THE THEORY OF p-LOCAL GROUPS: A SURVEY
CARLES BROTO, RAN LEVI, AND BOB OLIVER
The motivation for this project comes from the study of the p-local homotopy *
*theory
of classifying spaces of finite groups, or more generally of compact Lie groups*
*. By "p-
local homotopy theoryö f a space we mean the homotopy theory of its p-completi*
*on.
It turns out that there is a close connection between the p-local homotopy theo*
*ry
of BG and the "p-local structureö f the group G, by which we mean the fusion
(conjugacy relations) in a Sylow p-subgroup of G. This connection then suggeste*
*d to
us the construction of certain spaces (classifying spaces of "p-local finite gr*
*oupsä nd
"p-local compact groups") which have many of the same properties as have p-comp*
*leted
classifying spaces of finite and compact Lie groups.
A brief survey of the Bousfield-Kan p-completion functor will be given in Sec*
*tion 1.
For the purpose of this introduction it suffices to say that this is a functor *
*from spaces
to spaces, which focuses on the properties of a space which are visible through*
* its mod
p homology.
The p-fusion data of a finite group G consists of a Sylow p-subgroup S G, t*
*ogether
with information on how subgroups of S are related to each other via conjugatio*
*n in
G. This data includes an abundance of information about BG. For example, the
well known theorem of Cartan and Eilenberg [CE , Theorem XII.10.1], stating that
H*(G; Fp) is given by the subring of "stable elements" in H*(S; Fp), can be int*
*erpreted
as saying that mod p cohomology of finite groups is determined by p-fusion.
A much stronger version of the Cartan-Eilenberg theorem is given by the Marti*
*no-
Priddy conjecture [MP ]. The conjecture, recently proved by the third author, *
*states
roughly that the classifying spaces of two groups have the same p-local homotop*
*y type
if and only if the groups share the same p-fusion data. A more precise formula*
*tion
of this result, as well as a discussion of some of the ideas which go into its *
*proof, are
presented in Section 3.
Unfortunately, p-fusion is not quite sufficient to give all homotopy theoreti*
*c informa-
tion associated to the classifying space of a finite group G. For instance, the*
* topological
monoid of homotopy equivalences from BG^pto itself cannot easily be described u*
*sing
the fusion alone, and the work outlined below in Sections 2-3 was initially mot*
*ivated
by the need to obtain an algebraic description of this space. A partial answer *
*to this
problem, phrased in terms of p-fusion, appears in [BL ]; but the complete descr*
*iption
in [BLO1 ] requires the use of an additional structure, the "centric linking s*
*ystemö f
the finite group.
The introduction of this new concept led to several new results, as described*
* in detail
in Section 2. For instance, it gave a different condition for determining when *
*two finite
groups have equivalent p-completed classifying spaces: a condition which is les*
*s useful
than that given by the Martino-Priddy conjecture, but which is much easier to p*
*rove
(Theorem 2.2). It also made it possible to give a complete algebraic descriptio*
*n of the
___________
1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 55Q5*
*2, 20D20.
Key words and phrases. Classifying space, p-completion, finite groups, fusion.
C. Broto is partially supported by DGICYT grant PB97-0203.
B. Oliver is partially supported by UMR 7539 of the CNRS.
1
2 CARLES BROTO, RAN LEVI, AND BOB OLIVER
topological monoid of self homotopy equivalences of BG^pfor G finite. In additi*
*on, the
introduction of centric linking systems for groups paved the way to the more ge*
*neral
concept of p-local finite groups.
The p-fusion in a finite group G with a Sylow p-subgroup S can be made into a*
* cate-
gory, whose objects are the subgroups of S, and whose morphisms are those monom*
*or-
phisms induced by conjugation in G. This category is called the üf sion categor*
*y of G
at the prime p". Fusion systems can be defined more abstractly; they are catego*
*ries of
subgroups of a given p-group whose morphisms are certain monomorphisms between
them, and which satisfy certain axioms. The fusion systems we will be intereste*
*d in
satisfy certain extra conditions first formulated by Puig [Pu3 ],[Pu4 ], and ar*
*e known as
äs turated fusion systems".
Our first goal is to construct a classifying space for each saturated fusion *
*system.
This is motivated in part by representation theory: we refer to Section 5.3 for*
* a brief
discussion of the fusion system of a block as defined by Alperin & Brou'e. But *
*it also
provides a new class of spaces, which have many of the same properties of p-com*
*pleted
classifying spaces of finite groups, and which can be characterized in homotopy*
* the-
oretic terms. To construct these classifying spaces, we define what it means t*
*o be a
centric linking system associated to a saturated fusion system. Just like the *
*exam-
ples arising from finite groups, centric linking systems associated to abstract*
* saturated
fusion systems, if they exist, are categories which can be thought of as lifts *
*of the
respective fusion systems. The p-completed nerve of a centric linking system i*
*s the
desired classifying space of the corresponding fusion system. This does, howeve*
*r, lead
to the unsatisfactory situation that we do not yet know whether or not there ex*
*ists a
linking system associated to a given fusion system, and whether or not it is un*
*ique (see
Section 4.8).
A "p-local finite group" is now defined to be a triple consisting of a p-grou*
*p, a
saturated fusion system over it and an associated centric linking system. The c*
*lassifying
space of a p-local finite group is defined to be the p-completed nerve of its l*
*inking
system. When the triple consists of a Sylow p-subgroup of a finite group G, tog*
*ether
with its fusion and linking systems, then its classifying space has the homotop*
*y type
of BG^p.
A closely related topic, which provides extra motivation, is the study of p-c*
*ompact
groups. These objects were introduced in the 1980's by Dwyer and Wilkerson [DW3*
* ]
and extensively studied by them and several other authors. A p-compact group i*
*s a
loop space X (i.e., X ' (BX) for some pointed "classifying space" BX), such th*
*at
H*(X; Fp) is finite and BX is p-complete. The concept of a p-compact group was
designed to be a homotopy theoretic analogue of a classifying space of a compac*
*t Lie
group. For instance if G is a compact Lie group whose group of components is a*
* p-
group, then BG^pis the classifying space of a p-compact group. On the other han*
*d if
ß0(G) is not a p-group then the loop space (BG^p) is not generally Fp-finite. *
*In spite
of the fact that there are many common aspects to the homotopy theory of classi*
*fying
spaces of finite, compact Lie and p-compact groups, the techniques used in the *
*study of
p-compact groups fail or become hard to use if the Fp-finiteness condition is d*
*ropped.
Our approach suggests a way to fix this problem. The report on "p-local compa*
*ct
groups" given here is incomplete, as the research is still in progress at the t*
*ime of writing
this survey. However, the definition of p-local compact groups is a natural ext*
*ension of
that of p-local finite groups, the main change being the replacement of the fin*
*ite p-group
S in the definition of a p-local finite group by what we call a "discrete p-tor*
*al group"
(Section 6.1). Once we have defined saturated fusion systems and associated ce*
*ntric
THE THEORY OF p-LOCAL GROUPS: A SURVEY 3
linking systems over such groups, the definition of a p-local compact group bec*
*omes the
obvious generalization of that of p-local finite groups. We are then able to sh*
*ow that
the family of spaces obtained as classifying spaces of p-local compact groups c*
*ontains
all p-completed classifying spaces of compact Lie groups and p-compact groups.
The theory presented here suggests a vast ground for further exploration. As*
* one
example, we outline some results concerning extensions of p-local finite groups*
* (Section
5.4). Some of the other sections also discuss open problems and topics for fur*
*ther
investigation.
This survey is written in order to make it easier for interested readers to l*
*earn about
the subject. As the theory of p-local groups is still relatively new, the main *
*references
are contained in fairly large and quite technical papers. It is our hope that t*
*his survey
will be a valuable reference for readers to get the general picture before they*
* plunge into
more comprehensive articles. The paper is designed respectively, namely, we att*
*empted
to give good motivation to all statements, but most of them appear either with *
*a very
brief sketch of proof or with no proof at all. For all statements however, a p*
*recise
reference for where a proof can be found is listed.
Contents
1. The p-completion functor 4
2. Fusion and linking systems of finite groups *
* 5
2.1. Fusion systems of groups 5
2.2. Centric linking systems of groups *
* 5
2.3. Equivalences of p-completed classifying spaces *
* 6
2.4. Maps from classifying spaces of p-groups *
* 6
2.5. Fusion and linking systems for spaces *
* 7
2.6. Isotypical equivalences of fusion and linking categories *
* 8
2.7. Self homotopy equivalences of BG^p 9
3. The Martino-Priddy conjecture 9
3.1. Fusion preserving isomorphisms 9
3.2. Obstruction theory 10
3.3. Reduction to simple groups 11
3.4. The odd primary case 11
3.5. Obstruction groups at the prime 2 12
4. Abstract finite fusion and linking systems *
* 13
4.1. Fusion systems 14
4.2. Centric linking systems 14
4.3. p-local finite groups *
*15
4.4. Centralizers and mapping spaces 16
4.5. Homology decompositions 17
4.6. Cohomology of the classifying space 18
4.7. Fusion and linking systems determined by their classifying space *
* 20
4 CARLES BROTO, RAN LEVI, AND BOB OLIVER
4.8. Existence and uniqueness of linking systems or classifying spaces *
* 21
5. Examples and methods of construction 22
5.1. Construction of exotic p-local finite groups *
* 22
5.2. Fusion systems of a type studied by Solomon 24
5.3. The fusion system of a block 25
5.4. Extensions of p-local finite groups *
* 25
6. p-local compact groups 27
6.1. Discrete p-toral groups 27
6.2. Fusion and linking systems over discrete p-toral groups *
* 28
6.3. Reduction to finite subcategories *
*28
6.4. A homology decomposition of the classifying space *
*29
6.5. Reduced linking systems 29
6.6. Compact Lie groups 30
6.7. p-compact groups 31
References 31
1. The p-completion functor
We start with a brief description of the p-completion functor of Bousfield an*
*d Kan
[BK ], which we denote by (-)^p. It is a functor from the category of spaces t*
*o it-
self, and comes equipped with a natural transformation ~: Id --! (-)^p. A space
X is p-complete if ~X :X ---! X^pis a homotopy equivalence. One property of p-
completion (one of the few) which applies to all spaces is that a map f :X --! Y
induces a homotopy equivalence f^p:X^p---! Y ^pif and only if f induces an isom*
*or-
phism H*(X; Fp) ~=H*(Y ; Fp).
A space X is called p-good if ~X :X --! X^pinduces an isomorphism H*(X^p; Fp)*
* ~=
H*(X; Fp), or equivalently if X^pis p-complete. Spaces which are p-bad (i.e., *
*not
p-good) remain so permanently: repeated application of the completion functor w*
*ill
never produce a p-complete space. Simply connected spaces, or more generally ni*
*lpo-
tent spaces, are p-good. Spaces whose fundamental group is finite are p-good. M*
*ore
generally, any space X for which ß1(X) contains a p-perfect subgroup of finite *
*index
(i.e., a subgroup of finite index generated by its commutators and p-th powers)*
* is also
p-good. All spaces discussed in this article (at least, all spaces for which we*
* want to
take the p-completion) are p-good.
For any p-good space X, the map ~X :X ---! X^pis a final object among homotopy
classes of maps out of X which induce a mod-p homology isomorphism. If X and Y *
*are
any two p-good spaces, then their p-completions are homotopy equivalent if and *
*only
f g
if there exists some space Z, and maps X --! Z -- Y , such that f and g are bo*
*th
THE THEORY OF p-LOCAL GROUPS: A SURVEY 5
mod-p homology equivalences. We say that X and Y have the same p-local homotopy
type, or that they are mod p equivalent, if X^p' Y ^p.
Among our main objects of study here are the p-completed classifying spaces of
compact Lie groups. For any compact Lie group G, BG is p-good since its fundame*
*ntal
group is finite. Also, ß1(BG^p) ~= ß0(G)=Op(ß0(G)), where Op( ) is the maximal*
* p-
perfect subgroup of a finite group (equivalently, the smallest normal subgrou*
*p of
p-power index).
