GLOBAL STRUCTURE OF THE MOD TWO SYMMETRIC ALGEBRA, H*(BO; F2), OVER THE STEENROD ALGEBRA DAVID J. PENGELLEY AND FRANK WILLIAMS The first author dedicates this paper to his parents, Daphne M. and Eric T. Pengelley, in memoriam. Abstract. The algebra S of symmetric invariants over the field with two elements is an unstable algebra over the Steenrod alge- bra A, and is isomorphic to the mod two cohomology of BO, the classifying space for vector bundles. We provide a minimal presen- tation for S in the category of unstable A-algebras, i.e., minimal generators and minimal relations. From this we produce minimal presentations for various unstable A-algebras associated with the cohomology of related spaces, such as the BO(2m - 1) that classify finite dimensional vector bundles, and the connected covers of BO. The presentations then show that certain of these unstable A-algebras coalesce to produce the Dick- son algebras of general linear group invariants, and we speculate about possible related topological realizability. Our methods also produce a related simple minimal A-module presentation of the cohomology of infinite dimensional real projec- __ tive space, with filtered quotients the unstable modules F (2p- 1)=AA p-2, as described in an independent appendix. December 1, 2003 1. Introduction We continue our study [9] of invariant algebras as unstable algebras over the Steenrod algebra A by proving a structure theorem for the algebra S of symmetric invariants over the field F2. The algebra S is isomorphic to the mod two cohomology of BO, the classifying space for vector bundles [8], and we identify the two. We also make several applications to the cohomology of related spaces, which then reveal a relationship between S and the Dickson algebras [13]. ___________ 1991 Mathematics Subject Classification. Primary 55R45; Secondary 13A50, 16W22, 16W50, 55R40, 55S05, 55S10. Key words and phrases. Symmetric algebra, Steenrod algebra, unstable algebra, classifying space, Dickson algebra, BO, real projective space. 1 2 DAVID J. PENGELLEY AND FRANK WILLIAMS Our goal is to provide a minimal presentation for S = H*(BO; F2) in the category of unstable A-algebras [11], beginning with a minimally presented generating A-module and then introducing a minimal set of A-algebra relations. This reveals how a minimal set of A-module building blocks for S fit together in its A-algebra structure. In brief, our main result (Theorem 3.5) is that S = H*(BO; F2) is minimally presented in the category of unstable A-algebras as the free unstable A-algebra on the two-power Stiefel-Whitney classes w2k modulo rela- i tions expressing the fact that, for each i k - 2, Sq2 w2k differs from k-1 2i Sq2 Sq w2k-1by a decomposable. (By contrast, and at first seem- ingly paradoxically, we shall also see (Theorem 2.3) that while S is generated as an A-algebra by {w2k: k 0}, with relations linking the resulting algebra generators, in fact the A-submodule of S generated by {wm : m 0} is a free unstable A-module on all the Stiefel-Whitney classes.) We apply this structure theorem to characterize similarly the coho- mology images B*(n) for the connected covers of BO (Theorem 4.2) [3], which include the full cohomology algebras of BSO, BSpin, and BO <8>. We likewise characterize the quotients H*(BO(q); F2) for the classifying spaces of finite dimensional vector bundles [8], and in par- ticular (Theorem 4.3) we analyze H*(BO(2n+1 - 1); F2). Finally, we shall produce an A-algebra epimorphism from S = H*(BO; F2) to each of the mod two Dickson algebras (Theorem 4.4), which we characterized in [9] as unstable A-algebras. In fact we shall show that the (n + 1)-st Dickson algebra has the role of capturing precisely the quotient of S = H*(BO; F2) common to the cohomology of the n-th distinct connected cover BO and to BO(2n+1-1). We speculate about how this phenomenon may relate to spaces beyond the range in which Dickson algebras are directly realizable topologically. Our minimal A-algebra presentations for all the above objects will devolve naturally from our main presentation of S, and in that sense these A-algebras are all äp rallel" to the main presentation. In Appendix I, which can be read independently of the rest of the paper, we present_a related result, in which the unstable A-modules F (2p - 1)=AA p-2appear as the filtered quotients of a simple minimal A-presentation for H*(RP 1; F2). We thank Don Davis, Kathryn Lesh, and Haynes Miller for useful conversations regarding these modules. 