BEYOND THE HIT PROBLEM: MINIMAL PRESENTATIONS OF ODD-PRIMARY STEENROD MODULES, WITH APPLICATION TO CP (1) AND BU.
DAVID J. PENGELLEY AND FRANK WILLIAMS Abstract. We describe a minimal unstable module presentationover the Steenrod algebra for the odd-primary cohomology of infinitedimensional complex projective space and apply it to obtain aminimal algebra presentation for the cohomology of the classifying space of the infinite unitary group. We also show that there is aunique Steenrod module structure on any unstable cyclic module that has dimension one in each complex degree (half the topologi-cal degree) with a fixed alpha-number (sum of `digits') and is zero in other degrees.
1. Introduction The projective spaces RP (1) and CP (1) play a pivotal role in algebraic topology, and have an amazing combination of features. As Eilenberg-MacLane spaces they represent key cohomology groups. Contrastingly, in the past two decades we have learned that their cohomologies are unstable injective modules over the Steenrod algebra A (at p = 2 for RP (1), and at odd primes p for CP (1) when considering "complex" (i.e., evenly) graded modules) [1, 2, 6, 14]. This surprising feature has been key to solving famous problems like the Segal and Sullivan conjectures.
We might even imagine that by now we understand their cohomologies H* (RP (1); F2) and H* (CP (1); Fp) very well. As an algebra each is polynomial on a single generator with A-action determined by extremely simple formulas. And as an unstable A-algebra each is free on one generator. What could be simpler?
But how well do we understand their A-module structures, which are key to what they actually tell us about other spaces? In exactly what way are these remarkable A-modules built from generators and
1991 Mathematics Subject Classification. Primary 55R40; Secondary 55R45,55S05, 55S10. Key words and phrases. Steenrod algebra, unstable, Kudo-Araki-May algebra,complex projective space,
BU .
1
2 DAVID J. PENGELLEY AND FRANK WILLIAMS relations in order to produce all the amazing properties of projective spaces delineated above? If we ask first for minimal A-generators, this is the classical hit problem (i.e., which elements are not hit by the Aaction), and it is not hard to answer this. We shall describe a minimal set of A-module generators {u(s)|s ? 0} for H * (CP (1); Fp) . In fact, for each integer s ? 0 the complex degree of the generator u(s) is the least integer d for which ff(d) = s. (Here, and throughout this paper, "complex degree" will refer to one-half the topological degree, and the "alpha-number" of a nonnegative integer n, ff(n), will mean the sum of the p-ary digits of n.)
Going beyond the hit problem, the question of which A-relations are then necessary among these minimal A-generators in order to glue together precisely the cohomology of projective space is extremely delicate. Pleasantly, we find that they are not that great in number, are essentially unique, and can be written down explicitly.
We accomplished this for H * (RP (1); F2) in [9], and will now do so for H* (CP (1); Fp) (with p odd), where the answer has some fascinating extra twists but is still tractable. In so doing, we will analyze the cyclic subquotients of the A-filtration of H * (CP (1); Fp) given by alpha number of complex degree, and determine their minimal Arelations. The modules in this composition series are simple modules in the category U 0p/N il of evenly graded unstable A-modules modulo nilpotence described in [1, 5]. We also show that each of them is uniquely characterized as a cyclic unstable A-module just by having dimension one in each even degree with a fixed alpha number for the complex degree, and dimension zero in all other degrees. The generators of these cyclic modules are just the images in the filtered quotients of the elements u(s).
There are considerable similarities between the odd-primary and the mod-two cases. However, there are differences that are quite interesting. The filtered quotients in the mod two case are fairly well-knownA
-modules, i.e., they are isomorphic to the free unstable modules on single generators t2n-1-1 in degree 2n-1 - 1 subject to the A-relations Sq2
kt
2n-1-1 = 0 for 0 <= k <= n - 3 [9]. In the odd primary case, sincethe Steenrod algebra is concentrated in complex degrees divisible by
p - 1, any A-module splits as a direct sum of p - 1 A-modules, each in degrees with fixed residue mod (p - 1).
We shall see that the filtered quotients of H * (CP (1); Fp) are isomorphic to certain modules Mn,a on generators tapn-1-1 in (complex) degrees of the form apn-1 - 1, where 1 <= a <= p - 1, with a - 1 labeling the mod (p - 1) residue summand, and n further reflecting the filtration
BEYOND THE HIT PROBLEM 3 by alpha number within this summand. If we let s = ff(apn-1 - 1) = (a - 1) + (n - 1)(p - 1), then apn-1 - 1 is the smallest integer with ff-number s. Hence the generator of Mn,a is in the lowest degree with its alpha number. In this case the minimal A-relations include the expected Pp
kt
apn-1-1 = 0 for 0 <= k <= n - 3, but also include either oneor two additional relations that depend on a. These modules M
n,a are quite interesting: Since they have dimension one in degrees with alpha
number equal to that of apn-1 - 1, are zero in other degrees, and this uniquely characterizes them as cyclic A-modules, they may be regarded as basic building blocks of structures having to do with alpha-number. Moreover, they are analogs to an interesting phenomenon at the prime 2.
At p = 2, Franjou and Schwartz [3, 13, 14] considered the category Vn-1/Vn-2 (Vn-1 is the full subcategory of U /N il, i.e., modulo nilpotence, with objects the unstable A-modules of weight n - 1, i.e., trivial in degrees with alpha number greater than n - 1). They showed that Vn-1/Vn-2 is equivalent to the category of right modules over the group ring F2 [\Sigma n-1] on the symmetric group. The cyclic unstable Amodule we described above, on t2n-1-1 with relations Sq2
k t
2n-1-1 = 0for 0 <= k <= n-3, is of dimension one in each degree with alpha number
n - 1, and dimension zero in all other degrees, and thus corresponds under this equivalence of categories to the unique nontrivial rank one module over F2 [\Sigma n-1].
Our A-modules Mn,a are odd primary versions of these mod 2 Amodules, occupying a similar spot in the odd-primary analogue (see [1, p. 395] and [5]) of the theory in [3] of reduced unstable mod 2 Amodules. One of our theorems produces bases for the Mn,a that show that they are reduced and have weight exactly ff = ff (apn-1 - 1), i.e., lie in V0ff - V0ff-1 in the filtration of U 0p/N il. It would be interesting to find a direct proof, without using our basis theorem, that Mn,a is reduced and lies in V0ff. Such information might lead to an alternate proof of some of our results by invoking an odd-primary version of 2-primary results in [3].
Finally, in roughly the same way that our minimal A-presentation of H* (RP (1); F2) led to a minimal unstable A-algebra presentation of the symmetric algebra H*(BO; F2) [9], our minimal unstable A-module presentation for H*(CP (1); Fp) will lead to a minimal unstable Aalgebra presentation of the symmetric algebra H*(BU ; Fp).
Many of our methods will be the same ones we used in [7, 9, 10] to determine minimal relations for unstable A-modules and A-algebras. A fundamental element of our computations is the odd-primary even
4 DAVID J. PENGELLEY AND FRANK WILLIAMS topological Kudo-Araki-May algebra, K, whose definition and properties we developed in [11], and which is well-suited to studying unstableA
-modules. We summarize necessary ingredients from this material in an appendix to the present paper.
In the following section we shall list the principal results of this paper.
2. Definitions and principal results To set the stage, we work with coefficients in the field Fp, for p an odd prime. We consider only evenly graded modules over the Steenrod algebra A (with no Bocksteins), and generally use the complex degree (half the topological degree) throughout to describe the grading. N.B: Every A-module in this paper will be assumed unstable without further mention.
The hit problem for H*(CP (1)) = Fp [u] is easily solved. The formula for the action of the Steenrod algebra
Pkum = `mk 'um+(p-1)k quickly yields Proposition 2.1. A minimal set of A-module generators for H *(CP (1)) is given by the set {uap
n-1-1 | 1 <= a <= p - 1 and n >= 1}. As explained above, we can uniquely index these elements by the alpha number of their degrees, as {u(s) | s ? 0}, where s = ff(apn-1 - 1) = (a - 1) + (n - 1)(p - 1), and the degree apn-1 - 1 of each is the smallest integer with its alpha number.
Our main theorem is the following, whose proof will occupy most of this paper.
Theorem 2.2 (proven in section 5). A minimal set of A-module relations on the minimal module generators {uap
n-1-1 | 1 <= a <= p - 1 and
n >= 1} for H*(CP (1)) is given by the following equations: For 0 <= l <= n - 3 P
pluapn-1-1 = Papn-2 Ppluapn-2-1,
and for a >= 2
(a - 1)P(p-a+1)p
n-2 uapn-1-1 = `p - 1
a - 2'P
pn-1 Ppn-2uapn-2-1
and
(a - 1)Pp
n-1 Ppn-2 uapn-1-1 = aPpn-1+pn-2uapn-1-1,
BEYOND THE HIT PROBLEM 5 while for a = 1
2Pp
n-1+pn-2Ppn-2+pn-3uapn-1-1 = Ppn-1+pn-2+pn-3Ppn-2 uapn-1-1.
