DISCRETE MODELS FOR THE pLOCAL HOMOTOPY THEORY
OF COMPACT LIE GROUPS AND pCOMPACT GROUPS
CARLES BROTO, RAN LEVI, AND BOB OLIVER
Abstract. We define and study a certain class of spaces which includes p*
*completed
classifying spaces of compact Lie groups, classifying spaces of pcompact*
* groups,
and pcompleted classifying spaces of certain locally finite discrete gro*
*ups. These
spaces are determined by fusion and linking systems over "discrete ptora*
*l groups" _
extensions of (Z=p1 )r by finite pgroups _ in the same way that classify*
*ing spaces
of plocal finite groups as defined in [BLO2 ] are determined by fusion a*
*nd linking
systems over finite pgroups. We call these structures "plocal compact g*
*roups".
In our earlier paper [BLO2 ], we defined and studied a certain class of spac*
*es which
in many ways behave like pcompleted classifying spaces of finite groups. These*
* spaces
occur as "classifying spaces" of certain algebraic objects called plocal finit*
*e groups.
The purpose of this paper is to generalize the concept of plocal finite groups*
* to what
we call plocal compact groups. The motivation for introducing this family come*
*s from
the observation that pcompleted classifying spaces of finite and compact Lie g*
*roups,
as well as classifying spaces of pcompact groups [DW ], share many similar ho*
*motopy
theoretic properties, but earlier studies of these properties usually required *
*different
techniques for each case. Moreover, while pcompleted classifying spaces of fin*
*ite and,
more generally, compact Lie groups arise from the algebraic and geometric struc*
*ture
of the groups in question, pcompact groups are purely homotopy theoretic objec*
*ts.
Unfortunately, many of the techniques used in the study of pcompact groups fai*
*l for
pcompleted classifying spaces of general compact Lie groups. With the approach*
* pre
sented here, we propose a framework general enough to include pcompleted class*
*ifying
spaces of arbitrary compact Lie groups as well as pcompact groups.
The new idea here is to replace fusion systems over finite pgroups, as handl*
*ed in
[BLO2 ], by fusion systems over discrete ptoral groups. A discrete ptoral g*
*roup is a
group which contains a discrete ptorus (a group of the form (Z=p1 )r for finit*
*e r 0)
as a normal subgroup of ppower index. A plocal compact group consists of a tr*
*iple
(S, F, L), where S is a discrete ptoral group, F is a saturated fusion system *
*over S
(a collection of fusion data between subgroups of S arranged in the form of a c*
*ategory
and satisfying certain axioms), and L is a centric linking system associated to*
* F (a
category whose objects are a certain distinguished subcollection of the object *
*of F,
and of which the corresponding full subcategory of F is a quotient category). *
*The
linking system L allows us to define the classifying space of this plocal comp*
*act group
___________
1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 57T1*
*0.
Key words and phrases. Classifying space, pcompletion, fusion, compact Lie g*
*roups, pcompact
groups.
C. Broto is partially supported by MEC grant MTM200406686.
R. Levi is partially supported by EPSRC grant MA022 RGA0756.
B. Oliver is partially supported by UMR 7539 of the CNRS.
All three of the authors were partially supported by the MittagLeffler Insti*
*tute in Sweden, and
also by the EU grant nr. HPRNCT199900119.
1
2 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
to be the pcompleted nerve L^p. If S is a finite pgroup, then the theory re*
*duces to
the case of plocal finite groups as studied in [BLO2 ].
We hope that working with this setup will make it possible to prove results o*
*f in
terest in a uniform fashion for the entire family. In this paper (Theorem 7.1),*
* we give
a combinatorial description of the space of self equivalences of L^pin terms *
*of auto
morphisms of the category L, and a description of the group Out (L^p) of homo*
*topy
classes of self equivalences in terms of "fusion preserving automorphisms" of S*
*. We also
show that a plocal compact group (S, F, L) is determined up to isomorphism by *
*the
homotopy type of its classifying space L^p. One future goal is to show that t*
*he mod
p cohomology of the classifying space L^pof a plocal compact group (S, F, L)*
* can
always be described in terms of the fusion system F, as a ring of "stable eleme*
*nts" in
the cohomology of S. Other goals are to define connected plocal compact groups*
*, and
understand their properties and their relation to connected pcompact groups; a*
*nd to
characterize algebraically (connected) pcompact groups among all (connected) p*
*local
compact groups. Finally, a more general question which is still open is whethe*
*r the
pcompletion of the classifying space of every finite loop space is the classif*
*ying space
of a plocal compact group.
As one might expect, passing from a finite to an infinite setup introduces an*
* array of
problems one must deal with in order to produce a coherent theory. Some of the *
*basic
properties of fusion systems over discrete ptoral groups are analogous or even*
* identical
to the finite case, whereas other aspects are more delicate. Once the definiti*
*on of a
saturated fusion system over a discrete ptoral group is given and their basic *
*properties
are studied, one defines associated centric linking systems and plocal compact*
* groups
in a fashion more or less identical to the finite case. However, while in the *
*finite
case, any finite group G gives rise automatically to a saturated fusion system *
*and
an associated centric linking system, the corresponding construction for compac*
*t Lie
groups is less obvious. Similar complications present themselves when dealing w*
*ith the
fusion system and the centric linking system associated to a pcompact group. *
*It is
for that reason that the only aims of this paper are to establish the setup, st*
*udy some
basic properties, and prove that the classifying spaces which are the obvious c*
*andidates
to give rise to plocal compact groups indeed do so.
We proceed by describing the contents of the paper in some detail. In Sectio*
*n 1,
we define and list some properties of discrete ptoral groups. We show why this*
* class
of groups is a natural one to consider for our purposes, and study some of its *
*useful
properties. Then in Section 2, we define saturated fusion systems over discrete*
* ptoral
groups. The definitions in this section are very similar to those given in [BLO*
*2 ] for the
finite case, but some modifications are needed due to having given up finitenes*
*s.
Much of the work on plocal finite groups makes implicit use of the fact that*
* the
categories one works with are finite. If S is an infinite discrete ptoral grou*
*p, then any
fusion system over it will have infinitely many objects. In Section 3 we show t*
*hat any
saturated fusion system F over a discrete ptoral group S contains a full subca*
*tegory
with finitely many objects, which in the appropriate sense determines F complet*
*ely.
More precisely, we show that F contains only finitely many objects which are bo*
*th
centric and radical, and then prove the appropriate analog of Alperin's fusion *
*theorem.
The latter, roughly speaking, says that in a saturated fusion system, every mor*
*phism
can be factored into a sequence of morphisms each of which is the restriction o*
*f an
automorphism of a centric radical subgroup.
Carles Broto, Ran Levi, and Bob Oliver *
* 3
Linking systems associated to fusion systems over discrete ptoral groups are*
* defined
in Section 4. In fact, the definition is identical to that used when working ov*
*er a finite
pgroup, and the proof that the nerve L of a linking system is pgood is esse*
*ntially
identical to that in the finite case. The connection between linking systems as*
*sociated
to a given fusion system F and rigidifications of the homotopy functor P 7! BP *
*on
the orbit category Oc(F) is then studied.
Higher limits over the orbit category of a fusion system are investigated in *
*Section
5. We first describe how to reduce the general problem to one of higher limits*
* over
a finite subcategory, and then show how those can be computed with the help of *
*the
graded groups *( ; M) introduced in [JMO ]. These general results are then ap*
*plied
to prove the acyclicity of certain explicit functors whose higher limits appear*
* later as
obstruction groups.
Spaces of maps Map (BQ, L^p) are studied in Section 6, when Q is a discrete*
* ptoral
group and L^pis the classifying space of a plocal compact group, and the spa*
*ce of self
equivalences of L^pis handled in Section 7. In both cases, the descriptions w*
*e obtain
in this new situation (in Theorems 6.3 and 7.1) are the obvious generalizations*
* of those
obtained in [BLO2 ] for linking systems over finite pgroups. We also prove (T*
*heorem
7.4) that a plocal compact group is determined by the homotopy type of its cla*
*ssifying
space: if (S, F, L) and (S0, F0, L0) are plocal compact groups such that L^p*
*' L0^p,
then they are isomorphic as triples of groups and categories.
We finish with three sections of examples: certain infinite locally finite g*
*roups in
Section 8, including linear torsion groups; compact Lie groups in Section 9; an*
*d p
compact groups in Section 10. In all cases, we show that the groups in questio*
*n fit
into our theory: they have saturated fusion systems and associated linking syst*
*ems,
defined in a unique way (unique up to isomorphism at least), and the classifyin*
*g spaces
of the resulting plocal compact groups are homotopy equivalent to the pcomple*
*ted
classifying spaces of the groups in the usual sense.
The first and third authors would like to thank the University of Aberdeen fo*
*r it's
hospitality during several visits; in particular, this work began when we got t*
*ogether
there in January 2002. All three authors would like to thank the MittagLeffler*
* Insti
tute, where this work was finished over a period of several months.
1.Discrete ptoral groups
When attempting to generalize the theory of plocal finite groups to certain *
*infinite
groups, the first problem is to decide which groups should replace the finite p*
*groups
over which we studied fusion systems in [BLO2 ]. The following is the class of*
* groups
we have chosen for this purpose. Let Z=p1 ~=Z[1_p]=Z denote the union of the c*
*yclic
pgroups Z=pn under the obvious inclusions.
Definition 1.1. A discrete ptoral group is a group P , with normal subgroup P0*
* C P ,
such that P0 is isomorphic to a finite product of copies of Z=p1 , and P=P0 is *
*a finite
pgroup. The subgroup P0 will be called the identity component of P , and P wi*
*ll be
called connected if P = P0. Set ss0(P ) def=P=P0: the group of components of P .
The identity component P0 of a discrete ptoral group P can be characterized *
*as the
subset of all infinitely pdivisible elements in P , and also as the minimal su*
*bgroup of
4 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
finite index in P . Define rk(P ) = k if P0 ~=(Z=p1 )k, and set
P  def=(rk(P ), ss0(P )) = (rk(P ), P=P0) .
We regard the order of a discrete ptoral group as an element of N2 with the le*
*xico
graphical ordering. Thus P  P 0 if and only if rk(P ) < rk(P 0), or rk(P *
*) = rk(P 0)
and ss0(P ) ss0(P 0). In particular, P 0 P implies P 0 P , with eq*
*uality only if
P 0= P .
The obvious motivation for choosing this class is the role they play as "Sylo*
*w p
subgroups" in compact Lie groups and pcompact groups. But in fact, it seems d*
*if
ficult to construct fusion systems with interesting properties over any larger *
*class of
subgroups. The reason for this is that discrete ptoral groups are characteriz*
*ed by
certain finiteness properties, which are needed in order for fusion systems ove*
*r them
to be manageable, and for related homotopy theoretic phenomena to be controled *
*by
plocal information.
A group G is locally finite if every finitely generated subgroup of G is fini*
*te, and is a
locally finite pgroup if every finitely generated subgroup of G is a finite p*
*group. The
class of locally finite (p)groups is closed under subgroups and quotient group*
*s. It is
also closed under group extensions, since finite index subgroups of finitely ge*
*nerated
groups are again finitely generated.
A group G is artinian (satisfies the minimum condition in the terminology of *
*[W ]) if
every nonempty set of subgroups of G, partially ordered by inclusion, has a mi*
*nimal
element. Equivalently, G is artinian if its subgroups satisfy the descending ch*
*ain con
dition. The class of artinian groups is closed under taking subgroups, quotient*
*s, and
extensions. Every artinian group is a torsion group (since an infinite cyclic *
*group is
not artinian). If G is artinian and ' 2 Inj(G, G) is an injective endomorphism *
*of G,
then ' is an automorphism, since otherwise {'n(G)} would be an infinite descend*
*ing
chain. This is just one example of why it will be important that the groups we *
*work
with are artinian; the descending chain condition will be used in other ways la*
*ter.
It is an open question whether every artinian group is locally finite (see [K*
*W , pp.
3132] for a discussion of this). If one restricts attention to groups all of w*
*hose elements
have ppower order for some fixed prime p, then artinian groups are known to be*
* locally
finite if p = 2 [KW , Theorem 1.F.6], but this seems to be unknown for odd pri*
*mes.
However, any counterexample to these questions would probably be far too wild f*
*or
our purposes. Hence it is natural to restrict attention to locally finite group*
*s, and since
we are working with local structure at a prime p, to locally finite pgroups. T*
*he next
proposition tells us that in fact, this restricts us to the class of discrete p*
*toral groups.
It is included only as a way to help motivate this choice of groups to work wit*
*h.
Proposition 1.2. A group is a discrete ptoral group if and only if it is artin*
*ian and
a locally finite pgroup.
Proof.The group Z=p1 is clearly a locally finite pgroup and artinian. Since b*
*oth of
these properties are preserved under extensions of groups, they are satisfied b*
*y every
discrete ptoral group.
Conversely, assume that G is artinian and a locally finite pgroup. By [KW ,*
* Theorem
5.8], every locally finite artinian group is a ~Cernikov group; in particular, *
*it contains a
normal abelian subgroup with finite index. By [Fu , Theorems 25.1 & 3.1], every*
* abelian
artinian group is a finite product of groups of the form Z=qm where q is a prim*
*e and
Carles Broto, Ran Levi, and Bob Oliver *
* 5
m 1. Thus G is an extension of the form
1 ! A ! G ! ss ! 1,
where ss is a finite pgroup, and A is a finite product of groups Z=pm for m *
*1. The
subgroup of A generated by the factors Z=p1 is the subgroup of infinitely pdi*
*visible
elements, thus a characteristic subgroup of A, and a normal subgroup of G of p*
*power
index. It follows that G is a discrete ptoral group.
We next note some of the other properties which make discrete ptoral groups *
*con
venient to work with.
Lemma 1.3. Any subgroup or quotient group of a discrete ptoral group is a disc*
*rete
ptoral group. Any extension of one discrete ptoral group by another is a dis*
*crete
ptoral group.
Proof.These statements are easily checked directly. They also follow at once f*
*rom
Proposition 1.2, since the classes of locally finite pgroups and artinian grou*
*ps are both
closed under these operations.
Clearly, the main difficulty when working with infinite discrete ptoral grou*
*ps, in
stead of finite pgroups, is that they have infinitely many subgroups and infin*
*ite au
tomorphism groups. We next investigate what finiteness properties these groups*
* do
have.
Lemma 1.4. The following hold for each discrete ptoral group P .
(a)For each n 0, P contains finitely many conjugacy classes of subgroups of o*
*rder
pn.
(b)P contains finitely many conjugacy classes of elementary abelian psubgroups.
Proof.Clearly, for each n, P0 contains finitely many subgroups of order pn, sin*
*ce they
are all contained inside the pntorsion subgroup of P0 which is finite. So to p*
*rove (a),
it suffices, for each finite subgroup A P0 and each subgroup B = eB=P0 P=P0*
*, to
show that there are finitely many P conjugacy classes of subgroups Q P such *
*that
Q \ P0 = A and QP0 = eB. Let Q be the set of all such subgroups, and assume Q 6*
*= ?.
Then Q 2 Q if and only if Q=A \ P0=A = 1 and QP0=P0 = B; and this implies that
A C QP0 = eBand that Q=A is the image of a splitting of the extension
1 ___! P0=A ___! eB=A ___! B ___! 1.
In other words, Q is in onetoone correspondence with the set of splittings of*
* this
extension. The set of P0conjugacy classes of such splittings (if there are an*
*y) is in
onetoone correspondence with the elements of H1(B; P0=A) (see [Bw , Propositi*
*on
IV.2.3]). Since this cohomology group is finite, so is the set of conjugacy cl*
*asses of
such extensions.
This proves point (a). Point (b) follows from (a), together with the observat*
*ion that
for any elementary abelian subgroup E P , rk(E) rk(P ) + rkp(P=P0).
We next check what can be said about finiteness in automorphism groups.
Proposition 1.5. Let P be a discrete ptoral group.
(a)Any torsion subgroup of Aut (P ) is an extension of an abelian group by a fi*
*nite
group.
6 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
(b)Any torsion subgroup of Out(P ) is finite.
(c)For each Q P , OutP (Q) is a finite pgroup.
Proof.Assume first that P ~=(Z=p1 )r: a discrete ptorus of rank r 0. Then
Aut(P ) ~=GLr(bZp), and it is well known that the subgroup (1 + p2Mr(bZp))x of *
*ma
trices which are congruent modulo p2 to the identity is torsion free. This foll*
*ows, for
example, from the inverse bijections
log
(1 + p2Mr(bZp))x !p2Mr(bZp)
exp
defined by the usual power series: while logis not a homomorphism, it does sati*
*sfy the
relation log(Xr) = r log(X). So if H is a torsion subgroup of Aut(P ) (equivale*
*ntly, of
GLr(bZp)), then the composite
=p2 2
H ! GLr(bZp) ! GLr(Z=p )
is injective, and thus H is finite.
Now let P be an arbitrary discrete ptoral group with connected component P0 *
*and
group of components ss = P=P0. There is an exact sequence
0 ! H1(ss; P0) ! Aut(P )= AutP0(P ) ! Aut(P0) x Aut(ss)
(cf. [Sz, 2.8.7]), where Aut P0(P ) = {cx 2 Aut(P )  x 2 P0}. We have just s*
*een that
every torsion subgroup of Aut(P0) is finite, and H1(ss; P0) and Aut(ss) are cle*
*arly finite.
Hence every torsion subgroup of Aut(P )= InnP0(P ) is finite. This proves (b); *
*and also
proves (a) (every torsion subgroup of Aut(P ) is an extension of an abelian gro*
*up by a
finite group) since AutP0(P ) is abelian. Point (c) follows immediately from (b*
*), since
P is a torsion group all of whose elements have ppower order.
In the next section (in Definition 2.2), we will need some more precise bound*
*s on
the size of normalizers and centralizers.
Lemma 1.6. Let S be any discrete ptoral group, and set N = ss0(S)rk(S)+1. T*
*hen
for all P S,
ss0(CS(P )) N, ss0(NS(P )=P ) N, and ss0(NS(P )) N.ss*
*0(P ).
Proof.Set T = S0 for short,Qand set Q = P T=T . Let NQ :T ___! T be the norm m*
*ap
for the Q action: NQ(x) = gT2Qgxg1. The image of NQ is connected and central*
*izes
P , and thus Im(NQ) CS(P )0 = CT(P )0. If x 2 CT(P ), then
xQ= NQ(x) 2 CT(P )0.
Thus every element in CT(P )=CT(P )0 has order dividing Q, and it follows that
ss0(CT(P )) = CT(P )=CT(P )0 Qrk(S) ss0(S)rk(S).
Thus ss0(CS(P )) ss0(CT(P )).S=T  N.
If x 2 NT(P ), then
Y
xQ= NQ(x). [x, g] 2 CT(P )0.P NT(P )0.P.
gT2Q
Thus fi ffi fi
fiNT(P ) NT(P )0.(T \ P ) fi Qrk(S) ss0(S)rk(S),
and hence ss0(NS(P )=P ) N, by the same arguments as those used for ss0(CS(*
*P )).
The last inequality is now immediate.
Carles Broto, Ran Levi, and Bob Oliver *
* 7
Note that discrete ptoral groups are all solvable, but (in contrast to finit*
*e pgroups)
need not be nilpotent. For instance, the infinite dihedral group, a split exte*
*nsion of
Z=21 by Z=2, is a discrete 2toral group which is not nilpotent (since the nil*
*potency
class of D2n is n  1).
The following lemma contains some generalizations of a standard theorem about
automorphisms of finite pgroups: if ff 2 Aut (P ) is the identity on Q C P an*
*d on
P=Q, then it has ppower order.
Lemma 1.7. The following hold for any discrete ptoral group P and any automor
phism ff 2 Aut(P ).
(a)Assume, for some Q C P , that ffQ = IdQ and ff Id (mod Q). Then every
fforbit in P is finite of ppower order. If, in addition, [P : Q] < 1, then*
* ff has
finite order.
(b)ff has finite order if and only if ffP0 has finite order.
(c)Set P(1)= {g 2 P0 gp = 1}. If ffP(1)= Id and ff Id (mod P0), then each o*
*rbit
of ff acting on P has ppower order.
Proof.(a) The proof is identical to the proof for finite pgroups (see [Go , T*
*heorem
5.3.2]), and in fact applies whenever all elements of Q have ppower order. Fo*
*r any
g 2 P , ff(g) = gx for some x 2 Q (since ff Id(mod Q)), and ff(x) = x since f*
*fQ = Id.
k k *
* i
Thus ffn(g) = gxn for all n, and ffp (g) = g if p = x. Since the order of {*
*ff (g)}
depends only on the coset gQ, this also shows that ff is finite (and a power *
*of p) if
P=Q is finite.
(b) If ffP0 has finite order, then there is n 1 such that ffnP0 = Id and f*
*fn Id
(mod P0). Then ffn has finite order by (a), so ff also has finite order.
(c) For each m 1, let P(m) P0 be the pm torsion in P0. Fix g 2 P , and s*
*et
x = g1ff(g), pk = x, and Q = . The P(m)are all ffinvariant, and so*
* Q is also
ffinvariant since g1ff(g) 2 P(k). Also, ff acts via the identity on P(1)by as*
*sumption,
hence on P(i)=P(i1)for all 1 i k, and also on Q=P(k). So by (a) (and sinc*
*e Q
is a finite group), ffQ has ppower order. In particular, the fforbit of g ha*
*s ppower
order.
The next lemma is another easy generalization of a standard result about fini*
*te
pgroups.
Lemma 1.8. If P Q are distinct discrete ptoral groups, then P NQ(P ).
Proof.When [Q:P ] < 1, this follows by the same proof as for finite pgroups. M*
*ore
precisely, when Q=P is finite, the action of P on Q=P (defined by x(gP ) = xgP *
*for
x 2 P and g 2 Q) factors through a finite quotient group P=N of P . Also, P=N i*
*s a
pgroup since P is a ptorsion group. Thus
NQ(P )=P  = (Q=P )P=N Q=P  0 (mod p),
and so NQ(P )=P 6= 1.
Now assumenthat [Q:P ] is infinite; i.e., that P0 Q0. For each n, set An = *
*{x 2
Q0 xp = 1}. Then An C Q, and in particular is normalized by P . For n large
enough, An P , so P P An Q, P NPAn(P ) since [P An:P ] < 1, and thus
P NQ(P ).
8 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
We will also need the following well known result about finite subgroups of d*
*iscrete
ptoral groups.
Lemma 1.9. For any discrete ptoral group P , there is a finite subgroup Q P
such that P = QP0. There is also anSincreasing sequence Q1 Q2 Q3 . . .of
finite subgroups of P such that P = 1n=1Qn. More generally, for any finite su*
*bgroup
K Aut(P ), the Qi can be chosen to be Kinvariant.
Proof.Fix any (finite) set X of coset representatives for P0 in P , and set Q =*
* . Then Q is Kinvariant, Q is finite since P is locally finite, and P *
*= QP0
by construction. For each n 1, let Pn P0 be the pntorsionSsubgroup, and s*
*et
Qn = QPn. Then the Qn are also finite and Kinvariant, and P = 1n=1Qn.
To finish the section, we consider maps between the pcompleted classifying s*
*paces
of discrete ptoral groups. This following lemma is implicit in [DW ] and [DW2*
* ] (the
spaces in question are classifying spaces of pcompact groups). But it does not*
* seem
to be stated explicitly anywhere there.
Lemma 1.10. For any pair P, Q of discrete ptoral groups,
B :Rep (P, Q) ! [BP ^p, BQ^p]
is a bijection. In particular, any homotopy equivalence BP ^p'!BQ^pis induced*
* by an
isomorphism P ~= Q. Also, for any homomorphism ae: P ! Q, the homomorphism
(incl,ae)
CQ(ae(P )) x P ! Q induces a homotopy equivalence
BCQ(ae(P ))^p'!Map (BP ^p, BQ^p)Bae.
Proof.For any pair G, H of discrete groups,
[BG, BH] ~=Rep (G, H) and Map (BG, BH)Bae' BCH (ae(G))
for each ae 2 Hom (G, H). See, for example, [BKi , Proposition 7.1] for a proof.
By [DW2 , Proposition 3.1], the homotopy fiber of the map BQ ! BQ^pis a K(*
*V, 1)
for some bQpvector space V . Using this, together with standard obstruction th*
*eory and
the fact that eH*(BQ; Q) = 0, one checks that
[BP ^p, BQ^p] ~=[BP, BQ] ~=Rep (P, Q).
Now fix some ae 2 Hom (P, Q). By [DW , Propositions 5.1 & 6.22], Map (BP ^p,*
* BQ^p)Bae
is the classifying space of some pcompact group X, and in particular is pcomp*
*lete.
Since Map (BP, BQ)Bae' CQ(ae(P )) (P and Q are both discrete), we will be done *
*upon
showing that the completion map
Map (BP, BQ)Bae! Map (BP, BQ^p)Bae (1)
is a mod p homology equivalence.
Fix a sequence of finite subgroups P1 P2 . . .whose union is P . Since Q*
* is
artinian, CQ(ae(Pn)) = CQ(ae(P )) for n sufficiently large. Also, Map (BP, BQ^p*
*)Baeis the
homotopy inverse limit of the mapping spaces Map (BPn, BQ^p)Bae. So if (1) is a*
* mod
p equivalence upon replacing P by Pn for each n, it is also a mod p equivalence*
* for P .
In other words, it suffices to prove this when P is a finite pgroup.
Let X be the homotopy fiber of the completion map BQ ! BQ^p. As noted abov*
*e,
X is a K(V, 1) where V is a rational vector space. Since the map from Map (BP, *
*BQ)
Carles Broto, Ran Levi, and Bob Oliver *
* 9
to Map (BP, BQ^p) is a bijection on components, the homotopy fiber of the map i*
*n (1)
is XhP for a proxy action of P on X (in the sense of [DW ]) induced by ae.
Consider the fibration sequence
prO
XhP ! Map (BP, XhP){1}! Map (BP, BP )Id,
where prdenotes the projection of XhP to BP , and the total space is the set of*
* all maps
f :BP ! XhP such that prOf ' Id. Since XhP is the total space of a fibratio*
*n over
BP with fiber X, it is a K(ss, 1) where V C ss and ss=V ~=P . Since P is a fini*
*te pgroup
and V is a rational vector space, this extension splits, and the splitting is u*
*nique up
to conjugacy by elements of V .
It follows that
[BP, XhP]! [BP, BP ]
~=Rep(P,ss) ~=Rep(P,P)
is a bijection. Also, the induced map
ss1(Map (BP, XhP){1})!ss1(Map (BP, BP )Id)
~=Css(P) ~=Z(P)
is surjective, and its kernel V P (where the action of P on V is induced by the*
* action
on X) is a rational vector space.
Thus XhP ' K(V P, 1). It follows that XhP is mod p acyclic, and hence that (1*
*) is
a mod p equivalence. This finishes the proof.
2.Fusion systems over discrete ptoral groups
We now define saturated fusion systems over dicrete ptoral groups and study *
*their
basic properties. The definitions are almost identical to those in the finite c*
*ase ([BLO2 ,
x1]).
Definition 2.1. A fusion system F over a discrete ptoral group S is a category*
* whose
objects are the subgroups of S, and whose morphism sets Hom F (P, Q) satisfy th*
*e fol
lowing conditions:
(a)Hom S(P, Q) Hom F(P, Q) Inj(P, Q) for all P, Q S.
(b)Every morphism in F factors as an isomorphism in F followed by an inclusion.
Two subgroups P, P 0 S are called Fconjugate if IsoF(P, P 0) 6= ?.
Definition 2.2. Let F be a fusion system over a discrete ptoral group S.
o A subgroup P S is fully centralized in F if CS(P ) CS(P 0) for all P *
*0 S
which is Fconjugate to P .
o A subgroup P S is fully normalized in F if NS(P ) NS(P 0) for all P 0*
* S
which is Fconjugate to P .
o F is a saturated fusion system if the following three conditions hold:
(I)For each P S which is fully normalized in F, P is fully centralized in *
*F,
Out F(P ) is finite, and OutS (P ) 2 Sylp(Out F(P )).
10 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
(II)If P S and ' 2 Hom F(P, S) are such that '(P ) is fully centralized, an*
*d if we
set
N' = {g 2 NS(P )  'cg'1 2 AutS('(P ))},
_ _
then there is ' 2 Hom F(N', S) such that 'P = '.
(III)If P1S P2 P3 . . .is an increasing sequence of subgroups of S, with
P1 = 1n=1Pn, and if ' 2 Hom (P1 , S) is any homomorphism such that 'Pn 2
Hom F(Pn, S) for all n, then ' 2 Hom F(P1 , S).
By Lemma 1.6, there is a global upper bound for ss0(CS(P )) and ss0(NS(P )*
*), taken
over all subgroups P of any given S. In particular, for any given subgroup P *
* S,
CS(P 0) and NS(P 0) take on maximal values among all P 0which are Fconjuga*
*te
to P . This proves that the conjugacy class of P always contains fully centra*
*lized
subgroups and fully normalized subgroups.
It is very convenient, in the above definition, to be working with a class of*
* groups
where the concept of "order" of subgroups is defined. However, there are other *
*ways to
define fully normalized and fully centralized subgroups in a fusion system, and*
* hence
to define saturation; and this property was not a factor in our decision to res*
*trict
attention to fusion systems over discrete ptoral groups. The crucial propertie*
*s of these
groups, which seem to be needed frequently when developing the theory, are that*
* they
are artinian and locally finite.
When F is a saturated fusion system over the discrete ptoral subgroup S, the*
*n by
(I), OutF (P ) = AutF (P )= Inn(P ) is finite for fully normalized P S, and h*
*ence for all
P S. Since Inn(P ) is discrete ptoral (being a quotient group of P ), AutF (*
*P ) inherits
many of the properties of discrete ptoral groups. In particular, it is artinia*
*n, locally
finite, and contains a unique conjugacy class of maximal discrete ptoral subgr*
*oups.
This condition that OutF (P ) be finite does simplify slightly the definition o*
*f a saturated
fusion system, but it is in fact unnecessary, as is shown by the following prop*
*osition.
Proposition 2.3. Let F be a fusion system over the discrete ptoral group S. As*
*sume
that axiom (II) in Definition 2.2 holds, and that (I) holds for all finite full*
*y normalized
subgroups of S. Then OutF (P ) is finite for all P S.
m
Proof.Fix P S. For all m 1, set P(m) = {g 2 P0 gp = 1}. By Proposition
1.5(b), to show that Out F(P ) is finite, it suffices to show that Aut F(P ) is*
* a torsion
group.
Fix ff 2 Aut F(P ). We want to show that ff has finite order; by Lemma 1.7(b*
*),
it suffices to do this when P = P0 is connected. After replacing ff by ffn for*
* some
appropriate n 1, we can assume that ffP(1)= Id. Then by Lemma 1.7(c), ffm d*
*ef=
ffP(m)has ppower order for all m. For each m, there is 'm 2 Hom F (P(m), S)*
* such
that 'm (P(m)) is fully normalized, and by (I), 'm (P(m)) is fully centralized,*
* and 'm
can_be chosen such that 'm ffm '1m2 AutS ('m (P(m))). Also, 'm can be extended*
* to
'm 2 Hom F(S0, S) by (II), so 'm (P(m)) S0, and hence  AutS('m (P(m))) S*
*=S0.
Thus (ffm )S=S0= IdP(m)for each m, so ffS=S0= IdP, and ff has finite order.
In fact, one can show that in the definition of a saturated fusion system, it*
* suffices
to require that (I) holds for all finite fully normalized subgroups P S; it t*
*hen follows
that (I) holds for all fully normalized subgroups.
When F is a (saturated) fusion system over a discrete ptoral group S, we thi*
*nk of
the identity component S0 as the "maximal torus" of the fusion system, and thin*
*k of
Carles Broto, Ran Levi, and Bob Oliver *
* 11
AutF (S0) as its "Weyl group". The following lemma describes how morphisms betw*
*een
subgroups of the maximal torus are controlled by the Weyl group.
Lemma 2.4. Let F be a saturated fusion system over a discrete ptoral group S w*
*ith
connected component T = S0. Then the following hold for all P T .
(a)For every P 0 S which is Fconjugate to P and fully centralized in F, P 0 *
*T ,
and there exists some w 2 AutF (T ) such that wP 2 IsoF(P, P 0).
(b)Every ' 2 Hom F(P, T ) is the restriction of some w 2 AutF (T ).
Proof.We first prove the following statement.
(c)For each ' 2 Hom F (P, S) such that P 0def='(P ) is fully centralized in F, *
*there
exists w 2 AutF (T ) such that wP = '.
_
By assumption, P T CS(P_). By condition_(II) in Definition 2.2, there is '*
* 2
Hom F(CS(P ), S) such that 'P = '. Then '(T ) T since T is connected (infini*
*tely
_ def_
pdivisible), and so '(T ) = T since T is artinian. Thus w = 'T 2 AutF (T ) i*
*s such
that wP = '. This proves (c), and also proves (a) since P 0= w(P ) T .
Now fix any ' 2 Hom F (P, T ). Let Q be a fully centralized subgroup of S in*
* the
Fconjugacy class of P and '(P ), and choose _ 2 IsoF('(P ), Q). By (c), there*
* are
elements u, v 2 AutF (T ) such that uP = _ O' and v'(P)= _. So if we set w = *
*v1u,
then wP = '.
By Proposition 2.3, Out F(P ) is finite for every subgroup P S. The follo*
*wing
lemma extends this statement.
Lemma 2.5. Let F be a saturated fusion system over a discrete ptoral group S. *
*Then
for all P, Q S, the set Rep F(P, Q) def=Inn(Q)\ Hom F(P, Q) is finite.
Proof.As just noted, OutF (P ) is finite for all P S. Also, if ', '02 Hom F(P*
*, Q) and
Im(') = Im('0), then '0= ' Off for some ff 2 AutF (P ) by condition (b) in Defi*
*nition
2.1. So there is a bijection
~= 0 fi 0 ffi
Rep F(P, Q)= OutF(P ) ! P Q fiP Fconjugate to P (Qconjugacy),(1)
which sends the class of a homomorphism to the conjugacy class of its image.
By Lemma 2.4, the Fconjugacy class (P0) of P0 is just its orbit under the ac*
*tion
of Aut F(S0), and hence a finite set. By Lemma 1.4(a), for any given Q 2 (P0),
there are only finitely many NS(Q)=Qconjugacy classes of subgroups of order P*
*=P0
in NS(Q)=Q. Hence there are only finitely many Sconjugacy classes of subgroups
P 0 S which are Fconjugate to P and such that P00= Q. This shows that the tar*
*get
set in (1) is finite, and hence that RepF (P, Q) is also finite.
The definitions of centric and radical subgroups in a fusion system over a di*
*screte
ptoral group are essentially the same as those in the finite case.
Definition 2.6. Let F be a fusion system over a discrete ptoral group S. A sub*
*group
P S is called Fcentric if P and all its Fconjugates contain their Scentral*
*izers. A
subgroup P S is called Fradical if Op(Out F(P )) = 1; i.e., if Out F(P ) con*
*tains no
nontrivial normal psubgroup.
12 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
Notice that any Fcentric subgroup is fully centralized. Conversely, if P S*
* is fully
centralized and centric in S; that is, Z(P ) = CS(P ), then it is Fcentric. T*
*he next
proposition says that the set of Fcentric subgroups is closed under overgroups.
Proposition 2.7. Let F be a saturated fusion system over the discrete ptoral g*
*roup
S, and let P Q S be such that P is Fcentric. Then Q is also Fcentric.
Proof.Fix any Q0 which is Fconjugate to Q, choose ' 2 IsoF(Q, Q0), and set P 0=
'(P ). Then
CS(Q0) CS(P 0) P 0 Q0,
where the second inequality holds since P is Fcentric. So Q is also Fcentric.
The next proposition gives another important property of Fcentric subgroups;*
* one
which is much less obvious.
Proposition 2.8. Let F be a saturated fusion system over the discrete ptoral g*
*roup
S. Then for each P Q S such that P is Fcentric, and each ', '02 Hom F(Q, S)
such that 'P = '0P, there is some g 2 Z(P ) such that ' = '0Ocg.
Proof.The hypothesis implies that ' O'01'0(P)= Id'0(P), and we must show that
' O'01= Id'0(Q). It thus suffices to prove, for P Q S and ' 2 Hom F(Q, S) *
*where
P is Fcentric, that 'P = IdP implies ' = cg for some g 2 Z(P ).
Assume first that P C Q. Then for each x 2 Q, c'(x)P = cxP. Thus '(x) x (*
*mod
CS(P )), and CS(P ) P since P is Fcentric. In particular, this shows that '(*
*Q) = Q,
and thus that ' 2 AutF (Q). It also shows that ' induces the identity on Q=P . *
*Since
Q=P has finite order, ' has ppower order by Lemma 1.7(a).
Without loss of generality, we can replace Q by any other subgroup in its F
conjugacy class. In particular, we can assume that Q is fully normalized, and *
*hence
that Out S(Q) 2 Sylp(Out F(Q)). So every psubgroup of Aut F(Q) is conjugate t*
*o a
subgroup of AutS(Q). Thus there is O 2 AutF (Q) such that O O' OO1 = cy for so*
*me
y 2 NS(Q). Since 'P = IdP, cy acts as the identity on '(P ), which is also Fc*
*entric,
hence y 2 CS('(P )) = '(Z(P )). Set x = O1(y); then ' = cx.
Now assume P is not normal in Q. Let Q be the set of subgroups Q0 Q containi*
*ng
P such that 'Q0 = cgQ0 for some g 2 Z(P ). If P Q0 Q and Q0 2 Q, then
NQ(Q0) Q0 by Lemma 1.8, and NQ(Q0) 2 Q since the proposition holds for the
normal pair Q0C NQ(Q0). Hence if Q contains a maximal element, it must be Q its*
*elf.
S 1
Let Q1 Q2 . . .be any increasing chain in Q, and set Q1 = n=1Qn. Let
gn 2 Z(P ) be such that 'Qn = cgnQn. Since P is Fcentric, so are the Qn, and*
* thus
Z(Q1) Z(Q2) . .i.s a decreasing sequence of subgroups. Since S is artinian,*
* there
is some k such that Z(Qn) = Z(Qk) for all n k. This shows that gn gk (mod
Z(Qk)) for all n k, hence that 'Q1 = cgkQ1 , and hence that Q1 2 Q. Thus
by Zorn's lemma, Q contains a maximal element, so Q 2 Q, and this finishes the
proof.
3.A finite retract of a saturated fusion system
A fusion system F over a discrete ptoral group S generally has infinitely ma*
*ny
isomorphism classes of objects. In this section, we construct a subcategory Fo*
* of
F with only finitely many isomorphism classes of objects, together with a retra*
*ction
Carles Broto, Ran Levi, and Bob Oliver *
* 13
functor from F to Fo which is a left adjoint to the inclusion. This means that*
* in
many cases, it will suffice to work over the "finite" subcategory Fo rather tha*
*n the full
fusion system F. As a first application, we show that Ob (Fo) contains all Fce*
*ntric
Fradical subgroups, and hence that there are only finitely many conjugacy clas*
*ses
of such subgroups. A second application is Alperin's fusion theorem in this se*
*tting:
restriction to Fo allows us to repeat the same inductive argument as that used *
*for
fusion systems over a finite pgroup.
Following the group theorists' usual notation, whenever is a group of autom*
*or
phisms of a group G and H G, we write
C (H) = {fl 2  flH = IdH}.
The following definitions were motivated by some constructions of Benson [Be ],*
* which
he in fact used to prove a version of Alperin's fusion theorem for compact Lie *
*groups.
Definition 3.1. Let F be a saturated fusion system over a discrete ptoral grou*
*p S, let
T = S0 be the identity component of S, and set W = AutF (T ) = OutF (T ) (the "*
*Weyl
group"). Set
k
pm = exp(S=T ) def=min{pk xp 2 T 8 x 2 S}.
(a)For each P T , set
fi
I(P ) = T CW (P)= t 2 T fiw(t) = t 8 w 2 W such thatwP = IdP ;
and let I(P )0 be the identity component of I(P ).
m
(b)For each P S, let P [m]= T , and set
P o= P .I(P [m])0 def={gt  g 2 P, t 2 I(P [m])0}.
(c)Set
H(F) = {I(P )0 P T }, and Ho(F) = {P o P S};
and let Fo F be the full subcategory with object set Ho(F).
Thus for P T , I(P ) is the maximal subgroup of T such that for all w 2 W ,*
* wP =
Idif and only wI(P)= Id. In particular, for all v and w in W , vP = wP if an*
*d only if
vI(P)= wI(P). Together with Lemma 2.4(b), this implies that every ' 2 Hom F(P*
*, T )
extends to a unique I(') 2 Hom F(I(P ), T ), which is obtained by first extendi*
*ng ' to
T and then restricting to I(P ). In other words, every Fisomorphism ': P ___!*
* Q
between subgroups of T extends to a unique Fisomorphism I('): I(P ) ! I(Q).
For an arbitrary subgroup P S, P [m]is a subgroup of T , and the above argu*
*ments
apply. Since P [m]C P , any x 2 P normalizes P [m], and hence also normalizes I*
*(P [m]).
Thus P normalizes I(P [m])0, and this shows that the subset P odef=P .I(P [m])0*
* is a
group.
More generally, for any k m, we could define subgroups P ok P for each P *
* S
by setting P ok= P .I(P [k]). This can be different from P o, but P 7! P okhas*
* all of
the same properties which we prove here for P o. However, the only way in which
this generalization might be needed would be if we wanted to compare these "bul*
*let
functors" for two different fusion systems over two different discrete ptoral *
*groups,
and that will not be needed in this paper.
Lemma 3.2. The following hold for every saturated fusion system F over a discre*
*te
ptoral group S.
14 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
(a)The set H(F) is finite, and the set Ho(F) contains finitely many Sconjugacy
classes of subgroups of S.
(b)For all P S, (P o)o = P o.
(c)If P Q S, then P o Qo.
(d)If P S is Fcentric, then Z(P o) = Z(P ).
Proof.Let T = S0 C S be the identity component, and set W = AutF (T ) and pm =
exp(S=T ). Note that for any P Q T , CW (P ) CW (Q), and hence I(P ) I(*
*Q).
Also, CW (I(P )) = CW (P ) by definition, and hence I(I(P )) = I(P ).
(a) By definition, each subgroup in H(F) has the form I(P )0 = (T K)0 for some
P T , where K = CW (P ) W . Since the finite group W = Out F(T ) has a
finite number of subgroups, this shows that H(F) is finite. Also, for any P *
* S,
P0 P [m] I(P [m]), and so (P o)0 = I(P [m])0 2 H(F). In particular, there ar*
*e only
finitely many possibilities for identity components of subgroups in Ho(F).
Fix P S, and set K = CW (P [m]). Since P [m]is generated by all pm powers *
*in P
(and pm = exp(S=T )), P [m] T and
[P :P [m]] = [P :(P \T )].[(P \T ):P [m]] S=T .pm.rk(T).
Here, the last inequality holds since (P \ T )=P [m]is abelian with exponent at*
* most pm
and rank at most rk(T ). Also, since P [m].I(P [m])0 = P [m].(T K)0 T K,
ss0(P o) = ss0(P .(T K)0) ss0(P [m].(T K)0).P=P [m]
ss0(T K).P=P [m] ss0(T K).S=T .pm.*
*rk(T).
We have already seen that (T K)0 is the identity component of P o, and we have *
*just
shown that the number of components of P ois bounded by an integer which depends
only on K (and on S). Since NS((T K)0)=(T K)0 has only finitely many conjugacy
classes of finite subgroups of any given order (Lemma 1.4(a)), this shows that *
*there are
only finitely many conjugacy classes of subgroups in Ho(F) corresponding to any*
* given
K W ; and thus (since W is finite) only finitely many conjugacy classes of su*
*bgroups
in Ho(F).
(b) Fix P S. Since P normalizes I(P [m])0, for any g 2 P and any x 2 I(P [m]*
*)0,
m pm [m] [m] [m]
(gx)p 2 g .I(P )0 P .I(P )0. This proves the second inequality on the
following line:
P [m] (P o)[m] P [m].I(P [m])0 I(P [m]),
and the others are clear. Since I() is idempotent and preserves order, this sh*
*ows that
I((P o)[m]) = I(P [m]). Hence (P o)o = P o.I(P [m])0 = P o.
(c) If P Q, then P [m] Q[m], so I(P [m]) I(Q[m]), and hence P o Qo.
(d) For any P S, P P o. Thus if P is Fcentric, then so is P o, and Z(P o) *
* Z(P ).
To see that this is an equality, it suffices to show that every element in Z(P *
*) commutes
with I(P [m]). For all x 2 Z(P ), cx (as an element of W = AutF (T )) lies in C*
*W (P [m]),
[m])
hence commutes with all elements of I(P [m]) = T CW (P , and in particular with*
* all
elements of I(P [m])0.
We are now ready to prove the main, crucial, property of these subgroups P o.
Carles Broto, Ran Levi, and Bob Oliver *
* 15
Proposition 3.3. Let F be a saturated fusion system over a discrete ptoral gro*
*up
S. Fix P, Q S and ' 2 Hom F(P, Q). Then ' extends to a unique homomorphism
'o 2 Hom F(P o, Qo); and this makes P 7! P ointo a functor from F to itself.
Proof.The functoriality of P 7! P oand ' 7! 'o (i.e., the fact that (IdP)o = Id*
*Poand
(_ O')o = _o O'o) follows immediately from the existence and uniqueness of these
extensions. So this is what we need to prove.
As usual, we set T = S0 and W = AutF (T ). For all Q T , CW (Q) = CW (I(Q))*
* by
definition of I(). This will be used frequently throughout the proof.
We first check that there is at most one morphism 'o which extends '. Assume *
*that
_, _02 Hom F(P o, Qo) are two such extensions. By Lemma 2.4(b), there are eleme*
*nts
w, w02 W such that _P[m].I(P[m])0= wP[m].I(P[m])0and _0P[m].I(P[m])0= w0P[m*
*].I(P[m])0.
Since wP[m]= w0P[m], w1w0 2 CW (P [m]) = CW (I(P [m])), so wI(P[m])= w0I(P*
*[m])as
well. It follows that _ = _0, since they take the same values on P and on I(P [*
*m])0.
It remains to prove the existence of 'o. By Lemma 3.2(c), it suffices to prov*
*e this
when ' 2 IsoF(P, Q). Recall that P o= P .I(P [m])0. Fix u 2 W = AutF (T ) such *
*that
uP[m]= 'P[m]. Define 'o by setting, for all g 2 P and all x 2 I(P [m])0,
'o(gx) = '(g)u(x).
After two preliminary steps, we show in Step 3 that 'o is well defined and a ho*
*momor
phism, and in Step 4 that it is a morphism in F.
Step 1: Fix A, A0 T , and w 2 W such that w(A) = A0. We show here that
A B I(A), _ 2 Hom F(B, T ), _A = wA =) _ = wB ; (1)
and also that
A B A.I(A)0, _ 2 Hom F(B, S), _A = wA =) _(B) T and _ = wB . (2)
If _(B) T , then _ = w0B for some w0 2 W by Lemma 2.4(b), w1w0 2 CW (A) =
CW (I(A)), and thus _ = w0B = wB . This proves (1).
Now assume B A.I(A)0. By Lemma 2.4(a), there is w0 2 W such that w0(B) is
fully centralized in F. It thus suffices to prove (2) when B is fully centrali*
*zed. Set
B0= _(B) for short.
Now, B0 A0and B0 is abelian. So for all x 2 B0, if we regard cx as an elemen*
*t of
W = AutF (T ), then cx 2 CW (A0) = CW (I(A0)). Thus I(A0) = w(I(A)) CS(B0). By
axiom (II) (and since B = _1(B0) is fully centralized), _1 extends to an Fmo*
*rphism
defined on B0.CS(B0), and in particular to a morphism fi 2 Hom F(B0.I(A0), S). *
*Since
fiA0= w1A0 and fi(I(A0)0) T , fiA0.I(A0)0= w1A0.I(A0)0by (1).
Thus for all x 2 B0, fi(x) = _1(x) 2 B A.I(A)0 = fi(A0.I(A0)0). Since fi *
*is
injective, this shows that x 2 A0.I(A0)0 T . So B0 T , and (2) now follows f*
*rom (1).
Step 2: We next show that for all x 2 I(P [m]) and all g 2 P , the following i*
*dentity
holds:
u(gxg1) = '(g)u(x)'(g)1; (3)
or equivalently that c1'(g)OuOcg(x) = u(x). Set w = c1'(g)OuOcg 2 W for short*
*. Then (3)
holds for x 2 P [m]since 'P[m]= uP[m], and thus wP[m]= uP[m]. So wI(P[m])=*
* uI(P[m])
by (1), and this proves (3) for all x 2 I(P [m]).
16 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
Step 3: Recall that we defined 'o(gx) = '(g)u(x) for all g 2 P and x 2 I(P [m*
*])0.
By assumption, 'P[m]= uP[m]. Hence the restrictions of ' and u to P [m].(P \ *
*I(P0[m]))
are equal by (2), and this shows that 'o is well defined.
For all g, g02 P and all x, x02 I(P [m])0,
0 1 0 0
'o((gx)(g0x0))= '(gg0).u(g01xg0x0) = '(gg0). '(g ) u(x)'(g ) .u(x )
= '(g)u(x)'(g0)u(x0) = 'o(gx).'o(g0x0),
where the second equality follows from Step 2. Thus 'o is a homomorphism.
Step 4: It remains to show that 'o 2 IsoF(P o, Qo); i.e., that 'o is a morphi*
*sm
in the category F. By condition (III) in Definition 2.2, together with Zorn's l*
*emma,
there is a maximal subgroup P 0 P ocontaining P such that 'oP0 2 Hom F(P 0, Q*
*o).
Assume P 0 P o; and set '0 = 'oP0 and P 00= NPo(P 0) P 0. By condition (II*
*) in
Definition 2.2, '0 extends to some morphism _ 2 Hom F (P 00, S) (the existence *
*of the
homomorphism 'o shows that N'0 P 00). By (2) again, the restrictions of _, u, *
*and
'o to P 00\ (P [m].I(P [m])0) are equal. Since P 00= P .(P 00\ I(P [m])0), this*
* shows that
_ = 'oP00. This contradicts the maximality assumption about P 0; so P 0= P o, *
*and we
are done.
()o o
Note in particular that by Lemma 3.2(c), the functor F ! F of Proposition*
* 3.3
sends inclusions of subgroups to inclusions.
Corollary 3.4. The functor ()o is a left adjoint to the inclusion of Fo as a f*
*ull
subcategory of F.
Proof.Fix any P in F and any Q in Fo. Since Q = Qo by Lemma 3.2(b), every
' 2 Hom F (P, Q) extends to a unique 'o 2 Hom F (P o, Q) by Proposition 3.3. T*
*he
restriction map
Hom F(P o, Q) Res!HomF(P, Q)
is thus a bijection, and this proves adjointness.
Corollary 3.4 will later be extended to orbit and linking categories associat*
*ed to F
and Fo.
Corollary 3.5. Let F be a saturated fusion system over a discrete ptoral group*
* S.
Then all Fcentric Fradical subgroups of S are in Ho(F), and in particular the*
*re are
only finitely many conjugacy classes of such subgroups.
Proof.Assume P is Fcentric and Fradical. We claim that I(P [m])0 P , and th*
*us
that P = P o2 Ho(F).
Assume otherwise. Then P o P , and hence NPo(P ) P by Lemma 1.8. Thus
NPo(P )=P 6= 1, and since P is Fcentric, this group can be identified with a p*
*subgroup
of Out F(P ). By Proposition 3.3, any ff 2 Aut F(P ) extends to an automorphis*
*m of
P o, and in particular to an automorphism of NPo(P ). This shows that NPo(P )=P*
* C
OutF (P ), which contradicts the assumption that P is Fradical.
The last statement now follows since Ho(F) contains only finitely many conjug*
*acy
classes by Lemma 3.2(a).
As a third consequence of Proposition 3.3, we now prove Alperin's fusion theo*
*rem
in our context. This theorem was originally formulated for finite groups in [Al*
*], and
then for saturated fusion systems over finite pgroups by Puig [Pu ] (see also *
*[BLO2 ,
Carles Broto, Ran Levi, and Bob Oliver *
* 17
Theorem A.10]). Our approach here (and our definition of P o) is modelled on Be*
*nson's
proof of the theorem for fusion in compact Lie groups [Be ].
Theorem 3.6 (Alperin's fusion theorem). Let F be a saturated fusion system over
a discrete ptoral group S. Then for each ' 2 IsoF(P, P 0), there exist sequen*
*ces of
subgroups of S
P = P0, P1, . .,.Pk = P 0 and Q1, Q2, . .,.Qk,
and elements 'i2 AutF (Qi), such that
(a)Qi is fully normalized in F, Fradical, and Fcentric for each i;
(b)Pi1, Pi Qi and 'i(Pi1) = Pi for each i; and
(c)' = 'k O'k1O . .O.'1.
Proof.For each P S, let (P ) be the number of Fconjugacy classes of subgrou*
*ps
in Ho(F) which contain P . We prove the theorem by induction on (P ). Using
Proposition 3.3, we can assume that P, P 02 Ho(F). The claim is clear when (P *
*) = 1
(i.e., P = S).
Assume P S. Let P 00 S be any subgroup which is Fconjugate to P and fully
normalized in F, and fix _ 2 IsoF(P, P 00). The theorem holds for ' 2 IsoF(P, P*
* 0) if it
holds for _ and for _ O'1 2 IsoF(P 0, P 00). So we are reduced to proving the *
*theorem
when the target group P 0is fully normalized in F.
Since P 0is fully normalized, the psubgroup 'OAut S(P )O'1 of AutF(P 0) is *
*conjugate
to a subgroup of AutS(P 0). Let O 2 AutF (P 0) be such that_(OO')OAut S(P )O(OO*
*')1
AutS(P_0). By condition (II) in Definition 2.2, there is ' 2 Hom F(NS(P ), S) s*
*uch that
'P = O O'. Since NS(P_) P (since P S) and P 2 Ho(F), (NS(P )) < (P ), and
the theorem holds for ' (as an isomorphism to its image) by the induction hypot*
*hesis.
So it holds for ' if and only if it holds for O. Hence it now remains only to p*
*rove it
when P = P 0is fully normalized in F, P 2 Ho(F), and ' 2 AutF (P ).
In particular, P is fully centralized in F. So if P_is not Fcentric, then by*
* condition
(II) in Definition 2.2, ' extends to an automorphism ' 2 Aut F(CS(P ).P ). Si*
*nce
(CS(P ).P ) < (P ), the theorem holds for ' by the induction hypothesis.
Now assume that P is not Fradical. Let K Aut F(P ) be the subgroup such
that K= Inn(P ) = Op(Out F(P )) 6= 1. Since P is fully normalized in F, Out S(*
*P ) 2
Sylp(Out F(P )), and so K AutS(P ). In particular,
fi
NKS(P ) def=g 2 NS(P ) ficgP 2 K P
since K Inn(P ). Also, for each g 2 NKS(P ), 'cg'1 2 K (since K C Aut F(P )*
*),
and hence 'cg'1 = ch for some h 2 NKS(P ). So by condition (II) in Definition *
*2.2,
' extends to an automorphism of NKS(P ) P , and the theorem again holds for '*
* by
the induction hypothesis.
Finally, if ' 2 AutF (P ) and P 2 Ho(F) is a fully normalized Fcentric Frad*
*ical
subgroup of S, then the theorem holds for trivial reasons.
18 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
4. Linking systems over discrete ptoral groups
We are now ready to define linking systems associated to a fusion system over*
* a dis
crete ptoral group, and to study the relationship between linking systems and *
*certain
finite full subcategories.
Definition 4.1. Let F be a fusion system over the discrete ptoral group S. A c*
*entric
linking system associated to F is a category L whose objects are the Fcentric *
*subgroups
of S, together with a functor
ss :L ! Fc,
ffiP
and "distinguished" monomorphisms P ! AutL (P ) for each Fcentric subgroup P
S, which satisfy the following conditions.
(A) ss is the identity on objects and surjective on morphisms. More precisely, *
*for each
pair of objects P, Q 2 L, Z(P ) acts freely on Mor L(P, Q) by composition (u*
*pon
identifying Z(P ) with ffiP(Z(P )) AutL(P )), and ss induces a bijection
~=
MorL (P, Q)=Z(P ) ! Hom F(P, Q).
(B) For each Fcentric subgroup P S and each g 2 P , ss sends ffiP(g) 2 AutL(*
*P ) to
cg 2 AutF (P ).
(C) For each f 2 Mor L(P, Q) and each g 2 P , the following square commutes in *
*L:
f
P ______! Q
 
