1
SHARP HOMOLOGY DECOMPOSITIONS
FOR CLASSIFYING SPACES OF FINITE GROUPS
W. G. Dwyer
University of Notre Dame
x1. Introduction
Let G be a finite group and p a prime number. A homology decomposition for
the classifying space B G is a way of building B G up to mod p homology out of
classifying spaces of subgroups of G. More precisely, such a decomposition cons*
*ists
of a mod p homology isomorphism
(1.1) hocolim F = hocolimD F -~!pBG ;
where D is some small category, F is a functor from D to the category of spaces,
and, for each object d 2 D, F (d) has the homotopy type of B H for some subgroup
H of G. The operator "hocolim " is the homotopy colimit in the sense of Bousfie*
*ld
and Kan [4], which amounts to a homotopy invariant method of gluing together
the values of the functor F according to the pattern of the maps in D. There is*
* a
systematic study of some of the more common homology decompositions in [6] (see
1.3 below). Here we go further along the same lines. Bousfield and Kan associate
to 1.1 a first quadrant homology spectral sequence [4, XII.5.7]
(1.2) E2i;j= colimiH j(F ; Fp) ) H i+j(B G; Fp) :
In the description of this E2-term, colimi stands for the i'th left derived fun*
*ctor
of the colimit construction on the category of functors from D to abelian group*
*s.
Call a homology decomposition sharp if its spectral sequence collapses onto the
vertical axis in the sense that E2i;j= 0 for i > 0. The appeal of a sharp homol*
*ogy
decomposition is that it induces an isomorphism
~=
colim H*(F ; Fp) = colim0H *(F ; Fp) -! H *(B G; Fp)
and so gives a closed formula for the homology of B G in terms of the homology
of classifying spaces of subgroups of G. In this paper we do our best to determ*
*ine
which of the decompositions from [6] are sharp.
______________
The author was supported in part by the National Science Foundation
Typeset by AM S-T*
*EX
1
2 W. DWYER
1.3 Homology decompositions. Recall from [6] that a collection C of subgroups
of G is by definition a set of subgroups of G which is closed under conjugation,
in the sense that if H 2 C and g 2 G then gHg-1 2 C. Such a collection C is a
poset under the inclusion relation between subgroups. Let KC denote the associa*
*ted
simplicial complex [2, 6.2]: the vertices of KC are the elements of C, and the*
* n-
simplices (n 1) are the subsets of C of cardinality (n+1) which are totally or*
*dered
by inclusion. The action of G on C given by conjugating one subgroup to another
induces 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.
1.4 Definition. A collection C of subgroups of G is said to be ample if the map
KC ! * induces a mod p homology isomorphism (KC)hG ! (*)hG = BG.
Associated to any ample collection C are three homology decompositions of G,
called the centralizer decomposition, the subgroup decomposition, and the norma*
*l-
izer decomposition. The names signify that the constituents of the three homolo*
*gy
decomposition formulas are of the form B H, where H G is respectively, the cen-
tralizer in G of some element of C, an element of C itself, or an intersection *
*in G
of normalizers of elements of C. We will describe these three decompositions i*
*n a
general way; there are more details in [6] or at the end of x3.
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 .
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 . There is also a map
aC : hocolim ffC -! B G
which is a mod p homology isomorphism if and only if C is ample.
1.6 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
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 :
SHARP HOMOLOGY DECOMPOSITIONS 3
This map is a mod p homology isomorphism if and only if C is ample.
1.7 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)) There is also a map
dC : hocolim ffiC -! B G
which is a mod p homology isomorphism if and only if C is ample.
Sharp homology decompositions. Suppose that C is an ample collection of
subgroups of G. Some of the homology decompositions associated above to C might
be sharp, and others might not be. We need some terminology to describe this.
1.8 Definition. A collection C of subgroups of G is said to be centralizer-shar*
*p (resp.
subgroup-sharp, resp. normalizer-sharp) if C is ample and the centralizer func*
*tor
ffC (resp. the subgroup functor fiC, resp. the normalizer functor ffiC) gives a*
* sharp
homology decomposition of B G.
In this paper we determine the sharpness properties of a number of common
collections. Here are some examples.
Trivial examples. Suppose that C contains only the trivial subgroup of G. Then
C is both centralizer-sharp and normalizer-sharp, since both the centralizer and
normalizer decomposition diagrams reduce to the trivial diagram with B G as its
only constituent. By inspection, C is subgroup-sharp if and only if the mod p
homology of BG is trivial, which is the case if and only if the order of G is r*
*elatively
prime to p.
Suppose that C contains only G itself. Then C is both subgroup-sharp and nor-
malizer-sharp, since both the subgroup and normalizer decomposition diagrams
reduce to the trivial diagram with B G as its only constituent. Let Z be the ce*
*nter
of G. The category AC has one object whose group of self-maps is G=Z and the
functor ffC assigns to this object the space B Z. The Bousfield-Kan homology
spectral sequence for the associated homotopy colimit is the Lyndon-Hochschild-
Serre spectral sequence of the group extension Z ! G ! G=Z. From this it is not
hard to see that C is centralizer-sharp if and only if Z contains a Sylow p-sub*
*group
of G; this is the case if and only if G is the product of an abelian p-group an*
*d a
group of order prime to p.
1.9 Nontrivial p-subgroups. Let C be the collection of all nontrivial p-subgrou*
*ps
of G. As long as p divides the order of G, C is centralizer-sharp, subgroup-sha*
*rp,
and normalizer-sharp. This is proved in x7.
1.10 Elementary abelian subgroups. Recall that a finite abelian group is said t*
*o be
an elementary abelian p-group if it a module over Fp. Let C be the collection *
*of
4 W. DWYER
all nontrivial elementary abelian p-subgroups of G. If p divides the order of *
*G,
then C is both centralizer-sharp and normalizer-sharp; it cannot possibly be su*
*b-
group-sharp unless the mod p cohomology of G is detected on elementary abelian
p-subgroups. We will discuss this and related collections in x8.
1.11 p-centric subgroups. Recall that a p-subgroup P of G is said to be p-centr*
*ic
if the center of P is a Sylow p-subgroup of CG (P ). Let C be the collection o*
*f all
p-centric subgroups of G. Then C is subgroup-sharp (10.3).
1.12 Subgroups both p-centric and p-stubborn.. Recall that a p-subgroup P of G *
*is
said to be p-stubborn if NG (P )=P has no nontrivial normal p-subgroups. Let C *
*be
the collection of all subgroups of G which are both p-centric and p-stubborn. T*
*hen
C is subgroup-sharp (10.4).
1.13 A slight extension. Suppose that M is a module over Fp[G], so that M gives
rise in the usual way to a local coefficient system on BG. We will say a a coll*
*ection
C of subgroups of G is ample for M if the map (KC)hG ! (*)hG = BG induced by
KC ! * gives an isomorphism on the local coefficient homology groups H *(-; M).
For such a collection C there are local coefficient spectral sequences
E2i;j= colimiH j(F ; M) ) H i+j(B G; M) :
where F = ffC, fiC or ffiC. We will say that C is centralizer-sharp (resp. subg*
*roup-
sharp, resp. normalizer-sharp) for M if the spectral sequence for ffC (resp. fi*
*C, resp.
ffiC) collapses onto the vertical axis. In some of the examples described abov*
*e, we
will look at this more general notion of sharpness.
Organization of the paper. Section 2 discusses the "isotropy spectral sequence"*
* as-
sociated to a G-space, and x3 shows how to identify the Bousfield-Kan spectral
sequences of the above homology decompositions as isotropy spectral sequences.
Section 4 gives a homological interpretation of the E2-term of the isotropy spe*
*c-
tral sequence, and the next two sections develop techniques for showing that th*
*is
E2-term has appropriate vanishing properties. The most interesting technique is
probably the method of discarded orbits (6.8). The final sections apply these t*
*ech-
niques to study the homology decompositions associated to the collection of all
non-trivial p-subgroups (x7), certain collections of elementary abelian p-subgr*
*oups
(x8), and certain collections of subgroups which are p-centric and/or p-stubborn
(x10). There is a pause in x9 to look at the effect on a collection of "pruning*
*" (i.e.
removing) a single conjugacy class of subgroups.
