FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS CARLES BROTO AND JESPER M. MØLLER Abstract. We describe the spaces of homotopy fixed points of unstable Ad* *ams operations acting on p-compact groups and also of unstable Adams operations twisted* * with a finite order automorphism of the p-compact group. We obtain new exotic p-local finite* * groups. Contents 1. Introduction * * 1 2. p-compact groups * * 7 3. p-local finite groups * * 8 4. Recognition of classifying spaces of p-local finite groups * * 13 5. Homotopy fixed points p-compact groups * * 21 6. Homotopy fixed points of twisted unstable Adams operations * * 39 7. General structure of finite Chevalley versions of p-compact groups * * 44 8. Cohomology rings * * 49 9. Invariant theory * * 51 10. Finite Chevalley versions of Aguad'e exotic p-compact groups * * 60 11. Finite Chevalley versions of generalized p-adic Grassmannians * * 69 References * * 75 1. Introduction The main purpose of this paper is the description of the structure of the spa* *ces of ho- motopy fixed points of unstable Adams operations _q acting on p-compact groups * *and also of unstable Adams operations twisted by automorphisms of p-compact groups ø_q. * * In the classical case, where ø is an automorphism of a compact connected Lie group G a* *nd _q an unstable Adams operation of exponent a prime power q, coprime to p, results of * *Quillen [54] and Friedlander [27, 28], show that the space of homotopy fixed points is, up t* *o p-comple- tion, the classifying space of the finite twisted Chevalley group fiG(q). Here * *and throughout, p-completion is understood in the sense of Bousfield-Kan [8]. We will show that* * in case of ______________ 2000 Mathematics Subject Classification. 55R35, 55P15, 55P10. Key words and phrases. Chevalley group, p-compact groups, p-local finite grou* *ps. C. Broto is partially supported by MCYT grant BFM2001-2035. Both authors have been partially supported by the EU grant nr. HPRN-CT-1999-0* *0119. 1 2 CARLES BROTO AND JESPER M. MØLLER exotic p-compact groups we obtain classifying spaces of p-local finite groups, * *in some cases, new exotic examples. The concept of p-compact group was introduced by Dwyer and Wilkerson in [22] * *as a p-local homotopy theoretic analogue of compact Lie group. A p-compact group is * *a connected p-complete space BX where X = BX is Fp-finite; that is, H*(X; Fp) is finite. * *We will usually refer to a p-compact group simply as X. BX is then understood as its c* *lassifying space, a concrete loop space structure imposed in the underlying space X. If G * *is a compact connected Lie group, then the p-completion of its classifying space BG^pis a p-* *compact group. A p-compact group that cannot be obtained in this way is called exotic. We pos* *tpone till section 2 a more detailed description of the theory of p-compact groups. The concept of p-local finite group has been recently introduced in [11] as a* *lgebraic objects that are modeled on the p-local structure of finite groups and as such they hav* *e classifying spaces which are p-complete spaces. In turn, the classifying space of a p-loca* *l finite group determines its algebraic structure. We refer to section 3 for the precise defin* *ition and main properties of p-local finite groups. Theorem A. Let p be an odd prime. If X is a 1-connected p-compact group, q is a* * prime power, coprime to p, and ø is an automorphism of X of finite order coprime to p* *, then the space of homotopy fixed points of BX by the action of ø_q, denoted B fiX(q), is* * the classifying space of a p-local finite group. By analogy with the classical case, we will call p-local finite Chevalley gro* *up of type X to any finite p-local finite group X(q) obtained in Theorem A, with classifying sp* *ace BX(q). If X is obtained as p-completion of a compact Lie group G, BX(q) is homotopy eq* *uivalent to the p-completed classifying space of the Chevalley group G(q). For a prime number p, a prime power q, coprime to p, and a compact connected * *Lie group G, Friedlander shows a cohomological fibre square that becomes the homotopy pullba* *ck diagram f ^ BfiG(q)^p___________//_BGp f || || fflffl|(1,fi_q) fflffl| BG^p___________//BG^px BG^p after p-completion, and where fiG(q) is the twisted Chevalley group over Fq of * *type G and is the diagonal map. Unstable Adams operations can be defined over p-compact gr* *oups (see section 2), hence, following the above pattern, if X is a connected p-compact g* *roup and ø_q a twisted Adams operation, then the classifying space BfiX(q) is defined by the* * homotopy pullback square f BfiX(q)____________//BX f|| || |fflffl(1,fi_q) fflffl| BX __________//_BX x BX . This pullback square provides an alternative definition of the space of homotop* *y fixed points by the action of ø_q on BX (see section 6 for details). Our arguments concentrate in the exotic p-compact groups at odd primes, and n* *aturally break into two distinguished steps. One deals with actions of finite groups of* * order not divisible by p on p-compact groups and the results obtained have an independent* * interest FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 3 by their own. The other step deals with the action of unstable Adams operations* * _q where q 1 mod p and it is the one leading to the new exotic examples of p-local fin* *ite groups. Group actions will be understood in the weak sense of proxy actions; that is,* * we will say that an action of a group G on a space B is a fibration B ! BhG ! BG, [22].* * The total space BhG is referred to as the homotopy quotient and the space of homoto* *py fixed points BhG is the space of sections. When we specialize to p-compact groups X,* * an outer action of G is a homomorphism æ: G ! Out(X), where Out(X) is the group automorp* *hisms of the p-compact group X, in other words, homotopy classes of self-equivalences* * of BX. It turns out that if G has finite order coprime to p, then an outer action on a co* *nnected p- compact group X determines a unique action and the space of homotopy fixed poin* *ts is again a connected p-compact group: Theorem B. Let X be a connected p-compact group. If G is a finite group of orde* *r coprime to p and æ: G ! Out(X) an outer action, then (1) æ lifts to a unique action of G on X. (2) XhG is a connected p-compact group with H*(BXhG; Qp) ~=S[QH*(BX; Qp)G], the* * sym- metric algebra generated on the coinvariants QH*(BX; Qp)G. (3) (Harper splitting) XhG ! X is a p-compact group monomorphism, and X ' XhG x X=XhG , thus, in particular, X=XhG is an H-space. (4) If H*(BX; Fp) is a polynomial ring, then H*(BXhG; Fp) is also a polynomial * *ring. Here and throughout, H*(-; Qp) stands for H*(-; Zp) Q, and QH*(BX; Qp) deno* *tes the module of the indecomposables in H*(BX; Qp). This result is proved as a co* *rollary of Theorem 5.2 that establishes, more generally, that if G has order prime to p an* *d acts on a 1-connected p-complete space B then there exists a homotopy equivalence B -'! * *BhG x Fib (BhG ! B). Some interesting cases to which Theorem B applies are F4 at the prime 3 and E* *8 at the prime 5. In the first case, Friedlander's exceptional isogeny of F4 at the prim* *e 3 gives rise to an automorphims of order 2 and the homotopy fixed points p-compact group F4hC2i* *s the p- compact group X12= DI (2) whose cohomology realizes the Dickson algebra H*(BX12* *; F3) ~= F3[x12, x16] (subscripts indicate degrees). This case was already considered i* *n our previous work [13]. In thepsecond_case, a cyclic group of order 4 generated by the unst* *able Adams operation _i, i = -1, acts on E8. The homotopy fixed points p-compact group E* *hC48is the p-compact group X31 corresponding to the reflection group number 31 on the Clar* *k-Ewing list, and its mod 5 cohomology ring is H*(BX31; F5) = F5[x16, x24, x40, x48] (s* *ee 5.15). It turns out that X12 = DI(2) and X31 are the two exotic p-compact groups ori* *ginally constructed by Zabrodsky [63], and later included in the Aguad'e family [1]. Za* *brodsky used invariants by the action of these same automorphisms but only at the level of h* *omotopy groups of BF4 and BE8, respectively. The corresponding splittings are F4 ' DI (2)xF4 =DI (2) at the prime 3, first* * discovered by Harper [31], and E8 ' X31x E8=X31, that was obtained by Wilkerson [60]. Other e* *xamples appear in 5.3. Our next Theorem provides the necessary arguments in order to deduce the gene* *ral case of Theorem A from the two steps. 4 CARLES BROTO AND JESPER M. MØLLER Theorem C. Let p be an odd prime and X a connected p-compact group, ø an automo* *rphism of X of order prime to p and _q an unstable Adams operation, then: (1) If X is 1-connected and q 1 mod p, q 6= 1, then B fiX(q) ' BXhfi(q). (2) If q0 is another p-adic unit such that q q0mod p and p(1 - qr) = p(1 -* * q0r), where r is the order of q mod p, then BX(q) ' BX(q0). Since we can decompose a p-adic unit q as q = iq0 where i is a (p - 1)st-root* * of unity and q0 1 mod p, part (1) of the above Theorem will reduce the question of computi* *ng BX(q) to the case where q 1 mod p which turns out to be easier to handle in abstract c* *alculations and concrete examples. The second part of the Theorem tells us that BX(q) does only* * depend on the order r of q mod p and the p-adic valuation p(1 - qr), so we can change th* *e exact value of q at our convenience if we keep those parameters fixed. In particular, Theor* *em A, could more generally be stated for unstable Adams operations _q, with q a p-adic unit* * of infinite multiplicative order. Part (2) of Theorem C also explains the often observed fact that finite Cheva* *lley groups G(q) and G(q0) have same cohomology ring or identical p-local structure when q and q* *0 are prime powers, with qr q0r 1 mod p and p(1 - qr) = p(1 - q0r), for some r, 1 r * * p - 1. Our second step deals with the action of unstable Adams operations _q of expo* *nent q 1 mod p, q 6= 1, on connected p-compact groups X. The effect now is opposite in* * some sense to the case of finite groups of order prime to p. The spaces of homotopy fixed * *points BX(q) have the same p-rank as the original p-compact groups X, but the maximal tori T* * n' ((S1)n)^p are cut down to finite maximal tori T`n~=(Z=p`)n, ` = p(1-q) (the p-adic valua* *tion of 1-q), keeping, though, the same Weyl group (see 7.5, 7.6). We restrict our calculations in this part to p-compact groups for which the m* *od p co- homology ring H*(BX; Fp) is a polynomial ring. For simplicity, we will refer t* *o them as polynomial p-compact groups. At odd primes, these include all irreducible exoti* *c examples and will therefore suffice to our purposes. Theorem D. Let q be a p-adic unit such that q 1 mod p, q 6= 1. If X is an ir* *reducible 1-connected polynomial p-compact group, then BX(q) is the classifying space of * *a p-local finite group. Proof.The proof is based on the classification theorem for p-compact groups at * *odd primes [6]. The irreducible 1-connected p-compact groups with polynomial cohomology are (1) BSU(n)^p(family 1 in the Clark-Ewing list), (2) the Quillen generalized Grassmannians (family 2a in the Clark-Ewing list), (3) the non-modular p-compact groups, and (4) the Aguad'e family (numbers 12, 29, 31, and 34 in the Clark-Ewing list, at * *primes 3, 5, 5, and 7, respectively). The different cases are solved in 11.1, 11.4 , 9.7, and 10.3, respectively. In cases (1) and (3) we always obtain that BX(q) is the p-completed classifyi* *ng space of a finite group. The other two families contain the new exotic examples of p-local* * finite groups. A complete description of the structure of the p-local finite groups Xi(q), i* * = 12, 29, 31, 34, is obtained in section 10. For X12(q), p = 3, we obtain that if ` = 3(1 + 22n+* *1), then BX12' B(2F4(22n+1))^3(Example 10.7). For X31(q), p = 5, it turns out that if ` = 5(* *1 + 24m+2), FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 5 then BX31(q) ' BE8(22m+1)^5(Example 10.8). In particular, we can obtain the p-c* *ompact groups X12 and X31 as telescopes of a sequence of p-completed classifying space* *s of finite groups (see 10.9): BX12' hocolimB(2F4(22n+1))^3, n BX31' hocolimBE8(22m+1)^5. m The cases BX29(q) and BX34(q) at primes 5 and 7, respectively, are classifying * *spaces of exotic p-local finite groups (Example 10.6). Family 2a in the Clark-Ewing list consists of groups G(m, r, n) with r|m|(p -* * 1), defined as the pseudoreflection groups in GLn(Qp) generated the permutation matrices and t* *he diagonal matrices diag(a1, a2, . .,.an) with aim = 1 and (a1a2. .a.n)n=r= 1. We denote X* *(m, r, n) the p-compact group of rank n with Weyl group G(m, r, n). We also prove that BX(m, * *r, n)(q) is the classifying space of an exotic p-local finite group provided n p and r* * > 2 (Proposi- tion 11.5). q> Proof of Theorem A. We defined B fiX(q) = BXh h B fiX(q) = BXh Xh BXh and then, also, BXh n, as restrictions of Adams operations defined on BU. Then, ex* *tended by Wilkerson to all compact Lie groups [60]. In [32] it is shown that p-completed* * classifying spaces of compact connected Lie groups admit unstable Adams operations _q of ex* *ponent a p-adic unit q 2 Z*p. This is extended to p-compact groups for odd primes p in [* *48]. A p-compact group BX is irreducible if the reflection group (L Qp, W ) = (L Q* *p, W )(X) is irreducible. In this case, by Schur's lemma, AutZp[W](L) = Zxpconsists of sc* *alars only so that (1) takes the form 1 ! Zxp=Z(W ) ! NGL(L)(W )=W ! Outtr(W ) . * * (2) The group Out tr(W ) turns out to be trivial for most of the simple reflection * *groups, and in all known examples it consists of elements that lift to finite order element* *s in Out(X) = NGL(L)(W )=W . In those cases, Out(X) consists only of twisted Adams operations* * ff_q. The list of irreducible Zp-reflection groups can be derived [48, 11.18] from * *the Clark- Ewing [16] list of irreducible Qp-reflection groups. The simple p-compact grou* *ps, corre- sponding to the irreducible reflection groups, besides the Lie examples, are th* *e non-modular p-compact groups (including the Sullivan spheres), where p does not divide |W |* *, the Aguad'e p-compact groups [1], where p divides |W | exactly once, and the generalized Gr* *asmannians [52] [48, x7] corresponding to the second infinite family in the Clark-Ewing cl* *assification table. We refer to the surveys [42, 51, 18] for more information on p-compact groups. 3.p-local finite groups The concept of p-local finite group has been introduced in [11] (see also [12* *]). A p-local finite group is a triple (S, F, L) where S is a finite p-group, F a saturated f* *usion system over S, and L a centric linking system associated to F. We will state here agai* *n all necessary definitions for the convenience of the reader. A fusion system over a finite group S consists of a set Hom F(P, Q) of monomo* *rphisms for every pair of subgroups P , Q of S, such that it contains at least those monomo* *rphisms induced by conjugation by elements of S and all together form a category where every mo* *rphism FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 9 factors as an isomorphism followed by an inclusion. A fusion system is saturate* *d if it satisfies certain additional axioms formulated by L. Puig (see [11, x1] or the original s* *ource [53]). Two subgroups P , P 0of S are called F-conjugate if there is an isomorphism between* * them in F. Definition 3.1. Let F be a fusion system over a p-group S. (1) A subgroup P S is fully centralized in F if |CS(P )| |CS(P 0)| for all * *P 0 S which is F-conjugate to P . (2) A subgroup P S is fully normalized in F if |NS(P )| |NS(P 0)| for all P* * 0 S which is F-conjugate to P . (3) F is a saturated fusion system if the following two conditions hold: (i)For each P S which is fully normalized in F, P is fully centralized * *in F and AutS(P ) is a Sylow p-subgroup of AutF (P ). (ii)If P S and ' 2 Hom F(P, S) are such that 'P is fully centralized, an* *d if we set N' = {g 2 NS(P ) | 'cg'-1 2 AutS('P )}, _ _ then there is ' 2 Hom F(N', S) such that '|P = '. A subgroup P of S is centric if CS(P 0) P 0for every subgroup P 0 S which * *is F- conjugate to P . Fc denotes the full subcategory whose objects are the centric* * subgroups of S. Definition 3.2. 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 F-centric subgroups of S,* * together with a functor ß :L -! Fc, and distinguished monomorphisms ffiP:P -! AutL(P ) for each F-centric subgroup * *P S, which satisfy the following conditions. (A) ß is the identity on objects and surjective on morphisms. More precisely, * *for each pair of objects P, Q 2 L, Z(P ) acts freely on MorL (P, Q) by composition (upon id* *entifying Z(P ) with ffiP(Z(P )) AutL(P )), and ß induces a bijection ~= MorL (P, Q)=Z(P ) -! Hom F(P, Q). (B) For each F-centric subgroup P S and each g 2 P , ß sends ffiP(g) 2 Aut * *L(P ) to cg 2 AutF (P ). (C) For each f 2 Mor L(P, Q) and each g 2 P , the following square commutes in* * L: f P ____//_Q ffiP(g)|| |ffiQ(i(f)(g))| fflffl|fflffl|f P ____//_Q . The classifying space of the p-local finite group (S, F, L) is defined as the* * p-completion |L|^p of the nerve of the category L. The classifying space determines the p-local fi* *nite group in the sense that two p-local finite group are isomorphic if and only if they have* * homotopy equivalent classifying spaces. Actually, the complete structure of a p-local fi* *nite group can be recovered from its classifying space by homotopy theoretic methods. Finite groups are the main source of examples and motivation for p-local fini* *te group theory. If G is a finite group and S a Sylow p-subgroup, the monomorphisms from* * P to Q 10 CARLES BROTO AND JESPER M. MØLLER inducedfbyiconjugation in G, Hom G (P, Q) ~= NG(P, Q)=CG(P ), where NG(P, Q) = * * x 2 G fixP x-1 Q , form a saturated fusion system over S, FS(G). Furthermore, w* *e define LcS(G) as the category with objects all subgroups of S which are p-centric in G* *, and morphisms Hom L (P, Q) ~=NG(P, Q)=C0G(P ), where C0G(P ) is the p0-complement in CG(P ) o* *f the center of P , CG(P ) = Z(P ) x C0G(P ), which is well defined because P is p-centric. * * LcS(G) is a centric linking system associated to FS(G), and (S, FS(G), LcS(G)) is a p-local* * finite group with classifying space |LcS(G)|^p' BG^p(see [10, 11]). We call exotic those p-* *local finite groups that are not obtained in this way from any finite group. Examples of exo* *tic p-local finite groups are already shown in [11]. Recently, Levi and Oliver have obtain* *ed a family of exotic 2-local finite groups, B Sol(q) [34], based on fusion systems origina* *lly described by Solomon [58]. Definition 3.3. (a) For any saturated fusion system F over a p-group S, and any* * P S, fully centralized in F, the centralizer fusion system CF (P ) over CS(P ) is de* *fined by setting fi 0 0 Hom CF(P)(Q, Q0) = ('|Q) fi' 2 Hom F(P Q, P Q ), '(Q) Q , '|P = IdP for all Q, Q0 CS(P ). (b) For a p-local finite group (S, F, L) and P S is fully centralized in F, w* *e define the category CL(P ) whose objects are CF (P )-centric subgroups Q CS(P ) and where 0 fi 0 Mor CL(P)(Q, Q0) = ' 2 Hom L(P Q, P Q ) fiß(')|P = IdP, ß(')(Q) Q . It is proved in [11, x2] that if (S, F, L) is a p-local finite group and P * *S is fully centralized in F, then (CS(P ), CF (P ), CL(P )) is a p-local finite group. In [34] Levi and Oliver have obtained sufficient conditions for a fusion syst* *em to be sat- urated. We reproduce here their result for the convenience of the reader. We * *will write CF (x) = CF () for x 2 S. Proposition 3.4 ([34]). Let F be any fusion system over a p-group S. Then F is * *saturated if and only if there is a set X of elements of of order p in S such that the fo* *llowing conditions hold: (a) Each x 2 S of order p is F-conjugate to some element of X. (b) If x and y are F-conjugate and y 2 X, then there is some _ 2 Hom F(CS(x), C* *S(y)) such that _(x) = y. (c) For each x 2 X, CF (x) is a saturated fusion system over CS(x). Let F be a fusion system over a finite p-group S. A subgroup P S is called * *radical in F if OutF (P ) = AutF (P )= Inn(P ) is p-reduced, namely, it does not contain non* *-trivial normal p-subgroups. Alperin's fusion theorem for saturated fusion systems [11, A.10] establishes * *that morphisms in a saturated fusion system are composites of automorphisms of fully normalize* *d, centric, and radical subgroups of the system, or restrictions of those. Hence in order * *to describe a saturated fusion system F over a finite p-group S it is enough to describe Aut * *F(Qi) for a set Q1, . .,.Qr of fully normalized representatives of F-conjugacy classes of c* *entric, radical subgroups of S in F. This motivates the next construction. If F0 is a fusion system over S, Q1, . .,.Qr are subgroups of S and i is a g* *roup of automorphisms Inn(Qi) i Aut(Qi), for each i, then we denote by FQi( i) the* * fusion FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 11 system over Qi whose morphisms are restrictions of elements of i, and define F = the fusion system over S whose morphisms are composites of morphisms belonging * *to any of the generating fusion systems (cf. [11, x9]). Thus, in particular, if F is a saturated fusion system over a finite p-group * *S and Q1, . .,.Qr is a set of fully normalized representatives of F-conjugacy classes of centric,* * radical subgroups of S in F, then F = . Let G be a finite group and S a Sylow p-subgroup. Let FS(G) the fusion syste* *m of G over S. P is centric in FS(G) if and only if it is p-centric in G. A p-subgro* *up P of G is called p-radical if it is the maximal normal p-subgroup of NG(P ), P = Op(NG* *(P )), or, equivalently, if NG(P )=P is p-reduced. Notice though, that being radical in F* *S(G) means that OutFS(G)(P ) ~=NG(P )=P CG(P ) = OutG (P ) is p-reduced. If P S is centric and radical in FS(G), then it is p-centric and p-radical * *in G. Assume that P is not p-radical in G, then there is another p-subgroup Q with P / Q / N* *G(P ) and Q 6= P . Since P is p-centric, CG(P ) = Z(P ) x C0G(P ), where C0G(P ) is a p0* *-group, hence also C0G(P ) \ Q = 1, so, therefore P / Q / NG(P )=C0G(P ) and Q=P / NG(P )=P C* *0G(P ) = NG(P )=P CG(P ), hence OutG (P ) is not p-reduced. The converse it is not alway* *s true. We end this section by describing the fusion systems of GLp(q) and SLp(q) ove* *r the re- spective Sylow p-subgroups, where p is a prime number and q is a prime power q * * 1 mod p. This will be useful in later sections calculations. Example 3.5. We will describe the fusion system of GLp(q) over a Sylow p-subgro* *up, for p a prime and q a prime power such that q 1 mod p. We can use Alperin and Fong de* *scription of p-radical subgroups of general linear groups [4]. The p-primary part of the multiplicative group of units F*qis isomorphic to Z* *=p`. Call T`p~=(Z=p`)p the maximal finite torus. S~~= Z=p`o Z=p is the Sylow p-subgroup o* *f GLp(q). We can choose ~Sgenerated by T`p, diagonal matrices of p-power order, and the c* *ycle 0 1 0 0 . . . 