pcompact groups as framed manifolds
Tilman Bauer
Department of Mathematics, Rm. 2492, Massachusetts Institute of Technology,
Cambridge (MA) 02139
_____________________________________________________________________________
Abstract
We describe a natural way to associate to any pcompact group an element of
the plocal stable stems, which, applied to the pcompletion of a compact Lie g*
*roup
G, coincides with the element represented by the manifold G with its leftinvar*
*iant
framing. To this end, we construct a ddimensional sphere SG with a stable G
action for every ddimensional pcompact group G, which generalizes the onepoi*
*nt
compactification of the Lie algebra of a Lie group. The homotopy class represen*
*ted
by G is then constructed by means of a transfer map between the Thom spaces of
spherical fibrations over BG associated with SG .
Key words: pcompact groups; ThomPontryagin construction; adjoint
representation; transfer
_____________________________________________________________________________
1 Introduction
Let G be a compact Lie group with (real) Lie algebra g = TeG. Left mul
tiplication with an element g 2 G gives an isomorphism g ~= TgG, and by
choosing a basis for g, we thus obtain a framing L of the manifold G, called
the leftinvariant framing. The PontryaginThom construction produces from
this data an element in ßsd(S0), where d = dim G. Computations of homo
topy classes that arise in this way have been made by Smith [Smi74 ], Wood
[Woo76 ], Knapp [Kna78 ], and others. The most extensive table of homotopy
classes represented by Lie groups can be found in [Oss82 ].
This construction is intimately related to the transfer map for the universal
bundle over the classifying space of the Lie group G. More generally, for every
i
subgroup inclusion H ,! G of Lie groups, there is a transfer map in the stable
i! 1 h g
homotopy category 1 BGg ! BH . Here, BG stands for the Thom
space of the bundle associated to the adjoint representation of G on g. This
Preprint submitted to Elsevier Science May 1, 2002
map is a twisted version of the wellknown Umkehr map for the fibration
G=H ! BH !i BG,
BG+ ! BH , (1.1)
where stands for the normal bundle along the fibers of p. Note that the
tangent bundle along the fibers of p is g=h and hence = h  g as virtual
vector bundles. By taking Thom spaces with respect to the bundle g resp. p*g
on both sides in (1.1), we obtain the desired map.
Lemma 1 The homotopy class represented by the ddimensional compact Lie
group G is given by the following composite of maps:
i! 1 0
Sd ! 1 BGg ! EG+ ' S .
Here the left hand map is the inclusion of the bottom cell into BGg, and i is
the inclusion of the trivial subgroup into G.
Note that we can factor this map through any BHh, where H < G. For H = T
a maximal torus in a semisimple G, this leads to an explicit way of computing
the corresponding element in ßsd.
In this paper, we go one step further and show that the transfer functor
()! can be extended to the class of all pcompact groups. A pcompact group
([DW94 ]) G is a H*(; Z=p)local space BG such that G =def BG has totally
finite modp homology. Prominent examples are given by HZ=plocalizations
of compact Lie groups. Dwyer and Wilkerson have worked out an extensive
Lie theory of pcompact groups [DW94 ]. It turns out that the classification of
pcompact groups, at least at odd primes, boils down to the classical classifi
cation of complex reflection groups by Sheppard and Todd [ST54 ], refined by
Clark and Ewing [?] to padice reflection groups. These groups occur as "Weyl
groupsö f pcompact groups, and the pcompact groups themselves have been
constructed on a casebycase basis; no general method to construct them from
their Weyl groups is known so far.
The main results of this paper are
Theorem 2 (1) For every pcompact group G of Fphomological dimension
d, there is a HZ=plocal ddimensional sphere SG with a stable Gaction,
which in the case of the localization of a compact Lie group is equivalent
to the localization of the onepoint compactification of the Lie algebra g
with the adjoint action.
(2) For every monomorphism H < G of pcompact groups, there is a map
SG ! G ^H SH which is an isomorphism in Hd(; Fp).
2
Annoyingly, the morphism in (2) fails to be Gequivariant, but it does so in a
wellbehaved manner. In fact, there is an extension to EG+ ^ SG ! G ^H SH
that is Gequivariant.
Theorem 3 There is a contravariant functor t from the category of pcompact
groups and monomorphisms to the stable homotopy category with the following
properties:
(1) The spectrum
BGg := t(G) = EG+ ^G SG
is Z=plocal and connective, and H*(BGg; Zp) is a free module over H*(BG; *
*Zp)
on a Thom class in dimension d, the dimension of G.
(2) The functor t makes the following diagram commute:
SG ______//G ^H SG
 
 
fflffl fflffl
BGg _______//BHh
(3) The composition t O Lp, defined on the category of compact Lie groups
and monomorphisms (where Lp is HZ=p localization), is equivalent to
the functor Lp O ()!.
A few explanations are in order. A monomorphism of pcompact groups is, by
f
definition, a pointed map BH ! BG whose homotopy fiber has finite mod
p homology. Hence, even for Lie groups, we allow additional maps such as
_k x
unstable Adams operations LpBU(n) ! LpBU(n) (k 2 Zp ). Of course, in
that case the map is a homotopy equivalence and will not yield an interesting
transfer map.
Theorem 3 enables us, by means of Lemma 1, to associate to any pcompact
group an element in the stable stems, which one might provocatively call "the
pcompact group in its invariant framing".
Table 1 shows pcompact groups with the homotopy classes they represent.
Notation. The symbol Zp denotes the padic integers. All homology and coho
mology theories in this paper are assumed to be reduced, and all spaces to be
compactly generated weak Hausdorff.
3
_________________________________________________________________________
 Name  dim  rank  ST number  prime homotopy class 
____________________________________________________________________
 An n(n + 2)  n  1 any  , ... 
      
 X(m, q, n)  *  n  2a 1 (m) ? 
      
 I2m 2m + 2  2  2b  1 (m) 0 
      
 ~m 2m  1  1  3 1 (m) ff1 for m = p  1 
___________________________________________________________________ 
  18  2  4 1 (3) 0 
      
  34  2  5 1 (3) 0 
      
  30  2  6 1 (12) 0 
      
  46  2  7 1 (12) 0 
      
  38  2  8 1 (4) fi1 for p = 5 
      
  62  2  9 1 (8) 0 
      
  70  2  10 1 (12) 0 
      
  94  2  11 1 (24) 0 
      
 Za2  26  2  12 1, 3 (8) 0 y 
      
  38  2  13 1 (8) 0 
      
  58  2  14 1, 19 (24) 0 
      
  70  2  15 1 (24) 0 
      
  98  2  16 1 (5) 0 
      
  158  2  17 1 (20) 0 
      
  178  2  18 1 (15) 0 
      
  238  2  19 1 (60) 0 
      
  82  2  20 1, 4 (15) 0 
      
  142  2  21 1, 49 (60) 0 
      
  62  2  22 1, 9 (20) 0 
___________________________________________________________________ 
  33  3  23 1, 4 (5) 0 
      
 DW3  45  3  24 1, 2, 4 (7)wfor p = 2? 
      
  51  3  25 1 (3) 0 
      
  69  3  26 1 (3) 0 
      
  93  3  27 1, 4 (15) 0 
___________________________________________________________________ 
 F4  52  4  28 any ? 
      
 Za4  84  4  29 1 (4) 0 
      
  124  4  30 1, 4 (5) 0 
      
  124  4  31 1 (4) fi1fi2 for p = 5? 
      
