ADJOINT SPACES AND FLAG VARIETIES OF pCOMPACT GROUPS
TILMAN BAUER AND NATA`LIA CASTELLANA
Abstract.For a compact Lie group G with maximal torus T, Pittie and Smith
showed that the flag variety G/T is always a stably framed boundary. We *
*gener
alize this to the category of pcompact groups, where the geometric argu*
*ment is
replaced by a homotopy theoretic argument showing that the class in the *
*stable
homotopy groups of spheres represented by G/T is trivial, even Gequivar*
*iantly.
As an application, we consider an unstable construction of a Gspace mim*
*icking
the adjoint representation sphere of G inspired by work of the second au*
*thor and
Kitchloo. This construction stably and Gequivariantly splits off its to*
*p cell, which
is then shown to be a dualizing spectrum for G.
1. Introduction
Let G be a compact, connected Lie group of dimension d and rank r with max
imal torus T. Left translation by elements of G on the tangent space g = TeG
induces a framing of G. By the PontryaginThom construction, G with this fram
ing represents an element [G] in the stable homotopy groups of spheres; this has
been extensively studied for example in [Smi74, Woo76 , Kna78, Oss82].
The following classical argument shows that the flag variety G/T, while not
necessarily framed, is still a stably framed manifold: since every element in a*
* com
pact Lie group is conjugate to an element in the maximal torus, the conjugation
map G T ! G, (g, t) 7! gtg 1 is surjective, and furthermore, it factors throu*
*gh
c: G/T T ! G. An element s 2 T is called regular if the centralizer CG(s) ' T
equals T, or, equivalently, if cjG/T fsgis an embedding; it is a fact from Lie *
*the
ory that the set of irregular elements has positive codimension in T. Thus there
is a regular element s such that the derivative of c has full rank along G/T *
*fsg,
and by the tubular neighborhood theorem, it induces an embedding of G/T U,
where U is a contractible neighborhood of s in T. Thus the framing of G can be
pulled back to a stable framing of G/T.
Pittie and Smith showed in [Pit75, PS75] that G/T is always the boundary of
another framed manifold M, and moreover, that M has a Gaction which agrees
with the standard Gaction on G/T on the boundary. In terms of homotopy the
ory, this is saying that the class [G/T] 2 sssd rinduced by the PontryaginThom
construction is trivial.
The first main result of this paper generalizes this fact to pcompact groups.
Theorem 1.1. Let G be a Z/plocal, pfinite group with maximal torus T such that
dim(G) > dim(T). Then the PontryaginThom construction [G/T]: SG ! ST is G
equivariantly nullhomotopic, with G acting trivially on ST.
____________
Date: February 20, 2006.
N. Castellana is partially supported by MCYT grant MTM200406686.
Support by the Institut MittagLeffler (Djursholm, Sweden) is gratefully ack*
*nowledged.
1
2 TILMAN BAUER AND NATA`LIA CASTELLANA
The statement of this theorem requires some explanation. A pcompact group
[DW94 ] is a triple (G, BG, e) such that
fflG is pfinite, i. e., H (G; Fp) is finite;
fflBG is Z/plocal, i. e. whenever f :X ! Y is a modp homology equiva
lence of CWcomplexes, then f :[Y, BG] ! [X, BG] is an isomorphism;
ffle: G ! BG is a homotopy equivalence.
Clearly, G and e are determined by BG up to homotopy, making this definition
somewhat redundant. Although a priori G is only a loop space, we will henceforth
assume we have chosen a rigidification such that G is actually a topological gr*
*oup.
This is always possible, for example by using the geometric realization of Kan's
group model of the loops on a simplicial set [Kan56 ].
A Z/plocal, pfinite loop space is only slightly more general than a pcompa*
*ct
group in that the latter also requires ss0(G) to be a pgroup. We will have no *
*need
to assume this in Theorem 1.1.
By [DW94 ], every pcompact group has a maximal torus T; that is, there is a
monomorphism T ! G with T ' Lp(S1)r and r is maximal with this property.
By definition, a monomorphism of pcompact groups is a group monomorphism
H ! G such that G/H is pfinite (see [Bau04] for this slightly nonstandard point
of view). Dwyer and Wilkerson show that T is essentially unique. Since a maximal
torus is always contained in the identity component of a pcompact group, the
same works for Z/plocal, pfinite groups.
