The Eilenberg-Moore spectral sequence forK-theory; applications to p-compact

homogeneous spaces

A. Jeanneret & A. Osse∗

Abstract

We construct the Eilenberg-Moore spectral sequence for some gen-eralized cohomology theories, along the lines of Smith and Hodgkin.We prove its multiplicativity and give some sufficient conditions forits convergence to the desired target. As applications, we compute theK-theory of various spaces associated to p-compact groups.

AMS subject classification (1991): 55T20, 55N15, 55P35, 57T35.

Keywords: Spectral sequences, K-theory, completed tensor product,p-compact groups.

Introduction

The topology of homogeneous spaces constitutes an important aspect of Lietheory. 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 K-theory of homogeneous spaces,the Eilenberg-Moore type spectral sequence constructed by Hodgkin has beenthe most successful tool.

Recently Dwyer and Wilkerson [7] have introduced a homotopical gener-alization of the Lie groups, namely the p-compact groups. In this homotopyLie theory, the notions of Weyl group, maximal torus, homogeneous spacesare 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 p-adic cohomology. Its rational part can be

∗Supported by grant No 20-46899.96 of the Swiss National Fund for Scientific Research

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computed by combining a classical Eilenberg-Moore spectral sequence ar-gument with the result of [7]. At the moment, the main indications aboutthe torsion in p-adic cohomology are conjectural (see however the results of[16]). Because of the “lack of differentiability”, the K-theoretical construc-tions of Hodgkin do not apply to the p-compact setting. To circumvent tothis difficulty, we have to go back to the original definition of the spectralsequence, 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 othergeneralized 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 coefficientin a field, complex mod p K-theory and Morava K-theories. With our as-sumption, E∗(X) is a complete Hausdorff topological E∗(pt)-algebra for anyspace X. Moreover, E∗(−) satisfies the Kunneth isomorphism (with com-pleted tensor products). These two properties are the crucial ingredients forthe construction and the study of the Eilenberg-Moore spectral sequence forE∗(−). Our main result can be stated as follows:

Main Theorem Let B be a connected space and E∗(−) as above. For anypull-back diagram

X ×B Y - X

Y? f- B

?p

there exists a strongly convergent spectral sequence E∗,∗r (X, Y ); drr≥2 satis-fying the following properties:

1. The spectral sequence is multiplicative and compatible with the stableoperations of E∗(−).

2. Ei,∗2∼= Tor−iE∗(B)(E∗(X), E∗(Y )) as algebras, where Tor−iE∗(B)(−,−) is the

i-th derived functor of the completed tensor product.

3. If p : X −→ B is a fibration and E∗(ΩB) is an exterior algebra on odddegree generators, then the spectral sequence converges to E∗(X ×B Y ).

The third point of the Theorem desserves the following comment. Aseasily seen, if E∗(ΩB) is an exterior algebra, then E∗(ΩB) is connected; henceB is 1-connected. It is well known that this weaker property of B suffices to

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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 computethe K-theory of various p-compact homogeneous spaces. These computationslead to a new proof of the main result in [12]. We refer to Section 5 for moreprecise 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 methodsare slightly different from those in [18] and [11]. The multiplicative structureis discussed in Section 3. The reader will notice that this is the most delicatepart of the paper; we assure her/him that we did our best to present the ideasas clearly as possible. The convergence questions are treated in Sections 2and 4. Proposition 2.2 of Section 2 is already stated in Hodgkin’s paper, butthe proof presented here is new and hopefully simpler. Section 4 is crucial forthe application to the p-compact groups. In contrast to Hodgkin, we are ablehere to get rid of the “differentiability hypothesis”. In addition, the geomet-rical nature of our constructions readily implies that the spectral sequence iscompatible with the stable operations of E∗(−). We mention that the resultsof Section 1 to Section 4 are stated for sectionned spaces over B and implythe main theorem, via the basepoint adjunction trick discussed in Section 1.The applications are discussed in Section 5. We end with an appendix onprofinite rings and modules. The theme of the appendix is well known andhas been considered by some topologists (see [3] and [23]). We have decidedto discuss it in some details because of its central role in our arguments andalso because we didn’t find any reference suitable to our needs.

Aknowledgements. We would like to thanks U. Suter and U. Wurglerfor helpful discussions. We also thank J.M. Boardman for sending us a copyof [2].

1 The geometric cobar resolution

For the main construction of this section, we need some recollections aboutfiberwise topology. Until further notice, B is a fixed pointed and connectedspace; Top/B will denote the category of spaces over B with fibre maps asmorphisms. The base point inclusion pt → B, viewed as an element ofTop/B, will simply be denoted pt. The categorical product in Top/B is thefamiliar fibred product.

In the sequel we will work in the pointed version of this category, namelythe category (Top/B)∗ of sectionned spaces over B. The objects are triples

3

Bs−→ X

p−→ B such that p s = id and morphisms are commutativediagrams

X1

s1 > ZZZ

p1~

B BZZZs2~

p2

>

X2

?

φ

When there is no danger of confusion, we simply write X (respectively

Φ : X1 −→ X2 ) instead of Bs−→ X

p−→ B (respectively the diagram above).As the reader may have noticed, we recover the categories Top of topologicalspaces and Top∗ of pointed topological spaces by setting B = pt. It iswell known that all the usual homotopic theoretic constructions can be per-formed in the category (Top/B)∗. For instance, we have fiberwise homotopy,fiberwise mapping cone CB(−), fiberwise suspension ΣB(−), 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 spaceover B

Bs0−→ X ×B pr2−→ B

where pr2 is the projection on the second factor and s0(b) = (x0, b).

2. Let Xp−→ 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

Bs−→ X/BA

p−→ B

defined by:

• X/BA = (X∐B)/a ∼ p(a) ∀a ∈ A

• p(x) = p(x), ∀x ∈ X and p(b) = b ∀b ∈ B• s(b) = b, ∀b ∈ B.

In order to simplify the notation, we write X/A for X/BA and X+ forX/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 inclusionA+ → X+.

We turn to an important fact, also treated in [18] and [11]: for any mapΦ : X1 −→ X2 in (Top/B)∗ there is a Puppe sequence of maps in (Top/B)∗

X1Φ−→ X2

i(Φ)−→ CB(Φ)j(Φ)−→ ΣBX1

ΣB(Φ)−→ ΣBX2 −→ . . . (1)

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with the following properties:

1. For any Z in (Top/B)∗, the Puppe sequence of the map Φ ∧B id isobtained from (1) by smashing with Z.

2. If s1, s2, s respectively denote the sections of X1, X2 and CB(Φ) then

X1/s1(B)Φ−→ X2/s2(B)

¯i(Φ)−→ CB(Φ)/s(B)

is a cofibration sequence in the category Top∗.

Definition 1.1 A cohomology theory on (Top/B)∗ consists of a sequence

of contravariant functors hi : (Top/B)∗ −→ Abi∈ZZ (Ab is the category ofabelian groups) and a sequence of natural transformations δi such that:

1. δi : hi(X1) −→ hi+1(CB(Φ)) is defined for any cofibration sequence

X1Φ−→ X2

i(Φ)−→ CB(Φ) and the following sequence is exact:

. . . −→ hi−1(X1)δi−1

−→ hi(CB(Φ))i(Φ)∗−→ hi(X2)

Φ∗−→ hi(X1) −→ . . .

2. (Dold’s cylinder axiom) For any space X and any θ : X × [0, 1] −→ B,

we have h∗(X × [0, 1]/X) = 0, where the inclusion of X into X × [0, 1]is given by x 7→ (x, 0).