This paper is mostly independent of [6], although we sometimes refer to a re*
*sult
or argument which it did not seem worthwhile to reproduce.
Related work. There is a large literature about homology decompositions. For
instance, see [1, V.3] or [15] for the normalizer decomposition associated to t*
*he
collection of all nontrivial p-subgroups, [8] or [3] for centralizer decomposit*
*ions
associated to collections of nontrivial elementary abelian p-subgroups, and [9]*
* for
the subgroup decomposition associated to the collection of p-stubborn subgroups.
There is significant overlap between this paper and [14], although [14] takes a
different point of view. The reader can also find a great deal of interesting r*
*elated
material in [11], [12] and [13].
SHARP HOMOLOGY DECOMPOSITIONS 5
Notation and terminology. This paper is written with the convention that "space"
means "simplicial set" [4, VIII]. A map between spaces is said to be an equival*
*ence
or a weak equivalence if its geometric realization gives a homotopy equivalence
between topological spaces [10]. The letter G always stands for some finite gro*
*up.
If x belongs to a G-set, then Gx is the isotropy subgroup of x. If M is a G-mod*
*ule,
then H *(G; M) = H *(B G; M) stands for the group homology of G with coefficien*
*ts
in M, or equivalently for the homology of B G with coefficients in the local sy*
*stem
determined by M.
Throughout the paper p is a fixed prime number and Fp is the field with p el-
ements. The vector space over Fp generated by a set X is denoted Fp[X]. The
nerve of a category D is written |D| (cf. 3.3, [4, XI.2]).
The author would like to thank S. Smith for his suggestions and encouragemen*
*t.
x2. The isotropy spectral sequence
In this section we discuss a well-known spectral sequence, called the isotro*
*py
spectral sequence, associated to an action of G on a space. This spectral seque*
*nce
will be linked to homology decompositions in x3. The isotropy spectral sequence*
* is
a special case of the Leray spectral sequence.
2.1 The Leray spectral sequence. Recall that the n-skeleton skn B (n 0) of
a simplicial set B is the subobject of B generated by all simplices of dimensio*
*n n.
Let f : X ! B be a map of simplicial sets, and let Xn denote f-1 (skn B). The
Leray spectral sequence of f is the (mod p) homology spectral sequence associat*
*ed
to the filtration
X0 X1 . . .Xn . . .
of X; it is usually indexed so that E1i;j= H i+j(Xi; Xi-1; Fp). Since Xn conta*
*ins
skn X, E1i;j= 0 for j < 0 and this is a first quadrant, strongly convergent hom*
*ology
spectral sequence. In particular, the differential dr has bidegree (-r; r - 1)*
*. If
M is a local coefficient system on X there is also a Leray spectral sequence wi*
*th
coefficients in M. Here are a few examples.
Collapse map. If Y is a subspace of X, the E2-term of the Leray spectral sequen*
*ce of
f : X ! X=Y vanishes except for the groups E20;0= H 0(X; Fp), E20;j= H j(Y ; F*
*p)
(j > 0) and E2i;0= H i(X=Y ; Fp)) (i > 0). The various differentials running f*
*rom
the horizontal axis to the vertical axis give the connecting homomorphisms in t*
*he
long exact homology sequence of the pair (X; Y ).
Fibration. If f : X ! Y is a fibration, the Leray spectral sequence of f can be
identified with the Serre spectral sequence of f.
2.2 Homotopy colimit. Suppose that D is some small category and F : D ! Spaces
is a functor. The unique natural transformation from F to the constant functor *
with value the one-point space induces a map
f : hocolim F ! hocolim * ~=|D|
The Leray spectral sequence of f can be identified from E2 onwards with the
Bousfield-Kan spectral sequence [4, XII.5.7]
E2i;j= colimiH j(F ; Fp) ) H i+j(hocolim F ; Fp) :
6 W. DWYER
This can be seen by inspecting the definitions of the two spectral sequences.
2.3 The isotropy spectral sequence. We will be interested in a specific Leray
spectral sequence associated to a G-space X.
Definition. The isotropy spectral sequence of a G-space X is the Leray spectral
sequence of the map
f : XhG = (E G x X)=G -! X=G
More generally, suppose that M is a module over Fp[G]. The isotropy spectral
sequence of X with coefficients in M is the Leray spectral sequence of f with
coefficients in the local system on XhG derived from M.
2.4 Remark. Consider the diagram
{B Gx}
??
y
(2.5) X - ---! XhG ---q-! (*)hG = BG
?
f?y
X=G
Suppose that M is a module over Fp[G]. The horizontal lineup forms a fibration,
and the Leray spectral sequence of q is the usual Serre spectral sequence conve*
*rg-
ing to H *(XhG ; M). The vertical lineup is meant to suggest that the geometric
fibres of f in general differ from point to point, and that the fibre over a si*
*mplex
x 2 X=G can be identified up to homotopy with B Gx, where Gx is the isotropy
subgroup of a simplex x 2 X above x . The Leray spectral sequence of f with
coefficients in M is the isotropy spectral sequence of X with coefficients in M*
*, and
it too converges to H *(XhG ; M). These spectral sequences are special cases of*
* the
hyperhomology spectral sequences from [5, XVII]. Let Cn*Y denote the normal-
ized Fp-chain complex of a simplicial set Y . For each i 0, CniY is the quot*
*ient
Fp[Yi]=(s0Fp[Yi-1] + . .s.i-1Fp[Yi-1]); the boundary map CniY ! Cni-1Y is ob-
tained by taking an alternating sum of the face maps dk : Fp[Yi] ! Fp[Yi-1],
0 k i and observing that this homomorphism passes to the necessary quotient
groups. By the K"unneth theorem, Cn*(XhG ) is naturally chain homotopy equivale*
*nt
to Cn*(E G) Fp[G]Cn*X. This tensor product has two filtrations, {Fn} and {Fn0},
given by
Fn = Cn*(skn EG) Fp[G]Cn*X and Fn0= Cn*(E G) Fp[G]Cn*(skn X) :
From E2-onwards, the spectral sequence associated to {Fn} is the Serre spectral
sequence of XhG ! BG, and the spectral sequence associated to {Fn0} is the isot*
*ropy
spectral sequence of X.
2.6 Examples. If G acts freely on X then the map f of 2.5 is an equivalence with
contractible fibres, and the isotropy spectral sequence of X collapses onto the*
* hor-
izontal axis. More generally, suppose that K is a normal subgroup of G which
SHARP HOMOLOGY DECOMPOSITIONS 7
acts trivially on X, and that the quotient group W = G=K acts freely on X.
Then the map f of 2.5 is a fibration with fibre B K, and the isotropy spectral
sequence of X can be identified with the Serre spectral sequence of the fibrati*
*on
B K ! XhG ! XhW .
2.7 The E1-term of the isotropy spectral sequence. The E1-term of the isotropy
spectral sequence has E1i;j= H j(G; CniX). For each j 0 one can form a chain
complex {H j(G; Cn*X); d} which in dimension i 0 contains the group Hj(G; CniX)
and has boundary map d induced by the boundary maps CniX ! Cni-1X. The group
in position E2i;j-term of the isotropy spectral sequence is then the i'th homol*
*ogy
group of {H j(G; Cn*X); d}.
2.8 An unnormalized description of the E2-term. Let C*Y denote the unnormalized
chain complex of a simplicial set Y . For each i 0, Ci(Y ) is the vector space*
* Fp[Yi],
and the boundary map Ci(Y ) ! Ci-1(Y ) is obtained by taking an alternating
sum of the face maps dk : Fp[Yi] ! Fp[Yi-1], 0 k i. By [10, 22.2], the
kernel of the epimorphism C*Y ! Cn*Y is a natural summand of C*Y and has
a natural chain contraction. Let X as above be a G-space. For each j 0 one
can form a chain complex {H j(G; C*X); d} which in dimension i 0 contains
the group H j(G; CiX) and has boundary map d induced by the boundary maps
CiX ! Ci-1X. It follows from the above remarks that the maps C*X ! Cn*X and
{H j(G; C*X); d} ! {H j(G; Cn*X); d} (j 0) induce isomorphisms on homology.
