HOMOLOGY APPROXIMATIONS FOR CLASSIFYING SPACES
OF FINITE GROUPS
W. G. Dwyer
University of Notre Dame
x1. Introduction
Suppose that p is a prime number and that G is a finite group, a compact Lie
group, or even a p-compact group [12]. Recently there has been a lot of interes*
*t in
finding ways to construct the classifying space B G, at least up to mod p homol*
*ogy,
by gluing together classifying spaces of subgroups of G. In practice this means
finding a mod p homology isomorphism
(1.1) hocolim F -~!pBG
where D is some small category, F is a functor from D to the category of spaces,
and, for each object d of D, F (d) has the homotopy type of BH for some subgrou*
*p H
of G. An expression like 1.1 is sometimes called a homology approximation to BG*
* or
a homology decomposition of BG, and can be used either to make calculations with
B G or to prove general theorems about B G by induction. (Of course an induction
is likely to work only if the values of F are of the form BH for H a proper sub*
*group
of G!) For example, Jackowski and McClure [15] approximate B G by classifying
spaces of centralizers of non-trivial elementary abelian p-subgroups of G. The*
*ir
result had been anticipated for SU (2) (see [10]) and used to prove a homotopy
uniqueness theorem. The p-compact group version of their result [11] is exploit*
*ed
in [7]. Jackowski, McClure and Oliver [16] approximate B G (G compact Lie) by
classifying spaces of p-stubborn subgroups of G, and then use the approximation*
* to
make beautiful calculations about the space of self-maps of B G. Benson-Wilkers*
*on
[4] and Benson [3] use homology approximations to BG, where G is respectively t*
*he
Mathieu group M12 or Conway's group CO3, to obtain maps from BG to classifying
spaces of 2-compact groups.
One goal of this paper is to describe many different homology decomposition
formulas (including the ones mentioned above) in terms of a single invariant: *
*an
associated poset of subgroups of G. Although we expect to extend the results in
a future paper to compact Lie groups and p-compact groups, we concentrate here
on finite groups because there are fewer technicalities to get in the way of th*
*e basic
ideas. We also obtain what seems to be a new homology decomposition for finite
groups; this decomposition generalizes a classical theorem of Swan (see 1.21).
______________
The author was supported in part by the National Science Foundation.
Typeset by AM S-T*
*EX
1
2 W. DWYER
Many of the results in this paper were previously known, but the author's in-
tention is to gather them into a systematic account of how the usual homology
decompositions arise.
1.2 Homology decompositions. Suppose that G is a finite group. We will call
a set C of subgroups of G a collection if it is closed under the process of tak*
*ing
conjugates in G. Let C be such a collection, SC = (C; ) the poset given by elem*
*ents
of C with the inclusion relation, and KC the associated simplicial complex [2, *
*6.2].
The n-simplices of KC, for n 0, are the subsets {Hi} C of cardinality (n + 1)
which are totally ordered by inclusion. The group G acts on C by conjugation, a*
*nd
since this action preserves inclusion relationships it passes to an action of G*
* on KC.
Let E G be the universal cover of B G; if X is a G-space, the Borel construc*
*tion
or homotopy orbit space XhG is defined to be the quotient (E G x X)=G. Let *
denote the one-point space with trivial G action.
1.3 Definition. The collection C is said to be ample if the map
(1.4) qKC : (KC)hG -! (*)hG = BG
given by KC ! * induces an isomorphism on mod p homology.
One of the main conclusions of this paper is that giving a homology decompo-
sition of B G amounts in practice to describing an ample collection of subgroup*
*s of
G. In fact, there is a many-to-one correspondence: every ample collection provi*
*des
at least three homology decompositions. We will give a brief description of eac*
*h of
these decompositions, with details to follow later on. Assume as usual that C i*
*s a
collection of subgroups of G.
1.5 The centralizer decomposition. The C-conjugacy category AC is the category *
*in
which the objects are pairs (H; ), where H is a group and is a conjugacy class
of monomorphisms i : H -! G with i(H) 2 C. A morphism (H; ) -! (H0; 0)
is a group homomorphism j : H -! H0 which under composition carries 0 into
. One should probably restrict H in some way, for instance by requiring H to
be a subgroup of G, in order to force AC to be small; in any case, AC as defined
is equivalent to a small category. If H G is a subgroup, let CG (H) denote the
centralizer of H in G. There is a natural functor
ffC : (AC)op -! Spaces
which assigns to each object (H; ) a space which has the homotopy type of
B CG (i(H)) for any i 2 (see 3.1). There is also a natural map
aC : hocolim ffC -! B G :
1.6 Theorem. The map aC induces an isomorphism on mod p homology (that is,
aC gives a homology decomposition of B G) if and only if C is an ample collecti*
*on
of subgroups of G.
1.7 The subgroup decomposition. The C-orbit category OC is the category whose
objects are the G-sets G=H, H 2 C, and whose morphisms are G-maps. There is
HOMOLOGY APPROXIMATIONS 3
an inclusion functor J from OC to the category of G-spaces. Composing J with
the Borel construction (-)hG gives a functor
fiC : OC -! Spaces
whose value (G=H)hG at an object G=H has the homotopy type of BH. The natural
maps fiC(G=H) -! B G are compatible as G=H varies and induce a map
bC : hocolim fiC -! B G :
1.8 Theorem. The map bC induces an isomorphism on mod p homology (that is,
bC gives a homology decomposition of B G) if and only if C is an ample collecti*
*on
of subgroups of G.
1.9 The normalizer decomposition. Let sSC be the category of "orbit simplices" *
*for
the action of G on KC. The objects of sSC are the orbits oeof the action of G o*
*n the
simplices of KC, and there is one morphism oe! oe0if for some simplices oe 2 oe*
*and
oe0 2 oe0, oe0 is a face of oe. If H is a subgroup of G, let NG (H) denote the *
*normalizer
of H in G. There is a natural functor
ffiC : sSC -! Spaces
which assigns to the orbit of a simplex oe = {Hi} a space which has the homotopy
type of B (\iNG (Hi)) (see 3.3). There is also a map dC : hocolim ffiC -! B G.
1.10 Theorem. The map dC induces an isomorphism on mod p homology (that is,
dC gives a homology decomposition of B G) if and only if C is an ample collecti*
*on
of subgroups of G.
1.11 Examples of ample collections. Let G as above be a finite group. There
are quite a few ample collections of subgroups of G.
1.12 Trivial examples. If C is any collection of subgroups of G which contains *
*the
trivial subgroup {e}, then C is ample. This follows from the fact that the pose*
*t SC
has the trivial subgroup as a minimal element, so that KC is contractible (2.5,*
* 2.6)
and qKC (1.4) is a weak equivalence. Similar remarks apply if C contains G itse*
*lf. The
decomposition formulas associated to these collections are usually not interest*
*ing,
since in one way or another the formulas are circular, i.e., BG itself is hidde*
*n in the
homotopy colimit on the left hand side.
1.13 Nontrivial p-subgroups. Let C = P(G) be the collection of all nontrivial p-
subgroups of G. If p divides the order of G then C is ample. This can be proved*
* in
several ways (see 6.4). We give our own proof of a sharper result (6.3) due ori*
*ginally
to Jackowski-McClure-Oliver.
1.14 Nontrivial elementary abelian p-subgroups. Recall that an abelian group is*
* said
to be an elementary abelian p-group if it is a module over Fp. Let C be the col*
*lection
of all nontrivial elementary abelian p-subgroups of G. It is a theorem of Quil*
*len
that the inclusion map KC ! KP(G) is a homotopy equivalence [2, 6.6.1]. This
4 W. DWYER
implies that C has the same ampleness properties as P(G) (1.13). The centralizer
decomposition associated to C is the Jackowski-McClure decomposition [15].
1.15 Distinguished elementary abelian p-subgroups. Call a nontrivial elementary
abelian p-subgroup V of G distinguished if V is equal to the group of elements
of exponent p in the center of CG (V ). Let C be the collection of distinguish*
*ed
elementary abelian p subgroups of G. The inclusion KC ! KP(G) is a homotopy
equivalence [2, p. 231, exercise], so this collection also has the same amplene*
*ss
properties as P(G). The associated centralizer decomposition is a more economic*
*al
form of the Jackowski-McClure decomposition.