3. (Strong additivity) For any family Xββ∈J of sectionned spaces overB, we have

h∗(∨B

Xβ) ∼=∏

h∗(Xβ) .

It can be checked that these axioms imply that a cohomology theory h∗(−)on (Top/B)∗ is homotopy invariant and possesses a natural suspension iso-morphism, that is

h∗+1(ΣBX) ∼= h∗(X)

for any X in (Top/B)∗.Before giving examples, we would like to discuss product structures. A

cohomology theory h∗(−) on (Top/B)∗ is called multiplicative if it is equippedwith a natural pairing of graded abelian groups

κ : h∗(X)⊗ h∗(Y ) −→ h∗(X ∧B Y ), X, Y ∈ (Top/B)∗

which is associative and has a unit 1 ∈ h0((S0, ∗)B). The sectionned space(S0, ∗)B plays the role of a point in our category, since (S0, ∗)B∧BX = X for

5

any X in (Top/B)∗. Consequently h∗((S0, ∗)B) is a ring, called the coefficient

ring of h∗(−), and h∗(X) is a two sided module over it. In [11, Lemma 4.1]it is shown that the pairing κ factors through:

κ : h∗(X)⊗h h∗(Y ) −→ h∗(X ∧B Y ), X, Y ∈ (Top/B)∗

where h is a short hand for h∗((S0, ∗)B).

Example. Let E∗(−) be a multiplicative cohomology theory on the categoryof pointed spaces and E∗(−) the associated unreduced theory. Then

E∗B(−) : (Top/B)∗ −→ Ab, X 7→ E∗(X/s(B))

defines a multiplicative cohomology theory on (Top/B)∗. The coefficient

ring is E∗B((S0, ∗)B) ∼= E∗(B), so that any E∗B(X) is an E∗(B)-module. Moregenerally, for any fibration X −→ B the functor

(Top/B)∗ −→ Ab, Y 7→ E∗B(X+ ∧B Y )

is also a multiplicative cohomology theory on (Top/B)∗, whose coefficientring is E∗(X). The interested reader might consult the details in [5, Section3.4].

From now on, E∗(−) will denote a multiplicative cohomology theory onthe category Top∗ 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 E∗(−) satisfies theKunneth isomorphism (see [3]).

After all these preliminaries, we are now ready to present the geometriccobar resolution. To this end, we suppose that we are given a space X in(Top/B)∗. From this datum, we perform the following construction:

1. First we set X0 = X.

2. For each integer i ≤ 0 we suppose that Bs−→ Xi

p−→ B has beenconstructed and we successively define:

• Xi = (Xi/s(B))B

• φi : Xi −→ Xi , φi(x) = ([x], p(x))

• Xi−1 = CB(φi).

The most important properties of this construction are given in

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Lemma 1.2 With the notations above and for any integer i ≤ 0, the fol-lowing statements are true:

1. For any space Y ∈ (Top/B)∗, the spaces (Xi∧BY )/s(B) and Xi/s(B)∧Y/s(B) are homotopy equivalent in Top∗.

2. E∗B(Xi) ∼= E∗B(Xi)⊗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 φ∗i : E∗B(Xi) −→ E∗B(Xi) is 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 Xi/s(B) = Xi/s(B) ∧ B+. Itsuffices then to apply the Kunneth isomorphism. Concerning the third part,let

ψi : Xi/s(B) ∧B+ −→ Xi/s(B)

denote the collapsing of B to a point. Then ψi φi = id, whereφi : Xi/s(B) −→ Xi/s(B) is induced by φi. Consequently ψ∗i is a right in-verse for φ∗i and we are done.

Thanks to this Lemma, the long exact sequences of the φi’s break intoshort exact sequences:

0 −→ E∗B(Xi−1) −→ E∗B(Xi) −→ E∗B(Xi) −→ 0 .

As usual in homological algebra, we splice all these sequences together asfollows:

0 0ZZZ =

E∗B(X−1)

= ZZZ

0 E∗B(X) E∗B(X0) E∗B(X−1) E∗B(X−2) . . .ZZZ =

E∗B(X−2)

= ZZZ

0 0

As a result, we obtain a long exact sequence of E∗(B)-modules:

0←− E∗B(X)←− E∗B(X0)←− E∗B(X−1)←− . . . (2)

7

Fix i ≤ 0 and write E∗B(Xi) = E∗(Xi/s(B)) = lim←−

V αi , where each V α

i is

a finite dimensional E∗(pt)-module (see Example 1 in the Appendix). ByLemma 1.2 and Theorem 6.3.ii), we have

E∗B(Xi) ∼= ( lim←−

V αi )⊗E∗(pt)E

∗(B) ∼= lim←−

(V αi ⊗E∗(pt)E

∗(B)) .

Since each V αi ⊗E∗(pt)E

∗(B) is obviously a free E∗(B)-module, we can in-

voke Theorem 6.1.ii) to conclude that E∗B(Xi) is a projective E∗(B)-module.Consequently the sequence (2) is a projective resolution of the E∗(B)-module

E∗B(X).

2 The spectral sequence

The aim of this section is to construct the spectral sequence announced inthe introduction, to identify its E2-term and to discuss its convergence. Thisspectral sequence will be defined by means of derived exact couples. In orderto simplify the presentation, we will always replace finite sequences of mapsby inclusions in (Top/B)∗, via the fibrewise mapping cylinder construction.With this convention, the cofibre of a map f : A −→ C will simply be writtenC/A.

Let X, Y be spaces in (Top/B)∗ and let Xi −→ Xi −→ Xi−1i≤0 be thegeometric 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 φj’s toobtain the sequence Xi−1 −→ ΣBXi −→ Σ−i+1

B X0. The latter induces acofibration

ΣBWi/Wi−1βi−→ Σ−i+1

B W0/Wi−1αi−→ Σ−i+1

B W0/ΣBWi . (3)

To introduce the spectral sequence we define, for i ≤ 0

Di,j1 = E j+1

B (Σ−i+1B W0/Wi−1)

Ei,j1 = E j+1

B (ΣBWi/Wi−1) ,

and for i > 0Di,j

1 = Ei,j1 = 0 .

The cofibration (3) induces a long exact sequence in E∗B-cohomology whichcan be written as

. . . −→ Di+1,j−11

α∗i−→ Di,j1

β∗i−→ Ei,j1

γ∗i−→ Di+1,j1 −→ . . .

8

where γ∗i is induced by the Puppe map

γi : Σ−i+1B W0/ΣBWi −→ ΣB(ΣBWi/Wi−1) .

We splice all these exact sequences to form the following unraveled exactcouple (in the sense of Boardman [2])

· · · −→ Di+1,∗1

- Di,∗1

- Di−1,∗1

- . . .

I@@@

I@

@@

Ei,∗1 Ei−1

1

Observe that the bidegree of α∗i (respectively β∗i , γ∗i ) is (−1, 1) (respect-

ively (0, 0), (1, 0)). By standard techniques this yields a spectral sequenceE∗,∗r (X, Y ); drr≥1 (or simply E∗,∗r ; drr≥1 when X and Y are understood),with differentials dr : Ei,j

r −→ Ei+r,j+1−rr . As usual, for any r ≥ 2, the cycles

and boundaries are defined as

Z∗,∗r = Ker(dr−1), B∗,∗r = Im(dr−1) .

Since E∗,∗r = Z∗,∗r /B∗,∗r , it is important to have a geometric description of thegroups Z∗,∗r and B∗,∗r . For this purpose, we write

• ι for the natural inclusion (Wi−1; ΣBWi) ⊂ (Wi−1; ΣrBWi+r−1)

• δ = δ1 σ, where δ1 is the coboundary of the triple(Wi−r,Σ

r−1B Wi−1,Σ

rBWi) and σ the suspension isomorphism

E j+rB (ΣrBWi/Σ

r−1B Wi−1) ∼= E j+1

B (ΣBWi/Wi−1) .

