HOMOTOPY FIXED POINT SETS AND ACTIONS ON
HOMOGENEOUS SPACES OF pCOMPACT GROUPS
by
Kenshi Ishiguro and HyangSook Lee
Abstract. We generalize a result of Dror Farjoun and Zabrodsky on the
relationship between fixed point sets and homotopy fixed point sets, which
is related to the generalized Sullivan Conjecture. As an application, we
discuss extension problems considering actions on homogeneous spaces of
pcompact groups.
Introduction.
For a group ß, if X is a ßspace, the homotopy fixed point set Xhß is the
set of ßmaps mapß (Eß, X). The fixed point set Xß embeds in Xhß as the
subspace of constant maps. In [6], Dror Farjoun and Zabrodsky consider the
relationship between Xß and Xhß , when ß is a finite group and X is a finite
ßsimplicial complex. Our work has been motivated by a result of theirs.
Namely, if ß is a finite pgroup, they show that Xß is an empty set if and
only if Xhß is empty. We observe that, when the finiteness condition on X
is replaced by a plocal one, the corresponding result still holds. Recall [23,
p557] that the mod p cohomological dimension of X, denoted by cdp(X),
means the supremum of the integer m such that there exists a sheaf F of
Z=pZmodules with Hm (X; F ) 6= 0. If X is the pcompletion of a finite
complex, then cdp(X) < 1.
Theorem 0. For a finite pgroup ß, suppose a ßspace X is Fpcomplete
and cdp(X) is finite. Then Xß is an empty set if and only if the homotopy
fixed point set Xhß is empty.
Our proof is analogous to the one given by Dror Farjoun and Zabrodsky
[6]. Their argument uses Miller's theorem (Sullivan Conjecture) [19], and we
use its pcompact group version [9]. Our result is related to the generalized
Sullivan Conjecture. Some results on this matter can be found in [4] and
[5].
Typeset by AM STEX
1
2
A pcompact group, [8], is a loop space X such that X is Fpfinite
and that its classifying space BX is Fpcomplete. The pcompletion of a
compact Lie group G is a pcompact group if ß0(G) is a pgroup. Suppose
a map i : X ____Y is a monomorphism of pcompact groups so that the
homotopy fiber Y =X of the delooped map BX ____BY is Fpcomplete
and Fpfinite. For a map f : Bß ____BY , there is an extension map
fe: Bß ____BX if and only if (Y =X)hß 6= ;, [8, 3.4 Actions on homogeneous
spaces].
BX
~fj3 Bi
j j ?
Bß _______fBY
We consider what follows if (Y =X)hß 6= ;, and what conditions guarantee
the existence of an extension map respectively.
Theorem 1. Let X ____Y be a monomorphism of pcompact groups and
let ß be a finite pgroup. Suppose the pcompact group X is abelian and
f : Bß ____BY is induced from a monomorphism. If there is an extension
map Bß ____BX, the group ß is abelian.
Theorem 2. Suppose V is an elementary abelian pgroup and T ____Y is
a maximal torus of a connected pcompact group Y with Weyl group W (Y ).
If the map BT ____BY induces H* (BY ; Fp) = H* (BT ; Fp)W (Y ), any map
f : BV ____BY has an extension map BV ____BT .
Let G be a compact Lie group and let H be a subgroup of G with inclusion
i : H ____G. In [6, Example D1], it is shown that, for a homomorphism
æ : ß ____G, the group æ(ß) is conjugate in G to a subgroup of H if and
only if Bæ : Bß ____BG lifts (up to homotopy) to Bß ____BH. Of course
the condition is equivalent to (G=H)æ(ß) 6= ;. We consider a generalization
of this result.
We recall that the Gaction on G=H is based on the following two actions.
