by [JMO , Proposition 5.4] and an app* *ropri- ate finite filtration of F0, it suffices to prove that *(NG(P )=P ; H*(BP ; Fp* *)) = 0 for each p-toral subgroup P in OXS(G) which is not p-centric. For each such P* * , CG(P ).P =P ~= CG(P )=Z(P ) is a finite group of order a multiple of p which a* *cts trivially on H*(EG=P ; Fp) ~=H*(BP ; Fp), and hence *(NG(P )=P ; H*(BP ; Fp)) * *= 0 by [JMO , Proposition 5.5]. We are now ready to construct a rigidification of the homotopy functor B ^p. Proposition 9.8. Fix a maximal discrete p-toral subgroup S 2 Sylp(G), and set F* * = FS(G) for short. Let Fcs Fc be the full subcategory of subgroups P S which * *are p-centric in G and snugly embedded, and let Ocs(F) Oc(F) be its orbit categor* *y. Then there is a functor bB:Ocs(F) -----! Top which is a rigidification of the homotopy functor B ^p, and a homotopy equivale* *nce i j b : hocolim(Bb) ^ ------! BG^. -----! p p Ocs(F) Carles Broto, Ran Levi, and Bob Oliver * * 61 _ ___ Proof.Set S = S 2 Sylp(G). We will construct orbit categories and functors as indicated in the following diagram: ___cl__//_c ooopro_ c Ocs(F) O S(G) OS(G) HH vv HHH | vvv HHH |_Bc vvv (1) BcsHHHH || vvv Bc H##Hfflffl|--vvv Top, together with mod p homology equivalences cl* __ hocolim-----!(Bcs)_//hocolim(B c)_____//hocolim(B c) N -----! ' -----!p NNNN | pppp NNNN |_ pppp (2) csNNNNN | c pppp c NN | ppp NN''fflffl|xxpp BG . We then set bB= (Bcs)^p. The category OcS(G), together with_B cand c, were already constructed in Lem* *ma 9.7. We construct OcS(G), B c, and cin Step 1 (and prove the properties we ne* *ed); and then do the same for Ocs(F), Bcs, and csin Step 2. __ Step 1: Define O cS(G) by setting __ fi Ob (O cS(G)) = Ob (OcS(G)) = P S fiPp-toral and p-centric in,G and Mor_Oc (P , Q ) = Q \NG(P , Q )=CG(P ) ~=Rep G(P , Q ). S(G) __ Let_B cbe the left homotopy Kan extension of B calong the projection functor. L* *et cbe the composite of c with the standard homotopy equivalence __ ' :hocolim-----!(B c) -----! hocolim(B c) _ -----!c OcS(G) OS(G) __ of [HV , Proposition 5.5]. Thus __cis a mod p homology equivalence by Lemma 9.* *7, and it remains only to show that B cis a rigidification of B ^p(after p-complet* *ion). This means showing that the natural morphism of functors __ B c-----! B cOpr (natural up to homotopy) is a mod p homology equivalence on all objects. By def* *inition, for each P , __ B c(P ) = hocolim-----!(B cO ,). pr#P ff __c Here, pr#P is the overcategory whose objects are the morphisms Q ___! P in O * *S(G), and where a morphism from (Q , ff) to (R , fi) is a morphism ' 2 Mor OcS(G)(Q ,* * R) such that ff = fi Opr('). Also, , is the forgetful functor from pr#P to OcS(G). Consider the spectral sequence j i+j __ Ei,j2~=lim-iH (B cO ,(-); Fp) =) H (B c(P ); Fp) . (3) pr#P 62 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups For each Q in OcS(G), set prP K(Q ) = Ker AutOcS(G)(Q ) ---! Aut_Oc (Q ) ~=CG(Q )=Z(Q ). S(G) This is a finite group of order prime to p, and it acts trivially on H*(EG=Q ).* * Since K(Q ) acts trivially on the homology H*(B cO ,(Q , ff); Fp) = H*(EG=Q ; Fp) for* * each object (Q , ff), this functor factors through the overcategory Id#P . The proje* *ction of pr#P onto Id#P satisfies the hypotheses of [BLO1 , Lemma 1.3] (in particular* *, the target category is obtained from the source by dividing out by these automorphi* *sm groups K(Q ) of order prime to p), and hence ( * i * H*(B c(P ); Fp)if i = 0 lim-iH (B cO ,(-); Fp) ~= lim-H (B cO ,(-); Fp) ~= pr#P Id#P 0 if i > 0. Here, the last isomorphism holds since Id#P_has final object (P , Id). The spe* *ctral sequence (3) thus collapses, and hence H*(B c(P ); Fp) ~=H*(B_c(P_); Fp) (and t* *he iso- morphism is induced by the natural inclusion of B c(P ) into B c(P )). Step 2: The "closure functor" __ Ocs(F) ---cl---!OcS(G), __ is defined to send P to P. It induces a bijection between isomorphism classes o* *f objects by definition of Fcs and Lemma 9.6(b,c), and induces bijections on morphism set* *s by Lemma 9.4(c). So this is an equivalence of categories. __ __ Set Bcs= B cOcl. Since B cis (after p-completion)_a rigidification of the hom* *otopy functor P 7! BP ^pby Step 1, and since BP ^p' BP ^pwhen P is snugly embedded (Proposition 9.2), Bcsis a rigidification of the homotopy functor B^p:P 7! BP ^* *p(again, up to p-completion). __ Now let csbe the composite of cwith the map __ hocolim-----!(Bcs) --cl*---!hocolim-----!(B c) Ocs(F) _OcS(G) __ induced by cl. Then hocolim-----!(B c) ' hocolim-----!(Bcs) since clis an equi* *valence of cate- __ gories, and thus csis a mod p homology equivalence since cis by Step 1. Now set bB= (Bcs)^pand let i j i j ( )^ b : hocolim(Bb) ^ ' hocolim(B ) ^ -----cs-p--!BG^ -----! p -----! cs p p Ocs(F) Ocs(F) be the completion of cs. Then bBis a rigidification of the homotopy functor B * *^p(see Proposition 9.2), and b is a homotopy equivalence. We also need the following result about snugly embedded subgroups: Lemma 9.9. For each discrete p-toral subgroup P G, P ois snugly embedded. Proof.Fix S 2 Sylp(G), and set T = S0 and pm = exp(S=T ). By Definition 3.1, m C (Q) P o= P .I(Q), where Q = P [m]= P0,_and_I(Q) = T W . Then I(Q) contains all elements of p-power order in I(Q) = T CW (Q), and hence is sn* *ugly embedded. Since [P o: I(Q)] is finite, P ois also snugly embedded. Carles Broto, Ran Levi, and Bob Oliver * * 63 We are now ready to prove the main result. Theorem 9.10. Fix a compact Lie group G and a maximal discrete p-toral subgroup S 2 Sylp(G). Then there exists a centric linking system LcS(G) associated to FS* *(G) such that (S, FS(G), LS(G)) is a p-local compact group with classifying space |LS(G)* *|^p' BG^p. Proof.Set F = FS(G) for short; a saturated fusion system by Lemma 9.5. Let Fcs Fc be the full subcategory with objects the set of all P S which are p-centri* *c and snugly embedded in G, and let Ocs(F) be its orbit category. By Lemma 9.9, Fcs * *Fco. By Proposition 9.8, there is a functor bB:Ocs(F) -----! Top which is a rigidification of the homotopy functor B ^p, and a homotopy equivale* *nce i j b : hocolim(Bb) ^ ------! BG^. -----! p p Ocs(F) By Proposition 5.9, there is a functor eB:Ocs(F) ---! Top which is a rigidifica* *tion of the homotopy functor B , and a natural transformation of functors O: eB---! Bb * *such that O(P ) is homotopic to the completion map for each P . By Proposition 4.6, * *there is a centric linking system LcsS(G) associated to Fcs whose nerve has the homotopy* * type of hocolim-----!(Be), and thus i j i j |LcsS(G)|^p' hocolim-----!(Be) ^p' hocolim-----!(Bb) ^p' BG^p. Ocs(F) Ocs(F) __ __ Now define Fc ---! Fcs by setting (P ) = P .(P 0)(p),_where (P 0)(p)denotes * *the subgroup of elements of p-power order in the torus P 0. By Lemma 9.4(c), for e* *ach P 2 Ob (Fc) and Q 2 Ob (Fcs), __ __ RepG (P, Q) ~=Rep G(P , Q) ~=Rep G( (P ), Q), * * __ and thus is left adjoint to the inclusion. Also, for each P , CG( (P )) = CG* *(P ) = CG(P ), and hence Z( (P )) = Z(P ). So if we define LcS(G) to be the pullback * *of LcsS(G) and Fc over Fcs, then it is a centric linking system, and |LcS(G)| ' |L* *csS(G)|. (Compare this argument with the proof of Proposition 4.5(b).) The above construction of the linking system of G has the disadvantage that i* *t seems rather arbitrary. We know, by Theorem 7.4, that there is (up to isomorphism) at* * most one linking system LcS(G) such that |LcS(G)|^p' BG^p, but we would really like * *to have a more obvious algebraic connection between LcS(G) and the group G itself. We e* *nd this section by showing that LcS(G) can, in fact, be obtained as a subquotient * *of the transporter category of G _ although not in a completely canonical way. Fix a compact Lie group G, and choose S 2 Sylp(G). The transporter category TSc(G) of G over S is the category whose objects are the subgroups of S that ar* *e p- centric in G, and where Mor TSc(G)(P, Q) = NG(P, Q), for each pair of objects P* * and Q of TSc(G). Let CG :Oc(F)op---! Ab be the functor which sends P to its centrali* *zer. For any subfunctor CG, TSc(G)= denotes the quotient category with the same objects as TSc(G), and where Mor TSc(G)=(P, Q) = Mor TSc(G)(P, Q)= (P ) = NG(P, Q)= (P ). For example, in this notation, FcS(G) = TSc(G)=CG. 64 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups For each P 2 Ob (Fc), there is a central extension _____ _____ 1 ---! Z(P ) -----! CG(P ) -----! CG(P )=Z(P ) ---! 