In particular, the group in position E2i;jof the isotropy spectral sequence for*
* X is
isomorphic to the i'th homology group of {H j(G; C*X); d}.
More generally, if M is an Fp[G]-module, it is possible to form chain comple*
*xes
{H j(G; M C*X); d} (j 0); here the tensor product is taken over Fp and the
action of G on M CiX is diagonal. The homology groups of these chain complexes
form the E2 term of the isotropy spectral sequence of X with coefficients in M.
x3. Homology decompositions and the isotropy spectral sequence
The goal of this section is to show that each of the homology decompositions
of x1 is tied to an associated G-space X, in such a way that the Bousfield-Kan
spectral sequence of the decomposition (1.2) is the isotropy spectral sequence *
*of X
(2.3). This will allow us to work with the Bousfield-Kan spectral sequence by
manipulating G-spaces.
3.1 A general construction. Let D be a small category and "Fa functor from D to
the category of transitive G-sets. Let X be the G-space hocolim "F. The assumpt*
*ion
that "F(d) is a transitive G-set means that there is an natural isomorphism
(3.2) X=G = (hocolim F")=G = hocolim(F"=G) ~=hocolim * = |D| :
Let F = "FhG. The values of F have the homotopy type of BH for various subgroups
H of G; in fact, F (d) has the homotopy type of B Gx, where Gx is the isotropy
subgroup in G of any element x 2 F"(d). The natural maps F"(d) ! * induce
maps F (d) ! (*)hG = B G which are compatible as d varies and give a map
hocolim F ! B G. If this map induces an isomorphism on mod p homology, then
F provides a homology decomposition of B G.
8 W. DWYER
Parallel to 3.2 is an isomorphism
XhG = (hocolim F")hG ~= hocolim(F"hG) = hocolim F
The natural map hocolim F ! |D| from 2.2 corresponds to the map f : XhG !
X=G from 2.3. It follows from 2.2 that the Bousfield-Kan spectral sequence of F
can be identified with the isotropy spectral sequence associated to the action *
*of G
on X; this is true either with trivial coefficients Fp or with coefficients in *
*a general
Fp[G]-module M.
3.3 Remark. We briefly recall that if D is a small category, the nerve |D| is t*
*he
simplicial set in which the n-simplices (n 0) are the chains
d0 -f1!d1 -f2!. .-.fn!dn
of composable morphisms in D. The face maps di : |D|n ! |D|n-1 act by dropping
f1 if i = 0, composing fi+1 with fi if 0 < i < n, and dropping fn if i = n.
The degeneracy maps si : |D|n ! |D|n+1 act by inserting appropriate identity
morphisms. A functor between two categories induces a map between their nerves;
a natural transformation between functors gives a homotopy between maps. If the
group H acts on D, then the fixed point set of the induced action on |D| is the
nerve of the fixed subcategory: |D| H = |DH |. One explanation for the usefulne*
*ss
of the nerve construction is that it is sometimes easier to manipulate functors*
* and
natural transformations than maps and homotopies.
Note by [6, 2.11] that the space X = hocolim F above is isomorphic to the ne*
*rve
of the category X whose objects consist of pairs (d; a) where d is an object of*
* D
and a 2 F (d); a map (d; a) ! (d0; a0) in X is a map f : d ! d0 in D such that
F (f)(a) = a0. The action of G on X is induced by the action on X (via functors)
given by g . (d; a) = (d; ga),
We now describe how the homology decompositions of 1.3 can be constructed by
the method of 3.1 above. (In fact, any homology decomposition can be constructed
in this way.) Let C be a collection of subgroups of G.
3.4 Building the centralizer decomposition. Let AC be the C-conjugacy category
(1.5) associated to C, and "ffCthe contravariant functor on ffC which assigns t*
*o a
pair (H; ) the set itself, i.e., the set of all monomorphisms H ! G contained *
*in
the conjugacy class . The group G acts transitively on "ffC(H; ) by conjugation.
If C is ample the functor ffC = ("ffC)hG gives as above the centralizer decompo*
*sition
of B G associated to C [6, 3.1]. Let XffCdenote hocolim "ffC. By 3.1, the Bousf*
*ield-
Kan homology spectral sequence associated to the centralizer decomposition is t*
*he
isotropy spectral sequence (2.3) of the action of G on XffC.
The space XffCis the nerve of the category XffC, whose objects consist of pa*
*irs
(H; i) where H is a group and i : H ! G is a monomorphism with i(H) 2 C. A
morphism (H; i) -! (H; i0) is a group homomorphism j : H ! H0 with i0j = i.
In order to make this a small category, the groups H should be restricted to li*
*e in
some set which is large enough to include up to isomorphism all of the elements
of C. The action of G on XffCis induced by the action of G on XffCgiven by
g . (H; i) = (H; gig-1 ). The n-simplices of XffCcorrespond to diagrams
(3.5) H0 -! H2 -! . .-.!Hn -! G
SHARP HOMOLOGY DECOMPOSITIONS 9
in which all of the maps are monomorphisms and all of the composite maps Hi ! G
have image contained in C.
3.6 Building the subgroup decomposition. Let OC be the orbit category (1.6) as-
sociated to C, and "fiCthe inclusion functor OC to the category of G-spaces. By
construction, G acts transitively on "fiC(G=H). According to [6, 3.2], if C is*
* am-
ple the functor fiC = (f"iC)hG gives a homology decomposition of B G; this is *
*the
subgroup decomposition provided by C. Let XfiCdenote hocolim "fiC. By 3.1, the
Bousfield-Kan homology spectral sequence associated to the subgroup decomposi-
tion is the isotropy spectral sequence of the action of G on XfiC. In [6] the s*
*pace XfiC
was denoted E C.
The space XfiCis the nerve of the category XfiCwhose objects consist of pairs
(x; G=H), where H 2 C and x 2 G=H. A morphism (x; G=H) ! (x0; G=H0) is a
G-map f : G=H ! G=H0 with f(x) = x0. The action of G on XfiCis induced by
the action of G on XfiCgiven by g . (x; G=H) = (gx; G=H).
3.7 Building the normalizer decomposition. Let sSC be the category of orbit sim-
plices (1.7) for the action of G on the simplicial complex KC, and "ffiCthe fun*
*ctor
which assigns to a particular object oeof sSC the set oeitself, considered as a*
* set
of simplices in KC. By construction, the group G acts transitively on "ffiC(oe*
*). Ac-
cording to [6, 3.3], if C is ample the functor ffiC = ("ffiC)hG gives a homolo*
*gy de-
composition of B G; this is the normalizer decomposition provided by C. Let sdX*
*ffiC
denote hocolim "ffiC. By 3.1, the Bousfield-Kan spectral sequence associated t*
*o the
normalizer decomposition is the isotropy spectral sequence of the action of G on
sdXffiC.
Let XffiCbe the nerve of the poset category XffiCwhose objects are the subgr*
*oups H
of G with H 2 C. In this category there is one morphism H ! H0 if H H0, and
no other morphisms. There is an action of G on XffiCinduced by the action of G *
*on
XffiCgiven by g . H = gHg-1 . Since sdXffiCis just the barycentric subdivision *
*of XffiC,
the Bousfield-Kan spectral sequence associated to the normalizer decomposition
can be identified with the isotropy spectral sequence of the action of G on Xff*
*iC. The
space XffiCis the simplicial set which corresponds to the (ordered) simplicial *
*complex
KC from 1.3.
3.8 Relationships among the three decompositions. There are G-equivariant
functors
XffCu-!XffiCv-XfiC
given by setting u(H; i) = i(H) G and v(x; G=H) = Gx. In light of 3.1, the
following proposition explains why the centralizer, subgroup, and normalizer di*
*a-
grams for a collection C act together when it comes to success or failure at be*
*ing
homology decompositions.
3.9 Proposition. The G-maps XffC|u|-!XffiC|v|-XfiCare weak equivalences.
3.10 Remark. The maps |u| and |v| are not necessarily weak G-equivalences (4.6).
This explains (4.8) why the centralizer, subgroup and normalizer diagrams can h*
*ave
different sharpness properties.
