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# 186 Class Note for MATH M0070 at Purdue

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EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES BOB OLIVER ABSTRACT We prove here the MartinoPriddy conjecture for an odd prime p the p completions of the classifying spaces of two groups G and G are homotopy equivalent if and only if there is an isomorphism between their Sylow psubgroups which preserves fusion A second theorem is a description for odd p of the group of homotopy classes of self homotopy equivalences of the pcompletion of BC in terms of automorphisms of a Sylow p subgroup of G which preserve fusion in G These are both consequences of a technical algebraic result which says that for an odd prime p and a nite group G all higher derived functors of the inverse limit vanish for a certain functor 33 on the p subgroup orbit category of G In an earlier paper BLOI in collaboration with Carles Broto and Ran Levi7 we reduced certain problems involving equivalences between p completed classifying spaces of finite groups to a question of whether certain obstruction groups vanish The main technical result of this paper is that these groups do always vanish when p is odd The proof of this result depends on the classification theorem for finite simple groups Fix a prime p and a finite group G For any pair of subgroups P Q 3 G7 let NgP Q denote the transporter NgP Q as E G l xPx l S Q The p subgroup orbit category of G is the category OPG whose objects are the p subgroups of G7 and where M0r0pltaP7 Q QNGP7Q g MaPGGP7 GQ A p subgroup P S G is called p centric if ZP is a Sylow p subgroup of GgP7 or equivalently if GgP ZP gtlt Gal for some subgroup Gap of order prime to p Let 2G 0170 Ab denote the functor EGO ZP if P is p centric in G7 and EGO 0 otherwise We refer to BLOI7 6 for more details on how this is made into a functor This paper is centered around the proof of the following theorem Theorem A For any odd prime p and any nite group G gaze 0 for all 2 2 1 we 1991 Mathematics Subject Classi cation Primary 55R35 Secondary 55R40 20D05 Key words and phrases Classifying space pcompletion nite simple groups Partially supported by UMR 7539 of the CNRS 1 2 BOB OLIVER Theorem A is proven as Theorem 45 below It was motivated by applications for studying equivalences between p completed classifying spaces of nite groups Let G and G be nite groups and let S S G and S S G be Sylow p subgroups An isomorphism go S i S is called fusion preserving if for all P Q S S and all P a 1 i Q oz is conjugation by an element of G if and only if 9013 90Q 1s conjugation by an element of G The Martino Priddy conjecture states that for any prime p and any pair GG of finite groups BGQ 2 BG Q if and only if there is a fusion preserving isomorphism between Sylow p subgroups of G and G The only if77 part of the conjecture was proved by Martino and Priddy MP and follows from the bijection RepP G d f HomP G lnnG BR 309 for any p group P and any finite group G cf BLOL Proposition 21 Conversely by BLOL Proposition 61 given a fusion preserving isomorphism between Sylow p subgroups of G and G the obstruction to extending it to a homotopy equivalence BGQ 2 BG Q lies in ZQ ZCQ Hence Theorem A implies Theorem B Martino Priddy conjecture at odd primes For any odd prime p and any pair G and G of nite groups with Sylow p subgroups S S G and S S G BGQ 2 BG Q if and only if there is a fusion preserving isomorphism S Q S We next turn to the question of self equivalences of BGQ For any space X let OutX denote the group of homotopy classes of self homotopy equivalences of X For any finite group G any prime p and any Sylow p subgroup S S G let AutfusS be the group of fusion preserving automorphisms of S let AutgS be the group of automorphisms induced by conjugation by elements of G ie elements of NgS and set OutfusS AutfusS Theorem A when combined with BLOL Theorem 62 gives the following description of OutBGQ Theorem C For any odd prime p and any nite group G with Sylow p subgroup S g G OutBG 2 Outfusw Theorem A should be a special case of a more general vanishing result formulated here as Conjecture 22 where orbit categories of groups are replaced by orbit categories of arbitrary saturated fusion systems77 in the sense of Puig Pu We refer to BLO2 1 and to the summary in Section 2 below for definitions of saturated fusion systems Conjecture 22 would in particular imply the existence and uniqueness of linking systems hence of classifying spaces associated to an arbitrary saturated fusion system over a pgroup This has motivated us to state results here as far as possible in the context of abstract saturated fusion systems It is only at the end that we translate our partial results to a condition on simple groups which is then checked in the individual cases EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 3 When p 2 Theorem A is not true since VZGQ can be nonzero The simplest counterexamples occur for G PSL2q when 1 E i1 mod 8 Recently we have proved that 0 for all i Z 2 when p 2 and G is an arbitrary nite group This means that the Martino Priddy conjecture does hold for p 2 but that Theorem C is not true as formulated above in this case The proof for p 2 not only requires the classification theorem for finite simple groups but also in its current form requires a long detailed case by case check when handling the simple groups of Lie type in odd characteristic as well as the sporadic groups For this reason we have not tried to incorporate it into this paper but will write it up separately Section 1 contains general material about higher limits over orbit categories of finite groups and Section 2 some results about saturated fusion systems and higher limits over their orbit categories Concrete criteria for proving the acyclicity of Z are then set up in Section 3 where the problem is reduced to a question about simple fusion systems Proposition 38 Also at the end of Section 3 there is a discussion of what further results would be necessary to prove Conjecture 22 for odd primes Finally in Section 4 we restrict attention to fusion systems of finite groups and apply the classification theorem for finite simple groups to finish the proof of Theorem A I would like to thank George Glauberman for his encouragement and his efforts to prove a result about p groups Conjecture 39 below which would have led to a proof of Conjecture 22 for odd p and in particular to a classification free77 proof ofTheorem A I also want to point out the importance to this work of Jesper Grodal7s techniques in Cr for computing higher limits of functors on orbit categories His main theorem while not used here directly was used in many of the computations which led to this proof Finally I thank Carles Broto and Ran Levi not only for their collaboration in the papers BLOI and BLOZ which are closely connected to this one but also for introducing me to this problem in the first place General notation We list for easy reference the following notation which will be used throughout the paper 9 SylpG denotes the set of Sylow p subgroups of G o OpG is the maximal normal p subgroup of G 0 SMP lt9 6 P l 917 1 for angroup P o NGHK as E