Then we have

Zi,jr∼= Imι∗ : E j+1

B (ΣrBWi+r−1/Wi−1) −→ E j+1

B (ΣBWi/Wi−1)Bi,jr∼= Imδ : E j+r−1

B (Σr−1B Wi−1/Wi−r) −→ E j+1

B (ΣBWi/Wi−1) .

With these identifications the differentials dr are induced by the composite

E j+1B (Σr

BWi+r−1/Wi−1)δ2- E j+2

B (Σ−i+1B W0/Σ

rBWi+r−1)

E j+2B (Σr+1

B Wi+r/ΣrBWi+r−1)

?β∗i+r

E j−r+2B (ΣBWi+r/Wi+r−1) ,

∼=?σ

9

where δ2 is the coboundary of the triple (Wi+1,ΣrBWi+r+1,Σ

−i+1B W0) and β∗i+r

is as above. The proof of these facts is standard; the interested reader mightwant to perform it by chasing on diagrams like the one on the top of page661 in [22].

Everything is now in place for the identification of the E2-term of thatspectral sequence. This is based on the following

Lemma 2.1 For any i ≤ 0 and any space Y in (Top/B)∗, the product in

E∗B-cohomology induces an isomorphism

E∗B(Xi)⊗E∗(B)E∗B(Y ) ∼= E∗B(Xi ∧B Y ) .

Proof. Consider the diagram

E∗B(Xi)⊗E∗(B)E∗B(Y ) - E∗B(Xi ∧B Y )

(E∗(Xi/s(B))⊗E∗(pt)E∗(B))⊗E∗(B)E

∗(Y/s(B))

?

∼=

- E∗(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 anisomorphism (this is the ordinary Kunneth isomorphism in E∗-cohomology).This complete the proof of the lemma.

Let us now deal with the differential d1 : Ei−1,∗1 −→ Ei,∗

1 . By definition itis the composite β∗i γ∗i−1. As easily checked, the following diagram commutes

Σ−i+2B W0

Σ2BWi/ΣBWi−1

- Σ2BWi−1

6

- ΣB(ΣBWi−1/Wi−2)

@@@βiR

γi−1

Σ−i+2B W0/ΣBWi−1

6

The horizontal composition is equivalent to

ΣBXi+1 ∧B Yφi+1∧Y−→ Σ2

BXi ∧B Yφi∧Y−→ Σ2

BXi ∧B Y .

10

This observation and Lemma 2.1 imply that the complex

0←− E0,∗1

d1←− E−1,∗1

d1←− E−2,∗1

d1←− . . .

is isomorphic to the one obtained by applying the functor −⊗E∗(B)E∗B(Y )

to the complex0←− E∗B(X0)←− E∗B(X−1)←− . . .

We have thus shown that

Ei,∗2∼= Tor−iE∗(B)(E∗B(X), E∗B(Y )) ,

where Tor−iE∗(B)(−,−) is the i-th derived functor of the completed tensor

product (see the Appendix).

We now turn to the convergence questions. Let us introduce the gradedgroup H∗(X, Y ) by taking Hr(X, Y ) as the direct limit of the sequence

D0,r1 −→ D−1,r+1

1 −→ . . . −→ Di,r−i1 −→ . . .

The group H∗(X, Y ) is filtered (as graded group) by

FiHr(X, Y ) = ImDi,r−i1 −→ 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 convergesto H∗(X, Y ); that is,

Ei,r−i∞∼= FiHr(X, Y )/Fi+1Hr(X, Y ) , for all i .

The expected target group is E∗B(W0) = E∗B(X ∧B Y ). In this section we will

give conditions under which H∗(X, Y ) is isomorphic to E∗B(X ∧B Y ).For each i ≤ 0, we consider the exact triangle induced in E∗-cohomology

by the cofibration Wi−1 −→ Σ−i+1B W0 −→ Σ−i+1

B W0/Wi−1. The direct limitof these triangles is the exact triangle

H∗(X, Y )Φ - E∗B(W0)

I@@@

G∗(X, Y )

(4)

where Gr(X, Y ) is the direct limit of the sequence

Er+1B (W−1) −→ Er+2

B (W−2) −→ . . . −→ Er−i+1B (Wi−1) −→ . . .

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If the group G∗(X, Y ) vanishes, then the map Φ of the triangle (4) will be anisomorphism so that our spectral sequence will have the expected abutment.The following proposition describes the main property of this obstructiongroup and gives a sufficient condition for the convergence to E∗B(X ∧B Y ).

Proposition 2.2 Let X be an element of (Top/B)∗ such that the projec-tion p : X −→ B is a fibration. The functor Y 7→ G∗(X, Y ) is a cohomologytheory on (Top/B)∗. Consequently, if G∗(X, pt+) = 0, then G∗(X, Y ) = 0 forall spaces Y in (Top/B)∗.

Proof. The first axiom of Definition 1.1 is easily checked using the exactnessof the direct limits and the properties of the smash product and cofibrationsin the category (Top/B)∗. The cylinder axiom requires the extra assumptionon the projection p. By Proposition 4.8 of [18] and Section 3.4 of [5], the

functors Y 7→ E∗B(Xi ∧B Y ) are cohomology theories for each i ≤ 0. Weinvoke one more time the exactness of the direct limits to conclude. We areleft to show that G∗(X, Y ) is strongly additive. Let Yββ∈J be a family ofspaces in (Top/B)∗. We want to prove that the natural injections Yβ →∨B Yβ induce an isomorphism

G∗(X,∨B Yβ) ∼=

∏G∗(X, Yβ) .

We start with three observations:

1. W0(X,∨B Yβ) is homotopy equivalent to

∨BW0(X, Yβ).

2. E∗B(∨BW0(X, Yβ)) is naturally isomorphic to

∏E∗B(W0(X, Yβ)).

3. The following triangle is exact (the direct product preserves exactness)∏H∗(X, Yβ)

Φ -∏E∗B(W0(X, Yβ))

I@@@

∏

G∗(X, Yβ)

Because of these observations, it is sufficient to prove that

H∗(X,∨B Yβ) ∼=

∏H∗(X, Yβ) .

From the definition of the spectral sequence and exactness of the directproduct, one checks easily that

∏E∗,∗r (X, Yβ);

∏drr≥1 is a spectral se-

quence. Moreover, the natural injections Yβ →∨B Yβ induce a morphism of

spectral sequences

ψr : E∗,∗r (X,∨B Yβ) −→

∏E∗,∗r (X, Yβ) . (5)

12

As the completed tensor product commutes with the direct products, themorphism

ψ2 : Tor∗E∗(B)(E∗B(X), E∗B(∨B Yβ)) −→

∏Tor∗E∗(B)(E∗B(X), E∗B(Yβ))

is an isomorphism. Thus the spectral sequences of (5) are isomorphic.We filter the group

∏H∗(X, Yβ) by setting Fi

∏H∗(X, Yβ) =

∏FiH∗(X, Yβ).

This filtration is Hausdorff and complete. We observe that the natural injec-tions Yβ →

∨B Yβ induce a continuous homomorphism

ψ0 : H∗(X,∨B Yβ) −→

∏H∗(X, Yβ) . (6)

Its associated graded homomorphism is nothing but the isomorphism

ψ∞ : E∗,∗∞ (X,∨B Yβ) −→

∏E∗,∗∞ (X, Yβ) .