10 W. DWYER
Proof of 3.9. By the remarks in 3.3, it is enough to find functors u-1 : XffiC!*
* XffC
and v-1 : XffiC! XfiCsuch that the composites uu-1 , u-1 u, vv-1 and v-1 v
are connected to the appropriated identity functors by natural transformations.
To obtain these, set u-1 (H) = (H; i), where i : H ! G is the inclusion, and
set v-1 (H) = (eG=sg; G=H). Note (cf. 3.10) that u-1 and v-1 are not G-
equivariant.
x4. Bredon Homology
The previous two sections identify the Bousfield-Kan spectral sequence deriv*
*ed
from a homology decomposition of B G as the isotropy spectral sequence of an
associated G-space X. The E2-term of this isotropy spectral sequence is a kind
of homological functor of X, and in this section we describe some of its formal
properties. It is convenient to work in a setting slightly more abstract than t*
*hat of
x2-3.
4.1 Definition. A coefficient functor for G is a functor H from the category of
Fp[G]-modules to the category of Fp vector spaces which preserves arbitrary dir*
*ect
sums. If K is a subgroup of G, H|K denotes the coefficient functor for K given*
* by
H|K (A) = H(Fp[G] Fp[K] A).
4.2 Example. If M is an Fp[G]-module, there are associated coefficient functors
HMj given by HMj(A) = H j(G; M A). Here the tensor product is over Fp and
the G-action is diagonal. These are the coefficient functors we will intereste*
*d in,
particularly for the trivial module M = Fp. Note (by Shapiro's lemma) that if K
is a subgroup of G and H = H j(G; M -), then H|K = H j(K; M -).
4.3 Definition. Suppose that H is a coefficient functor for G and that X is a G-
space. Let (CG*(X; H); d) be the chain complex with CGn(X; H) = H(Fp[Xn]) and
with boundary d induced by the alternating sum of the face maps in X. The Bredon
homology groups of X with coefficients in H, denoted H G*(X; H), are defined to*
* be
the homology groups of CG*(X; H).
4.4 Example. Suppose that X is a G-space and that M is a module over Fp[G],
with associated coefficient functors HMj as in 4.2. The E2 term of the isotropy
spectral sequence for X with coefficients in M is then (2.8) given by
E2i;j= H Gi(X; HMj) :
Remark. It is possible to use coefficients for Bredon homology more general than
the functors we have allowed above; in fact, any functor from the orbit categor*
*y of
G to abelian groups can be used to construct a Bredon homology theory. We will
not use this greater generality.
4.5 Properties of H G*(X; H). The groups H G*(X; H) have a certain basic invar*
*i-
ance property.
4.6 Definition. A map f : X ! Y of G-spaces is said to be a weak G-equivalence
if fH : XH ! Y H is a weak equivalence for every subgroup H of G.
SHARP HOMOLOGY DECOMPOSITIONS 11
4.7 Remark. If X is a G-space, let Iso(X) denote the set of subgroups of G which
appear as isotropy groups of simplices of X. By [6, 4.1], a map f : X ! Y
of G-spaces is a weak G-equivalence if and only if it induces a weak equivalence
fH : XH ! Y H for all H 2 Iso(X) [ Iso(Y ).
4.8 Proposition. Suppose that f : X ! Y is a a weak G-equivalence and that H
is as in 4.1. Then f induces isomorphisms H G*(X; H) ~=H G*(Y ; H).
For completeness, we will sketch a proof. If (X; A) is a pair of G-spaces (i*
*.e., A is a
subspace of X), let CG*(X; A; H) denote the quotient complex CG*(X; H)=CG*(A; H)
and HG*(X; A; H) its homology. It is clear that there is a long exact sequence
relating HG*(A; H), HG*(X; H), and HG*(X; A; H).
Let K be a normal subgroup of G. A pair (X; A) is said to be relatively free
mod K if K acts trivially on the simplices of X not in A and G=K acts freely on
these simplices. Let R = Fp[G=K]. If (X; A) is relatively free mod K, then the
relative simplicial chain complex C*(X; A; Fp) is a chain complex of free R-mod*
*ules,
and there is an evident isomorphism CG*(X; A; H) ~=H(R) R C*(X; A). The next
lemma follows from basic homological algebra.
4.9 Lemma. Suppose that f : (X; A) ! (Y; B) is a map between pairs of G-spaces
which are relatively free mod K, and that f induces an isomorphism H *(X; A; Fp*
*) ~=
H *(Y; B; Fp). Then f induces an isomorphism HG*(X; A; H) ~=HG*(Y; B; H).
Proof of 4.8. Pick representatives {Ki}mi=0for the conjugacy classes of subgrou*
*ps of
G and label the representatives in such a way that if Ki is conjugate to a subg*
*roup
of Kj then i j. If Z is a G-space, write Z(n) for the subspace of Z consisting
of all z 2 Z such that Gz is conjugate to one of the groups Ki for i n. Let the
height of Z be the least integer n such that Z = Z(n). The proof of 4.8 will be
by induction on the heights of the spaces X and Y involved. The result is easy*
* to
check if G acts trivially on X and Y , i.e., if the heights of these spaces are*
* 0.
Assume by induction that the statement of 4.8 is true if the G-spaces involv*
*ed
have height n - 1. Suppose that X and Y are G-spaces of height n and that
f : X ! Y is a map which induces weak equivalences XH ! Y H for all subgroups
H of G. Let A = X(n-1), B = Y (n-1), K = Kn and N = NG (K). We must prove
that the map HG*(X; H) ! HG*(Y ; H) is an isomorphism.
It is easy to check that there is a map of pushout squares
G xN AK ----! A G xN BK ----! B
?? ? ? ?
y ?y -! ?y ?y
G xN XK ----! X G xN Y K ----! Y
and that in these squares the vertical arrows are monic. By 4.7 the map A ! B
is a weak G-equivalence, so by induction and a long exact sequence argument
it is enough to show that the map HG*(X; A; H) ! HG*(Y; B; H) is an isomor-
phism. Given the above diagram of squares, this is equivalent to showing that
the map HN*(XK ; AK ; H|N ) ! HN*(Y K; BK ; H|N ) is an isomorphism. Since the
maps AK ! BK and XK ! Y K are weak equivalences of spaces, the map
H *(XK ; AK ; Fp) ! H *(Y K; BK ; Fp) is an isomorphism. The desired result now
12 W. DWYER
follows from 4.9, since the N-space pairs (XK ; AK ) and (Y K; BK ) are relativ*
*ely
free mod K, because all of the simplices which are added in going from A to X or
from B to Y have isotropy group conjugate to K.
The Bredon homology groups also have a gluing property. Say that a commuta-
tive square
X0 ---u-! X
? ?
(4.10) f0?y f?y
Y 0---v-! Y
of G-spaces is a homotopy pushout square if for each subgroup H of G the induced
diagram of H-fixed subspaces is a homotopy pushout square of spaces. If 4.10 is
a pushout square of G-spaces and at least one of the maps u or f0 is monic, then
4.10 is a homotopy pushout square. In general, let W be the double mapping cone
of u and f0. Then 4.10 is a homotopy pushout square if and only if the natural
map W ! Y is a weak G-equivalence. Alternatively, 4.10 is a homotopy pushout
square of G-spaces if and only if for each subgroup H of G the induced square of
H-fixed-point sets is a homotopy pushout square of spaces.
4.11 Lemma. Suppose that 4.10 is a homotopy pushout square of G-spaces, and
that H is a coefficient functor for G. Then there is a long exact sequence
. .!.H Gi(X0; H) ! H Gi(X; H) HGi(Y 0; H) ! H Gi(Y ; H) ! H Gi-1(X0; H) ! . *
*. .
Proof. By 4.8 we can replace Y by the double mapping cone of u and f0 and thus
assume, for instance, that u is monic. The result then follows from the fact t*
*hat
there is a chain complex short exact sequence:
0 -! CG*(X0; H) ! CG*(X; H) CG*(Y 0; H) ! CG*(Y ; H) ! 0 :
x5. The transfer
All of our techniques for dealing with the G-spaces from x3 are based in one*
* way
or another on the transfer. We assume in this section that H is a coefficient f*
*unctor
(4.1), in practice a functor given by some homology construction (4.2).