A left action G x X ____X and a right action X x K ____X give us the G
space X=K. In the case of G=H, take X = G and K = H. In this particular
case, the composite map G ____G x X ____X deloops to BG ____BX, and
similarly BK ____BX, since the maps are induced by homomorphisms of
groups. We consider the case when X is a sphere S2n1 . The unitary
group U (n) acts on the sphere S2n1 from both left and right in the usual
3
way so that S2n1 = U (n)=U (n  1). Let a finite pgroup ß is a subgroup
of U (n) and K = S1 = U (1) ,! U (n). We note that S1 is the maximal
torus of the mod p finite loop space S2n1 when n divides p  1. If the
map ß ____(S2n1 )^p is a homomorphism of pcompact groups, Theorem
0 implies that the delooped map lifts to Bß ____B(S1)^p if and only if
(S2n1 =S1)ß 6= ;.
Theorem 3. If a finite pgroup ß which is a subgroup of U (n) acting on
S2n1 as above is abelian, the fixed point set (S2n1 =S1)ß is nonempty.
If X ____Y is a monomorphism of pcompact groups, and a finite pgroup
ß acts on both BX and BY , then (Y =X)hß is Fpcomplete. A result of [8]
implies that (Y =X)hß is Fpfinite. We consider the space (Y =X)hß which
is obtained by the induced fibration over Bß with fiber Y =X from a map
Bß ____BY .
Y =X=== = = = = =Y =X
 
? ?
Eß xß (Y =X) _______BX
 
? ?
Bß ____________BY
Theorem 4. Let (Y =X)hß be the homotopy fixed point set obtained as above
for a monomorphism X ____Y of pcompact groups. If (Y =X)h~ is Fpgood
for any subgroup ~ of the finite pgroup ß, then (Y =X)hß is Fpfinite.
Any nilpotent space is Fpgood, [3]. Assuming a nilpotency condition,
the second author [18] shows the Fpfiniteness of a finite complex with an
action of a pcompact toral group.
The authors would like to thank Fukuoka University and Ewha Womans
University for financial support. The first author would like to thank Em
manuel Dror Farjoun for his comments. The second author was supported
by KOSEF 97070102015, partially supported by the MOST through R
& D Program M1002204000401G050900310.
4
0. A plocal generalization of the result of Dror Farjoun and
Zabrodsky.
The argument in this section is very similar to the one used in [6]. It can
be considered as a plocal version.
A compact topological group G is said to have the extended homotopy
fixed point property (EHFPP) if for every Fpcomplete Gspace X with
cdp(X) < 1, one has XG = ; if and only if XhG = ;. If G has no
EHFPP, that is, there exist an Fpcomplete Gspace X with cdp(X) < 1
and XG = ;, and an equivariant map EG ____X, we say that G is extended
compressible.
The following is an extended version of [6, Theorem B], and to show that
(iii) implies (i), the result of [13, p45 Cor. 1] can be used as mentioned in
[6].
Proposition 0.1. For an elementary abelian pgroup V , suppose a V 
space X is Fpcomplete and cdp(X) is finite. Then the following are equiv
alent.
(i)XV 6= ;
(ii)XhV 6= ;
(iii)The classifying map Ø : EV xV X ____BV induces a monomorphism
on mod p cohomology.
The following is a plocal version of [6, Lemma 2.1]
Lemma 0.2. Suppose the kernel H of an epimorphism G ____G0 is a com
pact Lie group such that the loop space (BH)^p is a pcompact group. Let
X be an Fpcomplete and Fpfinite G0complex. Then the natural compo
sition mapG0 (EG0, X) ____mapG (EG, X) is a weak equivalence of spaces.
Proof. The argument is very similar to the one used in [6]. Notice that
mapG (EG, X) = mapG (EG=H, X) = mapG (BH, X), where H has a triv
ial action on X and EG is a free contractible Hspace. We consider the
diagram
mapG (EG, X) _______=mapG (EG=H, X)
O6E =
 _ ?
mapG0 (EG0, X) _______mapG (BH, X)
To show OE is a weak equivalence, it suffices to show that _ is a weak equiv
alence. Taking the full function space from _, we get ~_: map(EG0, X) ____
map(BH, X). According to the result of [9, Theorem 9.3] we can see that
5
map(BH, X) ' X, since (BH)^p is a pcompact group and X is Fp
complete and Fpfinite. Now _~ is a Gmap, and a weak homotopy equiv
alence when the Gactions are ignored. Therefore _~ induces a homotopy
fixed point equivalence. Since EG0 and BH = EG=H are free G0space, we
see that mapG0 (BH, X) ' mapG0 (EG0, map(BH, X)) ' mapG0 (EG0, X)
by [6, Lemma 2.2]. Consequently ~_must be a weak G0equivalence. There
fore _ is weakly equivalent.
