· 216 kiyoshi igusa proof. it su ces to take n=0sincez[0] maps to z[n] for all n. if p is empty...

TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 352, Number 1, Pages 209–242S 0002-9947(99)02447-2Article electronically published on September 8, 1999

A MULTIPLICATION IN CYCLIC HOMOLOGY

KIYOSHI IGUSA

Abstract. We define a multiplication on the cyclic homology of a commuta-tive, cocommutative bialgebra H with “superproduct.” In the case when H isa field of characteristic zero the cyclic homology becomes a polynomial algebrain one generator. (The Loday-Quillen multiplication is trivial in that case.)

In this paper we define an associative multiplication on the cyclic homology ofcertain commutative algebras H . The definition is based on the Cartesian productof sets instead of the shuffle product as in the case of [LQ] and [Ga]. When Hcontains Q our product is compatible with the Hodge decomposition of HC∗(H) soit has the form:

HC(i)n (H)⊗HC(j)

m (H) → HC(i+j)n+m (H).

The existence of this multiplication in the case H = K is classical since it is justgiven by the tensor product of complex line bundle. However its significance cameto our attention in joint work with John Klein as a consequence of our study offiberwise Morse theory on circle bundles [IK] and in other joint works with R. Pennerand R. Hain. By doing Morse theory on circles John Klein and I realized that therewas a rationally nontrivial associative multiplication on the cyclic homology of theintegral group ring on any cyclic group and that it had important consequences forpseudoisotopy theory. This will be explained in a later paper.

In this paper we show that this multiplication in cyclic homotopy is defined firstin the case when the algebra is H = KM = K[M ] where M is the underlying addi-tive group of a ring, then in the case when H is a Hopf algebra with “superproduct”in the sense of Gaucher [Ga] and finally in the more general case of any “µ-algebra”(Example 12.4). In fact we do not need additive inverses in M nor do we need anantipode for H so we can take M to be a monoid and H to be a bialgebra. WhenH = K, i.e. M = 1, the cyclic homology becomes a divided power algebra whichis of course isomorphic to a polynomial algebra in one generator if K is a field ofcharacteristic zero.

We assume throughout the paper that K is a commutative ring with 1 and thatall tensor products are taken over K. This includes those tensor products whichappear in the definition of cyclic homology. We also use Connes’ original definitionthat a cyclic object in a category C is a covariant functor Λ → C where Λ is Connes’cyclic category. Since Λ ≈ Λop this is the same as a contravariant functor Λ → C.

An outline of the paper is as follows. In the first section we give a topological def-inition of the multiplication in HC∗(KM). The construction is so elementary that

Received by the editors April 14, 1994.1991 Mathematics Subject Classification. Primary 18G60; Secondary 16W30.This research was supported by NSF grant no. DMS 90 02512.

c©1999 American Mathematical Society

209

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210 KIYOSHI IGUSA

it must have been noticed before. For example when M = {1} the multiplication isinduced by the map BU(1)×BU(1) → BU(1) given by tensor product of universalline bundles. Coupled with the well-known isomorphism HC∗(K) ≈ H∗(BU(1);K)we get a multiplication on HC∗(K) with all the properties that we have claimed.

In §§2–6 we give an algebraic definition of this multiplication. The basic ideais as follows. Given generators (x0, . . . , xn) and (y0, . . . , ym) we get a “cylindrical”array of (n+1)(m+1) elements xi ∗ yi (given a pairing ∗ on M). We take all mapsφ of this cylinder into a circular array of n+m+1 points and multiply the elementsxi∗yj which go to the same point. The sum of the resulting elements (z0, . . . , zn+m)with appropriate coefficients is the product of (x0, . . . , xn) and (y0, . . . , ym). Thealgebraic description has the advantage that it generalizes to bialgebras.

In §§7–9 we verify some of the basic properties of the multiplication µ. In §§10–13 we show that µ is compatible with the power operations of [L] and [FT]. Finally,in the last five sections (14–18) we show that the algebraic definitions of µ and thepower operations agree with the topological definitions given in section 1 in thecase H = KM when both are defined.

I would like to thank Tom Goodwillie for some very incisive comments on thisresearch and for explaining to us his version of the power operations in cyclichomology. I would also like to thank J. Klein, R. Penner and R. Hain.

1. A topological description of the multiplication

If M is a multiplicative monoid then let BM be the space of all configurations inS1 consisting of a finite subset Y of S1 = U(1) together with a “labelling” functionλ : Y →M \ 1. We topologize BM so that it is a CW complex with one n-cell foreach (n + 1)-tuple x = (x0, x1, . . . , xn) in M where x1, . . . , xn are not equal to 1.The open cell ex corresponding to x is the set of all configurations (Y, λ) so that:

(a) Y \ 1 has exactly n elements.(b) λ(yi) = xi for 1 ≤ i ≤ n where y1, . . . , yn are the elements of Y \ 1 in

counterclockwise order.(c) 1 ∈ Y if and only if x0 6= 1.(d) λ(1) = x0 if x0 6= 1.

As we go to the boundary of ex the elements of Y either come together or go to 1 ∈S1. In the limiting configuration we should multiply the labels in counterclockwiseorder and delete points with resulting label equal to 1.

The spaceBM together with its cell decomposition is the geometric realization ofa simplicial set F (M) given by Fn(M) = Mn+1 with face and degeneracy operationsgiven by

∂i(x0, . . . , xn) = (x0, . . . , xixi+1, . . . , xn) for 0 ≤ i ≤ n− 1,

∂n(x0, . . . , xn) = (xnx0, x1, . . . , xn−1),

si(x0, . . . , xn) = (x0, . . . , xi, 1, xi+1, . . . , xn) for 0 ≤ i ≤ n.

The homeomorphism |F (M)| ∼= BM is given by sending (x, t) ∈Mn+1 ×∆n tothe configuration of points Y = (y0, . . . , yn) given by

yi = exp(2π√−1(t0 + · · ·+ ti−1)) ∈ S1

with labels λ(yi) = xi.Let GM be the space of all pairs (C,X) where C ∈ BU(1) and X is a finite

configuration in C with labels in M \ 1. Since C is a coset of U(1) = S1 in the


A MULTIPLICATION IN CYCLIC HOMOLOGY 211

contractible topological abelian group EU(1) it is an oriented circle. Then GM =BM ×U(1) EU(1). Therefore for any commutative ring K we have: Hn(GM ;K) ∼=HCn(KM). (See [J], Theorem 3.1.)

Now suppose that the multiplicative monoid M is commutative and has a bi-multiplicative pairing ∗ : M ×M → M so that 1 ∗ x = x ∗ 1 = 1 for all x ∈ M .Then we obtain a continuous map µ : GM ×GM → GM as follows.

µ((C,X), (D,Y )) = (CD,X ∗ Y )

where CD = {cd ∈ EU(1)|c ∈ C, d ∈ D} ∈ BU(1) and X ∗ Y is the configuration{xy ∈ CD|x ∈ X, y ∈ Y } together with the corresponding labels x∗y. We multiplythe labels when points coincide.

If ∗ is associative or commutative then so is µ. Also if ∗ has a two-sided neutralelement e then (S1, E) is a two-sided identity for µ where E is the configuration inS1 consisting of the one point 1 ∈ S1 with label e.

Theorem 1.1. If ∗ is associative with e then HC∗(KM) becomes a graded ring.Furthermore

(a) This multiplication is natural with respect to maps M → M ′ preservingmultiplication, ∗, 1 and e and with respect to change of rings K → K ′.

(b) In the case M = {1} = {e}, HC∗(K) is a divided power algebra, i.e. theproduct of the generators in degrees a, b is given by uaub =

(a+ba

)ua+b.

Proof. The homology ring H∗(BU(1)) is easy to compute and the answer is wellknown to be a divided power algebra.

Corollary 1.2. Suppose that M is a finite group and K is a field of characteristic0. Then the multiplication in HC∗(KM) is given by [x]i[y]i =

(i+ji

)[x ∗ y]i+j where

[x]i is the generator of HC2i(KM) corresponding to x ∈M .

Remark. If (M, ·) is a group then 〈M, ·, ∗〉 is a ring and KM is a ring object in thesense of Husemuller [H] or equivalently a Hopf algebra with superproduct in thesense of Gaucher [Ga].

Theorem 1.3. If K contains Q, then multiplication in HC∗(KM) is compatiblewith the Hodge decomposition of HC∗(KM). Thus we get

HC(i)n (KM)⊗HC(j)

m (KM) → HC(i+j)n+m (KM).

Proof. The k-th power operation 1/kΨk = (−1)k−1λk is given by the map Lk :GM → GM given by Lk(C,X) = (Ck, Lk(X)) where Ck ∈ BU(1) is the k-foldproduct Ck = CC · · ·C using the abelian group structure on BU(1) and Lk(X) isthe configuration of points (xi)k ∈ Ck given by the k-fold product in the abeliangroup EU(1). The label on the point z ∈ Lk(X) is the product of the labels ofthose points xi ∈ C for which z = (xi)k.

Since CkDk = (CD)k and Lk(X) ∗ Lk(Y ) = Lk(X ∗ Y ) we see that the productoperation ∗ strictly commutes with Lk in the sense that

Lk(C,X) ∗ Lk(D,Y ) = Lk(CD,X ∗ Y ) = Lk((C,X) ∗ (D,Y )).

Consequently if a ∈ HC(i)n (KM) and b ∈ HC(j)

m (KM) then 1/kΨk(a)∗1/kΨk(b) =kia ∗ kjb = ki+ja ∗ b = 1/kΨk(a ∗ b) so a ∗ b ∈ HC(i+j)

n+m (KM).


212 KIYOSHI IGUSA

2. Acyclic carriers

Let (C∗(KM), b) be the standard Hochschild complex given by Cn(KM) =KM⊗(n+1) ≈ KMn+1 and b(x0, . . . , xn) = (x0x1, x2, . . . , xn)− (x0, x1x2, . . . , xn)+· · ·+ (−1)n(xnx0, x1, . . . , xn−1) and let T (KM) be the total space of the followingstandard bicomplex whose homology is HC∗(KM).

b��

b��

C1(KM)

b

��

C1(KM)B

ggOOOOOOOOOOOO

b

��

· · ·B

eeKKKKKKKKKKK

C0(KM) C0(KM)B

ffMMMMMMMMMM

· · ·B

ccHHHHHHHHHH

Here B is given in a standard way by

B(x0, . . . , xn) =n∑i=0

[(−1)ni(1, xi, . . . , xn, x0, . . . , xi−1)

+ (−1)n(i+1)(xi−1, 1, xi, . . . , xn, x0, . . . , xi−2)].

We will construct an acyclic carrier for a map µ : T (KM) ⊗ T (KM) → T (KM)which induces the multiplication on HC∗(KM) defined in the last section. Thepurpose of this is to obtain (the existence of) an algebraic formula for this mul-tiplication and to use this formula (without knowing what it is) to generalize themultiplication to the case of HC∗(H) where H is a bialgebra over K with “super-product.” The purpose of this section is to describe the nature of the acyclic carrierthat we will be using to specify both µ and the power operations Ψk.

If C is a free K-complex then a basis for C forms the set of objects of a categorywhich we call a “basis category” for C.

Definition 2.1. A basis category (with coefficients in K) is defined to be a smallcategory B together with a nonnegative grading on the set of objects deg : Ob(B) →N and an incidence coefficient rα ∈ K for each morphism α : x → y of degree−1(deg(α) = deg(y)− deg(x)) so that the following two conditions hold.

(1) For each object x in B there are only finitely many morphisms x → y ofdegree −1 with nonzero incidence coefficient.

(2) For any morphism γ in B of degree −2 we have∑rβrα = 0 where the sum

is taken over all factorizations γ = βα of γ into a composition of two morphisms ofdegree −1.

We say that B is a basis category for a free K-complex C if Bn (the set of objectsof B of degree n) is a free basis for Cn and if the coefficient of y ∈ Bn−1 in ∂x forx ∈ Bn is equal to the sum

∑rα of all incidence coefficients of all morphisms

α : x→ y. Since C is uniquely determined by B we sometimes write C = KB.An augmentation ε : C0 → K for a free complex C corresponds to an augmenta-

tion for the basis category B. This is defined to be a system of coefficients rb ∈ Kfor each b ∈ B0 satisfying the obvious conditions similar to (2) above.

If B is an augmented basis category over K and D is an augmented K-complexthen an acyclic carrier fromKB toD is defined to be a contravariant functor C fromB to the category of acyclic augmented complexes over D. An augmented chain



map f : KB → D is said to be carried by the carrier C if for each b ∈ Bn, f(b) ∈ Dn

lifts to some f(b) ∈ Cn(b) so that, if n 6= 0, ∂f(b) =∑rαα ∗ f(tα) where the sum

is taken over all morphisms α : b→ tα of degree −1 in B and εf(b) = rb if n = 0.