5.1 The transfer. Suppose that f : S ! T is a map of G-sets. There is an
induced map Fp[S] ! Fp[T ], also denoted f, as well as a map
f* = H(f) : H(Fp[S]) ! H(Fp[T ]) :
Say that f is finite-to-one if for each x 2 T the set f-1 (x) is finite. For su*
*ch an f
therePis a G-map o (f) : Fp[T ] ! Fp[S], called the pretransfer, which sends x *
*2 T
to y2f-1(x)y. The induced map
o*(f) : H(Fp[T ]) ! H(Fp[S])
SHARP HOMOLOGY DECOMPOSITIONS 13
is the transfer associated to f.
5.2 Example. Suppose that H K are subgroups of G, let M be an Fp[G]-module
and let H be the functor H j(G; M -) (cf. 4.2). Suppose that f : G=H ! G=K is
the projection map. By Shapiro's lemma there are isomorphisms
H(Fp[G=H] ) ~=H j(H; M) H(Fp[G=K] ) ~=H j(K; M) :
Under these identifications, f* : H j(H; M) ! H j(K; M) is the map induced by
the inclusion H K and o*(f) : H j(K; M) ! H j(H; M) is the associated group
homology transfer map.
The transfer has the following basic properties, which are easy to verify by
calculations with pretransfers. Recall that H is assumed to commute with direct
sums.
5.3 Lemma. Suppose that f1 : S1 ! T1 and f2 : S2 ! T2 are maps of G-
sets. If f1 and f2 are finite-to-one, the so is f1 q f2 : S1 q S2 ! T1 q T2, a*
*nd
o*(f1 q f2) = o*(f1) o*(f2).
5.4 Lemma. Suppose that f1 : S1 ! T and f2 : S2 ! T are maps of G-sets. If
f1 and f2 are finite-to-one, the so is f1 + f2 : S1 q S2 ! T , and o*(f1 + f2) =
(o*(f1); o*(f2))
5.5 Remark. It follows from 5.2, 5.3, and 5.4 that if f : S ! T is a map of G-s*
*ets
which is finite-to-one, then o*(f) can be computed in terms of a sum of transfe*
*rs
associated to the projections G=Gx G=Gf(x), x 2 S. We will call such a transfer
the transfer associated to the inclusion Gx ! Gf(x).
5.6 Lemma. Suppose that f : S ! T and g : T ! R are maps of G-sets. If f and
g are finite-to-one then so is g . f, and o*(g . f) = o*(f) . o*(g).
5.7 Lemma. Suppose that
S0 ---s-! S
? ?
f0?y f?y
T 0---t-! T
is a pullback square of G-sets (i.e. a commutative diagram which induces an is*
*o-
morphism from S0 to the pullback of S and T 0over T ). Then if f is finite-to-o*
*ne,
so is f0, and o*(f) . t* = s* . o*(f0).
5.8 Definition. A map f : S ! T of G-sets is said to be an even covering mod p
if it is finite-to-one and the cardinality mod p of f-1 (x) does not depend on *
*the
choice of x 2 T . The common value mod p of these inverse image cardinalities *
*is
called the degree of f and denoted deg(f).
5.9 Lemma. Suppose that f : S ! T is a map of G-sets which is an even covering
mod p. Then the composite f* . o*(f) is the endomorphism of H(Fp[T ]) given by
multiplication by deg(f).
14 W. DWYER
5.10 Example. Suppose that H is a subgroup of G and that S is a G-set. The
action map a : G xH S ! S is finite-to-one and has degree given by the index of
H in G. Moreover, if S0 ! S is a map of G-sets, the diagram
G xH S0 ----! G xH S
? ?
a?y a?y
S0 ----! S
is a pullback square. It follows that the maps o*(a) give a natural map
H(Fp[S]) -o*(a)--!H(Fp[G H S])
on the category of G-sets. Moreover, the composite a* . o*(a) is the endomorphi*
*sm
of H(Fp[S]) given by multiplication by the index of H in G.
x6. Acyclicity for G-spaces
In this section we explicitly translate the question of whether a homology d*
*e-
composition of B G is sharp into a question about the Bredon homology the an
associated G-space X. We then study this second question. The symbol H denotes
a coefficient functor (4.1) for G.
6.1 Definition. A G-space X is said to be acyclic for H if the map X ! * induces
an isomorphism H G*(X; H) ! H G*(*; H).
6.2 Remark. Note that H Gi(*; H) vanishes for i > 0 and H G0(*; H) = H(Fp). Let
M be a module over Fp[G] and C a collection of subgroups of G. By x3 and 4.4,
the centralizer, subgroup and normalizer decompositions associated to C are sha*
*rp
for M (1.13) if and only if the G-spaces XffC, XfiC, and XffiC(respectively) ar*
*e acyclic
for the functors H i(G; M -), i 0.
We have three ways to show that a G-space X is acyclic for H.
6.3 The direct transfer method. This uses the fact that if K is a subgroup of G
of index prime to p then the transfer exhibits HG*(X; H) as a retract of HK*(X;*
* H|K ).
6.4 Theorem. Suppose that X is a G-space, H is a coefficient functor for G, and
K is a subgroup of G of index prime to p. If X is acyclic as a K-space for H|K ,
then X is acyclic as a G-space for H.
Proof. The transfers (5.1) associated to the maps q : GxK Xn ! Xn provide a map
t : CG*(X; H) ! CK*(X; H|K ) (4.1). By 5.10 this map commutes with differential*
*s,
and the index assumption implies that the composite CG*(X; H) -t!CK*(X; H|K ) -*
*q!
CG*(X; H) is an isomorphism. By naturality, then, the map H G*(X; H) ! H G*(*; *
*H)
is a retract of H K*(X; H|K ) ! H K*(*; H|K ), and the theorem follows from the*
* fact
that a retract of an isomorphism is an isomorphism.
6.5 The Mayer-Vietoris method. This gives a way to study G-spaces which
are constructed by gluing.
SHARP HOMOLOGY DECOMPOSITIONS 15
6.6 Theorem. Suppose that
X0 ---u-! X
? ?
f0?y f?y
Y 0---v-! Y
is a homotopy pushout square of G-spaces (4.8) and that either the space X or t*
*he
space Y is acyclic for H. Then the other member of the pair {X; Y } is acyclic*
* for
H if and only if the map f0 induces an isomorphism H G*(X0; H) ! H G*(Y 0; H).
Proof. This is clear from the exact sequence of 4.11.
6.7 The method of discarded orbits. This a more sophisticated version of the
direct transfer method which exploits the fact that K-orbits can be discarded if
they do not contribute to the transfer.
6.8 Theorem. Let X be a G-space, K a subgroup of G of index prime to p, and
Y a subspace of X which is closed under the action of K. Assume that Y is acycl*
*ic
for H|K , and that for each simplex x 2 X \ Y the transfer map H(Fp[G=Gx]) -!
H(Fp[G=Kx]) is zero (cf. 5.5). Then X is acyclic for H.
Proof of 6.8. The transfers associated to the maps q : G xK Xn ! Xn provide
a map t : CG*(X; H) ! CK*(X; H|K ) (4.1). By 5.10 this map commutes with
differentials, and the index assumption implies that the composite CG*(X; H) -t!
CK*(X; H|K ) -q!CG*(X; H) is an isomorphism. The transfer hypothesis shows that
the image of t is actually in the subcomplex CK*(Y ; H|K ) of CK*(X; H|K ). By
naturality, the homology map H G*(X; H) ! H G*(*; H) is a retract of H K*(Y ; H*
*|K ) !
H K*(*; H|K ), and the theorem follows from the fact that a retract of an isomo*
*rphism
is an isomorphism. .
6.9 Example. Let X be a G-space, K a subgroup of G of index prime to p, and Y
a subspace of X which is closed under the action of K. Suppose that M is a G-
module, and that Y is acyclic for the functors H j(K; M -), j 0. Assume final*
*ly
that for each x 2 X \ Y the transfer map H *(Gx; M) ! H *(Kx; M) is zero. In
light of 4.2, Proposition 6.8 implies that X is acyclic for the functors H j(G;*
* M -),
j 0.