Denote by S0[X] the suspension spectrum of a space X with a disjoint base poi*
*nt
added.
Definition ([Kle01]). Let G be a topological group. Define SG, the dualizing sp*
*ec
trum of G, to be the spectrum of homotopy fixed points of the right action of G*
* on
its own suspension spectrum. That is, SG = (S0[G])hGopas left Gspectra.
In [Bau04], the first author showed that for a connected, ddimensional pcom
pact group G, SG is always homotopy equivalent to a Z/plocal sphere of dimen
sion d. Furthermore, there is a Gequivariant logarithm map S0[G] ! SG, where
G acts on the left by conjugation. If G is the Z/plocalization of a connected *
*Lie
group, then SG is canonically identified with the suspension spectrum of the on*
*e
point compactification of the Lie algebra of G. Thus we may call SG the adjoint
(stable) sphere of G.
In a spectacular case of shortsightedness, [Bau04] restricts its scope to con*
*nected
pcompact groups where everything would have worked for Z/plocal, pfinite
groups G as well. In this case, SG has the modp homology of a ddimensional
sphere. Similarly, the proof of the following was given in [Bau04, Cor. 24] f*
*or
connected groups, but immediately generalizes.
Let DM be the SpanierWhitehead dual of a finite CWspectrum M.
Lemma 1.2. Let H < G be a monomorphism of Z/plocal, pfinite groups. Then ther*
*e is
a relative Gequivariant duality weak equivalence
i j
G+ ^H SH ' D S0[G/H] ^ SG.
For any space X, there is a canonical map ffl: S0[X] ! S0 given by applying t*
*he
functor S0[ ] to X ! . If T < G is a subtorus in a Z/plocal, pfinite group *
*then
ADJOINT SPACES AND FLAG VARIETIES OF pCOMPACT GROUPS 3
there is a stable Gequivariant map
i j
(1.3)[G/T]: SG id^Dffl!SG ^ D S0[G/T]
' G+ ^T ST ' S0[G/T] ^ ST ffl!ST
Lemma1.2
where the homotopy equivalence on the right hand side holds because ST has a
homotopy trivial Taction as T is homotopy abelian. The first map is studied in
[Bau04]. This is the map referred to in Theorem 1.1; it generalizes the Pontrya*
*gin
Thom construction.
In the second part of this paper, as an application of Theorem 1.1, we study *
*the
relationship between two notions of adjoint objects of pcompact groups. It is *
*an
interesting question to ask whether the action of G on SG actually comes from an
unstable action of G on Sd. We will not be able to answer this question in this
paper. However, there is an alternative, unstable construction of an adjoint ob*
*ject
for a connected pcompact group G inspired by the following:
Theorem 1.4 ([CK02 , Mit88]). Let G be a semisimple, connected Lie group of ran*
*k r.
There exist subgroups GI < G for every I ( f1, : :,:rg and a homeomorphism of G
spaces
AG := hocolimG/GI ! g [ f1g
I(f1,:::,rg
to the onepoint compactification of the Lie algebra g of G.
In the second part of this paper, we define a Gspace AG for every connected
pcompact group G and show:
Theorem 1.5. For any connected pcompact group G, there is a Gequivariant spli*
*tting
S0[AG] ' SG ` R for some finite Gspectrum R.
This result links the two notions of adjoint objects together. Thus stably, t*
*he
adjoint sphere is a wedge summand of the adjoint space.
Unfortunately, AG is in general not a sphere.
Acknowledgements. We would like to thank the Institut MittagLeffler for its
support while finishing this work, and Nitu Kitchloo for helpful discussions.
2.The stable pcomplete splitting of complex projective space
2.1. Stable splittings from homotopy idempotents. Let p be a prime. We denote
by Lp the localization functor on topological spaces with respect to modp homo*
*l
ogy, which coincides with pcompletion on nilpotent spaces [BK72 ]. Let S = LpS1
be the pcomplete 1sphere, and set P = S0[BS]. It is a classical result that
p`2
(2.1) P ' Ps
s=0
for certain (2i 1)connected spectra Pi. In this section, we will investigate*
* this
splitting and its compatibility with certain transfer maps.
Let X be a spectrum, e 2 [X, X] and define
eX = hocolimfX !e X !e g.