GlccHx l S K for HK S G 9 CI denotes conjugation by cc 9 H xgcc l o HomgH K for H K S G is the set of homomorphisms from H to K induced by conjugation in G o RepH K lnnKHomH K and RepGH K lnnKHomgH K o AutgH HomgH H and OutgH RepGH H AutgH lnnH o A functor F COP 1gt Ab is called acyclic if 0 for all i gt 0 4 BOB OLIVER 1 HIGHER LIMITS OVER ORBIT CATEGORIES OF GROUPS We rst collect some tools for computing higher limits of functors over the orbit category ofa nite group G Very roughly these reduce to two general techniques One is to lter a functor by a sequence of subfunctors such that each of the subquotients vanishes except on one conjugacy class of p subgroups of G Proposition 11 then gives some tools which are very effective when computing the higher limits of these subquotients The other method is to reduce computations to a situation described in Proposition 13 where the functor extends to a Mackey functor and hence is acyclic by a theorem of Iackowski and McClure Fix a prime p a nite group G and a ZltpG module M Let FM be the functor on OPG de ned by setting FMP MP the xed submodule and de ne MG M mm MG These graded groups were shown in IMO to be very effective tools when computing higher limits over functors on orbit categories We rst summarize the properties of the A which will be needed here Proposition 11 Fix a prime p Then the following hold a For any nite group G and any functor F GAGquotp Zltp mod which vanishes except on subgroups conjugate to some given p subgroup P S G F ANGPPFP one b If G is a nite group H lt1 G is a normal subgroup which acts trivially on the ZltpG module M and pllHl then AGM 0 c IfG is a nite group and H lt1 G is a normal subgroup of order prime to p which acts trivially on the ZltpG module M then AG M E AGH d If G is a nite group and OpG 7E 1 if G contains a nontrivial normal p subgroup then AG M 0 for all ZltpG modules M Proof See IMO Propositions 54 55 amp 61 1 The idea now is to lter an arbitrary functor F OPG Zltp mod in such a way that all quotient functors vanish except on one conjugacy class and hence are described via Proposition 11a We next look for some conditions on a pair of nite groups H S G and a functor F on OPG which reduce the computation of to one of higher limits of a functor over OPNHH In general for any small categories C and D and any functors C L D L Ab there is an induced homomorphism gm L ch EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 5 de ned as follows Let 1 be an injective resolution of F in the category of functors D Ab7 and let 1 be an injective resolution of F o Q Since 1 o Q is a resolution of F o Q though not injective7 there is a chain homomorphism 1 o Q L 1 which extends the identity on F 0 Q7 and which is unique up to chain homotopy Then Q is the homology of the composite homomorphism 139 A Elm mm c mm unwed D c Where the rst map is induced by the universal property of inverse limits over C Lemma 12 Fix a nite group G and a p subgroup Q S G Then there is a well de ned functor lt1gt cps amenQ 0130 such that QPQ P for all PQ S NgQQ Let T be the set of all p subgroups P S G with the property Q lt1 P7 and Q lt1 xPafl for to E G implies to E NgQ Then for any functor F 910G p Zltp mod which vanishes except on subgroups G conjugate to elements of T the induced homomorphism mm Foch lt1 MG OpltNGltQgtQgt is an isomorphism Proof Clearly7 Q is well de ned on objects To see that it is well de ned on morphisms7 recall rst that MOTOMGWP P P NGP7 P Where NgPP is the set of all to E G such that xPx l S P Hence for any pair of objects PQ and P Q in 910NgQQ7 MOTOPltNGltQgtQgtPQ7 PVC P QNNltQgtQPQ7 PVC PNNQP7P g P NGP P Mor0pltaP7 P and Q is de ned on morphism sets to be this inclusion Composition with Q is natural in F and preserves short exact sequences of functors Hence if F Q F is a pair of functors from OPG to Zltp mod and the lemma holds for F and for FF 7 then it also holds for F by the 5 lemma Hence it su ices to prove that 1 is an isomorphism when F vanishes except on the G conjugacy class of one subgroup P E 7 When P Q7 then 1 is precisely the isomorphism E ANQQ of Proposition 11a Now let P E T be arbitrary By condition gt0 Q lt1 P7 NaP S NgQ7 and F o Q vanishes except on the OpNgQQ isomorphism class of PQ Let xv ng OmanP GamaQ 6 BOB OLIVER be the functor RP RQ for p subgroups R S NaP S NaQ containing P Then the following square commutes mfg L F 0 ch 0 a 0p ltNltQgtQgt beryl wig ANGPPFP ANGPPFP and the vertical maps are isomorphisms by Proposition 11a see the proof of HMO Lemma 54 for the precise description of the isomorphisms This shows that CDquot is an isomorphism D The next proposition describes a different condition which implies the acyclicity of a functor on the orbit category of a nite group Proposition 13 Fix a nite group G a prime p and a ZltpG module M and let HOM 049quotp Zltp mod be the functor de ned by setting H MP H PM MP Let F 910G0p S Zltp mod be any subfunctor of HOM thus FP 3 MP for all P which satis es the following relative norm property for each pair of p subgroups P S Q S G mam d f Z 995 cc 6 FP MP g FQ l sPEQP Then F is acyclic 0 for alli gt 0 Proof The relative norms 91 make F into a proto Mackey functor in the sense of JM and hence it is acyclic by HM Proposition 514 D The following application of Proposition 13 plays an important role in Section 3 If G is a nite group and S E SylpG then 93G Q OPG denotes the full subcate gory Whose objects are the subgroups of S the inclusion is clearly an equivalence of categories As usual a subgroup T S S is called strongly closed in S with respect to G if no element of T is G conjugate to any element of ST Recall that for any p group P and any n 2 1 LAP denotes the subgroup generated by all to E P such that 5an 1 Proposition 14 Fizz a nite group G a Sylow subgroup S E SylpG and a subgroup T S S which is strongly closed in S with respect to G Let M be a nite Z9 Gl module and let F mayp Ab EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 7 be a subfunctor of HOM particular FP 3 MP for all P which satis es the relative norm property mFltPgt FQ ltgt for each pair of subgroups P S Q S S Let F1 Q F be the subfunctor FltPgti FP ifP T1 1 T 0 otherwise Assume WZltT91M 0 Then F1 0 for alli Z 1 039 Proof Note rst that F1 is a functor if P P S S are G conjugate then P O T 1 if and only if P O T 1 since T is strongly closed Assume 7E 0 for some n 2 1 We must prove that Zngl 7E 0 This will be shown by induction on it Let QkF Q F be the pk torsion subfunctor in F ie Q for each P Then for some ls 7E 0 This functor QkFQk1F satis es all of the hypotheses ofthe proposition with respect to the l plGl rnodule Q It thus su ices to prove the proposition when M 91M ie when pM 0 Without loss of generality we can assume M De ne a functor F Q F by setting FP NW N mPnT39Ml for all P S S We claim that F still satis es condition above To see this x subgroups P S Q S S set P POT lt1 P and Q Q T lt1 Q and set Q PQ Then P P Q so coset representatives for Q P are also representatives for Q P Hence 913059 mg FP m mpr g mam m mgmp39M g FltQgt m mgM M2 Thus upon replacing F by F without changing F1 we can assume that FP mpM for all P S T By Proposition 13 0 for all i gt 0 Hence 1FF1 7E 0 since 