  164  4  32 4 1 (3) fi2 for p = 7? 
_________________________________________________1_________________ 
  95  5  33 1 (3) 0 
___________________________________________________________________
 Ag6  258  6  34 1 (3) 0 
      
 E6  78  6  35 any fi3 for p = 3? p = 2? 
_________________________________________________2_________________ 
 E7  133  7  36 any ? 
___________________________________________________________________
 E8  248  8  37 any ? 
___________________________________________________________________
* m(n2  n + 2n_
q )  n
y does not vanish for purely dimensional and filtration reasons.
Table 1.1
pcompact groups and the homotopy classes they represent.
2 HZ=plocal equivariant spectra
2.1 HZ=plocalization and pcompletion
In [Bou79 ], a localization functor X ! XE is constructed for every spectrum
E with the property that X ! XE is the terminal E*equivalence out of
X. We will need this for E = HZ=p, the EilenbergMacLane spectrum with
coefficients in Z=p. If X is connective, this functor is very wellbehaved:
Lemma 4 (Bousfield [Bou79 ]) Let X be a connective spectrum. Then lo
calization with respect to HZ=p is equivalent to localization with respect to
M(Z=p), the Moore spectrum for Z=p. This localization can be constructed
explicitly as the pcompletion of X, i.e.
n o
XM(Z=p) = X^p= holim..!.X ^ M(Z=p3) ! X ^ M(Z=p2) ! X ^ M(Z=p) .
I will denote the HZ=plocalization functor by Lp.
For a finite spectrum X, smashing with X commutes with homotopy limits,
and therefore
LpX = X ^ LpS0.
Let S be the full subcategory of HZ=plocal spectra, i.e. of spectra X such
that X ! LpX is a weak equivalence. This category has all homotopy limits,
homotopy colimits, smash products and function spectra if we compose the
usual construction with the functor Lp. (In fact, a homotopy limit of Elocal
spectra is already Elocal.) The smash product is associative up to homotopy,
with unit object LpS0. When working in S, I will omit any mention of Lp and
also write S0 for the unit of the smash product.
2.2 Gspectra
To construct the transfer map t, we will need to work in a pointset category of
equivariant spectra. For our purposes, it is enough to work in the category of
socalled naive Gspectra. I will drop the word än ive" since it will make this
work appear so puny. Let GS be the category whose objects are HZ=plocal
spectra E, together with a (left) Gaction on every space En (n 2 Z), such
that the structure maps En ! En+1 are Gequivariant homeomorphisms.
Morphisms are defined as usual. This category has again all homotopy limits
and colimits, smash products, and function spectra. The unit is given by LpS0
with the trivial Gaction. It may be worth pointing out that the Gaction on
5
a smash product is the diagonal one, whereas the Gaction on map (X, Y ) is
given by conjugation.
There are at least two notions of equivariant equivalences in GS, and it is
important to distinguish between them.
Definition 5 I will call a Gequivariant map f : X ! Y between Gspectra
a coarse Gequivalence if it is a weak equivalence of underlying spectra.
It is called a Ghomotopy equivalence if there is an inverse map up to
homotopies through Gequivariant maps.
For a Lie group G, a coarse equivalence f that also induces an equivalence
on Hfixed points for every closed subgroup H is sometimes called a weak
Gequivalence.
By the equivariant Whitehead theorem for spaces with for a Lie group action
of G (cf. [Ada84 ], [LMS86 ]), a weak Gequivalence between GCW complexes
is a Ghomotopy equivalence; this need not be true for coarse Gequivalences
in general. For example, the obvious coarse Gequivalence EG ! * does not
have an equivariant inverse.
Define a free GCW spectrum to be a Gspectrum which is built from cells of
the form Sn ^ G+ .
Lemma 6 If E is a free GCW spectrum and X ! Y is a coarse Gequivalence
of Gspectra, then it induces weak equivalences
map G(E, X) ~! map G(E, Y ) and E ^G X !~ E ^G Y.
PROOF. Both equivalences are clear if E is a single cell Sn ^ G+ because in
that case,
map G(E, ) = map (Sn, ) and E ^G X = Sn ^ X.
It follows for finite spectra by induction and the fivelemma, and in general
by a direct limit argument.
2
For a Gspectrum X, define
XhoG = EG+ ^G X = (EG+ ^ X)=G and XhoG = map G (EG+ , X)
where map G denotes Gequivariant based maps.
6
The spectrum 1 EG+ is a free GCW spectrum. Therefore Lemma 6 implies
in particular that a coarse Gequivalence f : X ! Y induces weak equivalences
fhoH : XhoH ! Y hoH and fhoH : XhoH ! YhoH for any subgroup H < G. If
H is normal in G then these maps are coarse G=Hequivalences.
2.3 Duality
For a nonequivariant spectrum X, let DX =defmap (X, S) be its dual. This
spectrum DX will not have good duality properties in general. For instance,
there is no guarantee that D(DX) ' X. We call X strongly dualizable if there
''
is a map S ! X ^ DX such that the following diagram commutes up to
homotopy:
______''__//
S X ^ DX (2.1)
' fi
fflffl fflffl
map (X, X) oo___DX ^ X
Here, ø is the flip involution, ' is adjoint to the identity map X ! X, and
is the map adjoint to
X ^ DX ^ X eval^idX!X.
The existence of such a map j is equivalent to being a homotopy equivalence.
It implies that D(DX) ' X. Cf. [May96 ].
It turns out that the category GS contains very few strongly dualizable ob
jects, i.e. objects for which in the above diagram, there is an equivariant map
j, or equivalently, is a Ghomotopy equivalence. This is mainly due to the
fact that we are considering naive Gspectra. For example, if M is a com
pact Gmanifold, we usually construct a duality morphism j by embedding
M equivariantly into some Grepresentation V , use the ThomPontryagin
construction to get an equivariant map SV ! M ^ M+ , and desuspend by
SV . This last step is impossible in the category of naive Gspectra unless V is
a trivial representation, i. e., unless M has a trivial Gaction.
If G is a Lie group, and we work in the category of nonnaive Gspectra, it
is known that a GCW spectrum is strongly dualizable if and only if it is a
wedge summand of a finite GCW spectrum. It seems plausible that if one
succeeded to set up the "right" category of nonnaive Gequivariant spectra
for a pcompact group G, all the objects in this work that are nonequivariantly
dualizable but do not appear to be strongly dualizable in GS would actually
have a strong dual in that category. From a philosophical point of view, this
7
would be desirable and make some cumbersome technical problems disappear.
However, in my opinion, the effort needed for setting up such a category is
not warranted by the purposes of the present work.
Suppose that X is a Gspectrum that, as a nonequivariant spectrum, is strongly
dualizable. Then the map : DX ^ X ! map (X, X), which is always G
equivariant by naturality, is a coarse Gequivalence, and j exists but is not
necessarily Gequivariant. As should be expected, X will have about half of
all the good properties of a strongly dualizable object. For instance, there is
a weak equivalence
map G (A, B ^ DX) ! map G (A ^ X, B) (2.2)
given by
A ^ X ! B ^ DX ^ X id^eval!B
but in general no such map
map G(A ^ DX, B) 6! map G (A, X ^ B).
A spectrum or space X is called pfinite if H*(X; Fp) is totally finite.
The following lemma has a rather long history of my advisor suggesting a proof
using the Adams spectral sequence and me rejecting it and finding another
(erroneous) proof without it. Eventually, I caved in. Here's his proof. Kudos
for Mike.
Lemma 7 For every connective, HZ=plocal, pfinite spectrum X, there is a
finite spectrum X0 and pequivalence X0 ! X.
Remark. This association is not claimed to be functorial.
PROOF. Let k 2 Z be minimal with Hk(X; Fp) 6= 0. We proceed by induc
