THE SIMPLE CONNECTIVITY OF BSol (q)
ANDREW CHERMAK, BOB OLIVER, AND SERGEY SHPECTOROV
Abstract. A p-local finite group is an algebraic structure which includes*
* two cat-
egories, a fusion system and a linking system, which mimic the fusion and*
* linking
categories of a finite group over one of its Sylow subgroups. The p-compl*
*etion of the
geometric realization of the linking system is the classifying space of t*
*he finite group.
In this paper, we study the geometric realization, without completion, of*
* linking sys-
tems of certain exotic 2-local finite groups whose existence was predicte*
*d by Solomon
and Benson, and prove that they are all simply connected.
A p-local finite group consists of a finite p-group S together with a pair of*
* categories F
and L _ the fusion system and the centric linking system _ with auxiliary struc*
*tures
which relate F and L. The idea is to mimic the structure of a finite group G ha*
*ving S
as a Sylow p-subgroup, by first providing, by means of the fusion system F, a c*
*ollection
of maps between subgroups of S which are consistent with the notion of conjugat*
*ion by
elements of G, and then, with the linking system L, providing a collection of c*
*andidates
for the G-normalizers of a large class of subgroups of S. The resulting object *
*(S, F, L)
should be indistinguishible from such a finite group G, at least from an algebr*
*aic
point of view which takes only "p-local structure" into account. From the homot*
*opy-
theoretic viewpoint, the p-completion |L|^pof the topological realization of L *
*should
be indistinguishible from the p-completion of a classifying space BG. In the ca*
*se that
these structures really do arise from a finite group G with Sylow p-subgroup S,*
* we may
denote the system (S, F, L) by GS(G). If no such G exists, one says that L and*
* the
p-local finite group G = (S, F, L) are exotic.
This paper concerns the family Sol(q) of exotic 2-local finite groups - q an *
*arbitrary
odd prime power - constructed by Ran Levi and the second named author in [LO ].
These objects were prefigured in a paper of David Benson [Be ] and, earlier sti*
*ll, in work
of Ron Solomon [So]; and they are the only exotic 2-local finite groups that ar*
*e known
to exist. They are called the "Solomon 2-local finite groups" in recognition th*
*at it was
Solomon [So] who first discovered that there was a collection of group-like dat*
*a which
was internally consistent from a 2-local point of view, and which was not deriv*
*able
from any finite group.
The classifying space of a p-local finite group (S, F, L) is defined to be th*
*e space |L|^p:
the p-completion of the geometric realization of the category L. This was orig*
*inally
motivated by the observation in [BLO1 , Proposition 1.1] that when L is the li*
*nking
system of a finite group G, then |L|^phas the homotopy type of BG^p; and also b*
*ecause
whether or not L is associated to a group, |L|^pshares many of the homotopy the*
*oretic
properties of p-completed spaces of finite groups. However, interest has recent*
*ly been
growing in the geometric realization |L| without p-completion, and in particula*
*r in its
___________
1991 Mathematics Subject Classification. Primary 55R35. Secondary 20D06, 20D2*
*0.
Key words and phrases. Classifying space, finite groups, fusion.
B. Oliver is partially supported by UMR 7539 of the CNRS. Part of this work w*
*as done during a
stay at the Mittag-Leffler institute in Sweden.
S. Shpectorov was partially supported by an NSA grant.
1
2 ANDREW CHERMAK, BOB OLIVER, AND SERGEY SHPECTOROV
fundamental group, as an invariant of a p-local finite group (S, F, L). This ha*
*s been
spurred on by questions and conjectures formulated by Jesper Grodal.
Two general references for the geometric realization of a category are Segal'*
*s original
paper [Se, x1-2], and the more recent book of Srinivas [Sr, Chapter 3]. In gen*
*eral,
when C is a discrete, small category, c0 2 Ob (C), and I is a set of morphisms *
*in
C which includes exactly one morphism between c0 and each other object, then the
fundamental group ss1(|C|) can be described algebraically as the group generate*
*d by
Mor (C), modulo the relations given by composition, and modulo the relations gi*
*ven
by setting morphisms in I equal to the identity. In the case of a linking syste*
*m L, we
take c0 to be the "Sylow subgroup" S 2 Ob (C), and take I to be a set of "inclu*
*sion"
morphisms to S.
When L is the linking system associated to a finite group G, then in many cas*
*es,
ss1(|L|) is either isomorphic to G or surjects onto G. This is discussed briefl*
*y in Section
1, and several other examples will be given in the paper [GO ] now in preparati*
*on. This
connection with the underlying finite group, when there is one, made it natural*
* to look
at the fundamental groups of exotic linking systems.
The principal aim of this paper is to study the topological realizations of t*
*he linking
systems of the Solomon 2-local groups, and to show that they are simply connect*
*ed.
More precisely, we show:
Theorem A. For every odd prime power q, the geometric realization of the linking
system LcSol(q) is simply connected.
This will be proven as Theorem 5.1. These are the first (and only) examples *
*we
know of linking systems whose nerves are simply connected. In fact, these are t*
*he only
examples we know where the automorphism groups in L do not all map injectively *
*into
ss1(|L|).
In [LO ], an infinite "linking system" LcSol(p1 ) was constructed for all odd*
* primes p,
roughly as the union of the LcSol(pn) (taken over all n), and its 2-completed n*
*erve was
shown to have the homotopy type of the Dwyer-Wilkerson space BDI(4) [DW ]. One
consequence of Theorem A is that |LcSol(p1 )| is also simply connected (Corolla*
*ry 5.6).
When proving Theorem A, the first step is to show that if |LcSol(q)| is simpl*
*y con-
nected, then for all n 1, |LcSol(qn)| is also simply connected. This is fairl*
*y straightfor-
ward and simple. The following theorem then allows us to reduce the proof to sh*
*owing
that the topological realization of the linking system for Sol(3) is simply con*
*nected.
Theorem B. Let q and q0 be odd prime powers. Then the fusion systems FSol(q) and
FSol(q0), and also their associated linking systems LcSol(q) and LcSol(q0), are*
* isomorphic
if and only if q2 - 1 and q02- 1 have the same 2-adic valuation.
Theorem B will be shown below as Theorem 3.4, where we give a purely algebraic
proof of the result. It also follows from a result of Broto and Moller [BM , *
*Theorem
C], when combined with [BLO2 , Theorem A] which says that the homotopy type of
the classifying space of a p-local finite group determines its homotopy type. H*
*owever,
Broto and Moller state this result only for odd fusion (the general result foll*
*ows by
the same argument and will appear in a later paper), and their proof uses some *
*deep
results in homotopy theory. Hence our decision to include a purely algebraic p*
*roof
here.
An easy induction argument shows that if a is an odd integer such that v2(a *
* 1) =
k
m 2, then v2(a2 -1) = m+k for all k 1. Hence another consequence of Theorem*
* B
THE SIMPLE CONNECTIVITY OF BSol(q) 3
is that the methods in [AC ] apply to construct all of the Solomon 2-local fini*
*te groups:
since in that paper, the fusion and linking systems FSol(q) and LcSol(q) are de*
*fined only
when q is a power of a prime p 3, 5 (mod 8).
As mentioned above, when L is a linking system, ss1(|L|) is the free group on*
* the
morphisms in L modulo certain relations given (roughly) by composition and incl*
*usions.
Thus the main problem when proving Theorem A is to find enough relations among *
*the
morphisms to show that they all vanish. In [AC ], LcSol(3) (or its fundamental *
*group) is
shown to contain a certain amalgam of three maximal subgroups of the sporadic s*
*imple
group Co3. This allows us to reduce the proof of Theorem A to the following res*
*ult,
which is proven by using computer computations to show that a certain simplicial
complex is simply connected:
Theorem C. Let H1, H2, and H3 be the three maximal overgroups of a fixed Sylow
subgroup S 2 Syl2(Co3), and let G be the amalgam formed by the Hi and their int*
*er-
sections. Then colim(G) ~=Co3.
Theorem C is proven as Proposition 4.1.
We would like to thank Jesper Grodal for first getting us interested in this *
*question;
this paper is in some sense an offshoot of the paper [GO ] by Grodal and the se*
*cond au-
thor. Particular thanks go to the mathematics department at Cal Tech, and espec*
*ially
Michael Aschbacher, for their hospitality in giving the first two authors, and *
*later the
first and third authors, a chance to meet and discuss these problems. Some of t*
*he key
ideas in this paper were developped there. The second author would also like to*
* thank
the Mittag-Leffler Institute for providing ideal conditions for him to finish h*
*is share of
the work on this paper.
1.Background
We first recall the definition of a (saturated) fusion system. This definitio*
*n is orig-
inally due to [Pg ], although it is presented here in the simpler, but equivale*
*nt, form
given in [BLO2 ].
We first fix some general notation. For any group G, and any pair of subgrou*
*ps
H, K G, we set
NG (H, K) = {x 2 G | xHx-1 K},
let cx denote conjugation by x on the left (cx(g) = xgx-1), and set
fi
Hom G(H, K) = cx 2 Hom (H, K) fix 2 NG (H, K) ~=NG (H, K)=CG (H).
By analogy, we also write AutG (H) = Hom G(H, H) ~=NG (H)=CG (H).
A fusion system over a finite p-group S is a category F, where Ob (F) is the *
*set of all
subgroups of S, where each morphism set Hom F(P, Q) is a set of group monomorph*
*isms
from P to Q which contains Hom S (P, Q), and where each ' 2 Hom F(P, Q) is the
composite of an isomorphism in F followed by an inclusion. Two subgroups P, Q *
* S
are said to be F-conjugate if they are isomorphic as objects of the category F.*
* A
subgroup P S is fully centralized in F if |CS(P )| |CS(P 0)| for all P 0 S*
* which is
F-conjugate to P . Similarly, a subgroup P S is fully normalized in F if |NS(*
*P )|
|NS(P 0)| for all P 0 S which is F-conjugate to P .
A fusion system F is called saturated if the following two conditions hold:
4 ANDREW CHERMAK, BOB OLIVER, AND SERGEY SHPECTOROV
(I) For each P S which is fully normalized in F, P is fully centralized in F *
*and
AutS(P ) 2 Sylp(Aut F(P )).
(II)If P S and ' 2 Hom F(P, S) are such that 'P is fully centralized, and if *
*we set
N' = {g 2 NS(P ) | 'cg'-1 2 AutS('P )},
_ _
then there is ' 2 Hom F(N', S) such that '|P = '.
If G is a finite group and S 2 Sylp(G), then by [BLO2 , Proposition 1.3], th*
*e category
FS(G), defined by letting Ob (FS(G)) be the set of all subgroups of S and setti*
*ng
Mor FS(G)(P, Q) = Hom G(P, Q), is a saturated fusion system.
Again let F be an abstract saturated fusion system over a p-group S. A subgro*
*up
P S is F-centric if CS(P 0) = Z(P 0) for all P 0 S which is F-conjugate to P*
* . A
subgroup P S is F-radical if Out F(P ) is p-reduced; i.e., if Op(Out F(P )) =*
* 1. Let
Fc F denote the full subcategory whose objects are the F-centric subgroups of*
* S.
If F = FS(G) for some finite group G, then P S is F-centric if and only if
P is p-centric in G (i.e., Z(P ) 2 Sylp(CG (P ))), and P is F-radical if and *
*only if
NG (P )=(P .CG (P )) is p-reduced. Thus in this situation, a subgroup being F-r*
*adical is
not the same as its being a radical p-subgroup of G.
Alperin's fusion theorem in a version for abstract saturated fusion systems w*
*as first
formulated and proven by Puig [Pg ]. Since we need to use it several times in *
*what
follows, we state the following version of the theorem, which is proven in [BLO*
*2 ,
Theorem A.10].
Theorem 1.1. Let F be a saturated fusion system over a p-group S. Then each
morphism in F is a composite of restrictions of morphisms between subgroups of S
which are F-centric, F-radical, and fully normalized in F. More precisely, for*
* each
P, P 0 S and each ' 2 IsoF(P, P 0), there are subgroups P = P0, P1, . .,.Pk = *
*P 0,
subgroups Qi (i = 1, . .,.k) which are F-centric, F-radical, and f*
*ully
normalized in F, and automorphisms 'i2 AutF (Qi), such that 'i(Pi-1) = Pi for a*
*ll i
and ' = ('k|Pk-1) O. .O.('1|P0).
Again let F be a fusion system over the p-group S. A centric linking system a*
*sso-
ciated to F is a category L whose objects are the F-centric subgroups of S, tog*
*ether
ffiP
with a functor ss : L ---! Fc, and "distinguished" monomorphisms P --! AutL(P )
for each F-centric subgroup P S, which satisfy the following conditions.
(A) ss is the identity on objects. For each pair of objects P, Q in L, Z(P ) ac*
*ts freely
on Mor L(P, Q) via composition and ffiP, and ss induces a bijection
~=
MorL(P, Q)=Z(P ) ------! Hom F (P, Q).