While p-completion does not change the mod-p homology of BG when G (or ß0(G))
is finite, the space BG^pis not generally aspherical. In fact, for finite G, B*
*G^pis
aspherical only if G contains a normal subgroup of p-power index and order prim*
*e to p;
in all other cases BG^phas infinitely many non-trivial homotopy groups. The hom*
*otopy
theory of spaces of this form has some fascinating aspects. For a survey on som*
*e of the
classical homotopy theory associated with BG^p, the reader is referred to [CL ].
2. Fusion and linking systems of finite groups
The main idea in this section is to describe how the homotopy theory of BG^pis
related to certain categories, which we call the fusion and centric linking sys*
*tems of
G at the prime p (we emphasize "centric" since there is a notion, presently les*
*s useful,
of a more general linking system, which will not be mentioned here). Throughout*
* the
section, we fix a finite group G and a prime p.
2.1. Fusion systems of groups. The fusion system Fp(G) of G is the category who*
*se
objects are the p-subgroups of G, and where for any pair of p-subgroups P, Q *
*G,
Mor Fp(G)(P, Q) = Hom G(P, Q) def={ff 2 Hom (P, Q) | ff = cx, some x 2 G }.
Thus, if we define
NG(P, Q) = {x 2 G | xP x-1 Q}
(the transporter), then
Mor Fp(G)(P, Q) ~=NG(P, Q)=CG(P ).
If S is any Sylow p-subgroup of G, then FS(G) Fp(G) will denote the full su*
*bcate-
gory whose objects are the subgroups of S. Since every p-subgroup of G is conju*
*gate to
a subgroup of any given S 2 Sylp(G), the inclusion of FS(G) in Fp(G) is an equi*
*valence
of categories.
2.2. Centric linking systems of groups. For any finite group H, let Op(H) denote
the minimal normal subgroup of p power index, or equivalently the maximal normal
p-perfect subgroup of H. A p-subgroup P G is p-centric if Z(P ) 2 Sylp(CG(P )*
*), or
equivalently if CG(P ) = Z(P ) x Op(CG(P )) and Op(CG(P )) has order prime to p*
*. For
such subgroups we write Op(CG(P )) = C0G(P ) for short.
The centric linking system Lcp(G) of G is the category whose objects are the *
*p-centric
subgroups of G, and where
Mor Lcp(G)(P, Q) = NG(P, Q)=C0G(P )
for any pair of objects.
6 CARLES BROTO, RAN LEVI, AND BOB OLIVER
The category Lcp(G) is the same as what Puig [Pu1 , xVI.1] called the Ö -loca*
*lit'eö f
G (when restricted to p-centric subgroups).
If S is any Sylow p-subgroup of G, then LcS(G) Lcp(G) will denote the full *
*subcate-
gory whose objects are the subgroups of S which are p-centric in G. As was the *
*case for
the corresponding inclusion of fusion systems, this inclusion is always an equi*
*valence
of categories.
The following theorem helps to explain the usefulness of centric linking syst*
*ems when
studying the homotopy theory of BG^p.
Theorem 2.1. For any finite group G and any prime p, |Lcp(G)|^p' BG^p.
Proof.See [BLO1 , Proposition 1.1]. The idea of the proof is to construct a l*
*arger
category eLcp(G), with the same objects as Lcp(G), but where MorLecp(G)(P, Q) =*
* NG(P, Q)
for all P, Q. Let B(G) be the category with one object oG, and where EndB(G)(oG*
*) = G.
One then shows that the maps
|i| c effG
|Lcp(G)| ------ |Lep(G)| ------! |B(G)| ~=BG,
are Fp-homology equivalences, where ß is the obvious surjective functor, and wh*
*ere effG
is induced by the functor which is the inclusion on all morphism sets NG(P, Q) *
* G.
2.3. Equivalences of p-completed classifying spaces. The Martino-Priddy con-
jecture states roughly that the homotopy type of BG^pdepends only on the fusion*
* of
G in S 2 Sylp(G), or equivalently on the fusion system Fp(G). This will be disc*
*ussed
in more detail in Section 3, and stated precisely in Theorem 3.1.
In this section, we state a weaker result, which says that the homotopy type *
*of BG^p
depends only on its linking system Lcp(G). This does provide a (finite) combina*
*torial
condition for two p-completed classifying spaces to be homotopy equivalent. The*
* condi-
tion is, however, more complicated to check than the one stated in the Martino-*
*Priddy
conjecture, and hence less satisfactory. The proof of this statement is however*
* much
easier than the proof of the Martino-Priddy conjecture.
Theorem 2.2. For any pair G, G0of finite groups and any prime p, BG^p' BG0^pif
and only if Lcp(G) ' Lcp(G0).
Proof.See [BLO1 , Theorem A]. The "if" part of this theorem follows immediatel*
*y from
Theorem 2.1. For the ö nly if partä construction of a centric linking system*
* for a
space is required. See Section 2.5 for more details.
2.4. Maps from classifying spaces of p-groups. One of our goals is to describe *
*in
many cases maps between p-completed classifying spaces of finite groups. The si*
*mplest
case to consider is that where the source is the classifying space of a p-group.
If H and K are any two discrete groups then the space of (unpointed) maps from
BH to BK is very simple to describe, and the result is classical. Set
Rep (H, K) = Hom (H, K)= Inn(K),
the set of conjugacy classes of homomorphisms ("representations") from H to K. *
*Then
the natural map
B :Rep (H, K) -------! [BH, BK]
THE THEORY OF p-LOCAL GROUPS: A SURVEY 7
is a bijection. Also, for each æ: H --! K, the homomorphism
incl.j
CK (æ) x H -------! K,
where CK (æ) def=CK (Im (æ)), induces a map of spaces BCK (æ) x BH ---! BK, who*
*se
adjoint
~=
BCK (æ) ------! Map (BH, BK)Bj
is a homotopy equivalence. The simplest way to see this result is to first sho*
*w that
[BH, BK]* ~=Hom (H, K) (i.e., pointed homotopy classes of pointed maps) and th*
*at
each component of the pointed mapping space Map *(BH, BK) is contractible; and
then examine the fibration
Map *(BH, BK) ------! Map (BH, BK) ------! BK.
The following öf lk theorem" describes one situation in which this result can*
* be
generalized from classifying spaces to p-completed classifying spaces.
Theorem 2.3. For any finite p-group P , and any finite group G, the p-completion
map BG ___! BG^pinduces a (weak) homotopy equivalence
Map (BP, BG)^p---'---!Map (BP, BG^p).
In particular, the map (æ 7! Bæ^p) defines a bijection
Rep (P, G) ---B--!~[BP, BG^p].
=
For each æ: P ____! G, the induced product map CG(æ) x P ____! G induces (aft*
*er
taking adjoints) a homotopy equivalence
BCG(æ)^p-----! Map (BP, BG^p)Bj .
In particular, the mapping space Map (BP, BG^p) is p-complete.
Proof.See [BL , Proposition 2.1], but this theorem was known to the experts wel*
*l before
we wrote down a proof.
This result will be generalized in Section 3 (Theorem 4.2).
2.5. Fusion and linking systems for spaces. The ö nly if" part of Theorem 2.2 f*
*ol-
lows from another construction: a functor from spaces to categories which in pa*
*rticular
sends BG^pto LcS(G). This is described as follows.
Fix a space X, a p-group S, and a map f :BS --! X. We define FS,f(X) to be the
category whose objects are the subgroups of S, and where for P, Q S,
fi
Mor FS,f(X)(P, Q) = ff 2 Hom (P, Q) fif|BP ' f|BQ OBff .
This is clearly a fusion system over S (though not necessarily saturated!), and*
* can be
thought of as the fusion system of the space X with respect to the pair (S, f).
We next define the analogous linking system for X with respect to (S, f). Let*
* LS,f(X)
be the category whose objects are the subgroups of S, and where for P, Q S,
fi
Mor LS,f(X)(P, Q) = (ff, [OE]) fiff 2 Hom (P, Q), [OE] a homotopy class of p*
*aths
in Map (BP, X) from f|BP to f|BQ OBff.
A morphism of LS,f(X) from P to Q thus consists of a morphism ff 2 Hom FS,f(X)(*
*P, Q),
together with a homotopy class of homotopies from f|BP to f|BQ OBff.
8 CARLES BROTO, RAN LEVI, AND BOB OLIVER
There is an obvious öf rgetful" functor
ßS,f(X): LS,f(X) ------! FS,f(X)
which is the identity on objects and sends a morphism (ff, [OE]) to ff. Note th*
*at the set
of morphisms in LS,f(X) sitting over any given ff 2 Hom FS,f(X)(P, Q) is in bij*
*ective
correspondence with ß1(Map (BP, X)f|BP).
We also want a centric linking system in this situation. Let LcS,f(X) be the*
* full
subcategory of LS,f(X) whose objects are the subgroups P S such that CS(P 0) =
Z(P 0) for all P 0isomorphic to P in FS,f(X).
The ö nly if" part of Theorem 2.2 now follows easily from the following:
Theorem 2.4. For any finite group G and any prime p, fix S 2 Sylp(G), and let
` :BS --! BG^pbe the inclusion. Then there are equivalences of categories
FS,`(BG^p) ~=FS(G) and LcS,`(BG^p) ~=LcS(G).
Proof.See [BLO1 , Proposition 2.7]. The first isomorphism says that for any P,*
* Q S
and any ff 2 Hom (P, Q), ff 2 Hom G(P, Q) (i.e., ff is induced by conjugation i*
*n G) if and
only if the composite BP -Bff-!BQ -incl--!BG is homotopic to the inclusion BP *
* BG,
and this follows from Theorem 2.3.
The isomorphism between the two linking categories is slightly more delicate.*
* Note
first that by definition (and the isomorphism of fusion systems), both categori*
*es have
as objects the subgroups of S which are p-centric in G. For any ff 2 Hom G (P,*
* Q)
(any P, Q S), the number of morphisms in FS(G) which cover ff is equal to |Z(*
*P )|,
and the number of morphisms in FS,`(BG^p) which cover ff is equal to the order *
*of
ß1(Map (BP, BG^p)incl) ~=Z(P ) (see Theorem 2.3). This suggests that there shou*
*ld be
a natural bijection between these morphism sets, and such a bijection is constr*
*ucted
by mapping the transporter NG(P, Q) surjectively to both sets and showing that *
*one
has the same identifications in the two cases.
Theorem 2.4 suggests a way to extend the centric linking system of a finite g*
*roup
G to all p-subgroups. Let Lp(G) be the category whose objects are the p-subgrou*
*ps of
G, and where Mor Lp(G)(P, Q) = NG(P, Q)=Op(CG(P )). This is easily seen to be w*
*ell
defined on morphisms. As usual, for S 2 Sylp(G), LS(G) Lp(G) denotes the full
subcategory whose objects are the subgroups of S. Then the argument just sketch*
*ed
also shows that LS(G) ~=LS,`(BG^p).
2.6. Isotypical equivalences of fusion and linking categories. Let C be any of
the categories Fp(G), Lcp(G), FS,f(X), or LS,f(X), or any subcategory of one of*
* these
categories. Let C --ffl!Grbe the forgetful functor to the category of groups. *
* A self
equivalence ': C --! C is called isotypical if there is a natural isomorphism o*
*f functors
from ffl O' to ffl.
When C is one of the above categories, we let Aut (C) and Aut typ(C) denote t*
*he
monoids of self equivalences and isotypical self equivalences, respectively of *
*C. More
generally, we let Auttyp(C) be the strict monoidal category whose objects are t*
*he iso-
typical self equivalences of C, and whose morphisms are the natural isomorphism*
*s of
functors. We can then define Outtyp(C) to be the group of automorphisms modulo *
*nat-
ural isomorphisms; i.e., the group of connected components of the nerve |Auttyp*
*(C)|.
