CONTINUITY OF ß-PERFECTION FOR COMPACT
LIE GROUPS
HALVARD FAUSK AND BOB OLIVER
Abstract. Let G be a compact Lie group, and let ß be any prime
or set of primes. We construct a "ß-perfection map": a continuous
function from the space of conjugacy classes of all closed subgroups
of G to the space of conjugacy classes of ß-perfect subgroups with
finite index in their normalizer. We use this to show that the
idempotent elements of the Burnside ring of G localized at ß are
in bijective correspondence with the open and closed subsets of the
space of conjugacy classes of ß-perfect subgroups of G with finite
index in their normalizer.
1.ß-perfection
Let ß be a collection of primes, and let ß0 denote its complement. A
discrete group H is ß-perfect if it has no nontrivial solvable quotient
ß-groups. Any finite group H contains a unique maximal ß-perfect
subgroup, which we denote here Hi. Equivalently, Hi is the minimal
normal subgroup of H such that H=Hi is a solvable ß-group. It is easy
to see that Hi is the terminal subgroup in the decreasing sequence
of subgroups defined by setting H0 = H, and letting Hn be the sub-
group generated by the commutator [Hn-1, Hn-1] and all ß0-elements
of H. All groups are ?-perfect, while the {all primes}-perfect groups
are exactly the perfect groups in the usual sense.
A compact Lie group H will be called ß-perfect if the group ß0(H) =
H=HO of its connected components is ß-perfect. Hence the maximal
ß-perfect subgroup H0iof an arbitrary compact Lie group H is the
preimage in H of the maximal ß-perfect subgroup of H=HO. When H
is a closed subgroup in a compact Lie group G, there is a variant of
this construction with better properties, where we replace H0iby an
associated subgroup Hi of G with finite index in its normalizer.
____________
1991 Mathematics Subject Classification. Primary 55P91.
Key words and phrases. Burnside ring.
The second author was partially supported by UMR 7539 of the CNRS.
1
2 HALVARD FAUSK AND BOB OLIVER
Let G be a compact Lie group. We give the space of closed subgroups
of G the Hausdorff topology induced by any metric on G consistent with
its topology. The topology is compact Hausdorff and independent of
the metric.
Definition 1.1. Let (G) be the space of conjugacy classes of closed
subgroups of G, regarded as a quotient space, with the quotient topology,
of the space of all closed subgroups of G. Let (G) be the subspace of
(G) consisting of conjugacy classes of subgroups of G with finite index
in their normalizer.
Both (G) and (G) are countable compact metric spaces, and
hence totally disconnected. For any closed subgroup H G, we let
(H) 2 (G) denote its conjugacy class.
Given a subgroup H G, there is a canonical way (up to conjugacy)
to include H into a subgroup K G with finite index in its normalizer
such that the quotient group K=H is a torus.
Definition 1.2. Define
! : (G) ------! (G)
as follows. For any H G, let K=H be a maximal torus in NG (H)=H,
and set !(H) = (K).
By [3, 5.7.5(ii)], the preimage in NG (H) of a maximal torus in
NG (H)=H has finite index in its normalizer. So ! is well defined.
The map ! is continuous (see the remarks after Lemma 2.2), and is a
retraction of (G) onto (G).
Definition 1.3. The ß-perfection of a closed subgroup H in a compact
Lie group G is Hi def=!(H0i).
We denote the ß-perfection map by Pi : (G) --! (G). Note that
Hi depends on the ambient group G, not only on H and ß.
The map (G) --! (G) given by sending H to its maximal ß-
perfect subgroup H0iis not continuous. The main result of this paper
is the following theorem.
Theorem 1.4. Let G be a compact Lie group. The ß-perfection map
Pi : (G) ------! (G)
is a continuous map.
CONTINUITY OF ß-PERFECTION FOR COMPACT LIE GROUPS 3
Section 2 is devoted to a proof of this theorem. In Section 3, we
give an application of the theorem to the Burnside ring A(G) of a com-
pact Lie group G. The G-equivariant cohomology theories are natural
modules over A(G). An idempotent element in A(G) localized at ß
universally splits off a summand of all the G-equivariant cohomology
theories localized to invert the set ß of primes [7, XVII]. It is therefore
important to describe the idempotent elements in A(G) localized at
ß. Theorem 1.4 implies the following useful Lie group theoretic de-
scription of these elements. The idempotent elements in the Burnside
ring A(G), after localizing at ß, are in bijective correspondence with
open and closed subsets of the space of conjugacy classes of ß-perfect
subgroups of G with finite index in its normalizer.