1, _____ _____ where Z(P ) is abelian and p-toral and CG(P )=Z(P ) is finite of order prime to* * p (by definition of p-centric). Hence the set of elements of CG(P ) of finite order * *prime to p forms a subgroup, which we denote here p0(P ). Also, Z(P ) and p0(P ) are * *both normal subgroups of CG(P ), and the quotient group CG(P )=(Z(P ) x p0(P )) is * *a Q- vector space. As earlier, we write Z(P ) = Z(P ), and regard Z, p0, and Z x * *p0as subfunctors of CG. Lemma 9.11. The extension pr c TSc(G)=(Z x p0) ------! F is split, by a splitting which sends Out P(P ) to P=Z(P ) for each object P ; a* *nd such a splitting is unique up to natural isomorphism of functors. Proof.For all P, Q 2 Ob (Fc), choose maps ffi oeP,Q: Hom F(P, Q) -----! NG(P, Q) Z(P ) x p0(P ) = Mor TSc(G)=(Zx(p0)P, * *Q) which split the natural projection. This can be done in such a way that for ea* *ch ' 2 Hom F(P, Q) and each g 2 Q, oeP,Q(cg O') = [g] OoeP,Q(') (define it first o* *n orbit representatives for the action of Inn(Q) and then extend it appropriately). Als* *o, when Q = P , we let oeP,P(IdP) be the class of 1 2 NG(P ). The "deviation" of {oeP,Q} from being a functor is a 2-cocycle with values in* * the functor CG=(Z x p0), and the assumption that they commute with the Inn(Q)-acti* *ons implies that we get a cocycle over the orbit category Oc(F). If, furthermore, * *this cocycle is a coboundary, then the oeP,Q can be replaced by maps oe0P,Qwhich def* *ine a splitting functor. The obstruction to the existence of such a splitting thus* * lies in lim-2(CG=(Z x p0)). In a similar (but simpler) way, the obstruction to unique* *ness is Oc(F) seen to lie in lim-1(CG=(Z x p0)). Oc(F) We will show that both of these groups vanish, using Lemma 5.7 (and an argume* *nt similar to that used to prove Proposition 5.8). Let F be the functor CG=(Z x * *p0). As in_the proof of Proposition 5.8, set T = S0, Q = CS(T ), and = Out F(Q). * *Set M = T=(torsion), regarded as a Q[ ]-module. Let : Op( )op-----! Z(p)-mod be the functor ( ) = M for all p-subgroups . Then F (P ) ~= (Out P(Q)) (functorially)_for all P S containing Q, and Out Q(P ) acts trivially on F (* *P ) ~= Z(P )=(torsion) for each P . The hypotheses of Lemma 5.7 thus hold, and so lim-i(F~)=lim-i( ) Oc(F) Op( ) for all i. Since is a Mackey functor, these groups vanish for all i 1 by [* *JM , Proposition 5.14]. We are now ready to construct a more explicit linking system LcS(G), and prov* *e it is isomorphic to the one already constructed in Theorem 9.10. Carles Broto, Ran Levi, and Bob Oliver * * 65 Proposition 9.12. Let G be a compact Lie group, and choose S 2 Sylp(G). Fix a pr c c splitting s of TSc(G)=(Z x p0) ---! F , and define LS(G) to be the pullback ca* *tegory in the following pullback diagram ~s c LcS(G) ______! TS (G)= p0 | | ss| |pr # # s c FcS(G) ___! TS (G)=(Z x p0) . Then LcS(G) is a centric linking system associated to FcS(G), and is isomorphic* * to the centric linking system of Theorem 9.10. In particular, LcS(G) --~s-!TSc(G)= p0d* *escribes the linking system LcS(G) as a subquotient of the transporter catgeory. Proof.We will first show that the pullback category LcS(G) is a centric linking* * system associated to FS(G). Since s and prare the identity on objects, we can as well * *assume that the pullback category LcS(G) has the same objects, and that ~sand ss are t* *he identity on objects. Then for any pair of objects P, Q S p-centric in G, we h* *ave Mor LcS(G)(P, Q) = fi (', _) fi' 2 Mor FcS(G)(P, Q), _ 2 Mor TSc(G)=(p0P, Q), and ss(') = p* *r(_) Now, for each P S which is p-centric in G, we have P NG(P )= p0(P ) and the* *n we can define distinguished homomorphisms ffiP :P -----! AutLcS(G)(P ) by setting ffiP(g) = (cg, g). Conditions (A), (B), and (C) in the definition of* * a centric linking system are easily checked. Next we will find a map |LcS(G)| --! BG^pthat commutes with the respective na* *t- ural maps from BS. To do this, we first lift LcS(G) to a subcategory LecS(G) o* *f the transporter category TSc(G), defined via the pullback square: LecS(G)______incl!TSc(G) | | | | # # ~s c LcS(G) ____! TS (G)= p0 We will then construct the maps in the following commutative diagram: oBS MM ooo | MMMM oooo | MMMM wwoooo fflffl| M&&M |LcS(G)|^po'o_ |LecS(G)|^p___//BG^p. We proceed in two steps. (a) A map |LecS(G)| --! |LcS(G)| commuting with the respective natural maps fr* *om BS is induced by the functor LecS(G) --! LcS(G). We will show that it is a mod* * p homology equivalence. By definition of eLcS(G), for all P, Q S centric, we have that 0p(P ) acts* * freely on Mor eLcS(G)(P, Q) and the orbit set if Mor LcS(G)(P, Q). In particular, h i 0p(P ) = Ker Aut eLcS(G)(P ) ----! AutLcS(G)(P.) 66 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups Recall that p0(P ) is the subgroup of elements of CG(P ) of finite order pri* *me to p. It sits in an extension p0(P_)0_-! p0(P ) -! C0G(P ), where p0(P )0 is the set_* *of_elements_ of the maximal torus of Z(P ) of finite order prime to p and C0G(PS) = CG(P )=Z* *(P ). Therefore p0(P ) is locally finite and can be written as union m 0 p0(P )m o* *f finite groups of order prime to p. A generalized version of [BLO1 , Lemma 1.3] now ap* *plies to the constant functor defined on LcS(G) and the result follows. In fact, [BLO1 , Lemma 1.3] generalizes to allow that the kernels K(-) be co* *untable increasing unions of finite groups of order prime to p. Proving this requires s* *howing, for any such K, that H0(K; -) is an exact functor on the category of Z(p)[K]-mo* *dules. But if K is the union of a sequence of subgroups K1 K2 . . .each of which is finite, then H0(Ki; -) is exact for each i, H0(K; M) ~=colim---!iH0(Ki; M) for * *each M, and hence H0(K; -) is exact since direct limits of this type are exact. (b) A map |LcS(G)| --! BG that commutes up to homotopy with the respective maps from BS is defined as follows. Compose the inclusion eLcS(G) -incl--!TSc(G* *) with the functor TSc(G) --! B(G). Here, B(G) is the topological category with one ob* *ject and the Lie group G as morphisms (and all other categories are discrete), and t* *he functor sends the morphism g 2 NG(P, Q) to g 2 G for all objects P, Q of TSc(G). The nerve of B(G) is the topological bar construction B G ' BG, and the composi* *te functor induces a map |LecS(G)| ---! |B(G)| ' BG. Finally, Theorem 9.10 defines the centric linking system of G over S and show* *s that the p-completed nerve is homotopy equivalent to BG^p. This combines with the map constructed above, so that Lemma 7.3 implies that the pullback category LcS(G) * *is isomorphic to the centric linking system of Theorem 9.10, and then, also, that * *the map |LecS(G)| ---! BG constructed in step (b) is actually a homotopy equivalence af* *ter p-completion. 10. p-compact groups A p-compact group is a p-complete version of a finite loop space. As defined * *by Dwyer and Wilkerson in [DW ], a p-compact group is a triple (X, BX, e), where X is a* * space such that H*(X; Fp) is finite, BX is a pointed p-complete space, and e: X --! * *(BX) is a homotopy equivalence. If G is a compact Lie group such that the group of c* *om- ponents ss0(G) is a finite p-group, then upon setting BGb = BG^pand bG= (BGb),* * the triple (Gb, BGb, Id) is a p-compact group. For general references on p-compact * *groups, we refer to the original papers by Dwyer and Wilkerson [DW ] and [DW2 ], and * *also to the survey article by Jesper Moller [Mo ]. When T ~= (S1)r is a torus of rank r, then the p-completion bT= (BT ^p) of T* * is called a p-compact torus of rank r. Both BTb ' K((bZp)r, 2) and bT' K((bZp)r, 1* *) are Eilenberg-MacLane spaces. A p-compact toral group is a p-compact group (Pb, BPb* *, e) such that ss1(BPb) is a p-group, and the identity component of bPis a p-compact* * torus with classifying space the universal cover of BPb. If X is either a discrete p-toral group or a p-compact group, and Y is a p-co* *mpact group, a homomorphism f :X ! Y is by definition a pointed map Bf :BX ! BY . Two homomorphisms f, f0:X ! Y are conjugate if Bf and Bf0 are freely homotopic; i.e., via a homotopy which need not preserve basepoints. Given a homomorphism Carles Broto, Ran Levi, and Bob Oliver * * 67 f :X ! Y , the homotopy fibre of Bf is denoted Y=f(X), or just Y=X if f is unde* *rstood from the context. With this notation, f is called a monomorphism