STABILIZING THE LOWER OPERATIONS FOR MOD 2 COHOMOLOGY TERRENCE P. BISSON, DAVID J. PENGELLEY, AND FRANK WILLIAMS Abstract. In [BJ ] and [PW ] we studied a bialgebra K which underlies both the Steenrod algebra and the Dyer-Lashof algebra. Its elements act as lower-indexed operations in both the mod 2 cohomology of spaces and the mod 2 homology of E1 -spaces. The algebra K can be defined explicitly by generators and relations (as in [PW ]), or it can be defined as the algebra of operations in the theory of Q-modules (as in [BJ ]). In [PW ] a connection between K and the Steenrod algebra A of stable cohomology operations was established by means of a sheared algebra bijection between A and a new algebra K(1), which is a stabilized version of K. In [BJ ] the extended Milnor Hopf algebra M is used (among other purposes) to define a convolution algebra containing both K and A. In this paper we establish a connection between these two approaches. May 19, 1998 1. Introduction In [BJ ] and [PW ] we studied a bialgebra K, and developed many of its properties. The elements of K act as operations in both the mod 2 cohomology of spaces and the mod 2 homology of E1 -spaces, and K underlies both the Steenrod algebra and the Dyer-Lashof algebra. The relationship between K and the Dyer-Lashof algebra is especially close, as is discussed at length in [BJ ] and [PW ]. The connection between the structure of K and that of the Steenrod algebra A of stable cohomology operations is more intricate. In [BJ ] the connection between K and A is described in terms of the extended Milnor Hopf algebra M, which is a semi-direct extension of the dual Steenrod algebra. An algebra inclusion j : K ! [M+ ; F2] is given, where M+ is a certain sub-bialgebra of M and [M+ ; F2] is the convolution algebra dual to the coalgebra M+ . ____________ 1991 Mathematics Subject Classification. Primary 55S99; Secondary 16W30, 16W50, 55S10, 57T05. Key words and phrases. Steenrod algebra, Kudo-Araki, Q-ring, Q-module, sheared algebra map, Hopf algebra. 1 2 T. BISSON, D. PENGELLEY, AND F. WILLIAMS In [PW ] the connection between K and A is described in terms of a new algebra K(1), which we call "the algebra of stable lower opera- tions". It is constructed as an inverse limit of subalgebras of K, and is related to A by a sheared algebra bijection 1 : K(1) ! A. Our goal in this paper is to understand the relationship between these two maps j and 1 , and thereby obtain a tighter connection between K and A. In section 2 we recall the definitions of K and describe the dual bialgebra W. In section 3 we recall the construction of the maps j and 1 in formats that will enable us to develop the relationship between them in section 4. We conclude in section 5 by developing this relationship on the duals as well. We have slightly modified some notations from [BJ ] and [PW ] in order to avoid notational collisions. 2. Background Let us recall the basic definition of K. Definition 2.1. Let K be the F2 algebra (with identity) generated by elements Di (i = 0; 1; :::) subject to the (Adem) relations X k - 1 - j DiDj = Di+2j-2kDk; (i > j) k 2k - i - j (Note: D0 is not the identity). Furthermore, K is a bialgebra with comultiplication defined on gen- erators by the (Cartan) formula Xi OE(Di) = Dk Di-k. k=0 Remark 2.2. In [BJ ] K is developed as the algebra of operations in the theory of Q-modules, which are vector spaces with a suitable total operation. Precisely, a Q-module is a vector space V (over the field F2) together with a total operation Qt : V ! V [[t]] that satisfies simple axioms of additivity and symmetry: Qt(a + b) = Qt(a) + Qt(b) and Qt(Qs(a)) is symmetric in t and s. Here V [[t]] is the vector space of formal power series in t with coefficients in V , and Qt is extended over V [[s]] by Qt(s) = s(s + t). Then Qt is the total operation for individual operations (denoted here by Di), and the symmetry is equivalent to the Adem relations on the individual operations. The Cartan comul- tiplication in K