UNSTABLE K -COHOMOLOGY ALGEBRA IS FILTERED λ -RING

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UNSTABLE K -COHOMOLOGY ALGEBRA IS FILTERED λ -RING

DONALD YAU Received 3 August 2002

Boardman, Johnson, and Wilson (1995) gave a precise formulation for an unstable algebra over a generalized cohomology theory. Modifying their definition slightly in the case of complexK-theory by taking into account its periodicity, we prove that an unstable algebra for complexK-theory is precisely a filteredλ-ring and vice versa.

2000 Mathematics Subject Classification: 55N20, 55N15, 55S05, 55S25.

1. Introduction. Lambda operations in complexK-theory were first intro- duced by Grothendieck. These operations should be thought of as exterior power operations; in fact, for an elementαinK(X)that comes from an actual vector bundle onXwithXa finite complex,λⁱ(α)is the element represented by theith exterior power of that vector bundle. It was realized early that these operations generate allK-theory operations (see, e.g., [8]). Aλ-ring is, roughly speaking, a commutative ring with operationsλⁱwhich behave exactly likeλ- operations onK-theory of spaces. It is, therefore, natural to think thatλ-rings have all the algebraic structures to capture the unstableK-theory algebra of spaces. There is, however, a more precise notion of an unstable algebra, which we now recall.

In their seminal article [4], Boardman et al. gave a precise definition for an unstableE^∗-cohomology algebra, whereE^∗is a generalized cohomology the- ory, such asK-theory, satisfying some reasonable freeness conditions. Given E^∗, denote byE_kthekth space in theΩ-spectrum representingE^∗. Of course, operations onE^k(−)are just the elements ofE^∗E_k. There are functors

U^k(−)=FAlg

E^∗E_k,−

:FAlg→Set (1.1)

from the categoryFAlgof complete Hausdorff filteredE^∗-algebras and con- tinuousE^∗-algebra homomorphisms to sets. Thanks to the extra structures on the spacesE_k, the functorUwhose components areU^kto graded sets be- comes a comonad on the categoryFAlg. Then, these authors define anunstable E-cohomology algebrato be aU-coalgebra. This definition applies toK-theory in particular.

The purpose of this paper is to show that the notions of λ-rings and of unstable algebras forK-theory (almost) coincide. This is perhaps not surprising and is even intuitively obvious, but the author feels that it is still worthwhile to record this result.

We need to modify the above definition of an unstableE-cohomology algebra slightly in the case ofK-theory by taking into account its 2-periodicity. Now, the base point component of the 0th space in theΩ-spectrum representingK- theory is the classifying spaceBUof the infinite unitary group. SinceK⁰(pt)= Z, it makes sense to useZas the ground ring and to consider the functor

U(−)=U⁰(−)=FRingK(BU),−

:FRing →Set (1.2) from the categoryFRingof complete Hausdorff filtered rings and continuous ring homomorphisms. This functor can again be lifted to a comonad on the category of filtered rings. In what follows, anunstableK-cohomology algebra is by definition aU-coalgebra for this comonadU.

We will precisely define what afilteredλ-ring is below (seeDefinition 2.1).

This is basically a filtered ring with aλ-ring structure for which, in the expres- sionλⁱ(r ), both theλ-variable and ther-variable are continuous.

We are now ready to state the main result of this paper.

Theorem1.1. For a commutative filtered ringRthat is complete and Haus- dorff, there is a canonical bijection from the set of unstableK-cohomology al- gebra structures onRto the set of filteredλ-ring structures onR.

One advantage of having a result like this is that, in order to study unstable algebras in the sense of Boardman, Johnson and Wilson, we have to be able to unravel the enormous amount of information encoded in aU-coalgebra and should compare it to more familiar structures whenever possible.Theorem 1.1 does this forK-theory, identifyingU-coalgebras with the well-studiedλ-rings.

1.1. Organization of the paper. The rest of this paper is organized as fol- lows.

InSection 2, we recall the definitions of filtered rings andλ-rings. Then, we describe the filtered ringK(BU). The main aims of the section are to define a filteredλ-ring and to observe that the completion of theK-theory of a space is such.

Section 3begins with recalling the notions of comonads and their coalge- bras which are necessary in order to define unstableK-cohomology algebra.

