ON
λ
-RINGS AND TOPOLOGICAL REALIZATION
DONALD YAUReceived 19 July 2005; Accepted 12 April 2006
It is shown that most possibly truncated power series rings admit uncountably many filteredλ-ring structures. The question of how many of these filteredλ-ring structures are topologically realizable by theK-theory of torsion-free spaces is also considered for truncated polynomial rings.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
Aλ-ring is, roughly speaking, a commutative ringRwith unit together with operations λi,i≥0, on it that act like the exterior power operations. It is widely used in algebraic topology, algebra, and representation theory. For example, the complex representation ringR(G) of a groupGis aλ-ring, whereλiis induced by the map that sends a represen-tation to itsith exterior power. Another example of aλ-ring is the complexK-theory of a topological spaceX. Here,λiarises from the map that sends a complex vector bundleη overXto theith exterior power ofη. In the algebra side, the universal Witt ring W(R) of a commutative ringRis aλ-ring.
The purpose of this paper is to consider the following two interrelated questions: (i) classify theλ-ring structures over power series and truncated polynomial rings; (ii) which ones and how many of theseλ-rings are realizable as (i.e., isomorphic to)
theK-theory of a topological space?
The first question is purely algebraic, with no topology involved. One can think of the second question as aK-theoretic analogue of the classical Steenrod question, which asks for a classification of polynomial rings (over the field ofpelements and has an action by the modpSteenrod algebra) that can be realized as the singular modpcohomology of a topological space.
In addition to being aλ-ring, theK-theory of a space is filtered, makingK(X) a fil-teredλ-ring. Precisely, by a filteredλ-ring we mean a filtered ring (R,{R=I0⊃I1⊃ ···}) in whichRis aλ-ring and the filtration idealsInare all closed under theλ-operationsλi (i >0). It is, therefore, more natural for us to consider filteredλ-ring structures over
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 91267, Pages1–21
filtered rings. Moreover, we will restrict to torsion-free spaces, that is, spaces whose inte-gral cohomology is Z-torsion-free. The reason for this is that one has more control over theK-theory of such spaces than non-torsion-free spaces.
A discussion of our main results follows. Proofs are mostly given in later sections. Our first result shows that there is a huge diversity of filteredλ-ring structures over most power series and truncated polynomial rings.
Theorem 1.1. Letx1,...,xnbe algebraically independent variables with the same (arbitrary but fixed) filtrationd >0, and letr1,...,rn be integers≥2, possibly∞. Then the possibly truncated power series filtered ring Z[[x1,...,xn]]/(xr11,...,xnrn) admits uncountably many isomorphism classes of filteredλ-ring structures.
Herexi∞ is by definition equal to 0. In particular, this theorem covers both finitely generated power series rings and truncated polynomial rings. The caseri= ∞(1≤i≤n) is proved in [11]. In this case, there are uncountably many isomorphism classes that are topologically realizable, namely, by the spaces in the localization genus of (BS3)×n. The remaining cases are proved by directly constructing uncountably many filteredλ-rings. In general, there is no complete classification of all of the isomorphism classes of filtered λ-ring structures. However, such a classification can be obtained for small truncated poly-nomial rings, in which case we can also give some answers to the second question above. This will be considered below after a brief discussion of the Adams operations.
The Adams operations. The results below are all described in terms of the Adams oper-ations. We will use a result of Wilkerson [9] on recovering theλ-ring structure from the Adams operations. More precisely, Wilkerson’s theorem says that ifRis a Z-torsion-free ring which comes equipped with ring endomorphismsψk (k≥1) satisfying the condi-tions, (1)ψ1=Id andψkψl=ψkl=ψlψk, and (2)ψp(r)≡rp(modpR) for each prime pandr∈R, thenRadmits a uniqueλ-ring structure with theψkas the Adams opera-tions. The obvious filtered analogue of Wilkerson’s theorem is also true for the filtered rings considered inTheorem 1.1. Therefore, for these filtered rings, in order to describe a filteredλ-ring structure, it suffices to describe the Adams operations.
When there are more than oneλ-ring in sight, we will sometimes writeψRnto denote the Adams operationψninR.
Truncated polynomial rings. Consider the filtered truncated polynomial rings (Z[x]/(xn),
|x| =d) and (Z[x]/(xn), |x| =d), withxin filtration exactlyd >0 andd>0,
respec-tively. Let Λd denote the set of isomorphism classes of filtered λ-ring structures over (Z[x]/(xn), |x| =d). DefineΛdsimilarly. Then it is obvious thatΛdandΛdare in one-to-one correspondence, and there is no reason to distinguish between them. Indeed, for R∈Λd one can associate to itR∈Λd such thatψRk(x)=ψRk(x) as polynomials for all
k, and this construction gives the desired bijection betweenΛdandΛd. Therefore, using
the above bijections, we will identify the setsΛd ford=1, 2,..., and writeΛ(Z[x]/(xn)) for the identified set. Each isomorphism class of filteredλ-ring structures on Z[x]/(xn) with|x| ∈ {1, 2,...}is considered an element inΛ(Z[x]/(xn)).
We start with the simplest casen=2, that is, the dual number ring Z[x]/(x2).
Theorem 1.2 ([10, Corollary 4.1.2]). There is a bijection betweenΛ(Z[x]/(x2)) and the set of sequences (bp) indexed by the primes in which the componentbpis divisible byp. The filteredλ-ring structure corresponding to (bp) has the Adams operationsψp(x)=bpx.
Moreover, such a filteredλ-ring is isomorphic to theK-theory of a torsion-free space if and only if there exists an integerk≥1 such thatbp=pkfor allp.
Therefore, in this case, exactly countably infinitely many (among the uncountably many) isomorphism classes are topologically realizable by torsion-free spaces. Indeed, theK-theory of the even-dimensional sphereS2krealizes the filteredλ-ring withb
p=pk for all p. On the other hand, ifX is a torsion-free space withK(X)=Z[x]/(x2), then ψp(x)=pkxfor allpifxlies in filtration exactly 2k[2, Corollary 5.2].
In what follows we will use the notationθp(n) to denote the largest integer for which pθp(n)dividesn, wherepis any prime. By convention we setθ
p(0)= −∞.
To study the casen=3, we need to consider the following conditions for a sequence (bp) of integers indexed by the primes.