As the two groups involved in (6) are Hausdorff and complete, ψ0 has to bean isomorphism. The last assertion follows from the comparison theorem 4.1of [5].

3 The multiplicative structure

The discussion of the multiplicative structure of the Eilenberg-Moore spectralsequence requires more general definitions than those given in Section 2.Since this material will be used only in the present section, we have chosento introduce it here.

Definition 3.1 Let m ≥ 1 and X ∈ (Top/B)∗. A negative m-filtrationof X is a sequence U∗ of spaces Uii≤0 and maps ψi : Ui−1 −→ Σm

BUi in(Top/B)∗, with U0 = X. A morphism f∗ : U∗ −→ V∗ of negative m-filtrationsis a sequence of maps fi : Ui −→ Vi making the obvious diagrams commutat-ive.

Examples.

1. For X in (Top/B)∗ and m ≥ 1 the geometric cobar resolution of degreem− 1 of X is inductively defined by setting X0 = X and for i ≤ 0,

• Xi = (Xi/s(B))B

• φi : Xi −→ Xi , φi(x) = ([x], p(x))

• Xi−1 = CB(Σm−1B φi).

13

We set Xi(m) := Xi and take ψi : Xi−1(m) −→ ΣmBXi(m) to be the

next map in the Puppe sequence of the cofibration

Σm−1B Xi(m) −→ Σm−1

B Xi(m) −→ Xi−1(m) .

The resulting negative m-filtration will be denoted X∗(m) and calledthe cobar m-filtration. When m = 1, we recover the 1-filtration associ-ated to the geometric cobar resolution of Section 1; we will then simplywrite X∗ := X∗(1).

2. Let U∗ be a negative m-filtration of X ∈ (Top/B)∗ and let Y ∈(Top/B)∗. One constructs a negative m-filtration U∗ ∧ Y by smash-ing all the constituents of U∗ by Y . In particular, we suspend negativefiltrations by smashing them with Y = (Sk; ∗)B.

We recall here a construction of Hodgkin, since it will play a centralrole in our discussion. It might be illuminating to view this constructionas the geometric counterpart of the tensor product of chain complexes inhomological algebra. To get into work, we fix X ∈ (Top/B)∗ and let U∗ bea negative m-filtration of U0 = X. For any integer i ≤ 0 we consider thesequence

Uiψi+1−→ Σm

BUi+1ψi+2−→ . . .

ψ0−→ Σ−imB U0 . (7)

We may and will assume (in accordance with our convention) that all themaps in this sequence are inclusions. Let Zi be the subspace in (Top/B)∗ ofΣ−imB U0

∧B Σ−imB U0 defined by

Zi =⋃k+l=i

Σ−kmB Ui−k ∧B Σ−lmB Ui−l .

By comparing the sequence (7) for the index i and i−1, Hodgkin constructeda natural map χi : Zi−1 −→ Σ2m

B Zi and showed (see [11, page 24]) that

Σ2mB Zi/Zi−1 '

∨k+l=i

Σ−(i−k)mB (Σm

BUk/Uk−1) ∧B Σ−(i−l)mB (Σm

BUl/Ul−1) . (8)

As in [11], the negative 2m-filtration Zi;χii≤0 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 E∗B(−) a cohomology theory on (Top/B)∗. We fix an integer

14

i ≤ 0 and extract the part of the sequence (7) given by the inclusions

Ui−1 → ΣmBUi → Σ

(−i+1)mB U0. This sequence induces a cofibration

ΣmBUi/Ui−1 −→ Σ(−i+1)m

B U0/Ui−1 −→ Σ(−i+1)mB U0/Σ

mBUi .

With respect to this cofibration we define

Di,j1 = E j+1+(−i+1)(m−1)

B (Σ(−i+1)mB U0/Ui−1)

Ei,j1 = E j+1+(−i+1)(m−1)

B (ΣmBUi/Ui−1) .

We extend to all integers by setting, for i > 0

Di,j1 = Ei,j

1 = 0 .

As in Section 2 we obtain an unravelled exact couple; the associated spec-tral sequence is written Ei,j

r (U∗); drr≥1. Here also, the result of Board-man [2] implies that the spectral sequence strongly converges to H∗(U∗) :=lim−→

(D0,∗1 → D−1,∗+1

1 → . . . ). The latter is related to the desired abutment

via a natural map Φ : H∗(U∗) −→ E∗B(U0).

Lemma 3.2 Let X, Y ∈ (Top/B)∗ and m ≥ 1. Write X∗(m) (respectivelyX∗) for the cobar m-filtration (respectively 1-filtration) of X. Then there isa 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 φi : Xi(m) −→ Σ−i(m−1)B Xi by taking

φ0 = id and requiring the commutativity of the diagram

Σm−1B Xi(m) - Σm−1

B Xi(m) - Xi−1(m)

Σ(−i+1)(m−1)B Xi

φi?- Σ

(−i+1)(m−1)B Xi

φi?- Σ

(−i+1)(m−1)B Xi−1

φi−1?

where φi is the composite

Σm−1B Xi(m) −→ Σm−1

B (Σ−i(m−1)B Xi/s(B)×B) −→ Σ

(−i+1)(m−1)B Xi .

15

Then the map φi ∧ id : Xi(m) ∧B Y −→ Σ−i(m−1)B Xi ∧B Y induces the de-

sired homomorphism of spectral sequences. As in Section 1, one constructsa projective E∗(B)-resolution

0←− E∗B(X)←− E∗B(X0(m))←− E∗B(X−1(m))←− . . .

The comparison theorem of projective resolutions now implies that φi ∧ idinduces an isomorphism of E2-terms 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 ∈ (Top/B)∗ and set W∗ = X∗ ∧ Y , where X∗is the cobar 1-filtration of X. For 1 ≤ r ≤ ∞, there exist associative pairings

ψr : Ei,jr (W∗)⊗ Ep,q

r (W∗) −→ Ei+p,j+qr (W∗ ⊗W∗), a⊗ b 7→ a · b

satisfying the following properties:

1. ψ1 is induced by the multiplication of E∗B(−) .

2. ψr+1 is induced by ψr, via the isomorphism Er+1∼= H(Er).

3. For all a ∈ Ei,jr (W∗) and b ∈ Ep,q

r (W∗), we have

dr(a · b) = dr(a) · b+ (−1)j+1a · dr(b) .

Proof. We begin with some notation. For any negative m-filtration U∗, weset:

• Ai,jr (U∗) = E j+1+(−i+1)(m−1)B ((Σrm

B Ui+r−1/Ui−1)).

• For s ≤ r, αr,s : Ai,jr (U∗) −→ Ai,js (U∗) is the morphism induced by theinclusion (Ui−1,Σ

smB Ui+s−1) → (Ui−1,Σ

rmB Ui+r−1).

• ∆i,jr : Ai,jr (U∗) −→ Ai+r,j−r+1

r (U∗) is the coboundary operator of the triple(Ui−1,Σ

rmB Ui+r−1,Σ

2rmB Ui+2r−1).

Let us observe that Ai,j1 (U∗) = Ei,j1 (U∗), the first term of the spectral sequence

associated to U∗. By proceding as in the case m = 1 (see Section 2), we seethat

Zi,jr∼= Imαr,1 : Ai,jr (U∗) −→ Ai,j1 (U∗)

Bi,jr∼= Imδ : Ai−r+1,j+r−2

r−1 (U∗) −→ Ai,j1 (U∗) ,

16

where δ is defined as in Section 2. We set ν = j + 1 + (−i + 1)(m − 1)and consider the commutative diagram (obtained by playing around withadequate triples):

Eν+1B ((Σ

(−i+1)mB U0/Σ

rmB Ui+r−1))

δ > ZZZ

β∗i+r~

EνB((ΣrmB Ui+r−1/Ui−1)) Eν+1

B ((ΣmBUi+r/Ui+r−1))

ZZZ∆~

α >

Eν+1B ((Σ2rm

B Ui+2r−1/ΣrmB Ui+r−1))

?