1.16 The Benson collection. Let P be a Sylow p-subgroup of G, and E the smallest
subset of G which contains the elements of exponent p in the center of P , is c*
*losed
under conjugation in G, and is closed under the process of taking products of
commuting elements. Let C be the collection of all nontrivial elementary abeli*
*an
p-subgroups of G which are subsets of E. If p divides the order of G, then C is
ample [3, 3.2].
1.17 p-stubborn subgroups. Recall that a p-subgroup P of G is said to be p-stub*
*born
if the quotient NG (P )=P has no nontrivial normal p-subgroups. Any collection*
* C
of p-subgroups of G which contains all p-stubborn subgroups is ample. This foll*
*ows
from 1.6 and the theorem of Jackowski-McClure-Oliver [16] that for such a colle*
*ction
the subgroup decomposition map dC (1.8) is a mod p homology isomorphism. We
prove something slightly sharper in x7. These p-stubborn collections seem to be*
* of
limited usefulness for finite groups. In many cases, e.g. if G is simple, the*
* trivial
subgroup is p-stubborn (1.12); see also 7.3.
1.18 The Bouc collection. Let C be the collection of all nontrivial p-stubborn *
*sub-
groups of G. It is a theorem of Bouc [2, 6.6.6] that the inclusion KC ! KP(G) is
a homotopy equivalence. Therefore C has the same ampleness properties as P(G)
does (1.13).
1.19 p-centric subgroups. A p-subgroup P of G is said to be p-centric if the ce*
*ntral-
izer CG (P ) is the product of the center of P and a group of order prime to p.*
* This
is equivalent to the condition that the center of P be a Sylow p-subgroup of CG*
* (P ).
The collection C of all p-centric subgroups of G is ample; a slight generalizat*
*ion of
this is proved in x8.
1.20 Subgroups which are both p-stubborn and p-centric. The collection C of all
subgroups of G which are both p-stubborn and p-centric is also ample; see 8.10.
1.21 An example illustrating the three decompositions. Suppose that G has
an abelian Sylow p-subgroup P . In this case the p-centric subgroups of G are e*
*xactly
the conjugates of P . By 1.19, the collection C of conjugates of P is ample. *
*The
three associated homology decompositions are easy to figure out. Up to equivale*
*nce
of categories the C-conjugacy category has only one object, the inclusion H ! G;
the automorphisms of this object are NG (P )=CG (P ) and the space assigned to *
*the
object by the functor ffC is equivalent to B CG (P ). Let Q = NG (P )=CG (P ). *
*The
centralizer decomposition (1.6) gives a mod p homology isomorphism
hocolimQ B CG (P ) ' BCG (P )hQ ' BNG (P ) ! BG
HOMOLOGY APPROXIMATIONS 5
(see 2.16). Up to equivalence of categories the C-orbit category OC also has ju*
*st one
object, represented by the orbit G=P . The self-maps of this object are NG (P )*
*=P ,
and the space assigned to the object by fiC is equivalent to BP . Let W = NG (P*
* )=P .
The subgroup decomposition (1.8) gives a mod p homology isomorphism
hocolim W B P ' (B P )hW ' BNG (P ) -! B G :
The category sSC of 1.9 has only one object, which has no nonidentity self-maps.
The space assigned to this object by ffiC is equivalent to B NG (P ), and so th*
*e nor-
malizer decomposition degenerates into a mod p homology isomorphism
BNG (P ) -! B G :
All three decompositions say the same thing, but they express it in different w*
*ays.
In each case the content of the message is the theorem of Swan stating that if *
*G has
an abelian Sylow p-subgroup P , then the map B NG (P ) ! B G is an isomorphism
on mod p homology. Both 1.19 and 1.20 can be viewed as extensions of this theor*
*em
to cases in which the Sylow p-subgroups are not abelian.
1.22 Spectral sequences and sharp decompositions. Consider a homology decompo-
sition of the form 1.1. Bousfield and Kan [5, XII.5.7] give a first quadrant ho*
*mology
spectral sequence
E2i;j= colimiH j(F ; Fp) ) H i+j(B G; Fp) ;
where colimidenotes the i'th left derived functor of the colimit construction o*
*n the
category of functors from D into abelian groups. Call the homology decomposition
sharp if this spectral sequence has E2i;j= 0 for i > 0. A sharp homology decom-
position thus gives an isomorphism H *(B G; Fp) ~=colim H*(F ; Fp); in other wo*
*rds,
it gives a formula for H *(B G; Fp) in terms of the homology of certain subgrou*
*ps
of G. For example, the Jackowski-McClure decomposition is sharp [15]. In a futu*
*re
paper we intend to look at the question of which of the homology decompositions
discussed above are sharp.
Organization of the paper. Section 2 describes some homotopical category theory;
this is used in x3 to prove the three decomposition theorems. Sections 4 and 5
have brief discussions of, respectively, G-spaces and finite p-groups. The fina*
*l three
sections contain proofs that various collections of subgroups of a finite group*
* G are
ample, or even M-ample for some G-module M (6.1).
Motivation. This paper was originally motivated by a study of [16] and a subseq*
*uent
attempt to find some common ground between sections 2 and 5 of that paper. In
a sense, in the last three sections of this paper we prove decomposition theore*
*ms
like the one that follows from [16, 2.14] with algebraic techniques like the on*
*es from
[16, x5].
1.23 Notation and terminology. This paper is written with the convention that t*
*he
word "space" by itself means "simplicial set" [17] [5, VIII]. For instance, for*
* the rest
of the paper BG stands for the usual simplicial classifying space of the group *
*G [17,
6 W. DWYER
x21]. A map between spaces is an equivalence or weak equivalence if it becomes a
weak equivalence of topological spaces upon passing to geometric realizations [*
*17].
Throughout the paper, p is a fixed prime number and Fp is the field with p e*
*l-
ements. A space is Fp-acyclic if it has the Fp-homology of a point, and a map is
an Fp-equivalence if it induces an isomorphism on Fp-homology. The results of t*
*he
paper are stated for finite groups, but some of them, for instance the decompos*
*ition
results from x3, hold unchanged for infinite discrete groups.
The author would like to thank S. Smith for many suggestions.
x2. Homotopy colimits and the Grothendieck construction
One of the main techniques of this paper is to use categories as models for
spaces; this makes it possible, for instance, to prove that two maps are homoto*
*pic
by finding a natural transformation between associated functors. The advantage
of this is that it is easier to understand a natural transformation between fun*
*ctors
than to understand the formidable amount of data that goes into the construction
of an explicit simplicial homotopy [17, x5].
Nerves. The fundamental space associated to a (small) category D is its nerve
|D|. See [5, XI x2] or 4.7 for the definition; there are some examples below. *
* Let
Cat denote the category whose objects are small categories and whose morphisms
are functors between them.
2.1 Proposition. [5, XI x2] The nerve construction gives a functor
|-| : Cat ! Spaces :
This construction carries a natural transformation between two functors f and f0
into a (simplicial) homotopy between |f| and |f0|.
Proposition 2.1 has the following immediate consequence.
2.2 Proposition. If f : D ! D0 is an equivalence of categories, then |f| : |D|*
* !
|D0| is a weak equivalence of spaces.
Proof. If f0 : D0 ! D is an inverse equivalence, then the composites ff0 and f0f
are naturally equivalent to the appropriate identity functors.
2.3 The nerve of a groupoid. Suppose that G is a group. Associated to G is a
category G with one object *,and with the monoid of self-maps of * isomorphic to
G. A functor from G to some category M amounts to an object of M together
with an action of G on it. The nerve of G is isomorphic to the classifying spa*
*ce
B G.