We define the modified comonadUforK-theory, taking into account its peri- odicity, and define an unstableK-cohomology algebra as a coalgebra over this comonad. Particular attention is paid as to how this comonad arises from the extra structure of the spaceBU. We also observe that the completion of the K-theory of a spaceXis an unstableK-cohomology algebra.

InSection 4, the main result, that is,Theorem 1.1, is proved.

2. Filteredλ-ring. All rings considered in this paper are assumed to be com- mutative, associative, and have a unit.

2.1. Filtered ring. More information about filtered objects can be found in Boardman [3, Sections 3 and 6].

Afiltered ringis a ring R equipped with a directed system of ideals I^aR.

Directedness means that, for every I^aR and I^bR, there exists an I^cR which is contained in the intersection I^aR∩I^bR. The filtration induces a filtration topology onR, which is Hausdorff (resp., complete) if and only if the natural map

c:R →lim←R/I^aR (2.1)

is injective (resp., surjective). IfRis not already complete Hausdorff, we can al- ways complete it by taking the inverse limitR^∧=limR/I^aR, which is complete and Hausdorff with the induced filtrations. The ringR^∧is called the completion ofR.

We mention two elementary but very useful facts about the completion.

(i) The mapc has the following universal property. If f :R→S is a con- tinuous ring homomorphism to a complete Hausdorff filtered ringS, thenf uniquely factors throughcby a continuous ring homomorphismf^∧:R^∧→S.

(ii) The image ofRin its completion is dense.

From now on,all filtered rings will be required to be complete and Hausdorff unless otherwise specified.

A filtered ring homomorphism is a ring homomorphism which is continuous with respect to the filtration topology. The category of complete Hausdorff filtered rings and continuous ring homomorphisms is denoted byFRing.

When we discuss the comonadU, we need to use the not-at-all obvious fact that the completed tensor product is the coproduct in the category of complete Hausdorff filtered rings. More precisely, suppose thatR=(R,{I^aR})andS= (S,{I^bS})are complete Hausdorff filtered rings. Then, theircompleted tensor productis defined as

R⊗S=lim←_a,b

R⊗S ker

R⊗S→ R/I^aR

⊗

S/I^bS. (2.2)

Then,R⊗Sis a complete Hausdorff filtered ring and is the coproduct ofRand Sin the categoryFRing(see [3, Lemma 6.9] for a proof).

2.2.λ-ring. More information aboutλ-rings can be found in [1,2,6]. Consult [5] for a more algebraic-geometric viewpoint to the subject ofλ-rings.

Aλ-ringis a ringRequipped with functions

λⁱ:R →R (i≥0) (2.3)

which satisfy the following conditions. For any integersi,j≥0 and elements randsinR,

(i) λ⁰(r )=1, (ii) λ¹(r )=r,

(iii) λⁱ(1)=0 fori >1, (iv) λⁱ(r+s)=i

k=0λ^k(r )λ^i−k(s),

(v) λⁱ(r s)=Pi(λ¹(r ),...,λⁱ(r );λ¹(s),...,λⁱ(s)), (vi) λⁱ(λ^j(r ))=Pi,j(λ¹(r ),...,λ^ij(r )).

The polynomials P_i and P_i,j are defined as follows. Consider the variables ξ1,...,ξi andη1,...,ηi. Denote bys1,...,si andσ1,...,σi, respectively, the el- ementary symmetric functions of the ξ’s and the η’s. The polynomial Pi is defined by the requirement that the expressionP_i(s1,...,s_i;σ1,...,σ_i)be the coefficient oftⁱin the finite product

i m,n=1

1+ξmηnt

. (2.4)

Similarly, if s1,...,sij are the elementary symmetric functions of ξ1,...,ξij, then the polynomial Pi,j is defined by the requirement that the expression P_i,j(s1,...,s_ij)be the coefficient oftⁱin the finite product

l1<···<lj

1+ξ_l1···ξ_ljt

. (2.5)

Aλ-ring map is a ring map which commutes with theλ-operations.

An example of aλ-ring is the degree-0 partK(X)of theK-theory of a pointed, connected CW spaceX. It is also a filtered ring with filtration ideals

I^a(X)=kerK(X)→KX_a, (2.6)

whereX_a runs through all the finite subcomplexes of X. This is sometimes called theprofinite filtrationonK(X)(cf. [3, Definition 4.9]). Although the fil- tration onXis not unique, the isomorphism type of the resulting filtered ring structure onK(X)is well defined.