A sketch of the proof is as follows. It is straightforward to check Proposition 2.3. Our claimed set of relations is satisfied in H *(CP (1)). Proof. Left to the reader. \Lambda
Thus the proof of the presentation will primarily involve showing that this set of relations is sufficient, i.e., that H *(CP (1)) has the same graded rank as the quotient M of the free (unstable) A-module on generators in the specified degrees by the sub-A-module generated by the set of relations. To analyze this, we shall filter the two modules compatibly over A, in a fashion related to the alpha number of (complex) degree. We shall first describe a basis for each filtered quotient of H*(CP (1)), using monomials from the Kudo-Araki-May algebra K applied to a generating class. Although any element described using K can in principle also be described using A (and vice-versa) by iterating the conversion formula
(-1)j dju = Pq-ju, where u is a module class of (complex) degree q, in practice it seems that K, whose algebra structure is dramatically different from that ofA
, is often much more transparent for describing bases of unstable Amodules. In this case, we will provide a basis for H *(CP (1)) expressed in terms of K-monomials we call "chosen". Then we show that these same chosen monomials produce a basis for the corresponding filtered quotient of M, by showing that every element of M can be expressed in terms of chosen monomials. This determination that chosen monomials suffice to span M, based on its abstract defining relations, is the lengthy part of the proof. We refer the reader to our appendix for a summary of relevant information about K . Finally, the proof will also verify that the relations we give are all necessary.
We begin the process of describing our "chosen" monomials in K with some definitions. The fact that these particular monomials will have something special to do with the alpha number of degrees is not at all obvious, and will emerge in our proofs.
Notation 2.4. For an integer j = P jlpl, in p-ary representation, we shall write this representation as
j = (. . . , jl, . . . , j0).
6 DAVID J. PENGELLEY AND FRANK WILLIAMS Definition 2.5. A generator dj of K is called chosen if the p-ary digits of j are non-decreasing from left to right, i.e., if j = (. . . , jl, . . . , j0), then jl+1 <= jl, for l >= 0.
Definition 2.6. A 2-fold monomial didj 2 K is called chosen provided that di and dj are both chosen, that each digit of i is less than or equal to the corresponding digit of j, and that if jl 6= p - 1, then il+1 = 0 (in these circumstances the last condition is equivalent to i < p*p(j+1)+1, where *p is the exponent of p-divisibility).
Definition 2.7. An arbitrary monomial dI = di1di2 * * * dik (for k >= 0) is called chosen provided that each dil is chosen and for each l, 1 <= l <= k - 1, the monomial dildil+1 is chosen.
We also recall from the appendix the following definition in K. Definition 2.8. A monomial dI = di1 di2 * * * dik is called admissible provided that for each l, il <= il+1.
We recall that the admissibles are a basis for K, and note that every chosen monomial is admissible. Also notice that the definition of chosen refers only to monomials in K, having nothing directly to do with the degree of a class of application in a module.
We also recall here that a basis for the free unstable A-module on a class in degree m consists of applying all admissibles in A of (complex) excess less than or equal to m, or, equivalently, admissibles dI 2 K with final index less than m.
We are almost ready to describe H*(CP (1)) in terms of the monomials we have labeled as chosen.
Notation 2.9. Let FsH*(CP (1)) denote the direct sum of H 2k(CP (1)) for all k such that ff(k) <= s, where ff(k) denotes the alpha number of the integer k. One can check from the equation at the beginning of this section that this is a filtration by sub-A-modules.
The subquotient module FsH*(CP (1))/Fs-1H*(CP (1)) is concentrated in, and of rank one, in precisely those (complex) degrees with ff-number s. As we noted above, if we write s = q(p - 1) + r, where 0 <= r <= p - 2, then the smallest degree with this ff-number has p-ary representation (r, p - 1, . . . , p - 1), and is (r + 1)pq - 1, precisely the degree of a minimal A-module generator already noted above for H*(CP (1)). So there is a correspondence between the filtered quotients and the minimal generators. It is not surprising that the quotients turn out to be cyclic modules over A (equivalently K) on these generators. The important content of the following theorem is the
BEYOND THE HIT PROBLEM 7 explicit systematic identification of a single chosen monomial in K representing a basis element for each degree with a given alpha number. The theorem does this by melding the definition of chosen monomials in K with the unstable conditions of an A-module, obeyed by the cohomology of any space, that if x is a class of complex degree m, then dix = 0 for i > m, and dmx = (-1)m x, so that dix represents new elements only when i < m. For this reason we make the following definition.
Definition 2.10. We call the application of a monomial dI to a class x of degree m unstable if the the righthand factor dl of dI satisfies l < m.
Theorem 2.11 (proven in section 3). The filtered quotient
FsH*(CP (1))/Fs-1H*(CP (1)) ' M
ff(k)=s
H2k(CP (1))
has as a vector space basis the set of all unstable dIu(r+1)p
q-1 where d
I
is chosen.
Remark 2.12. In particular, this theorem tells us that the chosen monomials in K, applied unstably to any element in a degree of the form apn-1 - 1 (for a <= p - 1), land in precisely one-to-one fashion in all degrees with the same ff-number. This will be true in any module; the theorem indicates that in H *(CP (1)) they all represent nonzero elements as well.
Now we define the abstract module we claim presents H *(CP (1)). Notation 2.13. Let F be the free A-module on abstract classes tapn-1-1, for 1 <= a <= p - 1 and n >= 1 (where subscripts indicate the complex degree of each class, here and in the future). Let J be the sub-A-module generated by the following relations: For 0 <= l <= n - 3 P
pl tapn-1-1 = Papn-2 Ppltapn-2-1,
and for a >= 2
(a - 1)P(p-a+1)p
n-2 t
apn-1-1 = `p - 1a - 2'Pp
n-1 Ppn-2t
apn-2-1
and
(a - 1)Pp
n-1 Ppn-2 t
apn-1-1 = aPp
n-1+pn-2t
apn-1-1,
while for a = 1
2Pp
n-1+pn-2Ppn-2+pn-3 t
apn-1-1 = Pp
n-1+pn-2+pn-3Ppn-2 t
apn-1-1.
Define M to be the quotient A-module F/J.
8 DAVID J. PENGELLEY AND FRANK WILLIAMS
Next we wish to filter M in a fashion compatible with the filtration of H*(CP (1)). We again use the correspondence between natural numbers s and the smallest degree with s for its ff-number, described by writing s = q(p - 1) + r (with 0 <= r <= p - 2), yielding smallest degree (r + 1)pq - 1 = (r, p - 1, . . . , p - 1) with this ff-number. Notice that the correspondence is monotonic.
Notation 2.14. Let FsM denote the sub-A-module of M generated by all t's of degree less than or equal to the degree (r + 1)pq - 1 with ff-number s. Let Mn,a denote the cyclic subquotient module of this filtration with generator in degree apn-1 - 1 (obtained by letting r = a - 1 and q = n - 1 determine s, and defining Mn,a = FsM/Fs-1M).
As we did for H*(CP (1)), we now wish to analyze the filtered quotients Mn,a, and ultimately to see that they agree with those of H*(CP (1)).
Remark 2.15. We note that Mn,a has a single A-generator tapn-1-1, subject to the following A-relations: For 0 <= l <= n - 3 P
pltapn-1-1 = 0,
and for a >= 2 P
(p-a+1)pn-2 tapn-1-1 = 0
and
(a - 1)Pp
n-1 Ppn-2 t
apn-1-1 = aPp
n-1+pn-2t
apn-1-1,
while for a = 1
2Pp
n-1+pn-2Ppn-2+pn-3 t
apn-1-1 = Pp
n-1+pn-2+pn-3Ppn-2 t
apn-1-1.
We define a map from F to H*(CP (1)) by taking each tapn-1-1 to uap
n-1-1. By Proposition 2.3, this map carries the submodule J to
zero, so there is an induced map M ! H *(CP (1)). This is the map we shall show is an isomorphism.
Remark 2.16. From our earlier results and discussion it is clear that this map is well-defined, respects the filtrations on M and H *(CP (1)), and is an epimorphism.
Theorem 2.2 will result from seeing that this map induces A-isomorphisms on the filtered quotients. This is ensured by the next theorem, with proof occupying the bulk of the paper, determining a basis for Mn,a analogous to that of Theorem 2.11 for the filtered quotients of H *(CP (1)).
Theorem 2.17 (proven in section 4). A vector space basis for Mn,a consists of the set of all unstable dItapn-1-1 where dI is chosen.
BEYOND THE HIT PROBLEM 9 Corollary 2.18. From this theorem and Theorem 2.11, the map M!
H*(CP (1)) defined above induces isomorphisms
Mn,a = FsM/Fs-1M ' F sH*(CP (1))/Fs-1H*(CP (1)) where s = ff (apn-1 - 1). In particular, Mn,a is therefore concentrated in degrees with alpha number s, and always has rank one there.
The presentation of H*(CP (1)) in Theorem 2.2 is now essentially immediate. We make this explicit, and verify the minimality of the relations, in section 5.
We established above that the abstract modules Mn,a are cyclic unstable A-modules that have dimension one in degrees with ff-number equal to that of apn-1 - 1 and are zero in other degrees. Their importance is underscored by the fact that this characterizes them uniquely:
Theorem 2.19 (proven in section 6). Let ff0 >= 1. Let M be a cyclic (unstable) module over the Steenrod algebra A, p odd, such that dim(Ml) = 1 if ff(l) = ff0 and dim(Ml) = 0 if ff(l) 6= ff0. Then, asA
-modules, M ,= Mn,a, where ff(apn-1 - 1) = ff0.
We now move to our minimal presentation of H *(BU ) as an unstableA -algebra. Our intention is to identify a minimal sub-A-module (itself minimally presented as an A-module) that generates H *(BU ) as anA
-algebra, form the free unstable A-algebra on this module, and then impose a minimal set of A-algebra relations to obtain H *(BU ).
Remark 2.20. There is a map S2 ^ CP (1)+ -! BU that induces an epimorphism on integral cohomology. (Here CP (1)+ denotes the union of CP (1) with a disjoint basepoint.) The map classifies the virtual bundle (j1 -1)\Omega (j1 [0), where j1 and j1 denote the canonical line bundles over S2 = CP (1) and CP (1), respectively. A computation using the Chern character verifies that this map induces the desired epimorphism on integral cohomology. For details, see, e.g., the monograph [4, p. 73]. As all products vanish in the cohomology of S2 ^ CP (1)+ since it is a suspension, the induced map on indecomposables is an isomorphism. (An alternative way to see that QH *(BU ) is A-isomorphic to \Sigma 2H*(CP (1)) is via the mod p Wu formulas in H*(BU ); see [12, 15].)
