Contemporary Mathematics
MinamiWebb type decompositions for
compact Lie groups
John Martino and Stewart Priddy
1. Introduction
Let p be a fixed prime number. We extend to compact Lie groups
some stable classifying space decompositions of Minami [M ], following
Webb [W ]. One notable feature of [W ] is the use of a combinatorial
Möbius function to encode plocal information about the cohomology
of a finite group. We wish to show similar phenomena hold for com
pact Lie groups. However, for a compact Lie group G one is faced with
the problem of an infinite number of conjugacy classes of ptoral sub
groups, that is, extensions of tori by finite pgroups. These groups are
the analogs of pgroups for finite groups. We circumvent this problem
by considering a certain finite Gcomplex which allows us to introduce
combinatorial methods in the compact Lie group case. This complex
is based on the notion of pstubborn subgroups which arose earlier in
modular representation theory of finite groups (where they were called
pradical groups) in connection with Alperin's conjecture [A ], [Bouc ],
in group cohomology [W ], and in the study of homotopy classes of
maps between classifying spaces of compact Lie groups [JMO ]. We
also derive a decomposition based on the corresponding complex for el
ementary abelian psubgroups. Several examples are given to illustrate
the various decompositions.
2. Main Results
We begin by defining the Gsets used to construct our stable de
compositions. Let = (G) be the poset of nontrivial ptoral sub
groups P G with finite Weyl groups WG (P ) = NG (P )=P . Let
cO0000 (copyright holder)
1
2 JOHN MARTINO AND STEWART PRIDDY
( ) = Nerve( ) be the geometric realization of the nerve of
viewed as a category. Thus ( ) is a simplicial complex associated
with . The nsimplices oe = (P0, P1, ..., Pn) of ( ) are sequences of
inclusions P0 < P1 < ... < Pn of elements of . G acts on ( ) via
conjugation. Our goal is to replace ( ) by a finite Gcomplex which
can be used to study BG. We shall need a compact Lie group version
of seminal results of Quillen [Q ] (following Brown [Bwn ]) regarding
this complex and certain subcomplexes.
Throughout sums of classifying spaces will be taken in the Grothendieck
group of spectra under wedge sum.
The complex (S) of pstubborn subgroups:
Let S = S(G) (G) be the poset of pstubborn subgroups of G.
Thus P 2 S if and only if 1)P is ptoral with WG (P ) finite, 2) P =
Op(NG (P )), i.e., WG (P ) has no nontrivial normal psubgroups. For
finite groups homological properties of these subgroups were studied
by Bouc [Bouc ]. In the compact Lie group case JackowskiMcClure
Oliver showed that up to conjugacy in G, S is finite [JMO ]. Let
(S) = Nerve(S). As we shall see (S) has desirable homotopical
properties.
Lemma 2.1. The complex (S)=G is finite.
The proof is given in Section 5.
If oe = (P0, P1, ..., Pn) is a simplex of (S) then the isotropy sub
group
Gff= NG (P0) \ ... \ NG (Pn).
Theorem 2.2. Stably
X
BG^p' (1)dim(ff)(BGff)^p.
_ff2 (S)=G
Since (S)=G is a finite complex by Lemma 2.1, the summation is
finite. The proof of Theorem 2.2, given in Section 6, is a topological
variation of the proof of Webb's Theorem A [W2 ] for Mackey functors
from the category of finite Gsets to a category of modules.
MINAMIWEBB TYPE DECOMPOSITIONS 3
The complex (A) of elementary abelian psubgroups:
By considering elementary abelian groups we obtain another de
composition. Let A = A(G) be the poset of nontrivial elementary
abelian psubgroups of G and (A) = Nerve(A) the associated sim
plicial complex.
Lemma 2.3. The complex (A)=G is finite.
The proof is given in Section 5.
As in the case of finite groups we have
Lemma 2.4. If P is a nontrivial ptoral subgroup of G then the
fixed point complex (A)P is contractible.
Proof. The proof given in [Q , 4.4] works equally well for compact
Lie groups.
With this lemma the proof of Theorem 2.2 applied to (A) proves
Theorem 2.5. Stably
X
BG^p' (1)dim(ff)(BGff)^p.