if H*(Y=f(X); * *Fp) is finite. By [DW , Proposition 9.11], a homomorphism f is a monomorphism if * *and only if H*(BX; Fp) is a finitely generated H*(BY ; Fp)-module via H*(f; Fp). If bPis an arbitrary p-compact toral group, a discrete approximation to bPis * *a pair (P, f), where P is a discrete p-toral group and Bf :BP ! BPbinduces an isomorph* *ism in mod p cohomology. By [DW , Proposition 6.9], every compact p-toral group ha* *s a discrete approximation. Each discrete p-toral group P is a discrete approximati* *on of (Pb, BPb, Id), where BPb = BP ^pand bP= (BPb). Hence every monomorphism f :P ! bf X from a discrete p-toral group to a p-compact group factors as P ___! bP___! * * X: a discrete approximation followed by a monomorphism of p-compact groups. Lemma 1.10 says, among other things, that any two discrete approximations of a p-comp* *act toral group are isomorphic. If f :X ! Y is a homomorphism of p-compact groups, the centralizer of f in Y * *is defined to be the triple (CY (X, f), BCY (X, f), Id), where BCY (X, f) = Map (BX, BY )Bf and CY (X, f) = (BCY (X, f)). Whenever f is understood, we simply write CY (X) for CY (X, f). A discrete p-toral subgroup of a p-compact group X is a pair (P, f), where P * *is a f discrete p-toral group and Pb___! X is a monomorphism. We write BCX (P, f) = BCX (Pb, f) = Map (BP, BX)Bf and CX (P, f) = CX (Pb, f) for short. By [DW , x* *x5- 6], CX (P ) is a p-compact group, and the homomorphism CX (P ) --! X (induced by evaluation at the basepoint of BP ) is a monomorphism. The subgroup (P, f) is c* *alled central if this monomorphism CX (P ) --! X is an equivalence. Proposition 10.1. Let X be any p-compact group. f u (a)X has a maximal discrete p-toral subgroup S --! X. If P --! X is any other discrete p-toral subgroup of X, then Bu ' Bf OB_ for some _ 2 Hom (P, S); and (P, u) is maximal if and only if p - O(X=u(Pb)). Here, Euler characteristic* *s are taken with respect to homology with coefficients in Fp. f (b)The centralizer CX (P, f) of any discrete p-toral subgroupSP --! X is again* * a p- compact group, and a subgroup of X. Also, if P = 1n=1Pn, then BCX (P ) ' BCX (Pn) for n large enough. f (c)A discrete p-toral subgroup P --! X is central if and only if there is a ma* *p BP x BX --! BX whose restriction to BP x * is Bf and whose restriction to * x BX is the identity. When this is the case, then P is abelian, and there is a f* *ibration f sequence BP ^p--! BX --! B(X=P ) where B(X=P ) is the classifying space of a p-compact group X=P . Proof.Point (a) follows mostly from [DW2 , Propositions 2.10 & 2.14] together * *with Lemma 1.10. If (P, u) is not maximal, then since u factors through S, O(X=u(Pb)* *) = O(X=f(Sb)) . O(Sb=Pb), and the last factor is a multiple of p. Point (b) is shown in [DW , Proposition 5.1 & Theorem 6.1]. In point (c), a * *central subgroup is abelian by [DW2 , Theorem 1.2], while the other two claims are sho* *wn in [DW , Lemma 8.6 & Proposition 8.3]. 68 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups As in other contexts, the maximal discrete p-toral subgroups of a p-compact g* *roup X will be referred to as Sylow p-subgroups of X. The fusion system of a p-compact group is easily defined: it is just the fusi* *on system of the space BX, as defined in [BLO2 , Definition 7.1]. f Definition 10.2. For any p-compact group X with Sylow p-subgroup S --! X, let FS,f(X) be the category whose objects are the subgroups of S, and where for P, * *Q S, fi MorFS,f(X)(P, Q) = Hom X(P, Q) def=' 2 Hom (P, Q) fiBf|BQ OB' ' Bf|BP . We next want to show that FS,f(X) is saturated. Before doing this, we need to* * define and study normalizers of discrete p-toral subgroups of p-compact groups. We als* *o need to establish an "adjointness" relation which corresponds to the equivalence (fo* *r groups) between homomorphisms Q ---! NG(P ) and homomorphisms P o Q ---! G. Fix a p-compact group X and a Sylow p-subgroup f :S ---! X. For any subgroup P S and any discrete p-toral subgroup K AutX (P ), set ^ BNKX(P ) = EK xK BCX (P ) p, where K acts on BCX (P ) = Map (BP, BX)Bffvia the action on P . Set NKX(P ) = (BNKX(P )). Since the action of K on BP fixes the basepoint, evaluation at the basepoint of BP defines a map ^ ev: BNKX(P ) = EK xK Map (BP, BX)Bf p ------! BX. _ If Q is any discrete p-toral group, and ae2 Hom (Q, K), then any homomorphism EQ xQ BP ~=B(P o_aeQ) ------! BX is adjoint to a Q-equivariant map EQ ------! Map (BP, BX)f|BP = BCX (P ) , _ where Q acts on BCX (P ) via the action on P defined by ae(and via the trivial * *action on BX). After taking the Borel construction, this defines a map ^ BQ ------! BNKX(P ) = EK xK BCX (P ) p . In particular, when Q is the group NKS(P ) = {g 2 NS(P ) | cg 2 K}, and f B(P o NKS(P )) ------! BS ------! BX is induced by the inclusions and f, then this construction is denoted BflKP:BNKS(P ) ------! BNKX(P ) . f Lemma 10.3. Fix a p-compact group X, a Sylow p-subgroup S -! X, and subgroups P S and K AutX (P ) where K is discrete p-toral. Then the induced sequence BCX (P ) -----! BNKX(P ) ---o--!BK^p (1) is a fibration sequence. If Q is another discrete p-toral group, then for any h* *omomor- phism ae: Q ___! NKX(P ), there is a fibration sequence Map (B(P o_aeQ), BX)f,ae__! Map (BQ, BNKX(P ))Bae___! Map (BQ, BK^p)Ba_e, _ _^ where ae2 Hom (Q, K) is any homomorphism such that Baep' o OBae, P o_aeQ is the _ae semidirect product for the action Q ___! K Aut(P ), and the fiber is the spa* *ce of all Carles Broto, Ran Levi, and Bob Oliver * * 69 Bf| maps B(P o_aeQ) ___! BX which restrict (up to homotopy) to BP ___!P BX and are adjoint to Bae in the sense described above. Proof.The action of K on each cohomology group Hi(BCX (P ); Fp) factors through* * a finite quotient group of K, thus through the p-group ss0(K), and hence is nilpo* *tent. So by [BK , II.5.1], the usual fibration sequence BCX (P ) -----! EK xK BCX (P ) -----! BK for the Borel construction over BK is still a fibration sequence after p-comple* *tion. Thus (1) is a fibration sequence. _ Since [BQ, BK^p] ~= [BQ, BK] ~= Rep (Q, K) (Lemma 1.10), ae2 Hom (Q, K) is uniquely determined up to conjugacy by ae. _ For any fixed homomorphism ae:Q ___! K, (1) induces a fibration sequence BCX (P )hQ ___! Map (BQ, BNKX(P ))eae__! Map (BQ, BK^p)Ba_e^p, where eaedenotes the set of connected components of Map (BQ, BNKX(P )) that map* * into Map (BQ, BK^p)Ba_e^p; and (if eae6= ?) BCX (P )hQ is the homotopy fixed point s* *et of the Ba_e^p action of Q induced by the pullback of (1) over BQ ----! BK^p. We need to iden* *tify 'oOBae _ this action of Q on BCX (P ) with that induced by the action of Q on P via ae. * *This follows by comparing the fibrations BCXw(P )__! EK xK BCX (P ) _____! BK ww | | ww | | # # BCX (P )______! BNKX(P ) ________!BK^p, Ba_e^p since the action of Q on BCX (P ) induced by BQ -----! BK^pin the bottom fibra- _ tion coincides with that induced by aein the fibration sequence of the top row.* * By construction, the action of K on BCX (P ) induced by the top row is just the ac* *tion of K on BCX (P ) = Map (BP, BX)Bf|P induced by the original action of K on P . Now set fP = f|P :P ---! X for short. We can identify hQ BCX (P )hQ = Map (BP, BX)BfP = Map Q(EQ, Map (BP, BX)BfP) ' Map Q(BP x EQ, BX)fe' Map (BP xQ EQ, BX)fe' Map (B(P o_aeQ), BX)fe, where efis the set of connected components of maps whose restriction to BP is h* *o- motopic to BfP. Here, BP xQ EQ ' B(P o_aeQ) because the action of Q in BP is induced from the action described above of Aut(P ) on BP , and this has a fixed* * point, providing a section of the fibration BP ___! BP xAut(P)E Aut(P ) ___! B Aut(P ) . Finally, upon restricting to one component of Map (BQ, BNKX(P ))eae, we obtain * *the fibration in the statement of the proposition. Notice that in the particular case where K = 1, Map (BQ, BK) is contractible,* * and the fibration of Lemma 10.3 reduces to the equivalence Map (BP x BQ, BX) ' Map (BQ, Map (BP, BX)). 70 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups f Proposition 10.4. Let X be a p-compact group, let S --! X be a Sylow p-subgroup, and set F = FS,f(X) for short. Fix a subgroup P S, and a discrete p-toral gro* *up of automorphisms K AutF (P ). Then the following hold. (a)BNKX(P ) is the classifying space of a p-compact group which we denote NKX(P* * ), and flKP K NKS(P ) ---! NX (P ) is a discrete p-toral subgroup. Furthermore, the square BflKP K BNKS(P ) ___! BNX (P ) | | B incl| ev| (1) # # f BS ________!BX commutes up to pointed homotopy. (b)There is ' 2 Hom F(P, S) such that '(P ) is fully 'K'-1-normalized in F. (c)P is fully K-normalized in F if and only if NKS(P ) is a Sylow p-subgroup of* * NKX(P ). Proof.