Since the indecomposable quotient QH*(BU ) is isomorphic as anA -module to the double (topological) suspension of H *(CP (1)), then by Proposition 2.1 the set of Chern classes {capn-1 | 1 <= a <= p - 1 and n >= 1} is a minimal set of A-algebra generators for H *(BU ). Hence up to algebra decomposables we will find in H *(BU ) the double
10 DAVID J. PENGELLEY AND FRANK WILLIAMS suspension of the relations in the module M that minimally presents H*(CP (1)). We list these analogous relations in H *(BU ):
Remark 2.21. Since the relations in Theorem 2.2 were verified in Proposition 2.3 to hold in H *(CP (1)), we have the following relations in H*(BU ):
For 0 <= l <= n - 3P
pl capn-1 = Papn-2 Pplcapn-2 + D1(a, n, l),
and for a >= 2
(a - 1)P(p-a+1)p
n-2 c
apn-1 = `p - 1a - 2'Pp
n-1 Ppn-2c
apn-2 + D2(a, n)
and
(a - 1)Pp
n-1 Ppn-2 c
apn-1 = aPp
n-1+pn-2c
apn-1 + D3(a, n),
while for a = 1
2Pp
n-1+pn-2Ppn-2+pn-3c
apn-1 = Pp
n-1+pn-2+pn-3Ppn-2 c
apn-1 + D4(a, n).
Here D1(a, n, l), . . . , D4(a, n) are decomposable polynomials in the elements {PJ capk-1 | 1 <= a <= p - 1 and k >= 1} that may, in principle, be computed using the mod p Wu formulas [12, 15].
We shall prove that these form a minimal set of relations for H *(BU ) as an unstable algebra over the Steenrod algebra.
Since there are decomposables to contend with amongst the equations connecting these Chern classes, we do not immediately imposeA
-module relations on our abstract generators imitating the generating Chern classes.
Notation 2.22. Let N be the free (unstable) A-module on abstract classes o/apn-1 , for 1 <= a <= p - 1 and n >= 1 .
Let U (N ) be the free unstable A-algebra on the A-module N (the odd-primary analogue of [16, pp. 28-29]). Let I be the A-ideal in U (N ) generated by the following relations: For 0 <= l <= n - 3P
plo/apn-1 = Papn-2 Pplo/apn-2 + D1(a, n, l),
and for a >= 2
(a - 1)P(p-a+1)p
n-2 o/
apn-1 = `p - 1a - 2'Pp
n-1 Ppn-2o/
apn-2 + D2(a, n)
and
(a - 1)Pp
n-1 Ppn-2o/
apn-1 = aPp
n-1+pn-2o/
apn-1 + D3(a, n),
BEYOND THE HIT PROBLEM 11 while for a = 1
2Pp
n-1+pn-2 Ppn-2+pn-3o/
apn-1 = Pp
n-1+pn-2+pn-3Ppn-2 o/
apn-1 + D4(a, n).
(Here D1, . . . , D4 are the polynomials in the preceding remark with the Chern classes capn-1 replaced by the elements o/apn-1 .) Finally, letG
=U (N )/I, a quotient A-algebra of U (N ).
We define a map U (N ) ! H *(BU ) by taking o/apn-1 to the Chern class capn-1 . Since the ideal I is taken to zero by this map, we obtain a map OE : G ! H*(BU ). We shall check that the induced map QOE on indecomposable quotients is an isomorphism. Since H *(BU ) is a polynomial algebra, we obtain our presentation for H *(BU ).
Theorem 2.23 (proven in section 6). The map OE : G ! H *(BU ) is an isomorphism of unstable A-algebras.
Theorem 2.24. Our presentation for H *(BU ) is minimal, in the sense that the module N injects into H*(BU ) and our set of relations is minimal.
It is clear from previous results that our set of relations imposed on G is minimal, so Theorem 2.24 is an immediate consequence of the following theorem, which shows that in H*(BU ) there are no A-module relations amongst any Chern classes (even though for injectivity of N we only need this for those in degrees apn-1).
Theorem 2.25 (proven in section 6). Let R denote the free A-module on abstract classes tm, for m >= 1. Then the map R ! H *(BU ) defined by taking each o/m to the corresponding Chern class cm is a monomorphism.
3. Proof of Theorem 2.11 Fix the nonnegative integer s. Let k0 be the smallest nonnegative integer for which ff(k0) = s. We shall define a bijection between the set of chosen monomials di * * * dl with l < k0 and a basis forL
ff(k)=s H2k(CP (1)), by assigning to di * * * dl the element di * * * dlxk
0.
Of course we will need to show that this assignment is valid, nonzero, and creates a one-to-one correspondence between the chosen monomials and the degrees k with ff(k) = s.
We begin with a short calculation showing that the assignment always lands in the correct degrees. Let didj * * * dl be a chosen monomial with l < k0. There exist a (with 1 <= a <= p - 1) and n such that
k0 = apn-1 - 1 = (a - 1, p - 1, . . . , p - 1).
12 DAVID J. PENGELLEY AND FRANK WILLIAMS Write i = (iq-1, . . . , i0), where iq-1 6= 0 and q <= n. Since didj * * * dl is a chosen monomial, then j, . . . , l are of the form
j = (jt, . . . , jq-1, p - 1, . . . , p - 1). If k is the degree of dj * * * dlxk0 , by iteration we have k of the form (see the appendix to recall calculation of degrees involving K)
k = (ks, . . . , kq, jq-1, p - 1, . . . , p - 1). If we set m = pk - (p - 1)i, then m is the degree of di * * * dlxk0 and we calculate
m = (ms+1, . . . , m0)
= ksps+1 + * * * + (kq-1 - iq-1)pq + [(p - 1) - iq-2 + iq-1]pq-1+* * *
+ [(p - 1) - i0 + i1]p + i0.
An induction based on this formula yields the following lemma. Lemma 3.1. If didj * * * dl is a chosen monomial with l < k0 and m is the degree of di * * * dlxk0 , then ff(m) = s.
Next we will reverse this calculation, beginning solely with a degree m with ff(m) = s, and finding a chosen monomial didj * * * dl that produces a basis element in H 2m(CP (1)) when applied to xk0 . The formulas above will be our guide. We note that in the formula for m we had
q-1X
r=0
mr = (q - 1)(p - 1) + iq-1 > (q - 1)(p - 1),
and qX
r=0
mr = (q - 1)(p - 1) + kq-1 <= q(p - 1).
This motivates us to define q from m as follows: Definition 3.2. Let q >= 0 be the least integer such that Pqr=0 mr <= q(p - 1).
With q in hand we can continue to solve from the equations above for k and i.
Definition 3.3. Let q be as just defined, and set
kr = 8!:
p - 1, for 0 <= r <= q - 2P q
l=0 ml - (q - 1)(p - 1), for r = q - 1m
r+1, for r >= q.
BEYOND THE HIT PROBLEM 13 Note that from the definition of q, for r >= 0 we have 0 <= kr <= p - 1, and that for 0 <= r <= q - 1 we have
0 < m0 + * * * + mr - r(p - 1) <= p - 1. Set
i =
q-1X
r=0
[m0 + * * * + mr - r(p - 1)]pr.
Note that if m = apn-1 - 1, then i = k = m, and that otherwise i < k.
Next we check that for the i we have defined, xm is actually in the image of di. In H*(CP (1), we have (see appendix), for any i and k, the formula
dixk = (-1)i `ki 'xpk-(p-1)i.
Using this, we compute that for our defined m, k, and i,
m = pk - (p - 1)i and dixk = (unit) * xm. Now we continue this prescription backwards to produce a chosen monomial connecting xk0 to xm. We note that the digits of i are nondecreasing, so it is chosen. Further, we note that if we start with k as just defined in place of m and iterate the process to find a j by the same recipe we used to define i, we obtain
jr = p - 1, for r <= q - 2 and
jq-1 = iq-1 + mq >= iq-1.
So, inductively, we can start with an integer m such that ff(m) = s, and produce a chosen admissible didj * * * dl, with l < k0, such that
xm = (unit)didj * * * dlxk0 , where k0 is the least positive integer such that ff(k0) = s. This shows that assigning to a monomial didj * * * dl with l < k0 the element didj * * * dlxk0 is surjective from the set of such monomials to a basis forL
ff(k)=s H2k(CP (1))It remains only to see that there is a unique such monomial for each
degree m with ff(m) = s. But it is clear from our displayed formulas that began this section that if we start with a chosen monomial didj * * * dl with l < k0, and consider only the degree m of the element didj * * * dlxk0 , then apply our backwards algorithm above to find a new value for i, that the algorithm produces the same value for i that we began with in the chosen monomial.
14 DAVID J. PENGELLEY AND FRANK WILLIAMS
Summing up, we have shown that the set of elements didj * * * dlxk0, ranging over the chosen monomials didj * * * dl with l < k0, forms a basis for Lff(k)=s H2k(CP (1)). This proves Theorem 2.11.
4. Proof of Theorem 2.17. From Theorem 2.11 and the remark following it, the map of filtered quotients from Mn,a to the corresponding filtered quotient of H*(CP (1)) provides a nonzero representation of all the chosen unstable monomials on xap
n-1-1 in H*(CP (1)), with exactly one chosen
in each degree with the same alpha number as apn-1 - 1, and none elsewhere. Thus to prove that the chosen monomials applied unstably to the generator of Mn,a provide a basis for the module, we need only show that they span Mn,a. Our strategy will be to show that each admissible monomial that applies unstably to the generator um of Mn,a can be expressed in Mn,a as a multiple of a chosen unstable monomial on um. Here m = apn-1 - 1 is the degree of the generator, with 1 <= a <= p - 1.