1 B 1 0 0C B C C = BB0 1 0CC . @ ... ... ...A 0 0 . . .1 0 The center of GLp(q) is Z` ~=Z=p` embedded diagonally in T`p. Let i be a primit* *ive pth root of unity in F*q, and define the diagonal matrix B = diag(1, i, i2, . .,.ip-1) a* *nd the subgroup ` generated by Z`, together with the matrices B and C. This is a central product * *of the center Z` and an extraspecial group 1 of order p3 and exponent p, generated by A = diag(* *i, i, . .,.i), B, and C. There is an standard inclusion F*qp GLp(q), obtained by letting F*qpact on F* *qp by mul- tiplication and considering Fqp as Fq-vector space. We define U`+1 as the imag* *e in GLp(q) of the cyclic group Z=p`+1 F*qp, (q 1 mod p and ` = p(1 - q)) of p-power ro* *ots of unity in Fqp. 12 CARLES BROTO AND JESPER M. MØLLER With this notation and according to [4], if R is a p-radical subgroup of GLp(* *q), q 1 mod p, ` = p(1 - q), then R is conjugated to one of the following subgroups: ___________________________________________ __R__________NGLp(q)(R)_______Out_GLp(q)(R)_ Z` GLp(q) 1 T`p (F*q)p o p p S~ (F*q)p o (Z=p o Z=p - 1) Z=p - 1 * * (3) ` (F*q) . `. SL2(p) SL2(p) _U`+1________F*qpo_Z=p____________Z=p._____ Notice that Z` is clearly non-centric, but the other are all centric in GLp(q). It is now easy to extract from (3) the centric radical subgroups in the fusio* *n system of GLp(q) over ~S: __________________ _R___Out_GLp(q)(R)_ T`p p ~S Z=p - 1 * * (4) __`_____SL2(p).___ Example 3.6. We proceed now by describing the fusion system of SLp(q) over a Sy* *low p-subgroup, for p a prime and q a prime power such that q 1 mod p. We first show that every p-radical subgroup of SLp(q) is the intersection Q \* * SLp(q) of a p-radical subgroup Q of GLp(q) with SLp(q). For a given p-radical p-subgroup P * *of SLp(q) define Q = Op(NGLp(q)(P )). Q \ SLp(q) is a normal subgroup of NSLp(q)(P ) and * *since P is the maximal normal p-subgroup of NSLp(q)(P ), we have Q \ SLp(q) P . Same ar* *gument with NGLp(q)(P ) shows that P Q and therefore Q \ SLp(q) P . Every element g 2 GLp(q) normalizes SLp(q), so if g normalizes Q it also norm* *alizes Q \ SLp(q) P , so NGLp(q)(Q) NGLp(q)(P ). But, by definition of Q, this is* * normal in NGLp(q)(P ), hence we actually have NGLp(q)(Q)=NGLp(q)(P ). So, therefore, Q=Op* *(NGLp(q)(Q)) is p-radical. Fix the Sylow p-subgroup S = ~S\ SLp(q) of SLp(q). Assume that P S is centr* *ic and radical in the fusion system FS(SLp(q)), q 1 mod p, ` = p(1 - q). Then P is* * p-centric and p-radical in SLp(q). In particular P = Q \ SLp(q) where Q is p-radical in* * GLp(q), hence conjugate by an element g 2 GLp(q) to a p-subgroup in the list (3). Amon* *g those intersections, only S = ~S\ SLp(q), T`(p-1)= S \ T`p, and 1 = S \ ` are also * *p-centric. Hence the complete list of conjugacy classes of p-centric and p-radical subgrou* *ps of SLp(q), is obtained by conjugating these three subgroups by elements g 2 GLp(q): _______________________________________________ ___P____Out_SLp(q)(P_)Conditions_______________ T`(p-1) p p > 3 S Z=p - 1 * * (5) 1(,r) SL2(p) r = 0 if ` = 1, p = 3; r = 0, 1, . .,.p - 1 if ` > 1 or _____________________p_>_3,____________________ where 1(,r), r = 0, 1, . .,.(p - 1) is the conjugated subgroup of 1 by the di* *agonal matrix diag(,r, 1, . .,.1), , a (q - 1)st root of unity. Notice that for g 2 GLp(q), * *gSg-1 lies in S if and only it it is exactly S and the same happens with T`(p-1). In the case * *of 1 we just FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 13 need to check which of the subgroups 1(,r) are conjugated in SLp(q). In fact,* * Alperin's fusion theorem [11, A.10], together with the list of p-radical p-centric subgro* *ups that we have obtained so far, tells us that if two subgroups 1(,r) and 1(,s) are conjugate* *d in SLp(q) they are already conjugated in NSLp(q)(S), hence we obtain the list (5) by dire* *ct calculation. 4. Recognition of classifying spaces of p-local finite groups In [11] it is shown that a p-local finite group can be completely recovered f* *rom its classifying space by homotopy theoretic methods. Also, a recognition principle for classify* *ing spaces of p-local finite groups is provided in [11, Thm. 7.5]. We will briefly describe t* *hese methods and derive an inductive method that will be useful in our situation. We will first recall how a fusion system F(S,f)(X) or a linking system L(S,f)* *(X) are attached to a space X equipped with a map f :BS ! X, where S is a finite p-group. Then,* * the basic tool in order to show that these systems define a p-local finite group wi* *th classifying space X is [11, Thm. 7.5]. In order to apply the theorem we are generally faced* * to two main difficulties, namely, to show that the nerve of L(S,f)(X) is homotopy equivalen* *t to X and to show that F(S,f)(X) is a saturated fusion system. We will overcome these diffic* *ulties by an inductive method mainly based on the centralizers decomposition of p-local fini* *te groups that we develop in this section. Definition 4.1. Given spaces X and Y , we say that a map ff: X ! Y is a homoto* *py monomorphism at p if the homotopy fibre of ff, F , over any connected component* * of Y , is p- quasi-finite; that is, the inclusion F ! Map (BZ=p, F ) as constant maps is a w* *eak homotopy equivalence. It is not hard to prove that a composition of homotopy monomorphisms at p is * *again a homotopy monomorphism at p. Definition 4.2. Let X be a space. A finite p-subgroup of X is a pair (P, f), wh* *ere P is a finite p-group and f :BP ! X a homotopy monomorphism at p. A p-subgroup (S, f) * *of X is called a Sylow p-subgroup of X if for any other p-subgroup (Q, g) of X, g :BQ !* * X factors through f :BS ! X, up to homotopy. If (P, f) is a p-subgroup of X, then we denote BCX (P, f) = Map (BP, X)f. We will need later the next technical lemma. Lemma 4.3. Assume that X and Y are spaces for which Map (BZ=p, X)ct ' X and Map (BZ=p, Y )ct' Y . Let f :X ! Y be a homotopy monomorphism at p and ~: BP ! X a finite p-subgro* *up of X, then each map in the diagram BCX (P, ~)__ev__//X | f]|| |f| fflffl| fflffl| BCY (P, f O ~)ev_//_Y is a homotopy monomorphism at p. Proof. Let F be the homotopy fibre of the evaluation map BCX (P, ~) = Map (BP, X)~ ev-!X . 14 CARLES BROTO AND JESPER M. MØLLER There is an induced fibration Map (BZ=p, F ) ! Map (BZ=p, Map (BP, X)~)~ct! Map (BZ=p, X)ct where ~ctstands for all components mapping down to the component of the constan* *t map in Map (BZ=p, X). Since Map (BZ=p, X)ct' X, also Map (BZ=p, Map (BP, X)~)~ct' Map (BP, Map (BZ=p, X)ct)~~' Map (BP, X)~ , and therefore Map (BZ=p, F ) ' F ; that is, F is p-quasi finite and ev :CX (P, * *~) ! X is a ho- motopy monomorphism at p. Similarly, ev :BCY (P, f O~) ! Y is a homotopy monomo* *rphism at p and then, it is easy to obtain that also f] is a homotopy monomorphism at * *p. For any space X we denote Fp(X) the category in which the objects are finite * *p-subgroups (P, f) of X, and the morphisms are defined fi Mor Fp(X)((P, f), (Q, g)) = ' 2 Hom (P, Q) fif ' g O B' . Next construction appears already in [10]. We denote Lp(X) the category in whic* *h objects are p-subgroups (P, f) of X and morphisms are defined as fi Mor Lp(X)((P, f), (Q, g)) = (', [H]) fi' 2 Hom (P, Q) and [H] is the homotopy class of a homotopy from f to g O* * B'. We denote Lcp(X) the full subcategory whose objects are p-subgroups (P, f) wher* *e f is a p-centric map; that is, the induced map f]: Map (BP, BP )Id ! Map (BP, X)f is a* * mod p homology equivalence. If (S, f) is a p-subgroup of a space X we can define a fusion system over S, * *F(S,f)(X), by declaring fi Hom F(S,f)(X)(P, Q) = ' 2 Hom (P, Q) fif|BP ' f|BQ O B' BiP f for all P, Q S, where f|BP denotes the composition BQ --! BS -! X. Notice tha* *t if (S, f) is a Sylow p-subgroup of X, then, as categories, F(S,f)(X) is equivalent to Fp(* *X). Next, we define the category L(S,f)(X) that has objects the subgroups of S and fi Mor L(S,f)(X)(P, Q) = (', [H]) fi' 2 Hom (P, Q) and [H] is the homotopy class of a homotopy from f|BP to f|BQ O* * B', and teh full subcategory Lc(S,f)(X) whose objects are F(S,f)(X)-centric subgrou* *ps P S. There is also a natural functor ß :Lp(X) ! Fp(X), obtained by forgetting the co* *ncrete homotopy classes [H] in morphisms sets. The important question and the aim of the rest of this section consists in fi* *nding sufficient conditions on a space X and a p-subgroup (S, f) under which (S, F(S,f)(X), Lc(S,f)(X)) is a p-local finite group and X is its classifying space |Lc(S,f)(X)|^p' X. One first important case is that of X = |L|^p, the classifying space itself o* *f a given p- local finite group (S, F, L). In this case there is a natural inclusion ffiS :* *BS ! |L|^pand (S, F(S,ffiS)(|L|^p), Lc(S,ffiS)(|L|^p)) is isomorphic to the original (S, F, L* *). This is how a p-local finite group is completely recovered from its classifying space. FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 15 In [11, x7] it is also considered the case of an arbitrary p-complete space X* * equipped with a p-subgroup (S, f). The main argument establishes sufficient conditions on X * *and (S, f) under which (S, F(S,f)(X), Lc(S,f)(X)) is a p-local finite group. The question* * of whether or not X is the classifying space is left to directly checking if there is a homot* *opy equivalence |Lc(S,f)(X)|^p' X. There seems to be no natural way to construct a map between X and |Lc(S,f)(X)* *| in either direction. This problem was solved in [10] by means of an auxiliary simplicial * *space Mco(X) that comes equipped with a natural simplicial map øX :Mco(X) -! No(Lcp(X)) which induces a homotopy equivalence |øX |: |Mco(X)| ! |Lcp(X)|, provided th* *e spaces Map (BP, X)ffare aspherical for any p-centric subgroup ff: BP ! X (see [10, Le* *mma 4.2] and its proof). The geometric realization |Mco(X)| admits an evaluation map evX :|Mco(X)| ! X thus |øX | and evX can be used in order to connect geometrically X and |Lcp(X)|* *, or equivalently |Lc(S,f)(X)|. Proposition 4.4. There is a natural map Mf :BS ! |Mco(X)| that makes the diagram BS H |ffiS|rrrr|HHHfH rrr Mf | HHH xxrrr fflffl|HH## |Lcp(X)|o'o_|Mco(X)|_____//_X commutative up to homotopy. Proof. Proposition 2.7, Lemma 4.2 and Lemma 4.3 of [10] provide homotopy equiva* *lences |Lcp(S)|'__//_|Lcp(BS)|'o|Mco(BS)|o_'_//_BS * * (6) hence the Proposition could be proven by showing a map |Mco(BS)| ! |Mco(X)| mak* *ing commutative the necessary diagrams. * * '1 Mco(X) is a simplicial space where n-simplices are maps j : (P) ! X, where P * *= (P0 ! '2 'n P1 ! . .!. Pn) is a sequence of p-subgroups of S and monomorphisms, and (P) c* *an be B'1 B'2 B'n regarded as the homotopy colimit of the sequence BP0 -! BP1 -! . .-.! BPn, wi* *th the condition that the restriction of j to and BPi is a centric p-subgroup of X (se* *e [10, x4] for details). This last condition is what prevents the obvious construction of a map |Mco(B* *S)| ! |Mco(X)| from being well defined. In fact a subgroup P S which is centric in* * S, need f not be centric when regarded as a p-subgroups of X as BP Bincl-!BS -! X. 16 CARLES BROTO AND JESPER M. MØLLER We will have to restrict Mco(BS) to the subspace MSo(BS) of simplices j : (P)* * ! BS of Mco(BS) where every group in the sequence (P) is S itself. Accordingly, we cal* *l LSp(BS) the full subcategory of Lcp(BS) with objects the homotopy equivalences g :BS ! * *BS. With this notation we have a diagram of homotopy equivalences ' evBS |BS| __'__//_|LSp(BS)|oo|MSo(BS)|__'__//_BS * * (7) || ' || ' || ' || || fflffl| fflffl| fflffl|ev |||| |Lcp(S)|'__//_|Lcp(BS)|'o|Mco(BS)|o_'__BS//_BS where same arguments as in [10] for the sequence (6) are used. g f Now, for every equivalence g :BS ! BS, the composition BS ! BS ! X defines * *a centric p-subgroup of X, and then f induces a well defined map of simplicial spaces MSo* *(BS) ! Mco(X), that makes commutative the diagram ' evBS |BS| _'__//_|LSp(BS)|oo|MSo(BS)|__'__//_BS * * (8) II | III | | | III |f] |f] f| I$$Ifflffl| |fflfflev fflffl| |Lcp(X)|oo'___|Mco(X)|__'_X//_X from which the proposition follows. The next is a useful result that provides conditions on the space X and a Syl* *ow p- subgroup (S, f) under which the fusion system F(S,f)(X) is saturated. An eleme* *nt x 2 S of order p determines a homomorphism ix: Z=p ! S an then a map f O Bix: BZ=p ! * *X. We write BCX (x) = Map (BZ=p, X)x, the connected component that contains the map f* * O Bix, and fx: BCS(x) ! BCX (x) the map induced by f. Proposition 4.5. Let X be a space, (S, f) a Sylow p-subgroup of X, and X a set * *of elements of order p in S. Assume that: (1) Map (BZ=p, X)ct' X. (2) For all x 2 X, the natural map fx: BCS(x) ! BCX (x) is a Sylow p-subgroup f* *or BCX (x). (3) For all x 2 X, F(CS(x),fx)(BCX (x)) is a saturated fusion system over CS(x). (4) For all x 2 S of order p, there is ' 2 Hom F(S,f)(X)(, S) such that '(x)* * 2 X. Then F(S,f)(X) is a saturated fusion system over S and CF(S,f)(X)(x) * * coincides with F(CS(x),fx)(BCX (x)) as fusion systems over CS(x), for all x 2 X. Proof.Write F = F(S,f)(X) for short. Clearly, F is a fusion system over S. Cond* *ition (a) of Proposition 3.4 holds by (4); and it remains to show that conditions (b) and (c* *) of 3.4 hold. Condition (b) of 3.4: Fix x, y 2 S of order p such that y 2 X, and such th* *at there is _0 2 Hom F(, ) with _0(x) = y. We must show that _0 extends to some * *_ 2 Hom F (CS(x), CS(y)). FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 17 Since x and y are F-conjugate, [f O Bix] = [f O Biy] 2 [BZ=p, X], and thus Map (BZ=p, X)x = Map (BZ=p, X)y. Since CS(y) is a Sylow p-subgroup* * of Map (BZ=p, X)y by (2), the natural map BCS(x) ! Map (BZ=p, X)x factors through* * BCS(y). In other words, there is some _ 2 Hom (CS(x), CS(y)) such that the following sq* *uare com- mutes up to homotopy fOB(inclxix) BCS(x) x BZ=p _____________//X * * (9) || B_xId || |||| fflffl| fOB(inclxiy)|| BCS(y) x BZ=p _____________//_X . Thus _ 2 Hom F(CS(x), CS(y)). If æ, æ02 Hom (CS(x) x Z=p, S) denote the homomor* *phisms æ(g, t) = gxt and æ0(g, t) = _(g)yt, then f O Bæ ' f O Bæ0by (9), and hence Ker* *(æ) = Ker(æ0) by [11, Proposition 5.4(d)] (and point (1)). And this implies that _(x) = y. Condition (c) of 3.4: Fix some x 2 X; we must show that CF (x) is a saturated f* *usion system. By (3), the fusion system F0 def=F(CS(x),fx)(BCX (x)) is saturated, so it suffi* *ces to show that these two fusion systems over CS(x) are equal. * * __ To see this,_fix P, Q CS(x), and let ' 2 Hom (P, Q) be any monomorphism. S* *et P = P .and Q = Q.. Let æ 2 Hom (P x Z=p, S) and æ02 Hom (Q x Z=p, S) be defin* *ed by æ(g, t) = gxt and æ0(g, t) = gxt. Then ' 2 Hom F0(P, Q) if and only if the foll* *owing square commutes up to homotopy fOBj BP x BZ=p _________//X * *(10) || B'xId || |||| fflffl| fOBj0 || BQ x BZ=p _________//X . By (1) and [11, Proposition 5.4(d)], this holds if and only if K def=Ker(æ) = K* *er(æ0O (' x Id)) and the induced maps from B((P x Z=p)=K) to X are homotopic. The kernels are eq* *ual if _ __ __ and only if ' extends to a monomorphism ' from P to Q which sends x to itself. * *And in this_ case, the induced maps on B((P x Z=p)=K) are homotopic if and only if f|BP_' f|* *BQ_O B' , if and only if ' 2 Hom CF(x)(P, Q). Now, Proposition 3.4 implies that F(S,f)(X) is a saturated fusion system over* * S and the argument for condition (c) already contains the proof that CF (x) coincides* * with F0 = F(CS(x),fx)(BCX (x)) as fusion systems over CS(x). We derive now another characterization that will be useful in the specific ca* *ses in which we are interested or more generally in cases in which there is a good knowledge* * of elementary abelian p-subgroups of X and of its centralizers. 18 CARLES BROTO AND JESPER M. MØLLER Theorem 4.6. Let X be a p-complete space and (S, f) a p-subgroup of X. Assume t* *hat (1) Map (BZ=p, X)ct' X, and (2) for each non-trivial element x 2 S of order p (a)BCX (x) is the classifying space of a p-local finite group, and (b)if (H, g) is a Sylow p-subgroup for BCX (x), there is a group homomorphi* *sm æ: H ! S that makes the diagram Bj BH ______//_BS g|| |f| fflffl|ev |fflffl BCX (x) _____//X commutative up to homotopy, then, (S, f) is a Sylow p-subgroup for X and (S, F(S,f)(X), Lc(S,f)(X)) is a p-local finite group. Furthermore, X ' |L(S,f)(X)|^pif and only if the natural map induced by evalu* *ation hocolim Map (BE, X)f|BE- ! X Fe(S,f)(X)op is a mod p homology equivalence. Here Fe(S,f)(X) denotes the full subcategory * *of F(S,f)(X) consisting of non-trivial fully centralized elementary abelian p-subgroups of S. Proof.The proof is divided in four steps. First, we prove that (S, f) is a Sylo* *w p-subgroup for X. Next, that the fusion system of X over (S, f), F(S,f)(X) is saturated. * * In the third step we show that for each F(S,f)(X)-centric subgroup P S the map f|BP is p-c* *entric. These two last steps are the hypothesis (a) and (c) of [11, Theorem 7.5]. Ac* *cording to the remarks after the proof of this theorem in [11], this suffices in order to * *conclude that (S, F(S,f)(X), Lc(S,f)(X)) is a p-local finite group. This is the first part of* * the theorem. The second part states that X ' |L(S,f)(X)|^pif and only if the natural map i* *nduced by evaluation hocolimFe(S,f)(X)opMap(BE, X)f|BE- ! X is a mod p homology equivalen* *ce. This is proves in step 4. Notice that X ' |L(S,f)(X)|^pis condition (b) in [11, Theo* *rem 7.5]. Hence this second part of the theorem gives a necessary and sufficient condition for * *X to be the classifying space of the p-local finite group (S, F(S,f)(X), Lc(S,f)(X)). Step 1: (S, f) is a Sylow p-subgroup for X. Let (P, ~) be a finite p-subgroup o* *f X. Choose a central element x of order p in P . It determines a homomorphism ix: Z=p ! P * *for which CP(Z=p) = P , and a map ~ O Bix: BZ=p ! X. According to our hypothesis, BCX (x* *) is the classifying space of a p-local finite group, and if (H, g) is its Sylow p-s* *ubgroup, there are homomorphisms æ: H ! S and ': CP(Z=p) ! H that make the diagram B' ____________________________________________* *___________________________________________________________________________@ ________________________________________________* *___________________________________________________________________________@ ____________________________((____________________* *______________________________________________________~]g BCP(Z=p) ____//_BCX (Z=p, ~ O Bix)ooBH_ '|ev| ev|| |Bj| fflffl| ~ fflffl|f fflffl| BP ______________//_Xoo_________ BS FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 19 commutative up to homotopy. Hence, æ O ': P = CP(Z=p) ! S provides the factoriz* *ation of (P, ~) through (S, f). Step 2: The fusion system of X over (S, f), F(S,f)(X) is saturated. This part o* *f the proof will be based on Proposition 4.5. Define fi X = x 2 S fix of order p and fx: BCS(x) ! BCX (x) is a Sylow p-subgroup for BCX (* *x). Notice now that conditions (1) and (2) of Proposition 4.5 are satisfied by our * *hypothesis and by definition of the class X. Condition (3) is easily verified, too. In fact, b* *y hypothesis, for each x 2 X, BCX (x) is the classifying space of a p-local finite group and sinc* *e fx: BCS(x) ! BCX (x) is a Sylow p-subgroup for BCX (x), the fusion system F(CS(x),fx)(BCX (x* *)) is satu- rated. It remains to verify condition (4); that is, that every element x 2 S of orde* *r p is F(S,f)(X)- conjugate to an element of the class X. Assume that x 2 S has order p. It gives a homomorphism ix: Z=p ! S and a map f O Bix: BZ=p ! X. There is an evaluation map ev :BZ=p x BCX (x) ! X. Let (H,* * g) be a Sylow p-subgroup of BCX (x). Since (S, f) is a Sylow p-subgroup of X, the* *re is a homomorphism æ: Z=p x H ! S making the diagram Bj BZ=p x BH ______//_BS | 1xg || f|| fflffl| ev fflffl| BZ=p x BCX (x) ____//_X commutative up to homotopy. Let ' = æ|Z=pthe restriction of æ to the first component Z=p. From the above * *diagram we deduce that ' 2 Hom F(S,f)(X)(Z=p, S). Let y = '(x). Then, æ induces B~j fy ev BH -! BCS(y) -! BCX (y) -! X where all maps are homotopy monomorphisms at p. The first one because ~æis a mo* *nomor- phism, the others by Lemma 4.3. Now, ' induces a homotopy equivalence BCX (y) ' BCX (x), hence also an isomor* *phism between the respective Sylow p-subgroups. Since (H, g) is a Sylow p-subgroup fo* *r CX (x), it follows from the above sequence of maps that (CS(y), fy) is a Sylow p-subgroup * *for CX (y). Hence y = '(x) 2 X. Step 3: f|BP is a p-centric map for each F(S,f)(X)-centric subgroup P S. Sup* *pose that P S is F(S,f)(X)-centric. Choose a central element x 2 S or order p. Since P * *is centric, x 2 P and we have a sequence of homotopy monomorphisms at p fx ev BP -Bincl--!BS -! BCX (x) -! X . By hypothesis, BCX (x) is the classifying space of a p-local finite group, and * *from the above sequence of maps we easily obtain that (S, fx) is a Sylow p-subgroup for BCX (x* *). Further- more, P is also FS,fx(BCX (x))-centric, and then fx|BP is a p-centric map. Ther* *e is a sequence 20 CARLES BROTO AND JESPER M. MØLLER of equivalences Map (BP, BP )Id' Map (BP, BCX (x))fx|BP ' Map (BP x BZ=p, X)f|BPOBm' Map (BP, X)f|BP where m: P x Z=p ! P denotes multiplication by x, the generator of Z=p = . T* *he last equivalence is implied by the Zabrodsky's lemma applied to the fibration BZ=p -* *! BP x BZ=p Bm--!BP . The above composition shows that f|BP is a p-centric map. Step 4: X ' |L(S,f)(X)|^pif and only if the natural map hocolim Map (BE, X)f|BE- ! X Fe(S,f)(X)op induced by evaluation is a mod p homology equivalence. Since the categories Lcp* *(X) and Lc(S,f)(X) are equivalent, we can write the diagram of of Proposition 4.4 as th* *e homotopy commutative triangle BS C * *(11) ` ssss CCfCC ss CC yysss C!!C |Lc(S,f)(X)|__h________//X . It induces an equivalence of fusion systems over S: F(S,`)(|Lc(S,f)(X)|) = F(S,f)(X) and a natural map jP :Map (BP, |Lc(S,f)(X)|)`|BP-! Map (BP, X)f|BP * *(12) for every P S. Moreover, the diagram ''P Map (BP, |Lc(S,f)(X)|)`|BP_//_Map(BP, X)f|BP evfflffl|| |ev h fflffl| |Lc(S,f)(X)|_______________//_X is strictly commutative, with vertical maps induced by evaluation at the base p* *oint. As a consequence, we obtain a map between the corresponding homotopy colimits togeth* *er with compatible maps induced by evaluation: '' hocolim Map (BE, |Lc(S,f)(X)|^p)`|BE_//hocolimMap(BE, X)f|BE * *(13) Fe(S,f)(X)op Fe(S,f)(X)op evfflffl|| ev| h fflffl| |Lc(S,f)(X)|^p______________________//_X where Fe(S,f)(X) is the full subcategory of F(S,f)(X) consisting of non-trivial* * elementary abelian subgroups of E S that are fully centralized. Then problem is then reduced to showing that every map jP in (12) is a homoto* *py equiva- lence. In fact, the map j in the diagram (13) would be a homotopy equivalence, * *too. The left vertical map of (13) is also a homotopy equivalence by [11, 2.6 and 6.3]. The t* *heorem would follow as the right vertical map ev in (13) would be a homotopy equivalence if * *and only if h is a homotopy equivalence. FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 21 We will show that jP in (12) is a homotopy equivalence by induction on the or* *der of the group P . If P = , for some x 2 S of order p, then BCX (x) = Map (BP, X)f|B* *P is the classifying space of a finite p-local group, by hypothesis. According to Step 2* * above, we can assume without loss of generality that x 2 X, and so, the induced map fx: BCS(x* *) ! BCX (x) is the inclusion of a Sylow p-subgroup, and the fusion system F(CS(x),fx)(BCX (* *x)) coincides with CF(S,f)(X)(x) by Proposition 4.5. Now, diagram (11) induces the new homotopy commutative diagram BCS(x) * *(14) `]kkkkkk KKKKfxK kkkk KKK uukkkkk KK%% Map (BP, |Lc(S,f)(X)|^p)`|BP___''P______//_BCX (x) where, according to [11, 6.3], the map `] is the inclusion of a Syl* *ow p-subgroup of Map (BP, |Lc(S,f)(X)|^p)`|BPwhich is the classifying space of a centralizer p-* *local finite group with fusion system CF(S,f)(X)(x). Furthermore, jP induces an equivalence of fus* *ion systems, and therefore a homotopy equivalence. For an arbitrari non-trivial subgroup P S, we fix an element x of order p i* *n the center of P . Again, we can assume that x belongs to X. Then, the map jP, factors as t* *he composition Map (BP, |Lc(S,f)(X)|)`|BP-! Map (BP x B , |Lc(S,f)(X)|)`|BPOBm - ! Map (BP, Map (B , |Lc(S,f)(X)|)Bincl)`|BP - ! Map (BP, Map (B , X)x)f|BP - ! Map (BP x B , X))f|BPOBm - ! Map (BP, X)f|BP where all arrows are homotopy equivalences. That concludes the proof that jP i* *n equa- tion (12) is a natural mod p homology equivalence for subgroups P S. Notice also, that, reciprocally, if X is the classifying space of a p-local f* *inite group with Sylow p-subgroup (S, f), then all conditions of Theorem 4.6 are satisfied accor* *ding to [11, x7]. 5. Homotopy fixed points p-compact groups Let M be a space and G a group. It will be convenient for our purposes, to de* *fine an action of G on M as a fibration p M ____//_MhG___//BG * *(15) and, accordingly, a G-equivariant map between M and another space M0 supporting* * an action of G is a map f :M ! M0 that extends to a map fhG :MhG ! M0hGover BG. No* *tice that this is not actually a real action G x M ! M, but only a proxy action ([22* *, x10]). It determines a homotopy action of G on M, that is a homomorphism æ: G ! [M, M], which is obtained as the homomorphism induced on fundamental groups by the clas* *sifying map ': BG ! B aut(M). Thus, for a fixed homotopy action æ: G ! [M, M], a map ': BG ! B aut(M) with ß1(') = æ is interpreted as a lifting of æ to a proxy act* *ion, while a lifting to an actual action would be a homomorphism ~æ:G ! aut(M) whose compo* *sition with the projection aut(M) ! [M, M] is æ. The total space MhG of the fibration* * (15) is 22 CARLES BROTO AND JESPER M. MØLLER the homotopy quotient of the action and the homotopy fixed point set MhG is def* *ined as the space of sections of MhG ! M. Similarly, if X is a loop space with classifying space BX, we will say that a* *n action of the group G on X is a split fibration _p__//_ BX ___i//_BXhGsoo_BG . The section guarantees an induced action of G on X, compatible with the loop st* *ructure. The homotopy quotient for this action on X is defined as the pullback space in * *the diagram p~ XhG ______//BG * *(16) p~|| s|| fflffl| fflffl| BG __s_//_BXhG . This diagram turns out to be a diagram of spaces over BG. The homotopy fibre of* * ~pis X, and it has a canonical section ~sdefined by the pullback diagram (16) that we c* *an interpret as the homotopy constant loop ~i _~p_//_ X ____//_XhGoo_BG_. ~s The action of G on X depends on the section s: BG ! BXhG, and for this action w* *e obtain that the homotopy fixed point space XhG is a loop space with classifying space * *B(XhG) ' (BX)hGs, the connected component of (BX)hG with base point the section s. Furth* *ermore, the evaluation map XhG ! X is seen to be the loop map of the evaluation map (BX)hGs* *! BX, thus we have a sequence of fibrations ev hG hGev XhG ____//_X__//_X=X ____//_(BX)s____//_BX where we write X=XhG for the homotopy fibre of the evaluation map (BX)hGs! BX. By analogy with discrete group theory, we will write Out (X) = [BX, BX] and w* *ill say that an outer action of G on X is a homomorphism of groups æ: G ! Out (X). Thu* *s, an action of G on BX, classified by a map ': BG ! B aut(BX), gives rise to an oute* *r action, obtained as æ = ß1('): G ! Out (X). Equivalently, we say that a fibration over* * BG with fibre BX induces the outer action æ: G ! Out (X) if it is classified by a lifti* *ng of Bæ to B aut(BX): B aut(BX)88 'rrrrr | rrr | rr fflffl| BG _Bj_//_B Out(X) . As we have explained, the fibration over BG with fibre BX is not yet an action * *of G on X. An action of G on X inducing the given action on BX is classified by a further * *lifting B aut*(BX)88 _qqqqq | qqq | qqq fflffl| BG __'__//B aut(BX) . FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 23 We can also lift æ directly to an action of G on X, independently of the given * *action on BX: B aut*(BX)88 _qqqqq | qqq | qqq fflffl| BG _Bj__//_B Out(X) . The above classifying spaces fit together in a diagram of fibrations BX _______//_BXad_______//B2Z(X) * *(17) || | | || | | || fflffl| fflffl| BX ____//_B aut*(BX)__//_B aut(BX) | | | | fflffl| fflffl| B Out(X) _______B Out(X) that will enable us to compute the obstructions to the different liftings. Consider the fibration (15) with homotopy action æ: G ! [M, M]. The homotopy* * ac- tion determines an action of G on the group of path-components of M and ß0(M)G,* * or H0(G; ß0(M)), denote the set of path-components of M that remain fixed under th* *is action. With base point m 2 ß0(M)G in a G-invariant path-component of M there is a shor* *t exact sequence 1 ! ß1(M, m) ! ß1(MhG, m) ! ß1(BG, b) ! 1 * *(18) of fundamental groups, where b = p(m). If m 2 ß0(M)G happens to be in the image* * of the evaluation map i0(ev) G ß0(MhG ) ---! ß0(M) * *(19) then s(b) = m for some homotopy fixed-point s 2 MhG and then (18) does have a s* *ection, namely ß1(s). Since ß1(MhG, m) acts on the homotopy groups ßi(M, m) of the fib* *re, also G = ß1(BG, b) acts on ßi(M, m) through ß1(s). We let ßi(M, m)s*G, i 1, denot* *e the fixed-point group for this action. Recall that, if the exact sequence (18) splits, then we can identify the set * *of ß1(M, m)-conju- gacy classes of sections ß1(BG, b) ! ß1(MhG, m) with the cohomology group H1(G;* *ß1(M, m)). (We refer to [55] for the definition and properties of group cohomology with no* *n-abelian co- efficients.) Lemma 5.1. Suppose that G is a finite group of order prime to p and that ßi(M, * *m) is a module over the ring Z(p)of p-local integers for all i 2 and all base points * *m 2 ß0(M)G. (1) A point m 2 ß0(M)G is in the image of the evaluation map (19) if and only i* *f the exact sequence (18) splits. (2) If m 2 ß0(M)G is in the image of the evaluation map (19), then there is an * *exact sequence of pointed sets i0(ev) G * ! H1(G; ß1(M, m)) -!ß0(MhG ) ---! ß0(M) where m is the base point of ß0(M)G. (3) If s 2 MhG is a homotopy fixed-point with s(b) = m then ßi(MhG , s) ~=ßi(M, m)s*G for all i 1. 24 CARLES BROTO AND JESPER M. MØLLER Proof.The Postnikov functors Pr, defined as nullification with respect to Sr-1 * *(see [17]), determine a tower of fibrations MhG ! . .!.PrMhG ! Pr-1MhG ! . .!.P1MhG ! BG so that MhG is the homotopy inverse limit of a sequence . .!.PrMhG ! Pr-1MhG ! . .!.P1MhG of Postnikov homotopy fixed-point spaces. Note that ß0(P1MhG) = ß0(MhG) and that each path-component of P1MhG is aspher* *ical with fundamental group ß1(P1MhG, m) = ß1(MhG, m) for all m 2 P1M. It is now eas* *y to see that H1(G; ß1(M, m)) is indeed the fibre over m 2 ß0(P1M)G = ß0(M)G of the eval* *uation map ß0(P1MhG ) ! ß0(M)G and also that ß1(P1MhG , s) = ß1(M, m)s*G for any s 2 P* *1MhG with s(b) = m, cf. [41, x6]. Obstruction theory implies that ß0(MhG ) = ß0(P1Mh* *G ). Suppose that the homotopy fixed-point space is non-empty and let s 2 MhG be a* * homotopy fixed-point. Then the component MhG , s containing s is the homotopy inverse * *limits of the corresponding components hG hG hG . .!. PrM , sr ! Pr-1M , sr-1 ! . .!. P1M , s1 of the Postnikov homotopy fixed-point spaces. To finish the proof, observe [41,* * 3.1] that the fibre of PrMhG , sr ! Pr-1MhG , s-1r is the Eilenberg-Mac Lane space K(ßr(M* *, m)s*G, r). For an alternative formulation, let (M, m) denote the path-component of M con* *taining m 2 M. If the path component (M, m) 2 ß0(M) is G-invariant, then (M, m) is sub-* *G-space of M in the sense that the inclusion of (M, m) into M is a G-map; that is, the * *fibration M ! MhG ! BG contains a fibration of the form (M, m) ! (M, m)hG ! BG as a sub-fibration over BG. The homotopy fixed-point space [ MhG = (M, m)hG m2i0(M)G is a disjoint union of the homotopy fixed-point spaces (M, m)hG where (M, m) ru* *ns through the set of G-invariant path-components in M. Since (M, m) by its very definitio* *n is a path- connected G-space the homotopy groups of its homotopy fixed-point spaces are ( H1(G; ß1(M, m)) i = 0 ßi(M, m)hG = ßi(M, m)s*G i > 0 by the lemma. Theorem 5.2. Let B be any simply connected p-complete space, G a finite group o* *f finite order prime to p, and B ! BhG ! BG an action of G on B. There exists a homotopy equivalence B '-! BhG x Fib(BhG ! B) In particular, the fibre Fib(BhG ! B) of the evaluation map BhG ! B is an H-spa* *ce. FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 25 Proof. By obstruction theory, the space of sections BhG is non-empty. We will * *show first how to turn this action with a homotopy fixed point into a honest action of G o* *n a space homotopy equivalent to B and with a fixed point. The pullback diagram B _____//_EGoo``` | | | | fflffl| fflffl| BhG ____//_BGoo``` realizes B ! BhG as a regular covering space with G acting on B. Liftings of s* *ections BG ! BhG provide G-equivariant maps EG ! B. Let B=EG = B[C(EG) be the homotopy cofibre of any such G-map. Then B ! B=EG is a G-equivariant homotopy equivalenc* *e and the G-action on B=EG has a fixed point. Now, we can assume that there is a honest G-action on B with a fixed point. * *Let B denote the loop space based at any G-fixed point. It suffices to construct a h* *omotopy left inverse for the inclusion BhG ! B. Q Define tr: B ! B to be the map that takes any loop ! to the product g! * *of the loops g! where g runs through the elements of G in some fixed order. The image * *of the induced map tr*:ß*( B) ! ß*( B) on homotopy groups is contained in the fixed group ß*(* * B)G and the composition ß*( B) G ! ß*( B) ! ß*( B) G is an isomorphism. This implie* *s that the composition BhG ! B ! T , where T is the mapping telescope of B tr-! B t* *r-!. .,. is a (weak) homotopy equivalence and we have the left inverse we were looking f* *or. Proof of Theorem B. Fix a finite group G of order prime to p, and æ: G ! Out(X)* * an outer action of G on a connected p-compact group X. Recall that we have a fibration s* *equence B2Z(X) ! Baut(BX) ! BOut (X) and that the center of X, Z(X) is p-local. By obstruction theory we obtain a un* *ique lifting of æ to an action ': BG ! B aut(BX). Furthermore, since ß1(BX) = 1, Lemma 5.1 i* *mplies that ß0(BXhG) = *; that is, æ lifts to a unique action of G on X BX _____//BXhG___//_BGo.o_ * *(20) This is part (1) of the theorem. Now, Theorem 5.2 provides the splitting X ' * *XhG x X=XhG. It follows that X=XhG is an Fp-finite H-space, XhG is a loop space with * *classifying space BXhG and it is also Fp-finite. Furthermore, BXhG is p-complete because B* *X is p- complete [22, 11.13], hence XhG is a connected p-compact group. The rational cohomology algebra H*(BY ; Qp) is polynomial for any connected p* *-compact group Y and it follows that the Hurewicz homomorphism induces an isomorphism QH*(BY ; Qp) ! ß*(BY )_ Q between the indecomposables and the rationalized dual (ß_ = Hom Zp(ß, Zp)) of t* *he homotopy groups of the simply connected space BY . For the connected fixed-point p-comp* *act group BXhG, in particular, we have _ * QH*(BXhG; Qp) ~=ß*(BXhG)_ Q ~= ß*(BX) Q G ~= QH (BX; Qp) G for ß*(BXhG) = ß*(BX)G as the order of G is prime to p. This proves points (2) * *and (3). We finish by proving point (4). If X is a polynomial p-compact group H*(X; Fp) ~=H*(XhG; Fp) H*(X=XhG; Fp) 26 CARLES BROTO AND JESPER M. MØLLER is an exterior algebra, hence H*(XhG; Fp) is an exterior algebra, too. Therefor* *e, H*(BXhG; Fp) is a polynomial algebra.. Example 5.3. At any odd prime, let C2 act on E6 through the unstable Adams oper* *a- tion _-1. Since the fixed point p-compact group BEhC26is the p-compact group BF* *4 (5.15), there is a splitting E6 ' F4x E6=F4 of homogeneous spaces. This splitting is due to Harris [30]. Also, BP EhC26' BF* *4, where P E6 is the adjoint form of E6, (5.15), thus there is also a splitting P E6 ' F4x P * *E6=F4. Let p be an odd prime and m a divisor of p - 1 so that the cyclic group Cm of* * order m acts on BSU (mn + s), 0 s < m, through unstable Adams operations. Since the * *fixed point p-compact group BSU (mn+s)hCm is (5.12) the generalized Grassmannian BX(m* *, 1, n) with polynomial cohomology H*(BX(m, 1, n); Fp) = Fp[xm , . .,.xnm ], |xim| = 2i* *m, there is a splitting SU (mn + s) ' X(m, 1, n) x SU (mn + s)=X(m, 1, n) of homogeneous spaces. This splitting is originally due to Mimura, Nishida, and* * Toda [39]), although the recognition of X(m, 1, n) as a loop space is due to Quillen [54] (* *see also [59, 63, 15]). The case m = 2 is the classical splitting SU(2n) ' Sp(n) x SU(2n)=Sp(n).* * Similar splittings for central quotients of SU(n) can be worked out. Similarly, at p = 5, let C4 act on BE8 through unstable Adams operations. Sin* *ce (5.15) the fixed point p-compact group BEhC48is the p-compact group BX(G31) correspond* *ing to reflection group number 31 on the Clark-Ewing list, H*(BX(G31); Fp) = Fp[x16, x* *24, x40, x48] where subscripts indicate degrees, there is a splitting E8 ' X(G31) x E8=X(G31) of homogeneous spaces, that was obtained in [60]. At p = 3, BF4 admits an exceptional isogeny of order 2 and the fixed point gr* *oup BF4hC2 is [13] the p-compact group BDI (2) whose cohomology realizes the Dickson algebra * *F3[x12, x16]. The corresponding splitting F4' DI (2) x F4=DI (2) was first obtained in [31]. Later proofs of this splitting were obtained indep* *endently by Wilkerson and by Kono, using Friedlander's exceptional isogeny of F4 localized * *away from two. In these last two cases, it was Zabrodsky [63, 4.3], who first recognized the* * factors DI (2) and X(G31) as loop spaces. Later, Aguad'e gave a nice uniform construction of a* * family of modular p-compact groups included these cases [1]. Our next objective is to obtain a recognition principle for the homotopy fixe* *d point p- compact group BXhG. Let N ! X be the maximal torus normalizer for the p-compact group X. Again, t* *he short exact sequence of topological monoids BZ(N) = aut(BN)1 ! aut(BN) ! Out(N) induces a fibration sequence B2Z(N) ! Baut(BN) ! BOut (N) and we may write B2Z(N)hOut(N)= Baut(BN) for the classifying space for BN-fibra* *tions. FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 27 Since G is a finite group of order prime to p, we see from this that equivale* *nce classes of BX-fibrations over BG is in one-to-one correspondence with [BG, B2Z(X)hOut(X)] = [BG, BOut (X)] = Hom (G, Out(X)) and that equivalence classes of BN-fibrations over BG is in one-to-one correspo* *ndence with [BG, B2Z(N)hOut(N)] = [BG, BOut (N)] = Hom (G, Out(N)) . However, Out(X) ~=Out (N) and therefore there is a bijective correspondence bet* *ween BX- fibrations over BG and BN-fibrations over BG. We shall now make this correspon* *dence more explicit. Turn the maximal torus normalizer Bj :BN ! BX into a fibration. Write aut* *(Bj) for the group-like topological monoid of commutative diagrams BN ____//_BN Bj|| |Bj| fflffl| fflffl| BX ____//_BX where both horizontal arrows are homotopy equivalences. Lemma 5.4. Assume that p is odd. The forgetful homomorphisms aut(BN) oo__aut(Bj)_____//aut(BX) are homotopy equivalences. Proof. The group homomorphisms ß0aut(BN) ß0aut(Bj) ! ß0aut(BX) are injecti* *ve because X has N-determined automorphisms [48, 6]. The group homomorphism to the* * left is surjective because X is N-determined and the one to the right is surjective * *because any self-homotopy equivalence of BX lifts to a self-homotopy equivalence of BN [46,* * x3]. The identity components fit into a map of fibrations [23, 11.10] autBX (Bj)1_____//aut(Bj)1_______//aut(BX)1 || | | || | '| || fflffl| fflffl| autBX (Bj)1____//_aut(BN)1____//Map(BN, BX)Bj where the right vertical map, defined by composition with Bj, is a homotopy equ* *ivalence [23, 7.5, 1.3] [22, 9.1] [46, 3.4]. The fibre, consisting of the space of maps BN ! * *BN over BX and vertically homotopic to the identity map of BN, is (one component) of the space* * (X=N)hN which is contractible [44, 5.1]. Thus we have bijections [B, Baut(BN)] = [B, Baut(Bj)] = [B, Baut(BX)] for any space B and this means BN-fibrations and BX-fibrations over B are in bi* *jective correspondence. 28 CARLES BROTO AND JESPER M. MØLLER Proposition 5.5. Let X be a connected p-compact group with maximal torus normal* *izer N ! X. If G is a finite group of order prime to p, then any outer action æ: G !