1m 1 y assumption or eac su group S et e t e unctor on 7 F 0 b F h b Q T l FQ b h f 93G de ned by F P 7 FP if P HT is G conjugate to Q Q T 0 otherwise There is an obvious ltration of FF1 whose quotients are all isomorphic to FQ for various 1 7E Q S T Hence there is some Q S T such that mn 1FQ 0 1 me Since we can replace Q by any other subgroup of S in its G conjugacy class we can assume that NSQ E SylpNgQ 8 BOB OLIVER If n 1 then 1 implies that FQS 7E 0 Hence S T Q so Q T and 0 7E FTS Q FTT 91TM In particular WZltTM 7E 0 Now assume n gt 1 Set G NgQQ S NSQQ E SylpG and T NTQQ Then T is strongly closed in S with respect to G no element of NTQ is NgQ conjugate to any element of NSQNTQ since no element ofT is G conjugate to any element of ST Define functors F F1 OSG Ab by setting FTPQ mm and FWDQ FQP Thus Fl PQ F PQ whenever P T Q equivalently whenever PQ T 1 and FPQ 0 otherwise Consider the set 7BP SlP TQ If FQP 7E 0 and P S S then P HT is G conjugate to Q P is G conjugate to some P such that Q S P S NSQ since NSQ E SylpNgQ P O T is G conjugate to P O T since T is strongly closed and thus P O T Q and P E 76 By a similar argument ifP E 76 and Q lt1 xPas l then yPya 1 S NSQ for some y E NaQ yPya 1 O T Q and so cc 6 NgQ This shows that 70 is contained in the set T de ned in Lemma 12 and thus that each subgroup of G for which FQP 7E 0 is G conjugate to a subgroup in T Hence by Lemma 12 F1 2 Mae 03 GI Os G In particular 7E 0 by All of the conditions of the proposition are satis ed with G s T F and M replaced by G 3 T F and M d f FQ gigs1 So by the induction hypothesis if we set Z Q ZT then mZM mzltTmQM mzltTgtM 7e 0 and hence WZltTM 7E 0 since ZT 3 Z 1 2 HIGHER LIMITS OVER ORBIT CATEGORIES OF FUSION SYSTEMS We first brie y recall some definitions We refer to BLOZ 1l or Pu for more details A fusion system over a finite p group S is a category 7 whose objects are the subgroups of S and whose morphisms satisfy the following conditions 9 HomSPQ Q MOTquot3P7Q Q InjP Q for all P Q S S and 0 each morphism in 7 is the composite of an F isomorphism followed by an inclusion To emphasize that the morphisms in 7 are all homomorphisms of groups we write HomP Q MOTquot3P Q for the morphism sets Two subgroups of 7 are called T eonjugate if they are isomorphic in 7 A subgroup P S S is called fully centralized EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 9 in 7 fully normalized in 7 if lGSPl Z lGSP l Z lNSP l for all P in the f conjugacy class of P The fusion system 7 is saturated if I for each fully normalized subgroup P S S P is fully centralized and AutSP E SylpAutfP and H for each 90 E HomP S whose image is fully centralized in 7 if we set Nw I E N303 l mail 6 AutssoP7 then 90 extends to a morphism 927 E Hom NW S If G is a finite group and S E SylpG then we let 75G denote the category whose objects are the subgroups of S and where MOTfSltGP HomgP g It is not hard to see BLOZ Proposition 13 that 75G is a saturated fusion system over S and that a subgroup P S S is fully centralized fully normalized if and only if GSP E SylpGgP NSP E SylpNgP By analogy with the orbit category of a finite group when 7 is a saturated fusion system over a p group S we let 90 the orbit category of 7 be the category with the same objects and with morphism sets Mor0ltfP Q Rep P Q d f lnnQHomP Q If 75G for some finite group G then 90 is a quotient category of 93G the full subcategory of OPG whose objects are the subgroups of S but its morphism sets are much smaller in general More precisely if P and Q are two p subgroups of G then MOTOPGP7Q g QNGP7Q7 while MOTOltfpltGgtgtltP7 g QNgP Thus there is a natural projection functor OPG L O7 G which is the identity on objects and a surjection on all morphism sets but these maps of morphism sets are very far in general from being bijections However the next lemma shows that if one restricts to p centric subgroups of G then p local functors over these two categories have the same higher limits If 7 is a saturated fusion system over S then a subgroup P S S is f eentrie if GSP ZP for all P f conjugate to P If 7 73G then a subgroup is 7 centric if and only if it is p centric in G see BLOL Lemma A5 Let 75 Q 7 and C73 Q 90 be the full subcategories whose objects are the f centric subgroups of S Similarly for any finite group G and any S E SylpG we let OG Q OPG and 959 Q 95G be the full subcategories whose objects are the p centric subgroups of G and those contained in S respectively Lemma 21 Fix a prime p and a nite group G Let F O7 fG p Zltp mod be any functor and let lt1 OG O7 G 10 BOB OLIVER be the projection funetor De ne F 97G0p Zltp mod by setting lo a F o I and F who we lt1 0 9 0 G 0p G 0 tfP is not p eentm e in G Then Proof For any pair of p centric subgroups PQ S G write CgP ZP gtlt Cap where C CP has order prime to p For any to E NgP Q and a E ZP sea xax l E Q95 since xaas l E Q by definition of the transporter Thus M0TOfGP7 Q QNGP7QOGP QNGP7 QgtO GltPgt 2 Morotaxp QgtC GltPgt Since C CP has order prime to p the first isomorphism in 1 now follows as an immediate consequence of BLOL Lemma 13 The second isomorphism holds since if vanishes on all p subgroups which are not p centric and since every p subgroup of G which contains a p centric subgroup is also p centric D For any saturated fusion system 7 over a finite p group S let 3f 9050p Ab denote the functor ZAP ZP for all f centric subgroups P S S If go 6 HomP Q then Z go is the composite ZltQgt ZMP ZltPgt If 7 73G for some finite group G with Sylow p subgroup S then Zglosltg is the composite of Z with the projection between orbit categories So Lemma 21 implies as a special case that Ef e g ENET MG WP What we would like to prove is the following conjecture of which Theorem A is just the special case where 7 75G and p is odd Conjecture 22 Fix a prime p and let 7 be a saturated fusion system over ap group S Then ags 0 OM tfp is odd andi Z 1 ortfp2 andi Z2 Conjecture 22 would imply that each saturated fusion system over a p group S has a unique associated linking system in the sense of BLO2 1 and hence a unique associated classifying space see BLO2 Proposition 31 The vanishing of gulps would also imply when p is odd a description of the group of homotopy classes of self equivalences of the classifying space similar to the description of OutBGQ in Theorem C Throughout this section and the next we will be developping tools for computing higher limits of functors on centric orbit categories of saturated fusion systems in EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 11 particular those with connections to Conjecture 22 Only in the last section do we again return to the special case of fusion systems of finite groups and finish the proof of Theorem A If 7 is any saturated fusion system over a p group S and Q S S is fully normalized in 7 then NQ is defined to be the fusion system over NSQ whose morphisms are defined by the formula HomeltQPP alp l a e HomfPQP Q aP g P aQ Q By BLOZ Proposition A6 this is a saturated fusion system over NSQ We also let OEQ N Q denote the full subcategory of the orbit category of NQ whose objects are the subgroups which contain Q Lemma 23 Fix a saturated fusion system 7 over ap group S and afully normalized f