The EilenbergMoore spectral sequence for
Ktheory; applications to pcompact
homogeneous spaces
A. Jeanneret & A. Osse*
Abstract
We construct the EilenbergMoore spectral sequence for some gen
eralized cohomology theories, along the lines of Smith and Hodgkin.
We prove its multiplicativity and give some sufficient conditions for
its convergence to the desired target. As applications, we compute the
Ktheory of various spaces associated to pcompact groups.
AMS subject classification (1991): 55T20, 55N15, 55P35, 57T35.
Keywords: Spectral sequences, Ktheory, completed tensor product,
pcompact groups.
Introduction
The topology of homogeneous spaces constitutes an important aspect of Lie
theory. The celebrated works of Borel, Bott, Baum, Smith and others com
pletely describe their rational cohomology and provide important informa
tions on their integral cohomology. For the Ktheory of homogeneous spaces,
the EilenbergMoore type spectral sequence constructed by Hodgkin has been
the most successful tool.
Recently Dwyer and Wilkerson [7] have introduced a homotopical gener
alization of the Lie groups, namely the pcompact groups. In this homotopy
Lie theory, the notions of Weyl group, maximal torus, homogeneous spaces
are well defined, even if there is still no "differentiable structure" in sight.
The present work originates in understanding the topology of the p
compact homogeneous spaces of Dwyer and Wilkerson. An important in
variant of these spaces is their padic cohomology. Its rational part can be
______________________________
*Supported by grant No 2046899.96 of the Swiss National Fund for Scientific*
* Research
1
computed by combining a classical EilenbergMoore spectral sequence ar
gument with the result of [7]. At the moment, the main indications about
the torsion in padic cohomology are conjectural (see however the results of
[16]). Because of the "lack of differentiability", the Ktheoretical construc
tions of Hodgkin do not apply to the pcompact setting. To circumvent to
this difficulty, we have to go back to the original definition of the spectral
sequence, by basing all the constructions on the geometric cobar resolution.
This has led us to rewrite, adapt and simplify many classical arguments.
This approach turns out to be successful and allows us to deal with other
generalized cohomology theories. To be more precise, let E*( ) be a gener
alized multiplicative cohomology theory such that E*(pt) is a graded field.
Examples of such theories are given by singular cohomology with coefficient
in a field, complex mod p Ktheory and Morava Ktheories. With our as
sumption, E*(X) is a complete Hausdorff topological E*(pt)algebra for any
space X. Moreover, E*( ) satisfies the K"unneth isomorphism (with com
pleted tensor products). These two properties are the crucial ingredients for
the construction and the study of the EilenbergMoore spectral sequence for
E*( ). Our main result can be stated as follows:
Main Theorem Let B be a connected space and E*( ) as above. For any
pullback diagram
X xB Y ____X
 
