PAIRINGS OF pCOMPACT GROUPS AND
HSTRUCTURES ON THE CLASSIFYING
SPACES OF FINITE LOOP SPACES
by
Kenshi Ishiguro
In [8], the author investigated certain pairing problems for classifying
spaces of compact Lie groups. The main work in this paper can be regarded
as a pcompact group version. DwyerWilkerson [3] defined a pcompact
group and studied its properties. The purely homotopy theoretic object
appears to be a good generalization of a compact Lie group at the prime p. A
pcompact group has rich structure, such as a maximal torus, a Weyl group,
etc. A note of Moller [12] summarizes their work. Further development on
the homotopy theory of pcompact groups can be seen, for example, in
[4], [13] , [14] , [2] and [17]. We first recall some basic things about the
pcompact groups and pairing problems, and then state our main results.
A pcompact group, [3], 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. For
an odd dimensional sphere S2n1 , it is known that its pcompletion has a
loop structure if n divides p  1. This is an example of pcompact groups
other than compact Lie groups. More examples are known as ClarkEwing
pcompact groups, [12, x2].
For pcompact groups X and Y , a pointed map f : BX ____BY is
called a homomorphism. Let Y =X denote the homotopy fibre of f . The
homomorphism f is called a monomorphism if Y =X is Fpfinite, and an
epimorphism if the loop space (Y =X) is a pcompact group.
The centralizer of f is the loop space of the component containing f of the
mapping space of unpointed maps, denoted by map(BX; BY )f . A homo
morphism is called central if the evaluation map, ev : map(BX; BY )f ____
BY , is a homotopy equivalence. According to [4], any pcompact group X
has a unique maximal central subgroups that is called the center of X and
denoted by C(X). It is also shown in [4] that BC(X) ' map(BX; BX)id
where id : BX ____BX is the identity homomorphism.
Typeset by AM STEX
1
2
Next we recall pairing problems for pcompact groups and compact Lie
groups , [8] and [16]. Suppose that X, Y and Z are pcompact groups,
and that ff : BX ____BZ and f : BY ____BZ are homomorphisms. The
homotopy class of ff is said to be contained in the set of the homotopy classes
of axes f ?(BX; BZ) if there is a map (called a pairing) : BX x BY ____
BZ with restrictions (axes) BX ' ff and BY ' f . In other words, if
ff 2 f ?(BX; BZ), we have the following homotopy commutative diagram:
BY
 H H f
? HHj
BX x BY _______BZ
6 ff*

BX
We note that f ?(BX; BZ) is a subset of the homotopy set [BX; BZ]. For
a weak epimorphism f of the classifying spaces of connected compact Lie
groups, the set of the homotopy classes of axes has been determined in [8].
In this paper we will obtain analogous results for pcompact groups.
In [9], for connected compact Lie groups L and G, a map BL ____BG
or BL^p ____BG^p is called a weak epimorphism, if there exists a fibration
F ____BL ____BG or F ____BL^p ____BG^p such that H* (F ; Q) is a
finite dimensional Qmodule or that H* (F ; Z^p)Q is a finite dimensional
Q^pmodule, respectively. The second condition of the following theorem
requires a similar assumption for a homomorphism of connected pcompact
groups f : BY ____BZ. By the way, the connectivity is not assumed in the
first condition.
Theorem 1. Suppose X is a pcompact group. If either
(i)f : BY ____BZ is an epimorphism of pcompact groups, or
(ii)f : BY ____BZ is a homomorphism of connected pcompact groups
such that H* ((Z=Y ); Z^p) Q is a finite dimensional Q^pvector
space
then the following hold:
(1) If ff 2 f ?(BX; BZ), then the map ff factors through the classifying
space of the center of Z, denoted by C(Z), up to homotopy.
(2) Moreover, we have f ?(BX; BZ) = [BX; BC(Z)] .
Here we make a remark analogous to the one in [8]. Taking Y = Z and
f = id , our problem asks possible BXactions on BY . A consequence
3
of Theorem 1 shows that such an action under ff exists if and only if the
orbit map ff : BX ____BY is central. We see, for instance, that there
are no nontrivial BXactions on B(S2n1 )^p for n 3, since the center
C((S2n1 )^p) is contractible.
A connected pcompact group Y is called semisimple if ss1(Y ) is finite,
[13]. In this case, the center C(Y ) is a finite abelian pgroup, [14]. If X is
connected and Y is semisimple, the homotopy set [BX; BC(Y )] is trivial.
Consequently, there are likewise no nontrivial BXactions on BY .
