EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT
ODD PRIMES
BOB OLIVER
Abstract. We prove here the Martino-Priddy conjecture for an odd prime p:*
* the p-
completions of the classifying spaces of two groups G and G0are homotopy *
*equivalent
if and only if there is an isomorphism between their Sylow p-subgroups wh*
*ich preserves
fusion. A second theorem is a description for odd p of the group of homot*
*opy classes
of self homotopy equivalences of the p-completion of BG, in terms of auto*
*morphisms
of a Sylow p-subgroup of G which preserve fusion in G. These are both con*
*sequences
of a technical algebraic result, which says that for an odd prime p and a*
* finite group
G, all higher derived functors of the inverse limit vanish for a certain *
*functor ZG on
the p-subgroup orbit category of G.
In an earlier paper [BLO1 ] in collaboration with Carles Broto and Ran Levi,*
* we
reduced certain problems involving equivalences between p-completed classifying*
* spaces
of finite groups to a question of whether certain obstruction groups vanish. Th*
*e main
technical result of this paper is that these groups do always vanish when p is *
*odd. The
proof of this result depends on the classification theorem for finite simple gr*
*oups.
Fix a prime p and a finite group G. For any pair of subgroups P, Q G, let N*
*G(P, Q)
denote the transporter:
NG(P, Q) = {x 2 G | xP x-1 Q}.
The p-subgroup orbit category of G is the category Op(G) whose objects are the*
* p-
subgroups of G, and where
MorOp(G)(P, Q) = Q\NG(P, Q) ~=Map G(G=P, G=Q).
A p-subgroup P G is called p-centric if Z(P ) is a Sylow p-subgroup of CG(P )*
*, or
equivalently if CG(P ) = Z(P ) x C0G(P ) for some subgroup C0G(P ) of order pri*
*me to p.
Let
ZG :Op(G) ------! Ab
denote the functor ZG(P ) = Z(P ) if P is p-centric in G, and ZG(P ) = 0 otherw*
*ise.
We refer to [BLO1 , x6] for more details on how this is made into a functor.
This paper is centered around the proof of the following theorem.
Theorem A. For any odd prime p and any finite group G,
lim-i(ZG) = 0 for all i 1.
Op(G)
___________
1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 20D0*
*5.
Key words and phrases. Classifying space, p-completion, finite simple groups.
Partially supported by UMR 7539 of the CNRS.
1
2 BOB OLIVER
Theorem A is proven as Theorem 4.5 below. It was motivated by applications for
studying equivalences between p-completed classifying spaces of finite groups. *
*Let G
and G0 be finite groups, and let S G and S0 G0 be Sylow p-subgroups. An
~= *
* ff
isomorphism ': S --! S0is called fusion preserving if for all P, Q S and all *
*P --!~
*
* =
'ff'-1
Q, ff is conjugation by an element of G if and only if '(P ) ---!~ '(Q) is conj*
*ugation
=
by an element of G0.
The Martino-Priddy conjecture states that for any prime p, and any pair G, G0*
* of
finite groups, BG^p' BG0^pif and only if there is a fusion preserving isomorphi*
*sm
between Sylow p-subgroups of G and G0. The ö nly if" part of the conjecture was
proved by Martino and Priddy [MP ], and follows from the bijection
~= ^
Rep (P, G) def=Hom(P, G)= Inn(G) ------! [BP, BGp]
for any p-group P and any finite group G (cf. [BLO1 , Proposition 2.1]). Conve*
*rsely,
by [BLO1 , Proposition 6.1], given a fusion preserving isomorphism between Syl*
*ow p-
subgroups of G and G0, the obstruction to extending it to a homotopy equivalence
BG^p' BG0^plies in lim-2(ZG). Hence Theorem A implies:
Theorem B (Martino-Priddy conjecture at odd primes).For any odd prime p, and
any pair G and G0 of finite groups with Sylow p-subgroups S G and S0 G0,
~=
BG^p' BG0^pif and only if there is a fusion preserving isomorphism S --! S0.
We next turn to the question of self equivalences of BG^p. For any space X, *
*let
Out(X) denote the group of homotopy classes of self homotopy equivalences of X.
For any finite group G, any prime p, and any Sylow p-subgroup S G, let Autfus*
*(S)
be the group of fusion preserving automorphisms of S, let Aut G(S) be the group*
* of
automorphisms induced by conjugation by elements of G (i.e., elements of NG(S))*
*, and
set
Outfus(S) = Autfus(S)= AutG(S).
Theorem A, when combined with [BLO1 , Theorem 6.2], gives the following descri*
*ption
of Out(BG^p).
Theorem C. For any odd prime p and any finite group G, with Sylow p-subgroup
S G,
Out (BG^p) ~=Out fus(S).
Theorem A should be a special case of a more general vanishing result, formul*
*ated
here as Conjecture 2.2, where orbit categories of groups are replaced by orbit *
*categories
of arbitrary äs turated fusion systems" in the sense of Puig [Pu ]. We refer to*
* [BLO2 ,
x1], and to the summary in Section 2 below, for definitions of saturated fusion*
* systems.
Conjecture 2.2 would, in particular, imply the existence and uniqueness of link*
*ing
systems, hence of classifying spaces, associated to an arbitrary saturated fusi*
*on system
over a p-group. This has motivated us to state results here, as far as possible*
*, in the
context of abstract saturated fusion systems. It is only at the end that we tra*
*nslate our
partial results to a condition on simple groups, which is then checked in the i*
*ndividual
cases.
EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 3
When p = 2, Theorem A is not true, since lim-1(ZG) can be nonzero. The simple*
*st
counterexamples occur for G = P SL2(q), when q 1 (mod 8). Recently, we have
proved that lim-i(ZG) = 0 for all i 2, when p = 2 and G is an arbitrary finit*
*e group.
This means that the Martino-Priddy conjecture does hold for p = 2, but that The*
*orem
C is not true (as formulated above) in this case. The proof for p = 2 not only *
*requires
the classification theorem for finite simple groups, but also (in its current f*
*orm) requires
a long, detailed case-by-case check when handling the simple groups of Lie type*
* in odd
characteristic as well as the sporadic groups. For this reason, we have not tr*
*ied to
incorporate it into this paper, but will write it up separately.
Section 1 contains general material about higher limits over orbit categories*
* of finite
groups, and Section 2 some results about saturated fusion systems and higher li*
*mits
over their orbit categories. Concrete criteria for proving the acyclicity of ZF*
* are then
set up in Section 3, where the problem is reduced to a question about "simple" *
*fusion
systems (Proposition 3.8). Also, at the end of Section 3, there is a discussion*
* of what
further results would be necessary to prove Conjecture 2.2 for odd primes. Fin*
*ally,
in Section 4, we restrict attention to fusion systems of finite groups, and app*
*ly the
classification theorem for finite simple groups to finish the proof of Theorem *
*A.
I would like to thank George Glauberman for his encouragement, and his effort*
*s to
prove a result about p-groups (Conjecture 3.9 below) which would have led to a *
*proof
of Conjecture 2.2 for odd p, and in particular to a "classification free" proof*
* of Theorem
A. I also want to point out the importance to this work of Jesper Grodal's tech*
*niques
in [Gr ] for computing higher limits of functors on orbit categories. His main *
*theorem,
while not used here directly, was used in many of the computations which led to*
* this
proof. Finally, I thank Carles Broto and Ran Levi, not only for their collabora*
*tion in
the papers [BLO1 ] and [BLO2 ] which are closely connected to this one, but a*
*lso for
introducing me to this problem in the first place.
General notation: We list, for easy reference, the following notation which w*
*ill be
used throughout the paper.
o Sylp(G) denotes the set of Sylow p-subgroups of G
o Op(G) is the maximal normal p-subgroup of G
n
o n(P ) = (for a p-group P )
o NG(H, K) = {x 2 G | xHx-1 K} (for H, K G)
o cx denotes conjugation by x (g 7! xgx-1)
o Hom G (H, K) (for H, K G) is the set of homomorphisms from H to K induc*
*ed
by conjugation in G
o Rep (H, K) = Inn(K)\Hom (H, K) and RepG(H, K) = Inn(K)\Hom G (H, K)
o Aut G(H) = Hom G(H, H), and OutG(H) = RepG (H, H) = AutG (H)= Inn(H)
o A functor F : Cop ___! Ab is called acyclic if lim-i(F ) = 0 for all i > 0.
4 BOB OLIVER
1. Higher limits over orbit categories of groups
We first collect some tools for computing higher limits of functors over the *
*orbit
category of a finite group G. Very roughly, these reduce to two general techniq*
*ues. One
is to filter a functor by a sequence of subfunctors, such that each of the subq*
*uotients
vanishes except on one conjugacy class of p-subgroups of G. Proposition 1.1 th*
*en
gives some tools which are very effective when computing the higher limits of t*
*hese
subquotients. The other method is to reduce computations to a situation, descri*
*bed in
Proposition 1.3, where the functor extends to a Mackey functor, and hence is ac*
*yclic
by a theorem of Jackowski and McClure [JM ].
Fix a prime p, a finite group G, and a Z(p)[G]-module M. Let FM be the funct*
*or on
Op(G) defined by setting FM (P ) = MP (the fixed submodule), and define
*(G; M) = lim-*(FM ).
Op(G)
These graded groups were shown in [JMO ] to be very effective tools when compu*
*ting
higher limits over functors on orbit categories. We first summarize the proper*
*ties of
the * which will be needed here.
Proposition 1.1. Fix a prime p. Then the following hold.
(a)For any finite group G and any functor F : Op(G)op --! Z(p)-mod which vanish*
*es
except on subgroups conjugate to some given p-subgroup P G,
lim-*(F ) ~= *(NG(P )=P ; F (P )).
Op(G)
(b)If G is a finite group,fHiC G is a normal subgroup which acts trivially on t*
*he
Z(p)[G]-module M, and pfi|H|, then *(G; M) = 0.
(c)If G is a finite group, and H C G is a normal subgroup of order prime to p w*
*hich
acts trivially on the Z(p)[G]-module M, then
*(G; M) ~= *(G=H; M).
(d)If G is a finite group, and Op(G) 6= 1 (if G contains a nontrivial normal p-
subgroup), then *(G; M) = 0 for all Z(p)[G]-modules M.