centric subgroup Q S S Consider the functor 1 I39S OEQUVHQ OoucsltQgtOutfQ de ned by setting P OutpQ and I P L P alQ Then I is an isomorphism of categories Hence there is a functor ltIgt cps Oplt0utfltc2gtgt Om unique up to natural isomorphism whose restriction to OoutSltQOutQ is equal to 171 Proof Write F Out Q and S OutSQ for short Since Q is fully normalized in 7 S is a Sylow p subgroup of F condition I in the definition of a saturated fusion system and so the inclusion OSF Q Op is an equivalence of categories Now 3 NSQQ since Q is f centric in S So I defines a bijection between objects of OEQNQ and objects of OSF sending P to OutpQ E PQ Fix subgroups P P S NSQ containing Q and consider the function I RPz RepNFltQgtP P MorOltNFltQP P Mor0pltp0utpQ OutpQ which sends the class oz for oz 6 HomNFltQP P to the class of ab 6 Out Q F For any such oz the following square commutes 135 P CQJ lama P a P for all g E P so oz lies in the transporter NpOutpQ OutpQ and the map I RP is well defined If B E Aut Q is such that conjugation by B E F Out Q sends OutpQ into OutpQ then cg l E AutpQ for all g E P so extends to some oz 6 HomP P by condition II in the definition of a saturated fusion system and I RP sends oz to Thus I RP is onto lf 041042 6 HomNFltQPP are such that lung041 lung042 in CAP then allQ c9 0 Olle for some 9 E Q hence 041 c9 0 042 o CZ for some Z 6 ZQ by BLOZ Proposition A8 and so 041 a2 in 12 BOB OLIVER RepfP P Thus PJD is a bijection for each pair of objects P P and this nishes the proof that I is an isomorphism of categories The last statement now follows by letting I be the composite of a retraction of OPF onto OSF followed by I 1 followed by the inclusion of OEQNQ into C73 D The next proposition describes how higher limits over C73 can be reduced in certain cases to higher limits over the orbit category of OutQ for some subgroup Q Note its similarity with Lemma 12 in both the statement and the proof By analogy with the usual definition for subgroups of finite groups for any saturated fusion system 7 over a p group S a subgroup P S S is called weakly f closed or weakly f closed in S if P is not f conjugate to any other subgroup of S Proposition 24 Fix a saturated fusion system 7 over a p group S and a fully nor malized f centric subgroup Q S S and let lt1gt o5 Op0utfQ 0P be the functor of Lemma 23 Let T be the set of all subgroups P S S such that Q lt1 P and Q lt1 aP for oz 6 HomP S implies aQ Q Then for any functor F C73quotp S Zltp mod which vanishes except on subgroups f conjugate to elements of T the induced homomorphism mm L Foch lt1 0W Oplt0utrltQgtgt is an isomorphism In particular ifQ is weakly f closed in S then 1 holds for any functor F which vanishes except on subgroups which contain Q Proof Composition with I is natural in F and preserves short exact sequences of functors If F Q F is a pair of functors from C73 to Zltp mod and the lemma holds for F and for FF then it also holds for F by the 5 lemma Hence it suf ces to prove that 1 is an isomorphism when F vanishes except on the f conjugacy class of one subgroup P E 7 Fix P E T and set P OutpQ S OutfQ By condition Q lt1 P so P PQ and F0 1 vanishes except on the OpOutfQ isomorphism class of Outp Q PQ Also by again H2 H2 OutfP g OutN Q P and Outwo p AutopltNxltQgtgtP Autououmm s NOHtle133 by Lemma 23 Let it cpgggggggz momma mommy EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 13 be the functor RP RQ for p subgroups R S NaP S NaQ containing P Then the following square commutes mm V Foch OM 0plt0utrltQgtgt obyf wJg Out pb pll i AN0utfltQ3P FltPlly and the vertical maps are isomorphisms by BLOZ7 Proposition 32 and its proof and Proposition 11a It follows that CDquot is an isomorphism The last statement follows since if Q is weakly f closed in S7 then T P S S l P 2 Q every subgroup which contains Q satisfies 1 The following lemma describes how quotient fusion systems are obtained by dividing out by weakly f closed subgroups Lemma 25 Let 7 be a saturated fusion system over ap group S and let Q lt1 S be a weakly F closed subgroup Let FQ be the fusion system over SQ de ned by setting H0mfQPQ7 PVC SDQ l 90 E Horn4P P for all P P S S which contain Q Then fQ is saturated Also for any PQ S SQ P Q is fully normalized in 7 Q if and only if P is fully normalized in 7 while P is fully centralized in 7 whenever P Q is fully centralized in 7 Q Proof For each P S S which contains Q7 set KP KerAutP AUEfQUDQll and K2 KerAutSP AutsQPQll Then lOSQPQl lePllKl allQl and lNSQPQll lePllQl 1 By the second formula7 PQ is fully normalized in fQ if and only if P is fully normalized in 7 Assume PQ is fully normalized in fQ Then P is fully normalized in 7 so by condition I in the definition of a saturated fusion system applied to 7 P is fully centralized in 7 and AutSP E SylpAutP This last condition implies that K e Syipmp and AutSQPQ e SylpAutfQPQ Thus lCSPl and lKgl both take the largest possible values among subgroups in the f conjugacy class of P7 and hence PQ is fully centralized by This finishes the proof that condition 1 holds for fQ It also shows that if PQ is fully centralized in 7Q7 then lCSPl and lKgl must both take the largest possible values among subgroups in the F conjugacy class of P7 and in particular P is fully centralized in 7 To prove condition ll7 fix a morphism goQ E HOmfQPQ7SQ such that goPQ is fully centralized in 7Q7 and set N4 9 E N503 l 80099071 E KMP AUESWP 14 BOB OLIVER Then litQ NW d f M e NsPQ l WM WQ e AutsQSDPQ7 and we must show that goQ extends to KayQ Set P 90P for short P is fully centralized in 7 since P Q is fully centralized in fQ Since goAuthPgo 1 S KpAutSP 7 where Kp lt1 Aut P 7 AutSP E SylL7AutP 7 and the left hand side is a p group7 there is 1 6 Kg such that WW39AUENJPleDW S Auts yl So by condition II for the saturated fusion system 7 140 extends to a homomorphism 927 E HomNWS7 and gZQ is an extension of goQ to NWQ D 3 REDUCTION TO SIMPLE FUSION SYSTEMS In this section7 we establish a suf cient condition for proving the acyclicity of 3 a criterion which in the case 7 FAG will depend only on the simple components in the decomposition series of the finite group G Recall that for any p group P and any n 2 17 LAP denotes the subgroup of P generated by pn torsion elements n2 If H and K are two subgroups of a group G usually normal subgroups and W Kl 17 then we write HKn for the n fold iterated commutator HK1 lH7K2l llHyKhKly and H7Kn1l llHmelyKl De nition 31 For any p group S S denotes the largest subgroup ofS for which there is a sequence 1Q0 Q1 quot393Qnx5lt5 0f Subgroups all normal in S such that 91OSQ 717Q 1771 1 1 foreachi1n It is easy to see that there always is such a largest subgroup If 1 so am 1m are two sequences of normal subgroups of S which satisfy condition I in Definition 317 then the sequence 1maadgaagmgau also satisfies the same condition When p 27 S 0501S for any finite Z group S In particular7 S ZS if S is generated by elements of order 2 So these subgroups are not very interesting in that case We first note some elementary properties of these subgroups S EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 15 Lemma 32 pr is odd and S is a p group then S Z A for every normal abelian subgroup A lt1 S In