tion on the size of H*(X; Fp).
We will first show that there is a nontrivial map
f : ßk(X) ! Hk(X) ! Hk(X; Fp).
This would be a simple application of the Hurewicz theorem relative to a Serre
class if the class of groups that vanish when tensored with Fp were actually a
Serre class, which it is not.
Since X is connective and HZ=plocal, its HZ=pnilpotent completion and its
HZ=plocalization agree (Lemma 4) and are equal to X, hence the classical
8
s,t
HZ=pbased Adams spectral sequence converges to ß*(X). Since ExtA* (H*(X; Fp), *
*Fp) =
0 for t  s < k, the Hurewicz map f : ßk(X) ! Hk(X; Fp) has to be nonzero.
Let fi : Sk ! X be a map such that f([fi]) 6= 0. Let F be the HZ=plocalization
of the homotopy fiber of fi. F is pfinite, HZ=plocal and the size of its Z=p
homology is smaller than that of X, hence by induction, there is a finite
spectrum F 0and a pequivalence F 0! F . Let X0 be the cofiber of F 0!
F ! Sk; X0 is a finite spectrum and comes with a map X0 ! X which is a
pequivalence.
2
Corollary 8 Let X be a connective, HZ=plocal, pfinite spectrum. Then X
has a strong dual in S.
PROOF. By Lemma 7, there is a finite spectrum X0 and a pequivalence
X0 ! X. Hence there is a pequivalence of HZ=plocal spectra LpX0 ! LpX =
X, which therefore is a weak equivalence. It remains to show that Lp(D(X0))
is a strong dual of LpX0 for a finite spectrum X0.
We need to show that
Lp(map (X0, S)) = Lp map (LpX0, LpS).
Indeed,
Lp map (LpX0, LpS) ' map (X0, LpS) ' DX0^ LpS ' Lp(DX0).
Now j : S ! X ^ DX induces a duality map
Lpj : LpS ! Lp(X0^ DX0) = LpX0^ LpDX0 = LpX0^ D(LpX0),
which shows that Lp(D(X0)) is a strong dual.
2
3 pcompact groups
This section will provide some background about pcompact groups, a topic
that has become very popular starting in the early nineties, largely due to
some beautiful work of Dwyer and Wilkerson [DW94 ] and Dwyer, Miller, and
Wilkerson [?].
9
3.1 Definition and examples
Definition 9 ([DW94 ]) A pcompact group is a triple (X, BX, e) where
BX is a HZ=plocal space, X is an Fpfinite space, and e : X ! BX is a
homotopy equivalence.
As noted in the introduction, the HZ=plocalization LpG of a Lie group G
gives rise to a pcompact group (LpG, Lpe, LpBG) for every prime p. Here
e : G ! BG is the canonical equivalence.
To illustrate how to obtain other pcompact groups, it is instructive to recall
the connection between spaces with polynomial cohomology rings and finite
loop spaces. If X is a space such that
H*(X; Fp) ~=Fp[oe1, oe2, . .,.oer] with oei 2 Hdi(X; Fp), di even,
then by the EilenbergMoore spectral sequence,
^
H*( X; Fp) ~= {ø1, ø2, . .,.ør} with øi 2 Hdi1( X; Fp),
and øi is the image of oei under the transgression. In particular, H*( X; Fp) is
finite, and LpX is a pcompact group. The reader should be warned that not
all pcompact groups are polynomial in this sense.
A large class of pcompact groups, called the nonmodular groups, can be
constructed as follows:
First pick a finite group W < GLr(Zp) (a "Weyl" group for the pcompact
group); W acts on Zrpand hence also on K(Zrp, 2) = Lp(CP 1)r. Define a space
i j
BG =defLp K(Zrp, 2)hoW .
We want to determine what restrictions on W we have to make to ensure
that BG is a space with polynomial cohomology. There is a spectral sequence
converging to H*(BG; Fp) whose E2 term is
Er,s2= Hr(BW ; Hs(K(Zp, 2); Fp)) = Hr(BW ; Fp[[t1, . .,.tr]])
If p does not divide W then Er,s2= 0 for r > 0, and
E0,s2= Fp[[t1, . .,.tr]]W = Hs(BG; Fp)
Theorem 10 (SheppardTodd, ClarkEwing [ST54 ,?])
Let W < GLr(Fp) be finite.
If Fp[t1, . .,.tr]W is polynomial then W is a pseudoreflection group, i.e., i*
*t is
generated by a finite set of finite order elements that fix a hyperplane in Frp.
10
The converse is true if (but not only if) p does not divide the order of W . 2
Moreover, in the nonmodular case, every representation of W over Fp can be
lifted to a representation over Zp.
We can thus construct a pcompact group BG for every pseudoreflection
group defined as a subgroup of GLr(Zp) such that p does not divide the order
of W . All such groups are classified [ST54 ,?], and Table 1 lists some statist*
*ics
about them. In that table, all exotic groups of rank bigger than 1 that are
given a name are nonmodular.
Definition 11 A morphism BH ! BG of pcompact groups is just a pointed
map Bf : BH ! BG. It is a monomorphism if its homotopy fiber is Fp
finite, and an epimorphism if its homotopy fiber is a pcompact group.
Two morphisms BH ! BG are called conjugate if they are freely homotopic.
For Lie groups H and G, being conjugate in the pcompact sense is indeed the
same as being conjugate as Lie group homomorphisms.
3.2 Maximal tori
In the nonmodular case considered in the previous subsection, BG naturally
comes with a map
BT := K(Zrp, 2) ! Lp(K(Zrp, 2)hoW ) = BG
given by the inclusion of the fiber of the bundle BG ! BW . Call a monomor
phism of pcompact groups BT ! BG a maximal torus if BT = K(Zrp, 2)
for some r, and it does not factor through a larger torus. One of the main
results of [DW94 ] is that such tori also exist in the nonmodular case:
Theorem 12 (DwyerWilkerson [DW94 ])
(1) For every connected pcompact group BG, there is a maximal torus BT !
BG, unique up to conjugacy.
(2) The monoid map BG (BT, BT ) of endomorphisms of BT over BG is homo
topy equivalent to a finite group W acting as a group of pseudoreflections
on H2(BT ; Zp) ~=Zrp.
(3) H*Qp(BG+ ) ~=H*Qp(BT+ )W , and H*Qp(BT+ ) is a free module over H*Qp(BG+ ).
2
11
Here, H*Qp(X) =def H*(X; Zp) Zp Qp. (Note that H*(X; Qp) would be an
unreasonably large group; whereas Hom (Zp, Zp) = Zp, we have
Hom (Zp, Qp) = Hom (Qp, Qp) = Q@2p.)
Corollary 13 The pcompact flag variety G=T = hofib(BT ! BG) has
H*Qp(G=T+ ) = H*Qp(BT+ ) = (H*Qp(BT )W ).
PROOF. There is an EilenbergMoore spectral sequence
s,t * *
Es,t2= dTorH*Q(BG )(HQ (BT+ ), Qp) =) HQ (G=T+ ),
p + p p
where Tdorsis the sth derived functor of the completed tensor product ^ . In
this spectral sequence, Es,t2= 0 for s > 0 because H*Qp(BT+ ) is free, hence
flat, over H*Qp(BG+ ), and
E0,t2= H*Qp(BT+ ) ^H*Qp(BG+)Qp = H*Qp(BT+ ) = (H*Qp(BT )W ).
2It will become important in calculations to know exactly what the degree
of the map
c : H*(BT+ ; Zp) = (H*(BG; Zp)) ,! H*(G=T+ ; Zp)
is in the top dimension.
Lemma 14 Let p > 2 or G of Lie type or G = DW3. Then the cohomology
ring H*(G=T ; Zp) is concentrated in even dimensions and torsion free.
PROOF. This is a result that follows from Schubert calculus in the case
where G is a Lie group. For polynomial pcompact groups, we have
H*(G=T+ ; Zp) ~=H*(BT+ ; Zp) = (H*(BG; Zp))
by the same argument as in Corollary 13, and the assertion holds. Now the
only nonpolynomial pcompact groups for odd p are [KM97 ,?,?]:
o Type An with a fundamental group that is a pgroup;
o types F4, E6, E7, E8 for p = 3; and
o type E8 for p = 5.
In particular, they are all Lie groups.
2
12
Remark 1. Since the classification of 2compact groups in not finished at this
time, we cannot claim Lemma 14 holds for p = 2. However, the only known
nonLie 2compact group is DW3, which is polynomial [DW93 ]. It is conjec
tured that it actually is the only one.
Remark 2. It would be much more satisfying to find a proof that does not
rely on the accidental fact that all nonpolynomial pcompact groups are of
Lie type. For example, it would be exciting to produce a Schubert calculus for
pcompact flag varieties.
I am grateful to Nitu Kitchloo for pointing out to me the implication (i)) (iii)
in the following proposition:
Proposition 15 Let G, p be as in Lemma 14 and G be simply connected.
Then the following are equivalent:
(i) c is an isomorphism in the top dimension;
(ii) c is an isomorphism in all dimensions;
(iii)H*(G=T+ ; Zp) is generated by degree 2 classes;