2. Proof of the main theorem
In this section, we prove Theorem 1.4. We will need to refer to the
following well known facts about compact Lie groups.
Proposition 2.1. Let G be any compact Lie group.
(1) (Montgomery & Zippin [8]) For any sequence {Hi} of subgroups
of G which converges to some H G, there are elements gi 2 G
such that gi ! e and giHig-1i H for all i sufficiently large.
(2) If there is a sequence {Hi} of finite subgroups of G which con-
verges to H G, then HO is a torus.
(3) The group Out (G) of outer automorphisms of G is discrete.
(4) For any H E G such that G=H is a torus, CG (H)O is a torus, and
G = CG (H)O.H.
Proof. (1) This follows, for example, from [4, I.5.9]: for any neighbor-
hood U G of e, there is a neighborhood V of H such that K V
implies gKg-1 H for some g 2 U.
(2) By (1), we can assume Hi H for all i. By Jordan's theorem
[4, IV.6.4], there is some j = j(H) such that every finite subgroup
of H contains a normal abelian subgroup of index at most j. Choose
abelian normal subgroups Ai E Hi of index less or equal to j. By the
compactness of the space of subgroups of G, there is a subsequence
{Aij} which converges to A in the space of subgroups of H. Then A
is an abelian normal subgroup of H, and [H : A] j. So HO A is a
torus.
4 HALVARD FAUSK AND BOB OLIVER
(3) Let fi 2 Aut (G) be a sequence of automorphisms converging
to an automorphism f. Let Gfi, Gf G x G denote the graphs of
these maps. The sequence {Gfi} converges to Gf, so by (1), Gfi is
subconjugate (hence conjugate) to Gf for i sufficiently large. Hence fi
and f are equal in Out (G) for i large enough.
(4) Since H E G, the group G=(CG (H).H) is contained in Out (H),
which is discrete by (3). Hence G and CG (H).H have the same identity
component. Since G=H is connected, this implies
G=H = (CG (H).H)=H ~= CG (H)=Z(H) ~=CG (H)O=(Z(H) \ CG (H)O).
So G = CG (H)O.H, CG (H)O=Z(H)O is a connected finite covering group
of a torus and hence a torus, and CG (H)O is an extension of a torus by
a torus and hence itself a torus.
Lemma 2.2. Let G be a compact Lie group, and let H G be any
closed subgroup. Then !(H) = (T H) for any maximal torus T in
CG (H).
Proof. By definition, !(H) = (K) for any K NG (H) such that K=H
is a maximal torus of NG (H)=H. Since K=H is a torus, Lemma 2.1(4)
implies that CK (H)O is a torus and K = CK (H)O.H.
Let T be any maximal torus in CG (H)O. Then CK (H)O is a torus
in CG (H)O and hence subconjugate to T , while T H=H is a torus in
NG (H)=H and hence subconjugate to K=H. This shows that K =
CK (H)O.H is conjugate to T H, and hence that (T H) = (K) = !(H).
The continuity of ! follows easily from Lemma 2.2. For any sequence
{Hi} of subgroups of G converging to some H G, we can assume
Hi H by (1), and hence CG (Hi) CG (H). The sequence of central-
izers {CG (Hi)} converges to CG (H), since otherwise (after passing to
a subsequence, using the compactness of G) there would be elements
gi 2 CG (Hi) converging to g =2 CG (H), which is impossible. Proposi-
tion 2.1(1) then implies that CG (Hi) = CG (H) for i sufficiently large.
Hence for any maximal torus T of CG (H), !(Hi) = (T Hi) (i large) by
Lemma 2.2, and the sequence {(T Hi)} converges to (T H) = !(H).
Lemma 2.3. Let G be a compact Lie group, and let K E H G be a
pair of closed subgroups such that H=K is a torus. Then !(H) = !(K).