2. Motivation, first steps, and a plan The unstable A-algebra of symmetric invariants S = H*(BO; F2) is a polynomial algebra F2[wm : m 0, w0 = 1], with each elementary STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 3 symmetric function (Stiefel-Whitney class) wm having degree m [8]. The action of the Steenrod algebra is completely determined from the Wu formulas [3, 12, 14] Xj ` ' m - j + l - 1 Sqjwm = wj-lwm+l l=0 l and the Cartan formula on products [11]. To ease into our categorical point of view, and to illustrate our approach and methods, let us begin by seeing that abstract Stiefel- Whitney classes, taken all together as free unstable A-algebra genera- tors, along with imposed üW formulas", actually "present" S. This is something one might easily take for granted, but should actually prove, since in principle there might be ö ther" relations lurking in S beyond those inherent in the Wu formulas. To avoid confusion from notational abuse, we build from abstract classes tm which will correspond to the actual Stiefel-Whitney classes under an isomorphism. Proposition 2.1 (Wu formulas present S). The unstable A-algebra S = H*(BO; F2) is isomorphic to the quotient of the abstract free unstable A-algebra on classes tm in each degree m 1, modulo the left A-ideal generated by abstract üW formulas" formed by writing t's in place of w's in the Wu formulas above. Proof.Iterating the abstract Wu formulas via the Cartan formula shows that the abstract classes {tm : m 1} actually generate the abstract A-algebra quotient considered merely as an algebra, i.e., its (algebra) indecomposable quotient has rank at most one in each degree. On the other hand, by its construction the abstract A-algebra quotient must map onto S by sending each tm to wm , since the respective Wu formu- las correspond. Thus the two must be isomorphic, since S is free_as_a commutative algebra. |__| Notice, however, that this presentation of S is far from minimal in the category of unstable A-algebras, since it used vastly more generators than needed. What we seek instead is to achieve three features for a minimal presentation: Step 1: Find a minimal A-submodule of S that will generate S as an A-algebra. Step 2: Find a minimal presentation of this A-submodule, i.e., with min- imal generators and minimal relations. Step 3: Form the free unstable A-algebra U on this module, and find minimal relations on U so that its A-algebra quotient produces S. 4 DAVID J. PENGELLEY AND FRANK WILLIAMS To begin, let us find a minimal set of A-algebra generators for S. Consider the (algebra) indecomposable quotient QS, i.e., the vector space with basis {wm : m 1} and induced A-action ` ' m - 1 Sqjwm = wm+j . j m-1 Since j is always zero mod two when m + j is a two-power, and never zero when m is a two-power and j is less than m, we see that the A-module indecomposables of QS have basis exactly {w2k: k 0}. Since our philosophy is to begin the presentation at the A-module level, with minimal A-algebra generators and minimal module rela- tions, we thus start with Definition 2.2. Let M be the free unstable A-module on abstract classes {t2k : k 0}, where subscripts indicate the topological degree of each class. We wish to map M to S via t2k! w2k, and need first to ask whether M injects. In other words, is the A-submodule of S = H*(BO; F2) generated by {w2k : k 0} free? Or are there, to the contrary, A- relations amongst the two-power Stiefel-Whitney classes, which will compel us to introduce module relations on M in order to complete steps 1 and 2 above? The Wu formulas appear to suggest that no such relations exist. In fact we can prove something even stronger. Theorem 2.3 (Stiefel-Whitney classes inject freely).The A-submodule of S = H*(BO; F2) generated by {wm : m 0} is free unstable on these classes. The proof is in Section 5. Remark 2.4. The proof also shows that in H*(BO(q); F2) ~=H*(BO; F2)= (wm : m > q), the A-submodule generated by {wm : 0 m q} is free unstable on these classes. Remark 2.5. The fact that the free unstable A-module Fm on a sin- gle class in degree m injects into H*(BO(m); F2) on the class wm is clear from the already known result [4, p. 55] that Fm is isomor- m phic to the invariants F1m , which clearly inject naturally into (H*(RP 1; F2) m ) m ~=H*(BO(m); F2) on wm . Theorem 2.3 general- izes this by handling all Fm simultaneously, showing that they do not interfere when simultaneously perched on the Stiefel-Whitney classes in the symmetric algebra S = H*(BO; F2). STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 5 Corollary 2.6. The A-submodule of S = H*(BO; F2) generated by {w2k: k 0} is free unstable, so M injects naturally into S. This completes steps 1 and 2 of our goal, and we can begin step 3. Definition 2.7. Let U be the free unstable A-algebra on M, in other words, U is the free unstable A-algebra on abstract classes {t2k: k 0}. Clearly U maps via t2k ! w2k onto the desired A-algebra S, but the map has an enormous kernel, since QS is the vector space F2{wm : m 1}, while QU is much larger. Our goal in step 3 is to describe a minimal set of A-algebra relations producing S from