4 TILMAN BAUER AND NATA`LIA CASTELLANA
If e is idempotent, this is a homotopy theoretic analog of the image of e. Any
such idempotent e yields a stable splitting X ' eX ` (1 e)X. If fe1, : :,:eng*
* are
a complete set of orthogonal idempotents (this means that each eiis idempotent,
eiej' , and idX' e1+ + en), then they induce a splitting X ' e1X ` ` enX.
Example 2.2. Let p be an odd prime. Denote by _ :P ! P the map induced by
multiplication with a (p 1)st root of unity i. Define es:P ! P by
_ l 1 !
es= __1__p X 1i is_i.
i=0
It is straightforward to check that fe0, : :,:ep 2g are a complete set of ortho*
*gonal
idempotents in [P, P]. They induce the splitting (2.1)by defining Ps= esP.
Setting H (P) = Zpfxjg with jxjj = 2j, we have that (ei) : H (P) ! H (P) is
given by
(
(2.3) (ei) (xj) = xj; j j i (mod p 1)
0; otherwise.
2.2. Transfers as splittings. Let 1 ! H i! G ! W ! 1 be an extension of
compact Lie groups. Then associated to the fibration W ! BH ! BG there are
two versions of functorial stable transfer maps [BG75 , BG76]:
(1)The BeckerGottlieb transfer __o:S0[BG] ! S0[BH]
(2)The stable Umkehr map o :BGg ! BHh of Thom spaces of the adjoint
representation of the Lie groups.
Both versions can be generalized to a setting where the groups involved are not
Lie groups but only Z/plocal and pfinite [Dwy96 , Bau04]. For such a group G,
BGg is defined to be the homotopy orbit spectrum of G acting on the dualizing
spectrum SG; since H (SG) = H (Sd; Zp), we have a (possibly twisted) Thom
isomorphism between_H (BG) and H (BGg).
Note that ofactors through o in the following way:
0 comult:
(2.4)S0[BG] o! BH ! BH ^BG S0[BH]
id^! BH ^ 0 0 eval^id 0
BG S [BH] ^BG S [BH] ! S [BH]
where = h i g is the normal fibration along the fibers of BH ! BG, o0 is o
twisted by g, and the right hand side evaluation map is defined by identifying
BH with the fiberwise SpanierWhitehead dual of BH over BG.
Proposition 2.5. Let W = Clbe a finite cyclic group acting freely on S, with l *
*j p 1.
Denote by N = S o W the semidirect product with respect to this action. Then th*
*e Becker
Gottlieb transfer map __o:S0[BN] ! P factors through fP ! P for some idempotent
f :P ! P which induces the same map in homology as e0+ el+ + ep 1 l, and the
induced map S0[BN] ! fP is a modp homology equivalence.
Proof.Since p  jWj, the Serre spectral sequence associated to the group extens*
*ion
S !i N ! W is concentrated on the vertical axis and shows that
H (BN; Zp) ,=H (BS; Zp)W ,=Zp[zl] ,! Zp[z] ,=H (BS; Zp).
ADJOINT SPACES AND FLAG VARIETIES OF pCOMPACT GROUPS 5
In this case, the BeckerGottlieb transfer is nothing but the usual transfer fo*
*r finite
coverings, therefore i ffi __ois multiplication by jWj = l 2 Zp . Setting I = l*
* 1i: P !
S0[LpBN], we thus get orthogonal idempotents in [P, P]:
f = __offi I and e = idP f.
Clearly, e ffi __o' , thus __ofactors through fP and induces an isomorphism
S0[LpBN] ! fP, in particular a modp homology isomorphism between S0[BN]
and fP. The computation of the homology of BN together with (2.3)implies that
f = (e0+ el+ + ep 1 l) .
Corollary 2.6. Let S, N, W be as above. Then the stable Umkehr map
BNn ! BSs ' P
factors through fP ! P for some f :P ! P which induces the same morphism
p_1_
in homology as Pi=l0e(i+1)l.1The induced map BNn ! fP is a modp homology
equivalence.
Proof.This follows from a similarly simple homological consideration. The S
fibration n is not orientable, thus we have a twisted Thom isomorphism
H"n+1(BNn) ,=Hn(BN; H1(S; Zp))
where ss1(BN) = Z/l acts on H1(S; Zp) ,=Zp by multiplication by an lth root of
unity. Thus
(
Hi(BNn; Zp) = Zp; i j 1 (mod l)
0; otherwise.