(B) For each F-centric subgroup P S and each x 2 P , ss(ffiP(x)) = cx 2 AutF *
*(P ).
(C) For each f 2 Mor L(P, Q) and each x 2 P , the following square commutes in *
*L:
f
P ______! Q
| |
ffiP(x)| |ffiQ(ss(f)(x))
# #
f
P ______! Q.
THE SIMPLE CONNECTIVITY OF BSol(q) 5
A p-local finite group is defined to be a triple (S, F, L), where S is a fini*
*te p-group,
F is a saturated fusion system over S, and L is a centric linking system associ*
*ated to
F. The classifying space of the triple (S, F, L) is the p-completed nerve |L|^p.
For any finite group G with Sylow p-subgroup S, a category LcS(G) was defined*
* in
[BLO1 ], whose objects are the p-centric subgroups of G, and whose morphism se*
*ts are
defined by
Mor LcS(G)(P, Q) = NG (P, Q)=Op(CG (P )).
Since CG (P ) = Z(P ) x Op(CG (P )) when P is p-centric in G, LcS(G) is easily *
*seen
to satisfy conditions (A), (B), and (C) above, and hence is a centric linking s*
*ystem
associated to FS(G). Thus (S, FS(G), LcS(G)) is a p-local finite group, with cl*
*assifying
space |LcS(G)|^p' BG^p(see [BLO1 , Proposition 1.1]).
The following lifting lemma for linking systems helps to motivate some of the*
* con-
structions made here.
Lemma 1.2. Let (S, F, L) be a p-local finite group. Fix F-centric subgroups P, *
*Q, R
S, and let ' 2 Mor L(P, R) and _ 2 Mor L(Q, R) be morphisms such that Im (ss('))
Im(ss(_)). Then there is a unique morphism O 2 Mor L(P, Q) such that ' = _ OO.
Proof.By definition of a fusion system, there is f 2 Hom F (P, Q) such that ss(*
*') =
ss(_) Of in Hom F (P, R). Fix any O0 2 ss-1(f). By (A), there is a unique g 2*
* Z(P )
such that ' = _ OO0OffiP(g), and we set O = O0OffiP(g). This proves existence, *
*and the
proof uniqueness is similar (again using (A)). (See [BLO2 , Lemma 1.10].)
When working with a p-local finite group (S, F, L), we always assume we have *
*chosen
"inclusion morphisms" 'P 2 Mor L(P, S) for each P ; i.e., morphisms which are s*
*ent to
the inclusion of P in S under the functor ss : L ---! F (and where 'S = IdS). T*
*hen
by Lemma 1.2, for each P Q S in L, there is a unique "inclusion" morphism
'P,Q 2 Mor L(P, Q) such that 'P = 'Q O'P,Q. Moreover, for each ' 2 Mor L(P, Q),*
* and
each P0 P and Q0 Q such that ss(')(P0) Q0 and P0, Q0 2 Ob (L), there is a
unique "restriction" '|P0,Q02 Mor L(P0, Q0) such that 'Q0,QO '|P0,Q0= ' O'P0,P.
Again fix (S, F, L), let |L| be the nerve (geometric realization) of the cate*
*gory L,
and let * 2 |L| be the vertex corresponding to the object S. Let
J = JL : Mor (L) ---! ss1(|L|, *)
be the map which sends ' 2 Mor L(P, Q) to the loop in |L| formed by the edges [*
*'],
['P], and ['Q ]. In particular, J sends each of the inclusions ['P] to the iden*
*tity element
in the fundamental group. Also, J sends composites to products, and hence can *
*be
thought of as a functor J : L ---! B(ss1(|L|, *)).
Proposition 1.3. Let (S, F, L) be a p-local finite group. For any group , and*
* any
map of sets
b: Mor (L) ------!
which sends composites to products and sends inclusion morphisms to the identit*
*y, there
is a unique homomorphism : ss1(|L|, *) ---! such that b = OJ. In other wor*
*ds,
ss1(|L|, *) is the free group generated by the morphisms in L, modulo relations*
* defined
by composition and inclusions.
Proof.Let B( ) be the category with one object * and morphism group . Then b
extends to a functor : L ---! B( ), and this in turn induces a map
| |: |L| ------! |B( )| = B
6 ANDREW CHERMAK, BOB OLIVER, AND SERGEY SHPECTOROV
between the geometric realizations. Set
= ss1(| |): ss1(|L|, *) ------! ss1(|B( )|, *) = .
The relation b = OJ is clear by construction. The uniqueness of holds si*
*nce
every element of ss1(|L|, *) can be represented by a loop which follows along t*
*he edges
of |L| (corresponding to morphisms in L), and any such loop can be factored as a
composite of loops in Im(J).
Now, in the above situation, we let
o = oL : S ---! ss1(|L|, *)
denote the composite J OffiS. If g 2 P S, then by axiom (C) (applied with Q =*
* S
and f = 'P), 'P OffiP(g) = ffiS(g) O'P. Thus o(g) = J(ffiS(g)) = J(ffiP(g)). In*
* other words,
o(g) can be defined using any ffiP as long as g 2 P .
Proposition 1.4. Fix a p-local finite group (S, F, L), a (possibly infinite) gr*
*oup ,
and an epimorphism
: ss1(|L|, *) --- --i .
Then the following hold.
(a) Ker( Oo) is strongly F-closed in S.
(b) If Oo is the trivial homomorphism, then OJ restricts to a surjective ho*
*momor-
phism from AutL (S)=ffiS(S) ~=Out F(S) onto .
Proof. For any isomorphism ' 2 IsoF(P, Q) in F between F-centric subgroups, and
any g 2 P , o(g) and o('(g)) are conjugate in ss1(|L|, *) (since ' lifts to an *
*isomor-
phism in L); and hence either both lie in Ker( ) or neither does. By Alperin's *
*fusion
theorem (Theorem 1.1), any pair of F-conjugate elements of S is linked by a seq*
*uence
of isomorphisms between F-centric subgroups, and hence (a) holds.
Point (b) is basically a consequence of [BCGLO2 , Lemma 3.4], but because i*
*t's hard
to fit this situation precisely into that setting, we repeat the argument here.*
* Assume
Oo is the trivial homomorphism. In particular, OJ factors through a map
J0: Mor (Fc) ------!
in this case, since OJ(Z(P )) = 1 for all P . We must show that J0|AutF(S)is*
* onto.
Assume otherwise. By Alperin's fusion theorem again, is generated by the sub-
groups J0(Aut F(P )) for P S F-centric, F-radical, and fully normalized; we f*
*ix such
a subgroup P S which is maximal among all P S such that J0(Aut F(P ))
J0(Aut F(S)). Choose ' 2 AutF (P ) such that J0(') =2J0(Aut F(S)).
0
Now, J0(Aut S(P )) = 1 and Aut S(P ) 2 Sylp(Aut F(P )); hence Op (Aut F(P ))
Ker (J0). Set K = 'Aut S(P )'-1. Since K and Aut S(P ) are both Sylow p-subgrou*
*ps
0 p0
of Op (Aut F(P )), there is O 2 O (Aut F(P )) such that O' normalizes AutS (P *
*). Thus
J0(O') = J0('), and by axiom_(II) in the definition of_a saturated fusion syste*
*m, O'
extends to an automorphism ' 2 Aut_F(NS(P )). But J0(' ) = J0(O') (since J sen*
*ds
inclusions to the identity), J0(' ) 2 J0(Aut F(S)) by the maximality of P , and*
* this is a
contradiction. This finishes the proof of (b).
For any n 0, we write n_= {1, 2, . .,.n}. Let C(n) denote the category who*
*se
objects are the nonempty subsets I n_, with a unique morphism I ! J whenever
I J. By an amalgam of groups of rank n, we mean a functor A from C(n)op to the
category of groups and monomorphisms. A faithful completion of the amalgam A is*
* a
THE SIMPLE CONNECTIVITY OF BSol(q) 7
collection of monomorphisms fI: A(I) ---! G for all ? 6= I n_which commute wi*
*th
the monomorphisms induced by A, such that
G = .
The following properties of an amalgam of groups are well known; we include t*
*hem
here for ease of later reference.
Proposition 1.5. Fix n 3, let A be an amalgam of groups of rank n, and let G *
*be
a faithful completion of A. Write GI = A(I), Gi = G{i}, Gij= G{i,j}, etc. for s*
*hort,
and regard these as subgroups of G for simplicity.` Let X be the corresponding *
*coset
complex: the simplicial complex with vertex set ni=1(G=Gi), with edges the un*
*ion of
the G=Gij, etc. Then X is connected, and there is a short exact sequence of gro*
*ups:
1 ---! ss1(X) -----! colim(A) -----! G ---! 1.
In particular, the natural homomorphism from colim(A) to G is an isomorphism if*
* and
only if X is simply connected.
Proof.This follows from [T , Proposition 1]. Alternatively, it follows from the*
* following
argument which applies van Kampen's theorem to the Borel construction on X.
Consider the Borel construction on X:
ffi
XhG def=EG xG X = EG x X ~.
Here, EG is a contractible space upon which G acts freely on the right, and we *
*identify
(yg, x) ~ (y, gx) for all y 2 EG, g 2 G, and x 2 X. Thus EG x X is a covering
space of XhG, and is also homotopy equivalent to X. By the standard properties*
* of
fundamental groups in covering spaces, this yields an exact sequence
1 ---! ss1(X) -----! ss1(XhG) -----! G,
where the last homomorphism is surjective if and only if X is connected.
For each i = 1, . .,.n, let Xi X be the union of the orbit G=Gi togetherSwi*
*th all
orbits of open simplices which have this orbit as a vertex. Thus X = ni=1Xi. *
*Also,
Xi has one connected component for each vertex in G=Gi (and the components are
contractible), and so ss1((Xi)hG) ~= Gi. Similarly, for each i 6= j, Xi\ Xj ha*
*s one
connected component for each element of G=Gij, and ss1((Xi\ Xj)hG) ~=Gij. So by
van Kampen's theorem, ss1(XhG) ~=colim(A). Since G is generated by the Gi, this*
* also
proves that ss1(XhG) surjects onto G, and hence that X is connected.
Proposition 1.6. Fix a finite group G, a prime p, and a Sylow subgroup S 2 Sylp*
*(G).
Assume P1, . .,.Pn S are all centric in G (i.e., CG (Pi) Pi), and are all w*
*eakly
closed in S with respect to G. Define, for each I n_,
"
PI = and GI = NG (PI) = NG (Pi).
i2I
Let L LcS(G) be the full subcategory with Ob (L) = {PI| ? 6= I n_}. Then
ss1(|L|) ~=colim(A),
where A denotes the amalgam of rank n defined by setting A(I) = GI.
Proof.Let X be as in Proposition 1.5: the simplicial complex with vertex set t*
*he
disjoint union of the G=Gi, edges the disjoint union of the G=(Gi\ Gj), etc. Si*
*nce the
PI are weakly closed and GI = NG (PI), X is equivalent (as a simplicial complex*
* with
8 ANDREW CHERMAK, BOB OLIVER, AND SERGEY SHPECTOROV
G-action) to the poset of all subgroups of G which are conjugate to some PI. He*
*nce
by [BLO1 , Lemma 1.2] (or its proof),
|L| ' EG xG X.
So as in Proposition 1.5, ss1(|L|) ~=colim(A). (Note, however, that in this cas*
*e, |L| is
connected only if G = .)
The following examples will be needed later.
Proposition 1.7. Fix a prime p, a finite group G, and a Sylow p-subgroup S G.
(a) Assume G is a simple group of Lie type in characteristic p of Lie rank 3,*
* or a
quasisimple group in characteristic p of Lie rank 3 with center a p-group*
*. Then
ss1(|LcS(G)|) ~=G. Also, for any S 2 Sylp(G), G is the colimit of the diag*
*ram of
parabolic subgroups of G which contain S.
(b) Assume G is p-constrained. Then ss1(|LcS(G)|) ~=G=Op0(G).
Proof. (a) By the Borel-Tits theorem [GLS , Corollary 3.1.6], together with [G*
*r , Re-
mark 4.3], there is a bijection of posets from the poset of parabolic subgroups*
* of G to
the opposite poset of the poset of radical p-centric subgroups of G, defined by*
* sending
P 7! Op(P), and where NG (Op(P)) = P.
We claim that for each S 2 Sylp(G) and each parabolic subgroup P S, Op(P)
is weakly closed in S with respect to G. The following argument is taken from *
*[AS ,
Lemma I.2.5]. Assume otherwise, and let Q = Op(P) be maximal among subgroups
of this form which are not weakly closed in S. By Alperin's fusion theorem (The*
*orem
1.1), there is a radical subgroup Q0 S such that Q0 Q _ hence Q0= Op(P0) for
some other parabolic subgroup P0$ P _ and an element x 2 NG (Q0) = P0such that
xQx-1 6= Q. But this is impossible, since P0 P = NG (Q).
