**ON**

_{λ}

_{λ}

**-RINGS AND TOPOLOGICAL REALIZATION**

DONALD YAU
*Received 19 July 2005; Accepted 12 April 2006*

It is shown that most possibly truncated power series rings admit uncountably many
filtered*λ-ring structures. The question of how many of these filteredλ-ring structures*
are topologically realizable by the*K*-theory of torsion-free spaces is also considered for
truncated polynomial rings.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

**1. Introduction**

A*λ-ring is, roughly speaking, a commutative ringR*with 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
ring*R(G) of a groupG*is a*λ-ring, whereλi*_{is induced by the map that sends a }
represen-tation to its*ith exterior power. Another example of aλ-ring is the complexK*-theory of
a topological space*X. Here,λi*_{arises from the map that sends a complex vector bundle}* _{η}*
over

*X*to the

*ith exterior power ofη. In the algebra side, the universal Witt ring W(R) of*a commutative ring

*R*is a

*λ-ring.*

The purpose of this paper is to consider the following two interrelated questions:
(i) classify the*λ-ring structures over power series and truncated polynomial rings;*
(ii) which ones and how many of these*λ-rings are realizable as (i.e., isomorphic to)*

the*K-theory of a topological space?*

The first question is purely algebraic, with no topology involved. One can think of the
second question as a*K*-theoretic analogue of the classical Steenrod question, which asks
for a classification of polynomial rings (over the field of*p*elements and has an action by
the mod*p*Steenrod algebra) that can be realized as the singular mod*p*cohomology of a
topological space.

In addition to being a*λ-ring, theK*-theory of a space is filtered, making*K(X) a *
fil-tered*λ-ring. Precisely, by a filteredλ-ring we mean a filtered ring (R,{R=I*0* _{⊃}_{I}*1

_{⊃ ···}}_{)}in which

*R*is a

*λ-ring and the filtration idealsIn*

_{are all closed under the}

*(i >0). It is, therefore, more natural for us to consider filtered*

_{λ-operations}_{λ}i*λ-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**
the*K-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 filtered*λ-ring structures over most power*
series and truncated polynomial rings.

*Theorem 1.1. Letx1,...*,x*nbe 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*]]/(x

*r*11,...,x

*nrn) admits uncountably many*

*isomorphism classes of filteredλ-ring structures.*

Here*x _{i}∞* is by definition equal to 0. In particular, this theorem covers both finitely
generated power series rings and truncated polynomial rings. The case

*ri= ∞*(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 filtered

*λ-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 the*λ-ring structure from the*
Adams operations. More precisely, Wilkerson’s theorem says that if*R***is a Z-torsion-free**
ring which comes equipped with ring endomorphisms*ψk* _{(k}_{≥}_{1) satisfying the }
condi-tions, (1)*ψ*1_{=}_{Id and}_{ψ}k_{ψ}l_{=}_{ψ}kl_{=}_{ψ}l_{ψ}k_{, and (2)}_{ψ}p_{(r)}_{≡}_{r}p_{(mod}_{pR) for each prime}*p*and*r∈R, thenR*admits a unique*λ-ring structure with theψk*_{as 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
filtered*λ-ring structure, it su*ﬃces to describe the Adams operations.

When there are more than one*λ-ring in sight, we will sometimes writeψ _{R}n*to denote
the Adams operation

*ψn*

_{in}

_{R.}**Truncated polynomial rings. Consider the filtered truncated polynomial rings (Z[**x]/(xn_{),}

*|x| =d) and (Z[x]/(xn*_{),} _{|}_{x}_{| =}_{d}_{), with}_{x}_{in filtration exactly}_{d >}_{0 and}_{d}_{>}_{0, }

respec-tively. Let Λd denote the set of isomorphism classes of filtered *λ-ring structures over*
**(Z[x]/(x***n*_{),} _{|}_{x}_{| =}_{d). Define}_{Λd}_{}_{similarly. Then it is obvious that}_{Λd}_{and}_{Λd}_{}_{are in }
one-to-one correspondence, and there is no reason to distinguish between them. Indeed, for
*R∈*Λd one can associate to it*R∈*Λd such that*ψ _{R}k*(x)

*=ψ*(x) as polynomials for all

_{R}k*k, and this construction gives the desired bijection between*ΛdandΛd. Therefore, using

the above bijections, we will identify the setsΛd for*d=*1, 2,..., and writeΛ**(Z[x]/(x***n*_{))}
for the identified set. Each isomorphism class of filtered*λ-ring structures on Z[x]/(xn*_{)}
with*|x| ∈ {*1, 2,...*}*is considered an element inΛ**(Z[x]/(x***n*_{)).}

We start with the simplest case*n=***2, that is, the dual number ring Z[x]/(x**2_{).}

Theorem 1.2 ([10*, Corollary 4.1.2]). There is a bijection between*Λ**(Z[x]/(x**2_{)) 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,
the*K-theory of the even-dimensional sphereS*2*k*_{realizes the filtered}_{λ-ring with}_{b}

*p=pk*
for all *p. On the other hand, ifX* is a torsion-free space with*K(X)=***Z[***x]/(x*2_{), then}
*ψp*_{(x)}_{=}_{p}k_{x}_{for all}_{p}_{if}_{x}_{lies in filtration exactly 2k}_{[}_{2}_{, Corollary 5.2].}

In what follows we will use the notation*θp*(n) to denote the largest integer for which
*pθp*(*n*)_{divides}_{n, where}_{p}_{is any prime. By convention we set}_{θ}

*p*(0)*= −∞*.

To study the case*n=*3, we need to consider the following conditions for a sequence
(b*p*) of integers indexed by the primes.

(A)*b2 =*0,*bp≡*0 (mod*p) for all primes* *p, andbp*(b*p−*1)*≡*0 (mod 2*θ*2(*b*2)) for all
odd primes*p.*

(B) Let (b*p*) be as in (A). Consider a prime *p >*2 (if any) for which *bp =*0 and
*θp*(b*p*)*=*min*{θp*(b*q*(b*q−*1)) :*bq =*0*}*.

We will call them conditions (A) and (B), respectively. Notice that in (B),*θp*(b*p*)*=*
*θp*(b*p*(b*p−*1)), since*p*does not divide (b*p−*1). Moreover, there are at most finitely many
such primes, since each such*p*divides*b2(b2−*1), which is nonzero.