corresponds to the natural structure of Q-module on the tensor product of Q-modules. The mod 2 cohomology of spaces STABILIZING THE LOWER OPERATIONS 3 and the mod 2 homology of E1 -spaces have natural Q-ring structures, as we see from the usual lower-indexed operations defined in [St] and [AK ], etc. Remark 2.3. In [PW ] K is studied as the algebra of the lower-indexed operations (emerging from F2-equivariance and the quadratic construc- tion). These operations are given a unified treatment in [M ], where the Adem relations that they satisfy are derived. Hence in [PW ] K is called the Kudo-Araki-May algebra. The algebra of lower-indexed operations is also considered in [Sm ]. The vector space K = K*;*is bigraded, with the bidegree of elements in K defined inductively by the requirement that the multiplication be a map Km;i Kn;j! Km+n;i+2m j, with identity 1 2 K0;0and generators Di 2 K1;i. We may refer to ele- ments in Kn;ias having length n and dimension (or topological degree) i. Each Kn;iis a finite dimensional vector space, and K is a graded algebra with respect to length and a graded coalgebra with respect to dimension. Let Kn denote the vector space of elements of length n; then each Kn is a coalgebra, and the coalgebra K is isomorphic to the direct sum of the Kn. In both [BJ ] and [PW ] there is a complete description of the bialgebra dual to K. Precisely, let Wn denote the algebra Wn = F2[wi;j| i + j = n; i 0; j 0]=(wn;0- 1) and consider the bialgebra W = n0 Wn with comultiplication X p (wi;j) = w2i-p;j-q wp;q. p;q In [BJ ] and [PW ] it is shown that the algebra Wn is the graded dual of the coalgebra Kn, where wi;jis dual to the monomial Di0Dj1 with respect to a certain basis of admissibles. It is also shown that the comultiplication in W is dual to the multiplication in K. In this sense the bialgebra W is the bi-graded dual of K. The Nishida relations produce an action of Kop on K [PW ], which du- alizes to the A-action on the summands Wn that realizes the standard A-action on the Dickson algebras [Wi ]. 3. Definition and discussion of M and K(1) In this section we define and discuss the extended Milnor bialgebras M and M+ from [BJ ] and the stabilized algebra K(1) from [PW ]. 4 T. BISSON, D. PENGELLEY, AND F. WILLIAMS The positive extended Milnor bialgebra is M+ = F2[0; 1; 2; : :]: with comultiplication X p (i) = 2i-p p: The extended Milnor Hopf algebra is M = F2[0 ; 1; 2; : :]:, the result of adjoining -10to the bialgebra M+ . If we let A* denote the (graded) dual of the Steenrod algebra with its usual comultiplication, then A* is the quotient Hopf algebra A* = M=(0 - 1) = F2[1; 2; : :]:: In fact, M is a semi-direct extension of A* as Hopf algebras (see [BJ ] for instance). Remark 3.1. From [BJW , Theorem 16.2] we know M+ = Q(HF2)**, the indecomposables of the Hopf ring of unstable F2-homology. In this context, multiplication by 0 corresponds to the homology suspension. Remark 3.2. In [BJ ] a Q-ring structure on M is defined and used to give a general formulation of the Nishida relations. We will not consider this aspect here. Remark 3.3. As discussed in [BJ ], the bialgebras M+ ; M; A*; and W have beautiful interpretations within the setting of representable func- tors (algebraic groups). ThePbialgebra M+ , together with the generic i additive power series (x) = ix2 , represents the functor which as- signs to each commutative F2-algebra R the monoid of additive power series under composition. Similarly, the Hopf algebra M represents the subgroup A(R) of additive power series whose leading coefficient is invertible, and the dual Steenrod algebra A* represents the subgroup S(R) of additive power series whosePleading coefficient is equal to 1. j Moreover Wn, together with w(x) = wi;jx2 , represents the functor which assigns to each commutative F2-algebra the set of Ore polyno- mialsnof length n (the monic additive polynomials with highest term x2 ). The composition of Ore polynomials of length n and m is an Ore polynomial of length n + m, and this construction is represented by the comultiplication in W, which is dual to the multiplication in K. For any coalgebra C (over F2) we may consider the algebra of all linear functionals onPC, with multiplication givenPby the convolution product (f ? g)(c) = f(c0)g(c00) where c 7! c0 c00is the comulti- plication on C. We will denote this "full-dual" convolution algebra by [C; F2]. Following [BJ ], we consider the algebra morphism jn : M+ ! Wn given by jn(i) = wi;n-i(and jn(i) = 0 for i > n). Thus jn represents STABILIZING THE LOWER OPERATIONS 5 the inclusion of the Ore polynomials of length n into the additive power series. The maps jn define an algebra embedding of K into the dual algebra [M+ ; F2]. Precisely, we have j : K ! [M+ ; F2] defined by = <;Pjn()> for 2 Kn and 2 M+ . We easily calculate that j(Dk) = l0(k0l1)*, where (k0l1)* 2 [M+ ; F2] is the linear functional that is dual to k0l1(with respect to all monomials in M+ ). Following [PW ], considerPthe algebra morphism ff : K ! K dual to multiplication by w0 = n0 w0;n, which is a coalgebra morphism on W (the direct sum of the multiplications by w0;non each Wn). More precisely, = <; w0;nw> for 2 Kn and w 2 Wn. On the generators we have ff(Di) = Di-1. Remark 3.4. The map ff is related to the "suspension" functor on Q-modules (that is, on K-modules) given by s(M) = F2{s} M where F2{s} is the vector space with basis element s and Q-structure Qt(s) = st. For z 2 M we have Qt(s z) = s t Qt(z), and it follows that the action of K on s(M) is given by (s z) = s ff()(z) for 2 K and z 2 M. As in [PW ], let K(k)denote the subalgebra of K generated by {Di | 0 i < k}. In fact K(k)is a sub-bialgebra of K. Then ff restricts to give a sequence of algebra morphisms ffk (k-1) . . .! K(k)! K ! . . ., from which we define the limit K(1) = lim-K(k). ffk Recall from [PW ] that the algebra K(1) has generators D1-i for i > 0 (corresponding to the compatible sequences (Dk-i)k in the limit), with induced (Adem) relations X j - k - 1 D1-i D1-j = D1-(i+2j-2k)D1-k (i < j): k i + j - 2k The algebra K(1) is bigraded with respect to length and topological degree (with D1-i 2 K(1)1;-i). Further, there is an algebra homomor- phism ff1 : K(1) ! K(1) determined by ff1 (D1-i ) = D1-i-1 : 6 T. BISSON, D. PENGELLEY, AND F. WILLIAMS Note that on compatible sequences ff1 is given by (ak) 7! (ff(ak)). In [PW ] a linear bijection 1 : K(1) ! A is defined with 1 (D1-i ) = Sqi on the generators. But 1 is not an algebra morphism; it is shown in [PW ] to be a "sheared algebra morphism" with respect to the "shift- ing map" ff1 : Precisely, 0| 0 1 (zz0) = 1 (ff|z1(z)) 1 (z ) for z; z0 2 K(1) with z0 homogeneous of topological degree |z0|. Remark 3.5. Consider the action of K on the fundamental class for the Eilenberg-MacLane space K(F2; k). Recall from [PW ] that, for each k > 0, the admissible monomials in K involving only those Di for which 0 i < k correspond to a vector space basis for the coalgebra primitives P H*(K(F2; k)). The map ff corresponds to the cohomology suspension, so that the inverse limit under these maps realizes the linear bijection 1 from K(1) to the cohomology of the Eilenberg- MacLane spectrum HF2. Because of this topological interpretation, K(1) is called the algebra of stable lower operations. 4. Comparison of the maps j and 1 This section addresses our goal of relating 1 : K(1) ! A and j : K ! [M+ ; F2]. The map j embeds K as a subalgebra of [M+ ; F2]. We may also consider A as a subalgebra of [M+ ; F2] (via the dual of the Hopf algebra map M+ ! M ! A*), but the image of j has trivial intersection with the image of the Steenrod algebra A. On the other hand, 1 is bijective and has image the Steenrod algebra, but it is not an algebra map and its domain is not K. We will reconcile j and 1 by means of a map j1 : K(1) ! [M; F2]; an algebra embedding of K(1) into the convolution algebra [M; F2]. Q P To this end we first let cW = n Wn and consider wi = n0 wi;n2 cW . Then formally we have (wi) = P w2ji-j