It is wellknown that, for a compact Lie group K, the loop space (BK)^p
is pcompact if ß0(K) is a pgroup. The problem on the conditions of a
compact Lie group that its loop space of the pcompleted classifying space
be a pcompact group is considered in [15] and [16]. A result says that if
(BK)^p is a pcompact group, then ß0K must be pnilpotent.
Let G be a group. Recall [12] that the Frattini subgroup of G is the
intersection of maximal subgroups of G, denoted by (G). Let G0 = G= G
be Frattini factor group of G. Using Lemma 0.2, the argument of the proof
of [6, Theorem C] is applicable for the following result.
Proposition 0.3. A finite pgroup G is extended compressible if and only
if its Frattini factor group G0 is extended compressible.
Proof of Theorem 0. Since the Frattini factor of finite pgroup is an ele
mentary abelian pgroup, the desired result is immediate from Proposition
0.1 and Proposition 0.3.
Let G be a pcompact toral group. It is known [8, Proposition 6.9] that
any pcompact toral group G has a discrete approximation f : G1 ! G.
For the discrete toral group G1 , there exists an increasing chain Gn
Gn+1 . . .of finite subgroups of G1 such that G1 = [i n Gi. Then we
have the following result.
Corollary 0.4. Suppose X is an Fpcomplete space with the proxy action
of a pcompact toral group G and cdp(X) < 1. If G1 acts on X and
XGi = ; for some finite psubgroup Gi of G1 , then XhG is empty.
Proof. Since XhG1 is equivalent to the homotopy inverse limit of the tower
{XhGi } i n , if XGi = ; for some Gi then XhGi = ; by Theorem 0. This
implies XhG1 = ;. According to [8, Proposition 6.8], the discrete approxi
mation f induces a homotopy equivalence XhG ! XhG1 . Therefore XhG
is empty.
Corollary 0.5. Let G1 be a pdiscrete toral group. Suppose G1 space X
is Fpcomplete and cdp(X) < 1. Then XhG1 = ; if and only if XG1 = ;.
Proof. It suffices to show that XhG1 6= ; implies XG1 6= ;. So let XhG1 6=
;. Then there is m such that XhGi 6= ; for all i m. Theorem 0 says
XGi 6= ; for all i m. Therefore XG1 6= ;.
6
Let Xhß denote the Borel construction so that Xhß = Eß xß X. Assume
X satisfies the condition of Theorem 0. According to [8, Theorem 7.4]
together with Theorem 0, we immediately conclude that Xhß is Fpfinite if
and only if the fixed point set X~ is empty for any subgroup ~ of the finite
pgroup ß of order p.
1. Extension problems and the mod p cohomology.
In this section we consider extension problems, and prove Theorem 1 and
Theorem 2. Some results of mod p cohomology of classifying spaces will be
used.
Proof of Theorem 1. Recall that any abelian pcompact group is equiva
lent to the product of a pcompact torus and a finite abelian pgroup, so
that BX = (BG)^p for a compact abelian Lie group G, [9] and [21]. Thus
the extension map f~ : Bß ____BX is induced by a group homomorphism
æ : ß ____G, since ß is a finite pgroup, [11]. It is enough to show that this
group homomorphism æ is injective. Since f : Bß ____BY is a monomor
phism, the induced homomorphism f * : H* (BY ; Fp) ____ H* (Bß; Fp) is
finite, [8, Proposition 9.11]. This means that H* (Bß; Fp) is a finitely gen
erated module over f *(H* (BY ; Fp)). Consider the following commutative
diagram
H* (BX; Fp)
f~i* i i
ii) i 6
H* (Bß; Fp) oe_____f*H* (BY ; Fp)
Since f~*(H* (BX; Fp)) contains f *(H* (BY ; Fp)), we see that H* (Bß; Fp)
is a finitely generated module over f~*(H* (BX; Fp)), and therefore f~* =
((Bæ)^p)* is finite. So a result of Quillen [23] implies that the kernel of æ is
trivial.