The following result gives a complete classification forΛ**(Z[x]/(x**3_{)), the set of }
iso-morphism classes of filtered*λ-ring structures over the filtered truncated polynomial ring*

**Z[***x]/(x*3_{). As before we will describe a}_{λ-ring in terms of its Adams operations.}

*Theorem 1.3. LetRbe a filteredλ -ring structure on Z[x]/(x*3

_{). Then}_{R}_{is isomorphic to one}*of the following filteredλ-rings.*

(1)*S((cp*))*= {ψp*(x)*=cpx*2:*pprime}withc2≡1 (mod 2) andcp≡*0 (mod*p) forp >*
*2. Moreover, any such sequence (cp) gives rise to an element of*Λ**(Z[x]/(x**3*)), and two*
*such filteredλ-rings,S((cp)) andS((cp)), are isomorphic if and only if (cp*)*= ±*(c*p).*
(2)*S((bp*),k)*= {ψp*(x)*=bpx*+*cpx*2:*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(b*p*(b*p−1) : all primesp),*

(i) 1*≤k≤G/2;*

(ii)*k≡*0 (mod*p1···pn);*

(iii)*cp=kbp*(b*p−*1)/G*for all primesp.*

*Moreover, any such pair ((bp*),*k) gives rise to an element of*Λ**(Z[x]/(x**3*)). 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 (b*p*) satisfies condition (A), then it follows that there are exactly

*G*
4p1*···pn*

*=*

gcd*bpbp−*1: all*p*
4p1*···pn*

(1.1)

isomorphism classes of filtered*λ-ring structures over Z[x]/(x*3_{) with the property that}
*ψp*_{(x)}_{≡}_{b}

*px*(mod*x*2) for all primes*p. Heres*denotes the smallest integer that is greater
than or equal to*s. Applying formula (*1.1) to the cases (b*p*)*=*(p*r*), where*r∈ {*1, 2, 4*}*, we
see that there are a total of 64 elements inΛ**(Z[x]/(x**3_{)) satisfying}_{ψ}p_{(x)}_{=}_{p}r_{x}_{(mod}* _{x}*2

_{)}for all primes

*p. In fact, there is a unique such element whenr=*1 (k

*=*1), three such ele-ments when

*r=*2 (k

*∈ {*1, 3, 5

*}*), and sixty such elements when

*r=*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 filtered

*λ-ring structures over Z[x]/(x*3

_{) 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]/(x*3

_{).}*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*3

**-ring structures on Z[**x]/(x

_{) can be topologically realized by}*torsion-free spaces.*

Indeed, if*X*is a torsion-free space whose*K-theory filtered ring is Z[x]/(x*3_{), then by}
Adams’ result on Hopf invariant 1 [1], the generator*x*must have filtration exactly 2, 4, or
8. When the filtration of*x*is equal to 2r, one has*ψp*_{(x)}_{≡}_{p}r_{x}_{(mod}* _{x}*2

_{) for all primes}

*Therefore, byTheorem 1.3,*

_{p.}*K(X) must be isomorphic toS((pr*

_{),}

_{k) for some}_{r}_{∈ {}_{1, 2, 4}

*and some*

_{}}*k. The discussion preceding this corollary then implies that there are exactly 64*such isomorphism classes of filtered

*λ-rings.*

It should be remarked that at least 3 of these 64 isomorphism classes are actually
real-ized by spaces, namely, the projective 2-spaces*FP*2_{, where}_{P}_{denotes the complex }
num-bers, the quaternions, or the Cayley octonions. These spaces correspond to*r=*1, 2, and
4, respectively. Further work remains to be done to determine whether any of the other
61 filtered*λ-rings in*Corollary 1.4are topologically realizable.

Before moving on to the case*n=*4, we would like to present another topological
ap-plication ofTheorem 1.3, which involves the notion of Mislin genus. Let*X*be a nilpotent
space of finite type (i.e., its homotopy groups are all finitely generated and *πn*(X) is a
nilpotent*π1(X)-module for eachn≥2). The Mislin genus ofX, denoted by Genus(X), is*
the set of homotopy types of nilpotent finite type spaces*Y* such that the*p-localizations*
of*X*and*Y* are homotopy equivalent for all primes*p. Let H denote the quaternions. It is*
**known that Genus(HP***∞*) is an uncountable set [8]. Moreover, these uncountably many
homotopically distinct spaces have isomorphic*K-theory filtered rings [*11] but pairwise
nonisomorphic*K*-theory filtered*λ-rings [*7]. In other words, *K-theory filtered* *λ-ring*
**classifies the Mislin genus of HP***∞***. It is also known that Genus(HP**2_{) has exactly 4 }
ele-ments [6**]. We will now show that the genus of HP**2_{behaves very di}_{ﬀ}_{erently from that of}

Corollary 1.5. *K-theory filteredλ -ring does not classify the Mislin genus of HP*2

_{. In other}*words, there exist homotopically distinct spacesX*

*andY*

*2*

**in Genus(H**P

_{) 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 HP**2** _{all have Z[x]/(x}**3

_{) as their}

_{K-theory filtered ring, with}*x*in filtration exactly 4. Therefore, in each one of these

*λ-rings, we haveψp*

_{(x)}

*2*

_{≡}_{p}*(mod*

_{x}*x*2

_{) for all primes}

_{p. The corollary now follows, since by}_{Theorem 1.3}

_{, there are only}3 isomorphism classes of filtered

*λ-ring structures on Z[x]/(x*3

_{) of the form}

*2*

_{S((p}_{),}

*because*

_{k)}*k*must be 1, 3, or 5. It is still an open question as to whether

*K*-theory filtered

*λ-ring classifies the genus of HPn*

_{for 2}

_{< n <}_{∞}_{.}

We now move on to the case *n=*4, that is, the filtered truncated polynomial ring

**Z[***x]/(x*4_{). A complete classification theorem along the lines of Theorems}_{1.2} _{and} _{1.3}
has not yet been achieved for*n≥*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 the*K-theory of spaces.*

*Theorem 1.6. LetRbe an element of*Λ**(Z[x]/(x**4_{)).}

*(1) Ifψ _{R}p*(x)

*≡px*(mod

*x*2

_{) for all primes}_{p}_{, then}_{R}_{is isomorphic to the filtered}_{λ}_{-ring}*structure withψp*

_{(x)}

_{=}_{(1 +}

_{x)}p_{−}_{1 for all primes}_{p}_{.}*(2) Ifψ _{R}p*(x)

*≡p*2

_{x}_{(mod}

*2*

_{x}

_{) for all primes}_{p}_{, then}_{R}_{is isomorphic to one of the following}*60 mutually nonisomorphic filteredλ*4

**-ring structures on Z[**x]/(x

_{):}*Sk,d2=*

*ψp*(x)*=p*2_{x}_{+}*kp*2

*p*2_{−}_{1}

12 *x*

2_{+}_{d}*px*3

, (1.2)

*wherek∈ {*1, 5*},d2∈ {*0, 2, 4,*..., 58}, and*

*dp=* *p*

2* _{p}*4

_{−}_{1}

_{d2}60 +

*k*2* _{p}*2

*2*

_{p}

_{−}_{1}

*2*

_{p}

_{−}_{4}

360 (1.3)

