TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx XXXX, Pages 000-000 S 0002-9947(XX)0000-0 SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS FOR MOD 2 HOMOLOGY AND COHOMOLOGY DAVID J. PENGELLEY AND FRANK WILLIAMS Abstract. The mod 2 Steenrod algebra A and Dyer-Lashof al- gebra R have both striking similarities and differences, arising from their common origins in "lower-indexed" algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra K, whose module actions are equivalent to, but quite dif- ferent from, those of A and R. The exact relationships emerge as "sheared algebra bijections", which also illuminate the role of the cohomology of K. As a bialgebra, K* has a particularly attractive and potentially useful structure, providing a bridge between those of A* and R*, and suggesting possible applications to the Miller spectral sequence and the A structure of Dickson algebras. 1. Introduction Mod 2 "lower indexed" algebraic operations Di (i 0) arising via F2-equivariance and the quadratic construction were used by Steenrod [Stee1, Stee2, Stee3] to create "upper operations" Sqi (i 0) in the cohomology of spaces, and his approach was extended by Araki and Kudo [AK ] to create "upper operations" Qi (i 0) in the homology of iterated loop spaces. Adem [Ade ] deduced relations amongst the Sqi, giving us the Steenrod algebra A, and Dyer and Lashof [DL ] in- dicated how this could be done for the Qi, giving us the Dyer-Lashof algebra R, although these relations were first explicitly given by May [May2 ]. The relations amongst the underlying algebraic operations Di were first given, implicitly, by May in [May1 ], as part of his unified al- gebraic approach to A and R, but neither the explicit relations nor the (bi)algebra generated by these operations was studied until relatively recently. The first mention of this bigraded bialgebra, which we shall ___________ 1991 Mathematics Subject Classification. Primary 55S99; Secondary 16W30, 16W50, 55S10, 55S12, 57T05. Key words and phrases. Steenrod algebra, Dyer-Lashof algebra, bialgebras, sheared algebra map, Kudo-Araki-May algebra, Nishida relations. cO1997 American Mathematical Society 1 2 DAVID J. PENGELLEY AND FRANK WILLIAMS call the Kudo-Araki-May algebra K in recognition of their contributions just cited, occurs in a note by Smirnov [Sm ] in 1987, but more extensive study of it has not occurred until the mid-1990's, in independent work by Postnikov [Po ], Bisson-Joyal [BJ ], and the present authors. Perhaps one reason for this delay is that in K the gradings on products are not additive in the now familiar sense in topology, but rather skew-additive on the "topological degree" via a linear combination involving also the "length degree". We will use the term bigraded algebra in this more general sense. Note also that we call K a bialgebra, and not a Hopf algebra, since, like the Dyer-Lashof algebra, it is not connected and thus has no conjugation. Our point of view here, rather distinct from the historical develop- ment, will be to obtain the Steenrod and Dyer-Lashof algebras directly from the bialgebra K via the basic conversion formulas Sqixk = Dk-ixk Qiyk = Di-kyk on classes xk of degree -k in cohomology (negatively graded) and yk of degree k in homology. This process creates two new bialgebras com- pletely different from K , but with topological degree more simply linked to their actions on spaces. Each conversion involves a minor miracle from this point of view, namely a lack of entanglement between k and the relations between the new operations under composition. We intend to show that K and its dual K* are tractable, and have some clear advantages over A, R, and their duals, while providing a unifying view via the notion of "sheared algebra homomorphism". Moreover, the duality between K and K* has a number of very nice features. We believe that in certain circumstances, such as for the Miller spectral sequence [Mill] and the A structure of Dickson algebras, it may be more productive to consider the module actions available over K (or comodules over K*), rather than over A or R. For instance, our description can be used to provide a convenient way of explicitly identifying the indecomposables in the cohomology of iterated loop spaces on spheres in terms of the algebras of Dickson invariants, and this identification is used in [GP ] to study their A-module structures. We will also emphasize how the use of formal power series can motivate, illuminate, and help prove many of our results. In section 2 below, we shall describe the generators and relations of K and list the principal results of this paper. In section 3 we shall see how K acts on graded modules and graded algebras in topology, in- cluding its relation to the suspension isomorphism in cohomology, and the homology suspension on iterated loop spaces. We also examine the SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 3 bialgebra structures of K and K* and how they interact. In section 4 we study actions of K on itself and on K*, arising from the Nishida re- lations, including a description which may shed light on the A-algebra structure of the Dickson algebras, and we present a formula analogous to the Thom isomorphism theorem. In section 5 we formalize what we mean by a sheared algebra map, prove a general theorem for pro- ducing sheared algebra maps on Poincare-Birkhoff-Witt algebras, and apply it to show the exact relationship between K and the Dyer-Lashof algebra, and between their duals. In section 6 we provide the precise relationship between K and the Steenrod algebra via another sheared algebra bijection with domain an algebra K(1) constructed from K. In section 7 we use this to shed light on the tantalizing similarities and puzzling differences between the coproduct formulas in K* and A*. In section 8 we compute the cohomology of the algebra K and show how it is closely related to K(1) and A, again via sheared algebra bijections. We also examine the effect of sheared algebra maps in homology and cohomology, summarize how all our sheared algebra bijections are con- nected to each other, and mention possible applications via the Miller spectral sequence. Section 9 discusses future directions for research and applications. As an illustration, we show that a theorem of J. Lin about finite H-spaces can be reformulated elegantly in terms of K . We note that some previous work involves certain aspects of K, for in- stance in providing useful descriptions of the homology of iterated loop spaces [CCPS , CPS , CLM , MM ], and calculating the mod p homology of topological Hochschild homology [H ]. In [Sm ], K is characterized as the free radical Hopf algebra on a generator of dimension one. In [Po ], Theorem 2 gives the product and coproduct in K* , cf. Theorems B and C below. And [BJ ] includes the product, coproduct, and Nishida action in K as a main (and motivating) example in their study of Q- rings. Perhaps both K and the concomitant notion of sheared algebra homomorphism will find more widespread use. We thank Dennis Sjerve for conversations with the second author which were the genesis of this project. We also thank Bob Bruner, Vince Giambalvo, Chuck McGibbon, Haynes Miller, Jack Morava, Pe- ter May, Bill Singer, Jim Stasheff, and the referee for useful comments on earlier versions of this paper. 2. Statements of principal results For each principal result we indicate in parentheses which section contains its proof. 4 DAVID J. PENGELLEY AND FRANK WILLIAMS Definition 2.1. We define our bigraded bialgebra K = K*;*to be the F2-algebra with identity in bidegree (0; 0) and generated by elements Di2 K1;i(i = 0; 1; : :):, subject to the (Adem) relations X k - 1 - j DiDj = Di+2j-2kDk; (i > j): k 2k - i - j (Observe here that the binomial coefficient is zero unless i+j_2 k < i.) The bidegree of elements in K is defined inductively by the require- ment that the multiplication be a map Km;i Kn;j! Km+n;i+2m j. This definition is viable since the inductive formula is clearly consistent with the Adem relations and with associativity. We will call the first of the bidegrees the length, and the second the topological, degree. We shall frequently use the notation | | to denote topological degree. Finally, the coproduct of K is defined on generators by the usual Cartan formula Xi OE(Di) = Dt Di-t, t=0 and we shall see that the product and coproduct maps of K interact as in a bialgebra. We note that K is a cocommutative component coalgebra [CLM , p. 18] with counit Dn0in the n-th component Kn = Kn;*. The behavior of the coproduct on bidegrees is M OE : Kn;i! (Kn;t Kn;i-t) , t which one sees by induction on n. The `skew' additivity of multiplication on the topological degree is a very interesting feature, which we hope the reader will become con- vinced is well worth becoming comfortable with (see the relation to topology after Definition 3.4, and the categorical consequences of [Sm ]). In fact, Haynes Miller pointed out to us that the skew additivity can be interpreted as a grading of K over the nonabelian monoid constructed as the semidirect product of Z+ with Z+ using the homomorphism ' : Z+ ! End(Z+ ) given by '(t)(x) = 2tx. As with A and R, K also possesses a vector space basis of "admissible monomials" Di1. .D.inwhere i1 . . .in. We shall find it convenient to adopt the convention that Di = 0 for any i that is not a non- negative integer. As mentioned in the introduction, K, like R, has no conjugation since it is not connected. SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 5 We next move to the structure of K*. The first step is Lemma A (x3). The elements Di0Dj1(i + j = n) form a basis for the coalgebra primitives of Kn. From the lemma we see that K* is generated as an algebra by el- ements xi;jdual to the elements Di0Dj1with respect to the basis of admissible monomials. In fact, we prove Theorem B (x3). The algebra summand K*nis isomorphic to the poly- nomial algebra F2[xi;j| i + j = n; 0 i n - 1]: The identity element is xn;0. (Thus K*nis isomorphic to the Dickson algebra on n generators [W ].) We also determine the coalgebra structure of K*: Theorem C (x3). The coproduct in K* is given by the formula X p (xi;j) = x2i-p;j-q xp;q. p;q The reader may wonder at the similarities and differences between this formula and the coproduct formula on Milnor's polynomial generators i for A* [Miln][Stee3, p. 133] or the formula of Madsen [Mad ] for the coproduct in R*. These questions spurred much of our work, and are resolved in sections 5 and 7. There are Nishida relations in K, which give inductive formulas for a left (downward) action of the opposite algebra Kop on K, Kopm;i Kn;j! Kn;i+j_2m. We check that this Nishida action is a map of coalgebras, and thus the contragredient action K K*n! K*n makes K*ninto an unstable algebra over the bialgebra K. We shall see that this action yields a formula analogous to the ac- tion of the Steenrod squares on the Thom class of vector bundles. Specifically,Precall that if Sq, the total Steenrod square, is defined by Sq = i0 Sqi, if w is the total Stiefel-Whitney class of a vector bun- dle, and if U is its Thom class, then there is the formula Sq(U) = w . U: P P ToPstate our analogy, let x = i;j0xi;j, O = j0 x0;j, and D = i0 Di. Then we have 6 DAVID J. PENGELLEY AND FRANK WILLIAMS Theorem D. In the action of K on K*, there