(a) By Lemma 10.3, BCX (P ) --! BNKX(P ) --! BK^pis a fibration sequence. The loop spaces of the fiber and base of this sequence have finite mod p cohomo* *logy, so the same is true of NKX(P ) def= (BNKX(P )). Thus NKX(P ) is a p-compact gro* *up. The map BflKP K ^ BNKS(P ) ----! BNX (P ) = EK xK Map (BP, BX)Bf|P p is defined to be adjoint to the composite K B(incloincl) f B P o NS (P ) ---------! BS ----! BX. (2) Hence the composite of BflKPwith the evaluation map from Map (BP, BX)BfP to BX (evaluation at the basepoint of BP ) is equal to the restriction of (2) to BNKS* *(P ). This proves that (1) is commutative. If flKP factored through a quotient group NKS(P )=R for some R 6= 1, then the* * re- striction of Bf :BS ---! BX to BR would be homotopically trivial, but this cann* *ot f0 K K happen. So if S0 ---! NS (P ) is a maximal discrete p-toral subgroup, then flP * *factors through a monomorphism from NKS(P ) to S0 (Proposition 10.1(a)), and thus flKP * *is itself a monomorphism. (b,c) By Lemma 10.3, any discrete p-toral subgroup BQ ---! BNKX(P ) lifts to a fi map B(P o Q) --! BX which factors through a homomorphism P o Q --! S. Set P 0= fi(P o 1) S, ' = fi|Po1 2 IsoF(P, P 0), and K0 = 'K'-1 AutX (P 0). Then 0 0 K fi(Q) NKS(P ), and fi|1oQ is injective since otherwise BQ --! BNX (P ) would * *factor through a quotient group0of Q and hence wouldn't be a subgroup. Thus, the large* *st possible K-normalizer NKS(P 0) occurs when it is a Sylow p-subgroup of NKX(P ),* * so P 0 is fully K0-normalized in F, and P is fully K-normalized if and only if NKS(P )* * is a Sylow p-subgroup of NKX(P ). We are now ready to show that FS,f(X) is saturated. f Proposition 10.5. Let X be a p-compact group, and let S --! X be a Sylow p- subgroup. Then FS,f(X) is a saturated fusion system over S. Carles Broto, Ran Levi, and Bob Oliver * * 71 Proof.Write F = FS,f(X) for short. Proof of (I): Fix a subgroup P S which is fully normalized in F. Let K AutF* * (P ) be such that K Aut S(P ) and K= Inn(P ) 2 Sylp(Out F(P )). Then P is fully K- normalized, since it is fully normalized and NKS(P ) = NS(P ). So by Propositi* *on 10.4(c), NKS(P ) is a Sylow p-subgroup of NKX(P ). Set K0 = Aut S(P ) for short, and consider the following commutative diagram * *of connected spaces: BCS(P )^p_! BNKS(P )^p_! BK0^p | | f1| f2| f3| # # # BCX (P )__! BNKX(P )^p__! BK^p. Let Fi be the homotopy fiber of fi (for i = 1, 2, 3). Each row is a fibration * *se- quence before p-completion; and the actions of K0 on H*(BCS(P ); Fp) and of K on H*(BCX (P ); Fp) factor through finite p-group quotients and hence are nilpoten* *t. So the rows are still fibration sequences after p-completion by [BK , II.5.1]. Each of the maps fiis a monomorphism of p-compact groups, and hence H*(Fi; Fp* *) is finite for each i. Since BCX (P ) is connected, ss1(BNKX(P )^p) surjects onto s* *s1(BK^p) ~= ss0(K), and hence ss0(F2) surjects onto ss0(F3). Thus F1 is the homotopy fiber * *of the map F2 ---! F3, and so O(F2) = O(F1).O(F3). Since NKS(P ) 2 Sylp(NKX(P )), O(F2) is prime to p by Proposition 10.1(a). T* *hus O(F1) and O(F3) are both prime to p, and hence CS(P ) 2 Sylp(CX (P )) and (sinc* *e K is discrete p-toral) K0 = K. Hence OutS (P ) = K= Inn(P ) 2 Sylp(Out F(P )). Also,* * since CS(P ) 2 Sylp(CX (P )), we can again apply Proposition 10.4(c) (this time with * *K = 1), to show that P is fully centralized in F. This finishes the proof of (I). Proof of (II): Fix P S and ' 2 Hom F(P, S), and set P 0= '(P ). Assume that * *P 0 is fully centralized in F. Set N' = {g 2 NS(P ) | 'cg'-1 2 AutS(P 0)}, 0 0 0 and set K = AutN' (P ), K0 = 'K'-1 AutS (P 0), and N0'= NKS(P ). Then P is fully K0-normalized in F, since it is fully centralized and K0 AutS(P 0). Cons* *ider the following diagram: BflKP K proj ^ BN' ________! BNX (P )________! BKp ... | | B'_... ' Ec'x('*)-1| ~=|Bc' (1) ?.. BflK0 0 0# proj # BN0' ________P!BNKX(P 0) ________! BK0^p. ! Here, the composites in the two rows are induced by the epimorphisms N' --i K !0 0 0 and N0'--i K (exactly, not just up to homotopy). By Proposition 10.4(c), N' is 0 0 K _ a Sylow p-subgroup of NKX(P ) ' NX (P ), and hence there is a homomorphism ' 2 Hom (N', N0') which makes the left hand square commute up to homotopy. Since [BN', BK0^p] ~=Rep (N', K0) (Lemma 1.10), the homotopy commutativity of _ 0 0 _ 0 (1) implies that there is g 2_K such that c' O! =_c_gO! O '._Since ! is onto,* * there is g 2 N0'such that !0(g) = g; and upon replacing ' by cg O' we can assume that _ c' O! = !0O' . 72 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups Fix a homotopy H which makes the left hand square in (1) commute._ Then the composite projOH is a loop in Map (BN', BK0^p) based at B(!0O'), and this compo* *nent _ ^ has the homotopy type of BCK0(!0O' (N'))p by Lemma 1.10 again. So after replaci* *ng _ _ 0 0 ' by cg0O' for some appropriate g 2 N', and after modifying H using the homotopy _ _ 0 from B' to B(cg0O') determined by g , we can arrange that projOH is nullhomotop* *ic in Map (BN', BK0^p). We can now apply Lemma 10.3, to show that the following diagr* *am commutes up to homotopy: incloincl Bf B(P o N') _________!BS __________! BXw | ww |B('o_') w # w incloincl Bf B(P 0o N0')________! BS __________! BX . _ 0 In particular, ' 2 Hom F(N', N'). Also, the two homomorphisms from P o N' to S _ -1 induced by inclusions_and by ' o ' have the same kernel {(g, g ) | g 2 P }, an* *d this implies that ' = '|P. S 1 Proof of (III): Fix P = n=1Pn, where P1 P2 . .i.s an increasing sequence* * of subgroups. Let ' 2 Inj(P, S) be such that '|Pn 2 Hom F(Pn, S) for all n. Thus f* *or each n, (Bf OB')|BPn ' Bf|BPn. Also, by Proposition 10.1(b), Map (BPn, BX)Bf|BPn ' Map (BP, BX)Bf|BP for n sufficiently large. We can thus chooseShomotopies Hn fr* *om (Bf OB')|BPn to Bf|BPn such that Hn = Hn+1|BPnxI, and set H = Hn. This shows that Bf OB' ' Bf|BP , and hence that ' 2 Hom F(P, S). In [CLN ], a p-compact toral subgroup P of a p-compact group X is called cen* *tric if Bf the inclusion map BP --! BX is a centric map; i.e., if BfO- Map (BP, BP )Id-----! Map (BP, BX)Bf is an equivalence. We must check that this is equivalent to the concept of F-ce* *ntricity (applied to discrete p-toral subgroups) used here. f Lemma 10.6. Let X be a p-compact group, and let S ---! X be a Sylow p-subgroup. Set F = FS,f(X). Then for any subgroup P S, Bf|BP :BP ^p---! BX is a centric map if and only if P is F-centric. Proof.Assume P is F-centric. In particular, P is fully centralized in F. By Pro* *position 10.4(a,c) (applied with K = 1), CX (P ) is a p-compact group with Sylow p-subgr* *oup CS(P ) = Z(P ). Also, composition defines a map Map (BP, BP )Idx Map (BP, BX)Bf|BP ------! Map (BP, BX)Bf|BP. 'BZ(P) So by Proposition 10.1(c), Z(P ) is central in CX (P ), and there is a p-compac* *t group CX (P )=Z(P ) whose Euler characteristic is prime to p and a fibration sequence BZ(P )^p-----! BCX (P ) -----! B(CX (P )=Z(P )) . Then CX (P )=Z(P ) must be trivial, so B(CX (P )=Z(P )) ' *, BZ(P )^p' Map (BP, BX)Bf|BP, and hence Bf|BP is a centric map. Conversely, if Bf|BP is a centric map, then BCX (P ) ' BZ(P ) by Lemma 1.10, * *so CS(P 0) = Z(P 0) for all P 0 S which is F-conjugate to P , and P is F-centric. Carles Broto, Ran Levi, and Bob Oliver * * 73 It remains to construct a linking system associated to FS,f(X) whose p-comple* *ted nerve has the homotopy type of BX. This will be done using Proposition 4.6, tog* *ether with a construction by Castellana, Levi, and Notbohm in [CLN ]. f Theorem 10.7. Let X be a p-compact group, and let S --! X be a Sylow p-subgroup. Set F = FS,f(X) def=FS,Bf(BX) for short. Then there is a centric linking system L = LcS,f(X) associated to F such that |LcS,f(X)|^p' BX . In other words, (S, F, L) is a p-local compact group whose classifying space is* * homotopy equivalent to BX. Proof.By Proposition 10.5, the fusion system F is saturated. In [CLN ], the authors define a category Oc(F)+ by adding a final object to * *Oc(F); i.e., the category Oc(F)+ consists of Oc(F) together with an additional object * **, and a unique morphism from each object in Oc(F) to *. (The actual category they work * *with contains the same objects as Oc(F) by Lemma 10.6.) They then define a homotopy functor B +:Oc(F)+ ------! hoTop by setting B +(P ) = BP ^pfor all F-centric P S, and B +(*) = BX (with the ob* *vious maps between them). By Lemmas 1.10 and 10.6, this is a centric diagram in the s* *ense of [DK ]. Since Oc(F) has a final object, the Dwyer-Kan obstructions to rigidif* *ying B + to a functor to Top all vanish [DK ] (see also Corollary A.4), and so this func* *tor can be lifted. In particular, this restricts to a functor bBfrom Oc(F) to Top, togethe* *r with a map from hocolim-----!