*for odd primesp.*

*(3) In general, ifψ _{R}p*(x)

*≡bpx*(mod

*x*2

*) withbp =0 for all primesp, then there are only*

*finitely many isomorphism classes of filteredλ-ring structuresS*4

**over Z[**x]/(x

_{) such}*thatψSp*(x)

*≡bpx*(mod

*x*2

*) for all primesp.*

*(4) If (in the notation of the previous statement)b2=0, thenRis of the formS((cp*),
(d*p*))*= {ψp*(x)*=cpx*2+*dpx*3*}. Any such collection of polynomials gives rise to a*
*filteredλ-ring structure, provided thatψp*_{(x)}_{≡}_{x}p_{(mod}_{p) for all primes}_{p}_{. Two}*such filteredλ-rings,S((cp*), (d*p)) andS(( ¯cp*), ( ¯*dp)), are isomorphic if and only if*
(i) (c*p*)*= ±*( ¯*cp), and (ii) there exists an integerαsuch that ¯dp=dp*+ 2c*pαfor all*
*primesp.*

The first two statements of this theorem immediately leads to the following upper
bound for the number of isomorphism classes of filtered*λ-ring structures on Z[x]/(x*4_{)}
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]/(x*4

_{) can be realized as the}_{K}_{-theory of a }*torsion-free space.*

Indeed, if*X*is such a space, then the filtration of*x*must be exactly 2 or 4 [4, Corollary
4L.10], and therefore the linear coeﬃcient of*ψp*_{(x) for any prime} _{p}_{must be} _{p}_{(if the}
filtration of*x*is 2) or*p*2_{(if the filtration of}_{x}_{is 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-spaces*FP*3_{,}
where*F*is either the complex numbers or the quaternions. The*K*-theory of the former
space is case (1) inTheorem 1.6, while the latter space has*S(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 when*n >*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. Letn*be 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 the}* _{K-theory of torsion-free}*
spaces.

As mentioned above, the cases*n=*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. Letn*be any integer*≥*3 and let*{bp∈p***Z :***p*prime*}*be nonzero integers.
Then there exist only finitely many isomorphism classes of filtered*λ-ring structures on*

**Z[***x]/(xn*_{) with the property that}_{ψ}p_{(x)}_{≡}_{b}

*px*(mod*x*2) for all primes*p.*

In fact,Conjecture 1.8for*n≥*4 would follow from the cases, (b*p=p) and (bp=p*2),
ofConjecture 1.9. The*n=*3 case ofConjecture 1.9is contained inTheorem 1.3.
More-over, (1.1) gives the exact number of isomorphism classes in terms of the*bp*. One
plausi-ble way to prove this conjecture is to consider extensions of*λ-ring structures one degree*
at a time.

*Conjecture 1.10. Letn*be any integer*≥*3 and let*R*and*S*be isomorphic filtered*λ-ring*
**structures on Z[x]/(x***n*_{). Then there exists an isomorphism}_{σ}_{:}_{R}_{−→}∼=_{S}_{with the }
follow-ing property. If*R* is a filtered*λ-ring structure on Z[x]/(xn*+1_{) such that}_{ψ}p

*R*(x)*≡ψ*

*p*
*R*(x)
(mod*xn*_{) for all primes}_{p, then}

*S*def*=* *ψ _{S}*

_{}

*p*(x)

*=σ−*1

_{◦}_{ψ}p*R◦σ*

(x)*∈***Z[***x]*

*xn*+1:*p*primes

(1.4)

Here*σ(x) is considered a polynomial in both Z[x]/(xn*_{) and Z[x]/(x}*n*+1_{), and}* _{σ}−*1

_{(x) is}the (compositional) inverse of

*σ(x) in Z[x]/(xn*+1

_{). In the definition of}

_{ψ}p

*S*(x), the symbol

*◦*means composition of polynomials.

Observe that for any isomorphism*σ*:*R→S, one has that (ψ _{S}*

_{}

*p◦ψ*

_{S}_{}

*q*)(x)

*=*(ψ

_{S}q_{}

*◦ψ*

_{S}p_{})(x) for all primes

*p*and

*q. Thus, to prove*Conjecture 1.10, one only needs to show that

*ψ*

_{S}_{}

*p*(x)

*≡xp*

_{(mod}

_{p) for all primes}_{p. Furthermore, if}_{Conjecture 1.10}

_{is true, then}

_{σ}_{}in-duces an isomorphism

*R∼=S. Denote by*ΛR the set of isomorphism classes of filtered

*λ-ring structuresR*

**over Z[x]/(x**

*n*+1

_{) such that}

_{ψ}p
*R*(x)*≡ψ*

*p*

*R*(x) (modx*n*) for all primes*p,*
and defineΛSsimilarly with*S*replacing*R. Sinceψ _{S}*

_{}

*p*(x)

*≡ψSp*(x) (mod

*xn*) for all primes

*p*and any isomorphism

*τ*:

*R→S, it follows that*Conjecture 1.10implies that

*σ*induces 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*
*structuresR on Z[x]/(xn_{) satisfying}*

*ψ _{R}p*(x)

*≡ap*,1x+

*···*+

*ap*,

*n−*2x

*n−*2mod

*xn−*1 (1.5)

*for all primespis at most min{|an−*1

*p*,1 *−ap*,1*|*:*pprimes}.*

Notice that ifConjecture 1.10is true for*n≤N*for some*N, then, using*Theorem 1.11,
one infers thatConjecture 1.9is true for*n≤N*+ 1, which in turn impliesConjecture 1.8

for*n≤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,*ψ _{S}*

_{}

*p*(x)

*≡xp*

_{(mod}

*be proved for*

_{p), for all primes}_{p. In fact, this only needs to}*p < 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, thenψ _{S}*

_{}

*p*(x)

*≡xp*

_{(mod}

_{p)}*for all primesp≥n.*

Using this result, one can show thatConjecture 1.10is true for some small values of*n.*
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 for*n=*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 of*λ-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 of***λ***-rings**

The reader may refer to the references [3,5] for more in-depth discussion of basic
prop-erties of*λ-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 a***λ-ring, we mean a commutative ringR*that is equipped with functions

*λi*:*R−→R* (i*≥*0), (2.1)

called*λ-operations. These operations are required to satisfy the following conditions. For*
any integers*i,j≥*0 and elements*r*and*s*in*R, one has*

(i)*λ*0_{(r)}_{=}_{1;}
(ii)*λ*1_{(r)}_{=}_{r;}

(iii)*λi*_{(1)}_{=}_{0 for}_{i >}_{1;}
(iv)*λi*_{(r}_{+}_{s)}_{=}i

*k=*0*λk*(r)λ*i−k*(s);
(v)*λi*_{(rs)}_{=}_{P}

*i*(λ1(r),*...,λi*(r);*λ*1(s),*...*,λ*i*(s));
(vi)*λi*_{(λ}*j*_{(r))}_{=}_{P}

*i*,*j*(λ1(r),*...*,λ*i j*(r)).