As in Section 2 and with the identifications above, the differential dr is equalβ∗i+r δ. Consequently, dr is induced by the morphism ∆∗,∗r .

We now go back to our data and define pairings

φr : Ai,jr (W∗)⊗Ap,qr (W∗) −→ Ai+p,j+qr (W∗ ⊗W∗)

in the following manner. First we use the product of E∗B(−) and the suspen-sion isomorphism to send Ai,jr ⊗ Ap,qr into

E∗B(Σ−p+1B (Σr

BWi+r−1/Wi−1) ∧B Σ−i+1B (Σr

BWp+r−1/Wp−1)) .

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 verbatimto our situation. To be entirely honest, we should mention that Kulze’sproofs are based on two hypothesis (the axioms M1 and M2 (page 290) of[13]). Fortunately, these hypotheses are satisfied in our case (see Theorem9.10 (page 238) of [1]).

As in the classical case, the next step consists in comparing the spectralsequences of the negative filtrations W∗⊗W∗ and W∗, via the diagonal map.Unfortunately we have not been able to proceed directly. However, as wewill 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 anynegative m-filtration U∗, the suspension isomorphism induces an isomorphismof spectral sequences (This is easily checked at the level of exact couples)

σ : Ei,jr (U∗)

∼=−→ Ei,j+1r (ΣU∗). (9)

Lemma 3.4 Let U∗ be a negative m-filtration of U0 = V . As above, wewrite V∗(m) for the cobar m-filtration of V . There exists a negative m-filtration Z∗ of V and two morphisms of negative filtrations

ΣV∗(m)g∗←− Z∗

f∗−→ ΣU∗

such that f0 = g0 = id.

17

Proof. To begin with we set Z0 = V and f0 = g0 = id. Given fi : Zi −→ ΣBUiand gi : Zi −→ ΣBVi(m), let

U ′i = ΣmBUi/Ui−1, V

′i = Σm

B Vi(m) and Z ′i = Σm−1B Zi/s(B)× (U ′i ×B V ′i ) .

We define Zi−1 as the fiberwise cone of the obvious map Σm−1B Zi −→ Z ′i and

we construct fi−1 and gi−1 by requiring that the following diagram commutes:

ΣmBUi

- U ′i- ΣBUi−1

Σm−1B Zi

6fi

- Z ′i

6f ′i

- Zi−1

6fi−1

ΣmBVi(m)?gi

- V ′i

?g′i

- ΣBVi−1(m)?gi−1

Here f ′i and g′i are the obvious projections.

Theorem 3.5 Let X, Y ∈ (Top/B)∗ and set W∗ = X∗ ∧ Y , where X∗ isthe cobar 1-filtration of X. There exist associative pairings, for 2 ≤ r ≤ ∞,

µr : Ei,jr (W∗)⊗Ep,q

r (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)(E∗B(X), E∗B(Y )), then µ2 becomesthe usual internal product of Tor∗E∗(B)(−,−).

2. µr+1 is induced by µr, via the isomorphism Er+1∼= H(Er).

3. For all a ∈ Ei,jr (W∗), b ∈ Ep,q

r (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∗); drr≥1.

There is a product µ : H∗(X, Y )⊗H∗(X, Y ) −→ H∗(X, Y ) which in-duces µ∞. In addition, the natural homomorphism

Φ : H∗(X, Y ) −→ E∗B(X ∧B Y )

respects the products.

18

Proof. For the construction of the µr’s, we already have the composite

E∗,∗r (W∗)⊗E∗,∗r (W∗)ψr - E∗,∗r (W∗ ⊗W∗)

E∗,∗r ((X∗ ⊗X∗) ∧ (Y ∧B Y ))

∼=?τr

E∗,∗r (ΣB(X∗ ⊗X∗) ∧ (Y ∧B Y ))

∼=?σr

(10)

where ψr has been defined in Proposition 3.3, τr is induced by the twistsX ∧B Xj ' Xj ∧B X and σr 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 2-filtration Z∗ and two morphisms of spectral sequences

E∗,∗r (ΣBU∗ ∧ (Y ∧B Y ))f ∗ - E∗,∗r (Z∗ ∧ (Y ∧B Y ))

E∗,∗r (ΣBV∗(2) ∧ (Y ∧B Y )) .

g∗6

(11)

We claim that g∗ is an isomorphism for any r ≥ 2. This can be checked as inLemma 3.2, using property (8) and the construction of Z∗ (see Lemma 3.4).To obtain the pairing µr, we compose the diagrams (10) and (11) with thesequence

E∗,∗r (ΣBV∗(2) ∧ (Y ∧B Y ))σ−1- E∗,∗r (V∗(2) ∧ (Y ∧B Y ))

E∗,∗r (X∗(2) ∧ Y )

∆∗?

E∗,∗r (W∗)

∼=?

(12)

where σ−1 is the inverse of the suspension isomorphism (9), ∆∗ is inducedby the diagonals X −→ X ∧B X and Y −→ Y ∧B Y and the last arrowis the isomorphism of Lemma 3.2. Even though suspensions appear in theconstruction, we note that the µr’s are bigraded morphisms.

We now go through the claimed properties of the pairings. The first onefollows 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,q

1 (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 asequence of morphisms following the same pattern as in diagrams (10), (11)and (12). Of course one needs to check that

g∗ : H∗(ΣBV∗(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 ofthese constructions implies that Φ is multiplicative and µ induces µ∞.

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 ΩB will be denoted e : ΣΩB −→ B. As-sume that E∗(ΩB) ∼= Λ(ξ1, . . . , ξn) with ξi ∈ Eodd(B) for i = 1, . . . , n. Aneasy calculation with the Rothenberg-Steenrod spectral sequence implies thatE∗(B) ∼= E∗[[ρ1, . . . , ρn]] where ρi ∈ Eeven(B) is chosen so that e∗(ρi) = σξifor i = 1, . . . , n.

Theorem 4.1 Let B be a connected and pointed space, p : EB −→ B thepath space fibration and assume that E∗(ΩB) ∼= Λ(ξ1, . . . , ξn) with ξi ∈Eodd(B). Then the Eilenberg-Moore spectral sequence of the pull-back dia-gram

ΩB - EB

pt?⊂ - B

?

converges strongly to E∗(ΩB).

20

Following the notation of Section 1, we write Xi −→ Xi −→ Xi−1i≤0 forthe 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 inclusionpt+ → B+ = (S0, ∗)B):

W0- EB+ - W−1

i0- ΣBW0

X0

j0?

φ0

- X0

?- X−1

j1?

- ΣBX0 .?j0

By definition E∗B(W0) ∼= E∗(ΩB) and the contractibility of EB implies that

the composite E∗(ΣΩB) → E∗(ΣΩB) ∼= E∗B(ΣBW0)i∗0−→ E∗B(W−1) is an

isomorphism. By abuse of notation, we will identify the generators σξi ∈E∗(ΣΩB) to their images in E∗B(W−1). A similar argument shows that E∗B(X−1)∼= E∗(B); here also we will identify the ρi’s with their images in E∗B(X−1).

Lemma 4.2 With the notation and identifications above, the morphisminduced in E∗B-cohomology by the map j1 : W−1 −→ X−1 satisfies: j∗1(ρi) =−σξi, for i = 1, . . . , n.