THE THEORY OF p-LOCAL GROUPS: A SURVEY 9
2.7. Self homotopy equivalences of BG^p. Using Theorems 2.1 and 2.4, one can al*
*so
describe the monoid of self homotopy equivalences of BG^pin terms of automorphi*
*sms
of the category Lcp(G). For any space X, we let Aut(X) denote the topological m*
*onoid
of all self homotopy equivalences of X; and by analogy with the notation for se*
*lf
equivalences of categories, let Out (X) = ß0(Aut (X)) denote the group of homot*
*opy
classes of self equivalences.
For any finite group G, Op0(G) C G denotes the largest normal subgroup of G of
order prime to p.
Theorem 2.5. For any finite group G and any prime p, the connected components of
the space Aut(BG^p) are all aspherical, and there are isomorphisms
Out (BG^p) ~=Out typ(Lcp(G)) and ß1(Aut (BG^p)) ~=Z(G=Op0(G))^p.
Moreover, there is a homotopy equivalence
B Aut(BG^p) ~=B|Auttyp(Lcp(G))|.
Proof.See [BLO1 , Theorems B & C] and [BL , Theorem 1.1].
3. The Martino-Priddy conjecture
Originally, the results in Section 1 were motivated partly as a means of stud*
*ying
Aut(BG^p) (Theorem 2.5), but also partly as a means of finding algebraic or com*
*bi-
natorial conditions for two p-completed classifying spaces to be homotopy equiv*
*alent
(Theorem 2.2). What we really want is to describe both of these in terms of the*
* fusion
systems, or equivalently in terms of fusion among subgroups of a Sylow subgroup.
3.1. Fusion preserving isomorphisms. Fix a pair of finite groups G and G0, a pr*
*ime
' 0
p, and Sylow subgroups S 2 Sylp(G) and S0 2 Sylp(G0). An isomorphism S ---! S
is called fusion preserving if for all P, Q S and all P --ff!~Q, ff is induc*
*ed by
=
'ff'-1 *
* 0
conjugation in G if and only if '(P ) ----!~ '(Q) is induced by conjugation in *
*G .
=
Clearly, a fusion preserving isomorphism S --! S0 in the above situation indu*
*ces
an isomorphism of categories FS(G) ~=FS0(G0), and hence an equivalence of the l*
*arger
categories Fp(G) ' Fp(G0). Conversely, given an isotypical equivalence between*
* the
fusion systems of G and G0, it is not hard to construct a fusion preserving iso*
*morphism
between their Sylow p-subgroups [BLO1 , Lemma 5.1].
A classical result in group cohomology is a theorem by Cartan and Eilenberg (*
*see [CE ,
Theorem XII.10.1]), which states roughly that for any finite group G, the cohom*
*ology
ring H*(BG; Fp) is determined by S 2 Sylp(G) and fusion in S (see Section 4.6).*
* The
key point of the Martino-Priddy conjecture is the much stronger statement that *
*the
homotopy type of the p-completed classifying space BG^pis determined by p-fusio*
*n.
Theorem 3.1 (Martino-Priddy conjecture). For any pair G, G0 of finite groups and
any prime p, the following three conditions are equivalent:
(a)BG^p' BG0^p.
10 CARLES BROTO, RAN LEVI, AND BOB OLIVER
'
(b)There is a fusion preserving isomorphism S ----!~S0 between Sylow p-subgrou*
*ps
=
of G and G0.
(c)There is an isotypical equivalence Fp(G) ' Fp(G0).
The implications (a) =) (b) () (c) were proved by Martino and Priddy [MP ]. *
*In
particular, (b) and (c) are equivalent by the above remarks. The remaining impl*
*ication
was recently proven by the third author ([O1 ] and [O2 ]), using the classifica*
*tion theorem
for finite simple groups.
3.2. Obstruction theory. The first step in analyzing the Martino-Priddy conject*
*ure
is to reduce it to a problem of higher derived functors of inverse limits over *
*the orbit
category of G. For any finite group G and any prime p, the orbit category Op(G)*
* of G
is the category whose objects are the p-subgroups of G, and where for all p-sub*
*groups
P, Q G,
MorOp(G)(P, Q) = Map G(G=P, G=Q) ~=Q\NG(P, Q).
Let ZG :Op(G)op---! Ab be the functor
(
Z(P ) if P is p-centric
ZG(P ) =
0 otherwise.
By [BLO1 , Proposition 6.1], the obstruction (for S 2 Sylp(G) and S02 Sylp(G*
*0)) to
lifting a fusion preserving isomorphism
' 0
S ------!~S
=
to an equivalence
Lcp(G) ---'---!Lcp(G0)
lies in the group
lim-2(ZG).
Op(G)
This can be seen directly by choosing maps
Mor Lcp(G)(P, Q) ------! Mor Lcp(G0)('(P ), '(Q))
(for all P, Q S), and then examining the 2-cocycle in
Y
C2(Op(G); ZG) def= Z(P )
P!Q!R
which describes the failure of the images of commutative triangles in Lcp(G) to*
* commute
in Lcp(G0).
The Martino-Priddy conjecture was proven by showing:
Theorem 3.2. For any finite group G and any prime p, lim-i(ZG) = 0 if i 2, o*
*r if
Op(G)
p is odd and i 1.
It is the proof of this result which depends on the classification of finite si*
*mple groups.
THE THEORY OF p-LOCAL GROUPS: A SURVEY 11
3.3. Reduction to simple groups. To reduce the proof of Theorem 3.2 to a problem
about simple groups, consider a maximal normal series of G:
1 = H0 C H1 C H2 C . .C.Hn = G.
Then for each i, Hi=Hi-1is a product of copies of some fixed simple group Li(se*
*e [Go ,
Theorem 2.1.5]). Define subfunctors ZiG ZG by setting ZiG(P ) = Z(P ) \ Hi if *
*P is
p-centric, and ZiG(P ) = 0 otherwise. From the long exact sequences
. .-.--! lim-n(Zi-1G) ---! lim-n(ZiG) ---! lim-n(ZiG=Zi-1G)
Op(G) Op(G) Op(G)
---! lim-n-1(Zi-1G) ---! . .,.
Op(G)
we see that lim-n(ZG) = 0 (for any given n) if lim-n(ZiG=Zi-1G) = 0 for all i.
Op(G)
Higher limits of the functors ZiG=Zi-1Gare described in terms of higher limit*
*s of
certain other functors YL, defined when L is quasisimple (i.e., L is perfect an*
*d L=Z(L)
is simple) and Inn(L) Aut(L). For any such L and , let c: L --! denote
the homomorphism which sends g 2 L to conjugation by g, and define
YL :Op( )op-----! Ab
by setting
(
(c-1P )P=Z(L)P if P \ Inn(L) is p-centric in Inn(L)
YL (P ) =
0 otherwise .
For example, when L is simple and = Inn(L), then YL = ZL. In terms of these
functors, we get
Proposition 3.3 ([O2 , Theorem 3.3]). Assume Hi-1C HiC G are normal subgroups,
where Hi=Hi-1 is a minimal normal subgroup of G=Hi-1, and define Zi-1G ZiGas
above. If Hi=Hi-1is abelian, or if there is a p-subgroup Q G such that [Q, Hi*
*] Hi-1
and CHi(Q) Hi-1, then the quotient functor ZiG=Zi-1Gis acyclic (its higher li*
*mits
vanish in degrees 1). Otherwise, there are a quasisimple group L such that Hi*
*=Hi-1
is a product of copies of the simple group L=Z(L) and a subgroup Aut(L) con*
*taining
Inn(L), together with homomorphisms
lim-n(YL) -----! lim-n(ZiG=Zi-1G)
Op( ) Op(G)
which are onto when n = 1 and isomorphisms when n 2.
Proposition 3.3 describes how the vanishing of higher limits of ZG for arbitr*
*ary finite
groups G is reduced to a question involving finite simple groups. More precise*
*ly, it
involves the vanishing of higher limits of the functors YeL, when eLis a centra*
*l extension
of a simple group L and is an extension of L by outer automorphisms. In fact,*
* when
p is odd, a more direct approach was found which does not involve these functor*
*s YeL,
but they do still have to be studied when p = 2.
3.4. The odd primary case. When p is odd, then for any p-group P , we define
X(P ) P to be the largest subgroup for which there is a normal series
1 = Q0 C Q1 C . .C.Qn = X(P )
such that QiC P for all i, and such that
[ 1(CP(Qi-1)), Qi; p- 1] = 1 8 i = 1, . .,.n.
12 CARLES BROTO, RAN LEVI, AND BOB OLIVER
Here, for any p-group P , 1(P ) = . Also, for subgroups H, K *
* G,
[H, K; n] denotes the n-fold commutator [. .[.[H, K], K] . .,.K].
In the following proposition, when G is a group and S 2 Sylp(G), we say that a
subgroup P S is weakly Aut(G)-closed in S if there is no other subgroup P 6= *
*P 0 S
which lies in the Aut(G)-orbit of P . Also, Je(P ) denotes the Thompson subgrou*
*p: the
subgroup generated by all (maximal) elementary abelian p-subgroups of P of maxi*
*mal
rank.
Proposition 3.4 ([O1 , Propositions 4.1 & 3.7]). For any odd prime p and any fi*
*nite
group G, ZG is acyclic if for each nonabelian simple group L which occurs in t*
*he
decomposition series of G, and any S 2 Sylp(L), there is a subgroup Q X(S) wh*
*ich is
centric and weakly Aut(L)-closed in S. In particular, this always holds if Je(S*
*) X(S).
In fact, [O1 , Proposition 4.1] is slightly stronger, in that X(S) in the abo*
*ve statement
is replaced by a larger subgroup (but depending also on L). When p is odd, L is*
* simple,
and S 2 Sylp(L), then in almost all cases, either X(S) = S, or S contains a uni*
*que
elementary abelian subgroup of maximal rank (and clearly Je(S) X(S) when eith*
*er
of these happens). The only exceptions to this (i.e., the only cases where X(S)*
* S
and there is more than one elementary abelian subgroup of maximal rank) occur w*
*hen
p = 3 and L ~=P SUn(3k); in which case the more restrictive hypothesis of Propo*
*sition
3.4 holds. In fact, no examples are known of a p-group P (for p odd) for which *
*X(P )
does not contain Je(P ).
3.5. Obstruction groups at the prime 2. Since lim-1(ZG) can be nonzero when
p = 2, it is not surprising that this case is harder. We describe here some of*
* the
techniques used to prove, for L simple, that the higher limits of YeLvanish in *
*degrees
2 whenever eLis perfect, eL=Z(eL) ~=L, and Aut(eL) contains Inn(eL). To s*
*implify
the following discussion, we restrict attention to the case eL= L and = Inn(L*
*); i.e.,
the case where YeL= ZL.
In most cases, one first filters the functor ZL by subfunctors
0 = F0 F1 . . .Fk-1 Fk = ZL,
such that for each i there is a p-centric subgroup Pi L such that
(
Z(P ) if P conjugate Pi
(Fi=Fi-1)(P ) ~=
0 otherwise.
We say that "Pi contributes to lim-k(ZL)" if
lim-k(Fi=Fi-1) 6= 0.
Of course, via the long exact sequences which connect the higher limit of Fi, F*
*i-1, and
Fi=Fi-1, the contribution of one subgroup to lim-k(ZL) can äc ncel" the contrib*
*ution
of another subgroup to lim-k 1(ZL).
For simple groups L (and p = 2), there are some cases where a 2-subgroup cont*
*ributes
to lim-2(ZL). For example, this occurs when L is a projective special linear g*
*roup
P SLn(2) for n 4, an alternating group An for n 0, 1 (mod 8), the Mathieu g*
*roup
M24, or the Held group He. In all such cases, this contribution to lim-2(ZL) is*
* cancelled
by another subgroup contributing to lim-1(ZL). There do not seem to be any case*
*s in
which a 2-subgroup contributes to lim-i(ZL) for i 3.
