FINITE CHEVALLEY VERSIONS OF pCOMPACT GROUPS
CARLES BROTO AND JESPER M. MØLLER
Abstract. We describe the spaces of homotopy fixed points of unstable Ad*
*ams operations
acting on pcompact groups and also of unstable Adams operations twisted*
* with a finite order
automorphism of the pcompact group. We obtain new exotic plocal finite*
* groups.
Contents
1. Introduction *
* 1
2. pcompact groups *
* 7
3. plocal finite groups *
* 8
4. Recognition of classifying spaces of plocal finite groups *
* 13
5. Homotopy fixed points pcompact groups *
* 21
6. Homotopy fixed points of twisted unstable Adams operations *
* 39
7. General structure of finite Chevalley versions of pcompact groups *
* 44
8. Cohomology rings *
* 49
9. Invariant theory *
* 51
10. Finite Chevalley versions of Aguad'e exotic pcompact groups *
* 60
11. Finite Chevalley versions of generalized padic 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 pcompact groups *
*and also
of unstable Adams operations twisted by automorphisms of pcompact 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 pcomple
tion, the classifying space of the finite twisted Chevalley group fiG(q). Here *
*and throughout,
pcompletion is understood in the sense of BousfieldKan [8]. We will show that*
* in case of
______________
2000 Mathematics Subject Classification. 55R35, 55P15, 55P10.
Key words and phrases. Chevalley group, pcompact groups, plocal finite grou*
*ps.
C. Broto is partially supported by MCYT grant BFM20012035.
Both authors have been partially supported by the EU grant nr. HPRNCT19990*
*0119.
1
2 CARLES BROTO AND JESPER M. MØLLER
exotic pcompact groups we obtain classifying spaces of plocal finite groups, *
*in some cases,
new exotic examples.
The concept of pcompact group was introduced by Dwyer and Wilkerson in [22] *
*as a
plocal homotopy theoretic analogue of compact Lie group. A pcompact group is *
*a connected
pcomplete space BX where X = BX is Fpfinite; that is, H*(X; Fp) is finite. *
*We will
usually refer to a pcompact 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 pcompletion of its classifying space BG^pis a p*
*compact group.
A pcompact 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 pcompact groups.
The concept of plocal finite group has been recently introduced in [11] as a*
*lgebraic objects
that are modeled on the plocal structure of finite groups and as such they hav*
*e classifying
spaces which are pcomplete spaces. In turn, the classifying space of a ploca*
*l finite group
determines its algebraic structure. We refer to section 3 for the precise defin*
*ition and main
properties of plocal finite groups.
Theorem A. Let p be an odd prime. If X is a 1connected pcompact 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 plocal finite group.
By analogy with the classical case, we will call plocal finite Chevalley gro*
*up of type X to
any finite plocal finite group X(q) obtained in Theorem A, with classifying sp*
*ace BX(q).
If X is obtained as pcompletion of a compact Lie group G, BX(q) is homotopy eq*
*uivalent
to the pcompleted 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 pcompletion, 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 pcompact gr*
*oups (see
section 2), hence, following the above pattern, if X is a connected pcompact 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 pcompact 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 pcompact groups and the results obtained have an independent*
* interest
FINITE CHEVALLEY VERSIONS OF pCOMPACT 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 plocal 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 pcompact groups X,*
* an outer
action of G is a homomorphism æ: G ! Out(X), where Out(X) is the group automorp*
*hisms
of the pcompact group X, in other words, homotopy classes of selfequivalences*
* 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 pcompact group:
Theorem B. Let X be a connected pcompact 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 pcompact 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 pcompact group monomorphism, and
X ' XhG x X=XhG ,
thus, in particular, X=XhG is an Hspace.
(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
1connected pcomplete 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 pcompact 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 pcompact group E*
*hC48is the
pcompact group X31 corresponding to the reflection group number 31 on the Clar*
*kEwing
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 pcompact 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 pcompact group, ø an automo*
*rphism
of X of order prime to p and _q an unstable Adams operation, then:
(1) If X is 1connected and q 1 mod p, q 6= 1, then B fiX(q) ' BXhfi(q).
(2) If q0 is another padic 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 padic unit q as q = iq0 where i is a (p  1)stroot*
* 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 padic 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 padic 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 plocal 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 pcompact 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 prank as the original pcompact groups X, but the maximal tori T*
* n' ((S1)n)^p
are cut down to finite maximal tori T`n~=(Z=p`)n, ` = p(1q) (the padic valua*
*tion of 1q),
keeping, though, the same Weyl group (see 7.5, 7.6).
We restrict our calculations in this part to pcompact 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 pcompact groups. At odd primes, these include all irreducible exoti*
*c examples
and will therefore suffice to our purposes.
Theorem D. Let q be a padic unit such that q 1 mod p, q 6= 1. If X is an ir*
*reducible
1connected polynomial pcompact group, then BX(q) is the classifying space of *
*a plocal finite
group.
Proof.The proof is based on the classification theorem for pcompact groups at *
*odd primes [6].
The irreducible 1connected pcompact groups with polynomial cohomology are
(1) BSU(n)^p(family 1 in the ClarkEwing list),
(2) the Quillen generalized Grassmannians (family 2a in the ClarkEwing list),
(3) the nonmodular pcompact groups, and
(4) the Aguad'e family (numbers 12, 29, 31, and 34 in the ClarkEwing 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 pcompleted classifyi*
*ng space of a
finite group. The other two families contain the new exotic examples of plocal*
* finite groups.
A complete description of the structure of the plocal 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 pCOMPACT GROUPS *
* 5
then BX31(q) ' BE8(22m+1)^5(Example 10.8). In particular, we can obtain the pc*
*ompact
groups X12 and X31 as telescopes of a sequence of pcompleted 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 plocal finite groups (Example 10.6).
Family 2a in the ClarkEwing list consists of groups G(m, r, n) with rm(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
pcompact 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 plocal 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 pcompleted*
* classifying
spaces of compact connected Lie groups admit unstable Adams operations _q of ex*
*ponent a
padic unit q 2 Z*p. This is extended to pcompact groups for odd primes p in [*
*48].