(iv) H*(BG+ ; Zp) has no torsion;
(v) H*(BG+ ; Zp) is a polynomial algebra.
PROOF. If G is simply connected it follows from the Serre spectral sequence
associated to G=T ! BT ! BG that
~= 2
H2(BT+ ; Zp) ! H (G=T+ ; Zp).
This shows (iii), (ii)) (i). For (iv)) (ii), assume c fails to be an isomor
phism in dimension k. Then in the above Serre spectral sequence, a class x
in Hk(G=T+ ; Zp) has to support a nontrivial differential di. Since rationally,
H*Qp(G=T+ ) is always generated by degree 2 classes, di(x) has to be a torsion
class in
Hi(BG+ ; Hk+1i(G=T+ ; Zp)).
By Lemma 14, the latter group is isomorphic to Hi(BG+ ; Zp) ^Hk+1i(G=T+ ; Zp).
Since by the same lemma, H*(G=T+ ; Zp) is torsion free, there must be a torsion
class in Hi(BG+ ; Zp).
For (i)) (iv), assume y 2 Hj(BG+ ; Zp) is torsion with j minimal. By the
multiplicativity and Lemma 14, this implies that
y = dj(x) for some x 2 Hj1(G=T+ ; Zp).
Pick a generator g 2 Htop(G=T ; Zp). Now
0 = dj(gx) = dj(g)x gdj(x).
13
Since dj(x)g = yg 6= 0, di(g) cannot be trivial, hence g is not a permanent
cycle, and c is not an isomorphism in the top dimension.
For (iv), (v), note that if H*(BG+ ; Zp) has no torsion, it has to be concen
trated in even degrees since it injects into H*Qp(BG+ ). Hence H*(G+ ; Zp) is
a degreewise free Hopf algebra on odddimensional generators, which implies
that it is an exterior algebra. Hence, by the Serre spectral sequence for the
pathloopfibration on BG, H*(BG+ ; Zp) is a polynomial algebra.
2
3.3 A comment on rigidity
In the definition of a pcompact group (X, BX, e), the data X and e are
redundant and probably only classically included to provide some justification
for speaking of ä pcompact group Xä nd not the more accurate "BX". On
the other hand, it is always possible to choose a model for the loop space
X := BX such that X is actually a topological group and not just an H
space. A possible construction is the geometric realization of Kan's loop group
functor G as described in [Kan56 ].
Let S denote the category of simplicial sets and S0 the full subcategory of
reduced simplicial sets, i.e., simplicial sets X such that X0 = pt. Equip S0
with the projective model structure, i.e. weak equivalences and cofibrations
are shared with S. It turns out ([GJ99 ]) that a map X ! Y between fibrant
reduced simplicial sets is a fibration if and only if it induces a surjection on
fundamental groups.
Let s Gr denote the category of simplicial groups, carrying the injective model
structure (sharing weak equivalences and fibrations with the underlying sim
plicial sets).
Proposition 16 (Kan) There is a Quillen equivalence
___
W : s Gr $ S0 : G
Furthermore, there is a Quillen equivalence between the category S0 and the
category of connected, pointed simplicial sets, Sc, where the functor F : Sc !
S0 is given by
F (X)n = {x 2 Xn  i*(x) = * for everyi : [0] ! [n]}.
14
Passing to topological spaces, we also have Quillen equivalences
Sc $ {pointed connected topological spaces }
and
s Gr $ {topological groups}
This suggests the following alternative definition of a pcompact group:
Definition 17 (alternative) The category of pcompact groups is the full
subcategory of all HZ=plocal topological groups (compactly generated, weak
Hausdorff) whose objects are fibrant, cofibrant, Fpfinite, and such that ß0(G)
is a finite pgroup.
The condition on the group of components is necessary to ensure that BG is
still HZ=plocal. By the above Quillen equivalences, every map BH ! BG
is, up to homotopy, induced by a group homomorphism H ! G if H and G
are pcompact groups in this sense.
Moreover, a monomorphism BH ! BG in the sense of the original definition
is always, up to homotopy, induced by an injective group homomorphism
H ! G. In fact, we can functorially replace BH ! BG by a cofibration, and
Kan's functor G preserves cofibrations. Cofibrations of simplicial groups are
injective.
We will therefore work in the category of pcompact groups according to the
above alternative definition, and define monomorphisms as actual subgroup
inclusions.
4 Adjoint representations
Although much of Lie theory carries over to the more general setting of p
compact groups, the representation theory, and in particular the adjoint rep
resentation, does not seem to have a direct analogue for pcompact groups.
We do not know how to construct a vector bundle on a pcompact group BG
that plays that role, but we can manufacture something that, in the Lie cases,
looks like its Thom spectrum.
Definition 18 For any connected pcompact group G, define
op
SG = ( 1 G+ )hoG .
Note that G acts on 1 G+ by both left and right multiplication. We agree to
use the right action for the formation of this homotopy fixed point spectrum,
15
leaving us a left Gaction on SG .
The adjoint Thom spectrum of G is the spectrum
BGg =def(SG )hoG = EG+ ^G SG .
Klein [Kle01 ] has shown that this construction for G a (nonlocalized) con
nected compact Lie group indeed gives rise to the Thom spectrum of the
adjoint bundle. It is therefore reasonable to mimick this construction for a
pcompact group G. The main point of this section is to show that SG , defined
as above for a connected pcompact group G, is homotopy equivalent to a
sphere.
We will need two classical lemmas on finitedimensional Hopf algebras. All
cohomology and homology groups are with coefficients in Fp.
Lemma 19 If G is a topological group such that H*(G) is totally finite, then
H*( 1 G+ ) is a free H*(G)module on a generator in dimension dim G.
PROOF. Note that A = H*(G) is a Hopf algebra, and
H*( 1 G+ ) ~=A*.
by universal coefficients. The dual algebra A* is a Hopf algebra with antipode
c coming from inversion in the group G, and A is a right Hopf module over
A*: the module structure is given by
A A* ! A, the adjoint map of the coproduct _ : A ! A A,
and the comodule structure by
A ! A A*, the adjoint map of the product on A.
Let P (A) denote the Fpvector space of primitives of A as an A*comodule,
i.e. n fi o
P (A) = a 2 A fifiax = affl(x) for allx 2,A
where ffl is the augmentation H*(G+ ) ! H*(S0).
Then (cf [Par71 ]), we have a splitting
~= *
A ! P (A) A
16
as right A*Hopf modules, given by
____//_ * _id__//_ * * id_c_id// * * __~_id_// *
A [ [ [A[ [A[ A A A A A A A AOO (4.1)
[ [ [ [ 
[ [ [ [ [ [ 
[ [ [[ [ [ 
[ [ [ [ [ [P (A) A*
Since A is finite dimensional, it follows that dim P (A) = 1. The assertion of
Lemma 19 follows.
2
We will show later (Proposition 26) that for G a pcompact group, this map
is realizable as a map of spectra.
An algebra like H*(G; Fp), which, as a module, is isomorphic to a suspension
of its dual, is called a Frobenius algebra.
Lemma 20 (MoorePeterson [MP73 ]) If A is a finitedimensional Frobe
nius algebra over a field, then the class of its projective modules coincides w*
*ith
the class of its injective modules. 2
Proposition 21 SG is homotopy equivalent to a HZ=plocal sphere of dimen
sion d.
PROOF. It is enough to know that SG has the mod p homology of a sphere
because the proof of Lemma 7 produces a pequivalence Sd ! SG in that case.
To see that SG has the correct homology, we will use a spectral sequence
associated to the cosimplicial spectrum
op G 1
( 1 G+ )hoG = map (EG+ , G+ ),
where EG = map (, G) = 1 G is the usual simplicial space with Gn+1 in
dimension n. The E2term is given by
E2p,q= Hp(map G(G(q+1), 1 G+ ); Fp),
and by the Lemma below, this spectral sequence collapses at the E2term with
8
< 0; p 6= 0 or q 6= d
Ep,q2= :
Fp; otherwise.
17
and converges strongly. Therefore and H*(SG ) = H*(Sd).
2
This proves the first part of Theorem 2.
Lemma 22 Let k 2 N0[ {1}, and let EG(k)+be the Gequivariant kskeleton
of the simplicial space EG+ . Then
8
< Fp; n = d < k