We will frequently and often without mention use the Adem relations in the Steenrod algebra:
PaPb = X
t
(-1)a+t `(p - 1) (b - t) - 1a - pt 'Pa+b-tPt,
with Pt in complex degree (p - 1) t. And we will often use without mention that fact that the initial relations in Mn,a (in Remark 2.15) clearly make any terms Ptum = 0 for 0 < t < pn-2. The other relation in Mn,a involving a single Steenrod operation will be used in the proof for length one monomials, and the relations involving two-fold monomials will be used in Lemmas 4.19 and 4.21.
4.1. Length one monomials. We shall prove Theorem 2.17 in stages. We note that the set of monomials dJ um in Mn,a is filtered by the length of J = (j1, . . . .jr) , dJum = dj1 * * * djr um. In this subsection we shall deal with monomials of length one. Our first stage is the following Lemma.
Lemma 4.1. Fix n and a. In the module Mn,a, with generator um in complex degree apn-1 - 1, if dj is unchosen, then djum = 0.
Proof. By definition, dj is chosen if the digits of j are non-decreasing. I.e., if we write
j = (jn-1, . . . , j0),
BEYOND THE HIT PROBLEM 15 then this says that jn-1 <= * * * <= j0. Now write Ptum = (-1)j djum. We shall call Ptum chosen if dj is chosen. Now
deg(um) = (a - 1, p - 1, . . . , p - 1) So for Ptum to be chosen, we need, writing
t = (tn-1, . . . , t0)
= (a - 1 - jn-1, p - 1 - jn-2 . . . , p - 1 - j0),
that
tn-1 + p - a >= tn-2 >= tn-3 >= * * * >= t0,
so if Ptum is unchosen, we must have tn-1 < tn-2 - (p - a) or tr < tr-1 for some 1 <= r <= n - 2.
The requirement that Pp
iu
m = 0 for 0 <= i <= n-3 yields immediately
that Ptum = 0 for all 0 < t < pn-2, i.e., when tn-1 = tn-2 = 0. Further,
if tn-1 = 0, then the requirement that P(p-a+1)p
n-2 u
m = 0 tells us thatPt
um = 0 for all t such that (p - a + 1)pn-2 <= t < pn-1.
Henceforth assume that either tn-1 or tn-2 is nonzero. We shall induct on the degree of Ptum. Fix a value of t and suppose that all unchosen Psum = 0 for s < t.
Case 1. Suppose Ptum is unchosen and that there is an 1 <= r <= n-2 for which tr < tr-1. (Note that this must happen if a = 1.) Then one can check that Pt-p
r u
m is unchosen and in lower topological degree
than Ptum, so we may assume inductively that Pt-p
r u
m = 0. Suppose
there exists a least integer 0 <= s <= r - 2 such that ts 6= 0. ThenP
t-psum is unchosen and
0 = Pp
sPt-psu
m = (unit)Ptum.
So, without loss of generality, we may assume that tr-2 = * * * = t0 = 0. We have
0 = Pp
r Pt-pr u
m = -(tr-1 - tr)Ptum.
If a = 1, this is the only possible case, so this completes the proof when a = 1.
Case 2. Suppose that tn-1 < tn-2 - (p- a). (Then a > 1.) Without loss of generality, as above, we may take tn-3 = * * * = t0 = 0, so that t = tn-1pn-1 + tn-2pn-2. Noting that in this case Pp
n-2Pt-pn-2 u
m =
(tn-2)Ptum, we may assume that tn-2 = tn-1 + (p - a) + 1. When
tn-1 = 0, we have that Ptum = 0 is one of the defining relations for our module.
There remains only to consider tn-1 > 0. Using Adem relations, we get the following formulas, using the inductive hypothesis on degree to
16 DAVID J. PENGELLEY AND FRANK WILLIAMS eliminate needing to write down many terms:
Pp
n-1 Pt-pn-1u
m = -(p - a + 1)Ptum + Pt-p
n-2Ppn-2 u
m,
Pp
n-1+pn-2Pt-pn-1-pn-2 u
m =
(p - a)(p - tn-2)Ptum - (p - tn-2 + 1)Pt-p
n-2Ppn-2 u
m
+ Pt-p
n-2-pn-3 Ppn-2+pn-3u
m,
Pp
n-1+***+pn-sPt-pn-1-***-pn-su
m = -Pt-p
n-2-***-pn-sPpn-2+***+pn-su
m
+ Pt-p
n-2-***-pn-s-1Ppn-2+***+pn-s-1u
m
for 3 <= s <= n - 1,
and Pp
n-1+***+p+1Pt-pn-1-***-p-1u
m = -Pt-p
n-2-***-p-1Ppn-2+***+p+1u
m.
Since Pt-p
n-1 u
m through Pt-p
n-1-***-p-1u
m are unchosen and of lower
topological degree than Ptum, then inductively they are zero. We have
have a matrix equation M X = 0, where
M =
26666 6666664
-(p - a + 1) 1 0 0 * * * 0 0 0 (p - a)(p - tn-2) -(p - tn-2 + 1) 1 0 * * * 0 0 0
0 0 -1 1 * * * 0 0 0 0 0 0 -1 * * * 0 0 0.
.. ... ... ... ... ... ...
0 0 0 0 * * * -1 1 0 0 0 0 0 * * * 0 -1 1 0 0 0 0 * * * 0 0 -1
37777 7777775
and
X =
26666 6664
PtumP t-pn-2 Ppn-2um
...
Pt-p
n-2-***-pn-sPpn-2+***+pn-su
m.
..P t-pn-2-***-p-1Ppn-2+***+p+1um
37777 7775
This matrix is nonsingular, since its determinant mod p is (-a + 1 - tn-2) (-1)n , which is nonzero since tn-1 > 0. Hence all entries of X are zero. In particular Ptum = 0. \Lambda
BEYOND THE HIT PROBLEM 17 4.2. Length two monomials. The proof for length one monomials proceeded by induction on topological degree. The remainder of the proof that the chosen monomials span Mn,a will continue this way. Within each topological degree we also order the admissible unstable monomials on um as follows:
Definition 4.2. If dI and dJ are admissibles that are unstable and in the same topological degree when applied to um, we define dIum to be lower in order than dJ um provided that dI has shorter length than dJ. Further, if they have the same length, then we define dIum to be lower in order than dJ um provided that I is lower than J in lexicographical ordering, starting from the left.
Remark 4.3. Note that our ordering is only defined for admissible monomials that are unstable when applied to um (i.e., their final subscript is less than m). If an admissible is not unstable when applied to um, it will collapse to zero or to an admissible unstable monomial of shorter length when applied to um.
Remark 4.4. Although our ordering is only defined for admissible monomials, note that a K-Adem relation (see appendix) applied to an inadmissible always produces admissible terms of lower lexicographic order than the inadmissible. We may use this without mention in calculations.
Remark 4.5. In a fixed topological and length degree, the admissible unstable monomials applied to um are finite in number.
INDUCTIVE ASSUMPTION. Assume inductively that in topological degrees less than a given one, every unchosen admissible unstable monomial dI um of Mn,a is a sum of admissible monomials of lower order than itself. (Hence, in those lower degrees, Mn,a has the chosen unstable monomials on um as a basis, and is isomorphic to the corresponding filtered quotient of H *(CP (1))).
We proceed to the proof of the theorem for length two unstable admissibles on um in the given degree. We shall fix an unstable admissible didjum and inductively assume also, within its topological degree, that every unchosen unstable admissible dkdlum of lower order than didjum is a sum of admissibles of lesser order than itself. Our goal will be to show that if didj is unchosen, then didjum is a sum of two-fold (or 1-fold) admissibles of lower order than itself.
Definition 4.6. If dj is chosen, let i(j) denote the largest i for which didj is chosen.
18 DAVID J. PENGELLEY AND FRANK WILLIAMS Remark 4.7. Suppose that r is an integer for which jl = p - 1 for l < r and that jr 6= p - 1. Then
i(j) = (0, . . . , 0, jr, p - 1, . . . , p - 1). Remark 4.8. A very useful formula in the Kudo-Araki-May algebra is the following, for i <= j :
didj = dpj-(p-1)idi + X
t>=1
(-1)t-1 `(p - 1) (j - i - t) - 1pt 'di-ptdj+t .
For i < j this is simply a rewriting of the Adem relation for the inadmissible dpj-(p-1)idi (see appendix), but it switches the element di between appearance on the right and left in a two-fold monomial, and extracts an expression for the admissible of highest order in the relation.
This allows us to prove immediately that a large class of unchosen admissibles are sums of chosens.
Lemma 4.9. If the admissible didj is unchosen, and either i <= i(j), or di or dj is unchosen, then didjum can be expressed as a sum of lower order terms.
Proof. Checking the requirements of the definition for chosenness, we see that if i <= i(j), the only way didj can be unchosen is for either di or dj to be unchosen, so by the preceding remark and the result above for 1-folds, the lemma follows. \Lambda
So we only need to consider cases in which*
the monomial didj is admissible, j < m, and* both di and dj are chosen, and* the index i > i(j).