_ff2 (A)=G
A Möbius function and MinamiWebb type formula:
Let Zp denote the padic integers. If G is a finite group, H G
then uH denotes the permutation module Zp ZpH ZpG viewed as an
element of the Green ring which is the Grothendieck group (over Q) of
finitely generated indecomposable ZpG modules [W ]. We recall that a
cyclic modp group is a extension of a finite pgroup by a finite cyclic
p0group. Let C(G) be the collection of all cyclic modp subgroups of
G. A Möbius function f : C(G) ! Z is defined recursively by
X
f(K) = 1.
J K2C(G)
Computation of f is facilitated by P. Hall's observation [Ha ] that f
vanishes except on intersections of maximal subgroups.
By Webb's formula [W , Theorem D']
X f(H)
(1) uG = _________ uH
H2C(G)*WG (H)
where the sum is taken a set of representatives of conjugacy classes
C(G)* of cyclic modp subgroups of G and WG (H) =: NG (H)=H, the
4 JOHN MARTINO AND STEWART PRIDDY
Weyl group of H. From this Minami derives a corresponding formula
for classifying spaces [M , Theorem 6.6]
X f(H)
(2) BG^p' _________ BH^p
H2C(G)*WG (H)
In this formula we may omit any p0groups of C(G) since completed at
p their classifying space contributes nothing to the sum. However as
we shall see in Examples 2 and 3 of Section 3 it is necessary to leave
these groups in formula (1).
In the general compact Lie group case let C(S) be the set of all
cyclic modp extensions of pstubborn subgroups. Thus H 2 C(S) if
and only if H contains a normal subgroup P 2 S such that H=P is a
finite cyclic p0group.
Since the conjugacy classes of S are finite and each element has finite
Weyl group it follows that C(S) has a finite number of conjugacy classes.
Let cJK denote the number of conjugates of K which contain J. By
Lemma 5.2, cJK is finite for J, K 2 C(S). Thus we can define a Möbius
function f : C(S)  ! Z recursively by
X
(3) cJK f(K) = 1
J K2C(S)*
for fixed J 2 C(S) with Op(J) 6= 1. Here C(S) *is a set of representa
tives for the conjugacy classes of C(S) .
Theorem 2.6. Stably
X f(H)
BG^p' _________ BH^p.
H2C(S)*WG (H)
Here we have tensored the Grothendieck group of spectra with the
rationals Q. The proof is given in Section 7.
3. Applications
In these examples all spaces are stable and completed at p = 2.
Example 3.1.
G = SO(3). Then up to conjugacy S consists of {O(2), V } where
V = O(1) x O(1) O(2) is an elementary abelian 2group of rank 2.
NG (O(2)) = O(2) and NG (V ) is the octahedral group isomorphic to
4, the symmetric group on four letters. This is easily checked from
the information on the conjugacy classes of subgroups of G given in
MINAMIWEBB TYPE DECOMPOSITIONS 5
[TD2 ]. Let D8 be the dihedral group of order 8. Then up to conjugacy
C(S) consists of {V, A4, D8, O(2)}:
SO(3)

6 I@
 @
 @
 @
 O(2) (1)
 
 6
 
 
 
 
(1) A4 
 
6 
 
 
 
 
 D8 (0)

 `



(3) V
where arrows represent inclusion and the values of the Möbius func
tion f are given in parentheses. These are computed inductively from
equation (3). Since Q = O(2), A4 are maximal, f(Q) = 1. By Lemma
5.2, cD8,O(2)= 1 hence
f(D8) + cD8,O(2)f(O(2)) = 1
implies f(D8) = 0. Similarly cV,O(2)= 3, cV,A4 = 1 hence B
f(V ) + cV,A4f(A4) + cV,O(2)f(O(2)) = 1
implies f(V ) = 3. By Theorem 2.6 we have
1
(4) BSO(3) ' BO(2) + __(BA4  BV )
2
Similarly the elementary abelian 2subgroups of G fall into two
conjugacy classes {V, C} where C V has order two.
Lemma 3.2. Let C0 V , V 0 O(2) be subgroups of SO(3) iso
morphic to C, V respectively. Then
1) C0 is N(V )conjugate to C.
2) V 0is O(2)conjugate to V .
6 JOHN MARTINO AND STEWART PRIDDY
Proof. 1) N(V ) = V oGL2(F2). Thus any two nontrivial involutions
of V are conjugate by an element which normalizes V .