An argument analogous to the one used here shows the following result:
Theorem 1.1. Let X ____Y be a monomorphism of pcompact groups and
let ß be a finite pgroup. Suppose BX = (BG)^p for a compact Lie group
G and f : Bß ____BY is induced from a monomorphism. Assume there is
an extension map Bß ____(BG)^p so that this map is induced from a group
homomorphism æ : ß ____G. Then æ is injective.
Proof of Theorem 2. Since H* (BY ; Fp) = H* (BT ; Fp)W (Y ), using a result
of [2] one can show that there is a homomorphism OE : H* (BT ; Fp) ____
H* (BV ; Fp) which makes the following diagram commutative over the
Steenrod algebra:
7
H* (BT ; Fp)
OiEi i
ii)i 6
H* (BV ; Fp) oe_____f*H* (BY ; Fp)
Here OE factors through the polynomial part of H* (BV ; Fp). We note that
H* (B(Z=p)n ; Fp) = Fp[x1, . .,.xn ] (y1, . .,.yn ) for odd prime p where
each yi has dimension 1 and each xi = fiyi has dimension 2. Since V
is an elementary abelian pgroup, a result of [17] implies that there is a
homomorphism æ : V ____T such that OE = (Bæ)*, and the following diagram
is commutative:
BT
Bæ j3 
j j ?
BV _______fBY
This completes the proof.
For a connected compact Lie group G, we note that, for instance, if p is
odd and G is ptorsion free, then H* (BG; Fp) is isomorphic to the invariant
ring H* (BTG ; Fp)W (G) , where TG is a maximal torus, and W (G) denotes
the Weyl group. Analogous results hold for connected pcompact groups X
when H* (X; Z^p) is torsion free, [10] and [22]. Next we recall that SO(3)
contains Z=2 Z=2 as a subgroup. Considering Theorem 2 when p = 2 and
Y = SO(3)^2, we notice that H* (BSO(3); F2) 6~= H* (BS1; F2)Z=2, and that
there is no extension for the monomorphism B(Z=2 Z=2) ____BSO(3)^2,
since rank(SO(3)) = 1. Generally, we see that if rank(G) < 2rank(G),
then H* (BG; F2) is not isomorphic to the invariant ring H* (BTG ; F2)W (G) .
2. Actions on homogeneous spaces and fixed point sets.
We recall that a left action GxX ____X and a right action X xK ____X
give the Gspace X=K. Let [x] = xK 2 X=K. For g 2 G, the action is given
by g . [x] = [gx]. If [x0] 2 (X=K)G , for any g we can find k 2 K such that
gx0 = x0k. In the case G = U (n), X = S2n1 and K = S1 = U (1) ,! U (n)
as mentioned in the introduction, the equation of gx0 = x0k is expressed
in the matrix form. For n = 2, for instance, the expression is given by the
following:
` ' ` ' ~ ` ' ~T
a11 a12 x1 z 0
a21 a22 . x2 = ( x1 x2) . 0 1
8
where AT denotes the transpose of a matrix A.
Proof of Theorem 3. Since ß is abelian, all the irreducible representations
of ß have degree 1. Consequently there is oe 2 U (n) such that oe1 ßoe
is a subgroup of T n, where T n is the maximal torus of U (n) consisting
of diagonal matrices. Let g 2 ß and let g0 = oe1 goe. If g0x = xk for
some x 2 S2n1 , then gx0 = x0k where x0 = oex. It remains to find
such x 2 S2n1 . Since g0 is a diagonal matrix, it is easy to see that if
x = (1, 0, . .,.0) 2 S2n1 , then, as seen below,
0 i1 0 1 0 1 1 2 0 i 1 3T
1 0
BB i2 CC BB0 CC 66 BB 1 CC77
@ ... A . @ ...A= 4 (1 0 . . . 0 ). @ ... A 5
0 in 0 0 1
for any g0 there is k 2 S1 such that g0x = xk.