Theorem 2.2. Any acyclic carrier from KB to D carries at least one chain mapKB → D and any two such maps are chain homotopic.

One standard example of a basis category is the opposite simp(X)op of thecategory of simplices of a simplicial set X . (See [W], p. 337.) The following relatedexample comes from any cyclic set.

Let X be a cyclic set, i.e. a covariant functor Λ → Sets where Λ is Connes’ cycliccategory. Then we define the basic category B(X) in such a way that KB(X) isthe total complex of the bicomplex B∗(KX) as defined in [Go], II.2 where KX isthe cyclic free K-module generated by X .

Definition 2.3. For any cyclic set X the basis category B(X) is given as follows.(1) The objects of B(X) are all symbols Si(n, x) where i ≥ 0 and x ∈ Xn.(2) deg(Si(n, x)) = n+ 2i.(3) A morphism Si(n, x) → Sj(m, y) is a morphism α : [n] → [m] in Λ so that

y = α∗x.(4) The morphism ∂i : Sj(n, x) → Sj(n− 1, ∂ix) has incidence coefficient (−1)i.(5) stk : Si(n, x) → Si−1(n + 1, stkx) has incidence coefficient (−1)nk. Here

s = tsn so that s(x0, . . . , xn) = (1, x0, . . . , xn).(6) tstk : Si(n, x) → Si−1(n+ 1, tstkx) has incidence coefficient (−1)n(k+1).(7) All other morphisms have coefficient 0.(8) The augmentation of each element of the form S0(0, x) is 1.

As an example we take the cyclic set MΛ given by (MΛ)n = Mn+1 with theusual action of the cyclic operators: α#(x0, . . . , xn) = (y0, . . . , ym) where each yjis the product of xi for all i ∈ α−1(j). Then KB(MΛ) = T (KM).

A morphism of cyclic sets X → Y induces a morphism of basis categoriesB(X) → B(Y ) is defined below and thus a chain map KB(X) → KB(Y ).

Definition 2.4. A morphism of basis categories is a degree preserving functorf : A → B satisfying the following.

(a) For each object X in A and each morphism β : F (X) → Y in B there is aunique object Z in A and a unique morphism α : X → Z in A so that f(Z) = Yand f(α) = β.

(b) rα = ff(α) for all morphisms α in A.(c) rX = rf(X) for all objects X in A0.

It is easy to see that any morphism f : A → B induces an augmentation pre-serving chain map f∗ : KA→ KB.

3. Acyclic complex given by a cyclically ordered set

Tony Elmendorf and perhaps others have been observed that Connes’ category Λcan be expressed in terms of cyclically ordered sets. In this section we construct anacyclic augmented complex E(P ) associated to any cyclically ordered set P . Thesewill be used to construct acyclic carriers.

Definition 3.1. A cyclic partial ordering on a set P is defined to be a partialordering on the set P = P × Z so that:


214 KIYOSHI IGUSA

(a) (x, n) < (x, n+ 1) for all (x, n) ∈ P .(b) σ(x, n) = (x, n+ 1) gives an automorphism of the poset P .(c) For all x, y ∈ P there is an N ∈ Z so that (x,−N) < (y, 0) < (x,N).

We will sometimes use the notation σn(x) = (x, n). To avoid confusion we shouldpoint out that the term “cyclic poset” already has a (different) meaning, namely itis a cyclic object in the category of posets.

Some important examples of cyclically ordered sets are as follows.

Definition 3.2. (a) Z[n] = {0, 1, 2, . . . , n} with the cyclic total ordering 0 < 1 <· · · < n < σ(0).

(b) λkZ[n] = {0, 1, 2, . . . , n} with the cyclic partial ordering 0 < 1 < · · · < n <σk(0).

(c) Z[0, 1) is the subset [0, 1) of R with the cyclic total ordering (x, n) < (y,m)if x+ n < y +m.

Definition 3.3. A morphism of cyclically ordered sets φ : P → Q is defined to bean equivalence class of maps φ : P → Q which are order preserving (i.e. φ(a) ≤ φ(b)whenever a < b) and which commutes with σ. Two such maps are equivalent ifthey differ by a power of σ. (φ ∼ φ ◦ σk) For any morphism φ : P → Q there is aninduced set map φ : P → Q given by φ(x, n) = (φ(x),m).

Given two cyclically ordered set P,Q the cartesian product P ×Q has a productcyclic ordering given by (x, y, n) ≤ (p, q,m) if (x, i) ≤ (p, j) in P and (y, n − i) ≤(q,m − j) in Q for some i, j. Given morphisms φ : S → P, ψ : T → Q there is aninduced morphism φ×ψ : S×T → P ×Q given by φ× ψ(s, t, n) = (x, y, a+ b+n)if φ(s, 0) = (x, a), ψ(t, 0) = (y, b). Although P × Q is not the categorical productof P and Q (e.g. there is no natural morphism P ×Q→ P ) it does define a factorC × C → C where C is the category of cyclically ordered sets.

A morphism α : Z[n] → Z[m] can be represented by a sequence of wordsw0, w1, . . . , wm in the letters 0, 1, . . . , n so that the word w0w1 · · ·wm is equal tothe word (i)(i + 1) · · · (n)(0) · · · (i − 1) for some i. The correspondence is given byletting wj be the product of the elements of α−1(j) in the correct order. It shouldbe well known that the objects Z[n] together with the morphisms between themgive Connes’ category Λ.

To obtain the correct names for cyclic operators (morphisms induced by covariantfunctors on Λ) we make the following notation.

Definition 3.4. In the above notation the morphisms ∂i, si, s, t are given by:(a) ∂i = (0, 1, . . . , i(i+ 1), . . . , n) : Z[n] → Z[n− 1] if 0 ≤ i ≤ n− 1,(b) ∂n = (n0, 1, . . . , n− 1) : Z[n] → Z[n− 1],(c) si = (0, 1, . . . , i,∅, i+1, . . . , n) : Z[n] → Z[n+1] where ∅ is the empty word,(d) t = (n, 0, 1, . . . , n− 1) : Z[n] → Z[n],(e) s = tsn = (∅, 0, 1, . . . , n) : Z[n] → Z[n+ 1].

For any cyclically ordered set P let X(P ) denote the cyclic set given by Xn(P ) =Hom(P,Z[n]). If we represent and element φ of Hom(P,Z[n]) in the format



〈a0, . . . , an〉 where ai = φ−1(i) then the cyclic operators act on X(P ) in the ex-pected way by

∂i〈a0, . . . , an〉 = 〈a0, . . . , aj ∪ ai+1, . . . , an〉 for 0 ≤ i ≤ n− 1,

∂n〈a0, . . . , an〉 = 〈σ−1an, a0, . . . , an−1〉,si〈a0, . . . , an〉 = 〈a0, . . . , ai,∅, ai+1, . . . , an〉 for 0 ≤ i ≤ n,

s〈a0, . . . , an〉 = 〈∅, a0, . . . , an〉,t〈a0, . . . , an〉 = 〈σ−1an ∪ a0, . . . , an−1〉.

More generally, for any morphism ψ : Z[n] → Z[m] we have

ψ∗φ = 〈φ−1ψ−1(0), . . . , φ−1ψ−1(m)〉.We define E(P ) to be the augmented complex E(P ) = KBX(P ). It is clear from

the definitions that X,E are contravariant functors from the category of cyclicallyordered sets to the category of cyclic sets or augmented K-complexes respectively.Since the augmented complex E(P ) plays a crucial role in the rest of the paper wewill describe it explicitly.

Proposition 3.5. (1) The objects of the basis category BX(P ) are symbols of theform sa(n, φ) where φ : P → Z[n] is a morphism and a, n ≥ 0.

(2) A morphism α : Sa(n, φ) → Sb(m,ψ) is given by α : Z[n] → Z[m] so thatψ = α ◦ φ.

(3) In E(P ) we have

dSa(n, φ) =n∑i=0

[(−1)iSa(n, ∂iφ)

+(−1)niSa−1(n+ 1, stiφ) + (−1)ni+nSa−1(n+ 1, tstiφ)]

(i.e. d = b+B) where the undefined terms S−1(∗, ∗) are taken to be zero.

Remark. We will sometimes abbreviate this expression for dSa(n, φ) as∑α∈W (a,n)

sgn(α)S∗(∗, αφ)

whereW (a, n) is the set of all operations ∂i, sti, tsti appearing in the above sum.

The rest of this section is devoted to the proof of the following theorem.

Theorem 3.6. E(P ) is acyclic for any nonempty finite cyclically ordered set P .

Proof. From the topological viewpoint this is more or less obvious.Let M(P ) be the space of all maps f : P → R so that(1) f ◦ σ(x) = f(x) + 1 for all x ∈ P ,(2) f(x) ≤ f(y) if x < y in P .

Then M(P ) is convex and thus contractible. (By Lemma 3.7 below M(P ) isnonempty.) This means that the space of all morphisms P → Z[0, 1) is homotopyequivalent to M(P )/Z ' S1 = U(1) and so (M(P )/Z)×U(1)EU(1) is contractible.But H∗E(P ) is the homology of this space with coefficients in K so E(P ) is acyclic.An algebraic version of this proof is given below.

Lemma 3.7. If P is a finite cyclically ordered set then there is at least one mor-phism P → Z[n] for each n.


216 KIYOSHI IGUSA

Proof. It suffices to take n = 0 since Z[0] maps to Z[n] for all n.If P is empty or contains only one element then there is a unique morphism

P → Z[0] so suppose that P has at least two elements and the lemma hold forS = P \ x where x ∈ P is fixed. Let f : S → Z be a poset map representinga morphism S → Z[0]. Then we can extend f to a poset map g : P → Z byletting g(x, n) be the smallest value of f(y,m) as (y,m) ranges over all elements ofS greater than (x, n).

The algebraic proof of Theorem 3.6 starts with a special case.

Lemma 3.8. E(Z[m]) is acyclic for all m ≥ 0.

Proof. Since E(Z[m]) is a free K-complex with integer coefficients it is chain con-tractible in general if and only if it is contractible in the special case when K = Z.This in turn is equivalent to the condition that E(Z[m])⊗F is acyclic for any fieldF . Thus it suffices to prove the lemma in the case when K is a field.

The proof will be by induction on m using the cyclic homology of monomialrelation algebras over a field K. (See [IZ].) Suppose first that m = 0. ThenE(Z[0]) is isomorphic to the weight x component of the total complex T (K[x])of the polynomial algebra K[x]. The lemma in this case follows from an easycalculation starting with the observation that the only nondegenerate terms inC∗(K[x]) of weight x are x ∈ C0(K[x]) and 1⊗ x ∈ C1(K[x]).

Now suppose that m ≥ 1 and that the lemma holds for m− 1. Let Mm be thefree associative monoid generated by the symbols x0, x1, . . . , xm, i.e. Mm is the setof all words in these symbols. If we define an equivalence relation on Mm by sayingthat a, b ∈ Mm are cyclically equivalent when the word b is a cyclic permutationof the word a then the cyclic homology of KMn decomposes as HC∗(KMm) =⊕

cHC∗(KMm)c where c runs over all cyclic equivalence classes in Mm. Thisis because the Hochschild complex C∗(KMm) and the cyclic complex T (KMm)decomposes in the same way.

If we compare the definitions we see that the complex E(Z[m]) is isomorphic tothe complex T (KMm)c(m) where c(m) is the cyclic equivalence class of x0x1 · · ·xm.However there is a short exact sequence:

0 → T (KMm−1)c(m−1) → T (KMm)c(m) → T (KMm/J)c(m) → 0

where the first map is indicated by the morphism f : Mm−1 → Mm given byf(xi) = xi+1 for i < m − 1 and f(xm−1) = xmx0 and the second map is thenatural quotient map where J is the two-sided ideal in KMm generated by allmonomials of the form xixj where j ≤ i. Therefore by induction on m it suf-fices to show that T (KMm/J)c(m) is acyclic as a nonaugmented complex, i.e. thatHC∗(KMm/J)c(m) = 0.

Since J is generated by monomials in the noncommuting generators x0, . . . , xmand since c(m) = x0x1 · · ·xm is aperiodic we may use [IZ], Lemma 4.8 which impliesthat HC∗(KMm/J)c(m) ≈ H∗(CX,X ;K) where CX is the cone of the simplicialcomplex X (CX = ∗ if X is empty) and X is the finite simplicial with verticesgiven by all minimal “relations” (i.e. generators of J) which are subwords of wordsin the cyclic class c(m). In this case X has only one vertex xmx0 so we are done.However for the sake of completeness we give the general formula for the highersimplices of X . The vertices w0, w1, . . . , wk in X span a k-simplex in X if and onlyif they all occur as subwords of the same word xixi+1 · · ·xmx0 · · ·xi−1 ∈ c(m).