ffiP(g) ffiQ(ss(f)(g))
# #
f
P ______! Q .
More generally, if F0 Fc is any subcategory, then a linking system associated*
* to F0
is a category L0, together with a functor L0 ss0!F0 and distinguished monomor*
*phisms
ffiP
P ! AutL0(P ) for P 2 Ob (F0) = Ob (L0), which satisfy conditions (A), (B), *
*and
(C) above.
It is now clear, by analogy with the finite case, how to define plocal compa*
*ct groups.
Definition 4.2. A plocal compact group is a triple (S, F, L), where S is a dis*
*crete p
toral group, F is a saturated fusion system over S, and L is a linking system a*
*ssociated
to F. The classifying space of such a triple (S, F, L) is the pcompleted nerve*
* L^p.
The following very basic lemma about linking systems extends [BLO2 , Lemma 1*
*.10]
to this situation.
Lemma 4.3. Fix a plocal compact group (S, F, L), and let ss :L ! Fc be the p*
*ro
jection. Fix Fcentric subgroups P, Q, R in S. Then the following hold.
' _ c 1
(a)Fix any sequence P ! Q ! R of morphisms in F , and let _e 2 ssQ,R(_)
and f_' 2 ss1P,R(_') be arbitrary liftings. Then there is a unique morphis*
*m e'2
Mor L(P, Q) such that
e_Oe'= f_'; (1)
and furthermore ssP,Q('e) = '.
Carles Broto, Ran Levi, and Bob Oliver *
* 19
(b)If e', e'02 Mor L(P, Q) are such that the homomorphisms ' def=ssP,Q('e) and *
*'0 def=
ssP,Q('e0) are conjugate (differ by an element of Inn(Q)), then there is a u*
*nique
element g 2 Q such that e'0= ffiQ(g) Oe'in Mor L(P, Q).
Proof.Part (a) is an easy application of axiom (A) for a linking system. Part (*
*b) is
first reduced to the case where ' = '0using axiom (B), and this case then follo*
*ws from
(A) and (C). For more detail, see the proof of [BLO2 , Lemma 1.10].
We next show that the nerve of a linking system is pgood, and hence that the
classifying space of a plocal compact group is pcomplete.
Proposition 4.4. Let (S, F, L) be any plocal compact group at the prime p. The*
*n L
is pgood. Also, the composite
ss1(`) ^
S ! ss1(L) ! ss1(Lp),
induced by the inclusion BS `!L, factors through a surjection ss0(S) i ss*
*1(L^p).
Proof.For each Fcentric subgroup P S, fix a morphism 'P 2 Mor L(P, S) which
lifts the inclusion (and set 'S = IdS). By Lemma 4.3(a), for each P Q S, th*
*ere is
a unique morphism 'QP2 Mor L(P, Q) such that 'Q O'QP= 'P.
Regard the vertex S as the basepoint of L. Define
! :Mor (L) ! ss1(L)
by sending each ' 2 Mor L(P, Q) to the loop formed by the edges 'P, ', and 'Q (*
*in
that order). Clearly, !(_ O') = !(_).!(') whenever _ and ' are composable, and
!('QP) = !('P) = 1 for all P Q S. Also, ss1(L) is generated by Im(!) sinc*
*e any
loop in L can be split up as a composite of loops of the above form.
By Theorem 3.6 (Alperin's fusion theorem), each morphism in F, and hence each
morphism in L, is (up to inclusions) a composite of automorphisms of fully norm*
*alized
Fcentric subgroups. Thus ss1(L) is generated by the subgroups !(Aut L(P )) f*
*or all
fully normalized Fcentric P S.
Let K C ss1(L) be the subgroup generated by all infinitely pdivisible elem*
*ents. For
each fully normalized Fcentric P S, AutL (P ) is generated by its Sylow subg*
*roup
NS(P ) together with elements of order prime to p. Hence ss1(L) is generated *
*by K
together with the subgroups !(NS(P )); and !(NS(P )) !(S) for each P . This s*
*hows
that ! sends S surjectively onto ss1(L)=K, and hence (since the identity comp*
*onent
of S is infinitely divisible) factors through a surjection of ss0(S) onto ss1(*
*L)=K. In
particular, this quotient group is a finite pgroup.
Set ss = ss1(L)=K for short. Since K is generated by infinitely pdivisible*
* elements,
the same is true of its abelianization, and hence H1(K; Fp) = 0. Thus, K is pp*
*erfect.
Let X be the cover of L with fundamental group K. Then X is pgood and X^pis
simply connected since ss1(X) is pperfect [BK , VII.3.2]. Also, since ss is a *
*finite pgroup,
it acts nilpotently on Hi(X; Fp) for all i. Hence X^p! L^p! Bss is a fi*
*bration
sequence and L^pis pcomplete by [BK , II.5.1]. So L is pgood, and ss1(L*
*^p) ~=ss is
a quotient group of ss0(S).
Recall, from Section 3, that for any saturated fusion system F, we defined a *
*finite
subcategory Fo such that the inclusion Fo F has a left adjoint ()o. We next *
*show
that we can do the same on the level of linking systems.
20 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
Proposition 4.5. Let F be a saturated fusion system over a discrete ptoral gro*
*up
S, and let Fco Fc be the full subcategory whose objects are the Fcentric sub*
*groups
contained in Ho(F).
(a)Let L be a centric linking system associated to F, and let Lo L be the full
subcategory with Ob (Lo) = Ob (Fco). Then the inclusion Lo ,! L has a left a*
*djoint,
which sends P to P ofor each Fcentric P S. In particular, the inclusion *
*Lo
L is a homotopy equivalence.
(b)Let Lo be a linking system associated to Fco. Let L be the category whose ob*
*jects
are the Fcentric subgroups of S, and where
o o fi o
MorL (P, Q) = ' 2 Mor Lo(P , Q ) fiss (')(P ) Q};
and let ffiP :P ! AutL(P ) be the restriction of Po. In other words, L*
* is the
pullback category in the following square:
L ______!Lo
 
ss sso
# #
()o co
Fc _____! F .
Then L is a centric linking system associated to F.
Proof.(a) For each Fcentric subgroup P S, fix a morphism 'P 2 Mor L(P, S)
such that ss('P) is the inclusion (and such that 'S = IdS). For any pair of Fc*
*entric
subgroups P Q S, the same group Z(P ) acts freely and transitively on the s*
*ets of
morphisms in L covering the inclusions P Q and P S, and hence there is a un*
*ique
morphism 'QP2 Mor L(P, Q) such that 'Q O'QP= 'P.
Now let ' 2 Hom F(P, Q) be any morphism in Fc. By Proposition 3.3, ' has a un*
*ique
extension to 'o 2 Hom F(P o, Qo). Also, by Lemma 3.2(d), Z(P o) = Z(P ). Hence *
*by
condition (A) in the definition of a linking system, restriction sends the morp*
*hisms in
ss1('o) bijectively to the morphisms in ss1('). Thus for any _ 2 Mor L(P, Q)*
* such
that ss(_) = ', there is a unique "extension" _o 2 Mor L(P o, Qo) of _; i.e., a*
* unique
o Qo
morphism such that _o O'PP = 'Q O_.
Thus, if we define ` :L ! Lo by setting `(P ) = P oand `(_) = _o, then ` is*
* well
defined. This also shows that Mor L(P, Q) = Mor L(P o, Q) when Q = Qo, and thus*
* that
` is a left adjoint functor to the inclusion. Since the inclusion has a left a*
*djoint, it
follows that it induces a homotopy equivalence Lo ' L.
(b) Since Z(P ) = Z(P o) for all Fcentric P S (Lemma 3.2(d) again), axiom (*
*A)
for L follows from the same axiom applied to Lo. Axioms (B) and (C) for L foll*
*ow
immediately from axioms (B) and (C) for Lo by restriction.
We finish the section with a description of the relation between linking syst*
*ems
associated to a given fusion system F0, and rigidifications of the homotopy fun*
*ctor
B :Oc(F0) ! hoTop defined by setting B (P ) = BP . Each linking system L0 i*
*n
duces a rigidification of B, which in turn defines a decomposition of L0 as a*
* homotopy
colimit. More precisely, by a "rigidification of the homotopy functor B" in the*
* following
proposition is meant a functor eB:O(F0) ! Top together with a natural homoto*
*py
equivalence of functors (in hoTop ) from B to ho OeB; i.e., a natural transform*
*ation of
functors to hoTop which defines a homotopy equivalence BP ! eB(P ) for each*
* P . A
Carles Broto, Ran Levi, and Bob Oliver *
* 21
natural homotopy equivalence of rigidifications from eBto eB0is a natural trans*
*formation
Be~! eB0
of functors to Top such that ho(~) commutes with the functors from B. Two rigid*
*ifica
tions eB1and eB2are equivalent if there is a third rigidification eB0and natura*
*l homotopy
equivalences eB1! Be0 eB2; this is seen to be an equivalence relation b*
*y taking
pushouts.
By a linking system L0 in the following proposition is always meant the categ*
*ory
L0 together with the projection to the associated fusion system and the disting*
*uished
monomorphisms. Hence an isomorphism of linking systems means an isomorphism of
the categories which is natural with respect to these other structures.
Proposition 4.6. Fix a saturated fusion system F over a discrete ptoral group *
*S,
and let F0 Fc be any full subcategory. Then there are mutually inverse biject*
*ions
8 9 8 9
< linking systems = ke < rigidifications O(F0) ___! Top=
associated to F0 ! of the homotopy functor B .
: up to isomorphism; ls : up to natural homotopy equivalence;
More precisely, the following hold for any linking system L0 associated to F0 a*
*nd any
rigidification eBof the homotopy functor B on O(F0).
(a)The left homotopy Kan extension ke(L0) of the constant functor L0 *!Top al*
*ong
the projection ess0:L0 ! O(F0) is a rigidification of B , and there is a h*
*omotopy
equivalence
L0 ' hocolim!(ke(L0)) . (1)
O(F0)
(b)There is a linking system ls(Be) associated to F0, and a natural homotopy eq*
*uiva
lence of functors
ke(ls(Be)) '!Be.
Furthermore, if Be0is another rigidification of B , any natural homotopy equ*
*iva
lence of rigidifications ~: eB! eB0induces an isomorphim ~]:ls(Be) ! ls*
*(Be0)
of linking systems.
(c)There is an isomorphism L0 ~=ls(ke(L0)) of linking systems associated to F0.
We define ke([L0]) = [ke(L0)] for each L0, and ls([Be]) = [ls(Be)] for each eB.
Proof.The left homotopy Kan extension is natural with respect to isomorphisms
L0 ! L00of linking systems. Thus ke sends isomorphic systems to natural homot*
*opy
equivalent functors O(F0) ! Top , these are rigidifications of B by (a), and *
*hence ke
is well defined. Point (b) implies that lsis well defined, and it also implies *
*that lsOke
is the identity. Finally, (c) implies that ke Ols is the identity. Hence the Pr*
*oposition
follows once we prove (a), (b), and (c).
(a) Fix L0, and set Be = ke(L0) for short. Recall that we write Rep F(P, Q)*
* =
Mor O(F)(P, Q). By definition, for each P in F0, eB(P ) is the nerve (homotopy *
*colimit
of the point functor) of the overcategory ess0#P , whose objects are pairs (Q, *
*ff) for Q in
L0 and ff 2 RepF (Q, P ), and where
fi
Mor ess0#P(Q, ff), (R, fi) = ' 2 Mor L(Q, R) fiff = fi Oess0(')(.2)
Since L0 ~=hocolim!L0(*), (1) holds by [HV , Theorem 5.5].
22 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
It remains to show that Be is a rigidification of the homotopy functor B . F*
*ix a
section eoe:Mor(O(F0)) ! Mor(L0) of ess0which sends identity morphisms to id*
*entity
morphisms. For each P , let B(P ) be the category with one object oP and morphi*
*sm
group P (so B(P ) ~=BP ), and define functors
`P P
B(P ) ! ess0#P ! B(P )
as follows. Let `P(oP) = (P, Id), and `P(g) = ffiP(g) (as a morphism in ess0#P*
* us
ing (2)) for all g 2 P . Set P(Q, ff) = oP; and let P send each morphism ' 2
Mor ess0#P((Q, ff), (R, fi)) to the unique element g 2 P (unique by Lemma 4.3(b*
*)) such
that the following square commutes:
'
Q _____! R
 
eoe(ff) eoe(fi)
# #
ffiP(g)
P _____! P .
Clearly, P O`P = IdB(P). As for the other composite, define f :Id ! `P O P*
* by
sending each object (Q, ff) to the morphism eoe(ff) 2 Mor L(Q, P ). This is cl*
*early a
natural transformation of functors, and thus
eB(P ) = ess0#P  ' B(P ) ' BP.
To finish the proof that eB is a rigidification of the homotopy functor B , w*
*e must
show, for any ' 2 Hom F (P, Q), that the following square commutes up to natural
transformation:
`P
B(P )_______! ess0#P
 _
B' 'O
# #
`Q
B(Q) _______! ess0#Q .
Here, ['] 2 Rep F(P, Q) denotes the class of '. This means constructing a natu*
*ral
transformation F1 ! F2 of functors B(P ) ! ess0#Q, where F1 = (['] O) O`P*
* and
F2 = `Q OB' are given by the formulas
F1(oP) = (P, [']), F1(g) = ffiP(g), and F2(oP) = (Q, Id), F2(g) = ff*
*iQ('(g)).
Let e'2 Mor L(P, Q) be any lifting of '. Then by condition (C), can be define*
*d by
sending the object oP to the morphism e'2 Mor ess0#P(P, [']), (Q, Id) .
(b) We first fix some notation. For any space X and any x, x02 X, ss1(X; x, x0)*
* denotes
the set of homotopy classes of paths in X (relative endpoints) from x to x0. Fo*
*r any
u 2 ss1(X; x, x0), u* denotes the induced isomorphism from ss1(X, x) to ss1(X, *
*x0). Also,
for any map of spaces f : X ! Y , f* denotes the induced map from ss1(X; x, x0)*
* to
ss1(Y ; f(x), f(x0)).
Now fix a rigidification Be:O(F0) ____! Top; we want to define a linking sys*
*tem
L0 = ls(Be) associated to F. Since eB is a rigidification of the homotopy func*
*tor B ,
fflP
we are given homotopy equivalences BP ! eB(P ) such that the following squa*
*re
commutes up to homotopy for each ' 2 Hom F(P, Q):
fflP
BP _______! eB(P )