Our general strategy now is to find didjum in the image of a Steenrod operation from a lesser topological degree, and to apply the inductive assumptions to see that it is a sum of terms of lower order than itself. The numerous cases that will need individual consideration stem from the fact that the Steenrod operation required depends on the p-ary representations of i and j. In particular, for the rest of the proof we will let r denote the greatest integer for which il = jl = p - 1 for 0 <= l < r. We note that if r = n - 1 (r = n - 2 if a = 1), then didjum is automatically chosen, so we may also assume henceforth that*
the index r satisfies 0 <= r <= n - 2. The lemmas in the rest of this section have as their common goal to show that if didj is unchosen and satisfies the four bulleted restrictions above, then didjum can be expressed as a sum of lower order terms.
BEYOND THE HIT PROBLEM 19 We thus combine the bullets as a set of common hypotheses for all lemmas that follow, and together the succeeding lemmas will cover all possibilities for the p-ary representations of i and j subject to these hypotheses. COMMON HYPOTHESES FOR ALL SUCCEEDING LEMMAS: We assume that didj is admissible and unchosen, j < m, both di and dj are chosen, and i > i(j). Also, with r denoting the greatest integer for which il = jl = p - 1 for 0 <= l < r, we assume that 0 <= r <= n - 2.
Before beginning to cover particular cases, we pause for two preparatory lemmas for showing that many types of terms in Mn,a are actually zero by climbing up inductively from lower degrees in which we already know that Mn,a is zero. These two lemmas do not depend on the common hypotheses.
Lemma 4.10. Suppose that ff(|didjum|) = ff(m). Then for 0 <= l <= r- 1, ff(|didjum| - (p - 1)pl) < ff(m). Hence, by the inductive assumption, all terms in degree |didjum| - (p - 1)pl in Mn,a are zero.
Proof. We have|
didjum| = p2(m - j) + p(j - i) + i
= (*, . . . , *, ir, p - 1, . . . , p - 1).
The result follows immediately. \Lambda
Lemma 4.11. Let 0 < s < pk. For any q >= 0, Pqp
k+s is in the right
ideal of A generated by {P1, Pp, . . . , Pp
k-1 }.
Proof. This is presumably well-known. It follows immediately from the Adem relations. \Lambda
Notice how these two lemmas can work together to show that a term is zero. If ff(|didjum|) = ff(m), and if we can see that didjum is in the
image of Pqp
k+s, where 0 < s < pk and k <= r (i.e., it is in the image of
Pl where l 6j 0 (mod pr)), then it is in the image of zero.
Now we begin covering particular cases of the form of the p-ary representations of i and j. Throughout we will continue frequently to use Adem relations on inadmissibles without explicit mention, as well as the basic relations in Mn,a that Ptum = 0 for 0 < t < pn-2, and the fact that 1-folds are of lower order than 2-folds, and our result already proven for 1-folds.
Lemma 4.12. Along with the common hypotheses above, suppose that jr 6= ir. Then didjum can be expressed as a sum of lower order terms.
20 DAVID J. PENGELLEY AND FRANK WILLIAMS Proof.
Part 1. We have
i = (in-1, . . . , ir+1, ir, p - 1, . . . , p - 1)
= Ipr+1 + (ir + 1)pr - 1 and
j = (jn-1, . . . , jr+1, jr, p - 1, . . . , p - 1)
= J pr+1 + (jr + 1)pr - 1 and
m = (a - 1, p - 1, . . . , p - 1)
= Apr+1 - 1. Then
djum = (-1)j Pffium,
where ffi = (A - J )pr+1 - (jr + 1)pr.
Further, |
djum| = (p - 1)ffi + m
= N pr+1 + (jr + 1)pr - 1. Subtracting i, we get
fl = M pr+1 + (jr - ir)pr, so that
didjum = (-1)i+j PflPffium.
We compute:
Pp
r d
i+pr (djum) = (-1)i+j+1 Pp
r Pfl-pr \Gamma Pffiu
m\Delta
= (-1)i+j+1
pr-1X
t=0
(-1)t+1`(p - 1)[M p
r+1 + (jr - ir - 1)pr - t] - 1
pr - pt 'P
fl-tPt \Gamma Pffium\Delta
= -(jr - ir)didjum +
pr-1X
t=1
ffltdi+ptdj-tum, for some fflt 2 Fp.
For 1 <= t <= pr-1 - 1, dj-tum is unchosen, and hence zero. The remaining two parts of the proof will analyze the orders of di+prdj-pr-1um andP
prdi+prdjum.
Part 2. Note that
di+prdj-pr-1um = (-1)i+j Pfl-p
r-1 Pffi+pr-1 u
m.
BEYOND THE HIT PROBLEM 21 We aim to hit this with Pp
r-1 from
Pfl-2p
r-1Pffi+pr-1u
m = (-1)i+j+1 di+pr+pr-1dj-pr-1um.
Now
i + pr + pr-1 = (in-1, . . . , ir+1, ir + 2, 0, p - 1, . . . , p - 1) (with possible abuse of notation since ir + 2 may not be a valid digit) and
j - pr-1 = (jn-1, . . . , jr+1, jr, p - 2, p - 1, . . . , p - 1) .
Write K = (in-1, . . . , ir+1, ir + 2) . The only possible chosen two-fold in the topological degree of di+pr+pr-1dj-pr-1um is di+pr+pr-1-Kprdj-pr-1+Kpr-1um, i.e., if dj-pr-1+Kpr-1um is chosen. Since K >= 3, we can compute that
the image of this term under Pp
r-1 is of lower order than d
idjum. Thus
the image under Pp
r-1 of d
i+pr+pr-1dj-pr-1um is of lower order thand
idjum.
We compute
Pp
r-1Pfl-2pr-1Pffi+pr-1 u
m = (-1)i+j+1 di+prdj-pr-1um
+ (-1)i+j di+pr+pr-1dj-pr-1-pr-2um + terms that are zero since dj-pr-1-tum is unchosen for values of t between 0 and pr-2.
Iterating this analysis creates a downward induction allowing us to conclude that di+pr dj-pr-1um is a sum of terms of lower order than didjum.
Part 3. Finally we consider di+prdjum. It is unchosen, so it is the sum of lower order terms. What can they be? Well,
i + pr = (in-1, . . . , ir+1, ir + 1, p - 1, . . . , p - 1) (as above, ir + 1 might not be a valid digit) and
j = (jn-1, . . . , jr+1, jr, p - 1, . . . , p - 1). For 0 < t <= pr-1, di+pr-ptdj+t is unchosen (since dj+t is unchosen), and so di+pr djum is a sum of terms of order lower than didj+pr-1um. Hence by calculation similar to Part 1, Pp
r d
i+pr djum is the sum of terms of
order lower than didjum. \Lambda
Lemma 4.13. Along with the common hypotheses, suppose that ir = jr, and either jr 6= p - 2, or i > i(j + pr). Then didjum can be expressed as a sum of terms of lower order than itself.
22 DAVID J. PENGELLEY AND FRANK WILLIAMS Proof. As above, write
i = Ipr+1 + (ir + 1)pr - 1, j = Jpr+1 + (jr + 1)pr - 1,
and m = Apr+1 - 1. As above, we may write
didjum = (-1)i+j PflPffium, where
ffi = (A - J)pr+1 - (jr + 1)pr
and
fl = M pr+1 + (jr - ir)pr.
We compute
Pp
r+1d
idj+prum = (-1)i+j+1 Pp
r+1Pfl+pr-pr+1Pffi-pr u
m
= lower order terms + (-1)j+1 diPp
r P(A-J)pr+1-(jr+2)pr u
m
= lower order terms + (jr + 1)didjum.
So, since ir = jr, by the definition of r we have jr 6= p - 1. Hence didjum is equal to a unit multiple of Pp
r+1d
idj+prum plus lower order
terms.
Now consider Pp
r+1d
idj+prum. We have
j + pr = (jn-1, . . . , jr+1, jr + 1, p - 1, . . . , p - 1).
If i > i(j +pr), then inductively didj+prum is a sum of lower order terms than itself and hence, using basic module relations and Adem relations,P
pr+1didj+prum is a sum of terms of lower order than didjum.
So the only values of i to consider are i(j) < i <= i(j + pr) and jr 6= p - 2. Here, we have jr 6= ir, so the preceding lemma applies. \Lambda
Remark 4.14. At this point, the cases still to check are those in which ir = jr = p - 2 and i <= i(j + pr).
Lemma 4.15. Along with the common hypotheses, suppose that r <= n - 3. Suppose also that there is an integer s such that r < s <= n - 2 and jl = p - 2 for s > l >= r, and that js 6= p - 2. Then didjum can be expressed as a sum of lower order terms.
Proof. We may write
j = Jps+1 + (js + 1)ps - ps-1 - * * * - pr - 1,
BEYOND THE HIT PROBLEM 23 and
m = Aps+1 - 1.
Then
djum = (-1)j Pffium,
where
ffi = (A - J )ps+1 - (js + 1)ps + ps-1 + * * * + pr.
We compute
Pp
s+1d
idj+psum = lower order terms + diPp
sd
j+psum.
Further calculation with the Adem relations gives
diPp
sd
j+psum = (js + 2)didjum.
Whence P
ps+1didj+psum = (js + 2)didjum + lower order terms.
What about didj+ps? Well,
j + ps = (jn-1, . . . , js + 1, p - 2, . . . , p - 2, p - 1, . . . , p - 1), so i(j + ps) = i(j). Hence didj+psum is unchosen. It is in lower topological degree, so it is a sum of terms of lower order than itself. Hence Pp
s+1 d
idj+psum is a sum of terms of lower order than didjum. \Lambda
Lemma 4.16. Along with the common hypotheses, suppose that r <= n - 3. Also suppose that i(j + pr) >= i > i(j). Suppose that ir = p - 2, that jr = * * * = jn-2 = p - 2, and that jn-1 6= a - 1. Then didjum can be expressed as a sum of lower order terms.
