NORMALIZERS OF MAXIMAL TORI
JESPER M. MOLLER
March 31, 1995
Abstract.Normalizers and p-normalizers of maximal tori in p-compact grou*
*ps
can be characterized by the Euler characteristic of the associated homog*
*eneous
spaces. Applied to centralizers of elementary abelian p-groups these cri*
*teria
show that the normalizer of a maximal torus of the centralizer is given *
*by
the centralizer of a preferred homomorphism to the normalizer of the max*
*imal
torus; i.e. that "normalizer" commutes with "centralizer".
1.Introduction
The purpose of this paper is to formulate recognition criteria for the normal*
*izer
and the p-normalizer of a maximal torus of a p-compact group.
Fix a prime p and consider a p-compact group X. Let N(i): N(T ) ! Xdenote
the normalizer and Np(i): Np(T ) ! Xthe p-normalizer [7, 9.8] of a maximal torus
[7, 8.9] i: T ! X. N(T ) is the middle term of a short exact sequence [7, 3.2]*
* of
loop spaces
T -! N(T ) -!WT(X)
where WT(X) is the Weyl group. Np(T ) is the middle term of a short exact seque*
*nce
of p-compact groups
T -! Np(T ) -!WT(X) p
where WT(X) p is a Sylow p-subgroup of WT(X) . Thus N(T ) is an extended p-
compact torus [6, 3.12] and Np(T ) is a p-compact toral group [7, 6.3]. The mor*
*phism
Np(i): Np(T ) ! Xis a monomorphism [7, 3.2] of p-compact groups. Before we can
say that N(i): N(T ) ! X is a monomorphism, too, we need to define what it
means for a morphism of an extended p-compact torus to a p-compact group to be
a monomorphism.
Let G be any extended p-compact torus and f :G ! X a morphism, i.e. a based
map Bf :BG ! BX . G fits into a short exact sequence
S -! G -!ss
where the identity component S = G0 is a p-compact torus and the component
group ss = ss0(G) is finite. Define o0(G) to be the kernel of the conjugation a*
*ction
ss -! Aut(ss1(S)) of the component group on the identity component.
____________
1991 Mathematics Subject Classification. 55P35, 55S37.
Key words and phrases. Monomorphism, p-compact group, normalizer, Weyl group.
1
2 J. M. MOLLER
Call G -!X a p-monomorphism if for some (hence (2.1) any) Sylow p-subgroup
Gp -! G, the restriction Gp -! G -! X is a monomorphism of p-compact groups.
A Sylow p-subgroup of G is a morphism Gp -! G of a p-compact toral group Gp
to G which restricts to an isomorphism on the identity components and induces a
monomorphism on component groups taking ss0(Gp) isomorphically onto a Sylow
p-subgroup of ss0(G). Np(T ) is a Sylow p-subgroup of N(T ) and N(i): N(T ) ! X
is a p-monomorphism.
Definition 1.1.The morphism f :G ! X is a monomorphism if it is a p-
monomorphism and o0(G) is a p-group.
With this definition, the p-monomorphism N(i): N(T ) ! Xis a monomorphism
for [10, 3.4.2] o0(N(T )) = ss0(CX (T )) is a p-group.
Let X=G denote the homotopy fibre of Bf :BG ! BX . If f is a monomorphism,
X=G is (3.1) Fp-finite so the Fp-Euler characteristic O(X=G) is defined. For in*
*stance
[7, 8.10, 9.5], O(X=T ) = |WT(X) |, O(X=Np(T )) = |WT(X) : WT(X) p|, and (3.3,
3.10) O(X=N(T )) = 1.
Theorem 1.2. ( 3.4, 3.5, 3.6) Let f :G ! X be a monomorphism from an ex-
tended p-compact torus G to X. Then f is conjugate to
(1) the maximal torus i if and only if O(X=G) 6= 0 and ss0(G) is trivial.
(2) the p-normalizer Np(i) if and only if O(X=G) 6= 0 mod p and ss0(G) is a
p-group.
(3) the normalizer N(i) if and only if O(X=G) = 1.
We say that two morphisms G -!X and H -! X of extended p-compact tori to
X are conjugate if there exists an isomorphism G -!H, i.e. a homotopy equivalen*
*ce
BG -!BH, making
BG F______'______//_BH
FF xxx
FFF xxx
FF"" --xx
BX
homotopy commutative.
Most work goes into proving (1.2(3)). Indeed, (1.2(1)) is already known [6, 2*
*.15]
and (1.2(2)) quickly follows from [7, 6].
An application of this recognition principle, shows that, in a certain sense,
centralizers commute with normalizers. To be more precise, let :V ! X a
monomorphism of an elementary abelian p-group V = (Z=p)d to X. Suppose
that : V ! N(T ) is a lift of over N(i) such that composition with N(i) in-
duces a morphism CN(i)(V ): CN(T)() ! CX ()of centralizers. Is CN(T)() the
normalizer of a maximal torus of CX ()? With a suitably chosen lift , it is.