Proof. Let ϕ : ΣΩB −→ B be the adjoint of the map inv : ΩB −→ ΩBwhich sends a loop α to its inverse α−1. We leave as an exercise to checkthat, in E∗-cohomology, ϕ∗(ρi) = −σξi, for i = 1, . . . , n.

We will construct two maps (in Top∗) F : X−1/s(B) −→ B andf : W−1/s(B) −→ ΣΩB making the following diagram commutative:

W−1- W−1/s(B)

f- ΣΩB

X−1

j−1?

- X−1/s(B)?

F- B .

?ϕ (13)

The space X−1 is by definition the cofiber of

φ0 : X0 = EB+ −→ X0 = X0/s(B)×B .

21

More precisely, X−1 = CB(X0)∐X0/(α; 0) ∼ φ0(α), where CB(X0) denotes

the reduced cone over B of X0 (see [18] for its definition). We construct amap X−1 −→ B by sending

(α; t) 7→α(1− t)α

if α ∈ EBif α ∈ B

([x]; b) 7→ b if ([x]; b) ∈ X0 .

We leave as an exercise to check that this map is well defined and inducesF : X−1/s(B) −→ B. It is also easily checked that the composite

(EB ×B)∐

pt = X0/s(B) −→ X−1/s(B)F−→ B

is the projection onto the second factor. This shows that F ∗(ρi) = ρi fori = 1, . . . , n.

Similarly W−1 is the cofiber of the map W0 = ΩB+ −→ EB+. ThusW−1 = CB(W0)

∐EB+/(α; 0) ∼ α. We construct a map W−1 −→ ΣΩB by

sending

(α; t) 7→

[(α, t)][(ε0, 0)]

if α ∈ ΩBif α ∈ B

α 7→ [(ε0, 0)] if α ∈ EB+ ,

where ε0 ∈ EB+stands for the trivial loop. It is easily checked that thismap is well defined and induces f : W−1/s(B) −→ ΣΩB. We also leave asan exercice that the following diagram commutes up to homotopy:

W−1/s(B)i0- ΣBW0/s(B)

ΣΩB

f?

ΣprΣ(ΩB

∐pt)

?=

This implies that f ∗(σξi) = σξi 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 W−1 → ΣBW0 andW−2 → ΣBW−1. For simplicity, we will use the following identifications

E∗B(ΣBW−1/W−2) ∼= E∗B(ΣBX−1 ∧B pt+) ∼= E∗(ΣB) ∼= E∗−1(B) .

The second isomorphism follows from Lemma 1.2 and the others are obvious.

22

Lemma 4.3 Let q1 be the projection ΣBW−1 −→ ΣBW−1/W−2. With theidentifications above, we have q∗1(ρi) = −σξi , for i = 1, . . . , n.

Proof. We consider the following commutative diagram in (Top/B)∗, ob-tained from the natural map X−2 → ΣBX−1:

ΣBW−1

q1- ΣBW−1/W−2

ΣBX−1

j1? q2- ΣBX−1/X−2 .

?j2

With our usual identifications, we have

E∗B(ΣBX−1/X−2) ∼= E∗B(ΣBX−1) ∼= E∗−1B (X−1) ∼= E∗(B)⊗ E∗(B) .

The two homomorphisms j∗2 and q∗2 send ρi⊗

1 to ρi. We note that thesehomomorphisms behave differently, for example on the elements ρi

⊗ρj .

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

Φ : H∗(EB+, pt+) −→ E∗B(W0) is a ring isomorphism. In our situation, theE2-term of the Eilenberg-Moore spectral sequence is isomorphic to

Tor∗E∗(B)(E∗(pt), E∗(pt)) ∼= Λ(y1, . . . , yn) ;

the argument is standard and involves the Koszul resolution (see [21]). Fordimensional reasons, the generators yi are permanent cycles. The multi-plicative properties of the spectral sequence imply that it collapses; that is,E2∼= E∞. Hence, H∗(EB+, pt+) is a free E∗(pt)-module of rank 2n. We next

consider the diagram

E∗B(W0)i∗0- E∗+1

B (W−1)i∗1 - E∗+2

B (W−2) - . . .

I@@@q∗1

E∗B(ΣBW−1/W−2)

Since Im(q∗1) = Ker(i∗1), Lemma 4.3 implies that the image of the generators

ξi ∈ E∗B(W0) ∼= E∗(ΩB) are zero in the obstruction group G∗(EB+, pt+) =

lim−→E∗B(Wi). Therefore the generators ξi lie in the image of Φ, 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∗(ΩB) ∼=Λ(ξ1, . . . , ξn), with ξi ∈ Eodd(B). Let X, Y in (Top/B)∗ and assume thatthe projection p : X −→ B is a fibration. Then the Eilenberg-Moore spectralsequence E∗,∗r (X, Y ); drr≥1 converges strongly to E∗B(X ∧B Y ).

Proof. Recall that G∗(−,−) denotes the obstruction to the “good beha-viour” of the Eilenberg-Moore spectral sequence (see Section 2). By The-orem 4.1, we have G∗(EB+, pt+) = 0, where p : EB −→ B is the pathspace fibration. It follows from Proposition 2.2 that G∗(EB+, X) is alsotrivial. The argument of Hodgkin (see pages 40-41 of [11]) shows that0 = 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; henceG∗(X,EB+) = G∗(X, pt+) = 0. We invoke one more time Proposition 2.2 toobtain G∗(X, Y ) = 0 and this concludes the proof.

Remark. If E∗(ΩB) is not an exterior algebra, the spectral sequence maynot converge to the desired target. For instance, let E∗(−) = K∗(−; IF2) bethe mod 2 complex K-theory and B = BSO(3), the classifying space of theLie group SO(3). As well known, we have

K∗(B; IF2) ∼= IF2[[ρ]] and K∗(ΩB; IF2) ∼= IF2[ξ]/(ξ4) .

For the path space fibration ΩB −→ EB −→ B, The limit of the Eilenberg-Moore spectral sequence is H∗(EB+, pt+) ∼= Λ(y), which is obvously differentfrom K∗(ΩB; IF2).

5 Applications

In this section, we will use our main theorem to describe the K-theory ofcertain spaces associated to p-compact groups. The basic references for thetheory of p-compact groups are [7] and [15]; we refer to these papers for therelevant facts about these objects.

Throughout this section, p is a fixed prime and R denotes either the fieldIFp of order p or the ring ZZp of p-adic integers. The complex K-theory withcoefficient R will be denoted K∗(−;R).

Our first result provides a large class of p-compact groups satisfying thehypothesis of the third point in the main theorem. The result might be wellknown to the experts, but we haven’t found any explicit reference.

Theorem 5.1 Let X be a connected p-compact group. Then K∗(X;R) isan exterior algebra on odd degrees generators if and only if π1(X) is torsionfree.

24

Proof. A Bockstein spectral sequence argument shows that K∗(X; IFp) isan exterior algebra on odd degrees generators if and only K∗(X; ZZp) is anexterior algebra, if and only if K∗(X; ZZp) is torsion free. This reduces ourproblem to the case R = ZZp.

Observe that π1(X) is a finitely generated ZZp-module, hence π1(X) ∼=ZZp

r ⊕ π where π is a finite abelian p-group. We combine the arguments ofTheorem 4.3 (page 63) in [14] and corollary 3.3 of [15] to show that X ishomotopy equivalent (only as a space) to Y × (S1

p)r, where Y is a connectedp-compact group with π1(Y ) ∼= π.