 
 p
? f ?
Y _______B
there exists a strongly convergent spectral sequence {E*;*r(X; Y ); dr}r2 sati*
*s
fying the following properties:
1. The spectral sequence is multiplicative and compatible with the stable
operations of E*( ).
2. Ei;*2~=ToriE*(B)(E*(X) ; E*(Ya))s algebras, where Tor iE*(B)(; )is the
ith derived functor of the completed tensor product.
3. If p : X ! B is a fibration and E*(B )is an exterior algebra on odd
degree generators, then the spectral sequence converges to E*(X xB Y ).
The third point of the Theorem desserves the following comment. As
easily seen, if E*(B )is an exterior algebra, then E*(B )is connected; hence
B is 1connected. It is well known that this weaker property of B suffices to
2
ensure the convergence of the spectral sequence, when E*( ) = H*(; K),
the singular cohomology with coefficient in a field K (see [18])
With respect to our original aim, we apply the main theorem to compute
the Ktheory of various pcompact homogeneous spaces. These computations
lead to a new proof of the main result in [12]. We refer to Section 5 for more
precise statements.
The paper is organized as follows. The construction of the spectral se
quence is given in Sections 1 and 2. The ideas are standard but our methods
are slightly different from those in [18] and [11]. The multiplicative structure
is discussed in Section 3. The reader will notice that this is the most delicate
part of the paper; we assure her/him that we did our best to present the ideas
as clearly as possible. The convergence questions are treated in Sections 2
and 4. Proposition 2.2 of Section 2 is already stated in Hodgkin's paper, but
the proof presented here is new and hopefully simpler. Section 4 is crucial for
the application to the pcompact groups. In contrast to Hodgkin, we are able
here to get rid of the "differentiability hypothesis". In addition, the geomet
rical nature of our constructions readily implies that the spectral sequence is
compatible with the stable operations of E*( ). We mention that the results
of Section 1 to Section 4 are stated for sectionned spaces over B and imply
the main theorem, via the basepoint adjunction trick discussed in Section 1.
The applications are discussed in Section 5. We end with an appendix on
profinite rings and modules. The theme of the appendix is well known and
has been considered by some topologists (see [3] and [23]). We have decided
to discuss it in some details because of its central role in our arguments and
also because we didn't find any reference suitable to our needs.
Aknowledgements. We would like to thanks U. Suter and U. W"urgler
for helpful discussions. We also thank J.M. Boardman for sending us a copy
of [2].
1 The geometric cobar resolution
For the main construction of this section, we need some recollections about
fiberwise topology. Until further notice, B is a fixed pointed and connected
space; T op=B will denote the category of spaces over B with fibre maps as
morphisms. The base point inclusion {pt} ,! B, viewed as an element of
T op=B, will simply be denoted pt. The categorical product in T op=B is the
familiar fibred product.
In the sequel we will work in the pointed version of this category, namely
the category (T op=B)* of sectionned spaces over B. The objects are triples
3
p
B s! X ! B such that p O s = id and morphisms are commutative
diagrams
X1
s1 ae>  Z p1
aeae  ZZ"
B OE B
Z Z  ae>
s2 Z" ?aeaep2
X2
When there is no danger of confusion, we simply write X (respectively
p
: X1  ! X2 ) instead of B s! X ! B (respectively the diagram above).
As the reader may have noticed, we recover the categories T op of topological
spaces and T op* of pointed topological spaces by setting B = pt. It is
well known that all the usual homotopic theoretic constructions can be per
formed in the category (T op=B)*. For instance, we have fiberwise homotopy,
fiberwiseWmapping cone CB (), fiberwise suspension B (), fiberwise wedge
 B, etc. We refer to [18] for the definitions. Two other constructions
play a role in what follows:
1. If (X; x0) is a pointed space, (X; x0)B will denote the sectionned space
over B
pr2
B s0!X x B ! B
where pr2 is the projection on the second factor and s0(b) = (x0; b).
p
2. Let X  ! B be a space over B and A a closed subspace of X. The
fiberwise collapse of X with respect to A is the sectionned space
p
B s! X=B A ! B
defined by:
`
o X=B A = (X B)=a ~ p(a) 8a 2 A
o p(x) = p(x); 8x 2 X and p(b) = b 8b 2 B
o s(b) = b; 8b 2 B.
In order to`simplify the notation, we write X=A for X=B A and X+ for
X=B/O = X B (base point adjunction). We mention in passing that X=A is
fiberwise homotopy equivalent to the fiberwise cone of the obvious inclusion
A+ ,! X+ .
We turn to an important fact, also treated in [18] and [11]: for any map
: X1  ! X2 in (T op=B)* there is a Puppe sequence of maps in (T op=B)*
i() j() B()
X1 ! X2 ! CB () ! B X1  ! B X2 ! : : : (1)
4
with the following properties:
1. For any Z in (T op=B)*, the Puppe sequence of the map ^B id is
obtained from (1) by smashing with Z.
2. If s1; s2; s respectively denote the sections of X1; X2 and CB () then
i()
X1=s1(B) ! X2=s2(B) ! CB ()=s(B)
is a cofibration sequence in the category T op*.
Definition 1.1 A cohomology theory on (T op=B)* consists of a sequence
of contravariant functors {ehi: (T op=B)* ! Ab }i2ZZ(Ab is the category of
abelian groups) and a sequence of natural transformations {ffii} such that:
1. ffii: ehi(X1)! ehi+1(CB ())is defined for any cofibration sequence
i()
X1 ! X2 ! CB () and the following sequence is exact:
i1 i i()* i * i
: ::! ehi1(X1) ffi!eh(CB ()) ! eh(X2) ! eh(X1) ! : : :
2. (Dold's cylinder axiom) For any space X and any : X x [0; 1]! B;
we have eh*(X x [0; 1]=X) = 0, where the inclusion of X into X x [0; 1]
is given by x 7! (x; 0).
3. (Strong additivity) For any family {Xfi}fi2Jof sectionned spaces over
B, we have _ Y
eh*( Xfi) ~= eh*(Xfi) :
B
It can be checked that these axioms imply that a cohomology theory eh*()
on (T op=B)* is homotopy invariant and possesses a natural suspension iso
morphism, that is
eh*+1(B X) ~=eh*(X)
for any X in (T op=B)*.
Before giving examples, we would like to discuss product structures. A
cohomology theory eh*() on (T op=B)* is called multiplicative if it is equipped
with a natural pairing of graded abelian groups
: eh*(X) eh*(Y )! eh*(X ^B Y ); X; Y 2 (T op=B)*
which is associative and has a unit 1 2 eh0((S0; *)B ). The sectionned space
(S0; *)B plays the role of a point in our category, since (S0; *)B ^B X = X for
5