A pcompact 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 Zpreflection groups can be derived [48, 11.18] from *
*the Clark
Ewing [16] list of irreducible Qpreflection groups. The simple pcompact grou*
*ps, corre
sponding to the irreducible reflection groups, besides the Lie examples, are th*
*e nonmodular
pcompact groups (including the Sullivan spheres), where p does not divide W *
*, the Aguad'e
pcompact groups [1], where p divides W  exactly once, and the generalized Gr*
*asmannians
[52] [48, x7] corresponding to the second infinite family in the ClarkEwing cl*
*assification table.
We refer to the surveys [42, 51, 18] for more information on pcompact groups.
3.plocal finite groups
The concept of plocal finite group has been introduced in [11] (see also [12*
*]). A plocal
finite group is a triple (S, F, L) where S is a finite pgroup, 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 pCOMPACT 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 Fconjugate if there is an isomorphism between*
* them in F.
Definition 3.1. Let F be a fusion system over a pgroup S.
(1) A subgroup P S is fully centralized in F if CS(P ) CS(P 0) for all *
*P 0 S which is
Fconjugate 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
Fconjugate 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 psubgroup 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 pgroup S. A centric linking*
* system
associated to F is a category L whose objects are the Fcentric subgroups of S,*
* together with
a functor
ß :L ! Fc,
and distinguished monomorphisms ffiP:P ! AutL(P ) for each Fcentric 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 Fcentric 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))
fflfflfflfflf
P ____//_Q .
The classifying space of the plocal finite group (S, F, L) is defined as the*
* pcompletion L^p
of the nerve of the category L. The classifying space determines the plocal fi*
*nite group in
the sense that two plocal finite group are isomorphic if and only if they have*
* homotopy
equivalent classifying spaces. Actually, the complete structure of a plocal 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 plocal fini*
*te group
theory. If G is a finite group and S a Sylow psubgroup, 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 x1 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 pcentric in G*
*, and morphisms
Hom L (P, Q) ~=NG(P, Q)=C0G(P ), where C0G(P ) is the p0complement in CG(P ) o*
*f the center
of P , CG(P ) = Z(P ) x C0G(P ), which is well defined because P is pcentric. *
* LcS(G) is a
centric linking system associated to FS(G), and (S, FS(G), LcS(G)) is a plocal*
* 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 plocal
finite groups are already shown in [11]. Recently, Levi and Oliver have obtain*
*ed a family
of exotic 2local 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 pgroup 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 plocal 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 plocal finite group and P *
*S is fully centralized
in F, then (CS(P ), CF (P ), CL(P )) is a plocal 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 pgroup 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 Fconjugate to some element of X.
(b) If x and y are Fconjugate 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 pgroup S. A subgroup P S is called *
*radical in F
if OutF (P ) = AutF (P )= Inn(P ) is preduced, namely, it does not contain non*
*trivial normal
psubgroups.
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 pgroup S it is enough to describe Aut *
*F(Qi) for a
set Q1, . .,.Qr of fully normalized representatives of Fconjugacy 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 pCOMPACT 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 pgroup *
*S and Q1, . .,.Qr
is a set of fully normalized representatives of Fconjugacy classes of centric,*
* radical subgroups
of S in F, then
F = .
Let G be a finite group and S a Sylow psubgroup. Let FS(G) the fusion syste*
*m of G
over S. P is centric in FS(G) if and only if it is pcentric in G. A psubgro*
*up P of G
is called pradical if it is the maximal normal psubgroup of NG(P ), P = Op(NG*
*(P )), or,
equivalently, if NG(P )=P is preduced. Notice though, that being radical in F*
*S(G) means
that OutFS(G)(P ) ~=NG(P )=P CG(P ) = OutG (P ) is preduced.
If P S is centric and radical in FS(G), then it is pcentric and pradical *
*in G. Assume
that P is not pradical in G, then there is another psubgroup Q with P / Q / N*
*G(P ) and
Q 6= P . Since P is pcentric, 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 preduced. 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 psubgroups, 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 psubgro*
*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 pradical subgroups of general linear groups [4].
The pprimary 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 psubgroup o*
*f GLp(q).
We can choose ~Sgenerated by T`p, diagonal matrices of ppower 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, . .,.ip1) 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 Fqvector 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 ppower ro*
*ots of unity
in Fqp.
12 CARLES BROTO AND JESPER M. MØLLER
With this notation and according to [4], if R is a pradical 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 noncentric, 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
psubgroup, for p a prime and q a prime power such that q 1 mod p.
We first show that every pradical subgroup of SLp(q) is the intersection Q \*
* SLp(q) of a
pradical subgroup Q of GLp(q) with SLp(q). For a given pradical psubgroup 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 psubgroup 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 pradical.
Fix the Sylow psubgroup 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*
* pcentric
and pradical in SLp(q). In particular P = Q \ SLp(q) where Q is pradical in*
* GLp(q),
hence conjugate by an element g 2 GLp(q) to a psubgroup in the list (3). Amon*
*g those
intersections, only S = ~S\ SLp(q), T`(p1)= S \ T`p, and 1 = S \ ` are also *
*pcentric.
Hence the complete list of conjugacy classes of pcentric and pradical 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`(p1) 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), *
*gSg1 lies in S
if and only it it is exactly S and the same happens with T`(p1). In the case *
*of 1 we just
FINITE CHEVALLEY VERSIONS OF pCOMPACT 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 pradical pcentric 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 plocal finite groups
In [11] it is shown that a plocal finite group can be completely recovered f*
*rom its classifying
space by homotopy theoretic methods. Also, a recognition principle for classify*
*ing spaces of
plocal 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 pgroup. Then,*
* the
basic tool in order to show that these systems define a plocal 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 plocal 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
quasifinite; 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 psubgroup of X is a pair (P, f), wh*
*ere P is a
finite pgroup and f :BP ! X a homotopy monomorphism at p. A psubgroup (S, f) *
*of X is
called a Sylow psubgroup of X if for any other psubgroup (Q, g) of X, g :BQ !*
* X factors
through f :BS ! X, up to homotopy.
If (P, f) is a psubgroup 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 psubgro*
*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 pquasi 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 *
*psubgroups
(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 psubgroups (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 psubgroups (P, f) wher*
*e f is a
pcentric 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 psubgroup 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) fifBP ' fBQ O B'
BiP f
for all P, Q S, where fBP denotes the composition BQ ! BS ! X. Notice tha*
*t if (S, f)
is a Sylow psubgroup 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 fBP to fBQ 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 psubgroup (S, f) under which
(S, F(S,f)(X), Lc(S,f)(X))
is a plocal 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 plocal
finite group is completely recovered from its classifying space.