U, i.e., minimal generators for the kernel as an A-ideal. Let us explore a prototype example in degree five, which is the first place a difference occurs. There QS has only w5, whereas Sq1t4 and Sq2Sq1t2 are distinct indecomposables in QU (recall that QU ~= M, and that a basis for M consists of the unstable admissible monomials on the A-generators t2k[11]). A few calculations with the Wu formulas show that in S we have Sq1w4 = w5 + w1w4 and Sq2Sq1w2 = w5 + w1w4 + w2w3 + w1w22+ w21w3 + w31w2. Thus to imitate S abstractly via U, we must impose an algebra relation on U decreeing that Sq1t4 = Sq2Sq1t2 + some decomposable, per the calculations above. One challenge in doing even this, though, is that it is not clear how to describe that needed decomposable difference in U, since there we have no name as yet for the element corresponding to w3. To remedy this, and to describe general formulas for relation- ships like the one we have just discovered, we wish to use the Wu formulas to focus our understanding as much as possible on both two- power Steenrod squares and two-power Stiefel-Whitney classes. Thus one of our formulas in the next section will express each Stiefel-Whitney class purely in this way (Lemma 3.2). While the plethora of algebra relations, such as the one above, needed to obtain S from U may appear intractable to specify, recall that our chosen task is actually somewhat different. Since we are work- ing in the category of A-algebras, we seek relations in U whose A- algebra consequences, not just their algebra consequences, will pro- duce S. We shall show that this requires only a much smaller and more tractable set of relations, for which our illustration in degree five serves as perfect prototype. Specifically, the relationship between i 2k-1 2i Sq2 w2k and Sq Sq w2k-1for every i k - 2 will be the key place 6 DAVID J. PENGELLEY AND FRANK WILLIAMS to focus attention. We shall impose one abstract relation on U for each such pair (k, i), and prove that these are precisely the minimal relations producing S = H*(BO; F2) in the category of A-algebras. Our general plan is as follows. Form our abstract presentation candi- date as just outlined; call it G. The construction of G will immediately provide a natural A-algebra epimorphism to S. The hard part now is showing that our (k, i)-indexed family of A-algebra relations leaves no remaining kernel, i.e., that we have put in enough relations to generate the kernel as an A-ideal. To achieve this we show that the epimorphism G ! S induces a monomorphism QG ! QS, on the indecomposable quotients, by computing a basis for QG. For this we appeal to our earlier understanding [9], via the Kudo-Araki-May algebra K [10] (see Appendix II), of bases for the unstable cyclic A-modules arising in the analogous structure theorem for the Dickson algebras. With QG ! QS an isomorphism, G ! S must be an isomorphism also, since S is a free commutative algebra. The minimality of the (k, i)-family of relations is then not hard to see by appropriate filtering. 3. Main theorem We first identify the key A-algebra relations in S = H*(BO; F2). Analysis of the binomial coefficients in the Wu formulas shows that if r 1, then j-1 (3.1) Sq2 wr2j= w2j-1wr2j+ w2j-1+r2j. This formula will serve two purposes. It will guide us below in how to specify any Stiefel-Whitney class from just the two-power ones, which is needed for creating our abstract presentation. But before this it will lead us to the key relations needed from S. To find these, recall from the previous section that we seek a relation i 2k-1 2i involving a decomposable difference between Sq2 w2kand Sq Sq w2k-1 for every i k - 2. We begin with a special case of equation (3.1): For i k - 2, we have i Sq2 w2k-1= w2iw2k-1+ w2k-1+2i. k-1 Applying Sq2 , we get k-1 2i 2k-1 2k-1 Sq2 Sq w2k-1= Sq (w2iw2k-1)+ Sq (w2k-1+2i). Using a Wu formula on the last term, analyzing the binomial coeffi- cients, and using (3.1) again, the reader may check that we obtain the following relations. STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 7 Proposition 3.1 (Key relations in S).For i k - 2, k-1 2i 2i (3.2) Sq2 Sq w2k-1= Sq w2k+ 2k-i-1-2X k-1 Sq2 (w2iw2k-1)+ w2k-1-2ilw2k-1+2i+2il, l=0 i 2k-1 2i These show explicitly how the elements Sq2 w2kand Sq Sq w2k-1 differ by a decomposable, and will guide us to the corresponding ab- stract relations needed in G. However, the relations we have found here involve non-two-power Stiefel-Whitney classes, which still have as yet no analogs in U. We remedy this problem now by extending equation (3.1). Mixing notations, we write (3.1) as j-1 w2j-1+r2j= (Sq2 + w2j-1)wr2j (i.e., (Sqm + wm )x means Sqm x + wm . x). The following lemma is then immediate. Lemma 3.2 (Expressing Stiefel-Whitney classes).Every Stiefel-Whitney class can be expressed in terms of two-power classes and two-power squares as follows: If we write any m = 2n1 + . .