The induced map in theK-theory of a map between spaces is, of course, a λ-ring map as well as a filtered ring map.

2.3. The filtered ringK(BU). The purpose of this section is to describe the filtered ringK(BU)in relation to theλ-operations.

Since theK-theory of a pointed, connected CW space is represented by the classifying spaceBUof the infinite unitary group, the Yoneda lemma implies

that theK-theory operations for such spaces can be identified with self-maps of BU, which in turn can be identified with the elements ofK(BU). More precisely, we consistently identify the following:

(i) a self-mapr:BU→BU,

(ii) the corresponding elementr∈K(BU), (iii) the operationr_∗:K(−)→K(−).

It is well known that theK-theory filtered ring ofBUis a power series ring K(BU)=Z

λ1,λ2,...

(2.7) in countably infinitely many variablesλi(i >0), corresponding to the opera- tionsλⁱinK-theory (see, e.g., [8, Theorem 4.15]). The variableλiexactly lies in filtration 2i, and the filtered ring structure onK(BU)is generated this way.

2.4. Filteredλ-ring

Definition2.1. Afilteredλ-ringis a complete Hausdorff filtered ringR= (R,{I^aR})with aλ-ring structure such that the following two conditions hold.

(i) Theλⁱ(i >0)is an equicontinuous family of functions. That is, for every filtration ideal I^aR, there exists anI^bR such that whenever r ∈I^bR, λⁱ(r )∈I^aRfor everyi >0.

(ii) For every elementr ∈Rand every filtration idealI^aR, there exists an integerN >0 (depending onranda) such that, wheneverk

l=1i_le_l≥N, k

l=1λ^il(r )^el∈I^aR.

Thus, a filteredλ-ring is essentially a filtered ring with aλ-ring structure in which the expressionλⁱ(r )is continuous in both theλ- and ther-variables.

A filteredλ-ring map is a continuous ring homomorphism which commutes with theλ-operations.

In order that these algebraic gadgets do model theK-theory of spaces (com- pleted if necessary), we have to show thatK(X)^∧is a filteredλ-ring for any CW spaceX.

Proposition2.2. For any pointed, connected CW spaceX, the completion K(X)^∧has a canonical filteredλ-ring structure for which the completion map c:K(X)→K(X)^∧is aλ-ring map.

Proof. The completionK(X)^∧is clearly a complete Hausdorff filtered ring.

Its universal property implies that the composite map

K(X) ^λi→K(X) ^c→K(X)^∧ (2.8)

uniquely factors through cvia a map that we call λⁱ. Once it is shown that theseλⁱmakeK(X)^∧a filteredλ-ring, it is automatically true thatcis a map ofλ-rings.

Now, since the image of K(X)in its completion is dense and sinceK(X) with itsλ-operations is aλ-ring, it follows immediately thatK(X)^∧with itsλⁱ is also aλ-ring. Still, we must prove that theseλⁱhave the required continuity properties inDefinition 2.1to makeK(X)^∧a filteredλ-ring. Again, sinceK(X) is dense inK(X)^∧, it suffices to show that theλ-operations onK(X)have these continuity properties.

To see that theλⁱonK(X)are equicontinuous, pick any filtration idealI^aX= ker(K(X)→K(Xa)) corresponding to a finite subcomplexXa and pick any elementα∈I^aX. Then,αis represented by a mapα:X→BUwhose restriction toXais nullhomotopic. Ifr∈K(BU)is any operation at all, represented as a mapr:BU→BU, then the composite r◦α:X→BU is still nullhomotopic when restricted toX_a. That is, the filtration idealI^aXis, in fact, closed under anyK-theory operations, including theλ-operations. This proves that{λⁱ}i>0

is an equicontinuous family of functions onK(X).

To demonstrate the other continuity property, letαbe an element ofK(X) and letI^aXbe a filtration ideal. The elementαis represented by a mapα:X→ BU, which induces a continuous ring homomorphism

α^∗:K(BU) →K(X), (2.9)

sending an elementr ∈K(BU)to the elementr (α)inK(X)represented by the composite

X ^α→BU ^r→BU. (2.10)

That the required continuity property holds follows now from the continuity ofα^∗and the filtered ring structure onK(BU)(seeSection 2.3).

Corollary2.3. Iff:X→Yis a map of pointed, connected CW spaces, then the induced mapf^∗∧:K(Y )^∧→K(X)^∧is a filteredλ-ring map.

