EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL BOB OLIVER AND JOANA VENTURA Abstract. We study extensions of p-local finite groups where the kernel i* *s a p- group. In particular, we construct examples of saturated fusion systems F* * which do not come from finite groups, but which have normal p-subgroups A C F such* * that F=A is the fusion system of a finite group. One of the tools used to do t* *his is the concept of a "transporter system", which is modelled on the transporter c* *ategory of a finite group, and is more general than a linking system. Let G be a finite group, with Sylow p-subgroup S 2 Sylp(G). The fusion system of G (at p) is the category FS(G) whose objects are the subgroups of G, and whe* *re MorFS(G)(P, Q) is the set of monomorphisms from P to Q induced by conjugation by elements of G. The transporter system of G at p is the category TS(G) with the * *same objects as FS(G), and with morphism sets Mor TS(G)(P, Q) = NG (P, Q): the set * *of elements of G which conjugate P into Q. A subgroup P S is called p-centric in* * G if CG (P ) = Z(P ) x C0G(P ) for a (unique) subgroup C0G(P ) of order prime to p; * *and the centric linking system of G is the category LcS(G) whose objects are the subgro* *ups of S which are p-centric in G, and where Mor LcS(G)(P, Q) = NG (P, Q)=C0G(P ). All o* *f these definitions are repeated in more detail at the beginning of Section 1. In sever* *al papers, such as [BLO1 ] and [O2 ], the fusion and linking systems of G are shown to pl* *ay a central role in describing homotopy theoretic properties of the p-completed cla* *ssifying space BG^p. Abstract fusion and linking systems have also been defined and studied, and a* *re shown in [BLO2 ] to have many of the same properties as the fusion and linking* * systems of finite groups. A p-local finite group is defined to be a triple (S, F, L), * *where S is a finite p-group, F is a saturated fusion system over S (Definitions 1.2 and 1.* *3), and L is a centric linking system associated to F (Definition 1.6). Normal and cen* *tral p-subgroups of fusion systems and linking systems are also defined (Definition * *1.4). Certain types of extensions of p-local finite groups, and in particular centr* *al exten- sions, were studied in [BCGLO2 ]. One hope was that extensions could provide * *a new way to construct exotic examples. But in the case of central extensions, this w* *as shown to be impossible. By [BCGLO2 , Theorem 6.13 and Corollary 6.14], if A is a c* *entral subgroup in (S, F, L), and (S=A, F=A, L=A) is induced by a group G, then (S, F,* * L) is induced by a group eGsuch that A Z(Ge) and eG=A ~=G. In this paper, we look at the more general situation of extensions with p-gro* *up kernel. Equivalently, given a p-local finite group (S, F, L) and a finite p-gr* *oup A, we want to find p-local finite groups (Se, eF, eL) such that A C F and (S, F, L* *) ~= (Se=A, eF=A, eL=A). One problem when doing this is that the fusion system F=A c* *ontains too little information: F cannot be described as an extension of F=A by A in an* *y sense. ___________ 2000 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 20D2* *0. Key words and phrases. Classifying space, p-completion, finite groups, fusion. B. Oliver is partially supported by UMR 7539 of the CNRS. J. Ventura is partially supported by FCT/POCTI/FEDER and grant PDCT/MAT/58497* */2004. Both authors were partially supported by the Mittag-Leffler Institute in Swed* *en. 2 BOB OLIVER AND JOANA VENTURA Another problem is that in general, when eLis a linking system and A C eL, then* * eLcan be thought of as an extension of eL=A by A, but eL=A need not be a linking syst* *em. As explained in Section 2, it can contain much more information than a linking sys* *tem does. Conversely, if we take a linking system L and try naively to extend it, t* *hen the resulting category will in general have too few objects to be a linking system.* * So we were forced to look at a larger class of categories to extend. A transporter system is a category whose objects are subgroups of a given p-g* *roup S, associated to a given fusion system, which satisfies axioms motivated by the tw* *o main examples: the transporter category of a finite group, and categories of the for* *m L=A when L is a linking system and A C L. A transporter system T for which Ob (T ) * *is the set of all subgroups of S is always the transporter category of the finite grou* *p AutT (1) (Proposition 3.12), so we are interested mainly in the cases where not all subg* *roups of S are objects. The precise definition is given at the start of Section 3. Nerves of transporter systems have many of the topological properties which a* *re already known for linking systems. For example, if T is a transporter system as* *sociated to the fusion system F, and Ob (T ) includes all F-centric subgroups, then T i* *nduces a centric linking system L associated to F and |T |^p' |L|^p(Proposition 4.6). * * As another example, if T r T is the full subcategory whose objects are the "T -r* *adical subgroups" (Definition 3.9), then |T r| ' |T | (Proposition 4.7). Extensions of transporter systems are defined and studied in Section 5. If T* * is a transporter system, and o :eT ! T is a functor which satisfies certain categ* *ory theoretic properties (Definition 5.1), then eTis also a transporter system and * *eT=A ~=T for a certain normal p-subgroup A C eT. Moreover, in this situation, |Te| ! |T* * | is a fibration with fiber BA. Once this has been established, then conditions are de* *scribed (Theorem 5.11) which imply that eTis in fact a centric linking system, or at le* *ast a full subcategory of a centric linking system which includes all subgroups which are * *centric and radical. Finally, in Section 6, we look at extensions 1 ! A ! eT ! T ! 1 of this type, when T is a full subcategory of the transporter category of a finite group G. W* *e first show that if the induced action of ss1(|T |) on A factors through G, then eT wi* *ll be a full subcategory of the transporter category of some group eGsuch that eG=A ~* *=G. Afterwards, we give examples (Example 6.2) of such extensions where the action * *does not factor through G, and where eTand its associated fusion system eFare exotic* * in in the sense that eFis not the fusion system of any finite group. We would like to thank the University of Aberdeen, the Universitat Aut`onoma * *de Barcelona, and especially the Bernoulli Center in Lausanne and the Mittag-Leffl* *er Institut near Stockholm for their hospitality, allowing the two authors to meet* * together and work on this project. We would also like to thank Albert Ruiz for his very * *timely discovery of some examples of exotic fusion systems which helped lead to our Ex* *ample 6.2. 1.Background: fusion and linking systems We first fix some notation. For any group G, and any x 2 G, cx denotes conjug* *ation by x (cx(g) = xgx-1). For H, K G, we write NG (H, K) = {x 2 G | xHx-1 K} EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 3 to denote the transporter set, and fi Hom G(H, K) = cx 2 Hom (H, K) fix 2 NG (H, K) ~=NG (H, K)=CG (H). We also set AutG (H) = Hom G(H, H) ~=NG (H)=CG (H). Definition 1.1. Fix a finite group G and a Sylow subgroup S 2 Sylp(G). (a) FS(G) and TS(G) denote the categories where Ob (FS(G)) = Ob (TS(G)) is the * *set of all subgroups of S, and where MorFS(G)(P, Q) = Hom G(P, Q) ~=NG (P, Q)=CG (P ) and Mor TS(G)(P, Q) = NG (P, Q). Let ae: TS(G) ----! FS(G) be the functor which is the identity on objects, * *and which sends x 2 NG (P, Q) to cx 2 Hom G(P, Q). (b) A p-subgroup P G is p-centric in G if Z(P ) 2 Sylp(CG (P )); equivalently* *, if CG (P ) = Z(P ) x C0G(P ) for some (unique) subgroup C0G(P ) of order prime* * to p. Define LcS(G) to be the category whose objects are the subgroups of S which* * are p-centric in G, and where Mor LcS(G)(P, Q) = NG (P, Q)=C0G(P ). We call FS(G) the fusion system (or fusion category) of G, TS(G) the transpor* *ter system, and LcS(G) the centric linking system. In this paper, we will be looki* *ng at abstract versions of all three of these systems, starting with fusion systems. Definition 1.2 ([Pg ], [BLO2 , Definition 1.1]). A fusion system over a finite* * p-group S is a category F, where Ob (F) is the set of all subgroups of S, and which sat* *isfies the following two properties for all P, Q S: o Hom S(P, Q) Hom F(P, Q) Inj(P, Q); and o each ' 2 Hom F(P, Q) is the composite of an isomorphism in F followed by an inclusion. Fusion systems as defined above are too general for our purposes, and additio* *nal axioms are needed for them to be very useful. When F is a fusion system over a * *finite p-subgroup S, then two subgroups P, Q S are said to be F-conjugate if they are isomorphic as objects of the category F. A subgroup P S is called fully centr* *alized in F if |CS(P )| |CS(P 0)| for all P 0 S which is F-conjugate to P . Simila* *rly, a subgroup P S is called fully normalized in F if |NS(P )| |NS(P 0)| for all * *P 0 S which is F-conjugate to P . Definition 1.3 ([Pg ], [BLO2 , Definition 1.2]). A fusion system F over a fini* *te p-group S is saturated if the following two conditions hold: (I)For all P S which is fully normalized in F, P is fully centralized in F * *and Aut S(P ) 2 Sylp(Aut F(P )). (II)If P S and ' 2 Hom F (P, S) are such that 'P is fully centralized, and i* *f we set N' = {g 2 NS(P ) | 'cg'-1 2 AutS('P )}, _ _ then there is ' 2 Hom F(N', S) such that '|P = '. 4 BOB OLIVER AND JOANA VENTURA If G is a finite group and S 2 Sylp(G), then FS(G) is a saturated fusion syst* *em. Axioms (I) and (II) follow mostly as consequences of the Sylow theorems (cf. [B* *LO2 , Proposition 1.3]). We next specify certain collections of subgroups relative to a given fusion s* *ystem. Definition 1.4. Let F be a fusion system over a finite p-subgroup S. o A subgroup P S is F-centric if CS(P 0) = Z(P 0) for all P 0 S which is F- conjugate to P . We let Fc F denote the full subcategory with objects the F* *-centric subgroups of S. o A subgroup P S is F-radical if Out F(P ) is p-reduced; i.e., if Op(Out F(P * *)) = 1. o A subgroup A S is normal in F (denoted A_C F) if for all P, Q S and_all f 2 Hom F(P, Q), f extends to a morphism f 2 Hom F(P A, QA) such that f(A) = * *A. o A subgroup A S is centralin_F if for all P, Q S_and all f 2 Hom F (P, Q),* * f extends to a morphism f 2 Hom F(P A, QA) such that f|A = IdA. If F = FS(G) for some finite group G, then P S is F-centric if and only if P is p-centric in G (i.e., Z(P ) 2 Sylp(CG (P ))), and P is F-radical if and * *only if NG (P )=(P .CG (P )) is p-reduced. However, P being F-radical is not the same a* *s being a radical p-subgroup. In fact, it turns out that saturated fusion systems defined only on the centr* *ic sub- groups are equivalent to saturated fusion systems defined on all subgroups. In * *other words, when constructing saturated fusion systems over a finite p-group S, we r* *eally need only define it on the centric subgroups of S, and check that it satisfies * *axioms (I) and (II) for those subgroups. The next theorem describes how a category constru* *cted in this way can then be extended in a unique way to a saturated fusion system o* *ver S. For any fusion system F over S, and any set H of subgroups of S which is clos* *ed under F-conjugacy, we say that F is H-saturated if conditions (I) and (II) in D* *efinition 1.3 are satisfied for all P 2 H. We say that F is H-generated if each morphism * *in F is a composite of restrictions of morphisms between subgroups in H. Theorem 1.5. Fix a p-group S and a fusion system F over S. (a) Assume F is saturated, and let H be the set of F-centric F-radical subgroups of S. Then F is H-generated. More precisely, for each P, P 0 S and each ' 2 IsoF(P, P 0), there are subgroups P = P0, P1, . .,.Pk = P 0, subgroups * *Qi (i = 1, . .,.k) which are F-centric, F-radical, and fully normal* *ized in F, and automorphisms 'i 2 Aut F(Qi), such that 'i(Pi-1) = Pi for all i and ' = ('k|Pk-1) O. .O.('1|P0). (b) Let F be a fusion system over a finite p-group S. Let H be a set of subgrou* *ps of S closed under F-conjugacy such that F is H-saturated and H-generated. Assume also that each F-centric subgroup of S not in H is F-conjugate to some subg* *roup P S such that Out S(P ) \ Op(Out F(P )) 6= 1. Then F is saturated. Proof. Part (a) is Alperin's fusion theorem for saturated fusion systems, in th* *e form shown in [BLO2 , Theorem A.10]. Part (b) is proven in [BCGLO1 , Theorem 2.2]. We now turn to linking systems associated to abstract fusion systems. Definition 1.6 ([BLO2 , Definition 1.7]). Let F be a fusion system over the p-* *group S. A centric linking system associated to F is a category L whose objects are * *the EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 5 F-centric subgroups of S, together with a functor ss :L ---! Fc, and distinguis* *hed ffiP monomorphisms P --! AutL(P ) for each F-centric subgroup P S, which satisfy the following conditions. (A) ss is the identity on objects. For each pair of objects P, Q 2 L, Z(P ) act* *s freely on Mor L(P, Q) by composition (upon identifying Z(P ) with ffiP(Z(P )) Aut L* *(P )), and ss induces a bijection ~= Mor L(P, Q)=Z(P ) ------! Hom F (P, Q). (B) For each F-centric subgroup P S and each x 2 P , ss(ffiP(x)) = cx 2 AutF * *(P ). (C) For each f 2 Mor L(P, Q) and each x 2 P , the following square commutes in * *L: f P ______//_Q | | | | |ffiP(x) |ffiQ(ss(f)(x)) | | fflffl|f fflffl| P ______//_Q . A p-local finite group is defined to be a triple (S, F, L), where S is a fini* *te p-group, F is a saturated fusion system over S, and L is a centric linking system associ* *ated to F. The classifying space of the triple (S, F, L) is the p-completed nerve |L|^p. For any finite group G with Sylow p-subgroup S, the category LcS(G) (Definiti* *on 1.1) is easily seen to satisfy conditions (A), (B), and (C) above, and hence is* * a centric linking system associated to FS(G). Thus (S, FS(G), LcS(G)) is a p-local finite* * group, with classifying space |LcS(G)|^p' BG^p(see [BLO1 , Proposition 1.1]). The following lifting lemma for linking systems is used frequently. Lemma 1.7. Let (S, F, L) be a p-local finite group. Fix F-centric subgroups P, * *Q, R ' _ in S, and let P --! Q --! R be any sequence of morphisms in F. Then the follow* *ing hold. (a) Let e_and f_'be arbitrary liftings to L of _ and _', respectively. Then the* *re is a unique morphism e'2 Mor L(P, Q) such that _eO'e= f_'; (1) and furthermore ssP,Q('e) = '. (b) Choose liftings f_'2 Mor L(P, R) and e'2 Mor L(P, Q) of _' and ', respectiv* *ely. Then there is a unique morphism e_2 Mor L(Q, R) such that e_Oe'= f_', and s* *uch that ss(_e) = _ Ocq for some q 2 '(Z(P )). Proof.(a) See [BLO2 , Lemma 1.10] or Lemma A.7(a). (b) Let ff 2 Mor L(Q, R) be a lifting of _, i.e., ss(ff) = _. Then by axiom (* *A) of a linking system, there is z 2 Z(P ) such that f_'= ff Oe'OffiP(z), and by axiom * *(C) we have e'OffiP(z) = ffiQ ('(z)) Oe'. Set e_= ff OffiQ ('(z)), and note that ss(_e* *) = _ Oc'(z). If _e0is another morphism satisfying the same conditions, then by assumption, ss(_e0) = ss(_e) Ocx for some x 2 '(Z(P )). Then by axiom (A), _e0= _eO ffiQ (* *'(y)) for some y 2 Z(P ) such that '(y) 2 x.Z(Q). Also, e_Oe'= f_'= e_0Oe'= e_OffiQ ('(y)) Oe'= e_Oe'OffiP(y), 6 BOB OLIVER AND JOANA VENTURA where the last equality follows from (C). Since the action of Z(P ) on Mor L(P,* * R) is free, it follows that y = 1, and hence that e_0= e_. The following is an easy corollary to Lemma 1.7(a). Corollary 1.8. Let F be a fusion system (not necessarily saturated) over a p-gr* *oup S, and let L be a centric linking system associated to F. For each F-centric su* *bgroup P S, choose an "inclusion" morphism 'P 2 Mor L(P, S) such that ss('P) = inclP* *,S. Then there are unique injections ffiP,Q: NG (P, Q) ---! Mor L(P, Q), for each p* *air of subgroups P, Q 2 Ob (L), with the property that 'Q OffiP,Q(g) = ffiS(g) O'P for* * all g 2 NG (P, Q). Also, ffiP is the restriction to P NS(P ) of ffiP,P for each P , a* *nd the ffiP,Q define a functor from the transporter category of S (restricted to the objects * *of L) to L. Proof. See [BLO2 , Proposition 1.11]. We finish the section by noting the following standard result in group theory* * which will be needed later. Lemma 1.9. (a) If Q C P are finite p-groups and ff 2 Aut(P ) is such that ff|* *Q = IdQ and ff=Q = IdP=Q, then ff has p-power order. (b) If S is a p-subgroup of the finite group G, and H C G, then S 2 Sylp(G) if * *and only if S \ H 2 Sylp(H) and SH=H 2 Sylp(G=H). Proof. Point (a) is shown in [Go , Corollary 5.3.3]. Point (b) follows since S=* *(S \ H) ~= SH=H and hence [G : S] = [H : S \ H] . [G=H : SH=H]. 