THE THEORY OF p-LOCAL GROUPS: A SURVEY 13
In general, there are graded abelian groups *( ; M), defined for each finite*
* group
and each Z(p)[ ]-module M, such that
lim-*(Fi=Fi-1) ~= *(NL(P )=P ; Z(P ))
[JMO , Lemma 5.4]. Thus P contributes to lim-i(ZL) if and only if i(NL(P )=P *
*; Z(P ))
is nonvanishing. Some of the very nice properties of these functors include:
(
M if * = 0
Proposition 3.5. (a)If p - | |, then *( ; M) =
0 if * > 0.
fi
(b) *( ; M) = 0 if pfi| Ker[ --! Aut(M)]|
(c) *( ; M) = 0 if Op( ) 6= 1
(d) *( ; M) ~=He*-1(St( ); M) (where the Steinberg complex St( ) is the nerve*
* of
the poset of nontrivial p-subgroups of )
(e)If |M| < 1 and n( ; M) 6= 0, then there are p-subgroups
1 = P0 P1 . . .Pn
such that Pi C Pn and Pi = Op N(P1) \ . .\.N(Pi) for all i, such that Pn 2
Sylp(N(P1) \ . .\.N(Pn-1)); and such that Fp[Pn] M (as a Pn-module). In
particular, if M is finite, then rkp(M) pn.
Proof.See [JMO , Proposition 6.1(i,ii)] for points (a)-(c), [Gr ] for point (d*
*), and [O2 ,
Proposition 4.4] for point (e).
As one simple example, let V ~= F22 be the faithful 2-dimensional representat*
*ion of
3. One can show that
2( 3 x 3; V 2) ~=Z=2,
and hence that P L contributes to lim-2(ZL) if N(P )=P ~= 3x 3 and Z(P ) ~=V*
* 2
as a module over N(P )=P . This situation does occur in some simple groups, suc*
*h as
P SLn(2) for n 4, and the sporadic groups M24 and He . On the other hand, the
simplest example where 3 6= 0 is the case
3( 3 x 3 x 3; V 3) 6= 0,
and this does not seem to occur as a pair (N(P )=P, Z(P )) in any simple group.
4. Abstract finite fusion and linking systems
The results of the preceding section suggest that it may be possible to formu*
*late
a more general context in which to study the homotopy theory of classifying spa*
*ces.
In other words, we would like to define an algebraic object for which the notio*
*n of
a classifying space makes sense, and in which the p-local homotopy theory of th*
*at
classifying space is intrinsic to the algebraic structure and vice versa.
The existence of these algebraic objects was predicted by Dave Benson in the *
*mid-
1990's. In the introduction to his paper [Be1 ], where he relates the Dwyer-Wil*
*kerson
space BDI(4) to what he calls the "classifying spacesö f certain ön nexistent *
*finite
simple groups" (see Section 5.2), he writes: "This prompts the speculation that*
* there
should exist a theory of `p-local groups' in which one only gives a Sylow p-sub*
*group
and a fusion pattern. The fusion pattern should obey a set of axioms which are *
*strong
14 CARLES BROTO, RAN LEVI, AND BOB OLIVER
enough to be able to build a p-completed classifying space.". Later, in some un*
*pub-
lished and privately distributed notes, Benson made this more precise by formul*
*ating
some of the axioms of what we now call centric linking systems, and predicting *
*that
their p-completed realizations should function as classifying spaces of the fus*
*ion pat-
terns.
The concept of a fusion system over a p-group S was in fact defined in the 19*
*90's by
Lluis Puig [Pu3 ] with an entirely different purpose in mind. Fusion systems of*
* finite
groups are particular cases of these more general objects. The version presente*
*d here
is a modification of his definition, but is completely equivalent to it.
In this section, we first present the definition of abstract (saturated) fusi*
*on systems,
and then define what it means to be a centric linking system associated to an a*
*bstract
fusion system. These concepts then lead to the definition of a p-local finite g*
*roup and
its classifying space.
4.1. Fusion systems. A fusion system F over a finite p-group S is a category wh*
*ose
objects are the subgroups of S, and whose morphism sets Hom F (P, Q) satisfy the
following 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.
The first requirement in the definition is intuitively obvious. It is less clea*
*r that the
second requirement has to be stated as an axiom. However, it clearly holds when*
* the
fusion system is that of a finite group, and it is essential for the theory. P*
*uig calls
fusion systems satisfying (b) "divisible" fusion systems.
The next definitions may appear quite mysterious and are quite hard to motiva*
*te.
The fusion system of a finite group turns out to satisfy an extra set of condit*
*ions, which
basically makes the theory work. Puig has managed to distil precisely the neces*
*sary
features in his definition of a saturated fusion system. Before we can explain *
*what this
means, we must distinguish certain collection of objects in F. By analogy with *
*groups,
two objects P and Q which are isomorphic in F are said to be F-conjugate.
A subgroup P S is fully centralized in F if |CS(P )| |CS(P 0)| for every *
*P 0 S
which is F-conjugate to P , and is fully normalized in F if |NS(P )| |NS(P 0)*
*| for all
P 0 S which is F-conjugate to P .
A fusion system F is saturated if the following two conditions hold:
(I)For each fully normalized subgroup P in F, P is fully centralized, and AutS(*
*P ) 2
Sylp(Aut F(P )).
(II)If P S and ' 2 Hom F(P, S) are such that 'P is fully centralized, and if *
*we set
N' = {g 2 NS(P ) | 'cg'-1 2 AutS('P )},
_ _
then there is ' 2 Hom F(N', S) such that '|P = '.
4.2. Centric linking systems. Before we can define a centric linking system ass*
*oci-
ated to a given saturated fusion system, we need to explain what it means to be*
* centric
in this context.
Let F be any fusion system over a p-group S. A subgroup P S is F-centric if
CS(P 0) = Z(P 0) whenever P 0is F-conjugate to P .
THE THEORY OF p-LOCAL GROUPS: A SURVEY 15
Let F be a fusion system over the p-group S. A centric linking system associ*
*ated
to F is a category L whose objects are the F-centric subgroups of S, together w*
*ith a
functor
ß :L ------! Fc,
ffiP
and "distinguished" monomorphisms P --! AutL (P ) for each F-centric subgroup
P S, which satisfy the following conditions.
(A) ß is the identity on objects and surjective on morphisms. More precisely, f*
*or 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 F-centric 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(i(f)(g))
# #
f
P ______! Q .
Condition (A) means that Fc is a quotient category of L, which is obtained by
dividing out a free action of the centers of source objects. Conditions (B) an*
*d (C)
ensure compatibility of the different ingredients with each other.
4.3. p-local finite groups. A p-local finite group is a triple (S, F, L), where*
* F is a
saturated fusion system over the p-group S and L is a centric linking system as*
*sociated
to F. The classifying space of the p-local finite group (S, F, L) is the space *
*|L|^p.
The first thing one wants to make sure of is that genuine finite groups give *
*rise to p-
local finite groups, which is indeed the case. For any finite group G and S 2 S*
*ylp(G) the
fusion system FS(G) is saturated and LcS(G) is an associated centric linking sy*
*stem.
Thus (S, FS(G), LcS(G)) is a p-local finite group whose classifying space |LcS(*
*G)|^pis
homotopy equivalent to BG^p. It is interesting to point out that another conseq*
*uence
of the proof of the Martino-Priddy conjecture is that the natural centric linki*
*ng system
of a finite group G is (up to equivalence) the only one associated to the fusio*
*n system
of G (see Section 4.8).
In the examples coming from finite groups it is clear that the nerve of the c*
*entric
linking system is p-good, so that its p-completion is p-complete. It is one cr*
*ucial
ingredient among what makes it possible to reconstruct both the fusion system a*
*nd
the centric linking system from the classifying space of the corresponding p-lo*
*cal finite
group. This is also the case for p-local finite groups; and in fact, we can als*
*o describe
explicitly the fundamental group of the classifying space.
For any saturated fusion system F over a p-group S, define
-1 fi p ff
OpF(S) = g ff(g) fig 2 P S, ff 2 O (Aut F(P )) C S.
This subgroup OpF(S) is the hyperfocal subgroup of F as defined by Puig [Pu4 ].*
* If
F = FS(G) is the fusion system of a finite group G with respect to S 2 Sylp(G),*
* then
the hyperfocal subgroup theorem [Pu2 , x1.1] says that OpF(S) = S \ Op(G) (where
Op(G) is the smallest normal subgroup of p-power index).
16 CARLES BROTO, RAN LEVI, AND BOB OLIVER
Theorem 4.1. Let (S, F, L) be any p-local finite group at the prime p. Then |L*
*| is
p-good. Also, the composite
i1(|`S|) ^
S --------! ß1(|L|) ----! ß1(|L|p),
`S
induced by the inclusion B(S) ---! L, is surjective, and induces an isomorphism
ß1(|L|^p) ~=S=OpF(S).
Proof.See [BLO2 , Proposition 1.11] and [BCGLO , x1].
4.4. Centralizers and mapping spaces. Our next task is to study mapping spaces
from the classifying space of a finite p-group to that of a p-local finite grou*
*p. The
result is a generalization of Theorem 2.3, which we now describe. If G is a fin*
*ite group
and Q a finite p-group, then the set of components [BQ, BG^p] is described in t*
*erms of
homomorphisms from Q to G modulo conjugation in G. This motivates us to define,
for any p-local finite group (S, F, L) and any p-group Q,
Rep(Q, F) = Hom (Q, S)=~ ,
where ~ is the equivalence relation defined by setting æ ~ æ0 if there is some *
*Ø 2
IsoF(æ(Q), æ0(Q)) such that æ0= Ø Oæ.
Components of mapping spaces Map (BQ, BG^p) are described in terms of p-com-
pleted classifying spaces of centralizers. Thus before we can state the analogo*
*us result
for p-local finite groups, we need to define what is meant by the centralizer f*
*usion and
linking systems. The original definition of the centralizer fusion system is du*
*e to Puig.
Fix a p-local finite group (S, F, L) and a subgroup Q S which is fully cent*
*ralized
in F. Define CF (Q) to be the category whose objects are the subgroups of CS(Q)*
*, and
where
0 _ 0 _ _
Hom CF(Q)(P, P 0) = ' 2 Hom F(P, P )|9' 2 Hom F(P Q, P Q), '|P = ', '|Q = 1Q*
* .
Define CL(Q) to be the category whose objects are the CF (Q)-centric subgroups *
*of
CS(Q), and where Mor CL(Q)(P, P 0) is the set of those ' 2 Mor L(P Q, P 0Q) who*
*se
underlying homomorphisms are the identity on Q and send P into P 0. By [BLO2 ,
Proposition 2.5], the triple (CS(Q), CF (Q), CL(Q)) is always itself a p-local *
*finite group
in this situation. When Q is fully normalized, there is an analogous constructi*
*on of a
ön rmalizer" p-local finite group (NS(Q), NF (Q), NL(Q)) [BLO2 , Lemma 6.2].
If G is a finite group, S 2 Sylp(G) and Q S is fully centralized in F = FS(*
*G) (i.e.,
CS(Q) 2 Sylp(CG(Q))), then CF (Q) and CL(Q) are isomorphic to FCS(Q)(CG(Q)) and
LcCS(Q)(CG(Q)) respectively. Also Rep(Q, F) ~=Rep (Q, G). With this in mind all*
* that
is left is to translate the statement of Theorem 2.3 to the new terminology.
Theorem 4.2. Let (S, F, L) be a p-local finite group, and let f :BS --! |L|^pbe*
* the
natural inclusion followed by completion. Then the following hold, for any p-gr*
*oup Q.
(a)The map
~= ^
Rep(Q, F) -----! [BQ, |L|p],
defined by sending the class of æ: Q --! S to f OBæ, is a bijection.