B'  eB(['])
# ffl #
BQ _______Q! eB(Q) .
Carles Broto, Ran Levi, and Bob Oliver *
* 23
Here, ['] 2 Rep F(P, Q) denotes the class of ' (mod Inn(Q)). For each P in F0,*
* let
*P 2 eB(P ) be the image under fflP of the base point of BP , and let
~=
flP :P ! ss1(Be(P ), *P)
be the isomorphism induced by fflP on fundamental groups.
Let L0 = ls(Be) be the category with Ob (L0) = Ob (F0), and with
fi
Mor L0(P, Q) = (', u) fi' 2 RepF (P, Q), u 2 ss1(Be(Q); eB'(*P), *Q) .
Composition is defined by setting
(_, v) O(', u) = (_', v . eB_*(u)),
where paths are composed from right to left. Let ss0: L0 ! F0 be the functor *
*which
is the identity on objects, and which sends (', u) 2 Mor L0(P, Q) to the compos*
*ite
flP Be'* u* fl*
*1Q
P !~ ss1(Be(P ), *P) ! ss1(Be(Q), eB'(*P)) ! ss1(Be(Q), *Q) !*
* Q.
= ~=
Also, for each P , define
ffiP :P ! AutL0(P ) by setting ffiP(g) = (IdP, flP(g)).
Axioms (A), (B), and (C) for a centric linking system are easily seen to hold f*
*or L0.
For example, (C) follows as an immediate consequence of the definition of ss0.
Now set B1 = ke(L0) = ke(ls(Be)): the left homotopy Kan extension along the
projection ess0:L0 ! O(F0) of the constant point functor on L0. Thus B1(P ) =
B1(P ) for each P , where B1(P ) is the category with objects the pairs (Q, f*
*f) for
ff 2 RepF (Q, P ), and with morphism sets
fi
Mor B1(P)(Q, ff), (R, fi) = b'2 Mor L0(Q, R) fiff = fi Oess0('b)
fi
= (', u) fi' 2 RepF (Q, R), ff = fi O', u 2 ss1(Be(R); eB'(*Q), *
**R) .
We define a natural homotopy equivalence of functors : B1 ! eBas follows. F*
*or
all P , maps P :B1(P ) ___! eB(P ) are defined inductively, one skeleton at a*
* time, (and
simultaneously for all P ) as follows.
o Each vertex (Q, ff) in B1(P ) = B1(P ) is sent to eB(ff)(*Q) 2 eB(P ).
(',u)
o For each edge oe = (Q, ') ! (P, Id) in B1(P ), where
' 2 RepF (Q, P ) and u 2 ss1(Be(P ); eB'(*Q), *P),
Poe= ^ufor some path ^uin the homotopy class of u.
(',u)
o For each edge oe = (Q, ') ! (R, fi) in B1(P ), where fi 6= IdP, write o*
*e0 =
(',u)
(Q, ') ! (R, Id) (an edge in B1(R)), and set Poe= eB(fi) O( Roe0).
o Consider a simplex of dimension m 2 in B1(P ) of the form
i j
oe = (Q0, ff0) ! (Q1, ff1) ! . ..!(Qm , ffm ) .
If (Qm , ffm ) = (P, Id), then let Poebe any singular simplex in eB(P ) wh*
*ose bound
ary is as already defined. Otherwise, let oe0 be the unique simplex in B1(Qm*
* ) rep
resenting a chain ending in (Qm , Id) such that oe = B1(ffm )(oe0), and set *
* Poe=
Be(ffm ) O( Qm oe0).
24 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
Since B1(P ) ' eB(P ) ' BP (where P is given the discrete topology), the above *
*con
struction is always possible, and defines a homotopy equivalence. It induces th*
*e identity
on fundamental groups, under their given identifications with P . By constructi*
*on, the
P form a natural morphism of functors from B1 to eB.
Let Be0, {ffl0P} be another rigidification of B , and let ~: eB! Be0be a*
* natural
homotopy equivalence of rigidifications. We have already chosen our basepoint **
*P =
fflP(*), where * 2 BP is a fixed basepoint, and we now set *0P= ffl0P(*). Fix, *
*for each
P , a homotopy HP between ~P OfflP and ffl0P. The restriction of HP to the base*
* point of
BP provides a canonical path in eB0(P ) from ~P(*P) to *0P, whose homotopy clas*
*s we
denote wP 2 ss1(Pe; ~P(*P), *0P). We now define
~]:L0 ! L00
to be the identity on objects, and for (', u) 2 Mor L0(P, Q),
~](', u) = (', wP . ~Q*(u) . eB0'*(wP)1) . (1)
It is straightforward to show that ~ is a well defined isomorphim of linking sy*
*stems;
i.e., an isomorphism of categories which is natural with respect to the project*
*ions to
F0 and the distinguished monomorphisms.
(c) Now assume that L0 is given; it remains to construct an isomorphism L0 ~=
ls(ke(L0)) of linking systems associated to F0. Set eB= ke(L0) and L1 = ls(Be)*
* for
short. By definition, L0 and L1 have the same objects, and a morphism in L1 fr*
*om
P to Q is a pair (', u), where ' 2 RepF (P, Q) and u 2 ss1(Be(Q); eB'(*P), *Q).*
* Also,
eB(P ) = ess0#P  where ess0is the projection of L0 onto O(F0); in particular,*
* we choose
*P to be the vertex of (P, Id). Define : L0 ___! L1 by sending each object to*
* itself,
and by sending ff 2 Mor L0(P, Q) to (ess0(ff), [ff]), where [ff] is the homotop*
*y class of ff,
regarded as an edge in ess0#Q from (P, ess0(ff)) = ess0ff(*P) to (Q, Id) = *Q*
*. This is easily
checked to be an isomorphism of categories, and to commute with the distinguish*
*ed
monomorphisms and the projections to F0.
5. Higher limits over orbit categories
If F is any fusion system over a discrete ptoral group S, then O(F) will den*
*ote its
orbit category: the category whose objects are the subgroups of S, and where
Mor O(F)(P, Q) = RepF (P, Q) def=Inn(Q)\ Hom F(P, Q).
Also, we write Oc(F) = O(Fc) to denote the full subcategory of O(F) whose objec*
*ts
are the Fcentric subgroups of S; and more generally write O(F0) to denote the *
*full
subcategory of O(F) corresponding to any full subcategory F0 of F.
By Lemma 2.5, the morphism sets in the orbit category are all finite. There *
*is
a canonical projection functor F ! O(F) which is the identity on objects and the
natural projection Hom F(P, Q) ! RepF (P, Q) on morphisms.
Throughout this section, when C is a category, we frequently write Cmod to *
*denote
the category of functors Cop ! Ab . This notation will not be used in the st*
*atements
of results here, but it is used in several of the proofs.
Carles Broto, Ran Levi, and Bob Oliver *
* 25
Lemma 5.1. Let F be a saturated fusion system over a discrete ptoral group S, *
*and
let F0 F be any full subcategory such that P 2 Ob (F0) implies P o2 Ob (F0). *
*Set
Fo0= F0 \ Fo. Then there are well defined functors
()o
Oc(F) !O(Fco) ,
incl
where ()o sends P to P oand ['] to ['o]. Also, ()o is a left adjoint to the i*
*nclusion.
Proof.This follows from Corollary 3.4. The only thing to check is that ()o is*
* well
defined on morphisms in the orbit category. If '1, '2 2 Hom F (P, Q) represent*
* the
same morphism in the orbit category, then '1 = cgO '2 for some g 2 Q, so 'o1= c*
*gO 'o2
by functoriality, and hence ['o1] = ['o2] in RepF (P o, Qo).
The following proposition shows that the problem of describing higher limits *
*over
the orbit categories we are considering can always be reduced to one over a fin*
*ite
subcategory.
Proposition 5.2. Let F be a saturated fusion system over a discrete ptoral gro*
*up S.
Let F0 F be any full subcategory such that P 2 Ob (F0) implies P o2 Ob (F0), *
*and
set Fo0= F0\ Fo. Then for any functor F :O(F0)op! Z(p)mod, restriction to F*
*o0
induces an isomorphism
lim*(F~)=lim*(F O(Fo0)).
O(F0) O(Fo0)
Proof.Consider the functors
R
O(F0)mod !O(Fo0)mod ,
T
where R is given by restriction and T by composition with the functor ()o. The*
*n T
is a left adjoint to R, since ()o is a left adjoint to the inclusion by Lemma *
*5.1. Also,
T and R are both exact functors, and R sends injectives to injectives since it *
*is right
adjoint to an exact functor.
Let Z_be the constant functor on O(Fo0) which sends all objects to Z. Then T *
*(Z_) is
the constant functor on O(F0), and hence for any functor F on O(F0),
lim(F )= Hom O(F0)mod(T (Z_), F ) ~=Hom O(Fo0)mod(Z_, R(F )) = lim(R(F*
* )).
O(F0) O(Fo0)
Since R is exact and sends injectives to injectives, it sends injective resolut*
*ions to in
jective resolutions, and thus induces an isomorphism between higher limits over*
* O(F0)
and over O(Fo0).
We next want to show that the techniques which we have already developped for
handling higher limits over orbit categories in the finite case [BLO2 , x3] al*
*so apply in
this new situation. The proof of this is similar to the proof in [BLO2 ] of th*
*e analogous
result for fusion systems over finite pgroups, and is in fact a special case o*
*f a very
general result which we prove here.
For any group (not necessarily finite), and any set H of subgroups of , we*
* define
OH ( ) to be the corresponding orbit category of : the category with Ob(OH ( )*
*) = H,
and with morphism sets
Mor OH( )(H, H0) = H0\N (H, H0) ~=Map ( =H, =H0).
26 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
Here, N (H, H0) is the transporter set:
N (H, H0) = {g 2  gHg1 H0}.
If 1 2 H, then for any Z[ ]module M, we define
*H( ; M) = lim*(FM ),
OH( )
where FM :OH ( )op! Ab is the functor FM (H) = 0 if H 6= 1 and FM (1) = M.
It is important to distinguish between the orbit category of a group and the *
*orbit
category of a fusion system. When G is a finite group and S 2 Sylp(G), the orb*
*it
category of the fusion system FS(G) is not the same as the orbit category OS(G)*
* (the
orbit category of G with objects the subgroups of S).
Proposition 5.3. Fix a category C, a group , a set H of subgroups of such th*
*at
1 2 H, and a functor
ff: OH ( ) ! C.
Set c0 = ff(1). For each object d in C, we regard the set Mor C(c0, d) as a s*
*et via ff
and composition. Assume that the following conditions hold:
(a)ff sends = AutOH( )(1) bijectively to End C(c0).
(b)For each d 2 Ob (C) such that d 6~= c0, all isotropy subgroups of the acti*
*on on
Mor C(c0, d) are nontrivial and conjugate to subgroups in H.
(c)For each , 2 Mor (OH ( )), ff(,) is an epimorphism in the categorical sense:*
* ' O
ff(,) = _ Off(,) implies ' = _.
(d)For any H 2 H, any d 2 Ob (C), and any ' 2 Mor C(c0, d) which is Hinvariant,
_ _ H
there is some ' 2 Mor C(ff(H), d) such that ' = ' Off(incl1).
Let
: Cop ! Ab
be any functor which vanishes except on the isomorphism class of c0. Then the n*
*atural
map
* * *
lim*( )ff!~lim( Off)= H ( ; (c0))
C = OH( )
is an isomorphism.
Proof.Consider the functors
ff*
OH ( )mod !Cmod ,
Rff
where ff* is composition with ffop, and Rffis the right Kan extension of ffop. *
*Specif
ically, for d 2 Ob (C), let ff#d be the overcategory whose objects are pairs (H*
*, ')
for ' 2 Mor C(ff(H), d), and where a morphism from (H, ') to (K, _) is a morphi*
*sm
O 2 Mor OH( )(H, K) such that _ Off(O) = '. Let ~d: ff#d ! OH ( ) be the for*
*getful
functor. Then (ff#d)op = d#ffop (the undercategory), and for F :OH ( )op! *
*Ab ,
Rff(F ) is defined by setting
Rff(F )(d) = lim(F O~dop).
(ff#d)op
Carles Broto, Ran Levi, and Bob Oliver *
* 27
On morphisms, Rff(F ) sends f 2 Mor C(d, d0) to the morphism induced by the fun*
*ctor
fO
ff#d ! ff#d0 . By [McL , xX.3, Theorem 1], Rffis right adjoint to ff*. In *
*particular,
since ff* preserves exact sequences, Rffsends injectives to injectives.
Fix H 2 H and d 2 Ob (C). Consider the map
~: Mor C(ff(H), d) ! Mor C(c0, d)
defined by composition with the "inclusion" morphism ff(inclH1). This map is in*
*jective
by (c), and Im (~) Mor C(c0, d)H by (d). Also, Im (~) is contained in Mor C*
*(c0, d)H
since inclH1Ox = inclH1for all x 2 H. Thus ~ induces a bijection
~0 H
MorC(ff(H), d) !~ Mor C(c0, d) . (1)
=
Fix representatives {'di}i2Idfor the orbits in Mor C(c0, d), and let di *
* be the
stabilizer subgroup of 'di. By (b), we can choose the 'disuch that di2 H for a*
*ll i. By
(1), each 'dihas a unique "extension" to _di2 Mor C(ff( di), d); i.e., there is*
* a unique
d
_disuch that 'di= _diOff(incl1i). Also, for any (H, O) in ff#d, there is a uniq*
*ue i 2 Id
and a unique morphism O0 2 Mor OH( )(H, di) such that O = _diOO0. So each obje*
*ct
( di, _di) is a final object in its connected component of the overcategory ff#*
*d. Thus for
any F in OH ( )mod , Y
Rff(F )(d) ~= F ( di). (2)
i2Id
In particular, Rffis an exact functor.
Let Z_denote the constant functor on Cop which sends each object to Z and each
morphism to the identity. Then ff*Z_is the constant functor on OH ( )op. If F*
* :
Cop ___! Ab is any functor, then
lim(F )~=HomCmod(Z_, F );
C
and similarly for functors in OH ( )mod .
Assume H 2 H is such that ff(H) ~=ff(1) = c0. Since all endomorphisms of c0 a*
*re
automorphisms (by (a)), Mor C(c0, ff(H)) contains only isomorphisms, and in par*
*ticular
ff(inclH1) is an isomorphism. Also, inclH1Ox = inclH1for all x 2 H, so ff(x) = *
*Idc0for all
x 2 H. By (a) again, this implies that H = 1.
The functor ff* = Offop : OH ( )op ___! Z(p)mod thus sends the object 1 *
*to
(c0) (with the given action of ), and sends all other objects to 0. Then Rffs*
*ends an
injective resolution I* of ff* to an injective resolution Rff(I*) of Rff(ff* )*
*. It follows
that
*
*H( ; (c0)) def=lim*(ff*~)=H* Mor OH( )mod(ff Z_, I*)
OH( )
~=H* Mor Cmod(Z_, Rff(I*)) ~=lim*(R (ff* )).
C ff
It remains only to show that Rff(ff* ) ~= . For each d 2 Ob (C), if d 6~= c0*
*, then
Mor C(c0, d) is a disjoint union of orbits = di, where 1 6= di2 H by (b). So *
*by (2),
Y
Rff(ff* )(d) = Rff( Off)(d) ~= (ff(Hi)) = 0;
i
where the last equality holds since we already showed that H 6= 1 implies ff(H)*
* 6~=
c0. If d ~= c0, then Mor C(c0, d) consists of one free orbit of (by (a)), an*
*d hence
Rff(ff* )(d) ~= (ff(1)) ~= (c0). This finishes the proof that Rff(ff* ) ~= .
28 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
Our first application of Proposition 5.3 is to the case where C is the orbit *
*category
of a saturated fusion system over a discrete ptoral group. As in [JMO ] and [*
*BLO2 ],
when is finite and H is the set of psubgroups of (or the set of subgroups *
*of a given
Sylow psubgroup), we write *( ; M) = *H( ; M) (and the prime p is understood*
*).
Proposition 5.4. Let F be a saturated fusion system over S. Let
: Oc(F)op! Z(p)mod
be any functor which vanishes except on the isomorphism class of some fixed Fc*
*entric
subgroup Q S. Then
lim*( )~= *(Out F(Q); (Q)).
Oc(F)
Proof.It suffices to do this when Q is fully normalized. Set = Out F(Q) and *
* =
OutS(Q) 2 Sylp( ), and let H be the set of subgroups of . Since ~= NS(Q)=Q,
each subgroup of has the form OutP (Q) for some unique P NS(Q) containing Q.
Define
ff: O ( ) ! Oc(F)
on objects by setting ff(Out P(Q)) = P for Q P NS(Q). If ' 2 AutF (Q) is su*
*ch
that ['] 2 N (Out P(Q), OutP0(Q)) (the set_of elements which conjugate OutP (Q)*
* into
OutP0(Q)),_then ' can be extended to some ' 2 Hom F(P, P 0) by axiom (II), the *
*class
of ' in the orbit category is uniquely_determined by ' by Proposition 2.8, and *
*ff sends
the class of ['] to the class of '.
We apply Proposition 5.3 to this functor ff. Condition (a) is clear, (c) hold*
*s for Oc(F)
by Proposition 2.8, and (d) holds by axiom (II) of a saturated fusion system. A*
*s for (b),
since every morphism in F is the composite of an isomorphism followed by an inc*
*lusion,
it suffices to prove that the stabilizer in of an inclusion inclPQ2 Hom F(Q, *
*P ), where
Q P , is a nontrivial psubgroup. But the stabilizer is OutP (Q) ~=NP(Q)=Q, w*
*hich
is nontrivial by Lemma 1.8. All of the hypotheses of Proposition 5.3 thus hold*
*, and
the result follows.
Using the terminology of [BLO2 ], we say that a category C has bounded limit*
*s at p
if there is k > 0 such that for any functor : Cop ! Z(p)mod, limi( ) = 0 *
*for all
i > k. The following is a first corollary of Proposition 5.4.
Corollary 5.5. Let F be a saturated fusion system over a discrete ptoral group*
* S,
and let F0 Fc be a full subcategory such that P 2 Ob (F0) implies P o2 Ob (F0*
*).
Then the orbit category O(F0) has bounded limits at p.
Proof.By Proposition 5.2, it suffices to prove this when F0 Fo; in particular*
*, when
F0 has only finitely many isomorphism classes. By [JMO2 , Proposition 4.11], f*
*or each
finite group , there is some k such that i( ; M) = 0 for all Z(p)[ ]modules*
* M
and all i > k . Let k be the maximum of the kOutF(P)for all P 2 Ob (F0). Then by
Proposition 5.4, for each functor : O(F0)op! Z(p)mod which vanishes exce*
*pt on
one orbit type, limi( ) = 0 for i > k. The same result for an arbitrary ploca*
*l functor
on O(F0) now follows from the exact sequences of higher limits associated to *
*short
exact sequences of functors.
In practice, when computing higher limits over orbit categories Oc(F), it is *
*useful
to combine Propositions 5.2 and 5.4, as illustrated by the following corollary.
Carles Broto, Ran Levi, and Bob Oliver *
* 29
Corollary 5.6. Let F be a saturated fusion system over a discrete ptoral group*
* S.
Let F :Oc(F)op ___! Z(p)mod be a functor with the property that for each Fce*
*ntric
subgroup P 2 Ho(F), *(Out F(P ); F (P )) = 0. Then lim*(F=)0.
Proof.Let F0: Oc(Fo)op ___! Z(p)mod be the restriction of F . By Proposition *
*5.2,
lim*(F~)= lim*(F0).
Oc(F) Oc(Fo)
Assume first that F0 vanishes except on the conjugacy class of one subgroup P 2
Ho(F). Let F 0be the functor on Oc(F) which takes the same value on the conjuga*
*cy
class of P and vanishes on all other subgroups. Then
lim*(F0)~=lim*(F 0)~= *(Out F(P ); F (P ))
Oc(Fo) Oc(F)
by Propositions 5.2 and 5.4, and this is zero by assumption.
By Lemma 3.2(a), the category Oc(Fo) contains only finitely many isomorphism
classes. Hence there is a sequence
0 = 0 1 . . . k = F0
of subfunctors defined on Oc(Fo), with the property that for each i, i= i1van*
*ishes
except on the conjugacy class of one subgroup P , and ( i= i1)(P ) ~=F (P ). W*
*e have
just seen that lim*( i= i1)= 0 for all i; and hence lim*(F0)= 0 by the relat*
*ive long
exact sequences of higher limits.
The following lemma will be useful in showing that certain functors on the or*
*bit
category are acyclic. As usual, when F is a fusion system over S, a subgroup P *
* S
will be called weakly closed in F if it is the only subgroup in its Fconjugacy*
* class.
Lemma 5.7. Let F be any saturated fusion system over a discrete ptoral group S*
*, and
let Q C S be any Fcentric subgroup which is weakly closed in F. Set = OutF (*
*Q),
and let F Q Fc be the full subcategory whose objects are the subgroups which *
*contain
Q. Define the functor
: O(F Q )op! Op( )
by sending an object P to OutP (Q) , and by sending a morphism ' 2 RepF (P, *
*P 0)
to the class of 'Q 2 N ( (P ), (P 0)). Then for any pair of functors
F :Oc(F)op! Z(p)mod and : Op( )op! Z(p)mod
such that O ~= F O(F Q), and such that Out Q(P ) ~= NPQ (P )=P acts trivial*
*ly on
F (P ) for all P S,
lim*(F~)=lim*( ).
Oc(F) Op( )
Proof.Define a functor
F 0:Oc(F)op! Z(p)mod
by setting F 0(P ) = F (P ) if P Q and F 0(P ) = 0 otherwise. Regard F 0as a *
*quotient
functor of F , and set F 00= Ker[F i F 0].
If P S is Fcentric and P Q, then Out Q(P ) ~= NPQ (P )=P 6= 1, and by
assumption this group acts trivially on F (P ) ~=F 00(P ). Hence the kernel of *
*the action
of Out F(P ) on F 00(P ) has order a multiple of p, and so *(Out F(P ); F 00(P*
* )) = 0 by
[JMO , Proposition 5.5]. Thus lim*(F 00)= 0 by Corollary 5.6, and hence
lim*(F )~=lim*(F.0)
Oc(F) Oc(F)
30 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
Recall that = OutF (Q). Since Q is fully normalized in F (it is the unique *
*subgroup
in its Fconjugacy class), (S) = Out S(Q) 2 Sylp( ). Also, defines a biject*
*ion
between subgroups of (S) ~= S=Q and subgroups of S which contain Q. For all
Q P, P 0 S,
()Q 0
RepF(P, P 0) ! Mor Op( )( (P ), (P ))
is injective by Proposition 2.8. If g 2 N ( (P ), (P 0)) is any element in th*
*e trans
porter, and g = ['] for ' 2 Aut F(Q), then for all x 2 P there is y 2 P 0such t*
*hat
'cx'1 = cy as automorphisms_of Q. Hence by condition (II)_in Definition 2.2, '
extends to a homomorphism ' 2 Hom F(P, P 0), and sends [' ] 2 RepF (P, P 0) t*
*o the
class of g.
This proves that induces bijections on all morphism sets, and thus is an eq*
*uivalence
of categories. Hence if is such that O ~=F O(F Q), then
lim*( )~= lim*(F O(F Q))~=lim*(F~0)=lim*(F.)
Op( ) O(F Q) Oc(F) Oc(F)
This can now be applied to prove the acyclicity of certain explicit functors.
Proposition 5.8. Let F be any saturated fusion system over a discrete ptoral g*
*roup
S. Define
F1, F2: Oc(F)op! Z(p)mod
on objects by setting F1(P ) = Z(P )0 and F2(P ) = ss2(B(Z(P ))^p). On morphism*
*s, each
Fi sends the class of ' 2 Hom F(P, P 0) to the homomorphism induced by the incl*
*usion
of Z(P 0) into Z('(P )) followed by '1Z('(P)). Then F1 and F2 are both acycli*
*c.
Proof.Set T = S0 (the "maximal torus" in F), Q = CS(T ) C S, and = OutF (Q).
Then Q is Fcentric, and is weakly closed in F since T is. Let
: O(F Q ) ! Op( )
be the functor of Lemma 5.7. For each psubgroup , regarded as a group of
automorphismsQof Q, let N be the norm map for the action of on T ; i.e., N (*
*t) =
fl2fl(t) for t 2 T . Define
1( ) = N (T ) and 2( ) = Hom (Z=p1 , T ) .
These define functors i:Op( )op ___! Z(p)mod.
For each P S which contains Q, NP=Q(T ) is connected (i.e., infinitely pdi*
*visible),
and has finite index in Z(P ) since Z(P ) \ T = T Pand T P=NP=Q(T ) has exponen*
*t at
most P=Q. Hence NP=Q(T ) is equal to the identity component Z(P )0, and we ha*
*ve
F1(P ) = Z(P )0 = NP=Q(T ) = 1( (P )).
In general, for any discrete ptoral group P ,
ss2(BP ^p) = [S2, BP ^p] ~=[BS1, BP ^p] ~=Hom (Z=p1 , P ).
Here, the last equivalence follows from Lemma 1.10, while the middle one follow*
*s by
obstruction theory (since ssi(BP ^p) = 0 for i > 2). Hence for any P S which *
*contains
Q,
F2(P ) = ss2(BZ(P )^p) ~=Hom (Z=p1 , Z(P )) ~=Hom (Z=p1 , T )P=Q = 2( (P )).
Carles Broto, Ran Levi, and Bob Oliver *
* 31
Thus iO ~=FiO(F Q) (for i = 1, 2). Also, for each P S, OutQ (P ) acts t*
*rivially
on Fi(P ) for i = 1, 2 since Q centralizes Z(P )0 T . So by Lemma 5.7,
lim*(Fi)~=lim*( i).
Oc(F) Op( )
The functors 1 and 2 are both Mackey functors on Op( ) (see [JM , Proposition*
* 5.14]
or [JMO , Proposition 5.2]), and hence are acyclic.
As in Section 4, when F is a saturated fusion system over S, we let B denote *
*the
homotopy functor B (P ) = BP , and by extension let B ^pdenote the functor B ^p*
*(P ) =
BP ^p. The following proposition is a first application of Proposition 5.8. It *
*shows that
there is a bijective correspondence between rigidifications of these two functo*
*rs.
Proposition 5.9. Let F be a saturated fusion system over a discrete ptoral gro*
*up S,
and let F0 Fc be any full subcategory which contains Fco. Let bB:O(F0) ! T*
*op be
any rigidification of the homotopy functor B ^p. Then there is a f*
*unctor
eB:O(F0) ! Top such that Be(P ) ' BP for all P , together with a natural tr*
*ans
formation of functors eB___! bBwhich is a homotopy equivalence after pcomplet*
*ion.
Moreover, there is a bijection between equivalence classes of rigidifications o*
*f B and
equivalence classes of rigidifications of B ^p.
Proof.Let O: B ! B ^pbe the natural transformation of homotopy functors which
sends BP to BP ^pby the canonical map. We want to apply Theorem A.3, which is
a relative version of the DwyerKan theorem [DK ] for rigidifying centric homot*
*opy
diagrams. We first check that O is relatively centric in the sense of Theorem A*
*.3. This
means showing, for each ' 2 Mor O(F0)(P, Q), that the square
B'O
Map (BP, BP )Id_______! Map (BP, BQ)B'
 