any X in (T op=B)*. Consequently eh*((S0; *)B ) is a ring, called the coefficie*
*nt
ring of eh*(), and eh*(X) is a two sided module over it. In [11, Lemma 4.1]
it is shown that the pairing factors through:
: eh*(X) h eh*(Y )! eh*(X ^B Y ) ; X; Y 2 (T op=B)*
where h is a short hand for eh*((S0; *)B ).
Example. Let eE*( )be a multiplicative cohomology theory on the category
of pointed spaces and E*( ) the associated unreduced theory. Then
eE*B(:)(T op=B)* ! Ab ; X 7! E*(X=s(B) )
defines a multiplicative cohomology theory on (T op=B)*. The coefficient
ring is eE*B((S0; *)B)~=E*(B), so that any eE*B(X)is an E*(B) module. More
generally, for any fibration X ! B the functor
(T op=B)* ! Ab; Y 7! eE*B(X+ ^B Y )
is also a multiplicative cohomology theory on (T op=B)*, whose coefficient
ring is E*(X) . The interested reader might consult the details in [5, Section
3.4].
From now on, eE*( )will denote a multiplicative cohomology theory on
the category T op* and E*( ) the associated unreduced theory. We will al
ways assume that E*(pt) is a graded field; that is, its non zero homogeneous
elements are invertible. This assumption implies that Ee*( )satisfies the
K"unneth isomorphism (see [3]).
After all these preliminaries, we are now ready to present the geometric
cobar resolution. To this end, we suppose that we are given a space X in
(T op=B)*. From this datum, we perform the following construction:
1. First we set X0 = X.
p
2. For each integer i 0 we suppose that B  s! Xi ! B has been
constructed and we successively define:
o eXi= (Xi=s(B))B
o OEi: Xi ! eXi, OEi(x) = ([x]; p(x))
o Xi1 = CB (OEi).
The most important properties of this construction are given in
6
Lemma 1.2 With the notations above and for any integer i 0, the fol
lowing statements are true:
1. For any space Y 2 (T op=B)*, the spaces (Xei^B Y )=s(B) and Xi=s(B)^
Y=s(B) are homotopy equivalent in T op*.
2. eE*B(Xei)~=eE*B(Xib)E*(pt)E*(B) as E*(B) modules, where the E*(B) 
action on the right hand side is given by right multiplication.
3. This induced morphism OE*i: eE*B(Xei)! eE*B(Xii)s surjective.
Proof. The first part can be safely left to the reader. For the second part,
take Y = (S0; *)B in the first part to obtain Xei=s(B) = Xi=s(B) ^ B+ . It
suffices then to apply the K"unneth isomorphism. Concerning the third part,
let
i: Xi=s(B) ^ B+  ! Xi=s(B)
denote the collapsing of B to a point. Then i O OEi = id, where
OEi: Xi=s(B) ! eXi=s(B) is induced by OEi. Consequently *iis a right in
verse for OE*iand we are done.
Thanks to this Lemma, the long exact sequences of the OEi's break into
short exact sequences:
0 ! eE*B(Xi1)! eE*B(Xei)! eE*B(Xi)! 0 :
As usual in homological algebra, we splice all these sequences together as
follows:
0 0
"ZZ ae
Z =aeae
eE*B(X1 )
ae "Z
=aeae ZZ
0 oe__ eE*B(X)oe__eE*B(Xe0)oe_____ eE*B(Xe1)oe______eE*B(Xe2)oe___ : : :
"ZZ ae
Z =aeae
eE*B(X2 )
ae "Z
=aeae ZZ
0 0
As a result, we obtain a long exact sequence of E*(B) modules:
0  Ee*B(X) Ee*B(Xe0) Ee*B(Xe1) : : : (2)
7
Fix i 0 and write eE*B(Xi )= eE*(Xi=s(B) )= lim Viff, where each Viffis
a finite dimensional E*(pt)module (see Example 1 in the Appendix). By
Lemma 1.2 and Theorem 6.3.ii), we have
Ee*B(Xei)~=( lim V ff) b * E*(B) ~= lim (V ffb * E*(B) ) :
 i E (pt)  i E (pt)
Since each ViffbE*(pt)E*(B) is obviously a free E*(B) module, we can in
voke Theorem 6.1.ii) to conclude that eE*B(Xei)is a projective E*(B) module.
Consequently the sequence (2) is a projective resolution of the E*(B) module
eE*B(X).
2 The spectral sequence
The aim of this section is to construct the spectral sequence announced in
the introduction, to identify its E2term and to discuss its convergence. This
spectral sequence will be defined by means of derived exact couples. In order
to simplify the presentation, we will always replace finite sequences of maps
by inclusions in (T op=B)*, via the fibrewise mapping cylinder construction.
With this convention, the cofibre of a map f : A ! C will simply be written
C=A.
Let X, Y be spaces in (T op=B)* and let {Xi ! eXi! Xi1}i0 be the
geometric cobar resolution of X. Set
Wi(X; Y ) = Wi = Xi^B Y ; for all i 0:
For a fixed integer i 0, we iterate the Puppe construction of the OEj's to
obtain the sequence Xi1 ! B Xi ! i+1BX0. The latter induces a
cofibration
fiii+1 ffii+1
B Wi=Wi1 ! B W0=Wi1 ! B W0=B Wi : (3)
To introduce the spectral sequence we define, for i 0
Di;j1= eEj+1B(i+1BW0=Wi1 )
Ei;j1= eEj+1B(B Wi=Wi1 ;)
and for i > 0
Di;j1= Ei;j1= 0 :
The cofibration (3) induces a long exact sequence in eE*Bcohomology which
can be written as
ff*ii;j fi*ii;j fl*ii+1;j
: ::! Di+1;j11! D1 ! E1 ! D1 ! : : :
8
where fl*iis induced by the Puppe map
fli: i+1BW0=B Wi ! B (B Wi=Wi1) :
We splice all these exact sequences to form the following unraveled exact
couple (in the sense of Boardman [2])
. ..! Di+1;*1__________Di;*1__________Di1;*1___ : : :
I@@ I@
@ @@
Ei;*1 Ei11
Observe that the bidegree of ff*i(respectively fi*i, fl*i) is (1; 1) (respe*
*ct
ively (0; 0), (1; 0)). By standard techniques this yields a spectral sequence
{E*;*r(X; Y ); dr}r1 (or simply {E*;*r; dr}r1 when X and Y are understood),
with differentials dr : Ei;jr! Ei+r;j+1rr. As usual, for any r 2, the cycles
and boundaries are defined as
Z*;*r= Ker(dr1); B*;*r= Im(dr1) :
Since E*;*r= Z*;*r=B*;*r, it is important to have a geometric description of the
groups Z*;*rand B*;*r. For this purpose, we write
o for the natural inclusion (Wi1; B Wi) (Wi1; rBWi+r1)
o ffi = ffi1 O oe, where ffi1 is the coboundary of the triple
(Wir; r1BWi1; rBWi) and oe the suspension isomorphism
eEj+rB(rBWi=r1BWi1 )~=eEj+1B(B Wi=Wi1 : )
Then we have
Zi;jr~=Im{* : eEj+1B(rBWi+r1=Wi1 )! eEj+1B(B Wi=Wi1 } )
Bi;jr~=Im{ffi : eEj+r1B(r1BWi1=Wir )! eEj+1B(B Wi=Wi1} :)
With these identifications the differentials dr are induced by the composite
eEj+1B(rBWi+r1=Wi1 )__ffi2eEj+2B(i+1BW0=rBWi+r1 )
 *
fii+r
?
Eej+2B(r+1BWi+r=rBWi+r1 )
~=oe
?
eEjr+2B(B Wi+r=Wi+r1 ;)
9
where ffi2 is the coboundary of the triple (Wi+1; rBWi+r+1; i+1BW0) and fi*i+r
is as above. The proof of these facts is standard; the interested reader might
want to perform it by chasing on diagrams like the one on the top of page
661 in [22].
Everything is now in place for the identification of the E2term of that
spectral sequence. This is based on the following
Lemma 2.1 For any i 0 and any space Y in (T op=B)*, the product in
eE*Bcohomology induces an isomorphism
Ee*B(Xei)b* eE*(Y )~=eE*(Xe ^ Y ):
E (B) B B i B
Proof. Consider the diagram
eE*B(Xei)b* eE*(Y )____________________eE*(Xe ^ Y )
E (B) B B i B
 