2.Quotients of linking systems In this section, we show that whenever (S, F, L) is a p-local finite group an* *d A C F (Definition 1.4), then we can define a quotient p-local finite group (S=A, F=A,* * (L=A)c*) as a quotient of (S, F, L) by A. We also show that L=A, defined as the quotient* * of the free action of A on L, is not in general a linking system; and this will motiva* *te the concept of a transporter system defined in the next section. We first consider quotients of fusion systems. Recall that for any fusion sys* *tem F over S, a subgroup A S is weakly closed in F if A is the only subgroup in its* * F- conjugacy class. Clearly, any normal subgroup in F is weakly closed, and any we* *akly closed subgroup is normal in S. When A is weakly closed in F, then we define F=A to be the fusion system over S=A where fi Hom F=A(P=A, Q=A) = f=A fif 2 Hom F(P, Q) . Lemma 2.1. Fix a saturated fusion system F over a p-group S. Then for any subgr* *oup A C S which is weakly F-closed in S, F=A is a saturated fusion system over S=A. Proof. This is shown, for example, in [O2 , Lemma 2.6]. But since it plays a c* *entral role in this paper, we repeat the proof here. Proof of (II): Fix ' 2 Hom F=A(P=A, S=A) such that Im (') is fully centralized* * in F=A. Set P 0=A = '(P=A), and N' = {g 2 NS=A(P=A) | 'cg'-1 2 AutS=A(P 0=A)}. EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 7 Choose Q S which is F-conjugate to P and P 0and fully normalized in F. By [BLO2 , Proposition A.2(b)], there is O 2 Hom F(NS(P 0), NS(Q)) _ 0 such that O(P 0) = Q._ Let O 2 Hom F=A(NS=A(P =A), NS=A(Q=A)) be the induced homomorphism. Thus O(CS=A(P 0=A)) = CS=A(Q=A) since P 0=A is fully centralized, and this restricts to an isomorphism CS(P 0)=A ~=CS(Q)=A. We have now proven Q fully normalized inF =) Q=A fully centralized in F=A. (1) _ Set _ = O O' 2 IsoF=A(P=A, Q=A) for short, and choose a lifting e_02 IsoF(P, * *Q) of _. Consider the subgroups eN_=A = N_ = {g 2 NS=A(P=A) | _cg_-1 2 AutS=A(Q=A)} and KQ = Ker[Aut F(Q) ---! Aut F=A(Q=A)]. Thus _e0AuteN_(P )_e-10 Aut S(Q).KQ . Since Q is fully normalized in F, KQ C AutS(Q).KQ and Aut S(Q) 2 Sylp(Aut S(Q).KQ ), and hence all Sylow p-subgroups of AutS(Q).KQ are conjugate by elements of KQ . In particular, there is ! 2 KQ su* *ch that (!_e0)Aut eN_(P )(!_e0)-1 AutS(Q). (2) Set e_= !_e0. This is also a lifting of _ since ! 2 KQ (!=A = IdQ=A); and by (2* *), fi Ne_ N_edef=g 2 NS(P ) fie_cge_-12 AutS(Q) . Since Q is fully centralized in F (by axiom (I) for F), axiom (II) for F now * *implies _ that _e extends to a morphism _b 2 Hom F(Ne_, NS(Q)), and hence that _ = O O ' extends to _ _ 2 Hom F=A(N_, NS=A(Q=A)). _ _ We claim that _(N') Im(O ). To see this, fix g 2 N', and let h 2 NS=A(P 0=A)* * be such that 'cg'-1 = ch. Then _ _-1 c__(g)= _cg_-1 = OchO = c_O(h)2 Aut(Q=A), _ _ _ so _(g) 2 O(h).CS=A(Q=A). We have_already seen that CS=A(Q=A) = O(CS=A(P 0=A)), _ and this finishes the proof that _(g) 2 Im(O ). Thus there is _ 0 ' 2 Hom F=A(N', NS=A(P =A)) _ _ _ _ such that O O' = _, and '|P=A = '. This finishes the proof of condition (II) fo* *r F=A. Proof of (I): Assume P=A is fully normalized in F=A. Since NS=A(P 0=A) = NS(P 0* *)=A for all P 0in the same F-conjugacy class, P is also fully normalized in F. Then* * P=A is fully centralized in F=A by (1). Also, Aut S=A(P=A) 2 Sylp(Aut F=A(P=A)) si* *nce AutS(P ) 2 Sylp(Aut F(P )) (by condition (I) again for F), and since a surjecti* *on of finite groups sends Sylow subgroups onto Sylow subgroups. Whenever (S, F, L) is a p-local finite group and A C S is normal in F, we let* * L=A be the category whose objects are the subgroups P=A S=A such that P is F-cent* *ric, and where Mor L=A(P=A, Q=A) = Mor L(P, Q)=A. 8 BOB OLIVER AND JOANA VENTURA Here, g 2 A acts on Mor L(P, Q) via composition with ffiP(g) 2 AutL (P ). Let (* *L=A)c be the subcategory of L=A whose objects are the F=A-centric subgroups of S=A. We will define (L=A)c*to be a certain quotient category of (L=A)c. Lemma 2.2. Fix a p-local finite group (S, F, L), and assume A C S is normal in * *F. Let F=A be the induced fusion system over S=A, and let (L=A)c be defined as abo* *ve. Then for each F=A-centric subgroup P=A S=A, P is F-centric, and there is a un* *ique subgroup E0(P ) AutL=A(P=A) of order prime to p such that CAutL=A(P=A)(P=A) = E0(P ) x Z(P=A). Let (L=A)c*be the category whose objects are the F=A-centric subgroups of S=A, * *and where Mor (L=A)c*(P=A, Q=A) = Mor L=A(P=A, Q=A)=E0(P ). Then (L=A)c*is a well defined category, and is a centric linking system associa* *ted to F=A. Proof. Fix an F=A-centric subgroup P=A S=A. For all P 0=A which is F=A-conjug* *ate to P=A, P 0is F-conjugate to P , and CS(P 0)=A CS=A(P 0=A) P 0=A. Thus CS(P* * 0) P 0for all such P 0, and this shows that P is F-centric. In particular, P 2 Ob * *(L). Set K0P= Ker Aut L=A(P=A) ---! Aut F=A(P=A) , (3) and let bffiP=Adenote the homomorphism bffiP=A= ffiP,P=A: NS=A(P=A) ------! AutL=A(P=A) . =NS(P)=A =AutL(P)=A If P is fully normalized in F, then Aut S(P ) 2 Sylp(Aut F(P )) =) ffiP,P(NS(P )) 2 Sylp(Aut L(P )) =) bffiP=A(NS=A(P=A)) 2 Sylp(Aut L=A(P=A)) . Hence by Lemma 1.9(b), K0P\bffiP=A(NS=A(P=A)) = bffiP=A(Z(P=A)) is a Sylow p-su* *bgroup of K0P. Since |Z(P=A)| and |K0P| are both invariant under F-conjugacy, this im* *plies that bffiP=A(Z(P=A)) 2 Sylp(K0P) whether or not P is fully normalized. For each f 2 AutL(P ) and each g 2 P , ffiP(ss(f)(g)) = f OffiP(g) Of-1 by axiom (C) for L. Upon passing to the quotient group Aut L=A(P=A), this shows that each element of K0Pcentralizes the subgroup bffiP=A(P=A). In particular, t* *he Sylow p-subgroup bffiP=A(Z(P=A)) is central in K0P, and hence K0P= E0(P ) x Z(P=A) (4) for a unique subgroup E0(P ) AutL=A(P=A). Now define (L=A)c*to be the category whose objects are the F=A-centric subgro* *ups of S=A, and where Mor (L=A)c*(P=A, Q=A) = Mor L=A(P=A, Q=A)=E0(P ). To see that composition is well defined, we must show, for each f 2 Mor L=A(P=A* *, Q=A) between F=A-centric subgroups, that E0(Q) Of f OE0(P ). EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 9 For each ff 2 E0(Q), there is by Lemma 1.7(a) (applied to L) a unique morphism fi 2 AutL=A(P=A) such that ff Of = f Ofi, and fi 2 E0(P ) since the induced squ* *are in F=A must commute. This shows that (L=A)c*is a well defined category. The distinguished monomorp* *hisms ffiP=A: P=A -----! Aut L=A(P=A) = AutL(P )=A are induced by the distinguished monomorphisms ffiP for L. Conditions (B) and * *(C) for (L=A)c follow directly from the corresponding conditions for L. It remains * *to prove condition (A), and this follows by (3)and (4). Note that by Proposition A.10, whenever (S, F, L) is a p-local finite group w* *ith normal p-subgroup A C F, then the sequence BA -----! |L| -----! |L=A| is a homotopy fibration sequence. It is easy to construct examples of linking systems L with normal subgroup A * *for which (L=A)c is not a centric linking system. As a rather trivial example, set * *p = 2, fix a finite group H of even order, and set G = A4xH. Let A C G be the normal subgr* *oup of order 4 in A4, and fix Sylow subgroups S0 2 Syl2(H) and S = A x S0 2 Syl2(G). Let F = FS(G), L = LcS(G), and L0 = LcS0(H). Then A is normal in F. For any P S which is F-centric and contains A, P = A x P0 for some P0 S0, AutL(P ) = A4 x AutL0(P0), and hence AutL=A(P=A) ~=C3 x AutL0(P0). Thus (L=A)c is not a linking system. Note also, in the above example, that the kernel of the map from Aut F(P ) to AutF=A(P=A) is not a p-group for any P containing A. This helps to motivate the following general criterion for L=A to be a linking system. Proposition 2.3. Let (S, F, L) be a finite p-group, and assume A C F. Then (L=A* *)c is a linking system associated to F=A if and only if Ker Aut F(P ) -! AutF=A(P=* *A) is a p-group for all P S such that P A and P=A is F=A-centric. Proof.Let P be the set of all subgroups P S such that P A and P=A is F=A- centric. Consider the following subgroups for all P 2 P: KP = Ker[Aut F(P ) -! AutF=A(P=A)] K0P = Ker[Aut L=A(P=A) -! AutF=A(P=A)] . By Lemma 2.2, K0P= E0(P ) x Z(P=A) for some subgroup E0(P ) of order prime to p, and (L=A)c is a linking system associated to F=A if and only if K0Pis a p-group* * for all P 2 P. Consider the diagram =A AutL(P )______////_AutL=A(P=A) =Z(P)|| |=K0P| fflfflfflffl|=K fflfflfflffl| AutF (P )_____////_PAutF=A(P=A) . Here, in all cases, "=H" means dividing out by the subgroup H. Since A and Z(P* * ) are both p-groups, this shows that KP is a p-group if and only if K0Pis a p-gro* *up, and thus (L=A)c is a linking system if and only if KP is a p-group for all P 2 P. 10 BOB OLIVER AND JOANA VENTURA The following proposition describes one more very simple way to construct such examples. Proposition 2.4. Fix a finite group G and Sylow subgroup S 2 Sylp(G). Assume there is a normal p-subgroup A C G which is centric in G; i.e., CG (A) = Z(A). * *Let LSA (G) LcS(G) be the full subcategory with objects those P S containing A.* * Then A is normal in FS(G), and LSA (G)=A ~=TS=A(G=A). Proof. Since A is centric in G, so is every subgroup which contains A. Thus CG * *(P ) = Z(P ) for every P 2 Ob (LSA (G)), and so LSA (G) is a full subcategory of the t* *ransporter category TS(G). It follows that LSA (G)=A ~=TS A(G) ~=TS=A(G=A). One easily finds examples of groups whose (centric) transporter category is n* *ot a linking category. For example, there are subgroups P A7 such that P ~= C22, * *and CA7(P ) ~=C22x C3. Thus P is 2-centric in A7, but for S 2 Syl2(A7), LcS(A7) is * *not a full subcategory of TS(A7). So if we set G = C72o A7, where A7 acts on A = C72by permuting a basis, then (LcS(G)=A)c is not a linking system for G=A ~=A7. 3. Transporter systems In the last section, we saw that for a centric linking system L with normal s* *ubgroup A, the quotient category (L=A)c need not be a linking system associated to any * *fusion system. This motivates us to define what we call "transporter systems": categ* *ories with extra structures satisfying some properties similar to those of the transp* *orter categories associated to finite groups, but without necessarily having such a g* *roup. Recall (Definition 1.1) that for any finite group G and any S 2 Sylp(G), TS(G) denotes the category with objects the subgroups of G, and with morphism sets the transporter sets NG (P, Q). For any set H of subgroups of S, we let TH (G) be t* *he full subcategory of TS(G) with object set H. Definition 3.1. Let F be a fusion system over a p-group S. A transporter system associated to a fusion system F is a nonempty finite category T , together with* * a pair of functors ae TOb(T )(S) ---"---!T ------! F , satisfying the following conditions: (A1) Ob (T ) Ob (F), and Ob (T ) is closed under F-conjugacy and overgroups. Also, " is the identity on objects and ae is the inclusion on objects. (A2) For each P, Q 2 Ob (T ), the kernel E(P ) def=Ker[aeP,P:Aut T(P ) -! AutF (P )] acts freely on Mor T(P, Q) by right composition, and aeP,Q is the orbit * *map for this action. Also, E(Q) acts freely on Mor T(P, Q) by left composition. (B) For each P, Q 2 Ob (T ), "P,Q:NS(P, Q) ---! Mor T(P, Q) is injective, an* *d the composite aeP,QO "P,Q sends g 2 NS(P, Q) to cg 2 Hom F(P, Q). EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 11 (C) For all ' 2 Mor T(P, Q) and all g 2 P , the diagram ' P ______//_Q "P,P(g)|| "Q,Q(ae(')(g))|| fflffl|' fflffl| P ______//_Q commutes in T . (I)"S,S(S) 2 Sylp(Aut T(S)). __ __ __ (II)Let '_2_IsoT(P, Q), P C P S, and Q C Q S be such that 'O"P,P(P )O'-1 _ __ __ _ "Q,Q(Q ). Then there is some ' 2 Mor T(P , Q) such that 'O"P,_P(1) = "Q,_* *Q(1)O'. We will often write "(S, F, T ) is a transporter system" to mean that T is a* * trans- porter system associated to the fusion system F over the p-group S. The above a* *xioms are clearly labelled to show their connection with axioms (A), (B), and (C) of * *a linking system, and axioms (I) and (II) of a saturated fusion system. Note, however, t* *hat the concepts of fully normalized and fully centralized subgroups do not appear * *in the axioms of a transporter system _ which does help simplify some of our proofs th* *at if one category is a transporter system then another one is too. For any transporter system (S, F, T ), axiom (A2) implies that the functor ae* *: T ! F is "source regular" in the sense of Definition A.5. So by Lemma A.6, a morphism* * in T is an isomorphism if and only if its image in F is an isomorphism. In particula* *r, T is an EI category (all endomorphisms are automorphisms) since F is one. We will show in Proposition 3.5 that transporter categories of finite groups,* * linking systems associated to saturated fusion systems, and (more generally) categories* * of the form L=A when L is a linking system and A a normal subgroup, are all examples of transporter systems. These examples provided our main motivation for the abo* *ve definition, and the axioms for a transporter system are clearly related to thos* *e for fusion and linking systems. We will prove soon, as Proposition 3.4(a), the following stronger form of axi* *om (I), which is more closely analogous to the axiom (I) in Definition 1.3: (I0) If P is fully normalized in F, then "P,P(NS(P )) 2 Sylp(Aut T(P )). The weaker axiom (I) we use here is motivated by the alternative set of axioms * *for a saturated fusion system due to Radu Stancu [St]. For all P Q S objects in T , we set 'P,Q = "P,Q(1), and think of these as* * the inclusion morphisms in T . By condition (B), ae sends inclusions in T to inclu* *sions in F. Whenever P0 P S and Q0 Q S are in Ob (T ), and '0 P0______//_Q0 'P0,P|| |'Q0,Q| fflffl|' fflffl| P _______//Q , is a commutative square in T , we say that '0 is a restriction of ' (and someti* *mes write '|P0,Q0= '0), and also that ' is an extension of '0. Thus axiom (II) gives cond* *itions under which a morphism can be extended. This terminology suggests that the restriction '|P0,Q0should always exist and* * be unique, whenever P0 and Q0 are objects in T and ae(')(P0) Q0. This is shown i* *n the 12 BOB OLIVER AND JOANA VENTURA following lifting lemma, which is the analog for transporter systems of Lemma 1* *.7(a). We will see later (Lemma 3.8) that extensions are also unique when they exist. Lemma 3.2. Let (S, F, T ) be a transporter system, and let ae : T -! F be the projection functor. (a) Fix morphisms ' 2 Hom F (P, Q) and _ 2 Hom F (Q, R), where P, Q, R 2 Ob (T * *). Then for any pair of liftings e_2 ae-1Q,R(_) and f_' 2 ae-1P,R(_'), there i* *s a unique lifting e'2 ae-1P,Q(') such that e_Oe'= f_'. (b) All morphisms in T are monomorphisms in the categorical sense. In other wor* *ds, for all P, Q, R 2 Ob (T ) and all '1, '2 2 Mor T(P, Q) and _ 2 Mor T(Q, R), _ O'1 = _ O'2 implies '1 = '2. (c) For every morphism ' 2 Mor T(P, Q), and every P0, Q0 2 Ob (T ) such that P0 P , Q0 Q, and ae(')(P0) Q0, there is a unique morphism '0 2 Mor T(P0, Q* *0) such that 'O'P0,P= 'Q0,QO'0. In particular, every morphism in T is the comp* *osite of an isomorphism followed by an inclusion. Proof. Since morphisms in F are all group monomorphisms, they are also monomor- phisms in the categorical sense. Hence points (a) and (b) are special cases of * *Lemma A.7(a,b). In the situation of (c), by definition of a fusion system, ae(')|P0,Q02 Hom F* *(P0, Q0). So the result follows from (a), with P, Q, R replaced by P0, Q0, Q, f_'by ' O'P* *0,P, etc. The last statement is the special case where P0 = P and Q0 = ae(')(P ). The following technical lemma can be thought of as a converse to axiom (II). * *It shows that the condition for extending a morphism, which is sufficent by axiom * *(II), is also necessary. __ __ Lemma 3.3. Fix a transporter system (S, F, T ), and objects P C P and Q Q of * *T . _ __ __ If ' 2 Mor T(P , Q) is an extension of ' 2 IsoT(P, Q), then ' P ____//_Q "P,P(x)|| |"Q,Q(ae(_')(x))| (1) fflffl|fflffl|' P ____//_Q __ commutes in T for all x 2 P. Proof. By axiom (C), the following square commutes in T : __ '_ __ P ____//_Q "_P,_P(x)|| |"_Q,_Q(ae(_')(x))| (2) fflffl|fflffl|__'_ P ____//_Q Each morphism in (1)is the restriction of the corresponding morphism in (2). So* * square (1) also commutes by the uniqueness of restriction morphisms (Lemma 3.2(c)). We next prove that axiom (I) can be replaced by the stronger axiom (I0) stated above. EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 13 Proposition 3.4. The following hold for any transporter system (S, F, T ) and a* *ny subgroup P S. (a) P is fully normalized in F if and only if "P,P(NS(P )) 2 Sylp(Aut T(P )). (b) P is fully centralized in F if and only if "P,P(CS(P )) 2 Sylp(E(P )). Proof.The "if" part in both (a) and (b) is clear, since |Aut T(P )| and |E(P )|* * depend only on the F-conjugacy class of P . (a) Suppose otherwise: that T does not satisfy axiom (I0). Let P be a maximal * *coun- terexample. Thus P is fully normalized, and "P,P(NS(P )) is not a p-Sylow subgr* *oup of Aut T(P ). Since axiom (I0) holds for the group S by assumption, P S, and* * so P NS(P ). Choose Q 2 Sylp(Aut T(P )) such that "P,P(NS(P )) Q. We know that "P,P(NS(P )) 6= Q, so "P,P(NS(P )) Q0 def=NQ ("P,P(NS(P ))). We thus have st* *rict inclusions of p-subgroups of AutT (P ): "P,P(P ) "P,P(NS(P )) C Q0 (3) Pick any morphism ' 2 Q0 not in "P,P(NS(P )). Then '-1O"P,P(x)O' 2 "P,P(NS(P * *)) for_all x 2 NS(P ) since "P,P(NS(P )) C Q0. So by axiom (II), there is an exte* *nsion ' 2 AutT (NS(P )) of '; i.e., a morphism such that the following square commute* *s: _' NS(PO)_____//NS(PO)OO 'P,NS(P)|| |'P,NS(P)| | | P ____'____//P . _ k k Set |'_| = p m where p-m. Choose r suchfthatir 0 (mod m) and r 1 (mod p ). Then 'r has order_pk and (since |'|fipk) is again an extension of 'r = '. We ca* *n thus assume that ' has p-power order. __ __ Choose P fully normalized and F-conjugate to NS(P_). Let fl_2 IsoT(NS(P ), P)* * be any lifting of an isomorphism fl02 IsoF(NS(P ),_P)._Since |P | = |NS(P_)| > |P * *| and P * * _ -1 is a maximal counterexample, we have "_P,_P(NS(P )) 2 Sylp(Aut T(P )). Hence fl* * O'Ofl __ _ is conjugate to "_P,_P(x) for some x 2 NS(P ), because ' has p-power order (and* * so does _ -1 fl O' Ofl ) and all_p-Sylow subgroups are conjugate. By replacing fl by an app* *ropriate _ -1 element of AutT (P ) Ofl, we can arrange that fl O' Ofl = "_P,_P(x). _ __ __ Now, ' 2 Aut T(P ) restricts to ' 2 Aut T(P ). Set R = ae(fl)(P ) P and f* *l0 = fl|P,R 2 IsoT(P, R) (using Lemma 3.2(c)). Then fl0'fl-102 Aut T(R) is a restri* *ction of "_P,_P(x). Hence ae("_P,_P(x)) = cx restricts to an automorphism of R (in F* *), which means that x 2 NS(R) and fl0'fl-10= "R,R(x). Also, |NS(R)| |NS(P )| since P i* *s fully normalized and R is F-conjugate to P , so ae(fl)(NS(P )) = NS(R), and x = ae(fl* *)(y) for some y 2 NS(P ). We thus have the following two commutative squares of isomorph* *isms in T : fl0 fl0 P ____//_R P ____//_R '|| |"R,R(x)| "P,P(y)|| |"R,R(x)| fflffl|fflffl|fl0 fflffl|fflffl|fl0 P ____//_R P ____//_R , where the second commutes by Lemma 3.3. Upon comparing the two squares, we finally get ' = "P,P(y), which contradicts our assumption that ' =2"P,P(NS(P )). 