Algebraic proof of Theorem 3.6. We will do induction on the complexity of P whichis defined to be the number of elements (x, y, k) in P × P × Z so that (x, 0) and(y, k) are unrelated in the poset P . By (c) in Definition 3.1 the integer k is boundedand thus the complexity if finite.

The complexity of P is zero if and only if P is isomorphic to some Z[m]. ByLemma 3.8 the theorem holds in this case. If the complexity of P is nonzero thenP contains two unrelated elements a, b. Let P+, P− be the cyclically ordered setsobtained from P by adding the condition that a < b, b < a respectively. Let P0

be the cyclically ordered sets obtained from P by coalescing a, b. Then E(P ) =E(P+) + E(P−) and E(P0) ≈ E(P+) ∩ E(P−). Since P+, P−, P0 have smallercomplexity than P the complexes E(P+), E(P−), E(P0) are acyclic by induction.Therefore E(P ) is acyclic.

4. The multiplication µ on HC∗(KM)

Since B(MΛ) is a basis category for the free complex T (KM) the product cat-egory B(MΛ) × B(MΛ) with the appropriate sign conventions for the incidencecoefficients will be a basis category for T (KM) ⊗ T (KM). Therefore we need toconstruct an acyclic complex over T (KM) for each object of B(MΛ)×B(MΛ) andshow that it has the required properties. We assume that M is a commutativemonoid and that ∗ : M×M →M is a bimultiplicative map so that 1∗x = x∗1 = 1for all x ∈ M , i.e. ∗ distributes over any finite product in M including the emptyproduct 1.

If x = (x0, . . . , xn), y = (y0, . . . , ym) which we consider as set maps x : Z[n] →M, y : Z[m] → M then x ∗ y is defined to be the set map Z[n,m] → M whichsends (i, j) to xi ∗ yj . (We use an abbreviation Z[n,m] = Z[n] × Z[m] sincethis product occurs quite often.) Given any morphism φ : Z[n,m] → Z[k] let(x ∗ y)#φ = φ#(x ∗ y) = (z0, . . . , zk) ∈ Mk+1 where zt =

∏{xi ∗ yj|φ(i, j) = t} isthe product in M of the images under x ∗ y of the elements of φ

−1(t) ⊆ Z[n,m].

Since M is commutative with unit (for the empty product) this is well defined and(x ∗ y)# gives a morphism of cyclic sets XZ[n,m] → MΛ. Let D(x ∗ y) be theaugmented K-complex E(Z[n,m]) = KB(XZ[n,m]) together with the chain mapto T (KM) = KB(MΛ) induced by the cyclic map (x ∗ y)#.

In order to be an acyclic carrier D needs to be a contravariant functor fromB(MΛ) × B(MΛ) into the category of acyclic complexes over T (KM). So sup-pose that α× β : (S∗(n, x), S∗(m, y)) → (S∗(α, α#x), S∗(b, β#y)) is a morphism inB(MΛ) × B(MΛ) given by two cyclic operators α : Z[n] → Z[a], β : Z[m] → Z[b].(We ignore the irrelevant subscripts in S∗.) Then α, β give a morphism α × β :Z[n,m] → Z[a, b] which induces a morphism

Hom(Z[a.b], Z[k]) → Hom(Z[n,m], Z[k]).

This gives a morphism of cyclic sets (α× β)∗ : XZ[a, b] → XZ[n,m].

Lemma 4.1. The following diagram commutes, i.e.

(α#x ∗ β#y)# = (x ∗ y)#(α× β)∗ :


218 KIYOSHI IGUSA

XZ[a, b](α×β)∗

//

(a#x∗b#y)#$$I

IIIIIIII

XZ[n,m]

(x∗y)#zzttttttttt

MΛ

Proof. Take φ : Z[a, b] → Z[k] in XkZ[a, b]. Then

(x ∗ y)#(α× β)∗(φ) = (x ∗ y)#(φ ◦ (α× β))

= (φ ◦ (α× β))#(x ∗ y) = φ#(α× β)#(x ∗ y)= φ#(α#x ∗ β#y) = (α#x ∗ β#y)#(φ)

where the crucial step (α × β)#(x ∗ y) = α#x ∗ β#y follows from the definition ofα#x, β#y given at the end of §2 and the fact that ∗ distributes over multiplication.

Lemma 4.1 implies that (α × β)∗ : D(α#x ∗ β#y) = E(Z[a, b]) → E(Z[n,m]) =D(x ∗ y) is a morphism of complexes over T (KM).

Definition 4.2. Let µ : T (KM)⊗ T (KM) → T (KM) be a chain map carried bythe acyclic carrier D.

5. A “formula” for µ

In order to show that the multiplication µ can be extended to the case of abialgebra H with “superproduct” and to prove many of the properties of µ we needa formula for µ satisfying certain properties. In this section we prove only theexistence of such a formula.

We note that if µ is carried by the carrier D of the previous section then for eachobject (Sa(n, x), Sb(m, y)) of B(MΛ)× B(MΛ) we have an element

µ(Sa(n, x), Sb(m, y)) ∈ Dk(x ∗ y) = E(Z[n,m])(1)

where k = n+m+ 2a+ 2b which is compatible with augmentation and boundaryoperators in a certain way, for example µ(S0(0, x), S0(0, y)) ∈ E0(x ∗ y) ∼= K musthave augmentation equal to 1 ∈ K.

If these elements (1) depend only on a, b, n,m (i.e. are independent of x, y) thenwe get a “uniform family” of elements Fab(n,m) ∈ Ek(Z[n,m]) for µ which givesthe following uniform formula for µ.

µ(Sa(n, x), Sb(m, y)) = (x ∗ y)#Fab(n,m).(2)

A family of elements Fab(n,m) is a uniform family for µ if it satisfies certain condi-tions which we will give below. Any two such families will be “uniformly homotopic”since E(Z[n,m]) is acyclic.

Definition 5.1. By a uniform family for the multiplication µ we mean a sequenceof elements Fab(n,m) ∈ Ek(Z[n,m]) where k = n+m+2a+2b for all n,m, a, b ≥ 0satisfying the following conditions.

(a) F00(0m0) ∈ E0(Z[0, 0]) is the unique element with augmentation 1.



(b)

dFab(n,m) =n∑i=0

[(−1)i(∂i × 1)∗Fab(n− 1,m)

+ (−1)ni(sti × 1)∗Fa−1,b(n+ 1,m)

+ (−1)ni+n(tsti × 1)∗Fa−1,b(n+ 1,m)]

+ (−1)nm∑j=0

[(−1)j(1× ∂j)∗Fab(n,m− 1)

+ (−1)mj(1× stj)∗Fa,b−1(n,m+ 1)

+ (−1)mj+m(1 × tstj)∗Fa,b−1(n,m+ 1)].

The undefined terms in (b) which occur when a, b, n or m = 0 are understood to bezero. Using the terminology of the remark following Proposition 3.5 the right-handside of (b) can be written as:∑

α∈W (a,n)

sgn(α)(α× 1)∗F∗b(∗,m)

+ (−1)n∑

β∈W (b,m)

sgn(β)(1× β)∗Fa∗(n, ∗).

We also use the expression Fdab(n,m) to denote the right-hand side of (b).

Proposition 5.2. Let F be a uniform family for µ then(a) The expression (2) defines a chain map µ = µF : T (KM) ⊗ T (KM) →

T (KM) carried by D.(b) µF is natural with respect to morphisms of monoids M → N .

Proposition 5.3. A uniform family F for µ exists. Furthermore F can be chosenso that

(a) Fa0(n, 0) = Sa(n, p1) for all a, n ≥ 0 where p1 : Z[n, 0] → Z[n] is the naturalisomorphism.

(b) F0b(0,m) = Sb(m, p2) for all b,m ≥ 0 where p2 : Z[0,m] → Z[m] is thenatural isomorphism.

Proof. Take the special case M = {1} and let µ1 : T (K)⊗T (K) → T (K) be a chainmap carried by an acyclic carrier D. Then Fab(n,m) = µ1(Sa(n, id), Sb(m, id)) isthe desired family of elements. (Definition 5.1 is equivalent to the condition that µ1

is carried by D.) The elements Fab(n,m) can be constructed by induction on thebidegree (n+ 2a,m+ 2b). We can start with formulas (a), (b). The first nontrivialcase if F00(1, 1).

Definition 5.4. By a uniform homotopy between two uniform families F, F ′ wemean a homotopy between the corresponding chain maps T (K)⊗ T (K) → T (K)which is carried by D.

Proposition 5.5. A uniform homotopy G : F ' F ′ is given by a family of elementsGab(n,m) ∈ Ek+1(Z[n,m]) satisfying the following.

(a) dG00(0, 0) = 0.


220 KIYOSHI IGUSA

(b) Fab(n,m)− F ′ab(n,m) = dGab(n,m) +Gdab(n,m)where

Gdab(n,m) =∑

α∈W (a,n)

sgn(α)(α × 1)∗G∗b(∗,m)

+ (−1)n∑

β∈W (b,m)

sgn(β)(1× β)∗Ga∗(n, ∗).

Proposition 5.6. Let µ, µ′ : T (KM)⊗T (KM)→ T (KM) be given by the uniformfamilies F, F ′ and let G : F ' F ′ be a uniform homotopy. Then there is a homotopyg : µ ' µ′ carried by D and this homotopy is given by:

g(Sa(n, x), Sb(m, y)) = Gab(n,m),(3)

g(Sa(n, x), Sb(m, y)) = (x ∗ y)#Gab(n,m).(4)

6. Extension of µ to bialgebras

We will define a multiplication on the cyclic homology of a commutative, cocom-mutative K-bialgebra with “superproduct.” But first we recall the definitions. AK-bialgebra H = (H,m, u,∆, ε) is a K-algebra (H,m, u) together with a coalgebrastructure (H,∆, ε) so that the diagonal (comultiplication) ∆ : H → H ⊗ H andaugmentation map ε : H → K are morphisms of K-algebras. (See, e.g., [S] fordetails.) In particular we have

∆(xy) = ∆(x)∆(y) =∑

(x),(y)

x(1)y(1) ⊗ x(2)y(2) (or “(xy)(i) = x(i)y(i)”),(1)

∆(1) = 1⊗ 1 where 1 = u(1),(2)

ε(x, y) = ε(x)ε(y),(3)

ε(1) = 1.(4)

Any monoid algebra KM has the structure of a K-bialgebra given by ∆(∑aixi)

=∑aixi ⊗ xi, ε(

∑aixi) =

∑ai. This comultiplication is obviously cocommuta-

tive.

Definition 6.1 ([Ga]). Suppose that H is a K-bialgebra. Then by a superproducton H we mean a K-linear map ∗ : H ⊗ H → H satisfying the following for allx, y, z ∈ H .

(a) ∆(x ∗ y) = ∆(x) ∗∆(y), i.e., “(x ∗ y)(i) = x(i) ∗ y(i).”Here ∗ : (H⊗H)⊗ (H⊗H) → H⊗H is given by (a⊗ b) ∗ (c⊗ d) = (a ∗ c)⊗ (b ∗ d),

(b) ε(x ∗ y) = ε(x)ε(y),(c) x ∗ 1 = 1 ∗ x = uε(x),(d) (left distributivity) x ∗ (yz) =

∑(x)(x(1) ∗ y)(x(2) ∗ z),

(e) (right distributivity) xy ∗ z =∑

(z)(x ∗ z(1))(y ∗ z(2)).Note that any bimultiplicative map ∗ : M ×M →M on a monoid M satisfying

x ∗ 1 = 1 ∗ x = 1 gives a superproduct on KM .

Definition 6.2 ([Ga]). A unit for (H, ∗) is an element e of H satisfying the fol-lowing.

(a) ∆(e) = e⊗ e.(b) ε(e) = 1.(c) x ∗ e = e ∗ x = x for all x ∈ H .



Note that if ∗ is associative with unit e, then conditions 6.1(a), (b) and 6.2(a),(b), (c) are equivalent to saying that (H, ∗, e,∆, ε) is a bialgebra.

Given a commutative, cocommutative K-bialgebraH with superproduct ∗ and auniform family F we would like to construct a chain map µ : T (H)⊗T (H) → T (H)with the formula

µ(Sa(n, x), Sb(m, y)) = (x ∗ y)#Fab(n,m)(5)

if x ∈ Cn(H) = H⊗(n+1) and y ∈ Cm(H) = H⊗(m+1). In order for this to bedefined we need is a chain map (x ∗ y)# : E(Z[n,m]) → T (H).