+.2ns, where n1 > . .>.ns, we have 2ns 2n2 (3.3) wm = Sq + w2ns . .S.q + w2n2 w2n1. We are now ready to define formally the abstract presentation G. Definition 3.3. In U, extend the set of generators {t2k, k 0}, to define elements tm for all m 1, by first writing m = 2n1+ . .+.2ns, where n1 > . .>.ns . Then by analogy with equation (3.3) set ns 2n2 tm = (Sq2 + t2ns) . .S.q + t2n2t2n1. Definition 3.4 (Abstract key relations).Imitating equation (3.2), let G be the the A-algebra quotient of U by the left A-ideal generated by the elements i 2k-1 2i (3.4) ` (k, i)= Sq2 t2k+ Sq Sq t2k-1+ 2k-i-1-2X k-1 Sq2 (t2k-1t2i)+ t2k-1-2ilt2k-1+2i+2il l=0 for i k - 2. Theorem 3.5 (Structure of S). The symmetric algebra S = H*(BO; F2) is isomorphic to G as an algebra over the Steenrod algebra. Moreover, 8 DAVID J. PENGELLEY AND FRANK WILLIAMS the relations (3.4) generating the A-ideal are minimal, i.e., nonredun- dant. The proof is in Section 5. 4. Applications and speculation We apply the main structure theorem to the cohomology images from the connected covers of BO, and to the cohomology of the spaces BO(q) for classifying finite dimensional vector bundles. Finally we shall see how these descriptions naturally converge into the Dickson invariant algebras. First we consider cohomology images from the connected covers. Definition 4.1. Following [3], let B*(n) be the cohomology image of the map induced by the projection BO ! BO, where BO is the n-th distinct connected cover of BO. That is, BO is (OE(n) - 1)-connected, where n = 4s + t, 0 t 3, and OE(n) = 8s + 2t. In particular, for n = 0, 1, 2, 3 the projections are surjective in co- homology, so the unstable A-algebras B*(n) are isomorphic to the co- homologies of BO, BSO, BSpin, and BO <8>[3]. In general, B*(n) is (2n - 1)-connected, and is the quotient of B*(0) = H*BO = S by the A-ideal generated by {w2k: k < n} [3]. Theorem 4.2 (Structure of connected cover images).An abstract pre- sentation of B*(n) is obtained from that of B*(0) = H*BO = S (The- orem 3.5) as the quotient by the A-ideal generated by {t2k : k < n}. This produces a minimal presentation as follows. Let Kn denote the direct sum of the A-module M(n, 0) on t2n with the free unstable A-module on the t2k, k n + 1. Here M(n, 0) is as defined in [9], namely the free unstable A-module on one generator t2n i modulo the left A-submodule generated by Sq2 t2n, i n - 2. Then B*(n) is isomorphic to the quotient of the free unstable A- algebra on Kn by the left A-ideal generated by the elements ` (k, i), k n + 1, i k - 2, subject to the requirement that all appearances in ` (k, i)of tm , 0 < m < 2n, are replaced by zero. The proof is in Section 5. For our second application, we note that the presentation for H*BO in our main theorem will immediately produce presentations for the cohomologies of the classifying spaces H*BO(q), since each is just the algebra quotient (actually also A-algebra quotient) of H*BO by the STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 9 ideal generated by {wm : m > q} [8], and wm corresponds to tm , which we defined in the presentation of H*BO. The resulting presentation becomes both tractable and useful for H*BO(2n+1 - 1). Theorem 4.3 (Structure of H*BO(2n+1 - 1)). An abstract presenta- tion of H*BO(2n+1 - 1) is obtained from that of B*(0) = H*BO = S (Theorem 3.5) as the quotient by the A-ideal generated by {t2k : k n + 1}. This produces a minimal presentation as follows. H*BO(2n+1 - 1) is presented by the free unstable A-algebra on ab- stract classes {t2k : 0 k n}, modulo the left A-ideal generated by the elements ` (k, i)for k n + 1, i k - 2, (using Definition 3.3 of tm for m < 2n+1), subject to the requirement that when k = n + 1, the i term Sq2 t2n+1 is replaced by zero for each i (all other terms involve only t's in degrees less than 2n+1). The proof is in Section 5. Finally, combining the relations on S = H*(BO; F2) from the two theorems above will produce the common A-algebra quotient of B*(n) and H*BO(2n+1 - 1). Since the first of these is (2n - 1)-connected, while the second is decomposable beyond degree 2n+1-1, we will obtain an A-algebra with algebra generators in the range 2n through 2n+1 - 1. Surprisingly, this much smaller quotient of S = H*BO turns out to be already familiar. We will show now that as an A-algebra it is isomorphic to the n-th Dickson algebra Wn+1 (see Figure 1). In this sense one can say that the Dickson algebra captures precisely the cohomology common to BO and BO(2n+1-1) from H*BO, i.e., it is the A-algebra pushout. Wn+1 H*BO(2n+1 - 1) " " B*(n) H*BO Figure 1 Theorem 4.4 (Convergence to Dickson algebras). The quotient of the symmetric algebra S by the left A-ideal generated by {w2k: k 6= n}is isomorphic to the n + 1-st mod 2 Dickson