3. UnstableK-cohomology algebra. The main purpose of this section is to define the comonadU(see (3.3)). Its coalgebras are, by definition, the unstable K-cohomology algebras. We also see that theK-theory of a space (completed if necessary) is aU-coalgebra.

We begin by recalling the concepts of comonads and their coalgebras.

3.1. Comonad and coalgebra. The reader can consult MacLane’s book [7]

for more information on this topic.

Acomonad on a categoryCis a functorF :C→Cequipped with natural transformationsη:F→Id and∆:F→F², called counit and comultiplication, satisfying the counital and coassociativity conditions

Fη◦∆=IdR=ηF◦∆, F∆◦∆=∆F◦∆. (3.1)

The natural transformations∆andηare often omitted from the notation, and we speak ofF as a comonad.

IfFis a comonad on a categoryC, then anF-coalgebra structureon an object XofCis a morphismξ:X→FXinC, called the structure map, satisfying the counital and coassociativity conditions

Fξ◦ξ=∆◦ξ, η◦ξ=IdX. (3.2) We sometimes abuse notation and say thatX is anF-coalgebra, leaving the structure mapξimplicit.

A map ofF-coalgebrasg:(X,ξX)→(Y ,ξY)consists of a morphismg:X→Y inCsuch thatξ_Y◦g=Fg◦ξ_X.

3.2. The comonadU forK-theory. The discussion in this section closely follows Boardman et al. [4, Section 8], except that we take into account the periodicity ofK-theory and consider only the degree-0 part. We first define the comonadUand then discuss its ring structure (when applied to a filtered ring), filtration, comultiplication, and counit. After that, we define unstableK- cohomology algebra and observe that the argument in Boardman, Johnson and Wilson shows that theK-theory of any space (completed if necessary) is such.

We will use the Yoneda lemma many times without explicitly mentioning it.

3.2.1. Definition ofU. Let,R=(R,{I^aR})be an arbitrary complete Haus- dorff filtered ring. Define the functorU(toSetonly at the moment) to be

UR=FRingK(BU),R, (3.3)

the set of continuous ring homomorphisms fromK(BU)toR.

3.2.2. Ring structure onUR. LetXbe an arbitrary pointed, connected CW space.

There are maps

µ,φ:BU×BU BU (3.4)

which induce the natural addition (byµ) and multiplication (byφ) structure on theK-theoryK(X)of a spaceX. These maps satisfy certain associativity, commutativity, and so on, conditions which makeK(X)a commutative ring with unit. Thanks to the Künneth isomorphismK(BU×BU)K(BU)⊗K(BU) [3, Theorem 4.19], they induce inK-theory the following maps:

µ^∗,φ^∗:Z

λ1,λ2,...

Z

λ1,λ2,...

⊗Z

λ1,λ2,...

. (3.5) Therefore, for each integerk≥1, we can write

µ^∗ λk

=

α

r_α⊗r_α (3.6)

for some elementsr_α andr_αinK(BU). By precomposition, this becomes the Cartan formula for a sum

λ^k(x+y)=

α

r_α(x)r_α(y)

x,y∈K(X)

=

i+j=k

λⁱ(x)λ^j(y), (3.7)

where we have used the usual conventionλ⁰(x)=1. Since (3.7) holds for an arbitrary spaceXand any elementsxandyinK(X), we must have

µ^∗ λk

=

i+j=k

λi⊗λj. (3.8)

A similar reasoning, using the property

λ^k(xy)=P_kλ¹x,...,λ^kx;λ¹y,...,λ^ky, (3.9) leads to the formula

φ^∗λ_k

=P_kλ1⊗1,...,λ_k⊗1; 1⊗λ1,...,1⊗λ_k. (3.10) Now, suppose thatf andg are elements ofUR=FRing(K(BU),R). Their sum and productf+gandf gboth have the form

K(BU)→K(BU)⊗K(BU) ^f⊗g→R⊗R multiplication

→R, (3.11)

where forf+g(resp.,f g), the left-hand map isµ^∗(resp.,φ^∗). Then, (3.8) and (3.10) imply that, on the elementλ_k∈K(BU), these maps can be expressed as the Cartan formulas

(f+g)λ_k

=

i+j=k

fλ_igλ_j, (f g)λ_k

=P_kfλ1,...,fλ_k

;gλ1,...,gλ_k.