FINITE CHEVALLEY VERSIONS OF pCOMPACT GROUPS *
* 15
In [11, x7] it is also considered the case of an arbitrary pcomplete space X*
* equipped with
a psubgroup (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 plocal 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 pcentric 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
ffiSrrrrHHHfH
rrr Mf  HHH
xxrrr fflfflHH##
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)'oMco(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 nsimplices are maps j : (P) ! X, where P *
*= (P0 !
'2 'n
P1 ! . .!. Pn) is a sequence of psubgroups 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 psubgroup 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 psubgroups 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)ooMSo(BS)__'__//_BS *
* (7)

'  '  '  
fflffl fflffl fflfflev 
Lcp(S)'__//_Lcp(BS)'oMco(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
psubgroup 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)ooMSo(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 psubgroup 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 psubgroup 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 pCOMPACT GROUPS *
* 17
Since x and y are Fconjugate,
[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 psubgroup*
* 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 fBP_' 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 psubgroups of X and of its centralizers.
18 CARLES BROTO AND JESPER M. MØLLER
Theorem 4.6. Let X be a pcomplete space and (S, f) a psubgroup of X. Assume t*
*hat
(1) Map (BZ=p, X)ct' X, and
(2) for each nontrivial element x 2 S of order p
(a)BCX (x) is the classifying space of a plocal finite group, and
(b)if (H, g) is a Sylow psubgroup for BCX (x), there is a group homomorphi*
*sm æ: H ! S
that makes the diagram
Bj
BH ______//_BS
g f
fflfflev fflffl
BCX (x) _____//X
commutative up to homotopy,
then, (S, f) is a Sylow psubgroup for X and
(S, F(S,f)(X), Lc(S,f)(X))
is a plocal finite group.
Furthermore, X ' L(S,f)(X)^pif and only if the natural map induced by evalu*
*ation
hocolim Map (BE, X)fBE ! 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 nontrivial fully centralized elementary abelian psubgroups of S.
Proof.The proof is divided in four steps. First, we prove that (S, f) is a Sylo*
*w psubgroup
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 fBP is pc*
*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 plocal 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)fBE ! 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 plocal finite group (S, F(S,f)(X), Lc(S,f)(X)).
Step 1: (S, f) is a Sylow psubgroup for X. Let (P, ~) be a finite psubgroup 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 plocal finite group, and if (H, g) is its Sylow ps*
*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 ~ fflfflf fflffl
BP ______________//_Xoo_________ BS
FINITE CHEVALLEY VERSIONS OF pCOMPACT 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 psubgroup 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 plocal finite group and sinc*
*e fx: BCS(x) !
BCX (x) is a Sylow psubgroup 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 psubgroup of BCX (x). Since (S, f) is a Sylow psubgroup 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 ' = æZ=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 psubgroups. Since (H, g) is a Sylow psubgroup fo*
*r CX (x), it
follows from the above sequence of maps that (CS(y), fy) is a Sylow psubgroup *
*for CX (y).
Hence y = '(x) 2 X.
Step 3: fBP is a pcentric 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 plocal finite group, and *
*from the above
sequence of maps we easily obtain that (S, fx) is a Sylow psubgroup for BCX (x*
*). Further
more, P is also FS,fx(BCX (x))centric, and then fxBP is a pcentric map. Ther*
*e is a sequence
20 CARLES BROTO AND JESPER M. MØLLER
of equivalences
Map (BP, BP )Id' Map (BP, BCX (x))fxBP
' Map (BP x BZ=p, X)fBPOBm' Map (BP, X)fBP
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 fBP is a pcentric map.
Step 4: X ' L(S,f)(X)^pif and only if the natural map
hocolim Map (BE, X)fBE ! 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)fBP *
*(12)
for every P S. Moreover, the diagram
''P
Map (BP, Lc(S,f)(X))`BP_//_Map(BP, X)fBP
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)fBE *
*(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 nontrivial*
* 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 pCOMPACT 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)fB*
*P is the
classifying space of a finite plocal 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 psubgroup, 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 psubgroup 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 nontrivial 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)fBP
 ! Map (BP x B , X))fBPOBm
 ! Map (BP, X)fBP
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 plocal f*
*inite group with
Sylow psubgroup (S, f), then all conditions of Theorem 4.6 are satisfied accor*
*ding to [11,
x7].
5. Homotopy fixed points pcompact 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 Gequivariant 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 pCOMPACT 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 pathcomponents of M and ß0(M)G,*
* or
H0(G; ß0(M)), denote the set of pathcomponents of M that remain fixed under th*
*is action.
With base point m 2 ß0(M)G in a Ginvariant pathcomponent 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 fixedpoint 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
fixedpoint 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*
*nabelian 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 plocal 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 fixedpoint 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 Sr1 *
*(see [17]),
determine a tower of fibrations
MhG ! . .!.PrMhG ! Pr1MhG ! . .!.P1MhG ! BG
so that MhG is the homotopy inverse limit of a sequence
. .!.PrMhG ! Pr1MhG ! . .!.P1MhG
of Postnikov homotopy fixedpoint spaces.
Note that ß0(P1MhG) = ß0(MhG) and that each pathcomponent 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 fixedpoint space is nonempty and let s 2 MhG be a*
* homotopy
fixedpoint. Then the component MhG , s containing s is the homotopy inverse *
*limits of the
corresponding components
hG hG hG
. .!. PrM , sr ! Pr1M , sr1 ! . .!. P1M , s1
of the Postnikov homotopy fixedpoint spaces. To finish the proof, observe [41,*
* 3.1] that the
fibre of PrMhG , sr ! Pr1MhG , s1r is the EilenbergMac Lane space K(ßr(M*
*, m)s*G, r).