The*Pi*and*Pi*,*j*are certain universal polynomials with integer coeﬃcients, and they are
de-fined using the elementary symmetric polynomials as follows. Given the variables*ξ1,...,ξi*
and*η1,...,ηi*, let*s1,...,si*and*σ1,...,σi*, respectively, be the elementary symmetric
func-tions of the*ξ’s and theη’s. Then the polynomialPi*is defined by requiring that the
ex-pression*Pi*(s1,...,s*i*;σ1,...,σ*i*) is the coeﬃcient of*ti*in the finite product

*i*

*m*,*n=*1

1 +*ξmηnt*

*.* (2.2)

The polynomial*Pi*,*j*is defined by requiring that the expression*Pi*,*j*(s1,...,*si j*) is the
coef-ficient of*ti*_{in the finite product}

*l*1*<···<lj*

1 +*ξl*1*···ξljt*

*.* (2.3)

A*λ-ring map is a ring map which commutes with all theλ-operations.*

*By a filtered ring, we mean a commutative ringR*together with a decreasing sequence
of ideals

*A filtered ring mapf* :*R→S*is a ring map that preserves the filtrations, that is, *f*(I* _{R}n*)

*⊆I*for all

_{S}n*n.*

*A filteredλ-ring is a* *λ-ringR* which is also a filtered ring in which each ideal*In* _{is}
closed under*λi* _{for}_{i}_{≥}_{1. Suppose that}_{R}_{and}_{S}_{are two filtered}_{λ-rings. Then a filtered}*λ-ring map* *f* :*R→S*is a*λ-ring map that also preserves the filtration ideals.*

**2.2. The Adams operations. There are some very useful operations inside a***λ-ringR, the*
so-called Adams operations:

*ψn*_{:}_{R}_{−→}_{R}_{(n}_{≥}_{1).} _{(2.5)}

They are defined by the Newton formula:

*ψn*(r)*−λ*1(r)ψ*n−*1(r) +*···*+ (*−*1)*n−*1*λn−*1(r)ψ1(r) + (*−*1)*nnλn*(r)*=*0. (2.6)

Alternatively, one can also define them using the closed formula:

*ψk _{=}_{Q}*

*k*

*λ*1_{,}_{...,λ}k_{.}_{(2.7)}

Here*Qk*is the integral polynomial with the property that

*Qk*

*σ1,...,σk*

*=xk*1+*···*+*xkk*, (2.8)

where the*σi*are the elementary symmetric polynomials of the*x’s. The Adams operations*
have the following properties.

(i) All the*ψn*_{are}_{λ-ring maps on}_{R, and they preserve the filtration ideals if}_{R}_{is a}
filtered*λ-ring.*

(ii)*ψ*1_{=}_{Id and}_{ψ}m_{ψ}n_{=}_{ψ}mn_{=}_{ψ}n_{ψ}m_{.}

(iii)*ψp*_{(r)}_{≡}_{r}p_{(mod}_{pR) for each prime}_{p}_{and element}_{r}_{in}_{R.}

A*λ-ring map* *f* is compatible with the Adams operations, in the sense that *f ψn _{=}_{ψ}n_{f}*
for all

*n.*

The following simple observation will be used many times later in this paper. Suppose
that*R*and*S*are*λ-rings withS***Z-torsion-free and that** *f* :*R→S*is a ring map satisfying
*f ψp _{=}_{ψ}p_{f}*

_{for all primes}

_{p. Then}_{f}_{is a}

_{λ-ring map. Indeed, it is clear that}_{f}_{is compatible}with all

*ψn*

_{by (ii) above. The Newton formula and the Z-torsion-freeness of}

_{S}_{then imply}that

*f*is compatible with the

*λn*

_{as well.}

As discussed in the introduction, Wilkerson’s theorem [9] says that if*R***is a **
Z-torsion-free ring equipped with ring endomorphisms*ψn*_{(n}_{≥}_{1) satisfying conditions (ii) and}
(iii) above, then there exists a unique*λ-ring structure onR*whose Adams operations are
exactly the given*ψn*_{. In particular, over the possibly truncated power series filtered ring}

**Z[[***x1,...,xn*]]/(x*r*11,...,x*rnn*) as in Theorem 1.1, a filtered*λ-ring structure is specified by*
power series*ψp*_{(x}

*i*) without constant terms,*p*primes and 1*≤i≤n, such that*

*ψp _{ψ}q_{x}*

*i*

*=ψq _{ψ}p_{x}*

*i*

, (2.9)

*ψpxi≡xip*(mod*p)* (2.10)

**2.3. Proof ofTheorem 1.1**

*Proof. Denote byR***the possibly truncated power series filtered ring Z[[x1,...,x***n*]]/(x*r*11,
*...,xrn*

*n*) as in the statement ofTheorem 1.1.

As was mentioned in the introduction, the case*ri= ∞*for all*i* is proved in [11]. It
remains to consider the cases when at least one*ri*is finite.

Assume that at least one*ri*is finite. Let*N*be the maximum of those*rj*that are finite.
For each prime*p≥N*and each index*j*for which*rj<∞*, choose an arbitrary positive
in-teger*bp*,*j∈p***Z such that***bp*,*j≥rj*and*bp*,*j =p. There are uncountably many such choices,*
since*N <∞*and there are countably infinitely many choices for*bp*,*j*for each*p≥N. *
Con-sider the following power series in*R:*

*ψp _{x}*

*i=*

⎧ ⎨ ⎩

1 +*xibp,i−*1 if*p≥N,ri<∞*,

1 +*xip−*1 otherwise.

(2.11)

Here*p*runs through the primes and*i=*1, 2,...,n. The collection of power series*{ψp*_{(x}
*i*) :
1*≤i≤n}*extends uniquely to a filtered ring endomorphism*ψp*_{of}_{R.}

We first claim that these endomorphisms*ψp*_{,}_{p}_{primes, are the Adams operations of a}
filtered*λ-ring structureS*on*R. SinceR***is Z-torsion-free, by Wilkerson’s theorem [**9], it
suﬃces to show that

*ψpψq=ψqψp* (2.12)

and that

*ψp*_{(r)}_{≡}_{r}p_{(mod}_{pR)}_{(2.13)}

for all primes *p* and*q*and 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 each*xi*and that
the*xi*are algebra generators of*R. Equation (*2.13) is true, since it is true for each*r=xi*.