*
* __
Proof.See [JMO , Propositions 5.4, 5.5, & 6.1]. *
* |__|
The idea now is to filter an arbitrary functor F : Op(G) ___! Z(p)-mod in su*
*ch a
way that all quotient functors vanish except on one conjugacy class, and hence *
*are
described via Proposition 1.1(a).
We next look for some conditions on a pair of finite groups H G, and a func*
*tor F
on Op(G), which reduce the computation of lim-*(F ) to one of higher limits of *
*a functor
over Op(N(H)=H) . In general, for any small categories C and D and any functors
C ------! D ---F---!Ab ,
there is an induced homomorphism
* *
lim-*(F ) ------! lim-(F O )
D C
EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 5
defined as follows. Let I* be an injective resolution of F (in the category of *
*functors
D ___! Ab ), and let bI*be an injective resolution of F O . Since I* O is a r*
*esolution
_* *
of F O (though not injective), there is a chain homomorphism I* O --! Ib which
extends the identity on F O , and which is unique up to chain homotopy. Then **
* is
the homology of the composite homomorphism
lim(_*) *
lim-(I*) ------! lim-(I* O ) ------! lim-(bI),
D C C
where the first map is induced by the universal property of inverse limits over*
* C.
Lemma 1.2. Fix a finite group G and a p-subgroup Q G. Then there is a well
defined functor
= GQ:Op(NG(Q)=Q) ------! Op(G)
such that (P=Q) = P for all P=Q NG(Q)=Q. Let T be the set of all p-subgroups
P G with the property
Q C P , and Q C xP x-1 for x 2 G implies x 2 NG(Q). (*)
Then for any functor F :Op(G) op___! Z(p)-mod which vanishes except on subgrou*
*ps
G-conjugate to elements of T , the induced homomorphism
* *
lim-*(F ) ------!~ lim- (F O ) (1)
Op(G) = Op(NG(Q)=Q)
is an isomorphism.
Proof.Clearly, is well defined on objects. To see that it is well defined on *
*morphisms,
recall first that
Mor Op(G)(P, P 0) = P 0\NG(P, P 0),
where NG(P, P 0) is the set of all x 2 G such that xP x-1 P 0. Hence for any *
*pair of
objects P=Q and P 0=Q in Op(NG(Q)=Q) ,
Mor Op(NG(Q)=Q)(P=Q, P 0=Q) = (P 0=Q)\NN(Q)=Q(P=Q, P 0=Q) ~=P 0\NN(Q)(P, P 0)
P 0\NG(P, P 0) = Mor Op(G)(P, P 0);
and is defined on morphism sets to be this inclusion.
Composition with is natural in F and preserves short exact sequences of fun*
*ctors.
Hence if F 0 F is a pair of functors from Op(G) to Z(p)-mod, and the lemma hol*
*ds
for F 0and for F=F 0, then it also holds for F by the 5-lemma. Hence it suffic*
*es to
prove that (1) is an isomorphism when F vanishes except on the G-conjugacy clas*
*s of
one subgroup P 2 T . When P = Q, then (1) is precisely the isomorphism lim-*(F *
*) ~=
*(N(Q)=Q; F (Q)) of Proposition 1.1(a).
Now let P 2 T be arbitrary. By condition (*), Q C P , NG(P ) NG(Q), and F O
vanishes except on the Op(NG(Q)=Q) -isomorphism class of P=Q. Let
= N(Q)=QP=Q:Op(NG(P )=P-)-----!Op(NG(Q)=Q)
6 BOB OLIVER
be the functor (R=P ) = R=Q for p-subgroups R NG(P ) NG(Q) containing P .
Then the following square commutes
* *
lim-*(F_)______! lim- (F O )
Op(G) Op(N(Q)=Q)
( O )*~=|| * |~=|
# #
*(NG(P )=P ; F (P=))= *(NG(P )=P ; F (P )) ,
and the vertical maps are isomorphisms by Proposition 1.1(a) (see the proof of *
*[JMO ,
Lemma 5.4] for the precise description of the isomorphisms). This shows that__**
* is an
isomorphism. |__|
The next proposition describes a different condition which implies the acycli*
*city of
a functor on the orbit category of a finite group.
Proposition 1.3. Fix a finite group G, a prime p, and a Z(p)[G]-module M, and l*
*et
H0M :Op(G) op------! Z(p)-mod,
be the functor defined by setting
H0M(P ) = H0(P ; M) = MP .
Let
F :Op(G) op---! Z(p)-mod
be any subfunctor of H0M (thus F (P ) MP for all P ) which satisfies the foll*
*owing
"relative norm property": for each pair of p-subgroups P Q G,
defn X fifi P o
NQPF (P ) = gx fix 2 F (P ) M F (Q). (*)
gP2Q=P
Then F is acyclic: lim-i(F ) = 0 for all i > 0.
Proof.The relative norms NQPmake F into a proto-Mackey functor in the sense of *
* __
[JM ], and hence it is acyclic by [JM , Proposition 5.14]. *
* |__|
The following application of Proposition 1.3 plays an important role in Secti*
*on 3. If
G is a finite group and S 2 Sylp(G), then OS(G) Op(G) denotes the full subcat*
*e-
gory whose objects are the subgroups of S (the inclusion is clearly an equivale*
*nce of
categories). As usual, a subgroup T S is called strongly closed in S with res*
*pect to
G if no element of T is G-conjugate to any element of Sr T .
Recall that for any p-groupnP and any n 1, n(P ) denotes the subgroup gene*
*rated
by all x 2 P such that xp = 1.
Proposition 1.4. Fix a finite group G, a Sylow subgroup S 2 Sylp(G), and a subg*
*roup
T S which is strongly closed in S with respect to G. Let M be a finite Z(p)[G*
*]-module,
and let
F :OS(G)op------! Ab
EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 7
be a subfunctor of H0M (in particular, F (P ) MP for all P ), which satisfie*
*s the
relative norm property:
NQPF (P ) F (Q) (*)
for each pair of subgroups P Q S. Let F1 F be the subfunctor
(
F (P )if P \ T = 1
F1(P ) =
0 otherwise.
Assume NZ(T). 1(M) = 0. Then lim-i(F1) = 0 for all i 1.
OS(G)
Proof.Note first that F1 is a functor: if P, P 0 S are G-conjugate, then P \ T*
* = 1 if
and only if P 0\ T = 1 since T is strongly closed.
Assume lim-n(F1) 6= 0 for some n 1. We must prove that NZ(T). 1(M) 6= 0. Th*
*is
will be shown by induction on n.
Let kF F be the pk-torsion subfunctor in F ; i.e., ( kF )(P ) = k(F (P ))*
* for each
P . Then for some k, lim-n( kF= k-1F ) 6= 0. This functor kF= k-1F satisfies a*
*ll of
the hypotheses of the proposition with respect to the Fp[G]-module k(M)= k-1(M)
1(M). It thus suffices to prove the proposition when M = 1(M); i.e., when pM *
*= 0.
__
Without loss of generality, we can assume M = F (1). Define a functor F F by
setting __
F (P ) = F (P ) \ (NP\T .M)
__
for all P S. We claim that F still satisfies condition (*) above. To see t*
*his, fix
subgroups P Q S, set P 0= P \ T C P and Q0= Q \ T C Q, and set Q00= P Q0.
Then P 0= P \Q0, so coset representatives for Q0=P 0are also representatives fo*
*r Q00=P .
Hence
__ Q Q Q0
NQPF (P ) = NP F (P ) \ NP0.M NP(F (P )) \ NP0(NP0.M)
__
F (Q) \ (NQ0.M) = F(Q).
__
Thus, upon replacing F by F (without changing F1), we can assume that F (P ) =
NP.M for all P T .
By Proposition 1.3, lim-i(F ) = 0 for all i > 0. Hence lim-n-1(F=F1) 6= 0, s*
*ince
lim-n(F1) 6= 0 by assumption. For each subgroup Q T , let FQ be the functor *
*on
OS(G) defined by
(
F (P )if P \ T is G-conjugate to Q
FQ(P ) =
0 otherwise.
There is an obvious filtration of F=F1 whose quotients are all isomorphic to FQ*
* for
various 1 6= Q T . Hence there is some Q T such that
lim-n-1(FQ) 6= 0. (1)
OS(G)
Since we can replace Q by any other subgroup of S in its G-conjugacy class, we *
*can
assume that NS(Q) 2 Sylp(NG(Q)).
8 BOB OLIVER
If n = 1, then (1) implies that FQ(S) 6= 0. Hence S \ T = Q, so Q = T , and
0 6= FT(S) FT(T ) = NT.M. In particular, NZ(T).M 6= 0.
Now assume n > 1. Set G0 = NG(Q)=Q, S0 = NS(Q)=Q 2 Sylp(G0), and T 0=
NT(Q)=Q. Then T 0is strongly closed in S0with respect to G0: no element of NT(Q*
*) is
NG(Q)-conjugate to any element of NS(Q)r NT(Q) since no element of T is G-conju*
*gate
to any element of Sr T . Define functors
F 0, F10:OS0(G0) ------! Ab
by setting
F 0(P=Q) = F (P ) and F10(P=Q) = FQ(P ).
Thus F10(P=Q) = F 0(P=Q) whenever P \T = Q, equivalently whenever (P=Q)\T 0= 1;
and F10(P=Q) = 0 otherwise.
Consider the set
T0 = {P S | P \ T = Q}.
If FQ(P ) 6= 0 (and P S), then P \ T is G-conjugate to Q, P is G-conjugate to*
* some
P 0such that Q P 0 NS(Q) (since NS(Q) 2 Sylp(NG(Q))), P 0\ T is G-conjugate
to P \ T since T is strongly closed, and thus P 0\ T = Q and P 02 T0. By a simi*
*lar
argument, if P 2 T0 and Q C xP x-1, then (yx)P (yx)-1 NS(Q) for some y 2 NG(Q*
*),
(yx)P (yx)-1 \ T = Q, and so x 2 NG(Q). This shows that T0 is contained in the *
*set
T defined in Lemma 1.2, and thus that each subgroup of G for which FQ(P ) 6= 0*
* is
G-conjugate to a subgroup in T . Hence by Lemma 1.2,
lim-*(F10) ~= lim-*(FQ).