2) V =< a, I2 . a > where
~ ~
1 0
a = 0 1
Let ~ ~
0 1
ø = 1 0
Since SO(2) does not contain an elementary abelian 2group of rank
2 we may assume V 0contains two generators a0, b0 2 O(2)1, the non
identity component of O(2). Thus V 0= where a0= ø x, b0= ø y
for some x, y 2 SO(2). However a0b0 = b0a0, hence xy1 = yx1.
Thus x = I2 . y and V 0= . Now since every nontrivial
involution of O(2)1 is O(2)conjugate to a and I2 is central, V 0is
O(2)conjugate to V .
By Lemma 3.2(1), (A)=G has a single 1simplex corresponding to
C V and two 0simplices corresponding to C, V . We have NG (C) =
O(2), NG (V ) = 4 and NG (C) \ NG (V ) = D8. Thus by Theorem 2.5
(5) BSO(3) ' BO(2) + B 4  BD8.
Using S and Lemma 3.2(2), Theorem 2.2 yields the same result.
These formulas are consistent with those of [MitchP ].
Example 3.3.
G = SU(2) = S3. There are two conjugacy classes of 2stubborn
subgroups, H = NG (S1) = and K = Q8. It is easy to check
that any pair of subgroups H0 K0 conjugate to H, K respectively
is simultaneously conjugate. (This type of argument is illustrated in
Lemma 3.2.) Thus (S)=G has a single one simplex corresponding to
K H and two zero simplices corresponding to H, K. Furthermore
NG (H) = H and_NG (K) is the binary octahedral group which is iso
morphic to 4, the twofold cover of 4 and N(H) \ N(K) = Q8. Thus
by the formula of Theorem 2.2
__
BS3 ' BNG (S1) + B 4 BQ8
Applying Theorem 2.6 we have the refinement
1
BS3 ' BNG (S1) + __B(Q8 o Z=3  BQ8)
2
MINAMIWEBB TYPE DECOMPOSITIONS 7
The interested reader can verify that this decomposition relates well to
that of BQ8 given in [MitchP ] i.e.,
1
1BS3=BNG (S1) ' __B(Q8 o Z=3  BQ8).
2
Example 3.4.
G = U(2) with standard maximal torus T . The center C T of
G, i.e., matrices of the form zI2, z 2 S1, is nontrivial so Theorem 2.5
does not yield a useful expression of BG. We could use Theorem 2.6.
However, more simply G=C = SU(2)=< I2> = SO(3) and C is in every
maximal subgroup. Therefore the Möbius functions for G and G=C
correspond. Hence we may pull back formula (4) to obtain
1
(6) BU(2) ' B(T o Z=2) + __[B(Q o Z=3)  BQ]
2
where
Q = ( S1 x Z=2) o Z=2,
i.e., Q is generated by the elements
~ ~
z 0 1
0 z , z 2 S
and the involutions a, ø defined in the proof of Lemma 3.2.
The action of Z=3 is given by conjugation with the element
~ ~
1 1  i 1  i
fi = __ .
2 1  i 1 + i
Another decomposition can be obtained from S which (up to con
jugacy) consists of {N(T ), Q} with Q N(T ). This follows from
Oliver's description of the pstubborn subgroups of the classical groups
[O ] where Q is denoted by U2.
Lemma 3.5. If Q0 N(T ) is conjugate to Q then Q0is T conjugate
to Q.
Proof. Since S1 = ZU(2), Q0= < (S1), a0, ø 0> where a0, ø 0cor
respond to a, ø under the given conjugation. Then a0, ø 0are non
commuting involutions. Since ø 02 N(T ) it has the form ø 0= ø ffl(z1, z2),
ffl = 0, 1.
Case 1: ffl = 1.
ø20= [ø (z1, z2)]2 = (z1z2, z1z2) = 1
8 JOHN MARTINO AND STEWART PRIDDY
Hence z2 = z11. Then ø 0= ø (z1, z11). However
(z1, 1)ø (z1, z11)(z11, 1) = ø (1, 1) = ø
Thus we may assume ø 0= ø . Now if a0 2 T , then a0 = a. If not
a0= ø (z1, z2), then arguing as above we have z2 = z11. Hence
1 = ø a0ø a0= (z12, z12)
Thus z1 = i and a0= (i, i)a. Thus either way Q0 is T conjugate to
Q.