14 BOB OLIVER AND JOANA VENTURA (b) Again fix P S, and let P 0be any subgroup which is F-conjugate to P and fully normalized in F. Then "P0,P0(NS(P 0)) 2 Sylp(Aut T(P 0)) by (a), and hen* *ce "P0,P0(CS(P 0)) = "P0,P0(NS(P 0)) \ E(P 0) is a Sylow p-subgroup of E(P 0) by L* *emma 1.9(b). Also, E(P ) ~=E(P 0), so P is fully centralized if and only if |CS(P )|* * = |CS(P 0)|; equivalently, "P,P(CS(P )) 2 Sylp(E(P )). We next check that the examples which motivated Definition 3.1 really are tra* *ns- porter systems. Proposition 3.5. (a) For any p-local finite group (S, F, L), L is a transporte* *r system associated to F. More generally, if A C F is a normal subgroup, then L=A is* * a transporter system associated to F=A. (b) For any finite group G and any S 2 Sylp(G), TS(G) is a transporter system a* *sso- ciated to FS(G). (c) Let (S, F, T ) be a transporter system, and let T0 T be any nonempty ful* *l sub- category such that Ob (T0) is closed under F-conjugacy and overgroups. Then* * T0 is also a transporter system associated to F. Proof. Point (c) follows immediately from Definition 3.1. We next check point (b). For finite G and S 2 Sylp(G), define ae T (S) ---"---!TS(G) ------! FS(G) to be the inclusion, and the functor g 7! cg, respectively. The axioms of Defin* *ition 3.1 are easily checked. It remains to prove (a). Let (S, F, L) be a p-local finite group. We prove he* *re only that L is a transporter system associated to F. The last statement, that L=A i* *s a transporter system for any A C F, will then follow as a special case of Proposi* *tion 3.11, to be shown later. Fix morphisms 'P 2 Mor L(P, S), for all P 2 Ob (L), such that ss('P) = inclP,* *S2 Hom F(P, S), and such that 'S = IdS. By Corollary 1.8, there is a unique funct* *or ": TOb(L)(S) -----! L such that "P,S(1) = 'P and ("P,P)|P = ffiP for all P . Furthermore, "P,Q is an * *injection of NS(P, Q) into Mor L(P, Q) for all P, Q 2 Ob (L) (again by Corollary 1.8). Fo* *r each P Q S such that P, Q 2 Ob (L), we set 'P,Q = "P,Q(1) 2 Mor L(P, Q). We think of these as the inclusion morphisms, and define restriction and extension in L * *with respect to them. We are now ready to check that the axioms of a transporter system hold for L. Axioms (A1) and (C) follow immediately from axioms (A) and (C) for a linking system. Axiom (A2) : By axiom (A) for a linking system, for any P, Q 2 Ob (L), E(P )* * = ffiP(Z(P )) acts freely on MorL (P, Q), and aeP,Qis the orbit map of that actio* *n. It remains to show that E(Q) = ffiQ (Z(Q)) acts freely on Mor L(P, Q). Assume f 2 Mor L(P,* * Q) and x 2 Z(Q) are such that ffiQ (x) Of = f. Then x centralizes ss(f)(P ), so x * *= ss(f)(y) for some y 2 Z(P ) since P is F-centric, f = ffiQ (x) Of = f OffiP(y) by axiom * *(C) for a linking system, and so y = 1 by axiom (A) of a linking system. Thus x = 1, and * *the action is free. EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 15 Axiom (B) : For any P, Q 2 Ob (L) and any x 2 NS(P, Q), 'Q O"P,Q(x) = "Q,S(1) O"P,Q(x) = "S,S(x) O"P,S(1) = ffiS(x) O'P . Since ss('P) and ss('Q ) are inclusions in F, this shows that ss("P,Q(x)) is th* *e restriction to Hom F (P, Q) of ss(ffiS(x)), and hence is conjugation by x by axiom (B) for * *L as a linking system. Thus ss("P,Q(x)) = cx, and this proves axiom (B) for L as a tra* *nsporter system. Axiom (I0) : Assume P is fully normalized in F. Then AutS (P ) ~=NS(P )=Z(P ) is a Sylow p-subgroup of AutF (P ) ~=Aut L(P )=ffiP(Z(P )) by axiom (I) for the* * saturated fusion system F (and axiom (A) for L as a linking system), and so "P,P(NS(P )) * *is also a Sylow p-subgroup of AutL (P ). __ __ Axiom (II) :_ Let f 2 IsoL(P,_Q), and P C P S and Q C Q S, be such that f O"P,P(P ) Of-1 "Q,Q(Q ). If Q is fully centralized in F, then axiom (* *II) for the saturated_fusion system F implies that ss(f)_extends to a homomorphism _ 2 Hom F(P , S), whose image must be contained in Q since ss(f) conjugates AutP_(P* * ) into _ __ __ Aut_Q(Q). Hence by Lemma 1.7(b), there is f 2 Mor L(P , Q) (not necessarily a l* *ifting of _) which extends f. Now assume Q is not fully centralized. Choose R which is F-conjugate to P and Q and fully normalized in F. Then "R,R(NS(R)) is a Sylow p-subgroup of Aut L(R) (by axiom (I0)), and hence contains every p-subgroup of AutL(R) up to conjugacy* *. For any isomorphism ' 2 IsoL(Q, R), ' O"Q,Q(NS(Q)) O'-1 is a p-subgroup of AutL(R),* * so there is O 2 AutL(R) such that '0def=O O' conjugates "Q,Q(NS(Q)) into "R,R(NS(R* *)). Since R is_fully centralized,_the result of the last paragraph implies that the* *re are _ __ _0 0 morphisms_f0 2 Mor L(P , NS(R)) and ' 2 Mor L(Q , NS(R)) such that f |P,R = ' O* * f and '|Q,R = '0. By axiom (C) for a linking system, _ __ _ _ __ _ __ _ * * _ __ f0O ffi_P(P ) Of0-1= ffiNS(R)(ss(f0)(P )) and ' Offi_Q(Q ) O'-1 = ffiNS* *(R)(ss(' )(Q )). After restriction, this shows that in AutT (R), _ __ __ "R,R(ss(f0)(P )) = ('0Of) O"P,P(P ) O('0Of)-1 __ _ __ '0O"Q,Q(Q ) O'0-1= "R,R(ss(' )(Q )), __ __ where_the_inequality holds since f O"P,P(P ) Of-1 "Q,Q(Q ) by assumption. Th* *us _ __ * *__ __ ss(f0)(P ) ss(' )(Q ). By definition of a fusion system, there is ~ 2 Hom F (* *P , Q) such _ _0 _ * * __ __ that ss(' ) = ss(f ) O~, and so Lemma 1.7(a) now implies that there is f 2 Mor * *L(P , Q) _ _0 _ 0 such_that ' = f Of. Upon restricting these_morphisms to P , this implies that '* * Of = '0Of |P,Q; and hence by Lemma 1.7(a) that f|P,Q = f. More generally, one can also show that any quasicentric linking system in the* * sense of [BCGLO1 , x3]) is a transporter system. Later, in Proposition 3.12, we prove a partial converse to Proposition 3.5(b)* *, by showing that any transporter system which "has enough objects" in a certain sen* *se to be made precise is a full subcategory of the transporter category of a finite g* *roup. Transporter systems were defined associated to arbitrary fusion systems. But * *in fact, the conditions on the definition are sufficiently restrictive that if a transpo* *rter system T is associated to a fusion system F, then F is saturated _ at least with resp* *ect to the objects of T . 16 BOB OLIVER AND JOANA VENTURA Proposition 3.6. Let F be a fusion system over a p-group S (not necessarily sat* *u- rated), and let T be a transporter system associated to F. Then F is Ob (T )-sa* *turated. If F is also Ob (T )-generated, and if Ob (T ) Ob (Fc), then F is saturated. * *More gen- erally, F is saturated if it is Ob (T )-generated, and every F-centric subgroup* * P S not in Ob (T ) is F-conjugate to some P 0such that Out S(P 0) \ Op(Out F(P 0)) * *6= 1. Proof. Assume P 2 Ob (T ). If P is fully normalized in F, then by Proposition 3* *.4(a), "P,P(NS(P )) 2 Sylp(Aut T(P )). Hence by Lemma 1.9(b), applied with G = Aut T(P* * ) and H = E(P ), "P,P(CS(P )) 2 Sylp(E(P )) (hence P is fully centralized by Prop* *osition 3.4(b)) and AutS(P ) 2 Sylp(Aut F(P )). Thus F satisfies axiom (I) for the subg* *roup P . Now fix f 2 IsoF(P, Q) such that Q is fully centralized in F, and let ' 2 Iso* *T(P, Q) be any isomorphism such that ae(') = f. Set Nf = {x 2 NS(P ) | f Ocx Of-1 2 AutS(Q)}. Then '("P,P(Nf))'-1 is a p-subgroup of E(Q)."Q,Q(NS(Q)). Since Q is fully cen- tralized, "Q,Q(CS(Q)) 2 Sylp(E(Q)) (Proposition 3.4(b) again), and so "Q,Q(NS(Q* *)) is a Sylow p-subgroup of E(Q)."Q,Q(NS(Q)). Thus there is O 2 E(Q) such that (O O')("P,P(Nf))(O O')-1 is contained in "Q,Q(NS(Q)). So by axiom (II) for T , * *O O' extends to a T -morphism defined on Nf, ae(O O') = ae(') = f, and hence f exten* *ds to an F-morphism defined on Nf. Thus F satisfies axiom (II) for the subgroup P . We have now shown that F is Ob (T )-saturated. The last two statements (F is saturated under additional hypotheses) follow from Theorem 1.5(b). For any transporter system (S, F, T ), we let T c T denote the full subcate* *gory whose objects are those P 2 Ob (T ) which are F-centric. We want to show that T determines a unique linking system L associated to F with object set Ob (T c). * *When doing this, we use the term "linking system" associated to F in a slightly more* * general way than previously: to refer to any category L which satisfies all of the cond* *itions in Definition 1.6, except that Ob (L) need not contain all F-centric subgroups. Proposition 3.7. Let (S, F, T ) be a transporter system. Let E(P ), for P 2 Ob * *(T ), be as in Definition 3.1. Then for every F-centric subgroup P S, E(P ) = E0(P )xZ* *(P ), where E0(P ) is the subgroup generated by all elements in E(P ) of order prime * *to p. We can thus define a centric linking system L associated to F by setting Ob (L) = * *Ob (T c), and by setting Mor L(P, Q) = Mor T(P, Q)=E0(P ) for all P, Q 2 Ob (L). Proof. By axiom (C), for all P 2 Ob (T ), E(P ) commutes with "P,P(P ) in Aut T* *(P ). Hence if P is F-centric, then "P,P(Z(P )) is central in E(P ), and is a Sylow p* *-subgroup by Proposition 3.4. This implies that E(P ) splits as a product Z(P ) x E0(P ),* * where E0(P ) consists of all elements in E(P ) of order prime to p. It is now straightforward to check that L, when defined as above, is a quotie* *nt category of T c(i.e., composition is well defined). Also, axioms (A), (B), and * *(C) for a transporter system imply that L satisfies the corresponding axioms for a link* *ing system, and thus is a linking system associated to F. We have already shown that every morphism in a transporter system is a monomo* *r- phism in the categorical sense. We now show that every morphism is also an epim* *or- phism. EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 17 Lemma 3.8. Let (S, F, T ) be a transporter system. Fix subgroups P, Q, R 2 T ,* * to- gether with morphisms _ 2 Mor T(P, Q) and '1, '2 2 Mor T(Q, R) such that '1 O_ = '2 O_. Then '1 = '2. In other words, all morphisms in T are epimorphisms in the categorical sense. Proof.Since _ is the composite of an isomorphism followed by an inclusion (Lemma 3.2(c)), it suffices to prove this when P Q and _ = 'P,Q is the inclusion. A* *lso, it suffices to do this when P C Q: otherwise it can be shown in several steps usin* *g a chain of subgroups linking P to Q, each normal in the following one. Set P 0= ae('1)(P ). By Lemma 3.2(c), '1 O'P,Q = '2 O'P,Q has a unique restri* *ction fi = '1|P,P0= '2|P,P02 IsoT(P, P 0). Fix x 2 Q, set yi = ae('i)(x), and consid* *er the following two squares: 'i fi Q _______//_R P _______//_P 0 | | "Q,Q(x)|| "R,R(yi)|| "P,P(x)|| |"P0,P0(yi)| fflffl|'i fflffl| fflffl|fi fflffl| Q _______//_R P _______//_P 0 The first square commutes by axiom (C), and the second square is defined to be a restriction of the first. Note that "P,P(x) is the restriction of "Q,Q(x) since* * " is a functor (and since 'A,B = "A,B(1) for all A B S). Hence the second square commutes * *by the uniqueness of restriction morphisms (Lemma 3.2(c)). Thus "P0,P0(y1) = "P0,P* *0(y2) = fi O"P,P(x) Ofi-1. Since "P0,P0is injective, this shows that y1 = y2. Since thi* *s holds for all x 2 Q, ae('1) = ae('2). By axiom (A2), we now get '2 = '1 Off for some ff 2 E(Q). Hence '1 Off O'P,Q = '2 O'P,Q = '1 O'P,Q, so ffO'P,Q = 'P,Qby Lemma 3.2 again, and ff = IdQsince E(Q) acts freely on MorT* * (P, Q) (axiom (A2)). It follows that '1 = '2. By analogy with the definition of radical and normal subgroups in a fusion sy* *stem, we define: Definition 3.9. Let (S, F, T ) be a transporter system. o A subgroup Q 2 Ob (T ) is called T -radical if "Q,Q(Q) = Op(Aut T(Q)). o An arbitary subgroup Q S (not necessarily an object in T ) is called normal* * in T , denoted_Q C T , if for every morphism ' 2 Mor_T(P, P 0) in T_, there is a mor* *phism ' 2 Mor T(P Q, P 0Q) such that 'P0,P0QO' = ' O'P,PQ and ae(' )(Q) = Q. If T is a full subcategory of TS(G) for some finite group G and S 2 Sylp(G), * *then for Q 2 Ob (T ), Q is T -radical if and only if Q is a radical p-subgroup of G in t* *he usual sense. If (S, F, L) is a p-local finite group and we regard L as a transporter * *system, then P 2 Ob (L) is L-radical if and only if it is F-centric and F-radical. More* * generally, if (S, F, T ) is an abstract transporter system and P S is F-centric and F-ra* *dical, then it is not hard to see that P is also T -radical, but not conversely. For e* *xample, assume G = H o S where H has order prime to p, S 2 Sylp(G), and CG (H) H. Set F = FS(G) and T = TS(G). Then every subgroup of S is T -radical, but no proper subgroup of S is F-radical (nor F-centric if S is abelian). The following proposition is the version for transporter systems of Alperin's* * fusion theorem. 18 BOB OLIVER AND JOANA VENTURA Proposition 3.10. Let (S, F, T ) be a transporter system. For each P, P 02 Ob * *(T ) and each ' 2 IsoT(P, P 0), there are subgroups P = P0, P1, . .,.Pk = P 0 and Qi (i = 1, . .,.k) where each Qi is T -radical and fully normalized in F, and also automorphisms f* *fi 2 Aut T(Qi) and isomorphisms 'i 2 IsoT(Pi-1, Pi), such that 'i = ffi|Pi-1,Pifor e* *ach i, and ' = 'k O. .O.'1. Proof. Fix ', and assume inductively that the result holds for all isomorphisms* * between larger subgroups of S. If P = P 0= S, there is nothing to prove, so we assume t* *hat P, P 0 S. Choose a fully normalized subgroup Q in the F-conjugacy class of P a* *nd P 0. Since "Q,Q(NS(Q)) is a Sylow p-subgroup in Aut T(Q) by axiom (I0), there * *are isomorphisms _ 2 IsoT(P, Q) and _02 IsoT(P 0, Q) such that _ O"P,P(NS(P )) O_-1 "Q,Q(NS(Q)) and similarly for _0. So by axiom (II), _ extends to a morphism fr* *om NS(P ) to NS(Q), and similarly for _0, and so the proposition holds for _ and _* *0both. Thus to prove the proposition for ', it suffices to prove it for _0O ' O_-1 2 A* *utT (Q). In other words, we are reduced to the case where P = P 0is fully normalized. If P is T -radical, then we are done. Otherwise, let R P be such that "P,P(* *R) = Op(Aut T(P )). By axiom (II) again, any ' 2 AutT (P ) extends to an automorphis* *m of R, and again we are done by the induction hypothesis. If (S, F, T ) is a transporter system and A C T , then we define the quotient* * category T =A by letting Ob (T =A) be the set of all P=A for A P 2 Ob (T ), and setting MorT =A(P=A, Q=A) = Mor T(P, Q)="P,P(A) = "Q,Q(A)\Mor T(P, Q). The equivalence between these two formulas for Mor T =A(P=A, Q=A) follows from * *axiom (C). We next show that T =A is itself a transporter system. Proposition 3.11. If (S, F, T ) is a p-local finite group, and A is a normal su* *bgroup in T , then T =A is a transporter system associated to the fusion system F=A. Proof. We will denote by [f] the morphism in Mor T =A(P=A, Q=A) represented by f in Mor T(P, Q), and by f=A the morphism in Mor F=A(P=A, Q=A) induced by f in Mor F(P, Q). So all morphisms in T =A have the form [f] for some f 2 Mor (T ), * *and all morphisms in F=A have the form f=A for some f 2 Mor (F). _ _ Let_ae: T =A -! F=A be the functor induced by ae : T - ! F, i.e., ae([f]) = a* *e(f)=A. Let " : TOb(T =A)(S=A) -! T =A be the functor which is the identity on objects,* * and where _ "P=A,Q=A:NS=A(P=A, Q=A) ------! MorT =A(P=A, Q=A) =NS(P,Q)=A =MorT(P,Q)=A _ is defined by setting "P=A,Q=A(gA) = ["P,Q(g)], for all g 2 NS(P, Q). _ Axiom (A1) holds by definition of ae. Axiom (A2): E(P=A) acts freely on Mor T =A(P=A, Q=A). Assume ' 2 Mor T(P, Q) and O 2 AutT (P ) are such that [']O[O] = ['] and [O] 2 E(P=A). Then 'OO = 'O"P* *,P(a) for some a 2 A, O = "P,P(a) by Lemma 3.2(b), and hence [O] is the identity in E* *(P=A). _ aeP,Qis the orbit map for the E(P )-action on Mor T =A(P=A, Q=A). Fix P_and Q* *, and_ ', _ 2 Mor T(P, Q). If [_]_= [']O[O]_for some [O] 2 E(P=A), then clearly ae([_]* *) = ae([']). Conversely, assume that ae([']) = ae([_]); we must show that they are in the sa* *me E(P=A)-orbit. We have _ _ ae(')(P )=A = Im(ae(['])) = Im(ae([_])) = ae(_)(P )=A, EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 19 and so ae(')(P ) = ae(_)(P ). Since every morphism in F factors as an F-isomorp* *hism followed by an inclusion, this implies that there is ff 2 Aut F(P ) such that a* *e(_) = ae(')_Off. So by Lemma 3.2(a), there is O 2 AutT (P ) such that _ = ' OO and ae* *(O) = ff. Then ae([O]) = ff=A = IdP=A, and hence [O] 2 E(P=A). E(Q) acts freely on Mor T(P, Q). Assume ' 2 Mor T(P, Q) and O 2 Aut T(Q) are such that [O] O['] = ['] and [O] 2 E(Q=A). Then for some a 2 A, O O' = ' O"P,P(a) = "Q,Q(ae(')(a)) O' where the second equality holds by axiom (C) for a transporter system; ae(')(a)* * 2 A since A is normal in F; O = "Q,Q(ae(')(a)) by Lemma 3.8; and hence [O] is the i* *dentity in E(Q=A). Axiom (B) for T =A follows directly from axiom (B) for T . Axiom (C) for T =A is a consequence of axiom (C) applied to T . More precisely* *, for each f 2 Mor T(P, Q) and each g 2 P , _ [f] O"P=A,P=A(gA)= [f] O["P,P(g)] = [f O"P,P(g)] = ["Q,Q(ae(f)(g)) Of] _ _ _ = "Q=A,Q=A(ae(f)(g)A) O[f] = "Q=A,Q=A(ae([f])(gA)) O[f] . Axiom (I): By axiom (I) for T_, "S,S(S) is a Sylow p-subgroup of AutT (S). So * *upon diving out by A, we get that "S=A,S=A(S=A) is a Sylow p-subgoup of AutT =A(S=A). __ Axiom_(II): Fix [f] 2 IsoT =A(P=A, Q=A), and let P=A C P =A S=A and Q=A C Q=A S=A be such that _ __ -1 _ __ [f] O"P=A,P=A(P =A) O[f] "Q=A,Q=A(Q =A). Then for any lifting of [f] to f 2 IsoT(P, Q), __ __ f O"P,P(P ) Of-1 "Q,Q(Q ). _ __ __ So f_extends to some_f 2 Mor_T(P , Q) by axiom (II) applied to T , and thus [f]* * extends to [f] in Mor T =A(P =A, Q=A). The next proposition shows that a transporter system over S which contains all subgroups of S as objects is the transporter category of a finite group. Proposition 3.12. Let T be a transporter system over a p-group S for which Ob * *(T ) is the set of all subgroups of S. Set G = AutT (1), and identify S as a subgrou* *p of G via "1,1. Then S 2 Sylp(G), and there is an isomorphism of 4-tuples T , F, ae, " ~= TS(G), FS(G), (g 7! cg), incl. Proof.Note first that S 2 Sylp(G) by axiom (I0) (Proposition 3.4), applied with* * P = 1. By (II) (applied with P = Q = 1), for each P, Q S and each g 2 NG (P, Q), there is some b"P,Q(g) 2 Mor T(P, Q) which extends g 2 AutT (1), and this exten* *sion is unique by Lemma 3.8. If R is another subgroup and h 2 NG (Q, R), then b"P,R(hg)* * = b"Q,R(h) Ob"P,Q(g) by the uniqueness of the construction. 