wj; in particular, w0 is formally a group-like element, so that multiplication by w0 gives a map of coalgebras. The product of the algebra maps jn : M+ ! Wn gives an algebra map j* : M+ ! Wc ; whose value on generators is j*(i) = wi; formally, j* preserves the comultiplication. Notice that j* and j are dual: = <; j*()>. Since 0 is a group-like ele- ment, multiplication by it produces a coalgebra endomorphism of M+ . Clearly j*(0x) = w0j*(x) for x 2 M+ . Recall that the algebra endo- morphism ff of K is dual to multiplication by w0 on W. Let fi be the algebra endomorphism of [M+ ; F2] that is dual to multiplication by 0 on M+ . We have by duality the equality of algebra maps j O ff = fi O j. STABILIZING THE LOWER OPERATIONS 7 Let j(k)denote the composition of j with the inclusion of K(k)into K. Then j(k-1)O ffk = fi O j(k), so that the maps j(k)induce an algebra inclusion j1 : lim-K(k)! lim-[M+ ; F2]: ffk fi To identify the target better, let ssk : [M; F2] ! [M+ ; F2] be the algebra morphism dual to the coalgebra map M+ ! M given by 7! -k0. These projections induce an algebra isomorphism [M; F2] = [lim-!M+ ; F2] = lim-[M+ ; F2]: .0 fi Altogether we now have an algebra embedding j1 : K(1) = lim-K(k)! lim-[M+ ; F2] = [M; F2]: ffk fi We may compute j1 on the generators D1-i in K(1) by chasing through the definitions, and we have Theorem 4.1. There is an algebra embedding j1 : K(1) ! [M; F2] given on generators by X j1 (D1-i ) = (-i0l1)*: l0 We now have a triangle j1 K(1) ______[M;-F2] | | | 1 | | fl1 | |? A in which fl1 denotes the natural embedding induced by the projection M ! A*. But this triangle is far from commutative; indeed since 1 is sheared, while j1 and fl1 are not, it couldn't be commutative. So at this point we have two embeddings j1 and fl1 O 1 of K(1) into [M; F2]. We begin the process of connecting these two maps by constructing a sheared algebra morphism * : [M; F2] ! [A*; F2]. We may define an algebra inclusion of the dual Steenrod algebra A* into the extended Milnor Hopf algebra M as the component of total exponent zero, given by l11. .l.nn7! -l0l11. .l.nn; 8 T. BISSON, D. PENGELLEY, AND F. WILLIAMS P where l = li. Then is a section of the quotient projection of Hopf algebras M ! A* given by 0 7! 1, but is not a coalgebra morphism. Instead, satisfies the identity X |y0| (y) = 0 iyi y0i; i P where y = iyi y0ifor y a homogeneous element in M. We may rephrase this as M O = fl O ( ) O A*: Here fl : M M ! M M is the linear map given by fl(z z0) = 0| |z0z z0 (where |z0| denotes the topological degree of z0 2 M). We may say that is a "cosheared coalgebra morphism" (see Remark 5.18 in [PW ]). It follows that * : [M; F2] ! [A*; F2] is a sheared algebra morphism. Remark 4.2. This coshearing has a nice interpretation in terms of the functors represented by M and A*. We have a semi-direct extension of groups (for any commutative ring R) S(R) ! A(R) ! U(R) where U(R) = Alg (F2[u ]; R) is the group of invertible elements in R and the group homomorphism A(R) ! U(R) is given by a(x) 7! a0(0). The splitting of A(R) ! U(R) comes from identifying u 2 U(R) with the additive series ux 2 A(R) (or identifying F2[u ] with the sub-Hopf algebra F2[0 ] in M). Consider the bijection A(R) ! S(R) x U(R) given by a(x) 7! (a(_x_a0); a0x) (where a0 = a0(0) is the lead coefficient of the additive series a(x)); then we have a(x) = a(_x_a0) O a0x. The algebra inclusion represents the projection onto the first factor A(R) ! S(R), with a(x) 7! a(_x_a0). The algebra map fl O ( ) represents the function A(R) x A(R) ! S(R) x S(R) given by b(a0x) a(x) (b(x); a(x)) 7! a-10_______; _____: b0 a0 This is related to the discussion of the "commutation operator" for a semi-direct group extension (Example 1 in Appendix C in [BJ ]). We now have a map * O j1 : K(1) ! [A*; F2]. Further, if we let fl2 denote the natural inclusion of A into [A*; F2], we have the rectangle STABILIZING THE LOWER OPERATIONS 9 j1 K(1) ______[M;-F2] | | | | | |* 1 | | | | | | |? fl2 |? A ________[A*;-F2] in which the horizontals are algebra embeddings and the verticals are sheared algebra morphisms. We wish to check the commutativity of this diagram. The monomials in the D1-i generate K(1) additively (in fact, the "admissible" monomials form a basis). We use shearing to calculate * and 1 on products, and we use the formulas we have for 1 ; j1; and * on basis elements. Precisely, we have *(j1 (D1-i1 . .D.1-in)) = *(j1 (D1-i1 ) . .j.1(D1-in )) ! X X = * (-i10a11)* . . . (-in0an1)* a1 an ! X = * (-i10a11)* . .