The factorization (2.4)of __othrough o
0 comult:
S0[BN] o! BS ! BS ^BN S0[BS]
id^!BS ^ 0 0 eval^id 0
BN S [BS] ^BN S [BS] ! S [BS]
simplifies considerably since i is the trivial 1dimensional fibration over B*
*S, and
the composition of the three right hand_side_maps is an equivalence.
In Prop. 2.5 it was shown that I ffi o = idS0[LpBN], thus the same holds after
twisting with n:
nIn n
idBNn:LpBNn o! BSs ! BSi ! LpBN .
n
If we denote the composition BSs ! BSi nI! LpBNn by I, overriding its previous
meaning, the argument now proceeds as in Prop. 2.5. Using the computation of
H (BNn; Zp), we find that LpBNn ' (o ffi I)P, and
p_1_
l
(o ffi I) = X (e(i+1)l)1
i=0
6 TILMAN BAUER AND NATA`LIA CASTELLANA
3.Framing pcompact flag varieties
Before proving Theorem 1.1, we need an alternative description of the Pontrya
ginThom construction (1.3)on G/T.
Lemma 3.1. The map [G/T] is Gequivariantly homotopic to the map
rffl
SG incl!BGg o! BTt ' rS0[BT] ! Sr,
where BGg, BTt, and o are as in Section 2.2, and all spectra except SG have a t*
*rivial
Gaction.
Proof.Applying homotopy Gorbits to (1.3), we get a Gequivariant diagram
SG ______//SG ^ D(S0[G/T]),__//_S0[G/T] ^ ST___//ST
incl  
fflffl o , fflffl 
BGg __________//_BTt__________//_S0[BT] ^_ST___//ST
which is commutative by the definition of o [Bau04, Def. 25].
In the proof of Theorem 1.1, certain special subgroups will play an important
role. In order to define them we need to recall certain facts about the Weyl gr*
*oup
of a pcompact group.
Dwyer and Wilkerson showed in their groundbreaking paper [DW94 ] that
given any connected pcompact group G with maximal torus T, there is an as
sociated Weyl group W(G), which is defined as the group of components of the
homotopy discrete space of automorphisms of the fibration BT ! BG. This gen
eralizes the notion of Weyl groups of compact Lie groups; they are canonically
subgroups of GL(H1(T)) = GLr(Zp), and they are socalled finite complex reflec
tion groups. This means that they are generated by elements (called reflections
or, more classically, pseudoreflections) that fix hyperplanes in Zrp. The comp*
*lete
classification of complex reflection groups over C is classical and due to Shep*
*hard
and Todd [ST54], the refinement to the padics is due to Clark and Ewing [CE74 *
*].
Call a reflection s 2 W primitive if there is no reflection s02 W of strictly*
* larger
order such that s = (s0)k for some k.
Denote by s 2 W a primitive reflection of minimal order l > 1. Let Ts < T be
the fixed point subtorus under s. Since s is primitive,
hsi = fw 2 W j wjTs= idTsg.
Definition. Given connected pcompact group G and a primitive reflection s 2
W(G) of minimal order l > 1, define Csto be the centralizer of Ts in G.
Since G is connected, so is the subgroup Cs[DW95 , Lemma 7.8]. Furthermore,
Cs has maximal rank because T < Cs by definition, and the inclusion Cs < G
induces the inclusion of Weyl groups hsi < W [DW95 , Thm. 7.6]. Since the Weyl
group of Csis Z/l, the quotient of Csby its pcompact center, Cs/Z(Cs), can have
rank at most 1. By the (almost trivial) classification of rank1 pcompact grou*
*ps,
we find that its rank is equal to 1 and
i j
(3.2) Cs,= Lp(S1)r 1 LpS2l 1 / ,
ADJOINT SPACES AND FLAG VARIETIES OF pCOMPACT GROUPS 7
where LpS2l 1is simply Lp SU(2) for l = 2, and the Sullivan group given by
i j
LpS2l 1= Lp Lp(BS1)hZ/l
for p odd, and is a finite central subgroup.
Proof of Thm. 1.1.By Lemma 3.1, showing equivariant nullhomotopy is equiva
lent to showing that the map
proj r
h(G/T): BGg o! BTt ! S
is null. Note that for any given subgroup H < G of maximal rank, there is a
factorization of o through BHh. In particular, we may assume that G is connecte*
*d.