(A)b2 =0,bp≡0 (modp) for all primes p, andbp(bp−1)≡0 (mod 2θ2(b2)) for all odd primesp.
(B) Let (bp) be as in (A). Consider a prime p >2 (if any) for which bp =0 and θp(bp)=min{θp(bq(bq−1)) :bq =0}.
We will call them conditions (A) and (B), respectively. Notice that in (B),θp(bp)= θp(bp(bp−1)), sincepdoes not divide (bp−1). Moreover, there are at most finitely many such primes, since each suchpdividesb2(b2−1), which is nonzero.
The following result gives a complete classification forΛ(Z[x]/(x3)), the set of iso-morphism classes of filteredλ-ring structures over the filtered truncated polynomial ring
Z[x]/(x3). As before we will describe aλ-ring in terms of its Adams operations.
Theorem 1.3. LetRbe a filteredλ-ring structure on Z[x]/(x3). ThenRis isomorphic to one of the following filteredλ-rings.
(1)S((cp))= {ψp(x)=cpx2:pprime}withc2≡1 (mod 2) andcp≡0 (modp) forp > 2. Moreover, any such sequence (cp) gives rise to an element ofΛ(Z[x]/(x3)), and two such filteredλ-rings,S((cp)) andS((cp)), are isomorphic if and only if (cp)= ±(cp). (2)S((bp),k)= {ψp(x)=bpx+cpx2:pprime}with (bp) satisfying condition (A) above and thecphaving the following form. Letp1,...,pnbe the list of all odd primes satis-fying condition (B) above. Then there exists an odd integerksuch that, writingGfor gcd(bp(bp−1) : all primesp),
(i) 1≤k≤G/2;
(ii)k≡0 (modp1···pn);
(iii)cp=kbp(bp−1)/Gfor all primesp.
Moreover, any such pair ((bp),k) gives rise to an element ofΛ(Z[x]/(x3)). Two such filteredλ-rings,S((bp),k) andS((bp),k), are isomorphic if and only ifbp=bp for all primespandk=k. NoS((cp)) is isomorphic to anyS((bp),k).
In case (2), if (bp) satisfies condition (A), then it follows that there are exactly
G 4p1···pn
=
gcdbpbp−1: allp 4p1···pn
(1.1)
isomorphism classes of filteredλ-ring structures over Z[x]/(x3) with the property that ψp(x)≡b
px(modx2) for all primesp. Heresdenotes the smallest integer that is greater than or equal tos. Applying formula (1.1) to the cases (bp)=(pr), wherer∈ {1, 2, 4}, we see that there are a total of 64 elements inΛ(Z[x]/(x3)) satisfyingψp(x)=prx(modx2) for all primesp. In fact, there is a unique such element whenr=1 (k=1), three such ele-ments whenr=2 (k∈ {1, 3, 5}), and sixty such elements whenr=4 (k∈ {1, 3,..., 119}). This simple consequence of Theorem 1.3 leads to the following upper bound for the number of isomorphism classes of filteredλ-ring structures over Z[x]/(x3) that are topo-logically realizable by torsion-free spaces.
Corollary 1.4. Let X be a torsion-free space whoseK-theory filtered ring is Z[x]/(x3). Then, using the notation ofTheorem 1.3,K(X) is isomorphic as a filteredλ-ring toS((pr),k) for somer∈ {1, 2, 4}and some k. In particular, at most 64 of the uncountably many iso-morphism classes of filteredλ-ring structures on Z[x]/(x3) can be topologically realized by torsion-free spaces.
Indeed, ifXis a torsion-free space whoseK-theory filtered ring is Z[x]/(x3), then by Adams’ result on Hopf invariant 1 [1], the generatorxmust have filtration exactly 2, 4, or 8. When the filtration ofxis equal to 2r, one hasψp(x)≡prx(modx2) for all primesp. Therefore, byTheorem 1.3,K(X) must be isomorphic toS((pr),k) for somer∈ {1, 2, 4} and somek. The discussion preceding this corollary then implies that there are exactly 64 such isomorphism classes of filteredλ-rings.
It should be remarked that at least 3 of these 64 isomorphism classes are actually real-ized by spaces, namely, the projective 2-spacesFP2, wherePdenotes the complex num-bers, the quaternions, or the Cayley octonions. These spaces correspond tor=1, 2, and 4, respectively. Further work remains to be done to determine whether any of the other 61 filteredλ-rings inCorollary 1.4are topologically realizable.
Before moving on to the casen=4, we would like to present another topological ap-plication ofTheorem 1.3, which involves the notion of Mislin genus. LetXbe a nilpotent space of finite type (i.e., its homotopy groups are all finitely generated and πn(X) is a nilpotentπ1(X)-module for eachn≥2). The Mislin genus ofX, denoted by Genus(X), is the set of homotopy types of nilpotent finite type spacesY such that thep-localizations ofXandY are homotopy equivalent for all primesp. Let H denote the quaternions. It is known that Genus(HP∞) is an uncountable set [8]. Moreover, these uncountably many homotopically distinct spaces have isomorphicK-theory filtered rings [11] but pairwise nonisomorphicK-theory filteredλ-rings [7]. In other words, K-theory filtered λ-ring classifies the Mislin genus of HP∞. It is also known that Genus(HP2) has exactly 4 ele-ments [6]. We will now show that the genus of HP2behaves very differently from that of
Corollary 1.5. K-theory filteredλ-ring does not classify the Mislin genus of HP2. In other words, there exist homotopically distinct spacesX andY in Genus(HP2) whose K-theory filteredλ-rings are isomorphic.
Indeed, an argument similar to the one in [11] shows that the 4 homotopically distinct spaces in the Mislin genus of HP2all have Z[x]/(x3) as theirK-theory filtered ring, with x in filtration exactly 4. Therefore, in each one of these λ-rings, we haveψp(x)≡p2x (modx2) for all primesp. The corollary now follows, since byTheorem 1.3, there are only 3 isomorphism classes of filteredλ-ring structures on Z[x]/(x3) of the form S((p2),k) becausekmust be 1, 3, or 5. It is still an open question as to whetherK-theory filtered λ-ring classifies the genus of HPnfor 2< n <∞.
We now move on to the case n=4, that is, the filtered truncated polynomial ring
Z[x]/(x4). A complete classification theorem along the lines of Theorems1.2 and 1.3 has not yet been achieved forn≥4. However, some sort of classification is possible if one imposes certain conditions on the linear coefficients of the Adams operations that usually appear in theK-theory of spaces.