particular S is centric in S Proof If A lt1 S is abelian then SAp71 S SAA 1 and so A S S by definition Now let A be maximal among the normal abelian subgroups of S If CSA f A then ACSAA is a nontrivial normal subgroup of SA and hence contains an element 50A 6 ZSA of order p But then ltAagt is a larger normal abelian subgroup of S which is a contradiction Thus A is centric in S and in particular S Z A is centric in S D The following lemma is useful when proving that certain subgroups of S are contained in Lemma 33 Fizz an odd prime p and ap group S Let Q lt1 S be any normal subgroup such that l91Z3537Qp1l1 1 Then S 2 Q Proof Set 35 S for short By definition there is a sequence 1Q0Q1quot39Qnx of subgroups normal in S such that 91CSQ1 Qp71 1 for each i If Q lt1 S is normal and satisfies condition 1 then since Z 04 by Lemma 32 we can set Q7 QQn and Q 3 Q7 3 35 by definition 1 The purpose of these subgroups S is to provide a tool for applying Proposition 14 when trying to show that the functors Z are acyclic These are most useful when applied to a filtration of these functors described as follows For any saturated fusion system 7 over a p group S a subgroup P S S is strongly F closed in S if no element of P is T conjugate to any element of SP If T S S is strongly T closed subgroup in S let 2 OUTCVP Zltp mod be the subfunctor of Z defined by setting ZAP ZP O T When 7 is a saturated fusion system over a p group S and T lt1 S is a strongly 7 closed subgroup then a fully T normalized subgroup P S T will be called flaw radical if OpOutfP OutTP 1 Lemma 34 Fizz an odd prime p a saturated fusion system 7 over a p group S and a pair T0 lt1 T lt1 S of subgroups strongly F closed in S Write TT0 XTO for short For any fully T normalized subgroup Q S T de ne ZQ C730p Zltp mod 16 BOB OLIVER by setting for T centrtc P S S ZgZEO P E if P O T is T conjugate to Q 0 otherwise ZQP Assume that Q 4 X 07 that Q is not centric in T 07 that Q is not flaw radical Then WZQ 0 00 Proof Since Q is fully F norrnalized for any Q S S which is T conjugate to Q there is some 90 E HornNSQ NSQ such that 90Q Q BLOZ Proposition A2c Hence each subgroup P S S for which ZQP 7E 0 is T conjugate to a subgroup P such that P O T Q Assume rst Q 4 T0 Then for each T centric subgroup P such that P O T Q NPT0PP 7E 1 and acts trivially on ZQP and so AOutPZQP 0 by Proposition 11b Thus WZQ 0 in this case If Q is not centric in T then for each T centric subgroup P such that P O T Q NPCTltQPP 7E 1 and acts trivially on ZQP and so AOutP ZQP 0 by Proposition 11b Again lgn Zd 0 in this case Now assume Q is centric in T but not TlT radical Set Q I E NTQ l Can 6 0p0utrQ QQ 7E 1 by assumption Let P g S be a T centric subgroup such that P O T Q Each element of AutfP leaves Q invariant since T lt1 S so we have a restriction map 0 AutfP AUEHQ gtlt AuWPQ and Ker0 is a p group by Go Corollary 533 Hence 0 1OpAutQ gtlt 1 is a normal p subgroup of AutfP Also P norrnalizes Q since it norrnalizes Q so 1 y QoQ 2 QQP since QQ 7E 1 and Q0 3 NSP since Q0P S Q by de nition For any cc 6 Q0Q Q NPP CI 6 0 1OpAutfQ gtlt 1 its class in OutP is nontrivial since P is 7 centric and cc g P and hence OpOutP 7E 1 Thus AOutPZQP 0 by Proposition 11b Since this holds for all T centric P with P T Q 3Q 0 in this case It remains to consider the case where Q 2 T0 Q is centric in T and Q 4 X This will be done in three steps In the rst two steps we show that is isomorphic to the higher limits of a certain functor over an orbit category of a group Only in Step 3 do we apply the assumption that Q 4 X Step 1 In this case we set Q QCsQ Then Q is T centric and Q Q T QAdoes not contain X Also Q is fully normalized in 7 since if Q is T conjugate to Q and fully norrnalized in 7 then there is some EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 17 04 E HomNSQNSQ with aQ Q see BLOZ Proposition A2c Hence 04c Q HT lt1 Nam and WMQM leQl Z leaQl Z leQ l since Q is fully normalized Let Q be the quotient functor of ZQ Where A ZQP if P contains a subgroup f conjugate to Q ZQUD 0 otherwise If QP 7E ZQP ie if QP 0 and ZQP 7E 0 then up to conjugacy P is f centric and P OT Q but P 4 Q Then NPQPP is a nontrivial p subgroup of Out P which acts trivially on ZQP so AOutP ZQP 0 in this case Thus Mei mWZQ 1 WP WP Step 2 Set F 011mg 5 011mg 6 SylpP and T 011mg for short Using the isomorphism 6 OEQltN ltQgtgt OsF HZ of Lemma 23 we see that T is strongly closed in S S with respect to P since no element of TQ can be N Q conjugate to any element of STQ By de nition each subgroup on which Q is nonvanishing is f conjugate to some P 2 Q such that POT Q In particular Q lt1 P since T lt1 S and so Q QCSQ lt1 P If P is any subgroup f conjugate to P which contains Q and 04 E lsoP P is any isomorphism then 04Q04PWTP HTHTQ and this is an equality since laQl Hence aQ Hypothesis of Proposition 24 is thus satis ed and hence may 0 v1 may lt2 0M1quot WP Set M1 ZltQgt Zltc gt HT and M0 ZltQgt mTo M H To and set M MlMO We regard these as ZltpP modules Let F OSPf p Zltp mod be the functor FP MlpMOP for all P S S This is clearly a subfunctor of HOM which satis es the relative norm condition in Propositions 13 and 14 Also for 18 BOB OLIVER P OutpQ g 5 ie Q lt1 P and P 2 PQ A g 2 FP if P m T Q 2 oxlrl P 3 P Q Qlt 0 otherwise and P m T Q if and only if P TQ Q if and only if P m T 1 Hence by Proposition 14 together with 1 and 2 aimyaw 0 implies WZQ 2 magma 0 3 0ltPgt 0M1 More precisely Proposition 14 only tells us that ZQ is acyclic But Q g T since it does not contain 35 so ZQS 0 and this implies OZQ 0 Step 3 By de nition of TT0 there are subgroups 1 QoTo S QlTo S S QnTO xTT07 all normal in TTO such that l91OTT0Qi21T07 QiTOW ll 1 4 for all i 1 n Let i S n Z 1 be the smallest integer such that Q 4 Qi Then QQ g Q so NQQQQ is nontrivial and is normal in NTQQ since Qi lt1 T Hence the xed subgroup NQQxelQlNTlQW IQ l x e NQQxQ lx7NTltQgtl Q is also nontrivial Fix some to E NQQQQ such that to E Qi and 50NTQ S Q In particular IQ E ZNTQQ 5 Since Q 2 QFl by assumption 91M795 P 1l S 91ZQT0793 P 1l S 91OTT0Ql21T07Q p ll 17 6 where the last equality holds by Now regard M additively as a ZltpPl module Then 6 translates to the statement that mltxgt91M12 xyrlngM 0 Also to E ZOutTQ by 5 so to E ZT by Hence mZltT91M 0 so WZQ 0 by 3 and this finishes the proof 1 Using Proposition 24 and Lemma 34 with T0 1 and T S it is not hard to show that for any saturated fusion system 7 over a pgroup S Z is acyclic if S contains a subgroup which is both centric and weakly T closed in S Since we are unable to prove directly that this holds for all 7 we instead filter Z via a maximal series of strongly T closed subgroups of S and use the following more general result Proposition 35 File a saturated fusion system 7 over a p group S and let T0 lt1 T lt1 S be a pair of subgroups strongly T closed in S Assume there is a subgroup XTO S TT0 which is centric in TTO and weakly TTO closed Then the quotient functor ZgZEO is acyclic EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 19 More generally let XfTT0 be the intersection of all subgroups QTO S TTO containing XTT0 such that Q is fully T normalized and flaw radical Assume there is a subgroup XTO S XfTT0 which is centric in TTO and