(.-in0an1)* a1;:::;an X n-2 -i -a a = *((-i1-a2-...-20 ana11)*) . .*.((0 n-1 n1n-1)*)*((-in0an1)*) a1;:::;an n-2in* in-1+in * in * = (i1+i2+...+21 ) . .(.1 ) (1 ) n-2in i +in in = fl2(Sqi1+i2+...+2 . .S.q n-1 Sq ) = fl2( 1 (D1-i1 . .D.1-in)): Thus our rectangle commutes and we have Theorem 4.3. Under the identification of the Steenrod algebra A with its image fl2(A) embedded in [A*; F2], the sheared algebra bijection 1 factors as 1 = * O j1 : K(1) ! A: So we have obtained our desired relation between 1 and j1 via the map *, which explains the shearing that occurs in 1 . 10 T. BISSON, D. PENGELLEY, AND F. WILLIAMS 5. Some dual considerations In [PW ] there is a discussion of the dual of the shearing bijection 1 : K(1) ! A. We review some of this here, working with W as the bi-graded dual of K. We begin by considering the quotient W ! W(k)that is dual to the inclusion K(k) ! K. We recall from [CPS ] or [PW ] that a basis for W(k)nis given by the images under the projection Wn ! W(k)nof all monomials in w0;n; : :;:wn-1;1of total exponent k - 1: Let us define W(1)nto be the direct (co)-limit of of the maps ff*k+1: W(k)n! W(k+1)ngiven by multiplication by w0;n: This process amounts to inverting w0;n; the projection Wn ! W(k)ninduces a map w-10;nWn = lim-!Wn ! lim-!W(k)n= W(1)n: .w0;n ff*k It is shown in [PW ] that a basis for W(1)nis given by "monomials" n-1X w-l0;nwl11;n-1.w.l.n-1n-1;12 W(1)n forl > li. i=1 We may view the coalgebra W(1) = nW(1)nas dual to the algebra K(1). ___ We have maps jn : M ! W(1)ndefined as follows. The map jn : M+ ! Wn and the projection Wn ! W(k)ndetermine ___ + (k) (1) jn: M = lim-!M ! lim-!Wn ! lim-!Wn = Wn : .0 .w0;n ff*k Composing with the co-shearing : A* ! M yields a map ___ * (1) n = jnO : A ! Wn : Q (1) ___ __ Let cW(1) = n Wn . The product of the maps jn gives j* : M ! cW(1) with __j*(i) = wi. Note that __j*is an extension of the map j* : M+ ! cW defined in section 4. Similarly, the product of the maps n gives : A* ! cW(1) ; with (l11. .l.nn) = w-l0wl11. .w.lnn: Remark 5.1. The components of this last expression can be difficult to interpret in terms of the known basis for W(1): If ln 1; its compo- nent in W(1)mfor m < n is zero, its component in W(1)nis an element of the known basis (above), but we know little about its components for m n; since they do not present themselves as basis elements. STABILIZING THE LOWER OPERATIONS 11 In [PW ] a leading term formula for *1 : A* ! W(1) was proven, namely P n L (1) Theorem 5.2 ([PW ]). Let l = i=1li: If ln 1; then modulo m>n Wm *1(l11. .l.nn) = w-l0;nwl11;n-1.w.l.n-1n-1;1: Comparing with Theorem 5.2 yields Theorem 5.3. For ln 1, Y (l11. .l.nn) *1(l11. .l.nn) mod W(1)m: m>n Since is built from and from j*, which is essentially dual to j, this theorem provides a relationship between j and 1 ; via , similar to that of Theorem 4.3. 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Williams, Sheared algebra maps and operation Hopf algebr* *as for mod 2 homology and cohomology, to appear. [Sm] V.A. Smirnov, Hopf radical algebras and the Steenrod algebra, Russian Math. Surveys 42:2 (1987), 301-302. [St] N.E. Steenrod, D.B.A. Epstein, Cohomology Operations, Princeton Univ. Press, 1962. [Wi] 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. 12 T. BISSON, D. PENGELLEY, AND F. WILLIAMS Canisius College, Buffalo, NY 14208 E-mail address: bisson@canisius.edu New Mexico State University, Las Cruces, NM 88003 E-mail address: davidp@nmsu.edu E-mail address: frank@nmsu.edu