Theorem 1.6. LetRbe an element ofΛ(Z[x]/(x4)).
(1) IfψRp(x)≡px(modx2) for all primesp, thenRis isomorphic to the filteredλ-ring structure withψp(x)=(1 +x)p−1 for all primesp.
(2) IfψRp(x)≡p2x(modx2) for all primesp, thenRis isomorphic to one of the following 60 mutually nonisomorphic filteredλ-ring structures on Z[x]/(x4):
Sk,d2=
ψp(x)=p2x+kp2
p2−1
12 x
2+d px3
, (1.2)
wherek∈ {1, 5},d2∈ {0, 2, 4,..., 58}, and
dp= p
2p4−1d2
60 +
k2p2p2−1p2−4
360 (1.3)
for odd primesp.
(3) In general, ifψRp(x)≡bpx(modx2) withbp =0 for all primesp, then there are only finitely many isomorphism classes of filteredλ-ring structuresSover Z[x]/(x4) such thatψSp(x)≡bpx(modx2) for all primesp.
(4) If (in the notation of the previous statement)b2=0, thenRis of the formS((cp), (dp))= {ψp(x)=cpx2+dpx3}. Any such collection of polynomials gives rise to a filteredλ-ring structure, provided thatψp(x)≡xp (modp) for all primes p. Two such filteredλ-rings,S((cp), (dp)) andS(( ¯cp), ( ¯dp)), are isomorphic if and only if (i) (cp)= ±( ¯cp), and (ii) there exists an integerαsuch that ¯dp=dp+ 2cpαfor all primesp.
The first two statements of this theorem immediately leads to the following upper bound for the number of isomorphism classes of filteredλ-ring structures on Z[x]/(x4) that can be topologically realized by torsion-free spaces.
(1) and (2) inTheorem 1.6. In particular, at most 61 of the uncountably many isomorphism classes of filteredλ-ring structures on Z[x]/(x4) can be realized as theK-theory of a torsion-free space.
Indeed, ifXis such a space, then the filtration ofxmust be exactly 2 or 4 [4, Corollary 4L.10], and therefore the linear coefficient ofψp(x) for any prime pmust be p(if the filtration ofxis 2) orp2(if the filtration ofxis 4). So the result follows immediately from
Theorem 1.6.
It should be noted that at least 2 of the 61 isomorphism classes in the first two state-ments ofTheorem 1.6are topologically realizable, namely, by the projective 3-spacesFP3, whereFis either the complex numbers or the quaternions. TheK-theory of the former space is case (1) inTheorem 1.6, while the latter space hasS(1, 0) as itsK-theory. It is still an open question as to whether any of the other 59 isomorphism classes are topologically realizable.
What happens whenn >4, as far as the two questions stated in the beginning of this in-troduction is concerned, is only partially understood. We will discuss several conjectures and some partial results in this general setting.
Concerning the problem of topological realizations, we believe that the finiteness phe-nomenon in Corollaries1.4and1.7should not be isolated examples.
Conjecture 1.8. Letnbe any integer≥3. Then, among the uncountably many isomor-phism classes of filtered λ-ring structures over the filtered truncated polynomial ring
Z[x]/(xn), only finitely many of them can be realized as theK-theory of torsion-free spaces.
As mentioned above, the casesn=3 and 4 are known to be true. Just like the way Corollaries1.4and1.7are proved, one way to approach this conjecture is to consider λ-rings with given linear coefficients in its Adams operations. More precisely, we offer the following conjecture.
Conjecture 1.9. Letnbe any integer≥3 and let{bp∈pZ :pprime}be nonzero integers. Then there exist only finitely many isomorphism classes of filteredλ-ring structures on
Z[x]/(xn) with the property thatψp(x)≡b
px(modx2) for all primesp.
In fact,Conjecture 1.8forn≥4 would follow from the cases, (bp=p) and (bp=p2), ofConjecture 1.9. Then=3 case ofConjecture 1.9is contained inTheorem 1.3. More-over, (1.1) gives the exact number of isomorphism classes in terms of thebp. One plausi-ble way to prove this conjecture is to consider extensions ofλ-ring structures one degree at a time.
Conjecture 1.10. Letnbe any integer≥3 and letRandSbe isomorphic filteredλ-ring structures on Z[x]/(xn). Then there exists an isomorphismσ :R−→∼= S with the follow-ing property. IfR is a filteredλ-ring structure on Z[x]/(xn+1) such thatψp
R(x)≡ψ
p R(x) (modxn) for all primesp, then
Sdef= ψSp(x)=σ−1◦ψp R◦σ
(x)∈Z[x]
xn+1:pprimes
(1.4)
Hereσ(x) is considered a polynomial in both Z[x]/(xn) and Z[x]/(xn+1), andσ−1(x) is the (compositional) inverse ofσ(x) in Z[x]/(xn+1). In the definition ofψp
S(x), the symbol
◦means composition of polynomials.
Observe that for any isomorphismσ:R→S, one has that (ψSp◦ψSq)(x)=(ψSq◦ψSp)(x) for all primes p and q. Thus, to prove Conjecture 1.10, one only needs to show that ψSp(x)≡xp(modp) for all primesp. Furthermore, ifConjecture 1.10is true, thenσ in-duces an isomorphismR∼=S. Denote by ΛR the set of isomorphism classes of filtered λ-ring structuresRover Z[x]/(xn+1) such thatψp
R(x)≡ψ
p
R(x) (modxn) for all primesp, and defineΛSsimilarly withSreplacingR. SinceψSp(x)≡ψSp(x) (modxn) for all primes pand any isomorphismτ:R→S, it follows thatConjecture 1.10implies thatσinduces an embeddingΛRΛS. In particular,Conjecture 1.9would follow fromConjecture 1.10
and the following finiteness result.
Theorem 1.11. Letnbe any integer≥3, and letap,ibe integers forpprimes and 1≤i≤ n−2 withap,1 =0 for every p. Then the number of isomorphism classes of filteredλ-ring structuresRon Z[x]/(xn) satisfying
ψRp(x)≡ap,1x+···+ap,n−2xn−2modxn−1 (1.5)
for all primespis at most min{|an−1
p,1 −ap,1|:pprimes}.