OS0(G0) OS(G)
In particular, lim-n-1(F10) 6= 0 by (1). All of the conditions of the proposit*
*ion are
satisfied (with G, S, T , F , and M replaced by G0, S0, T 0, F 0, and M0 def=F *
*(Q) = NQ.M).
So by the induction hypothesis, if we set Z0=Q = Z(T 0), then
0
NZ0.M = NZ(T0).NQ.M = NZ(T0).M 6= 0,
*
*__
and hence NZ(T).M 6= 0 since Z(T ) Z0. |*
*__|
2.Higher limits over orbit categories of fusion systems
We first briefly recall some definitions. We refer to [BLO2 , x1] or [Pu ] *
*for more
details.
A fusion system over a finite p-group S is a category F whose objects are the
subgroups of S, and whose morphisms satisfy the following conditions:
o Hom S(P, Q) Mor F(P, Q) Inj(P, Q) for all P, Q S; and
o each morphism in F is the composite of an F-isomorphism followed by an inclu*
*sion.
To emphasize that the morphisms in F are all homomorphisms of groups, we write
Hom F(P, Q) = Mor F(P, Q) for the morphism sets. Two subgroups of F are called
F-conjugate if they are isomorphic in F. A subgroup P S is called fully centr*
*alized
EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 9
in F (fully normalized in F) if |CS(P )| |CS(P 0)| (|NS(P )| |NS(P 0)|) for*
* all P 0in
the F-conjugacy class of P . The fusion system F is saturated if
(I)for each fully normalized subgroup P S, P is fully centralized and AutS (P*
* ) 2
Sylp(Aut F(P )); and
(II)for each ' 2 Hom F(P, S) whose image is fully centralized in F, if we set
N' = {x 2 NS(P ) | 'cx'-1 2 AutS('(P ))},
_
then ' extends to a morphism ' 2 Hom F(N', S).
If G is a finite group and S 2 Sylp(G), then we let FS(G) denote the category*
* whose
objects are the subgroups of S, and where
Mor FS(G)(P, Q) = Hom G(P, Q) ~=NG(P, Q)=CG(P ).
It is not hard to see [BLO2 , Proposition 1.3] that FS(G) is a saturated fusio*
*n system
over S, and that a subgroup P S is fully centralized (fully normalized) if an*
*d only
if CS(P ) 2 Sylp(CG(P )) (NS(P ) 2 Sylp(NG(P ))).
By analogy with the orbit category of a finite group, when F is a saturated f*
*usion
system over a p-group S, we let O(F) (the orbit category of F) be the category *
*with
the same objects, and with morphism sets
Mor O(F)(P, Q) = RepF (P, Q) def=Inn(Q)\Hom F (P, Q).
If F = FS(G) for some finite group G, then O(F) is a quotient category of OS(G)*
* (the
full subcategory of Op(G) whose objects are the subgroups of S), but its morph*
*ism
sets are much smaller in general. More precisely, if P and Q are two p-subgroup*
*s of G,
then
Mor Op(G)(P, Q) ~=Q\NG(P, Q),
while
MorO(Fp(G))(P, Q) ~=Q\NG(P, Q)=CG(P ).
Thus, there is a natural projection functor Op(G) --! O(Fp(G)) which is the id*
*entity
on objects and a surjection on all morphism sets, but these maps of morphism se*
*ts are
very far in general from being bijections. However, the next lemma shows that i*
*f one
restricts to p-centric subgroups of G, then p-local functors over these two cat*
*egories
have the same higher limits.
If F is a saturated fusion system over S, then a subgroup P S is F-centric *
*if
CS(P 0) = Z(P 0) for all P 0F-conjugate to P . If F = FS(G), then a subgroup i*
*s F-
centric if and only if it is p-centric in G (see [BLO1 , Lemma A.5]). Let Fc *
* F and
O(Fc) O(F) be the full subcategories whose objects are the F-centric subgroup*
*s of
S. Similarly, for any finite group G, and any S 2 Sylp(G), we let Ocp(G) Op(G*
*) and
OcS(G) OS(G) be the full subcategories whose objects are the p-centric subgro*
*ups of
G, and those contained in S, respectively.
Lemma 2.1. Fix a prime p and a finite group G. Let
F :O(Fcp(G))op------! Z(p)-mod
be any functor, and let
: Ocp(G) ------! O(Fcp(G))
10 BOB OLIVER
be the projection functor. Define
Fb:Op(G) op------! Z(p)-mod
by setting bF|Ocp(G)= F O , and bF(P ) = 0 if P is not p-centric in G. Then
lim-*(F ) ~= lim-*(F O ) ~= lim-*(Fb). (1)
O(Fcp(G)) Ocp(G) Op(G)
Proof.For any pair of p-centric subgroups P, Q G, write CG(P ) = Z(P ) x C0G(*
*P )
where C0G(P ) has order prime to p. For any x 2 NG(P, Q) and a 2 Z(P ), xa =
(xax-1)x 2 Qx, since xax-1 2 Q by definition of the transporter. Thus
Mor O(Fcp(G))(P, Q) = Q\NG(P, Q)=CG(P )
~=Q\NG(P, Q)=C0G(P ) ~=Mor Oc (P, Q)=C0(P ).
p(G) G
Since C0G(P ) has order prime to p, the first isomorphism in (1) now follows as*
* an
immediate consequence of [BLO1 , Lemma 1.3]. The second isomorphism holds since
bFvanishes on all p-subgroups which are not p-centric, and since every p-subgro*
*up of
*
* __
G which contains a p-centric subgroup is also p-centric. *
* |__|
For any saturated fusion system F over a finite p-group S, let
ZF :O(Fc)op------! Ab
denote the functor ZF (P ) = Z(P ) for all F-centric subgroups P S. If ' 2
Hom F(P, Q), then ZF (') is the composite
'-1
Z(Q) Z('(P )) ------! Z(P ).
If F = FS(G) for some finite group G with Sylow p-subgroup S, then ZG|OS(G)is t*
*he
composite of ZF with the projection between orbit categories. So Lemma 2.1 impl*
*ies
as a special case that
lim-*(ZG) ~= lim-*(ZF ).
Op(G) O(Fc)
What we would like to prove is the following conjecture, of which Theorem A is *
*just
the special case where F = FS(G) and p is odd.
Conjecture 2.2. Fix a prime p, and let F be a saturated fusion system over a p-*
*group
S. Then
lim-i(ZF ) = 0
O(Fc)
if p is odd and i 1, or if p = 2 and i 2.
Conjecture 2.2 would imply that each saturated fusion system over a p-group S*
* has
a unique associated "linking system", in the sense of [BLO2 , x1], and hence a*
* unique
associated classifying space (see [BLO2 , Proposition 3.1]). The vanishing of *
*lim-1(ZF )
would also imply (when p is odd) a description of the group of homotopy classes*
* of
self equivalences of the classifying space, similar to the description of Out (*
*BG^p) in
Theorem C.
Throughout this section and the next, we will be developping tools for comput*
*ing
higher limits of functors on centric orbit categories of saturated fusion syste*
*ms; in
EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 11
particular, those with connections to Conjecture 2.2. Only in the last section*
* do we
again return to the special case of fusion systems of finite groups, and finish*
* the proof
of Theorem A.
If F is any saturated fusion system over a p-group S, and Q S is fully norm*
*alized
in F, then NF (Q) is defined to be the fusion system over NS(Q) whose morphisms*
* are
defined by the formula
fi 0 0
Hom NF(Q)(P, P 0) = ff|P fiff 2 Hom F(P Q, P Q), ff(P ) P , ff(Q) = Q .
By [BLO2 , Proposition A.6], this is a saturated fusion system over NS(Q). We *
*also let
O Q (NF (Q)) denote the full subcategory of the orbit category of NF (Q) whose *
*objects
are the subgroups which contain Q.
Lemma 2.3. Fix a saturated fusion system F over a p-group S, and a fully norm*
*alized
F-centric subgroup Q S. Consider the functor
= FQ:O Q (NF (Q)) ------! OOutS(Q)(Out F(Q))
defined by setting
ff 0
(P ) = OutP (Q) and P ---! P = [ff|Q].
Then is an isomorphism of categories. Hence there is a functor
= FQ:Op(Out F(Q)) ------! O(Fc),
unique up to natural isomorphism, whose restriction to OOutS(Q)(Out F(Q)) is eq*
*ual to
-1.
Proof.Write = OutF (Q) and S0= OutS(Q) for short. Since Q is fully normalized*
* in
F, S0 is a Sylow p-subgroup of (condition (I) in the definition of a saturate*
*d fusion
system), and so the inclusion OS0( ) Op( ) is an equivalence of categories.
Now, S0 ~=NS(Q)=Q since Q is F-centric in S. So defines a bijection between
objects of O Q (NF (Q)) and objects of OS0( ), sending P to OutP (Q) ~=P=Q.
Fix subgroups P, P 0 NS(Q) containing Q, and consider the function
P,P0:Rep NF(Q)(P, P 0) = Mor O(NF(Q))(P, P 0) ------! Mor Op( )(Out P(Q), OutP*
*0(Q))
which sends the class [ff], for ff 2 Hom NF(Q)(P, P 0), to the class of [ff|Q] *
*2 OutF (Q) = .
For any such ff, the following square commutes
ff 0
P ______! P
| |
cg| |cff(g)
# #
ff 0
P ______! P
for all g 2 P , so ff lies in the transporter N (Out P(Q), OutP0(Q)), and the m*
*ap P,P0
is well defined. If fi 2 Aut F(Q) is such that conjugation by [fi] 2 = Out F*
*(Q)
sends OutP (Q) into OutP0(Q), then ficgfi-1 2 AutP0(Q) for all g 2 P , so fi ex*
*tends to
some ff 2 Hom F(P, P 0) by condition (II) in the definition of a saturated fusi*
*on system,
and P,P0sends [ff] to [fi]. Thus P,P0is onto. If ff1, ff2 2 Hom NF(Q)(P, P 0)*
* are such
that P,P0([ff1]) = P,P0([ff2]) in Op( ), then ff1|Q = cg Off2|Q for some g 2 *
*Q, hence
ff1 = cg Off2 Ocz for some z 2 Z(Q) by [BLO2 , Proposition A.8], and so [ff1] *
*= [ff2] in
12 BOB OLIVER
RepF (P, P 0). Thus, P,P0is a bijection for each pair of objects P, P 0, and t*
*his finishes
the proof that is an isomorphism of categories.