Hn(map G(EG(k)+, 1 G+ )) ~=:
0; n < k and n 6= d
No statement is made about the homology groups beyond degree k. By G
equivariant kskeleton we mean the truncation of the simplicial space EG+ at
the kth stage.
From now on, until the end of this section, all homology groups are taken with
coefficients in Fp.
PROOF. Since EGn = Gn+1, we have that
(map G(EG+ , 1 G+ ))n = map G (Gn+1+, 1 G+ ).
The evaluation map
map G(Gn+1+, 1 G+ ) ^ Gn+1+! 1 G+
induces a natural map
Hn(map G(Gn+1+, 1 G+ ) ! Hom nH*(G+)(H*(G+ ) (n+1), H*( 1 G+ )),
where Hom n stands for module homomorphisms that raise degree by n. This
map is an isomorphism because the following diagram commutes:
Hn(map G(Gn+1+, 1 G+ ))_____//HomnH*(G+)(H*(G+ ) (n+1), H*( 1 G+ ))
~ ~
fflffl fflffl
Hn(map (Gn+, 1 G+ )) Hom n(H*(G+ ) n , H*( 1 G+ ))
~ ~
fflffl fflffl
Hn((DG+ )^n ^ 1 G+ ) ______~___//(H*(DG+ ) n H*( 1 G+ )) n.
The coboundary operators are induced by the simplicial operators on H*(G+ ) o
from the bar construction on H*(G+ ). Hence Hn(map G(EG(k)+, 1 G+ )) is the
18
group of homomorphisms from a truncated projective resolution of k over
H*(G) to H*(G).
Associated to the tower
map G(EG(k)+, 1 G+ ) ! . .!.map G (EG(j)+, 1 G+ ) !
! . .!.map G (EG(0)+, 1 G+ ) = 1 G+ ,
we therefore obtain a spectral sequence with E2term
8
< Extp,q (Fp, H*(G+ )); q < k (k)
E2p,q= : H*(G+) =) Hp+q(map G(EG+ , 1 G+ ).
0; q > k
Because of Lemmas 19 and 20, H*(G+ ) is injective as a module over itself,
and hence E2p,q= 0 for q > 0 and q 6= k. On the other hand,
E2p,0= E1p,0= Hom qH*(G+)(Fp, H*(G+ )) = P (H*(G+ ))
with the notation of Lemma 19, and hence
8
< Fp; n = d
Hn(map G(EG(k)+, 1 G+ )) ~=:
0; n < k and n 6= d
2
5 Selfduality for pcompact groups
5.1 Two Lemmas on restricted homotopy fixed points
Lemma 23 For a subpcompact group H < G, there is a coarse Gequivalence
op
G ^H SH  ~! ( 1 G+ )hoH .
PROOF. First note that as (G x Hop)spectra,
1 G+ ' G ^H 1 H+ ,
where on the right hand side, H acts on the right factor from the right and G
acts on the left factor from the left.
19
We therefore have a map
op 1
G ^H SH = G ^H map H (EH+ , H+ )
op 1 1 hoHop
 ! map H (EH+ , G ^H H+ ) = ( G+ ) .
This map is clearly Gequivariant, and it is a weak equivalence because G and
H are nonequivariantly dualizable.
2
If X 2 (G x Gop)S and Y 2 HopS, we have Gequivariant homotopy equiva
lences (given by shearing maps)
G ^H X ' G=H+ ^ X
and
op
map H (Y ^ G+ , X) ' map (Y ^H G, X)
In particular, if Y 2 (G x Gop)S, we have
op
map H (Y ^ G+ , X) ' map (Y ^ G=H+ , X). (5.1)
Lemma 24 Let H < G be as above. Then there is a coarse Gequivalence
op ~
(DG+ ) hoH ! D(G=H+ ),
natural on subgroups of G.
PROOF. The map is the following composite of coarse Gequivalences, all
of which are natural:
op H
(DG+ )hoH = map (EH+ , DG+ )
' map H(EG+ , DG+ )
! map H(EG+ ^ G+ , S0)
f 0
! map (EG+ ^ G=H+ , S )
g
! D(G=H+ ).
20
For the first homotopy equivalence, we use that EG is a valid model for EH.
f is a Gequivariant homotopy equivalence by (5.1). Since EG+ has the usual
right action and a trivial left action, the map S0 ! EG+ is a left Ghomotopy
equivalence, and hence so is g.
2
5.2 Absolute Poincar'e duality
Denote by Gc the suspension spectrum of G with G acting by conjugation.
For G a Lie group, SG can be identified with the onepoint compactification
of a neighborhood of the identity in G; this identification is Gequivariant
if we equip G with the conjugation action. The following lemma shows that
such a öl garithmä lso exists for pcompact groups, at least up to a coarse
Gequivalence.
Lemma 25 For every pcompact group G, there is a Gspectrum E(G), a
natural coarse Gequivalence E(G) ! (Gc)+ , and a Gequivariant retraction
E(G) ! SG .
Remark: An equivariant retraction X ! Y means two equivariant maps
Y ! X ! Y
such that the composite is a coarse Gequivalence.
PROOF. The auxiliary spectrum E(G) is defined as
op 1
E(G) = map G (EG+ , G+ ^ DG+ ).
Consider 1 G+ as a (G x Gop)spectrum by left and right multiplication.
Then the diagonal map
1 G+ ! 1 G+ ^ 1 G+
is (G x Gop)equivariant and has an equivariant adjoint
1 G+ ^ DG+ ! 1 G+ . (5.2)
Similarly, the (G x Gop)equivariant projection map to the first factor
1 G+ ^ 1 G+ ! 1 G+
21
has an equivariant adjoint
1 G+ ! DG+ ^ 1 G+ .
The composite
1 G+ ! 1 G+ ^ DG+ ! 1 G+ (5.3)
is a weak equivalence.
Taking homotopy fixed points with respect to Gop = 1 x Gop G x Gop on
the left hand side of (5.2) yields
op 1 ~ Gop 1
E(G) = map G (EG+ , G+ ^ DG+ )  ! map (EG+ ^ G+ , G+ )
(2.2)
 ~! map (EG 1
+ , G+ )
 ~! map (S0, 1 G 1
+ ) ' G+
(5.4)
As in Lemma 24, the map induced by S0 ! EG+ is indeed a Ghomotopy
equivalence because the left Gaction on EG+ is trivial. In fact, all maps but
the first one are Ghomotopy equivalences.
We have to check that the Gaction on E(G) corresponds to the conjugate
action on 1 G+ .
op 1
The action of G on M = map G (EG+ ^ G+ , G+ ) is given by
(g.f)(x ^ fl) = gf(x ^ g1 fl) (g 2 G, f 2 M, x 2 EG+ , fl 2 G+ ).
The induced action of G on map (EG+ , 1 G+ ) is
(g.f)(x) = gf(xg)g1 (g 2 G, f 2 map (EG+ , 1 G+ ), x 2 EG+ )
since
op 1 1
map G (EG+ ^ G+ , G+ ) ! map (EG+ , G+ )
f 7! x 7! f(x, 1)
g.f 7! x 7! gf(x, g1 ) = gf(xg, 1)g1 .
The restricted Gaction on map (S0, 1 G+ ) becomes
(g.f)(x) = gf(x)g1
since S0 has the trivial Gaction, and the Gaction on 1 G+ is indeed by
conjugation.
22
op
Applying ()hoG to (5.3)yields the desired retraction E(G) ! SG .
2
Proposition 26 Regard the Gspectrum SG as a (G x Gop)spectrum with
trivial Gopaction. Then there is a coarse G x Gopequivalences
SG ^ DG+ '! 1 G+
On Gophomotopy fixed points, these maps make the following diagram com
mute:
op ' hoGop ' hoGop
SG = ( 1 G+ )hoG Zoo___(SG ^ DG+ ) oo___SG ^ (DG+ )
ZZZZZZZZZZZZZZZ
ZZZZZZZZZZZZZZZZZZ ' Lemma24
ZZZZZZZZZZZZZZZZZZ 
ZZZZZZZZZZZZZZZZfflffl
SG ^ S0
PROOF. We will have to deal with spectra with three Gactions, and for ease
of notation, for a (G x Gop)spectrum X, I will denote by aXbcthe spectrum
X with the left action a and the two right actions b and c, where a, b, c, are
one of the following:
o `O' denotes a trivial action
o `l' denotes the action from the left _ if this symbol appears on the right
then G acts by inverses from the left
o `r' denotes the action from the right _ if this symbol appears on the left
then G acts by inverses from the right
The main ingredient is a shearing map
l( 1 G+ )O l r sh l 1 O O l
r ^ (DG+ )O ! ( G+ )r ^ (DG+ )r. (5.5)
which is adjoint to
l( 1 G+ )O O 1 l l 1 O l 1 r
r ^ ( G+ )r ! ( G+ )r ^ ( G+ )O
g ^ h 7! g ^ hg.
This map is clearly a weak equivalence, and it is straightforward to check that
it is (G x Gop x Gop) equivariant as claimed.
23
By passing to homotopy orbits with respect to the O Ooaction of Gop in (5.5),
we obtain a (G x Gop)equivariant homotopy equivalence
i j hoGop i jhoGop
l( 1 G+ )O l r _____//l 1 O O l 0
r ^ (DG+O)OO ( G+ )r ^ (DG+ )r =: E (G)
'  ' 
 fflffl
l(SG )O ^ l(DG+ )r l( 1 G+ )r
The underlying spectrum of E0(G) is the spectrum E(G) of Lemma 25. It is
easy to see that with the remaining operations, the map E0(G) ! 1 G+ ,
described in (5.4), is (G x Gop)equivariant.
For the assertion about Gophomotopy orbits, observe that by changing the
order of taking Gophomotopy orbits, we have a large commutative diagram
___________________________________________*
*_____________________________________________________________________@
___________________________________________________*
*_____________________________________________________________________@
________________________________________________________*
*_____________________________________________________________________@
lSG = lGhoGop oo_____'_____lSG ^ O(DG)hoGop ___'__________________*
*_______//_lSG
OO
  