In Theorem 2, taking Y = (S2n1 )^p when n divides p  1, we obtain
((S2n1 =S1)^p)hV 6= ;, and Theorem 0 says ((S2n1 =S1)^p)V 6= ;. Note
that, in general, if ß is a finite pgroup and X is a finite ßcomplex, then
Xß 6= ; if and only if (X^p)ß 6= ;. This result follows from the following
diagram
Xß _________(X^p)ß
 
? ?
(Xß )^p _______(X^p)hß
and the result (Xß )^p' (X^p)hß , [20, Theorem 2]. Consequently it follows
that (S2n1 =S1)V 6= ;, which is a special case of Theorem 3 assuming the
map V ____(S2n1 )^p is a homomorphism of pcompact groups.
Next we consider the nonabelian case. Suppose Q8 denotes the quater
nion group in SU (2);
Q8 =< a, b  a4 = 1, a2 = b2, bab1 = a1 >
where
` ' ` '
a = i0 0i , b = 01 10
9
Taking x = (1, 0) as the base point of S3, the composite map Q8 ____
U (2) ____U (2) x S3 ____S3 is a homomorphism of groups. A direct calcu
lation shows (S3=S1)Q8 = ;. This result can be obtained from Theorem 1,
since Q8 is nonabelian. We have the following generalization.
Proposition 2.1. For n 2, let æ : ß ____U (n) be an irreducible repre
sentation for a nonabelian finite pgroup ß. Then (S2n1 =S1)ß = ;.
Proof. The center of a nontrivial finite pgroup contains more than one
element. Since the representation æ is irreducible, Schur's lemma tells us
that we can find a 2 ß such that æ(a) is a diagonal matrix
0 i 0 1
BB i CC
@ ... A
0 i
where i is a pth primitive root of unity. If (S2n1 =S1)ß 6= ;, then an
argument analogous to the one used in our proof of Theorem 3 shows that
the following equation should be satisfied:
0 i 01 0 z1 1 2 0 i 0 1 3T
BB i CC BBz2 CC 66 BB 1 CC77
@ ... A . @ ...A = 4 (z1 z2 . . . zn ). @ ... A 5
0 i zn 0 1
for suitable (z1, z2, . .,.zn ) 2 S2n1 . Consequently it follows that zi = 0
for i = 2, . .,.n. Now let x0 = (z1, 0, . .,.0). Using the equations gx0 =
x0k for all g 2 æ(ß), we see that all entries of the first column except
the (1, 1)entry of each matrix g are zero. This means that there would
be a 1dimensional invariant subspace. This is a contradiction, since the
representation is irreducible and n 2. Therefore (S2n1 =S1)ß = ;.
We have seen that the Gaction on G=H is based on the following two
actions: GxX ____X and X xK ____X. In the case X = G, the composite
of the based maps G ____G x X ____X deloops to BG ____BX. Here we
consider the deloopability problem for G = U (n) and X = S2n1 or SU (n).
Let _ : U (n) x S2n1 ____S2n1 be the U (n)action on S2n1 . For this
action, we will show that the pcompleted map (U (n))^p ____(S2n1 )^p is
not deloopable for any prime p.
10
_
Proposition 2.2. The composite map U (n) ____U (n) x S2n1 _____S2n1
is not deloopable at any prime p when n 2.
Our proof for the case n 3 will use admissible maps, [1]. The case
n = 2 is, however, treated separately. This is a special case of the following
U (n)action on SU (n). The action ~ : U (n) x SU (n) ____SU (n) is given
by 0 1
1 0
B ... C
~(A, B) = A . B . B@ CA
1
0 det A1
for A 2 U (n) and B 2 SU (n). This action is transitive, and the isotropy
subgroup at the identity is isomorphic to U (1).