Furthermore, if we take X = Y = Z and f = ff = id, the problem now
asks whether BX is an Hspace. A pairing : BX x BX ____BX could
be the Hmultiplication. Before stating our result, recall that a pcompact
group X is called abelian if ev : map(BX; BX)id ____BX is a equivalence.
Any abelian pcompact group is equivalent to the product of a pcompact
torus and a finite abelian pgroup, [4] and [14]. Corollary 2 stated in x2
implies that BX is an Hspace if and only if X is abelian. This result holds
when a pcompact group X is replaced by a finite loop space.
Theorem 2. Suppose X is a finite loop space. If its classifying space BX
is an Hspace, then X is equivalent to the product of a torus and a finite
abelian group.
The above result is a generalization of Corollary 2.4 in [8]: If G is a
compact Lie group and BG is an Hspace, then G is an abelian group.
Theorem 3 in x2 will give the pcompleted version of this result. Namely, if
(BG)^p is an Hspace, then G is pnilpotent in the sense of [6]. The group
G need not be abelian. We can find, however, an abelian compact Lie group
A such that (BG)^p' BA.
1. Mapping spaces and Proof of Theorem 1.
We will prove Theorem 1 in this section. To do so, we need a few basic
results about pcompact groups. The following lemma translates a setting
of groups to a homotopy setting of pcompact groups.
Lemma 1. Suppose j : BX ____BY and q : BY ____BZ are homo
morphisms of pcompact groups. If the composite map q . j is a homotopy
equivalence (isomorphism), then j is a monomorphism and q is an epimor
phism.
4
Sketch of Proof. We sketch the proof. From our assumption, one can show
that Y ' (Z=Y ) x Z and (Z=Y ) ' Y =X. Thus Y =X is Fpfinite, and
(Z=Y ) is a pcompact group. Therefore j is a monomorphism and q is an
epimorphism.
We recall [3, Theorem 9.7] that if a pcompact group X is connected,
the cohomology algebra H* (BX; Z^p) Q is a polynomial ring over Q^pcon
centrated in even degree. The number of the generators of the polynomial
algebra is called rank of X and denoted by rank(X). If n = rank(X),
it is known that the maximal torus of X is equivalent to (BT n)^p. *
* It
is also known that H* (BX; Z^p) Q is isomorphic to the invariant ring
(H* (BT n; Z^p) Q)W (X) , where W (X) is the Weyl group of X.
Proposition 1. Suppose either
(i)X, Y and Z are pcompact groups, i : BX ____BZ is a monomor
phism and f : BY ____BZ is an epimorphism, or
(ii)X, Y and Z are connected pcompact groups, i : BX ____BZ is a
monomorphism and f : BY ____BZ is a homomorphism such that
H* ((Z=Y ); Z^p) Q is a finite dimensional Q^pvector space.
If there is a map (extension) fe: BY ____BX with f ' i . ef,
BX
efj3 i
j j ?
BY _______fBZ
then BX is equivalent to BZ under the map i.
Proof. First assume the condition (i). It suffices to show that i : BX ____
BZ is an epimorphism. Recall that f : BY ____BZ lifts to fe if and
only if the homotopy fixed point (Z=X)hY is nonempty, [3, x 3.3]. Since
f : BY ____BZ is an epimorphism, by definition, the loop space (Z=Y )
is a pcompact group. Let U = (Z=Y ) so that BU ____BY ____BZ is a
fibration of pcompact groups. Then (Z=X)hY is homotopy equivalent to
((Z=X)hU )hZ . Notice here that the action of U on Z=X is trivial. Since the
Sullivan conjecture for pcompact groups holds, [4, Theorem 9.3], we see
(Z=X)hU ' Z=X. Consequently (Z=X)hY ' (Z=X)hZ . This means that
(Z=X)hZ is nonempty, and therefore the identity map 1BZ : BZ ____BZ
lifts to a map r : BZ ____BX so that i . r ' 1BZ .
5
BX
r j3 i
j j ?
BZ _______1BZBZ
From Lemma 1 the monomorphism i is also an epimorphism. Hence i is an
isomorphism.
Next assume the condition (ii). Since H* ((Z=Y ); Z^p) Q is finite
dimensional, we see that H* (Z=Y ; Z^p)Q is a finitely generated polynomial
algebra, and hence we have
H* (BY ; Z^p) Q ~= (H* (Z=Y ; Z^p) Q) (H* (BZ; Z^p) Q)
Thus we can find a homomorphism (left inverse) of polynomial algebras
r : H* (BY ; Z^p) Q ____H* (BZ; Z^p) Q with r . f *= id. Consequently
r . ef *. i* = id, since f ' i . ef. Hence i* is injective.
We claim that i* is surjective and hence this homomorphism is bijective.
It's enough to show that the composition ' = i* . r . ef *is bijective.