For an alternative formulation, let (M, m) denote the pathcomponent of M con*
*taining
m 2 M. If the path component (M, m) 2 ß0(M) is Ginvariant, then (M, m) is sub*
*Gspace
of M in the sense that the inclusion of (M, m) into M is a Gmap; that is, the *
*fibration
M ! MhG ! BG contains a fibration of the form
(M, m) ! (M, m)hG ! BG
as a subfibration over BG. The homotopy fixedpoint space
[
MhG = (M, m)hG
m2i0(M)G
is a disjoint union of the homotopy fixedpoint spaces (M, m)hG where (M, m) ru*
*ns through
the set of Ginvariant pathcomponents in M. Since (M, m) by its very definitio*
*n is a path
connected Gspace the homotopy groups of its homotopy fixedpoint 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 pcomplete 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 Hspa*
*ce.
FINITE CHEVALLEY VERSIONS OF pCOMPACT GROUPS *
* 25
Proof. By obstruction theory, the space of sections BhG is nonempty. 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 Gequivariant maps EG ! B. Let B=EG = B[C(EG) be the homotopy
cofibre of any such Gmap. Then B ! B=EG is a Gequivariant homotopy equivalenc*
*e and
the Gaction on B=EG has a fixed point.
Now, we can assume that there is a honest Gaction on B with a fixed point. *
*Let B
denote the loop space based at any Gfixed 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 pcompact 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 plocal. 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 Fpfinite Hspace, XhG is a loop space with *
*classifying
space BXhG and it is also Fpfinite. Furthermore, BXhG is pcomplete because B*
*X is p
complete [22, 11.13], hence XhG is a connected pcompact 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 fixedpoint pcomp*
*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 pcompact 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 pcompact group BEhC26is the pcompact 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 pcompact 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 pcompact group BEhC48is the pcompact group BX(G31) correspond*
*ing to
reflection group number 31 on the ClarkEwing 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 pcompact 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 pcompact 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 pcompact 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 BNfibra*
*tions.
FINITE CHEVALLEY VERSIONS OF pCOMPACT GROUPS *
* 27
Since G is a finite group of order prime to p, we see from this that equivale*
*nce classes of
BXfibrations over BG is in onetoone correspondence with
[BG, B2Z(X)hOut(X)] = [BG, BOut (X)] = Hom (G, Out(X))
and that equivalence classes of BNfibrations over BG is in onetoone 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 BNfibrations 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 grouplike 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 Ndetermined automorphisms [48, 6]. The group homomorphism to the*
* left
is surjective because X is Ndetermined and the one to the right is surjective *
*because any
selfhomotopy equivalence of BX lifts to a selfhomotopy 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 BNfibrations and BXfibrations over B are in bi*
*jective
correspondence.
28 CARLES BROTO AND JESPER M. MØLLER
Proposition 5.5. Let X be a connected pcompact 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 Gaction on BX and unique Gaction on BN. Moreover, these act*
*ions make
the map BN ! BX Gequivariant; 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 onetoone correspondenc*
*es between
the sets:
(1) ß0(BNhG ),
(2) lifts to a Gaction 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 nonempty, then they are also in onetoone 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 pCOMPACT 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 pcompact 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 Gequivar*
*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 onet*
*oone 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 pcompact group X and
__æ:G ! N
GL(L)(W )
is a lift of the given outer action. If Y is a connected pcompact 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 pcompact 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 pnormalizer Np(Y ), is a pmonomorphism. The homomorphi*
*sm __æ
corresponds (5.7) to compatible Gactions 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 pmonomorphism. Since the discrete approxim*
*a
tion to N(X), N(X)hG, and N(Y ) are semidirect_products [5], there is a pmono*
*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 padic 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 pCOMPACT 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 padic 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 pcompact groups for*
* actions
through unstable Adams operations on the pcompact groups of the three infinite*
* families
in the ClarkEwing classification table [16].
Proposition 5.11. Suppose that r and m divide p  1, m > 1, then
(
S2r1, m  r
(S2r1)hCm =
* , otherwise,
for the action through unstable Adams operations of exponent of Cm Z*pon the *
*pcompact
group S2r1.
Proof. Let ~ be a primitive mth root of unity, so that Cm = <~> Z*p. Accord*
*ing to
Theorem B, (S2r1)h<~>is a connected polynomial pcompact group. If m does not *
*divide r,
H2r(_~) = ~r is nontrivial, so that the vector space of covariants QH*(BS2r1; *
*Qp)<~>vanishes
in positive degrees, and the fixed point pcompact group is trivial. If m does *
*divide r, _~ acts
trivially on S2r1and the fixed point pcompact group is again S2r1.
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 pcompact 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(_~), mi
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(n1)+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
pcompact group U(mn + s)hCm = X(m, 1, n).
From the two short exact sequences of Zp mn+smodules [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+saction), 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 Zpreflection
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 pCOMPACT 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, . .,.xt1)
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  p1, be the simp*
*le polynomial
pcompact 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, . .,.xn1, 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 pcompact 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 Zpmodule L = Znpon which G(m, r, n) acts. *
* The free
Zpmodule
` 1 `f1f
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
i1 i1
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
` ` ` 1n=`
w = diag A(~ m, `m ), . .,.A(~ m,,`mA)(~ m, `m ) m 2 G(m, m, n)
____________z___________"
n=`m1
has order `. Note that ~1 is not an eigenvalue for A(~`m, `m )1n=`mbecause
A(~`m, `m )1n=`mhas eigenvalue ~1, A(~`m, `m )n=`m1has eigenvalue ~
, ~(`m)n=`m1= ~`m(n=`m1)`m, `  (`m )n=`m1+ `m (n=`m  1)`m
, `  n=gcd(`m , n=`m  1)) `  n ) `*
*m  n
which is not the case. Therefore the ~1eigenspace
` 1 ` 1ff
L<~w>= e1 + ~e2 + . .+.~ m e`m, . .,.e(n2`m)+1+ ~e(n2`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 )
i1
*
* 1
diag E,_._.,.E_z___", ~E, E, . .,.E*
*, ~ E ~i
i1
agree on their common domain A(gcd(`, m), 1, n=`m 1) and together with the mon*
*omorphism
" Ø "
n=`m1Ø__// n`m_//_ mthey define a homomorphism on the group A(lcm(`, m), 1, *
*n=`m 
1) o n=`m1such that the composition
*
*<~w>
A(lcm(`, m), 1, n=`m  1) o n=`m1,! 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 pCOMPACT 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_____"
(i1)`m n`m[n=`m]
`
diag E,_._.,.E_z___", ~E, E, . .,.E, ~__m*
*,_1,_..,.1z_____" ~i
(i1)`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, . .,.xn1 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 n1
G(m, r, n) m= A(m, r, n) o n1, L m= Zp
and the composition
" _ _jC _jC
A(m, 1, n  1) o n1Ø__//_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), n1 *
*,! n, identifies
the group to the right as the reflection group G(m, 1, n  1).