algebra, Wn+1. Specifically, using the notation of the presentation of Theorem 3.5, as an A-algebra it is minimally presented by the free unstable A-algebra on the mod- ule M(n, 0) (defined in Theorem 4.2), subject to the single A-algebra relation n 2n-1 2n-1 Sq2 Sq t2n = t2nSq t2n. 10 DAVID J. PENGELLEY AND FRANK WILLIAMS We proved in [9] that this precisely characterizes the Dickson algebra Wn+1. The proof is in Section 5. Let us speculate on how Figure 1 might fit in with something topo- logically realizable. It is known that Wn+1 is realizable precisely for n 3 [6], and that B*(n) H*BO is an isomorphism also pre- cisely in this range [3]. Thus for n 3 it is reasonable to expect that Figure 1 be realizable. For general n it is perhaps reasonable to hope for the existence of a space Xn and a homotopy commutative square (Figure 2) whose cohomology is compatible with Figure 1 in the sense of combining to produce the commutative diagram of Figure 3. Addi- tionally we would like Xn to have the property that the outer square in Figure 3 is also a pushout of unstable A-algebras. In other words, Xn does its best to realize a Dickson algebra, even when this is no longer possible. Xn ! BO(2n+1 - 1) # # BO ! BO Figure 2 H*Xn Wn+1 H*BO(2n+1 - 1) " " " H*BO B*(n) H*BO Figure 3 5. Proofs Proof of Theorem 2.3.Let Fm be the free unstable A-module (equiva- lently K module) on a generator tm in degree m. We shall show that the A-module map f : m 0 Fm ! H*BO determined by f(tm ) = wm is injective. From [10], basis elements for the domain of f consist of DJtm where J = (j1, . .,.js)and 0 j1 . . .js < m. (Appendix II recalls the features of the elements DJ in the Kudo-Araki-May algebra K essential to what follows.) STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 11 On the other side of f, basis monomials of the range H*BO can be written as . .w.n2wn1 with nondecreasing indices, i.e., labeled by finitely nonzero tuples (. .,.n2, n1)with 0 . . .n2 n1. We order the latter reverse lexicographically. Now for each basis element DJtm , we consider its image f (DJtm )= DJwm , and we claim that this element of H*BO has a "leading" mono- mial term, i.e., that s-1 2s-2 2 DJwm = w2m-jswm-js-1. .w.m-j2wm-j1wm + higher order terms. ______________-z_____________" z This will complete the proof, since distinct DJwm clearly produce dis- tinct leading monomials, with remaining terms always of higher order; so the DJwm are all linearly independent, and thus f is injective. We will use the following notation: As a subscript, "> k" (resp. "< k") denotes any index greater (resp. less) than k, each occurrence of an unsubscripted w denotes any element of H*BO, and expressions involving any of these mean any sum of expressions of such form. We prove our claim by induction on s, based on the Wu formula Djwm = Sqm-j wm = wm-j wm + higher order terms of formww>m . Clearly the claim holds for lengths 0 and 1. For the inductive step, consider DJbof length s + 1, and note that application of any nontrivially-acting DJ always increases the order of a monomial in H*BO. Now calculate, using the K-Cartan formula [10] as needed, and recalling that the leading term z was defined above: DJbwm = Dj1Dj2. .D.js+1wm = Dj1 Dj2. .D.js+1wm 0 1 s-1 = Dj1@ w2m-js+1. .w.m-j2wm + higher order terms thanxwm A _______-z______" x = x2wm-j1wm + x2ww>m + wDm + higher_order_terms_thanx-z__________"wm A v = z + ww>(m-j1)wm + higher order terms thanz 2 + ww>m + v wm-j1wm + ww>(m-j1)wm = z + higher order terms thanz, __ since the terms of v2 have higher order than x2. |__| 12 DAVID J. PENGELLEY AND FRANK WILLIAMS Proof of Theorem 3.5.There is a map of A-algebras U ! S obtained by taking t2k to w2k, and from Lemma 3.2 and Definition 3.3 this map takes each tm to wm . Since the relations (3.4) that define G map to those also satisfied in S (3.2), there is an induced A-algebra epimor- phism G ! S. We shall show that this map is monic by showing that the induced map on the indecomposable quotients is monic, essentially a counting argument. To start with, note that the indecomposables are I kff QU = Sq t2k: k 0, I admissible, of excess< 2 . Then QG is QU modulo the A-relations (degenerate versions of `(k, i) = 0) i 2k-1 2i Sq2 t2k= Sq Sq t2k-1, i k - 2. There is an A-module filtration I kff FpQU = Sq t2k: 0 k p, I admissible, of excess< 2, which induces an A-module filtration FpQG. Then FpQG=Fp-1QG = I pffn 2i o Sq t2p: I admissible, of excess< 2=A Sq t2p: i p - 2. This is the suspension of the module M(p, 1) analyzed in [9, Theorem 2.11]1, and the basis described there suspends to {DIt2p: I = (2a1, . .,.2al), where 0 a1 . . .al<.p} (As in the proof of Theorem 2.3, we refer the reader to Appendix II for essentials concerning the elements DI in the Kudo-Araki-May algebra K.) We shall finish the proof of isomorphism by showing that the above basis elements for p 0FpQG=Fp-1QG are in distinct degrees; in fact we claim there is exactly one in each positive degree (The appendix discusses the modules M(p, 1) in relation to the literature, and points out an alternative path for substantiating our claim.). Let m be a positive integer. Then m may be written uniquely in the form Xs m = 2r - 2bj, j=1 where s 0 and 0 b1 < . .