20 BOB OLIVER AND JOANA VENTURA We have thus defined a functor b"from TS(G) to T which is the identity on obj* *ects. Consider the following diagram: (g7!cg) TS(S) ___incl_//_TS(G)________//FS(G) | | | | | | |Id| b"| |Id fflffl| " fflffl||ae fflffl|| TS(S) __________//_T____________//F . By Lemma 3.3 (applied with P = Q = 1), for g 2 NG (P, Q), ae(b"P,Q(g)) is conju* *gation by g. This proves that FS(G) F, and that the right hand square in the diagram commutes. For g 2 NS(P, Q), "P,Q(g) is an extension of "1,1(g) (since " is a fu* *nctor). So " is the restriction of b"to TS(S), and thus the left hand square commutes. It remains to show that b"is an isomorphism of categories; it then follows fr* *om the surjectivity of ae that FS(G) = F. For all P, Q S and ' 2 Mor T(P, Q), there* * is a uniquely defined restriction '|1,12 AutT (1) = G by Lemma 3.2(c). Set g = '|1,1* *for short. By Lemma 3.3 again, gxg-1 = ae(')(x) 2 Q for all x 2 P (recall that we i* *dentify S as a subgroup of G via "1,1), and thus g 2 NG (P, Q). It follows that ' = b"P* *,Q(g); g is unique by the uniqueness of the restriction; and thus b"P,Qis a bijection. More generally, if (S, F, T ) is a transporter system, and there is some Q 2 * *Ob (T ) such that Q C T , then one can show that T is isomorphic to a full subcategory* * of TS(Aut T(Q)). We next describe another way to construct new transporter systems as quotients of other transporter systems; a construction which will be useful in Section 6.* * As in Definition 3.1, for a transporter system (S, F, T ), we let E(P ) be the kernel E(P ) = Ker Aut T(P ) ----! Aut F(P ) for all P 2 Ob (T ). We regard E as a functor E :T op---! Gps: for each ' 2 Mor T(P, Q) and each x 2 E(Q), E(')(x) 2 E(P ) is the unique morphism such that ' OE(')(x) = x O' 2 Mor T(P, Q). Proposition 3.13. Fix a transporter system (S, F, T ). Let E0: T op---! Gps b* *e a subfunctor of E which satisfies the following two conditions for each P, Q 2 Ob* * (T ) and each ' 2 Mor T(P, Q): (a) E0(P ) has order prime to p, and (b) E0(Q) = E(')-1(E0(P )). Let T =E0 be the quotient category which has the same objects as T , and where Mor T =E0(P, Q) = Mor T(P, Q)=E0(P ). Then T =E0 is a transporter system associated to F. Proof. The assumption that E0 is a subfunctor implies that T =E0 is a category * *(i.e., that composition is well defined). The structure functors 0 ae0 TOb(T )(S) ---"---!T =E0 ------! F are defined in the obvious way. Most of the axioms in Definition 3.1 are easily* * carried over from T to T =E0. The injectivity of "0, and axiom (I) ("0S,S(S) 2 Sylp(Aut* * T =E0(S))) follow from (a). The freeness of the action of E(Q)=E0(Q) on Mor T(P, Q)=E0(P )* * (part of axiom (A2)) follows from (b). EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 21 The only axiom which requires a little more explanation_is_(II). Fix '0_= [']* *_in_ IsoT =E0(P, Q) (where ' 2 IsoT(P, Q)). Assume P and Q are such that P C P, Q C * *Q, and __ __ '0O"0P,P(P ) O'0-1 "0Q,Q(Q ) AutT =E0(Q). Equivalently, in T , this means that __ __ ' O"P,P(P ) O'-1 "Q,Q(Q ).E0(Q). __ __ Since |E0(Q)| is prime to p, "Q,Q(Q ) is a_Sylow_p-subgroup of_"Q,Q(Q_).E0(Q). * *Hence there is ff 2 E0(Q) such that (ff') O"P,P(P ) O(ff')-1 "Q,Q(Q ). Thus ff' ca* *n be _ __ __ 0 extended to some ' 2 Mor T(P , Q) by axiom (II) for T , and so ' = ['] = [ff'] * *extends _0 _ __ __ to the morphism ' = [' ] 2 Mor T =E0(P , Q). The following is an example of how Proposition 3.13 will be applied in Sectio* *n 6. Fix a finite group G, a normal subgroup H C G, and a Sylow subgroup S 2 Sylp(G). Let H be the set of all subgroups P S such that (Z(P ) \ H) 2 Sylp(CH (P )). For * *each P 2 H, CH (P ) = (Z(P ) \ H) x C0H(P ), where C0H(P ) is the set (a subgroup) o* *f all elements of CH (P ) of order prime to p. Then C0His a subfunctor of CG which sa* *tisfies conditions (a) and (b) in Proposition 3.13, and hence TH (G)=C0His a transporte* *r system associated to FS(G). When H = G, then this is of course just the centric linkin* *g system LcS(G). The next proposition describes the opposite of this construction: it describe* *s general conditions for an extension of a transporter system to again be a transporter s* *ystem associated to the same fusion system. However, its main purpose is to show how* * to construct any centric transporter system associated to a given F _ any transpor* *ter system whose objects are the F-centric subgroups _ as an extension of a linking* * system associated to F. We refer to Definition A.5 for the definition of a source regular extension o* *f a category. For any source regular extension T 0---! T of T , we let eK:T 0op---! Gps denot* *e the "kernel functor", defined on objects by setting eK(P ) = Ker[Aut T(0P ) ---! Au* *t T(P )], and on morphisms by sending ' 2 Mor T(0P, Q) to the unique homomorphism eK(') 2 Hom (Ke(Q), eK(P )) such that ff O' = ' OeK(')(ff) for all ff 2 eK(Q). (Note th* *at eK as defined here is a functor from T 0to groups, as opposed to the functor K* from * *T to groups up to conjugacy defined in Lemma A.7(c).) Proposition 3.14. Fix a transporter system (S, F, T ) and a source regular exte* *nsion T 0---! T of T . Let eK:T 0op---! Gps be the kernel functor. Then T 0can be g* *iven a structure of a transporter system associated to F if the following conditions* * hold for all P 2 Ob (T 0) = Ob (T ): (a) eK(P ) is finite of order prime to p. (b) eK(') is a monomorphism for all ' 2 Mor (T ). (c) For all g 2 P , eK("P,P(g)) = IdeK(P). __ * * __ (d) If P P NS(P ) and x 2 eK(P ) are such that eK(ffiP,P(g))(x) = x for all* * g 2 P, then x 2 Im(Ke('P,_P)). ae * * 0 Proof.Let TOb(T )(S) --"-!T ---! F be the structure functors for T , and set a* *e = aeO . By (c) (applied with P = S), there is a unique homomorphism "0S,S:S ---! Aut T * *0(S) 22 BOB OLIVER AND JOANA VENTURA such that S,SO"0S,S= "S,S. For each P S, choose some "0P,S(1) 2 Mor T(0P, S)* * such that P,SO "0P,S(1) = "P,S(1). By Lemma A.7(a), for each P, Q 2 Ob (T 0) and e* *ach g 2 NS(P, Q), there is a unique morphism "0P,Q(g) 2 Mor T(0P, Q) such that "0Q,S(1) O"0P,Q(g) = "0S,S(g) O"0P,S(1), and this defines a functor "0such that O"0= ". Axiom (A1) for T 0follows immediately, (A2) by (b) and since is source regu* *lar, (B) since ae0O"0= aeO", and (I) from (a) and the corresponding axiom for the tr* *ansporter system T ). We next check axiom (C). Fix ' 2 Mor T 0(P, Q) and g 2 P , and set h = ae0(')* *(g) for short. By Lemma A.7(a), there is a unique automorphism fl 2 AutT 0(P ) such* * that P,P(fl) = "P,P(g), and such that the following square commutes in T 0: ' P ______//_Q | | fl| |"0Q,Q(h) | | fflffl|' fflffl| P ______//_Q . Set |h| = pk. By juxtaposing copies of the above square (and by the uniqueness* * in k -1 Lemma A.7(a)), flp = Id. By (c), P,P("P,P(P )) is the product of Ke(P ) with* * a p- group, hence "P,P(g) has a unique lifting to AutT 0(P ) of p-power order, and t* *his proves that fl = "0P,P(g). __ __ It remains to prove axiom (II)_for T 0. Fix_P C P S, Q C Q S, and ' 2 IsoT 0(P, Q) such that '"0P,P(P )'-1 "0Q,Q(Q ). Then (') extends to a morphi* *sm on __ __ __ P by axiom (II) for T , and hence there is _ 2 Mor T(0P, Q) such that (_|P,Q) * *= ('). Let ff_2_Ke(P ) be the unique element such that _|P,Q = ' Off; then ff normaliz* *es "0P,P(P ). Since ff 2 eK(P ) which is normal in AutT 0(P ), this shows that __ __ [ff, "0P,P(P )] 2 eK(P ) \ "0P,P(P ) = 1, where the intersection is trivial since one of the groups has order prime to p * *and the other is a p-group. We can now apply condition (d) to show that ff extends* * to _ __ _-1 __ ff2 AutT 0(P ), and hence _ Off is an extension of ' to P . 4. Homotopy properties of transporter systems We collect here some results about the fundamental group, the homotopy type, * *and the cohomology of nerves of transporter systems. Several results from the appe* *ndix will be needed when proving these. By Proposition A.3(a), for any transporter system (S, F, T ), ss1(|T |) is th* *e free group on the morphisms in T , modulo relations given by composition, and by setting i* *nclusion morphisms 'P,Q equal to the identity. More precisely, let ` :Mor (T ) -----! ss1(|T |) be the map which sends ' 2 Mor T(P, Q) to the loop (based at the vertex S) form* *ed by the edges 'Q,S.'.'P,S-1(composed from right to left). This clearly sends com* *posites to products and sends inclusions to the identity, and Proposition A.3 says that* * ` is universal among all maps defined on Mor (T ) with these properties. EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 23 If '0, ' 2 Mor (T ) are such that '0 is a restriction of ', then they differ * *by com- position with inclusion morphisms, and hence `('0) = `('). Thus by Proposition 3.10 (Alperin's fusion theorem for transporter systems), ss1(|T |) is generated* * by the subgroups `(Aut T(P )) for fully normalized T -radical subgroups P 2 Ob (T ). The following proposition describes one way to construct transporter subsyste* *ms, as "kernels" of homomorphisms defined on the fundamental group. Proposition 4.1. Let (S, F, T ) be a transporter system. Fix a finite group , * *and a homomorphism : ss1(|T |) ---! . We also identify with O`; i.e., as a func* *tion defined on Mor (T ) which sends composites to products and inclusion morphisms * *to the identity. For each subgroup H , set SH = ( O"S,S)-1(H), a subgroup of S, a* *nd assume that S1 2 Ob (T ). Let TH T be the subcategory defined by setting Ob (TH ) = {P 2 Ob(T ) | P SH } and Mor TH(P, Q) = {' 2 MorT (P, Q) | (')* * 2 H}. Let FH F be the fusion system over SH generated by ae(TH ), and let "H,H aeH TOb(TH)(SH ) -----! TH -----! FH be the restrictions of " and ae. Then the following hold for each H . (a) (Mor (T )) = (Aut T(S1)). (b) TH is a transporter system associated to the fusion system FH if and only* * if "S1,S1(SH ) 2 Sylp(Aut TH(S1)). In particular, TH is a transporter system* * when- ever H C . (c) Assume that "S1,S1(SH ) 2 Sylp(Aut TH(S1)); and also that for all P S ful* *ly cen- tralized in F, CS1(P ) P implies P 2 Ob (T ) ( is an admissible homomorp* *hism in the sense of Definition 5.10). Then FH is a saturated fusion system. (d) Assume P 2 Ob (T ) implies P \ S1 2 Ob (T ). Then |TH | has the homotopy ty* *pe of the covering space of |T | with fundamental group -1(H). Proof.In general, for any P S, we write P1 = P \ S1 for short. We first claim* *, for all P, Q S such that P1, Q1 2 Ob (T ), that there is a well defined restricti* *on map r = rP,Q:Mor T(P, Q) ------! Mor T(P1, Q1). In other words, for all _ 2 Mor T(P, Q), 'Q1,QO r(_) = _ O'P1,P for all _ 2 Mor T(P, Q). (1) By Lemma 3.2(c), this means proving that ae(_)(P1) Q1. For all g 2 P1, ("S,S(ae(_)(g))) = ("Q,Q(ae(_)(g))) = (_). ("P,P(g)). (_)-1 = 1 by axiom (C) for T , so ae(_)(g) 2 S1. Thus ae(_)(P1) Q1, and these restricti* *on maps are all defined. Note also that (1)implies (r(_)) = (_) for all _. (a) By Proposition 3.10 (Alperin's fusion theorem for transporter systems), (M* *or (T )) is contained in the subgroup generated by all (Aut T(P )) for fully normalized* * P 2 Ob (T ). So it suffices to show that (Aut T(P )) (Aut T(S1)) for all P 2 O* *b (T ) fully normalized in F. Fix such a P , and fix ff 2 AutT (P ). We can assume ind* *uctively that (Aut T(Q)) (Aut T(S1)) for all Q S such that Q P . If P S1, then rP,P(ff) 2 AutT (S1), and so (ff) = (r(ff)) 2 (Aut T(S1)). We are thus reduced to the case where P S1. Then P P S1, so P NPS1(P ) = P .NS1(P ). Since P is fully normalized, "P,P(NS(P )) 2 Sylp(Aut T(P )) by axio* *m (I0). Set Aut1(P ) = Ker Aut T(P ) ---! , 24 BOB OLIVER AND JOANA VENTURA so that "P,P(NS1(P )) = Aut1(P ) \ "P,P(NS(P )) 2 Sylp(Aut 1(P )). Since all Sylow p-subgroups of Aut1(P ) are conjugate, ff"P,P(NS1(P ))ff-1 = fi"P,P(NS1(P ))fi-1 for some fi 2 Aut 1(P ), and thus fi-1_Off normalizes "P,P(NS1(P_)). By axiom * *(II), fi-1_Off extends to an automorphism ff2_AutT (NPS1(P )); i.e., ff|P,P = fi-1 Of* *f. Then (ff) = (fi-1 Off) = (ff). Since (ff) 2 (Aut T(S1)) by the induction hypoth* *esis, this finishes the proof of (a). (b) The axioms (A1), (A2), (B), (C), and (II) of a transporter system for TH f* *ollow easily from the corresponding axioms for T , without any restrictions on H. Hen* *ce TH is a transporter system if and only if axiom (I) holds. Set A = AutT (S1) and AH = AutTH (S1) = {ff 2 A | (ff) 2 H} for short. Consi* *der the restriction map rSH,SH: AutT(SH ) ------! A = AutT (S1) as defined above; an injective map by Lemma 3.8. By axiom (II) (and Lemma 3.3), Im (rSH,SH) is the normalizer in A of "S1,S1(SH ). Thus "SH,SH(SH ) 2 Sylp(Aut TH(SH )) () "S1,S1(SH ) 2 SylpNAH ("S1,S1(SH )) . Since a proper subgroup of a p-group is properly contained in its normalizer, t* *his shows that axiom (I) holds if and only if "S1,S1(SH ) 2 Sylp(AH ). Now, S1 is fully normalized in F, since it is the only subgroup in its F-conj* *ugacy class. Hence "S1,S1(S) 2 Sylp(A) by axiom (I0). So if H C , then "S1,S1(SH ) i* *s a Sylow subgroup of AH by Lemma 1.9(b). (c) Now assume "S1,S1(SH ) 2 Sylp(Aut TH(S1)). Hence TH is a transporter sys* *tem by (b). Assume also that for all P S fully centralized in F, CS1(P ) P imp* *lies P 2 Ob (T ). In particular, if P SH is FH -centric, then P CSH(P ) CS1(P * *), and thus P 2 Ob (TH ). So FH is saturated by Proposition 3.6. (d) We must prove that |TH | is homotopy equivalent to a covering space of |T * *|. Let TH0 T be the subcategory with Ob (TH0) = Ob (T ), and where Mor TH0(P, Q) is t* *he set of all ' 2 Mor T(P, Q) such that (') 2 H. By Proposition A.4, |TH0| is homoto* *py equivalent to the covering space of |T | with fundamental group -1(H). Conditi* *on (1) in Proposition A.4 follows by point (a) here. It remains to prove that |TH0| ' |TH |. For all P, Q 2 Ob (TH0), all ' 2 Mor * *TH0(P, Q), and all g 2 P , ' O"P,P(g) O'-1 = "Q,Q(ae(')(g)) by axiom (C), and thus (') 2 H conjugates ("P,P(g)) to ("Q,Q(ae(')(g))). In particular, g 2 SH if and on* *ly if ae(')(g) 2 SH , and thus ae(')(P \ SH ) Q \ SH . So by Lemma 3.2(a), there i* *s a unique restriction map rHP,Q:MorTH0(P, Q) ------! Mor TH(P \ SH , Q \ SH ). Define a retraction functor rH :TH0---! TH by sending an object P to P \ SH* * , and sending _ 2 Mor T(P, Q) to rHP,Q(_). The inclusion morphisms define a natu* *ral transformation of functors from inclOrH to IdTH0, and thus |TH0| ' |TH |. We next look at the cohomology of the nerve of a transporter system T . By Pr* *opo- sition A.3(b), for any abelian group A with action of ss1(|T |), H*(|T |; A) ~=* * lim-*( ) * * T for a certain functor : T op---! Ab which sends all objects to A. So we must * *study EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 25 the higher limits of functors on T . A more general discussion of higher limits* * is given in the appendix. When (S, F, T ) is a transporter system, the orbit category O(T ) of T is def* *ined to be the category whose objects are the objects of T , and where Mor O(T()P, Q) = "Q,Q(Q)\Mor T(P, Q). When T = TS(G) is the category of a finite group G with Sylow subgroup S, then O(T ) is the usual orbit category of G: Mor O(T )(P, Q) = Q\NG (P, Q) ~=map G(G=P, G=Q). The relation between higher limits of p-local functors on T and on O(T ) (i.e.,* * functors which take values in Z(p)-modules) is described in the following lemma. Lemma 4.2. Fix a transporter system (S, F, T ), and let : T op---! Z(p)-mod be any functor. Then there is a spectral sequence Eij2= lim-i(Hj(-; (-))) =) lim-i+j( ). (2) O(T ) T Proof.This is a special case of Proposition A.11. The reason for working with higher limits over O(T ) instead of looking direc* *tly at higher limits over T is that certain techniques developed when studying higher* * limits over other orbit categories also apply to this situation. We refer to the appen* *dix for the definition of graded groups *(G; M), when G is any finite group and M is a Z(p)-module (and the prime p is understood). Their relation to limits over O(T* * ) is described by the following lemma. Lemma 4.3. Fix a transporter system (S, F, T ), and let : O(T )op ------! Z(p)-mod be any functor which vanishes except on the F-conjugacy class of one subgroup Q* * 2 Ob (T ). Then lim-*( ) ~= *(Aut O(T()Q); (Q)). O(T ) Proof.Since the result is independent of the choice of Q in its F-conjugacy cla* *ss, we can assume that Q is fully normalized. Set = AutO(T )(Q), and set = "Q,Q(NS(Q))* *=Q 2 Sylp( ). Let Op( ) be the p-subgroup orbit category: the category whose objects are th* *e p- subgroups of , and with morphism sets Mor Op( )(P, Q) = Q\N (P, Q). By definit* *ion, for any Z(p)[ ]-module M, ( M if P = 1 *( ; M) = lim-*(FM ) where FM (P ) = Op( ) 0 if P 6= 1. (See the discussion before Proposition A.2.) Let O ( ) Op( ) be the full subc* *ategory with objects the subgroups of ; this is clearly a subcategory equivalent to Op* *( ). Define a functor ff: O ( ) ------! O(T ) 26 BOB OLIVER AND JOANA VENTURA as follows. For each subgroup P=Q , set ff(P=Q) = P . For each P=Q, P 0=Q and each fl 2 AutT (Q) (i.e., Qfl 2 ) such that P 0fl 2 Mor O( )(P=Q, P 0=Q), * *fl extends to a unique morphism ' 2 Mor T(P, P 0) by axiom (II), and we set ff(Qfl) = P 0' 2 Mor O(T()P, P 0). We want to apply Proposition A.2 ([BLO3 , Proposition 4.3]) to this functor * *ff. Point (a) ( ~=End O(T )(Q)) holds by construction. Point (c) follows from Lemmas 3.8* * and A.8(b): all morphisms in O(T ) are epimorphisms in the categorical sense since* * all morphisms in T are epimorphisms. It remains to check the other two hypotheses. (b) For each P 2 Ob (O(T )) such that P and Q are not T -isomorphic, all isotro* *py subgroups of the -action on Mor O(T()Q, P ) (via right composition) are no* *ntrivial p-subgroups. For f 2 Mor T(Q, P ), the isotropy subgroup of [f] 2 Mor O(T()* *Q, P ) is the subgroup of all classes [ff] for ff 2 AutT (Q) such that f Off = "P,* *P(x) Of for some x 2 P . By Lemma 3.2, for any given x 2 P , there is at most one ff wh* *ich satisfies this equation, and there is such ff if and only if ae(f)(Q) = ae("P,P(x) Of)(Q) = x.ae(f)(Q).x-1. Thus the stabilizer subgroup of [f] is isomorphic to NP(ae(f)(Q))=Q, which * *is a nontrivial p-group since ae(f)(Q) P . (d) For any P=Q , any R 2 Ob (O(T_)), and any ' 2 Mor O(T )(Q, R)_which is P=Q-invariant, there is some ' 2 Mor O(T )(P, R) such that ' = ' O'Q,P. Th* *is follows from axiom (II). Thus, by Proposition A.2, the natural map * * * lim-*( ) ---ff---!