 
~= ~=
 
 
? ?
(Ee*(Xi=s(B) )bE*(pt)E*(B) ) bE*(B) eE*(Y=s(B) _)_eE*(Xi=s(B) ^ Y=s(B) )
We proceed as in [11, page 49] to show that it commutes. By Lemma 1.2,
the two vertical maps are isomorphism. The lower horizontal map is also an
isomorphism (this is the ordinary K"unneth isomorphism in E*cohomology).
This complete the proof of the lemma.
Let us now deal with the differential d1 : Ei1;*1! Ei;*1. By definition it
is the composite fi*iOfl*i1. As easily checked, the following diagram commutes
i+2BW0
6


2BWi=B Wi1 ________2BWi1 ________B (B Wi1=Wi2)
@@ fii 6
@R  fli1
i+2BW0=B Wi1
The horizontal composition is equivalent to
OEi+1^Y2 OEi^Y2
B eXi+1^B Y  ! B eXi^B Y ! B eXi^B Y :
10
This observation and Lemma 2.1 imply that the complex
0  E0;*1d1E1;*1d1 E2;*1d1 : : :
is isomorphic to the one obtained by applying the functor  bE*(B) eE*B(Y )
to the complex
0  Ee*B(Xe0) Ee*B(Xe1) : : :
We have thus shown that
Ei;*2~=ToriE*(B)(Ee*B(X); eE*B(Y));
where Tor iE*(B)(; )is the ith derived functor of the completed tensor
product (see the Appendix).
We now turn to the convergence questions. Let us introduce the graded
group H*(X; Y ) by taking Hr(X; Y ) as the direct limit of the sequence
D0;r1! D1;r+11! : ::! Di;ri1! : : :
The group H*(X; Y ) is filtered (as graded group) by
FiHr(X; Y ) = Im{Di;ri1! Hr(X; Y )} :
According to Boardman ([2, Section 6]), this filtration is Hausdorff and com
plete and the spectral sequence constructed in the previous section converges
to H*(X; Y ); that is,
Ei;ri1~=FiHr(X; Y )=Fi+1Hr(X; Y ) ; for alli :
The expected target group is eE*B(W0 )= eE*B(X ^B Y .)In this section we will
give conditions under which H*(X; Y ) is isomorphic to eE*B(X ^B Y .)
For each i 0, we consider the exact triangle induced in E*cohomology
by the cofibration Wi1 ! i+1BW0 ! i+1BW0=Wi1. The direct limit
of these triangles is the exact triangle
*
H*(X; Y ) ________eEB(W0 )
I@@ (4)
@
G*(X; Y )
where Gr(X; Y ) is the direct limit of the sequence
eEr+1B(W1 )! eEr+2B(W2 )! : ::! eEri+1B(Wi1)! : : :
11
If the group G*(X; Y )vanishes, then the map of the triangle (4) will be an
isomorphism so that our spectral sequence will have the expected abutment.
The following proposition describes the main property of this obstruction
group and gives a sufficient condition for the convergence to eE*B(X ^B Y .)
Proposition 2.2 Let X be an element of (T op=B)* such that the projec
tion p : X ! B is a fibration. The functor Y 7! G*(X; Y ) is a cohomology
theory on (T op=B)*. Consequently, if G*(X; pt+) = 0, then G*(X; Y )= 0 for
all spaces Y in (T op=B)*.
Proof. The first axiom of Definition 1.1 is easily checked using the exactness
of the direct limits and the properties of the smash product and cofibrations
in the category (T op=B)*. The cylinder axiom requires the extra assumption
on the projection p. By Proposition 4.8 of [18] and Section 3.4 of [5], the
functors Y 7! Ee*B(Xi^B Y )are cohomology theories for each i 0. We
invoke one more time the exactness of the direct limits to conclude. We are
left to show that G*(X; Y ) is strongly additive. Let {Yfi}fi2Jbe a family of
spacesWin (T op=B)*. We want to prove that the natural injections Yfi,!
B Yfiinduce an isomorphism
W Y
G*(X; BYfi) ~= G*(X; Yfi):
We start with three observations:
W W
1. W0(X; BYfi) is homotopy equivalent to B W0(X; Yfi).
W Q
2. eE*B( B W0(X; Yfi))is naturally isomorphic to eE*B(W0(X; Yfi)).
3. The following triangle is exact (the direct product preserves exactness)
Y Y
H*(X; Yfi)___________ eE*B(W0(X; Yfi))
I@@
@Y
G*(X; Yfi)
Because of these observations, it is sufficient to prove that
W Y
H*(X; BYfi) ~= H*(X; Yfi):
From the definition of the spectralQsequence andQexactness of the direct
product, one checks easily that { E*;*r(X; Yfi);W dr}r1 is a spectral se
quence. Moreover, the natural injections Yfi,! B Yfiinduce a morphism of
spectral sequences
W Q
r : E*;*r(X; BYfi)! E*;*r(X; Yfi): (5)
12
As the completed tensor product commutes with the direct products, the
morphism
W Q *
2 : Tor*E*(B)(Ee*B(X); eE*B( B Yfi))! Tor E*(B)(Ee*B(X); eE*B(Yfi))
is an isomorphism. ThusQthe spectral sequencesQof (5) are isomorphic.Q
We filter the group H*(X; Yfi) by setting Fi H*(X; Yfi)= FiH*(X; Yfi).
This filtrationWis Hausdorff and complete. We observe that the natural injec
tions Yfi,! B Yfiinduce a continuous homomorphism
W Q
0 : H*(X; BYfi) ! H*(X; Yfi) : (6)
Its associated graded homomorphism is nothing but the isomorphism
W Q
1 : E*;*1(X; BYfi)! E*;*1(X; Yfi):
As the two groups involved in (6) are Hausdorff and complete, 0 has to be
an isomorphism. The last assertion follows from the comparison theorem 4.1
of [5].
3 The multiplicative structure
The discussion of the multiplicative structure of the EilenbergMoore spectral
sequence requires more general definitions than those given in Section 2.
Since this material will be used only in the present section, we have chosen
to introduce it here.
Definition 3.1 Let m 1 and X 2 (T op=B)*. A negative mfiltration
of X is a sequence U* of spaces {Ui}i0 and maps i : Ui1 ! mBUi in
(T op=B)*, with U0 = X. A morphism f* : U* ! V* of negative mfiltrations
is a sequence of maps fi: Ui ! Vi making the obvious diagrams commutat
ive.
Examples.
1. For X in (T op=B)* and m 1 the geometric cobar resolution of degree
m  1 of X is inductively defined by setting X0 = X and for i 0,
o eXi= (Xi=s(B))B
o OEi: Xi ! eXi, OEi(x) = ([x]; p(x))
o Xi1 = CB (m1BOEi).
13
We set Xi(m) := Xi and take i : Xi1(m) ! mBXi(m) to be the
next map in the Puppe sequence of the cofibration
m1BXi(m) ! m1BeXi(m) ! Xi1(m) :
The resulting negative mfiltration will be denoted X*(m) and called
the cobar mfiltration. When m = 1, we recover the 1filtration associ
ated to the geometric cobar resolution of Section 1; we will then simply
write X* := X*(1).
2. Let U* be a negative mfiltration of X 2 (T op=B)* and let Y 2
(T op=B)*. One constructs a negative mfiltration U* ^ Y by smash
ing all the constituents of U* by Y . In particular, we suspend negative
filtrations by smashing them with Y = (Sk; *)B .
We recall here a construction of Hodgkin, since it will play a central
role in our discussion. It might be illuminating to view this construction
as the geometric counterpart of the tensor product of chain complexes in
homological algebra. To get into work, we fix X 2 (T op=B)* and let U* be
a negative mfiltration of U0 = X. For any integer i 0 we consider the
sequence
i+1m i+2 0 im
Ui ! B Ui+1 ! : ::! B U0 : (7)
We may and will assume (in accordance with our convention) that all the
maps inVthis sequence are inclusions. Let Zi be the subspace in (T op=B)* of
imBU0 B imBU0 defined by
[
Zi = kmBUik ^B lmBUil:
k+l=i
By comparing the sequence (7) for the index i and i1, Hodgkin constructed
a natural map Oi : Zi1 ! 2mBZi and showed (see [11, page 24]) that
_ (ik)m (il)m
2mBZi=Zi1 ' B (mBUk=Uk1) ^B B (mBUl=Ul1) : (8)
k+l=i
As in [11], the negative 2mfiltration {Zi; Oi}i0 will be noted U* U*.
We continue our recollection of Hodgkin's work, by explaining how neg
ative filtrations give rise to spectral sequences. Let U* be a negative m
filtration and eE*B( )a cohomology theory on (T op=B)*. We fix an integer
14
i 0 and extract the part of the sequence (7) given by the inclusions
Ui1 ,! mBUi ,! (i+1)mBU0. This sequence induces a cofibration
mBUi=Ui1 ! (i+1)mBU0=Ui1 ! (i+1)mBU0=mBUi :
With respect to this cofibration we define
Di;j1= eEj+1+(i+1)(m1)B((i+1)mBU0=Ui1 )
Ei;j1= eEj+1+(i+1)(m1)B(mBUi=Ui1: )
We extend to all integers by setting, for i > 0
Di;j1= Ei;j1= 0 :
As in Section 2 we obtain an unravelled exact couple; the associated spec
tral sequence is written {Ei;jr(U*); dr}r1 . Here also, the result of Board
man [2] implies that the spectral sequence strongly converges to H*(U* ) :=
lim!(D0;*1! D1;*+11! : :):. The latter is related to the desired abutment
via a natural map : H*(U* ) ! eE*B(U0 ).
Lemma 3.2 Let X; Y 2 (T op=B)* and m 1. Write X*(m) (respectively
X*) for the cobar mfiltration (respectively 1filtration) of X. Then there is
a natural morphism of spectral sequences
E*;*r(X*(m) ^ Y ) ! E*;*r(X* ^ Y )
which is an isomorphism for r 2:
Proof. We construct inductively maps OEi: Xi(m) ! i(m1)BXi by taking
OE0 = id and requiring the commutativity of the diagram
m1BXi(m) ______m1BeXi(m) ________Xi1(m)
  
OEi eOEi OEi1
? ? ?
(i+1)(m1)BXi___(i+1)(m1)BeXi___(i+1)(m1)BXi1
where eOEiis the composite
m1BeXi(m) ! m1B(i(m1)BXi=s(B) x B) ! (i+1)(m1)BeXi:
15
Then the map OEi^ id: Xi(m) ^B Y  ! i(m1)BXi^B Y induces the de
sired homomorphism of spectral sequences. As in Section 1, one constructs
a projective E*(B) resolution
0  Ee*B(X) Ee*B(Xe0(m) ) Ee*B(Xe1(m) ) : : :
The comparison theorem of projective resolutions now implies that OEi^ id
induces an isomorphism of E2terms and the lemma follows.
This is the right place to start the discussion of the multiplicative prop
erties of the spectral sequence.
Proposition 3.3 Let X; Y 2 (T op=B)* and set W* = X* ^ Y , where X*
is the cobar 1filtration of X. For 1 r 1, there exist associative pairings
r : Ei;jr(W*) Ep;qr(W*) ! Ei+p;j+qr(W* W*) ; a b 7! a . b
satisfying the following properties:
1. 1 is induced by the multiplication of eE*B( ).
2. r+1 is induced by r, via the isomorphism Er+1 ~=H(Er).
3. For all a 2 Ei;jr(W*) and b 2 Ep;qr(W*), we have
dr(a . b) = dr(a) . b + (1)j+1a . dr(b) :
Proof. We begin with some notation. For any negative mfiltration U*, we
set:
o Ai;jr(U*) = eEj+1+(i+1)(m1)B((rmBUi+r1=Ui1).)
o For s r, ffr;s: Ai;jr(U*)! Ai;js(U*)is the morphism induced by the
inclusion (Ui1; smBUi+s1) ,! (Ui1; rmBUi+r1):
o i;jr: Ai;jr(U*)! Ai+r;jr+1r(U*)is the coboundary operator of the triple
(Ui1; rmBUi+r1; 2rmBUi+2r1).
Let us observe that Ai;j1(U*) = Ei;j1(U*), the first term of the spectral seque*
*nce
associated to U*. By proceding as in the case m = 1 (see Section 2), we see
that
Zi;jr~=Im{ffr;1: Ai;jr(U*)! Ai;j1(U*)}
Bi;jr~=Im{ffi : Air+1;j+r2r1(U*)! Ai;j1(U*)} ;
16
where ffi is defined as in Section 2. We set = j + 1 + (i + 1)(m  1)
and consider the commutative diagram (obtained by playing around with
adequate triples):
eE+1B(((i+1)mBU0=rmBUi+r1) )
ffaei>  Z fi*i+r
aeae  ZZ"
eEB((rmBUi+r1=Ui1) )  eE+1B((mBUi+r=Ui+r1) )