'  '  
i  j O o i fflfflj O 
lGO O l ho ooo_'__ lO O hoGop ho o _'__//_l
r ^ (DG)rOO Gr ^ (DG)OOO SG

' sh ' sh 
i  j hoO o i  j hoO 
lGO l r ooo_'___ l l hoGop o__'__//l
r ^ (DG)OOO Gr ^ (DG)OOO SG

'  ' 
i  jhoO o  op
l(SG )O ^ l(DG)r oo_'____lSG_^ l(DG)hoG
For space reasons, the disjoint basepoint for 1 G and DG have been omitted
as well as the suspension functor 1 for G.
The important, if trivial, observation is that the shear map becomes homotopic
to the identity when passing to O oOhomotopy orbits on the DG+ factor. The
diagram claimed to be commutative in the proposition is the öb undaryö f
the diagram above.
2
24
5.3 Relative Poincar'e duality
Corollary 27 For any subpcompact group H of G, there is a zigzag of coarse
Gequivalences
G ^H SH oo'_//_____________D(G=H+ ) ^ SG
This zigzag is natural in the following sense: for any chain of pcompact groups
K < H < G, the following diagram commutes:
op ' '
( 1 G+ )hoH oo___G ^H SH oo__//____________D(G=H+ ) ^ SG
res D(proj)^id
fflfflop fflffl
( 1 G+ )hoK oo'__G ^K SK oo'_//_____________D(G=K+ ) ^ SG
PROOF. From Proposition 26, we have a coarse (G x Hop)equivalence
DG+ ^ SG ! 1 G+ .
Applying Hophomotopy orbits turns coarse (GxHop)equivalences into coarse
Gequivalences, and since the right actions on SG and SH are trivial, we obtain
op ' hoHop '
(DG+ ^ SG ) hoH oo___DG+ ^ SG ____//_D(G=H+ ) ^ SG
'
fflfflop
( 1 G+ )hoH oLem'mao23G_^H SH
For naturality, consider the following diagram:
op hoHop 1 hoHop
D(G=H+ ) ^ SG oo___DGhoH+ ^ SG _____//(DG+ ^ SG ) ____//_( G+ )
   
   
fflffl opfflffl fflfflop fflfflop
D(G=K+ ) ^ SG oo___DGhoK+ ^ SG _____//(DG+ ^ SG )hoK ____//_( 1 G+ )hoK
The left hand square commutes by Lemma 24, the other two for trivial reasons.
2
5.4 Definition of the transfer
Proof of Theorem 2: SG was constructed in Section 4. We obtain a (nonequiv
ariant) map
_ 1 hoGop 1 hoHop'
t: SG = ( G+ ) _____//( G+ ) __f_//_G ^H SH
25
coming from restricting from G to Hhomotopy fixed points. Here, f is the
nonequivariant homotopy inverse of the coarse Gequivalence given by Lemma
23.
By Lemma 6, there is also a Gequivariant map
~t: EG+ ^ SG  ! G ^H SH
such that the composite
SG ! EG+ ^ SG ! G ^H SH
is homotopic to ~t, and ~tis unique up to homotopy with this property.
To finish the proof of Theorem 2, we need to show that
~t*: Hd(SG ; Fp) ! Hd(G ^H SH ; Fp)
is an isomorphism for d = dim G. This now follows easily from Corollary 27:
The map ~tis by construction the composite
SG ! D(G=H+ ) ^ SG !~ G ^H SH .
Since the first map is an isomorphism in Hd(; Fp), so is the composite. 2
The first part of Theorem 3 claims that H*(BGg; Fp) is a Thom module over
H*(BG+ ; Fp). This follows from the spectral sequence
E2 = H*(BG+ ; H*(SG ; Zp)) =) H*(EG+ ^G SG ; Fp) = H*(BGg; Zp).
2
Definition 28 For a monomorphism H < G of pcompact groups, the trans
fer map tG,H is given by applying Ghomotopy orbits to the Gequivariant map
~t.
The domain of ~tis EG+ ^G (EG+ ^ SG ), which by Lemma 6 is homotopy
equivalent to BGg. For the functoriality, it is sufficient to notice that the
following diagram of Gequivariant maps commutes:
26
SG _____//MMM( 1 G+ )hoHopoo____G ^H SH
MMM 
MMM 
M&&M op fflfflop
( 1 G+ )hoKiiTT G ^H ( 1 H+O)hoKO
TTTTTT 
TTTTT 
TT 
G ^K SK
That is a less than remarkable statement since no two maps are composable.
But all of the maps going left or up or both are coarse Gequivalences, and the
diagram stays (nonequivariantly) homotopy commutative if we invert them.
By its definition, the commutativity of
SG _____//_G ^H SG
 
 
fflffl fflffl
BGg ______//_BHh
as claimed in Theorem 3 is immediate.
The next section will be devoted to identifying the transfer map on the cate
gory of HZ=plocalizations of compact Lie groups and monomorphisms.
6 Identification of the transfer map
Using the construction of t for {1} < G, we have a commutative diagram
coming from the natural transformation id ! ()hoG :
SG ______// 1 G+ (6.1)
 
 
fflfflt fflffl
BGg _____//B{1}+______S0
We will now identify t with the transfer map from the introduction in the case
where G is of Lie type.
First note that in (6.1), the composite map Sd ! S0 is indeed the same as the
ThomPontryagin construction on G if G is a Lie group: The ThomPontryagin
construction on G is given by the composition of maps
S0 ! DG+ ' Sd ^ G+ ! Sd,
27
where the first map is a desuspension of the map from the embedding sphere to
the Thom space of the normal bundle of G, which is DG; by Proposition 26 or
since the tangent bundle of G is trivial, this is equivalent to a desuspension *
*of
1 G+ ; and the second map is the projection of G+ to S0, the map classifying
the (trivial) stable normal bundle of G.
Using the Gequivariant isomorphism from Proposition 26, we have
()hoG
SG _______//KKO1OG+_____//S0
KKK _____
KKK _~___
KK%%fflffl___
SG ^ DG+
The bottom composition is the ThomPontryagin construction, the upper one
the map from (6.1).
6.1 An alternative construction of the transfer map
To show that t agrees with the Umkehr map not only on the bottom cell, we
will compare its definition to another, equally general construction, reminis
cent of Dwyer's construction of the BeckerGottlieb transfer in [Dwy96 ]. This
will be equivalent to the classical construction of the Umkehr map in the Lie
case.
Let H < G be pcompact groups. The quotient 1 G=H+ is dualizable, and
the projection D(G=H+ ) ! S0 onto the top cell is equivariant and has a
section ff; however, ff is not Gequivariant unless H = G. But we do get an
equivariant map if we "free up" the Gaction on S0: consider the following
diagram of Gequivariant maps:
1 EG+ ~___//_S0___//map( 1 G=H+O,O 1 G=H+ )
'''