Given x = (x0, x1, . . . , xn) ∈ Hn+1, y = (y0, y1, . . . , ym) ∈ Hm+1 let x∗y denotethe element of H⊗(n+1)(m+1) given by the (n+m+ 2)-fold iterated sum

x ∗ y =∑(x0)

· · ·∑(xn)

∑(y0)

· · ·∑(ym)

(xi(j) ∗ yj(i))(6)

where we use the standard notation∑

(xi)(xi(0), . . . , x

i(m)) for the m-fold diagonal

of xi and similarly for yj .

Lemma 6.3. (a) x ∗ y is K-bilinear in x, y and is therefore defined for any x ∈H⊗(n+1) and y ∈ H⊗(m+1).

(b) Given morphisms α : Z[n] → Z[a], β : Z[m] → Z[b] we have (α#x)∗ (β#y) =(α× β)#(x ∗ y) where (φ#z)j =

∏{zi|φ(i) = j} for any φ : P → Q and z ∈ H⊗P .

Proof. This follows from the cocommutativity of H and the fact that ∗ distributesover multiplication. (When α, β are not surjective we also need conditions (b), (c)of 6.1.)

Given any element (zij) of H(n+1)(m+1) we have an induced chain map (zij)# :E(Z[n,m]) → T (H) given on the generators Sa(b, φ : Z[n,m] → Z[b]) by

(zij)#Sa(b, φ) = Sa(b, φ#(zij)) = Sa(b, (c0, . . . , cb))

where each ck is given by ck =∏{zij|φ(i, j) = k}. Since H is commutative with

unit this is well-defined. The map (zij)# commutes with all cyclic operators (it isa morphism of cyclic K-modules) and therefore induces a chain map as required.Since (zij)# is K-multilinear the definition extends to any element of the tensorproduct H⊗(n+1)(m+1). Thus (x ∗ y)# : E(Z[n,m]) → T (H) is defined.

Proposition 6.4. The expression (5) defines an augmented chain map µ : T (H)⊗T (H) → T (H).

Proof. We compare dµ, µd using the definition of F .

dµ(Sa(n, x), Sb(m, y)) = (x ∗ y)#dF (n,m) =∑

±(x ∗ y)#(α× β)∗F ′,

µd(Sa(n, x), Sb(m, y)) =∑

±(α#x ∗ β#y)#F ′.

But as in Lemma 4.1 we have the formal identity (x∗y)#(α×β)∗ = [(α×β)#(x∗y)]#and this is equal to (α#x ∗ β#y)# by Lemma 6.3.

By the same argument we have the following.

Proposition 6.5. If µ, µ′ : T (H)⊗ T (H) → T (H) are given by two uniform fam-ilies F, F ′ then there is a uniform homotopy G : F ' F ′ and G gives a homotopyg : µ ' µ′ by

g(Sa(n, x), Sb(m, y)) = (x ∗ y)#Gab(n,m).(7)


222 KIYOSHI IGUSA

7. Associativity of µ for KM

We will now verify algebraically that the multiplication µ on the cyclic homologyof H satisfies the various conditions that we have claimed except for the computa-tion in Theorem 1.1b in the case H = K and Corollary 1.2. For these we must relyon the topological argument and the proof given at the end of the paper that thetopological and algebraic definitions of µ agree.

The first property that we will tackle is the associativity of µ in the case when ∗ isassociative. As in the case of Proposition 6.5 (the uniqueness of µ up to homotopy),a homotopy µ(µ×1) ' µ(1×µ) in the case of a bialgebra H results from a uniformhomotopy in the case H = KM which in turn comes from a specific homotopyin the case M = {1}. The key point therefore is to show that the chain mapsµ(µ × 1), µ(1 × µ) : T (KM) ⊗ T (KM) ⊗ T (KM) → T (KM) are carried by thesame acyclic carrier A which is functorial in M . In this section we construct thecarrier A and use it to show that µ(µ× 1), µ(1× µ) are uniformly homotopic. Weassume that the bimultiplicative pairing ∗ : M ×M →M is associative.

Let (Sa(n, x), Sb(m, y), Sc(l, z)) be an object of the basis category B(MΛ) ×B(MΛ)×B(MΛ) of T (KM)⊗T (KM)⊗T (KM). We define x∗y∗z to be the elementof M (n+1)(m+1)(l+1) with (i, j, k)-coordinate xi∗yj ∗zk and we define the augmentedcomplex A(x∗y∗z) to be the acyclic complex E(Z[n,m, l]) where Z[n,m, l] = Z[n]×Z[m]× Z[l] together with the augmented chain map (x ∗ y ∗ z)# : E(Z[n,m, l]) →T (KM) induced by the map of cyclic sets (x ∗ y ∗ z)# : X(Z[n,m, l]) →MΛ whichsends (φ : Z[n,m, l] → Z[p] ∈ XpZ[n,m, l]) to φ#(x ∗ y ∗ z) = (w0, . . . , wq) ∈M q+1

where each wt is the product wt =∏{xi ∗ yj ∗ zk|φ(i, j, k) = t}. As before we get

an acyclic carrier A.

Theorem 7.1. The acyclic carrier A carries both µ(µ×1) and µ(1×µ). Thereforeµ is homotopy associative on T (KM).

Remark. This is a special case of Theorem 13.1 below.

Proof. We will show that µ(µ× 1) is carried by A. The other case is similar.Recall that µ(Sa(n, x), Sb(m, y)) = (x ∗ y)#Fab(n,m) where Fab ∈ Ek(Z[n,m]),

k = n+m+ 2(a+ b). Let

Fab(n,m) =∑φ

rφSd(k−2d,φ)(1)

be an expression for Fab(n,m) as a linear combination of generators of Ek(Z[n,m]).Then

(x ∗ y)#Fab(n,m) =∑φ

rφSd(k − 2d, φ#(x ∗ y))

and

µ(µ(Sa(n, x), Sb(m, y)), Sc(l, z)) = µ((x ∗ y)#Fab(n,m), Sc(l, z))

=∑φ

rφ(φ#(x ∗ y) ∗ z)#Fdc(k − 2d, l).

But (φ#(x ∗ y) ∗ z)# = (x ∗ y ∗ z)#(φ× 1)∗ by Lemma 4.1 so we have

µ(µ× 1)(Sa(n, x), Sb(m, y), Sc(l, z)) = (x ∗ y ∗ z)#∑φ

rφ(φ× 1)∗Fdc(k − 2d, l)

(2)



where Gabc(n,m, l) =∑

φ rφ(φ× 1)∗Fdc(k − 2d, l) ∈ E(Z[n,m, l]) depends only ona, b, c, n,m, l. This is a uniform formula for µ(µ× 1) lifting it to the carrier A. Weneed only check the boundary conditions:

εG000(0, 0, 0) = 1,(3)

dGabc(n,m, l) =∑

α∈W (a,n)

sgn(α)(α × 1× 1)∗G∗c(∗,m, l)

+ (−1)n∑

β∈W (b,m)

sgn(β)(1 × β × 1)∗Ga∗c(n, ∗, l)

+ (−1)n+m∑

γ∈W (c,l)

sgn(γ)(1× 1× γ)∗Gab∗(n,m, ∗).

(4)

The verification of (3) is easy. Since F00(0, 0) = S0(0, p1), G000(0, 0, 0) =(p1×1)∗S0(0, p1) has augmentation 1. To verify (4) we begin by taking the bound-ary of (1):

dFab(n,m) =∑φ

rφ dSd(k − 2d, φ) =∑φ

rφ∑

ψ∈W (d,k−2d)

sgn(ψ)Sq(p, ψφ).(5)

For fixed c, l let the operator L : E(Z[n,m]) → E(Z[n,m, l]) be defined by

L(Sq(p, λ)) = (λ× 1)∗Fqc(p, l).

(L is not a chain map. It is a graded homomorphism of degree l + 2c.) ThenGabc(n,m, l) = LFab(n,m) so L transforms (5) and Fab(n,m) into the followingequation. ∑

φ

rφ∑

ψ∈W (d,k−2d)

sgn(ψ)(ψφ× 1)∗Fqc(p, l)

=∑

α∈W (a,n)

sgn(α)(α × 1× 1)∗G∗bc(∗,m, l)

+ (−1)n∑

β∈W (b,m)

sgn(β)(1 × β × 1)∗Ga∗c(n, ∗, l).

(6)

Compare this with

dGabc(n,m, l) =∑φ

rφ(φ× 1)∗ dFdc(k − 2d, l)

=∑φ

rφ(φ× 1)∗

∑ψ∈W (d,k−2d)

sgn(ψ)(ψ × 1)∗Fqc(p, l)

+ (−1)k∑

γ∈W (c,l)

sgn(γ)(1× γ)∗Fd∗(k − 2d, ∗)

=∑φ

rφ∑

ψ∈W (d,k−2d)

sgn(ψ)(ψφ× 1)∗Fqc(p, l)

+ (−1)k∑φ

rφ∑

γ∈W (c,l)

sgn(γ)(φ× γ)∗Fd∗(k − 2d, ∗).

(7)


224 KIYOSHI IGUSA

By (6) the∑

φ

∑ψ term in (7) agrees with the sum of the

∑α and

∑β terms in

(4). The following computation shows that the∑φ

∑γ term in (7) agrees with the∑

γ term in (4).

Gab?(n,m, ??) =∑φ

rφ(φ× 1)∗ dFd?(k − 2d, ??)

so

(−1)n+m∑

γ∈W (c,l)

sgn(γ)(1× 1× γ)∗Gab?(n,m, ??)

= (−1)k∑γ

sgn(γ)(1× 1× γ)∗(φ× 1)∗Fd?(k − 2d, ??)

= (−1)k∑γ

sgn(γ)(φ× γ)∗Fd?(k − 2d, ??).

8. Associativity of µ for bialgebras H

We now consider the case when H is a commutative, cocommutativeK-bialgebrawith associative superproduct ∗.Lemma 8.1. If x = (x0, . . . , xn), y = (y0, . . . , yn), z = (z0, . . . , zl) then (x∗y)∗z =x ∗ (y ∗ z) as elements of H⊗(n+1)(m+1)(l+1).

Proof. This is an easy computation:

(x ∗ y) ∗ z =∑

((xi(j) ∗ yj(i))(k) ∗ zk(i)(j)) (by definition)

=∑

((xi(j)(k) ∗ yj(i)(k)) ∗ zk(i,j)) (by Def. 6.1.1)

=∑

(xi(j)(k) ∗ (yj(i)(k) ∗ zk(i)(j))) (by associativity of µ)

=∑

(xi(j)(k) ∗ (yj(k)(i) ∗ zk(j)(i))) by cocommutativity of ∆)

=∑

(xi(j)(k) ∗ (yj(k) ∗ zk(j))(i)) (by Def. 6.1.1)

= x ∗ (y ∗ z).Theorem 8.2. µ : T (H)⊗ T (H) → T (H) is homotopy associative.

Proof. We recall that µ is given by µ(Sa(n, x), Sb(m, y)) = (x∗y)#Fab(n,m). Thus

µ(µ(Sa(n, x), Sb(m, y)), Sc(l, z)) = µ((x ∗ y)#Fab(n,m), Sc(l, z))

= µ

∑φ

rφ(x ∗ y)#Sd(k − 2d, φ), Sc(l, z)

= µ

∑φ

rφSd(k − 2d, φ#(x ∗ y)), Sc(l, z)

=∑φ

rφ(φ#(x ∗ y) ∗ z)#Fdc(k − 2d, l)

= ((x ∗ y) ∗ z)#∑φ

rφ(φ× 1)∗Fdc(k − 2d, l)

= ((x ∗ y) ∗ z)#Gabc(n,m, l)



where Gabc(n,m, l) is a uniform family for µ(µ× 1). Similarly

µ(Sa(n, x), µ(Sb(m, y), Sc(l, z))) = (x ∗ (y ∗ z))#G′abc(n,m, l)where G′abc(n,m, l) is a uniform family for µ(1 × µ). By Theorem 7.1 the uniformfamilies G,G′ are equivalent to chain maps g, g′ : T (K)⊗ T (K) ⊗ T (K) → T (K)carried by the carrier A. Therefore any homotopy h : g ' g′ carried by A isequivalent to a uniform homotopy G ' G′. Since (x ∗ y) ∗ z = x ∗ (y ∗ z) by Lemma8.1 we are done.

9. Commutativity and unit for µ

By an easy variation of the associativity argument we can prove the following.This general pattern of argument is later formalized in sections 12 and 13 althoughthe commutativity of µ is not a special case of Theorem 13.1.

Theorem 9.1. If ∗ is commutative then µ is homotopy skew commutative onHC∗(H).