Proof. Set S = CH (K)O for short; then S is a torus and H = KS by
Lemma 2.1(4). Let T CG (K) be any maximal torus which contains
CONTINUITY OF ß-PERFECTION FOR COMPACT LIE GROUPS 5
S. Then T is also a maximal torus of CG (H) = CG (KS), and !(H) =
(HT ) = (KT ) = !(K) by Lemma 2.2.
We are now ready to prove the main theorem.
Proof of Theorem 1.4. Since every element of (G) has a countable
neighborhood basis, it suffices to show, for every sequence {Hi} of
closed subgroups of G which converges to a subgroup H, that there
is a subsequence {Hij} such that {Pi(Hij)} converges to Pi(H). By
Lemma 2.1(1) again, we can assume that Hi H for all i.
The space of closed subgroups of G is a compact metric space, so
any sequence has an accumulation point. Hence after restricting to a
subsequence,_we can assume that {(Hi)0i} converges to some subgroup
H H. Since_(Hi)0iis normal in Hi for each i, it follows by taking
limits that H E H.
Clearly, Hi surjects onto H=HO for i sufficiently large,_and hence
(Hi)0isurjects onto the ß-perfect_group H0i=HO. So H surjects onto
H0i=HO, and in particular H0i=H is connected.
__ __
Since H is normal in H, Lemma 2.1(1) tells us that HOi H__for i
sufficiently large. In particular, the image Ki of Hi in H=H__is a finite
subgroup for i large, and the sequence_{Ki} converges to H=H . By
Lemma_2.1(2),_this implies that (H=H_)O is a torus, and hence (since
H0i=H is connected), that H0i=H is a torus.
__
Thus !(H ) = !(H0i) = Pi(H) by Lemma 2.3. By the continuity_of
!, the sequence {Pi(Hi)} = {!((Hi)0i)} converges to !(H ), and this
finishes the proof of the theorem.
3. Idempotents in the Burnside ring with ß0 inverted
For any ring R, we let R(i)= R Z Z(i)denote the localization of
R at the set of primes ß; i.e., R with the primes in the complement ß0
inverted. For example, R({p})= R(p): the localization of R at p.
The Burnside ring of a compact Lie group was defined by tom Dieck
[3]. It generalizes the Burnside ring of a finite group; and (additively)
can be regarded as the free group with basis the set of orbits G=K for
all (K) 2 (G); i.e., all conjugacy classes of subgroups K G which
have finite index in their normalizer. For each closed subgroup H G,
let OEH :A(G) --! Z be the homomorphism OEH (G=K) = Ø((G=K)H ).
Let C( (G), Z) be the ring of continuous integer valued functions on
6 HALVARD FAUSK AND BOB OLIVER
(G), and set
OE = OEH H2 (G) :A(G) ------! C( (G), Z).
Then OE is injective, and identifies A(G) as a subring of C( (G), Z).
For each H G, set q(H, 0) = OE-1H(0) and (for any prime p)
q(H, p) = OE-1H(pZ). If H0 E H and H=H0 is a torus, then clearly
OEH0 = OEH . Hence q(H, 0) = q(!(H), 0) and q(H, p) = q(!(H), p) for all
H. The minimal prime ideals of A(G)(i)are precisely the ideals q(H, 0)
for all conjugacy classes of subgroups H in (G), and q(H, 0) = q(H0, 0)
if and only if (H) = (H0) in (G). The maximal ideals of A(G)(i)are
the ideals q(H, p) for all conjugacy classes (H) 2 (G) and all p 2 ß.
Two maximal ideals q(H, p) and q(K, l) in A(G)(i)are equal if and only
if p = l and (Hp) = (Kp) in (G) (see [1, Prop. 8 & Theorem 4] or [7,
XVII 3.3]). These are the only prime ideals. The closure of q(H, 0) in
the Zariski topology consists of q(H, 0) and the q(H, p) for all p 2 ß.
It is well known that the idempotent elements of a commutative
unital ring R are in bijective correspondence with the open and closed
subsets of the prime ideal spectrum spec R. For any topological space
X, let 0(X) denote the space of components of X with the quotient
topology from X. This is a totally disconnected Hausdorff space.
Definition 3.1. Let i(G) denote the subspace of (G) consisting of
conjugacy classes of ß-perfect subgroups of G with finite index in its
normalizer.