Thus, by Proposition 1.6, ss1(|LcS(G)|) is isomorphic to the colimit of the a*
*malgam
A formed by the parabolic subgroups containing a given Sylow p-subgroup. By Pro*
*po-
sition 1.5, there is a short exact sequence
1 ---! ss1(X) -----! colim(A) -----! G ---! 1,
where X is the geometric realization of the poset of parabolic subgroups.
If G has Lie rank n, then by [Bw , xV.3], the geometric realization of the po*
*set of
its parabolic subgroups is a building of rank n, and hence by [Bw , Theorem IV.*
*5.2]
has the homotopy type of a bouquet of (n - 1)-spheres. Thus if n 3, the geome*
*tric
realization is simply connected, and colim(A) ~=G.
__ __
(b) Assume G is p-constrained,_and set G = G=Op0(G) and Q = Op(G ). Thus
C_G(Q) = Z(Q), and AutL (Q) ~=G . Let LrcS(G) LcS(G) be the full subcategory *
*with
__
objects the centric radical subgroups of G ; then |LrcS(G)| and |LcS(G)| have t*
*he_same
homotopy type by [BCGLO1 , Theorem B]._Since each centric radical subgroup of*
* G
contains Q, one easily sees that |LrcS(G )| contains as deformation retract the*
* nerve of
the subcategory with unique object Q. Thus
__ __ __
|LcS(G)| = |LcS(G )| ' |LrcS(G )| ' BAut L(Q) ' BG .
__
In particular, ss1(|LcS(G)|) ~=G .
THE SIMPLE CONNECTIVITY OF BSol(q) 9
2. The linking system of Spin7(q)
Let q be any prime power such that q 3 (mod 8). In this section, we descri*
*be
the fundamental group of LSol(q) as the colimit of a certain triangle of groups*
*. Before
doing this, we first need to look at the linking system of Spin7(q).
Set H = Spin7(q) for short, and fix S 2 Syl2(H). By [BCGLO1 , Theorem B],
|LcrS(H)| ' |LcS(H)| (the inclusion is a homotopy equivalence), and thus these *
*two
spaces have the same fundamental group. By [LO , Proposition A.12], every 2-sub*
*group
P H which is centric and radical in the fusion system FS(H) is in fact centri*
*c in
H; i.e., CH (P ) = Z(P ). (This also follows from the proof of Proposition 2.1 *
*below.)
Hence the linking system LrcS(H) is a full subcategory of the transporter categ*
*ory of
H: Mor LrcS(H)(P, Q) is the set of elements of H which conjugate P into Q. Thus*
* there
is a functor from LrcS(H) to B(H) _ the category with one object and morphism g*
*roup
H _ which sends a morphism to the corresponding element in H, and in particular
sends inclusions to the identity. Upon taking fundamental groups of the geomet*
*ric
realizations of these categories, this defines a homomorphism
~: ss1(|LcS(H)|) ~=ss1(|LrcS(H)|) ------! H.
Proposition 2.1. For any prime power q 3 (mod 8), there is an isomorphism
ss1(|Lc2(Spin7(q))|) ~=Spin7(Z[1_2])
which commutes with the natural homomorphisms
~ 1
ss1(|Lc2(Spin7(q))|) ------! Spin7(q) ------ Spin7(Z[_2]).
Proof.By [BCGLO2 , Theorem 6.8], this is equivalent to showing that
ss1(|Lc2( 7(q))|) ~= 7(Z[1_2]).
We work with 7(q) for simplicity.
Set V = Fq7, let q be its standard quadratic form, and fix an orthonormal ba*
*sis
{u1, . .,.un} of V . For each i = 1, 2, 3, set v2i-1= u2i-1+ u2iand v2i= u2i-1*
*- u2i.
Thus {v1, . .,.v6, u7} is an orthogonal basis of V , and q(vj) = 2 for all j = *
*1, . .,.6.
Set Wi = = (i = 1, 2, 3). Set Gb = GO(V, q) ~= GO7(q*
*) and
G = (V, q) ~= 7(q).
Let 4 (W1 W2, q) be the subgroup of those automorphisms ff of the form
ff(ui) = ffliuoe(i), where ffli= 1, ffl1ffl2ffl3ffl4 = 1, and oe 2 Alt4lies in*
* the normal subgroup
of order 4. Thus 4 ~=D8 xC2 D8 is an extraspecial 2-group of order 25. Conside*
*r the
following subgroups of G:
R = {ff 2 G | ff(ui) = ui for all i = 1, .}.,.7
R* = {ff 2 G | ff(vi) = vi for all i = 1, .}.,.6
Q = {ff 2 G | ff|W1 W2 2 4, ff(ui) = ui for i = 5, 6,}7.
Set S = RR*Q. By [GO , Proposition 10.1], S 2 Syl2(G), and the seven subgroups *
*R,
R*, Q, RR*, RQ, R*Q, and S = RR*Q are representatives for the distinct conjugacy
classes of 2-subgroups of G = 7(q) which are centric and radical in the fusion*
* system
F2(G). Also, R, R*, and Q are all weakly closed in S = RR*Q 2 Syl2(G), and hence
NG (P1P2) = NG (P1) \ NG (P2) for any pair P1, P2 of such subgroups.
Let L Lc2(G) be the full subcategory whose objects are the subgroups of G w*
*hich
are centric and radical in F2(G). By [BCGLO1 , Theorem 3.5], |L| ' |Lc2(G)|, *
*and hence
10 ANDREW CHERMAK, BOB OLIVER, AND SERGEY SHPECTOROV
they have the same fundamental group. By Proposition 1.6, ss1(|L|) is the colim*
*it of
the triangle of groups A with vertices NG (R), NG (R*), and NG (Q) and with edg*
*es
their pairwise intersections.
Now set = 7(Z[1_2]), and define bases {u1, . .,.u7} and {v1, . .,.v6, u7} *
*of Z[1_2]7
analogous to the above elements. Via these bases, we can lift R, R*, and Q to *
*, and
check directly that N (P ) ~=NG (P ) for P any of these three subgroups.
We want to compare colim(A) to a similar colimit of subgroups of studied by
Kantor in [Ka , xx5,7]. He constructs a certain 3-dimensional complex 7, toge*
*ther
with an action of which is transitive on 3-simplices. This action has four or*
*bits of
vertices
=N (R), =N (Q), =W +, =W -,
where W +~= W - are representatives of the two conjugacy classes of 7(2) in ,*
* and
W +\ W -= N (R*). (See [Ka , p.213].) By [Ka , Corollary 7.4], 7 is equivalent*
* to the
Euclidean building for 7(Q2), and hence contractible. So by Proposition 1.5 ag*
*ain, if
we let A7 denote the rank four amalgam consisting of the four stabilizer subgro*
*ups of
a 3-simplex and their intersections, then colim(A7) ~= .
We now construct group homomorphisms
colim(A) -------!-------colim(A7),
which will be inverses to each other. The first one is clear: is defined by *
*sending
NG (P ) to N (P ) for P = R and Q, and NG (R*) to W +\ W -. To define , we fir*
*st
note that by [Ka , p.213] again, W \ N (R), W \ N (Q), and N (R*) are the thr*
*ee
maximal parabolic subgroups of W ~=2Sp6(2) containing S 2 Syl2(W ), and the
colimit of these groups (together with their intersections) is W by Propositio*
*n 1.7(a).
This defines homomorphisms from W to colim(A), and together with the canonical
isomorphisms N (P ) ~=NG (P ) for P = R and Q these induce a homomorphism . It
is clear by construction that and are inverses, and thus colim(A) ~=colim(A*
*7) ~=
Spin7(Z[1_2]).
See also [GO , Theorem 10.2] for a slightly different argument.
We now set up some notation which will be used in this section and the next. *
*For
any odd prime power q, there is a homomorphism
_ _
! : SL2(F q)3 ------! Spin7(F q),
_ *
* _
with Ker(!)_= <(-I, -I,_-I)>, which arises from identifications Spin3(F q) ~=SL*
*2(F q)
and Spin4(F q) ~= SL2(F q)2. (See [LO_, Definition 2.2] for more details.) Th*
*e three
factors are ordered so that Z(Spin7(F q)) = . We write [[X1, X2, *
*X3]] =
!(X1, X2, X3) for short, and set U = <[[ I, I, I]]> ~=C22. By [LO , Propositi*
*on 2.5]
or [AC , Lemma 4.4(c)],
_ _
CSpin7(_Fq)(U) = !(SL2(F q)3) and NSpin7(_Fq)(U) = !(SL2(F q)3).
where o2 = 1 and o[[X1, X2, X3]]o-1 = [[X2, X1, X3]]. Finally, Im (!) \ Spin7*
*(q) is
generated by !(SL2(q)3), together with an element [[Y, Y, Y ]] for Y 2 NSL2(q2)*
*(SL2(q))
but not in SL2(q). This will be described in more detail in the next section, *
*in the
proof of Lemma 3.1.
___ _
We now restrict_to the case q = 3. Let SL 2(3) be the normalizer in SL2(F 3)*
* of
SL2(3). Thus SL 2(3) contains SL2(3) with index 2, and is the 2-fold central ex*
*tension
THE SIMPLE CONNECTIVITY OF BSol(q) 11
of Sym (4) whose Sylow 2-subgroup is quaternion of order 16. Set
___ ___
bK= (SL 2(3))3=<(z, z, z)>o Sym (3) and bB= (SL 2(3))3=<(z, z, z)>.*
* bK,
where o = (1 2) 2 Sym (3) acts by switching the first two coordinates._Let [X1,*
* X2, X3]
denote the class of a triple (X1, X2, X3). Choose any Y 2 SL 2(3)r SL2(3), and *
*set
3 ff
B1 = SL2(3) =<(z, z, z)> . [Y, Y, Y ]
___ 3 fi
= [X1, X2, X3] 2 (SL 2(3)) =<(z, z, z)> fiX1 X2 X3 (mod SL2(3)) .
Finally, define
K = B1o Sym (3) bK and B = bB\ K = B1,
_
and let ! : B ---! Spin7(3) be the homomorphism induced by !.
The following two propositions hold, in fact, for LcSol(q) for any q 3 (mo*
*d 8). We
state them here only for q = 3, since that simplifies somewhat the proofs, and *
*suffices
for our later applications.
_
Proposition 2.2. Set H = Spin7(Z[1_2]). Then ! lifts to an embedding ~: B ---! *
*H;
and there is an epimorphism
O: H * K -- ---i ss1(|LcSol(3)|),
B
where H *B K is the amalgamated free product defined by the amalgam
~ incl
H ---- B ----! K .
Proof.Fix S 2 Syl2(B) Syl2(Spin7(3)). By the constructions in [LO ] and [AC *
*],
LcSol(q) is generated by its two subcategories LcS(Spin7(3)) and LcS(K), which *
*intersect
in LcS(B). Also, ss1(|LcS(K)|) ~= K and ss1(|LcS(B)|) ~= B by Proposition 1.7(*
*b) (K
and B are 2-constrained and_O20(K) = O20(B) = 1). The inclusion of LcS(B) into
LcS(Spin7(3)) induced by ! now induces an inclusion of B into H ~=ss1(|LcS(Spin*
*7(3))|),
together with a homomorphism
O: H * K ------! ss1(|LcSol(3)|);
B
and O is surjective since by construction, all morphisms in LcSol(3) are compos*
*ites of
morphisms in these subcategories.
The following proposition will be needed in Section 5, in the proof that |LcS*
*ol(3)| is
simply connected.
Proposition 2.3. Again set H = Spin7(Z[1_2]), and
O: H * K -- ---i ss1(|LcSol(3)|).
B
be as in Proposition 2.2. Then there are subgroups H0 H, K0 K, and B0 =
H0 \ K0 B such that H0=Z ~= Sp6(2), [K:K0] = 3, and (H0 B0 K0) is an
amalgam of maximal subgroups of Co3. Furthermore, if ! 6= 1 (is not the trivial
homomorphism), then !|6= 1.
Proof.The inclusions of linking systems LcS(H0) LcS(Spin7(q)) and LcS(K0) L*
*cS(K)
(where S 2 Syl2(Spin7(q))) were constructed in [AC , Theorem B], in a way so th*
*at they
intersect in LcS(B0). Also, H0 ~=ss1(|LcS(H0)|) by Proposition 1.7(a), and the *
*analogous
result for K0 and B0 holds by Proposition 1.7(b). The inclusions H0 H and K0 *
* K
now follow upon taking fundamental groups.
12 ANDREW CHERMAK, BOB OLIVER, AND SERGEY SHPECTOROV
Now let N E H *B K be the normal closure of . To prove the last state*
*ment,
we must show that N = H *B K. Set G = Spin7(3), and fix S 2 Syl2(G), also
regarded as a subgroup of B. Since [B:B0] = 3, B0 contains S (up to conjugacy),
and hence N S. By [Ka , Corollary 7.4], H ~= Spin7(Z[1_2]) is generated by t*
*wo of
the subgroups W ~= 2Sp6(2) described in the proof of Proposition 2.1, which co*
*ntain
S by construction. (Note that in [Ka ], G7 is defined to be the subgroup of SO*
*7(Q)
generated by these two subgroups). Since both of these are quasisimple, N cont*
*ains
W +and W -since it contains S, and thus N H B. Since the normal closure of B
in K is K, this shows that N = H *B K.