D(G=H)+ ^ 1 G=H+ 1^i___//_D(G=H)+ ^ S0.
The map ß is the projection 1 G=H+ = 1 G=H _S0 ! S0, and j is a coarse
Gequivalence. Therefore, by Lemma 6, there exists an equivariant lifting
1 EG+ ! D(G=H+ ) (6.2)
which is nonequivariantly homotopic to the map
1 EG+ ! S0 ff!D(G=H+ ).
28
In the case where H and G are (localizations of) Lie groups, the homotopy
orbit space of D(G=H+ ) under this Gaction is the (localization of the) Thom
space of , the normal bundle along the fibers of BH ! BG. This follows
from the observation that BH = EG xG G=H, and that stably, the normal
bundle along the fiber is the fiberwise SpanierWhitehead dual, i.e. = EGxG
D(G=H). Hence its Thom spectrum is EG+ ^G D(G=H+ ), as claimed.
By passage to Ghomotopy orbits in (6.2), we therefore obtain a map
1 BG+ ' EG+ ^G 1 EG+ ! BHg=h,
where BHg=h denotes the Thom spectrum of the virtual inverse of the adjoint
bundle of G, pulled back to BH, modulo the adjoint bundle of H. This is the
Lie theoretic model of the normal bundle along the fibers.
Returning to the case of a general pcompact group, we now introduce a "twist
ing" by smashing source and target of the map with SG :
EG+ ^ SG _____//map( 1 G=H+ , S0) ^ SG __'__//D(G=H+ ) ^ SGCoo'or./27/_______*
*______G ^H SH
By Lemma 6 and since EG+ ^ SG is a free Gspectrum, we obtain a Gmap
(unique up to homotopy)
~t0: EG+ ^ SG ! G ^H SH ,
and passing to Ghomotopy orbits, we obtain:
t0: BGg = EG+ ^G SG ! EH+ ^H SH = BHh.
Lemma 29 Let H < G be pcompact groups. Then
~t' ~t0: EG+ ^ SG  ! G ^H SH
PROOF. We have to show that the following Gequivariant diagram com
mutes:
EG+ ^ SG __(6.2)//_D(G=H+O)O^_SG
____
~t ~ ______
fflffl~ fflffopl___
G ^H SH oo___(DG+/^/SG_)hoH___________
Since EG+ ^ SG is a free Gspectrum, there is by Lemma 6 a Gequivariant
map going diagonally
op
EG+ ^ SG ! (DG+ ^ SG )hoH
29
and making the upper right triangle of the diagram commute. The commuta
tivity of the lower left triangle then follows from the observation that in the
commutative diagram
~ op hoHop
SG oo___(DG+ ^ SG )hoG ____//_(DG+ ^ SG ) ,
  
  
fflffl fflfflop fflfflop
SG ________( 1 G+ )hoG ________//_( 1 G+ )hoH
the left hand map is the identity by Proposition 26.
2
Conclusion of the proof of Theorem 3 The previous lemma implies the
third part of the theorem (namely, that t O Lp ' Lp O ()! on compact Lie
groups and monomorphisms). Indeed, by applying Ghomotopy orbits to the
diagram in Lemma 29, the map induced by ~t0is homotopic to t, and the
preceding discussion shows that the former map is the classical Umkehr map
in the Lie case.
2
7 Computational methods
In this section, I will describe a general method for computing the homotopy
class represented by a pcompact group by constructing a representing cycle
in the Adams spectral sequence for a complex oriented cohomology theory E.
Let G be a simply connected ddimensional pcompact group of rank r with
maximal torus T . We want to identify the maps the following diagram induces
in the E2term for the Ecohomology ASS:
Sd ! BGg ! BT t! S0
7.1 The S1transfer
The righthand map is a suspension of the rfold smash product of the S1
transfer map
ø : CP+1 ! S1.
30
It is wellknown that the homotopy fiber of this transfer map is the spectrum
CP11, the Thom spectrum of the inverse of the universal line bundle on CP 1,
the fiber inclusion CP11! CP+1 being the obvious projection map onto the
0coskeleton.
For a complex oriented cohomology theory E and a finite spectrum X, there
is an AdamsNovikov spectral sequence
E2 = Ext(E*(X)) =) [X, LE S],
where, for an (E*, E*E)comodule A, Ext(A) is a shorthand for ExtE*E (E*, A).
We will now first restrict to the case of finite dimensional projective spaces
and study the map CP+m ! S1 as a map of E2terms of this ANSS
Ext(E*(S1)) _______+3_ß*(LE S1)
 
 
fflffl fflffl
Ext(E*(CP+m)) _____+3_[CP+m, LE S].
By a change of rings isomorphism, this spectral sequence is isomorphic to the
one associated to the Hopf algebroid
(Am , m ) = (E*(CPnm), (E ^ E)*(CPnm)) .
Note that Am represents the following functor:
8 9
>>> E* !ffR, >>
>>> fi >>>
< fifif is a function modulo degree m + n + 1 on the formal >=
R 7! > (ff, f) fifigroup law on R given by the image of the universal formal
>>> fi * * >>>
>>: group law under MU ! E ! R such that f vanishes to >>>;
the nth order at the identity.
Similarly, m represents isomorphisms of such data. Hence, for E = MU,
(Am , m ) classifies formal groups with an m + n + 1truncated function on it
that vanishes to the nth order at the identity. This interpretation makes it
easy to compute the structure maps of (Am , m ).
First assume that n = 0. Pick coordinates z such that
E*(CP0m) = E*[[z]]=(zm+1 ) and (E ^ E)*(CP01) = (E ^ E)*[[z]]=(zm+1 ).
Since there is a map of Hopf algebroids (E*, E*E) ! (Am , m ), we only need
to determine jL(z) and jR (z). We can make jL(z) = z by choice of coordinates;
31
then, jR (z) will be the image of the universal isomorphism
m1X
bizi+1 2 (MU ^ MU)*(CP0m)
i=0
in m .
As usual, MU* = Z[mi] and MU*MU = MU*[bi].
If n 6= 0, E*(CPnm) is still a free E*module, generated by {zn, zn+1 , . .,.zn*
*+m },
and the above formula for jR is correct when interpreted as jR (zi) = jR (z)i.
For our purposes, it would be easier to use BP theory instead of MU since we
are working in a plocal setting anyway. However, it only affects the complexity
of the computations, not the method.
Assembling all spectral sequences for varying m 0, we obtain towers
.. . .
. ..
 
 
fflffl fflffl
Ext m+1(Am+1 , Am+1 )_____+3_[CPnm+1, LE S]
 
 
fflffl fflffl
Ext m (Am , Am )________+3_[CPnm, LE S]
 
 
fflffl fflffl
.. .
. ..
The inverse limit of the left tower is not quite the Ext term associated to the
Hopf algebroid (A, ) = (E*(CP 1), (E ^ E)*(CP 1)). This is due to the fact
that
(E ^ E ^ E)*(CP 1) fi A
(the left hand side is a completion of the right hand side).
Similarly, the inverse limit of the tower on the right hand side is not quite
[CPn1, LE S]. It does not include the phantom maps.
Coming back to the problem of determining the induced map of the S1transfer
on E2terms, we look at the cobar construction functor
Bn(M) = M E* (E*E) E*n
for an (E*, E*E)comodule M.
32
0OO 0OO 0OO
  
  
  
Ext( 1S1)OOoo_______Ext(CP11)Ooo_______ExtO(CP01)OO
  
  
  
Z*(E* 1S1)OO oo_____Z*(E*CP11)OOoo_____Z*(E*CP01)OO oo____0
d1 d1 d1
  
0 oo___B*1(E* 1S1)OO oo___B*1(E*CP11)OOoo___B*1(E*CP01)OO oo___0
  
  
  
Z*1(E* 1S1)OO oo___Z*1(E*CP11)OOoo___Z*1(E*CP01)OO oo___0
  
  
  