The last statement now follows by letting be the composite of a retraction *
*of
Op( ) onto OS0( ), followed by -1, followed by the inclusion of O Q (NF (Q))_i*
*nto
O(Fc). |__|
The next proposition describes how higher limits over O(Fc) can be reduced in
certain cases to higher limits over the orbit category of OutF (Q) for some sub*
*group Q.
Note its similarity with Lemma 1.2, in both the statement and the proof.
By analogy with the usual definition for subgroups of finite groups, for any *
*saturated
fusion system F over a p-group S, a subgroup P S is called weakly F-closed (or
weakly F-closed in S) if P is not F-conjugate to any other subgroup of S.
Proposition 2.4. Fix a saturated fusion system F over a p-group S and a fully n*
*or-
malized F-centric subgroup Q S, and let
= FQ:Op(Out F(Q)) ------! O(Fc)
be the functor of Lemma 2.3. Let T be the set of all subgroups P S such that
Q C P , and Q C ff(P ) for ff 2 Hom F(P, S) implies ff(Q) = Q. (*)
Then for any functor F :O(Fc)op---! Z(p)-mod which vanishes except on subgroups
F-conjugate to elements of T , the induced homomorphism
* *
lim-*(F ) -------! lim- (F O ) (1)
O(Fc) Op(OutF(Q))
is an isomorphism. In particular, if Q is weakly F-closed in S, then (1) holds *
*for any
functor F which vanishes except on subgroups which contain Q.
Proof.Composition with is natural in F and preserves short exact sequences of
functors. If F 0 F is a pair of functors from O(Fc) to Z(p)-mod, and the lemma*
* holds
for F 0and for F=F 0, then it also holds for F by the 5-lemma. Hence it suffice*
*s to prove
that (1) is an isomorphism when F vanishes except on the F-conjugacy class of o*
*ne
subgroup P 2 T .
__ __
Fix P 2 T , and set P = OutP (Q) OutF (Q). By condition (*), Q C P (so P ~=
P=Q), and F O vanishes except on the Op(Out F(Q))-isomorphism class of OutP (Q*
*) ~=
P=Q. Also, by (*) again,
OutF (P ) ~=Out NF(Q)(P ),
and
__ __ __
Out NF(Q)(P ) = AutOp(NF(Q))(P ) ~=AutOp(OutF(Q))(P ) = NOutF(Q)(P )=P
by Lemma 2.3. Let
= OutF(Q)OutP(Q):Op(Out-F(P-))----!Op(Out F(Q))
EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 13
be the functor (R=P ) = R=Q for p-subgroups R NG(P ) NG(Q) containing P .
Then the following square commutes
* *
lim-*(F )_________! lim- (F O )
O(F) Op(OutF(Q))
( O )*~=|| * ~=||
# ~ #
= * __ __
*(Out F(P ); F (P_))! (NOutF(Q)(P )=P ; F (P )) ,
and the vertical maps are isomorphisms by [BLO2 , Proposition 3.2] (and its pr*
*oof) and
Proposition 1.1(a). It follows that * is an isomorphism.
The last statement follows since if Q is weakly F-closed in S, then T = {P *
*S |_P
Q}: every subgroup which contains Q satisfies (*). *
* |__|
The following lemma describes how quotient fusion systems are obtained by div*
*iding
out by weakly F-closed subgroups.
Lemma 2.5. Let F be a saturated fusion system over a p-group S, and let Q C S*
* be
a weakly F-closed subgroup. Let F=Q be the fusion system over S=Q defined by se*
*tting
Hom F=Q(P=Q, P 0=Q) = {'=Q | ' 2 Hom F(P, P 0)}
for all P, P 0 S which contain Q. Then F=Q is saturated. Also, for any P=Q S*
*=Q,
P=Q is fully normalized in F=Q if and only if P is fully normalized in F, while*
* P is
fully centralized in F whenever P=Q is fully centralized in F=Q.
Proof.For each P S which contains Q, set
KP = Ker[Aut F(P ) --i AutF=Q(P=Q)].
and
K0P= Ker[Aut S(P ) --i AutS=Q(P=Q)].
Then
ffi ffi
|CS=Q(P=Q)| = |CS(P )|.|K0P||Q| and |NS=Q(P=Q)| = |NS(P )||Q|. (1)
By the second formula, P=Q is fully normalized in F=Q if and only if P is fully
normalized in F.
Assume P=Q is fully normalized in F=Q. Then P is fully normalized in F, so by
condition (I) in the definition of a saturated fusion system applied to F, P is*
* fully
centralized in F and AutS(P ) 2 Sylp(Aut F(P )). This last condition implies th*
*at
K0P2 Sylp(KP) and AutS=Q(P=Q) 2 Sylp(Aut F=Q(P=Q)).
Thus |CS(P )| and |K0P| both take the largest possible values among subgroups i*
*n the
F-conjugacy class of P , and hence P=Q is fully centralized by (1). This finish*
*es the
proof that condition (I) holds for F=Q. It also shows that if P=Q is fully cent*
*ralized
in F=Q, then |CS(P )| and |K0P| must both take the largest possible values among
subgroups in the F-conjugacy class of P , and in particular P is fully centrali*
*zed in F.
To prove condition (II), fix a morphism '=Q 2 Hom F=Q(P=Q, S=Q) such that
'(P )=Q is fully centralized in F=Q, and set
eN'= {g 2 NS(P ) | 'cg'-1 2 K'(P).AutS('(P ))}.
14 BOB OLIVER
Then
fi
Ne'=Q = N'=Q def=gQ 2 NS(P )=Q fi('cg'-1)=Q 2 AutS=Q('(P )=Q) ,
and we must show that '=Q extends to Ne'=Q. Set P 0= '(P ) for short; P 0is fu*
*lly
centralized in F since P 0=Q is fully centralized in F=Q. Since
'.Aut eN'(P ).'-1 KP0.AutS(P 0),
where KP0 C Aut F(P 0), Aut S(P 0) 2 Sylp(Aut F(P 0)), and the left hand side i*
*s a p-
group, there is _ 2 KP0 such that
(_').Aut eN'(P ).(_')-1 AutS(P 0).
So by condition (II) for the saturated fusion system F, _' extends to a homomor*
*phism
_ _ *
* __
' 2 Hom F(Ne', S), and '=Q is an extension of '=Q to N'=Q. *
*|__|
3.Reduction to simple fusion systems
In this section, we establish a sufficient condition for proving the acyclici*
*ty of ZF : a
criterion which in the case F = FS(G) will depend only on the simple components*
* in
the decomposition series of the finite group G.
Recall that for any p-group P and any n 1, n(P ) denotes the subgroup of P
generated by pn-torsion elements.
If H and K are two subgroups of a group G (usually normal subgroups) and n
1, then we write [H, K; n] for the n-fold iterated commutator: [H, K; 1] = [H,*
* K],
[H, K; 2] = [[H, K], K], and [H, K; n+ 1] = [[H, K; n], K].
Definition 3.1.For any p-group S, X(S) denotes the largest subgroup of S for wh*
*ich
there is a sequence
1 = Q0 Q1 . . .Qn = X(S) S
of subgroups, all normal in S, such that
[ 1(CS(Qi-1)), Qi; p- 1] = 1 (1)
for each i = 1, . .,.n.
It is easy to see that there always is such a largest subgroup. If
1 = Q0 Q1 . . .Qn and 1 = Q00 Q01 . . .Q0m
are two sequences of normal subgroups of S which satisfy condition (1) in Defin*
*ition
3.1, then the sequence
1 = Q0 Q1 . . .Qn = Qn.Q00 Qn.Q01 . . .Qn.Q0n
also satisfies the same condition.
When p = 2, X(S) = CS( 1(S)) for any finite 2-group S. In particular, X(S) = *
*Z(S)
if S is generated by elements of order 2. So these subgroups are not very inter*
*esting
in that case.
We first note some elementary properties of these subgroups X(S):
EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 15
Lemma 3.2. If p is odd and S is a p-group, then X(S) A for every normal abe*
*lian
subgroup A C S. In particular, X(S) is centric in S.
Proof.If A C S is abelian, then [S, A; p- 1] [[S, A], A] = 1, and so A X(S)*
* by
definition.
Now let A be maximal among the normal abelian subgroups of S. If CS(A) A,
then A.CS(A)=A is a nontrivial normal subgroup of S=A, and hence contains an el*
*ement
xA 2 Z(S=A) of order p. But then is a larger normal abelian subgroup of*
* S,
which is a contradiction. Thus A is centric in S, and in particular X(S) A is*
*_centric
in S. |__|
The following lemma is useful when proving that certain subgroups of S are co*
*ntained
in X(S).
Lemma 3.3. Fix an odd prime p and a p-group S. Let Q C S be any normal subgro*
*up
such that
[ 1(Z(X(S))), Q; p- 1] = 1. (1)
Then X(S) Q.
Proof.Set X = X(S) for short. By definition, there is a sequence
1 = Q0 Q1 . . .Qn = X
of subgroups normal in S, such that [ 1(CS(Qi-1)), Qi; p- 1] = 1 for each i. If*
* Q C S
is normal and satisfies condition (1), then since Z(X) = CS(X) by Lemma 3.2, we*
*_can
set Qn+1 = Q.Qn, and Q Qn+1 X by definition. |_*
*_|
The purpose of these subgroups X(S) is to provide a tool for applying Proposi*
*tion
1.4, when trying to show that the functors ZF are acyclic. These are most usefu*
*l when
applied to a filtration of these functors, described as follows.
For any saturated fusion system F over a p-group S, a subgroup P S is stron*
*gly
F-closed in S if no element of P is F-conjugate to any element of Sr P . If T *
* S is
strongly F-closed subgroup in S, let
ZTF:O(Fc)op------! Z(p)-mod
be the subfunctor of ZF defined by setting ZTF(P ) = Z(P ) \ T .