* * Out(X), lifts to a unique G-action on BX and unique G-action on BN. Moreover, these act* *ions make the map BN ! BX G-equivariant; that is, the diagram BN ____//_BNhG____//_BG | | || | | || |fflffl fflffl| || BX ____//_BXhG____//_BG is homotopy commutative. Proof.Let us say that our input is an outer action æ: G ! Out(X) = W \NGL(L)(W ) = Out(N) * *(21) of the finite group G on X and N. The induced map Bæ 2 [BG, Baut(Bj)] = Hom (G, W \NGL(L)(W )) corresponds [19] to an iterated fibration BjhG BNhG ____//_BXhG___//_BG over BG. Next, we need to lift the action of G on BN and BX to and action on the loop * *spaces N and X, such that the inclusion N ! X is still equivariant. Again Lemma 5.1 applies to show that the fibration BX ! BXhG ! BG admits one * *and only one section; that is, there is a unique lifting of the action on BX to an * *action on X. However, ß1(BN) ~=W and then Lemma 5.1 does not ensure neither, the existence, * *nor the uniqueness of a lifting of the action of G on BN to an action of G on N. Instea* *d, it leads to the next description of the possible actions. Proposition 5.6. If a finite group G of order prime to p acts on BN with outer * *action æ: G ! W \NGL(L)(W ) ~=Out (N), then there are natural one-to-one correspondenc* *es between the sets: (1) ß0(BNhG ), (2) lifts to a G-action on N, and (3) W -conjugacy classes of lifts in the diagram NGL(L)(W ) _88____ _______|_ ________ | __________ fflffl| G __j_//_W \NGL(L)(W ) . If those sets are non-empty, then they are also in one-to-one correspondence wi* *th H1(G; W ). Proof.An action of G on BN is by definition a fibration BN ! BNhG ! BG , * *(22) and according to 5.5 this action of G on BN is uniquely determined by æ. FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 29 The map from ß0(BNhG ) to the set (2) is immediate because we can identify ß0* *(BNhG ) with the vertical homotopy classes of sections of (22), and a sectioned fibrati* *on is an action of G on N, by definition. Also, if _ :BG ! B aut*(BN) is a lift of æ to an acti* *on of G on N, then oe = ß1(_): G ! NGL(L)(W ) is an element in the set (3). Next, we can map ß0(BNhG ) directly to the set (3). Let ': BG ! B aut(BN) be* * a classifying map for the fibration (22). Thus, ' extends to a map of fibrations BN ______//_BNhG__________//BG || | |Bj || | | || fflffl| fflffl| BN ____//_Baut*(BN)____//Baut(BN) which on the level of fundamental groups [45, 5.2] [5, 3.3] induces a morphism W _____//_ß0(NhG)__________//_G * *(23) || || | |j || | | || fflffl| fflffl| W ____//_NGL(L)(W_)__//_W \NGL(L)(W ) of group extensions. We have seen (Lemma 5.1) that the existence of an action o* *f G on N lifting the action on BN is equivalent to the existence of a section of the exa* *ct sequence on the top row of (23), and the diagram shows that this is equivalent to the exist* *ence of a lifting of æ to a homomorphism oe :G ! NGL(L)(W ). This gives the bijection between ß0* *(BNhG ) and the set (3), and shows that all of the three sets are empty if one is empty. Finally, if they are non empty, then obstruction theory shows that all of the* *m are para- metrized by H1(G, W ), which coincides with both H1(G; ß1(BNad)) (that parametr* *izes (2), see diagram (17)) and H1(G; ß1(BN)) (that parametrizes (1), see Lemma 5.1). Proposition 5.7. Let X be a connected p-compact group with Weyl group W and max* *imal torus normalizer N ! X. If G is a finite group of order prime to p and æ: G ! Out(X) ~=W \NGL(L)(W ) an outer action, then æ lifts to a unique action of G on X, and each lift oe :G ! NGL(L)(W ) determines a unique action of G on N such that the inclusion N ! X is G-equivar* *iant. Proof. As we mentioned before, (1) follows directly from Lemma 5.1, and accordi* *ng to Propo- sition 5.6, the actions of G on N that lift the given outer action are in one-t* *o-one correspon- dence with lifts of æ to NGL(L)(W ). If we view one of these actions as a secti* *oned fibration BN _____//BNhG____//_BGoo_ it clearly induces an action on X that makes N ! X equivariant: BNhG H_____________//BXhG dHHHd_______ vvvv::_______ __HHH_____vvv_______ __H$$H_zzvv_____________ BG . The proposition follows because there is only one action of G on X inducing æ. 30 CARLES BROTO AND JESPER M. MØLLER Proposition 5.8. Let p be an odd prime and G a finite group of order prime to p* *. Assume that G acts on a connected p-compact group X and __æ:G ! N GL(L)(W ) is a lift of the given outer action. If Y is a connected p-compact group that s* *atisfies _ ___ _ _ (1) W_jGcontains_a_subgroup W , complementary to the kernel of W jG! GL (LjG), * *such that (W , L(X)jG) is a reflection group similar to (W (Y ), L(Y )), and (2) QH*(BY ; Qp) ~=QH*(BX; Qp)G, then BY = BXhG. Proof.By the classification theorem for p-compact groups at odd primes [48, 6],* * it suffices [47, 1.2] to find an map BN(Y ) ! BXhG that induces an isomorphism on H*(-; Qp) and * *restricts to monomorphism on the p-normalizer Np(Y ), is a p-monomorphism. The homomorphi* *sm __æ corresponds (5.7) to compatible G-actions BG ! BN(X)hG ! BXhG on N(X) and X. Taking homotopy fixed points we obtain a commutative diagram of loop space morp* *hisms N(X)hG ____//_XhG | | | | fflffl| fflffl| N(X) ______//_X which shows that N(X)hG ! XhG is a p-monomorphism. Since the discrete approxim* *a- tion to N(X), N(X)hG, and N(Y ) are semi-direct_products [5], there is a p-mono* *morphism N(Y ) ! N(X)hG for W (Y ) is a subgroup W jG= ß0N(X)hG by the first condition. * * By the second condition, H*(BY ; Qp) = H*(BN(Y ); Qp) and H*(BXhG; Qp) are abstrac* *tly iso- morphic graded vector spaces. Therefore, Y and XhG have the same rank [22, 5.9]* * so that T (Y ) ! N(X)hG ! XhG is a maximal torus and H*(BXhG; Qp) ! H*(BN(Y ); Qp) is injective [22, 9.7], hence bijective. A special case arises when G acts through unstable Adams operations so that t* *he action ß0æ: G ! Out(N) ! Out(W ) is trivial. Then the image of G in Out(N) = W \NGL(L)* *(W ) is contained in the subgroup Z(W )\CGL(L)(W ) [48, 3.16] and we have a morphism W ______//_ß0(NhG)____________//_G || || | Bj| || | | || fflffl| fflffl| W ____//_W.CGL(L)(W_)__//_Z(W )\CGL(L)(W ) of group extensions. The possible extensions occurring in the upper line, reali* *zing the trivial action G ! Out(W ), are classified by H2(G; Z(W )); they are all isomorphic to W ! Z(W )\(D x W ) ! G for some central extension Z(W ) ! D ! G [36, IV.x8]. If Z(W ) = 1 is trivial, * *ß0(NhG) = G x W and H1(G; W ) = Rep(G, W ). Assume that G = Cr is a cyclic group of order r, and the outer action of G on* * X, æ: Cr ! Out (X), is given by an Adams operation æ(~) = _~, where ~ 2 Zxpis a p-adic uni* *t of order r|(p - 1). We can lift _~ 2 Z(W )\CGL(L)(W ) to an element i 2 CGL(L)(W * *), that verifies ir 2 Z(W ). If there is a choice of i with ir = 1, then ~æ~ = i provid* *es a lifting of æ. FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 31 Assume, otherwise, that ir has order s in Z(W ). Since p is odd, Z(W ) has or* *der prime to p, hence s is prime to p. Now, even if there is no lift of the action of Cr * *on X to an action on N, we can reduce the problem by extending the action of Cr to an action of C* *sr on X determined by æ0(~) = _~ 2 Z(W )\CGL(L)(W ) Out(X), that now admits the lift * *~æ0(~) = i. Notice that Cs = <~r> acts trivially on X, so that BXhCs ' BX, and then BXhCsr~* *=BXhCr, so we can still determine BXhCr by analysing the equivariant action of Csron N * *and X. Notice also, that if W is irreducible, then CGL(L)(W ) consists of diagonal m* *atrices and therefore i is an Adams operation. Corollary 5.9. Let ~ 2 Zxpbe a p-adic unit of order r|(p - 1). Consider the out* *er action æ: Cr = <~>! W \NGL(L)(W ) through unstable Adams operations given by æ(~) = _* *~. Then, if æ admits a lift ~æ:Cr ! NGL(L)(W ), then all possible lifts are parametrized* * by H1(Cr; W ) = Rep (Cr, W ), the set of conjugacy classes of order r elements w of W , and _jC _jC <~w> (W ,rL r) = (CW (w), L ) for the lift __æ(~) = ~w corresponding to w. Proof. The lifts W.CGL(L)(W ) _66_____ _j_______|___ __________ | ______ fflffl| Cr = <~> _j__//_Z(W )\CGL(L)(W ) are given by __æ(~) = w_~ where w 2 W is any element of order r. We next apply the recognition principle (5.9) in some concrete cases. 5.10. The infinite families. We identify the fixed point p-compact groups for* * actions through unstable Adams operations on the p-compact groups of the three infinite* * families in the Clark-Ewing classification table [16]. Proposition 5.11. Suppose that r and m divide p - 1, m > 1, then ( S2r-1, m | r (S2r-1)hCm = * , otherwise, for the action through unstable Adams operations of exponent of Cm Z*pon the * *p-compact group S2r-1. Proof. Let ~ be a primitive mth root of unity, so that Cm = <~> Z*p. Accord* *ing to Theorem B, (S2r-1)h<~>is a connected polynomial p-compact group. If m does not * *divide r, H2r(_~) = ~r is nontrivial, so that the vector space of covariants QH*(BS2r-1; * *Qp)<~>vanishes in positive degrees, and the fixed point p-compact group is trivial. If m does * *divide r, _~ acts trivially on S2r-1and the fixed point p-compact group is again S2r-1. The next results, 5.12, 5.13, and 5.14, deal with complex and generalized gra* *smannians. The results of 5.12 and 5.14 were obtained by Castellana [15] using different m* *ethods. 32 CARLES BROTO AND JESPER M. MØLLER Proposition 5.12. Let p be an odd prime. Suppose that m|(p - 1), m > 1, and let* * Cm = <~> Zxpbe the cyclic group generated by a primitive mth root of unity acting * *through unstable Adams operations. Then ( X(m, 1, n) n > 0 X(mn + s)hCm = U(mn + s)hCm = * n = 0 for any p-compact group X(mn + s) locally isomorphic to SU (mn + s), 0 s < m. Proof.In H*(BU (mn+s); Qp)=Qp[c1, . .,.cmn+s] and H*(BX(mn+s);Qp)=Qp[c2, . .,.c* *mn+s] we have ci is preserved by H2i(_~), m|i and therefore QH*(BU (mn + s); Qp)Cm = Qp{cm , . .,.cmn } = QH*(BX(m, 1, n); Qp) = QH*(BX(mn + s); Qp)Cm . The Weyl group W = mn+s is the symmetric group in its natural representation o* *n L = Zmn+sp. Let e1, . .,.emn+s be the canonical basis vectors of L. The permutation w = (1 . .m.)(m + 1 . .2.m) . .(.m(n - 1) + 1 . .m.n) 2 mn+s has order m and (C mn+s(w), L<~w>) = (Cm o n x s, Zp{~e1 + ~2e2 + . .+.~m em , . .,.~em(n-1)+1+ . .+.* *~m emn }) contains the reflection group G(m, 1, n) = Cm o n as a a subgroup complementar* *y to the kernel, s, of the action of (C mn+s(w) on L<~w>. This means (5.9) that the fi* *xed point p-compact group U(mn + s)hCm = X(m, 1, n). From the two short exact sequences of Zp mn+s-modules [48, x10] 0 ! Zp -! L ! LP U(mn + s) ! 0, 0 ! LX(mn + s) ! LP U(mn + s) ! ~ß! 0 where is the diagonal and ~ßa subgroup of ß1(P U(mn + s)) = Zp=Zp(mn + s) (wi* *th trivial mn+s-action), we get that L<~w>= LP U(mn + s)<~w>= LX(mn + s)<~w> as ZpC mn+s(w)-modules. Let p be an odd prime and r 1 and m 2 natural numbers such that r | m | p* * - 1. Then the cyclic group Cm of order m is contained in the group of units Zxpfor Zp. Th* *e Zp-reflection group (G(m, r, n), Znp), n 2, is the group generated by all permutations of t* *he n coordinates and the diagonal matrices in A(m, r, n) = {diag(a1, . .,.an) 2 Cnm| (a1. .a.n)m=r = 1} which is an index r subgroup of A(m, 1, n) = Cnm. As abstract groups G(m, r, n)* * = A(m, r, n)o n. The proof of (5.13) will make use of these facts: o For arbitrary natural numbers m and n we write mn for m= gcd(m, n). Then* * mnn = lcm(m, n) and mnnm = lcm(m, n)= gcd(m, n). FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 33 o Clcm(q,m)= <~, ~|~q = 1, ~m = 1, ~~ = ~~, ~qm =>~mq. o Let A(a, t) 2 GL (Zp, t) denote the linear automorphism A(a, t)(x1, . .,.xt) = (axt, x1, . .,.xt-1) where a 2 Zxpis a unit. The ith power A(a, t)i has characteristic polyno* *mial (xti- ait)t=tiand A(a, t)t= aE. o If ~ 2 Zxp has order q, then A(~-qm, qm ) also has order q for A(~-qm, q* *m )qm = ~-qmE has order gcd(q, m). The ~-1 eigenspace of A(~-qm, qm ) has rank * *one and A(~-qm, qm )-1 acts on it as multiplication by ~. o In the exact sequence 1 ! A! CAoG (a, g) ! CG(g) the image in CG(g) c* *onsists of those h 2 CG(g) that fix a 2 A=(1 - g)A. Proposition 5.13. Let X(m, r, n), m 2, r 1, n 2, r | m | p-1, be the simp* *le polynomial p-compact group whose Weyl group is the imprimitive reflection group G(m, r, n)* *. Suppose that the natural number ` divides p - 1 and let the cyclic group C` Clcm(`,m)* * Zxpact on X(m, r, n) through unstable Adams operations. The homotopy fixed point grou* *p for this action is 8 >: X(lcm(`, m), 1, n=`m ) ` - mn where `m = `= gcd(`, m) and [n=`m ] is the biggest integer n=`m . (By convent* *ion, G(m, r, 1) is cyclic of order m=r and G(m, r, 0) is the trivial group.) Proof. Let ~ 2 Zxpbe a primitive `th root of unity. In the rational cohomology* * algebra H*(BX(m, r, n); Qp) ~=Qp[x1, . .,.xn-1, e] the degrees |xi| = 2im and |e| = 2m_* *rn so that xi is preserved by H2im(_~) = ~im, ` | im , `m | i m_n ~ m_n e is preserved by H2 r (_ ) = ~ r, ` | nm=r , `m=r | n and thus QH*(BX(m, r, n); Qp)C` is isomorphic to the indecomposables of the rat* *ional coho- mology algebra of the p-compact group on the right hand side of the equation. We have r` | mn , `m=r | n, ` | mn , `m | n, and `m | `m=r | `|p - 1. `m=r_|_n_____: The element -` -` w = diag A(~ m, `m ), . .,.A(~ m, `m2)G(m, r, n) ____________-z___________" n=`m has order `. Since ((~-`m)n=`m)m=r = ~-mn=r = 1 because `|(mn=r) by assumption,* * w does indeed belong to the index r subgroup G(m, r, n) of G(m, 1, n) = Cm o n. Let {* *e1, . .,.en} be the canonical basis for the free Zp-module L = Znpon which G(m, r, n) acts. * * The free Zp-module ` -1 `f-1f L<~w>= e1 + ~e2 + . .+.~ m e`m, . .,.e(n-`m)+1+ ~e(n-`m)+2+ . .+.~ m, en has rank n=`m. We shall now compute the centralizer of w. Let i be a generato* *r of the cyclic group Clcm(`,m) Zxpso that Cm = <~>and C` = <~>with ~ = i`m and ~ = im`* *. The homomorphisms A(`, 1, n=`m_)//_CG(m,1,n)(w)A(m,o1,on=`m_)defined by -` -1 ~i! diag E,_._.,.E_-z___", A(~ m, `m ) , E, . .,.E , diag E,_._.,.E_-z_* *__", ~E, E, . .,.E) ~i i-1 i-1 34 CARLES BROTO AND JESPER M. MØLLER combine to a homomorphism defined on A(lcm(`, m), 1, n=`m ) since they agree on* * their com- ffn=`m mon domain A(gcd(`, m), 1, n=`m ) = <~m`>n=`m= ~`m , Observe that (~a1, . .* *,.~an=`m) 2 A(`, 1, n=`m ) lies in the subgroup A(lcm(`, m), r, n=`m ) if and only if its i* *mage lies in G(m, r, n) and that (~a1, . .,.~an=`m) 2 A(m, 1, n=`m ) lies in the subgroup A(m, r, n=`m * *) if and only if its image lies in G(m, r, n). Together with the diagonal : n=`m! n given b* *y (oe)((i - 1)`m + j) = (oe(i) - 1)`m + j, 1 `i n=`m , 1 j `m , we obtain a group i* *somorphism ~= G(lcm(`, m), 1, n=`m ) -! CG(m,1,n)(w) that restricts to a group isomorphism G(lcm(`, m), r, n=`m ) ~=CG(m,r,n)(w) bet* *ween index r subgroups. This isomorphism identifies the pair (CG(m,r,n)(w), L<~w>) and the * *imprimitive reflection group (G(lcm(`, m), r, n=`m ), Zn=`mp). `m=r_-_n,_`m_|_n__: It will suffice to consider the case of G(m, m, n) where ` * *- n and `m | n. The element -` -` -` 1-n=` w = diag A(~ m, `m ), . .,.A(~ m,,`mA)(~ m, `m ) m 2 G(m, m, n) ____________-z___________" n=`m-1 has order `. Note that ~-1 is not an eigenvalue for A(~-`m, `m )1-n=`mbecause A(~-`m, `m )1-n=`mhas eigenvalue ~-1, A(~-`m, `m )n=`m-1has eigenvalue ~ , ~(`m)n=`m-1= ~-`m(n=`m-1)`m, ` | (`m )n=`m-1+ `m (n=`m - 1)`m , ` | n=gcd(`m , n=`m - 1)) ` | n ) `* *m | n which is not the case. Therefore the ~-1-eigenspace ` -1 ` -1ff L<~w>= e1 + ~e2 + . .+.~ m e`m, . .,.e(n-2`m)+1+ ~e(n-2`m)+2+ . .+.~ m e* *n-`m has rank n=`m - 1. The two monomorphisms A(`, 1, n=`m - 1) ! CG(m,m,n)(w) a* *nd CG(m,m,n)(w) A(m, 1, n=`m - 1) given by -` -1 -` ~i! diag E,_._.,.E_-z___", A(~ m, `m ) , E, . .,.E, A(~ m, `m ) i-1 * * -1 diag E,_._.,.E_-z___", ~E, E, . .,.E* *, ~ E ~i i-1 agree on their common domain A(gcd(`, m), 1, n=`m -1) and together with the mon* *omorphism " Ø " n=`m-1Ø__// n-`m_//_ mthey define a homomorphism on the group A(lcm(`, m), 1, * *n=`m - 1) o n=`m-1such that the composition * *<~w> A(lcm(`, m), 1, n=`m - 1) o n=`m-1,! CG(m,m,n)(w) i Im CG(m,m,n)(w) ! GL (L* * ) is an isomorphism with image similar to G(lcm(`, m), 1, n=`m - 1). `m_-_n____: It will suffice to consider the case of G(m, m, n). The element -` -` ` [n=` ] w = diag A(~ m, `m ), . .,.A(~ m,,`m~)m m, 1, . .,.12 G(m, m, n) ____________-z___________" _______-z______" [n=`m] n-`m[n=`m] has order `. Note that ~-1 is not an eigenvalue for ~`m[n=`m]because ~`m[n=`m]= ~-1 , ` | `m [n=`m ] + 1 , `m gcd(`, m) | `m [n=`m ] + 1 ) `* *m | 1 FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 35 which is not the case as `m > 1. Therefore the ~-1 eigenspace L<~w>has rank [n=* *`m ]. The two monomorphisms A(`, 1, [n=`m_])//_CG(m,m,n)(w)A(m,o1,o[n=`m_])given by -` -1 -` ~i! diag E,_._.,.E_-z___", A(~ m, `m ) , E, . .,.E, ~__m,_1,_.-.,.1z_____" (i-1)`m n-`m[n=`m] -` diag E,_._.,.E_-z___", ~E, E, . .,.E, ~__m* *,_1,_.-.,.1z_____" ~i (i-1)`m n-`* *m[n=`m] agree on their common domain A(gcd(`, m), 1, [n=`m ]) and together with the inc* *lusion " Ø " of permutation groups [n=`m]Ø_//_ `m[n=`m]//_,m they define a homomorphi* *sm on A(lcm(`, m), 1, [n=`m ]) o [n=`m]such that the composition * * <~w> A(lcm(`, m), 1, [n=`m ]) o [n=`m],! CG(m,m,n)(w) i Im CG(m,m,n)(w) ! GL * *(L ) is an isomorphism with image similar to the reflection group G(lcm(`, m), 1, [n* *=`m ]). The outer automorphism group of X(G(m, r, n)) is isomorphic to A(m, r, n)\Zxp* *A(m, 1, n) except in the cases (m, r, n) 2 {(2, 1, 2), (4, 2, 2), (3, 3, 3), (2, 2, 4)} [5* *2, x6] [48, 7.14]. The (exotic) homotopy action æ: Cm = <~>! Out(X(m, r, n)) ~=A(m, r, n)\ZxpA(m, 1, n) that takes the generator ~ of Cm to A(m, r, n)(~, 1, . .,.1) is distinct form t* *he actions through unstable Adams operations of (5.13) when gcd(r, n) > 1 [48, 7.14]. Proposition 5.14. Assume that m 2, r 1, n 2, and (m, r, n)62{(2, 1, 2), (4, 2,* * 2), (3, 3, 3), (2, 2, 4)}. Then the homotopy fixed point group X(m, r, n)hCm = X(m, 1, n - 1) for the above exotic homotopy action on X(m, r, n). Proof. The second assumption of (5.8) is clearly satisfied as the action preser* *ves the generators x1, . .,.xn-1 but does not preserve the generator e. To verify the first assum* *ption, take æ: _____C x __ m ! NGL(L)(G(m, r, n)) = ZpG(m, 1,tn)o be the obvious choice æ(~) = (~* *, 1, . .,.1). Then _jC _jC n-1 G(m, r, n) m= A(m, r, n) o n-1, L m= Zp and the composition " _ _jC _jC A(m, 1, n - 1) o n-1Ø__//_G(m, r, n)jCm////_ImG(m, r, n) m! GL (L m) where the first morphism is (~2, . .,.~m ) ! ((~2. .~.n)-1, ~2, . .,.~n), n-1 * *,! n, identifies the group to the right as the reflection group G(m, 1, n - 1). 5.15. The sporadic p-compact groups. We identify the fixed point p-compact grou* *ps for actions through unstable Adams operations on the p-compact groups corresponding* * to the 34 sporadic reflection groups of the Clark-Ewing classification table. These p-* *compact groups are determined by their rational Weyl groups except that the local isomorphism * *system of G35 contains two 3-compact groups E6 and P E6 [48, 11.18]. However, (E6)hC2 and (P * *E6)hC2 are identical so that in diagram (17) G35can mean either of these two. 36 CARLES BROTO AND JESPER M. MØLLER The relationship in terms of homotopy fixed point groups displayed in the dia* *gram C3037) = (G32, L32) meaning that that EhC38= X(G32). (2) (G37 = W (E8), C4, G31, p 1 mod 4) There is an element w 2 G37 of orde* *r 4 such that (CG37(w), L37) = (G31, L31) meaning that EhC48= X(G31). (3) (G37= W (E8), C5, G16, p 1 mod 15) There is an element w 2 G37 of orde* *r 5 and a primitive 5th root of unity ~ 2 Zxpsuch that (CG37(w), L<~w>37) = (G16, L16) meaning that EhC58= X(G16). (4) (G34, C4, G10, p 1 mod 12). There exists an element w 2 G34of order 4,* * a (index 4) subgroup G of CG34(w), and a primitive 4th root of unity ~ 2 Zxpsuch that (G, L<~w>34) = (G10, L10) meaning that X(G34)hC4 = X(G10). (5) (G32, C4, G10, p 1 mod 12) There is an element w 2 G32 of order 4 and * *a primitive 4th root of unity i 2 Zxpsuch that (CG32(w), L32) = (G10, L10) which means that X(G32)hC4 = X(G10). (6) (G32, C30, C5, p 1 mod 30) There is an element w 2 G32 of order 5 and * *a primitive 5th root of unity ~ 2 Zxpsuch that (CG32(w), L<~w>32) = (C30, Zp) which means that X(G32)hC5 = S59. (7) (G31, C3, G10, p 1 mod 12). There exists an element w 2 G31 of order * *3 and a primitive 3rd root of unity ~ 2 Zxpsuch that (CG31(w), L<~w>31) = (G10, L10) . FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 37 This means that X(G31)hC3 = X(G10). (The group that the computer finds * *is G10 and not G15(of the same rank and the same degrees) because the elements * *of order 8 square to central elements [56, p. 281].) (8)(G31, C8, G9, p 1 mod 24). There exists an element w 2 G31 of order 8* * and a primitive 8th root of unity ~ 2 Zxpsuch that the reflection group (CG31(w), L<~w>31) = (G9, L9) which means that X(G31)hC8 = X(G9). (9)(G10, C8, C24, p 1 mod 24) There is an element w 2 G10 of order 8 and * *a primitive 8th root of unity ~ 2 Zxpsuch that (CG10(w), L<~w>10) = (C24, Zp) which means that X(G10)hC8 = S47. (10) (G9, C3, C24, p 1 mod 24) There is an element w 2 G9 of order 3 and a * *primitive 3rd root of unity ~ 2 Zxpsuch that (CG9(w), L<~w>9) = (C24, Zp) which means that X(G9)hC3 = S47. (11) (G34, C9, C18, p mod 18) There is an element w 2 G34 of order 9 and a * *primitive 9th root of unity ~ 2 Zxpsuch that (CG34(w), L<~w>34) = (C18, Zp) which means that X(G34)hC9 = S37. The homotopy fixed point p-compact groups shown in C18Oo18o_G36CO G35D_5__//C5 * *(17) 18| 6zzzzz CC4CCC 2zzzzz DD3DDD | | __zzz C!!C 4 __zzz D""D G26 G8 oo___G28 G25 DD -- DDD zz DDD --- |8 DDD zzz 12| 12 DD""D""-12--fflffl|| 3 D""D __z2zz fflffl|| C12 C8 G5 __12_//C12 are justified by (5.9) and the following computer computations: (1)(G36 = W (E7), C6, G26, p 1 mod 6) There is an element w 2 G36 of orde* *r 6 and a primitive 6th root of unity ~ 2 Zxpsuch that (CG36(w), L<~w>36) = (G26, L26) which means that X(G36)hC6 = X(G26). (2)(G36 = W (E7),_C4, G8, p 1 mod 8). There is an element w 2 G36 of ord* *er 4, a subgroup W < CG36(w) of index 8, faithfully represented in L36, and* * a primitive 4th root of unity i 2 Zxpsuch that ___ (W , L36 ) = (G8, L8) ___ which means that X(G36)hC4 = X(G8). (The reflection group W contains el* *ements of order 8 with central square so it is not similar to G13[56, p. 281].) 38 CARLES BROTO AND JESPER M. MØLLER (3) (G36, C14, C14, p 1 mod 14) There is an element w 2 G36of order 14 and* * a primitive 14th root of unity ~ 2 Zxpsuch that (CG36(w), L<~w>36) = (C14, Zp) which means that X(G36)hC14= S27. (4) (G36, C18, C18, p 1 mod 18) There is an element w 2 G36of order 18 and* * a primitive 18th root of unity ~ 2 Zxpsuch that (CG36(w), L<~w>36) = (C18, Zp) which means that X(G36)hC18= S35. (5) (G26, C18, C18, p 1 mod 18) There is an element w 2 G26of order 18 and* * a primitive 18th root of unity ~ 2 Zxpsuch that (CG26(w), L<~w>26) = (C18, Zp) which means that X(G26)hC18= S35. (6) (G8, C12, C12, p 1 mod 12) There is an element w 2 G8 of order 12 and * *a primitive 12th root of unity ~ 2 Zxpsuch that (CG8(w), L<~w>8) = (C12, Zp) which means that X(G8)hC12= S23. (7) (G8, C8, C8, p 1 mod 8) There is an element w 2 G8 of order 8 and a pr* *imitive 8th root of unity ~ 2 Zxpsuch that (CG8(w), L<~w>8) = (C8, Zp) which means that X(G8)hC8 = S15. (8) (G35= W (E6), C2, G28= W (F4), p 1 mod 2) There is an element w 2 G35o* *f order 2 such that (CG35(w), L<-w>35) = (G28, L28) which means that EhC26= F4. (9) (G35, C3, G25, p 1 mod 3) There is an element w 2 G35 of order 3 and a* * primitive 3rd root of unity ~ 2 Zxpsuch that (CG35(w), L<~w>35) = (G25, L25) which means that X(G35)hC3 = X(G25). (10) (G35, C5, G25, p 1 mod 5) There is an element w 2 G35 of order 5 and a* * primitive 5th root of unity ~ 2 Zxpsuch that (CG35(w), L<~w>35) = (C5, Zp) which means that X(G35)hC5 = S9 (11) (G35, C4, G8, p 1 mod 4) There is an element w 2 G35 of order 4 and a * *primitive 4th root of unity i 2 Zxpsuch that (CG35(w), L35) = (G8, L8) which means that X(G35)hC4 = X(G8). FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 39 (12) (G25, C2, G5, p 1 mod 6) There is an element w 2 G25of order 2 such th* *at (CG25(w), L<-w>25) = (G5, L5) which means that X(G25)hC2 = X(G5). (13) (G28, C3, G5, p 1 mod 6) There is an element w 2 G28 of order 3 and a * *primitive 3rd root of unity ~ 2 Zxpsuch that (CG28(w), L<~w>28) = (G5, L5) which means that X(G28)hC3 = X(G5). (14) (G28, C4, G4, p 1 mod 4) There is an element w 2 G28 of order 4 and a * *primitive 4th root of unity i 2 Zxpsuch that (CG28(w), L28) = (G8, L8) which means that X(G28)hC4 = X(G8). (15) (G25, C12, C12, p 1 mod 12) There is an element w 2 G25of order 12 and* * a primitive 12th root of unity ~ 2 Zxpsuch that (CG25(w), L<~w>25) = (C12, Zp) which means that X(G25)hC12= S23. (16) (G55, C12, C12, p 1 mod 12) There is an element w 2 G25of order 12 and* * a primitive 12th root of unity ~ 2 Zxpsuch that (CG5(w), L<~w>5) = (C12, Zp) which means that X(G5)hC12= S23. 