is the formula D(O) = x . O: This is a special case of the following theorem, which completely determines the action of K on K*, and thus encodes the action on the Dickson algebras (cf. [HP , Mad , W ]). Since this description of the action is rather different from that of the Steenrod algebra on the dual Dyer-Lashof algebra, although equivalent to it, we hope it may be useful. P Theorem E (x4).P For fixed n, and 0 m n, let f+m;n= km xn-k;k and f-m;n= j 0, and 0 k 1, a vector space basis for the coalgebra primitives in H*kSm+k can be identified with the admissible monomials in K involving only Dl for 0 l < k, via the action on the fundamental class. When k is 8 DAVID J. PENGELLEY AND FRANK WILLIAMS finite, we will denote by K(k) the primitives in H*kSm+k under this identification: Definition 2.6. Define K(k) to be the vector subspace of K spanned by the admissible monomials of the form Di00Di11. .D.ik-1k-1. While it appears at first that K(k) is merely a vector subspace, in fact K(k) is a sub-bialgebra of K. We may now define a new algebra K(1) by K(1) = lim-K(k) k under algebra homomorphisms ff : K(k) ! K(k - 1) given on gener- ators by ff(Di) = Di-1: (Recall that since Dt = 0 whenever t is not a nonnegative integer, this formula gives ff(D0) = D-1 = 0:) Theorem H (from Theorem 6.11). There is a sheared algebra bijec- tion 1 : K(1) ! A: We move on to examine the dual map *1: A* ! K(1)*. Proposition I (from Proposition 7.3).A basis for K(1)* is given by "monomials" n-1X x-l0;nxl11;n-1.x.l.n-1n-1;12 K(1)*n forl > li. i=1 (Here n is a grading induced from the original length grading in K.) The notation x-l0;nxl11;n-1.x.l.n-1n-1;1is intended to suggest an algebra in which x0;nhas been inverted. In fact, in [BPW ] it is shown that K(1)* is a subspace of such an algebra. Finding a complete formula for *1 appears to be quite complicated and is related to the work of Campbell, Peterson, and Selick [CPS ]. However, we are able to compute the "leading terms," in the following sense. P n L Theorem J (x7). Let l = i=1li: If ln 1; then modulo m>n K(1)*m *1(l11. .l.nn) = x-l0;nxl11;n-1.x.l.n-1n-1;1: We caution that while K(1)* is endowed with a coproduct dual to the multiplication in K(1), we do not claim any algebra structure for K(1)*. We also do not know how to compute the coproducts of all elements of K(1)*; but we can compute certain ones which, combined with our leading term theorem, will shed light on the similarities and differences between the coproducts in A* and K*. SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 9 H*(^0) K(1) _________H*(K)-!______-H*(R(-1)) Q Q | |6 Q Q || || 1 Q QQs ||? ||? AL _________H*(opp)-oe Figure 1 The last group of results deals with cohomology. We use methods of [Pr] to compute the cohomology H*(K) = ExtK(F2; F2) of K, and then prove Theorem K (from Theorems 8.3 and 8.5). There exist sheared alge- bra bijections ! : K(1) ! H*(K) and : H*(K) ! A: Sheared algebra morphisms induce maps on homology and cohomol- ogy: Theorem L (x8). Let (K; ^K; F; d) be an algebra with shifting, M an algebra,_and__ : K !_M_a sheared algebra homomorphism. Then the map B ( ) : B (K) ! B (M) defined for the reduced bar constructions by __ d(z ...z ) d(z ) B ( )([z1| . .|.zn-1|zn]) = [ (F 2 (zn1)| . .|. (F n(zn-1))| (zn)] is a chain map, which then produces induced maps on homology and cohomology. Finally we summarize many of our results in a single commutative diagram. Before presenting it, recall from [KL1 , Pr, Mill, Lo] that there is an algebraic version R(-1) of the Dyer-Lashof algebra which has R as a quotient, is isomorphic to the opposite of the -algebra, and whose cohomology algebra H*(R(-1)) is isomorphic to AL; the Lie Steenrod algebra, which is obtained by replacing Sq0 = 1 by Sq0 = 0 in the defining relations for A. See Figure 1. All the maps in this diagram are degree-preserving isomorphisms of vector spaces, the labeled maps are sheared algebra bijections, and the unlabeled double-ended arrows are the algebra isomorphisms just mentioned. While ! and are both sheared algebra maps and 1 = 10 DAVID J. PENGELLEY AND FRANK WILLIAMS O !; it does not immediately follow that 1 will be a sheared algebra map. In fact 1 and ! do not even use the same degree function on their common domain K(1): This phenomenon can be explained by a more detailed examination of the categorical structure of sheared algebra morphisms which we shall treat in a subsequent paper. 3.Basic properties of K and K* We begin this section with some instances of the relations in Defini- tion 2.1. Example 3.1. We have D2iD0 = D0Di; D2i+1D0 = 0; DiDi-1= 0: We also can see, by checking binomial coefficients, that if the term D0D2a appears on the right hand side (RHS) of an Adem relation, then the left side must be D2a+1D0, and that DlDl does not appear on the RHS of any Adem relation. Finally, note that the RHS of the Adem relation for DiDj involves only those Dl with l < i. We believe that the nice features of these sample relations already promise that the K-Adem relations will sometimes be simpler to work with than those of A and R. To be certain that in our definition the product and coproduct in- teract as in a bialgebra, and that this is indeed the algebra of opera- tions generated by the "algebraic Steenrod operations" Di, we refer to [May1 ] where these operations are defined in a setting which applies to both cohomology of spaces and homology of iterated loop spaces (see [CLM ] for the latter). The symmetry that produces relations between iterated operations is expressed there by equation (e) in the proof of Theorem 4.7. The reader may find it satisfying to check that the some- what elaborate equation there is merely a detailed expansion of the attractive formal power series symmetry identity D(u)D((u + v)v) = D(v)D((u + v)u), P where D(x) denotes the formal sum i0 Dixi. We will call this iden- tity the symmetric K-Adem relations. Note that if we let u; v have formal topological degree -1, the skew-additivity of topological degree in K ensures that all the terms in the above relations are homogeneous of degree zero. The coproduct in K can be expressed in this context as D(x) = D(x) D(x): Clearly this coproduct respects the symmetric K-Adem relations, and thus does produce a bialgebra. SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 11 To extract the Adem relations of our Definition 2.1 from the sym- metric relations, we can use the clever residue method of [BM , Stei]. Letting w = (u + v)v and noting that for constant u, dw = udv (mod 2 of course), we use the symmetry identity in the second equality of what follows. D(u)D(w) du DiDj = res res __________dw ____ u=0 w=0 wj+1 ui+1 D(v)D((u + v)u) du = res res________________udv ___ u=0 v=0 ui((u + v)v)j+1 u " # X dv du = res res DlDkvl-juk-i+1(u + v)k-j-1______ u=0 v=0 v u k;l X k - j - 1 du = res DlDku(k-i+1)+(k-j-1)-(j-l)_ u=0 j - l u k;l X k - j - 1 = Di+2j-2kDk. k 2k - i - j Thus the relations in Definition 2.1 follow. We must note, however (as did [May1 , BM , Stei] in analogous situa- tions for the Steenrod and Dyer-Lashof operations), that these relations are valid not only for i > j, but for all i; j 0 (with cd interpreted as (-1)d -c+d-1d when c < 0). Is it not possible then, that the un- stated formulas for i j could impose additional relations beyond the "standard" ones in our definition? Neither Adem nor Steenrod in the case of the Steenrod algebra [Ade , Stee3], nor May in the case of the Dyer-Lashof algebra [May2 ], express concern about this issue when they deduce explicit relations. In each case this concern was presumably obviated by topological knowl- edge of the time; Adem knew [Ade , p. 231] from Serre's calculations of the cohomology of Eilenberg-MacLane spaces that the admissible monomials in Steenrod squares formed a basis for all operations, and May knew [May2 ] similarly from calculation of the homology of QS0 that the admissible monomials in Dyer-Lashof operations applied to the fundamental class are independent. Thus in each case no further relations were possible. In our case we may also appeal to the homology of QS0, since the isomorphism 0 of Theorem G (with K subject only to the "standard" relations of Definition 2.1) commutes with the actions of R and K on a zero-dimensional homology class. Thus the algebra of operations has no more relations than those in our definition. Having gone to all 12 DAVID J. PENGELLEY AND FRANK WILLIAMS this trouble, it would nonetheless be very nice to have purely algebraic reasoning for why the relations for i j must be totally redundant, but we know of none such at present. It is also very useful to have these asymmetric K-Adem relations expressed using formal power series, as follows. In the symmetric K- Adem relations identity, let w = (u + v)v, and seek to eliminate v from the identity. First write the quadratic v2 + uv + w = 0 (mod 2) as v_2 v_ w_ w_ u + u + u2= 0: Then v = uf u2 where f(x) is any mod 2 formal power series solution to f2 + f + x = 0: This leads to the asymmetric K-Adem relations identity i i w jj i i i w jjj D(u)D(w) = D uf ___ D u2 1 + f ___ u2 ! u2 i i w jj w = D uf ___ D ______ : u2 f w_u2 Working with this identity involves choosing a solution f(x) for the quadratic, and being able to identify the coefficients in its integer pow- ers as well. Of course there are many integral combinatorial choices for the mod 2 solutions. This important quadratic is well known and studied, and Lemma 4.7 of [GPR ] points to much of what one needs to know. For instance, one can obtain the K-Adem relations of our defi- nition by this approach, rather than by the Bullett-Macdonald residue method we used above. The behavior of the cohomology of a space or the homology of an infinite loop space in relation to K motivates the following definitions. Definition 3.2. A graded module M* over K is one satisfying the de- gree requirement Km;i Mj ! Mi+2m j . Note that K with its topological degree is thus a graded module over K. Definition 3.3. A graded module M over K, as above, is called un- stable provided that (1) Di: M-i ! M-i is the identity for i 1, and that (2) Di(Mk) = 0 (i > -k): Definition 3.4. If in addition to being an unstable graded module over K, as per Definition 3.3, M is also a graded algebra (with the usual SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 13 additive grading convention) over K (with respect to the coproduct OE), then M is called an unstable algebra over K if it satisfies the condition D0(y) = y2 for all y 2 M. Remark 3.5. In [May1 ] and [CLM ] it is implicit that the cohomology of a space (graded negatively) and the homology of an infinite loop space are functors to the category of unstable algebras over K. At this point, traditional treatments of homology and cohomology operations define "upper-indexed" Steenrod or Dyer-Lashof operations in terms of the generators of K, then develop the algebraic properties of the respective algebras of operations. We, however, shall continue to focus on the bialgebra K and its dual. Lemma 3.6. The elements Di1. .D.in, i1 . . .in, form a vector space basis for K (which we will call the basis of admissible monomials.) Proof.This follows in a standard way, cf. [Pr, p. 51], by using an appropriate ordering of the set of all multi-indices (i1; : :;:in), since_K is bigraded of finite type. |__| We next prove