When F is a saturated fusion system over a p-group S, and T C S is a strongly*
* F-
closed subgroup, then a fully F-normalized subgroup P T will be called F|T-ra*
*dical
if
Op(Out F(P )) \ OutT(P ) = 1.
Lemma 3.4. Fix an odd prime p, a saturated fusion system F over a p-group S, *
*and
a pair T0 C T C S of subgroups strongly F-closed in S. Write X(T=T0) = X=T0 for
short. For any fully F-normalized subgroup Q T , define
ZQ :O(Fc)op------! Z(p)-mod
16 BOB OLIVER
by setting, for F-centric P S,
( ffi
ZTF ZT0F(P ) ~=Z(P)\T_Z(P)\Tif P \ T is F-conjugate to Q
ZQ(P ) = 0
0 otherwise.
Assume that Q X, or that Q is not centric in T , or that Q is not F|T-radical*
*. Then
lim-*(ZQ) = 0.
O(Fc)
Proof.Since Q is fully F-normalized, for any Q0 S which is F-conjugate to Q, t*
*here
is some ' 2 Hom F(NS(Q0), NS(Q)) such that '(Q0) = Q [BLO2 , Proposition A.2(c*
*)].
Hence each subgroup P 0 S for which ZQ(P 0) 6= 0 is F-conjugate to a subgroup P
such that P \ T = Q.
Assume first Q T0. Then for each F-centric subgroup P such that P \ T = Q,
NPT0(P )=P 6= 1 and acts trivially on ZQ(P ), and so *(Out F(P ); ZQ(P )) = 0*
* by
Proposition 1.1(b). Thus lim-*(ZQ) = 0 in this case.
If Q is not centric in T , then for each F-centric subgroup P such that P \ T*
* = Q,
NP.CT(Q)(P )=P 6= 1 and acts trivially on ZQ(P ), and so *(Out F(P ); ZQ(P )) *
*= 0 by
Proposition 1.1(b). Again, lim-*(ZQ) = 0 in this case.
Now assume Q is centric in T , but not F|T-radical. Set
bQ= {x 2 NT(Q) | cx 2 Op(Out F(Q))};
bQ=Q 6= 1 by assumption. Let P S be a F-centric subgroup such that P \ T = Q.
Each element of AutF (P ) leaves Q invariant (since T C S), so we have a restri*
*ction
map
æ: Aut F(P ) ------! Aut F(Q) x Aut(P=Q),
and Ker(æ) is a p-group by [Go , Corollary 5.3.3]. Hence æ-1(Op(Aut F(Q)) x 1)*
* is a
normal p-subgroup of AutF (P ). Also, P normalizes bQ, since it normalizes Q, so
1 6= Q0=Q def=(Qb=Q)P
since bQ=Q 6= 1, and Q0 NS(P ) since [Q0, P ] Q by definition. For any x 2 *
*Q0r Q
N(P )r P , cx 2 æ-1(Op(Aut F(Q)) x 1), its class in OutF (P ) is nontrivial sin*
*ce P is F-
centric and x =2P , and hence Op(Out F(P )) 6= 1. Thus *(Out F(P ); ZQ(P )) = *
*0 by
Proposition 1.1(b). Since this holds for all F-centric P with P \ T = Q, lim-*(*
*ZQ) = 0
in this case.
It remains to consider the case where Q T0, Q is centric in T , and Q X. *
*This
will be done in three steps. In the first two steps, we show that lim-*(ZX) is *
*isomorphic
to the higher limits of a certain functor over an orbit category of a group. On*
*ly in Step
3 do we apply the assumption that Q X.
Step 1: In this case, we set
bQ= Q.CS(Q).
Then bQis F-centric, and bQ\ T = Q does not contain X. Also, bQis fully normali*
*zed
in F, since if Q0 is F-conjugate to bQand fully normalized in F, then there is *
*some
EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 17
ff 2 Hom F(NS(Qb), NS(Q0)) with ff(Qb) = Q0(see [BLO2 , Proposition A.2(c)]). *
*Hence
ff(Q) = Q0\ T C NS(Q0), and
|NS(Qb)| = |NS(Q)| |NS(ff(Q))| |NS(Q0)|
since Q is fully normalized.
Let bZQbe the quotient functor of ZQ where
(
bQ
bZQ(P ) = ZQ(P ) if P contains a subgroup F-conjugate to
0 otherwise.
If ZbQ(P ) 6= ZQ(P ) (i.e., if ZbQ(P ) = 0 and ZQ(P ) 6= 0), then up to conjuga*
*cy, P is
F-centric and P \ T = Q, but P bQ. Then NPQb(P )=P is a nontrivial p-subgroup
of Out F(P ) which acts trivially on ZQ(P ), so *(Out F(P ); ZQ(P )) = 0 in th*
*is case.
Thus
lim-*(ZbQ) ~= lim-*(ZQ). (1)
O(Fc) O(Fc)
Step 2: Set
= OutF (Qb), S0= OutS(Qb) 2 Sylp( ), and T 0= OutT(Qb)
for short. Using the isomorphism
Qb
O (NF (Qb)) -------!~ OS0( )
=
of Lemma 2.3, we see that T 0= (T bQ) is strongly closed in S0 = (S) with res*
*pect
to , since no element of T bQcan be NF (Q)-conjugate to any element of Sr T bQ.
By definition, each subgroup on which ZbQ is nonvanishing is F-conjugate to s*
*ome
P bQsuch that P \T = Q. In particular, Q C P since T C S, and so bQ= Q.CS(Q) C
P . If P 0is any subgroup F-conjugate to P which contains bQ, and ff 2 IsoF(P, *
*P 0) is
any isomorphism, then
ff(Q) = ff(P \ T ) = P 0\ T bQ\ T = Q,
and this is an equality since |ff(Q)| = |Q|. Hence ff(Qb) = Qb. Hypothesis (**
*) of
Proposition 2.4 is thus satisfied, and hence
lim-*(ZbQO -1) ~= lim-*(ZbQ). (2)
OS0( ) O(Fc)
Set
M1 = Z(Q) = Z(Qb) \ T and M0 = Z(Q) \ T0 = Z(Qb) \ T0;
and set M = M1=M0. We regard these as Z(p)[ ]-modules. Let
F :OS0( )op------! Z(p)-mod
be the functor F (P ) = M1P=M0P for all P S0. This is clearly a subfunctor of*
* H0M
which satisfies the relative norm condition (*) in Propositions 1.3 and 1.4. A*
*lso, for
18 BOB OLIVER
P 0= OutP (Qb) S0 (i.e., bQC P and P 0~=P=Qb),
( Z(P)\T
______~= F (P 0)if P \ T = Q
(ZbQO -1)(P 0) = bZQ(P ) = Z(P)\T0
0 otherwise;
and P \ T = Q if and only if P \ T bQ= Qb, if and only if P 0\ T 0= 1. Hence by
Proposition 1.4, together with (1) and (2),
NZ(T0). 1(M) = 0 implies lim-*(ZQ) ~= lim-*(ZbQO ) = 0. (3)
O(Fc) OS0( )
(More precisely, Proposition 1.4 only tells us that ZQ is acyclic. But Q T si*
*nce it
does not contain X, so ZQ(S) = 0, and this implies lim-0(ZQ) = 0.)
Step 3: By definition of X(T=T0), there are subgroups
1 = Q0=T0 Q1=T0 . . .Qn=T0 = X(T=T0),
all normal in T=T0, such that
[ 1(CT=T0(Qi-1=T0)), Qi=T0; p- 1] = 1 (4)
for all i = 1, . .,.n. Let i n (i 1) be the smallest integer such that Q *
*Qi. Then
QQi Q, so NQQi(Q)=Q is nontrivial, and is normal in NT(Q)=Q since QiC T . Hence
the fixed subgroup
NT(Q)=Q
NQQi(Q)=Q = {xQ | x 2 NQQi(Q), [x, NT(Q)] Q}
is also nontrivial. Fix some x 2 NQQi(Q)r Q such that x 2 Qiand [x, NT(Q)] Q.*
* In
particular,
xQ 2 Z(NT(Q)=Q). (5)
Since Q Qi-1by assumption,
[ 1(M), x; p- 1] [ 1(Z(Q=T0)), x; p- 1] [ 1(CT=T0(Qi-1=T0)), Qi; p- 1](=6*
*1,)
where the last equality holds by (4).
Now regard M additively as a Z(p)[ ]-module. Then (6) translates to the state*
*ment
that
N. 1(M) = (1 - x)p-1. 1(M) = 0.
Also, x 2 Z(Out T(Q)) by (5), so x 2 Z(T 0) by (3). Hence NZ(T0). 1(M) = 0, so*
* __
lim-*(ZQ) = 0 by (3), and this finishes the proof. *
* |__|
Using Proposition 2.4 and Lemma 3.4 (with T0 = 1 and T = S), it is not hard to
show that for any saturated fusion system F over a p-group S, ZF is acyclic if *
*X(S)
contains a subgroup which is both centric and weakly F-closed in S. Since we a*
*re
unable to prove directly that this holds for all F, we instead filter ZF via a *
*maximal
series of strongly F-closed subgroups of S, and use the following more general *
*result.
Proposition 3.5. Fix a saturated fusion system F over a p-group S, and let T0 C
T C S be a pair of subgroups strongly F-closed in S. Assume there is a subgro*
*up
X=T0 X(T=T0) which is centric in T=T0 and weakly F=T0-closed. Then the quotie*
*nt
functor ZTF=ZT0Fis acyclic.
EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 19
More generally, let XF (T=T0) be the intersection of all subgroups Q=T0 T=T0
containing X(T=T0) such that Q is fully F-normalized and F|T-radical. Assume th*
*ere
is a subgroup X=T0 XF (T=T0) which is centric in T=T0 and weakly F=T0-closed.
Then the quotient functor ZTF=ZT0Fis acyclic.
Proof.Write Z = ZTF=ZT0Ffor short. Assume X=T0 XF (T=T0) is centric in T=T0 a*
*nd
weakly F=T0-closed. In particular, X is weakly F-closed. Let ZX be the functo*
*r on
O(Fc) defined by setting, for all P S,
(
Z(P ) if P X
ZX(P ) =
0 otherwise.