Proof. Let w : T (H) ⊗ T (H) → T (H) ⊗ T (H) and ω : Z[n,m] → Z[m,n] be theswitching maps w(x, y) = (−1)deg x deg y(y, x) and ω(a, b, k) = (b, a, k). Then wewant to show that µ, µ ◦ w are uniformly homotopic for H = KM . First we notethat µ(Sa(n, x), Sb(m, y)) = (x ∗ y)#Fab(n,m) and

µw(Sa(n, x), Sb(m, y)) = (−1)nm(Sb(m, y), Sa(n, x)) = (−1)nm(y ∗ x)#Fba(m,n).

But F ′ab(n,m) = (−1)nmω∗Fba(m,n) is a uniform formula for µ and (y ∗ x)#ω∗ =(ω#(y ∗ x))# = (x ∗ y)#. Consequently a uniform homotopy F ' F ′ gives ahomotopy µ ' µ ◦ w.

In the bialgebra case we need to note that ω#(y∗x) = x∗y in H⊗(n+1)(m+1).

Theorem 9.2. Suppose that e ∈M acts as a two-sided unit for ∗. Then S0(0, e) ∈T0(KM) acts as a homotopy unit for µ. (S0(0, e) is a strict two-sided unit for µ ifF is chosen as in Proposition 5.3).

Proof. We may assume that F is as in Proposition 5.3 since it is unique up tohomotopy. Then:

µ(Sa(n, x), S0(0, e)) = (x ∗ e)#Fa0(n, 0) = x#Sa(n, p1) = Sa(n, x),

µ(S0(0, e), Sb(m, y)) = (e ∗ y)#F0b(0,m) = y#Sb(m, p2) = Sb(m, y).

Theorem 9.3. Suppose that H is a commutative, cocommutative bialgebra and ∗is a superproduct with unit e. Then S0(0, e) is a homotopy unit for µ.

Proof. Choose F as in Proposition 5.3. Then S0(0, e) is a strict two-sided unit forµ by the calculation above and the fact that x ∗ e = e ∗ x = x for all x ∈ H⊗(n+1)

which in turn follows from the assumption ∆(e) = e⊗ e.

10. The power operation λk

If P is a cyclically ordered set and k ≥ 1 then let λkP be the set P with anew partial ordering ≤k on P = P × Z given by (x, n) ≤k (y,m) if and only if(x, 0) ≤ (y, [(m − n)/k]) where [ ] denotes the greatest integer function. Sincethe new partial ordering ≤k on P is weaker than the original partial ordering theidentity map on P gives a morphism λkP → P . However this morphism is not


226 KIYOSHI IGUSA

natural. The reader can check that for P = Z[n] the definition of λkZ[n] agreeswith the original one given in Definition 3.3.

If φ : P → Q is a morphism of cyclically ordered sets then let λkφ : λkP → λkQbe given by λkφσn(x) = σnJkφ(x) where Jk is the operation Jk(y,m) = (y, km)and φ : P → Q is a representative of φ.

Proposition 10.1. (a) (x, n) < (y,m) in P if and only if (x, kn) < (y, km) inλkP , i.e., Jk : P → λkP is a poset embedding.

(b) Jkσ = σkJk (in particular Jk is not a morphism of cyclically ordered sets).(c) λk is a functor from the category of cyclically ordered sets to itself.(d) If φ : P → Q is a morphism of cyclically ordered sets, then the following

diagram commutes.

Pf

//

Jk

��

Q

Jk

��

λkPλkφ

// λkQ

Proof. We will verify that λkφ : λkP → λkQ is order preserving. The rest isobvious.

Suppose that (x, n) ≤ (y,m) in λkP . Then (x, 0) ≤ (y, a) in P where ak ≤ m−n.Thus φ(x) ≤ σaφ(y) in Q and Jkφ(x) ≤ σakJkφ(y) ≤ σm−nJkφ(y) in λkQ. Butthen λkφ(x, n) = σnJkφ(x) ≤ σmJkφ(y) = λkφ(y,m) in λkQ.

Proposition 10.2. For any two cyclically ordered sets P,Q there is a natural iso-morphism

λk(P ×Q) ∼= λkP × λkQ.

Proof. We first claim that the “identity map” gives an isomorphism λk(P ×Q) ∼=λkP × λkQ. To see this note that the following are equivalent.

(a) (a, b, n) ≤ (x, y,m) in λk(P ×Q).(b) (a, b, 0) ≤ (x, y, [(m− n)/k]) in P ×Q.(c) ∃i s.t. (a, 0) ≤ (x, i) in P and (b, 0) ≤ (y, [(m− n)/k]− i) in Q.(d) ∃i s.t. (a, 0) ≤ (x, ki) in λkP and (b, n) ≤ (y,m− ki) in Q.(e) (a, b, n) ≤ (x, y,m) in λkP × λkQ.Now suppose that φ : P → S, ψ : Q→ T are morphisms. Then

λk(φ × ψ)(x, y, 0) = Jk(φ× ψ)(x, t, 0) = (s, t, k(a+ b))

where φ(x, 0) = (s, a) and ψ(y, 0) = (t, b). But then λkφ(x, 0) = (s, ka) andλkψ(y, 0) = (t, kb) so

(λkφ× λkψ)(x, y, 0) = (s, t, ka+ kb) = λk(φ× ψ)(x, y, 0).

11. The power operations on HC∗(A)

We will give an acyclic carrier for the power operations Λk = 1/kΨk on T (A)for any commutative K-algebra A. This will enable us to show that Ψk commuteswith µ up to uniform homotopy. As before we will not need an explicit formula forΨk except to show that our operations agree with those of [L] and [FT].



As always we begin with the case A = KM where M is a commutative monoid.For each k ≥ 1 let Λk : T (KM) → T (KM) be any chain map carried by the acycliccarrier C given by C(Sa(n, x)) = E(λkZ[n]) together with the chain map x# :E(λkZ[n]) → T (KM) induced by the morphism of cyclic sets x# : X(λkZ[n]) →MΛ given by x#(φ) = φ# = (z0, . . . , zm) where zj =

∏{xi|φ(i) = j}. A morphismSa(n, x) → Sb(m,α#x) given by α : Z[n] → Z[m] induces a chain map (λkα)∗ :E(λkZ[m]) → E(λkZ[n]).

To verify that this is a carrier we note that (λkα)∗(λkβ)∗ = λk(βα)∗ andx#(λkα)∗ = λkα#(x)# = (α#x)# since α, λkα give the same map of underlyingsets Z[n] → Z[m].

As before we get a uniform formula for Ψk:

Λk(Sa(n, x)) = x#Lka(n)

where Lka(n) ∈ En+2a(λkZ[n]) is a family of elements depending only on a, n, k.

Proposition 11.1. The uniform family of elements Lka(n) is characterized (up touniform homotopy) by the following properties.

(a) εLk0(0) = 1,(b)∂Lka(n) =

n∑i=0

[(−1)i(λk∂i)∗Lka(n− 1) + (−1)ni(λksti)∗Lka−1(n+ 1)

+ (−1)ni+n(λktsti)∗Lka−1(n+ 1)].

With this characterization we can compare Lka(n) with the power operations Ψk

as described in [FT].

Theorem 11.2. An explicit formula for Lka(n) ∈ En+2a(λkZ[n]) modulo genera-cies is given as follows.

Lka(n) = ka∑I

sgn(φI)Sa(n, φI) + degenerate terms

where I = (I1, . . . , Ik) runs over all partitions of the set {1, 2, . . . , n} and φI :λkZ[n] → Z[n] is the morphism given by φI(0) = 0 and φI(i, 0) = (ai, π(i)) fori = 1, . . . , n where a1, a2, . . . , an are the elements of I1 in increasing order followedby the elements of I2 in increasing order etc. and π(i) = j if ai ∈ Ij.Proof. This is a computation which can be broken into two parts.

n∑i=0

(−1)i∂i∑I

sgn(φI)Sa(n, φI) =n∑j=0

(−1)j(λk∂j)∗∑J

sgn(φJ)Sa(n− 1, φJ ),

(1)

k

n∑i=0

(−1)nisti∑I

sgn(φI)Sa(n, φI)

=n∑j=0

(−1)nj(λkstj)∗∑L

sgn(φL)Sa−1(n+ 1, φL).

(2)


228 KIYOSHI IGUSA

Comparison of the remaining terms shows that they are not equal:

k

n∑i=0

(−1)ni+ntxti∑I

sgn(φI)Sa(n, φI)

6=n∑j=0

(−1)nj+n(λktstj)∗∑L

sgn(φL)Sa−1(n+ 1, φL).

(3)

However the expressions in (3) are all degenerate so Lka(n) = ka∑

I ±Sa(n, φI)satisfies 11.1.b up to degeneracies. (Since this formula for Lka(n) sends degeneraciesto degeneracies the degeneracies can be factored out.)

To verify equation (1) we rewrite it in the following formn∑i=0

∑I

(−1)isgn(φI)Sa(n− 1, ∂iφI)=n∑j=0

∑J

(−1)jsgn(φJ)Sa(n− 1, φJ ◦ ∂j).(4)

If 0 < i < n and i, i+1 belong to different parts of the partition I, then ∂iφi = ∂iφI ,where I ′ is obtained from I by interchanging i, i + 1. These terms occur withopposite sign on the left side of (4) and so they cancel. The terms with ∂0φI where1 ∈ I1 and ∂nφI where n ∈ Ik also cancel. The remaining terms are equal tocorresponding terms on the right hand side of (4).

Equation (2) can be rewritten as:

k

n∑i=0

∑I

(−1)nisgn(φI)Sa(n, stiφI)

=n∑j=0

∑L

(−1)njsgn(φL)Sa−1(n+ 1, φL ◦ λkstj).(5)

Since stiφI sends 0 to i+ 1 one can see that the morphisms stiφL are all differentand that each of them occurs exactly k times (with the same sign) as a morphismof the form φL ◦ λkstj .Corollary 11.3. kΛk = Ψk = (−1)kkλk.

Corollary 11.4. Let A be a commutative K-algebra. Then the power operationΨk = kΛk on T (A) is given by ΨkSa(n, x) = kx#Lka(n).

12. General operations on HC∗(H)

We want to show that the power operations Λk commute with the multiplicationµ. This will be a formal consequence of the isomorphism λkP × λkQ ∼= λk(P ×Q)in the category Cos of finite nonempty cyclically ordered sets.

In this section we will show how any functor f : Cosn → Cos gives rise toan operation on HC∗(KM) for any “f -monoid” M and on HC∗(H) for any “f -algebra” H .

Definition 12.1. Let f : Cos×· · ·×Cos→ Cos be a functor of n variables. We saythat f acts on the commutative monoid M (and we call M an f -monoid) providedthat for any n objects P1, P2, . . . , Pn of Cos there is an n-multiplicative map (i.e.multiplicative in each variable)

f∗ : MP1 × · · · ×MPn →Mf(P1,...,Pn)



so that for any n morphisms φi : Pi → Qi the following diagram commutes.

MP1 × · · · ×MPn

φ1#×···×φn

#

��

f∗// Mf(P1,...,Pn)

f(φ1,...,φn)#

��

MQ1 × · · · ×MQn

f∗// Mf(Q1,...,Qn)

(1)

i.e., f(φ1, . . . , φn)#f∗(x1, . . . , xn) = f∗(φ1#x

1, . . . , φn#xn). Here φ# : MP → MQ is

given by φ#(x) = y where y(j) =∏{x(i)|φ(i) = j}.

Note that there is a unique action of any functor f on the trivial monoid {1}.Suppose that f, g : Cosn → Cos are two functors acting on the monoids M,N

respectively. Then a morphism (M, f) → (N, g) of monoid-functor pairs consists ofa homomorphism of monoids h : M → N and a natural transformation η : f → gmaking the following diagram commute for all P1, . . . , Pn in Cos.

MP1 × · · · ×MPnf∗

//

h×···×h��

Mf(P1,...,Pn)

h◦η#��

NP1 × · · · ×NPng∗

// Ng(P1,...,Pn).

(2)

If η : f → f is the identity transformation we call h : M → N an f -morphism. Notethat for any f -monoid M there are unique f -morphisms M → {1} and {1} →M .

Examples 12.2. (a) f = λk, λi∗ = “identity” : MP → MλkP , in particular anycommutative monoid has the natural “trivial” structure as a λk-monoid.

(b) µ(P,Q) = P × Q,µ∗ : MP ×MQ → MP×Q given by µ∗(x, y) = x ∗ y, (x ∗y)(i, j) = x(i) ∗ y(j) where ∗ : M ×M → M is a bimultiplicative pairing so that1 ∗ x = x ∗ 1 = 1 for all x ∈M . (The commutativity of (1) in this case is given byLemma 4.1.)

By replacing products with tensor products these definitions can be extended toK-algebras.