O(P)O O(Q)O
# #
B'O ^
Map (BP, BP ^p)O(P)__! Map (BP, BQp)O(Q)OB'
is a homotopy pullback. By a classical result, the top row is a homotopy equiva*
*lence,
and both mapping spaces have the homotopy type of BZ(P ) (cf. [BKi , Propositi*
*on
7.1]). By Lemma 1.10, the second row is also a homotopy equivalence, and both
mapping spaces have the homotopy type of BZ(P )^p. So the square is a homotopy
pullback.
For each i 1, let fii:O(F0)op! Ab be the functor defined in Theorem A.*
*3,
where for each P ,
i O(P)O j
fii(P ) = ssi hofiberMap (BP, BP )Id! Map (BP, BP ^p)O(P) .
BZ(P) BZ(P)^p
By [DW2 , Proposition 3.1], this homotopy fiber is a K(V, 1) for some bQpvect*
*or space
V . In particular, the fiber is connected, fi1(P ) is abelian for all P , and f*
*ii = 0 for all
i 2. Also, by the homotopy exact sequence for the fibration, there is a short*
* exact
sequence of functors
0 ! F2 ! fi1 ! F1 ! 0,
where F1 and F2 are the functors of Proposition 5.8. By Proposition 5.2, for al*
*l i 1
and j = 1, 2,
limi(Fj) ~= limi(Fj) ~= limi(Fj),
O(F0) O(Fo) Oc(F)
32 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
where the last group vanishes by Proposition 5.8. Thus limi(fi1) = 0 for all i*
* 1.
The proposition now follows directly from Theorem A.3.
In Section 8, we will also need to work with higher limits over orbit categor*
*ies
of certain infinite groups. For any (discrete) group G, let Odpt(G) denote the*
* orbit
category of G whose objects are the discrete ptoral subgroups of G; and define*
* (for
any Z[G]module M),
(
M if P = 1
*dpt(G; M) = lim(FM ) where FM (P ) =
Odpt(G) 0 if P 6= 1.
We are now ready to give a second application of Proposition 5.3.
Lemma 5.10. Fix a group G, a discrete ptoral subgroup Q G, and a functor
: Odpt(G)op ___! Ab with the property that (P ) = 0 except when P is Gconju*
*gate
to Q. Let 0:Odpt(NG(Q)=Q)op ___! Ab be the functor 0(P=Q) = (P ). Then
lim*( ) ~= lim* ( 0) ~= *dpt(NG(Q)=Q; (Q)) . (1)
Odpt(G) Odpt(NG(Q)=Q)
Proof.We apply Proposition 5.3, where C = Odpt(G), = NG(Q)=Q, and H is the set
of discrete ptoral subgroups of . A functor
ff: Odpt( ) ! Odpt(G)
is defined by setting ff(P=Q) = P , and by sending the set (P 0=Q)\N (P=Q, P 0=*
*Q) to
P 0\NG(P, P 0) in the obvious way.
The hypotheses of Proposition 5.3 follow easily from the definition of the or*
*bit cate
gories, and so the isomorphisms between higher limits follow from the propositi*
*on.
The following very general lemma will help in certain cases to reduce computa*
*tions
of higher limits to those taken over finite subcategories.
Lemma 5.11. Let C be a (small) category, and let C1 C2 . . .be an increasing
sequence of subcategories of C whose union is C. Let F :Cop ! Ab be a functo*
*r such
that for each k, i j
lim1limk(F Ci) = 0.
i Ci
Then the homomorphism
~= i k j
limk(F ) ! lim lim(F Ci)
C i Ci
induced by the restrictions is an isomorphism for all k.
Proof.For any category D and any functor : Dop ___! Ab , lim*( ) is the homo*
*logy
of the chain complex (C*(D; ), d), defined by setting
Y
Cn(D; ) = (c0),
c0!...!cn
where the product is taken over composable ntuples of morphisms in D, and where
n+1X
d(,)(c0 ff!c1 ! . .!.cn+1) = ff*,(c1 ! . .!.cn+1) + ,(c0 ! . .b.ci.!.c.n+1*
*).
i=1
Carles Broto, Ran Levi, and Bob Oliver *
* 33
See, for example, [GZ , Appendix II, Proposition 3.3] or [Ol, Lemma 2]. If D0 *
* D is a
subcategory, then the restriction homomorphism from lim*( ) to lim*( D0) is *
*induced
D D0
by the obvious surjections C*(D; ) i C*(D0; ).
In the above situation, the chain complex (C*(C; F ), d) is the limit of an i*
*nverse
system of chain complexes (C*(Ci; F Ci), d) with surjections, where the invers*
*e system
of homology groups of these chain complexes has vanishing lim1(). Since lim*
*1()
vanishes for a (countable directed) inverse system with surjections, we conclud*
*e that
the cohomology of (C*(C; F ), d) is isomorphic to the inverse limit of the coho*
*mology
of the complexes (C*(Ci; F Ci), d).
The next lemma describes how, in some cases, the computation of *dpt(G; M) c*
*an
be reduced to the case where G is finite. When G is a finite group and M is a Z*
*[G]
module, we let *(G; M) denote the functor taken with respect to psubgroups *
*of
G.
Lemma 5.12. Let G be a locally finite group. Assume there is a discrete ptoral
subgroup S G such that every discrete ptoral subgroup of G is conjugate to a*
* subgroup
of S. Fix a Z[G]module M, and assume that for some finite subgroup H0 G,
*(H; M) = 0 for all finite subgroups H G which contain H0. Then *dpt(G; M) *
*= 0.
In particular, *dpt(G; M) = 0 if M is a Z(p)[G]module and the kernel of the a*
*ction of
G on M contains an element of order p.
Proof.By [JMO , Proposition 5.5], for any finite group H and any Z(p)[H]modul*
*e M
such that the kernel of the Haction on M has order a multiple of p, *(H; M) =*
* 0.
Hence the last statement follows as a special case of the first.
Fix a Sylow psubgroup S 2 Sylp(G), and let OS(G) Odpt(G) be the full subca*
*te
gory whose objects are the subgroups of S. Since each discrete ptoral subgroup*
*s of G
is Gconjugate to a subgroup of S, these categories are equivalent, and so we c*
*an work
over OS(G) instead. Define
(
M if P = 1
FM :Cop ! Ab by setting FM (P ) =
0 if P 6= 1.
By definition, *dpt(G; M) = lim*(FM ), and we must show that this vanishes in*
* all
degrees.
Step 1: To simplify the notation, we write C = OS(G), and let C0 C be the f*
*ull
subcategory whose objects are the finite subgroups of S. For each subgroup Q *
* S
and each abelian group A, let IAQin Cmod be the functor
Y
IAQ(P ) = Map (Mor C(Q, P ), A) ~= A.
MorC(Q,P)
For any F in Cmod , Hom Cmod(F, IAQ) ~=Hom Z(F (Q), A). Hence IAQis injective*
* if A
is injective as an abelian group, and each functor on C injects into a product *
*of such
injectives. Also, when Q is finite,
lim(IAQC0) ~=lim(IAQ) ~=A
C0 C
(where the second isomorphism holds for arbitrary Q S).
Choose a sequence of functors
0 ! FM d0!I0 d1!I1 d2!. .,. (1)
34 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
where each Ik is a product of injective functors IAQfor finite subgroups Q S *
*and
injective abelian groups A, and where (1) is exact after restriction to C0. We*
* claim
that this is an injective resolution of FM . In other words, the sequence
0 ! FM (P ) ! I0(P ) ! I1(P ) ! . .,. (2)
is exact for all finite P S, and we want to show it is exact for all P S. *
*Fix
an infiniteSsubgroup P S, and choose finite subgroups P1 P2 . . .such that
P = 1j=1Pj (Lemma 1.9). Then FM (P ) = 0 = limFM (Pj). For all finite Q S *
*and
j
all A,
IAQ(P ) = Map (Mor C(Q, P ), A) = limMap (Mor C(Q, Pj), A)
j
since Mor C(Q, P ) is the union of the Mor C(Q, Pj); and furthermore this is an*
* inverse
system of surjections. Hence (2) is the inverse limit of the corresponding exa*
*ct se
quences for the Pj, all restriction maps Ik(Pj+1) ! Ik(Pj) are surjective, a*
*nd so (2)
is also exact. Thus
*dpt(G; M) = lim*(FM ) ~=H*(lim(Ik), dk) ~=H*(lim(IkC0), dk) ~=lim*(FM(*
*C0).3)
C C C0 C0
StepS2: Fix a sequence S1 S2 S3 . . .of finite subgroups of S such that
S = 1j=1Sj (Lemma 1.9). We first construct inductively a sequence of finite s*
*ubgroups
H1 H2 . . .of G containing H0 such that for each j 1, Hj Sj, and Op(Hj)
contains the full subcategory with object set the psubgroups of Hj1. Fix j *
* 1,
and assume that Hj1 has been constructed. Let Cj be the full subcategory of Of*
*inp(G)
whose objects are the psubgroups of (a finite group since G is local*
*ly finite).
Choose a finite set of morphisms in Cj which generate it, let Xj G be a finit*
*e set of
elements which induce those morphisms, and set Hj = . Since G is locally fi*
*nite,
Hj is a finite subgroup. By construction, Op(Hj) Cj; and hence contains both *
*O(Sj)
and the full subcategory with the same objects as Op(Hj1).
S 1
Set C0def= j=1Op(Hj). This is a full subcategory of Ofinp(G) which contains a*
*ll finite
subgroups of S as objects. In particular, C0 is equivalent to C0, and hence lim*
**(FM ) ~=
C0
lim*(FM C0). Since lim*(FM Op(Hj)) = 0 for all j, lim*(FM C0) = 0 by Lem*
*ma 5.11.
C0 Op(Hj) C0
6.Mapping spaces
We now look at the spaces of maps from BQ to L^p, when Q is a discrete pto*
*ral
group and L is a linking system. In general, for any plocal compact group (S, *
*F, L)
and any discrete ptoral group Q, we define
Rep(Q, L) = Hom (Q, S)=~ ,
where ~ is the equivalence relation defined by setting ae ~ ae0 if there is som*
*e O 2
Hom F(ae(Q), ae0(Q)) such that ae0= OOae. We want to show that [BQ, L^p] ~=Re*
*p (Q, L).
The following lemma will be needed to reduce this to the case where Q is fini*
*te. The
functor ()o of Section 3 plays an important role when doing this.
Carles Broto, Ran Levi, and Bob Oliver *
* 35
Lemma 6.1. Fix a discrete ptoral groupSQ, and let Q1 Q2 . . .Q be a sequen*
*ce
of finite subgroups such that Q = 1n=1Qn. Let (S, F, L) be a plocal compact *
*group.
Then the following hold.
(a)The natural map
~=
R: Rep (Q, L) ! limRep(Qn, L) ,
n
induced by restriction, is a bijection.
(b)Assume Q S. Then for n large enough, Qon= Qo Q, and hence restriction
induces a bijection Hom F(Q, P ) ~=Hom F (Qn, P ) for all P 2 Ob (Fo).
Proof.In general, for any homomorphism ' 2 Hom (H, K), we let ['] denote its cl*
*ass
in Rep(H, K).
(a) Assume first that ', _ 2 Hom (Q, S) are such that R([']) = R([_]). Thus '*
*Qn
and _Qn are Fconjugate for each n; i.e., _Qn = ffn O'Qn for some unique ffn*
* 2
IsoF('(Qn), _(Qn)). In particular, Ker(')\Qn = Ker(_)\Qn for each n, so Ker(') =
Ker(_), and _ = ff O' for some unique ff 2 Iso('(Q), _(Q)). Then ffQn = ffn is*
* in F
for each n, so ff 2 IsoF('(Q), _(Q)) by axiom (III), and [_] = ['] 2 Rep(Q, F).
This proves the injectivity of R, and it remains to prove surjectivity. Fix s*
*ome
{['n]}n 1 2 limRep(Qn, L).
n
Thus for each n, 'n 2 Hom (Qn, S), and 'n+1Qn is Fconjugate to 'n. By Lemma
3.2(a), the set {'n(Qn)o n 1} contains finitely many conjugacy classes. Sin*
*ce for
all n, 'n(Qn) is Fconjugate to a subgroup of 'n+1(Qn+1), 'n(Qn)o is Fconjugat*
*e to
a subgroup of 'n+1(Qn+1)o by Lemma 3.2(b) and Proposition 3.3. Hence for some m,
'n(Qn)o is Fconjugate to 'm (Qm )o for all n m.
We now construct inductively homomorphisms '0n2 Hom (Qn, S) for all n > m such
that ['0n] = ['n] in Rep (Qn, L), and '0nQn1 = '0n1. Assume '0n1has been c*
*on
structed, and set ffn = 'n O'0n112 Hom F('0n1(Qn1), 'n(Qn)). By Proposition*
* 3.3
again, this extends to a unique morphism ffon2 Hom F ('0n1(Qn1)o, 'n(Qn)o), w*
*hich
must be an isomorphism since it is injective and the two groups are abstractly *
*isomor
phic and artinian. Set '0n= (ffon)1 O'n; then '0nQn1 = '0n1. Let ' 2 Hom (Q*
*, S) be
the union of the '0n; then ['] 2 R1({['n]}), and this proves the surjectivity *
*of R.
(b) Now assume Q S. By Lemma 3.2(a,b), for all n, Qon Qon+1 Qo, and the s*
*et
{Qon n 1} is finite. Hence Qon Q for n sufficiently large, and this implies*
* Qon= Qo.
If P = P o S, then every ' 2 Hom F(Qn, P ) extends to a unique 'o 2 Hom F(Qon,*
* P )
by Proposition 3.3, and thus Hom F(Q, P ) ~=Hom F (Qn, P ) whenever Qon= Qo.
For any linking system L and any discrete ptoral group Q, we let LQ be the c*
*ategory
whose objects are the pairs (P, ff) for P 2 Ob (L) and ff 2 Hom (Q, P ), and wh*
*ere
0 0 0 fi 0 *
* 0
MorLQ (P, ff), (P , ff ) = ' 2 Mor L(P, P ) fiff = ss(') Off 2 Hom (Q, *
*P ) .
We next show that Map (BQ, L^p) ' LQ^pin this situation.
Proposition 6.2. Fix a plocal compact group (S, F, L) and a discrete ptoral g*
*roup Q.
Let F0 Fc be any full subcategory which contains all Fcentric Fradical subg*
*roups
of S, and such that P 2 Ob (F0) implies P o2 Ob (F0). Let L0 L and LQ0 LQ
36 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
be the full subcategories where Ob (L0) = Ob (F0), and Ob (LQ0) is the set of p*
*airs
(P, ff) 2 Ob (LQ) such that P 2 Ob (L0). Then there is a bijection
~=
ss0(LQ0) ! Rep (Q, L) (1)
which sends a vertex (P, ff) to the class of ff as a homomorphism to S. If, fur*
*thermore,
we define : LQ0x B(Q) ! L0 by setting
i ' 0 0 j
(P, ff), oQ = P and (P, ff) ! (P , ff ) , x = ' OffiP(ff(x)*
*) ,
then the map
 0:LQ0^p! Map (BQ, L0^p) (2)
adjoint to   is a homotopy equivalence.
Proof.Every vertex (P, ff) in LQ0 is connected by an edge to the vertex (S, i*
*nclSPOff).
Furthermore, by the assumption that F0 contains all Fcentric Fradical subgrou*
*ps,
together with Alperin's fusion theorem (Theorem 3.6), two vertices (S, ff) and *
*(S, ff0)
in LQ0 are in the same connected component if and only if ff and ff0represent*
* the same
element of Rep(Q, L). This proves (1).
Since (', x) = ' OffiP(ff(x)) = ffiP0(ff0(x)) O' by condition (C), is a we*
*ll defined
functor. It remains to prove the homotopy equivalence (2). Step 1, where we han*
*dle
the case Q is finite, is essentially the same as the corresponding proof in [BL*
*O2 ]. In
Step 2, we extend this to the general case.
By assumption, for each P 2 Ob (L0), P o2 Ob (L0). So the functor ()o of Pro*
*po
sition 4.5 restricts to a functor from L0 to Lo0, and also induces a functor fr*
*om LQ0to
LoQ0. All of these are left adjoint to the inclusion functors, and hence induce*
* homotopy
equivalences between their geometric realizations. Thus, without loss of genera*
*lity, we
can assume that L0 = Lo0; i.e., that P = P ofor all P in L0. This assumption wi*
*ll be
needed at the end of each of Steps 1 and 2 below.
Step 1: Assume that Q is a finite pgroup. Let O(F0) Oc(F) be the full subca*
*te
gory with Ob(O(F0)) = Ob (F0) = Ob (L0), and let ess:L0 ! O(F0) be the projec*
*tion
functor. Let essQ:LQ0! O(F0) be the functor essQ(P, ff) = P and essQ(') = es*
*s('). Let
BeQ, eB:O(F0) ! Top
be the left homotopy Kan extensions over essQand ess, respectively, of the cons*
*tant
functors *. Then
L0 ' hocolim!(Be) and LQ0 ' hocolim!(BeQ)(3)
O(F0) O(F0)
(cf. [HV , Theorem 5.5]).
For each P in O(F0), eB(P ) is the nerve of the overcategory ess#P , whose ob*
*jects are
the pairs (R, O) for R 2 Ob (L0) = Ob (O(F0)) and O 2 RepF (R, P ), and where
0 0 0 0
Moress#P(R, O), (R , O ) = ' 2 Mor L0(R, R )  O = O Oess(') .
Let B0(P ) be the full subcategory of ess#P with the unique object (P, Id), and*
* with
morphisms the group of all ffiP(g) for g 2 P .
Similarly, BeQ(P ) is the nerve of the category essQ#P , whose objects are th*
*e triples
(R, ff, O) for R 2 Ob (L0) = Ob (O(F0)), ff 2 Hom (Q, R), and O 2 Rep F(R, P );*
* and
where
0 0 0 0 0 *
* 0
Mor essQ#P(R, ff, O), (R , ff , O ) = ' 2 Mor L0(R, R )  ff = ss(') Off, O *
*= O Oess(') .
Carles Broto, Ran Levi, and Bob Oliver *
* 37
Let B0Q(P ) be the full subcategory of essQ#P with objects the triples (P, ff, *
*Id) for ff 2
Hom (Q, P ).
Fix a section eoe:Mor(O(F0)) ! Mor(L0) which sends identity morphisms to i*
*den
tity morphisms. Retractions
Q 0
ess#P ! B0(P ) and essQ#P ! BQ(P )
are defined by setting
(R, O) = (P, Id) and Q(R, ff, O) = (P, sseoe(O) Off, Id);
and by sending ' in Mor ess#P((R, O), (R0, O0)) or Mor essQ#P((R, ff, O), (R0, *
*ff0, O0)) to the
automorphism ffiP(g) 2 AutL0(P ), where g 2 P is the unique element such that e*
*oe(O0) O
' = ffiP(g) Oeoe(O) in Mor L0(R, P ) (Lemma 4.3(b)). There are natural transfor*
*mations
Idess#P!inclO and IdessQ#P!inclO Q
of functors which send an object (R, O) to O 2 Mor ess#P((R, O), (P, Id)) and s*
*imilarly
for an object (R, ff, O). This shows that B0(P ) ess#P  and B0Q(P ) *
*essQ#P  are
deformation retracts.
We have now shown that for all P 2 Ob (L0),
Be(P ) ' B0(P ) ' BP and eBQ(P ) ' B0Q(P ) . (4)
All morphisms in B0Q(P ) are isomorphisms, two objects (P, ff, Id) and (P, ff0,*
* Id) are
isomorphic if and only if ff and ff0are conjugate in P , and the automorphism g*
*roup of
(P, ff, Id) is isomorphic to CP(ffQ). Thus
a
eBQ(P ) ' BCP(ffQ). (5)
ff2Rep(Q,P)
Let eB^pand eBQ^pbe the pcompletions of eBand eBQ; i.e., (Be^p)(P ) = (Be(P *
*))^pand
(BeQ^p)(P ) = (BeQ(P ))^p. By (3), and since the spaces eB(P ) and eBQ(P ) are *
*all pgood
by (4) and (5),
i j i *
* j
L0^p' hocolim!(Be^p) ^p and LQ0^p' hocolim!(BeQ^*
*p) ^p.
O(F0) O(F0)
Consider the commutative triangle
______________________//
LQ0x B(Q) rL0
PPP rrrr
PPPP rrr
essQOpr1((PPPPPyyressrrr
O(F0) .
The left homotopy Kan extension over essQOpr1of the constant functor * is the f*
*unctor
eBQx BQ, and so the triangle induces a natural transformation of functors
0:BeQx BQ ! eB.
The map e :BeQ! Map (BQ, eB) adjoint to 0 is also a natural transformation *
*of
functors from O(F0) to Top, and induces a commutative diagram
i hjocolim(e)i j!
hocolim!(BeQ)^p_^p_!hocolim!Map(BQ, eB^p)_^p!MapBQ, hocolim!(B*
*e)^p
O(F0) O(F0) O(F0)
' # ' 
#
 0 ^
LQ0^p________________________________________! Map (BQ, L0p) .
38 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
For each P S and Q0 Q, Lemma 1.10 (together with (4)) implies that each
component of Map (BQ0, B(P )^p) has the form BCP(ae(Q0))^pfor some ae 2 Hom (Q0*
*, P ).
So all such mapping spaces are pcomplete and have finite mod p cohomology in e*
*ach
degree. Also, O(F0) is a finite category (it has finitely many isomorphism cla*
*sses of
objects by Lemma 3.2(a) and has finite morphism sets by Lemma 2.5), and it has
bounded limits at p by Corollary 5.5. Hence ! is a homotopy equivalence by [BLO*
*2 ,
Proposition 4.2].
It remains only to show that e(P ) is a homotopy equivalence for each P 2 Ob *
*(L0).
By (4), this means showing that e(P ) restricts to a homotopy equivalence
e 0(P ): B0Q(P ) ! Map (BQ, B0(P )) .
Since B0(P ) ~=BP , and since e 0(P ) is induced by the homomorphisms (incl.f*
*f) from
CP(ff(Q)) x Q to P , this follows from (5).
Step 2: Now let Q be an arbitrary ptoral group. Let Q1 Q2 . .Q.be an
increasing sequence of finite subgroups whose union is Q (Lemma 1.9). Then
ss0(LQ0) ~=limss0(LQn0) ~=lim[BQn, L0^p] : (6)
n n
the first bijection holds by Lemma 6.1 and (1), and the second by Step 1.
Fix ' 2 Hom (Q, S), and set 'n = 'Qn. Let Map (BQ, L0^p)b'be the space of *
*maps
f :BQ ! L0^psuch that fBQn ' B'n for each n. (This contains the connected
component of B', but could, a priori, contain other components.) Let (LQ0)' *
*LQ0
and (LQn0)' LQn0be the full subcategories with objects those (P, ff) such tha*
*t ff is
Fconjugate to ' or to 'n, respectively. Thus (LQ0)' is the connected compone*
*nt of
LQ0 which contains (S, '), and (LQn0)' is the connected component which con*
*tains
(S, 'n).
Consider the following commutative diagram, for all n 1:
(LQ0)'^p_! Map (BQ, L0^p)b'
  (7)
# #
' ^
(LQn0)'^p_!Map(BQn, L0p)'n .
We want to show that the top row is a homotopy equivalence; the proposition then
follows by taking the union of such maps as ' runs through representatives of a*
*ll
elements of Rep (Q, L). The bottom row is a homotopy equivalence by Step 1. So *
*we
will be done if we can show that the vertical maps are homotopy equivalences fo*
*r n
large enough.
By Lemma 6.1(b), there is some m such that for all n m, '(Qn)o = '(Q)o, and
restriction induces a bijection Rep F('(Q), P ) ~=Rep F('(Qn), P ) for all P 2 *
*Ob (L0).
(Recall that we are assuming L0 = Lo0.) This implies that (LQ0)' ~=(LQn0)' *
*for all
n m. Hence the components Map (BQn, L0^p)B'n are all homotopy equivalent for
n m by Step 1, so Map (BQ, L^p)b'' Map (BQn, L^p)B'n for n m, and this *
*proves
that the vertical maps in (7) are equivalences.
The following theorem gives a more explicit description of the set [BQ, L^p*
*] of
homotopy classes of maps, as well as of the individual components in certain ca*
*ses.
Carles Broto, Ran Levi, and Bob Oliver *
* 39
Theorem 6.3. Let (S, F, L) be a plocal compact group, and let ` :BS ! L^pbe
the natural inclusion followed by completion. Then the following hold, for any *
*discrete
ptoral group Q.
(a)The natural map
~= ^
Rep(Q, L) ! [BQ, Lp]
is a bijection. Thus each map BQ ! L^pis homotopic to ` OBae for some ae*
* 2
Hom (Q, S). If ae, ae02 Hom (Q, S) are such that ` OBae ' ` OBae0as maps fro*
*m BQ
to L^p, then there is O 2 Hom F(ae(Q), ae0(Q)) such that ae0= O Oae.
(b)For each ae 2 Hom (Q, S) such that ae(Q) is Fcentric, the composite
incl.Bae ` ^
BZ(ae(Q)) x BQ ! BS ! Lp
induces a homotopy equivalence
BZ(ae(Q))^p'! Map (BQ, L^p)`OBae.
(c)The evaluation map induces a homotopy equivalence
Map (BQ, L^p)triv' L^p.
Proof.We refer to the category LQ, and to the homotopy equivalence
 0:LQ^p'!Map (BQ, L^p)
of Proposition 6.2. Point (a) is an immediate consequence of point (1) in the p*
*ropo
sition, and (c) holds since the component of LQ which contains the objects (P, *
*1) is
equivalent to L.
If ae 2 Hom (Q, S) is such that ae(Q) is Fcentric, then the connected compon*
*ent of
LQ which contains the vertex (ae(Q), ae) contains as deformation retract the *
*nerve of
the full subcategory with that as its only object. Since Aut LQ(ae(Q), ae) ~= *
*Z(ae(Q)),
this component has the homotopy type of BZ(ae(Q)), which proves point (b).
7. Equivalences of classifying spaces
We next describe the monoid Aut(L^p) of self homotopy equivalences of L^p*
*(The
orem 7.1); and also show that plocal compact groups which have homotopy equiva*
*lent
classifying spaces are themselves isomorphic (Theorem 7.4). There is some over*
*lap
between the proofs in this section and those of the corresponding results for p*
*local
finite groups in [BLO2 , Sections 8 & 7]; but they differ in some key respects*
*, mostly
due to the fact that we do not have a way to recover the category L from the sp*
*ace
L^pvia a functor from spaces to categories.
We first recall some notation used in [BLO1 ] and [BLO2 ]. For any space X,*
* Aut(X)
denotes the monoid of self homotopy equivalences of X, and Out(X) = ss0(Aut (X)*
*) is
the group of homotopy classes of self equivalences. For any discrete category C*
*, Aut(C)
is the category whose objects are the self equivalences of C and whose morphism*
*s are
the natural isomorphisms between self equivalences, and Out(C) = ss0(Aut(C)) *
*is the
group of isomorphism classes of self equivalences. We consider Aut(C) as a dis*
*crete
strict monoidal category, in the sense that composition defines a strictly asso*
*ciative
functor
Aut() x Aut() ! Aut()
40 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
with strict identity. The nerve of Aut(C) is thus a simplicial monoid, and its *
*realization
Aut(C) is a topological monoid.
Consider the evaluation functor
ev: Aut(C) x C ! C
which sends a pair of objects ( , c) to (c) 2 Ob (C), and which is defined on *
*morphisms
by setting
i O ' j i 0(')OO(c) j
ev ! 0, c ! d = (c) ! 0(d) .
=O(d)O (')
Upon taking geometric realizations, this defines a map of spaces from Aut(C) *
*x C to
C, which is adjoint to a homomorphism of topological groups
C: Aut(C) ! Aut(C).
Recall that part of the structure of a centric linking system L associated to*
* a fusion
ffiP
system is a homomorphism P ! AutL(P ) for each P in L. We write Pffi= Im(ff*
*iP),
which we think of as a "distinguished subgroup" of AutL (P ) which can be ident*
*ified
with P . For the purposes of this paper, an equivalence of categories L ! L w*
*ill be
called isotypical if for each P , P,Psends the subgroup Pffi AutL(P ) to the *
*subgroup
(P )ffi AutL( (P )). Let Auttyp(L) be the full subcategory of Aut(L) whose ob*
*jects
are the isotypical equivalences, and set Outtyp(L) = ss0(Auttyp(L)).
By [BLO2 , Lemma 8.2], when L is a linking system over a finite pgroup, an *
*equiva
lence : L ! L is isotypical if and only if the triangle involving and the*
* forgetful
functor from L to groups commutes up to natural isomorphism. The same proof ap
plies for linking systems over discrete ptoral groups, although we won't be us*
*ing that
here.
Clearly, any equivalence which is naturally isomorphic to an isotypical equiv*
*alence
is itself isotypical, and any inverse to an isotypical equivalence (inverse up *
*to natural
isomorphism of functors) is also isotypical. The subcategory Auttyp(L) is thus *
*a union
of connected components of Aut(L), and Outtyp(L) is a subgroup of Out(L).
The main result of this section is the following theorem:
Theorem 7.1. Fix a plocal compact group (S, F, L), and set = L. Then the
composite
()^p ^
^p:Auttyp(L) ! Aut (L) ! Aut (Lp)
induces a homotopy equivalence of topological monoids from Auttyp(L)^pto Aut(*
*L^p).
In particular, if we let ssi(BZ^p) denote the functor Oc(F)op! Ab which se*
*nds P
to ssi(BZ(P )^p) (each i 1), then
Out(L^p) ~=Out typ(L) , ssi(Aut (L^p)) ~= lim0(ssi(BZ^p))for i = *
*1,,2
Oc(F)
and ssi(Aut (L^p)) = 0 for i 3.
Proof.We prove the isomorphism between groups of components in Step 2, and the
homotopy equivalence between the individual components in Step 3. In Step 1, we
outline the general procedure for describing the mapping space Aut(L^p).
Assume we have fixed inclusion morphisms 'P 2 Mor L(P, S) for each P . If *
*is
an isotypical self equivalence of L, then clearly (S) = S, and hence S,S is an
Carles Broto, Ran Levi, and Bob Oliver *
* 41
automorphism of AutL(S) which sends Sffi(= Im(ffiS)) to itself. Set
_ = ffi1SO S,SSffiOffiS 2 Aut(S).
For each P 2 Ob (L), axiom (C) and the functoriality of imply that the follow*
*ing
diagram commutes for all g 2 P :
('P)
(P )_____________! S
 
(ffiP(g))2(P)ffi  (ffiS(g))=ffiS(_(g))
# #
('P)
(P )_____________! S .
Hence ss( ('P))( (P )) = _(P ) (by axiom (C) again). So ('P) = '_(P)O ffP for*
* a
unique ffP 2 IsoL( (P ), _(P )) by Lemma 4.3(a). Thus is naturally isomorphic*
* to an
automorphism 0of L such that 0S,S= S,S, and 0(P ) = _(P ) and 0P,S('P) = '*
*_(P)
for each P . This shows that every object in Auttyp(L) is isomorphic to an isot*
*ypical
automorphism of L which sends inclusions to inclusions, and from now on we rest*
*rict
attention to such automorphisms.
Step 1: Consider the decomposition
pr : hocolim!(Be) '!L
Oc(F)
of Proposition 4.6(a), where Be:Oc(F) ! Top is a rigidification of the homot*
*opy
functor P 7! BP . In the following constructions, we regard hocolim!(Be) a*
*s the union
of skeleta:
ian a j.
hocolim!(n)(Be) = Be(P0) x Di ~
Oc(F) i=0P0!...!Pn
where we divide out by the usual face and degeneracy relations.
Define functors Z, Z0: Oc(F)op! Ab and BZ^p:Oc(F)op! Top by setting
Z(P ) = Z(P ), Z0(P ) = Z(P )0, and BZ^p(P ) = BZ(P )^p,
and by sending ['] 2 Mor Oc(F)(P, Q) to '1Z(Q)or B '1Z(Q) ^p. For any eleme*
*nt
^
f= fP P2Oc(F)2 lim[B, Lp],
Oc(F)
let Map (L^p, L^p)fbe the union of the components of the mapping space whic*
*h restrict
to f. By [Wo ], the obstructions to this space being nonempty lie in the groups
^ i+1 ^
limi+1ssi(Map (B, Lp)f) ~= lim (ssi(BZp))
Oc(F) Oc(F)
for i 1; the functor vanishes for i > 2, and the higher limits vanish for i =*
* 2 by
Proposition 5.8. Also, if Map (L, L^p)f6= ?, then the filtration of the map*
*ping space
i j
Map (L^p, L^p) ' Map hocolim!(Be), L^p
Oc(F)
by the skeleta of the homotopy colimit defines a spectral sequence with E2term
^
E2i,j= limissj(Map (B, Lp)f) ,
Oc(F)
which converges to ssji Map (L, L^p)f .
42 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
By Theorem 6.3(b),
8
>:
0 if j 3.
Since ss2(BZ^p) is acyclic by Proposition 5.8, the only obstruction to Map (L*
*^p, L^p)f
being nonempty lies in lim2(Z=Z0); while the spectral sequence takes the form
8
>:
0 otherwise.
Step 2: Let Autfus(S) be the group of fusion preserving automorphisms of S; i.*
*e., the
group of those ff 2 Aut(S) which induce an automorphism of the fusion system F *
*by
sending P to ff(P ) and ' 2 Hom F(P, Q) to (ffQ) O' O(ffP)1 2 Hom F(ff(P ), *
*ff(Q)).
The proof that Out(L^p) ~=Out typ(L) is based on the following diagram:
~0 ~0 !0 2
1 _______! lim1(Z)____! Outtyp(L)____! Out fus(S)_____! lim(Z)
   