Lemma A from section 2. Proof of Lemma A. Clearly the elements Di0Dj1are primitive, since Di0 is grouplike (i.e. OE(Di0) = Di0 Di0) and D1D0 = 0. Since OE(D0DI) = (D0 D0)OE(DI); we need only examine admissible monomials DI in which the first sub- script is at least 1. So let DI = Di1. .D.inbe admissible, with i1 1 and in 2. Now OEDI = Dn1 (Di1-1. .D.in-1) + terms involving other basis elements. __ The lemma follows. |__| Note that Di0Dj1has bidegree (i + j; 2i(2j - 1)). Observe that K*;*is not of finite type in the second bidegree (the ele- ments Di0all have zero topological degree). With this in mind, we define K* to be the bigraded dual to K, i.e. K*n;-q= (Kn;q)*. We will con- tinue to refer to the dual bidegrees of K* as "length" and "topological" degrees, as in K. Since K is the direct sum of the cocommutative coal- gebras Kn, K* is the direct sum of the commutative algebras K*n. From Lemma A we saw that K* was generated as an algebra by elements 14 DAVID J. PENGELLEY AND FRANK WILLIAMS xi;jdual to the elements Di0Dj1with respect to the basis of admissible monomials. Of course the bidegree of xi;jis (i + j; -2i(2j - 1)). We now move to the proof of Theorem B. Proof of Theorem B. We note that the multiplication in K*nobeys the usual (non-skewed) degree convention (i.e. is additive in the second subscript; see the degree behavior of OE in Definition 2.1). Furthermore, note that on admissibles the correspondence Di1. .D.in! xi10;nxi2-i11;n-1.x.i.n-in-1n-1;1 provides a bijection of graded vector spaces between Kn and F2[xi;j| i + j = n; 0 i n - 1]. Since K*nis generated by {xi;j| i + j = n; 0 i n - 1}, the theorem_ follows. |__| We observe that since xi;jhas topological degree -2i(2j - 1), already the K*nrealize the graded Dickson algebras [W ], even before we consider their natural module structure over A or K arising from the Nishida relations. We now aim for a coproduct formula on the algebra generators xi;j2 K*. To that end, the next lemma gives another very useful property of the primitive elements Di0Dj1. Lemma 3.7. If DI and DJ are admissible monomials such that DIDJ = Di0Dj1, then there exist non-negative integers a and b such that DI = Di-a0Dj-b2a; DJ = Da0Db1: __ Proof.This follows from Example 3.1. |__| The algebra K has several interesting and useful self-maps. The next definition describes one of these. Definition 3.8. Let V : K ! K be the Verschiebung, the bialgebra map dual to the Frobenius (squaring) map on K*. Remark 3.9. Note that V is an epimorphism since the squaring map is one-to-one. Proposition 3.10. On algebra generators V is given by V (Di) = Di=2. (Recall our convention that Di= 0 for any i that is not a non-negative integer.) Proof.Since the Frobenius map on K**;*preserves the length degree and doubles topological degree, we must have V (Di) = Di=2since V_is onto. |__| SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 15 Remark 3.11. This shows that "halving", Di ! Di=2, extends to a valid algebra endomorphism on K, which is not immediately obvious from the explicit Adem relations, although it follows easily from the symmetric power series form of the relations since V (D(u)) = D(u2). We shall henceforth, for admissible elements y 2 K; use the notation y* for the dual element to y with respect to the basis of admissible monomials in K: Corollary 3.12. Let DJ be admissible. We have the formula a * (D*J)2 = D2aJ . Proof.By Proposition 3.10 we may compute, for DI admissible, a * a * * a * <(D*J)2 ; DI> = <(V ) (DJ); DI> = = ; which is nonzero if and only if DI=2a = DJ, or equivalently, DI =_ D2aJ. |__| We are now in a position to prove Theorem C. Proof of Theorem C. Let DI and DJ be admissibles. We compute <(xi;j); DI DJ> = , which is nonzero if and only if DIDJ = Di0Dj1. By Lemma 3.7 this means that DI = Di-p0Dj-q2p, and DJ = Dp0Dq1. So __ by Corollary 3.12 we see that the terms of (xi;j) are as claimed. |__| It is now interesting to observe that since clearly each Di is inde- composable in K, each xi0;1is a coalgebra primitive, which at first sight seems perplexing in an F2-bialgebra, but is in fact consonant with the world of multiple components. The K* coproduct has a very nice representation if we first define the formal power series X p p+q i i+j-q p (x)(u; v; w) = x2i-p;;j-q xp;qu2 -1v2 w2 -2 , 0pi 0qj and note that it determines the coproduct since the exponents of u; v; w determine i; j; p; q uniquely. Letting X n-l n x(v; w) = xn-l;lv2 w2 -1, 0ln we have 16 DAVID J. PENGELLEY AND FRANK WILLIAMS Corollary 3.13. The coproduct in K* is completely encoded by the identity (x)(u; v; w) = (1 x)((x 1)(v; w); u). Note here also that if we assign formal topological degrees |u|= |w|= 1 and |v|= -1, then both x(v; w) and (x)(u; v; w) are homogeneous of degree -1. We end this section with some further particularly interesting fea- tures of K and K*, some of which we shall use below. Theorem 3.14. For 0 < j1 < . .<.jr n, we have the formula xn-j1;j1. .x.n-jr;jr= (Dn-jr0Djr-jr-11.D.j.2-j1r-1Dj1r)*. We preface the proof of this theorem with two lemmas. Lemma 3.15. r X i2i-1= (r - 1)2r + 1: i=1 __ Proof.Easy induction. |__| Lemma 3.16. Let 0 < j1 < . . .< jr n. If DI 2 Kn;*; where I = (i1; : :;:in); and if |DI| = |Dn-jr0Djr-jr-11.D.j.2-j1r-1Dj1r|; and if in r - 1; then DI is not admissible. Proof.We note Xr |DI| |Dn-r0D1D2. .D.r| = 2n-r i2i-1= (r - 1)2n + 2n-r; i=1 by Lemma 3.15. But clearly, if DI were to be admissible, then |DI|__ in(2n - 1) (r - 1)(2n - 1): |__| Proof of Theorem 3.14.Let DI be admissible. Write x = xn-j1;j1. .x.n-jr;jr; x0= xn-j1;j1-1. .x.n-jr;jr-1; and I = (I0; in): From the coproduct formula for K* it is immediate that is zero if in > r: Furthermore, if in = r; we have = <(x); DI0 Dr> = = : If j1 - 1 > 0; then we may conclude inductively (on n) that this is nonzero if and only if DI0= D(n-1)-(jr-1)0.D.j.1-1r; whence DI = Dn-jr0. .D.j1r: If j1-1 = 0; i.e. j1 = 1; then = ; which, again inductively, is nonzero if and only if DI0= Dn-jr0. .D.j2r-1; whence again DI = Dn-jr0. .D.j1r: So if in = r; the only possible DI is the desired one. SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 17 Finally, Lemma 3.16 applies to show that there are no admissibles DI in this bidegree with in r - 1: Hence if DI is admissible, then is nonzero if and only if DI is as desired. __ |__| For more complex monomials in K*, the duality with admissibles in K becomes more elaborate. An algorithm for dealing with this question is given in [CPS ]. Let A be a bialgebra. An element g 2 A is called grouplike (cf. proof of Lemma A) if (g) = g g, where denotes the coproduct in A: Remark 3.17. If g 2 A is a grouplike element and if f : A ! A is given by f(a) = ga, then f is a morphism of coalgebras and (hence) f* : A* ! A* is a morphism of algebras. Some useful self-maps of K and K* are as follows: Example 3.18. In K, the coproductPformula shows that the element D0 and the formal sum D = i0 Diare grouplike. We will denote the algebra maps dual to left multiplication by D0 and D by * and *, respectively. From Lemma 3.7 their values on the algebra generators of K* are *(xi;j) = xi-1;j and *(xi;j) = xi-1;j+ xi;j-1 . P In K*, the sum O = j0 x0;jsatisfies (O) = O O, so ff : K ! K dual to multiplication by O is an algebra map. As with V *earlier, multiplication by O preserves the length degree. Then the epimorphism ff : K1;i! K1;i-1must be given by ff(Di) = Di-1: It is interesting to use the Adem relations to provide an alternative proof that this formula for ff yields an algebra endomorphism. In fact, this follows trivially from their symmetric power series form; since ff(D(x)) = xD(x); applying ff to the symmetric identity simply multiplies both sides by u(u + v)v. 18 DAVID J. PENGELLEY AND FRANK WILLIAMS Remark 3.19. The map ff has topological interpretations. If we regard K as operating on the cohomology of spaces and let : H*(X) ! H*(X) be the suspension isomorphism, x 2 H*(X); and DI be a monomial in K, then since is essentially a cup product with a one- dimensional class, one checks as in [Stee3, May1 ] that DI((x)) = (ff(DI)(x)): Also, if X is an infinite loopspace with K acting on it, z 2 H*(X), oe* is the homology suspension, and DI is a monomial in K, then it follows from [AK , DL , CLM ] that ff(DI)(oe*(z)) = oe*(DI(z)): The self-maps we have described will be quite useful in describing the relation between K and the Dyer-Lashof and Steenrod algebras, as will the following proposition, which illustrates their use. Proposition 3.20. If DI 2 Kn;*is admissible, then xa0;nD*I= D*I+(a;:::;a): Proof.We compute, for admissible DJ, = = ; which is non-zero if and only if ffa(DJ) = DI; i.e. when DJ = DI+(a;:::;a):_ |__| 4. Nishida actions In K, the Nishida relations [N , May2 ] [May1 , p. 209] [Mad , pp. 244-5] will give inductive formulas for a left (downward) action of the opposite algebra Kop on K, Kopm;i Kn;j! Kn;i+j_2m, which we shall denote by DI* DJ, to distinguish from multiplication in K. Traditionally, the Nishida relations mediate the interaction of the Aop and R actions on the homology of an infinite loop space. If X is an infinite loopspace and x 2 H*(X), then in terms of K they take the form X t+s_ Ds * (Dtx) = t-s2_ Dt+s_2-k(Dk * x): k 2 + k Here Ds*y is the left downward action of Kop on H*(X) that is equiva- lent to the standard left downward action of Aop on H*(X). To obtain SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 19 an action of Kop on K, we can use the action on a zero-dimensional homology class and define inductive formulas ( 1; if s = 0; Ds * 1 = 0; otherwise and X t+s_ Ds * (DtDJ) = t-s2_ Dt+s_2-k(Dk * DJ): k 2 + k The reader may again be pleased to check that this formula is merely the formal power series identity D(u) * (D(v) . ___) = D((u + v)v) . (D((u + v)u) * ___) P in different guise, where D(x)* denotes Dixi*. A straightforward computation, which we leave to the reader, shows that this formal power series formula for the action is compatible with (the opposite of) the formal power series symmetric K-Adem relations in section 3. This provides satisfying confirmation that we have really described a bona fide algebra action of Kopon K, underlying the traditional Nishida relations for Aop and R. It is also interesting to compare our formulas with (3a) of [Stei]. As in the formal power series identity for the coproduct in K*, where both sides were homogeneous of degree -1, in this case we find that both sides of our action formula have homogeneous degree zero if we let |u|= |v|= -1=2, where the degree of Ds * (Dt. ___) is (t + s)=2, and that of Dl. (Dk* ___) is l + k, since these are the amounts by which they increase the degrees they are applied to. We also wish to note at this point that by using the action based on fundamental classes in positive degrees, we obtain different actions of Kop on K. We shall consider these actions and their topological significance in a subsequent paper. Example 4.1. Taking t = 0, we obtain ( D0(Ds=2* DJ) if s is even; Ds * (D0DJ) = 0 if s is odd. Also, 8 >: 2 2 2 0 if s is even. 20 DAVID J. PENGELLEY AND FRANK WILLIAMS Further, Ds * (DtDJ) = 0 ifs; t have different parity. And finally, D0 * DJ = DJ=2; so we see that the action of D0 produces the Verschiebung map V of Definition 3.8. We next wish to observe, as pointed out in section 2, that dualizing from the Nishida action turns each K*ninto