(b)For any homomorphism æ: Q --! S such that æQ is fully centralized in F,
0L,jQ:|CL(æQ)|^p---'---!Map (BQ, |L|^p)Bj
is a homotopy equivalence.
THE THEORY OF p-LOCAL GROUPS: A SURVEY 17
In particular, Map (BQ, |L|^p) is p-complete.
Proof.See [BLO2 , Corollary 4.5 & Theorem 6.3].
Our original motivation for considering the centric linking system of a finit*
*e group
G was as a tool for describing the monoid of self homotopy equivalences of BG^p*
*. The
description in Section 2.7 can be extended directly to this more general situat*
*ion. We
refer to Sections 2.6 and 2.7 for definitions of isotypical equivalences, and t*
*he category
Aut(C) of self equivalences of a category C.
Theorem 4.3. For any p-local finite group (S, F, L), the topological monoids Au*
*t(|L|^p)
and |Auttyp(L)| are equivalent in the sense that their classifying spaces are h*
*omotopy
equivalent. In particular,
8
< lim0(Z) if i = 1
Out(|L|^p) ~=Out typ(L) and ßi(Aut (|L|^p)) ~= -F
: 0 if i 2.
Proof.See [BLO2 , Theorem 8.1].
4.5. Homology decompositions. One of the standard techniques when studying
maps between p-completed classifying spaces of finite groups is to replace them*
* by
(the p-completion of) a homotopy colimit of simpler spaces. There are many ways*
* of
decomposing BG^p[Dw ], of which the two most frequently used are the following:
o The subgroup decomposition: BG is mod p equivalent to the homotopy direct
limit, over the orbit category of G (Section 3.2), of the classifying spa*
*ces of
its p-radical subgroups [JMO ]. The same type of decomposition holds for*
* the
collections of the p-centric subgroups, or the p-centric and p-radical su*
*bgroups
of G.
o The centralizer decomposition: BG is mod p equivalent to the homotopy dir*
*ect
limit, over the fusion category of nontrivial elementary abelian p-subgro*
*ups
E G, of the classifying spaces of the centralizers CG(E) [JM ].
Both of these decompositions have analogs for classifying spaces of p-local fin*
*ite groups.
For any p-local finite group (S, F, L), the subgroup decomposition of |L|^pis*
* taken
over the orbit category of F. This is the category O(F) whose objects are the s*
*ubgroups
of S, and whose morphisms are defined by
Mor O(F)(P, Q) = RepF (P, Q) def=Inn(Q)\ Hom F(P, Q).
Also, Oc(F) denotes the full subcategory of O(F) whose objects are the F-centric
subgroups of S. If L is a centric linking system associated to F, then eßdenote*
*s the
composite functor
i c c
eß:L --- -i F --- -i O (F).
There is a difference between the orbit category of a fusion system and the o*
*rbit
category of a group. If G is a group and S 2 Sylp(G), then
Mor OS(G)(P, Q) ~=Q\NG(P, Q), while MorO(FS(G))(P, Q) ~=Q\NG(P, Q)=CG(P ).
In general, of course, these can be very different; but if P is p-centric, they*
* differ only
by the action of the group C0G(P ) which is of order prime to p.
18 CARLES BROTO, RAN LEVI, AND BOB OLIVER
Let Top denote the category of spaces. Let C and D be small categories and l*
*et
ffi F
C ___! D and C ___! Top be functors. The left homotopy Kan extension of F alo*
*ng OE
is a functor LffiF : D ___! Top with the property that
hocolim-----!F ' hocolim-----!(LffiF ).
C D
Details on the construction and properties of LffiF are given in [HV ].
Proposition 4.4. Fix a saturated fusion system F and an associated centric link*
*ing
system L, and let eß:L --! Oc(F) be the projection functor. Let
eB:Oc(F) ------! Top
be the left homotopy Kan extension along eßof the constant functor L -*-!Top . *
*Then
eBis a homotopy lifting of the homotopy functor P 7! BP , and
|L| ' hocolim-----!(Be). (1)
Oc(F)
Proof.See [BLO2 , Proposition 2.2]. The proof is mostly a formality, followin*
*g from
elementary properties and the construction of homotopy Kan extensions.
The centralizer decomposition of |L| is also mostly a formality in this conte*
*xt. Recall
the definition of centralizer fusion and linking systems in Section 4.4: for an*
*y p-local fi-
nite group (S, F, L) and any fully centralized subgroup Q S, (CS(Q), CF (Q), *
*CL(Q))
is again a p-local finite group. Also, for any fusion system F over a p-group S*
*, we let Fe
denote the full subcategory of F whose objects are the nontrivial elementary ab*
*elian
p-subgroups of S which are fully centralized in F.
Theorem 4.5. Fix a p-local finite group (S, F, L). Then there is a functor
Ce: Fe -------! Top,
which is a homotopy lifting of the homotopy functor E 7! |CL(E)|, such that
hocolim-----!(Ce) ' |L|.
E2Fe
Proof.See [BLO2 , Theorem 2.6]. The functor eC is defined explicitly there as*
* a left
Kan extension.
4.6. Cohomology of the classifying space. Arguably one of the most fundamen-
tal theorems in group cohomology is the statement that if G is a finite group a*
*nd
S 2 Sylp(G), then for every p-local G-module M, H*(G, M) is the module of "stab*
*le
elements" in H*(S, M); i.e., the module of elements stable with respect to fusi*
*on in
G. In our context, if we restrict attention to trivial p-local coefficients, th*
*en the same
statement holds for classifying space |L|^pof a p-local finite group. As one co*
*nsequence
of this result, H*(|L|^p; Fp) is noetherian for any p-local finite group (S, F,*
* L).
For any fusion system F over a p-group S, let H*(F; Fp) be the subring of H*(*
*BS; Fp)
consisting of those elements which are stable under all fusion in F; i.e.,
* fi *
H*(F; Fp) = x 2 H (BS; Fp) fiff (x) = x|BP , all ff 2 Hom F(P, S).
The proof that H*(|L|; Fp) ~= H*(F; Fp), when (S, F, L) is a p-local finite gro*
*up, is
based on the construction of a certain (S, S)-biset: a set which has left and *
*right
THE THEORY OF p-LOCAL GROUPS: A SURVEY 19
actions of S which commute with each other. If P S and ' 2 Hom (P, S), let
S x(P,')S denote the biset
S x(P,')S = (S x S)=~ , where (x, gy) ~ (x'(g), y) for x, y 2 S, g 2,P
and set
i '* trf j
[S x(P,')S] = H*(BS) ----! H*(BP ) ---P-!H*(BS) 2 End H*(BS; Fp) .
Here, trfPdenotes the transfer map. If B is a disjoint union of bisets Bi of th*
*is form,
we let [B] be the sum of the endomorphisms [Bi]. If B is an (S, S)-biset, then*
* for
P S and ' 2 Inj(P, S), we let B|(P,S)denote the restriction of B to a (P, S)-*
*biset,
and let B|(',S)denote the (P, S)-biset where the left P -action is induced by '.
Proposition 4.6 ([BLO2 , Proposition 5.5]). For any saturated fusion system F *
*over
a p-group S, there is an (S, S)-biset with the following properties:
(a) is a disjoint union of bisets of the form Sx(P,')S for P S and ' 2 Hom F*
*(P, S).
(b)For each P S and each ' 2 Hom F(P, S), |(P,S)and |(',S)are isomorphic as
(P, S)-bisets.
(c)| |=|S| 1 (mod p).
Furthermore, for any biset which satisfies these properties, [ ] is an idempo*
*tent in
End(H*(BS; Fp)), and
* [ ] * *
Im H (BS; Fp) ---! H (BS; Fp) = H (F; Fp).
It was Markus Linckelmann and Peter Webb who first formulated conditions (a),
(b), and (c) in the above proposition, and who saw the significance of finding *
*a biset
with these properties.
Theorem 4.7. For any p-local finite group (S, F, L), the natural homomorphism
~= *
H*(|L|^p; Fp) -------! H (F; Fp),
induced by the inclusion of BS in |L|, is an isomorphism. Furthermore, the ring
H*(|L|^p; Fp) is noetherian.
Proof.See [BLO2 , Theorem 5.8]. The idea is to use a certain decomposition the*
*orem
for unstable algebras over the Steenrod algebra, due to Dwyer and Wilkerson [DW*
*1 ,
Theorem 1.2]. Their theorem implies both that H*(F; Fp) is the inverse limit of*
* the
cohomology rings H*(|CL(E)|; Fp) as E runs over the elementary abelian p-subgro*
*ups
1 6= E S, and also that higher derived functors of this inverse system vanish*
*. Since
|L|^pis the homotopy direct limit of spaces homotopy equivalent to |CL(E)|^p(Th*
*eorem
4.5), we can then conclude from the spectral sequence of a homotopy colimit that
H*(F; Fp) ~=H*(|L|; Fp). One of the requirements when applying the Dwyer-Wilker*
*son
theorem is that the homomorphism
H*(F; Fp) ------! H*(BS; Fp)
be a split monomorphism, and the splitting is provided by Proposition 4.6.
The argument just sketched is carried out inductively, since we need to assum*
*e the
theorem holds for the centralizers. This means that a separate argument is need*
*ed for
fusion systems F with nontrivial center; i.e., those for which F = CF (E) (and *
*hence
L = CL(E)) for some 1 6= E S.
20 CARLES BROTO, RAN LEVI, AND BOB OLIVER
The (S, S)-biset of Proposition 4.6, associated to a given saturated fusion*
* system
F over S, can also be used to construct a spectrum associated to F which is equ*
*ivalent
to the suspension spectrum of any classifying space for F (if any exists). Any*
* finite
(S, S)-biset induces, via inclusions and transfer maps, a stable map from the s*
*uspension
spectrum 1 BS to itself. The properties of listed in Proposition 4.6 imply t*
*hat the
induced map [ ] is an idempotent in the ring of homotopy classes of maps from *
*1 BS
to itself, and that the image of H*([ ]; Fp) is equal to H*(F; Fp). The infinit*
*e mapping
telescope of [ ] is thus a stable summand of BS whose mod p cohomology is isomo*
*rphic
to H*(F; Fp). Hence by Theorem 4.7, if |L|^pis any classifying space for F, the*
*n this
stable summand of 1 (BS) is homotopy equivalent as a spectrum to 1 (|L|^p).
4.7. Fusion and linking systems determined by their classifying space. A
priori, one might think that the classifying space of a p-local finite group (S*
*, F, L)
should contain only part of the information given by the fusion and linking sys*
*tems
F and L. But in fact, one can recover both categories from the homotopy type of
|L|^p. This is done with the help of the functors from spaces to categories des*
*cribed in
Section 2.5.
Theorem 4.8. Fix a p-local finite group (S, F, L), and let f :BS --! |L|^pbe the
natural inclusion. Then there are equivalences of categories
F = FS,f(|L|^p) and L ~=LcS,f(|L|^p).
Proof.See [BLO2 , Proposition 7.3]. The proof is very similar to that of Theo*
*rem
2.4.
In other words, if (S, F, L) and (S0, F0, L0) are two p-local finite groups a*
*nd |L|^p'
|L0|^p, then (S, F, L) and (S0, F0, L0) are isomorphic as triples, via isomorph*
*isms of
groups and categories which commute with all of the structures which link them.*
* To
see how this follows from Theorem 4.8, note that by Theorem 4.2, for any homoto*
*py
_ 0 ^ ff 0
equivalence |L|^p___'! |L |p, there is an isomorphism S ___~=!S such that the f*
*ollowing
square commutes up to homotopy:
f ^
BS ________! |L|p
| |
Bff| _|
# #
f0 0 ^
BS0 _______! |L |p .
Here, f and f0 are the natural inclusions.