6. Homotopy fixed points of twisted unstable Adams operations Let X be a p-compact group and set ff: X ! X a p-compact group automorphism. * *The homotopy pullback diagram BF ff(X) __'____//_BX * *(18) '|| || fflffl|1xff fflffl| BX ______//_BX x BX serves as the definition of the space BF ff(X). If ff is homotopic to ff0, then* * one easily checks that BF ff(X)' BF ff0(X). The homotopy class of ff is an element ff 2 Out(X), and, in turn, this is rep* *resented by a loop ff: S1 ! B aut(BX), hence representing an action of Z on BX, BX ! BXhZ ! S* *1, in the sense of section 5 (see equation (15)). This fibration can also be obtained* * as the Borel contruction for the action of the positive integers, N, on BX, determined by Bf* *f: BX ! BX, thus BXhZ ' BX xN R+, hence, the homotopy fixed point space for this action is * *BXhZ ' Map N(R+, BX). This last can be easily identified with BF ff(X). In the special case where ff = ø_q is a twisted unstable Adams operation with* * q 2 Zp, q 6= 1, and q 6 0 mod p, we have BF ø_q(X) ' BfiX(q), or just BX(q), if ø = 1.* * For q = 1 we trivially obtain BX(1) ' (BX), the free loop space. Assume that ff represents an element of finite order r in Out(X), with r prim* *e to p, and X is a connected p-compact group. According to Theorem B, it defines an action * *of the cyclic 40 CARLES BROTO AND JESPER M. MØLLER group Cr on BX. Next proposition shows that the natural map BXhCr ! BF ff(X) i* *s a homotopy equivalence. Proposition 6.1. Assume that X is a connected p-compact group. If ff: BX ! BX * *rep- resents an element of Out (X) of finite order r, coprime to p, then BF ff(X) is* * homotopy equivalent to the space of free loops on BXhCr, where the action of the cyclic * *group Cr on BX is given by ff. Proof.According to Theorem B, ff defines an action of Cr on X, _p__//_ BX __i_//_BXhCrsooBCr_ and the space of homotopy fixed points is the homotopy fibre of the* * induced map Map (BCr, BXhCr)s ! Map (BCr, BCr)id, thus we have an adjoint map BXhCr x BCr ! BXhCr that produces a lifting to a map i: BXhCr ! BX that makes the triangle BX:: uuu | iuu | uuu | u | BXhCr |ff II | III | III | i I$$fflffl| BX commutative up to homotopy (BXhCr is simply connected by Lemma 5.1). Therefore* *, we can form a homotopy commutative diagram BXhCr Q________________//_QBXhCrTTT * *(19) | QQQQQQ | TTTTTTT | (( | )) | BXhCr ________________//_BXhCr x BXhCr | | | | | | fflffl| || fflffl| || BF ff(X) ________|________//BX T | QQQ | TTTTT | QQQQ | TTTTT | QQ((fflffl|(1,ff) T)) fflffl| BX ____________________//_BX x BX . We will show that BXhCr ! BF ff(BX) is a homotopy equivalence. According to * *Theo- rem 5.2, BXhCr is the classifying space of a connected p-compact group and by L* *emma 5.1 the natural map i: BXhCr ! BX induces an identification of the homotopy groups * *of BXhCr with the invariant elements in the homotopy groups of BX by the action of Cr: ß* *i(BXhCr) ~= ßi(BX)Cr ,! ßi(BX). There is a long exact sequence for the homotopy groups of B* *F ff(X): . .-.!ßi(BF ff(X)) -!ßi(BX) 1-ff*---!ßi(BX) -!ßi-1(BF ff(X)) -!. . . The same construction for the top square of diagram (19) degenerates to . .-.!ßi( BXhCr) -!ßi(BXhCr) 0-!ßi(BXhCr) -!ßi-1( BXhCr) -!. . . FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 41 Both long exact sequences together give 0 _____//_ßi+1(BX)Cr_____//ßi( BXhCr)_____//_ßi(BX)Cr_____//0 | | | | | | fflffl| fflffl| fflffl| 0 ____//_Coker{1 - ff*}_//_ßi(BF ff(X))__//_Ker{1 - ff*}__//0 . Now, Ker{1 - ff*} = ßi(BX)Cr and Coker{1 - ff*} ~= ßi+1(BX)Cr. Since r is co* *prime to p, and the homotopy groups ßi(BX) are Z(p)-modules for every i 2, the comp* *osition ßi+1(BX)Cr ! ßi+1(BX) ! ßi+1(BX)Cr is an isomorphism. Hence also the middle ver* *tical map ßi( BXhCr) ! ßi(BF ff(X)) is an isomorphism. Our next result will reduce, in many cases, the question of describing BF ff(* *X) to two separate steps. The computation of homotopy fixed points BXhCr, for elements ff* * of order r coprime to p, and the case in which ff = _q is an unstable Adams operation of e* *xponent q 1 mod p, q 6= 1 (see Theorem 2.2 and formula (2) in section 2). It is one * *of the two ingredients of Theorem C Proposition 6.2. Let X be a connected p-compact group. If ff is an automorphis* *m of X that factors ff = _qfi with (1) q 1 mod p, and (_q)*: H*(X; Fp) ! H*(X; Fp) is the identity, and (2) fi is an automorphism of X that represents an element of finite order r, co* *prime to p, in Out(X), then BF ff(X) ' BXhfi(q). Proof. Notice first that BXhfiis again a p-compact group, according to Theorem * *B, and ff restrict to _q on BXhfi. We will write BY = BXhfifor simplicity. With this no* *tation we have a homotopy commutative diagram BY (q)O______________//_BYP PP | OOOOO || PPPPP | O''O (1,_|q) P(( | BY _____________//_BY x BY | | | | | | fflffl| || fflffl| || BF ff(X) ______|______//_BX | OOO | PPPP | OOO | PPPP | O''fflffl|O(1,ff) P(( fflffl| BX _____________//_BX x BX where the top and bottom faces are homotopy pullback diagrams, and the front fa* *ce commutes up to homotopy because ff ' _q O fi and fi is homotopic to the identity when re* *stricted to BY . Consequently, the homotopy fibres of the vertical maps form another homoto* *py pullback diagram: E ___________//X=Y | | | | fflffl|(1,ff) fflffl| X=Y ______//_X=Y x X=Y where E = hofib(BY (q) ! BF ff(X)), and we still denote by ff the self-equivale* *nce of X=Y induced by ff: BX ! BX. Again, Theorem B implies that X=Y is a connected H-sp* *ace 42 CARLES BROTO AND JESPER M. MØLLER and then we can also describe E as the homotopy fibre of 1 - ff: X=Y ! X=Y , an* *d it also implies that the map (_q)*: H*(X=Y ; Fp) ! H*(X=Y ; Fp) can be read off the map* * (_q)* defined on H*(X; Fp), which, by hypothesis is the identity. This fact easily i* *mplies that (1 - ff)* = (1 - fi)*. According to Proposition 6.1, the homotopy fibre of 1 - fi is contractible, h* *ence (1 - fi)* is an automorphism of H*(X=Y ; Fp). Thus, a spectral sequence argument shows th* *at E is mod p acyclic. Finally, it is easy to see that E is p-complete, hence contracti* *ble, and therefore BY (q) ' BF ff(X). Remark 6.3. If X polynomial, the effect of _q, q 1 mod p, on mod p cohomology* * of X is determined by the effect on H*(BX, Fp) and this is in turn determined by the* * effect on H*(BTX ; Fp) which is multiplication by q, hence the identity. For X = F4, E6,* * E7, E8 at the prime 3 or X = E8 at the prime 5, we also obtain that _q, q 1 mod p, acts* * trivially on H*(X; Fp). The generators for this cohomology algebras either transgress to* * elements detected in the maximal torus or are linked to such elements by Steenrod operat* *ions (cf. [40, Ch7]). In particular, 6.2 applies to all 1-connected p-compact groups, p odd, a* *ccording to the classification theorem [6]. One further reduction is obtained by extending the action of Z on BX to an ac* *tion of Zp. We will se that this is possible for the action of unstable Adams operations _q* * of exponent q 1 mod p, and in this case we obtain that the homotopy fixed point space BX(* *q) = BXhZ depends only on the p-adic valuation p(1 - q). Let ff be an element of Out(X) = ß1B aut(BX) represented by a loop !ff:S1 ! B* * aut(BX) that classifies an action BX ! BXhZ ! S1. If the homomorphism ß1(!ff): Z ! Out* * (X) extends to Zp, then we can also extend !ffto a map ^!ff:^S1p! B aut(BX), which * *is an action of Zp on BX that extends the original action determined by ff. Lemma 6.4. Let X be a p-compact group. Assume that the action of Z on BX determ* *ined by an element ff 2 Out(X) extends to the p-adics, then BF ff(X)' BXhZ ' BXhZp. Proof.There is a map of fibrations BX ________BX | | | | fflffl| |fflffl BXhZ ____//_BXhZp | | | | fflffl| |fflffl BZ ______//_BZp where the right fibration is the p-completion of the left one. In fact, Zp can * *only act nilpo- tently on Hi(BX, Fp), which are finite Fp-vector spaces, hence the fibration on* * the right is preserved by p-completion. Since the base and the fibre are p-completed spaces,* * so is the total space BXhZp. The above diagram is a pullback diagram, so the left fibration is * *also preserved by p-completion, and then, since the top and bottom horizontal arrows are p-equ* *ivalences, so is the middle one. FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 43 The functor Map (S1, -) preserves pullback diagrams, thus, we have another pu* *llback dia- gram BXhZ ________________BXhZ | | | | fflffl| fflffl| Map (BZ, BXhZ)^1____//_Map(BZ, BXhZp)^1 | | | | fflffl| fflffl| Map (BZ, BZ)1 ______//_Map(BZ, BZp)1 and then, the map BZ ! BZp, which is a mod p equivalence, induces a diagram BXhZp ______'________//_BXhZ | | | | fflffl| fflffl| Map (BZp, BXhZp)^1_'__//_Map(BZ, BXhZp)^1 | | | | fflffl| fflffl| Map (BZp, BZp)1 __'___//_Map(BZ, BZp)1 where the middle and bottom horizontal maps are weak equivalences by [8, II,2.8* *], hence so is the top horizontal map and the lemma follows. Using to the description of Out (X) in section 2 we will see how those extens* *ions are obtained in case of actions of unstable Adams operations _q of exponent q 1 m* *od p. In order to compare actions of Z of Zp given by unstable Adams operations, we must* * analyse the diagram of group homomorphisms Hom (Zp, Zxp)__res__//_Hom(Z, Zxp) | | | | |fflffl res fflffl| Hom (Zp, Out(X))____//_Hom(Z, Out(X)) where the horizontal homomorphisms are given by restriction and the vertical on* *es by the inclusion of the Adams operations q 2 Zxp7! _q 2 Out(X). Recall that, for an odd prime p, Zxp~=Z=p - 1 x Zp, where Z=p - 1 correspond * *to the roots of unity contained in Zxpand Zp is identified with the subgroup of elements q 2* * Zxp, with q 1 mod p, via the exponential map: a 2 Zp 7! exp(pa) 2 Zxp (exp defined by the usual expansion exp(pa) = 1 + pa + . .).. Since there are n* *o non-trivial homomorphisms Zp ! Z=p - 1, the group Hom (Zp, Zxp) can be parametrized by Zp i* *n the following way m 2 Zp 7! !m 2 Hom (Zp, Zxp), defined !m (a) = exp(pma). Using the standard identification Hom (Z, Zxp) ~= Zxpgiven by evaluation at 1* * 2 Z, the restriction map is described by ~= res ~= Zp -! Hom (Zp, Zxp)--! Hom (Z, Zxp) -! Zxp m 7! !m 7! !m |Z 7! !m (1) = exp(pm) = 1 + pm + . . . 44 CARLES BROTO AND JESPER M. MØLLER It follows that the image of the restriction consists of actions given by unsta* *ble Adams operations _q with q 1 mod p, for which we can choose m = 1_plog(q). Now, we can prove the second ingredient of Theorem C. Proposition 6.5. If q, q0 2 Zxp, q q0mod p, both are of order r mod p, and * *p(1 - qr) = p(1 - q0r), then BX(q) ' BX(q0), for any connected p-compact group X. Proof.The proof is divided in two steps. First, we will consider0the case q q* *0 1 mod p (r = 1). In these cases, the actions of Z given by _q and _q , respectively, ex* *tend to actions of the p-adics described by mq = 1_plog(q) and mq0= 1_plog(q0), respectively. T* *he homotopy fixed points space BXhZp depends only of the image of the action Zp ! Out(X). T* *he image of the two actions are clearly the same if and only if mq and mq0differ by a p-* *adic unit; that is, if and only if p(mq) = p(mq0), if and only if p(1 - q) = p(1 - q0), in * *which case, we have q hZp h_q0 0 BX(q) ' BXh_ ' BX ' BX ' BX(q ) . In the general case, we can decompose q = i . q0 and q0= i . q00, where i is * *a primitive rth of unity and q0 q00 1 mod p, thus BX(q) ' BXh``(q0) ' BXh``(q00) ' BX(q0) . Remark 6.6. If q is a p-adic unit, we can find a prime number q0 such that q * *q0 mod p and p(1 - qr) = p(1 - qr0), where r is the order of q mod p, and then, BX(q) ' BX(q0) by Proposition 6.5. In fact, we can assume that q is an integer, otherwise change it by the sum o* *f enough first terms in its p-adic expansion. Then, by Dirichlet's theorem there is a prime nu* *mber q0 of the form q0 = pN c + q, with N > p(1 - qr), satisfying the above conditions. Proof of Theorem C.Part (1) follows from Proposition 6.2 and Remark 6.3. Part * *(2) is Proposition 6.5. 7.General structure of finite Chevalley versions of p-compact groups In this section we will study the first general properties of finite Chevalle* *y versions BX(q) of p-compact groups X. The main results being the identification of the maximal* * finite torus, the Weyl group, and the fusion category of elementary abelian p-subgroups. Proposition 7.1. Let X be a connected p-compact group and ff a self homotopy eq* *uivalence of X. Then (1) BF ff(X)is connected and p-complete. (2) ': BF ff(X)! BX is a homotopy monomorphism at p. (3) For any finite p-group P , Map (BP, BF ff(X))c ' BF ff(X). Proof.From the definition we obtain a fibration X -! BF ff(X)-'!BX where X and * *BX are p-complete, X is connected and BX is simply-connected. It follows that BF * *ff(X) is connected and p-complete. FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 45 For any finite p-group P , Map (BP, BX)c ' BX, and Map *(BP, X) ' X for any c* *hoice of base point. It then follows that ': BF ff(X)! BX is a homotopy monomorphism at * *p, and from the induced fibration Map (BP, X) ! Map (BP, BF ff(X))c ! Map (BP, BX)c it follows that Map (BP, BF ff(X))c ' BF ff(X). Lemma 7.2. Let X be a p-compact group, ff a self homotopy equivalence of BX, an* *d (P, ) an object of Fp(BX) fixed by ff up to homotopy; that is, ' ffO . If CX (P, ) is* * connected, then there is a unique lifting of :BP ! BX to a homotopy monomorphism g :BP ! BF f* *f(X), and Map (BP, BF ff(X))g____________//_Map(BP, BX) | | | | fflffl| 1xff] fflffl| Map (BP, BX) _______//Map(BP, BX) x Map (BP, BX) is a homotopy pullback diagram. Proof. Since (18) is a homotopy pullback diagram, there is at least a lifting o* *f , g :BP ! BF ff(X). The homotopy fibre of Map (BP, BX) -! Map (BP, BX) xMap (BP, BX) is CX (P, * * ) = Map (BP, BX) , hence pulling back along 1 x ff] we obtain a fibration, up to * *homotopy, '] CX (P, ) -!Map (BP, BF ff(X))^ -! Map (BP, BX) where Map (BP, BF ff(X))^ consists of all possible liftings of up to homotopy* *. The base space consists of just one connected component, hence if we assume that the fib* *re CX (P, ) is also connected, then the total space must be connected, and therefore any ot* *her lifting of is homotopic to g. The following lemma will help us determine the restriction of ff to the centr* *alizers. Lemma 7.3. Let X be a connected p-compact group and ff a self-equivalence of BX* *. Let T (ff) be a given restriction of ff to the maximal torus T = TX , and (P, ) an object* * of Fp(BX). Suppose that :BP ! BX admits a factorization ~: BP ! BT through the maximal torus j :BT ! BX. Then, the object (P, ) is fixed by ff if and only if T (ff)* *~ = w~ for an element w of the Weyl group. If this is the case, the restriction to the max* *imal torus of the induced self homotopy equivalence ff|CX(P, )of the centralizer CX (P, ) is* * T (ff|CX(P, )) = w-1 O T (ff). Proof. (P, ) is fixed by ff means that ' ff O B , and if factors as j O ~,* * that is to say, j O B~ ' ff O j O ~ ' j O T (ff) O ~, and according to [49, 4.1], [44, 3.4], th* *is is equivalent to the existence of w, in the Weyl group of X, such that w O ~ ' BT (ff) O ~. 46 CARLES BROTO AND JESPER M. MØLLER Now assuming the existence of such element w, we read from the commutative di* *agram T(ff) w BTOO__________________//_BToo________________BTOOOO ' |ev| '|ev| ' ev|| | T(ff)] | w] | Map (BP, BT )~_______//_Map(BP, BT )w~oo____ Map (BP, BT )~ jjjj ' |j]| '|j]| jjj'jjjjj fflffl| ff] fflffl|ttjjj]jj Map (BP, BX) ________//Map(BP, BX) that the restriction of ff|CX(P, )= ff] to the maximal torus of CX (V, ) is w-* *1 O T (ff). If the centralizer CX (V, ) is connected, this determines the restriction ff* *|CX (V, ) (see x2). Corollary 7.4. Let X be a p-compact group and :BV ! BX a toral elementary abe* *lian p- subgroup such that its centralizer CX (V, ) is connected. If _q is an unstable* * Adams operation of exponent q 1 mod p, q 6= 1, then (a) there is a unique lift of to g :BV ! BX(q), (b) _q|CX(V, )= _q is as well an unstable Adams operation of exponent q, and (c) the centralizer of (V, g) in X(q) is CX(q)(V, g) ~=CX (V, )(q). Proof.In particular, when :BV ! BX is a toral elementary abelian p-group in * *X and ff = _q is an Adams operation of exponent q 1 mod p, then we can write T (_q)* * = _q, the qth power map in the maximal torus T = TX and _q O ~ ' ~, where ~: BV ! BT is a* * lift to BT of :BV ! BX, so, by Lemma 7.3, there is a commutative diagram T(_q)=_q BT ________________//BT | | | | fflffl|_q|CX(V, ) fflffl| BCX (V, )__________//BCX (V, ) | | | | fflffl| _q fflffl| BX ________________//BX this proves (b), namely, _q|BCX(V, )is, as well, an unstable Adams map _q. Now, (a) and (c) follow from Lemma 7.2. We will now restrict our attention to cases with q 1 mod p, q 6= 1. Accord* *ing to Proposition 6.2, the general case can be reduced to this one, in the cases that* * are of interest to us (see Remark 6.3). Hence, essentially, there will be no loss of generality* * in our assumption. Proposition 7.5. Let X be a connected p-compact group, p an odd prime, and _q a* *n unstable Adams operation of exponent q 2 Z*p, with q 1 mod p, q 6= 1. Then the inclusi* *on :BtX ! BX of the subgroup of elements of order p in the maximal torus TX has a unique* * lift to g :BtX ! BX(q) and its centralizer is CX(q)(tX , g) = TX (q) . Proof.Since CX (tX , ) = TX and _q|TX = T (_q) = _q this follows from 7.3 (see* * 7.4). FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 47 The subgroup TX (q) ~= T`n ~=(Z=p`)n, where n is the rank of X and ` = p(q -* * 1), established in Proposition 7.5 will be referred to as the maximal finite torus * *of X(q). Then, we define the Weyl group of X(q) as the automorphism group WX(q)(T`n) = AutFp(BX(q))(T`n) of T`nin the category Fp(BX(q)). Proposition 7.6. Let X be a connected p-compact group, p an odd prime, and _q a* *n unstable Adams operation of exponent q 2 Z*p, with q 1 mod p, q 6= 1. If T`n~=(Z=p`)n * *is the maximal finite torus of X(q), then its Weyl group is WX(q)(T`n) ~=WX the Weyl group of X, with action on T`ngiven by the mod p` reduction of the p-a* *dic represen- tation of WX . The extension NX(q)(T`n) = T`no WX(q)(T`n) fits in the homotopy * *commutative diagram BNX(q)(T`n)____//_BN(TX ) | | | | fflffl| fflffl| BX(q) ________//_BX . Proof. It follows from the diagram Map *(BT`n, BT`n)^i'__//Map*(BT`n, BTX )c'Oi * *(20) | | | | fflffl| fflffl| Map (BT`n, BX(q))i_']_//_Map(BT`n, BX)'Oi, where ^iis the set of components that map down to the component of the inclusio* *n i: BT`n! BX(q) and similarly, d' Oiis the set of components that map down to the compone* *nt of ' O i. Now, Map (BT`n, BX)'Oi' BTX and by Lemma 7.2 the upper horizontal arrow in (20)* * induces a bijection on components, and hence an equivalence of homotopy discrete spaces. For X a p-compact group and ff a self equivalence, the inclusion ': BF ff(X)!* * BX induces a functor between the respective fusion categories ']:Fp(BF ff(X)) -! Fp(BX) and Lemma 7.2 above give some useful information in order to compare the morphi* *sm sets. Thus, for instance, Mor Fp(BFff(X))((P, g), (Q, h)) -! Mor Fp(BX)((P, ' O g), (Q, ' O * *h))(21) is a bijection provided CX (P, ' O g) is connected. It rarely happens that thos* *e centralizers are connected for a general p-group P , but it is not so unusual if we restrict to * *some particular classes of small groups. For a space Y , we denote Fep(Y ) the full subcategory* * of Fp(Y ) whose objects are the elementary abelian subgroups of Y . Corollary 7.7. Let p be an odd prime. If X is a connected polynomial p-compact * *group and ff a self homotopy equivalence, then the functor ']:Fep(BF ff(X)) -! Fep(BX) is both full and faithful. 