Notice that ifConjecture 1.10is true forn≤Nfor someN, then, usingTheorem 1.11, one infers thatConjecture 1.9is true forn≤N+ 1, which in turn impliesConjecture 1.8
forn≤N+ 1. We summarize this in the following diagram:
(Conjecture 1.10)n≤N+Theorem 1.11
=⇒(Conjecture 1.9)n≤N+1
=⇒(Conjecture 1.8)n≤N+1.
(1.6)
In view of these implications, even partial results aboutConjecture 1.10would be of in-terest.
It was mentioned above that in order to proveConjecture 1.10, one only needs to prove the congruence identity,ψSp(x)≡xp(modp), for all primesp. In fact, this only needs to be proved forp < n, since the following result takes care of the rest.
Theorem 1.12. Let the assumptions and notations be the same as in the statement of Conjecture 1.10. Ifσ:R−→∼= Sis any filteredλ-ring isomorphism, thenψSp(x)≡xp (modp) for all primesp≥n.
Using this result, one can show thatConjecture 1.10is true for some small values ofn. More precisely, we have the following consequence ofTheorem 1.12.
Corollary 1.13. Conjecture 1.10is true forn=3, 4, and 5. Therefore, Conjectures1.8and 1.9are true forn=3, 4, 5, and 6.
to go through forn=6 (see the discussion after the proof of this corollary). A more sophisticated argument seems to be needed to proveConjecture 1.10in its full generality.
Organization. The rest of this paper is organized as follows. The following section gives a brief account of the basics ofλ-rings and the Adams operations, ending with the proof of
Theorem 1.1. Proofs of Theorems1.3and1.6are in the two sections after the following section. The results concerning the three conjectures, namely, Theorems1.11and1.12
andCorollary 1.13, are proved in the final section.
2. Basics ofλ-rings
The reader may refer to the references [3,5] for more in-depth discussion of basic prop-erties ofλ-rings. We should point out that what we call aλ-ring here is referred to as a “special”λ-ring in [3]. All rings considered in this paper are assumed to be commutative, associative, and have a multiplicative unit.
2.1.λ-rings. By aλ-ring, we mean a commutative ringRthat is equipped with functions
λi:R−→R (i≥0), (2.1)
calledλ-operations. These operations are required to satisfy the following conditions. For any integersi,j≥0 and elementsrandsinR, one has
(i)λ0(r)=1; (ii)λ1(r)=r;
(iii)λi(1)=0 fori >1; (iv)λi(r+s)=i
k=0λk(r)λi−k(s); (v)λi(rs)=P
i(λ1(r),...,λi(r);λ1(s),...,λi(s)); (vi)λi(λj(r))=P
i,j(λ1(r),...,λi j(r)).
ThePiandPi,jare certain universal polynomials with integer coefficients, and they are de-fined using the elementary symmetric polynomials as follows. Given the variablesξ1,...,ξi andη1,...,ηi, lets1,...,siandσ1,...,σi, respectively, be the elementary symmetric func-tions of theξ’s and theη’s. Then the polynomialPiis defined by requiring that the ex-pressionPi(s1,...,si;σ1,...,σi) is the coefficient oftiin the finite product
i
m,n=1
1 +ξmηnt
. (2.2)
The polynomialPi,jis defined by requiring that the expressionPi,j(s1,...,si j) is the coef-ficient oftiin the finite product
l1<···<lj
1 +ξl1···ξljt
. (2.3)
Aλ-ring map is a ring map which commutes with all theλ-operations.
By a filtered ring, we mean a commutative ringRtogether with a decreasing sequence of ideals
A filtered ring mapf :R→Sis a ring map that preserves the filtrations, that is, f(IRn)⊆ISn for alln.
A filteredλ-ring is a λ-ringR which is also a filtered ring in which each idealIn is closed underλi fori≥1. Suppose thatRandSare two filteredλ-rings. Then a filtered λ-ring map f :R→Sis aλ-ring map that also preserves the filtration ideals.
2.2. The Adams operations. There are some very useful operations inside aλ-ringR, the so-called Adams operations:
ψn:R−→R (n≥1). (2.5)
They are defined by the Newton formula:
ψn(r)−λ1(r)ψn−1(r) +···+ (−1)n−1λn−1(r)ψ1(r) + (−1)nnλn(r)=0. (2.6)
Alternatively, one can also define them using the closed formula:
ψk=Q k
λ1,...,λk. (2.7)
HereQkis the integral polynomial with the property that
Qk
σ1,...,σk
=xk1+···+xkk, (2.8)
where theσiare the elementary symmetric polynomials of thex’s. The Adams operations have the following properties.
(i) All theψnareλ-ring maps onR, and they preserve the filtration ideals ifRis a filteredλ-ring.
(ii)ψ1=Id andψmψn=ψmn=ψnψm.
(iii)ψp(r)≡rp(modpR) for each primepand elementrinR.
Aλ-ring map f is compatible with the Adams operations, in the sense that f ψn=ψnf for alln.
The following simple observation will be used many times later in this paper. Suppose thatRandSareλ-rings withSZ-torsion-free and that f :R→Sis a ring map satisfying f ψp=ψpf for all primesp. Thenf is aλ-ring map. Indeed, it is clear thatf is compatible with allψnby (ii) above. The Newton formula and the Z-torsion-freeness ofSthen imply that f is compatible with theλnas well.
As discussed in the introduction, Wilkerson’s theorem [9] says that ifRis a Z-torsion-free ring equipped with ring endomorphismsψn(n≥1) satisfying conditions (ii) and (iii) above, then there exists a uniqueλ-ring structure onRwhose Adams operations are exactly the givenψn. In particular, over the possibly truncated power series filtered ring
Z[[x1,...,xn]]/(xr11,...,xrnn) as in Theorem 1.1, a filteredλ-ring structure is specified by power seriesψp(x
i) without constant terms,pprimes and 1≤i≤n, such that
ψpψqx i
=ψqψpx i
, (2.9)
ψpxi≡xip(modp) (2.10)
2.3. Proof ofTheorem 1.1
Proof. Denote byRthe possibly truncated power series filtered ring Z[[x1,...,xn]]/(xr11, ...,xrn
n) as in the statement ofTheorem 1.1.
As was mentioned in the introduction, the caseri= ∞for alli is proved in [11]. It remains to consider the cases when at least oneriis finite.