Definition 12.3. We say that the functor f : Cosn → Cos acts on a commutativeK-algebra H and we call H an f -algebra if for any P1, . . . , Pn ∈ Cos there is aK-linear map f∗ : H⊗P1⊗· · ·⊗H⊗Pn → H⊗f(P1,...,Pn) so that for any n morphismsφi : Pi → Qi the following “tensor analogue” of diagram (1) commutes.

H⊗P1 ⊗ · · · ⊗H⊗Pnf∗

//

φ1#⊗···⊗φn

#

��

H⊗f(P1,...,Pn)

f(φ1,...,φn)#

��

H⊗Q1 ⊗ · · · ⊗H⊗Qn

f∗// H⊗f(Q1,...,Qn)

(3)

Here φ# : H⊗P → H⊗Q is given on generators x ∈ HP by φ#(x) = y ∈ HQ, y(j) =∏{x(i)|φ(i) = j}.Examples 12.4. (a) Any commutativeK-algebra has a natural “trivial” λk-action.

(b) Any commutative, cocommutative K-algebra with superproduct is a µ-algebra where µ is the functor µ(P,Q) = P × Q. (This follows from Lemma6.3.b.)


230 KIYOSHI IGUSA

A morphism of algebra-functor pairs (H, f) → (H ′, g) is a K-algebra morphismh : H → H ′ and a natural transformation η : f → g making the obvious diagram(the tensor analogue of (2)) commute.

If f : Cosk → Cos is a functor which acts on the commutative monoid M let Afbe the acyclic carrier from T (KM)⊗ · · · ⊗ T (KM) to T (KM) given as follows.

(a) If a1, n1, . . . , ak, nk ≥ 0 and xi = (xi(0), . . . , xi(ni)) : Z[ni] → M are setfunctions then

Af (Sa1(n1, x1), . . . , Sak

(nk, xk)) = E(f(Z[n1], . . . , Z[nk]))

together with the map to T (KM) = KB(MΛ) induced by the morphism of cyclicsets

f∗(x1, . . . , xk)# : Xf(Z[n1], . . . , Z[nk]) →MΛ

given as before by f∗(x1, . . . , xk)#φ = φ#(f∗(x1, . . . , xk)). (Note that f∗(x1, . . . , xk)maps f(Z[n1], . . . , Z[nk]) into M and φ : f(Z[n1], . . . , Z[nk]) → Z[?].)

(b) If φi : Z[ni] → Z[mi], i = 1, 2, . . . , k, are morphisms then there is an inducedmap

f(φ1, . . . , φk)∗ : E(f(Z[m1], . . . , Z[mk])) → E(f(Z[n1], . . . , Z[nk])).

In order to show that this defines an acyclic carrier we need to show that thefollowing diagram commutes:

E(f(Z[m1], . . . , Z[mk]))f(φ1,...,φk)∗

//

f∗(φ1#x

1,...,φk#x

k)#))R

RRRR

RRRR

RRRR

RE(f(Z[n1], . . . , Z[nk]))

f∗(x1,...,xk)#vvlllllllllllll

T (KM).

This is a familiar calculation:

f∗(x1, . . . , xk)#f(φ1, . . . , φk)∗(ψ) = f∗(x1, . . . , xk)#(ψ ◦ f(φ1, . . . , φk))

= ψ#f(φ1, . . . , φk)#f∗(x1, . . . , xk)

= ψ#f∗(φ1#x

1, . . . , φk#xk)

= f∗(φ1#x

1, . . . , φk#xk)#(ψ).

We also recall that f(Z[n1], . . . , Z[nk]) is finite and nonempty by assumption.A uniform formula for a chain map θf carried by Af can be given by

θf (Sa1(n1, x1), . . . , Sak

(nk, xk)) = f∗(x1, . . . , xk)#Ff (a1, n1, . . . , ak, nk)(4)

where Ff (a1, n1, . . . , ak, nk) ∈ Ep(f(Z[n1], . . . , Z[nk])) with p = n1 + · · · + nk +2(a1 + · · ·+ ak) is a family of elements given uniquely up to uniform homotopy bythe following conditions.

εFf (0, 0, . . . , 0, 0) = 1,(5)



∂Ff (a1, n1, . . . , ak, nk)

=∑

φ1∈W (a1,n1)

sgn(φ1)f(φ1, 1, . . . , 1)∗Ff (∗, ∗, a2, n2, . . . , ak, nk)

+ (−1)n1∑

φ2∈W (a2,n2)

sgn(φ2)f(1, φ2, 1, . . . , 1)∗Ff (a1, n1, ∗, ∗, a3, n3, . . . , ak, nk)

+ · · ·+ (−1)n1+n2+···+nk

∑φk∈W (ak,nk)

sgn(φk)

× f(1, . . . , 1, φk)∗Ff (a1, n1, . . . , ak−1, nk−1, ∗, ∗).

(6)

Proposition 12.5. Suppose that f : Cosk → Cos acts on the K-algebra H. Then(4) above defines a chain map θf : T (H) ⊗ k → T (H) and thus an operationθf : HCn1(H)⊗· · ·⊗HCnk

(H) → HCn1+···+nk(H) which is natural with respect to

morphisms of f -algebras. Furthermore this map in cyclic homology is independentof the choice of the uniform family Ff .

Proof. Since f∗(x1, . . . , xk) is multilinear, (4) defines a linear map θf . Condi-tion (6) shows that θf commutes with boundary. We just need to check thatf∗(x1, . . . , xk)#f(φ1, . . . , φk)∗ = f∗(φ1

#x1, . . . , φk#x

k)#. But this follows from thecommutativity of (3).

13. Composition of uniform formulas

Let f : Coss → Cos be a functor of s variables and let gi : Costi → Cos be sfunctors. Then we get a composite functor fg = f ◦ (g1 × · · · × gs) : CosT → Coswhere T = t1+· · ·+ts. We will obtain a uniform family for this composite functor interms of the uniform family Ff for f and uniform families G1, . . . , Gs for g1, . . . , gs.

First we need some notation. For each i = 1, . . . , s we have

Gi(a1, n1, . . . , at, nt) =∑Ii

rIiSbI1(mI1 , ψ

Ii)(1)

where ψIi : gi(Z[n1], . . . , Z[nt]) → Z[mIi ] is a morphism, {Ii} is an index setdepending on i, (a1, n1, . . . , at, nt) and t = ti. We note that mIi + 2bIi =

∑nj +

2∑aj for each Ii. Since Gi is a uniform family we have

∂Gi(a1, n1, . . . , at, nt) =∑Ii

∑αi∈W (bIi

,mIi)

sgn(αi)rIiS∗(∗, αiψIi)

=∑

βi1∈W (a1,n1)

sgn(βi1)gi(βi1, 1, . . . , 1)∗Gi(∗, ∗, a2, n2, . . . , at, nt)

+ (−1)n1∑

βi2∈W (a2,n2)

sgn(βi2)gi(1, βi2, 1,. . .,1)∗Gi(a1, n1, ∗, ∗, a3, n3,. . ., at, nt)

+ · · ·+ (−1)n1+n2+···+nt−1∑

βit∈W (at,nt)

sgn(βit)

× gi(1, . . . , 1, βit)∗Gi(a1, n1, . . . , at−1, nt−1, ∗, ∗).

(2)


232 KIYOSHI IGUSA

We now construct Ffg.

θf (θ1, . . . , θs)/(Sa11(n11, x11), Sa12(n12, x

12), . . . , Sast(nst, xst))

= θf (g1∗(x

11, . . . , x1t)#G1(a11, n11, . . . , a1t, n1t), . . . ,

gs∗(xs1, . . . , xst)#Gs(as1, ns1, . . . , ast, nst)).

By (1) and the multilinearity of θf this is equal to:∑I1

· · ·∑Is

rI1 · · · rIsθf (g1∗(x

11, . . . , x1t)#SbI1(mI1 , ψ

I1), . . . ,

gs∗(xs1, . . . , xst)#SbIs

(mIs , ψIs))

=∑I1

· · ·∑Is

rI1 · · · rIsf∗(ψI1# g

1∗(x

11, . . . , x1t), . . . , ψIsgs∗(xs1, . . . , xst))#

× Ff (bI1 ,mI1 , . . . , bIs ,mIs)

=∑I1

· · ·∑Is

rI1 · · · rIsf∗(g1∗(x

11, . . . , x1t), . . . , gs∗(xs1, . . . , xst)#)

× f(ψI1 , . . . , ψIs)∗Ff (bI1 ,mI1 , . . . , bIs ,mIs)

= fg∗(x11, . . . , x1t, . . . , xs1, . . . , xst)#Ffg(a11, n11, . . . , ast, nst)

where

Ffg(a11, n11, . . . , ast, nst) ∈ E(f(g1(Z[n11], . . . , Z[n1t]), . . . , gs(Z[ns1], . . . , Z[nst])))

is given by

Ffg(a11, n11, . . . , ast, nst) =∑I1

· · ·∑Is

rI1 · · · rIsf(ψI1 , . . . , ψIs)∗

×Ff (bI1 ,mI1 , . . . , bIs ,mIs)(3)

where f(ψI1 , . . . , ψIs) : f(g1(Z[n11], . . . , Z[n1t]), . . . , gs(Z[ns1], . . . , Z[nst])) →f(Z[mI1 ], . . . , Z[mIs ]).

Theorem 13.1. Ffg is a uniform family for fg.

Proof. Since θf , θ1, . . . , θs, fg∗(x11, . . . , xst)# commute with augmentation we haveεFfg(0, . . . , 0) = 1. Thus we need only verify the following boundary condition.

∂Ffg(a11, n11, . . . , ast, nst)

=(?)

∑β11∈W (a11,n11)

sgn(β11)f(g1(β11, 1, . . . , 1), 1, . . . , 1)∗

× Ffg(∗, ∗, a12, n12, . . . , ast, nst)

+ (−1)n11∑

β12∈W (a12,n12)

sgn(β12)f(g1(1, β12, 1, . . . , 1), 1, . . . , 1)∗

× Ffg(a11, n11, ∗, ∗, a13, nst) + · · ·

(4)



where the indices of summation βij are as in equation (2). The actual boundary ofFfg(a11, . . . , nst) is obtained from (3):

∂Ffg(a11, . . . , nst) =∑I1

· · ·∑Is

rI1 · · · rIsf(ψI1 , . . . , ψIs)∗∂Ff (bI1 ,mI1 , . . . , bIs ,mIs)

=∑I1

· · ·∑Is

rI1 · · · rIsf(ψI1 , . . . , ψIs)∗

· ∑α1∈W (bI1 ,mI1 )

sgn(α1)f(α1, 1, . . . , 1)∗Ff (∗, ∗, bI2 , . . . ,mIs)

+(−1)mI1

∑α2∈W (bI2 ,mI2 )

sgn(α2)f(1, α2, 1, . . . , 1)∗

× Ff (bI1 ,mI1 , ∗, ∗, bI3, . . . ,mIs) + · · ·

(5)

where αi is as in (2). Thus it suffices to show that the RHS of (4) is equal to theRHS of (5). We will see that this is a formal consequence of (2).

The terms in (2) can be converted into the corresponding terms in (4), (5) viathe graded homomorphisms

Vi : E(gi(Z[ni1], . . . , Z[nit]))

→ E(f(t1(Z[n11], . . . , Z[n1t]), . . . , gs(Z[ns1], . . . , Z[nst])))

given for each c, k ≥ 0 and φ : gi(Z[ni1], . . . , Z[nit]) → Z[k] by

ViSc(k, φ) =∑I1

· · ·∑Ii−1

∑Ii+1

· · ·∑Is

rI1 · · · rIi−1rIi+1 · · ·

rIsf(ψI1 , . . . , ψIi−1 , φ, ψIi+1 , . . . , ψIs)∗

× Ff (bI1 ,mI1 , . . . , bIi−1 ,mIi−1 , c, k, bIi+1 ,mIi+1 , . . . , bIs ,mIs).

Although Vi is not a chain map (Vi∂ 6= ∂Vi) it is natural in the following sense. Ifφj : Z[n′ij ] → Z[nij ] are morphisms then

Vi ◦ gi(φ1, . . . , φt)∗ = f(1, . . . , 1, gi(φ1, . . . , φt), 1, . . . , 1)∗ ◦ V ′i

where V ′i is the same as Vi with each nij replaced by n′ij . If we compare the terms

in the second expression in (2) with those in the RHS of (5) we see that

s∑i=1

(−1)n11+···+ni−1,tVi(∂Gi(ai1, ni1, . . . , ait, nit)) = ∂Ffg(a11, n11, . . . , ast, nst).