an unstable algebra over Ka la Definition 3.4. First note that the Nishida action is a map of coalgebras, as may be verified straightforwardly by induction on length in K using the formal power series formulation above of the Nishida action. Thus the contragredient action K K*n! K*n makes K*ninto an algebra over the bialgebra K. Moreover, one checks by induction on length from the Nishida relations that * satisfies (the dual of) the conditions in Definition 3.3. Finally, using Proposition 3.10 to dualize the last equation of Example 4.1, we obtain the squaring condition of Definition 3.4. We shall prepare for the proof of Theorem E with a lemma illustrat- ing computations using the Nishida relations of Example 4.1. Lemma 4.2. Let 0 a < d a + b and c 0. Then D2a+b+2a-2d* (Da0Db1Dc2) = Dd-10Da+b+c-d+11: Proof.By repeated applications of the formulas in Example 4.1, we compute D2a+b+2a-2d* (Da0Db1Dc2) = Da0(D2b+1-2d-a* Db1Dc2) = Da0(D2d-a(2b-d+a-1)+1* Db1Dc2) = Dd-10(D2(2b-d+a-1)+1* Db-d+a+11Dc2) = Dd-10D1(D2b-d+a-1* Db-d+a1Dc2) = Dd-10D1+b-d+a1(D0 * Dc2) = Dd-10D1+b-d+a+c1: __ |__| SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 21 Proof of Theorem E. Since K acts unstably on K*n, and xn-m;m lies in topological degree -2n-m (2m - 1) in K*n, the element Dixn-m;m can be nonzero only if it lies in a degree from -2n-m+1 (2m - 1) through -2n-m (2m -1). In degrees from -(2n+1-2) to 0, a basis for K*nconsists of the elements xn-j;jxn-k;kfor 0 j k n, lying in the distinct degrees 2n-j+ 2n-k - 2n+1, together with the elements x2n-1;1xn-k;kfor 1 k n - 1, the latter lying respectively in the same degrees as x2n-k-1;k+1in the first list. Thus Dixn-m;m lands in a degree of zero rank unless i = 2n-j + 2n-k - 2n-m+1 for 0 j k n, in which degree the dual is spanned by Dn-k0Dk-j1Dj2, together with Dn-j+10Dj-21D3 if also j = k 2. We complete the proof by considering four cases discriminating primarily where m lies in relation to j and k. Case 1. If j < m k, then in the degree of Dixn-m;m , K*nhas rank one, and we compute = = = 1; taking a = n - k, b = k - j, c = j, and d = n - m + 1 in Lemma 4.2. Thus Dixn-m;m = xn-j;jxn-k;kas claimed. Case 2. If j = m = k, then D0xn-m;m = x2n-m;m, since K acts unstably on K*, producing the leading term in the formula claimed. Case 3. If j k < m, then for j 1, Dixn-m;m = 0 as claimed, since computing as in Case 1 using the formulas from Example 4.1, we obtain = = = 0, and, if also j = k 2, then similarly = 0: For j = 0 one checks that the result follows from instability. Case 4. If m j < k or m < j k, then i < 0, which cannot occur._ |__| 22 DAVID J. PENGELLEY AND FRANK WILLIAMS The action of K on K*; as described by Theorem E, lends itself to formulation with formal power series. For m 0, let X n-k n-m n-m n f+m(u; v; w)= xn-k;ku2 -2 v2 w2 -1 , mkn X n-j n-m n-m n f-m(u; v; w)= xn-j;ju2 -2 v2 w2 -1 , 0j 2j; i_2ti-j-1 2t - i P and with coproduct given by Qi= Qt Qi-t[May2 ]. We call a nonnegative sequence I = (i1; i2; : :;:in); and the corre- sponding monomial QI = Qi1. .Q.in, admissible provided it 2it+1 for each t, and recall [CLM , Mill] that the admissible monomials form a basis for R(-1). The excess of an admissible I and of QI is defined to be i1 - (i2 + . .+.in), and we recall that an admissible vanishes on all k-dimensional homology classes precisely if it has excess less than k [CLM , p. 17]. Following [CLM , Mill], we consider the ideal B(k) in R(-1) spanned by all admissibles of excess less than k, and form the bigraded algebras R(k) = R(-1)=B(k), with basis the admissibles of excess at least k. Then R, the "geometric" Dyer-Lashof algebra, is defined to be R(0), a quotient bialgebra of R(-1) [CLM , p. 17]. Before proceeding, we recall from Section 2 the shift map ffi : R(-1)n;i! R(-1)n;i-(2n-1)defined by n-1 i -2in-1 ffi(Qi1. .Q.in-1Qin) = Qi1-2 . .Q.n-1 Q on admissibles. Note that ffi reduces excess by one. For each fixed k 0, we inductively extend the conversion formula above for the action on k-dimensional classes, and abstract from this conversion on admissibles to define a map ^k: R(-1)n;i! Kn;i-(2n-1)k of vector spaces, commuting with the respective actions on Hk(Y ), given by the formula ^k(Qi1. .Q.in-1Qin) = Di1-i2-...-in-k.D.i.n-1-in-kDin-k: 24 DAVID J. PENGELLEY AND FRANK WILLIAMS Note that at this time we only know that this formula for ^k holds on admissibles. From our remarks above we know that ^k factors through R(k), creating k : R(k)n;i! Kn;i-(2n-1)k satisfying the same formula on admissibles, from which it is easily ver- ified that Proposition 5.1. For each k 0, (i) k+1 = ff O k and k+1 = k O ffi; (ii) k is an isomorphism of vector spaces; and (iii) 0 is an isomorphism of coalgebras. It follows that the dual of 0 is an algebra isomorphism. Combining this fact with Theorem B yields us Madsen's theorem [Mad ] on the algebra structure of R* . Further combining with Theorem E gives us l Madsen's computation of the A-action (of the Sq2 ) on generators of the Dickson algebras R*n. The bijections k will be some of our first examples of "sheared" al- gebra maps. While not algebra maps in the traditional sense, we see they respect products in a sheared sense (at least on admissible mono- mials), using the shift map ffi. For example, while k maps Qi to Di-k, it maps an admissible Qi1Qi2to Di1-i2-kDi1-k= k(ffii2(Qi1))k(Qi2). Thus as k passes through the product it is sheared by a shifting of the left factor. The shifting is by i2 applications of the shift map ffi, where i2 is the topological degree of the right factor. This is reminiscent in some respects of the notion of a semidirect product of groups. We will show that this phenomenon has both general structure and application. A sheared algebra map should behave as above on any product, not just on monomials of special form like admissibles. This is a pleasant circumstance, since it means the entire feature is compatible with (Adem) relations, and thus it is a general property of the map, unaffected by the representation of elements. In particular, the formula we gave above for ^k will hold even on inadmissibles! With these motivating examples in hand, we now proceed to describe the general setting for sheared algebra maps, and prove a theorem which can be applied to show that several of our constructions produce sheared algebra bijections. In addition to the maps k :R(k) ! K, these applications will include a map relating the cohomology of K and the Steenrod algebra, i.e. a sheared algebra bijection H*(K) ! A. This SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 25 will arise from sheared algebra maps involving a new graded algebra K(1) which we will construct from K. First we recall from Definition 2.4 what is meant by a sheared algebra homomorphism. Our general method for producing sheared algebra homomorphisms requires K to be a Poincare-Birkhoff-Witt algebra (P-B-W algebra for short), about which we briefly remind the reader, and refer to [Pr] or [Lo ]. The Lie Steenrod algebra and algebraic Dyer-Lashof algebra are examples [Lo , Mill, Pr], as well as our K and some other algebras we shall encounter. Begin with an augmented algebra with generating set K^ = {bi}i2I, I a subset of Z. Suppose further that K has only homogeneous 2- fold relations in these generators. Then K is called a homogeneous pre-Koszul algebra, and has a multiplicative length grading, which we denote by K*. (We are being somewhat more general than [Lo , Pr] by not yet considering or requiring a second, topological, grading.) Let I = (i1; : :;:in) denote a multi-index and call it the label of bI = bi1. .b.in, the corresponding monomial in generators. Suppose S is a set of multi-indices chosen so that B = {bI}I2S is a vector space basis for K. Then (B; S) is called a labeled basis for K. The monomials in B are called admissibles, the expression of any element in terms of them is called its admissible expression, and the admissible expression of any inadmissible 2-fold product of generators is called an admissible relation. Now the set of all possible multi-indices is ordered, first by length and then lexicographically. The labeled basis (B; S) is called a P-B-W basis (and then (K; ^K; B; S) a P-B-W algebra) provided: 1. If I; J 2 S then either the juxtaposition (I; J) is also in S or else the label of every monomial in the admissible expression for bIbJ is strictly greater than (I; J), and 2. For n > 2, (i1; : :;:in) 2 S if and only if for each j < n we have (i1; : :;:ij) 2 S and (ij+1; : :;:in) 2 S. Now we will describe a process which begins with a P-B-W alge- bra K, and constructs under three assumptions a particular type of degree function d and shift map F which interact nicely. The result- ing (K; ^K; F; d) will be called an "algebra with shifting", and we then prove a theorem enabling the creation of various sheared algebra ho- momorphisms with K as domain. We will apply this theorem first to ^k : R(-1)! K , and therefrom show that each k : R(k)! K is also a sheared algebra bijection. We will later apply the theorem in several other situations. 26 DAVID J. PENGELLEY AND FRANK WILLIAMS So we start with a P-B-W algebra (K; ^K; B; S) and describe first the construction of a degree function d under Assumption 1. Suppose b is a positive integer and suppose dP: ^K! Z+ is given satisfying the following condition: If k0k00= l0ml00mis an admissible relation, then for each m d(k0) + bd(k00) = d(l0m) + bd(l00m): Examples. For b = 1, any additively graded algebra, e.g. topolog- ical degree in R(-1), the negative of topological degree in A (i.e., positively graded), or length degree in R(-1) or K or AL (the Lie Steenrod algebra [Lo , Pr]). For b = 2, topological degree in K: To extend d to all monomials in K, consider first T (K^), the tensor algebra on K^: Definition 5.2. On the usual basis for T (K^) define d^(k1 . . .kn) = d(k1) + bd(k2) + b2d(k3) + . .+.bn-1d(kn): P Lemma 5.3. If kiki+1 = l0ml00mis an admissible relation, then for each m ^d(k1 . . .ki ki+1 . . .kn) = ^d(k1 . . .l0m l00m . . .kn): Proof.We need only check that bi-1d(ki) + bid(ki+1) = bi-1d(l0m) + bid(l00m), __ which follows from d(ki) + bd(ki+1) = d(l0m) + bd(l00m). |__| Corollary 5.4. If we define a map d on a typical monomial (of the form k1. .k.nwhere each ki2 ^K) by the formula d(k1. .k.n) = ^d(k1 . . .kn); then this yields a well-defined d on each element of KPwhich can be expressed as a monomial, in the sense that if k1. .k.n= l1m . .l.nm expresses k1. .k.nas a sum of admissibles, then for each m, d(k1. .k.n) = d(l1m . .l.nm): Remark 5.5. Thus d of a monomial is independent of the various pos- sible monomial representations of the element, even though d is defined and calculated from any monomial representation. We now state the main fact concerning d, whose easy proof we leave to the reader: Theorem 5.6. If z and z0 are monomials in K with z 2 Km , then d(zz0) = d(z) + bm d(z0): SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 27 Thus the degree d is in general "skew-additive" unless b = 1, which is the traditional additivity on products. Note that the algebra K*;*is now bigraded via length and degree. We next describe the construction of a shift map F , a vector space endomorphism of K, under Assumption 2. Suppose a is a positive integer, and suppose a mapP F : K^ ! K^[ {0} is given satisfying the condition: If k0k00= l0ml00m is an admissible relation, then X F a(k0)F (k00) = F a(l0m)F (l00m) is also an admissible relation. Examples. For a = 1, K any P-B-W algebra with F the restriction to ^Kof any algebra endomorphism that takes generators to generators or to 0 (e.g., K with ff). For a = 2, R(-1) with F = ffi. Definition 5.7. Define a linear transformation F^ : T (K^) ! K by setting F^(k1 . . .kn) = F an-1(k1) . .F.a(kn-1)F (kn) and ^F(1) = 1 on the usual basis elements. P Lemma 5.8. If kiki+1= l0ml00mis an admissible relation, then X ^F(k1 . . .ki ki+1 . . .kn) = F^(k1 . . .l0m l00m . . .kn): Proof.We begin by noting that for any positive integer e, X F ea(ki)F e(ki+1) = F ea(l0m)F e(l00m): We compute F^(k1 . . .ki ki+1 . . .kn) = n-1 an-i+1 an-i a F a (k1) . .F. (ki-1)F (ki) . .F.