ZZ  ffae>
Z" ? aeae
eE+1B((2rmBUi+2r1=rmBUi+r1) )
As in Section 2 and with the identifications above, the differential dr is equal
fi*i+rO ffi. Consequently, dr is induced by the morphism *;*r.
We now go back to our data and define pairings
OEr: Ai;jr(W*) Ap;qr(W*) ! Ai+p;j+qr(W* W*)
in the following manner. First we use the product of eE*B( )and the suspen
sion isomorphism to send Ai;jr Ap;qrinto
eE*B(p+1B(rBWi+r1=Wi1) ^B i+1B(rBWp+r1=Wp1) ):
Then we proceed as in [11, page 30] to send the latter into Ai+p;j+qr(W*W*).
We observe that the arguments in Hilfsatz 13 and 14 of [13] apply verbatim
to our situation. To be entirely honest, we should mention that Kulze's
proofs are based on two hypothesis (the axioms M1 and M2 (page 290) of
[13]). Fortunately, these hypotheses are satisfied in our case (see Theorem
9.10 (page 238) of [1]).
As in the classical case, the next step consists in comparing the spectral
sequences of the negative filtrations W* W* and W*, via the diagonal map.
Unfortunately we have not been able to proceed directly. However, as we
will see in a moment the suspensions of these filtrations can be compared.
This will be sufficient for our purpose. The reason is the following: For any
negative mfiltration U*, the suspension isomorphism induces an isomorphism
of spectral sequences (This is easily checked at the level of exact couples)
~= i;j+1
oe : Ei;jr(U*) ! Er (U*): (9)
Lemma 3.4 Let U* be a negative mfiltration of U0 = V . As above, we
write V*(m) for the cobar mfiltration of V . There exists a negative m
filtration Z* of V and two morphisms of negative filtrations
g* f*
V*(m)  Z* ! U*
such that f0 = g0 = id:
17
Proof. To begin with we set Z0 = V and f0 = g0 = id. Given fi: Zi ! B Ui
and gi: Zi ! B Vi(m) , let
U0i= mBUi=Ui1; Vi0= mBeVi(m) and Z0i= m1BeZi=s(B) x (U0ixB Vi0) :
We define Zi1 as the fiberwise cone of the obvious map m1BZi ! Z0iand
we construct fi1 and gi1 by requiring that the following diagram commutes:
mBUi ______U0i______B Ui1
6 60 6
fi fi fi1
  
m1BZi _____Z0i_______Zi1
  0 
gi gi gi1
? ? ?
mBVi(m) ____Vi0____B Vi1(m)
Here f0iand g0iare the obvious projections.
Theorem 3.5 Let X; Y 2 (T op=B)* and set W* = X* ^ Y , where X* is
the cobar 1filtration of X. There exist associative pairings, for 2 r 1,
r : Ei;jr(W*) Ep;qr(W*) ! Ei+p;j+qr(W*); a b 7! ab
satisfying the following properties:
1. If E*;*2(W*) is identified with Tor *E*(B)(Ee*B(X); eE*B(Y)), then 2 beco*
*mes
the usual internal product of Tor *E*(B)(; ).
2. r+1 is induced by r, via the isomorphism Er+1 ~=H(Er).
3. For all a 2 Ei;jr(W*), b 2 Ep;qr(W*) we have
dr(ab) = dr(a)b + (1)j+1adr(b) :
4. Let Hr(X; Y ) be the limit of the spectral sequence {Ei;jr(W*); dr}r1 .
There is a product : H*(X; Y ) H*(X; Y ) ! H*(X; Y ) which in
duces 1 . In addition, the natural homomorphism
: H*(X; Y ) ! eE*B(X ^B Y )
respects the products.
18
Proof. For the construction of the r's, we already have the composite
r *;*
E*;*r(W*) E*;*r(W*) ___________Er (W* W*)
~=or
?
E*;*r((X* X*) ^ (Y ^B Y )) (10)
~=oer
?
E*;*r(B (X* X*) ^ (Y ^B Y ))
where r has been defined in Proposition 3.3, or is induced by the twists
X ^B Xj ' Xj ^B X and oer is the suspension isomorphism discussed above
(see (9)). Next we apply Lemma 3.4 with U* := X* X* and V := X ^B X.
This yields a negative 2filtration Z* and two morphisms of spectral sequences
f* *;*
E*;*r(B U* ^ (Y ^B Y )) ______Er (Z* ^ (Y ^B Y ))
6
g*  (11)

E*;*r(B V*(2) ^ (Y ^B Y )) :
We claim that g* is an isomorphism for any r 2. This can be checked as in
Lemma 3.2, using property (8) and the construction of Z* (see Lemma 3.4).
To obtain the pairing r, we compose the diagrams (10) and (11) with the
sequence
oe1 *;*
E*;*r(B V*(2) ^ (Y ^B Y )) ____Er (V*(2) ^ (Y ^B Y ))
* 
?
E*;*r(X*(2) ^ Y ) (12)
~=
?
E*;*r(W*)
where oe1 is the inverse of the suspension isomorphism (9), * is induced
by the diagonals X  ! X ^B X and Y ! Y ^B Y and the last arrow
is the isomorphism of Lemma 3.2. Even though suspensions appear in the
construction, we note that the r's are bigraded morphisms.
We now go through the claimed properties of the pairings. The first one
follows from the definition of the internal product of Tor *E*(B)(; )(see [21,
page 65]). The next two properties are consequences of Proposition 3.3.
19
Let us deal with the multiplicative structure of H*(X; Y ) = H*(W* ) .
First we proceed as in Proposition 3.3 to define morphisms
Di;j1(W*) Dp;q1(W*) ! Di+p;j+q1(W* W*):
The direct limit of these morphisms yields a pairing
Hr(W* ) Hs(W* ) ! Hr+s(W* W* ):
To obtain the desired product on H*(W* ) , we compose this pairing with a
sequence of morphisms following the same pattern as in diagrams (10), (11)
and (12). Of course one needs to check that
g* : H*(B V*(2) ^ (Y ^B Y )) ) ! H*(Z* ^ (Y ^B Y )))
is an isomorphism, but this is true because the corresponding spectral se
quences are isomorphic (see the claim after diagram (11)). The naturality of
these constructions implies that is multiplicative and induces 1 .
4 An example of convergence
Let B be a pointed space and p : EB  ! B be the path space fibration.
The adjoint of the identity of B will be denoted e : B ! B . As
sume that E*(B )~= (1; : :;:n) with i 2 Eodd(B) for i = 1; : :;:n. An
easy calculation with the RothenbergSteenrod spectral sequence implies that
E*(B) ~=E* [[ae1; : :;:aen]] where aei 2 Eeven(B) is chosen so that e*(aei) = *
*oei
for i = 1; : :;:n.
Theorem 4.1 Let B be a connected and pointed space, p : EB  ! B the
path space fibration and assume that E*(B ) ~= (1; : :;:n) with i 2
Eodd(B) . Then the EilenbergMoore spectral sequence of the pullback dia
gram
B ____EB
 
 
 