By the dimension hypothesis of the theorem, W(G) is nontrivial. If H = Cs is
the subgroup associated to a primitive reflection s 2 W(G) of minimal order l >
1, then the map h(Cs/T) is the (r 1)fold suspension of h(LpS2l 1/S) by (3.2).
Therefore, it is enough to prove the theorem for those pcompact groups Cs.
We distinguish two cases.
First suppose that l = 2. By the classification of complex reflection groups
[ST54], and with the terminology of that paper, this is always the case except *
*when
W is a product of any number of groups from the list
fG4, G5, G16, G18, G20, G25, G32g.
This comment is only meant to intimidate the reader and is insubstantial for wh*
*at
follows.
In this case, the map h(Lp SU(2)/S) is null by Pittie [Pit75, PS75] since the
spaces involved are Lie groups, thus h(G/T) ' .
Now suppose that l > 2. This forces p > 2 as well, and since hsi acts faithfu*
*lly
on some line in H1(T; Zp) while fixing the complementary hyperplane, we must
have that it acts by an lth root of unity, and thus l j p 1. The proof is fin*
*ished if
we can show that
h(LpS2l 1/S) = 0
where S is the 1dimensional maximal torus in the Sullivan group G0:= LpS2l 1.
To see this, note that the inclusion S ! G0 factors through the maximal torus
normalizer NG0(S) ,=S o Z/l, and thus
0 o1 o2
h(G0/S): BG0g ! BNn ! S0[BS] ! S1.
Wp 2
If P ' i=0 eiP is any stable splitting of the pcompleted complex projective s*
*pace
P = S0[BS] induced by idempotents eias in the previous section, then the right
most projection map clearly factors through e0P, which is the part containing t*
*he
bottom cell. Since p > 2, Corollary 2.6 shows that there is an idempotent f 2 [*
*P, P]
such that o ' f ffi o and o ffi e0 = e0ffi o = 0, proving the theorem.
4.The adjoint representation
Let G be a ddimensional connected pcompact group with maximal torus T of
rank r. Choose a set fs1, : :,:sr0g of generating reflections of W = W(G) with *
*r0
minimal. The classification of pseudoreflection groups [ST54, CE74] implies th*
*at
for G semisimple, most of the time r = r0, but there are cases where r0= r + 1.
8 TILMAN BAUER AND NATA`LIA CASTELLANA
Example 4.1 (The group no. 7). Let p j 1 (mod 12). Let G7 be the finite group
generated by the reflection s of order 2 and the two reflections t, u of order*
* 3,
where s, t, u 2 GL2(Zp) are given by
` ' ` 7 ' ` 7 7 '
s = 10 01 , t = _1_p_ i i7 , u = _1_p_ i i .
2 i i 2 i i
Here i is a 24th primitive root of unity. Note that although possibly i 62 Zp,
_1_p_i 2 Z . In Shephard and Todd's classification, this is the restriction to *
*Z of the
2 p *
* p
complex pseudoreflection group no. 7. They show that even over C, G7 cannot be
generated by two reflections. The associated pcompact group is given by
Lp((BT2)hG7).
If G is not semisimple (i. e. it contains a nontrivial normal torus subgroup)*
*, then
r0may be smaller than r. Set ~ = r + 1 r0~ 0.
Let Ir0be the set of proper subsets of f1, : :,:r0g, and for I ` f1, : :,:r0g*
*, let TI
be the fixed point subtorus Thsiji2Iiand CI = CG(TI) be the centralizer in G, w*
*hich
is connected by [DW95 , Lemma 7.8].
Definition. Let G be a connected pcompact group. Define the adjoint space AG by
the homotopy colimit
AG = ~hocolim G/CI
I2Ir
with the induced left Gaction, and the trivial Gaction on the suspension coor*
*di
nates.
Theorem 1.4 shows that if G is a the pcompletion of a connected, semisimple
Lie group (in this case r = r0and ~ = 1), then AG is a ddimensional sphere G
equivariantly homotopy equivalent to g [ f1g. This holds more generally: if G is
a connected, compact Lie group with maximal normal torus Tk then
AG ,= kAG/Tk = (t [ f1g) ^ (g/t [ f1g) = g [ f1g.