(kn-1)F (kn): On the other hand, X ^F(k1 . . .l0m l00m . . .kn) = X n-1 n-i+1 n-i F a (k1) . .F.a (l0m)F a (l00m) . .F.a(kn-1)F (kn): 28 DAVID J. PENGELLEY AND FRANK WILLIAMS Hence the proof is completed by checking: n-i+1 an-i X an-i+10 an-i 00 F a (ki-1)F (ki) = F (lm )F (lm ); which follows from the remark at the beginning of the proof, taking_ e = an-i: |__| Corollary 5.9. We can extend F to an endomorphism of K by the formula F (k1. .k.n) = ^F(k1. .k.n). In particular, F is well-defined on the set of monomials in KPusing the formula from Definition 5.7, in the sense that if k1. .k.n= l1m .P.l.nmexpresses k1. .k.nas a sum of admissibles, then F (k1. .k.n) = F (l1m . .l.nm): Corollary 5.10. For each positive integer e, n-1e e F e(k1. .k.n) = F a (k1) . .F.(kn): From the above, we obtain the main theorem concerning F . Theorem 5.11. If e is a positive integer and z1 and z2 are monomials with z2 2 Kn;*; then ne e F e(z1z2) = F a (z1)F (z2): Assume now that d and F have been defined as above using Assump- tions 1 and 2. Suppose in addition that they interact as in Assumption 3. For k 2 ^K; d(F (k)) = d(k) + b - a: Then we are prepared for Definition 5.12. Let K be a P-B-W algebra satisfying Assumptions 1,2,3, with shift map F and (skew-additive) degree d as above. We call the quadruple (K; ^K; F; d) an algebra with shifting. Examples. Any additively graded algebra with a = b = 1; F the identity, and d the grading. Also R(-1) with a = 2, b = 1, F = ffi, and d the topological degree. We will have two other important examples soon. Remark 5.13. While K has both degree and shifting maps, they do not interact as in Assumption 3. Theorem 5.14. Let (K; ^K; F; d) be an algebra with shifting. If z is a monomial in Kn;*; then d(F (z)) = d(z) + bn - an: SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 29 Proof.Write z in monomial form k1. .k.n: We compute n-1 a d(F (k1. .k.n)) = d(F a (k1) . .F.(kn-1)F (kn)) n-1 an-2 n-2 a n-1 = d(F a (k1)) + bd(F (k2)) + . .+.b d(F (kn-1)) + b d(F (kn)) = (d(k1) + an-1(b - a)) + b(d(k2) + an-2(b - a)) + . . . +bn-2(d(kn-1) + a(b - a)) + bn-1(d(kn) + b - a) = (d(k1) + bd(k2) + . .+.bn-1d(kn)) + (an-1 + an-2b + . .+.abn-2 + bn-1)(b - a) = d(k1. .k.n) + bn - an: __ |__| We are now ready to prove the main theorem on algebras with shift- ing, Theorem F, which will enable us to produce various sheared alge- bra homomorphisms. We shall begin the proof with two preparatory lemmas. P Lemma 5.15. Let e be a positive integer and let k0k00= l0ml00mbe an admissible relation. Then under the hypotheses of the theorem, 00)+be0 e 00 X d(l00)+be0 e 00 (F d(k (k )) (F (k )) = (F m (lm )) (F (lm )): Proof.We compute, using Assumption 3, 00)+be0 e 00 d(k00)+be-aeae0 e 00 (F d(k (k )) (F (k )) = (F (F (k ))) (F (k )) e(k00))ae0 e 00 = (F d(F (F (k ))) (F (k )): By Assumption 2, X F ae(k0)F e(k00) = F ae(l0m)F e(l00m) is an admissible relation, so applying the hypothesis of the theorem, the last term above is X e 00 (F d(F (lmF))ae(l0m)) (F e(l00m)); which by reversing the above steps is X 00 (F d(lm )+be(l0m)) (F e(l00m)); __ as desired. |__| Lemma 5.16. For any monomial k1. .k.n; (k1. .k.n) = (F d(k2...kn)(k1)) . . .(F d(kn)(kn-1)) (kn): 30 DAVID J. PENGELLEY AND FRANK WILLIAMS Proof.Define ^ : T (K^) ! M by ^(k1 . . .kn) = (F d(k2...kn)(k1)) . . .(F d(kn)(kn-1)) (kn): From Lemma 5.15, Corollary 5.4, and TheoremP5.6, we see by taking e = d(ki+2. .k.n), that if kiki+1 = l0ml00mis an admissible relation, then ^(k1 . . .ki ki+1 . . .kn) = X ^(k1 . . .l0m l00m . . .kn): __ |__| Proof of Theorem F. Let z = k1. .k.mand z0= k01. .k.0nbe monomials. Then (zz0) = (k1. .k.mz0) 0) d(k ...km z0) d(z0) 0 = (F d(k2...km(zk1)) . . .(F i+1 (ki)) . . .(F (km )) (z ) (Lemma 5.1* *6) : m-id(z0) For each i, write fi= F a (ki): Now, we also compute 0) d(z0) (F d(z(z)) = (F (k1. .k.m)) m-1d(z0) am-id(z0) d(z0) 0 = (F a (k1) . .F. (ki) . .F. (km )) (Cor. 5.10 withe = d(z )) = (f1. .f.i. .f.m) = (F d(f2...fm()f1)) . . .(F d(fi+1...fm()fi)) . . .(fm ) (Lemma 5.16) : So to finish the proof, it will suffice to show that for each i 0) F d(fi+1...fm()fi) = F d(ki+1...km(zki); i.e. that m-id(z0) d(k ...km z0) F d(fi+1...fm()F a (ki)) = F i+1 (ki); which amounts to showing that d(fi+1. .f.m) + am-id(z0) = d(ki+1. .k.mz0): We have d(fi+1. .f.m) + am-id(z0) m-(i+1)d(z0) d(z0) m-i 0 = d(F a (ki+1) . .F. (km )) + a d(z ) SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 31 0) m-i 0 = d(F d(z(ki+1. .k.m)) + a d(z ) (Cor. 5.10) = d(ki+1. .k.m) + (bm-i - am-i)d(z0) + am-id(z0) (Theorem 5.14) = d(ki+1. .k.m) + bm-id(z0) = d(ki+1. .k.mz0) (Theorem 5.6): __ |__| As promised, we can now immediately apply Theorem F to prove the following theorem, which includes Theorem G as a special case. Theorem 5.17. For k 0, ^k : R(-1) ! K is a sheared algebra homomorphism, and k : R(k) ! K is a sheared algebra bijection. In particular, the defining formula for ^k holds on arbitrary products, not just admissibles. Proof.R(-1) is a P-B-W algebra, and an algebra with shifting using d = topological degree with b = 1 and F = ffi with a = 2, since d(ffi(Qi)) = d(Qi-1) = i - 1 = d(Qi) + b - a; verifying Assumption 3. Thus we can apply Theorem F by first defining ^k(Qi) = Di-kand then verifying that Adem relations are respected, as required by the hypothesis of the Theorem, which is a straightforward calculation. The earlier factorization of ^k through R(k) then yields the claim that k is a sheared algebra map (since d and ffi both pass to the quotient R(k); even without needing to know whether R(k) is a P-B-W algebra or algebra with shifting). We have seen earlier in __ Proposition 5.1 that each k is a bijection. |__| Remark 5.18. Dualizing Theorem 5.17 for k = 0 will shed light on the similarities and differences between the coproduct formulae in K* and R*, since *0: K* ! R* is an algebra isomorphism (Proposition 5.1) and a co-sheared coalgebra bijection (Theorem 5.17). We will label elements of R* using our description of K* (Theorem B). Our defining formulas show that ffi and ff correspond under k, i.e. k O ffi = ff O k; so we have the shearing formula 0 O R = K O (0 0) O fl = K O fi O (0 0); 0) 0 0 d(y0) 0 where fl(z z0) = ffid(z (z) z and fi(y y ) = ff (y) y . Since ff is dual to multiplication by O in K* (Example 3.18) we can dualize, 32 DAVID J. PENGELLEY AND FRANK WILLIAMS beginning with our formula (Theorem C) X p K*(xi;j) = x2i-p;j-q xp;q; p;q obtaining X p q p R*(xi;j) = O2 (2 -1)x2i-p;j-q xp;q p;q X 2p(2q-1) p = x0;i+j-p-qx2i-p;j-q xp;q: p;q This recovers Madsen's formula [Mad ] for the coproduct in R*. We see p(2q-1) that the extra complicating factors x20;i+j-p-qin the coproduct in R* are precisely those induced by the coshearing. The simpler coproduct in K* is one of the reasons why K may prove more useful than R in some applications. 6. Relationship between K and the Steenrod Algebra As mentioned in Section 2, while relating K to R was quite easy to accomplish, it is more challenging to relate K to the Steenrod algebra A via a sheared algebra bijection, but this is our aim.. Let X be a space, and grade cohomology negatively, as before (Remark 3.5). Let x 2 H-k(X), k 0. When one defines the action of A on H-k(X) by the conversion formula Sqj(x) = Dk-j(x); another minor miracle occurs and this produces a well-defined action of a Hopf algebra A on H*(X) [May1 , Stee3]. As in the previous section, any unstable algebra over K produces an unstable A-algebra and vice versa. We will also grade the Steenrod algebra A negatively. In formal power series we have Sq(r)x(t) = D(r-1)x(rt) with |r|= |t|= 1 (and the contragredient homology action 1_ 1_ Sq*(r)y(t) = D*(r-2 )y(r 2t) with |r|= 1, |t|= -1). As we did before with R(-1), for each k 0 we inductively extend the conversion formula and abstract from it on admissibles to produce a map, commuting with the actions on H*(X), OEk : A ! K, SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 33 given on an admissible monomial (al-1 2al> 0) by the formula OEk(Sqa1. .S.qan) = Db1. .D.bn2 Kn;(2n-1)k-j where Sqa1. .S.qan 2 A-j and bl= (k + an + . .+.al+1) - al: Remark 6.1. Some words of warning: It is curious that while this gen- eral conversion formula is correct even on inadmissibles when used to compare the two actions on HkX, it is nonetheless false for OEk it- self. For instance, while OE4(Sq1Sq2) = OE4(Sq3) = D1, Sq1Sq2x4 = D5D2x4, so D1 and D5D2 agree on every 4-dimensional class even though D1 6= D5D2, and the formula for OE4 defined above does not hold on Sq1Sq2, even though it does hold when followed by application to a cohomology class. This is due to the identification of Sq0 with 1 in the Steenrod algebra for topological reasons, which in particular prevents the Adem relations from being homogeneous, so A possesses no length grading. As a slightly different and more perplexing warn- ing example, Sq1Sq3 = 0, but Sq1Sq3x6 converts to D8D3x6. Now D8D3 = D0D7, which is nonzero in K, but of course is zero on every 6-dimensional class. This more subtle type of conversion failure can be avoided if one can arrange to encounter only "stable" Adem relations SqaSqb = . . ., those for which b a < 2b; they exhibit several very useful patterns and features. We could develop our analysis in this direction, but will take a different tack since in the end we will need to reverse the correspondence, ultimately mapping to, instead of from, the Steenrod algebra, with a sheared algebra bijection. This illustrates the noninvertibility of sheared algebra bijections. Despite the caution and pessimism advised by this remark, we intend to fit the OEk together to obtain a close relationship between A and K, ~= noting first that it is for k large (quite contrary to 0 : R ! K) that we should expect OEk to match A well with K. Given the warnings above, one might expect there will be no general map relating the multipli- cations in A and K in sheared fashion, but we will find, surprisingly, that there is. Recall from Definition 2.6 that K(k) is defined to be the vector sub- space of K spanned by the admissible monomials of the form Di00Di11. .D.ik-1k-1. We have Proposition 6.2. (i) K(k) is a sub-bialgebra of K; and 34 DAVID J. PENGELLEY AND FRANK WILLIAMS (ii) there is an exact sequence of vector spaces 0 ! K(k) ! K(k) ff!K(k - 1) ! 0.. Proof.Immediate from the definitions. (Recall the maps ff and from_ Section 3.) |__| The following proposition lists some information about OEk. Proposition 6.3. (i) The image of OEk is K(k). (ii) The kernel of OEk is the span of the admissible monomials in A with excess greater than k. (iii) OEk-1 = ff O OEk. Proof.Let SqA = Sqa1. .S.qan be an admissible monomial in A, i.e. al-1 2al> 0. Since an > 0, bn = k - an < k. Furthermore, bl- bl-1 = [(k + an + . .