Theorem 1.3. Let :V ! X be a monomorphism of an elementary abelian p-
group into X. There exists a lift : V ! N(T ) of such that
CN(i)(V ): CN(T)() ! CX ()
is conjugate to the normalizer of a maximal torus of CX (). If V has rank one
( 4.6), but not in general ( 4.11(3)), the lift is unique up to conjugacy
NORMALIZERS 3
This theorem is particularly useful (4.9) in connection with the centralizer *
*de-
composition [6, 8.1] of BX.
I would like to thank W. Dwyer for a most helpful conversation during the BCAT
94 and in particular for telling me how to exploit the result of [5] in the pro*
*of of
(1.3). I also benefitted from conversations with R. Kane and L. Smith. Special
thanks are due to the Centre de Recerca Matematica for warm hospitality and
support when part of this work was done.
2.Monomorphisms
Suppose that G and H are extended p-compact tori and X a p-compact group.
Note first of all, as tacitly required in the definition of a monomorphism, tha*
*t Sylow
p-subgroups of extended p-compact tori are essentially uniquely determined.
Lemma 2.1. Suppose that Gp -! G and G0p-! G are Sylow p-subgroups of G.
Then there exists an isomorphism Gp -!G0psuch that
BGp E______'______//_BG0p
EE yyy
EEE yyy
E""E__yy
BG
commutes up to homotopy.
Proof.By covering space theory, there exists a map BGp -! BG0p, necessarily a
homotopy equivalence, that makes the diagram homotopy commutative. __|_|
Let g :G ! H and h: H ! X be morphisms. Choose Sylow p-subgroups Gp -!
G and Hp -! H such that ss0(g)(ss0(Gp)) < ss0(Hp). Then g restricts to a mor-
phism gp: Gp ! Hp and further to a morphism g0: S ! T between the identity
components S = G0 and T = H0.
Call g :G ! H a p-monomorphism if gp: Gp ! Hp is a monomorphism of p-
compact toral groups and a 0-monomorphism if g0: S ! Tis a monomorphism of
p-compact tori. Any p-monomorphism is a 0-monomorphism.
Define the rank of G, rk(G), to be the rank [7, 6.3] of S. If g is a 0-monomo*
*rphism,
rk(G) rk(H) [7, 8.11] and equality holds if and only if g0 is an isomorphism.
Mapping BS into the fibration sequence BS -! BG -!Bss0(G) results in another
fibration sequence
map (BS; BS) -!map (BS; BG) -!map (BS; Bss0(G))
which leads to the short exact sequence
(2.2) S -! CG (S) -!o0(G)
of extended p-compact tori. Note in particular that ss0(CG (S)) = o0(G). The
short exact sequence (2.2) is not quite natural but composition with g :G ! H
4 J. M. MOLLER
and g0: S ! Tinduces a commutative diagram
S _____//CG (S)______//o0(G)
g0|| Cg(S)|| ||
fflffl| fflffl| fflffl|
T _____//CHO(S)___//_ss0(CHO(S))OO
|| C (g )| |
|| H 0| |
|| | |
T _____//CH (T_)_____//o0(H)
with exact rows where the two lower short exact sequences are isomorphic provid*
*ed
g0 is an isomorphism.
Lemma 2.3. Suppose that g is a 0-monomorphism. Then:
(1) kerss0(g) < o0(G).
(2) If rk(G) = rk(H), then o0(G) = ss0(g)-1(o0(H)).
(3) If rk(G) = rk(H) and g is a p-monomorphism, the order of kerss0(g) is
prime to p.
Proof.The first two statements follow from the commutative diagram
ss0(G) x ss1(S)__//_ss1(S)
ss0(g)xss1(g0)|| ss1(g0)||
fflffl| fflffl|
ss0(H) x ss1(T_)_//_ss1(T )
expressing naturality of conjugation actions. Here, ss1(g) = ss1(g0) is a monom*
*or-
phism if g is a 0-monomorphism [11, 3.4] and an isomorphism if also G and H have
the same rank.
Under the assumptions of point (3), ss0(g)p is a monomorphism. __|_|
It follows that there aren't any monomorphisms of extended p-compact tori to
p-compact toral groups besides the already known ones.
Proposition 2.4.Suppose that g :G ! H is a p-monomorphism into a p-compact
toral group H. Then g is a monomorphism if and only if G is a p-compact toral
group.
Proof.Suppose g is a monomorphism. Then ss0(G) is a p-group since (2.3)
kerss0(g) < o0(G) and imss0(g) < ss0(H) are p-groups. __|_|
The p-divisible group S = (ss1(S) Q)=ss1(S) ~=(Z=p1 )rk(G)is a discrete ap-
proximation [7, x6] to S.