(3.12)

It follows that the additive and multiplicative identities 0URand 1URofURare given by the maps

0UR:λ_k→0 (k >0), 1UR:λ_k→





1, ifk=1, 0, ifk >1.

(3.13)

The second equality follows from the fact thatP_k(x1,...,x_k; 1,0,...,0)=x_k. 3.2.3. Filtration onUR. The ringUR=FRing(K(BU),R)is filtered by the ideals

I^aUR=ker

UR →U

R/I^aR

=

f∈UR:f λk

∈I^aR∀k >0

. (3.14)

It is easy to see that the surjective mapUR→U(R/I^aR)has kernelI^aUR. Thus, sinceRis complete Hausdorff, it follows that

UR=FRingK(BU),R

=lim

a FRing

K(BU),R/I^aR

=lim

a (UR)/I^aUR.

(3.15)

That is,URis also a complete Hausdorff filtered ring. Note that the indexing set for the filtration ofURis the same as that forR.

So far we have seen thatUis a functor on the categoryFRingof complete Hausdorff filtered rings.

3.2.4. Comultiplication and counit. In order to make U a comonad on FRing, we still need the natural transformations ∆:U→U² andη:U→Id.

We begin with the former.

There is a filtered ring map ρ:K(BU) →U

K(BU)

, g→(f→f◦g). (3.16) Using the formulaλⁱλ^j(x)=P_i,j(λ¹x,...,λ^ijx), we can express the mapρin terms of the elementsλi∈K(BU)as follows:

ρλ_jλ_i

=λⁱ◦λ^j=P_i,jλ1,...,λ_ij. (3.17) Here,λⁱ◦λ^jis the element inK(BU)represented by the composition of the K-theory operationsλⁱandλ^j. Now, iff is an element inUR, then its image under the comultiplication map∆R:UR→U²Ris the composite map

∆Rf=(Uf )◦ρ:K(BU) →U K(BU)

→UR. (3.18)

As for the counitη:U→Id, it is defined by η_Rf=fλ1

; (3.19)

that is,η_Ris simply the evaluation map atλ1.

3.2.5. Unstable K-cohomology algebra. The proof in Boardman et al. [4, Theorem 8.8(a)] that theirUis a comonad on the category of complete Haus- dorff filteredE^∗-algebras carries over almost without change to show the fol- lowing.

Proposition 3.1. The functorU defined in (3.3), together with ∆ andη above, is a comonad on the categoryFRingof complete Hausdorff filtered rings.

Following Boardman et al. [4, Definition 8.9], we now give the following def- inition.

Definition3.2. AnunstableK-cohomology algebra is aU-coalgebra for the comonadUinProposition 3.1. A map of unstableK-cohomology algebras is a map ofU-coalgebras.

Now, given any pointed, connected CW spaceX, composition of maps yields a continuous map

◦:K(BU)×K(X) →K(X) (3.20) which gives, after completion, the map

K(BU)×K(X)^∧ →K(X)^∧. (3.21) Taking its adjoint, we obtain a map

ρX:K(X)^∧ →FRing

K(BU),K(X)^∧

=U K(X)^∧

. (3.22)

The argument of [4, Theorem 8.11(a)] in Boardman et al., which shows that their analogous mapρX:E^∗(X)^∧→U(E^∗(X)^∧)makesE^∗(X)^∧aU-coalgebra, now gives the following proposition.

Proposition3.3. The mapρ_X in (3.22) makesK(X)^∧an unstableK-coho- mology algebra.

This proposition is also a consequence ofProposition 2.2andTheorem 1.1.

Corollary3.4. Iff:X→Yis a map of pointed, connected CW spaces, then the induced map f^∗∧:K(Y )^∧ →K(X)^∧ is a map of unstable K-cohomology algebras.

4. Identifying unstableK-cohomology algebra with filteredλ-ring. In this final section, we proveTheorem 1.1.

So letR=(R,{I^aR})be an arbitrary complete Hausdorff filtered ring. Sup- pose thatRhas the structure of an unstableK-cohomology algebraξ:R→UR.

We must show that this gives a filteredλ-ring structure onR. Recall that UR=FRing

K(BU),R

=FRing Z

λ1,λ2,...

,R

. (4.1)

We define the operations

λⁱ:R →R (i≥0) (4.2)

by settingλ⁰≡1 and, fori >0,

λⁱ(r )^def=(ξr ) λi

(r∈R). (4.3)

We claim that these operations makeRinto a filteredλ-ring; that is, aλ-ring structure onR, together with the two continuity properties inDefinition 2.1.