(Bb) to BX. (See also Corollary A.5.) By [CLN , Theorem 8.5], this map from hocolim-----!(Bb) to BX induces a homo* *topy equivalence ^ hocolim-----!(Bb) p ' BX (the collection of F-centric subgroups of X is "subgroup ample"). Hence by Prop* *osition 5.9, there is a functor eB:Oc(F) ---! Top which is a rigidification of the homo* *topy functor B , and a natural transformation of functors O: eB---! Bb which is the * *com- pletion map on each object. Proposition 4.6 now applies to show that there is a* * centric linking system L = LcS,f(X) associated to F such that ^ ^ ^ |L|^p' hocolim-----!(Be) p ' hocolim-----!(Bb) p ' BXp. Oc(F) Oc(F) In fact, one can show that there is a unique centric linking system L associa* *ted to FS,f(X) such that |L|^p' BX^p, but we leave that for a later paper. Appendix A. Lifting diagrams in the homotopy category As elsewhere in the paper, we let Top denote the category of spaces, and hoTop the homotopy category. Let ho :Top ---! hoTop be the forgetful functor. When C 74 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups is a small category, a functor F from C to Top or hoTop is called centric if f* *or each morphism ' 2 Mor C(c, d), the natural map F(')O- Map (F (c), F (c))Id ------! Map (F (c), F (d))F(') ' is a homotopy equivalence. In [DK ], Dwyer and Kan identify the obstructions to rigidifying a centric functor F : C --! hoTop to a functor eF: C --! Top ; and* * also describe the space of such rigidifications. We prove here a relative version o* *f their result which is needed in Section 5. This result can, in fact, be derived from * *the main theorem in [DK ], but that argument is so indirect that we find it helpful to g* *ive a more direct, and also more elementary, proof. More precisely, a rigidification of F is a functor Fe:C ---! Top , together w* *ith a natural transformation of functors F ---! hoOFe which is a homotopy equivalenc* *e on each object. Two rigidifications eFand eF 0are equivalent if there is a third r* *igidifica- tion eF,00together with natural transformations of functors eF---! eF 00---eF * *0which commute with the natural transformations from F , and hence which define homoto* *py equivalences eF(c) ' eF 00(c) ' eF(0c) for each c 2 Ob (C). This is easily see* *n to be an equivalence relation by taking pushouts. The main idea here is to construct a rigidification of F :C ---! hoTop by fi* *rst constructing a space which looks like a "homotopy colimit" of F , and then show* * that this homotopy colimit automatically induces a rigidification eF. Recall that th* *e nerve of a small category C is defined by setting ia a j. BC = n ~ ; n 0 x0!...!xn and that the homotopy colimit of any functor F : C --! Top is the space ia a j. hocolim-----!(F ) = F (x0) x n ~ . C n 0 x0!...!xn Here, in both cases, we divide out by the usual face and degeneracy identificat* *ions. Let pF : hocolim-----!(F ) --! BC be the projection. It will be convenient to r* *efer to the "skeleta" of the homotopy colimit: let hocolim-----!(n)(F ) denote the union of* * the F (x0)x i for all i n (and all x0 ! . .!.xi in C). Now assume that F : C --! hoTop is a functor to the homotopy category instea* *d. We assume that for each f : x --! y in C, a concrete map F (f) : F (x) --! F (y* *) has been chosen. The 1-skeleton hocolim-----!(1)(F ) is defined in the same way as * *before: it is the union of the mapping cylinders of the F (f) taken over all f 2 Mor (C). It * *is also straightforward to define the 2-skeletion; but it is convenient at this stage t* *o replace f g 2 by a truncated triangle 2t. More preciesely, for each sequence x0 --! x1 --* *! x2, F (x0) x 2tis attached to hocolim-----!(1)(F ) via the following picture: F (g Of)_____ssF (g) OF (f) JJ J Id J F (f) J J J J _________________________Jss Id Id F (f) Carles Broto, Ran Levi, and Bob Oliver * * 75 where the small segment at the top is mapped using a homotopy between F (g Of) * *and F (g) OF (f). The first obstructions arise when constructing the 3-skeleton. For each x0 ! * *. .!. x3, we want to attach F (x0) x 3tto hocolim-----!(2)(F ), where 3t(the "trunc* *ated 3- simplex") is the cone over 2twith its vertex cut off. The attachment map is e* *asily defined, except on the "top" surface resulting from truncating the cone vertex.* * Hence, the obstruction to defining the attachment map lies in the group i j ss1 Map (F (x0), F (x3)) , F (x0 ! x3) . At this point, it becomes necessary to switch from the intuitive picture to f* *ormal def- initions, by replacing the truncated simplices ntby cubes In, and regarding si* *mplices as cubes modulo certain identifications. This correspondence will be made expl* *icit later. Let be the simplicial category, with objects the sets [n] = {0, . .,.n} for* * n 0, and morphisms the order preserving maps between sets. We let @i2 Mor ([n - 1],* * [n]) denote the i-th face map (with image [n]r {i}). Define a functor Io: ------! Top by setting Io([n]) = In+1 (where I is the closed interval I = [0, 1]), and i Y Y Y j Io(oe)(t0, . .,.tn) = ti, ti, . .,. ti i2oe-1(0) i2oe-1(1) i2oe-1(m) for oe 2 Mor j([n], [m]). Here, the product over the empty set is always 1. Let 1 0 be the subcategories with the same objects, where Mor 0 ([m], [n])= {oe 2 Mor ([m], [n]) | oe(0) = 0} Mor 1 ([m], [n])= {oe 2 Mor ([m], [n]) | oe(0) = 0, oe(m) = n}. For each n 0, let Io1([n]) Io0([n]) Io([n]) be the subspaces Io0([n])= {(0, x1, . .,.xn) 2 Io([n])} ~=In Io1([n])= {(0, x1, . .,.xn-1, 0) 2 Io([n])} ~=In-1 . Then for each j = 0, 1, Io| j restricts to a subfunctor Ioj: j ---! Top. Throughout the rest of this section, C denotes a fixed small category. For e* *ach n 0, define Mor n= Mor n(C) to be the set of all sequences c0 ! c1 ! . .!.cn * *of composable morphisms in C. In particular, Mor 0(C) = Ob (C) and Mor 1(C) = Mor * *(C). For oe 2 Mor ([n], [m]), oe*: Mor m(C) ---! Morn(C) is defined as usual by ta* *king compositions, inserting identity morphisms, and (if oe =2Mor ( 1)) dropping mor* *phisms at one or both ends of the chain. For example, @*i(from Mor n(C) to Mor n-1(C)* *) is defined by composing two morphisms in the sequence, or by dropping one of them * *if i = 0 or n. Also, for each , = (c0 ! . .!.cn) in Mor n(C) and each 0 i j * *n, we write ,ij= (ci! . .!.cj) 2 Mor j-i(C), O O* * O let ,ij2 Mor C(ci, cj) denote the composite of this sequence of maps, and set ,* *= ,0n. In order to simplify the notation in what follows, whenever F :C ---! hoTop * *is a functor and ' 2 Mor (C), we let F (') denote some chosen representative of the homotopy class of maps defined by F , not the homotopy class itself. 76 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups __ Definition A.1. Fix a functor F :C --! hoTop . An Rg 1-structure F on F consists __ (c) __ of a space F (c) and a homotopy equivalence F (c) ---! F (c), defined for eac* *h c 2 ' Ob (C); together with maps __ __ __ F (,): In-1 = Io1([n]) -----! Map F (c0), F(cn) , defined for each n 1 and each , = (c0 ! c1 ! . .!.cn) 2 Mor n(C), which satis* *fy the following relations. __ ' (a)For all ' 2 Mor C(c0, c1), F (c0 --! c1) O (c0) ' (c1) OF ('). (b)For all m, n 1, oe 2 Mor 1([m], [n]), , 2 Mor n(C), and t2 Im-1 , __ __ F(oe*,)(t) = F(,)(Io1(oe)(t)). (c)For all n 2, , 2 Mor n(C), 1 i n - 1, t12 Ii-1, and t22 In-i-1, __ __ __ F (,)(t1, 0, t2) = F(,in)(t2) OF (,0i)(t1). Schematically, relation (b) can be described via the following commutative di* *agram: Io1(oe) Im-1 __________________________//_NNIn-1_p NNFN(oe*(,))NN F (,)ppppp NNN ppppp NNN&&_ xxpp_ Map (F (c0), F(cn)) while relation (c) can be described via the diagram: t1,t27!(t1,0,t2) n-1 Ii-1x In-i-1____________________________! I _ _ | |_ F(,0i)xF(,in)#| #F|(,) __ __ __ __ composition __ __ Map (F (c0), F(ci)) x Map (F (ci), F(cn))|__|_|_!