Proof. We have, using the basic module relations, Pp
nd
idj+pn-1um = {terms of lower order than didj}+diPp
n-1 d
j+pn-1um
and
j = (jn-1, p - 2, . . . , p - 2, p - 1, . . . , p - 1).
By our inductive assumption, Mn,a is isomorphic to the corresponding filtered quotient of H *(CP (1)) in topological degrees below that of didjum. Hence we may calculate P p
n-1 d
j+pn-1um in H*(CP (1)), wherewe label the generator x. We have
Pp
n-1 d
j+pn-1xm = (-1)j+1 `pm - (p - 1) j - p
n + pn-1
pn-1 '`
m j + pn-1'x
M
and
djxm = (-1)j `mj 'xM ,
24 DAVID J. PENGELLEY AND FRANK WILLIAMS where M = pm - (p - 1) j We calculate that \Gamma pm-(p-1)j-p
n+pn-1
pn-1 \Delta = jn-1+2, that \Gamma mj+pn-1\Delta = (-1)n-r-1\Gamma a-1jn-1+1\Delta , and that \Gamma mj \Delta = (-1)n-r-1\Gamma a-1jn-1\Delta .
So P
pndidj+pn-1um = (a unit) * didjum + lower order terms.
Since didj+pn-1 is unchosen, we have by the inductive assumption that didj+pn-1um is a scalar multiple of the chosen in its degree, namely di-pr+1dj+pn-1+pr um. Hence Pp
nd
idj+pn-1um is a multiple of Pp
nd
i-pr+1dj+pn-1+pr umwhich we may check, using the inductive assumption, is the sum of
terms of lower order than didjum. \Lambda Lemma 4.17. Along with the common hypotheses, suppose that r <= n - 3. Also suppose that i(j + pr) >= i > i(j), and ir = p - 2, and that jr = * * * = jn-2 = p - 2, that jn-1 = a - 1, and that ir+1 6= p - 2. Then didjum can be expressed as a sum of lower order terms.
Proof. We have
i = (0, . . . 0, ir+1, p - 2, p - 1, . . . , p - 1)
= (ir+1 + 1)pr+1 - pr - 1 and
j = (a - 1, p - 2, . . . , p - 2, p - 1, . . . , p - 1).
So
didjum = (-1)i+j PflPffium,
where
fl = (a + 1)pn-1 - (ir+1 + 1)pr+1
and
ffi = pn-2 + pn-3 + * * * + pr.
We have
i + pr+1 = (0, . . . , ir+1 + 1, p - 2, p - 1, . . . , p - 1). Consider di+pr+1djum. There is a chosen monomial in this degree, it is didj+pr um. Using the inductive assumption, we may compute that
di+pr+1djum = \Gamma -1ir+1+1\Delta didj+prum. We now compute (-1)i+j+1 Pp
r+1 on
both sides of this equation. On the left-hand side, using the Adem relations, we obtain
(-1)i+j+1 Pp
r+1d
i+pr+1djum = Pp
r+1Pfl-pr+1Pffiu
m
= -(ir+1 + 1)PflPffium +
r-1X
s=0
^sPfl-p
r-1-***-psPffi+pr-1+***+psu
m
BEYOND THE HIT PROBLEM 25 for scalars ^s. The terms in the summation can be shown to be zero by Lemmas 4.10 and 4.11 (note that di-pr+1dj+pr is chosen, so ff(|didjum|) = ff(m) for applying Lemma 4.10). So we have
(-1)i+j+1 Pp
r+1d
i+pr+1djum = - (-1)i+j (ir+1 + 1)didjum.
Next, we have, again using the Adem relations, that
(-1)i+j+1 Pp
r+1d
idj+prum = Pp
r+1Pfl-pr+1+prPffi-pr u
m
= -(ir+1 + 2)Pfl+p
r Pffi-pr u
m + PflPffium
= (-1)i+j (-(ir+1 + 2)di-pr+1dj+prum + didjum) .
So
-(ir+1+1)didjum = ` -1i
r+1 + 1' (-(i
r+1 + 2)di-pr+1dj+prum + didjum) .
Since ir+1 6= 0 or p - 2, (ir+1 + 1)2 6= 1. Hence didjum is a multiple of di-pr+1dj+prum, a lower order term. \Lambda
Lemma 4.18. Along with the common hypotheses, suppose that 0 <= r < n - 3, and i(j + pr) >= i > i(j). Suppose that ir = jr and that jr = * * * = jn-2 = p - 2, that jn-1 = a - 1, and that ir+1 = p - 2. Then didjum can be expressed as a sum of lower order terms.
Proof. We have
i = (0, . . . 0, p - 2, p - 2, p - 1, . . . , p - 1) and
j = (a - 1, p - 2, . . . , p - 2, p - 1, . . . , p - 1).
So
didjum = (-1)i+j PflPffium,
where
fl = (a + 1)pn-1 - pr+2 + pr+1
and
ffi = pn-2 + pn-3 + * * * + pr.
We have
i + pr+2 = (0, . . . , 1, p - 2, p - 2, p - 1, . . . , p - 1). Consider di+pr+2djum. There is a chosen monomial in this degree; it is di-pr+1dj+pr+1+pr um. By the inductive assumption, di+pr+2djum =
Kdi-pr+1dj+pr+1+prum, for some K 2 Fp. We now compute Pp
r+2 on
26 DAVID J. PENGELLEY AND FRANK WILLIAMS both sides of this equation. On the left-hand side, we use the hypothesis that r < n - 3 and Lemmas 4.10 (note that di-pr+1dj+pr is chosen, so ff(|didjum|) = ff(m)) and 4.11 to obtain
Pp
r+2d
i+pr+2djum = (-1)i+j+1 Pp
r+2Pfl-pr+2Pffiu
m
= 2 (-1)i+j Pfl Pffium = 2didjum.
On the right-hand side, Pp
r+2d
i-pr+1dj+pr+1+prum consists of terms oflower order than d
idjum. \Lambda
Lemma 4.19. Along with the common hypotheses, suppose that r = n - 3, and i(j + pr) >= i > i(j). Suppose that ir = jr and that jr = * * * = jn-2 = p - 2, that jn-1 = a - 1, and that ir+1 = p - 2. Then didjum can be expressed as a sum of lower order terms.
Proof. We have
j = (a - 1, p - 2, p - 2, p - 1, . . . , p - 1) and
i = (0, p - 2, p - 2, p - 1, . . . , p - 1).
If a = 1, the conclusion is an immediate consequence of one of the defining relations for our module. So let a > 1.
Begin by noting that there is a chosen in the degree of didjum, it is di-pn-2dj+pn-3um. Hence the ff-number of this degree is ff(m), so we may use Lemmas 4.10, 4.11. Now consider di+pn-1djum. Inductively it is a multiple of the chosen in this topological degree, which is a onefold, di-pn-2um. We may then compute, as in the preceding lemmas,
that Pp
n-1 d
i+pn-1djum is of lower order than didjum. We also computeP
pn-1 di+pn-1djum = (-1)i+j+1 Ppn-1 P(a-1)pn-1+pn-2Ppn-2+pn-3um
= -(a - 1)didjum + (-1)i+j+1 Pap
n-1 P2pn-2+pn-3 u
m.
We shall show that the second term on the right is of order lower than didjum. We have
Pp
n-1 P(a-1)pn-1 P2pn-2+pn-3 u
m = aPap
n-1 P2pn-2+pn-3 u
m + Pap
n-1-pn-3 P2pn-2+2pn-3 u
m
+ 2Pap
n-1-pn-3-2pn-4P2pn-2+2pn-3+2pn-4 u
m
+ Pap
n-1-2pn-4P2pn-2+pn-3+pn-4u
m-
2Pap
n-1-pn-2P3pn-2+pn-3u
m.
The left side is of lower order for the same reason as P p
n-1 d
i+pn-1djumwas above. The third and fourth terms on the right are zero by Lemmas
BEYOND THE HIT PROBLEM 27 4.10 and 4.11. Further, note that|
didjum| = (a, 1, p - 2, p - 2, p - 1, . . . , p - 1) , so the terms Pap
n-1-3pn-3 P2pn-2+2pn-3 u
m and Pap
n-1-3pn-2 P3pn-2+pn-3u
m
lie in degrees with lesser ff-number and hence are zero by the inductive
assumption. We compute that
0 = P2p
n-3 Papn-1-3pn-3P2pn-2+2pn-3u
m = Pap
n-1-pn-3 P2pn-2+2pn-3u
m
+ terms that are zero by Lemmas 4.10 and 4.11.
If p = 3, P3p
n-2+pn-3u
m is unchosen. For p 6= 3, a calculation similar
to the preceding one shows that
0 = P2p
n-2 Papn-1-3pn-2 P3pn-2+pn-3u
m = Pap
n-1-pn-2 P3pn-2+pn-3u
m
+ terms that are zero by Lemma 4.10 and 4.11.
\Lambda Remark 4.20. We have now completed all cases in which 0 <= r <= n-3 or a = 1.
Lemma 4.21. Along with the common hypotheses, suppose that r = n - 2 and a > 1, that in-2 = jn-2 = p - 2, and that jn-1 = a - 1. Then in-1 6= 0, and
in-1didjum = (-1)j+1 adi-pn-1 um,
so didjum can be expressed as a sum of lower order terms.
Proof. We have
i = (in-1, p - 2, p - 1, . . . , p - 1)
= (in-1 + 1)pn-1 - pn-2 - 1 and
j = (a - 1, p - 2, p - 1, . . . , p - 1)
= apn-1 - pn-2 - 1. We have
didjum = (-1)i+j P(a-in-1)p
n-1Ppn-2 u
m
and
di-pn-1 um = (-1)i-1 P(a-in-1)p
n-1+pn-2u
m.