Proposition 2.3. The composite map U (n) ____U (n) x SU (n) _____~SU (n)
is not deloopable at any prime p.
Proof. There is a finite covering Z=n ____ SU (n) x S1 _____qU (n), where
qSU(n) is the inclusion SU (n) ,! U (n) and q(S1) is the center of U (n). If
the map U (n) ____SU (n) is deloopable at p, we obtain a map induced from
the composition
(BSU (n))^px (BS1)^p____(BSU (n))^p
The axis (BSU (n))^p____(BSU (n))^pis the identity map. According to [14,
Theorem 1], the other axis (BS1)^p ____(BSU (n))^p should factor through
(BZ=n)^p, where Z=n is the center of SU (n). This means that the map
(BS1)^p ____(BSU (n))^p would be a zero map. This is a contradiction,
since the map S1 ____SU (n) is a monomorphism. Thus we obtain the
desired result.
Lemma 2.4. The two U (2)spaces SU (2) and S3 are U (2)homeomorphic.
Proof.iAjhomeomorphism ø : SU (2) ____ S3 is given by a map sending
a ~b to a where a, b 2 C with a2 + b2 = 1. The desired result
b ~a b
follows from the following commutative diagram:
U (2) x SU (2) _______~SU (2)
1xø  ø 
? _ ?
U (2) x S3 ___________S3
This completes the proof.
Proof of Proposition 2.2. The case n = 2 is proved by Proposition 2.3
and Lemma 2.4. So we assume n 3. If the map U (n) ____S2n1 was
11
deloopable at p, we would have a map BU (n)^p ____B(S2n1 )^p. Notice
that the Lie group SU (n) is simple, and rank(SU (n)) 2. According to
[1, Proposition 2.12], the restriction of the delooped map on BSU (n)^p is
null homotopic.
BU (n)^p _______B(S2n1 )^p
6 *

BSU (n)^p
On the other hand, the restriction of U (n) ____S2n1 on U (1) is not null
homotopic.
BU (n)^p _______B(S2n1 )^p
6 6
 
BU (1)^p _________idB(S1)^p
Consequently the map BSU (n)^p____B(S2n1 )^p should be essential, since
U (1) ,! SU (2) ,! SU (n). This contradiction completes the proof.
3. Some properties of homotopy fixed point sets (Y =X)hß .
Suppose that X ____Y is a homomorphism of pcompact groups, that a
finite pgroup ß acts on classifying spaces BX and BY , and that BX ____
BY is a ßmap. According to [8, Lemma 10.6 and Proposition 5.8], if
BY hß 6= ;, then (Y =X)hß 6= ; and the space is Fpcomplete. If the map
X ____Y is a monomorphism, then Y =X is Fpfinite. We see, by [8, Theo
rem 4.6], that (Y =X)hß is Fpfinite.
As mentioned in the introduction, next we consider the space (Y =X)hß
which is obtained by the induced fibration over Bß with fiber Y =X from a
map Bß ____BY , [8, Lemma 10.4].
Y =X=== = = = = =Y =X
 
? ?
Eß xß (Y =X) _______BX
 
? ?
Bß ____________BY
A space X is said to be Fpgood, [3], if H*(X; Fp) ____H*(X^p; Fp)
induced from the Fpcompletion map X ____X^p is an isomorphism. For
instance, it is known [3, Ch VII Proposition 5.1] that if the fundamental
group ß1X is finite, then X is Fpgood for any prime p.
12
Proof of Theorem 4. Recall [8, Remark 11.13] that a space is Fpcomplete
if and only if X is both Fplocal and Fpgood. Since Y =X is Fplocal, so is
(Y =X)h~ for any subgroup ~ of the finite pgroup ß. From our assumption,
we see that each (Y =X)h~ is Fpcomplete. Since the map X ____Y is a
monomorphism of pcompact groups, the space Y =X is Fpfinite. Conse
quently [8, Theorem 4.6] implies that (Y =X)hß is Fpfinite.