Now suppose that ¯*S*is another filtered*λ-ring structure onR*constructed 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 which

*rj<∞*.) So in ¯

*S,ψp*(x

*i*) looks just like it is in (2.11), except that

*bp*,

*i*is replaced by ¯

*bp*,

*i*. Suppose, in addition, that there is a prime

*q≥N*such that

*bq*,*j*

* = _{b}*¯

_{q}_{,}

_{j}_{(2.14)}

as sets. We claim that*S*and ¯*S*are not isomorphic as filtered*λ-rings.*

To see this, suppose to the contrary that there exists a filtered*λ-ring isomorphism*

*σ*:*S−→ _{S.}*¯

_{(2.15)}

Let *j*be one of those indices for which*rj*is finite. Then, modulo filtration 2d, one has

*σxj*

for some*a1,...,an∈***Z, not all of which are equal to 0. If***ri= ∞*, we set ¯*bq*,*i=q. Equating*
the linear coeﬃcients on both sides of the equation

*σψq _{x}*

*j*

*=ψq _{σ}_{x}*

*j*

, (2.17)

one infers that

*bq*,*j·*

*aixi=*

*aib*¯*q*,*ixi.* (2.18)

If*ai =*0 (and such an*ai*must exist), then

*bq*,*j=b*¯*q*,*i* (2.19)

for some*i. 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 filtered*λ-ring structure on Z[x]/(x*3_{). We will continue to describe}* _{λ-rings in terms}*
of their Adams operations.

*Lemma 3.1. A collection of polynomials,{ψp*_{(x)}_{=}_{b}

*px*+*cpx*2:*pprime} , in Z[x]/(x*3

*)*

*ex-tends to (the Adams operations of) a filteredλ-ring structure if and only if the following three*

*statements are satisfied:*

(1)*bp≡*0 (mod*p) for all primesp;*

(2)*c2≡1 (mod 2) andcp≡*0 (mod*p) for all primesp >2;*
(3) (b2

*q−bq*)c*p=*(b2*p−bp*)c*qfor all primespandq.*

*Now suppose that these conditions are satisfied. Ifb2 =0, then (b*2

*p−bp*)*≡*0 (mod 2*θ*2(*b*2))
*for all odd primesp. Ifb2=0, thenbp=0 for all odd primesp.*

*Proof. The polynomialsψp*_{(x) in the statement above extend to a filtered}_{λ-ring }**struc-ture on Z[x]/(x**3_{) if and only if (}_{2.9}_{) and (}_{2.10}_{) are satisfied. Expanding}_{ψ}p_{(ψ}*q*_{(x)), one}
obtains

*ψpψq*(x)*=bp*

*bqx*+*cqx*2

+*cp*

*bqx*+*cqx*2
2

*=bpbq*

*x*+*bpcq*+*b*2*qcp*

*x*2* _{.}* (3.1)

Using symmetry and equating the coeﬃcients of*x*2_{, it follows that (}_{2.9}_{) in this case is}
equivalent to

*b*2

*q−bqcp=b*2*p−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, since*c2*is odd. If*b2=*0, then the right-hand
side of (3.2) when *p=*2 is equal to 0, and so*b*2

*q=bq* for all odd primes*q, sincec2 =*0.

But*bq =*1, so*bq=*0.

Next we need to know when two filtered*λ-ring structures over Z[x]/(x*3_{) are }
isomor-phic.

*Lemma 3.2. LetS= {ψp*_{(x)}_{=}_{b}

*px*+*cpx*2*}and ¯S= {ψp*(x)*=b*¯*px*+ ¯*cpx*2*}(where* *p* *runs*
*through the primes) be two filteredλ -ring structures over Z[x]/(x*3

_{). Then}_{S}_{and ¯}_{S}_{are }*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 exists}_{u}_{∈ {±}_{1}

_{}}_{such that}_{c}*p=uc*¯*pfor all primesp;*
*(b) ifb2= _{b2}*¯

_{ =}_{0, then there exist}_{u}_{∈ {±}_{1}

_{}}_{and}_{a}_{∈}_{Z such that}*ab2b2−*1*=c2−u¯c2.* (3.3)

*Proof. Suppose thatS*and ¯*S*are isomorphic, and let*σ*:*S→ _{S}*¯

_{be a filtered}

*isomor-phism. Then*

_{λ-ring }*σ*(x)*=ux*+*ax*2 _{(3.4)}

for some*u∈ {±*1*}*and integer*a. Applying the mapσψp*_{,}_{p}_{is any prime, to the generator}
*x, one obtains*

*σψp*_{(x)}_{=}_{ub}

*px*+*abp*+*cpx*2*.* (3.5)

Similarly, one has

*ψp _{σ}*

_{(x)}

*¯*

_{=}_{u}_{b}*px*+

*ab*¯2

*p*+*uc*¯*p*

*x*2_{.}_{(3.6)}

Recall from the previous lemma that*b2=*0 implies*bp=*0 for all odd primes *p. *
There-fore, the “only if ” part now follows by equating the coeﬃcients in the equation*σψp*_{(x)}_{=}*ψp _{σ}*

_{(x).}

Conversely, suppose that conditions (1) and (2)(a) in the statement of the lemma hold.
Then clearly the map*σ*:*S→ _{S}*¯

_{given on the generator by}

_{σ(x)}_{=}_{ux}_{is the desired }isomor-phism.

Now suppose that conditions (1) and (2)(b) hold. The polynomial*σ*(x)*=ux*+*ax*2
**extends uniquely to a filtered ring automorphism on Z[x]/x**3_{. The calculation in the first}
paragraph of this proof shows that if*bp=*0 for a certain prime*p, thenσψp*(x)*=ψpσ(x).*
If*bp =*0, then (3.2) in the proof ofLemma 3.1implies that

*a=c2−uc2*¯

*b*2
2*−b2*

*=cp−uc*¯*p*

*b*2
*p−bp* *.*

(3.7)

Therefore, the argument in the first paragraph once again shows that*σψp*_{(x)}_{=}_{ψ}p_{σ(x),}

*Proof ofTheorem 1.3. It follows immediately from Lemmas*3.1 and3.2that, in the
no-tation of the statement ofTheorem 1.3, the*S((cp*)) are all filtered *λ-ring structures on*

**Z[***x]/(x*3_{) and that two of them are isomorphic if and only if the stated conditions hold.}
Similar remarks apply to the*S((bp*),*k). Also noS((cp*)) is isomorphic to an*S((bp*),*k). It*
remains only to show that any filtered*λ-ring structureR***on Z[x]/(x**3_{) is isomorphic to}
one of them.