+.al+1) - al] - [(k + an + . .+.al) - al-1] = al-1- 2al 0: So OEk takes admissibles to admissibles, and Im (OEk) K(k). Now suppose that the excess of SqA is less than k + 1. Then b1 = k + (an + . .+.a2) - a1 = k - [a1 - (an + . .+.a2)] = k - excess(SqA ) 0: Since the n-tuple (b1; : :;:bn) completely determines (a1; : :;:an), the restriction of OEk to the space of admissibles of excess k is an iso- morphism. Finally, if the excess of SqA is greater than k, then the preceding calculation shows that b1 < 0, i.e. that Db1 = 0 . This __ proves part (ii). Part (iii) is immediate from the definitions. |__| Definition 6.4. We define K(1)*;*by K(1)n;-i= lim-K(k)n;(2n-1)k-i forn; i 0 k under the maps ff (recalling that ff : K(k)n;(2n-1)k-i! K(k-1)n;(2n-1)(k-1)-i). Note K(1)*;*has negative topological grading. The products in the K(k) combine under the bonding algebra maps ff to make K(1)*;*an algebra with length and topological degrees behaving as in K: 1 : K(1)m;i K(1)n;j! K(1)m+n;i+2m j : SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 35 Theorem 6.5. The algebra K(1) is described by generators D1-i 2 K(1)1;-i, for i > 0, with relations for i < j X j - p - 1 D1-i D1-j = D1-(i+2j-2p)D1-p : i + j - 2p Furthermore, the maps OEk induce a map OE1 : A ! K(1) that is an isomorphism of vector spaces and preserves topological de- grees (which are both negative). Proof.Since we defined K(1) as the bigraded limit of the K(k), we see that for every bidegree (n; i), K(1)n;iis isomorphic to K(k)n;(2n-1)k-i for k sufficiently large. Let D1-i be defined by the sequence {Dk-i | Dk-i 2 K(k)}. The relations will follow from taking representatives Dk-iand Dk-j in K(k) for k > max (i; j). Then if i < j, we have k - i > k - j, so X q - 1 - (k - j) Dk-iDk-j = D(k-i)+2(k-j)-2qDq: 2q - (k - i) - (k - j) Setting p = k - q, we get X j - p - 1 Dk-iDk-j = Dk-i-2j+2pDk-p; i + j - 2p T as desired. Since ker(OEk) = {0}, it follows that OE1 is an isomorphism._ |__| Remark 6.6. Given how we created K(1), it should be no surprise that the relations of Theorem 6.5 have a formal power series representation related to that for K. Indeed we have D1 (u-1)D1 ((u + v)-1v-1) = D1 (v-1)D1 ((u + v)-1u-1), P where D1 (x) denotes D1-i xi, as may be verified by the residue method of section 3. Remark 6.7. OE1 is given on admissibles by the formula OE1 (Sqi1. .S.qin) = D1-(i1-(i2+...+in)).D.1.-(in-1-in)D1-in: As usual, K(1) has a vector space basis of admissibles, where a mono- mial D1-i1 . .D.1-inis admissible provided that ik ik+1 for all k, and we see that OE1 is a bijection between both sets of admissibles. We shall relate the product in A to the product in K(1) by showing that the inverse of OE1 is a sheared algebra bijection. The map OE1 itself fails because of the presence of unstable Adem relations. 36 DAVID J. PENGELLEY AND FRANK WILLIAMS Definition 6.8. Let the inverse of OE1 : A ! K(1) be denoted by 1 : K(1)*;i! Ai. A short calculation based on Remark 6.7 yields the formula for 1 on admissibles: Lemma 6.9. 1 (D1-i1 . .D.1-in) = n-2in i +i +2ini +in in Sqi1+i2+2i3+...+2 . .S.q n-2 n-1 Sq n-1 Sq : Note already that 1 has the substantial virtue that the RHS of the above formula at least makes sense for any input (not necessarily admissible) on the LHS, while this failed for the OE1 formula. Remark 6.10. While we could have defined 1 directly without first producing OE1 via the OEk, we would not be able to have 1 arise directly from maps k out of the inverse limit, since maps out of an inverse limit are not generally forthcoming. The map ff induces an algebra endomorphism ff1 : K(1)n;-i! K(1)n;-i-(2n-1), with ff1 (D1-i ) = D1-(i+1), which we will use in our analogue of Theorem 5.17. Theorem H follows from Theorem 6.11. The map 1 : K(1) ! A is a sheared algebra bijec- tion, given on arbitrary products by the formula of Lemma 6.9. Proof.To apply Theorem F, we first note that K(1) is clearly a P- B-W algebra. It is also an algebra with shifting, taking d to be the negative of topological degree and b = 2, and F = ff1 with a = 1, since ff1 is an algebra map. Assumption 3 is verified by d(ff1 (D1-i )) = d(D1-(i+1)) = i + 1 = d(D1-i ) + b - a: We then apply Theorem F by first defining 1 (D1-i ) = Sqi, and verifying the hypothesis that Adem relations are respected, which is_ again a straightforward calculation. |__| 7.The relationship between K* and A* The relationship in Remark 5.18 between the coproduct formulas in K* and R* arose since 0 : R ! K was a sheared algebra bijection. We intend to shed an analogous light on why the coproduct formula in A* resembles that of K* with xi;jreplaced by i, a most curious relationship since this replacement puzzlingly converts the nonunits x0;ninto the SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 37 unit 0 = 1, and converts each unit xn;0into a nonunit n. We shall show how to make sense of this conundrum. It will be elucidated by the connection between the coproduct formulas in A* and K* given by the map *1 : A* ! K(1)* dual to the sheared algebra bijection 1 : K(1) ! A. As with K*; K(1)* is defined to be the bigraded dual to K(1); and we shall call the dual bidegrees the "length" and "topological" degrees. Thus K(1)*n;q= (K(1)n;-q)*: (We note that K(1)* appears in positive topological degrees.) We shall first describe a standard basis for K(1)*; then interpret *1 in terms of this basis and the standard monomial basis of A*: We begin by finding appropriate bases for the bialgebras K(k)*: Proposition 7.1. A basis for K(k)*n;*is given by the images under the projection K* ! K(k)* of all monomials in x0;n; : :;:xn-1;1of total exponent k - 1: Proof.We dualize the exact sequence of Proposition 6.2 and obtain * * * * 0 ! K(k - 1)*n;*ff!K(k)n;*! K(k)n-1;*! 0: The proposition follows from an induction on k + n based on the facts that K(k)*0;*has the single non-zero element x0;0; and that K(0) = 0: To see that the elementsPofPthe specified form are independent, we form in K*n;*a sum x = jXj+ ix0;nYi of distinct monomials, where the Xj are of exponent k - 1 and do not have x0;nas a factor, and the Yiare of exponent k - 2:We ask whetherPor not x projects to zero in K(k)*n;*: By Example 3.18, *(x) = jX0j;where X0jis Xj with the first indices of all factors reduced by one, and we know inductively that these projections are distinct basis elements of K(k)*n-1;*: So if x projects to zero, all of its summands must havePx0;nas a factor. In this case, ExampleP3.18 shows that x = ff*( iYi): Again, we know inductively that iYi projects nonzero to K(k - 1)*n;*: Since ff* is monomorphic on K(k - 1)*, we see that if x projects to zero, all the Yi must also be zero and hence so is x: Thus the images of these monomials are independent. Inductively the X0jand Yi form bases for the ends of the exact sequence, and thus the Xj and x0;nYi are a basis for K(k)*n;*,_ hence the proposition. |__| Remark 7.2. The algebra epimorphism K* ! K(k)* has been studied by [CPS ], in which they prove Proposition 7.1. They also describe an algorithm to resolve the rather complicated problem of determining the images of monomials with higher exponent. While in general thesea images can be complicated, it follows from Corollary 3.12 that x2i;j= j * a Di0D2a ; so x2i;j= 0 in K(k)* if 2a k: 38 DAVID J. PENGELLEY AND FRANK WILLIAMS We again recall from Example 3.18 that ff* : K(k)*n;*! K(k + 1)*n;*is given by multiplication by x0;n; which is monomorphic. As a consequence, K(1)* can be identified as the union of the K(k)* under the inclusions ff*: Thus a monomial xl00;nxl11;n-1.x.l.n-1n-1;1in K(k)* is identified with the monomial xl0+10;nxl11;n-1.x.l.n-1n-1;1in K(k + 1)*; and hence we obtain the following result, which includes Proposition I. Proposition 7.3. A basis for K(1)* is given by "monomials" n-1X x-l0;nxl11;n-1.x.l.n-1n-1;12 K(1)*n;t for l > li, i=1 P where t = l(2n-1)- li2i(2n-i-1): The "monomial" above represents the elements xk-l0;nxl11;n-1.x.l.n-1n-1;1in K(k)* for k l: As mentioned in Section 2, we do not know how to compute the coproducts of all elements of K(1)*; due to the challenge mentioned above involved in identifying the images of elements under the projec- tions K* ! K(k)*. But we can compute certain ones. For instance, if n m, then using the formula for the coproduct in K* and Remark 7.2, X m-j *K(p)(xm;n-m ) = x2j;k xm-j;(n-m)-k 0kn-m; max{-1;m-log2p} 0; then X *1(x-l0;n) = x-l0;k x-l0;n-k: Before proving our "leading term" theorem, Theorem J, we shall first see how it sheds light on the relationship between the coproducts in K* and A*: SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 39 We begin by noting, using Example 3.18, that the dual of ff1 : K(1) ! K(1), ff*1: K(1)*n;i! K(1)*n;i+(2n-1); takes the form ff*1(x-l0;nx) = x-l+10;nx. From Remark 2.5, the shearing property of 1 in Theorem 6.11 may be written as 1 O 1 = A O ( 1 1 ) O fl where 0) 0 -|z0| 0 fl(z z0) = F d(z(z) z = (ff1 ) (z) z and |z0|is the (negative) topological degree of z02 K(1)*;*. We dualize to obtain fl* O ( *1 *1) O *A= *1O *1; where fl* is given by (ff*1)t 1 on K(1)* K(1)**;t: We first see what is revealed by applying this equality to n: All calculations in K(1)* will be modulo length degree > n: We start with *1(n) = x-10;n; from Theorem J. Then by Lemma 7.4 we have Xn *1 *1(n) = x-10;j x-10;n-j j=0 for the right side of the equality. On the left side, Xn n-j *A(n) = 2j n-j; j=0 hence Xn n-j -1 ( *1 *1)*A(n) = x-20;j x0;n-j: j=0 Now x-10;n-j2 K(1)*n-j;2n-j-1; so Xn n-j-1 -2n-j -1 fl*( *1 *1)*A(n) = (ff*1)2 (x0;j ) x0;n-j j=0 Xn = x-10;j x-10;n-j; j=0 40 DAVID J. PENGELLEY AND FRANK WILLIAMS for the left side. So we see first that the coproduct formula on n is essentially forced by the apparently simpler coproduct formula on x0;n; together with the co-shearing feature of the map *1: Next we shall turn the tables around to see how, in the other direc- tion, the coproducts of the xi;jare essentially forced by the coproducts of the n and the co-shearing of *1: Let E = 2m and suppose that n m: We compute, using Theorem J and Lemma 7.4, that *1 *1(Enm ) = *1(x-(E+1)0;nxm;n-m ) X -(E+1) m-j -(E+1) = x0;j+k x2j;k x0;n-j-kxm-j;(n-m)-k: 0kn-m; 0jm On the other hand, X n-i m-j *A(Enm ) = 2i E2j En-im-j : 0in; 0jm The pairs of indices break into two cases: (1) i j and n - i m - j; and (2) otherwise. In case (2), the image of this under *1 *1 lands in total length degree > n: So modulo this filtration we get, again from Theorem J, ( *1 *1)*A(Enm ) = X -(2n-iE+2m-j) m-j -(E+1) x0;i x2j;i-j x0;n-i xm-j;(n-i)-(m-j): 0jm; 0i-jn-m The topological degree of the righthand factor, x-(E+1)0;n-ixm-j;(n-i)-(m-j), is (E +1)(2n-i-1)-2m-j (2(n-i)-(m-j)-1), which simplifies to 2n-iE + 2m-j - (E + 1): So fl*( *1 *1)*A(Enm ) = X -(E+1) m-j -(E+1) x0;i x2j;i-j x0;n-i xm-j;(n-i)-(m-j); 0jm; 0i-jn-m which, letting k = i - j; matches our formula above for *1 *1(Enm ). Thus we see the coproduct formula on xm;n-m emerge as essentially forced by the coproduct formulae on n and m . For us these sample calculations of mutual forcing via the cosheared feature of *1 provide a satisfying answer to the earlier puzzle of how the coproduct formulae in K* and A* seem so similar in some ways, and yet in other ways so different. We shall prove Theorem J using some related maps *k: A* ! K* which we now describe. SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 41 First fix a value for n. Let I = (i1; : :;:in) be an n-tuple of integers, and DI = Di1. .D.in. Definition 7.5. Let : Zn+1 ! Zn be given by (k; I) = (2n-1k - 2n-2in - . .