3. Identifying FSol(q) from its Sylow 2-subgroup
For any odd prime power q, let FSol(q) be the exotic fusion system constructe*
*d in
[LO ], over a 2-group S(q). Our aim in this section is to prove Theorem 3.4, w*
*hich
states that FSol(q) and FSol(q0) are isomorphic if and only if S(q) and S(q0) h*
*ave the
same order.
The results in this section will be used to reduce our main theorem _ the sim*
*ple
connectivity of |LcSol(q)| for all odd prime powers q _ to the case where q is *
*a power
of 3.
We first need some concrete information about the structure of the Sylow subg*
*roups
of these groups, and of their fusion.
Lemma 3.1. Let q be an odd prime power, set H = Spin7(q), and let S 2 Syl2(H).
Set F = FSol(q). Set n = v2(q2 - 1) (i.e., 2n is the highest power of 2 dividin*
*g q2 - 1),
and let Q2n be a generalized quaternion group of order 2n. Then the following h*
*old.
(a) There are unique normal subgroups U E S and E E S which are elementary abel*
*ian
of rank two and three, respectively.
(b) There is a unique abelian subgroup T S0 which is homocyclic of rank three*
* and
exponent 2n-1.
(c) There are exactly six normal subgroups_of_CS(U)_which are isomorphic to Q2n.
They can be labelled R1, R2, R3, R 1, R 2, R 3so as to have the following p*
*roperties:
__ __
(1)For each i = 1, 2, 3, URi = UR i, and Ri \ R iis cyclic of order 2n-1 and
contained in T .
__ __
(2)Of the Ri and R i, R3 and R 3are the only ones which are normal in S.
*
*__
(3)If P Ri is quaternion of order 8, then AutNH(U)(P ) = Aut(P ). If P *
*Ri is
quaternion of order 8, then AutNH(U)(P ) = AutS(P ).
(4)The three subgroups R1, R2, and R3 are NF (E)-conjugate.
(d) Let q0 be any other odd prime power such that v2(q2 - 1) = v2(q02- 1). Set
H0 = Spin7(q0),_fix S0 2 Syl2(H0), set F0, and let U0 E0 E S0 be as in (a*
*)._
Let R0i, R0i S0 be the subgroups which have the same properties as the Ri,*
* Ri S
~=
described in (c). Then any isomorphism ': S ---! S0 which induces an isomor-
phism of categories NF (E) ~=NF0(E0) and sends the Ri to the R0ialso induce*
*s an
isomorphism of fusion categories FS(NH (U)) ~=FS0(NH0(U0)).
THE SIMPLE CONNECTIVITY OF BSol(q) 13
Proof.We recall the notation used in Section 2. There is a homomorphism
_ _
! : SL2(F q)3 ------! Spin7(F q)
with Ker(!) = <(-I, -I, -I)>, and we write [[X1, X2, X3]] = !(X1, X2, X3). Set *
*U =
{[[ I, I, I]]}, and set B = NH (U), B0 = CH (U) = H \ Im(!), and S0 = CS(U).
Set L = SL2(q), and let bL SL2(q2) be the subgroup generated by L together with
p __ p __ x
the matrix diag( a, 1= a) for any a 2 Fq which is not a square (so [bL:L] = 2*
*). Then
fi 3 3ffi *
* ff
B0 = [[X1, X2, X3]] fiXi2 bL, X1 X2 X3 (mod L) !(bL) ~=bL (-I, -I, -I*
*) ;
and B = B0, where o2 = 1 and o[[X1, X2, X3]]o-1 = [X2, X1, X3].
Fix Sylow subgroups bQ2 Syl2(bL) and Q 2 Syl2(L); then Q ~=Q2n and bQ~= Q2n+1.
n*
*-2
Fix a pair of generators y, b 2 bQ, where |y| = 2n and |b| = 4, and set a = y2 *
* and
z = a2(= -I). Thus ~=Q8, and = Z(Qb). Since n 3, is the unique*
* cyclic
subgroup of bQof order 2n. Thus
fi
S0 = [[X1, X2, X3]] fiXi2 bQ, X1 X2 X3 (mod Q)
fi ff
= [[X1, X2, X3]] fiXi2 Q . [[y, y, y]] ,
and S = S0.
Set y1 = [[y, 1, 1]], y2 = [[1, y, 1]], y3 = [[1, 1, y]]; and similarly for b*
*i, ai, and zi. Also,
set by= [[y, y, y]] = y1y2y3, and similarly for bband ba. (By definition, [[z, *
*z, z]] = 1.) We
defined U = ~=C22, and now set
E = U = ba~=C32.
Let T S0 be the "toral" subgroup:
T = {[[yi, yj, yk]] | i j k (mod 2)}.
Then T ~=(C2n-1)3. If T 0 S0 is any subgroup such that T 0~=T , then T 0=(T \ *
*T 0)
S0=T is elementary abelian, so T 0 E (the 2-torsion subgroup of T ), T 0 CS0(*
*E) =
T .; and since bb= [[b, b, b]] inverts T it follows that T 0= T . This prov*
*es (b). (In fact,
T is the unique subgroup of S of its isomorphism type: this was shown in the pr*
*oof of
[LO , Proposition 2.9], and was shown in [AC , Lemma 4.9(c)] when n = 3.)
If V E S is a normal elementary abelian subgroup, then [V, T ] V \ T is an
elementary abelian subgroup of T . Fix v 2 V . If v =2T , then [v, by] *
*has order
2n-1 4; while if v 2 o., then [v, y21] has order 2n-1. Also, if v 2 bb*
*.T , then
[v, T ] E. Thus if rk(V ) 3, then V T , and hence V E (the 2-torsion su*
*bgroup
of T ). This shows that E is the unique such normal subgroup of rank 3. Also, s*
*ince
the four elements [[zia, zja, zka]] of Er U are all S-conjugate to each other, *
*U is the
unique such subgroup of rank 2. This proves (a).
For i = 1, 2, 3, set Ri= ~=Q2n. Thus R1 is the image in S0_of Q x 1*
* x 1, R2
is the image of 1 x Q x 1, etc. Also, for each i,_RiU ~=Q2nx C2. Let Ri RiU b*
*e the
unique subgroup isomorphic to Ri such_that Ri\ Ri T and is cyclic of order 2n*
*-1.
All six of these subgroups Ri and R iare normal in S0.
Now, S0=T ~= C32, with coset representatives the elements bi1bj2bk3for i, j, *
*k 2 {0, 1}.
Also, [bi, T ] is cyclic of order 2n-1 for each i = 1, 2, 3; while for any x 2 *
*S0 such that
xT 2={T, biT }, U [x, T ] and hence [x, T ] is not cyclic. If R E S0 is isomo*
*rphic to
Q2n, then since R and T are both normal, [R, T ] R \ T must be cyclic; and th*
*us
[R, T ] R T for some i. Hence R = [R, T ] for some g 2 T such th*
*at
14 ANDREW CHERMAK, BOB OLIVER, AND SERGEY SHPECTOROV
(gbi)2 = b2i; i.e.,_such that bigb-1i= g-1; and this implies that_g 2 [R, T ]U.*
* Hence
R = Ri or R = R i, and this finishes the proof that the Ri and R iare the unique
normal subgroups of S0 isomorphic to Q2n.
Points (c1), (c2), and (c3) now follow easily from the above descriptions of *
*these
subgroups of S. By the construction of F = FSol(q) in [LO ] or [AC ], there i*
*s an
element fi 2 AutF (S0) which permutes the subgroups R1, R2, and R3 cyclically. *
*Also,
fi(T ) = T by (b) (the uniqueness of T ), so fi normalizes E (the 2-torsion sub*
*group of
T ). Thus the three subgroups Riare conjugate in NF (E). This proves (c4), and *
*hence
finishes the proof of (b) and (c).
It remains to prove (d). Let q0be any other odd prime power such that v2(q2 -*
* 1) =
v2(q02- 1), and let H0= Spin7(q0), S02 Syl2(H0), and F0 = FSol(q0). Let U0 E0*
* S0
be the unique normal subgroups with U0 ~=U and E0~= E, and let R0ibe the subgro*
*ups
~=
of CS0(U0) with the same properties as the Ri E CS(U). Let ': S ---! S0 be an
isomorphism which induces an isomorphism of categories NF (E) ~=NF0(E0), and wh*
*ich
sends each Ri to some R0j,. In particular, '(R3) = R03by (c2), and hence ' sen*
*ds
{R1, R2} to {R01, R02}. Upon composing with conjugation by o, if necessary, we*
* can
assume that '(Ri) = R0ifor all i. By (c3), the Ri and R0iare contained in the f*
*actors
SL2(q) H, H0.
Now, the only subgroups of bQ~=Q2n+1whose automorphism groups are not 2-groups
are the quaternion subgroups of order 8. Hence FQ (L) is generated by FQ (Q) to*
*gether
with the groups Aut (P ) for all P Q quaternion of order 8. Also, if P Q*
*b is
quaternion of order 8 but not contained in Q, then Aut bL(P ) = Aut bQ(P ) sinc*
*e any
automorphism leaves P \ Q invariant. This shows that FS0(B0) is generated by FS*
*(S),
together with those automorphisms ff 2 Aut(PiRjRk) (where {i, j, k} = {1, 2, 3}*
* and
Q8 ~=Pi Ri) such that ff|RjRk = Idand ff|Pihas order 3. Hence FS(B) is generat*
*ed by
FS0(B0) and FS(S) together with all automorphisms of the form fi 2 Aut(P1P2R3)
for P1 R1 and P2 R2 both quaternion of order 8 and exchanged by o0 2 S0o, w*
*here
fi(o) = o, fi|R3 = Id, and fi|P1P2has order 3. This proves that ' sends B = NH *
*(U)-
fusion to NH0(U0)-fusion, and thus induces an isomorphism of fusion categories.
Recall that a subgroup H of a group G is strongly embedded in G (at the prime*
* 2)
if H is a proper subgroup of even order such that |H \ Hg| is odd for all g 2 G*
*r H.
A 2-subgroup P G is essential if Z(P ) 2 Syl2(CG (P )) and Out G(P ) has a st*
*rongly
embedded subgroup. In particular, if S 2 Syl2(G) and P S an essential 2-subgr*
*oup
of G, then P is centric and radical in the fusion system FS(G).
By the Alperin-Goldschmidt fusion theorem [Go ], every morphism in FS(G) is a
composite of restrictions of automorphisms of S, and of essential subgroups P *
* S such
that NS(P ) 2 Syl2(NG (P )) (i.e., are fully normalized in FS(G)). For this rea*
*son, we
need information about the essential 2-subgroups of Spin7(q), which means infor*
*mation
about the essential 2-subgroups of 7(q).
Lemma 3.2. Fix an odd prime power q, set G = Spin7(q), and fix S 2 Syl2(G). Let
U E E S be the unique elementary abelian subgroups which are normal in S and *
*of
rank two and three, respectively (see Lemma 3.1). If P S is an essential 2-su*
*bgroup
of G, then P is G-conjugate to a subgroup P 0 S such that either
(1) U E P 0is an AutG (P 0)-invariant subgroup; or
(2) E E P 0is an AutG (P 0)-invariant subgroup.
THE SIMPLE CONNECTIVITY OF BSol(q) 15
Proof.Set V = F7q, and let q be a quadratic form on V with orthonormal basis. We
__ _
identify G = Spin(V, q). Set Z = Z(G), G = G=Z = (V, q), and S = S=Z, and let u
be a generator of U=Z.
__
Let P S be an essential_2-subgroup of G = Spin(V, q). Then P = P=Z is an
essential 2-subgroup of G (cf. [LO , Lemma A.11(e)])._ Let V = V1 . . .Vm *
* be a
decomposition of V as a sum of pairwise orthogonal P -invariant subspaces, chos*
*en so
that m is as large as possible. This_decomposition can be chosen such that for*
* each __
i, either Vi is irreducible as a P -representation, or it is a sum of two irred*
*ucible P -
representations neither of which supports a nondegenerate_quadratic form (cf. *
*[O1 ,
Lemma 7.1]). In particular, each element of N_G(P ) leaves invariant the sum of*
* all of
the Vi of any given dimension.
Set di = dim(Vi), and assume the summands are ordered so that the sequence
= (d1, . .,.dm ) is non-increasing. This sequence may be written in abbrevia*
*ted
fashion, using exponents to indicate repeated dimensions. For example, (4, 13)*
* is an
abbreviation for one such sequence. By [LO , Lemma A.6], each diis a power of 2*
*, and
the discriminant of Vi is a square in Fxqif di> 1.