Also G itself has [6, x3] a discrete approximation: Note that H3(ss0(G); ss1(*
*S)) ~=
H2(ss0(G); S) for any action ss0(G) -! Aut (ss1(S)) ~= Aut(S ). Thus there is a
bijection between fibrations
BS -! BG -!Bss0(G);
classified by elements of H3(ss0(G); ss1(S)), and group extensions
S -! G -!ss0(G);
NORMALIZERS 5
classified by elements of H2(ss0(G); S)).
By naturality of this correspondence, the fibre map Bg :BG ! BH induces [6,
3.13] a morphism
S _____//G_____//ss0(G)
g0|| g|| ss0(g)||
fflffl||fflffl fflffl|
T _____//H_____//ss0(H)
of group extensions.
Lemma 2.5. Suppose that g :G ! H is a p-monomorphism and h: H ! X a
monomorphism where rk(G) = rk(H). Then the following conditions are equiv-
alent:
(1) h O g :G ! Xis a monomorphism.
(2) g:G ! H is a monomorphism.
(3) ss0(g): ss0(G) ! ss0(H)is a monomorphism.
(4) ss0(Cg(S)): ss0(CG (S)) ! ss0(CHi(S))s a monomorphism.
Proof.(1) ) (2): Note that the discrete approximation gp:Gp ! Hp to
the monomorphism gp: Gp ! Hp is [11, 3.4] a monomorphism and (2.3) that
kerss0(g) < o0(G). Since o0(G) is a p-group, so is kerss0(g) and we may then
assume that kerss0(g) < ss0(Gp). It follows that kerg = Gp \ kerg = kergp is
trivial.
(2) ) (3): Since G and H have the same rank, any monomorphism G -! H
induces a monomorphism ss0(G) -!ss0(H).
(3) ) (4): Obvious, since ss0(CG (S)) < ss0(G) and ss0(CH (S)) < ss0(H).
(4) ) (1): Being isomorphic to a subgroup of the p-group o0(H) = ss0(CH (S)),
o0(G) = ss0(CG (S)) is a p-group. __|_|
Let now f :G ! X be a 0-monomorphism. Then rk(G) rk(X) [7, 8.11] and
we say that f is of maximal rank if rk(G) = rk(X).
Lemma 2.6. Let f :G ! X be a 0-monomorphism of maximal rank. Then there
exist 0-monomorphisms, N(f)|S and N(f), unique up to conjugation, such that the
diagrams
N(T=)= N(T )
zz N(f) z< 0. Moreover, f :G ! X and the
p-normalizer Np(i): Np(T ) ! Xare conjugate if and only if O(X=G) 6=
0 mod p.
(2) if rk(G) < rk(X), then O(X=G) = 0.
Consequently,
The morphisms f and Np(i) are conjugate, O(X=G) 6= 0 mod p
, p 6 |O(X=G) > 0
, O(X=G) = |WT(X) : WT(X) p|
And even more special case arises when G is a p-compact torus. Then, in the
maximal rank case, O(X=G) = |WT(X) | since ss0(G) is trivial.
Proposition 3.6.(Cfr. [6, 2.15].) Let f :G ! X be a monomorphism of a non-
trivial p-compact torus G to X. Then rk(G) rk(X) and
(1) if rk(G) = rk(X), then O(X=G) > 0 and f :G ! X is conjugate to the
maximal torus i: T ! X.
(2) if cd(G) < rk(X), then O(X=G) = 0.
Consequently,
The morphisms f and i are conjugate, O(X=G) 6= 0 , O(X=G) > 0
, O(X=G) = |WT(X) |
Example 3.7. Let f :G ! H be a monomorphism between p-compact toral
groups. Then rk(G) rk(H). If rk(G) = rk(H), the restriction f0: G0 ! H0to the
identity components is an isomorphism, ss0(f): ss0(G) ! ss0(H)a monomorphism,
and the Euler characteristic O(H=G) = |ss0(H) : imss0(f)|. If rk(G) < rk(H), th*
*en
the Euler characteristic O(H=G) = 0. Consequently,
f :G ! H is an isomorphism, O(H=G) 6= 0 mod p , p 6 |O(H=G) > 0
, O(H=G) = 1:
Example 3.8. Let Z be a p-compact toral group and z :Z ! X a central mor-
phism. Then there exists [7, x7] a lift N(z), unique up to conjugacy, such that
N(T<)<
zz
N(z)zzz |N(i)|
zzz fflffl|
Z ___z__//_X
commutes up to conjugacy. The induced map W (z) := ss0(N(z)): ss0(Z) ! WT(X)
is central since the discrete approximation to N(z) is central.