weakly TTO closed Then the quotient functor ZgZEO is acyclic Proof Write Z ZgZ for short Assume XTO S XfTT0 is centric in TTO and weakly FTO closed In particular7 X is weakly F closed Let 335 be the functor on C73 de ned by setting7 for all P S S ZP if P 4 X 0 otherwise Note that since X is weakly closed7 if P 4 X7 then the same holds for all subgroups in its T conjugacy class We regard Z36 as a subfunctor of Z De ne ZQ as in Lemma 34 then WZQ 0 for all fully T norrnalized Q S T such that QTO 4 XTT07 or such that Q is not TlT radical In particular7 this applies to all Q 4 X Thus via the obvious ltration of 336 we get that 07 and hence that WZZae ENE 1 cm om Set Xquot Then Xquot is F centric and Xquot H T X since X is centric in T since XTO is centric in TTO If X S P S S and P 4 Xquot7 and P is centric7 then NxPPP OutxP is a nontrivial p subgroup of OutfP which acts trivially on ZP TZP To7 and so Alt0utfltPgt zltPgtgt o for such P Hence if we let F denote the functor FUD 7 ZP 25jS if P 2 Xquot for some P T conjugate to P T 0 otherwise then EMF 2 ENEle 2 WP WP Set M1 205 HT ZX and M0 295 m To Since X is weakly closed7 X dSf is both centric in S and weakly F closed So by Proposition 247 there is a functor i OpOutX Ab where ZP H T FUD36 ZP T0 2 M113 M0P for all PXquot E OutpX S OutSX E SylpOutX 20 BOB OLIVER and such that F 2 mm lt3 0plt0utflt3 WP Finally 0 for i gt 0 by Proposition 13 P E HOMlHOMO Together with 1 2 and 3 this nishes the proof of the proposition D It now remains to determine for each saturated fusion system 7 over a p group S p odd whether there always exists a sequence of strongly f closed subgroups for which Proposition 35 applies to each successive pair For convenience we de ne a subgroup Q S S to be universally weakly closed in S if for every saturated fusion system 7 over a p group S 2 S such that S is strongly f closed Q is weakly f closed in S Lemma 36 Fizz an odd prime p and a p group S Then a subgroup Q S S is uni versally weakly closed if for all P S S containing Q Q is a characteristic subgroup of P Proof Assume that Q S S is not universally weakly closed Then there exist a sat urated fusion system 7 over a pgroup S 2 S such that S is strongly f closed and such that Q is not weakly f closed in S By Alperin7s fusion theorem for saturated fusion systems BLOZ Theorem A10 there is a subgroup P S S containing Q and an automorphism oz 6 AutfP such that aQ 7E Q Set P T P Then aP P since T is strongly f closed and hence oz induces an automorphism of P S S which does not send Q to itself Thus Q is not a characteristic subgroup of P 1 The next lemma gives some simple conditions on the pgroup for being able to apply Proposition 35 For any p group S let JS denote Thompson7s subgroup the subgroup generated by all elementary abelian subgroups of S of maximal rank Proposition 37 Fizz an odd prime p and a p group S which satis es any of the following conditions a S Z JS b S contains a unique elementary abelian p subgroup E of maximal rank c SS is abelian Then there is a subgroup P S S which is centric and universally weakly closed in S Proof Write as S for short a Assume 35 Z JS Clearly JS is universally weakly closed in S however it need not be centric So instead consider the subgroup Q JSCSJS S S This is clearly normal and centric in S and is characteristic in any subgroup of S which contains it since JS is Thus Q is universally weakly closed in S by Lemma 36 It remains to check that Q S 35 Since JS S 35 every elementary abelian subgroup of S of maximal rank commutes with Z and thus contains 91Z since otherwise it would not be maximal Thus 91Z S ZJS so 91Z3 7Ql S lZJS7JSCsJSl 1 EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 21 Hence Q S X by Lemma 33 b If E S S is the unique elementary abelian subgroup of maximal rank then JS E and E S X by Lemma 32 The result thus follows from a 0 Assume that SX is abelian and that X is not universally weakly closed in S By Lemma 36 there is a subgroup P S S containing X and an automorphism 04 E AutP such that aX 7E X We claim that this is impossible Assume first that aZX f X and fix an element 9 E aZXX Then 91ZXg S aX since aX lt1 P and hence 91Z3 79l79lla3 79l1 since 9 E ZaX Set Q gXgt then 91ZXQ2 1 and Q lt1 S since SX is abelian Then Q E X by Lemma 33 and this contradicts the original assumption on 9 Now assume that aZX S X and thus that ZX S oflX and oz 1X 7E Fix a chain of subgroups 1Q0Q1quot39Qnx7 1 in Definition 31 Let i S n all normal in S hence in P which satisfy condition be such that Q i aX but QFl S aX Then 07191OPQ 71 91CP071Q 71 Z MCHf 91mg and hence 91Zx70471Qi P1l S 0471l91OPQi717Qtlpill 1 g by the assumption on the Qi Hence by Lemma 33 again X Qigt X which contra dicts the original assumption on Qi D We note the following immediate corollary to Propositions 37a and 35 Corollary 38 Let 7 be a saturated fusion system over ap group S and let 1 To 3 T1 3 3 T1 S be any sequence of subgroups which are all strongly f elosed in S Assume for all 1 S i S ls that X 1 Z JailTF1 Then gagT 0 for all 2 gt 0 El Corollary 38 motivates the following Conjecture 39 For any odd prime p and any p group P XP Z JP By Corollary 38 together with Lemma 25 in order to prove that Z is acyclic for all saturated fusion systems 7 it suf ces to prove Conjecture 39 for all p groups P which can occur as minimal strongly closed subgroups in saturated fusion systems However it seems to be very dif cult to prove or find a counterexample to this conjec ture even in this restricted form This also indicates that it will be very dif cult to find an example of a saturated fusion system 7 for which Z is not acyclic if there are any We finish this section with one other elementary result about the groups XS a result which will be useful in the next section 22 BOB OLIVER Proposition 310 Fizz an odd prime p and a p group S Then either rkZS 2 p or S S In particular S S ifrkS S p 71 Proof Set 3 S for short Assume rkZ 3 p71 and set E d f 91 lt1 S For each i Z 0 either E7Si1l llE7Sil7Sl E7Sil7 or ES 1 Since G k for Is 3 p 7 1 this shows that ESp71 1 and hence that S S by Lemma 33 1 4 THE ACYCLICITY OF 3G AT ODD PRIMES We are now ready to show for any nite group G and any odd prime p that all higher limits of ZG vanish when p is odd This will be based on the following proposition which gives for any nite group G a su icient condition for the acyclicity of ZG in terms of its simple composition factors When G is a nite group and S E SylpG set 3543 P g s l P 2 355 OpOutGP1NSPE Sy1NGP the intersection of all subgroups of S which contain S and are fully normalized and 7PG radical Proposition 41 For any prime p and any nite group G ZG is acyclic iffor each nonabelian simple group L which occurs in the decomposition series for G and any S E SylpL there is a subgroup Q S LS which is centric and weakly AutL closed in S In particular ZG is acyclic for each nite solvable group G Proof Fix a sequence of normal subgroups 1K0 K1 KnG such that each subquotient KiddK is a minimal normal subgroup of GKi We show that ZgwZgi is acyclic for each i Choose S E SylpG and set S S O K E SylpKi and 7 73G Assume that Q NQS S SHliS is centric in SHlS and that Q is fully 7 normalized ie NS Q E SylpNgQ If Q is