(Note that since X is weakly F-closed, if P X, then the same holds for all su*
*bgroups
in its F-conjugacy class.) We regard ZX as a subfunctor of Z. Define ZQ as in L*
*emma
3.4; then lim-*(ZQ) = 0 for all fully F-normalized Q T such that Q=T0 X(T=T*
*0),
or such that Q is not F|T-radical. In particular, this applies to all Q X. Th*
*us via
the obvious filtration of ZX, we get that lim-*(ZX) = 0, and hence that
lim-*(Z=ZX) ~= lim-*(Z) . (1)
O(Fc) O(Fc)
Set X* = X.CS(X). Then X* is F-centric; and X* \ T = X since X is centric in T
(since X=T0 is centric in T=T0). If X P S and P X*, and P is F-centric, t*
*hen
NX*P(P )=P ~=Out X*(P ) is a nontrivial p-subgroup of OutF (P ) which acts triv*
*ially on
(Z(P ) \ T )=(Z(P ) \ T0), and so
*(Out F(P ); Z(P )) = 0
for such P . Hence if we let F denote the functor
(
Z(P ) = Z(P)\T_Z(P)\Tif P 0 X* for some P 0F-conjugate to P
F (P ) = 0
0 otherwise;
then
lim-*(F ) ~= lim-*(Z=ZX). (2)
O(Fc) O(Fc)
Set
M1 = Z(X*) \ T = Z(X) and M0 = Z(X*) \ T0.
Since X is weakly F-closed, X* def=X.CS(X) is both centric in S and weakly F-cl*
*osed.
So by Proposition 2.4, there is a functor
__
F :Op(Out F(X*))------! Ab,
where
__ Z(P ) \ T
F (P=X*) = __________~=M1P=M0P
Z(P ) \ T0
for all
P=X* ~=OutP (X*) OutS(X*) 2 Sylp(Out F(X*));
20 BOB OLIVER
and such that
__
lim-* (F ) ~= lim-*(F ). (3)
Op(OutF(X*)) O(F*)
__ __
Finally, lim-i(F ) = 0 for i > 0 by Proposition 1.3 (F ~= H0M1=H0M0). Together *
*with __
(1), (2), and (3), this finishes the proof of the proposition. *
* |__|
It now remains to determine, for each saturated fusion system F over a p-grou*
*p S (p
odd), whether there always exists a sequence of strongly F-closed subgroups for*
* which
Proposition 3.5 applies to each successive pair. For convenience, we define a s*
*ubgroup
Q S to be universally weakly closed in S if for every saturated fusion system*
* F over
a p-group S0 S such that S is strongly F-closed, Q is weakly F-closed in S0.
Lemma 3.6. Fix an odd prime p and a p-group S. Then a subgroup Q S is uni-
versally weakly closed if for all P S containing Q, Q is a characteristic sub*
*group of
P .
Proof.Assume that Q S is not universally weakly closed. Then there exist a s*
*at-
urated fusion system F over a p-group S0 S such that S is strongly F-closed, *
*and
such that Q is not weakly F-closed in S0. By Alperin's fusion theorem for satur*
*ated
fusion systems [BLO2 , Theorem A.10], there is a subgroup P 0 S0 containing Q*
*, and
an automorphism ff 2 AutF (P 0) such that ff(Q) 6= Q. Set P = T \P 0. Then ff(P*
* ) = P
since T is strongly F-closed, and hence ff induces an automorphism of P S whi*
*ch __
does not send Q to itself. Thus Q is not a characteristic subgroup of P . *
* |__|
The next lemma gives some simple conditions on the p-group for being able to
apply Proposition 3.5. For any p-group S, let J(S) denote Thompson's subgroup: *
*the
subgroup generated by all elementary abelian subgroups of S of maximal rank.
Proposition 3.7. Fix an odd prime p, and a p-group S which satisfies any of the
following conditions.
(a)X(S) J(S).
(b)S contains a unique elementary abelian p-subgroup E of maximal rank.
(c)S=X(S) is abelian.
Then there is a subgroup P X(S) which is centric and universally weakly close*
*d in
S.
Proof.Write X = X(S) for short.
(a) Assume X J(S). Clearly, J(S) is universally weakly closed in S; however,*
* it
need not be centric. So instead, consider the subgroup Q = J(S).CS(J(S)) S. T*
*his
is clearly normal and centric in S, and is characteristic in any subgroup of S *
*which
contains it since J(S) is. Thus Q is universally weakly closed in S by Lemma 3.*
*6.
It remains to check that Q X. Since J(S) X, every elementary abelian subg*
*roup
of S of maximal rank commutes with Z(X), and thus contains 1(Z(X)) since other*
*wise
it would not be maximal. Thus 1(Z(X)) Z(J(S)), so
[ 1(Z(X)), Q] [Z(J(S)), J(S).CS(J(S))] = 1.
EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 21
Hence Q X by Lemma 3.3.
(b) If E S is the unique elementary abelian subgroup of maximal rank, then J(*
*S) =
E, and E X by Lemma 3.2. The result thus follows from (a).
(c) Assume that S=X is abelian, and that X is not universally weakly closed in *
*S. By
Lemma 3.6, there is a subgroup P S containing X, and an automorphism ff 2 Aut*
*(P )
such that ff(X) 6= X. We claim that this is impossible.
Assume first that ff(Z(X)) X, and fix an element g 2 ff(Z(X))r X. Then
[ 1(Z(X)), g] ff(X), since ff(X) C P , and hence
[[ 1(Z(X)), g], g] [ff(X), g] = 1
since g 2 Z(ff(X)). Set Q = ; then [ 1(Z(X)), Q; 2] = 1, and Q C S since *
*S=X is
abelian. Then Q 2 X by Lemma 3.3, and this contradicts the original assumption *
*on
g.
Now assume that ff(Z(X)) X, and thus that Z(X) ff-1(X) (and ff-1(X) 6= X).
Fix a chain of subgroups
1 = Q0 Q1 . . .Qn = X,
all normal in S (hence in P ), which satisfy condition (1) in Definition 3.1. L*
*et i n
be such that Qi ff(X) but Qi-1 ff(X). Then
-1
ff-1 1(CP(Qi-1)) = 1(CP(ff Qi-1)) 1(CP(X)) = 1(Z(X)),
and hence
[ 1(Z(X)), ff-1Qi; p- 1] ff-1 [ 1(CP(Qi-1)), Qi; p- 1] = 1
by the assumption on the Qi. Hence by Lemma 3.3 again, X, which contra*
*-__
dicts the original assumption on Qi. *
*|__|
We note the following immediate corollary to Propositions 3.7(a) and 3.5.
Corollary 3.8. Let F be a saturated fusion system over a p-group S, and let 1 =*
* T0
T1 . . .Tk = S be any sequence of subgroups which are all strongly F-closed i*
*n S.
Assume, for all 1 i k, that X(Ti=Ti-1) J(Ti=Ti-1). Then lim-i(ZF ) = 0 fo*
*r_all
i > 0. |__|
Corollary 3.8 motivates the following
Conjecture 3.9. For any odd prime p and any p-group P , X(P ) J(P ).
By Corollary 3.8 (together with Lemma 2.5), in order to prove that ZF is acyc*
*lic
for all saturated fusion systems F, it suffices to prove Conjecture 3.9 for all*
* p-groups
P which can occur as minimal strongly closed subgroups in saturated fusion syst*
*ems.
However, it seems to be very difficult to prove or find a counterexample to thi*
*s conjec-
ture, even in this restricted form. This also indicates that it will be very di*
*fficult to
find an example of a saturated fusion system F for which ZF is not acyclic, if *
*there
are any.
We finish this section with one other elementary result about the groups X(S)*
*, a
result which will be useful in the next section.
22 BOB OLIVER
Proposition 3.10. Fix an odd prime p and a p-group S. Then either rk(Z(X(S)))
p, or X(S) = S. In particular, X(S) = S if rk(S) p - 1.
Proof.Set X = X(S) for short. Assume rk(Z(X)) p-1, and set E def= 1(Z(X)) C S.
For each i 0, either
[E, S; i+ 1] = [[E, S; i], S] [E, S; i],
or [E, S; i] = 1. Since |E| ~=(Cp)k for k p - 1, this shows that [E, S; p- 1]*
*_= 1, and
hence that X(S) = S by Lemma 3.3. |__|
4.The acyclicity of ZG at odd primes
We are now ready to show, for any finite group G and any odd prime p, that al*
*l higher
limits of ZG vanish when p is odd. This will be based on the following proposi*
*tion,
which gives for any finite group G a sufficient condition for the acyclicity of*
* ZG in
terms of its simple composition factors. When G is a finite group and S 2 Sylp(*
*G), set
" fi
XG(S) = P S fiP X(S), Op(Out G(P )) = 1, NS(P ) 2 Sylp(NG(P )) :
the intersection of all subgroups of S which contain X(S), and are fully normal*
*ized and
Fp(G)-radical.
Proposition 4.1. For any prime p and any finite group G, ZG is acyclic if for e*
*ach
nonabelian simple group L which occurs in the decomposition series for G, and a*
*ny
S 2 Sylp(L), there is a subgroup Q XL(S) which is centric and weakly Aut(L)-c*
*losed
in S. In particular, ZG is acyclic for each finite solvable group G.
Proof.Fix a sequence of normal subgroups
1 = K0 K1 . . .Kn = G
such that each subquotient Ki+1=Kiis a minimal normal subgroup of G=Ki. We show
that ZKi+1G=ZKiGis acyclic for each i. Choose S 2 Sylp(G), and set Si = S \ Ki*
* 2
Sylp(Ki) and F = FS(G).
Assume that Q = Qe=Si Si+1=Si is centric in Si+1=Si and that Qe is fully F-
normalized (i.e., NS(Qe) 2 Sylp(NG(Qe))). If eQis F|Si+1-radical, then
proj
1 = Op(Out G(Qe)) \ OutKi+1(Qe) = Op(Out Ki+1(Qe)) ----i Op(Out Ki+1=Ki(Q)),
and hence Q is a radical p-subgroup of Ki+1=Ki. This proves that
XF (Si+1=Si) XKi+1=Ki(Si+1=Si).