As a special case, we notice [11, Lemma 2.3] that if the homotopy fiber
Y =X is nilpotent and mod pacyclic, then (Y =X)h~ is also nilpotent and
mod pacyclic for any finite psubgroup ~ of ß. Since any nilpotent space is
Fpgood, each (Y =X)h~ is Fpgood. This implies that (Y =X)hß is Fpfinite.
References
1. J.F. Adams AND Z.Mahmud, Maps between classifying spaces, Inventiones Math.
35 (1976), 141.
2. J.F. Adams and C.W. Wilkerson, Finite Hspaces and algebras over the Steenr*
*od
algebra, Ann. of Math. 111 (1980), 95143.
3. A. Bousfield and D. Kan, Homotopy limits, completions and localisations, SL*
*NM
304 (1972).
4. C. Broto and S. Zarati, Nillocalization of unstable algebras over the Stee*
*nrod algebra,
Math. Z. 199 (1988), 525537.
5. C. Broto and S. Zarati, On subA*palgebras of H*V , Proc. of 1990 Barcelon*
*a Conf.,
SLNM 1509 (1992), 3549.
6. E. Dror Farjoun and A. Zabrodsky, Fixed points and homotopy fixed points, C*
*om
ment. Math. Helv. 63 (2) (1988), 286295.
7. W.G. Dwyer and C.W. Wilkerson, Smith theory and the functor T , Comment. Ma*
*th.
Helv. 66 (1) (1991), 117.
8. W.G. Dwyer and C.W. Wilkerson, Homotopy fixedpoint methods for Lie groups *
*and
finite loop spaces, Ann. of Math. 139 (2) (1994), 395442.
9. W.G. Dwyer and C.W. Wilkerson, The center of a pcompact group, The C~ech
centennial (Boston, MA, 1993), Contemp. Math., AMS 181 (1995), 119157.
10. W.G. Dwyer, H. R. Miller and C.W. Wilkerson, Homotopical uniqueness of clas*
*sify
ing spaces, Topology 31 (1) (1992), 2945.
11. W.G. Dwyer and A. Zabrodsky, Maps between classifying spaces, Proc. of 1986
Barcelona conference, LNM 1298 (1987), 106119.
12. D. Gorenstein, Finite groups, second edition, Chelsce (1980).
13. W.Y. Hsiang, Cohomology theory of topological transformation groups, Sprin*
*ger
Verlag, New YorkHeidelberg (1975).
14. K. Ishiguro, Classifying spaces and homotopy sets of axes of pairings, Proc*
*. of AMS
124 (1996), 38973903.
15. K. Ishiguro, Toral groups and classifying spaces of pcompact groups, Conte*
*mp.
Math., 271, Amer. Math. Soc. (2001), 155167.
16. K. Ishiguro, Classifying spaces and a subgroup of the exceptional Lie group*
* G2,
Contemp. Math., 274, Amer. Math. Soc. (2001), 183193.
13
17. J. Lannes, Sur la cohomologie modulo p des pgroupes Abeliens elementaires,*
* öH 
motopy Theory, Proc. Durham Symp. 1985, Cambridge Univ. Press (1987), 9711*
*6.
18. H.S. Lee, Homotopy fixed point set for pcompact toral group, Bull. Korean*
* Math.
Soc. 38 (1) (2001), 143148.
19. H. Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of*
* Math.
120 (2) (1984), 3987.
20. H. Miller, The Sullivan conjecture and homotopical representation theory, P*
*roceed
ings of the ICM (Berkeley, Calif., 1986) (1987), 580589.
21. J. Mfiller and D. Notbohm, Centers and finite coverings of finite loop spac*
*es, J. reine
angew. Math. 456 (1994), 99133.
22. D. Notbohm, Spaces with polynomial modp cohomology, Math. Proc. Cambridge
Philos. Soc. 126 (2) (1999), 277292.
23. D. Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of *
*Math. 94
(2) (1971), 549572, 573602.
Fukuoka University, Fukuoka 8140180, Japan
email: kenshi@ cis.fukuokau.ac.jp
Ewha Womans University, Seoul, Korea
email: hsl@ mm.ewha.ac.kr