FlSM radical then proj 1 OUtKHiltQgt OPOutKi1Q OPOutKilKiQ7 and hence Q is a radical p subgroup of KiddKi This proves that xfsi1Si S xKi1Kisi1Si So by Proposition 35 and Lemma 21 to prove that Zgi Zgi is acyclic it su ices to show XKlKS1S contains a subgroup Q which is centric and weakly GKi I closed in SHlSi To simplify notation we replace G by GK so K 1 and set K KML and P SHL E SylpK Thus K is a minimal normal subgroup of G and we must nd EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 23 Q S 35KP which is centric and weakly G closed in P This is clear if K has order prime to p ie Q P 1 Since K is a minimal normal subgroup it is a product of finite simple groups isomor phic to each other cf G0 Theorem 215 If K is an elementary abelian p group then K and is centric and weakly closed in K So assume K L7L where L is simple and nonabelian and n 2 1 We can choose this identification in a way such that P P 7L for some xed P E SylpL Then P P 7L see Definition 31 and 35KP LP 7L since each radical p subgroup of K splits as a product ofn rad ical p subgroups of L HMO Proposition 16ii By assumption there is a subgroup Q S LP which is centric and weakly AutL closed in P Then Q d f Q 7L is centric in P and Q S KP It remains to show that Q is weakly AutK closed in P and hence weakly G closed in P Assume otherwise assume there is oz 6 AutL such that Q 7E aQ S P The n factors L are the unique minimal normal subgroups of L so each automorphism of L7L permutes these factors and hence AutL AutL 2 27 Thus oz o o 041 ozn for some 04 E AutL and some a E E7 regarded as an automorphism of Lquot and Q 7E ogQ S P for some i Which contradicts the assumption that Q is weakly AutL closed in P D We now prove that all finite nonabelian simple groups L satisfy the condition in Proposition 41 for any odd prime pllLl and any S E SylpL there is a subgroup Q S LS or Q S which is centric and weakly AutL closed in S We first consider some cases where this can be shown using Proposition 37b Proposition 42 Assume p is odd and let L be a simple group which is either an alternating group or a group of Lie type in characteristic di erent from p Then for S E SylpL JS S S and hence there is a subgroup Q S S which is centric and weakly AutL closed in S Proof If L 2 A7 then S contains a unique elementary abelian p subgroup E of max imal rank generated by a product of disjoint p cycles cf GL 10 5l Hence JS E S S by Lemma 32 and the result follows from Proposition 37a or W Now assume that L is a simple group of Lie type in characteristic p lf rkpL S 2 then S S by Proposition 310 So assume rkpL gt 2 Then by GL 10 21 each Sylow p subgroup of L contains a unique elementary abelian p subgroup of maximal rank and the result follows from Lemma 32 and Proposition 37ab again Note that all ofthe exceptional cases listed in GL 7 the simple groups A2 g 241211 G2 g 3D4g and 2F401 when p 3 7 have 3 rank at most 2 by GL 10 22 and Tables 101 and 102 D We next consider simple groups of Lie type in characteristic p We first summarize the structures in these groups which will be needed referring to Ca as a general reference Assume first that L is a Chevalley group L Gg where G is one ofthe groups A7 B7 etc defined over the finite field qu g pa For example Ang PSLn1qu Let lt1 Q V denote the root system of G where V is a real vector space Let ILr be the 24 BOB OLIVER set of positive roots thus I ir l r E Char Let I denote the set of primitive roots an lR basis of V To each root r E 1 corresponds a root subgroup X E qu in L 3a Then U d f 11914 X is a Sylow p subgroup of L Also B dSf NLU UgtltH the Borel subgroup where H is the subgroup of diagonal elements and has order prime to p Set N NLH then W E NH is the Weyl group of G and of the root system CD For example when L Anq PSLn1q then we can take VcElRquot1lZai0 ltIgteiiejli7 j where 61 76n1 is the standard basis of Rn ltIgteiiejliltj and 6i76i1 Then Xere is the subgroup of matrices which have 17s along the diagonal and are zero elsewhere except at entry ij U is the group of upper triangular matrices with 17s along the diagonal and B is the group of all upper triangular matrices Diagonal elements are represented by diagonal matrices N is the image of the subgroup of monomial matrices and W 2 27 We next x the notation for the twisted groups tGq Let T E AutV I 1 be an automorphism of the root system ofG of order t Set a 73093 6 AutltGq where iquot is induced by T and 925 is induced by go 6 Autqu of order t In most cases 1 qt and so qu is the xed sub eld of the automorphism go In all cases 7 E AutltIgt can be seen as an automorphism ofthe Dynkin diagram andt 23 Also 0XT X for each r E I and thus 0 leaves invariant the subgroups U H and N The twisted group L 3a is de ned to be the commutator subgroup of Gq or alternatively as the subgroup of Gq generated by U 5 and the analogous subgroup for the root groups of negative roots lts Borel subgroup is de ned to be B NLU U gtltHCr O L Proposition 43 Assume p is odd let L be a simple group ofLie type in characteristic p and x S E SylpL Then LS is weakly AutL closed in S Proof Write L tltGrq where q pa possibly t 1 We use the above notation In particular ltIgt denotes the set of positive roots and U E SylpL is the product of the root subgroups X for r E In For each T invariant subset J Q I consider the subgroups U H X In particular U0 U and U1 1 We claim that the following statement holds The subgroups U for T invariant subsets J Q I are the only subgroups of S U5r which are radical p subgroups ofL and they are all weakly L closed 1 in U 539 By a theorem of Borel and Tits see the corollary in BW every radical p subgroup of L is conjugate to one of the subgroups Uj39 and by Gr Lemma 42 if Ur is L conjugate to U for J J S I then J J But since we need to know that each EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 25 radical p subgroup of L contained in S is actually equal to one of the Uj39 we modify Grodal7s proof to show this Assume that P S S U5r is L conjugate to Uj39 We show that P Uj39 this proves that U3 is weakly L closed in S and hence using proves Since L S U 7 Ucr Ca Proposition 822 and since U lt1 U we have P useUj39x 1u 1 for some u E U5r and to E N and P Uj39 if and only if cUj39a 1 Uj39 So we can assume u 1 and P cUj39a 1 Now to permutes the root subgroups via the action ofw 05H 6 lVT on CD and so wlt1gtltJgt Q 1 Write A ltIgtltJgt for short this is closed in the sense that any r 6 CD which is a positive linear combination of elements of A also lies in A So wA has the same property This implies that all primitive roots for the system wiltJgt lt1gt are primitive roots in In and thus that wA ltIgtltJ gt for some J Q I After replacing w by its product with some element in the Weyl group of iltJ gt we can assume that it sends positive roots to positive roots and hence must be the identity So J J and P Uj39 Thus by 1 if Q S S is a radical p subgroup of L then Q Uj39 for some J Hence LS is the intersection of the subgroups Uj39 which contain Also U H UJ UJUJ7 since each element of U has a unique decomposition as product of elements ofthe root groups taken in an appropriate order Ca Theorem 533ii So any intersection of subgroups U3 is again of the same form