-.2i3 - i2 - i1; : :;:2k - in - in-1; k - in): Note that is linear in the (n + 1)-tuple (k; i1; : :;:in), and each (k; _) is an isomorphism. We define linear transformations k : K ! A for k 0 by Definition 7.6. If DI is admissible, we set k(DI) equal to the mono- mial string Sq (k;I), which represents zero in A if any indices are nega- tive, i.e. if in > k. (We will often need to focus on the specific monomial string Sq (k;I), not merely the element of A which it represents.) Remark 7.7. The formula for shows that k(DI) represents an ad- missible in A provided in < k, and that if in = k then it is an admissible string with some trailing Sq0s appended. Remark 7.8. We note that this definition gives the correspondence be- tween K and A acting on a cohomology class of degree -k: As such, if DI is an arbitrary (not necessarily admissible) monomial, then we can see, from a universal example such as K(Z=2; k), that it follows that k(DI) is given by the same recipe, provided it is then rewritten in terms of admissibles and those terms with excess greater than k dropped. We now prepare to prove a leading term theorem for *k: A* ! K*, from which the proof of Theorem J will follow. Definition 7.9. If SqJ = Sqj1. .S.qjn is admissible except possibly for trailing Sq0s, l 0; and 0 j n; set X V j(SqJ) = {SqJ-oe| oe is a(j; n - j)-shuffle of(2j-1; 2j-2; : :;:1; 0; : :;* *:0)}; and let Vlj(SqJ) denote V j(SqJ); expressed in terms of admissibles, deleting all summands of length less than n and all those of excess greater than l - 1: Definition 7.10. For DI admissible, define X W j(DI) = {DI- -11(oe)| oe is a(j; n - j)-shuffle of(2j-1; 2j-2; : :;:1; 0; * *: :;:0)}: 42 DAVID J. PENGELLEY AND FRANK WILLIAMS Lemma 7.11. If DI is admissible, then modulo the subspace of A with basis the set of admissibles of length less than n; Vkj( k(DI)) = k-1(W j(DI)): Proof.Let X = {oe | oe is a(j; n - j)-shuffle of(2j-1; 2j-2; : :;:1; 0; : :;:0)}: Then modulo the stated subspace of A, and using Definitions 7.6 and 7.10, Remark 7.8, and the linearity of , X k-1(W j(DI)) = { k-1(DI- -11(oe)) | oe 2 X} X -1 = {Sq (k-1;I- 1 (oe))| oe 2 X} (with summands of excess> k - 1 dropped after conversion to admissibles) X = {Sq (k;I)-oe| oe 2 X} (with summands of excess> k - 1 dropped after conversion to admissibles). This in turn is by Definition 7.9 and Remark 7.7 equal to Vkj(Sq (k;I)) if (k; I) is nonnegative, and also otherwise since both will then represent __ zero. Finally, Vkj(Sq (k;I)) = Vkj( k(DI)): |__| Theorem J will follow from P n TheoremL7.12. Let l = i=1li: If ln 1 and k l; then modulo * m>n Km;*, *k(l11: :l:nn) = xk-l0;nxl11;n-1.x.l.n-1n-1;1: Our proof of this will rely first on a special case in slightly stronger form. ae L xk-l ifk l Lemma 7.13. Let l 1. Modulo m>n K*m;*, *k(ln) = 0;n 0 otherwise. Proof.Let DI be admissible of length at most n. We evaluate < *k(ln); DI> = = SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 43 8 < 1 ifl-1( k(DI)) contains the term = (Sq2n-1. .S.q2Sq1)l when expressed using admissibles : 0 otherwise. But the only admissible SqJ of length n for which l-1(SqJ) n-1l l contains this term is Sq2 . .S.q , and since k(DI) contains this term in its admissible representation if and only if DI = Dnk-l, we * k-l have *k(ln) = Dnk-l modulo length n. By Proposition 3.20, x0;n= * * __ Dnk-l in K , whence the lemma. |__| Proof of Theorem 7.12.We shall freely use the following facts: If w = Sqi1. .S.qim is admissible with m < n; then w pairs to 0 with the ideal generated by n; n+1; : :.:If the excess of an admissible SqJ is greater than q; then SqJ pairs trivially with any monomial in the 's of total exponent q or less. Inductively, based on Lemma 7.13, which includes the case l = 1; assume the desired formula holds for total exponents less than l. Let j < n be the smallest j for which lj > 0 (the case where this smallest j equals n is already covered by Lemma 7.13). Write = lj-1jlj+1j+1:l:n:-1n-1and let x 2 K be defined correspondingly. Let DI be admissible of length n: If the length of DI is strictly less than n; then < *k(jlnn); DI> = 0 = , since k(DI) also has length < n. So assume that DI is admissible of length n: We compute: < *k(jlnn); DI> = = = by Remark 7.7 and Definition 7.9 = by Definition 7.9 = sincek l = from Lemma 7.11 = < *k-1(lnn); W j(DI)> = by induction, sincek - 1 l - 1, and sinceW j(DI) has length n. Finally, = from Definition 7.10, since using Lemma 3.7 one can check that the elements -11(oe) are precisely 44 DAVID J. PENGELLEY AND FRANK WILLIAMS Yk ______K(1)-fflk H H || H H 1 ae|k H H | H H |? HHj K(k) ______-Kj________-A k k Figure 2 the length n monomials which involve the admissible basis element __ Dj0Dn-j1(in fact they each equal it). |__| Proof of Theorem J.Choose k to be at least the topological degree of l11: :l:nn. Let Yk denote the subspace of K(1) with basis the set of admissibles {D1-I = D1-i1 . .D.1-iq| q 0 and i1 k}: We note first that in topological degrees -k; Yk is isomorphic to K(1): Second, let aek : K(1) ! K(k) denote the projection, and jk : K(k) ! K and fflk : Yk ! K(1) the inclusions. By following the bases we see that aek O fflk is an isomorphism, and that 1 O fflk = k O jk O aek O fflk by direct calculation using our formulas on admissibles for k and 1 (even though we caution that 1 6= k O jk O aek). We summarize this in Figure 2, in which the triangle is not commutative until composed with fflk. Therefore ffl*kO *1(l11. .l.nn) = ffl*kO ae*kO j*kO *k(l11. .l.nn): L Now *k(l11. .l.nn) = xk-l0;nxl11;n-1.x.l.n-1n-1;1modulo m>n K*m;*from The- orem 7.12 since k deg l11: :l:nn l. Thus, since fflk, aek, jk are each length-preserving, we have * * M ffl*kO ae*kO j*kO *k(l11. .l.nn) = ffl*kO ae*kO j*k(xk-l0;nxl11;n-1.x.l.n-1n-1* *;1) modulo K(1)*m;*; * * m>n which in turn equals ffl*k(x-l0;nxl11;n-1.x.l.n-1n-1;1) since the exponent of xk-l0;nxl11;n-1.x.l.n-1n-1;1is less than k: Now since_ffl* **k is an isomorphism in this degree, the result follows. |__| SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 45 8. The cohomology of K In this section we shall use methods of [Pr] to compute the cohomol- ogy ExtK(F2; F2) of K. (We suggest [Lo ] as a concise reference for the cohomology of P-B-W algebras.) We shall then construct a sheared algebra bijection to it from the algebra K(1) of Section 6. Finally, we shall construct a sheared algebra bijection from it to A, consider the effect of sheared algebra maps on homology and cohomology, summa- rize how our sheared bijections fit together, and suggest applications via the Miller spectral sequence. We begin by recalling the defining relations of K from Definition 2.1: X k - 1 - j DiDj = Di+2j-2kDk; (i > j): k 2k - i - j Following [Pr], we define m = i + 2j - 2k: We note that k - 1 - j k - 1 - j = : 2k - i - j j - m It is clear that K is a P-B-W algebra in our setting, with skew- additive topological degree with b = 2 in Assumption 1. The next proposition then follows from [Pr]. Before stating it, we shall make a brief remark on gradings: Remark 8.1. The cohomology of K is H*(K) = ExtK(F2; F2). As is pointed out in [Pr], since a P-B-W algebra is bi-graded, the cohomology is tri-graded. The elements in Hi;n;k(K) are defined by Priddy to be of cohomological degree i; length degree n; and internal degree k: For consistency with our convention of grading cohomology negatively, we define the topological degree of such elements to be -(i + k): Proposition 8.2. The cohomology of K, H*(K); is the algebra with generators oek 2 Ext1;1;k(k 0) and Adem relations, for m k, X k - 1 - j oem oek = oem+2k-2joej: j - m The internal degree of products behaves in analogous fashion to that of K; i.e. skew-additively with b = 2 in Assumption 1. As usual, H*(K) has a basis of admissibles, where a monomial oei1. .o.ein is admissible provided that il> il+1for all l. The relations for H*(K) 46 DAVID J. PENGELLEY AND FRANK WILLIAMS are encoded by uoe(u-1)oe((u + v)-1v-1) = voe(v-1)oe((u + v)-1u-1), which may again be verified by a residue calculation. We next proceed to develop a sheared algebra bijection ! : K(1) ! H*(K): We recall that K(1) is an algebra with shifting map ff1 : Previously we used the absolute value of the topological degree as degree map, but we now observe that the length degree n can also serve as a de- gree map, with b = 1; and n interacts with ff1 (with a = 1) in the desired way (Assumption 3) to again produce an algebra with shifting (K(1); ff1 ; n): In the now-standard way, we define ! on generators by the formula !(D1-i ) = oei-1 Theorem 8.3. The map ! is a sheared algebra bijection which pre- serves topological degree. Thus it is given by !(. .D.1-cD1-b D1-a ) = . .o.ec+1oeboea-1 on any monomial, not necessarily admissible. Proof.A direct calculation confirms that ! respects Adem relations as required by Theorem F. Alternatively, one may do a pleasant calcu- lation comparing the formal power series forms of the Adem relations. This yields the sheared algebra map and formula. It is then immediate __ that it preserves topological degree and is a bijection. |__| We now turn to the relationship between H*(K) and A: As we observed in Proposition 8.2, the internal degree of products in H*(K) behaves analogously to that in K and so internal degree will serve as a degree function for H*(K); with constant b = 2: To obtain a shifting map, we need only define s(oei) = oei+1, and verify Assumption 2. This is easily checked, with constant a = 1, by comparison of binomial coefficients, or by formal power series via the method mentioned for ff after Example 3.18, since here s(oe(x)) = x-1oe(x). Finally, s interacts with internal degree i as required by Assumption 3. We have Proposition 8.4. The map s is an algebra endomorphism, and we have an algebra with shifting (H*(K); s; i): SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 47 We intend to apply Theorem F to obtain yet another sheared algebra bijection, this time from H*(K) to A. But first, note that we could simply compose the isomorphisms !-1 : H*(K) ! K(1) and 1 : K(1) ! A to obtain an isomorphism of vector spaces, : H*(K) ! A; given on admissibles by (oei1. .o.ein) = n-2in+1 i +i +2in+1i +in+1 in+1 Sqi1+i2+2i3+...+2 . .S.q n-2 n-1 Sq n-1 Sq : It is immediate that preserves topological degree. It would be nice to claim now that this is a sheared algebra map, i.e. that this formula holds on any monomial, simply by combining Theorems 6.11 and 8.3, but unfortunately Theorem 8.3 does not yield adequate information about !