10 J. M. MOLLER
Define T \ Z to be the p-compact toral group that fits into the commutative
diagram
T \ Z ______//_Z_____//_imW (z)
| fflffl|
| | |
| | |
fflffl| fflffl| fflffl|
T ______//_Np(T_)__//_WT(X)p
|| fflffl|
|| | |
|| | |
|| fflffl| fflffl|
T _______//N(T_)____//WT(X)
with exact rows. Consider the induced morphisms
T=T \ Z -! Np(T )=Z -! N(T )=Z -! X=Z
where N(T )=Z is an extended p-compact torus with identity component T=T \ Z
and Sylow p-subgroup Np(T )=Z.
Since __X=Z__Np(T)=Z' X=Np(T ) is Fp-finite with Euler characteristic prime t*
*o p, the
monomorphism Np(i)=Z :Np(T )=Z ! X=Zis (3.5) the p-normalizer of the maximal
torus T=T \ Z -! X=Z.
Since the Fp-finite space _X=Z__N(T)=Z' X=N(T ) associated to the p-monomorph*
*ism
N(i)=Z :N(T )=Z ! X=Z has Euler characteristic 1, the induced group homomor-
phism W (N(i)=Z): ss0(N(T )=Z) ! WT=Z(X=Z)is (3.3) surjective. But the reg-
ular covering map X=T -! _X=Z__T=T\Zwith the p-group im W (z) as group of cove*
*r-
ing transformations shows [7, 4.14] that (3.6) the order O(__X=Z_T=T\Z) of the *
*Weyl
group WT=T\Z (X=Z) equals the order of the component group ss0(N(T )=Z) ~=
WT(X) = imW (z). Hence W (N(i)=Z) is in fact an isomorphism and N(i)=Z is
(3.4) the normalizer of the maximal torus of X=Z.
Example 3.9. Let G be a compact Lie group whose component group ss0(G) is
a finite p-group. Suppose that T -! G is a Lie theoretic maximal torus with
normalizer N(T ) -! G and p-normalizer Np(T ) -! G. Define BG^ = (BG)p,
BN^p(T ) = (BNp(T ))p, and BN^(T ) = (BN(T ))p where Kp is the partial p-
completion [3, VII,x6] that preserves the fundamental group and p-completes the
universal covering space of the pointed, connected space K. (If the fundamen-
tal group of K happens to be a finite p-group, the partial p-completion is the
p-completion.) Then ^Gis a p-compact group, ^Tis a p-compact torus, and ^N(T ) *
*is
an extended p-compact torus. Since also [3, II.5.3]
(G=T )p = ^G=T^; (G=Np(T ))p = ^G=N^p(T ); (G=N(T ))p = ^G=N^(T )
we see [3, VII.6.3] that ^T-!G^, ^Np(T ) -!G^, and ^N(T ) -!G^are p-monomorphis*
*ms
with O(G^=T^) = O(G=T ) 6= 0, O(G^=N^p(T )) = O(G=Np(T )) 6= 0 mod p, and
O(G^=N^(T )) = O(G=N(T )) = 1. It now easily follows from (3.4, 3.5, 3.6) that
^N(T ) -! ^Gis the normalizer and N^p(T ) -! ^Gthe p-normalizer of the maximal
torus ^T-! ^G.
We shall later need a little information on the special case where G = N(T ) *
*is
the normalizer of the maximal torus.
NORMALIZERS 11
The maximal torus i: T ! X for X factors through the identity component
T -! X0. The normalizer N0(T ) of this maximal torus for X0 is related, cfr. [1*
*1,
3.8], to the normalizer N(T ) of the maximal torus for X by a short exact seque*
*nce
N0(T ) -!N(T ) -!ss0(X)
of extended p-compact tori. The fibre map
BN0(T )_____//BN(T )____//Bss0(X)
| | ||
| | ||
fflffl| fflffl| ||
BX0 _______//_BX______//_Bss0(X)
shows that X=N(T ) ' X0=N0(T ) and thus that X=N(T ) is a connected homoge-
neous space with fundamental group ss1(X=N(T )) ~=WT(X0). (Use the fact that
ss1(N(T )) -!ss1(X) is [11, 5.6] surjective to get the expression for the funda*
*mental
group.)
Lemma 3.10. H*(X=N(T ); Qp)~=Qp.
Proof.As noted above, we may assume that X is connected. Then the induced
map H*(BN(i); Qp):H*(BX; Qp) ! H*(BN(T ); Qp) is an isomorphism [7, 9.7]
and the lemma follows from the Serre spectral sequence. __|_|
Finally, we turn to a somewhat different situation.
Let g :Y ! X be a monomorphism of some p-compact group Y to X. Then
rk(Y ) rk(X). In the maximal rank case, g induces (2.6) a normalizer morphism,
N(g), unique up to conjugacy, such that the diagram
N(g)
N(S) _____//N(T )
| |
| |
fflffl| fflffl|
Y ___g___//_X
commutes where N(S) -! Y is the normalizer in Y of a maximal torus S -! Y .
Define the Weyl homomorphism to be the group homomorphism
W (g) = ss0(N(g)): WS(Y ) ! WT(X)
induced by N(g).