and thus LS Uj39 for some T invariant subset J Q I Hence LS is weakly L closed in S by 1 again Each automorphism of L is congruent mod lnnL to some oz 6 AutL which sends S to itself and oz permutes the radical p subgroups of L contained in S and sends L to itself Thus each coset in OutL contains an automorphism which sends LS to itself and LS is weakly AutL closed in S since it is weakly L closed D In fact when L is simple of Lie type in characteristic p and p is odd as usual then for S E SylpL S S except when p 3 and and L E Cnq E PSp2nq n 2 2 01 L g 2AultQgt g PSUu1q2 n 2 We are now ready to consider the sporadic groups Proposition 44 Assume p is odd let L be a sporadic simple group and x S E SylpL Then there is a subgroup Q S XS which is Centric and weakly AutL closed in S Proof lfp Z 5 then rkpL lt p by GLS 56 and so S S by Proposition 310 So assume p 3 We consider several different cases a If L is one of the groups M11 M12 M22 M23 M24 J1 J2 J4 HS He or Ru then rk3L S 2 by GL p123 and so S S by Proposition 310 b Assume L is one of the groups J3 003 002 MCL Suz Ly O N or F5 In all of these cases S contains a normal elementary abelian 3 subgroup E of index 3 9 S 2 E by Lemma 32 so SS is abelian and S is weakly AutL closed by Proposition 37c More precisely there are the following inclusions of index prime to 3 26 BOB OLIVER L J3 003 002 M CL Suz Ly 0 N F5 E 0 03 0g cg 03 03 0g cg CgNOS 2 X M11 M10 M10 M11 2 X M11 order order See GL 5l for references In fact in all ofthe above cases E is the unique elementary abelian subgroup of S of maximal rank 0 Assume L g 001 By Cu p424 S is contained in a semidirect product C N2M12 and the elementary abelian subgroup 03 is generated by all elements of order 3 in S which lie in the conjugacy class 3A Thus S contains a unique elementary abelian 3 subgroup E of maximal rank and hence CSE S S is centric and weakly AutL closed in S by Proposition 37b e Assume L F3 By Ho or Pa see also As 142 there are subgroups DKMamp all normal in S such that K Cg is abelian CSK K and MK D ZM E Cg Also MK E Cg and NLDM E GL23 Thus M S S so rkZS S rkZM 2 and hence S S by Proposition 310 In the remaining cases for a p group R we use the notation ZnR lt1 R Z1R ZR and ZnRZn1R ZRZn1R The group R is of class n ifR ZnR g Zn1R Also following the notation of As we say that a subgroup H S L is of type H mtmt1 m1 if upon setting R OpH then HR E H ZR E Cg and ZRZi1R E Cg for all i We restrict for simplicity to the case where ZRZi1R is elementary abelian for all d Assume L E Fizz Then L contains a subgroup L0 2 973 with index prime to 3 cf As p26l Regard L0 as acting on V l g let W Q V be a maximal isotropic subspace dimW 3 and let H 3 L0 be the subgroup of elements which leave W invariant Then LozH is prime to 3 and hence we can assume that S S H One easily checks that H is of type SL3333 where R dSf 03H is the subgroup of elements whose restriction to W and to VWL is the identity and ZR is the subgroup of elements whose restriction to WL and to V W are the identity Also ZR RR and HR E SL33 acts on ZR as the group of 3 gtlt 3 antisymmetric matrices From this one quickly sees that R S If we set SOR ZSR E 03 then ZRS0S0 1 so S0 3 Hence rkZS S rkZS0 2 so S S by Proposition 310 Alternatively one can show that S contains a unique elementary abelian subgroup of maximal rank 5 and then apply Proposition 37b f Assume L Fizg or F2 By As p 33 there is an inclusion Fizg 3 F2 with index prime to 3 so these groups have isomorphic Sylow 3 subgroups By As p 27 amp 2087 209 there is a subgroup H S Fizg of index prime to 3 and of type SL333313 We can thus assume R d f 03H g S g H Also 2212 a cg 2312 OHZ2R 2 Q gtlt 03 where ZQ Q Q ZR and QZQ 2312 Z2R 2 03 and RZ3R acts on Z2R as the group of all automorphisms which are the identity on ZR and on Z2RZR Since Z3R Z3R Z3R 1 and Z2R RR 1 we see that R S S and hence S S by the same argument as was used for L E Fizz EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 27 g Assume L Fig4 By As pp 29 amp 2107211 there is a subgroup H S Fi 24 ofindex prime to 3 and oftype A5 gtlt SL23842 and we can assume R d f 03H S S S H Also RZ2R acts on Z2R C as the group of all automorphisms which are the identity on ZR and on Z2RZR and the actions of SL23 S HR on ZR and of A5 3 HR on Z2RZR E Cg are faithful Thus Z2R is centric in H and Z2RRR 1 It follows that R S S and hence since rkZR 2 that S S by Proposition 310 h Assume L F1 By As pp 35 amp 2117212 there is a subgroup H 3 F1 of index prime to 3 of type GL23 gtlt M111052 We can thus assume that R dSf 03H S S S H Also RZ2R acts on Z2R C as the group of automorphisms which are the identity on ZR and on Z2RZR and the actions of GL23 S HR on ZR and of M11 3 HR on Z2RZR are faithful Thus Z2R is centric in H and Z2RRR 1 It follows that R S S and hence by Proposition 310 since rkZR 2 that S S D We are now ready to prove Theorem A Theorem 45 For any odd prime p and any nite group G ZG is acyclic Proof Let L be a finite simple group and fix S E SylpL lf L is an alternating group or of Lie type in characteristic 74 p then by Proposition 42 there is a subgroup Q S S which is centric and weakly AutL closed in S If L is of Lie type in characteristic p then LS itself is centric and weakly AutL closed in S by Proposition 43 If L is a sporadic group then there is a subgroup Q S S which is centric and weakly AutL closed in S by Proposition 44 The theorem now follows from Proposition 41 together with the classification theorem for finite simple groups B REFERENCES As M Aschbacher Overgroups of Sylow subgroups in sporadic groups Memoirs Amer Math Soc 343 1986 BLO 1 C Broto R Levi amp B Oliver Homotopy equivalences of p completed classifying spaces of nite groups preprint BLOZ C Broto R Levi amp B Oliver The homotopy theory of fusion systems preprint BW N Burgoyne amp C Williamson On a theorem of Borel and Tits for nite Chevalley groups Arch Math Basel 27 1976 4897491 Ca R Carter Simple groups of Lie type Wiley 1972 Cu R Curtis On subgroups of 0 11 Local structure J Algebra 63 1980 4137434 GL D Gorenstein amp R Lyons The local structure of nite groups of characteristic 2 type Memoirs Amer Math Soc 276 1983 GLS D Gorenstein R Lyons amp R Solomon The classi cation of the nite simple groups nr 3 Amer Math Soc surveys and monogr 40 3 1997 Go D Gorenstein Finite groups Harper amp Row 1968 Gr J Grodal Higher limits via subgroup complexes preprint HM G Higman amp J McKay On Janko s simple group of order 50232960 Bull London Math Soc 1 1969 89794 Ho D Holt The triviality of the multiplier of Thompson s group F3 J Algebra 94 1985 3177323 28 BOB OLIVER JM S Jackowski amp J McClure Homotopy decomposition of classifying spaces Via elementary abelian subgroups Topology 31 1992 1137132 JMO S Jackowski J McClure amp B Oliver Homotopy classi cation of selfmaps of BC Via G actions Annals of Math 135 1992 1847270 MP J Martino amp S Priddy Unstable homotopy classi cation of BGQ Math Proc Cambridge Phil Soc 119 1996 119137 Pa D Parrott On Thompson s simple group J Algebra 46 1977 3897404 Pu Ll Puig unpublished notes LAGA INSTITUT GALILEE Av JB CLEMENT 93430 VILLETANEUSE FRANCE E mail address bonmathunivpari513 fr

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