-1 (recall that sheared algebra bijections are only one-way af- fairs). The following theorem, which emerges immediately in the now familiar way using Theorem F, accomplishes this goal of relating prod- ucts in H*(K) directly to those in A. Theorem 8.5. The map : H*(K) ! A is a sheared algebra bijection. We might equally well consider the Lie Steenrod algebra, AL [Lo , Pr],which is formed by replacing the relation Sq0 = 1 with the relation Sq0 = 0: Since the image under of any monomial in H*(K) produces a monomial in A for which the reduction to a sum of admissibles via Adem relations does not produce any terms with factors of Sq0 (cf. Re- mark 6.1), can equally well be regarded as a sheared algebra bijection from H*(K) to AL: Theorems 8.3 and 8.5 give Theorem K of Section 2. We now turn to the effect of sheared algebra morphisms on homology and cohomology, and note that the proof of Theorem L is straightforward using the definition of a sheared algebra homomorphism and Theorems 5.6, 5.11, and 5.14. Using the formulas of [Pr], it is easy to see that if K; M are both P- B-W algebras, and if carries generators to generators, one can write a formula for the induced map on cohomology. In particular, one can im- mediately see that the sheared algebra morphism ^0:iR(-1)j! K of Section 5 induces a vector space isomorphism H* ^0 on cohomology that is also a sheared algebra map. We hope to apply the results of this section to the setting of the Miller spectral sequence [KL1 , Mill]. The built-in "unstableness" of the action of K on the homology of infinite loopspaces should enable the use of 48 DAVID J. PENGELLEY AND FRANK WILLIAMS the standard Ext functors in place of the somewhat artificial Unext, and thereby result in shortened calculations. Also we feel that, with the simpler coproduct formula of K* as compared to R*, the range of applicability of the Miller spectral sequence can be extended, leading to more results in the direction of those of [KL2 ]. We are presently working on such computations. 9.Future directions As we have seen, the algebra K has a very rich internal structure. It may be regarded as being closer to the actual topology of spaces than either A or R, and we hope this, along with the "nice" coproduct formula in K* , may facilitate applications. We have also seen how K, and its cohomology, are related to A and R via various interrelated sheared algebra bijections. There are several avenues of further research and application that we are currently pursuing. We briefly list a few. (1) The work of Bisson-Joyal [BJ ] approaches K from a very different perspective. Combining our point of view with theirs is providing new insights as well as new results. In particular, in [BPW ] we use the extended Milnor Hopf algebra [BJ ] to explain the shearing that occurs in our map 1 : K ! A. (2) Although the present paper provides a unifying treatment of homology and cohomology operations through the use of K and its related algebras, it would be nice to see a direct connection between the Steenrod algebra and the Dyer-Lashof algebra. We treat this question in [PW1 ] in which we develop an inverse limit process, based on the way K(1) was formed from K, to produce the Steenrod algebra from the Dyer-Lashof algebra and vice versa. (3) As mentioned above, we plan to examine the categorical structure of sheared algebra morphisms in a future paper. We hope to relate this to the concept of gradings over various monoid structures on Z+ x Z+ that was discussed following Definition 2.1. (4) The "Nishida" actions of Kop on K based on fundamental classes in positive degrees, mentioned above in section 4, are closely related to the actions of the Steenrod algebra on the homology and cohomology of iterated loopspaces of spheres. We are interested in describing these actions through use of the maps ; ff, and V of this paper. (5) As mentioned in section 8, we also plan to use homological calcu- lations based on K to improve and extend the use of the Miller spectral spectral sequence [KL2 , Mill] in the computation of the homology of infinite loopspaces. SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 49 (6) In many cases the statements of theorems become more concise through use of K. For instance, the description of the homology of iterated loop spaces in terms of K is well-known [CLM , CPS , CCPS , May2 ]. Use of K in cohomology can also produce elegant results. In [PW2 ] we use K to describe bases for certain cyclic unstable A-modules. In particular, we show that the module M(n) which is the quotient of the free unstable A-module on a generator x2nof degree 2n by the action n-2 of A(Sq1; : :;:Sq2 ) can be expressed as a K-module with basis the set {DIx2n} where DI is admissible and I consists only of entries of the form 0 and powers of 2 that are less than 2n. We use this description to give the following characterization of the Dickson algebras (our version of a recent result of F. Peterson). The Dickson algebra K*n+1;*is the K- algebra quotient of the free unstable K-algebra on M(n) by the single additional relation D2n-1D2n-1x2n = x2nD2n-1x2n: Finally, we close with the following particularly elegant formulation of the theorem of J. Lin [Li] about the action of the Steenrod algebra in the cohomology of finite H-spaces. Let QH-iX denote the indecomposable quotient of H*X in de- gree -i (i 0). Let Nj be the natural numbersLwith j ones in their dyadic expansion, and let QH(j)X = i2NjQH-iX. Clearly L QHe*X = jQH(j)X. Then Theorem 9.1. Let X be a simply connected finite H-space, and let K = {Di11Di23: :D:ik2k-1.|.{.ik} is eventually zero}: Then for each j, j-1) QH(j)X = K . QH-(2 X: The theorem may be proved by a straightforward induction from Lin's theorem, which says that under the hypotheses on X, any inde- composablerin degree -(2r+1k + 2r - 1) for k > 0 is in the image of Sq2 k(which turns out to be just D2r-1). But the theorem is perhaps better understood as follows, and per- haps even stated a little more strongly. Elements of K acting on j-1) x 2 QH-(2 X retain exactly j dyadic ones in the resulting degree (indecomposability is not needed for this), and moreover every inde- composable y 2 QH(j)X arises in this way. In actual fact, for y in fixed degree, the element DI 2 K; for which y = DIx, is uniquely determined (provided we do not redundantly insert D2j-1, for which D2j-1x = x), and is easily calculated from the dyadic expansion. For instance, any indecomposable y in dyadic degree -101111100111 is the image of precisely D223-1D28-1on an indecomposable x in dyadic 50 DAVID J. PENGELLEY AND FRANK WILLIAMS degree -111111111. As the class y is traced back towards x by suc- cessive individual operations and intermediate classes, the number l of trailing ones in each intermediate dyadic degree dictates the subscript 2l- 1 of the next operation; and each operation always simply excises the rightmost zero in the expansion. Thus the lengths of blocks of con- secutive zeros in the expansion of the degree of y are the exponents, in reverse order, of DI. References [Ada] J.F. Adams, Infinite loop spaces, Princeton Univ. Press, 1978. [Ade] J. Adem, The relations on Steenrod powers of cohomology classes, in Alge- braic Geometry and Topology (ed. R.H. Fox), Princeton Univ. Press, 1957, pp. 191-238. [AK] S. Araki, T. Kudo, Topology of Hn-spaces and Hn-squaring operations, Mem. Fac. Sci. Kyusyu Univ. Ser. A 10 (1956), 85-120. [BJ] T.P. Bisson, A. Joyal, Q-rings and the homology of the symmetric groups, Contemporary Mathematics 202 (1997), 235-286. [BPW] T.P. Bisson, D.J. Pengelley, F. Williams, Stabilizing the lower operations for mod 2 cohomology, to appear. [BM] S.R. Bullett, I.G. Macdonald, On the Adem relations, Topology 21 (1982), 329-332. [CCPS] H.E.A. Campbell, F.R. Cohen, F.P. Peterson, P.S. Selick, Self-maps of loops spaces. II, Trans. Amer. Math. Soc. 293 (1986), 41-51. [CPS] H.E.A. Campbell, F.P. Peterson, P.S. Selick, Self-maps of loops spaces. I, Trans. Amer. Math. Soc. 293 (1986), 1-39. [CLM] F.R. Cohen, T.J. Lada, J. P. May, The homology of iterated loop spaces, Lecture Notes in Mathematics 533 (1976), Springer, New York. [DL] E. Dyer, R.K. Lashof, Homology of iterated loop spaces, Amer. J. Math. 84 (1962), 35-88. [GP] V. Giambalvo, F.P. Peterson, to appear. [GPR] V. Giambalvo, D.J. Pengelley, D.C. Ravenel, A fractal-like algebraic spli* *t- ting of the classifying space for vector bundles, Trans. Amer. Math. Soc. 307 (1986), 433-455. [HP] N.H.V. Hung, F.P. Peterson, A-generators for the Dickson algebra, Trans. Amer. Math. Soc. 347 (1995), 4687-4728. [H] T. Hunter, On the homology spectral sequence for topological Hochschild homology, Trans. Amer. Math. Soc. 348 (1996), 3941-3953. [KL1] D. Kraines, T. Lada, The cohomology of the Dyer-Lashof algebra, in proc., Northwestern Homotopy Theory Conference (Evanston, Ill., 1982), Con- temporary Math. 19 (1983), Amer. Math. Soc., 145-152. [KL2] D. Kraines, T. Lada, Applications of the Miller spectral sequence, in pro* *c. conf. Current Trends in Algebraic Topology (1981), Canadian Math. Soc. Conf. Proc. 2, Part 1, 479-497, American Mathematical Society, 1982. [Li] J.P. Lin, Steenrod connections and connectivity in H-spaces, Memoirs of the Amer. Math. Soc. 369 (1987), 1-87. [Lo] L. Lomonaco, A phenomenon of reciprocity in the universal Steenrod alge- bra, Trans. Amer. Math. Soc. 330 (1992), 813-821. SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 51 [Mad] I. Madsen, On the action of the Dyer-Lashof algebra in H*(G), Pacific J. Math. 60 (1975), 235-275. [MM] I. Madsen, R. J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Princeton Univ. Press, 1979. [May1]J.P. May, A general algebraic approach to Steenrod operations, in proc. conf., The Steenrod Algebra and its Applications (ed. F. Peterson), Lectu* *re Notes in Mathematics 168 (1970), Springer, New York, pp. 153-231. [May2]J.P. May, Homology operations on infinite loop spaces, in proc. conf., Al* *ge- braic Topology (Madison, 1970), Proc. Symp. Pure Math. 22, Amer. Math. Soc., 1971, pp. 171-185. [Mill]H. Miller, A spectral sequence for the homology of an infinite delooping, Pacific J. Math. 79 (1978), 139-155. [Miln]J. Milnor, The Steenrod algebra and its dual, Annals of Math. (2) 67 (195* *8), 150-171. [Mu] H. Mui, Modular invariant theory and cohomology algebras of symmetric groups, Jour. Fac. Sci. Univ. Tokyo, Sect. 1A, 22 (1975), 319-369. [N] G. Nishida, Cohomology operations in infinite loop spaces, Proc. Japan Acad. 44 (1968), 104-109. [PW1] D.J. Pengelley, F. Williams, Limits of algebras with shifting and a relat* *ion- ship between the Steenrod and Dyer-Lashof algebras, to appear. [PW2] D.J. Pengelley, F. Williams, Unstable cyclic modules over the Steenrod al- gebra, ideals in the Kudo-Araki-May algebra, and the Dickson algebras, to appear. [Pr] S.B. Priddy, Koszul Resolutions, Trans. Amer. Math. Soc. 152 (1970), 39- 60. [Po] M.M. Postnikov, The lower Steenrod algebra, Russian Math. Surveys 49:3 (1994), 192-193. [Sm] V.A. Smirnov, Hopf radical algebras and the Steenrod algebra, Russian Math. Surveys 42:2 (1987), 301-302. [Stee1]N.E. Steenrod, Homology groups of symmetric groups and reduced power operations, Proc. Nat. Acad. Sci. USA 39 (1953), 213-223. [Stee2]N.E. Steenrod, Cohomology operations derived from the symmetric group, Comment. Math. Helv. 31 (1957), 195-218. [Stee3]N.E. Steenrod, D.B.A. Epstein, Cohomology Operations, Princeton Univ. Press, 1962. [Stei]R. Steiner, Homology operations and power series, Glasgow Math. J. 24 (1983), 161-168. Corrigendum, Glasgow Math. J. 26 (1985), 105. [W] C. Wilkerson, A primer on the Dickson invariants, in proc., Northwestern Homotopy Theory Conference (Evanston, Ill., 1982), Contemporary Math. 19 (1983), Amer. Math. Soc., 421-434. New Mexico State University, Las Cruces, NM 88003 E-mail address: davidp@nmsu.edu E-mail address: frank@nmsu.edu