Corollary 3.11.(Cfr. [11, 3.12].) Let g :Y ! X be a monomorphism of p-
compact groups inducing an epimorphism ss0(g): ss0(Y ) ! ss0(X)of component
groups. Then rk(Y ) rk(X) and
(1) if rk(Y ) = rk(X), then W (g): WS(Y ) ! WT(X) is injective and the Euler
characteristic O(X=Y ) = |WT(X) : imW (g)|.
(2) if rk(Y ) < rk(X), then O(X=Y ) = 0.
12 J. M. MOLLER
Consequently,
g :Y ! X is an isomorphism) N(g): N(S) ! N(T )is an isomorphism
, O(X=Y ) 6= 0 and W (g) is an epimorphism
, O(X=Y ) = 1
The first implication can be reversed provided ss0(g) is an isomorphism.
Proof.The homogeneous space X=Y is connected and the action of the fundamental
group ss1(X=Y ) on H*(Y=N(S); Fp) associated to the fibration of Fp-finite spac*
*es
Y=N(S) -!X=N(S) -!X=Y
is nilpotent because [7, 11.6] it factors through the finite p-group ss0(Y ). H*
*ence [6,
11.6]
O(X=N(S)) = O(Y=N(S)) . O(X=Y ) = 1 . O(X=Y ) = O(X=Y )
by (3.10).
If rk(Y ) = rk(X), then W (g) is (2.5) injective and (3.3) the Euler characte*
*ristic
O(X=N(S)) = |WT(X) : imW (g)|; otherwise (3.3) O(X=N(S)) = 0. This proves
(1) and (2).
Suppose that O(X=Y ) = 1. Then Y and X have the same rank and
W (g) is bijective. Assuming ss0(g) is bijective, the Weyl homomorphi*
*sm
W (g0): WS(Y0) ! WT(X0), induced by the restriction g0: Y0 ! X0 to the iden-
tity components, is [11, 3.8] bijective too. Thus g0 is both a rational isomorp*
*hism
[7, 9.7] and a monomorphism, hence [11, 3.7] [6, 4.7] an isomorphism.
The remaining statements are easily proved. __|_|
The Euler characteristic conditions formulated in this section have seemingly*
* not
played any significant role in classical Lie group theory. In the next section,*
* they
will be applied to analyze the centralizer of an elementary abelian p-group.
4.Centralizers of elementary abelian p-groups
The content of this section constitutes a proof of (1.3).
Let :V ! X be a monomorphism of an elementary abelian p-group V to the
p-compact group X. The centralizer CX () = CX (V ) is again a p-compact group
and CX () -! X is a monomorphism [7, 5.1, 5.2]. The aim here is to identify the
normalizer of a maximal torus of CX () using the recognition principle of (3.4).
Consider lifts
BNp(TD)D
|Bjp|
Bp fflffl|
BN(T:):
uu
Buuuu |BN(i)|
uuu fflffl|
BV __B___//_BX
of B to BNp(T ) and BN(T ). According to [6, 2.14] such lifts always exist. Note
that V acts on T and, by conjugation, on WT(X) through the homomorphism
W () := ss0(): V ! WT(X) induced by .
NORMALIZERS 13
The fibration
BT hV-! map (BV; BN(T )) -!map (BV; BWT(X) )
shows that the homotopy groups of BCN(T)() are concentrated in degrees 2.
The fundamental group,
(4.1) ss1(CN(T)()) ~=ss2(BT hV; B) ~=ss1(T )V ;
is a free, finitely generated module over Zp. For the component group there is a
short exact sequence
(4.2) 0 -!H1(V; ss1(T )) -!ss0(CN(T)()) -!CWT(X) (im W ()) -! 1
where the group to the right is the isotropy subgroup at B for the action of the
fundamental group ss1(map (BV; BWT(X) ); BW ()) ~=CWT(X) (im W ()) on the
set ss0(BT hV) ~=H2(V; ss1(T )).
Corollary 4.3.Let : V ! N(T ) be a lift of :V ! X . Then the central-
izer CN(T)() is an extended p-compact torus and CN(T)() -! CX () is a p-
monomorphism.
Proof.The above computation of the homotopy groups shows that CN(T)() is an
extended p-compact torus. Moreover, CNp(T)(p) is a Sylow p-subgroup of CN(T)()
for a suitable lift p: V ! Np(T ) of (4.4) and the composite CNp(T)(p) -!
CN(T)() -!CX () is [6, 2.5] a monomorphism. __|_|
Lemma 4.4. Any morphism : V ! N(T ) admits a lift p: V ! Np(T ) such that
CNp(T)(p) -!CN(T)() is a Sylow p-subgroup.