H* (BX; Z^p) Q _______'H* (BX; Z^p) Q
fe* 6i*
? 
H* (BY ; Z^p) Q _______rH* (BZ; Z^p) Q
Since i : BX ____BZ is a monomorphism and i* is injective, we see
rank(X) = rank(Z). Hence the Krull dimension of the image of ' is
equal to rank(X). Thus, at each degree, ' is an injective linear selfmap
of a finite dimensional Q^pvector space, and therefore this linear map is
bijective.
Consequently the monomorphism i is a rational isomorphism. According
to [13, Lemma 2.5(1)], we see that BX is equivalent to BZ under the map
i.
Proof of Theorem 1. (1) : We will show that if ff 2 f ?(BX; BZ), the
composite map BX _____ffBZ ____B(Z=C(Z)) , say qff, is null homotopic.
BC(Z)

?
BX _________ffBZ
H H
qffHHj ?
B(Z=C(Z))
6
Using a result of Moller [13, Theorem 6.1], it's enough to prove that qff. ' 0
for any homomorphism : BZ=pn ____BX and any n 1. Since ff 2
f ?(BX; BZ), according to [8, Proposition 1.1], we see ff . is contained in
f ?(BZ=pn ; BZ). So f factors through map(BZ=pn ; BZ)ff., which is the
classifying space of the centralizer of ff . . A result of DwyerWilkerson [3],
[12, Theorem 5.1] shows that map(BZ=pn ; BZ)ff. is a pcompact group
and ev : map(BZ=pn ; BZ)ff. ____BZ is a monomorphsim, since Z=pn is a
pcompact toral group. If : BXxBY ____BZ is a pairing with restrictions
(axes) BX ' ff and BY ' f , then the map f : BY ____BZ is expressed
as the following composition:
__
BY _______map(BZ=pn ; BZ)ff.
P P P
f P PPq ?ev
BZ
where __ is induced by the adjoint map. In fact, for any y 2 BY , we see
ev O __(y) = __(y)(*) = ((*); y) ' f (y). Since ev is a monomorphsim, by
the assumption of f , Proposition 1 implies:
map(BZ=pn ; BZ)ff. ' BZ
Thus ff . is central. Hence the map qff : BX ____B(Z=C(Z)) is null
homotopic. Consequently, the map ff : BX ____BZ factors through BC(Z).
(2) : Using [4, Theorem 9.3], one can show that the map of homotopy
sets
[BX; BC(Z)] ____[BX; BZ]
is injective, since its kernel [BX; Z=C(Z)] is trivial. The image of the map is
included in f ?(BX; BZ). We have just seen in part (1) that [BX; BC(Z)]
maps onto f ?(BX; BZ). Consequently, f ?(BX; BZ) = [BX; BC(Z)].
As seen in [8, Proposition 1.1], there is a strong relationship between
pairing problems and mapping spaces. The following result shows that, for
the homomorphism f : BY ____BZ in Theorem 1, no pcompact groups
find a difference between BC(Z) and map(BY; BZ)f . The proof uses the
uniqueness of the pairing in our case.
Corollary 1. Let f : BY ____BZ be as in Theorem 1. For any pcompact
group X, the map of homotopy sets
[BX; BC(Z)] ____[BX; map(BY; BZ)f ]
7
is bijective, where the above map is induced by the canonical map
BC(Z) = map(BZ; BZ)id ____map(BY; BZ)f :
Proof. First notice that there is a map
j : [BX; map(BY; BZ)f ] ____f ?(BX; BZ)
induced by adjoints. In fact, a map BX ____map(BY; BZ)f induces a
pairing BX x BY ____BZ, and one of its axes is contained in f ?(BX; BZ).
Thus we get the following commutaive diagram:
[BX; BC(Z)] _______[BX; map(BY; BZ)f ]
P P P
P PPq ?j
f ?(BX; BZ)
By [4, Lemma 5.3], for ff 2 f ?(BX; BZ), there is a unique pairing :
BX x BY ____BZ with BX ' ff and BY ' f . Hence j is bijective.
Theorem 1 shows [BX; BC(Z)] ____f ?(BX; BZ) is bijective. Therefore
the desired result holds.
Remark. This result seems to indicate that map(BY; BZ)f can be homo
topy equivalent to BC(Z) for such an f . For instance, if map(BY; BZ)f
were shown to be a pcompact group, the statement would be true. When
f : BY ____BZ is an epimorphism, a result of DwyerWilkerson [4, Lemma
10.3] implies map(BY; BZ)f ' BC(Z).
2. Hstructures on the classifying spaces.
In this section we will prove Theorem 2 using the following result, which
is an easy consequence of Theorem 1.