__
Assume first that there is an N_G(P )-invariant orthogonal decomposition V = *
*V 0 V 00,
where dim(V 0) =_4 and dim(V_00) = 3._Let u0be the involution_(-Id)V_0 IdV 00.*
* Then
u0 centralizes P , so u0 2 P since P is 2-centric, and N_G(P ) C_G(u0). Al*
*so, u0 is
__ _ __ _ __ __
G-conjugate_to_u. Since u_2 Z(S ), there is P0 S which is G-conjugate to P an*
*d such
that u 2 Z(P 0) and N_G(P 0) C_G(u), and we are thus in the situation of case*
* (1).
__
Next assume = (23, 1). Let Q G be the group of elements which are Id on
each of V1, V2, and V3 (i.e., on the 2-dimensional summands), are the identity *
*on V4,
and_which negate an even number of summands._ Thus Q is_a fours group,_and is *
* __
G-conjugate to E=Z. Also, Q_centralizes P , so Q Z(P ) since P is 2-centric *
*in G ,
and every element of AutG_(P ) leaves Q invariant. So by the same reasoning as *
*in the
last paragraph, we are in the situation of case (2).
By inspection, we are now left only with the cases where = (2, 15) or (17).*
* Let n+
be the number of 1-dimensional summands Vi= such that q(v) is a square, and*
* let
n- be the number of Vi = such that q(v) is not a square. Then n- is even (s*
*ince
V itself has square discriminant), and n+ is odd.
__ *
* __
If = (2, 15),_then P O2(q) x C52, and O2(q) is a dihedral group. Also, si*
*nce P
is 2-centric in G , it contains every involution_which negates four of the 1-di*
*mensional
summands. So_the Viare pairwise_distinct as P-representations, and thus are per*
*muted
by Aut _G(P ). Also, N_G(P ) contains elements whose projections to O(V1, q) re*
*present
__ __ __ *
* __
all cosets of G = (V, q) in O(V, q), so Out _G(P ) = Out O(V,q)(P ). Thus Ou*
*t _G(P ) ~=
__
A x Sym (n+) x Sym (n-), where |A| 2. Since O2(Out _G(P )) = 1, this implies *
*that
__
A = 1, n+ = 5, and Out _G(P ) ~=Sym (5), which is impossible since Sym (5) does*
* not
contain a strongly embedded subgroup.
Finally, if = (17), then_similar (but simpler) arguments show that (n+, n-)*
* = (7, 0)
or (1, 6), and that Out _G(P ) is one of the groups Sym (7), Sym (6), Alt(7), o*
*r Alt(6) _
none of which contains a strongly embedded subgroup.
16 ANDREW CHERMAK, BOB OLIVER, AND SERGEY SHPECTOROV
Recall that if F is a fusion system over a p-group S, and is a set of F-mor*
*phisms,
then one says that F is generated by , and writes F = < >, if every morphism O*
*E in
F is a composite of restrictions of morphisms in . That is, there is no fusion*
* system
over S whose set of morphisms contains , and which is properly contained in F.
Proposition 3.3. Fix an odd prime power q, set H = Spin7(q), let S 2 Syl2(H),
and let U E S be the unique normal elementary abelian subgroups of ranks two
and three, respectively. Let F0 F be the fusion systems over S: F0 = FS(H) a*
*nd
F = FSol(q). Then
F0 = and F = .
Proof. By the Alperin-Goldschmidt fusion theorem [Go ], for any finite group G,*
* any
prime p, and any S 2 Sylp(G), the fusion system FS(G) is generated by the autom*
*or-
phism groups Aut G(P ) for P = S, and for subgroups P S which are essential in
FS(G).
Now set G = Spin7(q) and Z = Z(G), and set F00= . Assume the
lemma is false for F0; i.e., that F00$ F0. Let P be a maximal essential subgrou*
*p for
which Aut G(P ) is not contained in F00. By Lemma 3.2, P is G-conjugate to some*
* P 0
such that either U or E is contained in P 0and is AutG (P 0)-invariant. Thus Au*
*tG (P 0)
is in F00; while by the maximality assumption, P is conjugate to P 0by an isomo*
*rphism
in F00. It follows that AutG (P ) is in F00, a contradiction; and thus F0 = F00.
By construction in [LO ], FSol(q) is generated by FS(Spin7(q)) together with *
*one
morphism of order three which normalizes U, and which also can be chosen to nor*
*malize
E. So the result for F follows from that for F0.
Recall that v2(-) denotes the 2-adic valuation of an integer: v2(n) = k if k *
*is the
largest integer such that 2k|n.
Theorem 3.4. For any pair of odd prime powers q and q0, FSol(q) ~=FSol(q0) _ and
hence LcSol(q) ~=LcSol(q0) _ if and only if v2(q2 - 1) = v2(q02- 1).
Proof. By [LO , Lemma 3.2], together with the obstruction theory in [BLO2 , Pr*
*opo-
sition 3.1], FSol(q) has a unique associated linking system. Hence the equival*
*ence of
linking systems follows from the equivalence of fusion systems.
Set H = Spin7(q) and H0 = Spin7(q0), fix S 2 Syl2(H) and S0 2 Syl2(H0), and s*
*et
F = FSol(q) and F0 = FSol(q0). If v2(q2 - 1) 6= v2(q02- 1), then |S| 6= |S0|, a*
*nd hence
F 6~=F0.
Now assume v2(q2 - 1) = v2(q02- 1); we prove that F ~= F0. Let U E S and
U0 E0 S0 be the normal subgroups of ranks two and three (Lemma 3.1(a)). Set
B = NH (U) and B0= NH0(U0). Then S ~=S0 by Lemma 3.1(d). We use the notation
of the proof of Lemma 3.1, when needed, to describe elements of S.
Set n = v2(q2 - 1). By Lemma 3.1(b), there is a unique homocyclic subgroup
T S0 = CS(U) of rank 3 and exponent 2n-1. In particular, T is weakly closed a*
*nd
centric in any fusion system over S. Hence by [BLO1 , Proposition A6], NF (T *
*) is a
2-constrained, saturated fusion system; so by [BCGLO2 , Proposition 4.3], the*
*re is a
group L such that S 2 Syl2(L), F *(L) = O2(L), and FL = NF (T ). By construction
[LO , Section 2], we have CL(E) = CS(E) = T , where bbacts on T by invertin*
*g; and
L=CL(E) ~=GL3(2) with the obvious action on E.
Now let T 0E S0 and L0 S0 be the corresponding groups for the fusion system *
*F0.
We next show that L ~=L0via an isomorphism which sends S onto S0. Basically, th*
*is is
THE SIMPLE CONNECTIVITY OF BSol(q) 17
done by showing that L is the unique extension of T by L=T ~=GL3(2) x C2 for wh*
*ich
GL3(2) has the standard action on E while the C2 factor acts on T by inverting,*
* and
GL3(2) splits over T while L=T does not split.
Let L0 E L be the subgroup of index two such that L0=T ~=GL3(2). Set S0 = S \*
*L0,
and let S00 L00be the corresponding subgroups of L0. Choose isomorphisms
~= 0 ~= 0 0
'0: E -----! E and _ : L0=T -----! L0=T
such that _(S0=T ) = S00=T 0, and '0 commutes with the conjugation actions of L*
*0=T ~=
L00=T 0when identified via _. By [G , Proposition 6.4], the action of L0=T ~=GL*
*3(2) on
E ~=(Z=2)3 has a unique lifting to T ~=(Z=2n-1)3: unique up to an automorphism *
*of
~=
T . Thus '0 extends to an isomorphism '1: T ---! T 0which still commutes with *
*the
conjugation actions of L0=T ~=L00=T 0.
We next claim that L0 splits over T , and similarly for L00. Let T 2 T be t*
*he
subgroup of squares in T . Then CL(E)=T 2= T=T 2x ~=C42, and the quotient *
*group
S=T 2splits over CL(E)=T 2via (for example) the subgroup
o.T 2~=S=CL(E) ~=D8.
Hence by Gasch"utz's theorem (i.e., since H2(GL3(2); C42) injects into H2(D8; C*
*42)),
L=T 2splits as a semidirect product (CL(E)=T 2).R0 for some subgroup R0 ~=GL3(2*
*).
In particular, L0=T 2is a semidirect product of T=T 2by GL3(2). By [G , Theorem
6.5], the surjection of T onto T=T 2induces an isomorphism in group cohomology *
*from
H2(L0=T ; T ) to H2(L0=T ; T=T 2), and thus L0 also splits as a semidirect prod*
*uct over
T .
~=
Fix subgroups L1 L0 and L01 L00, both isomorphic to GL3(2). Let '2: L0 -! *
*L00
be the isomorphism which extends '1 by sending L1 to L01via _.
Now fix elements d 2 CS(E)r T and d0 2 CS0(E0)r T 0. Thus d inverts T and L =
L0; and similarly for d0. Since H1(L0=T ; T ) ~=Z=2 [G , Theorem 6.5], L0 co*
*ntains
two T -conjugacy classes of subgroups isomorphic to GL3(2). If (L1)d is T -conj*
*ugate to
L1, then some element of dT centralizes L1, and L would be split over T . Since*
* U T
is centralized by b3 L=T , this would imply that S contains some C52,*
* which
is impossible since S has rank four (cf. [LO , Proposition A.8] or [AC , Theore*
*m A]).
Thus conjugation by d switches the two T -conjugacy classes of subgroups GL3(*
*2)
*
* 0
L0, and similarly for d0. So there is some t 2 T 0such that '2((L1)d) = (L01)t*
*d; and
~=
we can now extend '2 to an isomorphism '3: L ----! L0by setting '3(d) = td0. By
construction, '3(S) = S0.
Set ' = '3|S. Since ' extends to L, and FS(L) = NF (E) by construction, '
defines an isomorphism from NF (E) to NF0(E0). Set FU = FS(NH (U)) and F0U=
FS0(NH0(U0)). Since F = by Proposition 3.3, and similarly for F0,*
* we
will be done if we can show that ' induces an isomorphism FU ~=F0U. By Lemma 3.*
*1(d),
this means showing that ' sends the set R = {R1, R2, R3} to R0= {R01, R02, R03}.
__ __ __ __
Assume otherwise. Then by Lemma 3.1(c4), ' sends R = {R 1, R2, R3} to R0. We
claim_there_is an automorphism ff 2 Aut(L) such that ff(S) = S and ff exchanges*
* R
with R . Once we have shown this, then we can replace ' by ' Off|S, and we are *
*done.
As seen above, there are two T -conjugacy classes of subgroups GL3(2) in L0, *
*and the
two classes are exchanged by elements of the coset dT . Furthermore, by [G , Th*
*eorem
6.5] again, the inclusion of E into T induces an isomorphism from H1(L0=T ; E) *
*to
H1(L0=T ; T ); and hence the two classes are both represented in the subgroup E*
*L1.
18 ANDREW CHERMAK, BOB OLIVER, AND SERGEY SHPECTOROV
We can thus choose d 2 CS(E)r T such that [d, L1] E. Let ff 2 Aut (L) be the
automorphism such that ff|EL1 is conjugation by d, and ff|CS(E) is the identity*
*. Set
V = E, and regard it as a 4-dimensional L1-representation.
Clearly, CE (L1) = 1, and hence CV (L1) = 1 since otherwise L would split ove*
*r T _
which we already know is not the case. Since L1 is generated by three involutio*
*ns (and
all of its involutions are conjugate), this means that |CV (g)| = 4 for each in*
*volution
g 2 L1. Also, [V, g] CV (g) (since g2 = 1), and hence [V, g] = CV (g) also ha*
*s order 4.
Recall the notation set up in Section 2 for elements of S. In particular, S *
* =
T .o, CS(E) = T ., and Ri = (Ri \ T ). For each i = 1*
*, 2, 3,
let si be the unique element of L1 in the coset biCS(T ) (an involution). Sinc*
*e si 2
biCS(E) CS(U), [V, si] = CV (si) U, and thus [V, si]_= U since it has order*
* 4.
Recall that U# = {z1, z2, z3}, where = Z(Ri) = Z(R i). Since siCS(E) = biC*
*S(E),
we have [E, si] = [E, bi] = <[a1a2a3, bi]> = , and so [ff, si] = [d, si] 2 *
*Ur . Since
ff|CS(E)= Id(and since CS(E)si= CS(E)bi), we now get [ff,_bi] 2 Ur {zi}. As zii*
*s the
unique involution in Ri~= Q2n, we conclude that (Ri)ff= Ri for all i. This comp*
*letes
the proof.
4. The Co3 geometry
Let G be the rank three 2-local geometry of G = Co3 constructed in [A ]. It c*
*an be
described as follows. There are two conjugacy classes of involutions in G, of w*
*hich 2A
denotes the class of central involutions (those in centers of Sylow 2-subgroups*
*). The
elements of G are the 2A-pure elementary abelian subgroups of G of rank 1, 2, o*
*r 4,
and incidence is given by symmetrized containment. By [Fi, Lemmas 5.8 & 5.9] (w*
*here
the conjugacy class 2A is denoted 21), G acts transitively by conjugation on th*
*e set of
such subgroups of a fixed order. Furthermore, if E 2 G has rank 4, then Aut G(E*
*) is
the full automorphism group GL4(2). It follows that G acts flag transitively on*
* G; i.e.,
it acts transitively on the set of all maximal flags X Y E (where rk(X) = 1*
* and
rk(Y ) = 2).