The argument is divided into six steps, the first one for the continuity proper- ties and the rest for theλ-ring structure.

Step1. We first check the continuity properties inDefinition 2.1. To see that the family{λⁱ}i>0is equicontinuous, letI^aRbe a filtration ideal. We must show that there exists anI^bRsuch thatλⁱ(I^bR)⊂I^aRfor everyi >0. Sinceξ is continuous, givenI^aUR, there existsI^bRsuch thatξ(I^bR)⊂I^aUR. That is, ifr∈I^bR, thenλⁱ(r )=(ξr )(λ_i)∈I^aRfor everyi >0. This shows that{λⁱ}i>0

is an equicontinuous family of functions onR.

To check the second continuity property inDefinition 2.1, letrbe an element inRand letI^aRbe a filtration ideal. The elementξr∈URis a continuous ring homomorphism from K(BU) to R. Thus, given I^aR, there exists an integer N >0 such that wheneverα∈K(BU)has a filtration strictly greater thanN, then(ξr )(α)∈I^aR. Now, ifk

l=1i_le_l≥N, then the filtration of the element k

l=1λ^e_il^l∈K(BU)isk

l=12i_le_l> N, and so we have k

l=1

λ^il(r )^el=(ξr )



^k

l=1

λ^e_il^l



∈I^aR. (4.4)

This proves the desired continuity property.

Step2. We check thatλ¹is the identity map onR. Recall that the counit η_R:UR→Ris the evaluation map atλ1. Since there is an equality Id_R=η_Rξ, it follows that, for any elementrinR, we have

λ¹(r )=(ξr )λ1

=η_Rξ(r )=r . (4.5) Soλ¹is the identity map onR.

Step3. We check thatλⁱ(1)=0 for anyi >1. Denoting the multiplicative identity ofURby 1UR(see (3.13)), we have, fori >1,

λⁱ(1)=(ξ1) λi

=1UR λi

=0 (4.6)

as desired.

Step4. Now, we show the Cartan formula for a sum of any two elementsx andyinR. Using the additivity ofξand (3.12), we calculate that

λ^k(x+y)=ξ(x+y) λk

=(ξx+ξy) λk

=

i+j=k

(ξx)λ_i(ξy)λ_j

=

i+j=k

λⁱ(x)λ^j(y).

(4.7)

This proves the Cartan formula for a sum.

Step5. The Cartan formula forλ^k(xy)is similarly proved using the mul- tiplicativity ofξand (3.12).

Step6. Finally, we show thatλⁱλ^j(x)=Pi,j(λ¹x,...,λ^ijx). Letxbe an ele- ment inR. Then, using the coassociativity ofξ(see (3.2), (3.17), and (3.18)), we calculate that

λⁱλ^j(x)=λⁱ ξx

λj

= ξ

ξx λj

λi

=

∆R(ξx)λ_jλ_i

=U(ξx)◦ρλ_jλ_i

=(ξx)Pi,j

λ1,...,λij

=Pi,j

λ¹x,...,λ^ijx (4.8) as desired.

We have shown that the operationsλⁱ in (4.3) makeR into aλ-ring which also satisfies the two continuity properties in Definition 2.1. Therefore, the unstableK-cohomology algebra structure onRgives a filteredλ-ring structure onR. This proves half ofTheorem 1.1.

The above argument can easily be reversed to show that a filteredλ-ring structure onRyields, via (4.3), an unstableK-cohomology algebra structure onR. Indeed, the two continuity properties in the definition of a filteredλ- ring make sure that both the proposed structure mapξ:R→URandξ(r ): K(BU)→R for anyr ∈R are continuous. The Cartan formulas forλⁿ(x+ y)andλⁿ(xy)imply the additivity and multiplicativity, respectively, of the structure mapξ, and the property aboutλⁱλ^j(x)leads to the coassociativity ofξ. Thatξis counital follows from the condition thatλ¹is the identity onR.

The proof ofTheorem 1.1is complete.

Acknowledgments. The author would like to thank the anonymous ref- erees for their helpful suggestions. The author would also like to thank Haynes Miller for several stimulating discussions on this subject.

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[8] H. Toda,A survey of homotopy theory, Advances in Math.10(1973), 417–455.

Donald Yau: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801, USA

E-mail address:[email protected]