A special case of the above is due to Webb [15] and Adem-Milgram [1, V.3].
6.10 Corollary. Let X be a G-space, P a Sylow p-subgroup of G, and M a module
over Fp[G]. Suppose that for every nonidentity subgroup Q of P the fixed point *
*space
XQ is contractible, and that for any x 2 X there is an element of order p in Gx
which acts trivially on M. Then X is acyclic for the functors H j(G; M -), j *
*0.
Proof. Let Y be the P -subspace of X consisting of simplices which are fixed b*
*y a
nonidentity element of P . By 4.7 the map Y ! * is a weak P -equivalence, and
so by 4.8 the space Y is acyclic for H j(P ; M -). Moreover, for any j 0 and
x 2 X \ Y the transfer map
H j(Gx; M) ! H j(Px; M) = H j({e}; M)
16 W. DWYER
is trivial. This is true for j > 0 becausePthe target group is zero, and true f*
*or j = 0
because by assumption the norm map g2Gxg : M ! M is trivial. The result
follows from 6.8 (cf. 6.9).
x7. Nontrivial p-subgroups
Let C be the collection of all nontrivial p-subgroups of G. Assume that C is
nonempty, i.e., that the order of G is divisible by p. In this section we will*
* show
that the three homology decompositions derived from C are sharp for suitable Fp*
*[G]-
modules M. Let P denote a Sylow p-subgroup of G. In all three cases we use the
method of Webb and Adem-Milgram (6.10) to show that the spaces XffiC, XfiC, and
XffCare acyclic for the coefficient functors H j(G; M -) (see 6.2).
The following fact is from [6, pf. of 6.5].
7.1 Lemma. Suppose Q is a p-subgroup of G and that M is an Fp[G]-module with
the property that the kernel of the action map G ! Aut (M) has order divisible
by p. Then there exists an element x in G of order p such that x commutes with Q
and x acts trivially on M.
7.2 The normalizer decomposition. Suppose that M is as in 7.1. We will show that
XffiCis acyclic for the coefficient functors H j(G; M - ). The first step is to*
* show that
for each nonidentity subgroup Q of P the space (XffiC)Q is contractible. By 3.*
*7 and
3.3, (XffiC)Q is the nerve of the full subcategory D of XffiCgenerated by the *
*objects
H of XffiC(equivalently, elements H 2 C) such that Q NG (H). The inclusions
H H . Q Q
provide a zigzag of natural transformations between the identity functor of D a*
*nd
the constant functor with value Q. The existence of this zigzag implies that |D*
*| is
contractible (3.3).
A typical n-simplex Q0 . . . Qn (Qi 2 C) of XffiChas isotropy subgroup
\iNG (Qi). The fact that there is an element of order p in this isotropy subgro*
*up
which acts trivially on M follows from 7.1. Now use 6.10.
7.3 The centralizer decomposition. Suppose that M is as in 7.1. We again show
that XffCis acyclic for the coefficient functors H j(G; M -). We first prove t*
*hat for
any nonidentity subgroup Q of P the space (XffC)Q is contractible. By 3.4 and 3*
*.3,
(XffC)Q is the nerve of the full subcategory D of XffCgenerated by the objects *
*(H; i)
with the property that Q CG (i(H)). Let Z be the center of Q and j : Z ! G
the inclusion. For an object (H; i) of D, let H0 denote the image of the produ*
*ct
map H x Z ! G and i0: H0 ! G the inclusion. The maps
(H; i) -! (H0; i0)- (Z; j)
give a zigzag of natural transformations between the identity functor of D and *
*the
constant functor with value (Z; j). As above, then, |D| is contractible.
The isotropy subgroup of a typical simplex (3.5) of XffChas the form CG (Q) *
*for
some Q 2 C. It follows from 7.1 that such an isotropy subgroup contains an elem*
*ent
of order p which acts trivially on M. Now use 6.10.
SHARP HOMOLOGY DECOMPOSITIONS 17
7.4 The subgroup decomposition. In this case we deal only with the trivial mod*
*ule
M = Fp, and show that XfiCis acyclic for the coefficient functors H j(G; -). As
above the first problem is to show that for any nontrivial subgroup Q of P the
space (XfiC)Q is contractible. By 3.6 and inspection (XfiC)Q is the nerve of *
*the full
subcategory D of XfiCgenerated by the pairs (x; G=H) with Q Gx. The category
D has (eQ; G=Q) as an initial element; in other words, for any object (x; G=H)
of D there is a unique map (eQ; G=Q) ! (x; G=H). These maps give a natural
transformation between the identity functor of D and the constant functor with
value (eQ; G=Q), which as above implies that |D| is contractible.
A typical simplex of XfiChas as its isotropy subgroup a group of the form Q *
*for
some Q 2 C. Any such isotropy subgroup contains an element of order p (which of
acts trivially on Fp). Now, again, use 6.10.
x8. Elementary abelian p-subgroups
In this section we prove several sharpness statements about collections of e*
*le-
mentary abelian p-subgroups of G.
Nontrivial elementary abelian p-subgroups. Let C be the collection of all
nontrivial elementary abelian p-subgroups of G. We will show that C is both both
centralizer-sharp and normalizer-sharp for any Fp[G]-module M such the the kern*
*el
of the action map G ! Aut (M) has order divisible by p. The arguments mimic
the ones in x7. Let P be a Sylow p-subgroup of G. In each case we use the method
of Webb and Adem-Milgram (6.10) to show that the spaces XffiCand XffCare acyclic
for the functors H j(G; M -).
The normalizer decomposition. We follow 7.2. The first step is to show that for
any nontrivial subgroup Q of P the space (XffiC)Q is contractible. By 3.7 and*
* 3.3,
(XffiC)Q is the nerve of the full subcategory D of XffiCdetermined by the obje*
*cts H
of XffiC(equivalently, elements H 2 C) such that Q NG (H). Let Z be the group
of elements of exponent p in the center of Q, and given an object H of D, let H0
be the group of elements of exponent p in the center of QH. The inclusions
H H \ H0 H0Z Z
give a zigzag of natural transformations between the identity functor of D and *
*the
constant functor with value Z. (See [6, 5.2] for the fact that H \ H0 is nontri*
*vial.)
By 3.3, |D| is contractible.
As in 7.2, the isotropy subgroup of any simplex of XffiCcontains an element *
*of
order p which acts trivially on M.
8.1 The centralizer decomposition. [8] We follow 7.3. The first step is to show*
* that
if Q is a nontrivial subgroup of P , then (XffC)Q is contractible. By 3.4 and*
* 3.3,
(XffC)Q is the nerve of the full subcategory D of XffCgenerated by objects (H;*
* i)
such that Q CG (i(H)). Let Z denote group of elements of exponent p in the
center of Q and j : Z ! G the inclusion. For an object (H; i) of D, let H0 deno*
*te
the image of the product map H x Z ! G and i0 : H0 ! G the inclusion. The
maps
(8.2) (H; i) -! (H0; i0)- (Z; j)
18 W. DWYER
provide a zigzag of natural transformations between the identity functor of D a*
*nd
the constant functor with value (Z; j). As above, |D| is contractible.
As in 7.3, the isotropy subgroup of any simplex of XffCcontains an element of
order p which acts trivially on M.
Smaller collections. If C is a collection of subgroups of G and K is a subgrou*
*p of
G, let C \ 2K denote the set of all elements of C which are subgroups of K. Cl*
*early
C \ 2K is a collection of subgroups of K. We are aiming at the following theor*
*em.
8.3 Theorem. Let K be a subgroup of G of index prime to p, and C a collection
of elementary abelian p-subgroups of G. If C \ 2K is centralizer-sharp (as a *
*K-
collection) then C is centralizer-sharp (as a G-collection).