? ?
pt _____B
converges strongly to E*(B ).
20
Following the notation of Section 1, we write {Xi ! Xei ! Xi1}i0 for
the cobar resolution of X0 = EB+ and we set
Wi := Xi^B pt+ ; (i 0) :
The first steps of this construction are summarized in the following com
mutative diagram (where the vertical maps are induced by the inclusion
pt+ ,! B+ = (S0; *)B ):
i0
W0 ____EB+ ___W1 ___B W0
   
j0  j1 j0
? ? ? ?
X0 _____eX0____X1 ___B X0 :
OE0
By definition eE*B(W0 )~=E*(B )and the contractibility of EB implies that
i*0 *
the composite Ee*(B ) ,! E*(B ) ~= eE*B(B W0 )! eEB(W1 )is an
isomorphism. By abuse of notation, we will identify the generators oei 2
eE*(B )to their images in eE*B(W1. )A similar argument shows that eE*B(X1 )
~= eE*(B); here also we will identify the aei's with their images in eE*B(X1.)
Lemma 4.2 With the notation and identifications above, the morphism
induced in Ee*Bcohomology by the map j1 : W1 ! X1 satisfies: j*1(aei) =
oei, for i = 1; : :;:n.
Proof. Let ' : B ! B be the adjoint of the map inv : B ! B
which sends a loop ff to its inverse ff1. We leave as an exercise to check
that, in E*cohomology, '*(aei) = oei, for i = 1; : :;:n.
We will construct two maps (in T op*) F : X1=s(B)  ! B and
f : W1=s(B)  ! B making the following diagram commutative:
f
W1 ___W1=s(B) ___B
 
j1  ' (13)
? ? ?
X1 ___X1=s(B) ____B :
F
The space X1 is by definition the cofiber of
OE0: X0 = EB+  ! eX0= X0=s(B) x B :
21
`
More precisely, X1 = CB (X0) Xe0=(ff; 0) ~ OE0(ff), where CB (X0) denotes
the reduced cone over B of X0 (see [18] for its definition). We construct a
map X1 ! B by sending
ae
ff(1  t) ifff 2 EB
(ff; t) 7!
ff ifff 2 B
([x]; b)7! b if([x]; b) 2 eX0:
We leave as an exercise to check that this map is well defined and induces
F : X1=s(B)  ! B . It is also easily checked that the composite
a F
(EB x B) pt = eX0=s(B) ! X1=s(B) ! B
is the projection onto the second factor. This shows that F *(aei) = aei for
i = 1; : :;:n.
Similarly W1`is the cofiber of the map W0 = B +  ! EB+ . Thus
W1 = CB (W0) EB+ =(ff; 0) ~ ff. We construct a map W1  ! B by
sending ae
[(ff; t)] ifff 2 B
(ff; t)7!
[(ffl0; 0)]ifff 2 B
ff 7! [(ffl0; 0)]ifff 2 EB+ ;
where ffl0 2 EB+ stands for the trivial loop. It is easily checked that this
map is well defined and induces f : W1=s(B)  ! B . We also leave as
an exercice that the following diagram commutes up to homotopy:
i0
W1=s(B) _____B W0=s(B)
 
f =
? a ?
B oe______(B pt)
pr
This implies that f*(oei) = oei for i = 1; : :;:n.
The commutativity of the diagram (13) is straightforward and implies the
lemma.
To proceed further we need to consider one more step in the cobar res
olution. More precisely, we will study the inclusions W1 ,! B W0 and
W2 ,! B W1. For simplicity, we will use the following identifications
eE*B(B W1=W2 ~)=eE*B(B eX1^B pt+ )~=eE*(B ~)=eE*1(B):
The second isomorphism follows from Lemma 1.2 and the others are obvious.
22
Lemma 4.3 Let q1 be the projection B W1 ! B W1=W2 . With the
identifications above, we have q*1(aei) = oei ; fori = 1; : :;:n.
Proof. We consider the following commutative diagram in (T op=B)*, ob
tained from the natural map X2 ,! B X1:
q1
B W1 ___B W1=W2

j1 j2
? q2 ?
B X1 ___B X1=X2 :
With our usual identifications, we have
eE*B(B X1=X2 )~=eE*B(B eX1~)=eE*1B(Xe1)~=eE*(B) E*(B) :
N
The two homomorphisms j*2and q*2send aei 1 to aei. We note thatNthese
homomorphisms behave differently, for example on the elements aei aej.
Lemma 4.2 and the commutative diagram above now implies the assertion.
Proof (of Theorem 4.1). We will show, as claimed, that the map
: H*(EB+ ; pt+) ! eE*B(W0 ) is a ring isomorphism. In our situation, the
E2term of the EilenbergMoore spectral sequence is isomorphic to
Tor *E*(B)(E*(pt); E*(pt))~=(y1; : :;:yn) ;
the argument is standard and involves the Koszul resolution (see [21]). For
dimensional reasons, the generators yi are permanent cycles. The multi
plicative properties of the spectral sequence imply that it collapses; that is,
E2 ~=E1 . Hence, H*(EB+ ; pt+)is a free E*(pt)module of rank 2n. We next
consider the diagram
* i*
eE*B(W0 )__i0eE*+1B(W1__)___________1eE*+2B(W2__)_ : : :
I@@q*1
@
eE*B(B W1=W2 )
Since Im(q*1) = Ker(i*1), Lemma 4.3 implies that the image of the generators
i 2 eE*B(W0 )~=E*(B )are zero in the obstruction group G*(EB+ ; pt+)=
lim!eE*B(Wi.)Therefore the generators i lie in the image of , showing the
surjectivity of the latter. A dimension count now implies the claim.
23
Theorem 4.4 Let B be a connected and pointed space such that E*(B )~=
(1; : :;:n), with i 2 Eodd(B) . Let X, Y in (T op=B)* and assume that
the projection p : X ! B is a fibration. Then the EilenbergMoore spectral
sequence {E*;*r(X; Y ); dr}r1 converges strongly to eE*B(X ^B Y .)
Proof. Recall that G*( ; ) denotes the obstruction to the "good beha
viour" of the EilenbergMoore spectral sequence (see Section 2). By The
orem 4.1, we have G*(EB+ ; pt+) = 0, where p : EB  ! B is the path
space fibration. It follows from Proposition 2.2 that G*(EB+ ; X) is also
trivial. The argument of Hodgkin (see pages 4041 of [11]) shows that
0 = G*(EB+ ; X) = G*(X; EB+ ). Since p : X ! B is a fibration, its ho
motopy fiber is homotopy equivalent to the fiber over the base point; hence
G*(X; EB+ ) = G*(X; pt+) = 0. We invoke one more time Proposition 2.2 to
obtain G*(X; Y )= 0 and this concludes the proof.
Remark. If E*(B )is not an exterior algebra, the spectral sequence may
not converge to the desired target. For instance, let E*( ) = K*( ; IF2)be
the mod 2 complex Ktheory and B = BSO(3), the classifying space of the
Lie group SO(3). As well known, we have
K*(B; IF2) ~= IF2[[ae]] and K*(B ; IF2)~=IF2[]=(4) :
For the path space fibration B ! EB ! B, The limit of the Eilenberg
Moore spectral sequence is H*(EB+ ; pt+)~= (y), which is obvously different
from K*(B ; IF2).
5 Applications
In this section, we will use our main theorem to describe the Ktheory of
certain spaces associated to pcompact groups. The basic references for the
theory of pcompact groups are [7] and [15]; we refer to these papers for the
relevant facts about these objects.
Throughout this section, p is a fixed prime and R denotes either the field
IFp of order p or the ring ZZ^pof padic integers. The complex Ktheory with
coefficient R will be denoted K*(; R).
Our first result provides a large class of pcompact groups satisfying the
hypothesis of the third point in the main theorem. The result might be well
known to the experts, but we haven't found any explicit reference.
Theorem 5.1 Let X be a connected pcompact group. Then K*(X; R) is
an exterior algebra on odd degrees generators if and only if ss1(X) is torsion
free.
24
Proof. A Bockstein spectral sequence argument shows that K*(X; IFp) is
an exterior algebra on odd degrees generators if and only K*(X; ZZ^p) is an
exterior algebra, if and only if K*(X; ZZ^p)is torsion free. This reduces our
problem to the case R = ZZ^p.
Observe that ss1(X) is a finitely generated ZZ^pmodule, hence ss1(X) ~=
ZZ^pr ss where ss is a finite abelian pgroup. We combine the arguments of
Theorem 4.3 (page 63) in [14] and corollary 3.3 of [15] to show that X is
homotopy equivalent (only as a space) to Y x (S1^p)r, where Y is a connected
pcompact group with ss1(Y ) ~=ss.
If ss1(X) is torsion free, then Y is a 1connected pcompact group and we
can invoke a theorem of Kane and Lin (see [12, Theorem 1.2]) to conclude
that K*(X; ZZ^p) is an exterior algebra. The argument for the converse is
implicit in [10]. Suppose that ss is non trivial and choose a cyclic subgroup
i : ZZ=p ,! ss. Let Y < 1 > be the universal cover of Y and M(p) = S1 [p e2
be the 2skeleton of BZZ=p. Consider the pullback diagram
Y < 1 > = Y < 1 > = Y < 1 >
  