Case 2: ffl = 0. We have 1 = det(ø 0) = z1z2 and 1 = ø 02=
(z1z2, z1z2). Hence z2 = z11, z1 = 1. Thus ø 0= a. In this case
a02= T since a0 and ø 0do not commute. Hence a0= ø (z1, z2) and so a0
is T conjugate to ø . This implies Q0 is T conjugate to Q.
By Lemma 3.5, (S) has one 1simplex and two 0simplices. Com
puting normalizers we have N(N(T )) = N(T ) = T o Z=2, N(Q) =
Q o 3 where the 3 action is generated by {ff, fi} with
~ ~
1 1 1
ff = ____p_ .
2 1 1
Then N(T ) \ N(Q) = Q o 2. Thus by Theorem 2.2 we have
(7) BU(2) ' B(T o Z=2) + B(Q o 3)  B(Q o 2)
We note that the same formula is obtained by pulling back equation
(5). This gives another description of Q o 3. Finally we note that
equation (7) transforms to equation (6) by simplifying B(Qo 3). This
is done by pulling back Webb's formula (1) for 3 to obtain
1
B(Q o 3) ' B(Q o 2) + __[B(Q o Z=3 ) + BQ].
2
Example 3.6.
G = SU(3). There are three conjugacy classes of 2stubborn sub
groups. By [O ] they are represented by the subgroups {T, NU(2)(T ), Q}
of U(2) defined in Example 3.4. We consider these as subgroups of
SU(3) by the usual monomorphism U(2) ! SU(3). Computing nor
malizers we find
NSU(3)(T ) = T o 3,
NSU(3)(NU(2)(T )) = NU(2)(T )
NSU(3)(Q) = Q o 3.
MINAMIWEBB TYPE DECOMPOSITIONS 9
From the inclusions T NU(2)(T ) Q we have
NSU(3)(T ) \ NSU(3)(NU(2)(T )) = T o 3
NSU(3)(NU(2)(T )) \ NSU(3)(Q) = Q o 2
By Lemma 3.5, Q NU(2)(T ) is unique up to conjugation (even
in U(2)), while T is the unique maximal torus of NU(2)(T ). Hence
(S)=G has exactly two 1simplices corresponding to T NU(2)(T ),
Q NU(2)(T ) and three 0simplices. Thus by Theorem 2.2
(8) BSU(3) ' B(T o 3) + B(Q o 3) + B(Q o 2).
Theorem 2.6 yields
1 1
(9) BSU(3) ' B(T oZ=2)+ __[B(T oZ=3)BT ]+ __[BQoZ=3BQ].
2 2
Alternatively, the conjugacy classes of elementary abelian 2subgroups
of G are represented by W of rank 2 generated by the diagonal matrices
with entries
(a, b, (ab)1), a, b = 1
and by Z of rank one generated by (1, 1, 1). Then NG (Z) = U(2),
NG (W ) = T o 3, NG (Z) \ NG (W ) = T o Z=2. Hence Theorem 2.5
gives
(10) BSU(3) ' BU(2) + B(T o 3)  B(T o Z=2).
We can simplify B(T o 3) by pulling back Webb's formula (1) for
3, as in Example 3.4, to obtain
1
B(T o 3) ' B(T o Z=2) + __[B(T o Z=3)  BT ]
2
Thus
1
BSU(3) ' BU(2) + __[B(T o Z=3)  BT ]
2
which combined with (6) yields another derivation of (9). Similarly
(10) combined with (7) gives (8).
4. A Ghomotopy equivalence of complexes
We shall need the following result in the proof of Theorem 2.2
Proposition 4.1. 1) Inclusion i : (S) ! ( ) is a Ghomotopy
equivalence.
2) Suppose P 2 is nontrivial ptoral subgroup then (S)P is
contractible.
10 JOHN MARTINO AND STEWART PRIDDY
Lemma 4.2. If P is a nontrivial ptoral subgroup then ( ) P is
contractible.
Proof. We adapt Quillen's method. Let Q 2 P then P
NG (Q). Then P Q NG (Q) is a compact Lie group which is a fi
nite extension of Q since WG (Q) is finite. Let S, T be the maximal
torus of P , Q respectively. Then S E P , T E Q since P , Q are ptoral.