Lemma 4.2. Let Irbe the poset category of proper subsets of f1, : :,:rg and
F, G :Ik ! ffinite CWcomplexes or finite CWspectrag.
be two functors. Then
(1)If F is has the property that dimF(;) > dim F(I) for every I 6= ;, then
dimhocolim F = dim F(;) + k 1.
(2)If f :F ! G is a natural transformation of two such functors such that
,=
f (;): HdimF(;)(F(;)) ! HdimG(;)(G(;)),
then f induces an isomorphism
hocolim f : HdimhocolimF(hocolimF) ! HdimhocolimG(hocolimG).
(3)Let F :Ir! Topbe the functor given by F(;) = Sn, F(I) = for I 6= ;. Then
hocolimIrF ' Sn+r 1.
ADJOINT SPACES AND FLAG VARIETIES OF pCOMPACT GROUPS 9
Proof.The first two assertions follow from the MayerVietoris spectral sequence
[BK72 , Chapter XII.5],
M
E1p,q= Hq(F(I)) =) Hp+q(hocolimF),
I2Ik, jIj=k 1 p
along with the observation that under the dimension assumptions of (1), E1p,q= 0
for q ~ dim F(;) except for E1k 1,dimF(;)= HdimF(;)(F(;)). In particular, this
group cannot support a nonzero differential and thus
Hi(F(;)) ,=Hi+k 1(hocolimF) fori ~ dim F(;).
The third one is an immediate consequence of the MayerVietoris spectral se
quence.
Corollary 4.3. For any connected pcompact group G, AG is a ddimensional Gspa*
*ce.
Proof.This follows from Lemma 4.2. Indeed, since any CI (I 6= ;) is connected
and has the nontrivial Weyl group WI, its dimension is greater than dim T. So t*
*he
condition
dim F(;) = dim G/T > dim F(I)
is satisfied, and
dimhocolim F = d r + r0 1 = d ~.
As mentioned at the end of the introduction, for pcompact groups G, AG is not
usually a sphere, as the following example illustrates.
Example 4.4. Let p ~ 5 be a prime, and letiG = S2pj3 be the Sullivan sphere,
whose group structure is given by BG = Lp BShCp 1, where Cp 1 ` Zp acts on
BS = K(Zp, 2) by multiplication on Zp. Clearly, G has rank 1, and I1 consists o*
*nly
of a point, thus AG = G/T ' Lp CPp 2. Since p ~ 5, this is not a sphere.
For the proof of Theorem 1.5 we need a preparatory result.
Proposition 4.5. Let P be a pcompact subgroup of maximal rank in a pcompact g*
*roup
G. Denote by T a maximal torus of P (and thus also of G). Then the following co*
*mposition
is Gequivariantly nullhomotopic:
fG,P:SG ^ DST ! S0[G/T] ! S0[G/P].
The second map is the canonical projection, whereas the first map is given by u*
*sing the
duality isomorphism
SG ^ DST ! SG ^ D(S0[G/T]) ^ DST ,! G+ ^T ST ^ DST
' S0[G/T] ^ ST ^ DST id^ev!S0[G/T].
Proof.In [Bau04, Cor. 24] it was shown that the relative duality isomorphism fr*
*om
Lemma 1.2 is natural in the sense that the following diagram commutes:
(S0[G])hPop_,__//G+ ^P SP___,//_D(S0[G/P]) ^ SG
res D(proj)^id
fflfflop, , fflffl
(S0[G])hT _____//G+ ^T ST____//D(S0[G/T]) ^ SG
10 TILMAN BAUER AND NATA`LIA CASTELLANA
Taking duals and smashing with DSG, we find that the map of the proposition is
the left hand column in the diagram
SG ^ D(G+O^POSP)___,____//S0[G/P]OO
 
 
 , 
SG ^ D(G+O^TOST) _______//_S0[G/T]
, 

SG ^ D(S0[G/T])O^ODST



SG ^ DST
Thus we need to show that the composition
G+ ^P SP ! G+ ^T ST ' S0[G/T] ^ ST ! ST
is Gequivariantly trivial, or equivalently, that
SP ! P+ ^T ST ! ST
is Pequivariantly trivial. But this map is exactly the homotopy class represen*
*ted
by [P/T], thus the assertion follows from Theorem 1.1.