||_M|_||ap(F (c0), F(c* *n)) . These relations are more easily understood when one thinks of Io1([n]) ~=In-1* * as the space of all (t0, . .,.tn) 2 Io([n]) ~= In+1 such that t0 = 0 = tn. Each coord* *inate in Io([n]) corresponds to one of the objects in the chain , = (c0 ! . .!.cn). When ti= 1 for some 0 < i < n, ti and ci can be removed, giving the face relation __ __ F (,)(t1, . .,.ti-1, 1, ti+1, . .,.tn-1) = F(@i,)(t1, . .,.ti-1, ti+1, .* * .,.tn-1). __ When ti = 0 for 0 < i < n, then F (,)(t) can be split as a composite at the obj* *ect ci (relation (c)). If one of the morphisms in , is an identity, then one can remov* *e it and multiply the coordinates corresponding to its two objects. For instance, when m = 2 and n = 1 (and oe is one of the surjections), condit* *ion (b) says that __ Id ' __ ' Id F(c0 --! c0 --! c1) and F(c0 --! c1 --! c1) __ __ __ __ __ are_both the constant maps to F ('). In particular, F (') OF(Idc0) = F(') = F(I* *dc1) O F('). __ ' _ __When_n_= 2, condition_(c) says that F c0 --! c1 --! c2 is a homotopy from F(_) OF (') to F (_ O'). More generally, when , 2 Mor n(C) for n 2, __ __ __ __ __ * *__O F (,)(0, . .,.0) = F(,n-1,n) O. .O.F(,12) OF (,01) and F(,)(1, . .,.1) = * *F(,). Carles Broto, Ran Levi, and Bob Oliver * * 77 * * __ O At the other vertices of In-1, we get all of the other possible composites of t* *he F (,ij). An Rg 1-structure on F is thus a collection of higher homotopies connecting giv* *en homotopies F (_) OF (') ' F (_ O'). From this point of view, one sees that when defining an Rg 1-structure on F ,* * it suffices to define it on all nondegenerate sequences , 2 Mor n(C) (i.e., those_* *containing no identity morphisms), inductively for increasing n, where at each step F (,) * *has already been defined on @In-1 and_must be extended in some way to In-1. The sta* *rting point can be any choice of maps F('), for all ' 2 Mor (C), in the given homotop* *y class determined by F ('), such that __ __ __ __ __ F (') OF (Idc) = F(') = F(Idd) OF (') for each morphism ' 2 Mor C(c, d) in C. __ __ __ __ If F and F0 are both Rg1 -structures_on F_, then a morphism : F ---! F 0cons* *ists of homotopy equivalences `(c): F(c) --'-!F 0(c) (for each c 2 Ob (C)) such that* * `(c) O (c) ' 0(c), and such that for each , = (c0 ! . .!.cn) and each t 2 In-1, __ __ __ __ `(cn) OF (,)(t) = F0(,)(t) O`(c0) 2 Map (F (c0, F0(cn)). Two Rg 1-structures on F are equivalent if there is a third to which they both * *have morphisms. One easily sees that (homotopy) pushouts exist for morphisms of Rg * *1- structures on F , and hence that this defines an equivalence relation among Rg * *1- structures. __ For any given_Rg 1-structure F on F : C --! hoTop , we define its "homotopy colimit" Sp (F ) to be the space __ ia a __ j . Sp(F ) = F (c0) x In ~ (In = Io0([n])) n 0 c0!...!cn where the following identifications_are made for each n 1, each , = (c0 ! . .* *!. cn) 2 Mor n(C), and each x 2 F(c0): o x; I0(oe)(t)~[,]x; t [oe*,] (oe 2 Mor 0([m], [n]), t 2 I* *m ) __ x; (t1, 0, t2)~[,]F(,0i)(t1)(x); t2[,in](1 i n, t12 Ii-1, t22 In-i.) f g 2 For example, in the case of a sequence , = (c0 -! c1 -! c2) in Mor (C), the c* *orre- sponding square Io0([2]) is attached to the 1-skeleton in the following way: __ __ __ __ __ F(g Of) F(,) F(g) OF (f) F(c2) |______________|ss s |(1,0) (0,0)| J | | J | | J | __ |__ __ J __ | 2 | F(c )x I F (c )x I Id| F (c0)x I |F(f) -------! 0 J 1 | | J | | J | | |(1,1) (0,1)| J _______________||ss_ __ ____________________Jss__ Id Id F(f) F (c0) F (c0)x I F (c1) __ The_labels in the first picture describe the maps by which a vertex F (c0) or a* *n edge F(c0)xI is attached to the space represented by the second picture. Thus the tr* *apezoid in the earlier picture has now been replaced by a square. One way to understand these relations and their connection with those in Defi* *nition A.1 is to think of Io0([n]) ~= In as the subspace of all (n + 1)-tuples (0, t1,* * . .,.tn) in 78 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups Io([n]) ~=In+1. For , 2 Mor n(C), each coordinate in Io([n]) corresponds to one* * of the objects in the chain , = (c0 ! . .!.cn). When ti= 1 for 0 < i n, ti and ci ca* *n be removed, giving the face relation x; (t1, . .,.ti-1, 1, ti+1, . .,.tn) [,]~ x; (t1, . .,.ti-1, ti+1, . * *.,.tn) [@i,]. When_ti_= 0 for 0 < i n, then , splits as a composite at the object ci, we ap- ply F (,0i)(t1, . .,.ti-1) to x, and get the second of the above relations. If* * one of the morphisms in , is an identity, then we get a degeneracy relation by removing it* * and multiplying the two corresponding coordinates. Consider the maps oen : In --! n defined by i * * j oen(t1, . .,.tn) = t1t2. .t.n, (1 - t1)t2. .t.n, (1 - t2)t3. .t.n, . .,.(1 - t* *n-1)tn, 1 - tn . __ When F is a functor to Top and F is the corresponding locally constant Rg1 -str* *ucture __ O (i.e., for each ,, F(,)_is the constant map with value F (,)), then the oen def* *ine a home- omorphism from Sp (F ) to the usual homotopy colimit hocolim-----!(F ). More g* *enerally, __ when F is an arbitrary Rg1 -structure, then there is a map __ pr_F:Sp (F ) ------! |C| __ defined on each subspace F (c0) x In by first projecting to the In and then to * * n via oen. __ __ We now define a functor Rg (F ) : C --! Top by letting Rg (F )(c) be the pul* *lback space __ __ Rg(F )(c)______!Sp (F ) | | | |pr_F # # |C#c|__________!|C| (the ordinary pullback, not the homotopy pullback). A morphism ' 2 Mor C(c, d) induces a map from |C#c|_to |C#d| via_composition_with ' in the usual way, and * *hence induces a map from Rg (F )(c) to Rg (F )(d). Equivalently, __ ia a __ j . Rg (F )(c) = F (c0) x In ~ (In = Io0([n])) n 0 c0!...!cn!c __ where the identifications_are analogous to those used to define Sp (F ). This * *clearly makes Rg (F ) into a functor from C to Top. __ __ For each c, F(c) can be identified as a subspace of Rg (F )(c): the inverse i* *mage under (c) __ the projection to |C#c| of the vertex (c -Id!c). The composite F (c) ---! F * *(c) __ __ ' Rg (F )(c) defines a natural transformation F --! hoORg (F ) of functors C --!* * hoTop . The following_proposition now shows that this is a natural equivalence, and hen* *ce that Rg (F ) is a rigidification of F . __ Proposition_A.2. For any Rg 1-structure F on F_:C ---! hoTop , for_each c 2 Ob (C), F (c) is a deformation retract of Rg (F )(c). Thus Rg (F ) is a rigidi* *fication of F . Carles Broto, Ran Levi, and Bob Oliver * * 79 __ __ Proof.Define : Rg (F )(c) x I ---! Rg (F )(c) by setting (x; t)[,!c], s = (x; (t, s)) Id [,!c-! c] __ for all ,_2 Mor n(C), t 2 In, and s 2_I. Then (u, 1) = u and (u, 0) 2 F (c) f* *or all u 2_Rg_(F )(c) by definition_of Rg (F )(c). Furthermore, the homotopy is the id* *entity on F (c), and thus F (c) is a deformation retract. For any given F :C ---! hoTop , let Rigid(F ) be the set of equivalence class* *es of rigidifications of F , and let Rg1 (F ) be the set of equivalence classes of Rg* *1 -structures on F . A rigidification of F_can_be regarded as a "locally_constant" Rg1 -struc* *ture on F ; i.e., an Rg1 -structure F where each of the maps F(,) (for , 2 Mor n(C)) is* * constant on In-1. We thus have maps const Rigid(F ) -------!-------Rg1(F ). Rg One easily checks that for any rigidification Fe, there is a natural transforma* *tion of functors from Rg (const(Fe)) to eF, and hence these are equal in Rigid(F ). We * *do not know whether the other composite is the identity on Rg 1(F ), but that will not* * be needed here. A natural transformation O: F ---! F 0of functors F, F 0:C ---! hoTop will * *be called relatively centric if for each morphism ' 2 Mor C(c, d) in C, the homoto* *py com- mutative square F(')O- Map (F (c), F (c))Id_____! Map (F (c), F (d))F(') | | O(c)O-| O(d)O-| # # F0(')O- 0 Map (F (c), F 0(c))O(c)_! Map (F (c), F (d))F0(')OO(c) is a homotopy pullback. For example, when F 0is the functor which sends every o* *bject to a point, then O is relatively centric if and only if the functor F defines a* * centric di- agram. Assume we are given a relatively centric natural transformation O: F ---* *! F 0 where F 0is a functor to Top, and assume furthermore that for each c 2 Ob (C), * *the homotopy fiber i O(c)O- j (c) def=hofiberMap(F (c), F (c))Id----! Map (F (c), F 0(c))O(c) is connected. We claim that this determines functors fii:Cop ------! Ab (all i 1) such that fii(c) ~=ssi( (c)) for all c. To show this, we can assume without los* *s of general- ity that O(c) is a fibration for all c, and let (c) be the space of all f 2 Ma* *p (F (c), F (c)) such that O(c) Of = O(c). Then (c) is a monoid under composition, and in part* *ic- ular, ss1( (c)) is abelian. For each morphism ' 2 Mor C(c, d) in C, we can cho* *ose a representative F (') such that the following square commutes: F(') F (c)____! F (d) | | O(c)| O(d)| # # F0(') 0 F 0(c)_____!F (d) . 