Let T 0be a maximal torus of (P Q)0 which contains S. Since P Q is a
finite extension of Q , T 0= T . Thus ST = T and so S T . It follows
easily that ß =: P Q=T is a finite pgroup generated by Q=T and P=S.
Hence P Q is ptoral.
We claim P Q 2 P . Since P NG (P Q) it remains to show
WG (P Q) is finite. If not there is a torus T 00 NG (P Q) such that
T 00 P Q Then T 00normalizes P Q but acts trivially on ß since Aut(ß)
is finite. Thus T 00acts trivially on the quotient P Q=Q = ß=(Q=T ).
Thus T 00normalizes Q and hence T 00 Q since WG (Q) is finite. This
contradicts the existence of T 00proving the claim.
Thus we have P P Q Q in . This proves ( )P is conically
contractible by [Q , 1.5].
We recall a result of Th'evenazWebb and some generalizations.
From here through the proof of Proposition 4.5 we will identify a poset
with its geometric realization.
If Y is a Gposet then
Y y = {z 2 Y  z y }
Y y = {z 2 Y  z y }.
Theorem 4.3 (Th'evanazWebb). [TW , Th. 1] Let G be a group,
let X, Y be Gposets and let OE : X ! Y be a map of Gposets. Suppose
either
1) OE1(Y y) is Gycontractible for all y 2 Y
2) OE1(Y y) is Gycontractible for all y 2 Y
Then OE is a Ghomotopy equivalence.
Lemma 4.4. Let P 2 . Then >P is NG (P )contractible if and
only if P =2S.
Proof. This is [TW , Lemma 2.1]; the proof applies equally well
to compact Lie groups.
We shall also need the following generalization of another result of
[TW ] extended to infinite groups.
MINAMIWEBB TYPE DECOMPOSITIONS 11
T
Proposition 4.5. Let X Y be Gposets. Asssume X = Yi
where Y0 = Y and Yi+1 is obtained by deleting from Yi the minimal
elements of Yi X. If y 2 Y  X implies Y>y is Gycontractible then
the inclusion X ! Y is a Ghomotopy equivalence.
Proof. Let OEi : Yi+1 ! Yi be the inclusion. If y 2 Yi+1 then
OEi1((Yi) y) = (Yi+1) y which has y as a minimal element. Thus
(Yi+1) y is the cone on y which is Gy fixed. Hence (Yi+1) y is Gy
contractible. If y 2 Yi Yi+1 then OEi1((Yi) y) = Y>y since y and no
element above y is deleted when forming Yi+1. Y>y is Gy contractible by
hypothesis, thus OEi is a Ghomotopy equivalence by Theorem 4.3. Now
for each closed subgroup H there is the usual Milnor exact sequence
0 ! lim1 ß*+1(YiH ) ! ß*(XH ) ! lim ß*(YiH ) ! 0
Since OEi*: ß*(Yi+1H ) ! ß*(YiH ) is an isomorphism, the lim1 term is
zero and ß*(XH ) ! ß*(Y H) is an isomorphism and so X ! Y is a
Ghomotopy equivalence.
Proof of Proposition 4.1. 1) Let S0 = S [ pos where pos denote the
elements of of positive dimension. We will show the inclusions of
posets S0 and S S0 induce G homotopy equivalences.
(11) (S0) ! ( )
(12) (S) ! (S0)
We wish to apply Prop. 4.5. Let X = S0, Y = . Then the
elements ß 2 Y  X are finite pgroups By consideringTtheir order ß,
one sees that ß 2 Yi then ß pi. Thus X = Yi. Hence Lemma 4.4
and Prop. 4.5 imply (11) is a Ghomotopy equivalence. For (12) we
induct on dimension. Since this induction is finite the full strength of
Prop. 4.5 is not needed.
Part 2) follows from Part 1) and Lemma 4.2
5. Proof of Lemmas 2.1 and 2.3
The following result of Bredon [Brd , Cor. II 5.7] will be useful.
Lemma 5.1. Let K H G be compact Lie groups. Then the
orbit space
(G=H)K =WG (K)
of the right translation action of WG (K) is finite.