  
  
? ? ?
Y (p) ______Y 0________Y
 
eq  q
? j ? Bi ?
M(p) ____BZZ=p _____Bss :
Since Y < 1 > is 2connected, obstruction theory tells us that the fibration
eqis trivial, hence eqinduces an injection in Ktheory. As well known, the
map Bi induces a surjection in Ktheory; an easy computation shows that
the same is true for the map j. All these facts imply that there exists a class
6= 0 in the image of q* : K*(Bss; ZZ^p)! K*(Y ; ZZ^p). A Chern character
argument shows that is a non trivial torsion class in K*(Y ; ZZ^p)and this
implies that K*(X; ZZ^p)has non trivial torsion.
Proposition 5.2 Let X be a connected pcompact group such that ss1(X) is
torsion free and let i : T  ! X be a maximal torus. The ring homomorphism
Bi* : K*(BX; R) ! K*(BT ; R) is injective and makes K*(BT ; R) into a
free and finitely generated K*(BX; R) module.
Proof. The arguments of the proof of Theorem 2.7 in [12] are valid in this
more general situation. The injectivity is due to the equality of the Krull
dimensions.
25
Everything is now in place for the main result of this section.
Theorem 5.3 Let X be a connected pcompact group such that ss1(X) is
torsion free. Let i : T  ! X be a maximal torus and X=T the associated
homogeneous space; that is, X=T is the homotopy fibre of Bi : BT ! BX :
Then the inclusion X=T ,! BT induces an isomorphism
K*(X=T ; R) ~=K*(BT ; R) K*(BX;R) K*(pt; R) ;
where the K*(BX; R)module structure on K*(BT ; R) (respectively K*(pt; R))
is given by the induced map Bi* (respectively the augmentation map).
Proof. Let us first deal with the case R = IFp. We may and we will assume
that the map Bi : BT ! BX is a fibration. Thanks to Theorem 5.1, we
can apply our main theorem to the pullback diagram
X=T ___BT
 
 Bi
 
? ?
pt ____BX :
Consequently, there is a strongly convergent EilenbergMoore spectral se
quence
Ei;*2= T oriK*(BX ; IF (K*(BT ; IFp); K*(pt; IFp)) ) K*(X=T ; IFp):
p)
Proposition 5.2 above implies that the spectral sequence is trivial, i.e. E*;*2=
E0;*2= E*;*1and the claim follows.
The case R = ZZ^pis a consequence of the preceding one, the universal
coefficients theorem and Nakayama's lemma.
For a general pcompact group and even if the EilenbergMoore spectral
sequence does not behave as expected, we still have the following qualitative
result.
Corollary 5.4 Let X be a connected pcompact group, i : T  ! X a
maximal torus and W the corresponding Weyl group. Then K1(X=T ; R) = 0
and K0(X=T ; R) is a free Rmodule of rank W .
Proof. Let X < 1 > be the universal cover of X. As we have seen above,
X < 1 > is a pcompact group. In the proof of corollary 5.6 of [15], it is shown
that X=T is homotopy equivalent to X < 1 > =S, where S  ! X < 1 > is
maximal torus for X < 1 >. Since the latter is 1connected, Theorem 5.3
26
applies. We invoke Proposition 9.5 of [7] to obtain the assertion about the
rank.
One of the main conjecture in the theory of pcompact groups states that
H*(X=T ; ZZ^p)is torsion free and concentrated in even degrees. The inter
ested reader is refered to [16] for some partial results about this conjecture.
The corollary above can be viewed as a positive solution of its Ktheoretical
version.
As a second application, we will now give a slightly different proof of the
main result in [12]. With this new method, we obtain an analoguous result
for mod p Ktheory. Even in the case of Lie groups, we are not aware of this
mod p Ktheory statement in the litterature.
Corollary 5.5 Let X be a connected pcompact group, i : T  ! X a
maximal torus and W the corresponding Weyl group. The map Bi induces a
ring isomorphism
K*(BX; R) ~=K*(BT ; R)W :
Proof. By proceeding as in Section 3 of [12], we may and we will assume that
X is 1connected. Theorem 5.1 and a RothenbergSteenrod spectral sequence
argument imply that K*(BX; R) ~= R[[ae1; : :;:aen]] with aei 2 K0(BX; R)
and n equal to the rank of X. Similarly, K*(BT ; R) ~= R[[o1; : :;:on]] with
oi 2 K0(BT ; R). For simplicity we set
SX = R[[ae1; : :;:aen]]; ST = R[[o1; : :;:on]]
and we will identify SX with its image in ST (this is justified because Bi* is
injective by Proposition 5.2).
By construction SX is contained in the ring of invariants SWT. To show
the equality, we consider the following diagram
ST ____F rac(ST )
6 6
 