Assume that at least oneriis finite. LetNbe the maximum of thoserjthat are finite. For each primep≥Nand each indexjfor whichrj<∞, choose an arbitrary positive in-tegerbp,j∈pZ such thatbp,j≥rjandbp,j =p. There are uncountably many such choices, sinceN <∞and there are countably infinitely many choices forbp,jfor eachp≥N. Con-sider the following power series inR:
ψpx i=
⎧ ⎨ ⎩
1 +xibp,i−1 ifp≥N,ri<∞,
1 +xip−1 otherwise.
(2.11)
Herepruns through the primes andi=1, 2,...,n. The collection of power series{ψp(x i) : 1≤i≤n}extends uniquely to a filtered ring endomorphismψpofR.
We first claim that these endomorphismsψp,pprimes, are the Adams operations of a filteredλ-ring structureSonR. SinceRis Z-torsion-free, by Wilkerson’s theorem [9], it suffices to show that
ψpψq=ψqψp (2.12)
and that
ψp(r)≡rp(modpR) (2.13)
for all primes p andqand elements r∈R. Both of these conditions are verified easily using (2.11). Equation (2.12) is true because it is true when applied to eachxiand that thexiare algebra generators ofR. Equation (2.13) is true, since it is true for eachr=xi.
Now suppose that ¯Sis another filteredλ-ring structure onRconstructed in the same way with the integers{b¯p,j}. (Here again p runs through the primes ≥N and j runs through the indices for whichrj<∞.) So in ¯S,ψp(xi) looks just like it is in (2.11), except thatbp,iis replaced by ¯bp,i. Suppose, in addition, that there is a primeq≥Nsuch that
bq,j
=b¯q,j (2.14)
as sets. We claim thatSand ¯Sare not isomorphic as filteredλ-rings.
To see this, suppose to the contrary that there exists a filteredλ-ring isomorphism
σ:S−→S.¯ (2.15)
Let jbe one of those indices for whichrjis finite. Then, modulo filtration 2d, one has
σxj
for somea1,...,an∈Z, not all of which are equal to 0. Ifri= ∞, we set ¯bq,i=q. Equating the linear coefficients on both sides of the equation
σψqx j
=ψqσx j
, (2.17)
one infers that
bq,j·
aixi=
aib¯q,ixi. (2.18)
Ifai =0 (and such anaimust exist), then
bq,j=b¯q,i (2.19)
for somei. In particular, it follows that{bq,j}is contained in{b¯q,j}. Therefore the two sets are equal by symmetry. This is a contradiction.
This finishes the proof ofTheorem 1.1.
3. Proof ofTheorem 1.3
First we need to consider when a collection of polynomials can be the Adams operations of a filteredλ-ring structure on Z[x]/(x3). We will continue to describeλ-rings in terms of their Adams operations.
Lemma 3.1. A collection of polynomials,{ψp(x)=b
px+cpx2:pprime}, in Z[x]/(x3) ex-tends to (the Adams operations of) a filteredλ-ring structure if and only if the following three statements are satisfied:
(1)bp≡0 (modp) for all primesp;
(2)c2≡1 (mod 2) andcp≡0 (modp) for all primesp >2; (3) (b2
q−bq)cp=(b2p−bp)cqfor all primespandq.
Now suppose that these conditions are satisfied. Ifb2 =0, then (b2
p−bp)≡0 (mod 2θ2(b2)) for all odd primesp. Ifb2=0, thenbp=0 for all odd primesp.
Proof. The polynomialsψp(x) in the statement above extend to a filteredλ-ring struc-ture on Z[x]/(x3) if and only if (2.9) and (2.10) are satisfied. Expandingψp(ψq(x)), one obtains
ψpψq(x)=bp
bqx+cqx2
+cp
bqx+cqx2 2
=bpbq
x+bpcq+b2qcp
x2. (3.1)
Using symmetry and equating the coefficients ofx2, it follows that (2.9) in this case is equivalent to
b2
q−bqcp=b2p−bpcq. (3.2) It is clear that (2.10) is equivalent to conditions (2) and (3) together, since xp=0 for p >2.
left-hand side. The assertion now follows, sincec2is odd. Ifb2=0, then the right-hand side of (3.2) when p=2 is equal to 0, and sob2
q=bq for all odd primesq, sincec2 =0.
Butbq =1, sobq=0.
Next we need to know when two filteredλ-ring structures over Z[x]/(x3) are isomor-phic.
Lemma 3.2. LetS= {ψp(x)=b
px+cpx2}and ¯S= {ψp(x)=b¯px+ ¯cpx2}(where p runs through the primes) be two filteredλ-ring structures over Z[x]/(x3). ThenSand ¯Sare iso-morphic filteredλ-rings if and only if the following two conditions are satisfied simultane-ously:
(1)bp=b¯pfor all primesp;
(2)(a) ifb2=b2¯ =0, then there existsu∈ {±1}such thatc
p=uc¯pfor all primesp; (b) ifb2=b2¯ =0, then there existu∈ {±1}anda∈Z such that
ab2b2−1=c2−u¯c2. (3.3)
Proof. Suppose thatSand ¯Sare isomorphic, and letσ:S→S¯be a filteredλ-ring isomor-phism. Then
σ(x)=ux+ax2 (3.4)
for someu∈ {±1}and integera. Applying the mapσψp,pis any prime, to the generator x, one obtains
σψp(x)=ub
px+abp+cpx2. (3.5)
Similarly, one has
ψpσ(x)=ub¯ px+
ab¯2
p+uc¯p
x2. (3.6)
Recall from the previous lemma thatb2=0 impliesbp=0 for all odd primes p. There-fore, the “only if ” part now follows by equating the coefficients in the equationσψp(x)= ψpσ(x).
Conversely, suppose that conditions (1) and (2)(a) in the statement of the lemma hold. Then clearly the mapσ:S→S¯given on the generator byσ(x)=uxis the desired isomor-phism.
Now suppose that conditions (1) and (2)(b) hold. The polynomialσ(x)=ux+ax2 extends uniquely to a filtered ring automorphism on Z[x]/x3. The calculation in the first paragraph of this proof shows that ifbp=0 for a certain primep, thenσψp(x)=ψpσ(x). Ifbp =0, then (3.2) in the proof ofLemma 3.1implies that
a=c2−uc2¯
b2 2−b2
=cp−uc¯p
b2 p−bp .