0 0 0
Fig. 7.1. This diagram admits a snake map.
Since E*, E*E, and H*(CP 1) are concentrated in even dimensions, and since
B is an exact functor on flat E*modules, we have a short exact sequence of
B(E*, E*E)modules
0 B(E* 1S1)) B(E*(CP11)) B(E*(CP01)) 0.
If we denote by Z*(X) B*(X) the cycles under the cosimplicial differential
d1, we have a diagram as shown in Figure 7.1. It follows from the snake lemma
that the kernel of the top right map is the image under the snake map
Z*1(E*S2) ! Ext(CP01),
which is, by following through the diagram, the image of d1E*{z1}.
Now if T is a pcompact torus, the transfer map in cohomotopy is simply the
rfold smash product of the map represented at the E2level by dE*{z1}.
7.2 The map Sd ! BGg ! rBT
We will first study the effect of this map on rational cohomology. By Theorem
12, H*Qp(BG) = H*Qp(BT )W(G) is always a polynomial algebra.
Proposition 30 For a pcompact groups G with maximal torus T , the fol
33
lowing diagram commutes:
H*Qp(BT+ )__proj_//H*Qp(BT+ ) = (H*Qp(BG))Cor._13_H*Qp(G=T+ )
rt* 
fflffl fflffl
H*Qp( rBGg) _____~=___//H*Qp(BG+ ){ø }__fi7!'__//H*Qp(Sdr),
where ø is the Thom class of BGg, and ' is the generator in H*(Sdr).
This allows us to compute the effect of the map
Sd ! BGg ! rBT fiber
in cohomology (this is the bottom composition of maps in the diagram) by
simply evaluating at the image in H*Qp(BT+ ) = (H*Qp(BG)) of the fundamental
class of G=T .
Proof of the proposition
By the construction of the transfer map, we have a commutative diagram
G ^TOSrO_____// rBT+OO,
 
 
 
Sd _________//BGg
and by Theorem 2(2), the left hand map Sd ! rG=T+ is an isomorphism in
the top homology group. Desuspending r times and applying H*Qpyields the
commutativity of the diagram of the proposition.
2
Now let E be a HZ=plocal complex oriented torsion free cohomology theory.
Denote by EQp the cohomology theory X 7! E*(X) Zp Qp.
We have E*(CP 1) = E*[[z]] ,! E*Qp(CP 1), and the same is true for E ^
E. Hence for computing a cobar representative, we can work with rational
coefficients and always hope that in the end of our computations, everythings
turns out integral. To compute this, we can use the Chern characters
exp : E*Qp(X) ! H*Qp(X) ^E*
and
exp : (E ^ E)*Qp(X) ! H*Qp(X) ^ß*(E ^ E)
34
which, for X = CP+1, is the exponential map for the formal group law associ
ated with E and an isomorphism, and for X = BT+ , a tensor power thereof.
The smash product is formed in the HZ=plocal category, as always.
This induces an isomorphism of (EQp *, (E ^ E)Qp *)modules
B(exp ) : B(E*QpBT+ ) ! B(E*) ^H*Qp(BT+ ).
We have a commutative diagram
B(E*QpBT+ ) oo___________B(E*BT+o)o
m PPPP
mmmm   PPPP
mmmm   PPP
vvmmmm   PP((
B(E*Qp) oo__________________________________________________B(E*)oo
hhQQQ   nnn66
QQQQ   nnn
QQQQQ   nnnn
Q fflffl fflfflnnn
B(E*Qp) H*(BT+ ) oo___B(E*)oo H*(BT+ )
So, to evaluate the class in B(E*(BT )) computed in the first part, we apply
B(exp ) to it and obtain a class in B(E*) H*Qp(BT ), which we then eval
uate at the image of the fundamental class [G=H] in (HQp )dr(G=H+ ) !
(HQp )dr(BT+ ). This class then must actually be integral.
8 The family no. 3 of groups ~m
The pcompact group ~m , for p ~=1 (m), has rank 1 and Weyl group W Zxp
a cyclic subgroup of order m of the padic units. It is a nonmodular group and
can therefore be constructed as
B~m = Lp (K(Zp, 2) xW EW ) .
Therefore, H*(B~m ; Zp) = Zp[[z]]W , where a W acts on z by multiplication.
This shows that
H*(B~m ; Zp) = Zp[[zm ]] ,! Zp[[z]].
The fundamental class
[~m =T ] 2 H*Qp(BT )=(H*Qp(B~m )) = Qp[z]=(zm )
is zm1 , and we conclude that ~m has dimension 1 + 2(m  1). It is straight
forward to see that for m < p  1, ~m cannot represent a nontrivial ho
motopy class in the pstems because (ßsn)(p)= 0 for 0 < n < 2p  3. But
(ßs2p3)(p)= Z=p{ff1}, and we will see that ~p1 represents this class.
35
In the pcompleted BP spectral sequence for ß*(CP11),
jR (z) = z + t1zp + O(zp+1), hence jR (z1 ) = z1  t1zp2 + O(zp1).
Applying the Chern character to this power series does not change it up to
O(zp1), and hence [~p1] is the coefficient of zp2 of this series, which is t*
*1.
Lying in filtration 1, t1 represents the homotopy class ff1.
9 Some exceptional cases
9.1 The 5compact group no. 8
The pseudoreflection group G which is no. 8 in Shephard and Todd's list has
order 96 and is generated, as a complex reflection group, by the two reflections
0 1 0 1
1_ i_ 1_ i_
B@i 0CA and B@2 2 2 + 2 CA.
0 1 1_2 i_21_2 i_2
The ring of invariants Z5[x1, x2]G is polynomial because 5 does not divide the
order of G; a straightforward calculation shows that it is generated by the
polynomials
~ = x81+ 14x41x42+ x82
and
= x121 33x81x42 33x41x82+ x122.
Hence H*(BG; Z5) = Z5[~, ], and by Proposition 15 the cohomology of G=T
is given by
H*(G=T+ ; Z5) = H*(BT+ ; Z5) = (H*(BG; Z5)).
A Gröbner basis calculation shows that the top class in H36(G=T ; Z5) is
1 3 15 1 15 3
x71x112= x111x72= ___x1x2 = ___x1 x2. (9.1)
13 13
We will use the 5primary BP spectral sequence for ß*(CP11) to determine
the homotopy class G represents. In this spectral sequence,
36
i j i j
jR (z) = z  t1z5 + 5 t12+ t1v1 z9 + 35 t13 12 t12v1 t1v12 z13
i j
+ 285 t14+ 137 t13v1+ 21 t12v12+ t1v13 z17
+O(z21)
and hence
i j i j i j
d1(z1 ) =  t1z3 + 4 t12+ t1v1 z7 + 26 t13 10 t12v1 t1v12 z11
i j
+ 204 t14+ 106 t13v1+ 18 t12v12+ t1v13 z15
+O(z19).
Applying the Chern character to this class yields
` 8 t v ' _ 78 t2 v 78 t v 2!
f(z) = t1z3 + 4 t12+ ___1_1_ z7 + 26 t13 ____1__1_ ____1_1_ z11
5 5 25
_ !
816 t13v1 1224 t12v12 816 t1v13 15
+ 204 t14+ _________ + ___________ + _________ z
5 25 125
+O(z19).
We need to evaluate the class f(z) f(z) at the classes given in (9.1)and add
them up. This yields:
[G]= 204t1 t41 808t21 t31 1208t31 t21 604t41 t1
160v1t1 t31 480v1t21 t21 320v1t31 t1
48v21t1 t21 48v21t21 t1.
By adding a suitable boundary, namely
d1(4t51+ 45v1t41+ 34v21t31+ 10v31t21+ v41t1),
we see that this class is homologous to
t1 t41+ t41 t1 + 2(t21 t31+ t31 t21),
which is the representative of fi1 in the ANSS.
37
9.2 The 3compact group Za2 (no. 12)
The Weyl group W of the modular group Za2 constructed by Zabrodsky
[Zab84 ] is generated by the two matrices
0 1 0 1
_1_p_i_p_ 1_ i_ 1_ i_
B@ 2 2CA and B@ 2 + 2 2 + 2CA.
_i_p_21_p_2 1_2+ i_21_2 i_2
Although 3  #W , the ring of invariants Z3[x1, x2]W is polynomial, generated
by the polynomials
~ = x81+ 14x41x42+ x82
and
= x51x2  x1x52.
We find that the top class in H36(G=T+ ; Z3) is
1 12 1 12
x41x82= x81x42= ___x1 = ___x2 . (9.2)
15 15
In the 3primary BP ANSS, logarithm, exponential, and universal isomor
phism are all odd power series; hence, evaluation at the class above yields 0
without further computations.
This means that [Za2] is of filtration at least 3; however, the only 38dimensi*
*onal
class in the AdamsNovikov E2term is fi3=2 in filtration 2.
Acknowledgements
I would like to thank my advisor Michael Hopkins for his constant support
and encouragement, for much input on this paper, and for explaining to me so
much of what I did not understand. I am also grateful for helpful discussions
with Haynes Miller and Jean Lannes.
38
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