Fix such a maximal flag X Y E in G. The maximal parabolics corresponding
to this flag are the three maximal subgroups of G containing a given Sylow subg*
*roup
S: L = N(E) ~= 24.GL4(2), M = N(Y ) ~= 22+6.32.D12, and N = N(X) ~= 2.Sp6(2)
(see [A ]). Notice that S has index three in the Borel subgroup B = L \ M \ N o*
*f order
210.3.
We will identify the elements of G as follows. We will call the conjugates of*
* X points,
the conjugates of Y lines, and the conjugates of E 3-spaces (for the lack of a*
* better
name; note that 3 here represents the projective dimension).
Let |G| be the flag complex of the geometry G: the simplicial complex with o*
*ne
vertex for each element of G (each point, line, and 3-space), and a simplex for*
* each
flag in G (each set of elements of G which are pairwise incident). A geometry i*
*s called
simply connected if it has no (proper) covering geometries, and this is the cas*
*e if and
only if its flag complex is simply connected as a space. We refer to [Pn , x8.3*
*] for more
details about coverings of geometries.
Equivalently, |G| is the coset complex for the three orbits G=L, G=M, and G=N.
Since G is generated by L, M, and N, the geometry G is connected; and in fact
THE SIMPLE CONNECTIVITY OF BSol(q) 19
residually connected (the link of each vertex in |G| is connected) since each o*
*f L, M,
and N is generated by its intersections with the other two subgroups.
The following proposition is the main result to be proven in this section.
Theorem 4.1. The geometry G (or its realization |G|) is simply connected. Hence*
* for
any complete flag X Y E in G, the colimit of the triangle of groups involv*
*ing
NG (X), NG (Y ), NG (E) and their intersections is isomorphic to G = Co3.
The last statement in Proposition 4.1 follows from the simple connectivity of*
* |G|
together with Proposition 1.5 (the standard argument involving Tits' Lemma).
Let be the graph whose vertex set is the set of points in G (i.e., the cent*
*ral invo-
lutions in Co3), and where two vertices are adjacent whenever they are colinear*
* in G
(whenever their product is also a point in G). Since the product of two commuti*
*ng cen-
tral involutions in Co3 is again a central involution (this follows from [Fi, L*
*emma 4.7]),
two vertices of are adjacent if and only if the corresponding involutions com*
*mute.
Thus, coincides with the commutation graph on the central involutions of G.
A cycle in (i.e., a loop) is called geometric if all of its vertices are in*
*cident to a
common 3-space.
Proposition 4.2. Assume every cycle in can be decomposed as a product of geom*
*etric
cycles. Then Theorem 4.1 holds.
Proof.This is a standard argument in diagram geometry (cf. [Pn , x12.6]), but *
*we
repeat it here. We regard a cycle fl in as a sequence fl = (x0, x1, . .,.xn *
*= x0) of
vertices (points in G) which are pairwise adjacent; i.e., such that *
*is a line
in G (or a point if xi-1 = xi) for each i. For each such cycle fl, set yi = ,
and let bflbe the cycle in |G| defined by the sequence (x0, y1, x1, y2, x2, . .*
*,.yn, xn). If
fl decomposes as a product of cycles ffi1 and ffi2, then bfldecomposes as the p*
*roduct of
the cycles bffi1and bffi2. If fl is geometric _ if the xi (i = 0, . .,.n) are a*
*ll contained in
some 3-space V _ then every vertex in bflis adjacent to the vertex V in |G|, an*
*d so bfl
is homotopic to a trivial loop.
Thus, under the hypotheses of the proposition, for every cycle fl in , bflis*
* homotopic
to the trivial loop. It remains to check that every cycle in |G| is homotopic t*
*o one of
this form.
Fix a cycle in |G|, regarded as a sequence (V0, V1, . .,.Vn = V0) of elements*
* of G such
that each pair (Vi, Vi+1) is incident. For each i, let xibe a point which is in*
*cident to Vi
(and set xn = x0). For each i = 1, . .,.n, is contained in Vi-1or Vi*
*(whichever
is larger), and hence xi-1and xiare adjacent in . Set fl = (x0, x1, . .,.xn = *
*x0), a cycle
in , and set yi= . For each i, the paths (xi-1, yi, xi) and (xi-1, V*
*i-1, Vi, xi) are
homotopic in |G| (relative to endpoints) since all of the vertices involved are*
* adjacent
to Vi-1 or Vi. Thus bflis homotopic to the loop (V0, V1, . .,.Vn = V0) we start*
*ed with,
and this is what we had to prove.
The next lemma shows that it suffices to decompose each cycle in as a produ*
*ct of
cycles of length three.
Lemma 4.3. Every 3-cycle in (i.e., every cycle of length three) is geometric.
Proof.Fix a 3-cycle in ; i.e., a sequence of points (x, y, z) in G any two of *
*which are
incident to a line. Thus, if we regard x, y, and z as central involutions in G*
*, they
generate an elementary abelian subgroup of rank two or three, all involutions i*
*n which
20 ANDREW CHERMAK, BOB OLIVER, AND SERGEY SHPECTOROV
are still central (see [Fi, Lemma 4.7] again). Since each 2A-pure elementary a*
*belian
subgroup of G is contained in one of rank four [Fi, Lemma 5.9], zis cont*
*ained
in some 3-space in the geometry G, and so the cycle (x, y, z) is geometric.
This also follows directly from the analysis given below of all pairs and tri*
*ples of
involutions of class 2A in G (see Figure 1). For example, Figure 1 classifies *
*pairs
of central involutions by the conjugacy class of their product, and shows that *
*if the
product is an involution then it must again be central.
It remains to show that every cycle in is a product of 3-cycles. This has *
*been
shown computationally, using the computer algebra system GAP [GAP ]. We realiz*
*e G
in GAP in its primitive action of length 276. This action can be found in a sta*
*ndard
library of GAP, namely, in the library of primitive permutation groups. Below,*
* we
provide an account of the computation.
The first task is to classify the orbits of G on the pairs of central involut*
*ions. Equiv-
alently, we need the orbits of the centralizer C = C(s) of a fixed central invo*
*lution s
on the set of central involutions (that is, the orbits of the stabilizer of the*
* vertex s on
the vertex set of ). Every group in GAP comes with a distinguished set of gene*
*rators.
As it turns out, the first generator of our copy of G is an element of order 4 *
*and its
square is a central involution, which we choose to be s. By taking random conju*
*gates
sg of s and by computing the double stabilizers C(~~) = CC (sg) we soon fi*
*nd the
following orbits:
o O2 = sC2of size 630; s and s2 commute and ss2 is again a central involution;
o O3 = sC3of size 1920; ss3 is of order 3 (class 3C);
o O30= sC30of size 8960; ss30is of order 3 (class 3B);
o O4 = sC4of size 30240; ss4 is of order 4;
o O5 = sC5of size 48384; ss5 is of order 5; and
o O6 = sC6of size 80640; ss6 is of order 6.
We notice that the lengths of these orbits sum to 170774 = [G:C]-1 (where the m*
*issing
1 clearly represents s itself), and so our count of orbits is complete. We also*
* remark
that every representative si comes with the conjugating element gi, such that s*
*i= sgi.
We store the elements gi for future use, alongside si.
It follows that the orbits of G on the set of pairs (a, b) of central involut*
*ions can be
distinguished by the order of ab, except when |ab| = 3. In the latter situation*
* the two
orbits with |ab| = 3 can be distinguished by the order of the centralizer C() equal
to 1512 if ab is in class 3C and equal to 324 if ab is in class 3B. Since the e*
*dges of
correspond to the pairs of commuting involutions, we conclude that has degree*
* 630
(each vertex is adjacent to 630 other vertices).
With this information, it is now easy, for each t 2 {s, s2, s3, s30, s4, s5, *
*s6}, to find the
630 neighbors of t and then determine the orbits of the double stabilizer C(~~~~) =
CC (t) on the edges starting from t. Indeed, the set of neighbors of s coincide*
*s with O2,
while the set of neighbors of each si is Ogi2= {xgi| x 2 O2}. Once the orbits o*
*f CC (t),
t 2 {s, s2, s3, s30, s4, s5, s6}, on the neighbors of t are determined, we can *
*place each of
these orbits in a particular Oj by checking the order of sx, where x is a repre*
*sentative
of the orbit, as described above. The results of this computation are presented*
* in the
distance distribution diagram of shown in Figure 1.
THE SIMPLE CONNECTIVITY OF BSol(q) 21
Figure 1. Distance distribution diagram of
For example, this diagram indicates that each element of O2 is incident to 37*
* other
elements of O2, in two C(~~~~)-orbits of orders 1 and 36. Hence G has two o*
*rbits
on the set of 3-cycles. One of the orbits consists of triples of points incide*
*nt to a
common line (i.e., the three involutions in a subgroup of rank 2). The other co*
*nsists of
noncollinear triples of points which generate an elementary abelian subgroup of*
* rank
three.
We now start decomposing cycles. An n-cycle in means a cycle of length n. *
* A
cycle is called isometric if the distance between two vertices of the cycle is *
*the same
when it is computed in the cycle and in . If a cycle is not isometric then it*
* can
be decomposed as a product of two shorter cycles. Thus, we only need to deal w*
*ith
isometric cycles. Since the diameter of is 3, there are no isometric cycles o*
*f length
more than 7.
We start with 4-cycles. Suppose abcd is an isometric 4-cycle. Clearly, d(a,*
* c) = 2,
and b and d are common neighbors of a and c. Since s and s6 have only one common
neighbor, the pair (a, c) is conjugate to (s, s30) or (s, s4). We start with th*
*e second case.
In this case i = (ac)2 is a central involution which commutes with all four inv*
*olutions
a, b, c, and d. Thus, the 4-cycle can be decomposed as a product of four 3-cycl*
*es.
Now suppose that (a, c) is conjugate to (s, s30). Without loss of generality *
*a = s and
c = s30. Let X = X(a, c) be the set of common neighbors of a and c. Then |X| = 9
and the double stabilizer C30= C() acts on X transitively. Clearly, b, d*
* 2 X.
Checking the orders of xy for x, y 2 X we see that all pairs (x, y) are conjuga*
*tes of
(s, s30). So the graph induced on X has no edges, and we need a new idea if we *
*want
to decompose these 4-cycles.
According to Figure 1, C30has two orbits (cf. 812; i.e. two orbits of length *
*81) on
the neighbors of c in O4. Checking representatives of these two orbits, we find*
* that one
of them (call it e) has the following properties:
o |ex| 2 {4, 6} for all x 2 X; and
o 5 elements of X have neighbors in Y , where Y = X(a, e) is the set of common
neighbors of a = s and e.
22 ANDREW CHERMAK, BOB OLIVER, AND SERGEY SHPECTOROV
If b and d are among these 5 elements of X then abcd can be decomposed. Indeed,*
* let b
be adjacent to f 2 Y and d be adjacent to h 2 Y . Then abcd is a product of abf*
*, fbce,
adh, hdce, and (if f 6= h) afeh. Notice that the 4-cycles used in this decompos*
*ition
have a pair of opposite vertices with product 4, hence these 4-cycles are decom*
*posable.
Consider the equivalence relation on X defined by setting x ~ y if if axcy is*
* de-
composable. Since C30acts transitively on X, this splits X as a union of equiva*
*lence
classes of the same order (which must divide 9). We have just shown that there *
*is an
equivalence class of order at least 5, and hence the relation is transitive. Th*
*us, we have
verified the following.
Lemma 4.4. All 4-cycles in are decomposable.
We now turn to 5-cycles. Suppose abcde is an isometric 5-cycle. For x and y*
* at
distance two from each other let X(x, y) denote, as above, the set of common ne*
*ighbors
of x and y (the so-called ~-graph of x and y). Notice that b 2 X(a, c) and e 2 *
*X(a, d).
If we substitute b by any other vertex b02 X(a, c) then the new 5-cycle ab0cde *
*differs
from abcde by a 4-cycle. Hence, by Lemma 4.4, abcde is decomposable if and only*
* if
ab0cde is. Similarly, e can be substituted by any other vertex e02 X(a, d). It *
*means
that we can only keep track of one vertex, a, and of the edge, cd, opposite tha*
*t vertex.
Without loss of generality, we can assume that a = s, in which case cd is an ed*
*ge
between two vertices at distance two from s. According to Figure 1, there are 5*
*0 C-
orbits of such edges, and so we have 50 cases to consider. The representative o*
*f all these
50 orbits were collected and stored, when the orbits of Ci = CC (si) on the nei*
*ghbors
of si, i 2 {30, 4, 6}, were determined.