Suppose that D is a groupoid, that is, a small category in which every morph*
*ism
is invertible. For any object x of D let Gx denote the group of self-maps of x*
* in
D; the associated category Gx can be identified with the full subcategory of D
generated by the object x. If D is connected in the sense that any two objects
can be joined by an arrow, then for any object x of D the inclusion Gx ! D is
an equivalence of categories. In this case it follows from 2.2 that the induced*
* map
B Gx = |Gx| ! |D| is a weak equivalence. In general |D| is weakly equivalent to
HOMOLOGY APPROXIMATIONS 7
a disjoint union of spaces B Gx, where x ranges over a set of representatives f*
*or
isomorphism classes of objects in D.
2.4 The nerve of a poset. Let P = (P; ) be a partially ordered set, for short a
poset. Associated to P is a simplicial complex KP with vertex set P , in which *
*the
simplices are the finite subsets of P which are totally ordered by the relation*
* .
The poset P can be viewed as a category with object set P , with one morphism
x ! y whenever x y, and with no other morphisms.
2.5 Proposition. For any poset P, the topological space KP is homeomorphic in
a natural way to the geometric realization of |P|.
Proof. Check that the nondegenerate simplices of the simplicial set |P| corresp*
*ond
exactly to the (geometric) simplices of KP .
2.6 Proposition. If the poset P = (P; ) has a minimal object or a maximal
object, then |P| is contractible.
Proof. Let x be a minimal (resp. maximal) object. The unique maps x ! y (resp.
y ! x) for y 2 P give a natural transformation between the identity functor of P
and the constant functor with value x. Now apply 2.1.
The homotopy orbit space functor (1.2) can be defined either simplicially or
topologically. In both forms it preserves weak equivalences, and the geometric
realization functor carries one form to the other. Thus 2.5 allows the notion *
*of
ampleness to be reformulated in terms of the nerve of the poset SC.
2.7 Proposition. Suppose that G is a finite group, and that C is a collection *
*of
subgroups of G. Then C is ample if and only if the map
qC : |SC|hG ! (*)hG = BG
induced by |SC| ! * is an Fp-equivalence.
Categorical models for homotopy colimits. By definition [5, XII x2], the nerve
of a category D is isomorphic to the homotopy colimit of the functor on D which
sends every object to a one-point space. It is very useful to have a method for
representing more complex homotopy colimits as nerves. The most general tool we
know of for doing this is the Grothendieck Construction.
2.8 Definition. Suppose that D is a small category, and that f : D ! Cat is
a functor. The Grothendieck Construction on f, denoted Gr (f), is the category
whose objects are the pairs (d; x) where d is an object of D and x is an object*
* of
f(d). An arrow (d; x) ! (d0; x0) in Gr (f) is a pair (u; v), where u : d ! d0 *
*is a
morphism in D and v : (f(u))(x) ! x0 is a morphism in f(d0). Arrows compose
according to the rule (u; v) . (u0; v0) = (u00; v00), where u00is the composite*
* u . u0 and
v00is the composite of v with the image of v0 under the functor f(u).
Thomason discovered the following remarkable property of this construction.
8 W. DWYER
2.9 Theorem. [19, 1.2] Suppose that D is a small category and f : D ! Cat is a
functor. Let Gr (f) be the Grothendieck Construction on f. Then there is a natu*
*ral
weak equivalence
hocolim |f| -'!|Gr (f)| :
2.10 Variations. If D is a small category, then the nerve of its opposite categ*
*ory
Dop is weakly equivalent to |D| in a natural way. Even better, Quillen exhibits
a category D0 , depending functorially on D, together with functors D0 ! D and
D0 ! Dop which induce weak equivalences on nerves [18, p. 94]. Suppose that
f : D ! Cat is a functor. Let fop denote the composite of f with the "opposite"
construction Cat ! Cat ; note that fop is again a functor D ! Cat . It follows
from the above remarks and the homotopy invariance of homotopy colimits [5,
p. 335] that the four categories Gr (f), Gr (f)op, Gr (fop) and Gr (fop)op all *
*have
nerves which are weakly equivalent in a natural way to hocolim |f|.
2.11 Remark. Suppose that f is a functor from D to the category of sets. We
can treat the values of f as discrete categories, i.e., categories with no noni*
*dentity
morphisms, and think of f as a special type of functor D ! Cat . For such an f *
*it
is easy to check that hocolim f is in fact isomorphic to |Gr (f)|.
The following propositions illustrate how 2.9 is used (in the form 2.11). If*
* G is
a finite group and C is a collection of subgroups of G, let EC denote the homot*
*opy
colimit of the inclusion J : OC ! Spaces (1.7). The space E Cobtains a G-action
from the fact that J actually takes values in the category of G-spaces.
2.12 Proposition. Suppose that G is a finite group. Then for any collection C *
*of
subgroups of G there is a G-map E C! |SC| (2.4) which is a homotopy equivalence
of spaces.
Proof. Consider J as a functor from OC to discrete categories (2.11), and let U*
* be
the corresponding Grothendieck construction. The objects of U are pairs (O; x),
where O is a G-orbit of the form G=H, H 2 C, and x 2 O. A morphism (O; x) !
(O0; x0) is a G-map f : O ! O0 such that f(x) = x0. The nerve |U| is isomorphic*
* in
a natural way to EC (2.11). The group G acts on U, with an element g 2 G giving
the functor from U to itself which sends an object (O; x) to (O; gx). Upon pass*
*age
to nerves this map induces the action of G on E C.
Consider the functor F : U ! SC which assigns to an object (O; x) the isotro*
*py
subgroup Gx of x. This functor commutes with the actions of G on the two cate-
gories, since translating x 2 O by g 2 G has the effect of conjugating the isot*
*ropy
subgroup of x by g. It is also clear that F is an equivalence of categories; an*
* inverse
equivalence is obtained by sending a subgroup H 2 C to the pair (G=H; eH). By
2.2, applying the nerve construction to F gives the desired map E C! |SC|.
2.13 Remark. Essentially the same argument gives the following more elaborate
result. Suppose that C is a collection of subgroups of G. If H G is a subgroup,
not necessarily in C, let SC(H) denote the poset consisting of all elements of *
*C which
contain H.
2.14 Proposition. Let G be a finite group, C a collection of subgroups of G, a*
*nd
H G a subgroup. Then there is a map
(E C)H ! |SC(H)|
HOMOLOGY APPROXIMATIONS 9
which is a homotopy equivalence of spaces and is equivariant with respect to the
natural actions of NG (H)=H on the spaces involved.
Proof. Observe that (E C)H = hocolim(J H) and repeat the argument above.
2.15 Corollary. Let G be a finite group and C a collection of subgroups of G.
Then for any H 2 C, (E C)H is contractible.
Proof. By 2.14, (E C)H is homotopy equivalent to |SC(H)|. This nerve is contrac*
*tible
by 2.6, since SC(H) is a poset with H itself as a minimal object.
2.16 Homotopy orbit spaces as homotopy colimits. Let G be a group which acts on
a space X, and fX : G ! Spaces the corresponding functor (2.3). It is easy to
check from the definitions that XhG (1.2) is isomorphic to hocolim fX . Suppo*
*se
that X = |D| for some category D, and that the action of G on X is induced by an
action of G on D. Let fD : G ! Cat be the functor given by this categorical ac*
*tion.
According to 2.9, XhG is weakly equivalent to |Gr (fD )|. We will call Gr (fD *
*) the
Grothendieck construction of the action of G on D.
For example, if G is a finite group and C is a collection of subgroups of G,*
* the
action of G on |SC| arises from an action of G on the poset (2.4) SC. By Propos*
*ition
2.5, the question of whether or not C is ample can be studied by looking at the*
* nerve
of the Grothendieck construction on the action of G on SC.
x3. Three homology decompositions
In this section we will prove the three homology decomposition theorems de-
scribed in x1. Throughout the section, G denotes some particular finite group.
Our approach is to use Grothendieck constructions to find explicit categoric*
*al
models for the homotopy colimits which appear as the domains of the decompositi*
*on
maps aC (1.6), bC (1.8) and dC (1.10). Inspecting the categories involved then *
*reveals
that each of these domains is weakly equivalent to |SC|hG in a way which respec*
*ts
the natural maps from these spaces to B G.