!1~=# ss0( ^p) !2~= !3~= (1)
# # #
~ ^ ~ ! 2
1 ______! lim1(Z=Z0)__! Out(Lp)__! limIRep(, F)_! lim(Z=Z0) .
Here, IRep(P, F) Rep(P, F) denotes the set of classes of injective homomorphi*
*sms.
All limits are taken over Oc(F), and !1 and !3 are induced by the natural surje*
*c
tion of functors from Z onto Z=Z0. They are isomorphisms since limi(Z0) = 0 f*
*or
all i 1 (Proposition 5.8). Also, !2 is induced by the inclusion of Out fus(S*
*) =
Autfus(S)= AutF(S) into IRep(S, F) = Aut(S)= AutF(S), and Im(!2) = limIRep(, *
*F)
(thus !2 is a bijection) by definition of fusion preserving. It remains to defi*
*ne the two
rows, and prove that they are exact and the diagram commutes. It will then fol
low immediately that ss0( ^p) is an isomorphism. Note that this does not requi*
*re us
to know that limIRep(, F) is a group or that !0 is a homomorphism; only that
Im(~) = !1(0), Im(~0) = !01(0), and the inverse image under ~ of each element*
* in
the target is a coset of Im(~).
We first consider the top row, where ~0is defined by restricting an isotypica*
*l equiva
lence of L to the image of ffiS. Any_fusion preserving automorphism ff 2 Autfus*
*(S) de
fines an isotypical automorphism_ffof F, and !0(ff) is the obstruction of [BLO2*
* , Propo
sition 3.1] to lifting ffto an automorphism of L. (The proof in [BLO2 ] applie*
*s without
change to the case of a linking system over a discrete ptoral group.) Finally*
*, the
description of Ker(~0) is identical to that shown in [BLO1 , Theorem 6.2]. Mor*
*e specifi
cally, a reduced 1cocycle " 2 Z1(Oc(F); Z) sends each morphism ['] 2 Mor Oc(F)*
*(P, Q)
to "(') 2 Z(P ) (where "(IdP) = 1), and ~0(["]) is represented by the automorph*
*ism
A" 2 Aut(L) defined by setting A"(P ) = P for all P , and A"(_) = _ OffiP("([ss*
*(_)]))1
for all _ 2 Mor L(P, Q). This proves the exactness of the top row.
As for the bottom row in (1), let ~ be the homomorphism defined by restrictio*
*n:
~: Out (L^p) Res![BS, L^p] ~=IRep(S, F).
We want to compare
i j
Map (L^p, L^p) ' Map hocolim!(Be), L^p
Oc(F)
Carles Broto, Ran Levi, and Bob Oliver *
* 43
with
lim[B(), L^p] ~= limIRep(, F).
Oc(F) Oc(F)
By Step 1, the only obstruction to extending any given ff in this last set to a*
*n automor
phism of L^plies in lim2(Z=Z0), while if there are liftings, then the set of*
* homotopy
classes is in bijective correspondence with lim1(Z=Z0). This proves the exact*
*ness of
the bottom row in the sense explained above.
The second square in (1) clearly commutes. To prove that the first square com*
*mutes,
fix some " 2 Z1(Oc(F); Z). Then ~0(["]) = [A"] where A" 2 Aut(L) is the automor
phism defined above; and A" 2 Aut(L) sends each BP B(Aut L(P )) L to *
*L
by the inclusion. For each ' 2 Hom Fc(P, Q), let C' L be the subcategory with*
* two
objects P and Q, whose morphisms are those morphisms in L which get sent to [Id*
*P],
[IdQ], or ['] in Oc(F). Then C' L is homeomorphic to the mapping cylinder*
* of
B': BP ! BQ; and A" sends C' to itself by a map which differs from the *
*iden
tity via a loop in Map (BP, BQ)B' ' BZ(P ) which represents "([']) 2 Z(P ). Af*
*ter
taking the pcompletion, this shows that [A"^p] = ~([~"]), where ~"2 Z1(Oc(F)*
*; Z=Z0)
is the class of " modulo Z0. This proves that the first square in (1) commutes.
Fix ff 2 Outfus(S), and let bffbe the automorphism of the fusion system F ind*
*uced
by ff. Choose maps
ff*P,Q
MorL(P, Q) ! Mor L(ff(P ), ff(Q))
which lift those defined by bff; then !0(ff) is the class of the 2cocycle fi 2*
* Z2(Oc(F); Z)
which measures the deviation of the ff*P,Qfrom defining a functor. These same l*
*iftings
ff*P,Qallow us to define a map of spaces
ff*: hocolim!(1)(Be) ! L,
Oc(F)
and the obstruction to extending this to hocolim!(2)(Be) is precisely the *
*class of the same
2cocycle fi: but regarded as a 2cocycle with coefficients in
Z=Z0 ~=ss1(Map (B, L^p)ff).
This proves that the third square commutes, and finishes the proof that ss0( ^p*
*) is an
isomorphism.
Step 3: Set Z(F) = lim(Z), regarded as a subgroup of S. Let
~: B(Z(F)) x L ! L
'
be the functor which sends (x, P ! Q) to ' OffiP(x). This is adjoint to a *
*functor
from B(Z(F)) to Aut(L), which in turn induces a map j0:BZ(F) ! Aut(L)Id
upon taking geometric realizations. On the other hand, if we first take geomet*
*ric
realizations, then pcomplete, and then take the adjoint, we get a map j from B*
*Z(F)^p
to Aut(L^p)Id. These maps now fit together in the following commutative squar*
*e:
j0
BZ(F) _____'!Aut(L)Id
 
()^p  (2)
# #
j ^
BZ(F)^p ____'! Aut(Lp)Id.
Since we are restricting attention to automorphisms of L (as opposed to worki*
*ng
with all equivalences), Aut(L) is a groupoid, and so ss1(Aut(L)) is the group*
* of
44 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
natural isomorphisms of functors from IdL to itself. A natural equivalence ff *
*sends
each object P to an element ff(P ) 2 Aut L(P ), such that for each ' 2 Mor L(P,*
* Q),
' Off(P ) = ff(Q) O'. In particular, upon restricting to the case P = Q and ' 2*
* ffiP(P ),
we see that ss(ff(P )) = IdP for each P , and thus ff(P ) 2 ffiP(Z(P )) ~= Z(P *
*). The
other relations are equivalent to requiring that ff 2 lim0(Z) = Z(F). This pro*
*ves that
ss1(Aut(L)) ~=Z(F); and since Aut(L)Idis aspherical, shows that j0 is a hom*
*otopy
equivalence.
The E2term of the spectral sequence for maps defined on a homotopy colimit w*
*as
described in Step 1: it vanishes except for the row coming from lim*(Z=Z0), an*
*d the po
sition E20,2~=lim0(ss2(BZ^p)). Hence from the spectral sequence, one sees imme*
*diately
that for i 1,
ssi(Aut (L^p)) ~= lim(ssi(BZ^p)) ~=ssi(BZ(F)^p).
Oc(F)
By naturality, these isomorphisms are induced by j, and thus j is a homotopy eq*
*uiva
lence.
It now follows from (2) and from Step 1 that ^pinduces a homotopy equivalence
Aut(L^p) ' Auttyp(L)^p.
We also note here the following result, which was shown while proving Theorem*
* 7.1.
Proposition 7.2. For any plocal compact group (S, F, L), there is an exact seq*
*uence
0 ! lim1(Z=Z0) ! Out (L^p) ! Out fus(S) ! lim2(Z=Z0),
Oc(F) Oc(F)
where Z0 Z :Oc(F)op! Ab are the functors Z(P ) = Z(P ) and Z0(P ) = Z(P )*
*0.
In Section 9, we will show that for any compact Lie group G, there is a ploc*
*al
compact group (S, F, L) = (S, FS(G), LcS(G)) such that L^p' BG^p. Hence when
G is connected, the exact sequence of Proposition 7.2 gives a new way to descri*
*be
Out(BG^p), which is different from but closely related to the descriptions in [*
*JMO ] and
[JMO3 ].
We now turn our attention to maps between pcompleted nerves of different lin*
*king
systems. We first look at the case where the linking systems in question are as*
*sociated
to the same fusion system. As usual, when we talk about an isomorphism of linki*
*ng
systems, we mean an isomorphism of categories which is natural with respect to *
*the
projections to the fusion system and with respect to the distinguished monomorp*
*hisms.
Lemma 7.3. Let F be a saturated fusion system over a discrete ptoral group S, *
*and
let F0 Fc be any full subcategory which contains Fco. Let L0 and L00be two li*
*nking
systems associated to F0. Assume that there is a map f :L0^p! L00^psuch *
*that
the triangle
BS
` @ `0
@@R
f 0 ^
L0^p________! L0p
is homotopy commutative. Here, ` and `0 are the maps induced by the inclusion *
*of
B(S) into L0 or L00. Then L0 and L00are isomorphic linking systems associated *
*to
~=
F0. Furthermore, we can choose an isomorphism L0 ! L00of linking systems that
induces f on pcompleted nerves.
Carles Broto, Ran Levi, and Bob Oliver *
* 45
Proof.Let ke(L0) and ke(L00) be the left homotopy Kan extensions of the constant
point functors along the projections "ss0:L0 ! O(F0) and "ss00:L00! O(F0) res*
*pec
tively. Let ~P :ke(L0)(P ) ! L0^pbe induced by the forgetful functor from es*
*s0#P to
L0, and similarly for ~0P:ke(L0)(P ) ! L00^p. Then ` and `0 factor through ~*
*S and
~0S, and we have a homotopy commutative diagram
~S ^
ke(L0)(P9)______//ke(L0)(S)_____//L0p97733hhh
ssss oooo hhhhhhhhh 
'sss 'oooo hhhhh 
sss oooohhhhhh` 
ss ooohhhhhhhh 
BP _________//_KBSOVVVV f
KKKK OOOOOVVVVVVVV0 
KKK OOO VVV`VVV 
' KKK ' OOO VVVVVV 
K%% O''O VVVVVV++Vfflffl~0
ke(L00)(P )_____//ke(L00)(S)____//L00^p. S
Hence the maps fP :ke(L0)(P ) ! L00^pand f0P:ke(L00)(P ) ! L00^p, defined*
* as the
obvious composites shown in the above diagram satisfy:
fP 0 ^ ' 0 f0P 0 ^
(a)The composites BP '! ke(L0)(P ) ! L0p and BP ! ke(L0)(P ) ! L0p
are homotopic, and are centric after pcompletion by Theorem 6.3(b).
(b)fQ Oke (L0)(') ' fP and f0QOke (L00)(') ' f0Pfor each morphism ': P ! Q of
O(F0).
Thus ke(L0)^pand ke(L00)^pare equivalent rigidifications of B ^pby Corollary A.*
*5; and
so ke(L0) and ke(L00) are equivalent rigidifications of B by Proposition 5.9. H*
*ence by
Proposition 4.6, L0 and L00are isomorphic linking systems associated to F0.
More precisely, there is a third rigidification eBof B , and a commutative di*
*agram of
natural transformations between functors O(F0) ! Top of the following form:
_ _0 0
ke(L0) _____________________! Be _______ ke(L0)
  
  
# # #
f 0 ^ f1 f2 0 ^
L0^p______! L0p________'!X ________'L0p .
Here, _(P ) and _0(P ) are homotopy equivalences for each P ; X is some space h*
*omotopy
equivalent to L00^p; all functors in the bottom row of the diagram are consta*
*nt functors
on O(F0) (sending all objects to the given space and all morphisms to the ident*
*ity); and
f1 and f2 are homotopy equivalences. Upon taking homotopy colimits of the funct*
*ors
in the top row, we get the homotopy commutative diagram:
' ' 0
hocolim!ke(L0)________________! hocolim!(Be)_hocolim!ke(L0)
O(F0) O(F0) O(F0)
'   ' 
# # #
f 0 ^ f1 f2 0 ^
L0^p__________! L0p____________'!X ____________'L0p .
Here, the left and right vertical maps are homotopy equivalences by Proposition*
* 4.6(a).
This proves that f is a homotopy equivalence. The last statement (an isomorphi*
*sm
L0 ~=L00can be chosen to induce f) now follows since by Theorem 7.1, every homo*
*topy
equivalence from L00^pto itself is induced by some self equivalence of L00.
46 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
An isomorphism (S, F, L) ! (S0, F0, L0) of plocal compact groups consists *
*of a
triple (ff, ffF , ffL), where
ffF 0 ffL 0
S ff!S0, F ! F , and L ! L
are isomorphisms of groups and categories which satisfy the following compatibi*
*lity
conditions:
(a)ffF (P ) = ffL(P ) = ff(P ) for all P S;
(b)ffF and ffL commute with the projections ss :L ! F and ss0:L0! F0; and
(c)ffL commutes with the distinguished monomorphisms ffiP :P ! AutL (P ) and
ffi0P:P ! AutL0(P ).
We are now ready to show that the isomorphism class of a plocal compact group *
*is
determined by the homotopy type of its classifying space. This was shown for p*
*local
finite groups in [BLO2 , Theorem 7.4].
Theorem 7.4. If (S, F, L) and (S0, F0, L0) are two plocal compact groups such *
*that
L^p' L0^p, then (S, F, L) and (S0, F0, L0) are isomorphic as plocal compac*
*t groups.
f 0 ^
Proof.If L^p ! L p is a homotopy equivalence, then by Theorem 6.3(a), t*
*here
'
are homomorphisms ff 2 Hom (S, S0) and ff02 Hom (S0, S) such that the squares
Bff 0 Bff0
BS ________!BS ________! BS
  