(3.7)
Therefore, the argument in the first paragraph once again shows thatσψp(x)=ψpσ(x),
Proof ofTheorem 1.3. It follows immediately from Lemmas3.1 and3.2that, in the no-tation of the statement ofTheorem 1.3, theS((cp)) are all filtered λ-ring structures on
Z[x]/(x3) and that two of them are isomorphic if and only if the stated conditions hold. Similar remarks apply to theS((bp),k). Also noS((cp)) is isomorphic to anS((bp),k). It remains only to show that any filteredλ-ring structureRon Z[x]/(x3) is isomorphic to one of them.
Writeψp(x)=b
px+cpx2 for the Adams operations inR. Ifb2=0, then so isbp for each odd prime p. Then (cp) satisfies condition (2) inLemma 3.1, andRis isomorphic (in fact, equal) toS((cp)).
Suppose thatb2 =0. Then byLemma 3.1the sequence (bp) satisfies condition (A). First consider the case whenbp=0 for all odd primes p. In this case, condition (3) in
Lemma 3.1implies that cp=0 for all odd primes p. Let rdenote the remainder ofc2 modulob2(b2−1). Sinceris also an odd integer, there exists a unique odd integerkin the range 1≤k≤b2(b2−1)/2 that is congruent (modb2(b2−1)) to eitherror−r. By
Lemma 3.2,Ris isomorphic toS((bp),k).
Finally, consider the case whenb2 =0 and there is at least one odd prime pfor which bp =0. If
c2=qb2b2−1+r (3.8)
for some integersqandrwith 0≤r < b2(b2−1), thenrmust be an odd integer, sinceb2 is even andc2is odd. Define
¯ c2= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
r if 1≤r≤b2
b2−1
2 ,
b2b2−1−r ifr >b2
b2−1
2 ,
¯ cq=
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
cq−qbq
bq−1
if 1≤r≤b2
b2−1
2 ,
(1 +q)bq
bq−1
−cq ifr >b2
b2−1 2
(3.9)
forq >2. The three conditions (1)–(3) inLemma 3.1are all easily verified for the poly-nomialsS= {ψp(x)=b
px+ ¯cpx2}, and soSis a filtered λ-ring structure on Z[x]/(x3). Observe that 1≤c2¯ ≤b2(b2−1)/2. Moreover, byLemma 3.2,Sis isomorphic toR. Equa-tion (3.2) implies that, ifpis a prime for whichbp =0, then
¯ c2≡0
mod b2
b2−1 gcdb2b2−1,bp
bp−1
. (3.10)
Therefore, we have
¯ c2≡0
mod lcm
b2b2−1 gcdb2b2−1,bp
bp−1
: all primesp
Now observe that
lcm
b2b2−1
gcdb2b2−1,bpbp−1: all primesp
=b2
b2−1
G , (3.12)
where
G=gcdbp
bp−1
: all primesp. (3.13)
From the last assertion ofLemma 3.1, we see that 2θ2(b2)is a factor ofG, and sob2(b2− 1)/Gis odd. If
¯ c2=lb2
b2−1
G , (3.14)
then it follows thatlmust be odd with 1≤l≤G/2 and that
¯ cp=lbp
bp−1
G (3.15)
for all primes p. Let p1,...,pnbe the list of all (if any) odd primes for which condition (B) holds for the sequence (bp). For eachp=pi, 1≤i≤n, we have
θpc¯p=θp lb
pbp−1 G
=θp(l)≥1, (3.16)
and so
l≡0 (modp). (3.17)
It follows that
l≡0modp1···pn
. (3.18)
We have shown thatRis isomorphic toS=S((bp),l), as desired.
4. Proof ofTheorem 1.6
4.1. Proof ofTheorem 1.6(1)
Proof. First we make the following observation. Consider a collection of polynomials
R=ψp(x)=bpx+cpx2+dpx3:pprimes
(4.1)
in Z[x]/(x4). ThenRextends to (the Adams operations of) a filteredλ-ring structure if and only if (i) thebpandcpsatisfy conditions (1)–(3) ofLemma 3.1, (ii)
b3
q−bq
dp=
b3 p−bp
dq+ 2cpcq
bp−bq
for all primespandq, and (iii)
dp≡ ⎧ ⎨ ⎩
0 (modp) ifp =3,
1 (mod 3) ifp=3. (4.3)
In fact, (4.2) and (4.3) are obtained from (2.9) and (2.10), respectively, by comparing the coefficients ofx3.
Now restrict to case (1) ofTheorem 1.6, that is, when bp=p for all primes p. By condition (3) inLemma 3.1, we have that inR,
ψp(x)=px+c2p(p−1)
2 x
2+d
px3 (4.4)
for each primep. Applying (4.2) to this present case, we obtain
dp=16p(p−1)
(p+ 1)d2+c22(p−2)
∈Z. (4.5)
LetSdenote the filteredλ-ring structure on Z[x]/(x4) with
ψp(x)=(1 +x)p−1 (4.6)
for all primesp. Defineσ(x) by
σ(x)=
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
−x+c2+ 1 2 x
2−
c2+ 1c2+ 2+d2
6 x
3 ifc2≡0 (mod 3),
x+c2−1 2 x
2+
c2−1c2−2+d2
6 x
3 otherwise.
(4.7)
Extendσ(x) to a filtered ring automorphism on Z[x]/(x4). Using (4.4) and (4.5), it is now straightforward to check thatσ is compatible with the Adams operationsψp inRandS (i.e.,σψRp(x)=ψSpσ(x)), and so it gives an isomorphismR∼=S.
4.2. Proof ofTheorem 1.6(2)
Proof. Using the casen=3 ofCorollary 1.13(i.e.,Conjecture 1.10forn=3), which will be proved below, one infers thatRis isomorphic to some filteredλ-ring structureT on
Z[x]/(x4) with the following property: there exists an integerk∈ {1, 3, 5}such that
ψTp(x)≡ψSp(x)modx3 (4.8)
for all primesp, whereS=S((p2),k) is as inTheorem 1.3(2). In other words,Rmust be isomorphic to a “λ-ring extension” to Z[x]/(x4) of one of the threeS((p2),k). Therefore, to proveTheorem 1.6(2), it suffices to prove the following three statements.
(1) The sixtyS(k,d2) listed in the statement ofTheorem 1.6are all filtered λ-ring structures on Z[x]/(x4).