3.1 The centralizer decomposition. Let C be a collection of subgroups of G, and
AC the C-conjugacy category described in 1.5. Associated to an object (H; ) of *
*AC
is the groupoid "ffC(H; ) whose objects consist of all homomorphisms i : H ! G
with i 2 ; a morphism i ! i0 in this groupoid is an element g 2 G such that
gig-1 = i0. This construction gives a functor from AopCto the category of group*
*oids.
It is clear that "ffC(H; ) is a connected groupoid in which the automorphism gr*
*oup
of an object i is the centralizer CG (i(H)), so the nerve |"ffC(H; )| is equiva*
*lent to
B CG (i(H)) (see 2.3). We let ffC = |"ffC|; this is a functor AopC! Spaces of*
* the
type promised in 1.5.
Let G be the category of the group G (2.3). For each object (H; ) of AC
there is a functor f"fC(H; ) ! G which assigns to a morphism gig-1 = i0 of
"ffC(H; ) the element g 2 G which determines it. These functors combine to give*
* a
natural transformation from "ffCto the constant functor on AopCwith value G, and
consequently a natural transformation from ffC to the constant functor with val*
*ue
|G| = BG. This gives a map [5, p. 329]
aC : hocolim ffC -! B G :
10 W. DWYER
In order to prove Theorem 1.6 it is enough to show that aC can be identified up*
* to
weak equivalence with the map qC of 2.7.
Let U be the category Gr (("ffC)op)op (see 2.10). The objects of U are pairs
(H; i) such that H is a group and i : H ! G is a monomorphism with i(H) 2 C.
A morphism (H; i) ! (H0; i0) is a pair (j; g) where j : H ! H0 is a homomorphism
and g 2 G is an element such that g(i0j)g-1 = i. By 2.10, |U| is weakly equival*
*ent
to hocolim ffC. The functor U ! G which sends a morphism (j; g) to g induces a
map |U| ! |G| = BG which corresponds (by naturality) to aC.
Let V be the Grothendieck construction of the action of G on SC (2.16). The
objects of V are the subgroups H of G. A morphism H ! H0 in this category
consists of an element g 2 G such that gHg-1 H0. By 2.9, |V| is weakly
equivalent to |SC|hG . There is a functor |V| ! G which sends a morphism g :
gHg-1 H0 to the element g 2 G which determines it; this induces a map |V| !
|G| = BG which corresponds to qC.
Consider the functor F : U ! V which sends an object (H; i) to the subgroup
i(H) G. It is easy to see that F is an equivalence of categories; an inverse
equivalence is given by the functor V ! U which sends a subgroup H to the pair
(H; ) where : H ! G is the inclusion. It follows that F induces a weak equival*
*ence
on nerves. The proof of 1.6 is completed by observing that F commutes with the
functors from U and V to G.
3.2 The subgroup decomposition. Let J : OC ! G-Spaces be the inclusion
functor and E C= hocolim J the space of 2.12. Since homotopy colimits (cf. 2.16)
commute with one another, there is a natural weak equivalence
hocolim fiC = hocolim(JhG ) ' (hocolim J )hG = (E C)hG :
Under this equivalence the map bC corresponds to the map
b0C: (E C)hG -! B G
induced by the G-map E C ! *. By 2.12, there is a weak equivalence E C ! |SC|
which is G-equivariant; this map induces a weak equivalence (E C)hG ! |SC|hG
which commutes with the respective maps b0Cand qC from these spaces to BG. This
implies that qC is an Fp-equivalence if and only if b0Cor equivalently bC is. B*
*y 2.7,
this proves 1.8.
3.3 The normalizer decomposition. The category sSC of simplices in KC has
as objects the simplices oe of KC, that is, the finite subsets oe of C which ar*
*e totally
ordered by inclusion:
(3.4) oe = {Hi| 0 i n(oe); H0 ( H1 ( . .(.Hn(oe)} :
There is exactly one morphism oe ! oe0 in sSC if oe0 oe, and there are no other
morphisms. It may be a little surprising that we have chosen to have morphisms
correspond to reverse inclusions, but this simplifies a construction below. The*
* con-
jugation action of G on C induces an action of G on sSC. The geometric realizat*
*ion
of |sSC| is the barycentric subdivision of KC (cf. 2.5), and in fact |sSC| is *
*weakly
HOMOLOGY APPROXIMATIONS 11
equivalent to |SC| in a way which respects the actions of G on these spaces. F*
*or
instance, such a weak equivalence is induced given by the functor sSC ! SC which
sends the object 3.4 of sSC to the object H0 of SC; see [9, x5] for more detail*
*s.
Let U be the Grothendieck construction of the action of G on sSC. The objects
of U are the simplices oe of KC; a morphism oe ! oe0 is an element g 2 G such t*
*hat
goe0g-1 oe. There is a functor s"qC: U ! G which sends a morphism goe0g-1 oe
to the element g which determines it. By 2.7, 2.16, and the homotopy invariance
of homotopy colimits, |U| is equivalent to |SC|hG in such a way that the map
(3.5) sqC = |s"qC| : |U| ! |G| = BG
corresponds to qC (2.7).
The category sSC of "orbit simplices" has as objects the equivalence classes*
* oeof
objects of sSC under the conjugation action of G. More explicitly, an object of*
* sSC
is an equivalence class
oe= <{Hi| 0 i n(oe); H0 ( H1 ( . .(.Hn(oe)}>
of totally ordered subsets of C, where two subsets {Hi} and {H0i} are considered
equivalent if there is a single element g 2 G such that gHig-1 = H0i, 0 i n(o*
*e).
Suppose that oe 2 oeand oe0 2 oe0. Then there is exactly one morphism oe! oe0in
sSC if there exists g 2 G such that goe0 oe. There are no other morphisms.
For each object oeof sSC, let "ffiC(oe) denote the groupoid whose objects ar*
*e the
elements oe 2 oe. A morphism oe ! oe0in "ffiC(oe) is an element g 2 G such that*
* goe = oe0.
It is clear that this groupoid is connected (2.3), and that |"ffiC(oe)| is equi*
*valent to
B Goe, where Goeis the isotropy subgroup in G of an element oe = {Hi} 2 oe. The
group Goecan be calculated by inspection:
n(oe)"
Goe= NG (Hi) :
i=1
We will now describe how the construction f"fiCgives a functor from sSC to
groupoids. Suppose that oe, oe0are objects of sSC and that f : oe! oe0is a map *
*in
sSC. Let {Hi| 0 i n(oe)} be an object of "ffiC(oe). It is not hard to check*
* that
there is a unique list (ik | 0 k n(oe0)) of distinct integers, 0 ik n(oe), *
*such
that the subset {Hik| 0 k n(oe0)} belongs to oe0. (Uniqueness follows from the
fact that a subgroup of G cannot be conjugate to a proper subgroup of itself.) *
*The
functor "ffiC(f) then takes the object {Hi} of "ffiC(oe) to the object {Hik} of*
* "ffiC(oe0).
The effect of "ffiC(f) on morphisms of "ffiC(oe) is more or less evident.
Let ffiC(oe) = |"ffiC(oe)|, so that ffiC : sSC ! Spaces is a functor of the*
* type promised
in 1.9. Let G be the category of the group G. For each object oeof sSC there *
*is
a a functor ffiC(oe) -! G which sends a morphism determined by an element g 2 G
to the morphism g of G. These functors combine to give a natural transformation
from "ffiCto the constant functor on sSC with value G. Passing to nerves gives*
* a
natural transformation from ffiC to the constant functor with value |G| = BG, a*
*nd
hence a map dC : hocolim ffiC ! BG, as required 1.9.
12 W. DWYER
In order to prove Theorem 1.10, it is enough to show that dC can be identifi*
*ed
up to weak equivalence with the map qC of 2.7, or even with the map sqC of 3.5.
Let V be the Grothendieck construction of "ffiC. An object of V is a simplex o*
*e of
KC, and a morphism oe ! oe0 is an element g 2 G such that goe0 oe. By 2.9 and
2.2, it is enough to check that the category V is equivalent to the category U *
*of
3.5 in a way which respects the relevant functors from these two categories to *
*G.