~lim-( Off) = ( ; (Q)) O(T ) = O ( ) is an isomorphism. Lemma 4.3 can now be combined with results in [JMO ] about the functors *(-* *; -), to obtain the following corollary. Corollary 4.4. Fix a transporter system (S, F, T ). (a) Assume : O(T )op ------! Z(p)-mod has the property that for all P 2 Ob (T ) such that (P ) 6= 0, there is an* * element of order p in AutO(T )(P ) which acts trivially on (P ). Then lim-*( ) = 0. O(T ) (b) Assume : T op---! Z(p)-mod has the property that for every fully centralized subgroup P 2 Ob (T ) such* * that (P ) 6= 0, there is g 2 CS(P )r P such that "P,P(g) acts trivially on (P * *). Then lim-*( ) = 0. T Proof. By [JMO , Proposition 5.5], for any finite group and any Z(p)[ ]-modu* *le M such that some element g 2 of order p acts via the identity on M, *( ; M) = * *0. In particular, by the hypothesis in (a), *(Aut O(T()P ); (P )) = 0 for each P 2 * *Ob (T ). EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 27 Point (a) now follows from Lemma 4.3, together with an appropriate filtration of and the long exact sequences in higher limits for extensions of functors on O(T* * ). Point (b) follows from (a), together with Lemma 4.2. We can now look at consequences for the homology of |T |. Proposition 4.5. Fix a finite group G, a Sylow subgroup S 2 Sylp(G), and a full* * sub- category T TS(G) whose set of objects is closed under G-conjugacy and overgro* *ups. Let A be an Z(p)[G]-module, regarded also as a functor from T to Z(p)-mod which* * sends all objects to A and which sends a morphism g 2 NG (P, Q) to the action of g on* * A. Let S1 S be the subgroup of elements which act on A via the identity. Assume,* * for all P S such that CS1(P ) P , that P 2 Ob (T ) (the action of T on A is adm* *issible in the sense of Definition 5.10). Then H*(|T |; A) ~=lim-*(A) ~=H*(G; A); T where the first group means cohomology with coefficients twisted via the natura* *l homo- morphism ss1(|T |) ---! G. Proof.The first isomorphism follows from Proposition A.3(b). To prove the seco* *nd, let 0 : TS(G) ---! Z(p)-mod be the functors ( A if P =2Ob (T ) (P ) = A for all P S and 0(P ) = 0 if P 2 Ob (T ). In both cases, g 2 NG (P, Q) acts on A via the Z(p)[G]-module structure. Let T{1}(G) TS(G) be the full subcategory with the trivial subgroup as the * *only object. We claim that lim-*( ) ~= lim-*( ) ~=H*(G; A) and lim-*( 0) = 0. TS(G) T{1}(G) TS(G) The first isomorphism holds because TS(G) contains T{1}(G) as a deformation ret* *ract (every morphism restricts to a unique automorphism of 1), and the second because functors from T{1}(G) to abelian groups are the same as Z[G]-modules. The last * *iso- morphism follows from Lemma 4.4(b), since whenever P =2Ob (T ), CS1(P ) P , a* *nd for any g 2 CS1(P )r P , "P,P(g) acts trivially on A. It now follows that lim-*(A) ~= lim-*( = 0) ~= lim-*( ) ~=H*(G; A). T TS(G) TS(G) Here, the first isomorphism holds since C*(T ; A) ~=C*(TS(G); = 0) (the chain * *com- plexes defined in Lemma A.1). As a second application of the results in this section, we show that the nerv* *e of a transporter system (S, F, T ) has the same p-completed homotopy type as the ner* *ve of the associated linking system, in the sense of Proposition 3.7. Proposition 4.6. Fix a transporter system (S, F, T ) such that Ob (T ) contains* * all subgroups P S which are F-centric and F-radical. Let T cbe the full subcateg* *ory with objects the F-centric subgroups which are in Ob (T ), and let L be the lin* *king system associated to T c. Then the inclusion T c,! T and the projection T c--* * i L induce homotopy equivalences |T |^p' |T c|^p' |L|^p. 28 BOB OLIVER AND JOANA VENTURA Proof. By [BLO1 , Lemma 1.3], if o :eC---! C is any source regular functor (se* *e Defi- nition A.5) such that Ker(oec,ec) is finite of order prime to p for each ec2 Ob* * (Ce), then |o| induces an isomorphism from H*(|Ce|; Fp) to H*(|C|; Fp), and hence a homotopy e* *quiv- alence |Ce| ' |C|. This is the situation for the projection of T conto L (by ax* *iom (A2) for T ), and hence |T c| ' |L|. Let : T op---! Ab be the constant functor which sends all objects to Fp and* * all morphisms to the identity. Let 0 be the subfunctor ( (P ) if P =2Ob (T c) 0(P ) = 0 if P 2 Ob (T c). Then H*(|T |; Fp) ~=lim-*( ) and H*(|T |, |T c|; Fp) ~=lim-*( 0). T T For each P 2 Ob (T )r Ob (T c) which is fully centralized, and each g 2 CS(P )* *r P , "P,P(g) acts trivially on (P ). So lim-*( 0) = 0 by Lemma 4.4(b). It follows* * that the inclusion of |T c| into |T | is a mod p homology equivalence, and hence ind* *uces a homotopy equivalence of p-completions. Since a space is p-good if its p-completion is p-good [BK , Proposition I.5.2* *], and the p-completion of a linking system is p-good by [BLO2 , Proposition 1.12], Propo* *sition 4.6 implies that |T | is p-good whenever Ob (T ) contains all F-centric subgrou* *ps. In fact, the same argument as that used in the proof of [BLO2 , Proposition 1.12]* * can be used in this situation (together with Proposition 3.10) to show directly that f* *or any transporter system T , |T | is p-good. We finish the section by describing one situation where an inclusion of trans* *porter systems actually induces a homotopy equivalence between their uncompleted nerve* *s. Let (S, F, T ) be a transporter system. Recall that a subgroup P 2 Ob (T ) i* *s T - radical if Op(Aut T(P )) = "P,P(P ). We let T r T denote the full subcategory* * whose objects are the T -radical subgroups of S. The following proposition generalizes [BCGLO1 , Theorem 3.5]. Proposition 4.7. Let (S, F, T ) be a transporter system. Then |T r| ' |T |. M* *ore generally, |T 0| ' |T | for any full subcategory T 0 T which contains T r. Proposition 4.7 is an immediate consequence of the following lemma, whose pro* *of is modelled on that of [BCGLO1 , Proposition 3.11]. Lemma 4.8. Let (S, F, T ) be a transporter system. Let T0 T be any full subca* *tegory such that Ob (T0) is closed under F-conjugacy. Let P S be maximal among those subgroups in Ob (T )r Ob (T0), and let T1 T be the full subcategory whose ob* *jects are the objects in T0 together with all subgroups F-conjugate to P . Assume further* *more that P is not T -radical. Then the inclusion of nerves |T0| |T1| is a homotopy equ* *ivalence. Proof. Throughout this proof, "extensions" and "restrictions" of morphisms are * *taken as usual with respect to the inclusions 'P,Q for P Q S. Also, for ' 2 Mor T* *(Q, Q0) and R Q, we write '(R) = ae(')(R) Q0. We must show that the inclusion functor I :T0 ! T1 induces a homotopy equival* *ence |T0| ' |T1|. By Quillen's Theorem A (see [Q ]), it will be enough to prove tha* *t the undercategory P 0#I has contractible nerve (i.e., |P 0#I| ' *) for each P 0in T* *1. This is clear when P 0is not isomorphic to P (since P 0#I has initial object (P 0, I* *d) in that EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 29 case), so it suffices to consider the case P 0= P . Since P was arbitrarily cho* *sen in its isomorphism class, we can also assume that P is fully normalized. By assumption, P is not T -radical, and thus Op(Aut T(P )) "P,P(P ). Define bP= "-1P,P(Op(Aut T(P ))) . Since P is fully normalized, "P,P(NS(P )) 2 Sylp(Aut T(P )) contains Op(Aut T(* *P )). Hence "P,P(Pb) = Op(Aut T(P )), and so bP P . Recall that the objects in P #I are the pairs (Q, ') for Q 2 Ob (T0) and ' 2 MorT (P, Q); and that Mor P#I((Q, '), (R, _)) is the set of all O 2 Mor T(Q, R)* * such that O O' = _. Consider the full subcategories C2 C1 C0 P #I, defined by setting Ob (C0)= {(Q, ') | '(P ) C Q} Ob (C1)= {(Q, ') | '(P ) = P C Q} Ob (C2)= {(Q, ') | '(P ) = P C Q, Pb Q} . We will prove that |P #I| ' |C0| ' |C1| ' |C2| ' *. Let r1: P #I ---! C0 be the retraction which sends (Q, ') to (NQ ('(P )), '0)* *, where '0 2 Mor T(P, NQ ('(P ))) is the restriction of ', and which sends a morphism t* *o its restriction. All of these restrictions are well defined by Lemma 3.2(a). Ther* *e is a natural transformation of functors from inclOr1 to the identity on P #I which s* *ends an object (Q, ') to the inclusion 'NQ('(P)),Q, and thus |P #I| ' |C0|. Now let (Q, ') be an object in C0, and set P 0= '(P ) C Q and '0 = '|P,P02 IsoT(P, P 0). Since P is fully normalized, "P,P(NS(P )) 2 Sylp(Aut T(P )), and * *hence the p-subgroup '0-1"P0,P0(Q)'0is conjugate in AutT (P ) to a subgroup of "P,P(NS(P * *)). Let ff 2 AutT (P ) be such that (ff'0-1)"P0,P0(Q)(ff'0-1)-1 is contained in "P,P(NS* *(P )). By axiom (II) for T , ff'0-1 extends to some morphism _ 2 Mor T(Q, NS(P )) such th* *at _(P 0) = P . Then (_(Q), _ O') is isomorphic to (Q, '); and (_(Q), _ O') is in * *C1 since _'(P ) = P . Thus every object in C0 is isomorphic to an object in C1, and this* * proves that |C0| ' |C1|. Now define a retraction r2: C1 ---! C2 as follows. On objects, we set r2(Q, * *') = (QPb, 'Q,QPbO'). Fix O 2 Mor C1((Q, '), (R, _)); then O(P ) = P since '(P ) = P* * = _(P ), so the restriction O0 def=O|P,P 2 Aut T(P ) is defined. Since O0 normalizes "P* *,P(Pb) = Op(Aut T(P )), and since O0 conjugates "P,P(Q) to "P,P(R) by Lemma 3.3, there i* *s an extension r2(O) 2 Mor T(QPb, RPb) of O0 by axiom (II), which is unique (and als* *o an extension of O) by Lemma 3.8. Thus r2(O) is a morphism in C2 from r2(Q, ') to r2(R, _), and the uniqueness of this extension implies that r2 is a functor. Th* *ere is a natural transformation of functors from inclOr2 to IdC1which sends an object (Q* *, ') to the morphism 'Q,QPb, and hence |C1| ' |C2|. It remains to show that |C2| is contractible. For any object (Q, ') in C2, ax* *iom (II) _ can again be used to construct an extension of ' to ' 2 Mor T(Pb, Q), thus a mo* *rphism in C2 from (Pb, 'P,Pb) to (Q, '), and this morphism is unique by Lemma 3.8. He* *nce (Pb, 'P,Pb) is an initial object in C2, and |C2| ' *. 30 BOB OLIVER AND JOANA VENTURA 5.Extensions of transporter systems We are now ready to study extensions of transporter systems and the resulting extensions of fusion systems. We first give a precise definition of what we me* *an by an extension of a transporter system. Afterwards, we show that any such extensi* *on is itself a transporter system, give conditions for when it is a linking system, a* *nd look at the obstruction theory for such extensions. Definition 5.1. Let (S, F, T ) be a transporter system. An_extension_T consist* *s of a category eTand a functor o : eT-! T , such that for all P , Q 2 Ob (Te), (a) o is a bijection on objects; __ __ (b) K_Pdef=Kero_P,_P:AuteT(P ) ---! Aut T(o(P )) is a finite p-group; __ __ (c) K_Pacts freely on Mor eT(P , Q) by right composition and o is the orbit map* * of this action; and __ __ (d) K_Qacts freely on Mor eT(P , Q) by left composition, and o is the orbit map* * of this action. Conditions (a), (c), and (d) above are equivalent to saying that o :eT---! T* * is a source and target regular extension, in the sense of Definition A.5. We want to show that every extension of a transporter system in this sense is* * again a transporter system; moreover, that an extension of T is equivalent to a tran* *sporter system eT with normal subgroup A C eTsuch that eT=A ~=T . In order to show that Te is a transporter system, we need first to associate a finite p-group eS, and* * a fusion system eFover eS, to the category eT. Define eSto be the pull-back of o~S,~Sand "S,S: e"eS eS______//_AuteT(S~) | | q || o~S,~S|| (1) fflffl|"S,S fflffl| S ______//_AutT(S) . Thus eS= {(g, f) 2 S x AutTe(S~) | "S,S(g) = o(f)}. Let q : eS- ! S and e"eS: * *eS- ! Aut eT(S~) be the structure maps of the pull-back, so q is surjective and e"eSi* *s injective. Set A = Ker(q), and set eP= q-1(P ) for each P S. So A and ePare subgroups of* * eS, and each ePcontains A. In fact, for each P 2 Ob (T ), the group ePis an extensi* *on of the form q|Pe 1 ---! A ---i--!Pe-----! P ---! 1 (including the case of S), where i is the inclusion map. Since the functor o is* * a bijection on objects and we defined a group eP eSfor each P 2 Ob (T ), we will consider * *that the objects in eTare subgroups of eS(all containing A as a normal subgroup) and* * will denote by ePthe object in eTsuch that o(Pe) = P . The pull-back square (1)also shows that e"eSrestricts to an isomorphism from * *A to Ker (oeS). In particular, A is a p-group by condition (b) in Definition 5.1, an* *d so eSis also a p-group. EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 31 The following lifting lemma is the analogy for extensions of transporter syst* *ems to Lemmas 1.7 and Lemma 3.2. Lemma 5.2. Let T be a transporter system and let eT be an extension of T by t* *he ' _ p-group A. Fix morphisms P ---! Q ---! R in T , and let f_'2 Mor eT(Pe, eR) be* * any lifting of _ O'. (a) For any lifting e'2 Mor eT(Pe, eQ) of ', there is a unique morphism e_2 Mor* * eT(Qe, eR) such that o(_e) = _ and e_Oe'= f_'. (b) For any lifting e_2 Mor eT(Qe, eR) of _, there is a unique morphism e'2 Mor* * eT(Pe, eQ) such that o('e) = ' and e_Oe'= f_'. Proof.These are special cases of Lemmas A.7(a) and A.8(a). Next, we define the functors eae TOb(eT)(Se) ---e"---!eT------! Gps which will give eTthe structure of a transporter system. Notice that we already* * have a group monomorphism e"eS: eS-! AutTe(Se), and we will be using it in the proof* *s of Lemmas 5.3 and 5.4. For each eP, choose a lifting e'Pe2 Mor eT(Pe, eS) of 'P 2 Mor T(P, S). These* * are chosen arbitrarily, except that we require e'eS= IdeSin eT. (Recall that 'P = "P,P(1).) Lemma 5.3. For each Pe, eQ2 Ob (Te) and each ex 2 NeS(Pe, eQ), there is a unique morphism e"Pe,Qe(ex) in eTthat makes the following diagram commute: e'Pe eP______//_eS | | e"Pe,Qe(ex)|| e"eS(ex)| (2) fflffl|e'fflffl||eQ eQ______//_eS. Furthermore, e"Pe,Pe(A) = Ker(oPe,Pe), and thus Ker(oPe,Pe) ~=A. Proof.The first statement follows immediately from lifting Lemma 5.2(b), applie* *d with e_= e'Qe, f_'= e"eS(ex) Oe'Pe, and ' = "P,P(q(ex)). ~= By Lemma 5.2, there is a unique bijection O: Ker(oPe,Pe) ---! Ker (oeS,eS) su* *ch that the square e'Pe Pe_______//eS | | ff|| O(ff)|| fflffl|e'fflffl|eQ Pe_______//eS. commutes for all ff 2 Ker (oPe,Pe). Since O(e"Pe,Pe(a)) = e"eS(a) for a 2 A by* * (2), and e"eS(A) = Ker(oeS,eS) by definition of e"eS, this shows that e"Pe,Pe(A) = Ker(o* *Pe,Pe). We thus get a well-defined map from Mor (TOb(eT)(Se)) to Mor (Te) which sends* * exto e"Pe,Qe(ex). By putting together two squares of the form (2) we see that this * *defines a 32 BOB OLIVER AND JOANA VENTURA functor e":TOb(eT)(Se) ------! eT which is the identity on objects. Notice that e"Pe,Qeis defined so that the squ* *are e"Pe,Qe NeS(Pe, eQ)______//_MoreT(Pe, eQ) q|| |oPe,Qe| (3) fflffl|"P,Q fflffl| NS(P, Q) ________//_MorT(P, Q) commutes in eT, i.e., e"Pe,Qeis a lifting of "P,Q. Lemma 5.4. For all Pe, eQ2 Ob (Te), the map e"Pe,Qe: NeS(Pe, eQ) - ! Mor eT(Pe,* * eQ) is injective. Proof. Let ex, ey2 NeS(Pe, eQ) be such that e"Pe,Qe(ex) = e"Pe,Qe(ey). Then e"eS(ex) Oe'Pe= e'QeOe"Pe,Qe(ex) = e'QeOe"Pe,Qe(ey) = e"eS(ey) * *Oe'Pe by (2). Also, by (3), q(ex) = q(ey), so e"eS(ex) and e"eS(ey) are liftings of t* *he same morphism "S,S(q(ex)). Thus e"eS(ex) = e"eS(ey) by Lemma 5.2(a). But e"eSis injective by * *construction (see (1)), so ex= ey. Lemma 5.5. Given a morphism