5.15. The sporadic pcompact groups. We identify the fixed point pcompact grou*
*ps for
actions through unstable Adams operations on the pcompact groups corresponding*
* to the
34 sporadic reflection groups of the ClarkEwing classification table. These p*
*compact groups
are determined by their rational Weyl groups except that the local isomorphism *
*system of G35
contains two 3compact 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 pCOMPACT 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 pcompact 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""12fflffl 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 pCOMPACT 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 pcompact group and set ff: X ! X a pcompact group automorphism. *
*The
homotopy pullback diagram
BF ff(X) __'____//_BX *
*(18)
' 
fflffl1xff 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 pcompact 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 pcompact 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 pcompact 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) 1ff*!ßi(BX) !ßi1(BF ff(X)) !. . .
The same construction for the top square of diagram (19) degenerates to
. ..!ßi( BXhCr) !ßi(BXhCr) 0!ßi(BXhCr) !ßi1( BXhCr) !. . .
FINITE CHEVALLEY VERSIONS OF pCOMPACT 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 pcompact 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 pcompact 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''fflfflO(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 selfequivale*
*nce of X=Y
induced by ff: BX ! BX. Again, Theorem B implies that X=Y is a connected Hsp*
*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 pcomplete, 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 1connected pcompact 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 padic 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 pcompact group. Assume that the action of Z on BX determ*
*ined
by an element ff 2 Out(X) extends to the padics, 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 pcompletion of the left one. In fact, Zp can *
*only act nilpo
tently on Hi(BX, Fp), which are finite Fpvector spaces, hence the fibration on*
* the right is
preserved by pcompletion. Since the base and the fibre are pcompleted spaces,*
* so is the total
space BXhZp. The above diagram is a pullback diagram, so the left fibration is *
*also preserved
by pcompletion, and then, since the top and bottom horizontal arrows are pequ*
*ivalences, so
is the middle one.
FINITE CHEVALLEY VERSIONS OF pCOMPACT 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 nontrivial
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 pcompact 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 padics 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 padic 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 padic 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 pcompact groups
In this section we will study the first general properties of finite Chevalle*
*y versions BX(q)
of pcompact groups X. The main results being the identification of the maximal*
* finite torus,
the Weyl group, and the fusion category of elementary abelian psubgroups.
Proposition 7.1. Let X be a connected pcompact group and ff a self homotopy eq*
*uivalence
of X. Then
(1) BF ff(X)is connected and pcomplete.
(2) ': BF ff(X)! BX is a homotopy monomorphism at p.
(3) For any finite pgroup 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 pcomplete, X is connected and BX is simplyconnected. It follows that BF *
*ff(X) is
connected and pcomplete.
FINITE CHEVALLEY VERSIONS OF pCOMPACT GROUPS *
* 45
For any finite pgroup 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 pcompact 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 pcompact group and ff a selfequivalence 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 ffCX(P, )of the centralizer CX (P, ) is*
* T (ffCX(P, )) =
w1 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] fflfflttjjj]jj
Map (BP, BX) ________//Map(BP, BX)
that the restriction of ffCX(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 pcompact 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) _qCX(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 pgroup 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_qCX(V, ) fflffl
BCX (V, )__________//BCX (V, )
 
 
fflffl _q fflffl
BX ________________//BX
this proves (b), namely, _qBCX(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 pcompact 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 _qTX = T (_q) = _q this follows from 7.3 (see*
* 7.4).
FINITE CHEVALLEY VERSIONS OF pCOMPACT 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 pcompact 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 pa*
*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 pcompact 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 pgroup 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 pcompact *
*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 pcompact group, then centralizers of elem*
*entary
abelian psubgroups are connected and Lemma 7.2 applies. In fact, if (E, ) is*
* an elemen
tary abelian psubgroup of X, then the centralizer CX (E, ) is also a polynomi*
*al pcompact
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 psubgroups (P, g)*
* and (Q, h)
of BF ff(X).
Corollary 7.8. Let p be an odd prime. If X is a connected polynomial pcompact *
*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
pcompact group every elementary abelian subgroup is toral.
Let X be a polynomial pcompact group with trivial center and q 2 Z*pa padic*
* 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 (_qCX(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 pcompact 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 psubgroup E T , write W*
* (E) for the
pointwise stabilizer subgroup of E.
Proposition 7.9. Let X be a polynomial pcompact group with trivial center, p o*
*dd, and
q 2 Z*pa padic 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, gBE ); Fp) ~=H*(BTX (q); Fp)W(E)
for any nontrivial, subgroup E of tX . Then (22) is an Fpequivalence.
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 psubg*
*roup of W .
Proof.This follows from the BousfieldKan spectral sequence because the functor
E ! H*(BCX(q)(E, gBE ); 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 pCOMPACT GROUPS *
* 49
8.Cohomology rings
This section is devoted to the proof of Theorem E. The EilenbergMoore 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 pcompact 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 padic unit q, the homotopy type of BU(n*
*)(q),
or BSU(n)(q), does only depend on the padic 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 EilenbergMoore spectral sequence associated to the pullback diag*
*ram
BX(q) ___'___//_BX *
*(23)
' 
fflffl1x_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 E2page 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)
' 
fflffl1x_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 EilenbergMoore 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 E2pages, 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 E2pages
**
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 E2page.
FINITE CHEVALLEY VERSIONS OF pCOMPACT 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 nontrivial,
` '
@xi
J = det ___ 6= 0,
@ti
by [61]. Since both spectral sequences collapse at the E2page, it *
* follows that
i*: H*(BX(q); Fp) ! H*(BT`n; Fp) is also injective.
Remark 8.1. The argument with the EilenbergMoore 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 pcompact 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
mdi, if m is the order of q mod p, and 2di 1 = degyi.
Notice that we can write q = iq0 where i is an mroot of one in Zp and q0 1*
* mod p.