8.4 Example. Theorem 8.3 can be used as a substitute for the argument of 8.1 in
showing that the collection C of all nontrivial elementary abelian p-subgroups *
*of G
is centralizer-sharp (for the trivial module Fp). To see this, let P be a Sylo*
*w p-
subgroup of G. It is enough to prove that the collection C0 = C \ 2P of all non*
*trivial
elementary abelian p-subgroups of P is centralizer-sharp as a P -collection. *
*We
can derive this from 4.8 by showing that XffC0is P -equivariantly contractible.*
* Let
j : Z ! P be the inclusion of the group of elements of exponent p in the center
of P . If (H; i) is an object of XffC0, let H0 denote the image of the product*
* map
H x Z ! P and i0: H0 ! P the inclusion. The maps
(8.5) (H; i) -! (H0; i0)- (Z; j)
provide a zigzag of natural transformations between the identity functor of Xff*
*C0and
the constant functor with value (Z; j). This zigzag respects the action of P on*
* XffC0,
and so gives an equivariant contraction of XffC0.
8.6 Example. [3] It is possible to do better than the above. Let P be a Sylow *
*p-
subgroup of G, and let Z be any nontrivial central elementary abelian p-subgroup
of P . Let C be the smallest collection of subgroups of G which contains Z and
has the property that if V 2 C and V commutes with Z then 2 C. An
argument virtually identical to the one in 8.4 shows that if C0 = C \ 2P , then*
* XffC0is
P -equivariantly contractible. It follows from 8.3 and 4.8 that C is centralize*
*r-sharp.
The proof of 8.3 depends on the following observation.
8.7 Lemma. Suppose that K is a subgroup of G and that V is an elementary
abelian subgroup of G not entirely contained in K. Then the transfer map
H*(CG (V ); Fp) ! H *(CG (V ) \ K; Fp)
associated to CG (V ) \ K ! CG (V ) (cf. 5.5) is zero.
Proof. Let C1 = CG (V ) \ K and C2 = CG (V ). Choose v 2 V with v =2 K, and
let C01~= C1 x be the subgroup of C2 generated by C1 and v. The inclusion
C1 ! C2 factors as the composite of f0 : C01! C2 with f : C1 ! C01, so the
transfer in question factors (5.6) as a parallel composite o*(f)o*(f0). However*
*, the
map o*(f) is zero; this follows from the fact that the map
f* : H *(C1; Fp) ! H *(C01; Fp)
SHARP HOMOLOGY DECOMPOSITIONS 19
is a monomorphism (f has a left inverse) and the fact that the composite o*(f) *
*. f*
is multiplication by p (5.9).
Proof of 8.3. Let X be the G-space XffC= |XffC|; we have to show that X is acyc*
*lic
for the functors H i(G; -). The strategy is to use the method of discarded orb*
*its
(6.7). Let Y be the full subcategory of XffCdetermined by the objects (H; i) su*
*ch
that i(H) is a subgroup of K, and let Y = |Y|, so that Y is a subspace of Xff*
*C.
The action of G on XffCrestricts to an action of K on Y , and it is clear that *
*Y is
equivalent as a K-space to XffC0, where C0 = C \2K . In particular, Y is by hyp*
*othesis
acyclic for the functors H i(K; -), i 0. Let x as in 3.5 be a simplex of X \ *
*Y ,
and let V be the image of Hn in G. Since V is not contained in K, Lemma 8.7
guarantees that the homology transfer map associated to the inclusion
Kx = CG (V ) \ K ! Gx = CG (V )
is zero. The desired result follows from 6.9.
x9. Pruning a collection
Let C0 be a collection of subgroups of G, K an element of C0, and C the coll*
*ection
obtained by deleting from C0 all conjugates of K. We say that C is obtained from
C0 by "pruning" the subgroup K. Our goal in this section is to obtain relations*
*hips
between the sharpness properties of C0 and those of C.
9.1 Definition. If H and K are two subgroups of G, then H is said to be compara*
*ble
to K if H K or H K.
9.2 Pruning the subgroup decomposition. Let linkfiC0(K) denote the full sub-
category of XfiC0given by the objects (x; G=H) such that Gx is comparable to K
but Gx 6= K. Denote the nerve of this category by linkfiC0(K). The action of G *
*on
XfiC0induces an action of NG (K) on linkfiC0(K).
Recall that if H is a group, EH denotes the universal cover of BH; in partic*
*ular,
E H is a contractible space on which H acts freely. We say that a map between
spectral sequences is an E2-isomorphism if it induces an isomorphism between E2-
terms.
9.3 Proposition. Suppose that C0 is a collection of subgroups of G, and that C*
* is
obtained from C0 by pruning K. Let N = NG (K) and W = N=K. Suppose that
either C or C0 is subgroup-sharp. Then the other collection of the pair {C; C0*
*} is
subgroup-sharp if and only if the projection E W x linkfiC0(K) ! E W (which is*
* a
map of N-spaces) induces an E2-isomorphism of isotropy spectral sequences.
The proof depends on a lemma. Let star fiC0(K) denote the full subcategory
of XfiC0given by the objects (x; G=H) with Gx comparable to K, and starfiC0(K)
its nerve. There is a natural action of NG (K) on starfiC0(K), and an equivari*
*ant
inclusion linkfiC0(K) starfiC0(K).
20 W. DWYER
9.4 Lemma. In the situation of 9.3 there is a homotopy pushout square (4.10) of
G-spaces
G xN (E W x linkfiC0(K)) ----! XfiC
? ?
(9.5) ?y ?y
G xN (E W x starfiC0(K)) ----! XfiC0
Proof. By examining simplices it is easy to check that there is a pushout squar*
*e of
G-spaces
G xN linkfiC0(K) ----! XfiC
?? ?
y ?y
G xN starfiC0(K) ----! XfiC0
which is in fact a homotopy pushout square because the left vertical arrow is m*
*onic.
To complete the proof of the proposition it is enough to show that the square
G xN (E W x linkfiC0(K)) ----! G xN linkfiC0(K)
?? ?
y ?y
G xN (E W x starfiC0(K)) ----! G xN starfiC0(K)
is also a homotopy pushout square. Here the horizontal arrows are induced by
projections. This is equivalent to the assertion that the square
E W x linkfiC0(K)----! linkfiC0(K)
?? ?
y ?y
E W x starfiC0(K)----! starfiC0(K)
is a homotopy pushout square of N-spaces. Let H be a subgroup of N and consider
the square of fixed point sets
(E W x linkfiC0(K))H----! linkfiC0(K)H
? ?
(9.6) ?y ?y
(E W x starfiC0(K))H----! starfiC0(K)H
If H is not contained in K, the spaces in the right hand column of 9.6 are empt*
*y,
and the left vertical map linkfiC0(K)H -! starfiC0(K)H is by inspection an is*
*omor-
phism. If H K, then H acts trivially on E W and the horizontal maps in 9.6
are equivalences. In either case, the square 9.6 is a homotopy pushout square *
*of
spaces.
Proof of 9.3. Suppose for concreteness that C is subgroup-sharp. By 6.2 and 6.6*
*, C0
is subgroup-sharp if and only if the left vertical map in 9.5 gives an E2-isomo*
*rphism
SHARP HOMOLOGY DECOMPOSITIONS 21
of isotropy spectral sequences. By Shapiro's lemma, though, the isotropy spectr*
*al
sequence of an N-space Y is naturally isomorphic to the isotropy spectral seque*
*nce
of the associated G-space G xN Y . It follows that C0 is subgroup-sharp if and
only if the map of N-spaces E W x linkfiC0(K) -! E W x starfiC0(K) induces an E*
*2-
isomorphism of isotropy spectral sequences. To complete the proof it is enough
to show that the map f : E W x starfiC0(K) ! E W gives an E2-isomorphism of
isotropy spectral sequences. We do this by proving that f is in fact a weak N-
equivalence (cf. 4.8). Let J be a subgroup of N. If J is not contained in K, th*
*en
both the domain and range of fJ are empty. If J K, then the range of fJ is the
contractible space E W , and the domain is E W x starfiC0(K)J, so that it is en*
*ough
to prove that starfiC0(K)J is contractible. The space starfiC0(K)J is the nerve*
* of the
subcategory D of starfiC0(K) consisting of pairs (x; G=H) such that J Gx. Let
F : D ! D be the functor given by the F (x; G=H) = (eK; G=K) if Gx K and
F (x; G=H) = (x; G=H) otherwise. The unique maps
(x; G=H) -! F (x; G=H)- (eK; G=K)
provide a zigzag of natural transformations connecting the identity functor of *
*D to
the constant functor with value (eK; G=K). By 3.3, |D| is contractible.