f 2 Mor eT(Pe, eQ), there is a unique group homorph* *ism ' 2 Hom (Pe, eQ) such that the diagram f eP____//_eQ e"Pe,Pe(x)|||e"Qe,Qe('(x))| (4) fflffl|fflffl|f eP____//_eQ commutes for all x 2 eP. Proof. Set ~x= q(x). By axiom (C) of the transporter system T , o(f) O"P,P(~x) = "Q,Q(ae(o(f))(~x)) Oo(f) , for all ~x2 P . So the result follows from the lifting Lemma 5.2(a). Now we can define the functor eae: eT -! Gps to be the identity on objects, * *and to send a morphism f 2 Mor eT(Pe, eQ) to the unique group homomorphism defined * *in Lemma 5.5. Again, by putting together commutative squares of the form (4), one * *sees that eaeis indeed a functor. Let Fe be the fusion system generated by the image of the functor eae. Thus * *when Pe, eQ2 Ob (Te), Mor eF(Pe, eQ) is the image under eaeof Mor eT(Pe, eQ). For ar* *bitrary sub- groups R, R0 eS, Mor eT(R, R0) Inj(R, R0) is the set of all monomorphisms wh* *ich are composites of restrictions of eF-morphisms between objects in eT. Proposition 5.6. The category eT, together with the maps e"Pe,Qeand the functor* * eaede- fined in Lemmas 5.3 and 5.5, define a transporter system associated to eF. In p* *articular, Fe is an Ob (Te)-saturated fusion system. Furthermore, A C eT, and (Se=A, eF=A,* * eT=S) ~= (S, F, T ). EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 33 Proof.The last statement is clear, and the saturation of eFfollows from Proposi* *tion 3.6, once we know that eTis a transporter system. So we need only check that eTsatis* *fies the axioms of Definition 3.1. Axiom (A1) follows from the conditions in Definit* *ion 5.1 for the extension eT, and axiom (C) holds by construction of eae. For each eP, eQ eS, the injectivity of e"Pe,Qewas shown in Lemma 5.4. The co* *mposite eaePe,QeOe"Pe,Qesends g 2 NeS(Pe, eQ) to cg 2 Hom Fe(Pe, eQ) by the constructio* *n of eaeusing Lemma 5.5 (applied with f = "Pe,Qe(g)). This proves axiom (B). We next check axiom (A2). For each eP2 Ob (Te), set eE(Pe) = Ker Aut eT(Pe) ---! Aut eF(Pe) . Then oPe,Pe(Ee(Pe)) E(P ). So the free actions of E(P ) and E(Q) on Mor T(P, * *Q) lift to free actions of eE(Pe) and eE(Qe) on Mor eT(Pe, eQ). If f, f0 are two morphi* *sms such that eae(f) = eae(f0), then ae(o(f)) = ae(o(f0)), so o(f0) = o(f) Off for some ff 2 * *E(P ) by axiom (A2) for T . By the lifting Lemma 5.2(b), ff lifts to a unique eff2 AutTe(Pe) s* *uch that f0 = f Oeff, and eff2 eE(Pe) since eae(f0) = eae(f). Axiom (I) for eT(e"eS,eS(Se) 2 Sylp(Aut eT(Se))) follows from axiom (I) for T* * . Axiom (II) is a consequence of the same axiom applied to T . Given a morphism ' 2 Mor eT(Pe, eQ), and subgroups eP eP 0 NeS(Pe) and eQ eQ0 NeS(Qe) such* * that ' Oe"Pe,Pe(Pe0) O'-1 e"Qe,Qe(Qe0), apply the functor o : eT-! T to the last i* *nclusion. Then axiom (II) for T implies that there is some f 2 Mor eT(Pe0, eQ0) such that o(f* *)|P,Q = o('). Hence by Lemma 5.3 (Ker (oPe,Pe) = e"Pe,Pe(A)), there is a 2 A such that * *f|Pe,Qe= _ def -1 ' Oe"Pe,Pe(a). So ' = f Oe"Pe0,Pe0(a ) is an extension of '. We now look at the action of T on A induced by an extension of transporter sy* *stems. Lemma 5.7. Fix a transporter system (S, F, T ), and an extension 1 ---! A ---! Te --o-!T ---! 1. Then this defines an action : Mor (T ) -! Out(A) of Mor (T ) on A as follows.* * For each f 2 Mor T(P, Q), choose any ef2 Mor eT(Pe, eQ) such that o(fe) = f, and set e (fe) = eae(fe)|A 2 Aut(A). Then the class (f) = [e (fe)] 2 Out(A) is well defined, independently of the c* *hoice of lifting ef, and satisfies the following conditions. (a) (f) O (f0) = (f Of0) for any composable pair of morphisms f, f0 in T . (b) For any lifting ef2 Mor (Te) of f, the following square commutes for all a * *2 A: ef eP______//_eQ e"Pe,Pe(a)|| |e"Qe,Qe(e(fe)(a))| (5) fflffl|effflffl| eP______//_eQ. Proof.If ef0is another lifting of f, then there is some a 2 A such that ef0= ef* *Oe"Pe,Pe(a). Then eae(fe0) = eae(fe) Oca, and so [eae(fe0)|A] = [eae(fe)|A] in Out(A). This * *proves that (f) is well defined in Out(A), independently of the choice of lifting. 34 BOB OLIVER AND JOANA VENTURA Point (a) is immediate from the definition, and (b) follows from axiom (C) fo* *r eT. Extensions of transporter systems in the sense of Definition 5.1 give one cla* *ss of examples of extensions of categories which are both source regular and target r* *egular, in the sense of Definition A.5. By that definition, an extension of the transporte* *r system T is exactly the same as a source regular extension of T by a functor F :T op---!* * Gps* such that F sends all objects in T to finite p-groups and sends all morphisms* * to isomorphisms. The general obstruction theory for constructing extensions of categories whic* *h are source and target regular and have abelian kernels is described in Proposition * *A.9 (with some remarks on the nonabelian case). The following proposition deals with the * *special case of extensions of transporter systems. Since any extension of a transporter* * system by a finite p-group can be factored as the composite of a sequence of extension* *s by abelian p-groups, the assumption that the kernel be abelian is not a major rest* *riction. Proposition 5.8. Fix a transporter system (S, F, T ). Then extensions of T by a* * given finite abelian p-group A are in one-to-one correspondence with actions of ss1* *(|T |) on A, together with elements of H2(|T |; A) (with coefficients twisted by ). Proof. By Proposition A.3(a), an action of ss1(|T |) on A correspond to maps fr* *om Mor (T ) to Aut (A) which send composites to products and inclusions to the ide* *ntity. Hence the action of ss1(|T |) on A is described in Lemma A.7(c); and is the sam* *e as that defined by Lemma 5.7 by point (b) of that lemma. By Proposition A.9, the s* *et of extensions which realize a given action of ss1(|T |) on A are in one-to-one cor* *respondence with lim-2(K), where K(P ) ~=A for all P , and this group is isomorphic to H2(|* *T |; A) by Lemma A.3(b). We now examine conditions for an extension of T to be a linking system, and a* *lso for its associated fusion system to be exotic. We first give an example of the comp* *lications which can arise if one doesn't impose extra conditions. Set p = 2, and let L b* *e the centric linking system of A6. Since ss1(|L|) ~= 4 * 4, it surjects onto A6 ~=P* * SL2(9) D8 and onto P SL2(7); in fact, onto P SL2(q) for any q 7 (mod 16) which is a pr* *ime or the square of a prime. Let |L| act on A = C72via the faithful action of A6 o* *n C42 and that of P SL2(7) ~=GL3(2) on C32. The resulting extension of L by A is an e* *xotic transporter system; but not very interesting, since it has too few objects to g* *enerate a saturated fusion system. We now want to define the concept of an "admissible" action or extension of a transporter system, in order to avoid such examples. We first need one more lem* *ma to motivate the precise conditions in the definition. When o :eT---! T is an extension of T by A, we let P,Q : Mor T(P, Q) -----! Out (A) and P : E(P ) -----! Out (A) be the restrictions of the action to the corresponding sets of morphisms. Re* *call that E(P ) = Ker(aeP,P) AutT (P ), so P is a group homomorphism. We also wri* *te Ee(Pe) = Ker(eaePe,Pe). Lemma 5.9. Fix a transporter system (S, F, T ), and an extension 1 ---! A -----! Te---o--! T ---! 1. EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 35 o q Let eF---F-! F and eS----! S be the extensions of fusion systems and p-groups a* *sso- ciated to eT. Set "S,S S,S S1 = q(CeS(A)) = Ker S ----! Aut T(S) ----! Out (A) . Then the following hold for any subgroup P 2 Ob (T ), where eP= q-1(P ). (a) If P is fully normalized in F, then eE(Pe) e"Pe,Pe(CeS(Pe)) if and only i* *f Ker( P) is a p-group. (b) If Pe eS is any subgroup which is Fe-centric and Fe-radical, or more gene* *rally any Fe-centric subgroup such that Out eS(Pe) \ Op(Out eF(Pe)) = 1, then eP * * A and CS1(P ) P . (c) If CS1(P ) P , then ePis eF-centric. Proof.(a) Consider the following commutative diagram: eae 1________//eE(Pe)_____//_AuteT(Pe)________//AuteF(Pe) | | | | oPe,Pe| _| | | | fflffl| fflfflfflffl|( ,ae) fflffl| 1______//_Ker( P)_____//_AutT(P_)____//_Out(A) x AutF (P ) where _(f) = (f|A, f=A) for f 2 Aut eF(Pe). By Lemma 1.9(a), Ker(_) is a p-gro* *up. k Hence for each x 2 Ker ( P), xp 2 oPe,Pe(Ee(Pe)) for some k; and this implies * *that oPe,PeeE(Pe) has p-power index in Ker( P). Since Ker(oPe,Pe) ~=A is a p-group,* * it now follows that eE(Pe) is a p-group if and only if Ker( P) is a p-group. In parti* *cular, if eE(Pe) "Pe,Pe(CeS(Pe)), then Ker( P) is a p-group. Now assume conversely that Ker( P) is a p-group. We have just shown that eE(P* *e) is a p-group. Since P is fully normalized, ePis also fully normalized, and so a* *xiom (I0) for eT implies that e"Pe,Pe(NeS(Pe)) is a Sylow p-subgroup of Aut eT(Pe). Sinc* *e a normal p-subgroup of a finite group is contained in all of its Sylow p-subgroups, this* * shows that eE(Pe) e"Pe,Pe(NeS(Pe)). Furthermore, by axiom (C) for eT, every element of eE(Pe) commutes with every e* *lement of e"Pe,Pe(Pe), and thus eE(Pe) e"Pe,Pe(CeS(Pe)). (b) Assume eP eSis Fe-centric and Out eS(Pe) \ Op(Out eF(Pe)) = 1, and consid* *er the following diagram with exact rows: conj 1 ______//_CeS(Pe)____//NeS(Pe)________//_AuteF(Pe) | | | ~|| |q| |_| fflffl| fflfflfflffl|conj fflffl| 1 ______//_CS1(P_)____//_NS(P_)____//_Out(A) x AutF (P ) . Then fi g 2 NeS(Pe) ficg 2 Ker(_) CeS(Pe) eP : (6) 36 BOB OLIVER AND JOANA VENTURA the first inclusion since Ker(_) Op(Aut eF(Pe)), and the second since ePis Fe* *-centric. In particular, A eP. Also, for all g 2 CS1(P ), there is eg2 NeS(Pe) such tha* *t q(eg) = g, eg2 ePby (6), and so g 2 P . (c) If CS1(P ) P , then q(CeS(Pe)) P , and so ePis eF-centric. Conditions (b) and (c) above help to motivate the following definition. Definition 5.10. Fix a transporter system (S, F, T ). (a) A homomorphism : ss1(|T |) ---! is called admissible if, upon setting S1* * = Ker ( O"S,S), P S fully centralized in F and CS1(P ) P imply P 2 Ob (T* * ). (b) An action : ss1(|T |) ---! Out (A) of T on a p-group A is called admissibl* *e if the homomorphisms is admissible. (c) An extension 1 ! A ! eT! T ! 1 is called admissible if the action of T on A defined in Lemma 5.7 is admissible. Recall, for any fusion system F over S and any A S, the centralizer fusion * *system CF (A) is the fusion system over CS(A)_where for each P, Q, Hom CF(A)(P,_Q) is * *the set of those f 2 Hom F(P, Q) which extend to f 2 Hom F(P A, QA) such that f|A = IdA (s* *ee [BLO2 , x2]). As a consequence of Lemma 5.9, when applied to admissible extens* *ions, we now get: Theorem 5.11. Fix a transporter system (S, F, T ) and an admissible extension 1* * ! A ! Te ! T ! 1, and let Se and Fe be the p-group and fusion system associated to eT. Let : ss1(|T |) ---! Out (A) be the action defined in Proposition 5.7,* * and let (S1, F1, T1) be the "kernel" transporter system of in the sense of Propositio* *n 4.1. Then the following hold. (a) Ob (Te) contains all subgroups of eSwhich are eF-centric and eF-radical. Mo* *reover, every Fe-centric subgroup Pe eSnot in Ob (Te) is Fe-conjugate to some Pe0* *such that Out eS(Pe0) \ Op(Out eF(Pe0) 6= 1. (b) Fe is a saturated fusion system. (c) Let eLbe the full subcategory of eTwhose objects are those eF-centric subgr* *oups of eS in Ob (Te). Then eLis a linking system if and only if Ker[ P :E(P ) ---! Ou* *t (A)] is a p-group for all P 2 Ob (T ) such that eP2 Ob (Le). (d) If Fe is the fusion system of a finite group Ge, then F is the fusion syste* *m of NGe(A)=A, and F1 is the fusion system of CGe(A)=A. Thus if F1 is exotic (no* *t the fusion system of any finite group), then so is eF. Proof. Point (a) follows immediately from Lemma 5.9(b), and point (b) then foll* *ows from Proposition 3.6. By definition, eLis a linking system associated to eFif and only if eE(Pe) * *e"Pe,Pe(Z(Pe)) for all eP2 Ob (Le), and Z(Pe) = CeS(Pe) for all such eP. Also, if this conditi* *on holds for any subgroup in the Fe-conjugacy class of eP, it also holds for eP. So (c) foll* *ows from Lemma 5.9(a). It remains to prove (d). By construction, A C eF, eF=A ~=F, and CFe(A)=A ~=F1* *. If Fe ~=FeS(Ge) for some finite group eG, then since A C eF, eFis also the fusion * *system of EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 37 NGe(A). Thus F is the fusion system of NGe(A)=A. Also, by definition, a morphis* *m in eFlies in CFe(A) if and only if it is conjugation by an element in CGe(A). Henc* *e CFe(A) is the fusion system of CGe(A), and so F1 is the fusion system of CGe(A)=A. We finish the section with some tools which are useful when computing the obs* *truc- tion groups which appear in Proposition 5.8. Lemma 5.12. Fix a transporter system (S, F, T ), a finite abelian p-group A, an* *d an admissible action : T ---! Aut(A). Set S1 = Ker( O"S,S) C S, and assume that P 2 Ob (T ) implies P \ S1 2 Ob (T ). Let T1 T be the transporter subsystem* * of Proposition 4.1: Ob (T1) is the set of subgroups of S1 which are in Ob (T ), an* *d Mor (T1) is the set of morphisms _ in T between objects of T1 such that (_) = 1. Then * *there is a spectral sequence Eij2= Hi( ; Hj(|T1|; A)) =) Hi+j(|T |; A) , where = Im( ) Aut(A). Here, H*(|T1|; A) is ordinary cohomology with untwist* *ed coefficients. The action of on Hj(|T1|; A) is induced by the action of on A* *, together with the homotopy action of ~=Aut T(S1)=Aut T1(S1) on |T1| via restrictions. Proof.This is the spectral sequence induced by the fibration sequence eX------! E x eX------! B , where eXdenotes the covering space of |T | with fundamental group Ker( ). By Pr* *opo- sition 4.1(d), Xe ' |T1|. The action of on Xe induces the homotopy action on* * |T1| described above via the construction in the proof of Propositions 4.1(d) and A.* *4. In the above situation, if one wants to determine the action of on H*(|T1|;* * A) in concrete situations where the actions on |T1| and on A are both nontrivial, * *then more precision is needed when formulating the above statement. But this formula* *tion suffices for our purposes here. The following is an immediate corollary to Lemma 5.12. Corollary 5.13. Let (S, F, T ), : T ---! Aut(A), S1 C S, and T1 T be as in Lemma 5.12, and set = Im( ) Aut(A). Assume, furthermore, that Hi(|T1|; Fp) = 0 for i = 1, 2. Then Hi(|T |; A) ~= Hi( ; A) for i = 1, 2. Thus there is a bi* *jective correspondence between extensions of T by A and group extensions of by A (wit* *h the given actions of T and on A). 6. Examples We first look at conditions which imply that an admissible extension of the t* *rans- porter category of a finite group (or one of its full subcategories) is again t* *he transporter category of a finite group. Afterwards, we give some examples of "exotic" exten* *sion, where the target is the transporter category of a finite group but the source i* *s not. Recall that for any transporter system (S, F, T ), ` :Mor (T ) ------! ss1(|T |) is the map which sends ' 2 Mor T(P, Q) to the loop 'Q,S.'.'P,S-1(composed from * *right to left). When T = TS(G) for some finite group G and some S 2 Sylp(G), there is* * also a map Mor (T ) ---! G induced by the inclusions of the transporter sets Mor T(P* *, Q) = NG (P, Q) into G. This map clearly sends composites to products and inclusions * *1 2 38 BOB OLIVER AND JOANA VENTURA NG (P, S) to the identity. Hence by the universality property of ` (Proposition* * A.3(a)), there is a unique homomorphism jG :ss1(|TS(G)|) ------! G such that jG (`(x)) = x for all P, Q S and all x 2 NG (P, Q). To keep the not* *ation simple, we also write jG for the analogous homomorphism defined on ss1(|T |) wh* *en T is a full subcategory of TS(G) which contains S. Proposition 6.1. Fix a finite group G, a Sylow subgroup S 2 Sylp(G), and a set H of subgroups of S which is closed under G-conjugacy and overgroups. Let 1 ---! A -----! Te---o--! TH (G) ---! 1 be an admissible extension where A is a finite p-group. Assume furthermore that* * the homomorphism : ss1(|TH (G)|) -----! Out (A) of Lemma 5.7 factors through G: that = 0O jG for some 0 2 Hom (G, Out(A)). Then there is an extension of groups O 1 ---! A -----! Ge-----! G ---! 1 such that eTis a full subcategory of TO-1(S)(Ge). Proof. Assume first that A is abelian. In this case, by Proposition 5.8, the ex* *tensions of T = TH (G) which realize the given action are in bijective correspondence* * with elements of H2(|T |; A). By Proposition 4.5, jG induces an isomorphism ~= * j*G:H*(G; A) ------! H (|T |; A). Furthermore, by the construction of the cohomology invariants, j*Gsends the ele* *ment in H2(G; A) which describes a group extension to the element of H2(|T |; A) ~=l* *im-2(A) T which describes the extension of categories. Thus each extension of categories * *is induced by an extension of groups, which is what we wanted to prove. __ Now assume A is nonabelian, and set T = eT=Z(A). We can assume, by induction on |A|, that the proposition holds for the extension __ o 1 ---! A=Z(A) -----! T -----! TH (G) ---! 1 . __ _ __ Thus there is an extension G of G by A=Z(A), a Sylow subgroup S 2 Sylp(G ), and* * a __ _ __ __ set H of subgroups of S, such that T ~=T_H(G ). Consider the following diagram: __ proj Res ss1(|T_H(G )|)______//ss1(|TH (G)|)_____//_Out(A)________//Aut(Z(A)) | | 0 ppp77p j_G|| jG || ppppp (1) | | pppp fflffl|_ proj fflffl|ppppp G _________________//_G . The triangle commutes_by assumption, and the square by the naturality of jG . * *The action of ss1(|T |) on Z(A) defined by the extension __ 1 ---! Z(A) -----! Te---o--! T_H(G ) ---! 