Suppose an edge cd represents one of the 50 cases. We will call this case ea*
*sy if
X(a, c) and X(a, d) either intersect, or have an edge connecting them. If this*
* is the
case then all 5-cycles containing a and cd are decomposable as a product of 3- *
*and
4-cycles. It turns out that 30 of the 50 cases are easy.
Most of the remaining 20 cases can be handled using an additional trick. Supp*
*ose the
distance between X(a, c) and X(a, d) is two, but there is a choice of b 2 X(a, *
*c) such
that the edge agbg (where g is selected to satisfy s = a = dg) represents an ea*
*sy case
(or more generally, a previously handled case of 5-cycles). Then, for any e 2 X*
*(a, d),
the cycle dgegagbgcg is decomposable and hence abcde is decomposable, too. This*
* trick
can be used iteratively, as more and more cases are settled, and eventually it *
*helps
decompose 5-cycles in 18 out of 20 "hard" case.
The remaining 2 orbits have been disposed of via a further trick, which proba*
*bly
applies in many other cases, as well. Namely, suppose we find a vertex f among *
*the
common neighbors of c and d, such that f is at distance 2 from a and, furthermo*
*re, cf
and df both fall into the previously decomposed cases. Then, clearly, we can de*
*compose
abcde as a product of cdf, abcfg, and aedfg, where g is an arbitrary vertex fro*
*m X(a, f).
Thus, abcde is also decomposable.
This concludes the verification of the following statement.
Lemma 4.5. All 5-cycles in are decomposable.
Once all cycles up to length 5 are decomposed, the 6-cycles and 7-cycles are *
*an easy
gain. For t = s3, s5 construct the set X = {x | d(t, x) = 1 and d(a, x) = 2} by*
* selecting
among the neighbors x of t the involutions belonging to O30, O4, and O6. Using*
* the
package GRAPE [GAP ], we then define a graph on X via commutation (so it is the
subgraph of induced on X) and check that this graph is connected for both cho*
*ices
THE SIMPLE CONNECTIVITY OF BSol(q) 23
of t. Connectivity means that all 6-cycles can be decomposed as products of 3-c*
*ycles
and 5-cycles.
Finally, according to Figure 1, there are 9 cases of isometric 7-cycles. (As *
*was the
case for 5-cycles, we only need to keep track of one vertex, say a = s, and the*
* edge,
say de, opposite it.) In each of these case d and e have a common neighbor that*
* is at
distance 2 from a, and so the 7-cycle can be decomposed as a product of a 3-cyc*
*le and
two 6-cycles. So the following is true.
Lemma 4.6. All 6- and 7-cycles in are decomposable.
Thus, all isometric cycles in are decomposable, and this finishes the proof*
* of
Theorem 4.1.
5. The fundamental group of BSol(q) and BDI(4)
In this section, we prove the following theorem.
Theorem 5.1. For each odd prime power q, the geometric realization of the linki*
*ng
system LcSol(q) is simply connected.
Theorem 5.1 will follow fairly easily from results in the first two sections,*
* once we
have shown the special case q = 3. So we first set up notation which will be us*
*ed to
prove this case.
Set H = Spin7(Z[1_2]) for short, and let
! : H * K -- ---i ss1(|LcSol(3)|)
B
be the surjective homomorphism of Proposition 2.2. Fix S 2 Syl2(B) (thus also a
Sylow 2-subgroup of Spin7(3)), and let U E S be the unique normal subgroup of o*
*rder
4. Then B is a finite subgroup of order 210.33, and has index 3 in K.
__
__Set G = H_*B_K and G = !(G) for short._Also, for any subgroup R G, we write
R = !(R) G. Since ! is surjective, G ~=ss1(|LcSol(3)|).
______ __
Lemma 5.2. Set Z = Z(H) ~=C2. Then CG (Z)= H .
______ __
Proof.Since H CG (Z), we need only show that CG (Z) H . Fix g 2 CG (Z); we
_ __
must show that g = !(g) 2 H .
Let be the standard tree for G, and set ff = H and fi = K as vertices of .*
* Thus
Gff= H, Gfi= K, each vertex of is in the orbit of ff or of fi, and G acts tra*
*nsitively
on the set of edges of . In particular, H acts transitively on the set of vert*
*ices adjacent
to ff, and K acts transitively on the set of vertices adjacent to fi.
Let
ff = ff0, fi1, ff1, . .,.fik, ffk = g(ff)
be the geodesic in from ff to g(ff), where each ffi is in the G-orbit of ff a*
*nd each
fii in the G-orbit of fi. Since H acts transitively on the set of vertices adja*
*cent to ff,
fi1 = g1(fi) for some g1 2 H. Then g-11(ff1) is adjacent to fi, and hence there*
* is g2 2 K
such that g-11(ff1) = g2(ff) and thus ff1 = g1g2(ff). Upon continuing in this *
*way, we
obtain a sequence of elements gi for i = 1, . .,.2k, where gi2 H for i odd and *
*gi2 K
for i even, and such that fii = g1. .g.2i-1(fi) and ffi = g1. .g.2i(ff) for eac*
*h i. Set
24 ANDREW CHERMAK, BOB OLIVER, AND SERGEY SHPECTOROV
bgi= g1. .g.ifor_each i. Then g-1bg2k(ff) = ff, so bg2k2 gH, it suffices to pr*
*ove that
!(bg2k) 2 H , and we can thus assume that g = bg2k= g1. .g.2k.
Now, Z H = Gff, and Z gHg-1 = Gg(ff). Since the fixed point set of the Z
action is a tree, this means that Z fixes the entire geodesic from ff to g(ff).*
* Thus for
each i, Z Gfii= bg2i-1Kbg-12i-1and Z Gffi= bg2iHbg-12i. So if we set Zi = *
*bg-1iZbgi,
then for each i = 1, . .,.2k, Zi K (if i is odd) or Zi H (if i is even), and
Zi= g-1iZi-1gi2 H \ K = B.
Now, each Ziis H-conjugate to a subgroup of U (this follows from [LO , Propos*
*ition
A.8], since B is the same as a subgroup of H = Spin7(Z[1_2]) or of Spin7(3)); a*
*nd each
subgroup of order 2 in U is K-conjugate to Z. Thus there is ti 2 HK such that
t-1iZiti = Z. Using this, we can write g as a product of elements in KHKHK, each
of which centralizes Z. So it suffices to prove the lemma for such g. In other *
*words,
we are reduced to the case where k = 3 and g1 = 1. We can regard this situation
schematically as follows.
g2 g3 g4 g5 g6
Z = Z1 ---! Z2 ---! Z3 ---! Z4 ---! Z5 ---! Z6 = Z.
(K) (H) (K) (H) (K)
Assume first that gi 2 B = H \ K for some i. If i = 3, 4, 5, then g 2 KHK.
If g2 2 B, then g2g3 2 H, Z3 = Z, and we need only consider the product g4g5g6.
Similarly, if g6 2 B, then we need only consider the product g2g3g4. Thus in al*
*l cases,
we can relabel the elements and assume that g5 = g6 = 1 (and Z4 = Z). Also,
Z2 = Z if and only if Z3 = Z, since Z3 = g-13Z2g3 and g3 2 H. If Z2 = Z3 = Z,
then g2, g4 2 CK (Z) = B, so g 2 H, and the result follows. If Z2 6= Z 6= Z3, *
*then
U = ZZ2 = ZZ3, so g3 2 NH (U) = B, g 2 K and centralizes Z, so g 2 H.
Now assume that none of the gi lies in B. Thus g2, g6 =2H, so Z2, Z5 U and *
*are
distinct from Z. Hence U = ZZ2 = ZZ5. Also, g3 2 Hr K implies ZZ3 = g-13ZZ2g3 6=
U, and hence that Z3 U. Similarly, Z4 U.
Let Ei CH (U) (all 1 i 6) be the rank three elementary abelian subgroups
defined by the requirements that E3 = UZ3, E4 = UZ4, and g-1iEi-1gi = Ei. Thus
U = ZZ5 g-15E4g5 = E5 since [g5, Z] = 1, and U E6 since g6 2 K normalizes U.
Via similar considerations for E1 and E2, we see that U Ei for all 1 i 6,*
* and
hence that Ei CH (U).
Set R = CH (U) for short. Then CS(U) 2 Syl2(R), so each Ei is R-conjugate to a
subgroup E0isuch that CS(E0i) 2 Syl2(CR(E0i)). Hence after composing with appro*
*priate
elements of R B, we can assume that CS(Ei) 2 Syl2(CR(Ei)) for each i, and that
g-1iCS(Ei-1)gi= CS(Ei) for each i. The subgroups CS(Ei) are all FSol(3)-centric*
*, and
thus g defines an isomorphism in CLcSol(3)(Z) from CS(E1) to CS(E6).
Now, CS(E) is centric in both H and K. The easiest way to see this is to note*
* that it
contains a subgroup C42which is self-centralizing in K, and also in H = Spin7(Z*
*[1=2])
since its eigenspaces in (Z[1=2])7 are all 1-dimensional.
Let L = LcSol(3) for short, and set LH = CL(Z) and LK = NL(U). Let
JH :Mor (LH ) ----! H ~=ss1(|LH |)
JK :Mor (LK ) ----! K ~=ss1(|LK |)
JL :Mor (L) ----! ss1(|L|)
be the maps defined in Section 1. For each i, gi2 X where X = H or X = K depend*
*ing
on the parity of i, and cgilifts to some morphism fi2 IsoLX(CS(Ei-1), CS(Ei)). *
*Then
g-1iJX (fi) 2 CX (CS(Ei-1)) = Ei-1 since CS(Ei-1) is centric in X, and we can t*
*hus
THE SIMPLE CONNECTIVITY OF BSol(q) 25
choose fi such that gi= JX (fi). Hence
!(g) = !(g6) . .!.(g2) = !(JK (f6)).!(JH (f5)) . .!.(JK (f2)) = JL(f) 2 ss1*
*(|L|)
where f 2 IsoL(CS(E1), CS(E6)) is the composite of the fi. Since f centralizes *
*Z, it is
a morphism in LH , and thus !(g) = !(JH (f)) where JH (f) 2 H.
By Proposition 2.3, there are subgroups H0 H and K0 K such that H0=Z ~=
Sp6(2), [K:K0] = 3, and (H0 B0 K0) is an amalgam of maximal subgroups of Co*
*3.
In the terminology of Section 4, H0 is the stabilizer of a point in the geometr*
*y G, and
K0 is the stabilizer of a line. Set G0 = G.
__ ___ ___ __
Lemma 5.3. If G 6= 1, then H0 ~=H0, K0 ~=K0, and G0 ~=Co3.
Proof.The normalizer N0 in LcSol(3) of a rank four subgroup in B0 is an extensi*
*on of C42
by GL4(2), the stabilizer of a 3-space in G. In other words, ! defines a_homomo*
*rphism_
from the amalgam {H0, K0, N0} of stabilizers_of a complete flag in G to G , and*
* the
images of these subgroups generate G0. Since the colimit of this amalgam_is iso*
*morphic
to Co3_by Proposition 4.1, this defines a surjection_of Co3 onto G0. Since Co3 *
*is simple,
and G 06= 1 by Proposition 2.3 again, we have G 0~=Co3.
We are now ready to prove a special case of the main theorem.
Proposition 5.4. |LcSol(3)| is simply connected.
__
Proof.As we have already noted,_! is onto, and hence G ~= ss1(|LcSol(3)|)._Assu*
*me_by_
way of contradiction that_G_6= 1. In particular, by Lemma 5.3, G0 ~=Co3,_and_S *
* B
is a Sylow 2-subgroup of G0. We also identify U and Z as subgroups of G0 G.
We refer to [Fi, x4] for information about the involutions of Co3 and their n*
*ormalizers.
In particular, Co3 has two classes of involutions, of which those of type 2A ar*
*e in centers
of Sylow subgroups. Fix an involution o0 2 Co3 of type 2B. Then CCo3(o0) = L0x *
*
where L0~= M12. By well known properties of M12 (see Lemma 5.5 below), there are
elementary abelian subgroups Z0 U0 L0of rank one and two, such that AutL0(U0*
*) =
Aut(U0) and L0= . By [Fi, Lemma 5.1], AutCo3(V ) has order th*
*ree
for any 2B-pure fours subgroup V Co3, so the involutions in U0 must have type*
* 2A.
Since Co3 contains a unique conjugacy class of 2A-pure subgroup of rank 2 [Fi, *
*Lemma
~= __
5.8], there is an isomorphism fl : Co3 ---! G0 such that fl(U0) = U and fl(Z0) *
*= Z.
Furthermore, since C_S(U) is a Sylow subgroup of C__G0(U), we can choose fl to *
*send
_ 0 _
into C_S(U). Set o = fl(o ) 2 S , the image of some o 2 S G, and s*
*et
L = fl(L0). Thus C__G0(o) = L x , L ~=M12, and L = .