Write*ψp*_{(x)}_{=}_{b}

*px*+*cpx*2 for the Adams operations in*R. Ifb2=*0, then so is*bp* for
each odd prime *p. Then (cp*) satisfies condition (2) inLemma 3.1, and*R*is isomorphic
(in fact, equal) to*S((cp*)).

Suppose that*b2 =*0. Then byLemma 3.1the sequence (b*p*) satisfies condition (A).
First consider the case when*bp=*0 for all odd primes *p. In this case, condition (3) in*

Lemma 3.1implies that *cp=*0 for all odd primes *p. Let* *r*denote the remainder of*c2*
modulo*b2(b2−*1). Since*r*is also an odd integer, there exists a unique odd integer*k*in
the range 1*≤k≤b2(b2−*1)/2 that is congruent (mod*b2(b2−*1)) to either*r*or*−r. By*

Lemma 3.2,*R*is isomorphic to*S((bp*),*k).*

Finally, consider the case when*b2 =*0 and there is at least one odd prime *p*for which
*bp =*0. If

*c2=qb2b2−*1+*r* (3.8)

for some integers*q*and*r*with 0*≤r < b2(b2−*1), then*r*must be an odd integer, since*b2*
is even and*c2*is odd. Define

¯
*c2=*
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩

*r* if 1*≤r≤b2*

*b2−*1

2 ,

*b2b2−*1*−r* if*r >b2*

*b2−*1

2 ,

¯
*cq=*

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

*cq−qbq*

*bq−*1

if 1*≤r≤b2*

*b2−*1

2 ,

(1 +*q)bq*

*bq−*1

*−cq* if*r >b2*

*b2−*1
2

(3.9)

for*q >*2. The three conditions (1)–(3) inLemma 3.1are all easily verified for the
poly-nomials*S= {ψp*_{(x)}_{=}_{b}

*px*+ ¯*cpx*2*}*, and so*S*is a filtered *λ-ring structure on Z[x]/(x*3).
Observe that 1*≤c2*¯ *≤b2(b2−*1)/2. Moreover, byLemma 3.2,*S*is isomorphic to*R. *
Equa-tion (3.2) implies that, if*p*is a prime for which*bp =*0, then

¯
*c2≡*0

mod *b2*

*b2−*1
gcd*b2b2−*1,b*p*

*bp−*1

*.* (3.10)

Therefore, we have

¯
*c2≡*0

mod lcm

*b2b2−*1
gcd*b2b2−*1,b*p*

*bp−*1

: all primes*p*

Now observe that

lcm

*b2b2−*1

gcd*b2b2−*1,b*pbp−*1: all primes*p*

*=b2*

*b2−*1

*G* , (3.12)

where

*G=*gcd*bp*

*bp−*1

: all primes*p.* (3.13)

From the last assertion ofLemma 3.1, we see that 2*θ*2(*b*2)_{is a factor of}* _{G, and so}_{b2(b2}_{−}*
1)/Gis odd. If

¯
*c2=lb2*

*b2−*1

*G* , (3.14)

then it follows that*l*must be odd with 1*≤l≤G/2 and that*

¯
*cp=lbp*

*bp−*1

*G* (3.15)

for all primes *p. Let* *p1,...,pn*be the list of all (if any) odd primes for which condition
(B) holds for the sequence (b*p*). For each*p=pi*, 1*≤i≤n, we have*

*θpc*¯*p=θp*
_{lb}

*pbp−*1
*G*

*=θp*(l)*≥*1, (3.16)

and so

*l≡*0 (mod*p).* (3.17)

It follows that

*l≡*0mod*p1···pn*

*.* (3.18)

We have shown that*R*is isomorphic to*S=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*+*cpx*2+*dpx*3:*p*primes

(4.1)

**in Z[x]/(x**4_{). Then}_{R}_{extends to (the Adams operations of) a filtered}* _{λ-ring structure if}*
and only if (i) the

*bp*and

*cp*satisfy conditions (1)–(3) ofLemma 3.1, (ii)

*b*3

*q−bq*

*dp=*

*b*3
*p−bp*

*dq*+ 2c*pcq*

*bp−bq*

for all primes*p*and*q, and (iii)*

*dp≡*
⎧
⎨
⎩

0 (mod*p)* if*p =*3,

1 (mod 3) if*p=*3. (4.3)

In fact, (4.2) and (4.3) are obtained from (2.9) and (2.10), respectively, by comparing the
coeﬃcients of*x*3_{.}

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 in*R,*

*ψp*_{(x)}_{=}_{px}_{+}*c2p(p−*1)

2 *x*

2_{+}_{d}

*px*3 (4.4)

for each prime*p. Applying (*4.2) to this present case, we obtain

*dp=*1_{6}*p(p−*1)

(p+ 1)d2+*c*22(p*−*2)

*∈***Z***.* (4.5)

Let*S*denote the filtered*λ-ring structure on Z[x]/(x*4_{) with}

*ψp*_{(x)}_{=}_{(1 +}_{x)}p_{−}_{1} _{(4.6)}

for all primes*p. Defineσ(x) by*

*σ*(x)*=*

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

*−x*+*c2*+ 1
2 *x*

2_{−}

*c2*+ 1*c2*+ 2+*d2*

6 *x*

3 _{if}_{c2}_{≡}_{0 (mod 3),}

*x*+*c2−*1
2 *x*

2_{+}

*c2−*1*c2−*2+*d2*

6 *x*

3 _{otherwise.}

(4.7)

Extend*σ*(x) to a filtered ring automorphism on Z[x]/(x4_{). Using (}_{4.4}_{) and (}_{4.5}_{), it is now}
straightforward to check that*σ* is compatible with the Adams operations*ψp* _{in}_{R}_{and}* _{S}*
(i.e.,

*σψ*(x)

_{R}p*=ψ*(x)), and so it gives an isomorphism

_{S}pσ*R∼=S.*

**4.2. Proof ofTheorem 1.6(2)**

*Proof. Using the casen=*3 ofCorollary 1.13(i.e.,Conjecture 1.10for*n=*3), which will
be proved below, one infers that*R*is isomorphic to some filtered*λ-ring structureT* on

**Z[***x]/(x*4_{) with the following property: there exists an integer}_{k}_{∈ {}_{1, 3, 5}_{}}_{such that}

*ψ _{T}p*(x)

*≡ψ*(x)modx3

_{S}p_{(4.8)}

for all primes*p, whereS=S((p*2_{),}_{k) is as in}_{Theorem 1.3}_{(2). In other words,}_{R}_{must be}
isomorphic to a “λ-ring extension” to Z[x]/(x4_{) of one of the three}* _{S((p}*2

_{),}

*to proveTheorem 1.6(2), it suﬃces to prove the following three statements.*

_{k). Therefore,}(1) The sixty*S(k,d2) listed in the statement of*Theorem 1.6are all filtered *λ-ring*
**structures on Z[x]/(x**4_{).}