48 CARLES BROTO AND JESPER M. MØLLER Proof.If X is a connected polynomial p-compact group, then centralizers of elem* *entary abelian p-subgroups are connected and Lemma 7.2 applies. In fact, if (E, ) is* * an elemen- tary abelian p-subgroup of X, then the centralizer CX (E, ) is also a polynomi* *al p-compact group, hence H1(BCX (E, ); Fp) = 0 and therefore CX (E, ) is connected (see [* *24, 1.3]) and the map (21) is a bijection for every elementary abelian p-subgroups (P, g)* * and (Q, h) of BF ff(X). Corollary 7.8. Let p be an odd prime. If X is a connected polynomial p-compact * *group and _q an unstable Adams operation of exponent q 2 Z*p, with q 1 mod p, then ']:Fep(BX(q)) -! Fep(BX) is an equivalence of categories. Proof.By Corollary 7.7 we only have to check that '] induces in this case a bij* *ection between isomorphism classes of objects, and this follows from Proposition 7.5, because * *in a polynomial p-compact group every elementary abelian subgroup is toral. Let X be a polynomial p-compact group with trivial center and q 2 Z*pa p-adic* * unit with q 1 mod p, q 6= 1. Putting BCX(q)(V, g) = Map (BV, BX(q))g for any object (V* *, g) of Fep(BX(q)) we get a functor from Fep(BX(q))opto topological spaces. There is na* *tural map hocolim BCX(q)! BX(q) * *(22) Fep(BX(q))op from the homotopy colimit of this functor. When CX (V, g) is connected, we have BCX(q)(V, g) ' BF (_q|CX(V,g))(CX (V, ' O g)) ' BCX (V, ' O g)(q) according to Lemma 7.3 and Remark 7.4. Let TX be the maximal torus and WX the Weyl group of a p-compact group X, p o* *dd. As usually, we denote by tX the group of all elements of order p in TX , and g * *:BtX ! X(q) the inclusion. For any nontrivial elementary abelian p-subgroup E T , write W* * (E) for the point-wise stabilizer subgroup of E. Proposition 7.9. Let X be a polynomial p-compact group with trivial center, p o* *dd, and q 2 Z*pa p-adic unit with q 1 mod p, q 6= 1. Assume that H*(BX(q); Fp) ~=H*(BTX (q); Fp)WX and that H*(BCX(q)(E, g|BE ); Fp) ~=H*(BTX (q); Fp)W(E) for any nontrivial, subgroup E of tX . Then (22) is an Fp-equivalence. A similar statement holds with Fep(BX(q)) replaced by the full subcategory ge* *nerated by all objects of the form (tX )P where P runs through the subgroups of a Sylow p-subg* *roup of W . Proof.This follows from the Bousfield-Kan spectral sequence because the functor E ! H*(BCX(q)(E, g|BE ); Fp) = H*(BTX (q); Fp)W(E) is exact with limit H*(BTX (q); Fp)WX = H*(BX(q); Fp), [21, 8.1] [48, 2.16]. This result motivates the research in next sections of the cohomology rings H* **(BX(q); Fp) and the invariant rings H*(BTX (q); Fp)WX . FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 49 8.Cohomology rings This section is devoted to the proof of Theorem E. The Eilenberg-Moore spectr* *al sequence is used in order to get a hold of the cohomology rings of the spaces BX(q) of f* *ixed points of unstable Adams operations acting on polynomial p-compact groups BX. We foll* *ow the arguments of [57] that already contain the first part of the theorem. We end this section with an application to the unitary groups BU(n), BSU(n), * *in which we show that at the prime p, and for a p-adic unit q, the homotopy type of BU(n* *)(q), or BSU(n)(q), does only depend on the p-adic valuation p(1 - qm ), where m is * *the order of q mod p. Proof of Theorem E. Part (1) is due to L. Smith [57]. We will sketch his argume* *nts here and then will continue with the proof of the second part of the theorem. There is an Eilenberg-Moore spectral sequence associated to the pullback diag* *ram BX(q) ___'___//_BX * *(23) |'| || fflffl|1x_q fflffl| BX _____//_BX x BX . This is a second quadrant spectral sequence with Es,t2~=Tors,tH*(BX;Fp)(2H*(BX; Fp), H*(BX; Fp)) =) Hs+t(BX(q); Fp) converging to a graded ring associated of H*(BX(q); Fp). For simplicity, we will write P [xi] = P [x1, . .,.xn] ~=H*(BX; Fp). The Kosz* *ul complex E(xi) = P [xi] P [xi] E[sx1, . .s.xn] with bideg(sxi) = (-1, 2di) and d(sxi) = xi 1 - 1 xi, is a free resolution o* *f P [xi] as (P [xi] P [xi])-module, with module structure given by the multiplication m =* * *. Then, Tor **P[xi] P[xi](P [xi], P [xi]) is the homology of the complex P [xi] P[xi] P[xi]E(xi) ~=P [xi] E[sx1, . .s.xn] where now the action of P [xi] P [xi] on the left hand side term P [xi] in gi* *ven by the algebra map (1 x _q)*, hence one obtains the expression d(sxi) = xi- qdixi for the diff* *erential, but since q 1 mod p, we actually have d(sxi) = 0 for all i = 1, . .,.n. This yiel* *ds E**2~=Tor**P[xi] P[xi](P [xi], P [xi]) ~=P [x1, . .,.xn] E[sx1, * *. .,.sxn] and, since the algebra generators appear in filtration degrees 0 and -1, the sp* *ectral sequence collapses at the E2-page and then we can find elements yiin H*(BX(q); Fp) repre* *senting sxi in the graded associated ring, with H*(BX(q); Fp) ~=P [x1, . .,.xn] E[y1, . .,.yn] . Let TX be the maximal torus of X and WX the Weyl group. Since X is polynomial* *, the mod p cohomology ring of BX coincides with the invariants by the action of the * *Weyl group on the mod p cohomology of BTX , H*(BTX ; Fp)WX ~=H*(BX; Fp) ~=P [x1, . .,.xn]. 50 CARLES BROTO AND JESPER M. MØLLER According to 7.4, 7.5, the classifying space of maximal finite torus of X(q) * *is BT (q) ~=BT`n and it is obtained from a pullback diagram BT`n ___'___//BT * *(24) |'| || fflffl|1x_q fflffl| BT _____//BT x BT . Furthermore, the Weyl group is WX (7.6) hence, the restriction map i*: H*(BX(q); Fp) ! H*(BT`n; Fp) has image in the invariant subring by the action of the Weyl group, WX . It rem* *ains to show that this restriction map is injective. The pullback diagram (24) yields another Eilenberg-Moore spectral sequence: __s,t s,t * * s+t n E 2 ~=TorH*(BT;Fp) 2(H (BT ; Fp), H (BT ; Fp)) =) H (BT` ; Fp) . * * __** We will pay special attention to the map between the two spectral sequences i* **: E**r! Er induced by the natural map from diagram (24) to diagram (23) given by inclusion* * of the maximal torus. In order to describe the induced map at the level of E2-pages, w* *e need some elementary algebraic considerations. Again for simplicity, we will write P [ti] = P [t1, . .,.tn] ~=H*(BTX ; Fp). * *The kernel of the multiplication m: P [ti] P [ti] ! P [ti] is a Borel ideal Ker m = (t1 1 - 1 t1, . .,.tn 1 - 1 tn) and then we can define derivations @i:P [ti] ! P [ti] for i = 1, . .,.n, in the following way. For any homogeneous polynomialPf 2 P [* *ti], f 1 - 1 f 2 Ker m, hence we can find an expression f 1 - 1 f = ici(f)(ti 1 -* * 1 ti), with coefficients ci(f) 2 P [ti] P [ti], and then define @i(f) = m(ci(f)) 2 P* * [ti]. A routine calculation shows: (1) @i is well defined and does not depend on the choice of coefficients c1(f),* * . .,.cn(f), (2) @i is a derivation of P [ti], and (3) @i(ti) = 1 and @i(tj) = 0 if j 6= i. These properties show that these are the partial derivatives: @f @i(f) = ___. @ti After these considerations we can easily describe the map_between the respectiv* *e E2-pages ** and show that it is injective. In order to compute the E2 , we define now the K* *oszul complex E(ti) = P [ti] P [ti] E[st1, . .s.tn] with bideg(sti) = (-1, 2) and d(sti) = ti 1 - 1 ti. As before, we obtain that __** ** E 2 ~=TorP[ti] P[ti](P [ti], P [ti]) ~=P [ti] P[ti] P[ti]E(ti) ~=P [ti] * *(E[st1,2.5.s.tn]) since the differential in this complex_turns out to be trivial, again, because * *q 1 mod p. Also ** as before, the algebra_generators_of E 2 appear in filtration degree 0 and -1 a* *nd therefore ** the spectral sequence E r collapses at the E2-page. FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 51 Now, the inclusion i*: P [xi] ! P [ti] extends to a map of Koszul complexes i*: E(xi) ! E(ti) which is a P [xi] P [xi]-module map defined by X i*(sxi) = ci(xi) stj j on generators. Then the induced map i*: Tor**P[xi] P[xi](P [xi], P [xi]) ~=P [xi] E[sx1, . .s.xn] -! Tor**P[ti](P[ti]P [ti], P [ti]) ~=P [ti] * * E[st1, . .s.tn] is determined by X X @xi i*(sxi) = @j(xi) stj = ___ stj. j j @tj Now, i* is injective because the jacobian determinant is non-trivial, ` ' @xi J = det ___ 6= 0, @ti by [61]. Since both spectral sequences collapse at the E2-page, it * * follows that i*: H*(BX(q); Fp) ! H*(BT`n; Fp) is also injective. Remark 8.1. The argument with the Eilenberg-Moore spectral sequence used in the* * proof of part (1) of the above Theorem applies more generally to the case of any unst* *able Adams operation _q of arbitrary exponent q 2 Z*pacting on a polynomial p-compact grou* *p (see [57]). Under these more general hypothesis we obtain that if H*(BX) ~= P [x1, . .,.xn]* * then the cohomology of BX(q) is H*(BX(q); Fp) ~=P [xi1, . .,.xik] E[yi1, . .,.yik] where the polynomial generators xijcorrespond to those xi with degree 2di = deg* *xi where m|di, if m is the order of q mod p, and 2di- 1 = degyi. Notice that we can write q = iq0 where i is an m-root of one in Zp and q0 1* * mod p. 0 `` `` Hence _q = _q O _ , and _ has finite order m as automorphism of the p-compact * *group X. i It follows from 6.2, 6.3, that BY (q0) ' BX(q) if BY = BXh_ . Moreover, by Theo* *rem B, i Y = Xh_ is again a polynomial p-compact group. According to Theorem E the coho* *mology of BY must be H*(BY ; Fp) ~=P [xi1, . .,.xik] . 9. Invariant theory Let X be a polynomial p-compact group of rank n and let q be a p-adic unit, q 1 mod p, q 6= 1, and ` = p(1 - q). In Theorem E(2) we obtained a monomorp* *hism i*: H*(BX(q); Fp) ,! H*(BT`n; Fp)WX , where T`nis the maximal finite torus of B* *X(q) and WX the Weyl group (see 7.5, 7.6). Whether or not i* is an isomorphism, H*(BX(q)* *; Fp) ~= H*(BT`n; Fp)WX , is now a question of invariant theory and this is the subject * *of this section. Continuing with the notation of the precedent section we write V = tX for the* * elements of order p in the maximal finite torus and identify the dual vector space with the* * two dimensional primitive elements in the cohomology of BT`n, V *~=P H2(BT`n; Fp). The Bockstei* *n operations 52 CARLES BROTO AND JESPER M. MØLLER provide a vector space isomorphism P H2(BT`n; Fp) ~=H1(BT`n; Fp), that we will * *denote as d: V *! dV *, of degree (-1). If P (V *) is the symmetric algebra on V *and E(* *dV *) the exterior algebra on dV *, we can describe the algebra structure of H*(BT`n; Fp)* * as K(V *) = P (V *) E(dV *) = P [x1, . .,.xn] E[dx1, . .,.dxn] , and d extends to an algebra derivation on K(V *). Moreover, any subgroup G GL* *(V ) of linear substitutions acts on K(V *) in a natural way that commutes with the der* *ivation d, hence K(V *)G is still a differential algebra. Assume that P (V *)G = P [æ1, . .,.æn] is a polynomial algebra; in particular* *, G is a pseu- doreflection group. Then dæ1, . .,.dæn are also invariant under the action of G* *. The purpose of the next theorem is to establish the cases in which {æ1, . .,.æn, dæ1, . .,.* *dæn} is a free system of generators for K(V *)G. P n If we write dæi = j=1aijdxj, the jacobian J = det(aij) 2 P (V *) is invaria* *nt relative to det-1; that is, for any g 2 G, g . J = det(g)-1J. The relative invariants form * *a free module over P (V *)G on one generator P (V *)Gdet-1= fdet-1. P (V *)G, for an element * *fdet-12 P (V *) which is unique up to an invertible of Fp (see [14]). It follows that fdet-1div* *ides J. Theorem 9.1 ([9]). Let V be a vector space of dimension n over a field of chara* *cteristic p 6= 2. Assume that G GL(V ) is a group of linear substitutions such that P (V *)G = * *P [æ1, . .,.æn] is a polynomial algebra, then K(V *)G = P [æ1, . .,.æn] E[dæ1, . .,.dæn] P n if and only if fdet-1has degree degfdet-1= i=1(deg æi- 2). Proof.Since P (V *)G = P [æ1, . .,.æn] is a polynomial ring of invariants, the * *Jacobian is non- zero, J 6=0 (see [61]), and this implies that the homomorphism P [æ1, . .,.æn] * *E[dæ1, . .,.dæn]! K(V *) defined from the free anticommutative algebra to the subalgebra of K(V ** *)G by map- ping the variable æito the polynomial æiof P (V *)G and dæito the differential * *of æiin K(V *) is injective. If I = (i1, . .,.ik) is an ordered sequence of integers 1 i1 < . . .< ik * *n, we write dæI = dæi1dæi2. .d.æikand also dxI = dxi1dxi2. .d.xik. Let F P (V *) be the gr* *aded field of fractions of P (V *). Then, F K(V *) = F P (V *) P(V *)K(V *) is a vector spac* *e over F P (V *) spanned by {dxI}I. And then, {dæI}I is also a base of F K(V *). P n Assume that degfdet-1= i=1(deg æi- 2). This is the degree of the Jacobian * *J, hence J = fdet-1,Pup to an invertible of Fp. Let w 2 K(V *)G be an arbitrary element.* * We can write w = IwIdæI, with wI 2 F P (V *) and then we will show that actually, for each* * index I, wI 2 P (V *). We choose I0 of minimal length such that wI06= 0. Let I00be the c* *omplementary sequence, then w dæI00= wI0dæI0dæI00= wI0dæ1. .d.æn = wI0Jdx1. .d.xn is an element of K(V *)G, and, since dx1. .d.xn is invariant relative to det, w* *I0J 2P (V *)Gdet-1= fdet-1P (V *)G. So, our assumption implies that wI0 2 P (V *)G. Now we can re* *peat the argument with w - wI0dæI02 K(V *)G. It follows that each wI belongs to P (V *)G* * and then w 2 P [æ1, . .,.æn] E[dæ1, . .,.dæn]. FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 53 P n Assume otherwise that degfdet-16= i=1(deg æi- 2); that is, J = 'fdet-1for s* *ome element ' 2 P (V *)G of positive degree, then dæ1. .d.æn w = __________= fdet-1dx1. .d.xn ' is an element of K(V *)G which does not belong to P [æ1, . .,.æn] E[dæ1, . .,* *.dæn]. Example 9.2 (G a non-modular group [3]). If G GL(V ) is a pseudoreflection gr* *oup of order not divisible by p, then it is known that P (V *)G = P [æ1, . .,.æn] is a polyn* *omial algebra and also that the degree of fdet-1is twice thePnumber of pseudoreflections of G.POn* * the other hand, the number of pseudoreflection is G is ni=1(degji_2- 1). Hence degfdet-1= n* *i=1(deg æi- 2) and then Theorem 9.1 implies K(V *)G = P [æ1, . .,.æn] E[dæ1, . .,.dæn] . For a group G GL(V ) we denote [x]Q= {gx | g 2 G} the orbit of an element x* * 2 V *. The coefficients ci of the polynomial y2[x](X - y) = Xm + c1Xm-1 + . .+.cm-1 * *X + cm * * Q are the Chern classes of the orbit [x] and belong to P (V *)G. The element cm * * = y2[x]y is also called the Euler element of [x]. If we choose just one element zL 2 L \* * [x] for eachQ 1-dimensional vector subspace L of V *that intersects the orbit [x] non trivial* *ly, E[x] = zL is the pre-Euler element of the orbit [x], defined up to a non-zero escalar. Th* *is is a relative invariant respect a linear character Ø of G that we can associate to the orbit * *[x] by the equation g(E[x]) = Ø(g) . E[x], for all g 2 G. (See [9, 14].) Example 9.3 (Family 1 in the Clark-Ewing list: n+1). The symmetric group n+1* * acts on the integral lattice of SU(n + 1) that we can describe as V = Z{(^t1- ^tn+1* *), (^t2- ^tn+1), . .,.(^tn- ^tn+1)} where n+1 permutes the letters ^t1, . .^.tn+1. Dual* *ly, V *is generated by classes t1, t2, . .,.tn, and n+1 permutes t1, t2, . .,.tn, tn+1 with the re* *lation t1 + t2 + . .+. tn + tn+1 = 0. The orbit of t1 is [t1] = {t1, t2, .Q.,.tn, tn+1}, and the Chern classes of t* *his orbit obtained as the coefficients of the polynomial n+1i=1(X - ti) are, up to a sign, the g* *enerators ci of the invariant ring P (V *) n+1= P [c2, . .,.cn+1]. The orbit of t1 - t2 is [t1 - t2]= {(ti- tj) | 1 i, j n + 1, i 6= j} = { (ti- tj) | 1 i fi j n + 1} = { (ti- tj) | 1 i fi j n} [ { (t1 + . .+.2ti+ . .+.tn) | 1 * *i n} , thus the pre-Euler element associated to this orbit is Y Y E = E[t1 - t2] = (ti- tj) (t1 + . .+.2ti+ . .+.tn) 1 ifij n 1 i n except in the particular case n = 2 at p = 3, in which case E[t1 - t2] = (t1 - * *t2). We can check that the linear character associated to the pre-Euler element is precisel* *y the determinant (det = det-1in this case) and also that the degree of E, n2+ n, coincides with * *the degree of the jacobian J in all cases except n = 2 at p = 3. Thus for (n, p) 6= (2, 3), w* *e have K(V *) n+1= P [c2, . .,.cn+1] E[dc2, . .,.dcn+1] . The particular case n = 2 at the prime 3 will be considered in next Example 9.4. 54 CARLES BROTO AND JESPER M. MØLLER Example 9.4 ( 3 at the prime 3). The integral lattice of SU(3) is ß2(TSU(3)) = * *Z{(^t1- ^t3), (^t2- ^t3)} with the action of 3 that permutes ^t1, ^t2, and ^t3. If 3* * is generated by the 3-cycle oe and the transposition ø, the representation afforded by ß2(TSU(3)) i* *s determined by ` ' ` ' -1 -1 0 1 oe 7! 1 0 , ø 7! 1 0 . The dual action in mod 3 cohomology V *= H2(BTSU(3); F3) = F3{t1, t2} gives P (* *V *) 3 ~= P [x4, x6], where x4 = t12+ t1t2 + t22and x6 = t1t2(t1 + t2). This is the part* *icular case of Example 9.3 with n = 2 at the prime 3. The action extends to K(V *)) = P [t1, t2] E[dt1, dt2] where we obtain inva* *riant elements y3 = dx4 = (t2 - t1)dt1 + (t1 - t2)dt2 y5 = dx6 = (t22- t1t2)dt1 + (t12- t1t2)dt2 and y4 = (t2 - t1)dt1dt2 so that y3y5 = (t12+ t1t2 + t22)(t2 - t1)dt1dt2 = x4y4. These elements together with the polynomial invariants generate the invariant r* *ing K(V *) 3: P [x4, x6] E[y3, y4, y5] K(V *) 3 ~=______________________. * *(26) (y3y5 - x4y4, y3y4, y4y5) The proof follows the method of Theorem 9.1. In this particular case 1, dt1, dt* *2, dt1dt2 is a basis of K(V *) as a free P (V *)-module, while 1, y3, y5, y3y5 or 1, y3, y4, y* *5 are basis of F K(V *) as graded F P (V *) vector spaces. Assume that w is any element of K(V *) 3. We can write w = w0 + w1y3 + w2y4 + w3y5. First, multiply this equality by y* *4: wy4 2 K(V *) 3 and wy4 = w0y4 = w0(t2 - t1)dt1dt2. Then, w0(t2 - t1) 2 P (V *)d3et-1* *= (t2 - t1)P (V *) 3, hence w0 2 P (V *) 3. Next w0 = w - w0 = w1y3 + w2y4 + w3y5 2 K(V *) 3. We now multiply this equali* *ty by x4-1y5 2 K(V *) 3: w0x4-1y5 2 K(V *) 3 and w0x4-1y5 = w1x4-1y3y5 = w1y4, and th* *en again the equality w1y4 = w1(t2 - t1)dt1dt2 2 K(V *) 3 implies that w1 2 P (V *) 3. Similarly, we obtain that also w2, w3 2 P (V *) 3, hence w belongs to the rin* *g generated by x4, x6, y3, y4, y5. This proves the isomorphism (26). Example 9.5 (Family 2a in the Clark-Ewing list: G = G(m, r, n), r|m|p-1, [9]). * *G(m, r, n) is the subgroup of GLn(Zp) generated by the permutation matrices and the diagonal * *matrices m_ diag(`1, . .,.`n), where `mi = 1 and (`1. .`.n) r = 1. In particular, G(m, 1, * *n) is isomor- phic to the semidirect product (Z=m)n o n.Q In this case we clearly have P (V * **)G(m,1,n)= P [æ1, . .,.æn], where 1 + æ1 + . .+.æn = ni=1(1 + xmi), if we write P (V *) * *= P [x1, . .,.xn]. Now, æn = (x1. .x.n)m is the Euler element associated to the orbit of x1, [x1].* * The pre-Euler element is E1 = E[x1] = x1. .x.n. It carries an associated linear character Ø1* *, defined by Ø1(diag(`1, . .,.`n))m= `1. .`.nand Ø1(oe) = 1 if oe 2 n is a permutation matr* *ix. Notice that __ G(m, r, n) = KerØ1r and m_ P (V *)G(m,r,n)= P [æ1, . .,.æn-1, E1 r] . FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 55 The orbit of (x1Q- x2) is [x1 - x2] = { `1xi- `2xj| `m1= `m2= 1 , i < j } and i* *ts pre-Euler element is E2 = i<* ` = 0 S2m-1(q) = S2m-1(1 + p`) 0 < ` < 1 >: S2m-1(1) ` = 1, where ` = p(1 - qm ), S2m-1(1 + p`) ' B(Z=p`o Cm )^p, and S2m-1(1) ' S2m-1. In fact, if we write q = i . q0, with q0 1 mod p and i a root of 1 in Zp, th* *en S2m-1(q) ~=(S2m-1)h<``>(q0) by Proposition 6.2. Since qm 1 mod p if and only if i 2 Cm is an mth root of * *unity, we have, (i)for 0 = p(1 - qm ), i =2Cm and (S2m-1)h<``>is contractible (see 5.11), (ii)for 0 < ` = p(1 - qm ) < 1, i 2 Cm , hence (S2m-1)h<``>~=S2m-1 (see 5.11)* *, and therefore S2m-1(q) ~=S2m-1(q0). Moreover, ` = p(1 - q0) and Theorem 9.7 i* *mplies that BS2m-1(q0) ' B(Z=p`oCm )^p. Notice that this result does only depend on ` * *= p(1-qm ), hence also BS2m-1(1 + p`) ' B(Z=p`o Cm )^p, (iii)and finally, if p(1 - qm ) = 1, we have 1 = qm : q = i 2 Cm is itself an* * mth root of 1, so q0= 1, and BS2m-1(q) ~=BS2m-1(1) ' BS2m-1. Example 9.10 (SU(3)(q) at the prime 3). Fix q a 3-adic integer with 0 < ` = 3(* *1-q) < 1. According to Theorem E H*(SU(3)(q); F3) ~=P [x4, x6] E[y3, y5] , with fi(`)(y3) = x4 and fi(`)(y5) = x6. According to propositions 7.5 and 7.6, T`2~=(Z=3`)2 is the maximal finite tor* *us of SU(3)(q) with Weyl group 3. Now, the invariant ring P [x4, x6] E[y3, y4, y5] H*(T`2; F3) 3 ~=______________________ (y3y5 - x4y4, y3y4, y4y5) computed in Example 9.4, turns out to differ from H*(SU(3)(q); F3). The natura* *l map H*(SU(3)(q); F3) ,! H*(T`2; F3) 3 (see Theorem E) has cokernel isomorphic to P * *[x6]y4. Some interesting examples involve the group SU(3) or the spaces BSU(3)(q) and* * in this cases Propostion 7.9 will not apply. Instead, we need to develope ad hoc techni* *ques in order to obtain mod p homology decompositions of such spaces. Given a finite group G and subgroups H1, H2, . .,.Hk G, we define a finite * *category I(k) with objects {0; 1, 2, . .,.k}, where G is the group of automorphisms of 0 and * *for each i > 0, Hi\G = Hom I(k)(i, 0) as G-sets and Aut I(k)(i) = NG(Hi)=Hi. We will write Ii * *for the full subcategory with objects 0 and i. Those categories appear in the context of th* *e Aguad'e p-compact groups and other compact Lie groups, as Quillen categories of element* *ary abelian subgroups (see [1, 48]). Next result is essentially contained in [1, 48]. 58 CARLES BROTO AND JESPER M. MØLLER Proposition 9.11. Let M be a given diagram of Zp-modules index by the category * *I(k). Assume that (a) Restriction gives an isomorphism Hj(G; A) ~=Hj(H1; A), for any Z(p)G-module* * A and j 1. (b) p - |NG(Hi)| and Mi= MHi0, for every i 2. Then, there is an exact sequence 0 ! lim-0M ! MNG(H1)=H11 MG0! MNG(H1)0! lim-1M ! 0 , I(k) I(k) and lim-jI(k)M = 0 if j 2. Proof.We consider a star-shaped category I(k) with k + 1 objects {0, 1, 2, . .,* *.k}. There is an exact sequence of the form [48] Y N (H )=H Y N (H ) 0 !lim0M ! MG0x MiG i i! M0G i i>0 i>0 Y Y ! lim1M ! H1(G; M0) x H1(NG(Hi)=Hi; Mi) ! H1(NG(Hi); M0) i>0 i>0 Y Y ! lim2M ! H2(G; M0) x H2(NG(Hi)=Hi; Mi) ! H2(NG(Hi); M0) i>0 i>0 ! lim3M ! . . . Under condition (b) this exact sequence reduces to the exact sequence 0 ! lim0M ! MG0x MNG(H1)=H11! MNG(H1)0 ! lim1M ! H1(G; M0) x H1(NG(H1)=H1; M1) ! H1(NG(H1); M0) ! lim2M ! H2(G; M0) x H2(NG(H1)=H1; M1) ! H2(NG(H1); M0) ! lim3M ! . . . Condition (a) implies that H1 and G have the same Sylow p-subgroup. Hence p* * - |NG(H1)=H1| and so therefore H*(NG(H1); A) ~=H*(H1; A)NG(H1)=H1. Now, in the di* *agram given by restrictions Hj(G; A) ! Hj(NG(H1); A) ! Hj(H1; A), j 1, the composit* *ion is an isomorphism and the second arrow is a monomorphism, hence both arrows are isomo* *rphisms: Hj(G; A) ~=Hj(NG(H1); A) ~=Hj(H1; A) , j 1 , and the Proposition follows. Example 9.12 (G2 at the prime 3). The exceptional Lie group G2 has rank two and* * the Weyl group is dihedral D12, listed in family 2b for m = 6 in the Clark-Ewing li* *st. The category Fe3(G2) of non-trivial elementary abelian 3-subgroups of G2 is equival* *ent to the category I(2), with G = D12, the Weyl group of G2, H1 = oe3, and H2 = 2. The c* *entralizer diagram for elementary abelian 3-subgroups is equivalent to ____________________________________________________________* *(_3)op\(D)op___________________________________________________________ Z=2__BSU399____________________________________________________* *_____________BTH2H________oo_opop//_BU2Z=2ee_______________________________@ _______________________________________* *______(_2)__\(D)______________________ _______________________________________* *_________________ (D)op FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 59 and it contains SU(3) as one of the centralizers, hence Propostion 7.9 does not* * apply to BG2(q) at the prime 3 (see Example 9.10). We will see how Proposition 9.11 * *instead, implies that the centralizers diagram for elementary abelian 3-subgroups (diagr* *am (27) be- low) is in fact a sharp homology decomposition for BG2(q) at the prime 3. The cohomology of BG2 at the prime 3 is H*(BG2; F3) ~=H*(BT 2; F3)D12~=P [x4,* * x12]. Fix q a 3-adic integer with 0 < ` = 3(1 - q) < 1. According to Theorem E H*(SU(3)(q); F3) ~=P [x4, x12] E[y3, y11] , with fi(`)(y3) = x4 and fi(`)(y11) = x12. On the other hand, according to Corollary 7.8 the categories of non-trivial e* *lementary abelian 3-subgroups of G2 and G2(q) coincide: Fe3(G2(q)) ~= Fe3(G2), and furthe* *rmore, for every object (E, ) of Fep(G2), BCG2(q)(E, ) ' BCG2(E, )(q), thus the central* *izer diagram of elementary abelian subgroups of G2(q) is equivalent to ___________________________________________________(_3)op\(D)* *op__________________________________________________ Z=2__BSU3(q)77__________________________________________________* *__________BT`2JJoo_//_BU2(q)Z=2gg__________________________________(27) _____________________________________(* *_2)op\(D)op___________________________ ______________________________________* *____________________ (D)op and there is a natural map hocolimFep(G2(q))opBCG2(q)! BG2(q). We will see by * *direct computation that this map is in fact a sharp homology decomposition. Notice that H*(BU(2)(q); F3) ~=H*(BT`2; F3) 2, and then, by Proposition 9.11,* * there is an exact sequence D12 0 ! lim-0 H*(BCG2(q); F3) ! H*(BSU(3)(q); F3)Z=2 H*(BT`2; F3) Fep(G2(q)) 3xZ=2 1 * ! H*(BT`2; F3) ! lim- H (BCG2(q); F3) ! 