1 (2) is the composite of the upper_row in (1)(see the definition of the action in Le* *mma 5.7), and it thus factors through G by the commutativity of (1). So by the abelian ca* *se of EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 39 __ the proposition, applied to (2), there is an extension eGof G by Z(A) such that* * eTis a fully subcategory of TeS(Ge). Also, eGis an extension of G by A, since TH (G) ~* *=eT=A. Thus, in order to find an exotic extension, we must find an admissible action* * of some TH (G) on a p-group A which does not factor through G. The following example was suggested to us by Albert Ruiz; it is just the simplest of a large family of su* *ch examples which he has constructed in [Rz ]. The example is stated in terms of the linkin* *g system of G, but it also holds if LcS(G) is replaced by the transporter system TSc(G) * *with the same objects. Example 6.2. Fix a prime p 5. Let q be another prime, let e be the order of q* * in Fxp(thus e|(p - 1)), and assume e > 2. Set G = GLep(q), fix S 2 Sylp(G), and s* *et F = FS(G) and L = LcS(G). Let A be cyclic of order p. Then there is a homomorph* *ism : ss1(|L|) = ss1(|LcS(G)|) ------! Aut (A), where |Im ( )| = e, which does not factor through G. Also, there is an admissi* *ble extension 1 ---! A -----! Le-----! LcS(G) ---! 1, where eLis a linking system associated to a saturated fusion system eFwhich is * *not the fusion system of any finite group. Proof.Set V = (Fqe)p (with the canonical basis). Fix an Fq-basis for Fqe, which* * also yields an Fq-basis for V . This allows us to identify GLp(qe) = AutFqe(V ) as a* * subgroup of GLpe(q) = AutFq(V ). Let Fxpbe the subgroup of order e generated by q. We identify this with Gal(Fqe=Fq) (field automorphisms) by identifying q 2 Fxpwith the automorphism x* * 7! xq. Set H = (Fqe)x o , regarded as a subgroup of GLe(q) = AutFq(Fqe) (vector s* *pace automorphisms). This defines an embedding of H o p = Hp o p into GLep(q). Now, ep-1Y epY |G| = (qep- qi) = qep(ep-1)=2.(qi- 1), i=0 i=1 where p|(qi- 1) only when e|i, and (qei- 1)=(qe - 1) i (mod p). We leave it a* *s an exercise to show that (qep- 1)=(qe - 1) is divisible by p but not by p2. Thus i* *f p` is the largest power of p dividing qe - 1, then pp`+1is the largest power of p div* *iding |G| and |H o p|; and this proves that H o p contains a Sylow p-subgroup of G. Fix 2 Sylp((Fqe)x), a cyclic group of order p`, and set S = o Cp 2 Sylp(H* * o p) Sylp(G). Set T = p S: the set of elements of S which are diagonal in GLp(qe). We now want to define a homomorphism 0 from ss1(|F|) onto , for which the kernel fusion system (in the sense of Proposition 4.1) is exotic. The existence* * of such a homomorphism is shown in [Rz ], but we give a more explicit construction here. Let V ~= (Fq)epbe the vector space upon which G acts. Consider the subset fi X = g 2 S fi|g| = p, Ker (g - IdV) = 0 ; i.e., the set of elements of S of order p whose action on V does not have 1 as * *eigenvalue. Every element of Sr T acts on V with characteristic polynomial (Xp - 1)e, which* * has 1 as root, and thus X T . So if we fix z 2 (Fqe)x of order p, and again iden* *tify GLp(qe) GLep(q) (and (Fqe)p ~=(Fq)pe), then X = {diag(zi1, . .,.zip) | i1, . .,.ip 2 Fxp}. 40 BOB OLIVER AND JOANA VENTURA By definition of X, any morphism in F = FS(G) sends elements of X to elements of X. Now, Aut F(T ) = Aut G( p) ~= o p: the subgroup generated by the field au* *to- morphisms = Aut(Fqe) Aut( ), and by all permutations of the p factors. Defi* *ne : AutF (T ) ---! to be the composite pr1 : AutF (T ) ~= p o p --- -i x p --- -i . Thus for all ff 2 Aut F(T ), if ff(diag(z, z, . .,.z)) = diag(zi1, . .,.zip), t* *hen (ff) = i1. .i.p2 Fxp. So (ff) depends only on ff|Z(S)\X. For each pair of subgroups P, Q S which are p-centric in G, define 0P,Q:Hom F(P, Q) ------! _ as follows. For ' 2 Hom F(P, Q), '|extends to some ' 2 AutF (T ) by axiom* * (II) (and since T is the unique abelian subgroup of index p in S), and we set 0P,Q(* *') = _ _ _ (' ). Since (' ) depends_only on ' |Z(S)\X and P Z(S), this is independent of_the choice of extension ' . If _ 2 Hom F(Q, R)_is another morphism in Fc, * *and _ _ 2 Aut F(T ) is an extension of _|, then _ O' is an extension of (_ O'* *)| (since '(P \ X) Q \ X) and thus 0P,R(_ O') = 0Q,R(_). 0P,Q('). We have thus defined a map 0 from Mor (Fc) to which sends composites to products and sends inclusions to the identity. So by Proposition A.3(a), 0 ca* *n be identified with a homomorphism from ss1(|Fc|) to . Set = 0Oss1(|ss|): ss1(|L|) ------! . Let (Se, eF, eL) be the semidirect product extension of (S, F, L) by A induce* *d by ; i.e., the extension which corresponds to the zero element in H2(|T |; A). In th* *e notation of Theorem 5.11, S1 = S, and so the extension is admissible. Hence eFis a satur* *ated fusion system by Theorem 5.11(b). Also, since L = LcS(G) is a linking system, E* *(P ) is a p-group for all P 2 Ob (L), and thus eLis a linking system by Theorem 5.11* *(c). Let L1 L and F1 F be the "kernel" linking and fusion systems: Mor (L1) = {' 2 Mor (L) | (') = 1} and Mor(F1) = {' 2 Mor (F) | 0(') = 1* *}. Thus by Proposition 4.1, F1 is a saturated fusion system over S and L1 is a lin* *king system associated to F1. By construction, Aut F1(T ) ~= p-1 o p. So by [BLO* *2 , Example 9.4] (or more directly by [BLO2 , Proposition 9.5]), F1 is not the fus* *ion system of any finite group. (This is where we need to assume that e > 2, and hence tha* *t p 5.) Hence by Theorem 5.11(d), eFis not the fusion system of any finite group. We next make some cohomology computations, to illustrate the use of Lemma 5.12 and Corollary 5.13, and to show that the extension in Example 6.2 is the only o* *ne (up to isomorphism) with that action on the kernel. Lemma 6.3. Let G = GLep(q), S 2 Sylp(G), A ~=Cp, and ss1(|LcS(G)|) ---! Aut (A) be as in Example 6.2, set F = FS(G) and L = LcS(G), and let F1 F and L1 L be the kernel subsystems. Then for i = 1, 2, Hi(|L|; A) ~=Hi(|L1|; Fp) = 0. Proof. Set H = (Fxqeo Ce) o p, regarded as a subgroup of G which contains S vi* *a the embedding (Fxqeo Ce) GLe(q). Let denote the composite pr1 x : H = (Fxqeo Ce) o p --- -i Ceo p --- -i Cex p --- -i Ce Fp, EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 41 where the second homomorphism takes the product of the p coordinates in Ce, and* * set H1 = Ker( ). Set __ H 1 = H1 \ (Fxqeo Ce)p ~=(Fxqe)p o (Ce)p-1. Then __ p-1 Hi(H 1; Fp) = Hi((Fxqe)p; Fp)(Ce) vanishes for i = 1, 2, and similarly for Hi( p; Fp) -mono----!Hi(Cp o Cp-1; Fp). So by the spectral sequence for the semidirect product, Hi(H1; Fp) = 0 for i = * *1, 2. By [BLO2 , Theorem B], the inclusion B(S) L1 induces a monomorphism H*(|L1|; Fp) ------! H*(S; Fp), and this factors through H*(H1; Fp). Hence Hi(|L1|; Fp) = 0 for i = 1, 2. So * *by Corollary 5.13, if ss1(|L|) acts on A via , then H2(|L|; A) ~=H2( ; A) = 0 (si* *nce has order prime to p). By Lemma 6.3 and Proposition 5.8, it now follows that the only extension of L* * by A (with the given action) is the "semidirect product" constructed in Example 6.* *2. Example 6.2 is interesting as an example of an exotic fusion system with norm* *al p- subgroup for which the quotient is the fusion system of a group. But it is "too* * simple" in some sense, and leaves a lot of questions as to what other sorts of extensio* *ns can occur. This motivates the following list of problems. In all cases, in the following list, 1 ! A ! eT! T ! 1 is an extension of tra* *nsporter systems, associated to fusion systems eFand F over p-groups eSand S (and where * *A is abelian). We always let denote the action of ss1(|T |) on A. The phrase "inte* *resting extension" is intentionally left imprecise, but at least means an extension of * *transporter systems 1 ! A ! eT! T ! 1 which is not induced by an extension of finite groups. (1) Find interesting extensions for which Im( ) has order a multiple of p. (2) Find interesting extensions where subgroups which are centric and radical i* *n Fe are not all sent to subgroups which are quasicentric in F. (3) Find an extension where F and F1 (the kernel subsystem) are both fusion sys* *tems of groups, but where Fe is exotic. Essentially, this means finding F1 C F w* *hich are fusion systems of groups, but not fusion systems of a pair of groups G1* * and G such that G1 C G with appropriate quotient. We refer to [BCGLO1 , Definition 3.1] for the definition of an F-quasicentr* *ic sub- group. When F = FS(G) for a finite group G, then a subgroup P S is F-quasicen* *tric if and only if CG (P ) has a normal subgroup of order prime to p and of p-power* * index. Question (2) is motivated by [BCGLO1 , Theorem B], which says that for any p-* *local finite group (S, F, L), there is a linking system Lq L for which Ob (Lq) is t* *he set of all F-quasicentric subgroups of S, and that for any full subcategory L0 Lq wh* *ich contains all F-centric F-radical subgroups of S, |L0|, |L|, and |Lq| all have t* *he same homotopy type. In the remaining part of this section, we describe some constructions which a* *re simple modifications of the one in Example 6.2, and which show that problems (1) and (2) have solutions if one keeps the requirements for "interesting extension* *s" to a 42 BOB OLIVER AND JOANA VENTURA minimum. What this really shows is that we want more restrictive conditions on * *what makes them "interesting". Let p, q, and e be as in Example 6.2: p is a prime with p 5, q is a prime p* *ower with p - q, and e 3 is the order of q in Fxp. Set H = GLpe(qp), and let G = H* * o Cp where Cp acts via the field automorphism of order p on Fqp. Fix S0 2 Sylp(H) a* *nd S = S0 o Cp 2 Sylp(G), and set L0 = LcS0(H). Set H = {P S | Z(P ) \ H 2 Sylp(CH (P ))}. For each P 2 H, CH (P ) = (Z(P ) \ H) x C0H(P ) where C0H(P ) is the subgroup of elements of order prime to p in CH (P ), and we set T = TH (G)=C0H. This is a transporter system by Proposition 3.13. Also, L0 is the kernel linkin* *g system of the homomorphism from ss1(|T |) to Cp induced by the surjection of ss1(|TH (* *G)|) onto G=H. Hence the homomorphism of Example 6.2 extends to a homomorphism b :ss1(|T |) -----! x Cp Aut(A), (A ~=Cp2) and the homomorphism is admissible by construction of T . The kernel linking sy* *stem of b is the same as the kernel system for ss1(|L0|) --0-! , which was shown to* * be exotic in Example 6.2. So by Theorem 5.11, for any extension 1 ---! A -----! Le-----! T ---! 1 which realizes this action of T on A, eLis a linking system associated to an ex* *otic fusion system. Using Lemmas 5.12 and 6.3, we see that H2(|L|; A) ~=H2(Cex Cp; A) = 0, * *so there is a unique extension of this type. A similar, but slightly more complicated example can be constructed by setting H = GLpe(q)p, G = GLpe(q) o p = H o p, S 2 Sylp(G), and T = TH (G)=C0Hfor an appropriate class H of subgroups of S. One can then define from ss1(|T |) to * * o p, where again is cyclic of order e, and look at the induced extensions of A = (* *Cp)p by T . As yet another example, let p, q, and e be as in Example 6.2, set G1 = GLpe(q* *), and set G2 = SL3(p). Fix Si2 Sylp(Gi), and set G = G1x G2 and S = S1x S2 2 Sylp(G). Set H = {P S | Z(P ) \ G1 2 Sylp(CG1(P ))}. For each P 2 H, CG1(P ) = (Z(P ) \ G1) x C0G1(P ), where C0G1(P ) consists of * *all elements in CG1(P ) of order prime to p. Set T = TH (G)=C0G1, a transporter sys* *tem by Proposition 3.13. Define : ss1(|T |) -----! ss1(|LcS1(G1)|) x ss1(|TS2(G2)|) -----! Aut (Cp) x Aut(* *C3p), where the first homomorphism is induced by the two projections, and the second * *is the product of the homomorphism of Example 6.2 with the canonical action of G2 = SL* *3(p) on C3p. Set A = Cp x C3p, let 1 ---! A -----! Le-----! T ---! 1 be any extension which realizes , and let eSand Fe be the associated p-group a* *nd fusion system. By Theorem 5.11 again, eFis a saturated fusion system over eS, b* *ut eF is not the fusion system of any finite group, since if it were the fusion syste* *m of some EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 43 eG, then CGe(A)=A would have as fusion system the subsystem of index e in FS1(G* *1). Also, for each P 2 Ob (T ), 0 0 Ker : CG (P )=CG1(P ) ---! Aut (A) CG1(P )=CG1(P ) is a p-group, and so the restriction of eLto eF-centric subgroups is a linking * *system by Theorem 5.11(c). If we now take P = S1 x 1 2 H, and let ePbe such that eP=A = P , then ePis ce* *ntric and radical in eF. However, P is not quasicentric in FS(G), since its centraliz* *er fusion system involves the fusion system of G2. Appendix A. General background on categories and higher limits We begin with a description of the "bar resolution" for computing higher deri* *ved functors of inverse limits of diagrams of abelian groups. For any small catego* *ry C, C-mod denotes the category of functors Cop ---! Ab . Lemma_A.1. Let C be any small category, and let F : Cop ---! Ab be any functor. Let C *(C; F ) be the normalized chain complex for F : __ Y Cn(C; F ) = F (c0), c0!...!cn where the_product is taken over all composable sequences of nonidentity morphis* *ms. For , 2 Cn(C; F ), define ' d(,)(c0 -!c1 ! . .!.cn+1) = F (')(,(c1 ! . .!.cn+1))+ Xn+1 (-1)i,(c0 ! . .b.ci.!.c.n+1) , i=1 where we set ,(c ! ._.!.c0)_= 0 for any sequence containing an identity morphis* *m. Then lim-*(F ) ~=H*(C *(C; F ), d). C Proof.The proof is the same as those in [GZ , Appendix II, Proposition 3.3] or * *[O1 , Lemma 2] (where the result is shown for the unnormalized chain complex). We ske* *tch the argument here. Let P-1 be the constant functor Z_. For each n 0, define Pn: Cop ---! Ab by letting Pn(c) be the free abelian group with basis the set * *of all ' '1 'n sequences c -! c0 -! . .-.! cn, modulo the subgroup generated by those sequenc* *es for which some 'i for i 1 is an identity morphism. Also, for _ 2 Mor (C), Pn(* *_) is defined by composing_the_first morphism with _. For all n 0 and all F in C-mo* *d , MorC-mod(Pn, F ) ~=C n(C; F ). Thus Pn is projective in C-mod for n 0. Bound* *ary morphisms @ :Pn ---! Pn-1 are defined by setting Xn @([c ! c0 ! . .!.cn]) = (-1)i[c ! c0 ! . .b.ci.!.c.n]. i=0 This is seen to be exact (hence a projective resolution of Z_) via the splittin* *g homomor- phisms Pn(c) ---! Pn+1(c) which send [c ! c0 ! . .!.cn] to [c Id-!c ! c0 ! . .!. 44 BOB OLIVER AND JOANA VENTURA cn]. Since lim-0(F ) ~=Mor C-mod(Z_, F ), C __ lim-*(F ) ~=Ext*C-mod(Z_, F ) ~=H*(Mor C-mod(P*, F ), @*) ~=H*(C *(C; F )* *, d). C For any finite group G, we let Op(G) be the p-subgroup orbit category of G: t* *he category whose objects are the p-subgroups of G, and where MorOp(G)(P, Q) = Q\NG (P, Q) ~=map G(G=P, G=Q). Here, NG (P, Q) is the transporter set: NG (P, Q) = {g 2 G | gP g-1 Q}. If S 2 Sylp(G), then OS(G) Op(G) denotes the full subcategory whose objects a* *re the subgroups of S. These two categories are clearly equivalent (every object i* *n Op(G) is isomorphic to an object in OS(G)). Note that OS(G) is not the same as the or* *bit category of the fusion system FS(G). For any Z[G]-module M, we define *(G; M) = lim-*(FM ), Op(G) where FM :Op(G)op ---! Ab is the functor FM (P ) = 0 if P 6= 1 and FM (1) = M. The role these groups play in computing higher limits of certain functors is il* *lustrated by the following proposition, which is a special case of [BLO3 , Proposition 4* *.3]. Proposition A.2. Fix a category C, a finite group G, a Sylow subgroup S 2 Sylp(* *G), and a functor ff: OS(G) ------! C. Set c0 = ff(1). For each object d in C, we regard the set Mor C(c0, d) as a G-s* *et via ff and composition. Assume that the following conditions hold: (a) ff sends G = AutOS(G)(1) bijectively to End C(c0). (b) For each d 2 Ob (C) such that d 6~= c0, all isotropy subgroups of the G-act* *ion on Mor C(c0, d) are nontrivial p-subgroups. (c) For each , 2 Mor (Op(G)), ff(,) is an epimorphism in the categorical sense:* * ' O ff(,) = _ Off(,) implies ' = _. (d) For each P S, each d 2 Ob (C), and each ' 2 Mor C(c0, d) which is fixed b* *y the _ _ P P -action, there is some ' 2 Mor C(ff(P ), 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) ~= (G; (c0)) C = OS(G) is an isomorphism. We now look at fundamental groups of nerves of categories. The next propositi* *on describes the relationship between an action of ss1(|C|) and an action of Mor (* *C). For any group , B( ) denotes the category with one object and with automorph* *ism group , and we identify |B(G)| = BG. This notation will be used in several of* * the proofs in this section. EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 45 Proposition A.3. Fix a small category C, and an object c0 in C. Assume, for ea* *ch c 2 Ob (C), that Mor C(c, c0) 6= ?, and fix some distinguished "inclusion" morp* *hism 'c 2 Mor C(c, c0) (where 'c0 = Idc0). For each morphism _ 2 Mor C(c, d), let `* *(_) 2 ss1(|C|, c0) be the element represented by the loop 'd._.'c-1 (where paths are * *composed from right to left). Then the following hold: (a) For any group , and any function F :Mor (C) ---! such that F ('c) = 1 fo* *r all c_and F (_ O') = F (_)F (') when __O' is defined, there is a unique homomor* *phsm F :ss1(|C|, c0) ---! such that F (`(_)) = F (_) for all _ 2 Mor (C). In * *other words, ss1(|C|, c0) is the free group on the morphisms in C modulo the rela* *tions given by composition and the 'c. (b) Let F :C ---! Ab be any functor which sends all morphisms in C to isomorph* *isms of groups. Set A = F (c0), and let F 0be the functor which sends each objec* *t to A, and where F 0(_) = F ('d) OF (_) OF ('c)-1 for _ 2 Mor C(c, d). Then F 0is * *naturally isomorphic to F . Also, lim-*(F ) ~=H*(|C|; A), C where the second group is cohomology twisted via the linear action of ss1(|* *C|) on A induced by F 0. Proof.