We also note here that for any given fusion system F over a p-group S, there *
*are
bijective correspondences
8 9 8 9 8 9
< linking = < classifying= < liftings of the=
systems ~= spaces ~= homotopy functor
: assoc. to F; : for F ; : P 7! BP ;
which are given as follows:
Leß(*)
L ___________________________________________________________*
*____________________________________________________________________/@
________|-|^p_____________________________________________*
*_____________________________________________________________________@
_______________________________________________________*
*________________mmm__________________________________________________@
_____________________________________________________*
*hocolimmmmmm_________________________________________________________@
LcS,`(-)____$$__________________________________________*
*___vvmmmm____________________________________________________________@
X
THE THEORY OF p-LOCAL GROUPS: A SURVEY 21
More precisely, the first set (from the left) consists of all linking systems a*
*ssociated to
F up to isomorphism (isomorphisms of categories which commute with the projecti*
*ons
to F and the distinguished monomorphisms). The second set contains all classify*
*ing
spaces (p-completed nerves of linking systems associated to F), modulo the rela*
*tion
that |L|^pand |L0|^pare equivalent if there is a homotopy equivalence between t*
*hem
which commutes (up to homotopy) with the natural inclusions of BS. The third set
consists of all functors from the orbit category Oc(F) to spaces which lift the*
* homotopy
functor P 7! BP , modulo natural homotopy equivalences of functors to Top . The
bijection between the first two sets follows from Theorem 4.8, and the commutat*
*ivity
of the triangle involving Kan extension, homotopy colimit, and | - |^pwas shown*
* in
Proposition 4.4. It remains to check that each lifting of (P 7! BP ) is the le*
*ft Kan
extension of some linking system, and this is shown in the proof of [BLO2 , Pr*
*oposition
2.3], where an explicit procedure is given for constructing a linking system as*
*sociated
to any given homotopy lifting.
It is also worth noting that the obstructions to lifting a given saturated fu*
*sion system
F to centric linking system L described in Section 4.8 below coincide with thos*
*e for
lifting homotopy functors on the orbit category associated to F described by Dw*
*yer and
Kan [DK ]. It was this observation which first suggested to us the above biject*
*ions, and
in fact which first suggested to us the idea that two groups have equivalent p-*
*completed
classifying spaces if and only if their linking systems are equivalent.
4.8. Existence and uniqueness of linking systems or classifying spaces. One
of the main open questions in this subject is that of the existence and uniquen*
*ess of
linking systems associated to a given saturated fusion system. The obstruction*
*s for
these problems are well understood, and are closely related to those for the Ma*
*rtino-
Priddy conjecture as discussed in Section 2.
Fix a saturated fusion system F over a p-group S. Define a functor
ZF :Oc(F)op-------! Ab
by setting ZF (P ) = Z(P ) = CS(P ) for each F-centric subgroup P S. The obst*
*ruc-
tion to the existence of a centric linking system associated to F lies in lim-3*
*(ZF ), and
the obstruction to its existence lies in lim-2(ZF ) [BLO2 , Proposition 3.1]. *
*This is com-
pletely analogous to the obstructions to the existence and uniqueness of an ext*
*ension of
a group G by another group K which acts on G via outer automorphisms: obstructi*
*ons
which lie in H3(K; Z(G)) and H2(K; Z(G)), respectively.
One way to see the obstruction to the existence of associated linking systems*
* is to
construct directly a äc tegory" which satisfies all of the conditions for a lin*
*king system,
except that the composition of morphisms need not be associative. This can be d*
*one
in such a way that the failure of associativity assigns to each triple of compo*
*sable
morphisms P0 ! P1 ! P2 ! P3 in Oc(F) an element of Z(P0), and these elements
combine to form a 3-cocycle with coefficients in ZF .
We do have some results about the existence and uniqueness of associated link*
*ing
systems in various special cases. For example,
Proposition 4.9. Let F be a saturated fusion system over a p-group S. Then there
exists a linking system associated to F if rk(S) < p3, and the linking system i*
*s unique
if rk(S) < p2.
22 CARLES BROTO, RAN LEVI, AND BOB OLIVER
Proof.[BLO2 , Corollary 3.4]. The point of this proof is that the correspondin*
*g higher
limit obstruction groups for existence and uniquness vanish under the hypothese*
*s of
the proposition.
Note that the Martino-Priddy conjecture is a special case of the problem of u*
*nique-
ness of centric linking systems. The proof of this conjecture, as sketched in S*
*ection 3,
shows in fact that there is exactly one linking system associated to the fusion*
* system of
a finite group. If we still hope to find a proof of this conjecture which is in*
*dependant of
the classification of finite simple groups, then this is not only for esthetic *
*reasons, but
also because it seems likely that any such proof could extend to a proof of the*
* existence
and uniqueness of linking systems (hence classifying spaces) associated to any *
*given
saturated fusion system.
5. Examples and methods of construction
One of the weak points in our investigations into the theory of p-local finit*
*e groups is
the problem of constructing "exotic" examples: examples which do not come from *
*finite
groups. The key difficulty seems to be that of showing that newly constructed f*
*usion
systems are saturated. Some attempts to develop general techniques for construc*
*ting
saturated fusion systems are described in Section 5.1. We then describe some w*
*hich
arise more än turally", notably those of the type studied by Solomon and Benson,
and those which come from block theory. We finish this section with a discussio*
*n of
"extensionsö f p-local finite groups: a subject which is still to a large exte*
*nt under
development.
We first note the following general result which is very useful when construc*
*ting
saturated fusion systems.
Proposition 5.1 ([BCGLO , Theorem 2.3]). Let F be a fusion system over a p-gr*
*oup
S. Assume the following hold:
(I)If P S is F-centric and fully normalized in F, then AutS(P ) 2 Sylp(Aut F(*
*P )).
(II)For each F-centric subgroup P S and each ' 2 Hom F(P, S), if we set
N' = {x 2 NS(P ) | 'cx'-1 2 AutS('(P ))},
_
then ' extends to some ' 2 Hom F(N', S).
Let F0 F be the subcategory with the same objects, and whose morphisms are the
composites of restrictions of morphisms in F between F-centric subgroups. Then *
*F0 is
saturated.
In fact, [BCGLO , Theorem 2.3] is formulated more generally, and deals with*
* fusion
systems which satisfy the axioms of saturation only on subgroups which are both*
* F-
centric and F-radical (though an extra axiom is then needed). But the case form*
*ulated
above seems to be the most important one.
5.1. Construction of exotic p-local finite groups. The following theorem is the
only result we know which gives a geometric criterion for showing that a fusion*
* system
f
is saturated. If S is a finite p-group, then we say that a map BS --! X is Sylo*
*w if
every map BP --! X, for a p-group P , factors through f up to homotopy.
THE THEORY OF p-LOCAL GROUPS: A SURVEY 23
Theorem 5.2. Fix a space X, a p-group S, and a map f :BS --! X. Assume that
(a)f is Sylow, and
(b)f|BP is a centric map for each FS,f(X)-centric subgroup P S.
Let F0 FS,f(X) be the subcategory with the same objects, and whose morphisms *
*are
the composites of restrictions of morphisms in FS,f(X) between F-centric subgro*
*ups.
Then F0 is saturated, and the triple S, F0, LcS,f(X) is a p-local finite grou*
*p.
Proof.See [BLO4 ]. One uses direct geometric arguments to show that FS,f(X) sa*
*tisfies
the axioms of saturation on centric subgroups, and then applies Proposition 5.1.
One might expect that it is difficult to find interesting examples of maps BS*
* --! X
which satisfy the centricity condition (b) above. But in fact, with the help of*
* [BLO2 ,
Proposition 4.2], which says that homotopy colimits commute with mapping spaces
Map (BP, -) under certain conditions, spaces X can be constructed which satisfy*
* the
above conditions without being completed classifying spaces of finite groups.
The following is one example of how Theorem 5.2 can be applied. If F0 is a fu*
*sion
system over a p-group S, and for each i = 1, . .,.m we are given subgroups Qi *
*S and
groups of automorphisms i Out(Qi), then we let
F def=
be the fusion system over S defined as follows. For each pair of subgroups P, P*
* 0 S,
Hom F(P, P 0) is the set of composites
'1 '2 'k-1 'k 0
P = P0 ---! P1 ---! P2 ---! . . .---!Pk-2 ---! Pk-1 ---! Pk = P ,
where for each i, either 'i 2 Hom F0(Pi-1, Pi); or Pi-1, Pi Qj for some j, an*
*d 'i =
ff|Pi-1for some ff 2 Aut(Qj) with [ff] 2 j.
Proposition 5.3. Fix a finite group G, a Sylow p-subgroup S G, and subgroups
Q1, . .,.Qm S such that no Qi is conjugate to a subgroup of Qj for i 6= j. *
*Set
Ki = OutG (Qi), and fix subgroups i Out(Qi) which contain Ki. Assume for each
i that
(1)p - [ i:Ki];
(2)Qi is p-centric in G, but for each P Qi there is ff 2 i such that ff(P ) *
*is not
p-centric in G; and
(3)for all ff 2 irKi, Ki\ ffKiff-1 has order prime to p.
Then the fusion system F def= is saturated, and*
* has an
associated centric linking system.
Proof.See [BLO4 ]. One first checks that for each i, there is a unique extensi*
*on Gi of
Qiby iwhich contains NG(Qi) (up to isomorphism). Let X be the union of BG and
the BGi, where each BGi is attached to BG along their common subspace BNG(Qi).
The proposition then follows from Theorem 5.2, applied to the space X^p. Condit*
*ion
(2) guarantees that the fusion system of X^pis generated by its restriction to *
*centric
subgroups; i.e., that F0 = FS,f(X^p) in the notation of Theorem 5.2.
Proposition 5.3 is a generalization of [BLO2 , Proposition 9.1]. Some more c*
*oncrete
applications of that result, for primes p 5, are given in [BLO2 , x9]: const*
*ructions
24 CARLES BROTO, RAN LEVI, AND BOB OLIVER
which can be thought of (very roughly) as mixing features of the fusion systems*
* of two
different groups.
The above proposition can also be applied when p = 3, to construct exotic 3-l*
*ocal
finite groups with Sylow subgroup the groups
fi -1 -1 3ff
S = a, b, x fi[a, b] = 1, xax = ab, xbx = ba
of order 81. We do not yet know whether it can be used to construct any exotic *
*2-local
finite groups.
In all of our examples, except those described in the next section, the proof*
* that a
saturated fusion system is not the fusion system of a finite group always invol*
*ves using
the classification of finite simple groups.
5.2. Fusion systems of a type studied by Solomon. Well before Puig formulated
the axioms defining a saturated fusion system, Ron Solomon had essentially foun*
*d one
which does not arise from the p-fusion system of any finite group. This was a b*
*i-product
of one step in the classification of finite simple groups [Sol]. Solomon consid*
*ered the
problem of classifying all finite simple groups whose Sylow 2-subgroups are iso*
*morphic
to those of the Conway group Co3. The end result of his paper was that Co3 is t*
*he only
such group. In the process of proving this, he needed to consider groups G in w*
*hich
all involutions are conjugate, and such that the centralizer of each involution*
* contains
a normal subgroup isomorphic to Spin7(q) with odd index, where q is an odd prime
power. Solomon showed that such a group G does not exist. However, the 2-local
structure that he found turned out to be perfectly consistent. It was only by a*
*nalyzing
its interaction with the p-local structure (where p is the prime of which q is *
*a power)
that he found a contradiction.
In a later paper [Be1 ], Dave Benson, inspired by Solomon's work, constructed*
* certain
spaces which can be thought of as the 2-completed classifying spaces which the *
*groups
studied by Solomon would have if they existed. Benson's construction was arguab*
*ly the
first indication of the existence of spaces which "behave" like p-completed cla*
*ssifying
spaces of finite groups, but which are themselves not of this form. To construc*
*t these
spaces he started with the spaces BDI(4) constructed by Dwyer and Wilkerson hav*
*ing
the property that
H*(BDI(4); F2) ~=F2[x1, x2, x3, x4]GL4(2)
(the rank four Dickson algebra at the prime 2). He then considered, for each o*
*dd
prime power q, the homotopy fixed point set of the Z-action on BDI(4) generated*
* by
an Ä dams operation" _q constructed by Dwyer and Wilkerson. Denote this homotopy
fixed point set by BDI4(q).