Proof.Consider the covering map
a
(WT(X) =WT(X) p)V -! BCNp(T)(p) -!BCN(T)()
jpOp'
obtained by applying map (BV; -) to the covering map BNp(T ) -! BN(T ). The
components of the total space are indexed by conjugacy classes of homomorphisms
p: V ! Np(T ) with jp O p conjugate to . Since the cardinality of the fibre is
congruent mod p to |WT(X) : WT(X) p|, there exists at least one p for which the
number of sheets of the covering BCNp(T)(p) -!BCN(T)() is prime to p. __|_|
In particular, the homogeneous space CX ()=CN(T)() is (3.1) Fp-finite for any
lift : V ! N(T ) of :V ! X .
In order to establish the base for an inductional proof of (1.3), we first co*
*nsider
the case where V has rank one.
The next lemma, which is the key observation, deals with the space (X=N(T ))hV
of all lifts to BN(T ) of B :BV ! BX .
Lemma 4.5. Assume that :V ! X is a monomorphism of a rank one elementary
abelian p-group V to X. Then the homotopy fixed point space (X=N(T ))hV is Fp-
finite with Euler characteristic O (X=N(T ))hV = 1.
14 J. M. MOLLER
Proof.Pull back along B :BV ! BX of the diagram
BNp(TI)______________//BN(T )
III vvvv
III vvv
I$$I zzvv
BX
produces a similar diagram of homotopy orbit spaces
(X=Np(T ))hV______________//(X=N(T ))hV
MM rr
MMM rrr
MMM rrr
MM&& xxrr
BV
where the horizontal map is a map over BV , i.e. a V -map between the V -spaces
X=Np(T ) and X=N(T ), and an |WT(X) : WT(X) p|-fold covering map. This hori-
zontal map induces [7, 10.6] another covering map
(WT(X) =WT(X) p)V -! (X=Np(T ))hV-! (X=N(T ))hV
of section, or homotopy fixed point, spaces. Using this map, build a commutative
diagram
BV x (X=Np(T ))hV _____//(X=Np(Th))V
| |
| |
fflffl| fflffl|
BV x (X=N(T ))hV _____//_(X=N(Th))V
where both vertical maps are covering maps and the horizontal ones are evaluati*
*on
maps. From [7, 4.11], using [7, 5.7] to verify the Fp-completeness hypothesis, *
*we
infer that the space (X=Np(T ))hV and the pair ((X=Np(T ))hV; BV x(X=Np(T ))hV)
are Fp-finite. By (3.2), the homotopy fixed point space (X=N(T ))hV and the pair
((X=N(T ))hV; BV x(X=N(T ))hV) are Fp-finite, too. In this situation, the Lefsc*
*hetz
number (X=N(T ); (X=N(T ))hV; V ) = 0 [7, 4.17]. Hence the Euler characteristic
of the trivial V -space (X=N(T ))hV is given by
O((X=N(T ))hV)= ((X=N(T ))hV; V ) = (X=N(T ); V )
= (Qp; V ) = 1
using the additive property [7, 4.12] of Lefschetz numbers and (3.10). __|_|
We can now prove (1.3) for elementary abelian p-groups of rank one.
Proposition 4.6.Assume that :V ! X is a monomorphism of a rank one ele-
mentary abelian p-group V to X. Then there exists up to conjugacy exactly one l*
*ift
: V ! N(T ) of such that
CN(T)()0 -!CN(T)() -!CX ()
is a maximal torus for the centralizer of . The normalizer of this maximal torus
is conjugate to CN(T)() -!CX ().
NORMALIZERS 15
Proof.In the fibration
a
(X=N(T ))hV-! map(BV; BN(T ))B -! map(BV; BX)B
N(i)O'
the components of the total space are indexed by conjugacy classes of homomor-
phisms : V ! N(T ) with N(i) O conjugate to . The fibre can also be described
as a finite disjoint union
a
(X=N(T ))hV' CX ()=CN(T)()
N(i)O'
of Fp-finite spaces. Hence
X
1 = O CX ()=CN(T)()
N(i)O'
by the Euler characteristic computation of (4.5). Since all terms of this sum a*
*re
nonnegative (3.3), exactly one of them must be equal to 1 and the rest equal to*
* 0.
Let now : V ! N(T ) denote the uniquely determined preferred lift of for
which the homogeneous space CX ()=CN(T)() has Euler characteristic equal to
1. This lift is characterized by the property that the identity component S =
CN(T)()0 is (3.4, 3.6) a maximal torus for CX (). It remains to show that the
normalizer of the maximal torus S is conjugate to CN(T)() = CN(T)(V ).
Consider the following statements
(1) CN(T)(S x V ) -!CX (S x V ) is an isomorphism.
(2) CN(T)(S) -!CX (S) is an isomorphism.
(3) CX (S) is a p-compact toral group.
(4) CX (S)=S is a p-compact toral group.