Lemma 4.1. Letk∈ {1, 5}be an integer and letd2be an even integer. Consider the collection of polynomials
S= ψp(x)=p2x+kp
2p2−1
12 x
2+ dpx3
, (4.9)
wheredp for p >2 is determined byd2 via (1.3). ThenSis a filteredλ-ring structures on
Z[x]/(x4).
Thus, thisSlooks just like one of thoseS(k,d2), except that inSwe only required2 to be even. In particular, this Lemma implies that eachS(k,d2) is indeed a filteredλ-ring structure on Z[x]/(x4).
Proof. We only need to check (4.2) and (4.3) inS, since it is clear that conditions (1)–(3) ofLemma 3.1are satisfied bybp=p2andcp=kp2(p2−1)/12. The condition (4.2) can be verified directly by using (1.3), which expresses eachdpin terms ofd2.
Now consider (4.3). The casep=2 is true by hypothesis. Ifp=3, then it follows from (1.3) that
d3=12d2+k2. (4.10)
Sincekis either 1 or 5, we must have thatd3≡1 (mod 3). Ifp=5, then
d5=260d2+ 35k2≡0 (mod 5). (4.11)
Suppose now thatp >5. We will show that both summands ofdpin (1.3) are divisible byp. First observe that 12=22·3 must divide (p−1)(p+ 1). If 5 divides either (p−1) or (p+ 1), then 60 divides (p2−1). If 5 divides neither one of them, thenp≡2 or 3 (mod 5). In either case, we have p2+ 1≡0 (mod 5), and therefore 60 divides (p2−1)(p2+ 1)= (p4−1). It follows that the first summand ofd
p, namely,p2(p4−1)d2/60, is an integer that is divisible by p(in fact, by p2). Now we consider the second summand. Sincep is odd, (p−1)p(p+ 1) is divisible by 23·3, and sop(p2−1)(p2−4) is divisible by 23·32. It is also clear that it is divisible by 5. Therefore, 23·32·5=360 dividesp(p2−1)(p2−4). It follows that the second summand ofdp is an integer that is divisible byp. This proves
the lemma.
Lemma 4.2. LetSandSbe two filteredλ-ring structures on Z[x]/(x4) of the form described inLemma 4.1. Writekanddpfor the coefficients ofx2inψ2
S(x) and ofx3inψSp(x), respec-tively. ThenSandSare isomorphic filteredλ-rings if and only if (i)k=kand (ii)d2≡d2 (mod 60).
In particular, it follows from this lemma that the sixtyS(k,d2) are mutually noniso-morphic and that anySas inLemma 4.1is isomorphic to one of theS(k,d2).
Proof. Suppose thatSandSare isomorphic. Since their reductions modulox3are also isomorphic filteredλ-rings and sincek,k∈ {1, 5}, it follows fromTheorem 1.3thatk=
k. Letσ:S→Sbe an isomorphism, and write
so thatu∈ {±1}. Consider the equality
ψ2 S◦σ
(x)=σ◦ψ2
S(x). (4.13)
(Recall that◦means composition of polynomials.) The coefficients ofx2on both sides of this equation give rise to the equality
4α+k=uk+ 16α, (4.14)
or, equivalently,
(1−u)k=12α. (4.15)
Since 3 does not dividek, the only solution isu=1 andα=0. The coefficients ofx3 in (4.13) then give rise to the equality
4β+d2=d2+ 64β. (4.16)
This proves the “only if ” part of the lemma.
Conversely, suppose thatk=k andd2=d2+ 60β for some integerβ. Letσ be the filtered ring automorphism on Z[x]/(x4) given by
σ(x)=x+βx3. (4.17)
Sincek=kandσ(x)≡x(modx3), it is clear that
ψSp◦σ
(x)≡σ◦ψSp
(x)modx3 (4.18)
for any primep. The coefficients ofx3in (ψp
S ◦σ)(x) and (σ◦ψSp)(x) are (p2β+dp) and
(dp+βp6), respectively. They are equal since
dp−dp=
p2p4−1d2−d
2
60 =p
2p4−1β. (4.19)
This shows thatσ:S→Sis an isomorphism of filteredλ-ring. This proves the lemma.
Lemma 4.3. Tis isomorphic to one of the sixtyS(k,d2) listed in the statement ofTheorem 1.6.
Proof. The first paragraph of the proof ofLemma 4.1also applies toT, and so the coeffi -cient ofx3inψp
T(x), call itdp, satisfies (1.3) for somek∈ {1, 3, 5}and some even integer d2. Since (4.10) holds inTas well, it follows thatk2≡1 (mod 3). Therefore,kmust be ei-ther 1 or 5. In oei-ther words,Tis of the form described inLemma 4.1, and, byLemma 4.2, it is isomorphic to one of the sixtyS(k,d2), as desired. This proves the lemma.
The proof ofTheorem 1.6(2) is complete.
4.3. Proof ofTheorem 1.6(3). This case will be proved below as part ofCorollary 1.13
(then=4 case ofConjecture 1.9, which follows from then=3 case ofConjecture 1.10
4.4. Proof ofTheorem 1.6(4)
Proof. Since the reduction ofRmodulox3 is a filteredλ-ring structure on Z[x]/(x3), it follows fromTheorem 1.3(1) that
ψp(x)=ψp R(x)≡0
modx2 (4.20)
for all primesp. Thus,Rhas the formS((cp), (dp)) as in the statement ofTheorem 1.6(4). Now letS=S((cp), (dp))= {ψp(x)=cpx2+dpx3}be a collection of polynomials sat-isfying ψp(x)≡xp (modp). In order to show thatS is a filtered λ-ring structure on
Z[x]/(x4), we only need to show that (ψp◦ψq)(x)=(ψq◦ψp)(x) for all primes p and q. This is true, since both sides are equal to 0 modulox4.
To prove the last assertion, let
¯ S=Sc¯p
,d¯p
=ψp(x)=c¯px2+ ¯dpx3
(4.21)
be another filteredλ-ring structure on Z[x]/(x4). Suppose first thatSand ¯Sare isomor-phic, and letσ:S→S¯be an isomorphism. Writeσ(x)=ux+αx2+βx3 withu∈ {±1}. Then we have
ψSp◦σ(x)=cpx2+
2cpuα+dpu
x3,
σ◦ψSp¯
(x)=uc¯px2+ud¯px3.