(6)

The signs match since ni1 + · · ·+ nit has the same parity as mIi . If we apply Vi toboth sides of (1) we get

ViGi(a1, n1, . . . , at, nt) =∑I1

· · ·∑Is

rI1 · · · rIsf(ψI1 , . . . , ψIs)∗

× Ff (bI1 ,mI1 , . . . , bIs ,mIs)

= Ffg(a11, n11, . . . , ast, nst).

(7)


234 KIYOSHI IGUSA

Now we can use (7) and the naturality of Vi to compute Vi∂Gi(ai1, ni1, . . . , ait, nit)using the last expression in (2):

Vi∂Gi(ai1, ni1, . . . , ait, nit)

=∑

βi1∈W (ai1,ni1)

sgn(βi1)f(1, . . . , 1, gi(βi1, 1, . . . , 1), 1, . . . , 1)∗

× ViGi(∗, ∗, ai2, ni2, . . . , ait, nit) + · · · .Comparing this with the RHS of (4) we see that

RHS(4) =s∑i=1

(−1)n11+···+ni−1,tVi(∂Gi(ai1, ni1, . . . , ait, nit))

= ∂Ffg(a11, n11, . . . , ast, nst).

Corollary 13.2. The following diagram commutes for any µ-algebra H and forall n ≥ 0, k ≥ 1. In particular this holds if H is a commutative cocommutativeK-bialgebra with superproduct :

HCn(H)⊗HCm(H)µ

//

Λk⊗Λk

��

HCn+m(H)

Λk

��

HCn(H)⊗HCm(H) µ// HCn+m(H)

Proof. When H = KM this follows from Theorem 13.1 and the fact that λk(P ×Q) ∼= λkP × λkQ (Proposition 10.2). For the general case we need to verify thatboth compositions µ(Λk,Λk),Λkµ are given by uniform formulas, i.e., we need toknow that the calculations leading to (3) above work for bialgebras. But this justboils down to checking that:

(a) (α#x ∗ β#y)# = (x ∗ y)# ◦ (α× β)∗,(b) (γ#(x ∗ y))# = (x ∗ y)# ◦ γ∗.More generally one needs to check that

f(φ1#g

1(x1), . . . , φs#gs(xs))# = f(g1(x1), . . . , gs(xs))# ◦ f(φ1, . . . , φs)∗.

But this is exactly Definition 12.3.

14. Topological carriers

In the last five sections of this paper we will show that the topological defini-tions of the operators µ,Λk on HC∗(KM) as given in §1 agree with the algebraicdefinitions given above. The idea is to show that the continuous maps defined in§1 are carried by “topological carriers” corresponding to the acyclic carriers usedabove. This section contains the definition of a topological carrier.

Definition 14.1. Suppose that (X, {ψe}) is a CW complex. (ψe : Dn → X is thecharacteristic map of the cell en.) Then a category of cells for X is a pair (E(X), L)where

(a) E(X) is a small category whose object set is the set of cells of X so that:(i) Each cell is the source of only finitely many morphisms.(ii) For each nonidentity morphism α : e→ e′, dim(e) > dim(e′).

(b) L is a contravariant functor from E(X) to the category of finite CW complexesover X so that:



(i) The cells of L(e) is in 1-1 correspondence with the morphisms in E(X)with source e. (The cell corresponding to α : e→ e′ is denoted eα.)

(ii) The map πe : L(e) → X sends the open cell eα homeomorphically ontothe open cell e′, in fact we assume that πe ◦ ψeα = ψe′ .

(iii) For each α : e→ e′ the induced map α∗ : L(e′) → L(e) satisfies α∗◦ψeβ=

ψeβα.

Proposition 14.2. If (E(X), L) is a category of cells for X then(a) X has the quotient topology with respect to

∐e L(e).

(b) If an orientation is chosen for each cell of X then the morphisms of E(X)of degree −1 acquire incidence numbers rα and E(X) becomes a basis category.

If Y is any space and X is as above then a topological carrier from X to Yis defined to be a contravariant functor F from E(X) to the category of weaklycontractible spaces over Y , i.e., spaces with the weak homotopy type of a point. Acontinuous map f : X → Y is said to be carried by the carrier F if for each cell ein X there is a lifting fe : L(e) → F (e) of the composition L(e) → X → Y so thatthe following diagram commutes for all morphisms α : e→ e′ in E(X).

L(e′) α∗//

fe′��

L(e)

fe

��

F (e′)α∗

// F (e)

Proposition 14.3. Every topological carrier from X to Y carries at least one mapand any two such maps are homotopic.

We say that the topological carrier F is regular if Y and each F (e) is a CWcomplex and all maps F (e) → Y and F (e′) → F (e) are regular in the sense that theysend open cells homeomorphically onto open cells. Since a regular map F (e) → Yinduces a well-defined chain map of cellular chain complexes C(F (e)) → C(Y ) weget an ordinary carrier “C(F )” from C(X) to C(Y ).

Proposition 14.4. Let F be a regular topological carrier from X to Y . Thenthere is a cellular map f : X → Y which is carried by F and the induced mapf∗ : C(X) → C(Y ) is carried by C(F ).

There are some CW complexes which have canonical cell categories. We saythat a CW complex X is weakly regular if for every pair of open cells e, e′ theinverse image ψ−1

e (e′) ⊆ Dn is locally path connected and the map ψ−1e (e′) → e′

is open. In particular this implies that each component of ψ−1e (e′) maps onto e′ so

the closure of each cell is a finite subcomplex of X . With a little thought one cansee that ψ−1

e (e′) has only finitely many components.

Definition 14.5. If X is a weakly regular CW complex then the canonical cellcategory for X is defined as follows.

(a) The objects of E(X) are the cells of X . A morphism en → e′ is defined to bea component of ψ−1

e (e′) ⊆ Dn.(b) For each cell en, L(e) is defined to be the quotient space of Dn modulo

the equivalence relation x ∼ y if ψe(x) = ψe(y) ∈ e′ ⊆ e and x, y lie in the samecomponent of ψ−1

e (e′). This means that each element of L(e) is uniquely representedby a pair (x, α) where α : e→ e′ is a morphism in E(X) and x ∈ e′.


236 KIYOSHI IGUSA

(c) πe : L(e) → X is given by πe(x, α) = x and α∗ : L(e′) → L(e) is given byα∗(x, β) = (x, βα).

(d) The composition of the quotient map Dm → L(e′) with the map α∗ : L(e′) →L(e) gives the characteristic map from the cell in L(e) corresponding to α : e→ e′.

As an example, the geometric realization of a simplicial set X is weakly regularand its canonical cell category resembles the category of simplices of X (withoutthe degeneracies).

15. The canonical cell category for VM ' BM

If M is an associative monoid then let VM be the space of all finite nonemptyconfigurations X ⊆ S1 together with labels in M . As in the case of BM in §1the points in X are allowed to merge with corresponding labels being multiplied inclockwise order. However we do not delete points with resulting label equal to 1.

There is a map VM → BM given by ignoring the points in X with label 1 andinserting 1 ∈ S1 with label 1 ∈M if it is not already in the configuration.

Proposition 15.1. VM ' BM .

Proof. Let V0M be the subspace of VM consisting of all configurations X contain-ing the point 1 ∈ S1. Then V0M is homeomorphic to the nondegenerate realization‖F (M)‖ of the simplicial set F (M) of §1 and thus V0M ' BM .

To see that V0M ' VM we note first that there is a retraction r : VM → V0Mgiven by inserting the point 1 ∈ S1 with label 1 ∈ M . A homotopy between r andthe identity on VM is given as follows. For any nonempty configuration X ⊆ S1

we detach one new point from each point of X with label 1 and move it clockwiseuntil it merges with the next point. When one of these new points passes the point1 ∈ S1 we split it in two and leave a fixed point at 1 with label 1.

Now we need a cell decomposition of VM so that V0M is a subcomplex. Ifx = (x0, . . . , xn) ∈ Mn+1 then let ex ⊆ V0M be the set of all configurations{y0, . . . , yn} in S1 written counterclockwise with y0 = 1 so that yi has label xi.The characteristic map ψex : ∆n → V0M of the n-cell ex is given in barycentriccoordinates by ψex(t) = {y0, . . . , yn} where yi = exp(2π

√−1(t0 + · · ·+ ti−1)) withappropriate labels. Then ηex : ∂∆n → V0M

n−1 is clearly regular so V0M becomesa weakly regular CW complex.

For each x = (x0, . . . , xn) ∈ Mn+1 let ux ⊆ VM be the set of all configurations{y0, . . . , yn} with yn < 1 < y0 in counterclockwise order with labels xi. Then uxis an (n + 1)-cell with characteristic map ψux : ∆n+1 → VM given by ψux(t) ={y0, . . . , yn} where yi = exp(2π

√−1(t0 + · · ·+ ti)). Again ηux : ∂∆n+1 → VMn isregular so VM is a weakly regular CW complex. Consequently there is a canonicalcell category E(VM). From the geometric description it is obvious that there is onemorphism ex → e′ in E(VM) for every simplex in ∆n and one morphism ux → e′′

for every simplex in ∆n+1. Furthermore the incidence numbers are all ±1.A morphism ex → ey where y = (y0, . . . , ym) is given by an epimorphism α :

Z[n] → Z[m] so that α(0) = 0 (these are dual to monomorphisms [m] → [n] in ∆)and so that y = α#x. There are no morphisms ex → uy. A morphism ux → uyis given by an epimorphism α : Z[n] → Z[m] in ∆ so that y = α#x. A morphismux → ey is given by an epimorphism α : Z[n] → Z[m] so that y = α#x and α−1(0)contains either 0 or n.



The incidence numbers for the morphisms in E(VM) of degree −1 are indicatedby the following proposition.

Proposition 15.2. In the cellular chain complex of VM the boundaries of thegenerators (n, x), u(n, x) corresponding to ex, ux have boundaries:

∂(n, x) =n∑i=0

(−1)i(n− 1, ∂ix),

∂u(n, x) = −n−1∑i=0

(−1)iu(n− 1, ∂ix) + (n, x)− (−1)n(n, tx).

16. The canonical cell category for WM ' GM

Let WM denote the space of all pairs (C,X) where C ∈ B(1) = CP∞ and X isa finite nonempty configuration in C with labels in M . We give WM the topologyanalogous to that of the previous configuration spaces VM,BM,GM . Then WMis a fiber bundle over BU(1) with fiber VM . There is a natural map WM → GMgiven by ignoring points with label 1. Since VM ' BM we conclude the following.

Theorem 16.1. WM ' GM .

To construct a cell decomposition for WM we need to start with the stan-dard cell decomposition of BU(1) = CP∞ which has one cell e2a in every evendimension so that e0 ∪ e2 ∪ · · · ∪ e2a = CP a. Thus the open cells are given bye2a = {[z0, . . . , za] ∈ CP a|za 6= 0}. The characteristic map ψ2a : D2a → CP a of e2a

is given by ψ2a(z0, . . . , za−1) = [z0, . . . , za−1, x] where x =√

1− ‖z‖2 is the uniquenonnegative real number so that (z0, . . . , za−1, x) ∈ S2a+1. This has an obviouslifting ψ2a : D2a → S2a+1 given by ψ2a(z0, . . . , za−1) = (z0, . . . , za−1, x).

We will use the notation 6c to denote the unique element 6c = (z0, . . . , za) of eachfiber of E(1) = S∞ → BU(1) = CP∞ so that the last nonzero coordinate za ispositive and real. This is the same as saying that 6c lies in the image of ψ2a(intD2a)for some a ≥ 0.

For each a, n ≥ 0 and x ∈Mn+1 let eSa(n, x) denote the set of all (C,X) ∈ WMso that C ∈ BU(1) lies in the open cell e2a and 6c ∈ X = {y0, . . . , yn} with elementsnumbered counterclockwise with y0 = 6 c and with corresponding labels x0, . . . , xn.Then eSa(n, x) is an (n+2a)-cell with characteristic map ψ : D2a×∆n →WM givenby ψ(z, t) = (ψ2a(z), {y0, . . . , yn}) where yi = exp(2π

√−1(t0 + · · ·+ ti−1))ψ2a(z).Similarly let eRa(n, x) denote the set of all (C,X) ∈ WM so that C ∈ e2a

but 6c /∈ X = {y0, . . . , yn} with corresponding labels x0, . . . , xn. Then eRa(n, x)is an (n + 2a + 1)-cell with characteristic map ψ : D2a × ∆n+1 → WM given byψ(z, t) = (ψ2a(z), {y0, . . . , yn}) where yi = exp(2pi

√−1(t0 + · · ·+ ti))ψ2a(z).