*Lemma 4.1. Letk∈ {*1, 5*}be an integer and letd2be an even integer. Consider the collection*
*of polynomials*

*S=* *ψp*(x)*=p*2*x*+*kp*

2* _{p}*2

_{−}_{1}

12 *x*

2_{+}
*dpx*3

, (4.9)

*wheredp* *for* *p >2 is determined byd2* *via (1.3). ThenSis a filteredλ-ring structures on*

**Z[***x]/(x*4_{).}

Thus, this*S*looks just like one of those*S(k,d2), except that inS*we only require*d2*
to be even. In particular, this Lemma implies that each*S(k,d2) is indeed a filteredλ-ring*
**structure on Z[x]/(x**4_{).}

*Proof. We only need to check (*4.2) and (4.3) in*S, since it is clear that conditions (1)–(3)*
ofLemma 3.1are satisfied by*bp=p*2and*cp=kp*2(p2*−*1)/12. The condition (4.2) can
be verified directly by using (1.3), which expresses each*dp*in terms of*d2.*

Now consider (4.3). The case*p=*2 is true by hypothesis. If*p=*3, then it follows from
(1.3) that

*d3=*12d2+*k*2_{.}_{(4.10)}

Since*k*is either 1 or 5, we must have that*d3≡*1 (mod 3). If*p=*5, then

*d5=*260d2+ 35k2*≡*0 (mod 5). (4.11)

Suppose now that*p >*5. We will show that both summands of*dp*in (1.3) are divisible
by*p. 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, then}_{p}_{≡}_{2 or 3 (mod 5).}
In either case, we have *p*2_{+ 1}_{≡}_{0 (mod 5), and therefore 60 divides (}* _{p}*2

_{−}_{1)(p}2

_{+ 1)}

*(p4*

_{=}

_{−}_{1). It follows that the first summand of}

_{d}*p*, namely,*p*2(p4*−*1)d2/60, is an integer
that is divisible by *p*(in fact, by *p*2_{). Now we consider the second summand. Since}_{p}_{is}
odd, (*p−*1)p(p+ 1) is divisible by 23_{·}_{3, and so}* _{p(p}*2

_{−}_{1)(p}2

_{−}_{4) is divisible by 2}3

_{·}_{3}2

_{. It}is also clear that it is divisible by 5. Therefore, 23

_{·}_{3}2

_{·}_{5}

_{=}_{360 divides}

*2*

_{p(p}

_{−}_{1)(p}2

_{−}_{4).}It follows that the second summand of

*dp*is an integer that is divisible by

*p. This proves*

the lemma.

*Lemma 4.2. LetSandSbe two filteredλ -ring structures on Z[x]/(x*4

_{) of the form described}*inLemma 4.1. Writekandd*2

_{p}for the coeﬃcients ofx*2*

_{in}_{ψ}*S*(x) and of*x*3*inψ _{S}p*(x),

*respec-tively. ThenSandSare isomorphic filteredλ-rings if and only if (i)k=kand (ii)d2≡d*2

*(mod 60).*

In particular, it follows from this lemma that the sixty*S(k,d2) are mutually *
noniso-morphic and that any*S*as inLemma 4.1is isomorphic to one of the*S(k,d2).*

*Proof. Suppose thatS*and*S*are isomorphic. Since their reductions modulo*x*3_{are also}
isomorphic filtered*λ-rings and sincek,k∈ {*1, 5*}*, it follows fromTheorem 1.3that*k=*

*k*. Let*σ*:*S→S*be an isomorphism, and write

so that*u∈ {±*1*}*. Consider the equality

*ψ*2
*S◦σ*

(x)*=σ◦ψ*2

*S*(x). (4.13)

(Recall that*◦*means composition of polynomials.) The coeﬃcients of*x*2_{on 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 divide*k, the only solution isu=*1 and*α=*0. The coeﬃcients of*x*3 _{in}
(4.13) then give rise to the equality

4β+*d2=d*2+ 64β. (4.16)

This proves the “only if ” part of the lemma.

Conversely, suppose that*k=k* and*d2=d*2+ 60β for some integer*β. Letσ* be the
**filtered ring automorphism on Z[x]/(x**4_{) given by}

*σ*(x)*=x*+*βx*3_{.}_{(4.17)}

Since*k=k*and*σ*(x)*≡x*(mod*x*3_{), it is clear that}

*ψSp◦σ*

(x)*≡σ◦ψSp*

(x)modx3 (4.18)

for any prime*p. The coe*ﬃcients of*x*3_{in (ψ}*p*

*S* *◦σ)(x) and (σ◦ψSp*)(x) are (p2*β*+*d _{p}*) and

(d* _{p}*+

*βp*6

_{), respectively. They are equal since}

*dp−dp=*

*p*2* _{p}*4

_{−}_{1}

_{d2}_{−}_{d}2

60 *=p*

2* _{p}*4

_{−}_{1}

_{β.}_{(4.19)}

This shows that*σ*:*S→S*is an isomorphism of filtered*λ-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 of*Lemma 4.1also applies to*T*, and so the coeﬃ
-cient of*x*3_{in}_{ψ}p

*T*(x), call it*dp*, satisfies (1.3) for some*k∈ {*1, 3, 5*}*and some even integer
*d2. Since (*4.10) holds in*T*as well, it follows that*k*2_{≡}_{1 (mod 3). Therefore,}_{k}_{must be }
ei-ther 1 or 5. In oei-ther words,*T*is of the form described inLemma 4.1, and, byLemma 4.2,
it is isomorphic to one of the sixty*S(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 of**Corollary 1.13

(the*n=*4 case ofConjecture 1.9, which follows from the*n=*3 case ofConjecture 1.10

**4.4. Proof ofTheorem 1.6(4)**

*Proof. Since the reduction ofR*modulo*x*3 _{is a filtered}* _{λ-ring structure on Z[x]/(x}*3

_{), it}follows fromTheorem 1.3(1) that

*ψp*_{(x)}_{=}_{ψ}p*R*(x)*≡*0

mod*x*2 _{(4.20)}

for all primes*p. Thus,R*has the form*S((cp*), (d*p*)) as in the statement ofTheorem 1.6(4).
Now let*S=S((cp*), (d*p*))*= {ψp*(x)*=cpx*2+*dpx*3*}*be a collection of polynomials
sat-isfying *ψp*_{(x)}_{≡}_{x}p_{(mod}_{p). In order to show that}_{S}_{is a filtered} _{λ-ring structure on}