We now have
_______________ _____
L = = CG (o).
___
where_the second equality holds since H0 and K0 are sent isomorphically to H0 a*
*nd
K0. Since is G-conjugate to_Z_(all_involutions_in S_are_conjugate_in_G), CG*
* (o) is G-
conjugate to CG (Z), and hence CG (o)is G-conjugate to CG (Z)= H by Lemma 5.2. *
*In
particular, M12is contained in H=N for some subgroup N normal in H ~=Spin7(Z[1_*
*2]).
We claim that this is impossible. By a theorem of Margulis [M , Theorem 2.4.6*
*], the
only normal subgroups of H are those which contain congruence subgroups, and th*
*ose
which are contained in Z(H). By a theorem of Kneser [Kn , 11.1] (see also the "*
*Zusatz
26 ANDREW CHERMAK, BOB OLIVER, AND SERGEY SHPECTOROV
bei der Korrektur"), the "congruence kernel" of H = Spin7(Z[1_2]) is central, w*
*hich
implies that every normal subgroup of finite index contains the commutator subg*
*roup
of a congruence subgroup. If N Z(H), then clearly M12 is contained in 7(Z=n)
for some odd n. If H=N is finite, and N contains the commutator subgroup of the
congruence subgroup for nZ[1_2], then since M12 is not abelian, it must be cont*
*ained in
some quotient group of 7(Z=n). From this, using the simplicity of M12 again, *
*and
also the simplicity of the groups 7(p), one sees that M12 is isomorphic to a s*
*ubgroup
of 7(p) for some odd prime p.
Since M12 has no faithful irreducible (complex) characters of degree less tha*
*n 8 (cf.
[Frb, x5]), p must divide |M12|. The odd primes dividing |M12| are 3,5, and 11*
*. For
p = 3 and p = 5, one finds that | 7(p)| is not divisible by 11. Suppose p = 11*
*. We
note that Alt(6) is a subgroup of M12, and that the only irreducible complex ch*
*aracter
degrees for Alt(6) which are less than 8 are 1 and 5. Thus Alt(6) centralizes a*
* 2-space
in any orthogonal representation of M12on a space V of dimension 7 over F11. A *
*Sylow
3-subgroup of M12 is extraspecial of order 33, so Alt(6) contains a central 3-e*
*lement r
from M12. Then [V, r] admits a faithful action by a group of order 27. Since 27*
* doesn't
divide | 5(11)|, we have a contradiction; and this completes the proof of Propo*
*sition
5.4.
The following lemma was needed in the above proof.
Lemma 5.5. Set L ~=M12. Then there are elementary abelian subgroups Z U L
of ranks one and two, such that AutL (U) = Aut(U) and L = .
Proof. It is very well known (see [Co , p. 235]) that Z and U can be chosen suc*
*h that
both normalizers are maximal subgroups in L. However, since we know of no publi*
*shed
proof of this, we give the following short argument (where in fact, the subgrou*
*p U which
we take is not in the same conjugacy class as the one whose normalizer is maxim*
*al).
Let X be a set of order 12 upon which L acts 5-transitively [G2 , Theorem 6.1*
*8], and
let Y X be any subset of order 10. By [G2 , Exercise 6.25.2], the subgroup L0*
* L
of elements which stabilize Y is isomorphic to Aut(Alt(6)) _ an extension of Sy*
*m (6)
by an outer automorphism of order 2. Let Z U L00= [L0, L0] ~=A6 be elementa*
*ry
abelian 2-subgroups of rank one and two. (Note that AutL(U) = AutL00(U) = Aut(U*
*).)
The two subgroups U, U0 NL00(Z) ~=D8 isomorphic to C22are conjugate in L0, and
L00is generated by their normalizers. From this, it is clear that L0 .
By 5-transitivity, L0 is a maximal subgroup of L, and it remains only to show*
* that
NL(Z) or NL(U) contains elements of Lr L0. Since a Sylow 2-subgroup S0 of L is
not elementary abelian, Z(S0) contain elements which are squares in S0 L A1*
*2.
Since a product of three 4-cycles is an odd permutation, this shows that Z(S0) *
*contains
involutions which have fixed points on X; and thus that M10contains involutions*
* which
are central in Sylow subgroups of L. Since M10rA6 contains no involutions, and*
* A6
contains a unique class of involutions, this shows that for the subgroups Z con*
*structed
above, CL(Z) contains a Sylow 2-subgroup of L, and thus (by counting) is not co*
*ntained
in L0. This finishes the proof that L = .
We can now prove the main theorem.
Proof of Theorem 5.1.By Theorem 3.4, for any odd prime power q, |LcSol(q)| is h*
*omo-
topy equivalent to |LcSol(3m )| for some m 1. So it suffices to prove the the*
*orem when
q = 3m . When m = 1, this is Proposition 5.4.
THE SIMPLE CONNECTIVITY OF BSol(q) 27
Let S(3) S(3m ) be the Sylow subgroups of the linking systems LcSol(3) and
LcSol(3m ). Let o : S(3m ) ---! ss1(|LcSol(3m )|) be the homomorphism of Propos*
*ition 1.4,
and let o0 be the corresponding homomorphism defined on S(3). We claim there is*
* a
homomorphism from ss1(|LcSol(3)|) to ss1(|LcSol(3m )|) which makes the followin*
*g square
commute:
incl m
S(3) ________! S(3 )
| |
o0| o)|
# #
1=ss1(|LcSol(3)|)_!ss1(|LcSol(3m )|) .
This follows from [LO , Lemma 4.1] and from [AC , Theorem C], using two very di*
*fferent
approaches. Hence
m o c m
S(3) K def=KerS(3 ) ----! ss1(|LSol(3 )|) ,
and K is strongly closed in FSol(3m ) by Proposition 1.4(a). From the descript*
*ion in
Lemma 3.1 of S(3m ) and its fusion, this implies that K contains the subgroups *
*Ri in
S(3m ), hence the subgroup T S(3m ) (since R1R2R3\ T has index 2 in T ), and *
*hence
that K = S(3m ).
Thus o is the trivial homomorphism. So by Proposition 1.4(b), Out FSol(3m)(S(*
*3m ))
surjects onto ss1(|LcSol(3m )|). Also,
OutFSol(3m)(S(3m )) = OutSpin7(3m)(S(3m )),
since FSpin7(3m)(S(3m )) is the centralizer of an involution. By [LO , Proposit*
*ion 1.9] (or
by its proof), S(3m ) contains a unique subgroup R0 ~=(C2k)3 (where 2k is the l*
*argest
power dividing 3m 1), CS(3m)(R0) = R0, and AutSpin7(3m)(R0) ~=C2x Sym 4. So e*
*very
element in NSpin7(3m)(S(3m )) acts on R0 and on S(3m )=R0 with 2-power order; t*
*his
implies that Out Spin7(3m)(S(3m )) is a 2-group (hence trivial), and thus that *
*|LcSol(3m )|
is simply connected.
For any odd prime p, let LcSol(p1 ) be the category constructed in [LO , Sect*
*ion 4],
as a "linking system" associated to the union FSol(p1 ) of the fusion systems F*
*Sol(pm ).
By [LO , Proposition 4.3], |LcSol(p1 )|^2' BDI(4): the classifying space of the*
* exotic
2-compact group constructed by Dwyer and Wilkerson. We can now show:
Corollary 5.6. For any odd prime p, |LcSol(p1 )| is simply connected.
Proof.By the construction in [LO , Section 4], the linking category LcSol(p1 ) *
*is the
union of subcategories LccSol(pm ), whose nerves have the homotopy type of |LcS*
*ol(pm )|
[LO , Lemma 4.1], and hence are simply connected by Theorem 5.1. Thus |LcSol(p1*
* )| is
simply connected.
References
[A] M. Aschbacher, Overgroups of Sylow subgroups in sporadic groups, Memoir*
*s of the Amer.
Math. Soc. 60 (1986)
[AC] M. Aschbacher & A. Chermak, A group-theoretic approach to a family of 2*
*-local finite
groups constructed by Levi and Oliver, preprint
[AS] M. Aschbacher & S. Smith, The classification of quasithin groups: I. St*
*ructure of strongly
quasithin K-groups, Amer. Math. Soc. (2004)
28 ANDREW CHERMAK, BOB OLIVER, AND SERGEY SHPECTOROV
[Be] D. Benson, Cohomology of sporadic groups, finite loop spaces, and the D*
*ickson invariants,
Geometry and cohomology in group theory, London Math. Soc. Lecture note*
*s ser. 252,
Cambridge Univ. Press (1998), 10-23
[BCGLO1] C. Broto, N. Castellana, J. Grodal, R. Levi, & B. Oliver, Subgroup fam*
*ilies controlling
p-local finite groups, Proc. London Math. Soc. (to appear)
[BCGLO2] C. Broto, N. Castellana, J. Grodal, R. Levi, & B. Oliver, Extensions o*
*f p-local finite
groups, Trans. Amer. Math. Soc. (to appear)
[BLO1] C. Broto, R. Levi, & B. Oliver, Homotopy equivalences of p-completed cl*
*assifying spaces
of finite groups, Invent. math. 151 (2003), 611-664
[BLO2] C. Broto, R. Levi, & B. Oliver, The homotopy theory of fusion systems, *
*Journal Amer.
Math. Soc. 16 (2003), 779-856
[BM] C. Broto & J. Moller, Finite Chevalley versions of p-compact groups (pr*
*eprint)
[Bw] K. Brown, Buildings, Springer-Verlag (1989)
[Co] J. Conway, Three lectures on exceptional groups, Higman-Powell: Finite *
*simple groups,
Academic press (1971)
[DW] W. Dwyer & C. Wilkerson, A new finite loop space at the prime two, J. A*
*mer. Math. Soc.
6 (1993), 37-64
[Fi] L. Finkelstein, The maximal subgroups of Conway's group C3 and McLaughl*
*in's group, J.
Algebra 25 (1973), 58-89
[Frb] F. G. Frobenius, "Uber de Charaktere der mehrfach transitiven Gruppen, *
*Sitzungsberichte
K"onig. Preuss. Akad. Wiss. zu Berlin (1904), 558-571; Gesammelte Abhan*
*dlungen III, 335-
348
[GAP] The GAP Group, GAP _ Groups, Algorithms, and Programming, Version 4.4 (*
*2006);
package GRAPE (http://www.gap-system.org)
[Go] D. Goldschmidt, A conjugation family for finite groups, J. Algebra 16 (*
*1970), 138-142
[GLS] D. Gorenstein, R. Lyons, & R. Solomon, The classification of the finite*
* simple groups, nr.
3, Amer. Math. Soc. (1998)
[G] R. Griess, Sporadic groups, code loops, and nonvanishing cohomology, J.*
* Pure and Applied
Alg. 44 (1987), 191-214
[G2] R. Griess, Twelve sporadic groups, Springer-Verlag (1998)
[Gr] J. Grodal, Higher limits via subgroup complexes, Annals of Math. 155 (2*
*002), 405-457
[GO] J. Grodal & B. Oliver, Fundamental groups of linking systems, in prepar*
*ation
[Ka] W. Kantor, Some exceptional 2-adic buildings, J. Algebra 92 (1985), 208*
*-223
[Kn] M. Kneser, Normalteiler ganzzahliger Spingruppen, J. Reine Angew. Math.*
* 311/312 (1979),
191-214
[LO] R. Levi & B. Oliver, Construction of 2-local finite groups of a type st*
*udied by Solomon and
Benson, Geometry and Topology 6 (2002), 917-990
[M] G. Margulis, Finiteness of quotient groups of discrete subgroups, Funct*
*ional Analysis and
Its Applications 13 (1979), 178-187
[O1] B. Oliver, Equivalences of classifying spaces completed at the prime tw*
*o, Memoirs Amer.
Math. Soc. (to appear)
[Pn] A. Pasini, Diagram geometries, Oxford (1994)
[Pg] Ll. Puig, Unpublished notes (ca. 1990)
[Se] G. Segal, Classifying spaces and spectral sequences, Publ. Math. I.H.E.*
*S. 34 (1968), 105-
112
[So] R. Solomon, Finite groups with Sylow 2-subgroups of type .3, J. Algebra*
* 28 (1974), 182-198
[Sr] V. Srinivas, Algebraic K-theory, 2nd ed., Birkh"auser (1996)
[T] J. Tits, Ensembles ordonn'es, immeubles et sommes amalgam'ees, Bull. So*
*c. Math. Belg.
S'er. A, 38 (1986), 367-387.
THE SIMPLE CONNECTIVITY OF BSol(q) 29
Department of Mathematics, Kansas State University, Manhattan, KS 66502, USA
E-mail address: chermak@math.ksu.edu
LAGA, Institut Galil'ee, Av. J-B Cl'ement, 93430 Villetaneuse, France
E-mail address: bobol@math.univ-paris13.fr
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15
2TT, UK
E-mail address: s.shpectorov@bham.ac.uk
~~