Proposition 16.2. This defines a weakly regular cell decomposition of WM . Theboundary map in the associated cellular chain complex is given by

∂Sa(n, x) =n∑i=0

(−1)iSa(n− 1, ∂ix) +n∑i=0

(−1)niRa−1(n, tix)

(i.e. ∂ = b+N),

(a)


238 KIYOSHI IGUSA

∂Ra(n, x) = −n−1∑i=0

(−1)iRa(n− 1, ∂ix) + Sa(n, x)− (−1)nSa(n, tx)

(i.e., ∂ = −b′ + 1− t),

(b)

where Sa(n, x) ∈ Cn+2a(WM) and Ra(n, x) ∈ Cn+2a+1(WM) are the generatorscorresponding to eSa(n, x) and eRa(n, x) and the undefined terms R−1(n, tix) areunderstood to be zero.

The canonical cell category E(WM) can be described as follows. For A,B =S or R there is a morphism eAa(n, x) → eBb(m, y) for every epimorphism α :Z[n] → Z[m] so that y = α#x provided that the open cell eBb(m, y) is containedin the closure of eAa(n, x). This geometric condition is equivalent to the followingcombinatorial conditions.

(a) b ≤ a in all cases,(b) b < a if (A,B) = (S,R).(c) Suppose that a = b then (as in the case of VM):

(i) α(0) = 0 in the case (A,B) = (S, S),(ii) α−1(0) contains either 0 or n in the case (A,B) = (R,S),(iii) α : [n] → [m] is a morphism in ∆ in the case (A,B) = (R,R).

This construction generalizes to the following.

Definition 16.3. If X is any cyclic set let A(X) be the basis category defined asfollows.

(1) The objects of A(X) are symbols Sa(n, x), Ra(n, x) where a, n ≥ 0 andx ∈ Xn.

(2) deg Sa(n, x) = n+ 2a, deg Ra(n, x) = n+ 2a+ 1.(3) A morphism Aa(n, x) → Bb(m, y) is an epimorphism α : Z[n] → Z[m] so

that y = α∗x and the combinatorial conditions (a), (b), (c) above are satisfied.(4) The incidence coefficients are as indicated in Proposition 16.2.

With this definition we have E(WM) = A(MΛ).

Proposition 16.4. For any cyclic set X there is a natural chain homotopy equiv-alence νX : KB(X) ∼−→ KA(X) which is carried by the acyclic carrier which sendsSa(n, x) to KA(XZ[n]) which maps to KA(X) by the cyclic map x# : XZ[n] → Xgiven by x#(α : Z[n] → Z[k]) = α∗(x) ∈ Xk.

Proof. νX(Sa(n, x)) = Sa(n, 1) +∑n

i=0(−1)niUa−1(n+ 1, sti).

17. A topological carrier for µ : WM ×WM →WM

We are now ready to show that the topological and algebraic definitions of themultiplication µ in HC∗(KM) agree. We are assuming that M is a commutativemonoid with a bimultiplicative pairing ∗ : M ×M → M so that 1 ∗ x = x ∗ 1 = 1for all x ∈M .

Let µ : WM ×WM → WM be given by µ((C,X), (D,Y )) = (CD,X ∗ Y ) asin §1. Then we will construct a regular topological carrier Fµ for µ. For each celle(Aa(n, x), Bb(m, y)) = eAa(n, x)×eBb(m, y) in WM ×WM where A,B representeither S or R let Fµ(e(Aa(n, x), Bb(m, y))) = Fµ(n,m) be the space of all pairs(C, φ) where C ∈ BU(1) and φ : Z[n,m] → C is a morphism of cyclically orderedsets with the map (x∗y)# : Fµ(n,m) →WM given by (x∗y)(C, φ) = (C, φ#(x∗y))



where φ#(x ∗ y) is the image of φ together with labels∏{xi ∗ yj|φ(i, j) = z} for

each z ∈ Imφ.For any morphism (α, β) : e(Aa(n, x), Bb(m, y)) → e(Xa′(n′, α#x), Yb′ (m′, β#y))

given by two morphisms α : Z[n] → Z[n′], β : Z[m] → Z[m′] let (α, β)∗ : Fµ(n′,m′)→ Fµ(n,m) be given by (α, β)∗(C, φ) = (C, φ ◦ (α× β)).

Since Fµ(n,m) is clearly contractible we have:

Lemma 17.1. Fµ is a topological carrier from WM ×WM to WM .

Lemma 17.2. µ : WM ×WM →WM is carried by Fµ.

Proof. Let µ : L(eAa(n, x))× L(eBb(m, y)) → Fµ(n,m) be given by

µ((C,X, α), (D,Y, β)) = (CD, φ)

where we view α, β as morphisms α : Z[n] → C, β : Z[m] → D with imagesX,Y having labels α#x, β#y and φ : Z[n,m] → CD is given by φ(i, j, k) =(α(i, 0), β(j, k), k). The following discussion shows that µ is a continuous function.

Let ν : Z[0, 1) × Z[0, 1) → Z[0, 1) be the morphism of cyclically ordered setsgiven by

ν(x, y, n) =

{(x+ y, n) if x+ y < 1,(x+ y − 1, n+ 1) if x+ y ≥ 1.

Then ν is rotationally invariant in the sense that ν ◦ (Ta × Tb) = Ta+b ◦ ν for alla, b ∈ R where Ta : Z[0, 1) → Z[0, 1) is given by Ta(x) = x+a (if we identify Z[0, 1)with R).

Since ν is rotationally invariant it induces a morphism ν : C ×D → CD for anyC,D ∈ BU(1). Furthermore the underlying set map ν of ν varies continuously with(C,D) ∈ BU(1)× BU(1). The map φ : Z[n,m] → CD factors as φ = ν ◦ (α× β) :Z[n,m] → C ×D → CD so it varies continuously with ((C,X, α), (D,Y, β)).

Lemma 17.3. Fµ is a regular topological carrier and the canonical cell category ofE(Fµ(n,m)) is naturally isomorphic to A(XZ[n,m]).

Proof. First we need a cell decomposition of Fµ(n,m). For each morphism φ :Z[n,m] → Z[k] and a ≥ 0 let eSa(k, φ) be the space of all pairs (C,α ◦ φ) whereC ∈ e2a ⊆ BU(1) and α : Z[k] → C is a morphism so that α(0) = 6c. Let eRa(k, φ)be the space of all pairs (C, β ◦ φ) where C ∈ e2a and β : Z[k] → C is a morphismso that 6c < β(0) < · · · < β(k) < σ(6c).

The characteristic maps

ψ : D2a ×∆n, D2a ×∆n+1 → Fµ(n,m)

for eSa(k, φ), eRa(k, φ) are given by ψ(z, t) = (ψ2a(z), α ◦ φ), (ψ2a(z), β ◦ φ) re-spectively where α, β are given by α(i) = exp(2π

√−1(t0 + · · · + ti−1))ψ2a(z) andβ(j) = exp(2π

√−1(t0 + · · ·+ tj))ψ2a(z).By comparing formulas it is clear that the maps (x ∗ y)# : Fµ(n,m) → WM

and (α × β)∗ : Fµ(n′,m′) → Fµ(n,m) are all regular. It follows that Fµ(n,m) isweakly regular and that (x∗y)# induces a morphism of basis categories. This forcesE(Fµ(n,m)) to be isomorphic to A(XZ[n,m]).


240 KIYOSHI IGUSA

Theorem 17.4. The topological and algebraic definitions of the multiplication

µ : HC∗(KM)⊗HC∗(KM) → HC∗(KM)

agree.

Proof. The regular topological carrier Fµ gives an operation µA on KA(MΛ). Theacyclic carrier E gives an operation µB on KB(MΛ) = T (KM). Thus it suffices toshow that the following diagram commutes up to homotopy where ν is the chainhomotopy equivalence given by Proposition 16.4.

KB(MΛ)⊗KB(MΛ)µB

//

ν⊗ν��

KB(MΛ)

ν

��

KA(MΛ)⊗KA(MΛ) µB

// KA(MΛ)

In order to do this we form the mapping cylinder Cyl(ν) of ν and construct anacyclic carrier for a map µC : Cyl(ν) ⊗ Cyl(ν) → Cyl(ν) making the followingdiagram commute.

KB(MΛ)⊗KB(MΛ)µB

//

��

KB(MΛ)

��

Cyl(ν)⊗ Cyl(ν)

��

µC// Cyl(ν)

��

KA(MΛ)⊗KA(MΛ)µA

// KA(MΛ)

(0)

The mapping cylinder Cyl(ν) is defined to be KC(MΛ) where C(X) is the basiscategory defined for any cyclic set X as follows.

The category C(X) is a union of three disjoint full subcategories A(X),B(X),and

∑B(X) where∑B(X) is the same as B(X) except that the degrees of all

objects are raised by 1 and the incidence coefficients change in sign (deg∑A =

degA+ 1, r∑α = −rα). There are no morphism between A(X) and B(X) nor fromA(X)

∐B(X) into∑B(X).

The morphisms from∑B(X) to B(X) are simply morphisms in B(X). Thus

C(X)(∑A,B) = B(X)(A,B) for all A,B in B(X). The identity morphisms

∑A→

A have incidence coefficient equal to 1. All other morphisms∑A → B have

coefficient zero. The morphisms from∑Sa(n, x) ∈

∑B(X) to Ab(m, y) ∈ A(X)(A = r or S) are defined to be the morphism α : Z[n] → Z[m] so that y = α#x.The only morphisms from B(X) to A(X) with nonzero incidence coefficient arethose which occur in the formula for νX . These are id :

∑Sa(n, x) → Sa(n, x) with

coefficient 1 and sti :∑Sa(n, x) → Ra−1(n + 1, stix) with coefficient (−1)ni for

0 ≤ i ≤ n. Composition is defined in the obvious way by composition of morphismsin Cos.

The inclusion functors A(X) → C(X) and B(X) → C(X) induce the chainmaps from KA(X), KB(X) into Cyl(ν) = KC(X) which appear in (0) above for



X = MΛ. To prove the theorem it now suffices to construct an acyclic carrier fora map µC : KC(X) ⊗ KC(X) → KC(X) which is compatible with µA, µB. Suchan acyclic carrier is given by sending (Aa(n, x), Bb(m, y)) in C(MΛ) × C(MΛ) forA,B = R,S,

∑S to the complex KC(XZ[n,m]) which maps to KC(MΛ) by the

cyclic map (x ∗ y)# : XZ[n,m] →MΛ.

18. A topological carrier for Λk : WM →WM

Theorem 18.1. The topological and algebraic definitions of the power operationsΛk = 1/kΨk on HC∗(KM) agree.

Remark. This result is essentially due to Goodwillie since our topological definitionsof Λk is just a rewording of this intuitive description of these operations.

Proof. This proof is analogous to, but easier than, the proof of Theorem 17.4. Tominimize repetition we give only the first half of the proof.

Let Λk : WM → WM be the continuous function Λk(C,X) = (Ck,Λk(X)).Then a regular topological carrier FΛ for Λk can be given as follows. For any celleAa(n, x) in WM where A = R or S let FΛ(eAa(n, x) = FΛ(n) be the space ofall pairs (C, φ) where C ∈ BU(1) and φ : λkZ[n] → C is a morphism of cyclicallyordered sets with the map x# : FΛ → WM given by x#(C, φ) = (C, φ#x). Anymorphism α : eAa(n, x) → eBb(m, y) is given by a morphism α : Z[n] → Z[m] sothat y = α#x. Let α∗ : FΛ(m) → FΛ(n) be given by α∗(C, φ) = (C, φ ◦ λkα).

To see that Λk is carried by FΛ let Λk : L(eAa(n, x)) → FΛ(n) be given byΛk((C,X), α) = (Ck, φ) where φ : λkZ[n] → Ck is the composition of λkα :λkZ[n] → λkC and the unique continuous morphism ρ : λkC → Ck so thatρ(x) = xk.

The remainder of the proof is analogous to the proof for Fµ, for example, inthe final step we use the acyclic carrier which sends Aa(n, x) ∈ C(MΛ) for A =R,S,

∑S to the complex KC(XλkZ[n]) which maps to KC(MΛ) by x#. This

defines a chain map ΛkC : KC(MΛ) → KC(Mk) compatible with the topologicallydefined operators ΛkA on KA(MΛ) and the algebraically defined operators ΛkB onKB(MΛ).

References

[C] A. Connes, Noncommutative differential geometry, Publ. IHES 62 (1985), 41–144. MR87i:58162

[FT] B. Feigin and B. Tsygan, Additive K-theory, K-Theory, Arithmetic and Geometry, Lec-ture Notes in Math. 1289, Springer, New York-Heidelberg-Berlin, 1987, pp. 67–209. MR89a:18017

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242 KIYOSHI IGUSA

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Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254-

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· 216 kiyoshi igusa proof. it su ces to take n=0sincez[0] maps to z[n] for all n. if p is empty...

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