12 JOHN MARTINO AND STEWART PRIDDY
Lemma 5.2. For subgroups J, K G the number, cJK , of conju
gates of K which contain J is finite if WG (J) is finite. Moreover if
WG (K) is also finite then
cJK = (G=K)J=WG (K)
Proof. Suppose WG (J) is finite. By definition cJK = {g 2 G :
gKg1 J}=N(K). On the other hand (G=K)J = {gK : gKg1
J} = {g : gKg1 J}=K is finite since it has a finite number of
WG (J) orbits by Lemma 5.1. Thus cJK is finite. Since WG (K) acts on
(G=K)J with orbit space {g 2 G : gKg1 J}=N(K) it follows that
cJK = (G=K)J=WG (K)
if WG (K) is also finite.
Proof of Lemma 2.1: Consider a simplex oe = (Q1, Q2, ..., Qn), Q1 <
Q2 < ... < Qn. Since there are only finitely many conjugacy classes
of subgroups in S there are only a finite number of choices of Q1 up
to conjugacy. Since the number of Gconjugacy classes of S is finite,
Lemma 5.2 implies there are only a finite number of choices of Q2 which
contain Q1. This process is finite and terminates after the conjugacy
classes of S have been used.
Proof of Lemma 2.3: Since G is compact there is a bound d for the rank
of all maximal elementary abelian subgroups of G. Then the dimension
of (A) < d. Let (E0, E1, ..., E(d1)) be a simplex of highest dimension.
Since G has only finitely many conjugacy classes of elementary abelian
subgroups [Q2 , Lemma 6.3], it follows that in (A)=G the subgroup
E(d1)ranges over a finite set. Hence (A)=G is a finite complex.
6. Proof of Theorem 2.2
Let {Spectra} be the Grothendieck group of spectra completed at
p. Let {Gspace} be the category of Gspaces with a continuous left
Gaction and define a functor F : {Gspace} ! {Spectra} by F (X) =
1 EG+ ^G X+ . In what follows, as elsewhere in the paper, we work
stably and omit the symbol 1 for suspension spectrum. Then F is a
Mackey functor [W2 ] with restriction and induction given by
res#HK = F (i) : F (K) ! F (H)
ind"HK = transfer : F (H) ! F (K)
MINAMIWEBB TYPE DECOMPOSITIONS 13
where i : K ! H is an inclusion of closed subgroups of G. At this point
we formulate a special case of Webb's theorems the proof of which is
applicable to the case of compact Lie groups.
Theorem 6.1. [Webb] Let G be a finite group, M a Mackey func
tor, X and Y collections of subgroups of G closed under conjugation
and taking subgroups, and a finite Gcomplex. Suppose
1) For every simplex oe of the vertices of lie in distinct Gorbits.
2) For every subgroup H 2 X  Y , H is contractible.
3) A Sylow psubgroup Gp 2 X and M*(pr) : M(Gp x T ) ! M(T )
is a split surjection natural in T .
4) For every Y 2 Y, M(Y ) = 0. Then
M
M(G) _ (1)dim(ff)M(Gff)
ff2 =G
Proof. [W2 ] Theorem A.
Proof of Theorem 2.2: The homotopy category Ho{Spectra} is an
additive category and Theorem 6.1 applies even though it is stated for
Mackey functors with Rmodules as target. Addition of stable maps
gives the hom sets the structure of abelian groups and direct sum is
given by the wedge product.
In order to define terms let = (S) , X equal the set of all ptoral
subgroups of G, and Y = {1}. Hypothesis (1) was observed in [W2 ].
For (2) we note H 2 X implies H is contractible by Lemma 4.2. For
(3) we recall Ø(G=Gp) is prime to p. Thus the transfer for the (space
level) fibration
pr
G=Gp ! EGx G(Gp x T ) ! EGx GT
implies
pr* + +
F (Gp x T ) = EG+ ^G (Gp x T )+  ! EG ^G T = F (T )
is a natural split surjection as required. We note X is closed under
conjugation and under taking subgroups. Thus the proof of Theorem
6.1 applies.
14 JOHN MARTINO AND STEWART PRIDDY
7. Proof of Theorem 2.6
Let G be a compact Lie group_which_we initially assume has a
normal maximal torus_T_. Let C(G ) be the set of cyclic modp subgroups
of the finite group G =: G=T . Let WG (H) =: NG (H)=H, the Weyl
group of H.