80 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups Since O is relatively centric, the fibers of the map (O(d) O-): Map (F (c), F (d))F(')------! Map (F (c), F 0(d))F0(')OO(c) have the homotopy type of (c) and hence are connected. Hence any two choices f* *or F (') differ by a path in the fiber over the point F 0(') OO(c); i.e., by a hom* *otopy {Ft(')}t2Isuch that O(d) OFt(') = F 0(') OO(c) for each t. For each ' 2 Mor C(c, d), consider the following diagram: -OF(') F(')O- Map (F (d), F (d))Id____=w1!Map(F (c), F (d))F(')____=w2Map(F (c), F (c))Id | | | O(d)O-=u1| O(d)O-=u2| O(c)O-=u3| # # # -OF(') 0 F0(')O- 0 Map (F (d), F 0(d))O(d)_!Map (F (c), F (d))O(d)F(')___Map (F (c), F (c))O(c) where the right hand square commutes by the assumption on F (') (and the other * *since composition is associative). Set (c) = u-13(O(c)) and fii(c) = ssi( (c), IdF(* *c)) (and similarly for d). By assumption, w2 sends (c) to u-12(F 0(') OO(c)) by a homo* *topy equivalence, and we let fii(') be the composite (w2O-)-1 ssi( (d), IdF(d))w1O------!ssi(u-12(F 0(') OO(c)), F (')) -------!~ ssi( (c)* *, IdF(c)). =fii(d) = =fii* *(c) By the above remarks, this is independent of the choice of map F ('). Hence th* *is defines a functor on Cop: the relations fii(_ O') = fii(_) Ofii(') follow using* * any choice of homotopy from F (_ O') to F (_) OF (') which covers F 0(_ O'). (Recall that * *we are assuming F 0is a functor to Top, so F 0(_ O') = F 0(_) OF 0(').) The following theorem is our main result giving a relative version of the Dwy* *er-Kan obstruction theory. The special case where F 0(c) is a point for all c 2 Ob (C)* * is the case shown by Dwyer and Kan in [DK ]. Theorem A.3. Fix functors F :C ---! hoTop and F 0:C ---! Top , and let O: F ---! hoO F 0 be a relatively centric natural transformation of functors. For each c 2 Ob (C)* *, assume that the homotopy fiber O(c)O- 0 (c) = hofiberMap (F (c), F (c))Id----! Map (F (c), F (c))O(c); is connected. Let fii:Cop ---! Ab (all i 1) be the functors defined above. T* *hen the eO 0 O 0 obstructions to the existence of a rigidification eF----! F of F ----! F lie* * in the groups lim-n+2(fin) for n 1; while the obstructions to the uniqueness of (Fe,* * eO) up to C equivalence of rigidifications lie in lim-n+1(fin) for n 1. C Proof.We use here the description of the higher limits of a functor ff: Cop ---* *! Ab as the homology groups of the normalized cochain complex __ Y Cn(C; ff) = ff(c0), c0!...!cn Carles Broto, Ran Levi, and Bob Oliver * * 81 where the_product is taken over all composable sequences of nonidentity morphis* *ms. For , 2 Cn(C; ff), define ' d(,)(c0 -! c1 ! . .!.cn+1) = F (')(,(c1 ! . .!.cn+1))+ n+1X (-1)i,(c0 ! . .b.ci.!.c.n+1) . i=1 __ Then lim-*(ff) ~=H*(C *(C; ff), d) (cf. [GZ , Appendix II, Proposition 3.3] or * *[Ol, Lemma C 2].) Proof of existence: As above, we replace each O(c) by a fibration, and replace* * each F (') (for '_2_Mor C(c, d) by a map such that O(d) OF (') = F 0(') OO(c). We a* *lso assume that F (Idc) = Id_F(c)for each c. Then (c) def=f 2 Map (F (c), F (c)) | O(c) Of = O(c) is a topological monoid under composition, and is connected by assumption. So * *we can ignore basepoints when working in the homotopy groups fii(c) = ssi( (c)). __ __ __We want to construct an Rg1 -structure F such that F (c) = F (c) for all c 2 * *Ob (C), F(') = F (') for all ' 2 Mor (C), and such that for each n 2 and each , = (c0* * ! . .!.cn) 2 Mor n(C), the following square commutes (exactly) for each t 2 In-1: _ F(,)(t) F (c0)____! F (cn) | | O(c0)| O(cn)| (1) # O # F0(,) 0 F 0(c0)___! F (cn) . By Proposition A.2, any such structure induces a rigidification eFof F , togeth* *er with a natural transformation of functors eOfrom eFto F 0. __ Assume, for some n 2, that F has been defined on Mor i(C) for all i < n. * *Fix , 2 Mor n(C), a composite of (nonidentity) maps from c0 to cn. Consider the fol* *lowing commutative square, which is a homotopy pullback by assumption: O F(,)O- Map (F (c0), F (cn)) O_______ Map (F (c0), F (c0))Id F(,)=w | O(cn)O-=u| O(c0)O-| (2) # O # F0(,)O- 0 Map (F (c0), F 0(cn)) O ____Map (F (c0), F (c0))O(c0). F0(,)OO(c0) __ Conditions (b) and (c) in Definition A.1 determine a map F (,)0 from @In-1 to O Map (F (c0), F (cn)) O whose image lies in u-1(F 0(,) OO(c0)). Hence the obstr* *uction to __ F(,) defining F (,) on In-1 is an element -1 0O wO- j(,) 2 ssn-2 u (F (,) OO(c0)), ----~ ssn-2( (c0)) = fin-2(c0). = If_one of the morphisms in the sequence , is an identity morphism, then we defi* *ne F(,)_using the appropriate formula in Definition A.1(b), and j(,) = 0. Thus j 2 Cn(C; fin-2). 82 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups We claim that dj = 0. Fix ! = (c0 ! . .!.cn+1) 2 Mor n+1(C). Consider the face maps on the n-cube ffiti:In-1 -----! In where ffiti(t1, . .,.tn-1) = (t1, . .,.ti-1, t,* * ti, . .,.tn-1) (for all i = 1, . .,.n and t = 0, 1). The conditions in Definition A.1(b,c) def* *ine a map __ F o(!): (In)(n-2)-----! Map (F (c0), F (cn+1))F(O!)' Aut(F (c0))1 = Map (F (c0), F (c0))Id, and hence Xn i __ __ j (-1)i [F o(!) Offi1i|@In-1] - [F o(!) Offi0i|@In-1] = 0 2 ss1(Aut((F3* *(c0))1).) i=1 __ Furthermore, F o(!) extends to the faces ffi0i(In-1) for 2 i n - 1 (again, * *by the conditions in Definition A.1(c)), and so those terms vanish in (3). So we are l* *eft with the equality __ Xn __ __ 0 = [F o(!) Offi01|@In-1] + (-1)i[F o(!) Offi1i|@In-1] + (-1)n+1[F o(!) * *Offi0n|@In-1] i=1 n+1X = F (!01)*(j(@0!)) + (-1)ij(@i!) = dj(!). i=1 Thus dj = 0, and so [j] 2 lim-n(fin-2). C __ If [j] = 0, then there is ae 2 C n-1(C; fin-2)_such that j = dae. Similar (bu* *t simpler) arguments to those used above now show that F can be "changed by ae" on element* *s of_ Mor n-1(C), in a way so that the obstruction j vanishes. We can thus arrange th* *at F can be_extended to Morn(C). Upon continuing this procedure, we obtain the Rg1 -stru* *cture F. Proof of uniqueness: Now assume that eF1--eO1---!F 0--eO2---eF2 are two rigidifications of O: F ---! F 0. In other words, we have a homotopy c* *ommu- tative diagram 1 F _______! ho OeF1 HH || HH O | 2# HH ho(eO1)| (4) ho(eOHHj)# ho OeF2_____!2ho OF 0 of functors C ! hoTop and natural transformations between them. We can assume that the maps eO(c) and eO0(c) are fibrations for each c 2 Ob (C); otherwise we* * replace them by fibrations using one of the canonical constructions. For each c 2 Ob (C), let `(c): eF1(c) ---! Fe2(c) be any map such that `(c) O* * 1(c) ' 2(c) as maps from Fe1(c) to Fe2(c). Using the homotopy commutativity of (4), * *and the homotopy lifting property for eO2(c), we can assume that eO2(c) O`(c) = Oe1* *(c) (exactly, not just up to homotopy). Let bF(c) be the mapping cylinder of `(c), * *and let bO(c): bF(c) ---! F 0(c) be the projection induced by eO1(c) and eO2(c). Regard eF1(c) and eF2(c) as subspaces of bF(c). We want to extend the locally* * finite Rg1 -structures Fe and Fe0to an Rg 1-structure Fb covering F 0. For each morph* *ism Carles Broto, Ran Levi, and Bob Oliver * * 83 ' 2 Mor C(c, d) in C, `(d) OeF1(') ' eF2(') O`(c), and hence eF1(') and eF2(') * *can be extended to a map bF(') from bF(c) to bF(d). Using the homotopy lifting propert* *y again, this can be chosen such that bO(d) ObF(') = F 0(') ObO(c). Assume, for some n 2, that Fb has_been_defined on Mor i(C) for i < n in a way so that (1) commutes (with F and F replaced by bF) for each ,. Then for each , = (c0 ! . .!.cn) in Mor n(C), bF(,) has been defined on i j i j (Fe1(c0) [ eF2(c0)) x In-1 [ bF(c0) x @In-1 , O and must be extended to bF(c0)xIn-1 while covering bO(cn)OFb(,) 2 Map (Fb(c0), * *F 0(cn)). So with the help of diagram (2) again, the obstruction to defining bF(,) is see* *n to be an_element o(,) 2 ssn-1( (c0)) = fin-1(c0). Together, these define a cochain o* * 2 Cn(C; fin-1). Just as in the proof of existence, one then shows that do = 0, * *and