This is clear.
x4. G-spaces and fixed point sets
In this section we will recall some facts about the relationship between a G*
*-space
X and the fixed point sets XH for various subgroups H of G. Throughout, G is a
fixed finite group. By our conventions a G-space is a simplicial set with an ac*
*tion
of G, but the results remain true for G-CW complexes.
A map Y ! X of G-spaces is said to be a weak G-equivalence (resp. a G-Fp-
equivalence) if for every subgroup H of G the induced map Y H ! XH is a weak
equivalence (resp. an Fp-equivalence). Let IsoG(X) or Iso(X) denote the collect*
*ion
of subgroups of G consisting of the isotropy subgroups of the action of G on X.
We will be interested in the following two fairly well-known results, which wil*
*l be
proved at the end of this section.
4.1 Proposition. Suppose that f : X ! Y is a map of G-spaces, and that C =
Iso(X)[Iso(Y ). If fH : XH ! Y H is a weak equivalence (resp. an Fp-equivalen*
*ce)
for all H 2 C, then f is a weak G-equivalence (resp. a G-Fp-equivalence).
4.2 Proposition. Suppose that X is a G-space and C is a collection of subgroups
of G with Iso(X) C. Then there exists a zigzag of G-maps
X- f X0 h-!EC
such that f is a weak G-equivalence. If XH is weakly contractible (resp. Fp-ac*
*yclic)
for all H 2 C, then h is a weak G-equivalence (resp. a G-Fp-equivalence).
4.3 Remark. These propositions will be applied mostly to spaces X which them-
selves are of the form EC0. For this it is necessary to have information about *
*Iso(E C).
The following statements are clear from the construction of the homotopy colimit
(see 4.7 or [5, p. 338]).
4.4 Lemma. For any collection C of subgroups of G, Iso(E C) = C.
4.5 Lemma. Suppose that C and C0 are collections of subgroups of G. Assume
that C C0 (so that E Cis a subspace of E C0) and that given H 2 C and K 2 C0\ *
*C,
H is not a subgroup of K. Then every simplex of E C0\ EC has isotropy subgroup
contained in C0\ C.
Results similar to 4.1 and 4.2 are proved in [16, Appendix] by an induction *
*on
orbit types which uses a pushout formula like the one below in Lemma 8.6 in the
inductive step. We will derive 4.1 and 4.2 by referring to a general method for
building G-spaces from fixed point data.
HOMOLOGY APPROXIMATIONS 13
4.6 Definition. Suppose that D is a small category, and that f : Dop ! Spaces
and g : D ! Spaces are functors. The double bar construction B(f; D; g) is the
simplicial space which in dimension k consists of the following coproduct (inde*
*xed
by strings of composable arrows in D)
a
g(dk) x f(d0) ;
dk!...!d0
and which has the usual face and degeneracy operators. See [5, XI x2], [13, x3]*
*, or
[8, x9] (which uses the notation N**(f; D; g)). The homotopy coend of f and g,
denote hocoend (f; g), is the realization or diagonal of B(f; D; g) [5, XII, 3.*
*4, 5.3].
See [14, x3] for the topological version of this construction.
4.7 Remark. Homotopy coends have many convenient properties. Sometimes they
can be obtained up to homotopy as the nerves of generalized Grothendieck constr*
*uc-
tions [8, x9]. Let * denote the constant one-point valued functor on D, conside*
*red
as necessary to be either covariant or contravariant. In the situation of 4.6 *
*there
are natural isomorphisms
hocolim f ~=hocoend (f; *) hocolim g ~=hocoend (*; g) |D| ~=hocoend (*; *)*
* :
If u : f ! f0 and v : g ! g0 are natural transformations such that, for each ob*
*ject
d of D, u(d) and v(d) are weak equivalences (resp. Fp-equivalences), then the m*
*ap
hocoend (u; v) : hocoend (f; g) ! hocoend (f0; g0) is a weak equivalence [5, XI*
*I, 4.3]
(resp. Fp-equivalence [5, XII, 5.7]).
Suppose that G is a finite group, that C is a collection of subgroups of G, *
*and
that X is a G-space. There is a fixed point functor XC : OopC! Spaces given by
XC(G=H) = Map G (G=H; X) (= XH ) :
as well as an inclusion functor J : OC ! G-Spaces . Let XC = hocoend (XC; J ).
The action of G on the values of J induces by naturality an action of G on XC.
4.8 Theorem. Let G be a finite group, C a collection of subgroups of G, and X a
G-space. Then there is a natural G-map XC ! X. If C contains Iso(X), this map
is a weak G-equivalence.
Results like this were first proved by Elmendorf [13] in the case in which C*
* is the
collection consisting of all subgroups of G.
Proof. According to the description of the homotopy coend in 4.6, XC is the rea*
*l-
ization of a simplicial space which in dimension k is a disjoint union of space*
*s of the
form G=Hk x Map G(G=H0; X), the union indexed by chains G=Hk -! . .-.!G=H0
in OC. Such a chain gives a composite map G=Hk ! G=H0, which can be com-
bined with the evaluation map G=H0 x Map G (G=H0; X) ! X to give a map
G=Hk x Map G (G=H0; X) ! X. These maps are compatible with the simplicial
operators and upon passage to the realization give a map XC ! X; for details
consult [13, x3].
14 W. DWYER
One can check directly that XC ! X is a weak equivalence if X = G=H for
some H 2 C. This follows for instance from the Reduction Theorem [14, 4.4] and
the fact that if X = G=H then XC is the representable functor Map (-; G=H) on
OC. There is an explicit argument in [13, x3]. The functor X 7! XC preserves we*
*ak
G-equivalences (4.7). It also preserves coproducts, pushouts in which one of t*
*he
maps is a monomorphism of simplicial sets, and sequential colimits. It follows *
*that
the map XC ! X is a weak equivalence whenever X can be constructed from the
collection of "G-cells" {G=H x [k] | H 2 C; k 0} by these three operations. Th*
*is
can be done if and only if Iso(X) C.
Proof of 4.1. By 4.8 and naturality there is a commutative diagram
XC - ---! X
?? ?
y f?y
YC - ---! Y
in which the the horizontal arrows are weak G-equivalences. By 4.7 the left ver*
*tical
arrow is a weak G-equivalence (resp. a G-Fp-equivalence).
Proof of 4.2. Let * be the contravariant functor on OC whose value is the one-p*
*oint
space. There is a unique natural transformation XC ! *, which gives rise to a m*
*ap
XC = hocoend (XC; J ) ! hocoend (*; J ) = hocolim J = EC :
The required zigzag is X- XC -! E C. It has the necessary properties by 4.8 a*
*nd
4.7.
x5. Finite p-groups
In this section we recall some properties of finite p-groups. The propertie*
*s are
elementary and very well-known, but since they are so important in what follows
it seems worthwhile to state them explicitly.
5.1 Proposition. Suppose that X is a finite set with an action of the finite p*
*-group
P . Then the cardinality of XP is congruent mod p to the cardinality of X.
Proof. All of the non-trivial orbits of the action of P have cardinality divis*
*ible
by p.
5.2 Proposition. Let P and Q be finite p-groups, and suppose that P acts on Q
via group automorphisms. Then there exists a nonidentity element x in the center
of Q such that x is fixed by the action of P .
Proof. Let G be the semidirect product of P with Q, so that the conjugation act*
*ion
of Q on itself combines with the given action of P on Q to give an action of G *
*on
Q. Since G fixes the identity element e 2 Q, counting (5.1) shows that G must f*
*ix
a nonidentity element x.
HOMOLOGY APPROXIMATIONS 15
5.3 Proposition. Suppose that P is a finite p-group and that M is an Fp[P ]-
module which is finite dimensional as a vector space over Fp. Then M has a fini*
*te
filtration by P -submodules with the property that P acts trivially on the filt*
*ration
quotients.
Proof. Use induction on the order of M. By 5.2 there exists a nonzero submodule
M0 M on which P acts trivially, and by induction the quotient module M=M0
has a filtration of the required type.
Remark. The finite-dimensionality hypothesis in 5.3 is not necessary. This foll*
*ows
from the fact that 5.3 applies to the universal case, i.e., the module given by*
* the
left action of P on Fp[P ].