Corollary 2. Suppose X is a pcompact group. If BX is an Hspace, then
X is abelian.
Proof. Since BX is an Hspace, we see (1BX )? (BX; BX) = [BX; BX].
Because, if m : BX x BX ____BX is the Hmultiplication, for any
ff 2 [BX; BX], a pairing is given by the composite map m O (ff x 1BX ).
Taking ff = 1BX in Theorem 1, we see that the identity map of BX factors
through BC(X). Proposition 1 implies BX ' BC(X), and therefore X is
abelian.
8
Remark. A double loop space is homotopy commutative, and McGibbon
[11] shows that G^pis homotopy commutative if p > 2nr where G is a simply
connected compact Lie group and G '0 S2n11 x . .x.S2nr1 with n1
. . . nr. The twice deloopability or the existence of an Hstructure on the
classifying space is, however, far different from the homotopy commutativity.
Corollary 2 implies BG^pis an Hspace if and only if G is a torus. We note
here a theorem of Hubbuck [7]; Namely T n is the only nontrivial finite
connected homotopy commutative Hspace.
Proof of Theorem 2. First consider a connected finite loop space X. At any
prime p, the pcompletion X^p is a pcompact group, and BX^p is an H
space. Corollary 2 says that there is a torus T n such that BX^p ' (BT n)^p,
where n = rank(X). Hence BX ' BT n.
Next consider the general case so that we begin with the fibration X0 ____
X ____ss0X where X0 denotes the identity component of X. Since BX is an
Hspace, then ss0X = ss1BX is abelian. Consequently, we have a fibration
BT n ____BX ____Bss0X. Notice [1] that this fibration is principal so
that it is preserved by the pcompletion. So the loop space BX^p is a
pcompact group. Corollary 2 says that there is a finite abelian pgroup
flp suchQthat BX^p ' (BT n)^px Bflp. We notice Bflp = (Bss0X)^p so that
ss0X = p flp, since ss0X is a finitely generated abelian group. Considering
the fiber square,
Q
BX __________ p(BX)^p
 
? Q ?
(BX)0 _______( p(BX)^p)0
we see that the splitting of each BX^p induces a section for the fibration
BT n ____BX ____Bss0X. Since this fibration is principal, the classifying
space BX also splits. Consequently BX ' BT nx Bss0X.
If a compact Lie group G is connected and the pcompletion of the clas
sifying space (BG)^pis an Hspace, then G must be abelian. When G is not
connected, however, the analogous result does not hold. A counterexample
is given by a pnilpotent group.
A finite group ss is called pnilpotent, if the subgroup of ss generated
by all elements of order prime to p does not contain any ptorsion element.
It is known that ss is the semidirect product o ssp where ssp is the p
Sylow subgroup. Consequently, if ssp is abelian, the pcompleted space
9
(Bss)^p ' Bssp is an Hspace (actually, an infinite loop space). Henn [6]
provides a generalized definiton of the pnilpotence for compact Lie groups.
Theorem 3. Suppose G is a compact Lie group and the pcompletion of
the classifying space (BG)^pis an Hspace. Then G is the product of a torus
T and a finite pnilpotent group oe whose pSylow subgroup oep is abelian,
and hence (BG)^p' (BT )^px Boep.
Proof. Suppose P is a maximal ptoral subgroup of G, [10]. The H
structure on (BG)^p induces a group homomorphism P x P ____P which
makes BP an Hspace, [5] and [15].
(BG)^px (BG)^p _______(BG)^p
6 6
 
BP x BP _ _________BP
According to [8, Corollary 2.4], we see that P is an abelian group. Let
N P be the normalizer of P in G and let W = N P=P . Since the maximal
ptoral subgroup P is abelian, the mod p cohomology H* ((BG)^p; Fp) is
isomorphic to the ring of invariants H* (BP ; Fp)W = H* (BN P ; Fp) and
therefore (BG)^p' (BN P )^p. Consequently (BN P )^p has an Hstructure:
: (BN P )^px (BN P )^p____(BN P )^p
and we obtain the following diagram
__
(BN P )^p _______map(BP; (BN P )^p)Bi
P P P
id P PPq ?ev
(BN P )^p
Notice [5] and [15] that map(BP; (BN P )^p)Bi ' BP , since the classifying
space of the centralizer of P in N P = P o W is pequivalent to BP .
Consequently (BN P )^p ' BP and hence (BG)^p ' BP . This implies that
the compact Lie group G is pnilpotent in the sense of [6]. By [6, Proposition
1.3 and Theorem 2.5], we can show the desired result.
10
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Fukuoka University, Fukuoka 81480, Japan
email: kenshi@ ssat.fukuokau.ac.jp