hence_that_o represents a class [o] 2 lim-n(fin-1). If [o] = 0, then o = dae f* *or some ae 2 C n-1(C; fin-1), and bFcan be modified on Mor n-1(C) using ae in such a wa* *y that it can then be extended to Mor n(C). Upon continuing this procedure, we constru* *ct an Rg1 -structure bFon F , together with a natural transformation to F 0and morphi* *sms of Rg1 -structures eF1-----! bF----- Fe2. So by Proposition A.2, eF1' Rg (Fe1) ' Rg (Fb) ' Rg (Fe2) ' eF2. We finish the section with two corollaries to Theorem A.3. The first is the * *main theorem of Dwyer and Kan in [DK ]. It is the "absolute case" of Theorem A.3: th* *e case where F 0is the constant functor which sends each object to a point. A functor F from C to Top or hoTop will be called centric if for each morphi* *sm ' 2 Mor C(c, d) in C, the induced map 'O- Map (F (c), F (c)Id------! Map (F (c), F (d))' is a homotopy equivalence. This is what Dwyer and Kan call a centric diagram. Corollary A.4. Fix a centric functor F :C ---! hoTop . Define ffi:Cop ---! Ab * * ' (all i 1) by setting ffi(c) = ssi Map (F (c), F (c))Id and by letting ffi c _* *__! d be the composite (-OF('))* (F(')O-)* ssi Map(F (d), F (d))Id ______! ssi Map(F (c), F (d))F(') ______~=ssi Map(F * *(c), F (c))Id . Then the obstructions to the existence of a rigidification Fe of F lie in the * *groups lim-n+2(ffn) for n 1; while the obstructions to the uniqueness of eFup to equ* *ivalence C of rigidifications lie in lim-n+1(ffn) for n 1. C The second corollary is a generalization of [CLN , Proposition B], and follo* *ws upon combining Corollary A.4 with an idea taken from the proof of that proposition. Corollary A.5. Fix a space X, and a centric functor F :C ---! hoTop . We also l* *et X denote the constant functor X :C ---! Top which sends each object to X and ea* *ch morphism to IdX. 84 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups (a)Assume there is a natural transformation of functors O: F ---! hoO X such t* *hat the map O(c): F (c) ---! X is centric for each c 2 Ob (C). Then there is a r* *igidi- fication eFof F , together with a rigidification eO:eF---! X of O. (b)Assume eF1and eF2are two rigidifications of F . Let _i:F ---! hoO eFibe nat* *ural equivalences, and let eOi:eFi---! X be natural transformations of functors s* *uch that for all c 2 Ob (C), eOi(c) 2 Map (Fei(c), X) is centric, and the square _1(c) F (c)_________! eF1(c) | _2(c)#| eO1(c)| (1) # eF2(c)_________eO2(c)!X commutes up to homotopy. Then Fe1and Fe2are equivalent rigidifications. Mo* *re precisely, there is a third rigidification eF0of F , natural transformations* * of functors eF1--e_1---!eF0e_2-----eF2 such that e_i(c) is a homotopy equivalence for each c 2 Ob (C), a space X0 t* *ogether with a natural transformation eO0:eF0---! X0 to the constant functor, and ho* *mo- topic homotopy equivalences f1 ' f2: X ---! X0, such that the following diag* *ram commutes for each c 2 Ob (C): eF1(c)______e_1(c)!eF(c)____e_2(c)eF(c) ' 0 ' 2 | | | eO1(c)| eO0(c)| eO2(c)| (2) # # # f1 f2 X _________! X0 _________X . Proof.Let C+ be the category C with an additional final object * added. For any functor ff: C+op ---! Ab , lim-i(ff) = 0 for all i 1 since C+op has an initia* *l object. A functor F+ :C+ ---! hoTop can be thought of as a triple F+ = (F, X, O), where F = F+|C is a functor from C to hoTop , X = F+(*) is a space, and O is a natural transformation of functors from F to the constant functor X. Functors from C+ t* *o Top are described in an analogous way. In the situation of (a), (F, X, O) is a functor from C+ to hoTop . The obstr* *uction groups of Corollary A.4 vanish, and hence it has a rigidification (Fe, eX, eO).* * Upon composing with an appropriate homotopy equivalence eX--'-! X, we can arrange th* *at eX= X. In the situation of (b), (Fe1, X, eO1) and (Fe2, X, eO2) are two functors fro* *m C+ to Top which are rigidifications of the same functor (F, X, eO1O_1) by the homotopy co* *mmu- tativity of (1). Since the uniqueness obstructions of Corollary A.4 all vanish,* * there is a third homotopy lifting (Fe0, X0, eO0), together with natural transformations of* * functors _e1 _e2 (Fe1, X, eO1) ----- (Fe0, X0, eO0) -----! (Fe2, X, eO2) which induce homotopy equivalences on all objects. Thus upon setting fi= e_i(*)* *, we obtain the commutative diagram (2), where all horizontal maps are homotopy equi* *v- alences. Finally, e_1(*) ' e_2(*), since they come from equivalences between li* *ftings of the same homotopy functor, and this finishes the proof of (b). Carles Broto, Ran Levi, and Bob Oliver * * 85 References [Al] J. Alperin, Sylow intersections and fusion, J. Algebra 6 (1967), 222-241 [Be] D. Benson, Fusion in compact Lie groups, Quart. J. Math. 47 (1996), 261-2* *68 [BK] P. Bousfield & D. Kan, Homotopy limits, completions, and localizations, L* *ecture notes in math. 304, Springer-Verlag (1972) [Br] G. Bredon, Introduction to compact transformation groups, Academic Press,* * New York- London (1972) [BKi] C. Broto & N. Kitchloo, Classifying spaces of Kac-Moody groups, Math. Z. * *240 (2002), 621- 649 [BLO1]C. Broto, R. Levi, & B. Oliver, Homotopy equivalences of p-completed clas* *sifying spaces of finite groups, Invent. math. 151 (2003), 611-664 [BLO2]C. Broto, R. Levi, & B. Oliver, The homotopy theory of fusion systems, Jo* *urnal Amer. Math. Soc. 16 (2003), 779-856 [Bw] K. Brown, Cohomology of groups, Springer-Verlag (1982) [CLN] N. Castellana, R. Levi, & D. Notbohm, Homology decompositions for p-compa* *ct groups, preprint [DK] W. Dwyer & D. Kan, Centric maps and realization of diagrams in the homoto* *py category, Proc. Amer. Math. Soc. 114 (1992), 575-584 [DW] W. Dwyer & C. Wilkerson, Homotopy fixed-point methods for Lie groups and * *finite loop spaces, Annals of math. 139 (1994), 395-442 [DW2] W. Dwyer & C. Wilkerson, The center of a p-compact group, The ~Cech cente* *nnial, Contemp. Math. 181 (1995), 119-157 [Fe] M. Feshbach, The Segal conjecture for compact Lie groups, Topology 26 (19* *87), 1-20 [Fu] L. Fuchs, Infinite abelian groups, vol. I, Academic Press (1970) [GZ] P. Gabriel & M. Zisman, Calculus of fractions and homotopy theory, Spring* *er-Verlag (1967) [Go] D. Gorenstein, Finite groups, Harper & Row (1968) [HV] J. Hollender & R. Vogt, Modules of topological spaces, applications to ho* *motopy limits and E1 structures, Arch. Math. 59 (1992), 115-129 [JM] S. Jackowski & J. McClure, Homotopy decomposition of classifying spaces v* *ia elementary abelian subgroups, Topology 31 (1992), 113-132 [JMO] S. Jackowski, J. McClure, & B. Oliver, Homotopy classification of self-ma* *ps of BG via G- actions, Annals of Math. 135 (1992), 184-270 [JMO2]S. Jackowski, J. McClure, & B. Oliver, Homotopy theory of classifying spa* *ces of compact Lie groups, Algebraic topology and its applications, M.S.R.I. Publ. 27, Sprin* *ger-Verlag (1994), 81-123 [JMO3]S. Jackowski, J. McClure, & B. Oliver, Self homotopy equivalences of clas* *sifying spaces of compact connected Lie groups, Fundamenta Math. 147 (1995), 99-126 [KW] O. H. Kegel & B. A. F. Wehrfritz, Locally finite groups. North-Holland Ma* *thematical Library, Vol. 3. North-Holland Publishing Co., Amsterdam-London; American Elsevier* * Publishing Co., Inc., New York, 1973. xi+210 pp. [McL] S. Mac Lane, Categories for the working mathematician, Springer-Verlag (1* *971) [Mo] J. Moller, Homotopy Lie groups, Bull. Amer. Math. Soc. 32 (1995), 413-428 [Ol] B. Oliver, Higher limits via Steinberg representations, Comm. in Algebra * *22 (1994), 1381-1393 [Pu] L. Puig, Unpublished notes [Seg] G. Segal, Categories and cohomology theories, Topology 13 (1974), 293-312 [Sz] M. Suzuki, Group theory I, Springer-Verlag (1982) [W] B. A. F. Wehrfritz, Infinite linear groups. Ergebnisse der Matematik und * *ihrer Grenzgebiete, Band 76. Springer-Verlag, New York-Heidelberg (1973) [Wo] Z. Wojtkowiak, On maps from holimF to Z, Algebraic topology, Barcelona, 1* *986, Lecture Notes in Math. 1298, Springer-Verlag (1987), 227-236 86 Discrete models for the p-local homotopy theory of compact Lie groups and * *p-compact groups Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, E-08193 Bel- laterra, Spain E-mail address: broto@mat.uab.es Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, U.K. E-mail address: ran@maths.abdn.ac.uk LAGA, Institut Galil'ee, Av. J-B Cl'ement, 93430 Villetaneuse, France E-mail address: bobol@math.univ-paris13.fr