Set k = a - in-1. Then we are to prove that
(a - k)Pkp
n-1 Ppn-2 u
m = aPkp
n-1+pn-2u
m,
Notice that in-1 6= 0 by unchosenness, and in-1 <= a-1 by admissibility, so we are considering 1 <= k <= a - 1.
28 DAVID J. PENGELLEY AND FRANK WILLIAMS
For k = 1, this equation is one of the defining relations for our module. Inductively assume that the relation holds for a fixed value of k. We shall apply Pp
n-1 to both sides of the equation and use Adem
relations. On the left hand side,P
pn-1 Pkpn-1Ppn-2 um = (k + 1)P(k+1)pn-1Ppn-2 um
+ P(k+1)p
n-1-pn-3Ppn-2+pn-3 u
m + 2P(k+1)p
n-1-pn-2P2pn-2 u
m.
On the right hand side,P
pn-1 Pkpn-1+pn-2um = kP(k+1)pn-1+pn-2 um + P(k+1)pn-1 Ppn-2 um.
We shall prove that P(k+1)p
n-1-pn-3 Ppn-2+pn-3 u
m = 0 = P(k+1)p
n-1-pn-2P2pn-2 u
m.
Granting this, we compute
(a-k)(k+1)P(k+1)p
n-1 Ppn-2 u
m = akP(k+1)p
n-1+pn-2u
m+aP(k+1)p
n-1 Ppn-2 u
m.
Combining terms, simplifying, and cancelling k, which is nonzero mod p, we obtain our goal:
(a - k - 1)P(k+1)p
n-1 Ppn-2 u
m = aP(k+1)p
n-1+pn-2 u
m.
Next, we note that P(k+1)p
n-1-pn-3Ppn-2+pn-3 u
m = 0 by Lemmas 4.10
and 4.11 (which apply in this degree because there is the chosen 1-fold
di-pn-1um here). Finally, consider P(k+1)p
n-1-pn-2 P2pn-2 u
m. Well,P
2pn-2 P(k+1)pn-1-3pn-2 P2pn-2 um = P(k+1)pn-1-pn-2P2pn-2 um
+ terms that are zero by Lemmas 4.10 and 4.11. The left-hand side is zero by ff-number. This completes the proof of the Lemma. \Lambda
Lemma 4.22. Along with the common hypotheses, suppose that r = n - 2 and a > 1, and let ir = jr = p - 2. Suppose that jn-1 < a - 1. Then didjum can be expressed as a sum of lower order terms.
Proof. We have
i = (in-1, p - 2, p - 1, . . . , p - 1)
= (in-1 + 1)pn-1 - pn-2 - 1 and
j = (jn-1, p - 2, p - 1, . . . , p - 1)
= (jn-1 + 1) pn-1 - pn-2 - 1. Define
k = (a - 1, p - 2, p - 1, . . . , p - 1)
= apn-1 - pn-2 - 1.
BEYOND THE HIT PROBLEM 29 By the previous Lemma, there is a unit A such that Adjum = dj+pn-1dkum and a unit B such that didkum = Bdi-pn-1um. So, using the preceding lemma and Remark 4.8, we compute
Adidjum = didj+pn-1dkum
= di+p(j+pn-1-i)didkum + X
1<=t<=i/p
ffltdi-ptdj+pn-1+tdkum.
Now for each term in this sum, dj+pn-1+tdkum is unchosen and in lower topological degree than didjum, hence inductively can be expressed using terms of lower order than itself. But from our result for 1- folds, the only d with index larger than k that can act nontrivially on um is dm, so dj+pn-1+tdkum collapses to a 1-fold, and thus each term di-ptdj+pn-1+tdkum can be expressed as a sum of admissibles of lower order than didjum. So we may continue with
Adidjum = di+p(j+pn-1-i)didkum + lower order terms than didjum
= Bdi+p(j+pn-1-i)di-pn-1 um + lower order terms than didjum = Bdi-pn-1 dj+pn-2um + lower order terms than didjum = lower order terms than didjum.
\Lambda
4.3. Higher-fold monomials. We now move to the proof for threeand higher-folds. In this part, we consider didjdkdLum, admissible and unchosen, with L possibly empty, and with the last subscript of the entire monomial less than m. The goal is to show that didjdkdLum can be expressed as a sum of lower order terms.
Assume inductively that every unchosen admissible of length degree less than that of didjdkdL, when applied to um, can be expressed in terms of lower order terms than itself. From our previous steps, we may assume that djdkdL is chosen. Hence didj is unchosen.
Lemma 4.23. Under these hypotheses, didjdkdLum is a sum of lower order terms.
Proof. Let the symbol j represent congruence modulo terms of order lower than didjdkdL applied to um. Using Adem relations (especially Remark 4.8 in both directions of the equality, and recalling that Adem relations on inadmissibles always reduce lexicographic order) and the inductive assumption, we obtain (where all monomial terms below have
30 DAVID J. PENGELLEY AND FRANK WILLIAMS length no greater than that of didjdkdL).
didjdkdLum = didj+(k-j)pdjdLum + X
t>0
"tdidj-ptdk+tdLum, for some "t 2 Fp
j didj+(k-j)pdjdLum = di+(j+(k-j)p-i)pdidjdLum + X
s>0
jsdi-psdj+(k-j)p+sdjdLum, for some js 2 Fp
j di+(j+(k-j)p-i)pdidjdLum = di+(j+(k-j)p-i)p * X (terms of order lower than didjdLum) j di+(j+(k-j)p-i)p * X
i0*= 2, the element P(p-a+1)p
n-2 t
apn-1-1 iteratesto Pp-a+1t
ap-1 2 M2,a. But M2,a has no lower degree relation thanPp-a+1 tap-1 = 0, which involves an unstable admissible basis element,
since a >= 2.
BEYOND THE HIT PROBLEM 31 When a = 1, we can iterate ^V on the given special relation to its lowest full incarnation 2Pp
2+p+1Ppu
p2-1 = Pp
2+pPp+1u
p2-1 in M3,1. InM
3,1 the only other generating relation is P1up2-1. We thus study the two terms of this lowest special relation as elements in the quotient of
the free unstable module on up2-1 by only the single relation P1up2-1. In fact we can choose to project these terms yet further to the free unstable module on up+1 of degree p + 1, subject to P 1up+1 = 0, since there is a natural epimorphism from the free unstable module on up2-1
to the one on up+1. There Pp
2+p+1Ppu
p+1 is zero, since Pp
2+p+1Pp has
excess 2p + 1. On the other hand, P p
2+pPp+1u
p+1 is nonzero there,
as the reader may easily verify: using the admissible basis for A, one
finds by direct calculation and the unstable admissible basis for the free module on up+1 that the submodule AP1up+1 of the free module
contains no element with Pp
2+pPp+1u
p+1 as a term. Thus the special
relation cannot hold in M3,1, so all the a = 1 special relations are
nonredundant. \Lambda
6. Proofs of Theorems 2.19, 2.23, and 2.25 We begin this section by proving our uniqueness theorem for cyclic modules.
Proof of Theorem 2.19. We shall begin by checking that the defining relations for Mn,a are satisfied in M. Let u 2 Mapn-1-1 be nonzero. We have ff0 = (n-1)(p-1)+(a-1), since apn-1-1 = (a-1, p-1, . . . , p-1). The next larger integer with its ff-number is (a, p - 2, p - 1, . . . , p - 1) = (a + 1)pn-1 - pn-2 - 1. Hence Pp
tu = 0, for 0 <= t <= n - 3.
Let a > 1. We have fififiP(p-a+1)p
n-2 ufififi = pn + (a - 1)pn-2 - 1, whose
ff-number is (n - 2) (p - 1) + a - 1. So P (p-a+1)p
n-2 u = 0. In this case
it remains to check the third relation.
Begin by noting that Pap
n-1+pn-2u = 0, by unstability. The degree
of this term is apn + (p - 1) pn-2 - 1. The next smaller integer having ff-number ff0 is apn - p. So by Lemma 4.11, Pap
n-1 Ppn-2 u is the only
possible nonzero admissible in its degree (note there are no possible nonzero admissibles of greater length in this degree). Since the ffnumber is ff0, it is non-zero. Let 0 <= l <= a - 2. We shall induct on l. Assume that lP(a-l)p
n-1 Ppn-2 u = aP(a-l)pn-1+pn-2 u (this is true for
l = 0 from above; note that this induction goes downward in topological degree). As in the proof of Lemma 4.21, under the hypotheses of the present theorem, the Adem relations yield
Pp
n-1 P(a-l-1)pn-1+pn-2u = (a-l-1)P(a-l)pn-1+pn-2u+P(a-l)pn-1 Ppn-2 u
32 DAVID J. PENGELLEY AND FRANK WILLIAMS and P
pn-1 P(a-l-1)pn-1 Ppn-2 u = (a - l)P(a-l)pn-1 Ppn-2 u.
The second equation, and the fact that P ap
n-1 Ppn-2 u is nonzero from
above, shows us inductively that in these degrees, left action by P p
n-1
is a (nonzero) isomorphism. We compute:
Pp
n-1 (aP(a-l-1)pn-1+pn-2u - (l + 1)P(a-l-1)pn-1 Ppn-2 u
= a(a - l - 1)P(a-l)p
n-1+pn-2u + aP(a-l)pn-1 Ppn-2 u - (l + 1)(a - l)P(a-l)pn-1 Ppn-2 u
= (a - l - 1)(aP(a-l)p
n-1+pn-2u - lP(a-l)pn-1 Ppn-2 u) = 0.
Hence aP(a-l-1)p
n-1+pn-2u - (l + 1)P(a-l-1)pn-1 Ppn-2 u 2 ker Ppn-1 =
0. So aP(a-l-1)p
n-1+pn-2u = (l + 1)P(a-l-1)pn-1 Ppn-2 u. Taking l =
a - 2, we obtain the desired relation.