<.bs < r - 1. The reader may check by induction on s that the unique basis element in degree m is DIt2p, where ___________ 1M(p, 1) is defined in [9] as the quotient of the free unstable A-module on a class in degree 2p - 1 modulo the action_of Sq2ifor i p - 2; in other words, * *in usual notation, M(p, 1) = F (2p- 1)=AA p-2. STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 13 p = r - s and I = (2a1, . .,.2as), with aj = bj - j + 1. With both QG and QS having rank one in each degree, QG ! QS is an isomorphism. Then since S is a free commutative algebra, the epimorphism G ! S must be an isomorphism also. That the relations are minimal (nonredundant) is clear from the fact that in FpQU=Fp-1QU, which is the suspension of the free unstable module on a class in degree 2p - 1, the induced relations are simply i __ Sq2 t2p= 0, for i p - 2, and these are all nonredundant. |__| Proof of Theorem 4.2.We have already mentioned that according to [3], B*(n) is isomorphic to the quotient of S by the A-ideal generated by {w2k: k n-1}. Hence the images under the projection S ! B*(n) of all wm , 1 m 2n-1, are certainly zero from Lemma 3.2. From [3] we also have that B*(n) is a polynomial algebra generated by certain remaining wm (see below). We denote the images of the wm in B*(n) by the same symbols wm . Let Hn denote the quotient of the free unstable A-algebra on Kn by the left A-ideal generated by the elements ` (k, i), for k n + 1, subject to the requirement that all appearances of tm , 0 < m < 2n, are replaced by zero, as in the statement of the theorem. We begin by defining a map from Kn to B*(n) by, as in the preced- ing proof, assigning t2k to w2k for k n. Since the defining relations for Kn are clearly satisfied in B*(n) (from equation (3.2)), this assign- ment extends to the desired map. And since the defining relations for the algebra Hn are also clearly satisfied in B*(n), this extends to an A-algebra map Hn ! B*(n). This map is epimorphic (since B*(n) is generated by certain wm with i 2n), so as in the preceding proof, we need only show the the induced map on indecomposables is monomor- phic. According to [3]2, the polynomial generators of B*(n) are the wm for which ff(m - 1), the number of ones in the binary representation of m - 1, is at least n. We filter QHn as in the proof of the previous theorem, I ff FpQHn= Sq t2k2 QHn : k p , and as in the previous proof the filtered quotient FpQHn=Fp-1QHn is the suspension of the module M(p, 1) for p n, and 0 for p < n. It is straightforward to check that the alpha numbers of one less than the degrees of the elements {DIt2p: I = (2a1, . .,.2al), where 0 a1 . . .al< p} ___________ 2Kochman describes degrees of generators in terms of ff(m) + (m) ( is the 2-divisibility), but we equivalently use ff(m - 1) = ff(m) + (m) - 1. 14 DAVID J. PENGELLEY AND FRANK WILLIAMS are exactly p n, so these are all in degrees where B*(n) has gener- ators. Since we showed in the previous proof that these elements are also in distinct degrees, this similarly completes the proof. Minimality_ follows as in the previous proof. |__| Proof of Theorem 4.3.It is clear that the presentation of S collapses in the manner stated. Minimality follows for most of the relations as in the previous proofs. We comment only that to confirm that the collapsed top relations n 2i 0 = ` (n + 1, i) Sq2 Sq t2n+ decomposables fori n - 1 are also all nonredundant, one can observe that there is a natural map of the new presentation without these final relations to the presentation n 2i for S, and compute that on indecomposables, each Sq2 Sq t2n maps to w2n+1+2i. Now from the Wu formulas, QH*BO is filtered over A by FpQH*BO = {wm : ff(m - 1) p}, and w2n+1+2iis in filtration exactly i + 1. Thus {w2n+1+2i: i n - 1}must be a minimal generating set for the A-submodule it generates in QH*BO. The same then must be true of {` (n + 1, i): i n - 1}in the indecomposables of the new __ presentation without these final relations; so they too are minimal. |__| Proof of Theorem 4.4.In [9] we proved that the (n + 1)-st Dickson al- gebra Wn+1 is isomorphic to the quotient of the free unstable A-algebra on the module M(n, 0) on generator x2n by the single A-algebra rela- tion n 2n-1 2n-1 Sq2 Sq x2n = x2nSq x2n, and that M(n, 0) injects into Wn+1 ([9], proof of Theorem 2.11). In other words, this is a minimal presentation in our sense. Now let us turn to the quotient of the symmetric