` `0 `
# # #
f 0 ^ f0 ^
L^p______! L p_______! Lp
commute up to homotopy, where f0 is any homotopy inverse to f. The composites
ff0Off and ff Off0are Fconjugate to IdSand IdS0by Theorem 6.3(a) again, and th*
*us ff
is an isomorphism.
By yet another application of Theorem 6.3(a), for any P, Q S,
fi
Hom F(P, Q) = ' 2 Inj(P, Q) fi`BQ OB' ' `BP .
From this, and the corresponding result for Hom F0(ff(P ), ff(Q)), we see that *
*ff induces
an isomorphism of categories from F to F0.
Upon replacing S0and F0by S and F, we can now assume that L and L0are two lin*
*k
f*
* 0 ^
ing systems associated to F, for which there is a homotopy equivalence L^p*
*! L p
such that f O` ' `0. Then L ~= L0 (as linking systems associated to F) by Lemma
7.3.
8. Fusion and linking systems of infinite groups
We now want to find some general conditions on an infinite group G which guar*
*an
tee that we can associate to G a plocal compact group (S, FS(G), LcS(G)) such *
*that
LcS(G)^p' BG^p. This will be done in as much generality as possible. For exam*
*ple,
we prove the saturation of the fusion system FS(G) in sufficient generality so *
*that the
result also applies to the case where G is a compact Lie group.
Carles Broto, Ran Levi, and Bob Oliver *
* 47
At the end of the section, to show that the theory we have built up does cont*
*ain
some interesting examples, we show that it applies in particular to all linear *
*torsion
groups in characteristic different from p.
We say that a group G "has Sylow psubgroups" if there is a discrete ptoral *
*subgroup
S G which contains all discrete ptoral subgroups of G up to conjugacy. For *
*any
such G, we let Sylp(G) be the set of such maximal discrete ptoral subgroups.
Lemma 8.1. Fix a group G, a normal discrete ptoral subgroup Q C G, and a subgr*
*oup
K G such that G = QK. Assume that K has Sylow psubgroups. Then G has Sylow
psubgroups, and
Sylp(G) = {QS  S 2 Sylp(K)}.
Proof.Let Syl0p(G) = {QS  S 2 Sylp(K)}. All subgroups in Syl0p(G) are Gconjug*
*ate
since all subgroups in Sylp(K) are Kconjugate. If P G is an arbitrary discr*
*ete
ptoral subgroup, then QP is also discrete ptoral (since Q and QP=Q are discre*
*te
ptoral), and
QP = QK \ QP = Q.(K \ QP ).
Thus P QP QS 2 Syl0p(G) for any S 2 Sylp(K) which contains K \ QP . This
shows that G has Sylow psubgroups, and that they are precisely the subgroups in
Syl0p(G).
We first establish some general conditions on an infinite group G with Sylow *
*p
subgroups, which imply that FS(G) is a saturated fusion system for S 2 Sylp(G).*
* The
following technical lemma will be needed when doing this.
Lemma 8.2. Fix a group G, and normal subgroups N, Q C G, with the following
properties:
(a)Q is a discrete ptoral group.
(b)G=QN is a finite group.
(c)For each H G such that H N and H=N is finite, H has Sylow psubgroups.
(d)each coset gN 2 G=N contains at least one element of finite order.
Then G has Sylow psubgroups. For any discrete ptoral subgroup P G, P 2 Sylp*
*(G)
if and only if P Q, P \ N 2 Sylp(N), and P N=QN 2 Sylp(G=QN).
Proof.Fix any G0 G such that G0 QN and G0=QN 2 Sylp(G=QN). For every
discrete ptoral subgroup P G, P QN=QN is conjugate to a subgroup of G0=QN,
hence P is Gconjugate to a subgroup of G0. Hence G has Sylow psubgroups if G0
does, and in that case, Sylp(G0) is the set of subgroups of G0which are in Sylp*
*(G). It
thus suffices to prove the lemma when G = G0; i.e., when G=QN is a finite pgro*
*up;
and we assume this from now on.
Step 1: Assume first that Q = 1, and thus that G=N is a finite pgroup. Then*
* G
has Sylow psubgroups by (c). Throughout this step, we fix some S 2 Sylp(G). We*
* first
prove that NS = G (hence NS=N 2 Sylp(G=N)) and S \ N 2 Sylp(N). Afterwards,
we prove the converse: P \ N 2 Sylp(N) and NP = G imply P is Gconjugate to S,
and hence P 2 Sylp(G).
If NS G, then NS=N G=N, where the latter is a finite pgroup. Since every
proper subgroup of a pgroup is contained in a proper normal subgroup, there is*
* a
48 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
proper normal subgroup Nb C G which contains NS. By (d), there is an element
g 2 Gr bN of finite order. Write g = mpk where p  m, and set g0 = gm . Then
g0 2 Gr bNsince Nb has ppower index, and g0 = pk. This means that is a f*
*inite
psubgroup of G which is not conjugate to a subgroup of S, which contradicts the
assumption that S 2 Sylp(G). Thus NS = G.
For all S02 Sylp(N), there are elements x 2 G and y 2 N such that xS0x1 S *
*\ N
and y(S \ N)y1 S0. Thus (yx)S0(yx)1 S0, and this must be an equality sinc*
*e S0
is artinian. It follows that S \ N = xS0x1 2 Sylp(N).
Now let P G be any subgroup such that P \N 2 Sylp(N) and P N = G. Fix x 2 G
such that xP x1 S. Then (xP x1)N = xP Nx1 = G, xP x1\N = x(P \N)x1
S \ N, and this last must be an equality since P \ N 2 Sylp(N). It follows that
G=N = xP x1.N=N = xP x1=(xP x1 \ N) S=(S \ N) = SN=N G=N;
so these are all equalities, and P = x1Sx 2 Sylp(G).
Step 2: Now consider the general case. By assumption, G=N is an extension
of the discrete ptoral group QN=N by the finite pgroup G=QN, and hence is dis
crete ptoral. So by Lemma 1.9, there is a finite psubgroup G0=N G=N such th*
*at
(G0=N).(QN=N) = G=N, and thus QG0 = G (since G0 N). Then G0 has Sylow
psubgroups by (c). Hence G has Sylow psubgroups by Lemma 8.1. Also, by Step 1
applied to the pair N C G0 (recall G0=N is a pgroup),
P 2 Sylp(G0) () P \ N 2 Sylp(N) and P N = G0.
Let P G be any discrete ptoral subgroup which contains Q, and set P0 = P \*
* G0.
In general, for any A, B G and C C G with C A, C.(A \ B) = A \ CB. Thus
QP0 = Q.(P \ G0) = P \ QG0 = P \ G = P
(1)
P0N = (P \ G0).N = P N \ G0.
Also, by Lemma 8.1 again, Sylp(G) = {QS  S 2 Sylp(G0)}. Hence
P 2 Sylp(G) () P0 = P \ G0 2 Sylp(G0)
() P0 \ N 2 Sylp(N) and P0N = G0 (Step 1)
() P \ N 2 Sylp(N) and P N = G;
where the last equivalence holds by (1) and since P0\ N = P \ (G0\ N) = P \ N.
Let G be any group which has Sylow psubgroups. For any S 2 Sylp(G), we let
FS(G) be the fusion system over S with objects the subgroups of S and morphisms
Hom FS(G)(P, Q) = Hom G(P, Q) .
Proposition 8.3. Let G be a group for which the following conditions hold:
(a)For each discrete ptoral subgroup P G, each element of AutG (P ) is conju*
*gation
by some x 2 NG(P ) of finite order.
(b)For each discrete ptoral subgroup P G, and each finite subgroup H=CG(P )
NG(P )=CG(P ), H has Sylow psubgroups.
(c)For each increasing sequence P1 P2 P3 . . .of discrete ptoral subgrou*
*ps of
G, there is some k such that CG(Pn) = CG(Pk) for all n k.
Carles Broto, Ran Levi, and Bob Oliver *
* 49
Then for each S 2 Sylp(G), FS(G) is a saturated fusion system. Furthermore, the
following hold for each subgroup P S.
(1)CG(P ) has Sylow psubgroups, and P is fully centralized in FS(G) if and onl*
*y if
CS(P ) 2 Sylp(CG(P )).
(2)NG(P ) has Sylow psubgroups, and P is fully normalized in FS(G) if and only*
* if
NS(P ) 2 Sylp(NG(P )).
Proof.Note first that G has Sylow psubgroups by (b), applied with P = 1.
Fix S 2 Sylp(G), and let P S be any subgroup. By (a), Aut G(P ) is a torsi*
*on
group, so Out G(P ) is a torsion group, and hence is finite by Proposition 1.5(*
*b). We
first claim that
P .CG(P ) NG(P ) =) has Sylow psubgroups; (3)
and that if we set S0 = S \ , then
S0.CG(P )
S0 2 Sylp( ) () CS(P ) 2 Sylp(CG(P )) and _________2 Sylp( =P .CG(P )).(4)
P .CG(P )
Points (3) and (4) follow from Lemma 8.2, applied with N = CG(P ) and Q = P .
Conditions (c) and (d) of Lemma 8.2 follow from conditions (b) and (a) above. N*
*ote
that =QN is finite since OutG (P ) ~=NG(P )=QN is finite.
We next prove (1) and (2). For all P S, (3) (applied with = NG(P )) impli*
*es that
there is Q 2 Sylp(NG(P )) such that NS(P ) Q NG(P ). Choose g 2 G such that
gQg1 S; then gQg1 is a Sylow psubgroup of gNG(P )g1 = NG(gP g1). Since
gQg1 S, gQg1 = S \ NG(gP g1) = NS(gP g1). Hence NS(gP g1) is a Sylow p
subgroup of NG(gP g1). If P is fully normalized, then NS(P ) NS(gP g1) *
*= Q.
Since NS(P ) Q, this implies that NS(P ) = Q 2 Sylp(NG(P )).
Conversely, suppose that NS(P ) 2 Sylp(NG(P )). Choose g 2 G such that gP g1*
* S
and is fully normalized in FS(G). Then NS(gP g1) 2 Sylp(NG(gP g1)), so NS(P )*
* ~=
NS(gP g1) since NG(P ) ~=NG(gP g1), and P is also fully normalized.
This proves (2). The proof of (1) (the condition for P to be fully centraliz*
*ed) is
similar, except that CG(P ) has Sylow psubgroups by (b).
We now prove that FS(G) is saturated.
(I) Assume that P S is fully normalized in FS(G). We have already seen that
OutG (P ) is finite (since it is a torsion group by (a)). Also, NS(P ) 2 Sylp(*
*NG(P ))
by (2). So by (4), applied with = NG(P ), CS(P ) 2 Sylp(CG(P )) (hence P is f*
*ully
centralized by (1)), and OutS(P ) 2 Sylp(Out G(P )).
(II) Let P S be an arbitrary subgroup, and let g 2 G be such that P 0def=gP*
* g1
S is fully centralized. Set = NS(P 0).CG(P 0), and define
fi 1 0 fi 1
N = x 2 NS(P ) ficg Ocx Ocg 2 AutS(P ) = x 2 NS(P ) figxg 2 .
Then CS(P 0) 2 Sylp(CG(P 0)) by (1), and so by (4), NS(P 0) is a Sylow psubgro*
*up of
= NS(P 0).CG(P 0) (S0.CG(P ) = in the notation of (4)). Since gNg1 is a di*
*screte p
toral subgroup of , it is conjugate to a subgroup of NS(P 0). Thus there are*
* elements
x 2 NS(P 0) and y 2 CG(P 0) such that (xyg)N(xyg)1 NS(P 0). Then (yg)N(yg)1
NS(P 0) S, and cyg2 Hom G(N, S) is an extension of cg 2 Hom G(P, S).
50 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
S (III) Let P1 P2 P3 . . .be a sequence of subgroups of S, and set P1 =
1
n=1Pn. Assume ' 2 Hom (P1 , S) is a monomorphism such that 'Pn 2 Hom G(Pn, S)
for each n. Fix elements gn 2 G for each n such that '(x) = gnxg1n, for x 2 Pn*
*. Then
for all 1 k < n, g1ngk 2 CG(Pk).
By (c), there is k such that CG(Pk) = CG(Pn) = CG(P1 ) for all n k. Hence f*
*or all
n k and all x 2 Pn, '(x) = gnxg1n= gkxg1k, and thus ' = cgk2 Hom G(P1 , S).
In general, for any group G, we define a pcentric subgroup of G to be a disc*
*rete
ptoral subgroup P G such that Z(P ) is the unique Sylow psubgroup of CG(P )
(i.e., every discrete ptoral subgroup of CG(P ) is contained in Z(P )). Equiva*
*lently, P
is pcentric if and only if CG(P )=Z(P ) has no elements of order p.
Proposition 8.4. Let G be any group which has Sylow psubgroups, and fix S 2
Sylp(G). Then a subgroup P S is FS(G)centric if and only if P is pcentric *
*in
G.
Proof.Assume that P is pcentric in G; i.e., that Z(P ) 2 Sylp(CG(P )). For ev*
*ery
g 2 G such that gP g1 S, CS(gP g1) is a discrete ptoral subgroup of CG(gP *
*g1) =
gCG(P )g1, and Z(gP g1) = gZ(P )g1 is a Sylow psubgroup (hence the unique o*
*ne)
of gCG(P )g1. It follows that Z(gP g1) = CS(gP g1) for all such g, and so P*
* is
FS(G)centric.
Conversely, suppose that P S is FS(G)centric. Let Q be any discrete ptora*
*l sub
group of CG(P ). Then QP is a discrete ptoral subgroup, and hence there is an *
*element
g 2 G such that g(QP )g1 S. Thus gP g1 S, and gQg1 S \ CG(gP g1) =
CS(gP g1). Since P is FS(G)centric, this shows that gQg1 Z(gP g1), and th*
*us
that Q Z(P ). In other words, every discrete ptoral subgroup of CG(P ) is co*
*ntained
in Z(P ), and so P is pcentric in G.
We now restrict attention to locally finite groups. For any such group G, fo*
*r the
purposes of this section, we define Op(G) C G be the subgroup generated by all *
*ele
ments of order prime to p. This clearly generalizes the usual definition of Op(*
*G) for
finite G (although it is not the only generalization).
Proposition 8.5. If G is locally finite, then a discrete ptoral subgroup P *
*G is
pcentric if and only if CG(P ) = Z(P ) x Op(CG(P )) and all elements of Op(CG(*
*P ))
have order prime to p.
Proof.By the above definition, a discrete ptoral subgroup P G is pcentric i*
*f and
only if CG(P )=Z(P ) has no elements of order p. So if P is not pcentric, then*
* either
Op(CG(P )) has ptorsion, or CG(P ) is not generated by Z(P ) and Op(CG(P )).
Conversely, assume that P is pcentric, and thus that CG(P )=Z(P ) has no pt*
*orsion.
Consider the universal coefficient exact sequence
2
0 ! Ext H1(CG(P )=Z(P )), Z(P ) ! H (CG(P )=Z(P ); Z(P ))
! Hom H2(CG(P )=Z(P )), Z(P ) ! 0.
By assumption, all elements of CG(P )=Z(P ) have order prime to p, all elements*
* of
Z(P ) have ppower order, and both groups are locally finite. Hence for i = 1,*
* 2,
Hi(CG(P )=Z(P )) is a direct limit of finite abelian groups of order prime to p*
*, and thus
a torsion group all of whose elements have order prime to p. This shows that al*
*l terms
in the above sequence vanish. Hence the central extension
1 ! Z(P ) ! CG(P ) ! CG(P )=Z(P ) ! 1
Carles Broto, Ran Levi, and Bob Oliver *
* 51
splits, so CG(P ) ~=Z(P ) x (CG(P )=Z(P )), and all elements of the group Op(CG*
*(P )) ~=
CG(P )=Z(P ) have order prime to p.
When working with fusion systems over discrete ptoral groups and their orbit*
* cate
gories, we are able to reduce certain problems to ones involving finite categor*
*ies using
the functor ()o constructed in Section 3. This is not a functor on the orbit c*
*ategory
of a group, and so we need a different way to make such reductions. For any gro*
*up G
with Sylow psubgroups, we let X = X(G) denote the set of all subgroups of G wh*
*ich
are intersections of (nonempty) subsets of Sylp(G). Since discrete ptoral gro*
*ups are
artinian, it makes no difference whether we require finite intersections or all*
*ow infinite
intersections.
Lemma 8.6. Let G be a group such that for each discrete ptoral subgroup P G,
NG(P ) has Sylow psubgroups. Assume, for every increasing sequence P(1) P(2)
P(3) . . .of discrete ptoral subgroups of G, that the union of the P(i)is aga*
*in a
discrete ptoral group, and that there is some k such that CG(P(n)) = CG(P(k)) *
*for all
n k. Then the set X(G) contains finitely many Gconjugacy classes.
Proof.For each discrete ptoral subgroup P G, we let P O P denote the inters*
*ection
of all Sylow psubgroups of G which contain P . We first prove
(1)For each discrete ptoral subgroup P G, there is a finite subgroup P 0 P *
*such
that P 0O= P O.
To see this, set pn = exp(ss0(S)) for S 2 Sylp(G). The discrete ptorus S0 is *
*the
union of an increasing sequence of finite psubgroups, and since centralizers s*
*tabilize
by assumption, therenis a finite subgroup Q P0 such that CG(Q) = CG(P0). Set
Q0 = {x 2 P0 xp 2 Q}: also a finite psubgroup. By Lemma 1.9, there is a fin*
*ite
subgroup P 0 P such that P 0 Q0and P 0P0 = P .
Fix S 2 Sylp(G) which contains P 0. Then S0 Q (since S Q0), and hence
S0 CG(Q) = CG(P0). Since S0 is a maximal discrete ptorus in G and S0.P0 is a*
*lso a
discrete ptorus, this implies that S0 P0. Hence S P 0P0 = P . Since this h*
*olds for
all S 2 Sylp(G) which contains P 0, we have shown that P 0O= P O; and this fini*
*shes the
proof of (1).
__
Let X (G) be the set of Gconjugacy classes of subgroups_in X(G). We let (P )
denote the conjugacy class of the subgroup P , and_make X(G) into a poset by se*
*tting
(P ) (Q) if P xQx1 for some x 2 G. Let P X (G) be the set of all classes
(P_) which are contained in infinitely many other classes. We claim that P = ?.*
* Since
X(G) contains a smallest element which is contained in all the others_(the clas*
*s of the
intersection of all Sylow psubgroups of G), P = ? implies that X(G) is finite,*
* which
is what we want to prove.
Assume otherwise: assume P 6= ?. We claim that P has a maximal element. For
any totally ordered subset P0 of P, upon restricting to those subgroups of maxi*
*mal
rank, we obtain a sequence of subgroups P(1) P(2) P(3) . . .whose conjugacy
classes are cofinal in P0. If thisSsequence is finite, then P0 clearly has a m*
*aximal
element. Otherwise, set P(1) = 1i=1P(i), and let P 0 P(1) be a finite subgro*
*up such
that P 0O= P(1) (apply (1)). Then P 0 P(k)for some k, and so (P(k)) = (P(1)) i*
*s a
maximal element in P0.
Thus by Zorn's lemma, P contains a maximal element (Q), and clearly Q =2Sylp(*
*G).
Since NG(Q) has Sylow psubgroups, there is some S 2 Sylp(G) such that every p*
*toral
52 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
subgroup of G containing Q with index p is Gconjugate to a subgroup of NS(Q); *
*and
hence by Lemma 1.4, there are finitely many Gconjugacy classes_of such subgrou*
*ps.
Hence since (Q) is contained in infinitely many classes in X(G), the same holds*
* for (Q0)
for some Q0 G such that Q C Q0with index p. Then (Q0O) 2 P, which contradicts
the maximality assumption about Q. So P contains no maximal element, hence must
be empty, and so X(G) has finitely many Gconjugacy classes.
Now, for any discrete group G which has Sylow psubgroups, let Lcp(G) be the *
*cate
gory whose objects are the pcentric subgroups of G, and where
Mor Lcp(G)(P, Q) = NG(P, Q)=Op(CG(P )).
For any S 2 Sylp(G), let LcS(G) Lcp(G) be the equivalent full subcategory who*
*se
objects are the subgroups of S which are pcentric in G.
It will be convenient, throughout the rest of this section, to use the term "*
*pgroup"
to mean any group each of whose elements has ppower order. It is not hard to s*
*how
that if G is locally finite, and has Sylow psubgroups in the sense described a*
*bove,
then every psubgroup of G is a discrete ptoral subgroup. Hence there is no l*
*oss of
generality to assume this in the following theorem.
Theorem 8.7. Let G be any group which satisfies the following conditions:
(a)G is locally finite.
(b)Each psubgroup of G is a discrete ptoral group.
(c)For any increasing sequence A(1) A(2) A(3) . .o.f finite abelian psubgro*
*ups
of G, there is some k such that CG(A(n)) = CG(A(k)) for all n k.
Then G has a unique conjugacy class Sylp(G) of maximal discrete ptoral subgrou*
*ps.
For any S 2 Sylp(G), (S, FS(G), LcS(G)) is a plocal compact group, with classi*
*fying
space LcS(G)^p' BG^p.
Proof.We first apply Proposition 8.3 to show that FS(G) is a saturated fusion s*
*ystem
over S. Once this has been checked, then it easily follows that LcS(G) is a cen*
*tric linking
system associated to FS(G): condition (A) in Definition 4.1 holds by Propositio*
*ns 8.5
and 8.4, and conditions (B) and (C) are immediate. It then will remain only to *
*show
that LcS(G)^p' BG^p.
By [KW , Theorem 3.4], conditions (a) and (c) above imply that all maximal p
subgroups of G are conjugate, and hence (by (b)) that G has Sylow psubgroups. *
*Since
these three conditions are carried over to subgroups of G, this also shows that*
* each
subgroup of G has Sylow psubgroups. This proves condition (b) in Proposition 8*
*.3,
and condition (a) holds since G is locally finite.
It remains to prove condition (c) in Proposition 8.3, which we state here as:
(d)For any increasing sequence P(1) P(2) P(3) . .o.f discrete ptoral subgro*
*ups
of G, there is some k such that CG(P(n)) = CG(P(k)) for all n k.
To see this, fix any such sequence, and let P(1) be its union. Let A = (P(1))0 *
*be the
identity component, and set A(i)= A \ P(i)for all i. Let r be such that P(i)sur*
*jects
onto ss0(P(1)) for all i r; equivalently, P(i).A = P(1) for all i r. ForSe*
*ach i, let
A0i A(i)be the finite subgroup of elements of order at most pi. Then A = 1i=*
*1A0i;
and so by (c) there is r such that CG(A) = CG(A0r). Hence CG(A(i)) = CG(A(r)) f*
*or
Carles Broto, Ran Levi, and Bob Oliver *
* 53
all i r (since A0r A(r) A(i) A). We can assume that r is chosen large enou*
*gh
so that P(r)surjects onto P(1)=A; i.e., such that P(r).A = P(1). Then for all *
*i r,
P(i)= A(i).P(r),
CG(P(i)) = CG(A(i)) \ CG(P(r)) = CG(A(r)) \ CG(P(r)) = CG(P(r)),
and this finishes the proof of (d).
We have now shown that the hypotheses of Proposition 8.3 hold, and thus that
FS(G) is a saturated fusion system over S. We have already seen that LcS(G) is*
* a
linking system associated to FS(G), and it remains only to show that LcS(G)^p*
*' BG^p.
As in Section 5, for any discrete group G, we let Op(G) be the category whose*
* objects
are the discrete ptoral subgroups P G, and where
MorOp(G)(P, Q) = Q\NG(P, Q) ~=Map G(G=P, G=Q).
Let OX(G) Op(G) be the full subcategory with object set X = X(G): the set of *
*all
intersections of subgroups in Sylp(G). For each discrete ptoral subgroup P G*
*, we let
P O2 X denote the intersection of all subgroups in Sylp(G) which contain P . Cl*
*early,
for any P and Q, NG(P, Q) NG(P O, QO), and so this defines a functor ()O from
Op(G) to OX(G). Since NG(P O, Q) = NG(P, Q) when Q 2 X(G), the two functors
incl
OX(G) !Op(G) (1)
()O
are adjoint.
Step 1: Let I and be the following functors from Op(G) to (G)spaces:
I(P ) = G=P and (P ) = EG xG I(P ) ~=EG=P.
Then for any full subcategory C Op(G),
ia1 a j.
hocolim!(I) = G=P0 x n ~
C n=0G=P0!...!G=Pn
is the nerve of the category whose objects are the cosets gP for all P 2 Ob (C)*
*, and
with a unique morphism gP ! hQ exactly when gP g1 hQh1. When C = OX(G),
this category has as initial object the intersection of all Sylow psubgroups o*
*f G, and
hence hocolim!OX(G)(I) is contractible. Since the Borel construction comm*
*utes with
homotopy colimits in this situation (being itself a special case of a homotopy *
*colimit),
i j
hocolim!( ) ~=EG xG hocolim!(I) ' BG. (2)
OX(G) OX(G)
Step 2: Fix some Q 2 X which is not pcentric. For each i 0, consider the fu*
*nctor
(
Hi(BP ; Fp) if P is Gconjugate t*
*o Q
Fi[Q]:Op(G)op ___! Ab where Fi[Q](P ) =
0 otherwise.
The subgroup CG(Q).Q=Q ~=CG(Q)=Z(Q) of AutOp(G)(Q) = N(Q)=Q acts trivially on
Fi[Q](Q), and contains an element of order p since Q is not pcentric. Hence by*
* Lemmas
5.10 and 5.12,
[Q]
lim*(Fi[Q]) ~= lim*(Fi[Q]) ~= * NG(Q)=Q; Fi (P ) = 0 for all,i
OX(G) Op(G)
where the first isomorphism follows from the adjoint functors (1).
54 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
Step 3: Now let OcX(G) OX(G) be the full subcategory with objects the pcent*
*ric
subgroups which lie in X. Let P1, P2, . .,.Pm S be representatives for thos*
*e G
conjugacy classes in X(G) which are not pcentric (a finite set by Lemma 8.6). *
* We
assume these are ordered such that Pi Pi+1 for each i.
For each i 0, consider the functor Fi:Op(G)op ___! Ab , defined by setting*
* Fi(P ) =
Hi(BP ; Fp) for all P . For all k = 0, . .,.m, define functors
(
0 if P ~ Pj, some j k
Fi,k:OX(G)op ___! Ab by setting Fi,k(P ) = G
Fi(P ) otherwise.
Here, "~ " means "Gconjugate", and these are all defined to be quotient functo*
*rs of
G
FiOX(G). In particular, Fi,0= FiOX(G) and Fi,m= FiOcX(G). Also, for all k,
[P ]
Ker Fi,ki Fi,k+1~= Fi kOX(G),
and the higher limits of this last functor vanish by Step 2. So there are isomo*
*rphisms
lim*(Fi) = lim*(Fi,0) ~= lim*(Fi,1) ~=. .~.=lim*(Fi,m) ~= lim*(F*
*i);
OX(G) OX(G) OX(G) OX(G) OcX(G)
whose composite is induced by restriction from OX(G) to OcX(G).
The spectral sequence for the cohomology of a homotopy colimit now implies th*
*at the
inclusion of OcX(G) into OX(G) induces a mod p homology isomorphism of homotopy
colimits of , and hence a homotopy equivalence
^ ' ^
hocolim!( ) p ! hocolim!( ) p . (3)
OcX(G) OX(G)
Also, the adjoint functors in (1) restrict to adjoint functors between OcX(G) a*
*nd Ocp(G),
and hence induce a homotopy equivalence
^ ^
hocolim!( ) p ' hocolim!( ) p . (4)
OcX(G) Ocp(G)
Step 4: Let Tpc(G) be the centric transporter category for G: the category wh*
*ose
objects are the pcentric subgroups of G, and where the set of morphisms from P*
* to Q
is the transporter NG(P, Q). By exactly the same argument as in [BLO1 , Lemma *
*1.2],
hocolim!( ) ' Tpc(G). (5)
Ocp(G)
The canonical projection functor Tpc(G) ___! Lcp(G) satisfies all of the hypot*
*heses of
the functor in [BLO1 , Lemma 1.3], except that we only know that
p
K(P ) def=KerAutTpc(G)(P ) i AutLcp(G)(P ) = O (CG(P ))
is a locally finite group all of whose elements have order prime to p (not nece*
*ssarily
a finite group). But this suffices to ensure that coinvariants preserve exact s*
*equences
of Z(p)[KP]modules, which is the only way this property of KP is used in the p*
*roof of
[BLO1 , Lemma 1.3]. Hence the induced map
Tpc(G) ! Lcp(G).
is a mod p homology equivalence. Together with (2), (3), (4), and (5), this sho*
*ws that
LcS(G)^p' Lcp(G)^p' Tpc(G)^p' BG^p.
Carles Broto, Ran Levi, and Bob Oliver *
* 55
We now finish the section by exhibiting a more concrete class of groups which*
* satisfy
the hypotheses of Theorem 8.7. A linear torsion group is a torsion subgroup of *
*GLn(k),
for any positive integer n and any (commutative) field k. These are also refer*
*red
to as "periodic linear groups", since their elements are all periodic transform*
*ations
(automorphisms of finite order) of a finite dimensional vector space.
The following facts about linear torsion groups are the starting point of our*
* work
here.
Proposition 8.8. The following hold for every field k, and every linear torsion*
* group
G GLn(k).
(a)G is locally finite.
(b)For p 6= char(k), every psubgroup of G is a discrete ptoral group.
Proof.Point (a) is a theorem of Schur, and is shown in [W , Corollary 4.9]. By *
*[W , 2.6],
every locally finite psubgroup of GLn(k) is artinian (when p 6= char(k)), and *
*hence is
discrete ptoral by Proposition 1.2.
In order to apply Theorem 8.7, it remains only to check that centralizers of *
*discrete
ptoral subgroups of linear torsion groups stabilize in the sense of Theorem 8.*
*7.
Proposition 8.9. Let A1 A2 A3 . .b.e an increasing sequence of finite abe*
*lian
psubgroups of a linear torsion group G GLn(k), where char(k) 6= p. Then ther*
*e is r
such that CG(Ai) = CG(Ar) for all i r.
Proof.Upon replacing k by its algebraic closure if necessary, we can assume tha*
*t k
is algebraically closed. Hence any representation over k of a finite abelian p*
*group A
splits as a sum of 1dimensional irreducible representations. Moreover, if A *
*GLn(k),
and kn = U1 . . .Um is the unique decomposition with the property that each Ui*
*is
a sum of irreducible modules with the same character and different Ui correspon*
*d to
different characters of A, then
Ym Ym
CGLn(k)(A) ~= Autk(Ui) ~= GLdi(k). (di= dim(Ui))
i=1 i=1
From this observation, it is clear that for any increasing sequence of such sub*
*groups Ai,
the centralizers CGLn(k)(Ai) stabilize for i sufficiently large, and hence the *
*stabilizers
CG(Ai) also stabilize.
Propositions 8.8 and 8.9 show that linear torsion groups satisfy all of the h*
*ypotheses
of Theorem 8.7. So as an immediate consequence, we get:
Theorem 8.10. Fix a linear torsion group G, a prime p different than the defini*
*ng
characteristic of G, and a Sylow subgroup S 2 Sylp(G). Then (S, FS(G), LcS(G)) *
*is a
plocal compact group, with classifying space LcS(G)^p' BG^p.
9. Compact Lie groups
Throughout this section, we fix a compact Lie group G and a prime p. Our main
result is to show that G defines a plocal compact group whose classifying spac*
*e has
the homotopy type of BG^p.
56 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
A compact Lie group P is called ptoral if its identity_component is a torus *
*and if
its group of components is a pgroup._ The closure P of a discrete ptoral sub*
*group
P G is a ptoral_group, since P0 is abelian and connected, hence a torus, and*
* has
ppower index in P. We will generally denote ptoral groups (including tori) by*
* P , Q ,
T , etc., to distinguish them from discrete ptoral groups P , Q, T , etc. Our *
*first task
is to identify the maximal (discrete) ptoral subgroups of G.
Definition 9.1. (a)For any ptoral group P , Sylp(P ) denotes the set of discre*
*te p
toral subgroups P P such that P .P0= P and P contains all ppower torsion *
*in
P 0.
__
(b)A discrete ptoral subgroup P G is snugly embedded if P 2 Sylp(P ).
___
(c)Sylp(G) denotes the set of all ptoral subgroups S G such that the identit*
*y com
ponent S0 is a maximal torus of G and S=S 02 Sylp(N(S 0)=S 0).
__
(d)Sylp(G) denotes the set of all discrete ptoral subgroups P G such that P*
* 2
___ __
Sylp(G) and P 2 Sylp(P ).
__
For any discrete_ptoral_subgroup_P_ G, P0 is_a torus, as noted above, and h*
*as
finite index in P . Hence P0 = (P )0, and ss0(P ) ~=P=P0. So P is snugly embedd*
*ed_in
G if and only if P0 is snugly embedded, and this holds exactly when rk(P ) = rk*
*(P ).
As an example of a subgroup which is not snugly embedded, one can construct a r*
*ank
one subgroup P ~=Z=p1 which is densely embedded in a torus (S1)r for r > 1.
__ __
Clearly, when rk(P ) < rk(P ), we cannot expect BP ^pand BP ^pto have the same
homotopy type. But we do get a homotopy equivalence when P is snugly embedded.
__
Proposition 9.2. If P G is snugly_embedded, then the inclusion of P in P indu*
*ces
a homotopy equivalence BP ^p' BP ^p.
__
Proof.This means showing that the inclusion of BP into BP induces an isomorphi*
*sm
on mod p cohomology. See, for example, [Fe, Proposition 2.3].
___
The following proposition is well known. It says that Sylp(G) is the set of m*
*aximal p
toral subgroups of G, that Sylp(G) is the set of maximal discrete ptoral subgr*
*oups of G,
and that each of these sets contains exactly one Gconjugacy class. Note in par*
*ticular
the case where G = P is ptoral: there is a unique conjugacy class of discrete *
*ptoral
subgroups snugly embedded in P , and every discrete ptoral subgroup of P is co*
*ntained
in a snugly embedded subgroup.
Proposition 9.3. The following hold for any compact Lie group G and any ptoral
group P .
___
(a)Any two subgroups in Sylp(G) are Gconjugate,_and each ptoral subgroup P G
is contained in some subgroup S 2 Sylp(G).
(b)Any two subgroups in Sylp(G) are Gconjugate, and each discrete ptoral subg*
*roup
P G is contained in some subgroup S 2 Sylp(G).
___
Proof.(a) The subgroups in Sylp(G) are clearly all_conjugate_to each other, si*
*nce
all maximal tori in G are conjugate. For any S 2 Sylp(G) with identity compone*
*nt
the maximal torus T = S 0, O(G=N(T )) = 1 (see [Br, Proposition 0.6.3]), and h*
*ence
Carles Broto, Ran Levi, and Bob Oliver *
* 57
O(G=S ) is prime to p. If Q is an arbitrary ptoral subgroup of G, then_O((G=S*
* )Q ) is
congruent mod p to O(G=S ), so (G=S )Q 6= ?, and hence Q gS g1 2 Sylp(G) for
some g 2 G.
(b) Assume first that G = P is ptoral. Set T = P 0, and let T T be the subg*
*roup
of elements of ppower torsion. By definition, Sylp(P ) is the set of all subgr*
*oups P T
such that P=T is the image of a splitting of the extension
1 ! T =T ! P=T ! P=T ! 1. (1)
The cohomology groups Hi(P =T ; T =T ) vanish for all i > 0, since P =T is a p*
*group
and T =T is uniquely pdivisible. Hence the extension (1) is split, and any two*
* splittings
are conjugate by an element of T =T . Thus Sylp(P ) 6= ?, and its elements are *
*conjugate
to each other by elements of T .
Now let Q P be an arbitrary discrete ptoral subgroup. Then QT is also a di*
*screte
ptoral subgroup (since T C P ), and QT=T is the image of a splitting of the ex*
*tension of
T =T by QT =T . We have seen that any two such splittings are conjugate by elem*
*ents
of T =T , and hence they all extend to splittings of the extension by P =T . I*
*n other
words, there is a subgroup P 2 Sylp(P ) which contains QT , and hence contains *
*Q.
_
Now let G be an arbitrary compact Lie group. For any S, S0 2 Sylp(G), S is G
__ _ ______ _
conjugate to S0 by (a), so S = gS0g1 for some g 2 G, and S, gS0g1 2 Sylp(S ).*
* We
_
have just shown that all subgroups in Sylp(S ) are conjugate, and hence S and S*
*0 are
*
* __
conjugate. If P G is an arbitrary discrete ptoral subgroup, then_its_closure*
* P is a
ptoral subgroup, and hence contained in some maximal subgroup S 2 Sylp(G) by (*
*a)
again. So there is some S 2 Sylp(S ) Sylp(G) which contains P .
We next need some information about the outer automorphisms of (discrete) pt*
*oral
subgroups of G.
Lemma 9.4. The following hold for all discrete ptoral subgroups P, Q G.
(a)If P Q, then Out _Q(P ) is a finite pgroup, and Out _Q(P ) = Out Q(P ) if*
* Q is
snugly embedded in G. In particular, OutQ_(Q) = 1 if Q is snugly embedded in*
* G.
__
(b)Out G(P ) and OutG (P ) are both finite.
(c)If Q is snugly embedded, then the natural map
~= __ __
Rep G(P, Q) ! Rep G(P , Q)
is a bijection.
__ __ __
Proof.(a) Choose Q0 Q such that Q0 = Q and Q0 is snugly embedded. Then
OutQ0(P ) is a finite pgroup by Proposition 1.5(c). The first statement thus *
*follows
from the second.
Now assume Q is snugly embedded. We must show that Out _Q(P ) = Out Q(P ); or
equivalently that AutQ_(P ) = AutQ (P ). Fix x 2 N_Q(P ), and set P  = pk.
__ __ __
Let Q =Q be the set of left cosets gQ for_g 2 Q ,_and let (Q =Q)P be the fixe*
*d point
set of the left P action._Then for g 2 Q , gQ 2 (Q =Q)P if and_only_if_g1P g *
* Q. In
particular, xQ 2 (Q =Q)P since x normalizes P and P Q. Since Q =Q = Q0=Q0 and
58 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
*
* __
the latter group is uniquely pdivisible (since Q is snugly embedded), there is*
* y 2 Q 0
k __ P
such that yp 2 xQ and yQ 2 (Q =Q) .
Set Y k Y k
by= (aya1) = yp . ((y1ay).a1) 2 yp Q = xQ;
a2P a2P
where the inclusion holds since P Q and y1P y Q. Then by2 C_Q(P ). Since*
* x
was arbitrary, this proves that N_Q(P ) = C_Q(P ).NQ(P ), and finishes the proo*
*f that
AutQ_(P ) = AutQ (P ). In the case P = Q, this shows that OutQ_(Q) = OutQ (Q) =*
* 1.
(b) The kernel of the homomorphism
__
OutG(P ) ! Out G(P ) (1)
is OutP_(P ). By (a), this is always finite, and is trivial if P is snugly embe*
*dded. If P
__ __ __ *
* __
is snugly embedded, i.e., if P 2 Sylp(P ), then NG(P ) = P.NG(P ) (any subgroup*
* of P
__
which is Gconjugate to P is also P conjugate to_P_), and_hence_the_map_in (1)*
* is also
surjective. Thus in this case, Out G(P ) ~=Out G(P ) ~=NG(P )=P .CG(P ) is a co*
*mpact_
Lie group, all torsion subgroups of which are finite by Theorem 1.5. Thus Out *
*G(P )
is finite (since otherwise it would contain S1). If P is an arbitrary discrete*
* ptoral
subgroup of G, then the kernel and the image of the map in (1) are finite, and *
*hence
OutG (P ) is also finite in this case.
(c) Assume P, Q S, where_Q_is snugly embedded. We must show that the map
from RepG (P, Q) to RepG (P , Q) which sends_a_homomorphism_to_its unique conti*
*nuous
extension is_a_bijection. For any_' 2 Hom G(P , Q), '(P ) is Q conjugate to a *
*subgroup
of Q 2 Sylp(Q ), and hence ' is Qconjugate to a homomorphism which sends P int*
*o Q.
This proves surjectivity._ To_prove injectivity, fix '1, '2 2 Hom G (P, Q) whic*
*h induce
the same class in Rep G(P , Q), and set Pi = Im ('i). Then '2 = O O'1 for some
O 2 Iso_Q(P1, P2). We must show that O 2 IsoQ(P1, P2), and it suffices to do th*
*is when
_
O 2 Iso_Q0(P1, P2). In this case, P1Q0 = P2Q0, so O extends to O 2 Aut _Q0(P1Q*
*0).
Also, P1Q0 is snugly embedded since Q is, so OutQ_0(P1Q0) = 1 by (b), and hence*
* O is
__
conjugation by an element of Q 0\ P1Q0 = Q0.
The fusion system of a compact Lie group is defined exactly as in Section 8. *
*For any
S 2 Sylp(G), FS(G) is the fusion system over S where for P, Q S,
Mor FS(G)(P, Q) = Hom G(P, Q) ~=NG(P, Q)=CG(P )
is the set of homomorphisms from P to Q induced by conjugation by elements of G.
Here, as usual,
NG(P, Q) = {x 2 G  xP x1 Q}
denotes the transporter set.
Lemma 9.5. For each maximal discrete ptoral subgroup S 2 Sylp(G), FS(G) is a
saturated fusion system over S. Also, a subgroup P S is fully centralized in *
*FS(G)
if and only if CS(P ) 2 Sylp(CG(P )).
Proof.We must show that conditions (a), (b), and (c) of Proposition 8.3 all hol*
*d.
For each discrete ptoral subgroup P G, Out G(P ) is finite by Lemma 9.4(b),*
* so
AutG (P ) ~=NG(P )=CG(P ) is a torsion group. Hence for each g 2 NG(P ), .CG*
*(P ) is
a finite extension of CG(P ), thus a closed subgroup, and so the coset gCG(P ) *
*contains
Carles Broto, Ran Levi, and Bob Oliver *
* 59
elements of finite order. Also, for each finite subgroup H=CG(P ) in NG(P )=CG(*
*P ), H
is a closed subgroup of G, and hence has Sylow psubgroups in the sense of Sect*
*ion 8.
If P1 P2 P3 . . .is an increasing sequence of discrete ptoral subgroups *
*of G,
then the centralizers CG(Pi) form a decreasing sequence of closed subgroups of *
*G, and
hence is constant for i sufficiently large.
Thus Proposition 8.3 applies: for any S 2 Sylp(G), FS(G) is a saturated fusi*
*on
system over S, and a subgroup P S is fully centralized in FS(G) if and only *
*if
CS(P ) 2 Sylp(CG(P )).
Recall from Section 8 that a discrete ptoral subgroup P G is called pcent*
*ric
in G if Z(P ) 2 Sylp(CG(P )). By analogy with this definition, a ptoral subgr*
*oup
___
P G is called pcentric if Z(P ) 2 Sylp(CG(P )). We next note some conditio*
*ns
which characterize ptoral and discrete ptoral subgroups of G which are pcent*
*ric.
Lemma 9.6. The following hold for any discrete ptoral subgroup P G.
__ __ __ __ __
(a)If P is pcentric in G, then NG(P )=P is finite, and CG(P )=Z(P ) is finite*
* of order
prime to p.
__
(b)If P is pcentric in G, then P is pcentric in G.
__
(c)If P is pcentric in G and P is snugly embedded, then P is pcentric in G.
Proof.(a) Assume P is pcentric in G, and consider the groups
CG(P )=Z(P ) ~=P .CG(P )=P and OutG (P ) ~=NG(P )=(P .CG(P )).
The first group is finite of order prime to p since Z(P ) is a maximal ptoral *
*subgroup
of CG(P ) which is also central. The second group is finite by Lemma 9.4(b). He*
*nce
NG(P )=P is also finite.
_____ __
(b) If P is_pcentric_in G,_then Z(P ) Z(P ) is a maximal ptoral subgroup in
CG(P ) = CG(P ), and hence P is also pcentric in G.
(c) Assume P 2 Sylp(P ). If x 2 Z(P ) has ppower order, then since [x, P0] = *
*1, the
only elements of ppower order in xP 0 are those in xP0. Since some element of *
*xP 0
lies in P and has ppower order, this shows that x 2 P , and hence that x 2 Z(P*
* ). In
other words, Z(P ) 2 Sylp(Z(P )). So if P is pcentric in G, then Z(P ) is a m*
*aximal
discrete ptoral subgroup of CG(P ) = CG(P ), and hence P is also pcentric in *
*G.
We want to apply Proposition 4.6, to construct a centric linking system LcS(G)
associated to FS(G), and to show that LcS(G)^p' BG^p. This means constructing
a rigidification of the homotopy functor B :P 7! BP ; which by Proposition 5.9*
* is
equivalent to constructing a rigidification of the homotopy functor B ^p:P 7! B*
*P ^p.
This last is closely related to the homotopy decomposition of BG constructed in*
* [JMO ].
___
For any S 2 Sylp(G), we let OS(G) denote the category whose objects are the p*
*toral
subgroups of S , and where
Mor OS(G)(P , Q ) = Q \NG(P , Q ).
Define B :OS(G) ! Top by setting
Qx .x1
B (P ) = EG=P and B(P ! Q ) = (EG=P ! EG=Q ).
60 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
Let
: hocolim!(B ) ! EG=G = BG
OS(G)
be the map induced by the obvious surjections from B (P ) = EG=P onto BG = EG=*
*G.
Lemma 9.7. Fix a maximal ptoral subgroup S 2 Sylp(G). Let OcS(G) OS(G) be
the full subcategory whose objects are those ptoral subgroups of S which are p*
*centric
in G, and let
Bc:OcS(G) ! Top and c: hocolim!(B c) ! BG
OcS(G)
be the restrictions of B and , respectively. Then c is a mod p homology equiv*
*alence.
Proof.Define
fi
X = P S ptoralfiNG(P )=P  < 1, Op(NG(P )=P ) = 1 or P is pcentric.
Let OXS(G) OS(G) be the full subcategory with object set X, and let
B X :OXS(G) ! Top and X :hocolim!(B X ) ! BG
OXS(G)
be the restrictions of B and .
By [JMO , Theorem 1.4], X is a mod p homology equivalence. So to prove the
proposition, we must show that the inclusion of hocolim!(B c) in hocolim*
*!(B X ) is a mod
p homology equivalence. Set F = H*(B X (); Fp), regarded as a functor on OXS(G*
*)op.
Let F0 F be the subfunctor defined by setting F0(P ) = 0 if P is pcentric in*
* G and
F0(P ) = F (P ) otherwise. We claim that lim*(F0)= 0. Assuming this, we see th*
*at
lim*(H*(B X (); Fp))~=lim*(F=F0)~=lim*(H*(B c(); Fp));
OXS(G) OXS(G) OcS(G)
the last step since there are no morphisms from any object of the subcategory to
any object not in the subcategory. This shows that the spectral sequences for *
*the
cohomology of hocolim!(B X ) and hocolim!(B c) have isomorphic E2ter*
*ms, and hence
that the inclusion is a mod p homology equivalence.
It remains to prove that lim*(F0)= 0. By [JMO , Proposition 1.6], X0 conta*
*ins
finitely many Gconjugacy classes. Hence by [JMO , Proposition 5.4] and an app*
*ropri
ate finite filtration of F0, it suffices to prove that *(NG(P )=P ; H*(BP ; Fp*
*)) = 0
for each ptoral subgroup P in OXS(G) which is not pcentric. For each such P*
* ,
CG(P ).P =P ~= CG(P )=Z(P ) is a finite group of order a multiple of p which a*
*cts
trivially on H*(EG=P ; Fp) ~=H*(BP ; Fp), and hence *(NG(P )=P ; H*(BP ; Fp)) *
*= 0
by [JMO , Proposition 5.5].
We are now ready to construct a rigidification of the homotopy functor B ^p.
Proposition 9.8. Fix a maximal discrete ptoral subgroup S 2 Sylp(G), and set F*
* =
FS(G) for short. Let Fcs Fc be the full subcategory of subgroups P S which *
*are
pcentric in G and snugly embedded, and let Ocs(F) Oc(F) be its orbit categor*
*y.
Then there is a functor
bB:Ocs(F) ! Top
which is a rigidification of the homotopy functor B ^p, and a homotopy equivale*
*nce
i j
b : hocolim(Bb) ^ ! BG^.
! p p
Ocs(F)
Carles Broto, Ran Levi, and Bob Oliver *
* 61
_ ___
Proof.Set S = S 2 Sylp(G). We will construct orbit categories and functors as
indicated in the following diagram:
___cl__//_c ooopro_ c
Ocs(F) O S(G) OS(G)
HH vv
HHH  vvv
HHH _Bc vvv (1)
BcsHHHH  vvv Bc
H##Hfflfflvvv
Top,
together with mod p homology equivalences
cl* __
hocolim!(Bcs)_//hocolim(B c)_____//hocolim(B c)
N ! ' !p
NNNN  pppp
NNNN _ pppp (2)
csNNNNN  c pppp c
NN  ppp
NN''fflfflxxpp
BG .
We then set bB= (Bcs)^p.
The category OcS(G), together with_B cand c, were already constructed in Lem*
*ma
9.7. We construct OcS(G), B c, and cin Step 1 (and prove the properties we ne*
*ed);
and then do the same for Ocs(F), Bcs, and csin Step 2.
__
Step 1: Define O cS(G) by setting
__ fi
Ob (O cS(G)) = Ob (OcS(G)) = P S fiPptoral and pcentric in,G
and
Mor_Oc (P , Q ) = Q \NG(P , Q )=CG(P ) ~=Rep G(P , Q ).
S(G)
__
Let_B cbe the left homotopy Kan extension of B calong the projection functor. L*
*et
cbe the composite of c with the standard homotopy equivalence
__ '
:hocolim!(B c) ! hocolim(B c)
_ !c
OcS(G) OS(G)
__
of [HV , Proposition 5.5]. Thus __cis a mod p homology equivalence by Lemma 9.*
*7,
and it remains only to show that B cis a rigidification of B ^p(after pcomplet*
*ion).
This means showing that the natural morphism of functors
__
B c! B cOpr
(natural up to homotopy) is a mod p homology equivalence on all objects. By def*
*inition,
for each P ,
__
B c(P ) = hocolim!(B cO ,).
pr#P
ff __c
Here, pr#P is the overcategory whose objects are the morphisms Q ___! P in O *
*S(G),
and where a morphism from (Q , ff) to (R , fi) is a morphism ' 2 Mor OcS(G)(Q ,*
* R) such
that ff = fi Opr('). Also, , is the forgetful functor from pr#P to OcS(G).
Consider the spectral sequence
j i+j __
Ei,j2~=limiH (B cO ,(); Fp) =) H (B c(P ); Fp) . (3)
pr#P
62 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
For each Q in OcS(G), set
prP
K(Q ) = Ker AutOcS(G)(Q ) ! Aut_Oc (Q ) ~=CG(Q )=Z(Q ).
S(G)
This is a finite group of order prime to p, and it acts trivially on H*(EG=Q ).*
* Since
K(Q ) acts trivially on the homology H*(B cO ,(Q , ff); Fp) = H*(EG=Q ; Fp) for*
* each
object (Q , ff), this functor factors through the overcategory Id#P . The proje*
*ction of
pr#P onto Id#P satisfies the hypotheses of [BLO1 , Lemma 1.3] (in particular*
*, the
target category is obtained from the source by dividing out by these automorphi*
*sm
groups K(Q ) of order prime to p), and hence
(
* i * H*(B c(P ); Fp)if i = 0
limiH (B cO ,(); Fp) ~= limH (B cO ,(); Fp) ~=
pr#P Id#P 0 if i > 0.
Here, the last isomorphism holds since Id#P_has final object (P , Id). The spe*
*ctral
sequence (3) thus collapses, and hence H*(B c(P ); Fp) ~=H*(B_c(P_); Fp) (and t*
*he iso
morphism is induced by the natural inclusion of B c(P ) into B c(P )).
Step 2: The "closure functor"
__
Ocs(F) cl!OcS(G),
__
is defined to send P to P. It induces a bijection between isomorphism classes o*
*f objects
by definition of Fcs and Lemma 9.6(b,c), and induces bijections on morphism set*
*s by
Lemma 9.4(c). So this is an equivalence of categories.
__ __
Set Bcs= B cOcl. Since B cis (after pcompletion)_a rigidification of the hom*
*otopy
functor P 7! BP ^pby Step 1, and since BP ^p' BP ^pwhen P is snugly embedded
(Proposition 9.2), Bcsis a rigidification of the homotopy functor B^p:P 7! BP ^*
*p(again,
up to pcompletion).
__
Now let csbe the composite of cwith the map
__
hocolim!(Bcs) cl*!hocolim!(B c)
Ocs(F) _OcS(G)
__
induced by cl. Then hocolim!(B c) ' hocolim!(Bcs) since clis an equi*
*valence of cate
__
gories, and thus csis a mod p homology equivalence since cis by Step 1.
Now set bB= (Bcs)^pand let
i j i j ( )^
b : hocolim(Bb) ^ ' hocolim(B ) ^ csp!BG^
! p ! cs p p
Ocs(F) Ocs(F)
be the completion of cs. Then bBis a rigidification of the homotopy functor B *
*^p(see
Proposition 9.2), and b is a homotopy equivalence.
We also need the following result about snugly embedded subgroups:
Lemma 9.9. For each discrete ptoral subgroup P G, P ois snugly embedded.
Proof.Fix S 2 Sylp(G), and set T = S0 and pm = exp(S=T ). By Definition 3.1,
m C (Q)
P o= P .I(Q), where Q = P [m]= P0,_and_I(Q) = T W . Then
I(Q) contains all elements of ppower order in I(Q) = T CW (Q), and hence is sn*
*ugly
embedded. Since [P o: I(Q)] is finite, P ois also snugly embedded.
Carles Broto, Ran Levi, and Bob Oliver *
* 63
We are now ready to prove the main result.
Theorem 9.10. Fix a compact Lie group G and a maximal discrete ptoral subgroup
S 2 Sylp(G). Then there exists a centric linking system LcS(G) associated to FS*
*(G) such
that (S, FS(G), LS(G)) is a plocal compact group with classifying space LS(G)*
*^p'
BG^p.
Proof.Set F = FS(G) for short; a saturated fusion system by Lemma 9.5. Let Fcs
Fc be the full subcategory with objects the set of all P S which are pcentri*
*c and
snugly embedded in G, and let Ocs(F) be its orbit category. By Lemma 9.9, Fcs *
*Fco.
By Proposition 9.8, there is a functor
bB:Ocs(F) ! Top
which is a rigidification of the homotopy functor B ^p, and a homotopy equivale*
*nce
i j
b : hocolim(Bb) ^ ! BG^.
! p p
Ocs(F)
By Proposition 5.9, there is a functor eB:Ocs(F) ! Top which is a rigidifica*
*tion of
the homotopy functor B , and a natural transformation of functors O: eB! Bb *
*such
that O(P ) is homotopic to the completion map for each P . By Proposition 4.6, *
*there is
a centric linking system LcsS(G) associated to Fcs whose nerve has the homotopy*
* type
of hocolim!(Be), and thus
i j i j
LcsS(G)^p' hocolim!(Be) ^p' hocolim!(Bb) ^p' BG^p.
Ocs(F) Ocs(F)
__ __
Now define Fc ! Fcs by setting (P ) = P .(P 0)(p),_where (P 0)(p)denotes *
*the
subgroup of elements of ppower order in the torus P 0. By Lemma 9.4(c), for e*
*ach
P 2 Ob (Fc) and Q 2 Ob (Fcs),
__ __
RepG (P, Q) ~=Rep G(P , Q) ~=Rep G( (P ), Q),
*
* __
and thus is left adjoint to the inclusion. Also, for each P , CG( (P )) = CG*
*(P ) =
CG(P ), and hence Z( (P )) = Z(P ). So if we define LcS(G) to be the pullback *
*of
LcsS(G) and Fc over Fcs, then it is a centric linking system, and LcS(G) ' L*
*csS(G).
(Compare this argument with the proof of Proposition 4.5(b).)
The above construction of the linking system of G has the disadvantage that i*
*t seems
rather arbitrary. We know, by Theorem 7.4, that there is (up to isomorphism) at*
* most
one linking system LcS(G) such that LcS(G)^p' BG^p, but we would really like *
*to have
a more obvious algebraic connection between LcS(G) and the group G itself. We e*
*nd
this section by showing that LcS(G) can, in fact, be obtained as a subquotient *
*of the
transporter category of G _ although not in a completely canonical way.
Fix a compact Lie group G, and choose S 2 Sylp(G). The transporter category
TSc(G) of G over S is the category whose objects are the subgroups of S that ar*
*e p
centric in G, and where Mor TSc(G)(P, Q) = NG(P, Q), for each pair of objects P*
* and Q
of TSc(G). Let CG :Oc(F)op! Ab be the functor which sends P to its centrali*
*zer.
For any subfunctor CG, TSc(G)= denotes the quotient category with the same
objects as TSc(G), and where
Mor TSc(G)=(P, Q) = Mor TSc(G)(P, Q)= (P ) = NG(P, Q)= (P ).
For example, in this notation, FcS(G) = TSc(G)=CG.
64 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
For each P 2 Ob (Fc), there is a central extension
_____ _____
1 ! Z(P ) ! CG(P ) ! CG(P )=Z(P ) ! 1,
_____ _____
where Z(P ) is abelian and ptoral and CG(P )=Z(P ) is finite of order prime to*
* p (by
definition of pcentric). Hence the set of elements of CG(P ) of finite order *
*prime to
p forms a subgroup, which we denote here p0(P ). Also, Z(P ) and p0(P ) are *
*both
normal subgroups of CG(P ), and the quotient group CG(P )=(Z(P ) x p0(P )) is *
*a Q
vector space. As earlier, we write Z(P ) = Z(P ), and regard Z, p0, and Z x *
*p0as
subfunctors of CG.
Lemma 9.11. The extension
pr c
TSc(G)=(Z x p0) ! F
is split, by a splitting which sends Out P(P ) to P=Z(P ) for each object P ; a*
*nd such a
splitting is unique up to natural isomorphism of functors.
Proof.For all P, Q 2 Ob (Fc), choose maps
ffi
oeP,Q: Hom F(P, Q) ! NG(P, Q) Z(P ) x p0(P ) = Mor TSc(G)=(Zx(p0)P, *
*Q)
which split the natural projection. This can be done in such a way that for ea*
*ch
' 2 Hom F(P, Q) and each g 2 Q, oeP,Q(cg O') = [g] OoeP,Q(') (define it first o*
*n orbit
representatives for the action of Inn(Q) and then extend it appropriately). Als*
*o, when
Q = P , we let oeP,P(IdP) be the class of 1 2 NG(P ).
The "deviation" of {oeP,Q} from being a functor is a 2cocycle with values in*
* the
functor CG=(Z x p0), and the assumption that they commute with the Inn(Q)acti*
*ons
implies that we get a cocycle over the orbit category Oc(F). If, furthermore, *
*this
cocycle is a coboundary, then the oeP,Q can be replaced by maps oe0P,Qwhich def*
*ine
a splitting functor. The obstruction to the existence of such a splitting thus*
* lies in
lim2(CG=(Z x p0)). In a similar (but simpler) way, the obstruction to unique*
*ness is
Oc(F)
seen to lie in lim1(CG=(Z x p0)).
Oc(F)
We will show that both of these groups vanish, using Lemma 5.7 (and an argume*
*nt
similar to that used to prove Proposition 5.8). Let F be the functor CG=(Z x *
*p0).
As in_the proof of Proposition 5.8, set T = S0, Q = CS(T ), and = Out F(Q). *
*Set
M = T=(torsion), regarded as a Q[ ]module. Let
: Op( )op! Z(p)mod
be the functor ( ) = M for all psubgroups . Then F (P ) ~= (Out P(Q))
(functorially)_for all P S containing Q, and Out Q(P ) acts trivially on F (*
*P ) ~=
Z(P )=(torsion) for each P . The hypotheses of Lemma 5.7 thus hold, and so
limi(F~)=limi( )
Oc(F) Op( )
for all i. Since is a Mackey functor, these groups vanish for all i 1 by [*
*JM ,
Proposition 5.14].
We are now ready to construct a more explicit linking system LcS(G), and prov*
*e it
is isomorphic to the one already constructed in Theorem 9.10.
Carles Broto, Ran Levi, and Bob Oliver *
* 65
Proposition 9.12. Let G be a compact Lie group, and choose S 2 Sylp(G). Fix a
pr c c
splitting s of TSc(G)=(Z x p0) ! F , and define LS(G) to be the pullback ca*
*tegory
in the following pullback diagram
~s c
LcS(G) ______! TS (G)= p0
 