0 `` ``
Hence _q = _q O _ , and _ has finite order m as automorphism of the pcompact *
*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 pcompact 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 pcompact group of rank n and let q be a padic 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
det1; 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 *)Gdet1= fdet1. P (V *)G, for an element *
*fdet12 P (V *)
which is unique up to an invertible of Fp (see [14]). It follows that fdet1div*
*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 fdet1has degree degfdet1= 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 degfdet1= i=1(deg æi 2). This is the degree of the Jacobian *
*J, hence
J = fdet1,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 *)Gdet1=
fdet1P (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 pCOMPACT GROUPS *
* 53
P n
Assume otherwise that degfdet16= i=1(deg æi 2); that is, J = 'fdet1for s*
*ome element
' 2 P (V *)G of positive degree, then
dæ1. .d.æn
w = __________= fdet1dx1. .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 nonmodular 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 fdet1is twice thePnumber of pseudoreflections of G.POn*
* the other hand,
the number of pseudoreflection is G is ni=1(degji_2 1). Hence degfdet1= 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 + c1Xm1 + . .+.cm1 *
*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
1dimensional vector subspace L of V *that intersects the orbit [x] non trivial*
*ly, E[x] = zL
is the preEuler element of the orbit [x], defined up to a nonzero 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 ClarkEwing 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 preEuler 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 preEuler element is precisel*
*y the determinant
(det = det1in 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
3cycle 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 *)d3et1*
*= (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
x41y5 2 K(V *) 3: w0x41y5 2 K(V *) 3 and w0x41y5 = w1x41y3y5 = 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 ClarkEwing list: G = G(m, r, n), rmp1, [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 preEuler
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, . .,.æn1, E1 r] .
FINITE CHEVALLEY VERSIONS OF pCOMPACT GROUPS *
* 55
The orbit of (x1Q x2) is [x1  x2] = { `1xi `2xj `m1= `m2= 1 , i < j } and i*
*ts preEuler
element is E2 = i<* ` = 0
S2m1(q) = S2m1(1 + p`) 0 < ` < 1
>:
S2m1(1) ` = 1,
where ` = p(1  qm ), S2m1(1 + p`) ' B(Z=p`o Cm )^p, and S2m1(1) ' S2m1.
In fact, if we write q = i . q0, with q0 1 mod p and i a root of 1 in Zp, th*
*en
S2m1(q) ~=(S2m1)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 (S2m1)h<``>is contractible (see 5.11),
(ii)for 0 < ` = p(1  qm ) < 1, i 2 Cm , hence (S2m1)h<``>~=S2m1 (see 5.11)*
*, and
therefore S2m1(q) ~=S2m1(q0). Moreover, ` = p(1  q0) and Theorem 9.7 i*
*mplies that
BS2m1(q0) ' B(Z=p`oCm )^p. Notice that this result does only depend on ` *
*= p(1qm ),
hence also BS2m1(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 BS2m1(q) ~=BS2m1(1) ' BS2m1.
Example 9.10 (SU(3)(q) at the prime 3). Fix q a 3adic integer with 0 < ` = 3(*
*1q) < 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 Gsets 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
pcompact 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 Zpmodules index by the category *
*I(k).
Assume that
(a) Restriction gives an isomorphism Hj(G; A) ~=Hj(H1; A), for any Z(p)Gmodule*
* A and
j 1.
(b) p  NG(Hi) and Mi= MHi0, for every i 2.
Then, there is an exact sequence
0 ! lim0M ! MNG(H1)=H11 MG0! MNG(H1)0! lim1M ! 0 ,
I(k) I(k)
and limjI(k)M = 0 if j 2.
Proof.We consider a starshaped 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 psubgroup. 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 ClarkEwing li*
*st. The
category Fe3(G2) of nontrivial elementary abelian 3subgroups 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 3subgroups is equivalent to
____________________________________________________________*
*(_3)op\(D)op___________________________________________________________
Z=2__BSU399____________________________________________________*
*_____________BTH2H________oo_opop//_BU2Z=2ee_______________________________@
_______________________________________*
*______(_2)__\(D)______________________
_______________________________________*
*_________________
(D)op
FINITE CHEVALLEY VERSIONS OF pCOMPACT 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 3subgroups (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 3adic 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 nontrivial e*
*lementary
abelian 3subgroups 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 ! lim0 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 limiFe BCG2(q)= 0 if i 2, where 3 x Z=2 ~=ND12( 3) = D12. It clearly*
* follows
p(G2(q))
that
lim0H*(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 BousfieldKan spectral sequence
limi 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) ~= lim0H*(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 nonmodular pcompact 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 ClarkEwing list. Let p be an odd prime and q a padic 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 prank 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 1dimensional
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 prank 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 pcompact groups
In [1], Aguad'e constructed the exotic pcompact 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 SheppardTodd and ClarkEwing 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 p1to 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(p1Gi)opdd______________________________________________@
FINITE CHEVALLEY VERSIONS OF pCOMPACT GROUPS *
* 61
Each Xi is a pcompact group with maximal torus TXi = T p1and 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 plocal 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 plocal finite groups. In par*
*ticular, cases i =
29, 34 provide new exotic examples of plocal finite groups.
The first results on the plocal structure of BXi(q) are given by proposition*
*s 7.5 and 7.6.
Set ` = p(1  q). The maximal elementary abelian psubgroup of Xi, (tXi, ), f*
*actors as a
psubgroup (tXi, g) of Xi(q), and the centralizer of this group
CXi(q)(tXi, g) ' T`p1~=(Z=p`)p1
is the maximal finite torus of Xi(q). All elementary abelian psubgroups of Xi*
*(q) factor
through this one. Moreover, the Weyl group is WXi(q)(T`p1) = Gi, and the norm*
*alizer
NXi(q)(T`p1) = T`p1o Gi sits in the maximal torus normalizer of Xi(q), making*
* homotopy
commutative the diagram
BNXi(q)(T`p1)____//_BNXi(T p1)
 
 
fflffl fflffl
BXi(q) ___________//BXi.