algebra that com- bines the relations from the previous two theorems, i.e., the quotient by the left A-ideal generated by {t2k, k 6= n}. Let us denote this quotient by Jn. In Jn, the relations ` (k, i)are all trivial except when k is n + 1 i or n. When k = n, they reduce to Sq2 t2n = 0, i n - 2, the defining relations for M(n, 0). When k = n + 1, we have the relations n 2i 2i 0 = ` (n + 1, i) Sq2 Sq t2n+ Sq t2n+1+ 2n-i-2X n Sq2 (t2nt2i)+ t2n-2ilt2n+2i+2il l=0 for i n - 1. These reduce to n 2i Sq2 Sq t2n = t2nt2n+2i. STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 15 Now since i i j i t2nt2n+2i= t2n Sq2 t2n+ t2it2n= t2nSq2 t2n, the relations can be rewritten as n 2i 2i Sq2 Sq t2n = t2nSq t2n. i Since Sq2 t2n = 0 for i < n - 1, these are trivial for i < n - 1, and yield n 2n-1 2n-1 Sq2 Sq t2n = t2nSq t2n for i = n - 1. This precisely matches the single relation (stated above) characterizing the Dickson algebra, so we obtain an isomorphism of A-algebras from Jn to Wn+1 by taking t2n 2 Jn to the generator x2n_2 Wn+1. |__| __ 6. Appendix I: The unstable modules F (2p - 1)=AA p-2 and a minimal A-presentation for H* (RP 1) For each p 0 , the module M(p, 1) is defined in [9] as the quotient of the free unstable A-module on a class x2p-1in degree 2p- 1 modulo i the action of Sq2 for i p - 2; in other words, in usual notation, __ M(p, 1) = F (2p - 1)=AA p-2. These modules are tractable, important, and interesting, and we shall show they are the filtered quotients of a simple minimal A-presentation for H*RP 1. In the proof of our primary Theorem 3.5 above, we appealed to our development in [9, Theorem 2.11] of bases for these modules. The proof used the bases to öc unt" that the direct sum of the modules (we were actually dealing with their suspensions in that theorem) has rank exactly one in each nonnegative degree. In fact we know the rank separately for each module: Theorem 6.1 (Rank of M(p, 1)). The module M(p, 1) has precisely a single nonzero element in each degree with alpha number p, i.e., with p ones in its binary expansion, and nothing else. Proof.The basis for M(p, 1) provided in [9, Theorem 2.11] is {DIx2p-1: the multi-indexI consists of nonnegative, nondecreasing entries of form2k - 1, k < p}. The reader may check that the degrees of these elements are precisely those with alpha number p (see Appendix II for a recollection of essen-_ tials regarding the elements DI in the Kudo-Araki-May algebra K). |__| 16 DAVID J. PENGELLEY AND FRANK WILLIAMS This suggests a connection to the cohomology of RP 1. Recall that ` ' l l+j (6.1) H*RP 1 ~=F2[y] with Sqjyl= y , j from which one sees that H*RP 1 is A-filtered by the number of ones in the binary expansion of degrees. Indeed it is now not hard to prove Theorem 6.2 (M(p, 1) and H*RP 1). The A-module M(p, 1) is iso- morphic to the p-th filtered quotient of H*RP 1. Proof.The module M(p, 1) clearly maps nontrivially to the p-thpfil- tered quotient of H*RP 1, since the quotient begins with y2 -1, and i 2p-1 Sq2 y lies in lower filtration for i p - 2. The map is onto because one sees from (6.1) that the p-th filtered quotient of H*RP 1 is gener- ated over A from degree 2p- 1. Now the previous theorem shows that __ the ranks agree, so the two are isomorphic. |__| Remark 6.3. This result also follows from [2], where it essentially appears in a stabilized form. Indeed, in [2] the A-modules p-1 n 2j o 2 A=A Sq : j 6= p - 1 are studied with stable purposes in mind. Each of these modules ob- viously maps onto the corresponding M(p, 1), and thus the two would clearly be isomorphic if it were known that the domain module is un- stable, which does not seem obvious. In fact, though, it is proven in [2] that these modules are isomorphic to the same filtered quotients of H*RP 1. Thus they are indeed unstable and isomorphic to the modules M(p, 1). The theorem follows. Remark 6.4. The modules M(p, 1) are also used in [5], where Remark 2.6 claims that in an unpublished manuscript [7], William Massey cal- culated that M(p, 1) is A-isomorphic to the p-th filtered quotient of H*RP 1, i.e., the theorem above. However, this does not actually seem to appear explicitly in [7]. Finally, we note that the filtered quotients of H*RP 1 arise again in [1, after Prop. 3.1] in a fashion closely related both to [5] and [7]. We are now equipped to show Theorem 6.5 (Minimal A-presentation of H* (RP 1)). There is a min- imal unstable A-module presentation of H* (RP 1; F2), as the quotient of the free unstable module on abstract classes s2k-1 in degrees 2k - 1 by the relations i 2k-1 2i Sq2 s2k-1= Sq Sq s2k-1-1, i k - 2. STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 17 Proof.There is an A-module map from the abstract quotient to H*RP 1, carrying each A-generator nontrivially, since the given relations are easily calculated also to