5.4 Proposition. Let Q be a finite p-group and P a proper subgroup of Q. Then
the normalizer NQ (P ) is strictly larger than P .
Proof. The quotient NQ (P )=P is the fixed point set of the action of P on Q=P .
Since P is a proper subgroup of Q, the cardinality of Q=P is divisible by p. T*
*he
identity coset eP is fixed by P , and so counting (5.1) shows that there are ot*
*her
fixed cosets.
x6. The collection of nontrivial p-subgroups
Let G be a fixed finite group. In this section we prove that if p divides t*
*he
order of G the collection of nontrivial p-subgroups of G is ample. We will actu*
*ally
be concerned with a something more delicate than this, which is described in the
following definition.
6.1 Definition. Let C be a collection of subgroups of G and M a G-module. The
collection C is said to be M-ample if the map qKC of 1.4 (equivalently, the map*
* qC
of 2.7) induces an isomorphism on (twisted) homology with coefficients in M.
6.2 Remark. If Fp denotes the trivial G-module, then C is Fp-ample in the sense
of 6.1 if and only if C is ample. An easy calculation shows that C is Fp[G]-amp*
*le if
and only if the spaces KC and |SC| are Fp-acyclic; in this situation C is M-amp*
*le
for any G-module M. The arguments of x3 show that C is M-ample if and only if
the three decomposition maps aC (1.6), bC (1.8) and dC (1.10) induce isomorphis*
*ms
on homology with coefficients in M. If |SC| is weakly contractible, then all t*
*hree
decomposition maps are weak equivalences, and C is M-ample for any G-module M.
For instance, this is the case if C is a trivial collection (1.12).
6.3 Theorem. Let C be the collection of all nontrivial p-subgroups of G. Suppo*
*se
that M is a module over Fp[G] such that p divides the order of the kernel of the
action map G ! Aut (M). Then C is M-ample.
Remark. Theorem 6.3 is a result of Jackowski-McClure-Oliver traveling in disgui*
*se.
Under the hypotheses of 6.3 they prove that certain groups *(G; M) vanish [16,
5.5], but it is not hard to see using the ideas in 3.2 that these groups are is*
*omorphic
to the relative groups H *(B G; |SC|hG ; M). We will give an independent proof *
*of 6.3,
since the argument in [16] is indirect and depends on results which we eventual*
*ly
want to prove ourselves,
16 W. DWYER
6.4 Remark. Theorem 6.3 implies that if p divides the order of G, then the coll*
*ection
of all nontrivial p-subgroups of G is ample. This was first proved by K. Brown *
*(cf.
[1, V.3.1]). There is a spectral sequence approach (which proves a lot more) d*
*ue
to P. Webb [1, V.3.2]. Finally, this can be proved by working backwards, using
1.6 and Quillen's theorem (1.14), from the theorem of Jackowski-McClure [15] th*
*at
for the collection of nontrivial elementary abelian p-subgroups of G, the centr*
*alizer
decomposition map (1.5) is an Fp-equivalence.
The key ingredient in the proof of 6.3 is the following lemma.
6.5 Lemma. Let M be as in 6.3, and let
P0 ( P1 ( . .(.Pn
be a chain of nontrivial p-subgroups of G. Then there exists an element g 2 G of
order p such that g normalizes each of the subgroups Pi and g acts trivially on*
* M.
Proof. In fact we can find an element g 2 G of order p such that g centralizes *
*Pn
and acts trivially on M. Let P be a Sylow p-subgroup of G which contains Pn,
and let H be the kernel of the action map G ! Aut (M). Since H is normal in G,
Q = H \ P 6= {e} is a Sylow p-subgroup of H and is normal in P . Choose g to be
some element of order p in Q which is fixed by the conjugation action of P on Q
(5.2).
Proof of 6.3. If C is a chain complex of Fp[G]-modules, in practice of finite l*
*ength,
we will let H *(G; C), H *(G; C) and ^H*(G; C) denote respectively the hyperhom*
*ol-
ogy, hypercohomology and Tate hypercohomology of G with coefficients in C [6,
XVII] [1, P. 164] [20, p. 166]. By the way in which complete (Tate) resolutions*
* are
constructed [1, II.7] there is a doubly infinite exact sequence
(6.6) . .-.!H i(G; C) -! H -i(G; C) -! ^H-i(G; C) -! H i-1(G; C) -! . . .:
Let CG (resp. CG ) be the chain complex obtained by applying H 0(G; -) (resp.
H 0(G; -)) dimensionwise to C. Further inspection of the recipe for a complete
resolution shows that the map in the above exact sequence can be factored as a
composite
Hi(G; C) -! H i(CG ) -(G-)*-!Hi(CG ) -! H -i(G; C) :
P
where G : CG -! CG is induced by the norm endomorphism G = g2G g of C.
In particular, if G is the trivial endomorphism of C, the map is trivial.
Let C be the normalized simplicial Fp-chain complex of |SC|, or equivalently
(2.5) the cellular chain complex of KC, and C" the kernel of the map C ! Fp
induced by the map from |SC| to a point. Denote by C M and "C M the chain
complexes obtained by taking tensor products over Fp and using diagonal G-actio*
*ns.
Recall that H *(G; C) is naturally isomorphic to H *(|SC|hG ; Fp) (cf. [1, p. *
*184]);
more generally, H *(G; C M) is naturally isomorphic to H *(|SC|hG ; M). Simila*
*rly,
H *(G; FpM) = H *(G; M) is naturally isomorphic to H*((*)hG ; M) = H *(B G; M).
Thus it is enough to show that the map C M ! Fp M induces an isomorphism
H *(G; C M) ~= H *(G; Fp M), or even, by a long exact sequence argument,
enough to show that H *(G; "C M) = 0.
HOMOLOGY APPROXIMATIONS 17
Let P G be a Sylow p-subgroup. The argument in [1, proof of V.3.1] shows
that C ! Fp induces an isomorphism ^H*(P ; C) ~=H^(P ; Fp). Essentially the same
argument shows that C M ! Fp M = M induces an isomorphism ^H*(P ; C
M) ! ^H*(P ; M). It is only necessary to check that if F is a free module over *
*Fp[P ]
then F M (with the diagonal action) is also a free module, and this is true for*
* any
Fp[P ]-module M. It follows that ^H*(P ; "C M) = 0, and from that by a transfer
argument that ^H*(G; "C M) = 0.
Let {mff} be an Fp-basis of M. The chain complex C M has an Fp-basis in
which each element is of the form oe mfffor some simplex oe = P0 ( P1 ( . .(.Pn
of KC. According to 6.5 each such basis element is fixed by an element in G of
order p, and so each basis element maps to zero under the norm endomorphism G .
(Keep in mind that C M is a chain complex of vector spaces over Fp.) It follows
that G is also the trivial endomorphism of C" M, and from the remarks above
that for each i the map : H i(G; "C M) ! H -i(G; "C M) is zero. Clearly, then,
the fact that ^H*(G; "C M) = 0 implies that H *(G; "C M) = 0.
We record some related results for future use.
6.7 Proposition. Let Y be a G-space, M a local coefficient system of exponent*
* p
on YhG , and C the collection of all nontrivial p-subgroups of G. Suppose that *
*there
is an element of order p in G which acts trivially on H *(Y ; M) (with respect *
*to the
Serre action 6.8). Then the map
(Y x |SC|)hG ! YhG
induced by the projection Y x |SC| ! Y induces an isomorphism on H *(-; M).
Remark. The G-action on Y x |SC| implicit in the statement of the proposition is
the diagonal one. The letter M is used to denote both the original local coeffi*
*cient
system on YhG and the systems on Y and on (Y x |SC|)hG pulled back over the
evident maps of these spaces to YhG .
Proof of 6.7. There are two Serre spectral sequences
E2*;*= H *(|SC|hG ; H*(Y ; M))) H *((Y x |SC|)hG ; M)
(6.8) 2
E*;*= H *(B G; H*(Y ; M)) ) H *(YhG ; M)
and a map between them which on abutments gives the homology map we are
interested in. By 6.3 the map on E2-terms is an isomorphism.