Proof of Thm. 1.5.Let G be a connected pcompact group whose Weyl group is gen
erated by a minimal set of r0reflections. Let F, A: Ir0! Topbe the functors giv*
*en
by F(;) = SG ^ DST, F(I) = for I 6= ;, and A(I) = G/CI. Note that, since G is
connected, CG(T) = T [DW94 , Proposition 9.1] and A(;) = G/T. There is a map
: F ! A of Ir0diagrams in the homotopy category of Gspectra which is fully
described by defining
(;) = fG,T:F(;) = SG ^ DST ! S0[G/T]
as the map given in Prop. 4.5. The strategy of the proof is to obtain a functor
F :Ir0! Top such that F(;) = SG ^ DST, F(I) ' for I 6= ;, and a map of
Ir0diagrams : F ! A in the category of Gspectra such that (;) = fG,T. From
this we get a Gequivariant map
0 1 ~ ~ 0 0
SG ' S~^ r SG ^ DST ' S ^ hocolimF ! hocolim S [G/CI] ' S [AG],
Ir0 Ir0
which will give us the splitting.
We will proceed by induction on the number of generating reflections r0. If
r0= 1 then AG is S~^ G/T and (;) = S~^ fG,T. We can construct the functor F
and the natural transformation step by step. Fix a subset I of cardinality k,*
* and
assume that F and have been defined for all vertices in the diagram correspon*
*d
ing to I0with jI0j < k.
Let P(I) be the poset category of all proper subsets of I. Since F and are
defined over P(I) by induction hypothesis, we can consider hocolimP(I)F '
k 1SG ^ DST ! S0[G/CI]. It is enough to show that this map is Gequivariantly
nullhomotopic. Then, we can fix a nullhomotopy and extend the map to the cone
ADJOINT SPACES AND FLAG VARIETIES OF pCOMPACT GROUPS 11
of hocolimP(I)F. Finally, we define F(I) = C(hocolimP(I)F) and (I) is the cor
responding extension.
Note that SG ^ DST ! S0[G/T] factors through G+ ^CISCI^ DST. By induc
tion, we know there is a map
k 1SCI^ DST ! S0[hocolimCI /CJ],
J2Ik
which splits the top cell. We get a factorization
(4.6) k 1SG ^ DST ! k 1G+ ^CISCI^ DST
! G+ ^CIS0[hocolimCI /CJ] ! G+ ^CIS0.
J2Ik
It thus suffices to show that in the Ikdiagram
SCI^ DST _________////_//_f gJ2Ikhf;gocolim//_ k 1SCI^ DST
  
  
fflffl // fflffl hocolim fflffl
S0[CI/T] _____//__//_fS0[CI/CJ]gJ2Ik_f;g//_S0[hocolimJ2IkCI/CJ]
  
  
fflffl______//_ fflffl hocolim fflffl
S0 ___________//_//_fS0gI2Ik_f;g___________//_S0
the right hand side composition k 1SCI^ DST ! S0 is CIequivariantly null
homotopic. In the latter diagram, it makes no difference whether the centralize*
*rs
are taken in CIor in G. But by Theorem 1.1, the left hand column is already nul*
*l
homotopic, thus, as a colimit of nullhomotopic maps over a contractible diagra*
*m,
so is the right hand column.
Conclusion and questions. In this paper, we have compared two imperfect no
tions of adjoint representations of a pcompact group G. One (SG) is a sphere, *
*but
has a Gaction only stably; the other (AG) is an unstable Gspace, but fails to*
* be a
sphere. The question remains whether there is an unstable Gsphere whose sus
pension spectrum is SG. It might even be true that AG splits off its top cell a*
*fter
only one suspension, yielding a solution to this problem in the cases where the
Weyl group of the rankr group G is generated by r reflections.
There are also a number of interesting open questions about the flag variety
G/T of a pcompact groups:
fflBy the classification of pcompact groups, H (G/T; Zp) is torsion free a*
*nd
generated in degree 2. Can this be seen directly?
fflIs there a manifold M such that LpM ' G/T, analogous to smoothings of
G [BKNP04 , BP06]? Is it a boundary of a manifold?
fflIf such a manifold M exists, can it be given a complex structure?
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Institut f"ur Mathematik
Westf"alische WilhelmsUniversit"at M"unster
Einsteinstr. 62
48149 M"unster, Germany
Email address: tbauer@math.unimuenster.de
Departament de Matem`atiques
Universitat Aut`onoma de Barcelona
08193 Bellaterra, Spain
Email address: natalia@mat.uab.es