(4.22)
Conditions (i) and (ii) in the statement ofTheorem 1.6(4) now follow by comparing the coefficients.
The converse is similar. Indeed, if (cp)=u( ¯cp) for someu∈ {±1}and ¯dp=dp+ 2cpα for some integerα, then the calculation above shows that an isomorphismσ:S→S¯is given by
σ(x)=ux+αx2, (4.23)
as desired. This provesTheorem 1.6(4).
The proof ofTheorem 1.6is complete.
5. Results about the conjectures
5.1. Proof ofTheorem 1.11
Proof. Consider a primeqsuch that an−1
q,1 −aq,1=minanp−,11−ap,1:pprimes. (5.1) LetSandT be two filteredλ-ring structures on Z[x]/(xn) satisfying (1.5) for all primes p, and letaandbbe the coefficients ofxn−1inψq
S(x) andψTq(x), respectively. We claim that if
a≡bmodan−1
thenSandTare isomorphic. Indeed, if (5.2) is true, then we can write
a+aq,1c=b+anq,1−1c (5.3)
for some integerc. Consider the filtered ring automorphismσon Z[x]/(xn) given by
σ(x)=x+cxn−1. (5.4)
Then we have that
ψSq◦σ(x)=
n−2
i=1
aq,ix+cxn−1i
+ax+cxn−1n−1
=
n−2
i=1 aq,ixi
+a+aq,1c
xn−1
=
n−2
i=1 aq,ixi
+b+anq−,11c
xn−1
=σ◦ψTq(x).
(5.5)
Now it follows from (5.5) and [10, Theorem 4.4] that
ψSp◦σ(x)=σ◦ψTp(x) (5.6)
for all primes p. (It is here that we are using the hypothesisap,1 =0 for all p.) That is, σ: S→Tis a filteredλ-ring isomorphism, as claimed. It is clear thatTheorem 1.11also
follows from this claim.
5.2. Proof ofTheorem 1.12
Proof. Writeσ(x)=n−1
k=1bkxk and σ−1(x)= n
i=1cixi. In particular, we haveb1=c1∈
{±1}. Let p be a prime with p≥n. WriteψRp(x)=n
j=1ajxj. Considering ψSp(x) as a polynomial in Z[x]/(xn+1), we have
ψSp(x)=σ−1◦ψp R◦σ
(x)=ψSp(x) +bn+1 1 an+α
xn, (5.7)
in whichαis a sum of integers, each summand having a factor ofaifor somei=1,..., n−1.
If p > n, then ψRp(x)≡xp=0 (modp), which implies that a
i≡0 (modp) for i= 1,...,n. SinceψSp(x)≡0 (modp) as well, one infers thatψSp(x)≡0=xp(modp).
If p=n, then we still haveψSp(x)≡0 (modp). But nowap≡1 (modp) andai≡0 (modp) fori < p. Since p+ 1 is an even integer, it follows thatψSp(x)≡xp (modp), as
5.3. Proof ofCorollary 1.13
Proof. Consider the casen=3 ofConjecture 1.10first. In view ofTheorem 1.12, we only need to show that
ψS2(x)≡x2(mod 2). (5.8)
Sinceψ2
S(x)≡x2(mod 2) already, it suffices to show that the coefficient ofx3inψS2(x) is even. We will use the same notations as in the proof ofTheorem 1.12. Observe that the congruence identity,ψ2
R(x)≡x
2 (mod 2), means thata1 anda3 are even and thata2 is odd. Computing modulo 2 andx4, we have
ψ2 S(x)≡
3
i=1 ci
3
j=1
ajb1x+b2x2j i
≡
3
i=1 ciai2
b1x+b2x22i.
(5.9)
It follows that the coefficient ofx3inψ2
S(x) is congruent (mod 2) toc1a2(2b1b2)≡0. This proves the casen=3 ofConjecture 1.10.
The proofs for the casesn=4 and 5 ofConjecture 1.10 are essentially identical to the previous paragraph. The only slight deviation is that, whenn=4, to check that the coefficient ofx4inψ2
S(x) is even, one needs to use the fact thatc2= −b1b2.
As discussed in the introduction, Conjectures 1.8 and 1.9 for n≤6 follow from
Conjecture 1.10forn≤5 andTheorem 1.11.
It should be pointed out that the argument above for the first three cases ofConjecture 1.10does not seem to go through forn=6. In fact, to use the same argument, one would have to check, among other things, that the coefficient ofx6 inψ2
S(x) is even. But this coefficient turns out to be congruent (mod 2) tob3. It may be true that one can always arrange to have an isomorphismσ in whichb3 is even, but this is only a speculation. Further works remain to be done to settle the three conjectures for higher values ofn.
References
[1] J. F. Adams, On the non-existence of elements of Hopf invariant one, Annals of Mathematics. Second Series 72 (1960), no. 1, 20–104.
[2] , Vector fields on spheres, Annals of Mathematics. Second Series 75 (1962), 603–632. [3] M. F. Atiyah and D. O. Tall, Group representations,λ-rings and theJ-homomorphism, Topology 8
(1969), no. 3, 253–297.
[4] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.
[5] D. Knutson,λ-Rings and the Representation Theory of the Symmetric Group, Lecture Notes in
Mathematics, vol. 308, Springer, Berlin, 1973.
[6] C. A. McGibbon, Self-maps of projective spaces, Transactions of the American Mathematical So-ciety 271 (1982), no. 1, 325–346.
[8] D. L. Rector, Loop structures on the homotopy type of S3, Symposium on Algebraic Topology
(Battelle Seattle Res. Center, Seattle, Wash, 1971), Lecture Notes in Math., vol. 249, Springer, Berlin, 1971, pp. 99–105.
[9] C. Wilkerson, Lambda-rings, binomial domains, and vector bundles over CP(∞), Communica-tions in Algebra 10 (1982), no. 3, 311–328.
[10] D. Yau, Moduli space of filteredλ-ring structures over a filtered ring, International Journal of
Math-ematics and Mathematical Sciences 2004 (2004), no. 39, 2065–2084.
[11] , On adic genus and lambda-rings, Transactions of the American Mathematical Society
357 (2005), no. 4, 1341–1348.
Donald Yau: Department of Mathematics, The Ohio State University Newark, 1179 University Drive, Newark, OH 43055, USA