9.7 Pruning the centralizer and normalizer decompositions. Suppose as
above that C is obtained from C0 by pruning the subgroup K. Let linkffC0(K) den*
*ote
the full subcategory of XffCgiven by the objects (H; i) such that i(H) is compa*
*rable
(9.1) to K but i(H) 6= K. Let linkffiC0(K) denote the full subcategory of XffiC*
*given
by the objects H of XffiCsuch that H is comparable to K but H 6= K. Denote the
nerves of these categories by linkffC0(K) and linkffiC0(K) respectively; the ca*
*tegories
and their nerves are naturally furnished with actions of NG (K). We will not u*
*se
the following results and we omit the proofs; they are similar to the proof of *
*9.3.
9.8 Proposition. Suppose that C0 is a collection of subgroups of G, and that C*
* is
obtained from C0 by pruning K. Let N = NG (K), C = CG (K), and W 0= N=C.
Suppose that either C or C0 is centralizer-sharp. Then the other collection of*
* the
pair {C; C0} is centralizer-sharp if and only if the projection E W 0x linkffC0*
*(K) !
E W 0(which is a map of N-spaces) induces an E2-isomorphism of isotropy spectral
sequences.
9.9 Proposition. Suppose that C0 is a collection of subgroups of G, and that C*
* is
obtained from C0 by pruning K. Let N = NG (K). Suppose that either C or C0 is
normalizer-sharp. Then the other collection of the pair {C; C0} is normalizer-s*
*harp
if and only if the map linkffiC0(K) ! * (which is a map of N-spaces) induces an
E2-isomorphism of isotropy spectral sequences.
Remark. The map linkffiC0(K) ! * of 9.9 induces an E2-isomorphism of isotropy
spectral sequences if and only if the N-space linkffiC0(K) is acyclic for the c*
*oefficient
functors H j(N; -), j 0.
9.10 Fixed point sets of normalizers acting on links. Propositions 9.3, 9.8
and 9.9 motivate studying the action of N = NG (K) on the spaces linkfiC0(K),
linkffC0(K) and linkffiC0(K). Let J be a subgroup of NG (K); we indicate how *
*to
22 W. DWYER
identify up to homotopy the fixed point sets of the action of J on these links.
The space linkfiC0(K)J is the nerve of the full subcategory of linkfiC0(K) give*
*n by
the objects (x; G=H) with Gx J; the functor v from 3.8 gives an equivalence
between this nerve and the nerve of the poset of all subgroups H of G such that*
* H
is comparable (9.1) to K, H 2 C, and H J. The space linkffC0(K)J is the nerve
of the full subcategory of linkfiC0(K) given by the objects (H; i) such that i(*
*H) is
centralized by J; the functor u from 3.8 gives an equivalence between this nerve
and the nerve of the poset of all subgroups H of G such that H is comparable to
K, H 2 C, and H is centralized by J. (The proofs of these statements are similar
to the proof of 3.9.) The space linkffiC0(K)J is the nerve of the poset of all *
*subgroups
H of G such that H is comparable to K, H 2 C, and H is normalized by J.
x10. p-centric collections
In this section we show that various collections involving p-centric subgrou*
*ps of
G are subgroup-sharp. We leave it to the reader to formulate a more elaborate
result involving "M-centric" collections as in [6, x8]. It takes us two stages*
* to
reach the collection of all p-subgroups P of G such that P is both p-centric an*
*d p-
stubborn. We use an upward induction on the size of the omitted group to discard
all subgroups which are not p-centric, and then a downward induction to elimina*
*te
all subgroups which are not in addition p-stubborn. This parallels the argument*
* in
[6, x8]. The starting point is this.
10.1 Proposition. The collection C of all p-subgroups of G is centralizer-shar*
*p,
subgroup-sharp and normalizer-sharp.
Proof. In fact, these collections are sharp for any Fp[G]-module M. This is pro*
*ved
as in x7, but substituting the direct transfer method (6.4) for the method of W*
*ebb
and Adem-Milgram.
10.2 Elimination of non-p-centric subgroups. The main result here is the
following. Note that it is a consequence of the definition of "p-centric subgr*
*oup"
that if P is a p-centric subgroup of G and Q is a p-subgroup of G which contains
P , then Q is also p-centric. In other words, the collection of all p-centric s*
*ubgroups
of G is closed under passage to p-supergroups.
10.3 Theorem. Let C be a collection of p-subgroups of G which contains all p-
centric subgroups and is closed under passage to p-supergroups. Then C is sub-
group-sharp.
Proof. The proof is by downward induction on the size of C or equivalently upwa*
*rd
induction on the size of the p-subgroups of G omitted from C. The case in which*
* C
is the collection of all p-subgroups of G is covered by 10.1. Suppose then that*
* C is as
described in the statement of the theorem and C does not contain all p-subgroups
of G. Let P G be a p-subgroup of G which is maximal (under the inclusion
ordering) with respect to the property that P 2= C, and let C0 be the union of C
with the set of all conjugates of P , so that C is obtained from C0 by pruning *
*P . The
group P is not p-centric, and we can assume by induction that C0 is subgroup-sh*
*arp.
In order to prove that C is sharp, we will use 9.3 with K = P . Let N = NG (*
*P ),
W = N=P , and L = linkfiC0(P ). Since C0 contains no proper subgroups of P , t*
*he
SHARP HOMOLOGY DECOMPOSITIONS 23
group P itself acts trivially on L and the action of N on L factors through an
action of W . Let P(W ) be the collection of all nonidentity p-subgroups of W .*
* It
follows from the argument in [6, pf. of 8.3] that the space L is weakly W -equi*
*valent
to the space XfiP(W). Briefly, it is clear by inspection and [6, 5.4] that the*
* set of
isotropy subgroups of L is the set of nontrivial p-subgroups of W , and by [6, *
*2.15]
or a calculation with 9.10 that LQ is contractible for each nontrivial p-subgro*
*up Q
of W . The fact that L is weakly W -equivalent to XfiP(W)is then a consequence *
*of
[6, 4.3]. By 4.8, we can replace L = linkfiC0(P ) by XfiP(W)in checking the iso*
*tropy
spectral sequence condition from 9.3.
As in 2.6, comparing the isotropy spectral sequence of E W x XfiP(W)to that *
*of
E W amounts to looking at the pullback square
(B P x XfiP(W))hW ----! B PhW = BN
?? ?
y ?y
(XfiP(W))hW ----! (*)hW = BW
and comparing the Serre spectral sequences of the two vertical fibrations. In f*
*act,
these two spectral sequences are isomorphic at E2. Let M be one of the W -modul*
*es
H j(B P ; Fp), j 0. Since P is not p-centric, there is an element of order p *
*in W
which acts trivially on M (cf. [6, 8.9]). By x7, P(W ) is normalizer-sharp for *
*M; in
particular, P(W ) is ample for M, so that the map (XffP(W))hW ! BW induces an
isomorphism ~
H *((XffP(W))hW ; M) -=!H *(B W ; M) :
As M varies through all of the groups H j(B P ; Fp), these isomorphisms provide*
* the
required isomorphism of spectral sequence E2-terms.
10.4 Elimination of all non-p-stubborn subgroups. Let C0 be the collection
of all p-centric subgroups of G, and C C0 the collection of all subgroups which
are in addition p-stubborn. By 10.3 the collection C0 is subgroup-sharp, and we
wish to show that C is subgroup-sharp too. By 4.8 and 4.4, it is enough to show
that the map XfiC! XfiC0is a weak G-equivalence. Since the isotropy subgroups of
these two spaces are contained in C0, it is enough to check that the fixed poin*
*t map
(XfiC)P ! (XfiC0)P is a weak equivalence for each P 2 C0. This is done in [6,*
* 8.10]
by descending induction on the size of P .
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University of Notre Dame, Notre Dame, Indiana 46556
Processed June 21, 1996
E-mail address: dwyer.1@nd.edu