ss pr
# #
s c
FcS(G) ___! TS (G)=(Z x p0) .
Then LcS(G) is a centric linking system associated to FcS(G), and is isomorphic*
* to the
centric linking system of Theorem 9.10. In particular, LcS(G) ~s!TSc(G)= p0d*
*escribes
the linking system LcS(G) as a subquotient of the transporter catgeory.
Proof.We will first show that the pullback category LcS(G) is a centric linking*
* system
associated to FS(G). Since s and prare the identity on objects, we can as well *
*assume
that the pullback category LcS(G) has the same objects, and that ~sand ss are t*
*he
identity on objects. Then for any pair of objects P, Q S pcentric in G, we h*
*ave
Mor LcS(G)(P, Q) =
fi
(', _) fi' 2 Mor FcS(G)(P, Q), _ 2 Mor TSc(G)=(p0P, Q), and ss(') = p*
*r(_)
Now, for each P S which is pcentric in G, we have P NG(P )= p0(P ) and the*
*n we
can define distinguished homomorphisms
ffiP :P ! AutLcS(G)(P )
by setting ffiP(g) = (cg, g). Conditions (A), (B), and (C) in the definition of*
* a centric
linking system are easily checked.
Next we will find a map LcS(G) ! BG^pthat commutes with the respective na*
*t
ural maps from BS. To do this, we first lift LcS(G) to a subcategory LecS(G) o*
*f the
transporter category TSc(G), defined via the pullback square:
LecS(G)______incl!TSc(G)
 
 
# #
~s c
LcS(G) ____! TS (G)= p0
We will then construct the maps in the following commutative diagram:
oBS MM
ooo  MMMM
oooo  MMMM
wwoooo fflffl M&&M
LcS(G)^po'o_ LecS(G)^p___//BG^p.
We proceed in two steps.
(a) A map LecS(G) ! LcS(G) commuting with the respective natural maps fr*
*om
BS is induced by the functor LecS(G) ! LcS(G). We will show that it is a mod*
* p
homology equivalence.
By definition of eLcS(G), for all P, Q S centric, we have that 0p(P ) acts*
* freely on
Mor eLcS(G)(P, Q) and the orbit set if Mor LcS(G)(P, Q). In particular,
h i
0p(P ) = Ker Aut eLcS(G)(P ) ! AutLcS(G)(P.)
66 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
Recall that p0(P ) is the subgroup of elements of CG(P ) of finite order pri*
*me to p. It
sits in an extension p0(P_)0_! p0(P ) ! C0G(P ), where p0(P )0 is the set_*
*of_elements_
of the maximal torus of Z(P ) of finite order prime to p and C0G(PS) = CG(P )=Z*
*(P ).
Therefore p0(P ) is locally finite and can be written as union m 0 p0(P )m o*
*f finite
groups of order prime to p. A generalized version of [BLO1 , Lemma 1.3] now ap*
*plies
to the constant functor defined on LcS(G) and the result follows.
In fact, [BLO1 , Lemma 1.3] generalizes to allow that the kernels K() be co*
*untable
increasing unions of finite groups of order prime to p. Proving this requires s*
*howing,
for any such K, that H0(K; ) is an exact functor on the category of Z(p)[K]mo*
*dules.
But if K is the union of a sequence of subgroups K1 K2 . . .each of which is
finite, then H0(Ki; ) is exact for each i, H0(K; M) ~=colim!iH0(Ki; M) for *
*each M,
and hence H0(K; ) is exact since direct limits of this type are exact.
(b) A map LcS(G) ! BG that commutes up to homotopy with the respective
maps from BS is defined as follows. Compose the inclusion eLcS(G) incl!TSc(G*
*) with
the functor TSc(G) ! B(G). Here, B(G) is the topological category with one ob*
*ject
and the Lie group G as morphisms (and all other categories are discrete), and t*
*he
functor sends the morphism g 2 NG(P, Q) to g 2 G for all objects P, Q of TSc(G).
The nerve of B(G) is the topological bar construction B G ' BG, and the composi*
*te
functor induces a map LecS(G) ! B(G) ' BG.
Finally, Theorem 9.10 defines the centric linking system of G over S and show*
*s that
the pcompleted nerve is homotopy equivalent to BG^p. This combines with the map
constructed above, so that Lemma 7.3 implies that the pullback category LcS(G) *
*is
isomorphic to the centric linking system of Theorem 9.10, and then, also, that *
*the map
LecS(G) ! BG constructed in step (b) is actually a homotopy equivalence af*
*ter
pcompletion.
10. pcompact groups
A pcompact group is a pcomplete version of a finite loop space. As defined *
*by Dwyer
and Wilkerson in [DW ], a pcompact group is a triple (X, BX, e), where X is a*
* space
such that H*(X; Fp) is finite, BX is a pointed pcomplete space, and e: X ! *
*(BX)
is a homotopy equivalence. If G is a compact Lie group such that the group of c*
*om
ponents ss0(G) is a finite pgroup, then upon setting BGb = BG^pand bG= (BGb),*
* the
triple (Gb, BGb, Id) is a pcompact group. For general references on pcompact *
*groups,
we refer to the original papers by Dwyer and Wilkerson [DW ] and [DW2 ], and *
*also to
the survey article by Jesper Moller [Mo ].
When T ~= (S1)r is a torus of rank r, then the pcompletion bT= (BT ^p) of T*
* is
called a pcompact torus of rank r. Both BTb ' K((bZp)r, 2) and bT' K((bZp)r, 1*
*) are
EilenbergMacLane spaces. A pcompact toral group is a pcompact group (Pb, BPb*
*, e)
such that ss1(BPb) is a pgroup, and the identity component of bPis a pcompact*
* torus
with classifying space the universal cover of BPb.
If X is either a discrete ptoral group or a pcompact group, and Y is a pco*
*mpact
group, a homomorphism f :X ! Y is by definition a pointed map Bf :BX ! BY .
Two homomorphisms f, f0:X ! Y are conjugate if Bf and Bf0 are freely homotopic;
i.e., via a homotopy which need not preserve basepoints. Given a homomorphism
Carles Broto, Ran Levi, and Bob Oliver *
* 67
f :X ! Y , the homotopy fibre of Bf is denoted Y=f(X), or just Y=X if f is unde*
*rstood
from the context. With this notation, f is called a monomorphism if H*(Y=f(X); *
*Fp)
is finite. By [DW , Proposition 9.11], a homomorphism f is a monomorphism if *
*and
only if H*(BX; Fp) is a finitely generated H*(BY ; Fp)module via H*(f; Fp).
If bPis an arbitrary pcompact toral group, a discrete approximation to bPis *
*a pair
(P, f), where P is a discrete ptoral group and Bf :BP ! BPbinduces an isomorph*
*ism
in mod p cohomology. By [DW , Proposition 6.9], every compact ptoral group ha*
*s a
discrete approximation. Each discrete ptoral group P is a discrete approximati*
*on of
(Pb, BPb, Id), where BPb = BP ^pand bP= (BPb). Hence every monomorphism f :P !
bf
X from a discrete ptoral group to a pcompact group factors as P ___! bP___! *
* X:
a discrete approximation followed by a monomorphism of pcompact groups. Lemma
1.10 says, among other things, that any two discrete approximations of a pcomp*
*act
toral group are isomorphic.
If f :X ! Y is a homomorphism of pcompact groups, the centralizer of f in Y *
*is
defined to be the triple (CY (X, f), BCY (X, f), Id), where
BCY (X, f) = Map (BX, BY )Bf and CY (X, f) = (BCY (X, f)).
Whenever f is understood, we simply write CY (X) for CY (X, f).
A discrete ptoral subgroup of a pcompact group X is a pair (P, f), where P *
*is a
f
discrete ptoral group and Pb___! X is a monomorphism. We write BCX (P, f) =
BCX (Pb, f) = Map (BP, BX)Bf and CX (P, f) = CX (Pb, f) for short. By [DW , x*
*x5
6], CX (P ) is a pcompact group, and the homomorphism CX (P ) ! X (induced by
evaluation at the basepoint of BP ) is a monomorphism. The subgroup (P, f) is c*
*alled
central if this monomorphism CX (P ) ! X is an equivalence.
Proposition 10.1. Let X be any pcompact group.
f u
(a)X has a maximal discrete ptoral subgroup S ! X. If P ! X is any other
discrete ptoral subgroup of X, then Bu ' Bf OB_ for some _ 2 Hom (P, S); and
(P, u) is maximal if and only if p  O(X=u(Pb)). Here, Euler characteristic*
*s are
taken with respect to homology with coefficients in Fp.
f
(b)The centralizer CX (P, f) of any discrete ptoral subgroupSP ! X is again*
* a p
compact group, and a subgroup of X. Also, if P = 1n=1Pn, then BCX (P ) '
BCX (Pn) for n large enough.
f
(c)A discrete ptoral subgroup P ! X is central if and only if there is a ma*
*p BP x
BX ! BX whose restriction to BP x * is Bf and whose restriction to * x BX
is the identity. When this is the case, then P is abelian, and there is a f*
*ibration
f
sequence BP ^p! BX ! B(X=P ) where B(X=P ) is the classifying space of a
pcompact group X=P .
Proof.Point (a) follows mostly from [DW2 , Propositions 2.10 & 2.14] together *
*with
Lemma 1.10. If (P, u) is not maximal, then since u factors through S, O(X=u(Pb)*
*) =
O(X=f(Sb)) . O(Sb=Pb), and the last factor is a multiple of p.
Point (b) is shown in [DW , Proposition 5.1 & Theorem 6.1]. In point (c), a *
*central
subgroup is abelian by [DW2 , Theorem 1.2], while the other two claims are sho*
*wn in
[DW , Lemma 8.6 & Proposition 8.3].
68 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
As in other contexts, the maximal discrete ptoral subgroups of a pcompact g*
*roup
X will be referred to as Sylow psubgroups of X.
The fusion system of a pcompact group is easily defined: it is just the fusi*
*on system
of the space BX, as defined in [BLO2 , Definition 7.1].
f
Definition 10.2. For any pcompact group X with Sylow psubgroup S ! X, let
FS,f(X) be the category whose objects are the subgroups of S, and where for P, *
*Q S,
fi
MorFS,f(X)(P, Q) = Hom X(P, Q) def=' 2 Hom (P, Q) fiBfBQ OB' ' BfBP .
We next want to show that FS,f(X) is saturated. Before doing this, we need to*
* define
and study normalizers of discrete ptoral subgroups of pcompact groups. We als*
*o need
to establish an "adjointness" relation which corresponds to the equivalence (fo*
*r groups)
between homomorphisms Q ! NG(P ) and homomorphisms P o Q ! G.
Fix a pcompact group X and a Sylow psubgroup f :S ! X. For any subgroup
P S and any discrete ptoral subgroup K AutX (P ), set
^
BNKX(P ) = EK xK BCX (P ) p,
where K acts on BCX (P ) = Map (BP, BX)Bffvia the action on P . Set NKX(P ) =
(BNKX(P )). Since the action of K on BP fixes the basepoint, evaluation at the
basepoint of BP defines a map
^
ev: BNKX(P ) = EK xK Map (BP, BX)Bf p ! BX.
_
If Q is any discrete ptoral group, and ae2 Hom (Q, K), then any homomorphism
EQ xQ BP ~=B(P o_aeQ) ! BX
is adjoint to a Qequivariant map
EQ ! Map (BP, BX)fBP = BCX (P ) ,
_
where Q acts on BCX (P ) via the action on P defined by ae(and via the trivial *
*action
on BX). After taking the Borel construction, this defines a map
^
BQ ! BNKX(P ) = EK xK BCX (P ) p .
In particular, when Q is the group
NKS(P ) = {g 2 NS(P )  cg 2 K},
and
f
B(P o NKS(P )) ! BS ! BX
is induced by the inclusions and f, then this construction is denoted
BflKP:BNKS(P ) ! BNKX(P ) .
f
Lemma 10.3. Fix a pcompact group X, a Sylow psubgroup S ! X, and subgroups
P S and K AutX (P ) where K is discrete ptoral. Then the induced sequence
BCX (P ) ! BNKX(P ) o!BK^p (1)
is a fibration sequence. If Q is another discrete ptoral group, then for any h*
*omomor
phism ae: Q ___! NKX(P ), there is a fibration sequence
Map (B(P o_aeQ), BX)f,ae__! Map (BQ, BNKX(P ))Bae___! Map (BQ, BK^p)Ba_e,
_ _^
where ae2 Hom (Q, K) is any homomorphism such that Baep' o OBae, P o_aeQ is the
_ae
semidirect product for the action Q ___! K Aut(P ), and the fiber is the spa*
*ce of all
Carles Broto, Ran Levi, and Bob Oliver *
* 69
Bf
maps B(P o_aeQ) ___! BX which restrict (up to homotopy) to BP ___!P BX and are
adjoint to Bae in the sense described above.
Proof.The action of K on each cohomology group Hi(BCX (P ); Fp) factors through*
* a
finite quotient group of K, thus through the pgroup ss0(K), and hence is nilpo*
*tent. So
by [BK , II.5.1], the usual fibration sequence
BCX (P ) ! EK xK BCX (P ) ! BK
for the Borel construction over BK is still a fibration sequence after pcomple*
*tion.
Thus (1) is a fibration sequence.
_
Since [BQ, BK^p] ~= [BQ, BK] ~= Rep (Q, K) (Lemma 1.10), ae2 Hom (Q, K) is
uniquely determined up to conjugacy by ae.
_
For any fixed homomorphism ae:Q ___! K, (1) induces a fibration sequence
BCX (P )hQ ___! Map (BQ, BNKX(P ))eae__! Map (BQ, BK^p)Ba_e^p,
where eaedenotes the set of connected components of Map (BQ, BNKX(P )) that map*
* into
Map (BQ, BK^p)Ba_e^p; and (if eae6= ?) BCX (P )hQ is the homotopy fixed point s*
*et of the
Ba_e^p
action of Q induced by the pullback of (1) over BQ ! BK^p. We need to iden*
*tify
'oOBae _
this action of Q on BCX (P ) with that induced by the action of Q on P via ae. *
*This
follows by comparing the fibrations
BCXw(P )__! EK xK BCX (P ) _____! BK
ww  
ww  
# #
BCX (P )______! BNKX(P ) ________!BK^p,
Ba_e^p
since the action of Q on BCX (P ) induced by BQ ! BK^pin the bottom fibra
_
tion coincides with that induced by aein the fibration sequence of the top row.*
* By
construction, the action of K on BCX (P ) induced by the top row is just the ac*
*tion of
K on BCX (P ) = Map (BP, BX)BfP induced by the original action of K on P .
Now set fP = fP :P ! X for short. We can identify
hQ
BCX (P )hQ = Map (BP, BX)BfP = Map Q(EQ, Map (BP, BX)BfP)
' Map Q(BP x EQ, BX)fe' Map (BP xQ EQ, BX)fe' Map (B(P o_aeQ), BX)fe,
where efis the set of connected components of maps whose restriction to BP is h*
*o
motopic to BfP. Here, BP xQ EQ ' B(P o_aeQ) because the action of Q in BP is
induced from the action described above of Aut(P ) on BP , and this has a fixed*
* point,
providing a section of the fibration
BP ___! BP xAut(P)E Aut(P ) ___! B Aut(P ) .
Finally, upon restricting to one component of Map (BQ, BNKX(P ))eae, we obtain *
*the
fibration in the statement of the proposition.
Notice that in the particular case where K = 1, Map (BQ, BK) is contractible,*
* and
the fibration of Lemma 10.3 reduces to the equivalence
Map (BP x BQ, BX) ' Map (BQ, Map (BP, BX)).
70 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
f
Proposition 10.4. Let X be a pcompact group, let S ! X be a Sylow psubgroup,
and set F = FS,f(X) for short. Fix a subgroup P S, and a discrete ptoral gro*
*up of
automorphisms K AutF (P ). Then the following hold.
(a)BNKX(P ) is the classifying space of a pcompact group which we denote NKX(P*
* ),
and
flKP K
NKS(P ) ! NX (P )
is a discrete ptoral subgroup. Furthermore, the square
BflKP K
BNKS(P ) ___! BNX (P )
 
B incl ev (1)
# #
f
BS ________!BX
commutes up to pointed homotopy.
(b)There is ' 2 Hom F(P, S) such that '(P ) is fully 'K'1normalized in F.
(c)P is fully Knormalized in F if and only if NKS(P ) is a Sylow psubgroup of*
* NKX(P ).
Proof.(a) By Lemma 10.3, BCX (P ) ! BNKX(P ) ! BK^pis a fibration sequence.
The loop spaces of the fiber and base of this sequence have finite mod p cohomo*
*logy,
so the same is true of NKX(P ) def= (BNKX(P )). Thus NKX(P ) is a pcompact gro*
*up.
The map
BflKP K ^
BNKS(P ) ! BNX (P ) = EK xK Map (BP, BX)BfP p
is defined to be adjoint to the composite
K B(incloincl) f
B P o NS (P ) ! BS ! BX. (2)
Hence the composite of BflKPwith the evaluation map from Map (BP, BX)BfP to BX
(evaluation at the basepoint of BP ) is equal to the restriction of (2) to BNKS*
*(P ). This
proves that (1) is commutative.
If flKP factored through a quotient group NKS(P )=R for some R 6= 1, then the*
* re
striction of Bf :BS ! BX to BR would be homotopically trivial, but this cann*
*ot
f0 K K
happen. So if S0 ! NS (P ) is a maximal discrete ptoral subgroup, then flP *
*factors
through a monomorphism from NKS(P ) to S0 (Proposition 10.1(a)), and thus flKP *
*is
itself a monomorphism.
(b,c) By Lemma 10.3, any discrete ptoral subgroup BQ ! BNKX(P ) lifts to a
fi
map B(P o Q) ! BX which factors through a homomorphism P o Q ! S. Set
P 0= fi(P o 1) S, ' = fiPo1 2 IsoF(P, P 0), and K0 = 'K'1 AutX (P 0). Then
0 0 K
fi(Q) NKS(P ), and fi1oQ is injective since otherwise BQ ! BNX (P ) would *
*factor
through a quotient group0of Q and hence wouldn't be a subgroup. Thus, the large*
*st
possible Knormalizer NKS(P 0) occurs when it is a Sylow psubgroup of NKX(P ),*
* so P 0
is fully K0normalized in F, and P is fully Knormalized if and only if NKS(P )*
* is a
Sylow psubgroup of NKX(P ).
We are now ready to show that FS,f(X) is saturated.
f
Proposition 10.5. Let X be a pcompact group, and let S ! X be a Sylow p
subgroup. Then FS,f(X) is a saturated fusion system over S.
Carles Broto, Ran Levi, and Bob Oliver *
* 71
Proof.Write F = FS,f(X) for short.
Proof of (I): Fix a subgroup P S which is fully normalized in F. Let K AutF*
* (P )
be such that K Aut S(P ) and K= Inn(P ) 2 Sylp(Out F(P )). Then P is fully K
normalized, since it is fully normalized and NKS(P ) = NS(P ). So by Propositi*
*on
10.4(c), NKS(P ) is a Sylow psubgroup of NKX(P ).
Set K0 = Aut S(P ) for short, and consider the following commutative diagram *
*of
connected spaces:
BCS(P )^p_! BNKS(P )^p_! BK0^p
 