If π1(X) is torsion free, then Y is a 1-connected p-compact group and wecan invoke a theorem of Kane and Lin (see [12, Theorem 1.2]) to concludethat K∗(X; ZZp) is an exterior algebra. The argument for the converse isimplicit in [10]. Suppose that π is non trivial and choose a cyclic subgroupi : ZZ/p → π. Let Y <1> be the universal cover of Y and M(p) = S1 ∪p e2

be the 2-skeleton of BZZ/p. Consider the pullback diagram

Y <1> = Y <1> = Y <1>

Y (p)?

- Y ′?

- Y?

M(p)

q? j- BZZ/p

? Bi- Bπ .?q

Since Y < 1> is 2-connected, obstruction theory tells us that the fibrationq is trivial, hence q induces an injection in K-theory. As well known, themap Bi induces a surjection in K-theory; an easy computation shows thatthe same is true for the map j. All these facts imply that there exists a classξ 6= 0 in the image of q∗ : K∗(Bπ; ZZp) −→ K∗(Y ; ZZp). A Chern characterargument shows that ξ is a non trivial torsion class in K∗(Y ; ZZp) and thisimplies that K∗(X; ZZp) has non trivial torsion.

Proposition 5.2 Let X be a connected p-compact group such that π1(X) istorsion free and let i : T −→ X be a maximal torus. The ring homomorphismBi∗ : K∗(BX;R) −→ K∗(BT ;R) is injective and makes K∗(BT ;R) into afree and finitely generated K∗(BX;R)- module.

Proof. The arguments of the proof of Theorem 2.7 in [12] are valid in thismore general situation. The injectivity is due to the equality of the Krulldimensions.

25

Everything is now in place for the main result of this section.

Theorem 5.3 Let X be a connected p-compact group such that π1(X) istorsion free. Let i : T −→ X be a maximal torus and X/T the associatedhomogeneous 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 assumethat the map Bi : BT −→ BX is a fibration. Thanks to Theorem 5.1, wecan apply our main theorem to the pullback diagram

X/T ⊂ - BT

pt?⊂ - BX .

?Bi

Consequently, there is a strongly convergent Eilenberg-Moore spectral se-quence

Ei,∗2 = Tori

K∗(BX; IFp)(K∗(BT ; IFp);K

∗(pt; IFp))⇒ K∗(X/T ; IFp) .

Proposition 5.2 above implies that the spectral sequence is trivial, i.e. E∗,∗2 =E0,∗

2 = E∗,∗∞ and the claim follows.The case R = ZZp is a consequence of the preceding one, the universal

coefficients theorem and Nakayama’s lemma.

For a general p-compact group and even if the Eilenberg-Moore spectralsequence does not behave as expected, we still have the following qualitativeresult.

Corollary 5.4 Let X be a connected p-compact group, i : T −→ X amaximal torus and W the corresponding Weyl group. Then K1(X/T ;R) = 0and K0(X/T ;R) is a free R-module of rank |W |.

Proof. Let X < 1> be the universal cover of X. As we have seen above,X<1> is a p-compact group. In the proof of corollary 5.6 of [15], it is shownthat X/T is homotopy equivalent to X < 1> /S, where S −→ X < 1> ismaximal torus for X < 1 >. Since the latter is 1-connected, Theorem 5.3

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applies. We invoke Proposition 9.5 of [7] to obtain the assertion about therank.

One of the main conjecture in the theory of p-compact groups states thatH∗(X/T ; ZZp) 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 K-theoreticalversion.

As a second application, we will now give a slightly different proof of themain result in [12]. With this new method, we obtain an analoguous resultfor mod p K-theory. Even in the case of Lie groups, we are not aware of thismod p K-theory statement in the litterature.

Corollary 5.5 Let X be a connected p-compact group, i : T −→ X amaximal torus and W the corresponding Weyl group. The map Bi induces aring isomorphism

K∗(BX;R) ∼= K∗(BT ;R)W .

Proof. By proceeding as in Section 3 of [12], we may and we will assume thatX is 1-connected. Theorem 5.1 and a Rothenberg-Steenrod spectral sequenceargument imply that K∗(BX;R) ∼= R[[ρ1, . . . , ρn]] with ρi ∈ K0(BX;R)and n equal to the rank of X. Similarly, K∗(BT ;R) ∼= R[[τ1, . . . , τn]] withτi ∈ K0(BT ;R). For simplicity we set

SX = R[[ρ1, . . . , ρn]], ST = R[[τ1, . . . , τn]]

and we will identify SX with its image in ST (this is justified because Bi∗ isinjective by Proposition 5.2).

By construction SX is contained in the ring of invariants SWT . To showthe equality, we consider the following diagram

ST ⊂ - Frac(ST )

SWT

∪

6

⊂ - Frac(SWT )∪

6

SX∪

6

⊂ - Frac(SX)∪

6

where Frac(−) 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 Frac(SX) →Frac(ST ) is a field extension of degree |W |. By Galois theory, Frac(SWT ) =

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Frac(ST )W → Frac(ST ) is a field extension of degree |W |. As consequence,the fields Frac(SX) and Frac(SWT ) coincide. Since ST is integrally closed (itis 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 MoravaK-theories. For n ≥ 1, K(n)∗(−) denotes the n-th Morava K-theory, itscoefficient ring is the graded field IFp[vn, v

−1n ] with |vn| = 2(pn − 1).

Let X be a connected p-compact group, i : T −→ X a maximal torusand W the corresponding Weyl group. If K(n)∗(X) is an exterior algebra onodd degrees generators, then the preceding arguments apply and we have:

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 p-compact groups X such thatK(n)∗(X) is an exterior algebra.

As well-known, K(n)∗(X) is an exterior algebra when H∗(X; ZZp) is torsionfree. Hence we recover Theorem 3.1 of [20]. In contrast to the paper justquoted, we can treat many spaces with torsion. For instance, our Theorem 5.1provides a complete answer to the question when n = 1. More interestingly,Maria Santos (private communication) has observed that K(2)∗(DI(4)) is anexterior algebra; here DI(4) is the exotic 2-compact group constructed byDwyer 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 gradedR-modules, where the morphisms are R-modules homomorphisms of degree0. 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 anyN ∈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 gradedrings and special subcategories of their modules categories. In the sequel, allinverse systems and limits are taken over direct sets.

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A profinite graded ring is an inverse limit of graded rings of finite length(i.e nœtheriean and artinian). We emphasize that, according to this defini-tion, every graded field is profinite; recall that a graded field is a graded ringwhose non zero homogeneous elements are all invertible. If R = lim

←−Rα is

a profinite ring and πα : R −→ Rα are the canonical projections, the familyof graded ideals Ker(πα) 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 theorysuch that E∗(pt) is a graded field. In other words, all graded E∗(pt)-modulesare free. Given a CW-complex X, let Xα be the direct system of all finiteCW-subcomplexes of X. Then we have ([3, Theorem 4.14])

E∗(X) ∼= lim←−E∗(Xα) ;

the corresponding topology will be called the profinite topology of E∗(X).Since the E∗(Xα) are finitely generated free E∗(pt)-modules, they are ringsof 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 lineartopology that an R-module of finite length can carry, the morphisms of F(R)are automatically continuous.

A profinite R-module is an inverse limit of of objects in F(R). Thus it car-ries a natural topology which makes it into a complete Hausdorff topologicalR-modules. Let Modprof(R) be the subcategory of Mod(R) whose objectsare profinite R-modules and whose morphisms are continuous R-module ho-momorphisms of degree 0.

Example 2 E∗(−) is as in example 1 above. Let f : X −→ B be a mapof CW-complexes. By the CW-approximation theorem, the induced mapE∗(f) : E∗(B) −→ E∗(X) is a continuous ring homomorphism (with respectto the profinite topologies). It follows that E∗(f) induces a profinite E∗(B)-module structure on E∗(X).