(5) CCX(S)=S(V ) is homotopically discrete.
pertaining to the homomorphism SxV -! CN(T)(V )xV -! N(T ) -!X. According
to (3.4), the first statement implies the proposition. Moreover, each statement
implies the one above it: The implication (3) ) (2) holds because, since S is a
p-compact torus, CN(T)(S) is known [10, 3.4.(3)] to be conjugate to the maximal
torus of CX (S) and (5) ) (4) holds because only in a p-compact toral group can
the centralizer of a cyclic p-group be homotopically discrete [5, 1.4]. Thus th*
*e proof
has been reduced to the verification of (5).
Let
(4.7) BShV -! map(BV; BCX (S)) -!map (BV; B(CX (S)=S))
be the fibration sequence obtained by mapping BV into the fibation sequence cor-
responding to the central extension [7, 8.2, 8.3]
S -! CX (S) -!CX (S)=S
of p-compact groups.
Each component of the fibre of (4.7) is homotopy equivalent to BS for BShV =
BS x BV and BShV = map(BV; BS) by centrality. Moreover, S -! CCX(S)(V ) =
CCX(V )(S) is the identity component of the centralizer of the maximal torus
S -! CX (V ) and hence the exact homotopy sequence of fibration (4.7) shows that
16 J. M. MOLLER
the base space component BCCX(S)=S(V ) is aspherical, i.e. that CCX(S)=S(V ) is
homotopically discrete. __|_|
The scene is now set foran inductive proof of Theorem 1.3.
Proof of Theorem 1.3.Let V be an elementary abelian p-group of rank at least tw*
*o.
Write V = V1 x V2 where V1 and V2 are elementary abelian p-groups and V1 has
rank one. Let 2 = |V2: V2 ! X be the restriction of and 1: V1 ! CX (2) the
adjoint of .
We may inductively assume that there exists a lift 2: V2 ! N(T )of 2: V2 ! X
such that CN(T)(2) -!CX (2) is conjugate to the normalizer of a maximal torus.
By (4.6), there exists a (unique) lift 1: V1 ! CN(T)(2) of 1: V1 ! CX (2) such
that CN(T)() -! CX (), where is adjoint to 1, is conjugate to the normalizer
of a maximal torus. __|_|
While I have been unable to find a characterization of the preferred lift in *
*general,
the rank one case is easily handled.
Proposition 4.8.Let X be a connected p-compact group and V ~=Z=p a rank
one elementary abelian p-group. Then : V ! N(T ) is the preferred lift of
= N(i) O : V ! X if and only if W () is trivial.
Proof.CX () -!X is [6, 4.3] a monomorphism of maximal rank and, since the Weyl
group is faithfully represented [7, 9.7] in ss1(T ), the rank of CN(T)() is max*
*imal if
and only if W () is trivial (4.1). __|_|
Example 4.9. Let f :X ! X be an automorphism of X whose normalizer mor-
phism N(f): N(T ) ! N(T )is conjugate to the identity and let :V ! X be a
monomorphism of an elementary abelian p-group to X. Choose a preferred lift
: V ! N(T ) of . Note that f restricts to an automorphism Cf(V ) of CX () for
f O = f O N(i) O = N(i) O N(f) O = N(i) O = . Moreover, the diagram
CN(f)(V )
CN(T)() _____//CN(T)()
CN(i)(V|)| |CN(i)(V|)
fflffl| fflffl|
CX () _Cf(V_)//_CX ()
commutes up to conjugacy. Thus also the normalizer morphism N(Cf(V )) =
CN(f)(V ) of Cf(V ) is conjugate to the identity.
Remark 4.10. Let T -!N (T ) -!WT(X) be a discrete approximation [6, 3.12] to
N(T ) and :V ! N (T )a discrete approximation to a preferred lift : V ! N(T )
of :V ! X . The component group homomorphism W (): V ! WT(X) deter-
mines an action of V on T (and on BT and ss1(T )). Let CWT(X) (im W ())
denote the subgroup of those elements w in the centralizer of imW () for which
w w-1 , where w 2 N (T ) is a lift of w, is T-conjugate to . Then the short ex*
*act
sequence (4.2) has the form
1 -!H1(V ; ss1(T )) -!W () -!CWT(X) (im W ()) -! 1
where W () denotes the Weyl group of CX ().
NORMALIZERS 17
Example 4.11. (1) For any monomorphism V -! T into the maximal torus, V -!
T -! N(T ) is the preferred lift of V -! T -! X, i.e. CN(T)(V ) -! CX (V ) is t*
*he
normalizer of the maximal torus T ~=CT(V ) -!CX (V ) [10, 3.4(3)].
(2) If p 6 | |WT(X) |, X is connected and it follows from [7, 9.5] [6, 2.14] [*
*10, 3.4(1)]
that [BV; BN(T )] ~=[BV; BX], so any monomorphism :V ! X lifts uniquely to
a monomorphism : V ! N(T ) which therefore must be the preferred lift of .