In [LO ], the second and third authors have shown that for each odd prime pow*
*er q,
there is a saturated fusion system F = FSol(q) over a Sylow 2-subgroup S = S(q)
Spin7(q), with the properties that all involutions in S are F-conjugate, and th*
*at when
z 2 Z(S) is the generator, then CF (z) is the fusion system of Spin7(q). The o*
*b-
struction theory described in Section 4.8 applies to show that there is a uniqu*
*e cen-
tric linking system LcSol(q) associated to FSol(q). We thus get a 2-local fini*
*te group
(S(q), FSol(q), LcSol(q)), which by Solomon's theorem (together with some other*
* group
theoretic results) cannot be associated to any finite group.
The classifying spaces BSol(q) def=|LcSol(q)|^2turn out to be equivalent to t*
*he spaces
constructed by Benson in [Be1 ]. For fixed q, one can take the union of the sp*
*aces
BSol(qn) for all n (strictly speaking this is not true as stated, but it can be*
* done up
to homotopy), and the 2-completion of this union is homotopy equivalent to BDI(*
*4).
THE THEORY OF p-LOCAL GROUPS: A SURVEY 25
Conversely, if _q 2 Aut (BDI(4)) is an Ä dams map" (its restriction to a maxima*
*l _
torus is induced by the Frobenius automorphism (x 7! xq) on the algebraic closu*
*re Fq),
then BSol(q) is homotopy equivalent to BDI4(q) _ the homotopy fixed point set of
_q when regarded as an action of the monoid N. For more details, see [LO , x4].
Other "exotic" p-local finite groups have been constructed by the first autho*
*r to-
gether with Møller [BM ]. Their method resembles Benson's construction of BDI4*
*(q).
Namely, they first take homotopy fixed point sets of actions of Adams maps on c*
*ertain
p-compact groups, and then show that these are the classifying spaces of p-loca*
*l finite
groups.
5.3. The fusion system of a block. Part of the motivation for constructing clas*
*sify-
ing spaces for fusion systems, and part of the reason in general for looking at*
* abstract
fusion and linking systems, comes from representation theory. Fix a finite gro*
*up G,
a prime p, and an algebraically closed field k of characteristic p. A block in *
*k[G] is a
factor in the maximal decomposition of k[G] as a product of rings, or equivalen*
*tly a
minimal central idempotent. Alperin and Brou'e [AB ] defined, for any block b, *
*inclusion
and conjugacy relations among the Brauer pairs associated to b (the "b-subpairs*
*"); and
Puig [Pu3 ] showed that these satisfy the axioms of a saturated fusion system o*
*ver the
defect group of the block. We refer to [AB ] or [Alp] for more details (includ*
*ing the
definitions of defect groups and Brauer pairs). The existence of a (unique or *
*canon-
ical) linking system associated to the fusion system of a block would thus impl*
*y the
existence of a classifying space for the block, which might in turn have implic*
*ations in
representation theory (see [Li]).
5.4. Extensions of p-local finite groups. We describe here some work in progress
which will appear in [BCGLO ]; work which in certain specialized situations a*
*llows us to
study extensions of p-local finite groups. It is still unclear how to define su*
*ch extensions
in general.
One of the reasons for this difficulty is that an extension of (finite) group*
*s need
not induce a fibration sequence of their p-completed classifying spaces. For ex*
*ample,
if 1 ! H ! G ! K ! 1 is an extension where K has order prime to p, then
BK^pis contractible, while BH^pand BG^pneed not be homotopy equivalent (since H
and G need not have the same cohomology mod p). Two cases where an extension
does induce a fibration sequence of p-completed classifying spaces are those wh*
*ere the
quotient group is a p-group, and the case of central extensions. It is this las*
*t case which
is the simplest to consider.
If F is a fusion system over a p-group S, then a subgroup A S_is called cen*
*tral if
CF (A) = F; i.e., if each ' 2 Hom F(P, Q) extends to a morphism ' 2 Hom F(P A, *
*QA)
which is the identity on A. Clearly, if A is central in F, then A Z(S). In *
*such a
situation, there is an obvious way to define a quotient fusion system F=A over *
*S=A,
by letting Hom F=A(P=A, Q=A) be the image in Hom (P=A, Q=A) of Hom F (P, Q), and
this is saturated by [BLO2 , Lemma 5.6]. If, furthermore, L is a centric linki*
*ng system
associated to F, then there is a canonical way to construct a centric linking s*
*ystem
L=A associated to F=A ([BLO2 , Lemma 5.6] again). A central extension of a p-l*
*ocal
finite group (S, F, L) by an abelian group A can now be defined as a p-local fi*
*nite
group (Se, eF, eL), together with an isomorphism A ~=A0 eSwhere A0is central i*
*n eF,
such that (Se=A, eF=A, eL=A) ~=(S, F, L).
26 CARLES BROTO, RAN LEVI, AND BOB OLIVER
Theorem 5.4. Central extensions of a p-local finite group (S, F, L) by a finite*
* abelian
p-group A are in one-to-one correspondence with principal fibrations
BA ! X ! |L|^p,
and also in natural one-to-one correspondence with H2(|L|^p; A).
Proof.See [BCGLO , x8].
We next look at extensions where the quotient group is a p-group.
Theorem 5.5. Let (S, F, L) be a p-local finite group. Let ~: ß1(|L|^p) -i ß b*
*e any
surjection of groups, and set S0 = Ker[S -i ß1(|L|^p) -i ß]. Let f :|L|^p--! *
*Bß be
the classifying map for ~, and let X be its homotopy fiber. Then there is a p-l*
*ocal finite
group (S0, F0, L0), where F0 is a subcategory of F, such that |L0|^p' X.
Conversely, assume that |L0|^p--! X --! Bß is a fibration sequence, where the
fiber is the classifying space of a p-local finite group (S0, F0, L0). Then X '*
* |L|^pfor
some p-local finite group (S, F, L), where S0 C S, F0 F, and S=S0 ~=ß.
Proof.See [BCGLO , x5].
Note that in the above proposition, we do not say that the centric linking sy*
*stem
L0 is a subcategory of L. In fact, it is not in general a subcategory, since F0*
*-centric
subgroups of S0 need not be F-centric. This helps to illustrate another of the*
* key
problems in this subject: the lack of a good concept of subobjects (or morphis*
*ms),
since even group inclusions and group homomorphisms do not in general send p-ce*
*ntric
subgroups to p-centric subgroups.
We can also describe "extensions" where the quotient is a group of order prim*
*e to
p. For the purpose of the following theorem, for any saturated fusion system F *
*over a
p-group S, we say that a saturated fusion subsystem F0 F over S has index pri*
*me
to p if
0
Aut F0(P ) Op (Aut F(P ))
0
for each P S. Here, for any group G, Op (G) is the largest normal subgroup of*
* index
prime to p; i.e., the subgroup generated by the Sylow p-subgroups of G.
Theorem 5.6. Fix a saturated fusion system F over a p-group S. Then there is a
normal subgroup Out0F(S) C OutF (S) of index prime to p, and a map
b`:Mor(Fc) ------! Out F(S)= Out0F(S)
with the following properties:
(a)b`(fi Off) = b`(fi).b`(ff) for each composable pair of morphisms fi, ff in F*
*c.
(b)The restriction of b`to AutF (S) is the natural surjection.
(c)b`sends inclusions to the identity.
(d)For each subgroup T OutF (S)= Out0F(S), the subcategory FT F with the sa*
*me
objects and whose morphisms are generated by restrictions of morphisms in b`*
*-1(T ),
is a saturated fusion subsystem of F of index prime to p.
(e)Each saturated fusion subsystem F0 F of index prime to p is equal to FT for
some subgroup T OutF (S)= Out0F(S).
THE THEORY OF p-LOCAL GROUPS: A SURVEY 27
(f)If L is a centric linking system associated to F with projection functor ß :*
*L --! F,
and if T OutF (S)= Out0F(S) is a subgroup, then LT def=ß-1(FT) L is a ce*
*ntric
linking system associated to FT.
Proof.See [BCGLO , x6], where the subgroup Out 0F(S) is defined explicitly. (*
*In fact,
it is the smallest subgroup of OutF (S) for which a map b`exists satisfying con*
*ditions
(a-c).) Note, in point (f), that if T C OutF (S)= Out0F(S) is a normal subgroup*
* with
quotient group oe, then |LT|^p--! |L|^p--! Boe is not in general a fibration se*
*quence.
The map b`of Theorem 5.6 cannot be extended to arbitrary morphisms in F, and
still satisfy conditions (a-c). For example, any such extension would have to s*
*end the
inclusion 1 --! S to the identity in Out F(S)= Out0F(S) _ and Condition (a) wou*
*ld
then imply that all automorphisms of S also get sent to the identity.
6. p-local compact groups
In very recent work still in progress, we have begun to extend the results on*
* p-local
finite groups to a more general situation, one which includes p-completed class*
*ifying
spaces of compact Lie groups and classifying spaces of p-compact groups. One of*
* our
hopes is that this will provide a new tool to study certain maps between these *
*spaces,
in particular self equivalences of these spaces.
6.1. Discrete p-toral groups. A p-toral group is a compact Lie group P whose id*
*en-
tity component is a torus T , such that P=T ~=ß0(P ) is a finite p-group. By co*
*ntrast,
a discrete p-toral group is a discrete group P , with normal subgroup P0 C P , *
*such
that P0 is isomorphic to a finite product of copies of Z=p1 (= Z[1_p]=Z) and P*
*=P0 is a
finite p-group. Any discrete p-toral group P contains a unique minimal subgroup*
* P0
of finite index, which we call its connected component. Also, P0 ~=(Z=p1 )r for*
* some
r def=rk(P ) = rk(P0). We define |P | = (rk(P ), |P=P0|) 2 N x N, and order the*
* elements
of N x N lexicographically. This ordering of the ö rdersö f discrete p-toral *
*groups
is what allows us to adapt with very little change the definition of a saturate*
*d fusion
system over a finite p-group to this new situation.
The motivation for studying fusion and linking systems over discrete p-toral *
*groups
comes from regarding them as "discrete approximationsö f p-toral groups. The p*
*-toral
subgroups of compact Lie groups play the same role as p-subgroups of finite gro*
*ups
(more on this below).
Proposition 6.1. Each p-toral group P contains a dense subgroup Pffi P which i*
*s a
discrete p-toral group of the same rank, and any two such subgroups are conjuga*
*te in
P . Furthermore, the inclusion induces an isomorphism H*(BP ; Fp) ~= H*(BPffi;*
* Fp),
and hence a homotopy equivalence (BPffi)^p' BP ^p.
Proof.See [DW3 , Proposition 6.9]. Let T P be the identity connected compone*
*nt,
and let Tffi T be the subgroup of elements of p-power order. Clearly, Tffiis t*
*he unique
discrete p-toral subgroup of T of the same rank, and there is a bijective corre*
*spondence
between dense discrete p-toral subgroups of P which contain Tffiand splittings *
*of the
group extension 1 ! T=Tffi! P=Tffi! P=T ! 1. Since P=T is a p-group and T=Tffi
28 CARLES BROTO, RAN LEVI, AND BOB OLIVER
is uniquely p-divisible, Hi(P=T ; T=Tffi) = 0 for all i > 0, and hence the abov*
*e group
extension has a splitting which is unique up to conjugation.
The proof that H*(BP ; Fp) ~=H*(BPffi; Fp) is easily reduced to the case wher*
*e P ~=
S1 and Pffi~=Z=p1 , and this result is classical.
One advantage in working with discrete p-toral groups rather than p-toral gro*
*ups is
that all subgroups of discrete p-toral groups are again discrete p-toral groups*
*, unlike
the case for compact p-toral groups. But it's also simpler to work with discret*
*e groups,
and not have to worry about their topology.