Note that ? (G) = (G). Since the ß-perfection map is continuous
and (G) is compact Hausdorff, we get the following.
Proposition 3.2. The space i(G) of conjugacy classes of ß-perfect
subgroups of G with finite index in their normalizer is a closed subspace
of (G).
We define a map fi : i(G) --! 0(spec A(G)(i)) by sending the
conjugacy class of H to the component of q(H, 0). As pointed out in
[7, XVII.5.5] the continuity of the ß-perfection map allows us to prove
the following proposition.
Proposition 3.3. The map
fi : i(G) ------! 0(spec A(G)(i))
is a homeomorphism.
Proof. We already noted that for all H G, q(H, 0) is in the same
component as q(H, p) for all primes p 2 ß. There is a sequence of
CONTINUITY OF ß-PERFECTION FOR COMPACT LIE GROUPS 7
normal extensions from H0ito H with p-group quotients for various
p 2 ß. From this, it follows that q(H, 0) is in the same component as
q(H0i, 0). Since Hi=H0iis a torus, q(Hi, 0) = q(H0i, 0).
The map ff0 : spec A(G)(i)--! i(G), defined by sending q(H, p)
and q(H, 0) to Hi, is well defined. The composite map
q ff0
(G) x specZ(i)-----! spec A(G)(i)-----! i(G)
is continuous since it is equal to the composite
pr1 Pß
(G) x specZ(i)-----! (G) -----! i(G)
of the projection and ß-perfection maps. The map
q : (G) x specZ(i)--! spec A(G)(i)
is an identification [6, V.5.7]. So ff0is continuous. Since i(G) is totally
disconnected, we get that ff0 factors through the space of components
of spec A(G)(i). This gives a continuous map
ff : 0(spec A(G)(i)) --! i(G)
that sends the component containing q(H, 0) to Hi. The maps ff and
fi are inverses of each other. Also, fi is continuous since q is continuous.
Hence ff and fi are mutually inverse homeomorphisms.
In the case ß = ?, this result was proved by tom Dieck [2]. Propo-
sition 3.3 gives the following description of the idempotent element of
A(G) localized at a set of primes.
Theorem 3.4. There is a bijection between open and closed subsets of
i(G) and idempotent elements of A(G)(i). Let eU denote the idempo-
tent element of A(G)(i)corresponding to an open and closed subset U
of i(G). The image of eU in C( (G), Z(i)) is described by OEH (eU ) = 1
if Hi 2 U, and OEH (eU ) = 0 if Hi 62 U.
Note that Lemma 2.1(1) implies that the conjugacy class of any
abelian subgroup of G with finite index in its normalizer is an open
and closed point in (G). This observation, together with Theorem
3.4, is used in [5].
References
[1]T. tom Dieck, The Burnside ring of a compact Lie group I, Math. Ann. 215
(1975), 235-250.
[2]T. tom Dieck, Idempotent elements in the Burnside ring, J. Pure Appl. Algeb*
*ra
10 (1977/78), no. 3, 239-247.
8 HALVARD FAUSK AND BOB OLIVER
[3]T. tom Dieck, Transformation groups and representation theory, Lecture Notes
in Math. Vol 766. Springer Verlag. 1979.
[4]T. tom Dieck, Transformation groups, Walter de Gruyter. 1987.
[5]H. Fausk, Generalized Artin and Brauer induction for compact Lie groups,
preprint 2001.
[6]L. G. Lewis, J. P. May, and M. Steinberger (with contributions by J. E. Mc-
Clure), Equivariant stable homotopy theory, Lecture Notes in Math. Vol 1213.
Springer Verlag. 1986.
[7]J. P. May, et. al., Equivariant homotopy and cohomology theories. CBMS
regional conference series no. 91, Amer. Math. Soc. 1996.
[8]D. Montgomery & L. Zippin, A theorem on Lie groups, Bull. Amer. Math.
Soc. 48 (1942), 448-452.
Department of Mathematics, University of Oslo, Norway
E-mail address: fausk@math.uio.no
LAGA, Institut Galil'ee, Av. J-B Cl'ement, 93430 Villetaneuse, France
E-mail address: bob@math.univ-paris13.fr