The following is a theorem of Quillen.
6.9 Theorem. Suppose that G has a nontrivial normal p-subgroup, and let C be
the collection of all nontrivial p-subgroups of G. Then |SC| is contractible, a*
*nd so
C is M-ample for any G-module M.
Proof. Let P be a nontrivial normal p-subgroup of G. For each Q 2 C there are
maps (inclusions) Q ! P Q P ; these give a zigzag of natural transformations
between the identity functor of SC and a constant functor. The result follows
from 2.1.
18 W. DWYER
To help orient the reader, we will prove that |SC| is Fp-acyclic in a way wh*
*ich
ties this fact into 6.3. Let P G be a nontrivial normal p-subgroup. By 6.3
the collection C is M-ample for M = Fp[G=P ]. For any G-space X there is a
natural isomorphism H *(XhG ; Fp[G=P ]) ~= H *(XhP ; Fp) (Shapiro's lemma), and
so it follows that the map |SC|hP ! B P is an Fp-equivalence. But Fp[P ] has a
finite filtration by P -submodules such that P acts trivially on the associated*
* graded
groups (5.3), so an induction using long exact sequence comparisons and the five
lemma shows that the map
H*(|SC|hP ; Fp[P ]) ! H *((*)hP ; Fp[P ])
is an isomorphism. By Shapiro's lemma again, the groups on the left are the mod*
* p
homology groups of |SC|, and those on the right are the mod p homology groups of
a point.
x7. p-stubborn collections
Let G be a finite group and let C be a collection of p-subgroups of G which
contains all p-stubborn subgroups. In this section we will show that C is M-amp*
*le
for all G-modules M (6.2). In fact, we will show something stronger. For the
statement, recall the definition of SC(H) (H a subgroup of G) from 2.13.
7.1 Theorem. Let C be any collection of p-subgroups of G which contains all p-
stubborn subgroups. Then for any p-subgroup P of G the nerve |SC(P )| is weakly
contractible.
7.2 Remark. If H is the trivial subgroup of G then |SC(H)| = |SC|.
7.3 Remark. Let C be the collection of p-stubborn subgroups of G. The fact that
|SC| is contractible is a simple consequence of Bouc's theorem (1.18). To see t*
*his, let
P be the maximal normal p-subgroup of G. If P = {e}, then {e} G is p-stubborn,
and |SC| is contractible by 1.12. If P 6= {e}, then C is the Bouc collection, a*
*nd |SC|
is contractible by a combination of Bouc's theorem (1.18) and Quillen's theorem
(6.9). We include the argument below because it is short, it proves something a
little more than the weak contractibility of |SC|, it sets the stage for x8, an*
*d it
suggests an approach which generalizes to the case of compact Lie groups.
Theorem 7.1 raises the disquieting suspicion that working with a p-stubborn
collection C of subgroups does not in any sense involve focusing on the structu*
*re
of the group G at p, since the decomposition maps associated with C are weak
equivalences (6.2). Indeed, the reader who is familiar with [16] may wonder why
Theorem 7.1 is so strong; the results of [16, x2] suggest that if C is the coll*
*ection of
all p-stubborn subgroups of G then |SC| should at best be Fp-acyclic, not weakly
contractible. The explanation for this is partly visible in the proof below. *
*A key
element in the analysis of |SC| is the study of W (P ) = NG (P )=P for p-subgro*
*ups P
which are not p-stubborn. In such a case W (P ) has a nontrivial normal p-subgr*
*oup,
and so Theorem 6.9 provides a certain contractibility statement for W (P ). Su*
*p-
pose now that P is a p-toral subgroup of a compact Lie group G, and that, in the
appropriate Lie group sense [16, 1.3], P is not p-stubborn. If W (P ) is finite*
*, The-
orem 6.9 applies to W (P ) and again provides a contractibility statement. If W*
* (P )
HOMOLOGY APPROXIMATIONS 19
is positive-dimensional, an analogous theorem applies (cf. [16, 2.11]), but pro*
*vides
only Fp-acyclicity. This difference propagates through the theory and accounts *
*for
the fact that p-stubborn decomposition theorems for general compact Lie groups
are slightly weaker than the corresponding ones for finite groups.
Proof of 7.1. The proof is by downward induction on the size of the subgroup P .
We have to check that the result is true if P is as large as possible, i.e., if*
* P is a
Sylow p-subgroup of G. In this case, though, P is p-stubborn, the poset SC(P ) *
*has
only one element, and |SC(P )| is a single point.
Let P G be a p-subgroup of G and assume that for all p-subgroups Q of
larger order, |SC(Q)| is weakly contractible. We will show that |SC(P )| is we*
*akly
contractible. If P 2 C, then P itself is a minimal element of SC(P ), so |SC(P *
*)| is
contractible (see 2.6) and we are done. Assume then that P =2C.
Let X = EC . According to 2.14 it is enough to show that the space XP is we*
*akly
contractible. Let N be the normalizer NG (P ) and W the quotient group N=P . The
group W acts on XP in a natural way and the isotropy subgroups of this action
are (4.4) of the form (Q \ N)=P for subgroups Q 2 C such that Q P . These
isotropy subgroups are p-subgroups of W and in fact nontrivial p-subgroups of W*
* :
to see this recall that P =2C, so that if Q 2 C and Q P , then Q ) P and, by 5*
*.4,
Q \ N ) P . Let Q be a nontrivial p-subgroup of W and Q N its preimage. By
the inductive hypothesis (and 2.14) the fixed point space (XP )Q = XQ is weak*
*ly
contractible. An application of 4.2 to the group W and the W -space XP shows th*
*at
XP is weakly W -equivalent to |SC0|, where C0 is the collection of all nontriv*
*ial p-
subgroups of W . Since P is not p-stubborn, W has a nontrivial normal p-subgroup
and |SC0| is contractible (6.9); this proves that XP is weakly contractible.
Remark. The main idea above is lifted from an argument of Quillen [2, 6.9.2] wh*
*ich
identifies up to homotopy the simplicial complex associated to the poset of all*
* p-
subgroups of G which properly contain a given p-subgroup. This argument was
reproduced in a different context by Jackowski-McClure-Oliver [16, 5.4].
x8. p-centric collections
In this section we prove generalizations of the results described in 1.19 an*
*d 1.20.
As usual, G is a fixed finite group. Suppose that H G is a subgroup and that
M is a G-module. Let CG (H; M) denote the subgroup of G consisting of elements
which both centralize H and act trivially on M.
8.1 Definition. Let M be a module over Fp[G]. A p-subgroup P of G is said to be
M-centric if CG (P; M) is the product of a subgroup of P and a finite group of *
*order
prime to p. Equivalently, P is M-centric if CG (P; M) \ P is a Sylow p-subgroup*
* of
CG (P; M).
8.2 Definition. Let M be a module over Fp[G]. A collection C of p-subgroups of G
is said to be M-admissible if
(1) C contains all M-centric subgroups, and
(2) C is closed under the process of passing to p-supergroups: if P 2 C and*
* Q
is a p-subgroup of G with Q P then Q 2 C.
20 W. DWYER
8.3 Theorem. Let M be a module over Fp[G] and C a collection of p-subgroups
of G which is M-admissible. Then C is M-ample.
Remark. We can recover 1.19 from 8.3 by observing that
(1) the collection of all M-centric subgroups of G is M-admissible, and
(2) a p-subgroup P is M-centric for M = Fp if and only if P is p-centric in*
* the
sense of 1.19.
The following properties of the homotopy orbit space construction are well
known.
8.4 Lemma. Suppose that that K G is a subgroup and that X is a G-space.
Then there is a natural weak equivalence XhK ' (G xK X)hG .
Proof. (GxK X)hG can be identified with (E GxX)=K. The natural map EK ! EG
is a map between contractible spaces on which K acts freely, and so induces a w*
*eak
equivalence (E K x X)=K ! (E G x X)=G.
To avoid clutter in the next statement, we define a proxy action of a group W
on a space X to be some associated space X0, a weak equivalence X ! X0, and a
(genuine) action of W on X0. Given such a proxy action, XhW stands for X0hW.