Theorem 7.1. Suppose G has a normal maximal torus T . Then
stably
X f(H)
BG^p' _________ BH^p
__H2C(__G)*WG (H)
___
where H = H=T runs over_a set of representatives of the_finite set of
conjugacy_classes C(G )* of cyclic modp_subgroups of G and f(H) =:
f(H ) is the Möbius function f : C(G ) ! Z satisfying
X
f(H) = 1.
__J __H2C(__G)
__ __
for fixed J 2 C(G ).
Proof. By a slight variation on Feshbach's construction [F ] one
can construct a nested sequence of finite subgroups_Gk Gk+1 G
with normal subgroups Tk T such that Gk=Tk = G and
colim H*(BGk; Z=p) H*(BG; Z=p)
__ ___
Let ßk : Gk ! G be projection and set Hk = ßk1(H ). Let uH denote
the permutation module Zp ZpH ZpG. Then from [W , Theorem D'
and Lemma 7.1]
X f(H)
uGk = _________ uHk
__H2C(__G)*WG (H)
From this and (2) it follows directly that
X f(H)
(BGk)^p' _________ (BHk)^p
__H2C(__G)*WG (H)
Passing to the colimit over k gives the desired result since BH^p'
colim(BHk)^p.
Proof of Theorem 2.6: Let oe = (P0, P1, ..., Pn) 2 (S). Let T0 denote
the normal and hence unique maximal torus of P0. Since WG (P0) is
finite T0 is a maximal torus of NG (P0) which is also normal in NG (P0).
Since P0 \NG (Pi) NG (P0), T0 is a normal maximal torus of Gff=
MINAMIWEBB TYPE DECOMPOSITIONS 15
__
\NG (Pi). Let G ff= Gff=T0. As indicated above we can apply Theorem
7.1 to decompose BGff. Thus Theorem 2.2 yields
X X X fff(K)
BG^p' (1)dim(ff)(BGff)^p' (1)dim(ff) __________BK^p
_ff2 (S)=G _ff2 (S)=G __K2C(___Goe)*WGoe(K)
Collecting terms in this last expression we have
X X fff(K) X f(H)
(1)dim(ff) __________ BK^p= ________ BH^p
_ff2 (S)=G __K2C(___Goe)*WGoe(K) H2C(R)* W (H)
Thus we obtain a formula for f(H)
X X WG (K)
f(H) = (1)dim(ff) fff(K)__________
_ff2 (S)=G __K2C(___G)* WGoe(K)
H Goe K~GH oe
Lemma 7.2. Let J 2 C. If K H G then
a g
(G=K)J = (H=K)J
g2(G=H)J
.
Proof. This follows from the bundle H=K ! G=K ! G=H using
the usual homeomorphism _ : G xH H=K ! G=K of Gspaces given
by OE(g, hK) = ghK.
It remains to show f(H) satisfies the requisite formula of the the
orem. Computing we have
X
cJH f(H) =
J H2C(R)*
X X X WG (K)
(G=H)J=WG (H) (1)dim(ff) fff(K)__________
J H2C(R)* _ff2 (S)=G __K2C(___Goe)*WGoe(K)
H Goe K~GH
X X X (Gff=K)Jg
= (1)dim(ff) fff(K)____________
_ff2 (S)=G g2(G=Goe)JJg K2C(___G)* WGoe(K)
J Goe oe
by Lemma 7.2. Summing over
__oe2 (S)=G, J G
ff
16 JOHN MARTINO AND STEWART PRIDDY
is equivalent to summing over oe 2 (S)J and dividing by (G=Gff)J
which is finite. We note for use below that (S)J is thus also finite by
Lemma 5.1. Therefore the last expression becomes
0 1
X (1)dim(ff) X X (Gff=K)Jg
= __________J @ fff(K)____________A
ff2 (S)J(G=Gff) g2(G=Goe)J Jg K2C(___Goe)* WGoe(K)
For each g the expression in parentheses is 1 by the defining property
of fff. Thus the entire expression simplifies to
X
= (1)dim(ff)= Ø( (S)J) = 1
ff2 (S)J
by Proposition 4.1(1).
MINAMIWEBB TYPE DECOMPOSITIONS 17
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Department of Mathematics and Statistics, Western Michigan Uni
versity, Kalamazoo, MI 49008
Email address: martino@mathstat.wmich.edu
Department of Mathematics, Northwestern University, Evanston,
IL 60208
Email address: priddy@math.nwu.edu