Now, we fix the Sylow psubgroup S = (Z=p`)(p1)o Z=p of NXi(q)(T`p1), gene*
*rated
by T`p1and a pcycle of p Gi. We will denote by f :BS ! BXi(q) the homotopy
monomorphism obtained as the composition BS ! BNXi(q)(T`p1) ! BXi(q). Then (S,*
* f)
is a psubgroup of BXi(q), and it will play the role of a Sylow psubgroup.
Since Xi, i = 12, 29, 31, 34, are polynomial pcompact 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 psubgroup (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 pcompact group whose Weyl group, the p*
*ointwise
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, gBZ ),
such that the centralizer is
CXi(q)(Z, gBZ ) ' SUp(q)
and contains (S, f):
BS M
MMMMf
Bincl MMMM
fflffl M&&
BSUp(q) ____//_BXi(q)
as Sylow psubgroup of SUp(q).
If E tXi represents another conjugacy class of elementary abelian psubgrou*
*ps, then
CXi(q)(E, gBE ) ' T`p1o WXi(E)
where the order of WXi(E) is not divisible by p. Furthermore, the diagram
____Bincl_//
BT`(p1) BS

Bincl f
fflffl j fflffl
BCXi(q)(E, gBE_)___//BXi(q)
is commutative up to homotopy, where j :BCXi(q)(E, gBE ) ! BXi(q) is the natur*
*al map
induced by evaluation.
Proof.For Z tXi, we have CXi(q)(Z, gBZ ) ~=SUp(q) by Corollary 7.4.
If E tXi be another subgroup, not conjugated to Z, then the centralizer in *
*Xi is the
connected nonmodular pcompact group BCXi(E, BE ) ' B(TXioWXi(E))^p, and the*
*n, first,
Corollary 7.4 implies that BCXi(q)(E, gBE ) ' BCXi(E, BE )(q), and secondly,*
* Theorem 9.7
gives BCXi(E, BE )(q) ' B(T`p1o 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`p1' BCS(tXi) ___________BCS(E) __________ BS
' f] f] f
fflffl fflffl fflffl
BCXi(q)(tXi, g)___//_BCXi(q)(E, gBE_)//_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 pCOMPACT 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`p1; 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 psubgroups of BXi(q) are described in*
* Proposi
tion 10.1. The centralizer, CXi(q)(E, gBE ), of an elementary abelian psubgr*
*oup E tXi
in Xi(q) is either SUp(q) or a nonmodular pcompact group.
In cases i = 29, 31, 34, H*(CXi(q)(E, gBE ); 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 pcompact 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 psubgroups, 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 BousfieldKan spectral sequence
Ei,j2~=limiHj(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 E2term 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 ! lim0H*(BCDI2(q); F3) ! H*(BSU(3)(q); F3)Z=2 H*(BT`2; F3)
I(1)
3xZ=2 1 *
! H*(BT`2; F3) ! limH (BCDI2(q); F3) ! 0 , *
* (30)
I(1)
and limiI(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 ! lim0H*(BCDI2(q); F3) ! ____________________________
I(1) (y11y15 x16y10, y10y11, y10y15)
~R Z=2 1 *
 ! P [x6]y4 ! limH (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
lim0H*(BCDI2(q); F3) ~=P [x12, x16] E[y11, y15]
I(1)
and limiI(1)BCDI2(q)= 0 if i 1, so, therefore the BousfieldKan 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) ~=lim0H*(BCDI2(q); F3) ~=P [x12, x16] E[y11, y15] .
I(1)
Theorem 10.3. (S, f) is a Sylow psubgroup for BXi(q), the fusion system F(S,f)*
*(BXi(q))
of the space BXi(q) over the psubgroup (S, f) is saturated, and
(S, F(S,f)(BXi(q)), L(S,f)(BXi(q)))
is a plocal 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 psubgroups is equival*
*ent to that
of the pcompact group Xi; in particular, every elementary abelian psubgroup i*
*s toral; that
is, Fconjugate to a subgroup of T`(p1). 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
psubgroup 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`(p1), 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`(p1) 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 pCOMPACT 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`p1is centric in F and*
* OutF (T`p1) ~=
Gi is preduced, hence T`p1is also radical in F. No other subgroup of T`p1is *
*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`p1____//_S___//Z=p
where Q0 = T`p1\ 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`(p1)) 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: Nonabelian 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: Nonabelian centric radical subgroups having noncyclic abelian charac*
*teristic sub
groups. Assume now that Q contains a noncyclic 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 psubgroup 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 psubgroup S itself is the only centric radical that c*
*ontains Q,
hence, there is Ø 2 Aut CF(Z)(S) with ØQ = _1=``O ', hence ' = _``O ØQ 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 nontrivial 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 ps*
*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 pCOMPACT 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 Fconjugate 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 Fauto*
*morphisms
of S are given by conjugation in the normalizer N`,iof the maximal finite torus*
* T`(p1). We
have seen already that the same is true for Q = T`(p1).
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 Fautomorphism 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`(p1)) ~=T`(p1)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
pcompleted classifying spaces of finite groups and which cases correspond to e*
*xotic examples
of plocal 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 pcompleted 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 Fconjugate to any element in S \ P . Now, if P is non trivial*
* it contains
at least an element of order p, and this is Fconjugate to an element of order *
*p in T`(p1).
Now, the maximal elementary abelian psubgroup t of T`(p1)turns out to be an i*
*rreducible
Gimodule, hence t P and since the cycle of order p generating S=T`(p1)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`(p1)___//_S___//Z=p
where t PT = P \ T . Now S=P ~= T`(p1)=PT is abelian. The abelianization of *
*S is seen
to be Z=p x Z=p, and then we obtain that T`(p1)=PT is either trivial or has or*
*der p. It
follows that all elements of order up to p`1 of T`(p1)belong to PT. Taking th*
*e quotient by
___ ______(p1)
this subgroup we obtain an inclusion of Gimodules PT T` , but again, this *
*last is an
___ ______(p1)
irreducible Gimodule, 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
plocal 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 pcompleted 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`(p1)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 3completed 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 3completed 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 5adic unit, q 2 Z*5, BX31(q) at p = 5 is the 5completed*
* 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 pCOMPACT GROUPS *
* 69
We fix a prime power q0 with q0 2 mod 5 and ` = 5( iq1) = 5(q041) = 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(q41).