(a) By elementary homotopy theory, any element of ss1(|C|, c0) is repres* *ented by a loop which follows the edges of |C|, and hence is a product of loops of th* *e form `(') for ' 2 Mor_(C). Thus ss1(|C|) = . This also proves the uniqueness* * of the homomorphism F , and it remains to prove its existence. Let eF:C ---! B( ) be the functor which sends all objects of C to the unique * *object of B( ), and which sends morphisms via F . Set __ F = ss1(|Fe|): ss1(|C|, c0) -----! . For each _ 2 Mor (C), __ __ F (`(_)) = F 'd._.'c-1 = F ('d).F (_).F ('c)-1 = F (_) since F ('c) = F ('d) = 1 by assumption. (b) The functor F 0is naturally isomorphic to F via the natural isomorphism wh* *ich sends each c 2 Ob (C) to 'c. The map from Mor_(C) to Aut (A) defined by F 0fac* *tors through ss1(|C|) by (a). The chain complex C *(C; F 0) of Lemma A.1 is precise* *ly the same as the chain complex C*(|C|; A) for cohomology with twisted coefficients, * *and thus lim-*(F ) ~=lim-*(F 0) ~=H*(|C|; A). C C The following proposition describes very general conditions for the nerve of * *a sub- category of C to have the homotopy type of a covering space of |C|. Proposition_A.4. Fix a small_category_C, a group , and a surjective homomorphi* *sm F :ss1(|C|) ---! . Set F = F O` :Mor (C) ---! , where ` is as in Proposition * *A.3. For each H , let CH C be the subcategory with the same objects, where for * *all _ 2 Mor (C), _ 2 Mor (CH ) if and only if F (_) 2 H. Assume that (1)for each c 2 Ob (C) and each g 2 , there is an object d 2 Ob (C) and an iso* *mor- phism _ 2 IsoC(c, d) such that F (_) = g. 46 BOB OLIVER AND JOANA VENTURA Then for each H , |CH_| is homotopy equivalent to the covering space of |C| * *whose fundamental group is F -1(H). Proof. For any set X with left -action, let E (X) be the category with object * *set X, and where MorE (X)(x, y) = {g 2 | gx = y}. Let eCHbe the pullback category in the following square: eCH_________//E ( =H) | | | | | |~ | | fflffl|bF fflffl| C ___________//_B( ) . Here, bFand ~ send all objects to the unique object of B( ); bFsends morphisms * *via F , and ~ sends g 2 Mor E ( =H)(aH, bH) (if gaH = bH) to g 2 Mor (B( )). Thus Ob (CeH) = Ob (C) x ( =H), and Mor (CeH) is the set of pairs of morphisms in C * *and E ( =H) which get sent to the same morphism in B( ). Then |CeH| is a covering_s* *pace of |C| by construction; and is the covering space with fundamental group F-1(H)* * since ss1(|E ( =H)|) = H. The subcategory CH can be identified with the full subcategory of eCHwhose ob* *jects are the pairs (c, 1H) for c 2 Ob (C). Under this identification, each object i* *n eCH is isomorphic to an object in CH by (1), and thus |CH | ' |CeH|. We now discuss extensions of categories. We want to look at the following ve* *ry general situation, which occurs in several different contexts throughout the pa* *per. Definition A.5. Let o :eC-----! C be a functor between categories which is a bi* *jection on objects and surjective on morphism sets. For each ec2 Ob (Ce), set K(ec) def=Keroec,ec:AuteC(ec) ----! Aut C(o(ec)) . (a) We say that o is source regular (or a source regular extension of C by K) i* *f for all ec, ed2 Ob (Ce), K(ec) acts freely on Mor eC(ec, ed) and oec,edis the o* *rbit map for this action. (b) We say that o is target regular (or a target regular extension of C by K) i* *f for all ec, ed2 Ob (Ce), K(de) acts freely on Mor eC(ec, ed) and oec,edis the o* *rbit map for this action. The next three lemmas list some of the elementary properties of these classes* * of functors. For simplicity in notation throughout the rest of this section, when * *working with functors eC---! C which are source or target regular, we identify each c 2* * Ob (Ce) with o(c) 2 Ob (C). Lemma A.6. If o :eC---! C is either source or target regular, then a morphism i* *n eC is an isomorphism if and only if its image in C is an isomorphism. In particula* *r, if all endomorphisms in C are automorphisms, then the same is true of eC. Proof. Clearly, o sends isomorphisms to isomorphisms, and we need only prove the converse. Assume e_2 Mor eC(c, d) is such that oc,d(_e) is an isomorphism, and * *choose 'e2 Mor eC(d, c) such that o('e) = o(_e)-1. Since oc,cis the orbit map for a fr* *ee action of EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 47 K(c), and oc,c('eOe_) = oc,c(Idc), e'Oe_2 K(c), and hence is an automorphism. S* *imilarly, e_Oe'2 K(d) AutCe(d), and thus e_and e'are both isomorphisms. Let Gps be the category of groups with homomorphisms. Let Gps *be the category with the same objects, and where Mor Gps*(G, H) = Rep(G, H) def=Inn(H)\Hom (G, H). Lemma A.7. The following hold for any source regular functor o :eC---! C. ' _ (a) For any sequence c0 -! c1 -! c2 of morphisms in C, and any pair of morphisms e_2 Mor eC(c1, c2) and f_' 2 Mor eC(c0, c2) such that o(_e) = _ and o(_f') * *= _', there is a unique morphism e'2 Mor eC(c0, c1) such that o('e) = ' and e_Oe'* *= f_'. (b) For each e_2 Mor (Ce), e_is a monomorphism in eCin the categorical sense (i* *.e., e_Oe'1= e_Oe'2implies e'1= e'2) if and only if o(_e) is a monomorphism in C. (c) There is a functor K*: Cop ----! Gps *, defined on objects by setting oc,c K*(c) = Ker Aut eC(c) ----! Aut C(c) for all c 2 Ob (Ce). On morphisms, for f 2 Mor eC(c, d), K*(o(f)) = [_], wh* *ere _ 2 Hom (K*(d), K*(c)) is the unique homomorphism such that the following square commutes in eCfor all ff 2 K*(d): f c _______//_d | | | | _(ff)| |ff (1) | | fflffl|f fflffl| c _______//d. If o is also target regular, then K* sends all morphisms in C to isomorphis* *ms. Proof.(a) Let b'2 Mor T(c0, c1) be any lifting of '. Then there is an element ,* * 2 K(c0) such that f_'= e_Ob'O,, and we set e'= b'O,. If e'0is any other lifting which s* *atisfies the same condition, then e'0= e'Oj for some j 2 K(c0), and j = 1 since e_Oe'= e* *_Oe'0 and K(c0) acts freely on Mor T(c0, c2). (b) Fix e_2 Mor eC(c, d), and set _ = oc,d(_e). Assume first that e_is a monom* *orphism, and let '1, '2 2 Mor C(c0, c) be such that _ O'1 = _ O'2. By (a), applied to an* *y given lifting of _ O'1, there are morphisms e'1, e'22 Mor eC(c0, c) such that e_O'e1=* * e_O'e2 and oc0,c('ei) = 'i. Then e'1= e'2since e_is a monomorphism, so '1 = '2, and _ * *is a monomorphism. Conversely, assume _ is a monomorphism, and let e'1, e'22 Mor eC(c0, c) be su* *ch that e_Oe'1= _eO e'2. Then oc0,c('e1) = oc0,c('e2) since _ is a monomorphism, and h* *ence e'1= e'2by the uniqueness of the lifting in (a). So e_is a monomorphism. (c) By (a), for each f 2 Mor eC(c, d) and each ff 2 K*(d), there is a unique _(* *ff) 2 K*(c) which makes square (1) commute. By juxtaposing such squares, one sees that the resulting function _ :K*(d) ---! K*(c) is a homomorphism which depends only on * *f, and that this defines a functor eK from eCto Gps . For each c 2 Ob (C), eK send* *s each morphism in K*(c) to an inner automorphism in K*(c) (by (1)again), and hence Ke factors through a functor K* from C to Gps *. 48 BOB OLIVER AND JOANA VENTURA If o is also target regular, then each _(ff) determines a unique ff in the si* *tuation of (1), and so K* sends each morphism to an isomorphism. The following lemma is proven in exactly the same way as Lemma A.7. Lemma A.8. The following hold for any target regular functor o :eC---! C. ' _ (a) For any sequence c0 -! c1 -! c2 of morphisms in C, and any pair of morphisms 'e 2 Mor eC(c0, c1) and f_' 2 Mor eC(c0, c2) such that o('e) = ' and o(_f')* * = _', there is a unique morphism e_2 Mor eC(c1, c2) such that o(_e) = _ and e_Oe'* *= f_'. (b) For each _e2 Mor (Ce), _e is an epimorphism in eCin the categorical sense (* *i.e., 'e1O_e= e'2Oe_implies e'1= e'2) if and only if o(_e) is an epimorphism in C. (c) There is a functor K*: C ----! Gps *, defined on objects by setting oc,c K*(c) = Ker Aut eC(c) ----! Aut C(c) for all c 2 Ob (Ce). On morphisms, for f 2 Mor eC(c, d), K*(o(f)) = [_], wh* *ere _ 2 Hom (K*(c), K*(d)) is the unique homomorphism such that the following squa* *re commutes in eCfor all ff 2 K*(c): f c_______//_d | | | | ff| _(ff)| | | fflffl|f fflffl| c_______//d. The obstruction theory for describing source or target regular extensions of * *C by a given functor K from C to Gps * has been studied by Hoff [Hf]. For simplicity * *here, we handle only the case where K takes values in abelian groups. The nonabelian * *case is handled in [Hf] by using higher limits of functors with values in nonabelian* * groups (when they exist). Alternatively, if one assumes that K restricts to a functor* * ZK (where ZK(c) = Z(K(c))), then there is an obstruction theory which involves the groups lim-i(Z(K)) for i = 2, 3. Now fix K :Cop ---! Ab , and let eC---o--!C be a source regular extension of * *C by K. Choose a section s: Mor(C) ---! Mor (Ce). ' _ For each pair of composable morphisms c0 ---! c1 ---! c2 in C, let !(', _) 2 K(* *c0) be the unique element such that s(_) Os(') = s(_') O!(', _) 2 Mor eC(c0, c2). __ This defines an element ! 2 C2(C; K) (see Lemma A.1). ' _ O We next check that d! = 0. Fix a triple c0 ---! c1 ---! c2 ---! c3 of composa* *ble morphisms. Associativity in eCgives the relation s(O) O s(_) Os(') = s(O_') O!(_', O) O!(', _) = s(O) Os(_) Os(')= s(O_') O!(', O_) OK(')(!(_, O)). This gives the relation !(_', O) O!(', _) = !(', O_) OK(')(!(_, O)) 2 K(c0). EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 49 __ This proves that d! = 0, and hence that ! 2 Z2(C; K). It remains to show that the class [!] 2 lim-2(K)_is independent of the choice* * of section s. Let s0 be another section, and let_fi 2 C 1(C; K) be such that s0(') = s(')* * Ofi(') for all ' 2 Mor C(c0, c1). Let !02 C2(C; K) be the cochain analogous to the coc* *hain ! already defined. Then for ' and _ as above, s0(_) Os0(')= s0(_') O!0(', _) = s(_') Ofi(_') O!0(', _) = s(_) Ofi(_) Os(') Ofi(') = s(_) Os(') OK(')(fi(_)) Ofi(') . Thus, in the abelian group K(c0), !(', _)-1 O!0(', _) = fi(_')-1 OK(')(fi(_)) Ofi(') = dfi(', _); and this proves that !0= !.dfi. We have now shown how to assign a unique element of lim-2(K) to each extensio* *n of C by K. It is easy to show, using similar manipulations, that all elements of l* *im-2(K) can be realized in this way, and that extensions corresponding to the same elem* *ent of lim-2(K) are isomorphic. This can all be summarized in the following propositio* *n: Proposition A.9. Fix a small category C, and a functor K :Cop ---! Ab . Then th* *ere is a bijective correspondence between isomorphism classes of source regular ext* *ensions of C by K and the group lim-2(K). C We next look at geometric realizations of source and target regular functors.* * The following, very general result can be used in many situations to prove that cer* *tain maps between linking or transporter systems are fibrations with fiber BA, for a* * certain p-group A. Proposition A.10. Fix a group G, and a functor o :eC---! C between small catego* *ries which is both source regular and target regular. Assume |C| is connected, fix c* *0 2 Ob (Ce), |o| and set G = Ker(oc0,c0). Then the map |Ce| ---! |C| of geometric realizations i* *s a fiber bundle with fiber BG. Proof.We prove this by constructing a topological group, together with a princi* *pal bundle over that group of which |o| is an associated bundle. For each c 2 Ob (* *C) = Ob (Ce), set K(c) = Ker(oc,c) C AutCe(c). Let G be the category of self equivalences of B(G), where morphisms in G are * *natural isomorphisms of functors. Thus Ob(G) = Aut(G) and Mor G(ff, fi) = {g 2 G | fi = cg Off}. Let P be the category of functors B(G) ---! Cewhich go isomorphically to one of* * the "fibers" of o; more precisely: fi Ob(P) = (c, ~) fic 2 Ob (C), ~ 2 Iso(G, K(c)) fi Mor P (c, ~), (d, ~)= ' 2 Mor eC(c, d) fi~ = c' O~ . Consider the following square of categories and functors: pr2 P x B(G) _______//P ev|| ~|| (2) fflffl| fflffl| eC_____o____//C ; 50 BOB OLIVER AND JOANA VENTURA where ~ is the forgetful functor ~(c, ~) = c, and ev is the "evaluation functor" ' 'O~(g) ev g, (c, ~) -! (d, ~) = c -----! d . =~(g)O' For each n 0, let Gn denote the set of n-simplices in the nerve of G, and s* *imilarly for the other categories. Then Gn is a group with multiplication g1 g2 h1 h2 g1.ff0(h1) g2.ff* *1(h2) ff0 -! ff1 -! . . .. fi0 -! fi1 -! . . .= ff0 Ofi0 -----! ff1 Ofi1 -----* *! . . ., and the face and degeneracy maps are all homorphisms. So |G| is a topological g* *roup. Furthermore, Gn has a free right action on Pn defined by ' g 'O~0(g) (c0, ~0) -! (c1, ~1) O ff0 -!ff1 = (c0, ~0 Off0) ----! (c1, ~1 Off* *1) ; and a left action on B(G)n defined by g h g.ff0(h) ff0 -!ff1 . * -! * = * ----! * ; actions which again commute with face and degeneracy maps. Together with the functors ~ and ev, these actions induce bijections Pn=Gn ~=Cn and Pn xGn B(G)n ~=eCn for all n. Hence the geometric realizations |ev| and |~| are principal |G|-fibr* *ations (see, e.g., [May , xx18-20] or [GJ , Corollary V.2.7]). In particular, ev induces a h* *omeomor- phism |Ce| ~=|P| x|G|BG, and hence |o| is a fiber bundle with fiber BG associat* *ed to |~| the principal bundle |P| -! |C|. We now finish by describing how the Grothedieck spectral sequence applies to * *de- scribe higher limits over a target regular extension. Proposition A.11. Fix small categories eCand C, and let o :eC-----! C be a targ* *et regular functor. For each c 2 Ob (C), set o K(c) = Ker Aut eC(c) ----! Aut C(c) . Let : eCop---! Ab be any functor. Then there is a spectral sequence j i+j Eij2= lim-iH (K(-); (-)) =) lim- ( ). (3) C eC Proof. This is shown using the Grothendieck spectral sequence (cf. [Wb , x5.8]* *). Con- sider the following triangle of categories and functors: Ro eC-modB____________//_C-mod BBB ____ BBB ___ (4) lim-BBB!!B""_lim-___ Ab . Here, Ro: eC-mod ---! C-mod is right Kan extension by o. Since Ro is right adj* *oint to an exact functor (composition with oop), it sends injectives to injectives. Let Riobe the i-th derived functor of Ro. Thus for all in eC-mod and all c * *2 Ob (C), Rjo( )(c) = lim-j( O~opc), (o#c)op EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL 51 where ~c:o# c ---! Ceis the forgetful functor. The Grothendieck spectral sequen* *ce for the triangle (4)takes the form j i+j Eij2= lim-iRo( ) =) lim- ( ). (5) C eC Now, o# c is the overcategory with objects the pairs (d, ') for d 2 Ob (Ce) a* *nd ' 2 MorC(o(d), c), and where 0 0 fi Moro#c (d, '), (d , _) = f 2 Mor eC(d, d ) fi_ Oo(f) = ' . Consider the functors B(K(c)) ---S--!o# c ---T--!B(K(c)), where S sends the unique object of B(K(c)) to (c, Id) and sends ff 2 K(c) to ff* * 2 Auto#c(c, Id); and where T is defined as follows. Fix an arbitrary map of sets s: Ob (o# c) -----! Mor (Ce) such that for all (d, '), s(d, ') 2 Mor eC(d, c) and od,c(s(d, ')) = '. For eac* *h morphism f in o# c from (d, ') to (d0, _), there is (since o is target regular) a unique* * element T (f) 2 K(c) such that T (f) Os(d, ') = s(d0, _) Of. This defines T . Furthermore, T OS = IdB(K(c)), while s defines a natural trans* *formation of functors from S OT to Ido#c. It now follows that for all c and j, Rjo( )(c) = lim-j( O~opc) ~= lim-j( O~cO S) ~=Hj(K(c); (c)). (o#c)op B(K(c)) The spectral sequence (3)now follows from this together with (5). References [BK] P. Bousfield & D. Kan, Homotopy limits, completions, and localizations, Le* *cture notes in math. 304, Springer-Verlag (1972) [BCGLO1] C. Broto, N. Castellana, J. Grodal, R. Levi, & B. Oliver, Subgroup fam* *ilies controlling p-local finite groups, Proc. London Math. Soc. (to appear) [BCGLO2] C. Broto, N. Castellana, J. Grodal, R. Levi, & B. Oliver, Extensions o* *f p-local finite groups, Trans. Amer. Math. Soc. (to appear) [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 [BLO3]C. Broto, R. Levi, & B. Oliver, Discrete models for the p-local homotopy * *theory of compact Lie groups, in preparation [GZ] P. Gabriel & M. Zisman, Calculus of fractions and homotopy theory, Springe* *r-Verlag (1967) [GJ] P. Goerss & R. Jardine, Simplicial homotopy theory, Birkh"auser (1999) [Go] D. Gorenstein, Finite groups, Harper & Row (1968) [Hf] G. Hoff, Cohomologies et extensions de categories, Math. Scand. 74 (1994),* * 191-207. [JMO]S. Jackowski, J. McClure, & B. Oliver, Homotopy classification of self-map* *s of BG via G- actions, Annals of Math. 135 (1992), 184-270 [May]P. May, Simplicial objects in algebraic topology, Van Nostrand (1967) [O1] B. Oliver, Higher limits via Steinberg representations, Comm. in Algebra 2* *2 (1994), 1381-1393 [O2] B. Oliver, Equivalences of classifying spaces completed at odd primes, Mat* *h. Proc. Camb. Phil. Soc. 137 (2004), 321-347 52 BOB OLIVER AND JOANA VENTURA [Pg] Ll. Puig, Unpublished notes (ca. 1990) [Q] D. Quillen, Algebraic K-theory I, Lecture notes in mathematics 341 (1973)* *, 77-139 [Rz] A. Ruiz, Exotic subsystems of finite index in the fusion systems of gener* *al linear groups over finite fields, preprint [St] R. Stancu, Equivalent definitions of fusion systems, preprint [Wb] C. Weibel, An introduction to homological algebra, Cambridge Univ. Press * *(1994) LAGA, Institut Galil'ee, Av. J-B Cl'ement, 93430 Villetaneuse, France E-mail address: bobol@math.univ-paris13.fr Departamento de Matem'atica, Instituto Superior T'ecnico, Av. Rovisco Pais, 1* *049- 001 Lisboa, Portugal E-mail address: jventura@math.ist.utl.pt