(3) Let ~=Z=2xZ=2 denote the diagonal subgroup of the normalizer N(T ) = O(2)
of the maximal torus T = SO(2) of SO (3). Then the inclusion : ! N(T ) is a
preferred lift of the inclusion : ! SO(3) for CN(T)(n) = n = CSO(2n+1)(n).
Let P be the permuation matrix of the cycle (1 2 3). Then P P -1 is another
preferred lift of (the conjugacy class of) which is not conjugate in N(T ) to .
(4) Write, in the above situation, = V V ? where V = \ T . Then the
restriction | V is the preferred lift of | V but | V ? is not the preferred *
*lift of
| V ? (4.8).
The third of these examples, which shows that (4.8) and the uniqueness part of
(4.6) do not hold in general, is understood to take place in the category of 2-*
*compact
groups by means of (3.9) and (4.12).
Lemma 4.12. Let G be a compact Lie group whose component group ss0(G) is a
p-group and let CG (V ) denote the centralizer of a homomorphism f :V ! G of an
elementary abelian p-group V into G. Then the component group of CG (V ) is [9,
A.4] a p-group and the adjoint to the p-completion of BV x BCG (V ) -!BG,
BC"G(V )-!BCG^(V );
is [8, 1.1] [4, 2.5] a homotopy equivalence.
The above discussion focused on the normalizer of the maximal torus. I close
this note with a few words on the analogous, but much easier, problem for the
p-normalizer.
With denoting a preferred lift of as in (1.3) and p a lift of as in (4.4),
CNp(T)(p) -! CX () is conjugate to the p-normalizer of a maximal torus of the
centralizer CX (). The existence of such a lift to the p-normalizer is, however*
*, more
easily proved directly (even in a more general situation).
Proposition 4.13.Let :G ! X be a monomorphism of a p-compact toral group
G into X. There exists a lift p: G ! Np(T ) of such that
CNp(i)(G): CNp(T)(p) ! CX ()
is conjugate to the p-normalizer of a maximal torus of CX ().
Proof.Let Np() -! CX () denote the p-normalizer of a maximal torus of the
centralizer CX (). The adjoint, G x Np() -! X, of this homomorphism admits
[6, 2.14] a lift G x Np() -! Np(T ) over Np(i). The restriction p: G ! Np(T ) to
G is a lift of with the property that the p-normalizer Np() -! CX () factors
through CNp(T)(p) -! CX (). Conversely, CNp(T)(p) -! CX () factors through
Np() -! CX () for general reasons [6, 2.14] and hence (3.7) these two p-compact
toral groups are isomorphic. __|_|
18 J. M. MOLLER
References
1.D.J. Benson, Polynomial invariants of finite groups, London Mathematical Soc*
*iety Lecture
Note Series, vol. 190, Cambridge University Press, 1994.
2.N. Bourbaki, Algebre, Chp. 10, Masson, Paris, 1980.
3.A.K. Bousfield and D.M. Kan, Homotopy limits, completions and localizations,*
* 2nd ed., Lec-
ture Notes in Mathematics, vol. 304, Springer-Verlag, Berlin-Heidelberg-New *
*York-London-
Paris-Tokyo, 1987.
4.W.G. Dwyer, H.R. Miller, and C.W. Wilkerson, The homotopy uniqueness of BS3,*
* Algebraic
Topology. Barcelona 1986. Lecture Notes in Mathematics, vol. 1298 (Berlin-He*
*idelberg-New
York-London-Paris-Tokyo) (J. Aguade and R. Kane, eds.), Springer-Verlag, 198*
*7, pp. 90-105.
5.W.G. Dwyer and J.M. Moller, Homotopy fixed points for cyclic p-group actions*
*, Preprint,
1994.
6.W.G. Dwyer and C.W. Wilkerson, The center of a p-compact group, Preprint, 19*
*93.
7._____, Homotopy fixed point methods for Lie groups and finite loop spaces, A*
*nn. of Math.
(2) 139 (1994), 395-442.
8.W.G. Dwyer and A. Zabrodsky, Maps between classifying spaces, Algebraic Topo*
*logy.
Barcelona 1986. Lecture Notes in Mathematics, vol. 1298 (Berlin-Heidelberg-N*
*ew York-
London-Paris-Tokyo) (J. Aguade and R. Kane, eds.), Springer-Verlag, 1987, pp*
*. 106-119.
9.S. Jackowski, J. McClure, and R. Oliver, Homotopy classification of self-map*
*s of BG via
G-actions, Part II, Ann. of Math. (2) 135 (1992), 227-270.
10.J.M. Moller, Rational isomorphisms of p-compact groups, Topology (to appear).
11.J.M. Moller and D. Notbohm, Centers and finite coverings of finite loop spac*
*es, J. reine
angew. Math. 456 (1994), 99-133.
Matematisk Institut, Universitetsparken 5, DK-2100 Kobenhavn O
E-mail address: moller@math.ku.dk