[ [
SWT ___F rac(SWT)
6 6
 
[ [
SX ___F rac(SX )
where F rac() stands for the fractions field. By Proposition 5.2 and Corol
lary 5.4, ST is a free SX module of rank W ; it follows that F rac(SX ) ,!
F rac(ST ) is a field extension of degree W . By Galois theory, F rac(SWT) =
27
F rac(ST )W ,! F rac(ST ) is a field extension of degree W . As consequence,
the fields F rac(SX ) and F rac(SWT) coincide. Since ST is integrally closed (it
is a power series ring over R) and the ring extension SX ,! ST is integral,
we obtain that SX = SWT.
Let us close this section by describing how our results extend to Morava
Ktheories. For n 1, K(n)*() denotes the nth Morava Ktheory, its
coefficient ring is the graded field IFp[vn; v1n] with vn = 2(pn  1).
Let X be a connected pcompact group, i : T  ! X a maximal torus
and W the corresponding Weyl group. If K(n)*(X) is an exterior algebra on
odd degrees generators, then the preceding arguments apply and we have:
N
1. K(n)*(X=T ) ~=K(n)*(BT ) K(n)*(BX)K(n)*(pt).
2. K(n)*(BX) ~=K(n)*(BT )W .
These statements naturally give rise to the following question:
For each integer n 1, find all the pcompact groups X such that
K(n)*(X) is an exterior algebra.
As wellknown, K(n)*(X) is an exterior algebra when H*(X; ZZ^p)is torsion
free. Hence we recover Theorem 3.1 of [20]. In contrast to the paper just
quoted, we can treat many spaces with torsion. For instance, our Theorem 5.1
provides a complete answer to the question when n = 1. More interestingly,
Maria Santos (private communication) has observed that K(2)*(DI(4)) is an
exterior algebra; here DI(4) is the exotic 2compact group constructed by
Dwyer and Wilkerson [6]. Are there any other examples of this type?
6 Appendix
Let R be a graded ring with 1 and denote by Mod(R) the category of graded
Rmodules, where the morphisms are Rmodules homomorphisms of degree
0. This category is abelian and possesses arbitrary direct and inverse limits
(perform all the relevant contructions degreewise). If R is also commutative,
then the graded tensor product yields a biadditive functor which is associat
ive, commutative and has R as a unit (up to coherence). Moreover, for any
N 2 Mod(R), the functor
 R N : Mod(R) ! Mod(R) ; M 7! M R N
is right exact. In our applications, we will be dealing with particular graded
rings and special subcategories of their modules categories. In the sequel, all
inverse systems and limits are taken over direct sets.
28
A profinite graded ring is an inverse limit of graded rings of finite length
(i.e noetheriean and artinian). We emphasize that, according to this defini
tion, every graded field is profinite; recall that a graded field is a graded r*
*ing
whose non zero homogeneous elements are all invertible. If R = lim Rffis
a profinite ring and ssff: R ! Rff are the canonical projections, the family
of graded ideals {Ker(ssff)} equipp R with a topology which is complete,
Hausdorff and compatible with the graded ring structure.
Example 1 Let E*( ) be a multiplicative (unreduced) cohomology theory
such that E*(pt) is a graded field. In other words, all graded E*(pt)modules
are free. Given a CWcomplex X, let {Xff} be the direct system of all finite
CWsubcomplexes of X. Then we have ([3, Theorem 4.14])
E*(X) ~= lim E*(Xff);
the corresponding topology will be called the profinite topology of E*(X) .
Since the E*(Xff) are finitely generated free E*(pt)modules, they are rings
of finite length; hence E*(X) is a profinite graded ring.
Let R be a profinite graded ring. We consider the full subcategory F(R)
of Mod(R) consisting of the objects M which are discrete topological R
modules and have finite length. Since the discrete topology is the only linear
topology that an Rmodule of finite length can carry, the morphisms of F(R)
are automatically continuous.
A profinite Rmodule is an inverse limit of of objects in F(R). Thus it car
ries a natural topology which makes it into a complete Hausdorff topological
Rmodules. Let Modprof(R) be the subcategory of Mod(R) whose objects
are profinite Rmodules and whose morphisms are continuous Rmodule ho
momorphisms of degree 0.
Example 2 E*( ) is as in example 1 above. Let f : X ! B be a map
of CWcomplexes. By the CWapproximation theorem, the induced map
E*(f) : E*(B)  ! E*(X) is a continuous ring homomorphism (with respect
to the profinite topologies). It follows that E*(f) induces a profinite E*(B) 
module structure on E*(X) .
p
Example 3 E*( )is as above and B s! X ! B is a sectionned space over
B. Fix a finite subcomplex Xffof X, set Bff= s1(s(B) \ Xff) and denote
the image of Xffin X=s(B) by (X=s(B))ff. In the commutative diagram
29
s
B _____ X ______X=s(B)
6 6 6
  
[ s [ [
Bff ____Xff ____(X=s(B))ff
the rows are cofibrations. Since p O s = id, the long exact sequences in
E*cohomology of these cofibrations reduce to the following commutative
diagrams with exact rows:
s* *
0 ______eE*(X=s(B) )____E*(X) ____E (B) _____0
  
   (14)
? ? * ?
s *
0 _____eE*((X=s(B))ff )__E*(Xff)___E (Bff)____0 :
It follows that E*(X) ~= eE*(X=s(B) ) E*(B) as profinite E*(B) modules,
showing that eE*B(X):= eE*(X=s(B) )is a profinite E*(B) module with respect
to the topology induced by E*(X) . Using diagram (14), one checks that
this topology is the same as the profinite topology. Hence we have shown
that eE*B(X)is always a profinite E*(B) module with respect to the profinite
topology.
Theorem 6.1 Let R be a profinite graded ring.
i) The category Modprof(R) of profinite Rmodules is abelian. It has
enough projective objects and exact inverse limits.
ii) Every inverse limit of projective objects of Modprof(R) is projective.
The profinite Rmodule R is projective.
Proof. As easily checked F(R) is abelian, artinian (i.e, every descending
chain of subobjects of any object of F(R) stabilizes) and equivalent to a
small category. Let P ro(F(R)) be the category of inverse systems in F(R)
(see [19, page 21] for the definition). By [17] and [9, page 356]), P ro(F(R))
is an abelian category with enough projective objects and exact inverse lim
its. Moreover every inverse system is isomorphic to a strict one (i.e, whose
transition morphisms are epimorphisms). Let us now consider the functor
: P ro(F(R))  ! Modprof(R) ; (Mff)ff2I7 ! lim Mff:
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The argument of Section 2.6 in [19] are easily adapted to our situation to
show that is an equivalence of categories. Consequently Modprof(R) enjoys
all the properties of P ro(F(R)) mentionned above, and we are done with the
first point of the theorem.
For the second part, we note that both P ro(F(R)) and Modprof(R) are
proartinian in the sense of [4, page 563]. The first assertion follows from
Corollaire 3.4 (page 567) in [4]. Finally R is projective because every morph
ism from R into a profinite Rmodule M is of the form r 7! r . m for some
m 2 M.
For the rest of this section, R denotes a profinite graded ring which is
commutative. Our next aim is to study the topological tensor product in
Modprof(R) . We start with
Lemma 6.2 If M and M0 are in F(R), so is M R M0.
Proof. Since M is finitely generated, its annihilator Ann(M) = {r 2
R; r . m = 0; 8m 2 M} is an open ideal in R (it is the intersection of
the annihilators of the members of a finite generating set of M). Similarly
Ann(M0) is an open ideal of R. We write R = lim Rff, with each projection
ssff: R ! Rff surjective (this is always possible).Then there exists ff such
that Ker(ssff) Ann(M) \ Ann(M0). Consequently M, M0 and M R M0
can be regarded as Rffmodules. These new structures are strongly related
to the former since any subset of M (respectively M0, M R M0) is a R
submodule if and only if it is a Rffsubmodule. We observe now that Rffis
a ring of finite length and M R M0 is a finitely generated Rffmodule; this
implies that M R M0 is a Rmodule of finite length, that is, an object of
F(R).
Let M = lim Mffand N = lim Nfibe two modules in Modprof(R) .
Their completed tensor product is defined as
M bR N = lim (MffR Nfi) :
The preceding lemma insures that M bR N lies in Modprof(R) . It can be
shown that M bR N is the completion of the ordinary tensor product M N
with respect to a certain filtration (see [3, page 603]). This last assertion is
very useful in proving that M bR N = M R N when N is a finitely generated
Rmodule.
31
Theorem 6.3 Let N be a profinite Rmodule.
i) The functor  bRN : Modprof(R) ! Modprof(R) is right exact and
commutes with inverse limits.
ii) Let Tor jR(; N) (j = 0; 1; 2; : :):denote the jth left derived functor
of  bRN (their existence follows from the preceding point). Then
TorjR(; N) commutes with inverse limits. Moreover it coincides with
the usual T orjR(; N) when N is a finitely generated Rmodule.
Proof.
i) This point is true because of the right exactness of R N and because
inverse limits are exact and commute with each other in Modprof(R) .
ii) The first assertion is standard homological algebra (see for example
Corollaire 3.8 (page 568) in [4]). The last assertion is a consequence of
the fact stated before the theorem.
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