Example 3 E∗(−) is as above and Bs−→ X

p−→ B is a sectionned space overB. Fix a finite subcomplex Xα of X, set Bα = s−1(s(B) ∩Xα) and denotethe image of Xα in X/s(B) by (X/s(B))α. In the commutative diagram

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Bs- X - X/s(B)

Bα

∪

6

s- Xα

∪

6

- (X/s(B))α∪

6

the rows are cofibrations. Since p s = id, the long exact sequences inE∗-cohomology of these cofibrations reduce to the following commutativediagrams with exact rows:

0 - E∗(X/s(B)) - E∗(X)s∗- E∗(B) - 0

0 - E∗((X/s(B))α)

?- E∗(Xα)

? s∗- E∗(Bα)?

- 0 .

(14)

It follows that E∗(X) ∼= E∗(X/s(B)) ⊕ E∗(B) as profinite E∗(B)-modules,

showing that E∗B(X) := E∗(X/s(B)) is a profinite E∗(B)-module with respectto the topology induced by E∗(X). Using diagram (14), one checks thatthis topology is the same as the profinite topology. Hence we have shownthat E∗B(X) is always a profinite E∗(B)-module with respect to the profinitetopology.

Theorem 6.1 Let R be a profinite graded ring.

i) The category Modprof(R) of profinite R-modules is abelian. It hasenough projective objects and exact inverse limits.

ii) Every inverse limit of projective objects of Modprof(R) is projective.The profinite R-module R is projective.

Proof. As easily checked F(R) is abelian, artinian (i.e, every descendingchain of subobjects of any object of F(R) stabilizes) and equivalent to asmall category. Let Pro(F(R)) be the category of inverse systems in F(R)(see [19, page 21] for the definition). By [17] and [9, page 356]), Pro(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, whosetransition morphisms are epimorphisms). Let us now consider the functor

Λ : Pro(F(R)) −→Modprof(R), (Mα)α∈I 7−→ lim←−

Mα .

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The argument of Section 2.6 in [19] are easily adapted to our situation toshow that Λ is an equivalence of categories. Consequently Modprof(R) enjoysall the properties of Pro(F(R)) mentionned above, and we are done with thefirst point of the theorem.For the second part, we note that both Pro(F(R)) and Modprof(R) areproartinian in the sense of [4, page 563]. The first assertion follows fromCorollaire 3.4 (page 567) in [4]. Finally R is projective because every morph-ism from R into a profinite R-module M is of the form r 7→ r ·m for somem ∈M .

For the rest of this section, R denotes a profinite graded ring which iscommutative. Our next aim is to study the topological tensor product inModprof (R). We start with

Lemma 6.2 If M and M ′ are in F(R), so is M ⊗RM ′.

Proof. Since M is finitely generated, its annihilator Ann(M) = r ∈R; r · m = 0, ∀m ∈ M is an open ideal in R (it is the intersection ofthe annihilators of the members of a finite generating set of M). SimilarlyAnn(M ′) is an open ideal of R. We write R = lim

←−Rα, with each projection

πα : R −→ Rα surjective (this is always possible).Then there exists α suchthat Ker(πα) ⊂ Ann(M) ∩ Ann(M ′). Consequently M , M ′ and M ⊗R M ′can be regarded as Rα-modules. These new structures are strongly relatedto the former since any subset of M (respectively M ′, M ⊗R M ′) is a R-submodule if and only if it is a Rα-submodule. We observe now that Rα isa ring of finite length and M ⊗R M ′ is a finitely generated Rα-module; thisimplies that M ⊗R M ′ is a R-module of finite length, that is, an object ofF(R).

Let M = lim←−

Mα and N = lim←−

Nβ be two modules in Modprof(R).

Their completed tensor product is defined as

M⊗RN = lim←−

(Mα ⊗R Nβ) .

The preceding lemma insures that M⊗RN lies in Modprof(R). It can beshown that M⊗RN is the completion of the ordinary tensor product M ⊗Nwith respect to a certain filtration (see [3, page 603]). This last assertion isvery useful in proving that M⊗RN = M⊗RN when N is a finitely generatedR-module.

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Theorem 6.3 Let N be a profinite R-module.

i) The functor −⊗RN : Modprof (R) −→Modprof (R) is right exact andcommutes with inverse limits.

ii) Let TorjR(−, N) (j = 0, 1, 2, . . . ) denote the j-th left derived functorof −⊗RN (their existence follows from the preceding point). ThenTorjR(−, N) commutes with inverse limits. Moreover it coincides withthe usual TorjR(−, N) when N is a finitely generated R-module.

Proof.

i) This point is true because of the right exactness of −⊗RN and becauseinverse limits are exact and commute with each other in Modprof(R).

ii) The first assertion is standard homological algebra (see for exampleCorollaire 3.8 (page 568) in [4]). The last assertion is a consequence ofthe fact stated before the theorem.

References

[1] J.F. Adams, Stable homotopy and generalized homology, Universityof Chicago Press (1974).

[2] J.M. Boardman, Conditionally convergent spectral sequences, pre-print 1981.

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[4] M. Demazure & P. Gabriel, Groupes algebriques (tome I), Masson& Cie - North Holland (1970).

[5] A. Dold, Chern classes in general cohomology, Symposia Math. vol.V (Geometria), Instituto Naz. di Alta Matematica, Roma (1971).

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[8] W.G. Dwyer & C.W. Wilkerson, The center of a p-compact group,Proceedings of the Cech centennial homotopy theory conference,AMS Contemporary Mathematics, vol. 181 (M. Cenkl and H. Miller,eds), 1995, pp.119-157.

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[11] L. Hodgkin, The equivariant Kunneth theorem in K-theory, SpringerLecture Notes in Math. No 496 (1975), 1-101.

[12] A. Jeanneret & A. Osse, The K-Theory of p-Compact Groups, pre-print (1996), to appear in Comment. Math. Helv.

[13] R. Kultze, Uber multiplikative Eigenschaften von spektralen Sequen-zen, Math. Annalen 158 (1965), 233-268.

[14] N. Mahammed, R. Piccinini & U. Suter, Some applications of topolo-gical K-theory, North-Holland Mathematical Studies No 45, North-Holland (1980).

[15] J.M. Møller & D. Notbohm, Centers and finite coverings of finiteloop spaces, J. reine angew. math. 456 (1994), 99-133.

[16] F. Neumann, On the cohomology of homogeneous spaces of finiteloop spaces and the Eilenberg-Moore spectral sequence, preprint1997.

[17] F. Oort, Commutative group schemes, Lecture Notes in Math. No15, Springer (1966).

[18] L. Smith, Lectures on the Eilenberg-Moore spectral sequence, Lec-ture Notes in Math. No 134, Springer (1970).

[19] J.-P. Serre, Groupes proalgebriques, Publ. Math. IHES 7 (1960),5-68.

[20] M. Tanabe, On Morava K-theories of Chevalley groups, Amer. J.Math. 117 (1995), 263-278.

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[21] C.A. Weibel, An introduction to homological algebra, CambridgeStudies in Advanced Mathematics No 38, Cambridge UniversityPress (1994).

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[23] U. Wurgler, Cobordism theories of unitary manifolds with singular-ities and formal group laws, Math. Zeit. 150 (1976), 239-260.

Mathematisches InstitutSidlerstrasse 5, CH-3012 BERN (Switzerland)

email: jeanner@math-stat.unibe.ch

Institut de MathematiquesRue Emile Argand 11, CH-2007 NEUCHATEL (Switzerland)

email: Akimou.Osse@maths.unine.ch

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