So by Proposition 3.5 (and Lemma 2.1), to prove that ZKi+1G=ZKiGis acyclic, it *
*suffices
to show
XKi+1=Ki(Si+1=Si) contains a subgroup Q which is centric and weakly G=Ki-
(1)
closed in Si+1=Si.
To simplify notation, we replace G by G=Ki (so Ki = 1), and set K = Ki+1 and
P = Si+12 Sylp(K). Thus, K is a minimal normal subgroup of G, and we must find
EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 23
Q XK (P ) which is centric and weakly G-closed in P . This is clear if K has *
*order
prime to p (i.e., Q = P = 1).
Since K is a minimal normal subgroup, it is a product of finite simple groups*
* isomor-
phic to each other (cf. [Go , Theorem 2.1.5]). If K is an elementary abelian p-*
*group,
then X(K) = K, and is centric and weakly closed in K. So assume K ~=Ln where L
is simple and nonabelian and n 1. We can choose this identification in a way *
*such
that P = (P 0)n for some fixed P 02 Sylp(L). Then X(P ) = X(P 0)n (see Definiti*
*on 3.1),
and XK (P ) = XL(P 0)n since each radical p-subgroup of K splits as a product o*
*f n rad-
ical p-subgroups of L [JMO , Proposition 1.6(ii)]. By assumption, there is a s*
*ubgroup
Q0 XL(P 0) which is centric and weakly Aut (L)-closed in P 0. Then Q def=(Q0*
*)n is
centric in P , and Q XK (P ). It remains to show that Q is weakly Aut(K)-clos*
*ed in
P , and hence weakly G-closed in P .
Assume otherwise: assume there is ff 2 Aut(Ln) such that Q 6= ff(Q) P . The*
* n
factors L are the unique minimal normal subgroups of Ln, so each automorphism o*
*f Ln
permutes these factors, and hence Aut(Ln) ~=Aut (L) o n. Thus ff = oe O(ff1, .*
* .,.ffn)
for some ffi 2 Aut (L) and some oe 2 n (regarded as an automorphism of Ln); and
Q0 6= ffi(Q0) P 0for some i. Which contradicts the assumption that Q0 is wea*
*kly_
Aut(L)-closed in P 0. |*
*__|
We now prove that all finite nonabelianfsimpleigroups L satisfy the condition*
* in
Proposition 4.1: for any odd prime pfi|L| and any S 2 Sylp(L), there is a subg*
*roup
Q XL(S) (or Q X(S)) which is centric and weakly Aut(L)-closed in S. We first
consider some cases where this can be shown using Proposition 3.7(b).
Proposition 4.2. Assume p is odd, and let L be a simple group which is either an
alternating group, or a group of Lie type in characteristic different from p. T*
*hen for
S 2 Sylp(L), J(S) X(S), and hence there is a subgroup Q X(S) which is centr*
*ic
and weakly Aut(L)-closed in S.
Proof.If L ~=An, then S contains a unique elementary abelian p-subgroup E of ma*
*x-
imal rank, generated by a product of [n=p] disjoint p-cycles (cf. [GL , 10-5])*
*. Hence
J(S) = E X(S) by Lemma 3.2, and the result follows from Proposition 3.7(a) or
(b).
Now assume that L is a simple group of Lie type in characteristic 6=p. If rkp*
*(L) 2,
then X(S) = S by Proposition 3.10. So assume rkp(L) > 2. Then by [GL , 10-2(1)]*
*, each
Sylow p-subgroup of L contains a unique elementary abelian p-subgroup of maximal
rank; and the result follows from Lemma 3.2 and Proposition 3.7(a,b) again. (N*
*ote
that all of the exceptional cases listed in [GL ] _ the simple groups A2(q), 2A*
*2(q), G2(q),
3D4(q), and 2F4(q) when p = 3 _ have 3-rank at most 2 by [GL , 10-2(2)] and Tab*
*les
__
10:1 and 10:2.) |_*
*_|
We next consider simple groups of Lie type in characteristic p. We first summ*
*arize
the structures in these groups which will be needed, referring to [Ca ] as a ge*
*neral
reference.
Assume first that L is a Chevalley group: L = G(q), where G is one of the gro*
*ups An,
Bn, etc., defined over the finite field Fq (q = pa). For example, An(q) ~=P SLn*
*+1(Fq).
Let V denote the root system of G, where V is a real vector space. Let + b*
*e the
24 BOB OLIVER
set of positive roots; thus = { r | r 2 +}. Let I denote the set of primitiv*
*e roots,
an R-basis of V.
To each root r 2 corresponds a root subgroup Xr ~= Fq in L = G(q). Then
Q def
U def= r2 +Xr is a Sylow p-subgroup of L. Also, B = NL(U) = Uo H (the Borel
subgroup), where H is the subgroup of diagonal elements (and has order prime to*
* p).
Set N = NL(H); then W ~=N=H is the Weyl group of G (and of the root system ).
For example, when L = An(q) = P SLn+1(q), then we can take
X
V = {x 2 Rn+1| xi= 0}, = {ei- ej| i 6= j}
(where {e1, . .,.en+1} is the standard basis of Rn+1),
+ = {ei- ej| i < j}, and I = {ei- ei+1}.
Then Xei-ejis the subgroup of matrices which have 1's along the diagonal and are
zero elsewhere except at entry (i, j), U is the group of upper triangular matri*
*ces with
1's along the diagonal, and B is the group of all upper triangular matrices. Di*
*agonal
elements are represented by diagonal matrices, N is the image of the subgroup of
monomial matrices, and W ~= n+1.
We next fix the notation for the twisted groups tG(q). Let ø 2 Aut(V, , +) *
*be an
automorphism of the root system of G of order t. Set oe = bøOb'2 Aut(G(q0)), wh*
*ere bøis
induced by ø and b'is induced by ' 2 Aut(Fq0) of order t. (In most cases, q0= q*
*t, and
so Fq is the fixed subfield of the automorphism '.) In all cases, ø 2 Aut( +) c*
*an be
seen as an automorphism of the Dynkin diagram (and t = 2, 3). Also, oe(Xr) = Xf*
*i(r)for
each r 2 , and thus oe leaves invariant the subgroups U, H, and N. The twisted*
* group
L = tG(q) is defined to be the commutator subgroup of G(q0)ff, or alternatively*
* as the
subgroup of G(q0) generated by Uffand the analogous subgroup for the root group*
*s of
negative roots. Its Borel subgroup is defined to be B = NL(Uff) = Uffo(Hff\ L).
Proposition 4.3. Assume p is odd, let L be a simple group of Lie type in charac*
*teristic
p, and fix S 2 Sylp(L). Then XL(S) is weakly Aut(L)-closed in S.
Proof.Write L = tG(q), where q = pa (possibly t = 1). We use the above notation*
*. In
particular, + denotes the set of positive roots, and U 2 Sylp(L) is the produc*
*t of the
root subgroups Xr for r 2 +.
For each ø-invariant subset J I, consider the subgroups
Y
UJ = Xr,
r2 +r
In particular, U; = U, and UI = 1. We claim that the following statement holds:
The subgroups UffJ, for ø-invariant subsets J I, are the only subgroups *
*of
S = Uffwhich are radical p-subgroups of L; and they are all weakly L-close*
*d(1)
in Uff.
By a theorem of Borel and Tits (see the corollary in [BW ]), every radical p-s*
*ubgroup
of L is conjugate to one of the subgroups UffJ, and by [Gr , Lemma 4.2], if Uff*
*Jis L-
conjugate to UffJ0for J, J0 I then J = J0. But since we need to know that ea*
*ch
EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 25
radical p-subgroup of L contained in S is actually equal to one of the UffJ, we*
* modify
Grodal's proof to show this.
Assume that P S = Uffis L-conjugate to UffJ. We show that P = UffJ; this
proves that UffJis weakly L-closed in S, and hence (using [BW ]) proves (1). *
*Since
L Uff.Nff.Uff[Ca , Proposition 8.2.2], and since UJ C U, we have P = ux(UffJ)*
*x-1u-1
for some u 2 Uffand x 2 Nff, and P = UffJif and only if x(UffJ)x-1 = UffJ. So w*
*e can
assume u = 1 and P = x(UffJ)x-1. Now, x permutes the root subgroups via the act*
*ion
of w = xH 2 W fion , and so w( +r ) +. Write = +r for short; this*
* is
closed in the sense that any r 2 which is a positive linear combination of el*
*ements of
also lies in . So w( ) has the same property. This implies that all primitiv*
*e roots
for the system w( )\ + are primitive roots in +, and thus that w( ) = +r <*
*J0>
for some J0 I. After replacing w by its product with some element in the Weyl*
* group
of , we can assume that it sends positive roots to positive roots, and hen*
*ce must
be the identity. So J0 = J, and P = UffJ.
Thus, by (1), if Q S is a radical p-subgroup of L, then Q = UffJfor some J.*
* Hence
XL(S) is the intersection of the subgroups UffJwhich contain X(S). Also, UJ \ U*
*J0=
UJ[J0, since each element of U has a unique decomposition as product of element*
*s of the
root groups taken in an appropriate order [Ca , Theorem 5.3.3(ii)]. So any inte*
*rsection
of subgroups UffJis again of the same form, and thus XL(S) = UffJfor some ø-inv*
*ariant
subset J I. Hence XL(S) is weakly L-closed in S by (1) again.
Each automorphism of L is congruent mod Inn(L) to some ff 2 Aut(L) which sends
S to itself, and ff permutes the radical p-subgroups of L contained in S and se*
*nds X(L)
to itself. Thus each coset in Out(L) contains an automorphism which sends XL(S)*
* to __
itself, and XL(S) is weakly Aut(L)-closed in S since it is weakly L-closed. *
* |__|
In fact, when L is simple of Lie type in characteristic p (and p is odd, as u*
*sual), then
for S 2 Sylp(L), X(S) = S except when p = 3 and and L ~=Cn(q) ~=P Sp2n(q) (n *
*2)
or L ~=2An(q) ~=P SUn+1(q2) (n 3).