6.2. Fusion and linking systems over discrete p-toral groups. A fusion system
F over a discrete p-toral group S is a category whose objects are the subgroups*
* of S,
and whose morphism sets Hom F(P, Q) satisfy the following 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.
A subgroup P S is fully centralized in F if |CS(P )| |CS(P 0)| for all P *
*0 S
which is F-conjugate to P , and is fully normalized in F if |NS(P )| |NS(P 0)*
*| for all
P 0 S which is F-conjugate to P .
The fusion system F is saturated if the following three conditions hold.
(I)For each P S which is fully normalized in F, P is fully centralized in F a*
*nd
Out S(P ) 2 Sylp(Out F(P )).
(II)If P S and ' 2 Hom F(P, S) are such that 'P is fully centralized, and if *
*we set
N' = {g 2 NS(P ) | 'cg'-1 2 AutS('P )},
_ _
then there is ' 2 Hom F(N', S) such that '|P = '.
S 1
(III)If P1 P2 P3 . . .is a sequence of subgroups of S, P1 = n=1Pn, and
' 2 Hom (P1 , S) is such that '|Pn 2 Hom F(Pn, S) for all n, then ' 2 Hom F(*
*P1 , S).
We note here that OutS(P ) is a finite p-group for any pair P S of discrete*
* p-toral
groups. So condition (I) includes the condition that Out F(P ) is finite for al*
*l P S.
The only real difference between this new definition and the definition of a sa*
*turated
fusion system over a finite p-group is thus condition (III), which can be thoug*
*ht of as
a öc ntinuity" condition.
When S is a discrete p-toral group and F is a saturated fusion system over S,*
* then
the definitions of F-centric subgroups of S, and of centric linking systems ass*
*ociated
to F, are identical to the definitions given in Section 4 above when S is a fin*
*ite p-
group. A p-local compact group is now defined to be a triple (S, F, L), where S*
* is a
discrete p-toral group, F is a saturated fusion system over S, and L is a linki*
*ng system
associated to F. The classifying space of such a triple (S, F, L) is the p-com*
*pleted
nerve |L|^p. Thus, as in the case for fusion and linking systems over finite p-*
*groups, a
linking system for a fusion system F over a discrete p-toral group S can be tho*
*ught of
as a means of associating a classifying space to F.
6.3. Reduction to finite subcategories. The main difficulty when generalizing r*
*e-
sults about p-local finite groups to the discrete p-toral case is that the cate*
*gories are
no longer finite. It is, however, possible to replace any linking system L asso*
*ciated to
THE THEORY OF p-LOCAL GROUPS: A SURVEY 29
a saturated fusion system F over a discrete p-toral group S, by a finite full s*
*ubcate-
gory L0 L such that |L0|^p' |L|^p. This is done by restricting L to those F-c*
*entric
subgroups which are also F-radical; i.e., those P S for which OutF (P ) conta*
*ins no
nontrivial normal p-subgroup.
Proposition 6.2. Let F be a saturated fusion system over a discrete p-toral gro*
*up S,
and let L be an associated linking system to F. Then S contains only finitely m*
*any F-
conjugacy classes of subgroups which are both F-centric and F-radical. Let L0 *
* L be
any finite full subcategory whose objects include F-conjugacy class representat*
*ives for
all F-radical F-centric subgroups of S. Then the inclusion |L0|^p |L|^pis a ho*
*motopy
equivalence.
Proof.See [BLO3 , Corollary 3.5 & x5].
6.4. A homology decomposition of the classifying space. The orbit category of
a fusion system F over a discrete p-toral group S is defined exactly as before:*
* it has
the same objects as F, and
Mor O(F)(P, Q) = RepF (P, Q) def=HomF(P, Q)= Inn(Q).
Also, we write Oc(F) to denote the full subcategory of O(F) whose objects are t*
*he
F-centric subgroups of S.
When F is a saturated fusion system over a discrete p-toral group S, then we *
*have
the same correspondence between linking systems associated to F, classifying sp*
*aces
associated to F, and liftings over Oc(F) of the homotopy functor (P 7! BP ) as *
*we
do when S is finite (Section 4.7). In particular, any classifying space for F *
*has a
decomposition as the homotopy direct limit of the corresponding homotopy liftin*
*g, as
described by the next proposition.
Proposition 6.3. Fix a saturated fusion system F over a discrete p-toral group *
*S,
and let F0 Fc be any finite full subcategory whose objects include F-conjugac*
*y class
representatives for all F-radical F-centric subgroups of S. Let L0 be any linki*
*ng system
associated to F0, and let eß0:L0 --! O(F0) be the projection functor. Let
eB:O(F0) ------! Top
be the left homotopy Kan extension over eß0of the constant functor L0 -*-!Top .*
* Then
eBis a homotopy lifting of the homotopy functor P 7! BP , and
i j
|L|^p' |L0|^p' hocolim-----!(Be) ^p. (1)
O(F0)
Proof.See [BLO3 , Proposition 5.7].
6.5. Reduced linking systems. Constructing centric linking systems over discrete
p-toral groups is not as straightforward as it sometimes is in the case of fini*
*te p-groups.
For this reason, we define another type of category which is intermediate betwe*
*en fusion
systems and linking systems.
Let F be a fusion system over a discrete p-toral group S, and let F0 Fc be
any full_subcategory. A reduced linking system associated to F0 consists of a *
*cat-
_ _
egory_L 0, together with a functor ß :L0 --! F0 and distinguished homomorphisms
ffiP
P --! AutL_0(P ), which satisfy axioms (A), (B), and (C) in Section 4.2, excep*
*t that
30 CARLES BROTO, RAN LEVI, AND BOB OLIVER
_
Ker(ffiP) = Z(P )0, and that the group Z(P )=Z(P )0 ~=ß0(Z(P )) (not Z(P ) itse*
*lf) acts
freely on morphism sets Mor _L(P, Q) and induces bijections
~=
Mor _L(P, Q)=ß0(Z(P )) ------! Hom F(P, Q).
Fortunately, we do know that each reduced linking system lifts to a unique li*
*nking
system.
Proposition 6.4. Let F be_a fusion system over the discrete p-toral group S. Th*
*en any
reduced linking system L associated to F lifts to a centric linking system L as*
*sociated
to F which is unique up to isomorphism.
Proof.See [BLO3 , Proposition 5.10].
6.6. Compact Lie groups. It was immediately obvious that every finite group can
be regarded as a p-local finite group. In fact, we were already studying the f*
*usion
system and linking system for finite groups long before we considered the more *
*abstract
definitions. The definitions of fusion systems, and of reduced linking systems*
*, for
compact Lie groups are also straightforward, but the definition of their linkin*
*g systems
is less obvious.
Fix a compact Lie_group G and a prime p. Let_T G be a maximal torus, and fi*
*x a
Sylow p-subgroup S=T 2 Sylp(NG(T )=T ). Then S is a maximal p-toral subgroup of*
* G,
_
and any other maximal p-toral subgroup of G is conjugate to S. These subgroups *
*play
the role of "Sylow p-subgroups" when working with compact Lie groups. For examp*
*le,
an arbitrary p-toral subgroup P G is maximal p-toral if and only if Ø(G=P ) i*
*s prime
to p, where Ø denotes as usual the Euler characteristic.
_ _
Now let S S be any discrete p-toral subgroup which is dense in S and has the
same rank. By Proposition_6.1, there is such a subgroup, and any two such subgr*
*oups
are conjugate in S. Since the closure of any discrete p-toral subgroup of G is *
*p-toral,
we now see that S is a maximal discrete p-toral subgroup of G, and that any oth*
*er
maximal discrete p-toral subgroup is G-conjugate to S.
Define the fusion category FS(G) exactly as was done for finite G: its object*
*s consist
of all subgroups of S, and Mor FS(G)(P, Q) is the set of all group homomorphism*
*s which
are induced by conjugation in G. A subgroup P S is FS(G)-centric if and only *
*if
Z(P ) is a_maximal_discrete p-toral subgroup of CG(P ), or equivalently if and *
*only if
CG(P ) = Z(P )x C0G(P ) for some (unique) subgroup C0G(P ) which is finite of o*
*rder
prime to p.
_
We now define the reduced linking system LcS(G) to be the category whose obje*
*cts
are the FS(G)-centric subgroups of S, and where
ffi_____ 0
Mor _Lc (P, Q) = NG(P, Q) Z(P )0x CG(P ) .
S(G)
*
* _
By Proposition 6.4, there is a unique centric linking system LcS(G) associated *
*to LcS(G).
Theorem 6.5. Fix a compact Lie group G and a maximal discrete p-toral subgroup
S G. Then (S, FS(G), LS(G)) is a p-local compact group, with classifying spa*
*ce
|LS(G)|^p' BG^p.
Proof.See [BLO3 , x6]. The proofs that FS(G) is a fusion system and LcS(G) an *
*asso-
ciated centric linking system are straightforward. The proof that |LS(G)|^p' BG*
*^pis
based on the homology decomposition of BG^pin [JMO ].
THE THEORY OF p-LOCAL GROUPS: A SURVEY 31
6.7. p-compact groups. A p-compact group is a triple (X, BX, e), where BX is a
pointed, connected, p-complete space, e: X --! (BX) is a homotopy equivalence,
and H*(X; Fp) is finite. These objects were introduced in the 1980's by Dwyer *
*and
Wilkerson [DW3 ], and extensively studied by them and many other authors. The
concept of a p-compact group was designed to be a homotopy theoretic analog of a
classifying space of a compact Lie group. For instance, if G is a compact Lie *
*group
whose group of components is a p-group, then BG^pis the classifying space of a *
*p-
compact group. In contrast, if ß0(G) is not a p-group, then the mod p cohomolog*
*y of
the loop space (BG^p) is not in general finite.
One of our original motivations for defining p-local compact groups was to pr*
*ovide
a wider framework for studying p-compact groups. This requires first checking *
*that
every p-compact group X can be considered as a p-local compact group; in partic*
*ular,
that BX ' |L|^pfor some linking system L associated to a saturated fusion syste*
*m F
over a discrete p-toral group.
By [DW3 , x8-9], for any p-compact group X, there is a unique maximal p-tora*
*l sub-
group, hence a unique maximal discrete p-toral subgroup f :BS --! BX (Propositi*
*on
6.1); and any other map from the classifying space of a discrete p-toral group *
*to BX
factors through f. We set
_ def
FS,f(X) def=FS,f(BX) and LcS,f(X) = LcS,f(BX),
where the fusion and linking systems of the space BX are defined as in Section *
*2.5.
Using the properties of the mapping spaces Map (BP, BX) described in [DW3 , x5*
*-6]
(for a discrete p-toral_group P ), we check that FS,f(X) is a saturated fusion *
*system
over S, and that LcS,f(X) is a reduced centric linking system associated to FS,*
*f(X). Let
_
LcS,f(X) be the centric linking system associated to LcS,f(X) (and hence to FS,*
*f(X))
by Proposition 6.4. We then show:
Theorem 6.6. For any p-compact group X and any maximal discrete p-toral subgroup
f :BS --! BX, (S, FS,f(X), LS,f(X)) is a p-local compact group with classifying*
* space
|LS,f(X)|^p' BX .
Proof.See [BLO3 , x7]. The proof that |LS,f(X)|^p' BX is based on the decompos*
*ition,
shown in [CLN ], of BX as a homotopy direct limit of BP 's for p-toral (or dis*
*crete p-
toral) subgroups P X.
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Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, E-08193 Bel-
laterra, Spain
E-mail address: broto@mat.uab.es
Department of Mathematical Sciences, University of Aberdeen, Meston Building
339, Aberdeen AB24 3UE, U.K.
E-mail address: ran@maths.abdn.ac.uk
LAGA, Institut Galil'ee, Av. J-B Cl'ement, 93430 Villetaneuse, France
E-mail address: bob@math.univ-paris13.fr