f1 f2 f3
# # #
BCX (P )__! BNKX(P )^p__! BK^p.
Let Fi be the homotopy fiber of fi (for i = 1, 2, 3). Each row is a fibration *
*se
quence before pcompletion; and the actions of K0 on H*(BCS(P ); Fp) and of K on
H*(BCX (P ); Fp) factor through finite pgroup quotients and hence are nilpoten*
*t. So
the rows are still fibration sequences after pcompletion by [BK , II.5.1].
Each of the maps fiis a monomorphism of pcompact groups, and hence H*(Fi; Fp*
*) is
finite for each i. Since BCX (P ) is connected, ss1(BNKX(P )^p) surjects onto s*
*s1(BK^p) ~=
ss0(K), and hence ss0(F2) surjects onto ss0(F3). Thus F1 is the homotopy fiber *
*of the
map F2 ! F3, and so O(F2) = O(F1).O(F3).
Since NKS(P ) 2 Sylp(NKX(P )), O(F2) is prime to p by Proposition 10.1(a). T*
*hus
O(F1) and O(F3) are both prime to p, and hence CS(P ) 2 Sylp(CX (P )) and (sinc*
*e K is
discrete ptoral) K0 = K. Hence OutS (P ) = K= Inn(P ) 2 Sylp(Out F(P )). Also,*
* since
CS(P ) 2 Sylp(CX (P )), we can again apply Proposition 10.4(c) (this time with *
*K = 1),
to show that P is fully centralized in F. This finishes the proof of (I).
Proof of (II): Fix P S and ' 2 Hom F(P, S), and set P 0= '(P ). Assume that *
*P 0
is fully centralized in F. Set
N' = {g 2 NS(P )  'cg'1 2 AutS(P 0)},
0 0 0
and set K = AutN' (P ), K0 = 'K'1 AutS (P 0), and N0'= NKS(P ). Then P is
fully K0normalized in F, since it is fully centralized and K0 AutS(P 0). Cons*
*ider the
following diagram:
BflKP K proj ^
BN' ________! BNX (P )________! BKp
...  
B'_... ' Ec'x('*)1 ~=Bc' (1)
?.. BflK0 0 0# proj #
BN0' ________P!BNKX(P 0) ________! BK0^p.
!
Here, the composites in the two rows are induced by the epimorphisms N' i K
!0 0 0
and N0'i K (exactly, not just up to homotopy). By Proposition 10.4(c), N' is
0 0 K _
a Sylow psubgroup of NKX(P ) ' NX (P ), and hence there is a homomorphism ' 2
Hom (N', N0') which makes the left hand square commute up to homotopy.
Since [BN', BK0^p] ~=Rep (N', K0) (Lemma 1.10), the homotopy commutativity of
_ 0 0 _ 0
(1) implies that there is g 2_K such that c' O! =_c_gO! O '._Since ! is onto,*
* there
is g 2 N0'such that !0(g) = g; and upon replacing ' by cg O' we can assume that
_
c' O! = !0O' .
72 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
Fix a homotopy H which makes the left hand square in (1) commute._ Then the
composite projOH is a loop in Map (BN', BK0^p) based at B(!0O'), and this compo*
*nent
_ ^
has the homotopy type of BCK0(!0O' (N'))p by Lemma 1.10 again. So after replaci*
*ng
_ _ 0 0
' by cg0O' for some appropriate g 2 N', and after modifying H using the homotopy
_ _ 0
from B' to B(cg0O') determined by g , we can arrange that projOH is nullhomotop*
*ic in
Map (BN', BK0^p). We can now apply Lemma 10.3, to show that the following diagr*
*am
commutes up to homotopy:
incloincl Bf
B(P o N') _________!BS __________! BXw
 ww
B('o_') w
# w
incloincl Bf
B(P 0o N0')________! BS __________! BX .
_ 0
In particular, ' 2 Hom F(N', N'). Also, the two homomorphisms from P o N' to S
_ 1
induced by inclusions_and by ' o ' have the same kernel {(g, g )  g 2 P }, an*
*d this
implies that ' = 'P.
S 1
Proof of (III): Fix P = n=1Pn, where P1 P2 . .i.s an increasing sequence*
* of
subgroups. Let ' 2 Inj(P, S) be such that 'Pn 2 Hom F(Pn, S) for all n. Thus f*
*or each
n, (Bf OB')BPn ' BfBPn. Also, by Proposition 10.1(b), Map (BPn, BX)BfBPn '
Map (BP, BX)BfBP for n sufficiently large. We can thus chooseShomotopies Hn fr*
*om
(Bf OB')BPn to BfBPn such that Hn = Hn+1BPnxI, and set H = Hn. This shows
that Bf OB' ' BfBP , and hence that ' 2 Hom F(P, S).
In [CLN ], a pcompact toral subgroup P of a pcompact group X is called cen*
*tric if
Bf
the inclusion map BP ! BX is a centric map; i.e., if
BfO
Map (BP, BP )Id! Map (BP, BX)Bf
is an equivalence. We must check that this is equivalent to the concept of Fce*
*ntricity
(applied to discrete ptoral subgroups) used here.
f
Lemma 10.6. Let X be a pcompact group, and let S ! X be a Sylow psubgroup.
Set F = FS,f(X). Then for any subgroup P S, BfBP :BP ^p! BX is a centric
map if and only if P is Fcentric.
Proof.Assume P is Fcentric. In particular, P is fully centralized in F. By Pro*
*position
10.4(a,c) (applied with K = 1), CX (P ) is a pcompact group with Sylow psubgr*
*oup
CS(P ) = Z(P ). Also, composition defines a map
Map (BP, BP )Idx Map (BP, BX)BfBP ! Map (BP, BX)BfBP.
'BZ(P)
So by Proposition 10.1(c), Z(P ) is central in CX (P ), and there is a pcompac*
*t group
CX (P )=Z(P ) whose Euler characteristic is prime to p and a fibration sequence
BZ(P )^p! BCX (P ) ! B(CX (P )=Z(P )) .
Then CX (P )=Z(P ) must be trivial, so B(CX (P )=Z(P )) ' *,
BZ(P )^p' Map (BP, BX)BfBP,
and hence BfBP is a centric map.
Conversely, if BfBP is a centric map, then BCX (P ) ' BZ(P ) by Lemma 1.10, *
*so
CS(P 0) = Z(P 0) for all P 0 S which is Fconjugate to P , and P is Fcentric.
Carles Broto, Ran Levi, and Bob Oliver *
* 73
It remains to construct a linking system associated to FS,f(X) whose pcomple*
*ted
nerve has the homotopy type of BX. This will be done using Proposition 4.6, tog*
*ether
with a construction by Castellana, Levi, and Notbohm in [CLN ].
f
Theorem 10.7. Let X be a pcompact group, and let S ! X be a Sylow psubgroup.
Set F = FS,f(X) def=FS,Bf(BX) for short. Then there is a centric linking system
L = LcS,f(X) associated to F such that
LcS,f(X)^p' BX .
In other words, (S, F, L) is a plocal compact group whose classifying space is*
* homotopy
equivalent to BX.
Proof.By Proposition 10.5, the fusion system F is saturated.
In [CLN ], the authors define a category Oc(F)+ by adding a final object to *
*Oc(F);
i.e., the category Oc(F)+ consists of Oc(F) together with an additional object *
**, and a
unique morphism from each object in Oc(F) to *. (The actual category they work *
*with
contains the same objects as Oc(F) by Lemma 10.6.) They then define a homotopy
functor
B +:Oc(F)+ ! hoTop
by setting B +(P ) = BP ^pfor all Fcentric P S, and B +(*) = BX (with the ob*
*vious
maps between them). By Lemmas 1.10 and 10.6, this is a centric diagram in the s*
*ense
of [DK ]. Since Oc(F) has a final object, the DwyerKan obstructions to rigidif*
*ying B +
to a functor to Top all vanish [DK ] (see also Corollary A.4), and so this func*
*tor can be
lifted. In particular, this restricts to a functor bBfrom Oc(F) to Top, togethe*
*r with a
map from hocolim!(Bb) to BX. (See also Corollary A.5.)
By [CLN , Theorem 8.5], this map from hocolim!(Bb) to BX induces a homo*
*topy
equivalence
^
hocolim!(Bb) p ' BX
(the collection of Fcentric subgroups of X is "subgroup ample"). Hence by Prop*
*osition
5.9, there is a functor eB:Oc(F) ! Top which is a rigidification of the homo*
*topy
functor B , and a natural transformation of functors O: eB! Bb which is the *
*com
pletion map on each object. Proposition 4.6 now applies to show that there is a*
* centric
linking system L = LcS,f(X) associated to F such that
^ ^ ^
L^p' hocolim!(Be) p ' hocolim!(Bb) p ' BXp.
Oc(F) Oc(F)
In fact, one can show that there is a unique centric linking system L associa*
*ted to
FS,f(X) such that L^p' BX^p, but we leave that for a later paper.
Appendix A. Lifting diagrams in the homotopy category
As elsewhere in the paper, we let Top denote the category of spaces, and hoTop
the homotopy category. Let ho :Top ! hoTop be the forgetful functor. When C
74 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
is a small category, a functor F from C to Top or hoTop is called centric if f*
*or each
morphism ' 2 Mor C(c, d), the natural map
F(')O
Map (F (c), F (c))Id ! Map (F (c), F (d))F(')
'
is a homotopy equivalence. In [DK ], Dwyer and Kan identify the obstructions to
rigidifying a centric functor F : C ! hoTop to a functor eF: C ! Top ; and*
* also
describe the space of such rigidifications. We prove here a relative version o*
*f their
result which is needed in Section 5. This result can, in fact, be derived from *
*the main
theorem in [DK ], but that argument is so indirect that we find it helpful to g*
*ive a more
direct, and also more elementary, proof.
More precisely, a rigidification of F is a functor Fe:C ! Top , together w*
*ith a
natural transformation of functors F ! hoOFe which is a homotopy equivalenc*
*e on
each object. Two rigidifications eFand eF 0are equivalent if there is a third r*
*igidifica
tion eF,00together with natural transformations of functors eF! eF 00eF *
*0which
commute with the natural transformations from F , and hence which define homoto*
*py
equivalences eF(c) ' eF 00(c) ' eF(0c) for each c 2 Ob (C). This is easily see*
*n to be an
equivalence relation by taking pushouts.
The main idea here is to construct a rigidification of F :C ! hoTop by fi*
*rst
constructing a space which looks like a "homotopy colimit" of F , and then show*
* that
this homotopy colimit automatically induces a rigidification eF. Recall that th*
*e nerve
of a small category C is defined by setting
ia a j.
BC = n ~ ;
n 0 x0!...!xn
and that the homotopy colimit of any functor F : C ! Top is the space
ia a j.
hocolim!(F ) = F (x0) x n ~ .
C n 0 x0!...!xn
Here, in both cases, we divide out by the usual face and degeneracy identificat*
*ions.
Let pF : hocolim!(F ) ! BC be the projection. It will be convenient to r*
*efer to the
"skeleta" of the homotopy colimit: let hocolim!(n)(F ) denote the union of*
* the F (x0)x i
for all i n (and all x0 ! . .!.xi in C).
Now assume that F : C ! hoTop is a functor to the homotopy category instea*
*d.
We assume that for each f : x ! y in C, a concrete map F (f) : F (x) ! F (y*
*) has
been chosen. The 1skeleton hocolim!(1)(F ) is defined in the same way as *
*before: it is
the union of the mapping cylinders of the F (f) taken over all f 2 Mor (C). It *
*is also
straightforward to define the 2skeletion; but it is convenient at this stage t*
*o replace
f g
2 by a truncated triangle 2t. More preciesely, for each sequence x0 ! x1 *
*! x2,
F (x0) x 2tis attached to hocolim!(1)(F ) via the following picture:
F (g Of)_____ssF (g) OF (f)
JJ
J
Id J F (f)
J
J
J
J
_________________________Jss
Id Id F (f)
Carles Broto, Ran Levi, and Bob Oliver *
* 75
where the small segment at the top is mapped using a homotopy between F (g Of) *
*and
F (g) OF (f).
The first obstructions arise when constructing the 3skeleton. For each x0 ! *
*. .!.
x3, we want to attach F (x0) x 3tto hocolim!(2)(F ), where 3t(the "trunc*
*ated 3
simplex") is the cone over 2twith its vertex cut off. The attachment map is e*
*asily
defined, except on the "top" surface resulting from truncating the cone vertex.*
* Hence,
the obstruction to defining the attachment map lies in the group
i j
ss1 Map (F (x0), F (x3)) , F (x0 ! x3) .
At this point, it becomes necessary to switch from the intuitive picture to f*
*ormal def
initions, by replacing the truncated simplices ntby cubes In, and regarding si*
*mplices
as cubes modulo certain identifications. This correspondence will be made expl*
*icit
later.
Let be the simplicial category, with objects the sets [n] = {0, . .,.n} for*
* n 0,
and morphisms the order preserving maps between sets. We let @i2 Mor ([n  1],*
* [n])
denote the ith face map (with image [n]r {i}). Define a functor
Io: ! Top
by setting Io([n]) = In+1 (where I is the closed interval I = [0, 1]), and
i Y Y Y j
Io(oe)(t0, . .,.tn) = ti, ti, . .,. ti
i2oe1(0) i2oe1(1) i2oe1(m)
for oe 2 Mor j([n], [m]). Here, the product over the empty set is always 1.
Let 1 0 be the subcategories with the same objects, where
Mor 0 ([m], [n])= {oe 2 Mor ([m], [n])  oe(0) = 0}
Mor 1 ([m], [n])= {oe 2 Mor ([m], [n])  oe(0) = 0, oe(m) = n}.
For each n 0, let Io1([n]) Io0([n]) Io([n]) be the subspaces
Io0([n])= {(0, x1, . .,.xn) 2 Io([n])} ~=In
Io1([n])= {(0, x1, . .,.xn1, 0) 2 Io([n])} ~=In1 .
Then for each j = 0, 1, Io j restricts to a subfunctor Ioj: j ! Top.
Throughout the rest of this section, C denotes a fixed small category. For e*
*ach
n 0, define Mor n= Mor n(C) to be the set of all sequences c0 ! c1 ! . .!.cn *
*of
composable morphisms in C. In particular, Mor 0(C) = Ob (C) and Mor 1(C) = Mor *
*(C).
For oe 2 Mor ([n], [m]), oe*: Mor m(C) ! Morn(C) is defined as usual by ta*
*king
compositions, inserting identity morphisms, and (if oe =2Mor ( 1)) dropping mor*
*phisms
at one or both ends of the chain. For example, @*i(from Mor n(C) to Mor n1(C)*
*) is
defined by composing two morphisms in the sequence, or by dropping one of them *
*if
i = 0 or n. Also, for each , = (c0 ! . .!.cn) in Mor n(C) and each 0 i j *
*n, we
write
,ij= (ci! . .!.cj) 2 Mor ji(C),
O O*
* O
let ,ij2 Mor C(ci, cj) denote the composite of this sequence of maps, and set ,*
*= ,0n.
In order to simplify the notation in what follows, whenever F :C ! hoTop *
*is
a functor and ' 2 Mor (C), we let F (') denote some chosen representative of the
homotopy class of maps defined by F , not the homotopy class itself.
76 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
__
Definition A.1. Fix a functor F :C ! hoTop . An Rg 1structure F on F consists
__ (c) __
of a space F (c) and a homotopy equivalence F (c) ! F (c), defined for eac*
*h c 2
'
Ob (C); together with maps
__ __ __
F (,): In1 = Io1([n]) ! Map F (c0), F(cn) ,
defined for each n 1 and each , = (c0 ! c1 ! . .!.cn) 2 Mor n(C), which satis*
*fy
the following relations.
__ '
(a)For all ' 2 Mor C(c0, c1), F (c0 ! c1) O (c0) ' (c1) OF (').
(b)For all m, n 1, oe 2 Mor 1([m], [n]), , 2 Mor n(C), and t2 Im1 ,
__ __
F(oe*,)(t) = F(,)(Io1(oe)(t)).
(c)For all n 2, , 2 Mor n(C), 1 i n  1, t12 Ii1, and t22 Ini1,
__ __ __
F (,)(t1, 0, t2) = F(,in)(t2) OF (,0i)(t1).
Schematically, relation (b) can be described via the following commutative di*
*agram:
Io1(oe)
Im1 __________________________//_NNIn1_p
NNFN(oe*(,))NN F (,)ppppp
NNN ppppp
NNN&&_ xxpp_
Map (F (c0), F(cn))
while relation (c) can be described via the diagram:
t1,t27!(t1,0,t2) n1
Ii1x Ini1____________________________! I
_ _  _
F(,0i)xF(,in)# #F(,)
__ __ __ __ composition __ __
Map (F (c0), F(ci)) x Map (F (ci), F(cn))____!_M_ap(F (c0), F(c*
*n)) .
These relations are more easily understood when one thinks of Io1([n]) ~=In1*
* as the
space of all (t0, . .,.tn) 2 Io([n]) ~= In+1 such that t0 = 0 = tn. Each coord*
*inate in
Io([n]) corresponds to one of the objects in the chain , = (c0 ! . .!.cn). When
ti= 1 for some 0 < i < n, ti and ci can be removed, giving the face relation
__ __
F (,)(t1, . .,.ti1, 1, ti+1, . .,.tn1) = F(@i,)(t1, . .,.ti1, ti+1, .*
* .,.tn1).
__
When ti = 0 for 0 < i < n, then F (,)(t) can be split as a composite at the obj*
*ect ci
(relation (c)). If one of the morphisms in , is an identity, then one can remov*
*e it and
multiply the coordinates corresponding to its two objects.
For instance, when m = 2 and n = 1 (and oe is one of the surjections), condit*
*ion (b)
says that
__ Id ' __ ' Id
F(c0 ! c0 ! c1) and F(c0 ! c1 ! c1)
__ __ __ __ __
are_both the constant maps to F ('). In particular, F (') OF(Idc0) = F(') = F(I*
*dc1) O
F(').
__ ' _
__When_n_= 2, condition_(c) says that F c0 ! c1 ! c2 is a homotopy from
F(_) OF (') to F (_ O'). More generally, when , 2 Mor n(C) for n 2,
__ __ __ __ __ *
*__O
F (,)(0, . .,.0) = F(,n1,n) O. .O.F(,12) OF (,01) and F(,)(1, . .,.1) = *
*F(,).
Carles Broto, Ran Levi, and Bob Oliver *
* 77
*
* __ O
At the other vertices of In1, we get all of the other possible composites of t*
*he F (,ij).
An Rg 1structure on F is thus a collection of higher homotopies connecting giv*
*en
homotopies F (_) OF (') ' F (_ O').
From this point of view, one sees that when defining an Rg 1structure on F ,*
* it
suffices to define it on all nondegenerate sequences , 2 Mor n(C) (i.e., those_*
*containing
no identity morphisms), inductively for increasing n, where at each step F (,) *
*has
already been defined on @In1 and_must be extended in some way to In1. The sta*
*rting
point can be any choice of maps F('), for all ' 2 Mor (C), in the given homotop*
*y class
determined by F ('), such that
__ __ __ __ __
F (') OF (Idc) = F(') = F(Idd) OF (')
for each morphism ' 2 Mor C(c, d) in C.
__ __ __ __
If F and F0 are both Rg1 structures_on F_, then a morphism : F ! F 0cons*
*ists
of homotopy equivalences `(c): F(c) '!F 0(c) (for each c 2 Ob (C)) such that*
* `(c) O
(c) ' 0(c), and such that for each , = (c0 ! . .!.cn) and each t 2 In1,
__ __ __ __
`(cn) OF (,)(t) = F0(,)(t) O`(c0) 2 Map (F (c0, F0(cn)).
Two Rg 1structures on F are equivalent if there is a third to which they both *
*have
morphisms. One easily sees that (homotopy) pushouts exist for morphisms of Rg *
*1
structures on F , and hence that this defines an equivalence relation among Rg *
*1
structures.
__
For any given_Rg 1structure F on F : C ! hoTop , we define its "homotopy
colimit" Sp (F ) to be the space
__ ia a __ j .
Sp(F ) = F (c0) x In ~ (In = Io0([n]))
n 0 c0!...!cn
where the following identifications_are made for each n 1, each , = (c0 ! . .*
*!.
cn) 2 Mor n(C), and each x 2 F(c0):
o
x; I0(oe)(t)~[,]x; t [oe*,] (oe 2 Mor 0([m], [n]), t 2 I*
*m )
__
x; (t1, 0, t2)~[,]F(,0i)(t1)(x); t2[,in](1 i n, t12 Ii1, t22 Ini.)
f g 2
For example, in the case of a sequence , = (c0 ! c1 ! c2) in Mor (C), the c*
*orre
sponding square Io0([2]) is attached to the 1skeleton in the following way:
__ __ __ __ __
F(g Of) F(,) F(g) OF (f) F(c2)
______________ss s
(1,0) (0,0) J
  J
  J
 __ __ __ J __
 2  F(c )x I F (c )x I
Id F (c0)x I F(f) ! 0 J 1
  J
  J
 
(1,1) (0,1) J
_______________ss_ __ ____________________Jss__
Id Id F(f) F (c0) F (c0)x I F (c1)
__
The_labels in the first picture describe the maps by which a vertex F (c0) or a*
*n edge
F(c0)xI is attached to the space represented by the second picture. Thus the tr*
*apezoid
in the earlier picture has now been replaced by a square.
One way to understand these relations and their connection with those in Defi*
*nition
A.1 is to think of Io0([n]) ~= In as the subspace of all (n + 1)tuples (0, t1,*
* . .,.tn) in
78 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
Io([n]) ~=In+1. For , 2 Mor n(C), each coordinate in Io([n]) corresponds to one*
* of the
objects in the chain , = (c0 ! . .!.cn). When ti= 1 for 0 < i n, ti and ci ca*
*n be
removed, giving the face relation
x; (t1, . .,.ti1, 1, ti+1, . .,.tn) [,]~ x; (t1, . .,.ti1, ti+1, . *
*.,.tn) [@i,].
When_ti_= 0 for 0 < i n, then , splits as a composite at the object ci, we ap
ply F (,0i)(t1, . .,.ti1) to x, and get the second of the above relations. If*
* one of the
morphisms in , is an identity, then we get a degeneracy relation by removing it*
* and
multiplying the two corresponding coordinates.
Consider the maps oen : In ! n defined by
i *
* j
oen(t1, . .,.tn) = t1t2. .t.n, (1  t1)t2. .t.n, (1  t2)t3. .t.n, . .,.(1  t*
*n1)tn, 1  tn .
__
When F is a functor to Top and F is the corresponding locally constant Rg1 str*
*ucture
__ O
(i.e., for each ,, F(,)_is the constant map with value F (,)), then the oen def*
*ine a home
omorphism from Sp (F ) to the usual homotopy colimit hocolim!(F ). More g*
*enerally,
__
when F is an arbitrary Rg1 structure, then there is a map
__
pr_F:Sp (F ) ! C
__
defined on each subspace F (c0) x In by first projecting to the In and then to *
* n via
oen.
__ __
We now define a functor Rg (F ) : C ! Top by letting Rg (F )(c) be the pul*
*lback
space
__ __
Rg(F )(c)______!Sp (F )
 
 pr_F
# #
C#c__________!C
(the ordinary pullback, not the homotopy pullback). A morphism ' 2 Mor C(c, d)
induces a map from C#c_to C#d via_composition_with ' in the usual way, and *
*hence
induces a map from Rg (F )(c) to Rg (F )(d). Equivalently,
__ ia a __ j .
Rg (F )(c) = F (c0) x In ~ (In = Io0([n]))
n 0 c0!...!cn!c
__
where the identifications_are analogous to those used to define Sp (F ). This *
*clearly
makes Rg (F ) into a functor from C to Top.
__ __
For each c, F(c) can be identified as a subspace of Rg (F )(c): the inverse i*
*mage under
(c) __
the projection to C#c of the vertex (c Id!c). The composite F (c) ! F *
*(c)
__ __ '
Rg (F )(c) defines a natural transformation F ! hoORg (F ) of functors C !*
* hoTop .
The following_proposition now shows that this is a natural equivalence, and hen*
*ce that
Rg (F ) is a rigidification of F .
__
Proposition_A.2. For any Rg 1structure F on F_:C ! hoTop , for_each c 2
Ob (C), F (c) is a deformation retract of Rg (F )(c). Thus Rg (F ) is a rigidi*
*fication
of F .
Carles Broto, Ran Levi, and Bob Oliver *
* 79
__ __
Proof.Define : Rg (F )(c) x I ! Rg (F )(c) by setting
(x; t)[,!c], s = (x; (t, s)) Id
[,!c! c]
__
for all ,_2 Mor n(C), t 2 In, and s 2_I. Then (u, 1) = u and (u, 0) 2 F (c) f*
*or all
u 2_Rg_(F )(c) by definition_of Rg (F )(c). Furthermore, the homotopy is the id*
*entity
on F (c), and thus F (c) is a deformation retract.
For any given F :C ! hoTop , let Rigid(F ) be the set of equivalence class*
*es of
rigidifications of F , and let Rg1 (F ) be the set of equivalence classes of Rg*
*1 structures
on F . A rigidification of F_can_be regarded as a "locally_constant" Rg1 struc*
*ture on
F ; i.e., an Rg1 structure F where each of the maps F(,) (for , 2 Mor n(C)) is*
* constant
on In1. We thus have maps
const
Rigid(F ) !Rg1(F ).
Rg
One easily checks that for any rigidification Fe, there is a natural transforma*
*tion of
functors from Rg (const(Fe)) to eF, and hence these are equal in Rigid(F ). We *
*do not
know whether the other composite is the identity on Rg 1(F ), but that will not*
* be
needed here.
A natural transformation O: F ! F 0of functors F, F 0:C ! hoTop will *
*be
called relatively centric if for each morphism ' 2 Mor C(c, d) in C, the homoto*
*py com
mutative square
F(')O
Map (F (c), F (c))Id_____! Map (F (c), F (d))F(')
 
O(c)O O(d)O
# #
F0(')O 0
Map (F (c), F 0(c))O(c)_! Map (F (c), F (d))F0(')OO(c)
is a homotopy pullback. For example, when F 0is the functor which sends every o*
*bject
to a point, then O is relatively centric if and only if the functor F defines a*
* centric di
agram. Assume we are given a relatively centric natural transformation O: F *
*! F 0
where F 0is a functor to Top, and assume furthermore that for each c 2 Ob (C), *
*the
homotopy fiber
i O(c)O j
(c) def=hofiberMap(F (c), F (c))Id! Map (F (c), F 0(c))O(c)
is connected. We claim that this determines functors
fii:Cop ! Ab (all i 1)
such that fii(c) ~=ssi( (c)) for all c. To show this, we can assume without los*
*s of general
ity that O(c) is a fibration for all c, and let (c) be the space of all f 2 Ma*
*p (F (c), F (c))
such that O(c) Of = O(c). Then (c) is a monoid under composition, and in part*
*ic
ular, ss1( (c)) is abelian. For each morphism ' 2 Mor C(c, d) in C, we can cho*
*ose a
representative F (') such that the following square commutes:
F(')
F (c)____! F (d)
 
O(c) O(d)
# #
F0(') 0
F 0(c)_____!F (d) .
80 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
Since O is relatively centric, the fibers of the map
(O(d) O): Map (F (c), F (d))F(')! Map (F (c), F 0(d))F0(')OO(c)
have the homotopy type of (c) and hence are connected. Hence any two choices f*
*or
F (') differ by a path in the fiber over the point F 0(') OO(c); i.e., by a hom*
*otopy
{Ft(')}t2Isuch that O(d) OFt(') = F 0(') OO(c) for each t.
For each ' 2 Mor C(c, d), consider the following diagram:
OF(') F(')O
Map (F (d), F (d))Id____=w1!Map(F (c), F (d))F(')____=w2Map(F (c), F (c))Id
  
O(d)O=u1 O(d)O=u2 O(c)O=u3
# # #
OF(') 0 F0(')O 0
Map (F (d), F 0(d))O(d)_!Map (F (c), F (d))O(d)F(')___Map (F (c), F (c))O(c)
where the right hand square commutes by the assumption on F (') (and the other *
*since
composition is associative). Set (c) = u13(O(c)) and fii(c) = ssi( (c), IdF(*
*c)) (and
similarly for d). By assumption, w2 sends (c) to u12(F 0(') OO(c)) by a homo*
*topy
equivalence, and we let fii(') be the composite
(w2O)1
ssi( (d), IdF(d))w1O!ssi(u12(F 0(') OO(c)), F (')) !~ ssi( (c)*
*, IdF(c)).
=fii(d) = =fii*
*(c)
By the above remarks, this is independent of the choice of map F ('). Hence th*
*is
defines a functor on Cop: the relations fii(_ O') = fii(_) Ofii(') follow using*
* any choice
of homotopy from F (_ O') to F (_) OF (') which covers F 0(_ O'). (Recall that *
*we are
assuming F 0is a functor to Top, so F 0(_ O') = F 0(_) OF 0(').)
The following theorem is our main result giving a relative version of the Dwy*
*erKan
obstruction theory. The special case where F 0(c) is a point for all c 2 Ob (C)*
* is the case
shown by Dwyer and Kan in [DK ].
Theorem A.3. Fix functors F :C ! hoTop and F 0:C ! Top , and let
O: F ! hoO F 0
be a relatively centric natural transformation of functors. For each c 2 Ob (C)*
*, assume
that the homotopy fiber
O(c)O 0
(c) = hofiberMap (F (c), F (c))Id! Map (F (c), F (c))O(c);
is connected. Let fii:Cop ! Ab (all i 1) be the functors defined above. T*
*hen the
eO 0 O 0
obstructions to the existence of a rigidification eF! F of F ! F lie*
* in the
groups limn+2(fin) for n 1; while the obstructions to the uniqueness of (Fe,*
* eO) up to
C
equivalence of rigidifications lie in limn+1(fin) for n 1.
C
Proof.We use here the description of the higher limits of a functor ff: Cop *
*! Ab as
the homology groups of the normalized cochain complex
__ Y
Cn(C; ff) = ff(c0),
c0!...!cn
Carles Broto, Ran Levi, and Bob Oliver *
* 81
where the_product is taken over all composable sequences of nonidentity morphis*
*ms.
For , 2 Cn(C; ff), define
'
d(,)(c0 ! c1 ! . .!.cn+1) = F (')(,(c1 ! . .!.cn+1))+
n+1X
(1)i,(c0 ! . .b.ci.!.c.n+1) .
i=1
__
Then lim*(ff) ~=H*(C *(C; ff), d) (cf. [GZ , Appendix II, Proposition 3.3] or *
*[Ol, Lemma
C
2].)
Proof of existence: As above, we replace each O(c) by a fibration, and replace*
* each
F (') (for '_2_Mor C(c, d) by a map such that O(d) OF (') = F 0(') OO(c). We a*
*lso
assume that F (Idc) = Id_F(c)for each c. Then
(c) def=f 2 Map (F (c), F (c))  O(c) Of = O(c)
is a topological monoid under composition, and is connected by assumption. So *
*we
can ignore basepoints when working in the homotopy groups fii(c) = ssi( (c)).
__ __
__We want to construct an Rg1 structure F such that F (c) = F (c) for all c 2 *
*Ob (C),
F(') = F (') for all ' 2 Mor (C), and such that for each n 2 and each , = (c0*
* !
. .!.cn) 2 Mor n(C), the following square commutes (exactly) for each t 2 In1:
_
F(,)(t)
F (c0)____! F (cn)
 
O(c0) O(cn) (1)
# O #
F0(,) 0
F 0(c0)___! F (cn) .
By Proposition A.2, any such structure induces a rigidification eFof F , togeth*
*er with
a natural transformation of functors eOfrom eFto F 0.
__
Assume, for some n 2, that F has been defined on Mor i(C) for all i < n. *
*Fix
, 2 Mor n(C), a composite of (nonidentity) maps from c0 to cn. Consider the fol*
*lowing
commutative square, which is a homotopy pullback by assumption:
O
F(,)O
Map (F (c0), F (cn)) O_______ Map (F (c0), F (c0))Id
F(,)=w

O(cn)O=u O(c0)O (2)
# O #
F0(,)O 0
Map (F (c0), F 0(cn)) O ____Map (F (c0), F (c0))O(c0).
F0(,)OO(c0)
__
Conditions (b) and (c) in Definition A.1 determine a map F (,)0 from @In1 to
O
Map (F (c0), F (cn)) O whose image lies in u1(F 0(,) OO(c0)). Hence the obstr*
*uction to
__ F(,)
defining F (,) on In1 is an element
1 0O wO
j(,) 2 ssn2 u (F (,) OO(c0)), ~ ssn2( (c0)) = fin2(c0).
=
If_one of the morphisms in the sequence , is an identity morphism, then we defi*
*ne
F(,)_using the appropriate formula in Definition A.1(b), and j(,) = 0. Thus j 2
Cn(C; fin2).
82 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
We claim that dj = 0. Fix ! = (c0 ! . .!.cn+1) 2 Mor n+1(C). Consider the face
maps on the ncube
ffiti:In1 ! In where ffiti(t1, . .,.tn1) = (t1, . .,.ti1, t,*
* ti, . .,.tn1)
(for all i = 1, . .,.n and t = 0, 1). The conditions in Definition A.1(b,c) def*
*ine a map
__
F o(!): (In)(n2)! Map (F (c0), F (cn+1))F(O!)' Aut(F (c0))1
= Map (F (c0), F (c0))Id,
and hence
Xn i __ __ j
(1)i [F o(!) Offi1i@In1]  [F o(!) Offi0i@In1] = 0 2 ss1(Aut((F3*
*(c0))1).)
i=1
__
Furthermore, F o(!) extends to the faces ffi0i(In1) for 2 i n  1 (again, *
*by the
conditions in Definition A.1(c)), and so those terms vanish in (3). So we are l*
*eft with
the equality
__ Xn __ __
0 = [F o(!) Offi01@In1] + (1)i[F o(!) Offi1i@In1] + (1)n+1[F o(!) *
*Offi0n@In1]
i=1
n+1X
= F (!01)*(j(@0!)) + (1)ij(@i!) = dj(!).
i=1
Thus dj = 0, and so [j] 2 limn(fin2).
C
__
If [j] = 0, then there is ae 2 C n1(C; fin2)_such that j = dae. Similar (bu*
*t simpler)
arguments to those used above now show that F can be "changed by ae" on element*
*s of_
Mor n1(C), in a way so that the obstruction j vanishes. We can thus arrange th*
*at F can
be_extended to Morn(C). Upon continuing this procedure, we obtain the Rg1 stru*
*cture
F.
Proof of uniqueness: Now assume that
eF1eO1!F 0eO2eF2
are two rigidifications of O: F ! F 0. In other words, we have a homotopy c*
*ommu
tative diagram
1
F _______! ho OeF1
HH
 HH O 
2# HH ho(eO1) (4)
ho(eOHHj)#
ho OeF2_____!2ho OF 0
of functors C ! hoTop and natural transformations between them. We can assume
that the maps eO(c) and eO0(c) are fibrations for each c 2 Ob (C); otherwise we*
* replace
them by fibrations using one of the canonical constructions.
For each c 2 Ob (C), let `(c): eF1(c) ! Fe2(c) be any map such that `(c) O*
* 1(c) '
2(c) as maps from Fe1(c) to Fe2(c). Using the homotopy commutativity of (4), *
*and
the homotopy lifting property for eO2(c), we can assume that eO2(c) O`(c) = Oe1*
*(c)
(exactly, not just up to homotopy). Let bF(c) be the mapping cylinder of `(c), *
*and let
bO(c): bF(c) ! F 0(c) be the projection induced by eO1(c) and eO2(c).
Regard eF1(c) and eF2(c) as subspaces of bF(c). We want to extend the locally*
* finite
Rg1 structures Fe and Fe0to an Rg 1structure Fb covering F 0. For each morph*
*ism
Carles Broto, Ran Levi, and Bob Oliver *
* 83
' 2 Mor C(c, d) in C, `(d) OeF1(') ' eF2(') O`(c), and hence eF1(') and eF2(') *
*can be
extended to a map bF(') from bF(c) to bF(d). Using the homotopy lifting propert*
*y again,
this can be chosen such that bO(d) ObF(') = F 0(') ObO(c).
Assume, for some n 2, that Fb has_been_defined on Mor i(C) for i < n in a
way so that (1) commutes (with F and F replaced by bF) for each ,. Then for each
, = (c0 ! . .!.cn) in Mor n(C), bF(,) has been defined on
i j i j
(Fe1(c0) [ eF2(c0)) x In1 [ bF(c0) x @In1 ,
O
and must be extended to bF(c0)xIn1 while covering bO(cn)OFb(,) 2 Map (Fb(c0), *
*F 0(cn)).
So with the help of diagram (2) again, the obstruction to defining bF(,) is see*
*n to be
an_element o(,) 2 ssn1( (c0)) = fin1(c0). Together, these define a cochain o*
* 2
Cn(C; fin1). Just as in the proof of existence, one then shows that do = 0, *
*and
hence_that_o represents a class [o] 2 limn(fin1). If [o] = 0, then o = dae f*
*or some
ae 2 C n1(C; fin1), and bFcan be modified on Mor n1(C) using ae in such a wa*
*y that
it can then be extended to Mor n(C). Upon continuing this procedure, we constru*
*ct an
Rg1 structure bFon F , together with a natural transformation to F 0and morphi*
*sms
of Rg1 structures
eF1! bF Fe2.
So by Proposition A.2,
eF1' Rg (Fe1) ' Rg (Fb) ' Rg (Fe2) ' eF2.
We finish the section with two corollaries to Theorem A.3. The first is the *
*main
theorem of Dwyer and Kan in [DK ]. It is the "absolute case" of Theorem A.3: th*
*e case
where F 0is the constant functor which sends each object to a point.
A functor F from C to Top or hoTop will be called centric if for each morphi*
*sm
' 2 Mor C(c, d) in C, the induced map
'O
Map (F (c), F (c)Id! Map (F (c), F (d))'
is a homotopy equivalence. This is what Dwyer and Kan call a centric diagram.
Corollary A.4. Fix a centric functor F :C ! hoTop . Define ffi:Cop ! Ab
*
* '
(all i 1) by setting ffi(c) = ssi Map (F (c), F (c))Id and by letting ffi c _*
*__! d be the
composite
(OF('))* (F(')O)*
ssi Map(F (d), F (d))Id ______! ssi Map(F (c), F (d))F(') ______~=ssi Map(F *
*(c), F (c))Id .
Then the obstructions to the existence of a rigidification Fe of F lie in the *
*groups
limn+2(ffn) for n 1; while the obstructions to the uniqueness of eFup to equ*
*ivalence
C
of rigidifications lie in limn+1(ffn) for n 1.
C
The second corollary is a generalization of [CLN , Proposition B], and follo*
*ws upon
combining Corollary A.4 with an idea taken from the proof of that proposition.
Corollary A.5. Fix a space X, and a centric functor F :C ! hoTop . We also l*
*et
X denote the constant functor X :C ! Top which sends each object to X and ea*
*ch
morphism to IdX.
84 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
(a)Assume there is a natural transformation of functors O: F ! hoO X such t*
*hat
the map O(c): F (c) ! X is centric for each c 2 Ob (C). Then there is a r*
*igidi
fication eFof F , together with a rigidification eO:eF! X of O.
(b)Assume eF1and eF2are two rigidifications of F . Let _i:F ! hoO eFibe nat*
*ural
equivalences, and let eOi:eFi! X be natural transformations of functors s*
*uch
that for all c 2 Ob (C), eOi(c) 2 Map (Fei(c), X) is centric, and the square
_1(c)
F (c)_________! eF1(c)

_2(c)# eO1(c) (1)
#
eF2(c)_________eO2(c)!X
commutes up to homotopy. Then Fe1and Fe2are equivalent rigidifications. Mo*
*re
precisely, there is a third rigidification eF0of F , natural transformations*
* of functors
eF1e_1!eF0e_2eF2
such that e_i(c) is a homotopy equivalence for each c 2 Ob (C), a space X0 t*
*ogether
with a natural transformation eO0:eF0! X0 to the constant functor, and ho*
*mo
topic homotopy equivalences f1 ' f2: X ! X0, such that the following diag*
*ram
commutes for each c 2 Ob (C):
eF1(c)______e_1(c)!eF(c)____e_2(c)eF(c)
' 0 ' 2
  
eO1(c) eO0(c) eO2(c) (2)
# # #
f1 f2
X _________! X0 _________X .
Proof.Let C+ be the category C with an additional final object * added. For any
functor ff: C+op ! Ab , limi(ff) = 0 for all i 1 since C+op has an initia*
*l object.
A functor F+ :C+ ! hoTop can be thought of as a triple F+ = (F, X, O), where
F = F+C is a functor from C to hoTop , X = F+(*) is a space, and O is a natural
transformation of functors from F to the constant functor X. Functors from C+ t*
*o Top
are described in an analogous way.
In the situation of (a), (F, X, O) is a functor from C+ to hoTop . The obstr*
*uction
groups of Corollary A.4 vanish, and hence it has a rigidification (Fe, eX, eO).*
* Upon
composing with an appropriate homotopy equivalence eX'! X, we can arrange th*
*at
eX= X.
In the situation of (b), (Fe1, X, eO1) and (Fe2, X, eO2) are two functors fro*
*m C+ to Top
which are rigidifications of the same functor (F, X, eO1O_1) by the homotopy co*
*mmu
tativity of (1). Since the uniqueness obstructions of Corollary A.4 all vanish,*
* there is a
third homotopy lifting (Fe0, X0, eO0), together with natural transformations of*
* functors
_e1 _e2
(Fe1, X, eO1)  (Fe0, X0, eO0) ! (Fe2, X, eO2)
which induce homotopy equivalences on all objects. Thus upon setting fi= e_i(*)*
*, we
obtain the commutative diagram (2), where all horizontal maps are homotopy equi*
*v
alences. Finally, e_1(*) ' e_2(*), since they come from equivalences between li*
*ftings of
the same homotopy functor, and this finishes the proof of (b).
Carles Broto, Ran Levi, and Bob Oliver *
* 85
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86 Discrete models for the plocal homotopy theory of compact Lie groups and *
*pcompact groups
Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, E08193 Bel
laterra, Spain
Email address: broto@mat.uab.es
Department of Mathematical Sciences, University of Aberdeen, Meston Building
339, Aberdeen AB24 3UE, U.K.
Email address: ran@maths.abdn.ac.uk
LAGA, Institut Galil'ee, Av. JB Cl'ement, 93430 Villetaneuse, France
Email address: bobol@math.univparis13.fr