0* * , Fep(G2(q)) and lim-iFe BCG2(q)= 0 if i 2, where 3 x Z=2 ~=ND12( 3) = D12. It clearly* * follows p(G2(q)) that lim-0H*(BCG2(q); F3) ~=H*(BSU(3)(q); F3)Z=2~=P [x4, x12] E[y3, y11] Fep(G2(q)) with x12= x62and y11= x6y5 in H*(BSU(3)(q); F3). The Bousfield-Kan spectral sequence lim-i Hj(BCG2(q); F3) =) Hi+j( hocolimeopBCG2(q); F3) Fep(G2(q)) Fp(G2(q)) collapses to the isomorphism H*( hocolim BCG2(q); F3) ~= lim0 H*(BCG2(q); F3) ~=P [x4, x12] E[y3, * *y11] ; Fep(G2(q))op Fe- p(G2(q)) in other words, hocolimFep(G2(q))opBCG2(q)! BG2(q) is a sharp homology decompos* *ition at the prime 3 and H*(G2(q); F3) ~= lim-0H*(BCG2(q); F3) ~=P [x4, x12] E[y3, y11] . Fep(G2(q)) 60 CARLES BROTO AND JESPER M. MØLLER Example 9.13 (G2 at odd primes). We will now complete the description of G2(q) * *at odd primes. In the previous example we have describe it at the prime 3. At primes b* *igger than 3, G2 turns out to be a connected non-modular p-compact group. Recall that the exc* *eptional Lie group G2 has rank two and the Weyl group is dihedral D12, listed in family * *2b for m = 6 in the Clark-Ewing list. Let p be an odd prime and q a p-adic unit. We will dis* *tinguish three cases: (1) q2 1 mod p, q2 6= 1: In this case Fep(G2 )(q) = Fep(G2 ), in particular t* *he p-rank of BG2(q) is two again, and its cohomology can be derived from Theorem E. (2) q2 6 1 mod p, q6 1 mod p, q6 6= 1: The element of order 3 in W (G2 ) has* * a 1-dimensional eigenspace of eigenvalue 2 in L(G2 ). Thus Z=3 x Z=2 = Z=6 acts on this eig* *enspace. We get an embedding N~(S11) ! N~(G2 ) and hence a monomorphism S11 ! G2 induc* *ing i * * 0 BS11 ' (BG2)h_ , where i is a 3rd primitive root of 1. Then, if we write q * *= iq , with i 0 11 0 11 q0 1 mod p, we have BG2(q) ' (BG2)h_ (q ) ' BS (q ) = BS (q), this last* * equality because i belongs to the Weyl group of S11. Now the p-rank of BG2(q) ' BS1* *1(q) is one and the cohomology ring follows from Theorem E. (3) q6 6 1 mod p: In this case q = iq0 with q0 1 mod p and i is a primitive * *root of one i whose order does not divide 6. It follows from Proposition 5.8 that (BG2)h_* * ' *, hence BG2(q) is as well contractible. In case q2 = 1, G2(q) is the free loop space G2, while for q2 6= 1 and q6 = 1,* * we have G2(q) ' S11. This result provides the geometric explanation of Kleinermann's computation o* *f cohomol- ogy rings of finite Chevalley groups of type G2 (see [33]). 10. Finite Chevalley versions of Aguad'e exotic p-compact groups In [1], Aguad'e constructed the exotic p-compact groups Xi, i = 12, 29, 31, 3* *4, with Weyl groups the groups G12 (rank 2, p = 3), G29 (rank 4, p = 5), G31 (rank 4, p = 5)* *, and G34 (rank 6, p = 7), on the Sheppard-Todd and Clark-Ewing lists, respectively. All * *four of them are obtained as the homotopy colimit of a diagram that we proceed by describing. Write Gi to denote one of the groups G12, G29, G31, or G34, and Z its center,* * namely, Z ~= Z=2 for G12, Z ~= Z=4 for G29, Z ~= Z=4 for G31, Z ~= Z=6 for G34, in all * *cases represented by diagonal matrices with entries p - 1 roots of unity. In all four* * cases we also fix a subgroup isomorphic to p. Then, the index category is the opposite categ* *ory of I(1), with two objects 0 and 1 and AutI(1)(0) = Gi, AutI(1)(1) = NGi( p)= p ~=Z , Mor I(1)(1, 0) = p\Gi, and Mor I(1)(0, 1) = ; . The functor assigns BT p-1to 0 and BSUp to 1, up to homotopy, where the center * *of Gi, Z, acts on BSUp via unstable Adams operations. The diagram is described in the * *following picture _______________________________________________________* *________________________________________________________( p)op\(Gi)op Z __BSUp88_______________________________________________* *________________BT(p-1Gi)opdd______________________________________________@ FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 61 Each Xi is a p-compact group with maximal torus TXi = T p-1and Weyl group WXi* * = Gi. The respective cohomology rings coincide with the invariant rings H*(BXi; * *Fp) ~= H*(BTXi; Fp)Gi, and these are the polynomial rings ([1, 2, 62]): H*(BX12; F3) ~=P [x12, x16] , H*(BX29; F5) ~=P [x8, x16, x24, x40] , H*(BX31; F5) ~=P [x16, x24, x40, x48] , H*(BX34; F7) ~=P [x12, x24, x36, x48, x60, x84] . Throughout this section we fix an unstable Adams operation _q of exponent q 2* * Z*pwith q 1 mod p, q 6= 1. We will describe the p-local structure of the spaces BXi(q* *), that have been defined by the homotopy pullback diagram BXi(q) ___'___//_BXi '|| || fflffl|(1,_q) fflffl| BXi ______//BXix BXi and will show that they are classifying spaces of p-local finite groups. In par* *ticular, cases i = 29, 34 provide new exotic examples of p-local finite groups. The first results on the p-local structure of BXi(q) are given by proposition* *s 7.5 and 7.6. Set ` = p(1 - q). The maximal elementary abelian p-subgroup of Xi, (tXi, ), f* *actors as a p-subgroup (tXi, g) of Xi(q), and the centralizer of this group CXi(q)(tXi, g) ' T`p-1~=(Z=p`)p-1 is the maximal finite torus of Xi(q). All elementary abelian p-subgroups of Xi* *(q) factor through this one. Moreover, the Weyl group is WXi(q)(T`p-1) = Gi, and the norm* *alizer NXi(q)(T`p-1) = T`p-1o Gi sits in the maximal torus normalizer of Xi(q), making* * homotopy commutative the diagram BNXi(q)(T`p-1)____//_BNXi(T p-1) | | | | fflffl| fflffl| BXi(q) ___________//BXi. Now, we fix the Sylow p-subgroup S = (Z=p`)(p-1)o Z=p of NXi(q)(T`p-1), gene* *rated by T`p-1and a p-cycle of p Gi. We will denote by f :BS ! BXi(q) the homotopy monomorphism obtained as the composition BS ! BNXi(q)(T`p-1) ! BXi(q). Then (S,* * f) is a p-subgroup of BXi(q), and it will play the role of a Sylow p-subgroup. Since Xi, i = 12, 29, 31, 34, are polynomial p-compact groups, according to C* *orollary 7.8, ': BXi(q) ! BXi induces an equivalence of categories ']:Fep(BXi(q)) -! Fep(BXi) . Thus, we obtain that every elementary abelian p-subgroup (E, ~) of BXi(q) fac* *tors as a subgroup of tXi: E tXi and ~ ' |BE . There is a distinguished subgroup Z=p ~* *=Z tXi such that, Z tXi TXi SUp ~=CXi(Z, |BZ ). If E tXi is not conjugate to * *Z in Xi, then the centralizer CXi(E, |BE ) is a p-compact group whose Weyl group, the p* *oint-wise stabilizer of E TXi, WXi(E), has order not divisible by p. In Xi(q), we obtai* *n: 62 CARLES BROTO AND JESPER M. MØLLER Proposition 10.1. There is one conjugacy class of elements of order p in Xi(q),* * (Z, g|BZ ), such that the centralizer is CXi(q)(Z, g|BZ ) ' SUp(q) and contains (S, f): BS M MMMMf Bincl|| MMMM fflffl| M&& BSUp(q) ____//_BXi(q) as Sylow p-subgroup of SUp(q). If E tXi represents another conjugacy class of elementary abelian p-subgrou* *ps, then CXi(q)(E, g|BE ) ' T`p-1o WXi(E) where the order of WXi(E) is not divisible by p. Furthermore, the diagram ____Bincl_// BT`(p-1) BS | Bincl|| |f| fflffl| j fflffl| BCXi(q)(E, g|BE_)___//BXi(q) is commutative up to homotopy, where j :BCXi(q)(E, g|BE ) ! BXi(q) is the natur* *al map induced by evaluation. Proof.For Z tXi, we have CXi(q)(Z, g|BZ ) ~=SUp(q) by Corollary 7.4. If E tXi be another subgroup, not conjugated to Z, then the centralizer in * *Xi is the connected non-modular p-compact group BCXi(E, |BE ) ' B(TXioWXi(E))^p, and the* *n, first, Corollary 7.4 implies that BCXi(q)(E, g|BE ) ' BCXi(E, |BE )(q), and secondly,* * Theorem 9.7 gives BCXi(E, |BE )(q) ' B(T`p-1o WXi(E))^p. f Finally, we use the inclusions BE ! BtXi ! BS -! BXi(q) in order to compare * *the centralizers of E and tXi in S and Xi(q): ' // // BT`p-1' BCS(tXi) ___________BCS(E) __________ BS ' f]|| |f]| |f| fflffl| fflffl| fflffl| BCXi(q)(tXi, g)___//_BCXi(q)(E, g|BE_)//_BXi(q) . Proposition 10.2. For i = 12, 29, 31, 34, the natural map hocolim BCXi(q)! BXi(q) * *(28) Fep(BXi(q))op is a mod p homology equivalence. FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 63 Proof. According to Theorem E the cohomology rings of BXi(q) are: H*(BX12(q); F3) ~=P [x12, x16] E[y11, y15] , H*(BX29(q); F5) ~=P [x8, x16, x24, x40] E[y7, y15, y23, y39] , H*(BX31(q); F5) ~=P [x16, x24, x40, x48] E[y15, y23, y39, y47] , H*(BX34(q); F7) ~=P [x12, x24, x36, x48, x60, x84] E[y11, y23, y35, y* *47, y59, y83] , and they embed in the invariant rings H*(BXi(q); Fp) H*(BT`p-1; Fp)Gi. These * *invariant rings are described in the Example 9.6. It turns out that the above inclusion i* *s an isomorphism if i = 29, 31, 34, but it is not surjective i = 12. The centralizers of elementary abelian p-subgroups of BXi(q) are described in* * Proposi- tion 10.1. The centralizer, CXi(q)(E, g|BE ), of an elementary abelian p-subgr* *oup E tXi in Xi(q) is either SUp(q) or a non-modular p-compact group. In cases i = 29, 31, 34, H*(CXi(q)(E, g|BE ); Fp) ~=H*(BTXi; Fp)W(E) is satis* *fied by Theo- rem E and examples 9.2 and 9.3, hence we meet the conditions of Proposition 7.9* * and the map (28) is a mod p homology equivalence. In the case i = 12, the Proposition 7.9 does not apply, so we will need a sep* *arate analysis. The p-compact group X12, p = 3, is also denoted DI 2= X12, because G12 ~=GL(2, * *3) and H*(BDI 2; F3) ~=H*(BT 2; F3)GL(2,3)~=F3[x12, x16] is the rank two Dickson algeb* *ra at p = 3. It admits two conjugacy classes of elementary abelian p-subgroups, one of rank * *one and another of rank two, hence so does BDI2(q), as well. We have equivalences of ca* *tegories Fep(BDI2) ~=Fep(BDI2(q)) ~=I(1) with AutI(1)(0) = GL(2, 3), AutI(1)(1) = NGL(2,3)( 3)= 3 ~=Z=2, where NGL(2,3)(* * 3) = 3 x Z=2, and Mor I(1)(1, 0) = 3\GL(2, 3), Mor I(1)(0, 1) = ;. The centralizers dia* *gram BCDI2(q) is described in the picture ________________________________________________________* *___________________________________________ op3\GL(2,3)op Z=2__BSU3(q)77_____________________________________________* *_______________BT`2GL(2,3)ophh______________________________________________@ The Bousfield-Kan spectral sequence Ei,j2~=lim-iHj(BCDI2(q); F3) =) Hi+j(hocolimopBCDI2(q); F3) I(1) I(1) computes the cohomology of the homotopy colimit hocolimI(1)opBCDI2(q). The computation of the E2-term follows from Proposition 9.11. Since NGL(2,3)(* * 3) ~= 3x Z=2 and H*(GL(2, 3); A) ~= H*(NGL(2,3)( 3); A) ~= H*( 3; A), for any GL(2, 3)-m* *odule A, there is an exact sequence GL(2,3) 0 ! lim-0H*(BCDI2(q); F3) ! H*(BSU(3)(q); F3)Z=2 H*(BT`2; F3) I(1) 3xZ=2 1 * ! H*(BT`2; F3) ! lim-H (BCDI2(q); F3) ! 0 , * * (30) I(1) and lim-iI(1)BCDI2(q)= 0 if i 2. The invariant rings H*(BT`2; F3)GL(2,3)and H*(BT`2; F3) 3as well as th* *e restriction R: H*(BT`2; F3)GL(2,3),! H*(BT`2; F3) 3have been described in examples 9.4 and* * 9.6. The cohomology of BSU(3)(q) is identified with the subalgebra P [x4, x6] E[y3* *, y5] of 64 CARLES BROTO AND JESPER M. MØLLER H*(BT`2; F3) 3. The cokernel of the inclusion is isomorphic to P [x6]y4, and t* *hen the ex- act sequence (30) is simplified to P [x12, x16] E[y10, y11, y15] 0 ! lim-0H*(BCDI2(q); F3) ! ____________________________ I(1) (y11y15- x16y10, y10y11, y10y15) ~R Z=2 1 * - ! P [x6]y4 ! lim-H (BCDI2(q); F3) !* * 0 , I(1) Z=2 and P [x6]y4 = P [x62](x6y4) which is in the image of ~R. It follows that lim-0H*(BCDI2(q); F3) ~=P [x12, x16] E[y11, y15] I(1) and lim-iI(1)BCDI2(q)= 0 if i 1, so, therefore the Bousfield-Kan spectral seq* *uence collapses to an isomorphism H*(hocolim BCDI2(q); F3) ~=lim0H*(BCDI2(q); F3) ~=P [x12, x16] E[y11, * *y15] ; I(1)op -I(1) that is, hocolimI(1)opBCDI2(q)! BG2(q) is a sharp homology decomposition at the* * prime 3 and H*(DI2(q); F3) ~=lim-0H*(BCDI2(q); F3) ~=P [x12, x16] E[y11, y15] . I(1) Theorem 10.3. (S, f) is a Sylow p-subgroup for BXi(q), the fusion system F(S,f)* *(BXi(q)) of the space BXi(q) over the p-subgroup (S, f) is saturated, and (S, F(S,f)(BXi(q)), L(S,f)(BXi(q))) is a p-local finite group with classifying space |L(S,f)(BXi(q))|^p' BXi(q) . Proof.It is a consequence of Theorem 4.6, using the above propositions 10.1 and* * 10.2. Now, we will go deeper into the structure of the fusion system F = F(S,f)(BXi* *(q)). We have seen that the fusion category of elementary abelian p-subgroups is equival* *ent to that of the p-compact group Xi; in particular, every elementary abelian p-subgroup i* *s toral; that is, F-conjugate to a subgroup of T`(p-1). If we denote Z = Z(S) the center of S* *, then (10.1) BCXi(q)(Z) = BSUp(q)^p' BSLp(q)^p, so, the centralizer fusion system CF (Z) ove* *r CS(Z) = S coincides with the fusion system of SLp(q) over S. Hence, we can identify S w* *ith the Sylow p-subgroup of SLp(q) and then use the notation of Example 3.6. Recall from 3.6* * that any centric radical subgroup of S in CF (Z) is conjugate to either S, T`(p-1), or a* *n extraspecial group 1(,r), r = 0, . .,.p - 1. Proposition 10.4. Any centric radical subgroup of S in F = F(S,f)(BXi(q)) is co* *njugate to one of the groups in the table: _______________________________________________ ___Q___________OutF_(Q)________Conditions______ T`(p-1) Gi S Z=(p - 1) x Z=(p - 1) * *(31) 1 GL2(p) __1(,)__________SL2(p)_________if_`_>_1_or_p_>_3. FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 65 Proof. The proof is divided in four steps, where we first determine a set of re* *presentatives for centric radical subgroups of S in F, and then refine it to a minimal set of* * representatives and compute their automorphisms groups in F. Step 1: Toral and non toral centric radical subgroups. T`p-1is centric in F and* * OutF (T`p-1) ~= Gi is p-reduced, hence T`p-1is also radical in F. No other subgroup of T`p-1is * *centric, so for any other centric and radical subgroup Q S in F, there is a morphism of exten* *sions Q0 _____//_Q___//_Z=p * *(32) | || || | | || fflffl| fflffl| || T`p-1____//_S___//Z=p where Q0 = T`p-1\ Q. We are assuming that Q is centric, hence the center Z ~= Z=p of S should be c* *ontained in Q0. But if Q0 = Z, then Q ~=Z=p x Z=p is elementary abelian and then toral i* *n F, hence it would not be centric. Thus Z 6= Q0 and the center of Q is Z(Q) = QZ=p0= Z. I* *n particular, every automorphism of Q restricts to an automorphism of Z, so we obtain a homom* *orphism Aut F(Q) ! AutF (Z). The kernel is composed of automorphisms of Q that restric* *t to the identity in Z; that is, automorphisms of Q in the centralizer fusion system CF * *(Z), hence we have an exact sequence 1 ! AutCF(Z)(Q) ! AutF (Q) ! AutF (Z) * *(33) where AutF (Z) Z=p - 1 lifts to AutF (T`(p-1)) and AutF (S) as unstable Adams* * operations (the center of Gi). Thus, if Q is radical in CF (Z), then it is radical in F. Step 2: Non-abelian centric radical subgroups, all of which abelian characteris* *tic subgroups are cyclic. Assume that all abelian characteristic subgroups of Q are cyclic, t* *hen a theorem of Hall implies that Q is the central product of an extraspecial group of exp* *onent p and a cyclic group C, where the elements or order p in C, 1(C), coincide with the * *center Z( ) of (cf. [29, Chap. 5, 4.9, 5.3]). The faithful irreducible representations of the central product of an extrasp* *ecial group or order p1+2rand a cyclic group of order p` over the algebraic closure of a fi* *eld of q elements, (q, p) = 1, have degree pr, and there are exactly p`-1(p - 1) inequivalent repr* *esentations in this degree. Hence, only the case r = 1 can appear in GLp(q). We denote 1 the extraspeci* *al group of order p3 and exponent p, and k the central product Z=pk O 1. The different* * irreducible faithful representations of k in GLp(q) are obtained by composing with the ext* *ension to k of the automorphisms of Z=pk, (Z=pk)*. Thus, there is at most one subgroup iso* *morphic to k in GLp(q), up to conjugation. A subgroup of GLp(q) isomorphic to 1 is de* *scribed in Example 3.5. Since CGLp(q)( 1) = Z(GLp(q)) ~=GL1(q), k is a subgroup of GLp(q)* * if and only if Z=pk < GL1(q). Hence `, ` = p(1 - q), is the biggest one that can occ* *ur in GLp(q) (see Example 3.5). Finally, the intersection of ` with SLp(q), and hence, of any conjugate of * *`, is isomorphic to 1, and there are exactly p conjugacy classes of such subgroups 1(,r) (see * *Example 3.6). These are radical in CF (Z), and so, therefore, they are also radical in F. 66 CARLES BROTO AND JESPER M. MØLLER Step 3: Non-abelian centric radical subgroups having non-cyclic abelian charac* *teristic sub- groups. Assume now that Q contains a non-cyclic abelian characteristic group. I* *f Q is radical in CF (Z), then it is radical in F. Now, we assume also that Q is not radical i* *n CF (Z). We can view Q S as subgroups of SLp(q) and GLp(q), for an appropriate prime* * power q such that S is the Sylow p-subgroup of SLp(q): ` = p(1 - q). Write N = NGLp(q)* *(Q). The arguments of [4, (4A)] show that (up to conjugacy in GLp(q)) Q N \ (Z=pk o Z=p) C N for some k `, or, taking the intersection with SLp(q) Q ~N\ Sk C ~N where Sk = (Z=pk o Z=p) \ SLp(q) S and N~= N \ SLp(q) = NSLp(q)(Q), an then InnQ (N~ \ Sk)=Z(Q) C AutCF(Z)(Q) where N~=CSLp(q)(Q) = Aut CF(Z)(Q). We will see that (N~ \ Sk)=Z(Q) is still n* *ormal in Aut F(Q). Assume that ' 2 AutF (Q) restricts to Z as the unstable Adams operation _``, * *i a (p - 1)st root of unity. If _1=``(Q) = Q0 S, then _1=``O': Q ! Q0is a morphism of F, tha* *t restricted to Z is trivial, hence a morphism of CF (Z). Since, we have assumed that Q is n* *ot radical in CF (Z), _1=``O' should be obtained as composition of restrictions of automorphi* *sms of centric radical subgroups of CF (Z), by Alperin fusion theorem [11, A.10]. This is the * *fusion system of SLp(q), and the Sylow p-subgroup S itself is the only centric radical that c* *ontains Q, hence, there is Ø 2 Aut CF(Z)(S) with Ø|Q = _1=``O ', hence ' = _``O Ø|Q extend* *s to an automorphism _``O Ø of AutF (S). Notice that _``(Sk) = Sk and also Ø(Sk) = Sk, * *hence, if g 2 Sk normalizes Q, we have ' O cg O '-1 = c'(g), with '(g) 2 ~N\ Sk. This pro* *ves that we have Inn Q (N~ \ Sk)=Z(Q) C AutF (Q) and since Q is radical in F, Q = Sk. We claim that only the case Sk = S is radical. First we compute the normaliz* *er of Z=pk o Z=p in GLp(q). The subgroup (Z=pk)p is a characteristic subgroup of Z=pk* * o Z=p, for it is the only abelian subgroup of index p, hence, NGLp(q)(Z=pk o Z=p) NGLp(q)((* *Z=pk)p). It is not difficult to compute NGLp(q)((Z=pk)p) = GL1(q) o p, the group of invertibl* *e matrices with only one non-trivial entry in each line and column. By direct computation one * *can obtain that NGLp(q)(Z=pk o Z=p) = GL1(q) . (Z=pk o N p(Z=p)), where GL1(q) is identifi* *ed with the subgroup of all diagonal matrices of GLp(q); that is, the center of GLp(q). Call Nk = NGLp(q)(Z=pk o Z=p) \ SLp(q). We have Nk ~=Bk o N p(Z=p), with pfi k p Bk = (z . x1, . .,.z . xp) 2 GL1(q) fixi2 Z=p , z x1. .x.p= 1 and NSLp(q)(Sk) = Nk. Notice that, when k < `, Sk has index p in the Sylow p-s* *ubgroup Bk o Z=p, and this is normal in Nk, hence only S = S` is radical in SLp(q). The centralizer of Sk in SLp(q) is CSLp(q)(Sk) = Z ~=Z=p and then Aut CF(Z)(* *Sk) ~= Aut SLp(q)(Sk) ~=Nk=Z. (Bk=Z)oZ=p is normal in Nk=Z, and, since the Adams opera* *tions _``, i a (p - 1)st root of unity, act internally in Bk, (Bk=Z) o Z=p is also a norma* *l of AutF (Sk): InnSk = Sk=Z=p C (Bk=Z=p) o Z=p C AutF (Sk) FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 67 thus, Sk is radical in the fusion system F if and only if k = `; that is, only * *the case Sk = S is radical. In this case we have obtained AutF (S) ~=N`=Z o Z=(p - 1), where Z=(p * *- 1) on the right is generated by the Adams operations of exponent a primitive (p - 1)st ro* *ot of unity, and OutF (S) ~=Z=(p - 1) x Z=(p - 1), given by the Adams operations and N p(Z=p* *)=Z=p. Step 4: Minimal set of representatives and automorphism groups. It remains to c* *heck which of those are F-conjugate to one of the others in the list and also to compute t* *heir F- automorphisms. For Q = S the restriction AutF (Q) ! AutF (Z) is split because unstable Adams* * operations extend to S. Moreover, since they are realized by the center of Gi, the F-auto* *morphisms of S are given by conjugation in the normalizer N`,iof the maximal finite torus* * T`(p-1). We have seen already that the same is true for Q = T`(p-1). Finally, we analyse the case Q = 1(,r), r = 0, . .,.p - 1. Assume that ' 2 * *Aut F(Q) and that the restriction to the center Z is the unstable Adams operation _z. Th* *is extends to an F-automorphism of S. Write Q0 = _z(Q). Then Ø = _z O '(-1):Q ! Q0 is a homomorphism of F that restricts to the identity in Z, hence it belongs to the * *centralizer fusion system CF (Z). In other words, every automorphism ' 2 AutF (Q) is the co* *mposite of an isomorphism Ø: Q ! Q0of CF (Z) and a unstable Adams operation _z. It is then enough to compute the effect of unstable Adams operations on the f* *amily of subgroups 1(,r). It turns out that unstable Adams operations restrict to autom* *orphisms of 1 = 1(,0) so that OutF ( 1) = GLp(q), while, for p > 3 or ` > 1, they conjuga* *te 1(,r) for r = 1, . .,.p - 1 to each other and OutF ( 1(,) = SLp(q). Corollary 10.5. The fusion system of BXi(q) is F(S,f)(BXi(q)) = < FN`,i(S) ; F 1(GL2(p)) , F 1(,)(SL2(p)) > , for p > 3 or ` > 1, and F(S,f)(BX12(q)) = < FN1,i(S) ; F 1(GL2(p)) >, for p = 3* * and ` = 1, where N`,i= NXi(q)(T`(p-1)) ~=T`(p-1)o Gi. Proof. It is a consequence of Proposition 10.4 and Alperin's fusion theorem for* * saturated fusion systems (see section 3). We end this section with a case by case study in order to determine which spa* *ces BXi(q) are p-completed classifying spaces of finite groups and which cases correspond to e* *xotic examples of p-local finite groups. We first observe that S contains no proper strongly closed subgroups in F = F* *(S,f)(BXi(q)) and so, according to [11, 9.2], if F BXi(q) is the p-completed classifying spac* *e of a finite group, this group is almost simple. In fact, a strongly closed subgroup of S in F is a normal subgroup P of S suc* *h that no element of P is F-conjugate to any element in S \ P . Now, if P is non trivial* * it contains at least an element of order p, and this is F-conjugate to an element of order * *p in T`(p-1). Now, the maximal elementary abelian p-subgroup t of T`(p-1)turns out to be an i* *rreducible Gi-module, hence t P and since the cycle of order p generating S=T`(p-1)is co* *njugate to an 68 CARLES BROTO AND JESPER M. MØLLER element of t, the extension of t by this cycle is in P . Thus we have a diagram* * of extensions PT ______//P____//_Z=p | || || | | || fflffl| fflffl| || T`(p-1)___//_S___//Z=p where t PT = P \ T . Now S=P ~= T`(p-1)=PT is abelian. The abelianization of * *S is seen to be Z=p x Z=p, and then we obtain that T`(p-1)=PT is either trivial or has or* *der p. It follows that all elements of order up to p`-1 of T`(p-1)belong to PT. Taking th* *e quotient by ___ ______(p-1) this subgroup we obtain an inclusion of Gi-modules PT T` , but again, this * *last is an ___ ______(p-1) irreducible Gi-module, hence PT = T` , and then P = S. Example 10.6. BX29(q) at p = 5 and BX34(q) at p = 7 are classifying spaces of e* *xotic p-local finite groups. We have seen that the Sylow subgroup does not contain any proper strongly clo* *sed subgroup in F(S,f)(BXi(q)), hence if this is the p-completed classifying space of a fini* *te group G, then G is almost simple [11, 9.2]. A complete list of almost simple groups with a Sy* *low subgroup of the characteristics of S is provided by [11, Proposition 9.5]. No one in the* * list contains G29 or G34 as automorphisms of T`(p-1)induced by conjugation in the group. Hence X* *29(q) at p = 5 and X34(q) at p = 7 are exotic. Example 10.7. BX12(q) at p = 3 is the 3-completed classifying space of a twiste* *d Chevalley group of type F4. More precisely, if ` = 3(q2- 1), there is a positive integer* * n such that also ` = 3(1 + 22n+1) and then BX12(q) ' B(2F4(22n+1))^3. The 3-completed classifying space of the twisted Chevalley groupn2F4(22n+1) c* *an be de- scribed at p = 3 as B(2F4(22n+1)) ' BF ff(F4), for ff = ' O _2 , where ' is the* * Friedlan- der's exceptional isogeny of F4. ' has the effect of reflecting the Dynkin dia* *gram of F4 by sending the short roots to the long roots and the long roots to 2 times shor* *t roots. Furthermore, '2 ' _2, and then we can choose i a square root of -2 in Z3 such t* *hat fi = ' O _1=``is a selfnequivalence of BF4 at p = 3 of order two and 2ni 1 mo* *d 3. We can write ff = fi O _2 ``, and then, by Proposition 6.2, BF ff(F4) ' (BF4)hfi(2* *ni). In [13] it is shown that (BF4)hfi' BX12, hence BX12(2ni) ' B(2F4(22n+1))^3. Since _-1 * *belongs to the Weyl group of X12, BX12(q) ' BX12(-q), and then, according to Theorem C,* * the homotopy type of BX12( q) does only depend on ` = 3(q2 - 1), thus, if we choos* *e n with ` = 3(q2 - 1) = 3(1 - 2ni) = 3(1 + 22n+1), then we have BX12(q) ' BX12(2ni) ' B(2F4(22n+1))^3. The local structure of 2F4(22n+1), also called Ree groups of characteristic two* *, was studied by Malle [37]. Example 10.8. For any 5-adic unit, q 2 Z*5, BX31(q) at p = 5 is the 5-completed* * classifying space of a Chevalley group of type E8, namely, BX31(q) ' BE8(22m+1)^5if 5(q4 -* * 1) = 5(1 + 24m+2). p ___ Let i = -1 be a primitive 4th root of unity. Since _i belongs to the Weyl g* *roup of X31, we can assume that q 1 mod 5 for otherwise we can multiply q by an appropriat* *e power of i and still have BX31(q) ' BX31(irq). Moreover, according to Theorem C, the * *homotopy type of BX31(q) will only depend on ` = 5(q4 - 1). FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 69 We fix a prime power q0 with q0 2 mod 5 and ` = 5( iq-1) = 5(q04-1) = 5* *(q02+1), where we choose +i or -i in order that the equality makes sense. We can write q0 = i . (-i . q0), where now -i . q0 1 mod p. Since _-1 belo* *ngs to the i Weyl group of E8, we can apply Proposition 6.2 and get BE8(q0) ' (BE8)h_ (-iq0)* *. Now we i have seen in Example 5.15(2), that (BE8)h_ ' BX31, so, therefore BE8(q0) ' BX31(-iq0) ' BX31(q0) , and this last is homotopy equivalent to BX31(q) by our choice of q0 with 5(q04* *-1) = 5(q4-1). Similar considerations can be made, more generally, at any prime p such that * *p 1 mod 4; that is, any prime at which X31 can be defined, and then obtain that BE8(q0) ' * *BX31(q0) for a prime power q0 with q02+ 1 0 mod p. The local structure of E8(q) was described in [35]. r pr+1 Remark 10.9. One can easily obtain natural maps BXi(qp ) ! BXi(q ), that at t* *he level of maximal finite torii induces an inclusion T`(p-1)+r T`(p-1)+r+1, and * *then obtain that the p-compact group Xi can be reconstructed by means of a telescope construction BXi' hocolimBXi(q) . q In particular, BX12= BDI2 and BX31are telescopes of classifying spaces of finit* *e Chevalley groups. 11. Finite Chevalley versions of generalized p-adic Grassmannians Let p be an odd prime, m 1, r 1, and n 1 with r|m|(p - 1). We denote by diag(a1, . .,.an) an nxn matrix with entries a1 through an in the diagonal and * *zero otherwise. Define the group fim m_ A(m, r, n) = diag(a1, . .,.an) fiai= 1, (a1. .a.n) r= 1 GLn(Zp) and G(m, r, n) = A(m, r, n) n GLn(Zp) where n is identified with the subgroup of permutation matrices in GLn(Zp). Ev* *ery group G(m, r, n) is a pseudoreflection group. Each group G(m, r, n) is realized as the Weyl group of a 1-connected p-compac* *t group X(m, r, n), whose cohomology is H*(BX(m, r, n); Fp) = H*(BT (X(m, r, n)); Fp)G(m,r,n)~=P [x1, . .,.xn-* *1, e] with deg(xi) = 2mi and deg(e) = 2mn_r. They are usually referred to as the ge* *neralized p-adic Grassmannians. This family of p-compact groups, as we have defined it, * *includes some classical Grassmannians, namely BX(1, 1, n) ' BU(n), BX(2, 2, n) ' BSO(2n)* *, and 2(m_r)-1 BX(2, 1, n) ' BSO(2n + 1). Furthermore, X(m, r, 1) ' ^Sp are the Sullivan * *spheres. The existence of X(m, r, n) for other values of m, r, and n is shown in [54, 52* *]. For m 2 and n 2, the groups G(m, r, n) form the family 2a in the list of * *Clark-Ewing. If n = 1, the groups G(m, r, n) ~=Z=(m_r) are cyclic and appear as family 3 in * *the Clark-Ewing list, the Weyl groups of the Sullivan spheres. For m = 1, G(1, 1, n) ~= n is th* *e symmetric group, which is a pseudoreflection group as Weyl group of GL(n, C), or U(n), bu* *t as a such 70 CARLES BROTO AND JESPER M. MØLLER it is not irreducible, hence it is not in the Clark-Ewing list. Family 1 in the* * list corresponds to n as Weyl group of SU(n). We are interested in the finite Chevalley versions of the generalized p-adic * *Grassmannians: the spaces BX(m, r, n)(q), defined by the the pullback diagram BX(m, r, n)(q)____'______//BX(m, r, n) * *(34) '|| || |fflffl(1,_q) fflffl| BX(m, r, n)______//BX(m, r, n) x BX(m, r, n) . Remark 11.1. Many cases already appear in the literature (cf. [25, 28, 54]). We* * can extract the following equivalences, up to p-completion, for a prime power q, coprime to* * p: (1) BSU(n + 1)(q) ' BSLn+1(q). (2) BU(n)(q) ' BX(1, 1, n)(q) ' BGLn(q). (3) BX(m, 1, n)(q) ' BGLmn (q). (4) BX(2, 2, n)(q) ' BSO(2n)(q) ' BSO+2n(q). According to Remark 6.6, we have that, also for any p-adic unit q, BSU(n + 1* *)(q), BX(m, 1, n)(q) and BX(2, 2, n)(q) are homotopy equivalent to classifying spaces* * of finite groups, up to p-completion. These also include the cases BX(m, 2, n)(q), that can be reduced to BX(2, 2, * *n)(q0) using propositions 5.13 and 6.2, so they are also equivalent, up to p-compeltion, to * *classifying spaces of orthogonal groups over finite fields. The above observations will be used as the starting point of the induction ar* *guments that we will develop in the rest of this section in order to study the structure of * *the finite Chevalley versions BX(m, r, n)(q), for q 1 mod p, q 6= 1, and general values of m, r, a* *nd n. Fix q 1 mod p, q 6= 1. The p-compact groups X(m, r, n) are polynomial, hen* *ce propo- sitions 7.5 and 7.6 apply. The maximal elementary abelian p-subgroup of X(m, r,* * n), (tX , ), factors as a p-subgroup, (tX , g), of X(m, r, n)(q), and the maximal finite tor* *us of X(m, r, n)(q) is BT`n' BCX(m,r,n)(q)(tX , g) where ` = p(q - 1). The Weyl group is WX(m,r,n)(q)(T`n) ~= G(m, r, n), and th* *e extension NX(m,r,n)(q)(T`n) ~=T`no G(m, r, n) sits in the maximal torus normalizer of X(m* *, r, n), making the following diagram homotopy commutative: BNX(m,r,n)(q)(T`n)__//_BNX(m,r,n)(T n) | | | | fflffl| ' fflffl| BX(m, r, n)(q)_______//BX(m, r, n) . Corollary 7.7 implies that the functor ']:Fep(X(m, r, n)(q)) ! Fep(X(m, r, n)) * *(35) is an equivalence of categories. The next result is a description of the centra* *lizers of elementary abelian p-subgroups. FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 71 Proposition 11.2. Let p be an odd prime, m 1, r 1, n 1 with r|m|(p - 1), * *and q 1 mod p, q 6= 1. Then, (1) any elementary abelian p-subgroup h: BE ! BX(m, r, n)(q), factors through t* *he maximal finite torus, and (2) for any subgroup E tx T`n, the centralizer of (E, g|BE ) in X(m, r, n)(* *q), BCX(m,r,n)(q)(E, g|BE ) ' BX(m, r, n0)(q) x BU(n1)(q) x . .x.BU(ns)(q)* * , n = n0 + n1 + . .+.ns, is determined by the point-wise stabilizer of E T`* *nin the Weyl group G(m, r, n), G(m, r, n)(E) ~=G(m, r, n0) x n1x . .x. ns. Proof. All elementary abelian p-subgroups of X(m, r, n) are toral, hence the sa* *me is true for X(m, r, n)(q) by the equivalence (35). If E tX , by Corollary 7.4, the re* *striction of _q to the centralizer of (E, g|BE ), is _q again, _q|CX(m,r,n)(q)(E,g|BE)= _q, and BCX(m,r,n)(q)(E, g|BE ) ' BCX(m,r,n)(E, |BE )(q) . The centralizers CX(m,r,n)(E, |BE ) are known to be connected p-compact groups* * of maximal rank, with Weyl group G(m, r, n0) x n1x . .x. ns, the point-wise stabilizer of* * E in T nby the action of the Weyl group G(m, r, n): BCX(m,r,n)(E, |BE ) ' BX(m, r, n0) x BU(n1) x . .x.BU(ns) , thus, BCX(m,r,n)(E, |BE )(q) ' BX(m, r, n0)(q) x BU(n1)(q) x . .x.BU(ns)(q) contains the same maximal finite torus T`nas X(m, r, n)(q), ` = p(q-1), n = n0* *+n1+. .+.ns and the Weyl group is G(m, r, n0) x n1x . .x. ns (see propositions 7.5 and 7.6* *). Proposition 11.3. Let p be an odd prime, m 1, r 1, n 1 with r|m|(p - 1), * *and q 1 mod p, q 6= 1. The natural map hocolim BCX(m,r,n)(q)-! BX(m, r, n)(q) Fep(X(m,r,n)(q))op is a mod p homology equivalence. Proof. According to Theorem E and Example 9.5 H*(BX(m, r, n)(q); Fp) ~=H*(BT`n; Fp)G(m,r,n)~=P [x1, . .,.xn-1, e] E[y1,* * . .,.yn-1, u] with deg(xi) = 2mi, deg(e) = 2mn_r, deg(yi) = 2mi - 1, and deg(u) = 2mn_r- 1. Since this is true for all values of m, r, n, we obtain from Proposition 11.2* * that also, for every elementary abelian p-subgroup E tX , H*(BCX(m,r,n)(q)(E, g|BE ); Fp) ~=H*(BT`n; Fp)G(m,r,n)(E) where G(m, r, n)(E) is the point-wise stabilizer of E in T`n, by the action of * *the Weyl group G(m, r, n). So, then, the result follows from Proposition 7.9. Fix a Sylow p-subgroup of NX(m,r,n)(q)(T`n), Sn,`~= Z=p` o Sn, where Sn is th* *e Sylow p- subgroup of the symmetric group n. Call f the composition BSn,`! BNX(m,r,n)(q)* *(T`n) ! BX(m, r, n)(q), Thus (Sn,`, f) is a p-subgroup of BX(m, r, n)(q). We will denote by F(m, r, n, q) = F(Sn,`,f)(BX(m, r, n)(q)) 72 CARLES BROTO AND JESPER M. MØLLER the fusion system of BX(m, r, n)(q) over (Sn,`, f) and by L(m, r, n, q) = L(Sn,`,f)(BX(m, r, n)(q)) , the associated centric linking system. Recall that the underlying category of F* *(m, r, n, q) is equivalent to Fp(BX(m, n, r)(q)). Theorem 11.4. If q is a p-adic unit such that q 1 mod p, q 6= 1, and ` = p(1* * - q), then, (Sn,`, f) is a Sylow p-subgroup for BX(m, r, n)(q) and (Sn,`, F(m, r, n, q), L(m, r, n, q)) is a p-local finite group with classifying space |L(m, r, n, q)|^p' BX(m, r, n)(q) . Proof.We proceed by induction on n, the p-rank of X(m, r, n)(q). For n < p, X(* *m, r, n) is a non-modular p-compact group, and then, X(m, r, n)(q) is the p-completed cl* *assifying space of a finite group (see 9.7). Also, for BX(1, 1, n) ' BU(n)^p, Remark 11.* *1 character- izes BX(1, 1, n)(q) as p-completed classifying spaces of finite groups. In all * *that cases, the conclusion of the theorem is clearly satisfied (see section 3). Assume, that n is large and that the theorem holds for every n0 < n. That is* *, for every n0 < n, BX(m, r, n0)(q) is the classifying space of the p-loca* *l finite group (Sn0,`, F(m, r, n0, q), L(m, r, n0, q)). The result about BX(m, r, n)(q) will * *follow from The- orem 4.6. We will show that the space BX(m, r, n)(q) and its p-subgroup (Sn,`, * *f) meet the conditions of 4.6. Condition (1) of 4.6 is satisfied by Proposition 7.1. Condition (2a) of 4.6 amounts to show that if E tX , then the * * centralizer BCX(m,r,n)(q)(E, g|BE ) is the classifying space of a p-local finite group. Thi* *s follows by the in- duction hypothesis. In fact, by 11.2, there is a homotopy equivalence BCX(m,r,n* *)(q)(E, g|BE ) ' BX(m, r, n0)(q) x BU(n1)(q) x . .x.BU(ns)(q), for n = n0 + n1 + . .n.s, a non-t* *rivial de- composition of n into positive summands, and by the induction hypothesis and [1* *1, 1.4] this is the classifying space of the p-local finite group defined as the product (Sn0,`, F(m, r, n0, q), L(m, r, n0, q)) x (Sn1,`, F(1, 1, n1, q), L(1, 1, n1, q)) x . .x.(Sns,`, F(1, 1, ns* *, q), L(1, 1, ns, q)) . Condition (2b) of 4.6 establishes that Sylow p-subgroups of centralizers of e* *lementary abelian subgroups of BX(m, r, n)(q) factor through (Sn,`, f). This is proved by* * reducing the question to unitary groups, obtained as centralizers of the center of Sn,`. Let Z ~=Z=p denote the diagonal elements of order p in T`n~= (Z=p`)n Sn,`. * *Then, the point-wise stabilizer of Z in T`nby the action of G(m, r, n) is n and therefor* *e, according to Proposition 11.2, BCBX(m,r,n)(q)(Z, g|BZ ) ' BU(n)(q). By naturality of the construction of the normalizer of the maximal finite tor* *us, we obtain a diagram BNU(n)(q)(T`n)___//_BNX(m,r,n)(q)(T`n) | | | | fflffl|Bjn fflffl| BU(n)(q) _______//_BX(m, r, n)(q) FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 73 hence a factorization of (Sn,`, f): BSn,`O * *(36) f0ssss OOOOfO sss OOOO yysss Bjn O'' BU(n)(q) _______________//BX(m, r, n)(q) . Choose any other subgroup E tX Sn,`. Assume that the point-wise stabiliz* *er of E in T`nby the action of G(m, r, n) is G(m, r, n)(E) ~=G(m, r, n0)x n1x. .x. ns. * *Define E0= Z .E tX , then, the point-wise stabilizer of E0will be G(m, r, n)(E0) ~= n0x * *n1x. .x. ns. The inclusions E E0 Z induce a commutative diagram of centralizers Bj]n BCX(m,r,n)(q)(E0, g|BE0)_//_BCX(m,r,n)(q)(E, g|BE ) * *(37) | | | | fflffl| Bjn fflffl| BCX(m,r,n)(q)(Z, g|BZ_)____//_BX(m, r, n)(q) . Now, BCX(m,r,n)(q)(E, g|BE ) ' BX(m, r, n0)(q) x BU(n1)(q) x . .x.BU(ns)(q) w* *ith Sylow p-subgroup Sn0,`x . .x.Sns,`while BCX(m,r,n)(q)(E0, g|BE0) ' BU(n0)(q) x BU(n1)* *(q) x . .x. BU(ns)(q) and from the above discussion we have a factorization B(Sn0,`x . .x.Sns,`)______//_BU(n0)(q) x BU(n1)(q) x . .x.BU(ns)(q) * *(38) XXXX XXXXXXX | ] XXXXXXX |Bjn'Bjn0x1x...x1 XXXXX,,X fflffl| BX(m, r, n0)(q) x BU(n1)(q) x . .x.BU(ns)(q) . Diagrams (36), (37), and (38) provide a homotopy commutative diagram ____________________________________________* *____________________________________________________________________________@ ____________________________________________________* *____________________________________________________________________________@ _______________________________________++________________* *____________________________________________________________________________@ B(Sn0,`x . .x.Sns,`)____//BCX(m,r,n)(q)(E0, g|BE0)//_BCX(m,r,n)(q)(E, g|* *BE ) ____ | | Bj _______ | | fflffl___ fflffl| Bjn fflffl| BSn,` _______________//_BCX(m,r,n)(q)(Z,_g|BZ/)/_BX(m, r, n)(q) ________________________________________22_________________* *____________________________________________________________________________@ _________________________________________________* *____________________________________________________________________________@ where the existence of the homomorphism æ: Sn0,`x . .x.Sns,`! Sn,`making homoto* *py commutative the left square is obtained because Sn,`is a Sylow p-subgroup of U(* *n)(q). We have proved that BX(m, r, n)(q) and (Sn,`, f) satisfy the conditions (1) a* *nd (2) of Theorem 4.6, and therefore, that (Sn,`, f) is a Sylow p-subgroup of BX(m, r, n)* *(q) and (Sn,`, F(m, r, n, q), L(m, r, n, q)) is a p-local finite group. Finally, BX(m, r, n)(q) is the classifying space |L(m, r, n, q)|^paccording t* *o Proposition 11.3 and Theorem 4.6. Proposition 11.5. X(m, r, n)(q) is a exotic p-local finite group if r > 2, n * *p. Notice that in the above hypothesis r|(p - 1), thus r > 2 can only occur with* * p 5, so that we are implicitely assuming also that p 5. 74 CARLES BROTO AND JESPER M. MØLLER Proof.We wil first reduce the question to the rank p-case. Then we classify the* * centric radical subgroups in the fusion system of BX(m, r, p)(q) and show that they coincide wi* *th the p-local finite groups of [11, Example 9.4]. There is an elementary abelian p-subgroup E tX , in X(m, r, n)(q), of rank * *n - p such that CX(m,r,n)(q)(E, g|BE ) ~=X(m, r, p)(q) x U(1)^p(q)n-p (see Proposition 11.2). Thus, if we assume that there is a finite group G su* *ch that BX(m, r, n)(q) ' BG^p, then the map Bg|BE :BE ! BX(m, r, n)(q) ' BG^pis induced by a homomorphism ': E ! G, and BCG('(E))^p' BX(m, r, p)(q) x BU(1)^p(q)n-p. Since BU(1)^p(q) ' BZ=p`, the projection BCG('(E))^p! BU(1)^p(q)n-p is the p-co* *mpletion of the map induced by a homomorphism æ: CG('(E)) ! (Z=p`)n-p. It has a section* *, also induced by a homomorphims oe :(Z=p`)n-p ! CG('(E)), hence æ is an epimorphism. * *There- fore, we have a short exact sequence Ker æ ! CG('(E)) ! (Z=p`)n-p and an induce* *d fi- bration B(Ker æ)^p! BCG('(E))^p! B(Z=p`)n-p, from which we obtain an equivalence B(Ker æ)^p' BX(m, r, p)(q). This reduces the question to showing that X(m, r, p* *)(q) is an exotic p-local finite group. We will show now that X(m, r, p)(q) coincide with the p-local finite groups c* *onstructed in [11, Example 9.4] in purelly algebraic terms. For this aim we will need to * *describe the centric and radical p-subgroups of X(m, r, p)(q). Recall that T`p~= (Z=p`)p is the maximal finite torus of X(m, r, p)(q) with W* *eyl group G(m, r, p) and they form a split extension T`p! NX(m,r,p)(q)(T`p) ! G(m, r, p) that contains Sp,`= T`po Z=p NX(m,r,p)(q)(T`p), a Sylow p-subgroup of X(m, r,* * p)(q). For simplicity we will denote F = F(m, r, p, q), the fusion system of BX(m, r, p)(q* *) over (Sp,`, f). The center of the Sylow p-subgroup is Z(Sp,`) ~=Z=p` embeded diagonally in T`* *p, and, if we write Z(tX ) for the elements of order p in Z(Sp,`), then we obtain BCX(m,r,* *p)(q)(Z(Sp,`) ' BCX(m,r,p)(q)(Z(tX )) ' BU(p)^p(q) (see Proposition 11.2). We also know (see Re* *mark 11.1) that BU(p)^p(q) ' BGLp(q0)^ppor a prime power q0 with ` = p(1 - q) = p(1 - q0* *), hence we conclude that the centralizer fusion system CF (Z(Sp,`)) coincides with the * *fusion system of GLp(q0), that has been described in Example 3.5. The Sylow p-subgroup Sp,`is clearly centric and radical. T`pis centric and O* *ut F(T`p) = G(m, r, p) hence it is also radical (p 5). Proper subgroups of T`pare not cen* *tric, so we will look at subgroups Q Sp,`not contained in T`p. such a subgroup fits in an exte* *nsion Q0 _____//Q_____//Z=p | | || | | || fflffl| fflffl| || T`p_____//Sp,`__//_Z=p where Q0 = Q \ T`n, and since Q is centric, Z(Sp,`) Q0. It turns out that thi* *s is actually a characteristic subgroup of Q, Hence there is an exact sequence of groups: 1 ! AutCF(Z(Sp,`))(Q) ! AutF (Q) ! AutF (Z(Sp,`)) FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS * * 75 where AutF (Z(Sp,`)) ~=Z=r is given by the action of the Adams operations of ex* *ponents a rth root of unity. Assume that Q is abelian. Then Q0 = Z(Sp,`) and Q is either Z=pxZ(Sp,`) or cy* *clic Z=p`+1. In the first case, Q is conjugated in F to a subgroup of T`n, hence it is not c* *entric while in the second case, it is conjugated to the group U`+1 described in Example reffusions* *ystemGLpq. Adams operations do not act internally in U`+1, hence OutF (U`+1) ~=Out CF(Z(Sp* *,`))(U`+1) ~= Z=p and then U`+1is not radical in F. Assume that Q is non-abelian. The same arguments as in 10.5 show that Q is ei* *ther Sp,` or `, and both are radical in CF (Z(Sp,`)). Thus we obtain that they complete* * the list of conjugacy classes of centric radical subgroups of Sp,`in F. In order to complete the picture it remains to compute the F-automorphisms of* * `. We have OutCF(Z(Sp,`))( `) ~=SL2(p). 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