8.5 Lemma. Suppose that K G is a normal subgroup with quotient group W =
G=K, and that X is a G-space. Then there is a natural proxy action of W on XhK
and a weak equivalence
(XhK )hW ' XhG :
If K acts trivially on X, then XhK is B K x X and the proxy action of W on
B K x X is a diagonal one.
Proof. The map
(E W x (E G x X)=K)=W ! ((E G x X)=K)=W = (E G x X)=G
is a weak equivalence because W acts freely on (E G x X)=K. This gives an equiv-
alence
((E G x X)=K)hW ' XhG :
As in the proof above, (E G x X)=K is weakly equivalent in a natural way to XhK*
* .
The last statement is clear.
8.6 Lemma. Let Y be a G-space, X Y a G-subspace, H G a subgroup, and
N = NG (H) its normalizer. Suppose that all of the simplices of Y which are n*
*ot
in X have isotropy subgroup conjugate to H. Then there is a (homotopy) pushout
diagram of G-spaces:
G xN XH ----! X
? ?
(8.7) ?y ?y :
G xN Y H ----! Y
HOMOLOGY APPROXIMATIONS 21
Proof. This amounts to the observation that all of the simplices of Y which are*
* not
in X lie in the G-orbit of Y H; the part of this orbit which lies in X is the o*
*rbit of
XH . The diagram is a homotopy pushout diagram because the left vertical arrow
is the inclusion of a G-subcomplex.
Proof of 8.3. The proof is by downward induction on the size of C. The statement
is clearly true (by 1.12) if C is the M-admissible collection of all p-subgroup*
*s of G.
Suppose then that C is M-admissible, C does not contain all p-subgroups of G, a*
*nd
that C0 is M-ample for all M-admissible collections of subgroups with C0 C. Let
P G be a p-subgroup of G which is maximal with respect to the property that
P =2C, and let C0 be the union of C with the set of all conjugates of P . Clear*
*ly C0
is M-admissible, and hence by induction M-ample. Since C0 contains all M-centric
subgroups of G, P is not M-centric.
Let X = EC and Y = EC0 (see 2.12), so that X can be considered as a subspace
of Y . By induction and 2.12 the natural map YhG ! B G induces an isomorphism
on H *(-; M); we must show that the same holds when Y is replaced by X. This
amounts to showing the map XhG ! YhG induces an isomorphism on H *(-; M).
Let N = NG (P ). The simplices of Y which do not lie in X all have isotropy
subgroup conjugate to P (4.5), and so by 8.6 there is a homotopy pushout diagra*
*m:
G xN XP ----! X
?? ?
y ?y :
G xN Y P ----! Y
We are interested in the spaces in the left hand column. By 2.15 the space Y P *
*is
contractible. Let W = N=P . The group W acts on XP and, as explained in the
proof of 7.1, Proposition 5.4 guarantees that the isotropy subgroups of this ac*
*tion
are nontrivial p-subgroups of W . Let Q be a nontrivial p-subgroup of W and Q N
its preimage. Since Q 2 C, it follows from 2.15 that (XP )Q = XQ is contractib*
*le.
Let |SP(W) | denote as usual the nerve of the poset P(W ) of nontrivial p-subgr*
*oups
of W . By 4.2 and 2.12, XP is weakly W -equivalent to |SP(W) |.
In the light of 8.4 and 8.5, applying the homotopy orbit space construction *
*to
8.7 gives a homotopy pushout diagram of the following form:
(B P x |SP(W) |)hW ----! XhG
? ?
(8.8) f?y ?y :
(B P )hW ----! YhG
Here we have used Y P ' * to give (Y P)hP ' BP . The action of W on B P in this
diagram is easily seen to correspond to the conjugation action of W on P (via o*
*uter
automorphisms). Since P is not M-centric, there is an element of order p in W
which acts trivially on H *(B P ; M) (see 8.9). By 6.7 the left vertical arrow*
* in 8.8
induces an isomorphism on H *(-; M), and so by Mayer-Vietoris the right vertical
arrow does too.
8.9 Remark. We continue to use the notation in the proof above. The reader
may be uneasy about the assertion that the fact that P is not M-centric implies
22 W. DWYER
that there is an element of order p in W which acts trivially on H *(B P ; M). *
* We
will show how to check this. Let Mod be the category whose objects are pairs
(K; A), where K is a group and A is a K-module. A map (K; A) ! (K0; A0) is a
pair (u; v), where u : K ! K0 is a group homomorphism and v : A ! A0 is an
abelian group map such that v(xa) = u(x)v(a), x 2 K, a 2 A. Homology gives a
functor (K; A) 7! H *(B K; A) from Mod to graded abelian groups. For any elem*
*ent
y 2 K let (cy; y) = (u; v) be the automorphism of (K; A) for which u(x) = yxy-1
and v(a) = ya. The usual argument that inner automorphisms of K act trivially
on H *(B K; Fp) generalizes to show that the automorphism (cy; y) of (K; A) acts
trivially on H *(K; A). In the situation of the proof above, N acts on the obj*
*ect
(P; M) of Mod by similar automorphisms (cy; y), y 2 N, and it follows that the
induced action of N on H *(B P ; M) factors through an action of W = N=P on the*
*se
homology groups. We will show that it is this action of W on H *(B P ; M) which
enters into 8.8. It is then clear that the image of CG (P; M) in W has order di*
*visible
by p (because P is not M-centric), and acts trivially on H *(B P ; M).
Write P for tensor product over Fp[P ], and let C*X denote the Fp-chain com-
plex of a simplicial set X. Consider the following diagram of chain complexes
C* EP P M- C* EP P (C* EN Fp M) -! Fp P (C* EN Fp M) :
The action of P on C* EN Fp M is the diagonal one. Since C* EP and C* EN
are free resolutions over Fp[P ] of the trivial module Fp, the maps in this dia*
*gram,
which are obtained from the augmentations C* EP ! Fp and C* EN ! Fp, in-
duce isomorphisms on homology. Pick x 2 N. Let c denote simultaneously the
automorphism of P given by conjugation with x and the automorphism of C* EP
induced by this conjugation. Let ` denote the automorphisms of M and of C* EN
given by left multiplication by x. The group N acts compatibly on the three cha*
*in
complexes above, with x acting respectively by c c`, c c(` 1 `) and 1 c(` 1 `).
The proof of 8.5 shows that it is the action of N on the right hand chain compl*
*ex
which induces the action of W on H *(B P ; M) figuring in 8.8. The action of N
on the left hand chain complex is the well-behaved one discussed in the precedi*
*ng
paragraph. The diagram shows how to identify these two actions with one another.
We will finish the section by proving the statement in 1.20. We leave it to *
*the
reader to formulate and prove a generalization involving a G-module M and an
appropriate notion of M-admissible collection.
8.10 Proof of 1.20. Let C0 be the collection of all p-centric subgroups of G, a*
*nd
C C0 the collection of subgroups which in addition are p-stubborn. By 2.12 and
8.3, it is enough to show that the natural map EC ! EC0 is a weak equivalence. *
*By
4.4 and 4.1, this will follow if we can prove that for all P 2 C0 the map (E C)*
*P !
(E C0)P is a weak equivalence. The target of this map is weakly contractible (2*
*.15),
so this amounts to showing that for all P 2 C0, (E C)P is weakly contractible.
The proof is by downward induction on the size of P . (Note that C0 is clos*
*ed
under passage to p-supergroups.) As in the proof of 7.1, the result is clear if*
* P is
of maximal size, i.e., if P is a Sylow p-subgroup of G. Suppose then that P 2 *
*C0
and that (E C)Q is weakly contractible for all p-subgroups Q of G which proper*
*ly
contain P . If P 2 C then (E C)Q is contractible by 2.15. Otherwise P is n*
*ot p-
stubborn, and the argument at the end of the proof of 7.1 shows that (E C)P is
weakly contractible.
HOMOLOGY APPROXIMATIONS 23
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University of Notre Dame, Notre Dame, Indiana 46556
Processed November 20, 1995
E-mail address: dwyer.1@nd.edu