We are now ready to consider the sporadic groups.
Proposition 4.4. Assume p is odd, let L be a sporadic simple group, and fix S 2
Sylp(L). Then there is a subgroup Q X(S) which is centric and weakly Aut (L)-
closed in S.
Proof.If p 5, then rkp(L) < p by [GLS , x5.6], and so X(S) = S by Proposition*
* 3.10.
So assume p = 3. We consider several different cases.
(a) If L is one of the groups M11, M12, M22, M23, M24, J1, J2, J4, HS , He, or *
*Ru, then
rk3(L) 2 by [GL , p.123], and so X(S) = S by Proposition 3.10.
(b) Assume L is one of the groups J3, Co 3, Co 2, McL , Suz, Ly, O'N , or F5. *
*In all
of these cases, S contains a normal elementary abelian 3-subgroup E of index *
*9,
X(S) E by Lemma 3.2, so S=X(S) is abelian, and X(S) is weakly Aut(L)-closed by
Proposition 3.7(c). More precisely, there are the following inclusions of index*
* prime to
3:
26 BOB OLIVER
__________________________________________________________________________
| L | |J | Co |Co |McL |Suz | Ly | O'N | F |
|__________|_|_3___|____3___|__2_|_____|____|________|_________|____5_____|
| E | |C3 | C5 | C4 | C4 |C5 | C5 | C4 | C4 |
|__________|_|_3___|___3___|__3__|__3__|_3__|___3___|_____3____|____3_____|
| N (E)=E | |C2o C |2 x M |M |M |M |2 x M |order 320 |order 1152 |
|__L______|_|_3___8_|_____11_|_10_|_10_|_11_|_____11_|__________|_________ |
See [GL , x5] for references. (In fact, in all of the above cases, E is the uni*
*que elementary
abelian subgroup of S of maximal rank.)
(c) Assume L ~=Co 1. By [Cu , p.424], S is contained in a semidirect product C6*
*3o2M12,
and the elementary abelian subgroup C63is generated by all elements of order 3 *
*in S
which lie in the conjugacy class (3A). Thus S contains a unique elementary abel*
*ian
3-subgroup E of maximal rank, and hence CS(E) X(S) is centric and weakly Aut(*
*L)-
closed in S by Proposition 3.7(b).
(e) Assume L = F3. By [Ho ] or [Pa ] (see also [As , 14.2]), there are subgroups
D K M S,
all normal in S, such that K ~= C53is abelian, CS(K) = K, and [M, K] = D =
Z(M) ~=(C3)2. (Also, M=K ~= C43and NL(D)=M ~= GL2(3).) Thus M X(S), so
rk(Z(X(S))) rk(Z(M)) = 2, and hence X(S) = S by Proposition 3.10.
In the remaining cases, for a p-group R, we use the notation Zn(R) C R: Z1(R)*
* =
Z(R), and Zn(R)=Zn-1(R) = Z(R=Zn-1(R)). The group R is of class n if R = Zn(R)
Zn-1(R). Also, following the notation of [As ], we say that a subgroup H L i*
*s of
type H0=mt=mt-1= . .=.m1 if upon setting R = Op(H), then H=R ~=H0, Z(R) ~=Cm1p,
and Zi(R)=Zi-1(R) ~= Cmipfor all i. (We restrict, for simplicity, to the case *
*where
Zi(R)=Zi-1(R) is elementary abelian for all i.)
(d) Assume L ~= Fi22. Then L contains a subgroup L0 ~= 7(3) with index prime
to 3 (cf. [As , p.26]). Regard L0 as acting on V = F73, let W V be a max*
*imal
isotropic subspace (dim (W ) = 3), and let H L0 be the subgroup of elements w*
*hich
leave W invariant. Then [L0:H] is prime to 3, and hence we can assume that S *
*H.
One easily checks that H is of type SL3(3)=3=3, where R def=O3(H) is the subgro*
*up
of elements whose restriction to W (and to V=W ?) is the identity, and Z(R) is *
*the
subgroup of elements whose restriction to W ? and to V=W are the identity. Als*
*o,
Z(R) = [R, R], and H=R ~=SL3(3) acts on Z(R) as the group of 3 x 3 antisymmetric
matrices. From this, one quickly sees that R X(S). If we set S0=R = Z(S=R) ~=*
*C3,
then [[Z(R), S0], S0] = 1, so S0 X(S). Hence rk(Z(X(S))) rk(Z(S0) = 2, so
X(S) = S by Proposition 3.10. (Alternatively, one can show that S contains a un*
*ique
elementary abelian subgroup of maximal rank 5, and then apply Proposition 3.7(b*
*).)
(f) Assume L = Fi23or F2. By [As , p. 33], there is an inclusion Fi23 F2 with *
*index
prime to 3, so these groups have isomorphic Sylow 3-subgroups. By [As , p. 27 &*
* 208-
209], there is a subgroup H Fi23of index prime to 3 and of type SL3(3)=3=3=1=*
*3.
We can thus assume R def=O3(H) S H. Also, Z2(R) ~=C43, Z3(R) = CH (Z2(R)) ~=
QxC3 where Z(Q) = [Q, Q] = Z(R) and Q=Z(Q) ~=Z3(R)=Z2(R) ~=C33, and R=Z3(R)
acts on Z2(R) as the group of all automorphisms which are the identity on Z(R) *
*and
on Z2(R)=Z(R). Since [[Z3(R), Z3(R)], Z3(R)] = 1 and [[Z2(R), R], R] = 1, we se*
*e that
R X(S); and hence X(S) = S by the same argument as was used for L ~=Fi22.
EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES 27
(g) Assume L = Fi024. By [As , pp. 29 & 210-211], there is a subgroup H Fi024*
*of index
prime to 3 and of type (A5xSL2(3))=8=4=2, and we can assume R def=O3(H) S H.
Also, R=Z2(R) acts on Z2(R) ~=C83as the group of all automorphisms which are the
identity on Z(R) and on Z2(R)=Z(R); and the actions of SL2(3) H=R on Z(R)
and of A5 H=R on Z2(R)=Z(R) ~=C43are faithful. Thus Z2(R) is centric in H and
[[Z2(R), R], R] = 1. It follows that R X(S), and hence (since rk(Z(R)) = 2) *
*that
X(S) = S by Proposition 3.10.
(h) Assume L = F1. By [As , pp. 35 & 211-212], there is a subgroup H F1 of in*
*dex
prime to 3 of type (GL2(3) x M11)=10=5=2. We can thus assume that R def=O3(H)
S H. Also, R=Z2(R) acts on Z2(R) ~= C73as the group of automorphisms which
are the identity on Z(R) and on Z2(R)=Z(R), and the actions of GL2(3) H=R on
Z(R) and of M11 H=R on Z2(R)=Z(R) are faithful. Thus Z2(R) is centric in H and
[[Z2(R), R], R] = 1. It follows that R X(S), and hence by Proposition 3.10_(*
*since_
rk(Z(R)) = 2) that X(S) = S. |__|
We are now ready to prove Theorem A.
Theorem 4.5. For any odd prime p and any finite group G, ZG is acyclic.
Proof.Let L be a finite simple group, and fix S 2 Sylp(L). If L is an alternati*
*ng group,
or of Lie type in characteristic 6= p, then by Proposition 4.2, there is a subg*
*roup Q
X(S) which is centric and weakly Aut(L)-closed in S. If L is of Lie type in cha*
*racteristic
p, then XL(S) itself is centric and weakly Aut(L)-closed in S by Proposition 4.*
*3. If L
is a sporadic group, then there is a subgroup Q X(S) which is centric and wea*
*kly
Aut(L)-closed in S by Proposition 4.4. The theorem now follows from Proposition*
* 4.1, __
together with the classification theorem for finite simple groups. *
* |__|
References
[As] M. Aschbacher, Overgroups of Sylow subgroups in sporadic groups, Memoirs A*
*mer. Math.
Soc. 343 (1986)
[BLO1]C. Broto, R. Levi, & B. Oliver, Homotopy equivalences of p-completed clas*
*sifying spaces of
finite groups (preprint)
[BLO2]C. Broto, R. Levi, & B. Oliver, The homotopy theory of fusion systems (pr*
*eprint)
[BW] N. Burgoyne & C. Williamson, On a theorem of Borel and Tits for finite Che*
*valley groups,
Arch. Math. Basel 27 (1976), 489-491
[Ca] R. Carter, Simple groups of Lie type, Wiley (1972)
[Cu] R. Curtis, On subgroups of .O II. Local structure, J. Algebra 63 (1980), 4*
*13-434
[GL] D. Gorenstein & R. Lyons, The local structure of finite groups of characte*
*ristic 2 type, Memoirs
Amer. Math. Soc. 276 (1983)
[GLS]D. Gorenstein, R. Lyons, & R. Solomon, The classification of the finite si*
*mple groups, nr. 3,
Amer. Math. Soc. surveys and monogr. 40 #3 (1997)
[Go] D. Gorenstein, Finite groups, Harper & Row (1968)
[Gr] J. Grodal, Higher limits via subgroup complexes, preprint
[HM] G. Higman & J. McKay, On Janko's simple group of order 50,232,960, Bull. L*
*ondon Math.
Soc. 1 (1969), 89-94
[Ho] D. Holt, The triviality of the multiplier of Thompson's group F3, J. Algeb*
*ra 94 (1985), 317-323
28 BOB OLIVER
[JM] S. Jackowski & J. McClure, Homotopy decomposition of classifying spaces vi*
*a elementary
abelian subgroups, Topology 31 (1992), 113-132
[JMO]S. Jackowski, J. McClure, & B. Oliver, Homotopy classification of self-map*
*s of BG via G-
actions, Annals of Math. 135 (1992), 184-270
[MP] J. Martino & S. Priddy, Unstable homotopy classification of BG^p, Math. Pr*
*oc. Cambridge
Phil. Soc. 119 (1996), 119-137
[Pa] D. Parrott, On Thompson's simple group, J. Algebra 46 (1977), 389-404
[Pu] Ll. Puig, unpublished notes
LAGA, Institut Galil'ee, Av. J-B Cl'ement, 93430 Villetaneuse, France
E-mail address: bob@math.univ-paris13.fr