Now take a = 1. We compute, using Lemma 4.11, that
P(p-2)p
n-2 Ppn-1+pn-2 Ppn-2+pn-3u = -Ppn-1+(p-1)pn-2 Ppn-2+pn-3 u
and that
P(p-2)p
n-2 Ppn-1+pn-2+pn-3Ppn-2 u = -2Ppn-1+(p-1)pn-2 Ppn-2+pn-3 u.
As above, this yields
Pp
n-1+pn-2+pn-3 Ppn-2 u = 2Ppn-1+pn-2 Ppn-2+pn-3u,
as desired, provided we know that P(p-2)p
n-2 is acting on the left as a
monomorphism between these degrees. This follows by checking, from the basic relations and Lemma 4.11, that P p
n-1+(p-1)pn-2 Ppn-2+pn-3u is
the only possible nonzero admissible in its degree, and thus is nonzero by the hypothesis of the theorem.
In all cases, the defining relations for Mn,a are satisfied in M. Hence taking tapn-1-1 to u defines an A-module map Mn,a ! M. Since M is cyclic, this map is surjective. By Theorems 2.11 (and its remark), 2.17, and the hypothesis on M, it must be an isomorphism. \Lambda
We next give the proof of the presentation for H *(BU ). Proof of Theorem 2.23. Recall the A-algebra map OE : G ! H *(BU ) defined by taking o/apn-1 to the Chern class capn-1 . By the definition ofG
, the following relations are satisfied in the indecomposable quotient QG:
For 0 <= l <= n - 3
Pp
l o/
apn-1 = Pap
n-2 Pplo/
apn-2 ,
BEYOND THE HIT PROBLEM 33 and for a >= 2
(a - 1)P(p-a+1)p
n-2 o/
apn-1 = `p - 1a - 2'Pp
n-1 Ppn-2o/
apn-2
and
(a - 1)Pp
n-1 Ppn-2o/
apn-1 = aPp
n-1+pn-2o/
apn-1 ,
while for a = 1
2Pp
n-1+pn-2Ppn-2+pn-3 o/
apn-1 = Pp
n-1+pn-2+pn-3Ppn-2 o/
apn-1 .
Since the analogous relations define the double (topological) suspension \Sigma 2M, there is a surjection of A-modules ss : \Sigma 2M ! QG given by taking \Sigma 2tapn-1-1 to o/apn-1 . By Theorem 2.2 and Remark 2.20 the map \Sigma 2M ! QH*(BU ) given by taking \Sigma 2tapn-1-1 to capn-1 is an isomorphism. Hence the induced A-module map QOE : QG ! QH *(BU ) is an isomorphism. Since H*(BU ) is a free commutative algebra, this guarantees that the algebra map OE is an isomorphism. \Lambda
Finally, we prove that there are no A-relations amongst Chern classes. Proof of Theorem 2.25. Since the symmetric algebra is an A-subalgebra of the polynomial algebra on variables {xi} of complex degree one, we shall work in the polynomial algebra. Identifying the Chern class cmwith the m-th elementary symmetric polynomial in the variables{
xi} , we see that cm has a term x1 * * * xm as a summand. Applying any element of K to this term results in a polynomial having all summands of the form ffxp
e1
1 * * * xp
em m , ff 2 Fp. By symmetry, atleast one of these terms satisfies the inequalities e
1 >= * * * >= em. We call a monomial of this form a basic monomial. For such a term, if
1 <= l <= e1+1, let rl be the number of occurrences of pe1-l+1 as an exponent in xp
e1
1 * * * xp
em m . Then we shall refer to the tuple (r1, r2, . . . , re1+1)
as the type of xp
e1
1 * * * xp
em m . We say that a tuple (m1, . . . , ma) has higherorder than a tuple (n
1, . . . , nb) provided that a > b, or if a = b, provided that it is greater in lexicographic order, ordering from the left. Let
dJ = dj1 * * * djs, js < m, be an admissible monomial in K. Then direct calculation reveals that dJ cm has a basic monomial summand whose type is (m - js, js - js-1, . . . , j2 - j1, j1). One can check that this term has the highest order of all basic monomials occurring in dJ cm. Since we can recover m and J from this type, this proves that the free moduleR
injects into H*(BU ). \Lambda
34 DAVID J. PENGELLEY AND FRANK WILLIAMS
7. Appendix : The Kudo-Araki-May algebra K We recall here just the bare essentials about K needed to understand the proofs in this paper. We refer the reader to [11] for much more extensive information about K.
The odd-primary (even) topological Kudo-Araki-May algebra K is the Fp-bialgebra (with identity) generated by elements {di : i >= 0} subject to homogeneous (Adem) relations
didj = X
l
(-1)pl-i `(p - 1) (l - j) - 1pl - i - (p - 1) j 'di+pj-pldl for all i, j >= 0,
with coproduct OE determined by the formula
OE(di) =
iX
t=0
dt \Omega di-t.
It is bigraded by length and complex topological degrees (|di| = (p - 1)i), which behave skew-additively under multiplication: |xy| = |x| + p |y|. Moreover, K is finite in each bidegree. A monomial di1 * * * din is called admissible provided that i1 <= * * * <= in. The admissible monomials provide a vector space basis for K.
The Fp-cohomology of any space concentrated in even dimensions is an unstable algebra over the Steenrod algebra A with no Bocksteins, and there is a correspondence between unstable A-algebras and unstable K-algebras, completely determined by iterating the conversion formulae:
(-1)j djuq = Pq-juq, where uq is a cohomology class of (complex) degree q. Since the degree of the element is involved in the conversion, and this changes as operations are composed, the algebra structures of A andK
are very different, and the skew additivity of the bigrading in K reflects this. Note for use in calculation that since P q-j has complex degree (p - 1) (q - j), the complex degree of djuq is pq-(p - 1) j. More generally, for a monomial dI = di1di2 * * * dil 2 K of length l, degrees in a K-module and in K itself are related by
|dIuq| = plq - |dI | . The requirements for an unstable K-algebra, corresponding to the nature and requirements of an unstable A-algebra, are: On any element xl of complex degree l,
dlxl = (-1)l xl, djxl = 0 for j > l, and d0xl = xpl .
BEYOND THE HIT PROBLEM 35 Finally, and used in our proofs, the K-algebra structure obeys the (Cartan) formula according to the coproduct OE in K:
di(xy) =
iX
t=0
dt(x)di-t(y).
References [1] D. Benson, V. Franjou, S'eries de d'ecompositions de modules instables et in-jectivit'e de la cohomologie du groupe Z
/2, Math. Zeit. 208 (1991), 389-399.
[2] G. Carlsson, G.B. Segal's Burnside ring conjecture for (Z/2)k, Topology 22(1983), 83-103.
[3] V. Franjou, L. Schwartz, Reduced unstable A-modules and the modular repre-sentation theory of the symmetric groups, Ann. Sci. Ec. Norm. Sup.
23 (1990),593-624.
[4] A. Kono, D. Tamaki, Generalized Cohomology, Translations of MathematicalMonographs 230, American Mathematical Society, 2006. [5] J. Lannes, S. Zarati, Sur les U-injectifs, Ann. Sci. Ec. Norm. Sup. 19 (1986),1-31. [6] H. Miller, The Sullivan conjecture on maps from classifying spaces, Annalsof Math.
120 (1984), 39-87; and corrigendum, Annals of Math. 121 (1985),605-609.
[7] D. Pengelley, F. Peterson, F. Williams, A global structure theorem for the mod2 Dickson algebras, and unstable cyclic modules over the Steenrod and KudoAraki-May algebras, Math. Proc. Camb. Phil. Soc. 129 (2000), 263-275.[8] D. Pengelley, F. Williams, Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology, Transactions of the American MathematicalSociety 352 (2000), 1453-1492. [9] D. Pengelley, F. Williams, Global structure of the mod two symmetric algebra,
H*(BO; F2), over the Steenrod Algebra, Algebraic & Geometric Topology 3(2003), 1119-1138.
[10] D. Pengelley, F. Williams, The global structure of odd-primary Dickson algebrasas algebras over the Steenrod algebra, Math. Proc. Camb. Phil. Soc. 129 (2000),
263-275.[11] D. Pengelley, F. Williams, The odd-primary Kudo-Araki-May algebra of algebraic Steenrod operations and invariant theory, to appear in Proceedings,International School and Conference in Algebraic Topology in honor of Huynh Mui, Vietnam National University, Hanoi, 2004, Geometry and Topology Pub-lications Monograph Series. [12] F. P. Peterson, A mod p Wu formula, Bol. Soc. Mat. Mexicana (2) 20 (1975),no. 2, 56-58. [13] L. Schwartz, Unstable modules over the Steenrod algebra, functors, and thecohomology of spaces, in Infinite Length Modules (Bielefeld, 1998), 229-249,
Trends in Mathematics, Birkh"auser, Basel, 2000.[14] L. Schwartz, Unstable modules over the Steenrod Algebra and Sullivan's fixed point conjecture, University of Chicago Press, 1994.
36 DAVID J. PENGELLEY AND FRANK WILLIAMS [15] P. Brian Shay, Mod p Wu formulas for the Steenrod algebra and the Dyer-Lashof algebra, Proceedings of the American Mathematical Society
63 (1977),339-347.
[16] N.E. Steenrod, D.B.A. Epstein, Cohomology operations, Princeton Univ. Press,1962.
New Mexico State University, Las Cruces, NM 88003E-mail address:
davidp@nmsu.edu
New Mexico State University, Las Cruces, NM 88003E-mail address:
frank@nmsu.edu*