**Z[***x]/(x*4_{), we only need to show that (ψ}*p _{◦}_{ψ}q*

_{)(x)}

_{=}_{(ψ}

*q*

_{◦}_{ψ}p_{)(x) for all primes}

_{p}_{and}

*q. This is true, since both sides are equal to 0 modulox*4

_{.}

To prove the last assertion, let

¯
*S=Sc*¯*p*

,*d*¯*p*

*=ψp*(x)*=c*¯*px*2+ ¯*dpx*3

(4.21)

be another filtered*λ-ring structure on Z[x]/(x*4_{). Suppose first that}_{S}_{and ¯}_{S}_{are }
isomor-phic, and let*σ*:*S→ _{S}*¯

_{be an isomorphism. Write}

_{σ}_{(x)}

_{=}_{ux}_{+}

*2*

_{αx}_{+}

*3*

_{βx}_{with}

_{u}_{∈ {±}_{1}

_{}}_{.}Then we have

*ψ _{S}p◦σ*(x)

*=cpx*2+

2c*puα*+*dpu*

*x*3_{,}

*σ◦ψ _{S}p*¯

(x)*=uc*¯*px*2+*ud*¯*px*3*.*

(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 (c*p*)*=u( ¯cp*) for some*u∈ {±*1*}*and ¯*dp=dp*+ 2c*pα*
for some integer*α, then the calculation above shows that an isomorphismσ*:*S→ _{S}*¯

_{is}given by

*σ*(x)*=ux*+*αx*2_{,} _{(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 primeq*such that
* _{a}n−*1

*q*,1 *−aq*,1*=*min*an _{p}−*,11

*−ap*,1:

*p*primes

*.*(5.1) Let

*S*and

*T*be two filtered

*λ-ring structures on Z[x]/(xn*

_{) satisfying (}

_{1.5}

_{) for all primes}

*p, and leta*and

*b*be the coeﬃcients of

*xn−*1

_{in}

_{ψ}q*S*(x) and*ψTq*(x), respectively. We claim
that if

*a≡b*mod*an−*1

then*S*and*T*are isomorphic. Indeed, if (5.2) is true, then we can write

*a+aq*,1c*=b*+*anq*,1*−*1*c* (5.3)

for some integer*c. Consider the filtered ring automorphismσ***on Z[x]/(x***n*_{) given by}

*σ(x)=x*+*cxn−*1_{.}_{(5.4)}

Then we have that

*ψ _{S}q◦σ*(x)

*=*

_{n}_{}_{−}_{2}

*i=*1

*aq*,*ix*+*cxn−*1*i*

+*ax*+*cxn−*1*n−*1

*=*

_{n}_{}* _{−}*2

*i=*1
*aq*,*ixi*

+*a*+*aq*,1c

*xn−*1

*=*

_{n}_{}* _{−}*2

*i=*1
*aq*,*ixi*

+*b*+*anq−*,11*c*

*xn−*1

*=σ◦ψ _{T}q*(x).

(5.5)

Now it follows from (5.5) and [10, Theorem 4.4] that

*ψ _{S}p◦σ*(x)

*=σ◦ψ*(x) (5.6)

_{T}pfor all primes *p. (It is here that we are using the hypothesisap*,1* =*0 for all *p.) That is,*
*σ*: *S→T*is a filtered*λ-ring isomorphism, as claimed. It is clear that*Theorem 1.11also

follows from this claim.

**5.2. Proof ofTheorem 1.12**

*Proof. Writeσ(x)=n−*1

*k=*1*bkxk* and *σ−*1(x)*=*
_{n}

*i=*1*cixi*. In particular, we have*b1=c1∈*

*{±*1*}*. Let *p* be a prime with *p≥n. Writeψ _{R}p*

_{}(x)

*=n*

*j=*1*ajxj*. Considering *ψSp*(x) as a
**polynomial in Z[x]/(x***n*+1_{), we have}

*ψ _{S}*

_{}

*p*(x)

*=σ−*1

_{◦}_{ψ}p*R◦σ*

(x)*=ψ _{S}p*(x) +

*bn*+1 1

*an*+

*α*

*xn*_{,} _{(5.7)}

in which*α*is a sum of integers, each summand having a factor of*ai*for some*i=*1,*...,*
*n−*1.

If *p > n, then* *ψ _{R}*

_{}

*p*(x)

*≡xp*

_{=}_{0 (mod}

_{p), which implies that}

_{a}*i≡*0 (mod*p) for* *i=*
1,*...,n. Sinceψ _{S}p*(x)

*≡*0 (mod

*p) as well, one infers thatψ*

_{S}p_{}(x)

*≡*0

*=xp*

_{(mod}

_{p).}If *p=n, then we still haveψ _{S}p*(x)

*≡*0 (mod

*p). But nowap≡*1 (mod

*p) andai≡*0 (mod

*p) fori < p. Since*

*p*+ 1 is an even integer, it follows that

*ψ*

_{S}_{}

*p*(x)

*≡xp*

_{(mod}

_{p), 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

*ψ _{S}*2

_{}(x)

*≡x*2(mod 2). (5.8)

Since*ψ*2

*S*(x)*≡x*2(mod 2) already, it suﬃces to show that the coeﬃcient of*x*3in*ψS*2(x) is
even. We will use the same notations as in the proof ofTheorem 1.12. Observe that the
congruence identity,*ψ*2

*R*(x)*≡x*

2 _{(mod 2), means that}_{a1}_{and}_{a3}_{are even and that}_{a2}_{is}
odd. Computing modulo 2 and*x*4_{, we have}

*ψ*2
*S*(x)*≡*

3

*i=*1
*ci*

_{3}

*j=*1

*ajb1x*+*b2x*2*j*
*i*

*≡*

3

*i=*1
*ciai*2

*b1x*+*b2x*22*i _{.}*

(5.9)

It follows that the coeﬃcient of*x*3_{in}* _{ψ}*2

*S*(x) is congruent (mod 2) to*c1a2(2b1b2)≡*0. This
proves the case*n=*3 ofConjecture 1.10.

The proofs for the cases*n=*4 and 5 ofConjecture 1.10 are essentially identical to
the previous paragraph. The only slight deviation is that, when*n=*4, to check that the
coeﬃcient of*x*4_{in}* _{ψ}*2

*S*(x) is even, one needs to use the fact that*c2= −b1b2.*

As discussed in the introduction, Conjectures 1.8 and 1.9 for *n≤*6 follow from

Conjecture 1.10for*n≤*5 andTheorem 1.11.

It should be pointed out that the argument above for the first three cases ofConjecture
1.10*does 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 of*x*6 _{in}* _{ψ}*2

*S*(x) is even. But this
coeﬃcient turns out to be congruent (mod 2) to*b3. It may be true that one can always*
arrange to have an isomorphism*σ* in which*b3* is even, but this is only a speculation.
Further works remain to be done to settle the three conjectures for higher values of*n.*

**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* *S*3_{, 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