hold amongst the nonzero classes in H*RP 1. Moreover this is epic, since H*RP 1 is generated over A from degrees one less than a two-power. To see that the two are isomorphic, we need merely show that these relations are enough, i.e., that the abstract quo- tient has only rank one in each degree. This we do by considering the A-filtration of the abstract quotient in which the p-th filtration is the A-submodule generated by {s1, . .,.s2p-1}. The p-th filtered quotient is clearly M(p, 1). That the union of these has rank one in each non- negative degree follows from either of the two previous theorems. Minimality of the presentation is clear. The nonzero classes in H*RP 1 in degrees one less than a power of two cannot be reached from below, so the generating set is minimal, and unique. The nonredundancy of all the relations is clear from the filtered quotients and the fact that two-power squares are minimal generators of A. An alternative proof would be to obtain this presentation simply by collapsing the relations (3.4) in the A-algebra presentation of H*BO in Theorem 3.5 to the indecomposable quotient, since H*RP 1 ~=_ QH*BO as A-modules (Wu formulas). |__| 7. Appendix II: 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 [10] for much more extensive information about K. The mod two Kudo-Araki-May algebra K is the F2-bialgebra (with identity) generated by elements {Di : i 0} subject to homogeneous (Adem) relations [10, Def. 2.1], with coproduct OE determined by the formula Xi OE(Di) = Dt Di-t. t=0 It is bigraded by length and topological degrees (|Di|= i), which be- have skew-additively under multiplication [10, Def. 2.1]. The F2-cohomology of any space is an unstable algebra over the Steenrod algebra, and there is a correspondence between unstable A- algebras and unstable K-algebras, completely determined by iterating the conversion formulae: On any element xl of degree l, and for all j 0, one has Djxl= Sql-jxl, equivalently,Sqjxl= Dl-jxl. 18 DAVID J. PENGELLEY AND FRANK WILLIAMS Since the degree of the element is involved in the conversion, and this changes as operations are composed, the algebra structures of A and K are very different, and the skew additivity of the bigrading in K reflects this. The requirements for an unstable K-algebra, corresponding to the nature and requirements of an unstable A-algebra, are: On any element xl of degree l, Dlxl= xl,Djxl= 0 forj > l, and D0xl= x2l. Finally, and used in our proofs, the K-algebra structure obeys the (Cartan) formula according to the coproduct OE in K: Xi Di(xy) = Dt(x)Di-t(y). t=0 References [1]A.K. Bousfield, D.M. Davis, On the unstable Adams spectral sequence for SO and U, and splittings of unstable Ext groups, Boletin de la Sociedad Matem- atica Mexicana (2) 37(1992), 41-53. [2]D. Davis, Some quotients of the Steenrod algebra, Proc. Amer. Math. Soc. 83 (1981), 616-618. [3]S. Kochman, An algebraic filtration of H*BO, in proc., Northwestern Ho- motopy Theory Conference (Evanston, Ill., 1982), Contemporary Math. 19 (1983), Amer. Math. Soc., 115-143. [4]J. Lannes, S. Zarati, Foncteurs d'eriv'es de la d'estabilisation, Math. Zei* *t. 194 (1987), 25-59. [5]K. Lesh, A conjecture on the unstable Adams spectral sequences for SO and U, Fundamenta Mathematicae 174 (2002), 49-78. [6]Mathematical Reviews #93k:55022, American Mathematical Society, 1993. [7]W. Massey, The mod 2 cohomology of certain Postnikov systems, unpublished handwritten manuscript (1978), 20 pages, courtesy of Haynes Miller. [8]J. Milnor, J. Stasheff, Characteristic Classes, Annals of Mathematics Studi* *es 76, Princeton Univ. Press, Princeton, NJ, 1974. [9]D. Pengelley, F. Peterson, F. Williams, A global structure theorem for the mod 2 Dickson algebras, and unstable cyclic modules over the Steenrod and Kudo-Araki-May algebras, Math. Proc. Camb. Phil. Soc. 129 (2000), 263-275. [10]D. Pengelley, F. Williams, Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology, Transactions of the American Mathematical Society 352 (2000), 1453-1492. [11]N.E. Steenrod, D.B.A. Epstein, Cohomology Operations, Princeton Univ. Press, 1962. [12]R. Stong, Determination of H*(BO(k, . .,.1), Z2) and H*(BU(k, . .,.1), Z2), Trans. Amer. Math. Soc. 107 (1963), 526-544. [13]C. Wilkerson, A primer on the Dickson invariants, in proc., Northwestern Homotopy Theory Conference (Evanston, Ill., 1982), Contemporary Math. 19 STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 19 (1983), Amer. Math. Soc., 421-434, as corrected at the Hopf Topology Archiv* *e, http://hopf.math.purdue.edu/pub/hopf.html. [14]W. Wu, Les i-carr'es dans une vari'et'e grassmannienne, C.R. Acad. Sci. Par* *is 230 (1950), 918-920. New Mexico State University, Las Cruces, NM 88003 E-mail address: davidp@nmsu.edu New Mexico State University, Las Cruces, NM 88003 E-mail address: frank@nmsu.edu