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`(p1)+r T`(p1)+r+1, and *
*then obtain that the
pcompact 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 padic Grassmannians
Let p be an odd prime, m 1, r 1, and n 1 with rm(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 1connected pcompac*
*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
padic Grassmannians. This family of pcompact 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 *
*ClarkEwing.
If n = 1, the groups G(m, r, n) ~=Z=(m_r) are cyclic and appear as family 3 in *
*the ClarkEwing
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 ClarkEwing 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 padic *
*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 pcompletion, 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 padic 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 pcompletion.
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 pcompeltion, 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 pcompact groups X(m, r, n) are polynomial, hen*
*ce propo
sitions 7.5 and 7.6 apply. The maximal elementary abelian psubgroup of X(m, r,*
* n), (tX , ),
factors as a psubgroup, (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 psubgroups.
FINITE CHEVALLEY VERSIONS OF pCOMPACT GROUPS *
* 71
Proposition 11.2. Let p be an odd prime, m 1, r 1, n 1 with rm(p  1), *
*and
q 1 mod p, q 6= 1. Then,
(1) any elementary abelian psubgroup 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, gBE ) in X(m, r, n)(*
*q),
BCX(m,r,n)(q)(E, gBE ) ' BX(m, r, n0)(q) x BU(n1)(q) x . .x.BU(ns)(q)*
* ,
n = n0 + n1 + . .+.ns, is determined by the pointwise 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 psubgroups 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, gBE ), is _q again, _qCX(m,r,n)(q)(E,gBE)= _q, and
BCX(m,r,n)(q)(E, gBE ) ' BCX(m,r,n)(E, BE )(q) .
The centralizers CX(m,r,n)(E, BE ) are known to be connected pcompact groups*
* of maximal
rank, with Weyl group G(m, r, n0) x n1x . .x. ns, the pointwise 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(q1), 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 rm(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, . .,.xn1, e] E[y1,*
* . .,.yn1, 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 psubgroup E tX ,
H*(BCX(m,r,n)(q)(E, gBE ); Fp) ~=H*(BT`n; Fp)G(m,r,n)(E)
where G(m, r, n)(E) is the pointwise 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 psubgroup 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 psubgroup 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 padic unit such that q 1 mod p, q 6= 1, and ` = p(1*
*  q), then,
(Sn,`, f) is a Sylow psubgroup for BX(m, r, n)(q) and
(Sn,`, F(m, r, n, q), L(m, r, n, q))
is a plocal finite group with classifying space
L(m, r, n, q)^p' BX(m, r, n)(q) .
Proof.We proceed by induction on n, the prank of X(m, r, n)(q). For n < p, X(*
*m, r, n)
is a nonmodular pcompact group, and then, X(m, r, n)(q) is the pcompleted 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 pcompleted 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 ploca*
*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 psubgroup (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, gBE ) is the classifying space of a plocal 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, gBE ) '
BX(m, r, n0)(q) x BU(n1)(q) x . .x.BU(ns)(q), for n = n0 + n1 + . .n.s, a nont*
*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 plocal 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 psubgroups 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
pointwise 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, gBZ ) ' 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)
 
 
fflfflBjn fflffl
BU(n)(q) _______//_BX(m, r, n)(q)
FINITE CHEVALLEY VERSIONS OF pCOMPACT 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 pointwise 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 pointwise 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, gBE0)_//_BCX(m,r,n)(q)(E, gBE ) *
*(37)
 
 
fflffl Bjn fflffl
BCX(m,r,n)(q)(Z, gBZ_)____//_BX(m, r, n)(q) .
Now, BCX(m,r,n)(q)(E, gBE ) ' BX(m, r, n0)(q) x BU(n1)(q) x . .x.BU(ns)(q) w*
*ith Sylow
psubgroup Sn0,`x . .x.Sns,`while BCX(m,r,n)(q)(E0, gBE0) ' 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, gBE0)//_BCX(m,r,n)(q)(E, g*
*BE )
____  
Bj _______  
fflffl___ fflffl Bjn fflffl
BSn,` _______________//_BCX(m,r,n)(q)(Z,_gBZ/)/_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 psubgroup 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 psubgroup of BX(m, r, n)*
*(q) and
(Sn,`, F(m, r, n, q), L(m, r, n, q)) is a plocal 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 plocal 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 pcase. 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 plocal
finite groups of [11, Example 9.4].
There is an elementary abelian psubgroup E tX , in X(m, r, n)(q), of rank *
*n  p such
that
CX(m,r,n)(q)(E, gBE ) ~=X(m, r, p)(q) x U(1)^p(q)np
(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 BgBE :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)np.
Since BU(1)^p(q) ' BZ=p`, the projection BCG('(E))^p! BU(1)^p(q)np is the pco*
*mpletion
of the map induced by a homomorphism æ: CG('(E)) ! (Z=p`)np. It has a section*
*, also
induced by a homomorphims oe :(Z=p`)np ! CG('(E)), hence æ is an epimorphism. *
*There
fore, we have a short exact sequence Ker æ ! CG('(E)) ! (Z=p`)np and an induce*
*d fi
bration B(Ker æ)^p! BCG('(E))^p! B(Z=p`)np, 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 plocal finite group.
We will show now that X(m, r, p)(q) coincide with the plocal 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 psubgroups 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 psubgroup 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 psubgroup 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 psubgroup 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 pCOMPACT 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 nonabelian. 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 Fautomorphisms of*
* `. We
have OutCF(Z(Sp,`))( `) ~=SL2(p). Now, the Adams operations act internally in *
*` and we get
Out F( `) ~=SL2(p).r.
By Alperin's fusion theorem, a fusion system over S is generated by the autom*
*orphisms of
its fully normalized centric radical subgroups in S. Since in our case al the a*
*utomorphisms
of T`pare induced by conjugation in NX(m,r,p)(q)(T`p), we can write
F(m, r, p, q) =
(see section 3) but this is precisely the definition of the fusion systems in [*
*11, Example 9.4].
The cases BX(m, r, n)(q) with r = 1, 2 or n < p, are homotopy equivalent to p*
*completed
classifying spaces of finite groups according to Theorem 9.7 and Remark 11.1.
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Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, 08193 Bella*
*terra,
Spain
Email address: broto@mat.uab.es
Matematisk Institut, Universitetsparken 5, DK2100 København
Email address: moller@math.ku.dk