DSpace at VNU: Spherical classes and the lambda algebra

Spherical Classes and the Lambda Algebra Author(s): Nguyễn H V Hung Source: Transactions of the American Mathematical Society, Vol 353, No 11 (Nov., 2001), pp 4447-4460 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/2693744 Accessed: 22/01/2015 17:39 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive We use information technology and tools to increase productivity and facilitate new forms of scholarship For more information about JSTOR, please contact support@jstor.org American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Transactions of the American Mathematical Society http://www.jstor.org This content downloaded from 128.235.251.160 on Thu, 22 Jan 2015 17:39:52 PM All use subject to JSTOR Terms and Conditions TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 11, Pages 4447-4460 S 0002-9947(01)02766-0 Article electronically published on May 22, 2001 SPHERICAL CLASSES AND THE LAMBDA ALGEBRA NGUYEN H V HUNG k r A be Singer's invariant-theoretic model of the ABSTRACT Let rA - Tor 4(F2,F2), dual of the lambda algebra with Hk(rA) where A denotes the mod Steenrod algebra We prove that the inclusion of the Dickson algebra, Dk, into rA is a chain-level representation of the Lannes-Zarati dual homomorphism wk >Tor4 (]F2, IF2) A~~~~~~~A : F2 (93Dk _-Hk (r/\ The Lannes-Zarati homomorphisms themselves, fOk, correspond to an associated graded of the Hurewicz map H: 7r (SO)- r*(QoSO) -?H* (QoSO) Based on this result, we discuss some algebraic versions of the classical conjecture on spherical classes, which states that Only Hopf invariant one and Kervaire invariant one classes are detected by the Hurewicz homomorphism One of these algebraic conjectures predicts that every Dickson element, i.e element in Dk, of positive degree represents the homology class in Tor (F2, F2) for k > We also show that (P* factors through F2 (0 Ker&k, where 09k: rkA k ~~~Ak ]p_ F2 denotes the differential of rA (0 Ker9k should be of interest Therefore, the problem of determining INTRODUCTION AND STATEMENT OF RESULTS Let QoS0 be the basepoint component of QS? = limn QnSn It is a classical unsolved problem to compute the image of the Hurewicz homomorphism H :7r (So) 7r*(Qo SO) -> H* (QoSO) Here and throughout the paper, homology and cohomology are taken with coefficients in F2, the field of two elements The long-standing conjecture on spherical classes reads as follows Conjecture 1.1 The Hopf invariant one and the Kervaire invariant one classes are the only elements in H* (QoSO) detected by the Hurewicz homomorphism (See Curtis [5], Snaith and Tornehave [22] and Wellington [23] for a discussion.) An algebraic version of this problem goes as follows Let Pk = F2 [x, , Xk] be the polynomial algebra on k generators x1, , Xk, each of degree Let the Received by the editors February 4, 1999 and, in revised form, November 4, 1999 2000 Mathematics Subject Classification Primary 55P47, 55Q45, 55S10, 55T15 Key words and phrases Spherical classes, loop spaces, Adams spectral sequences, Steenrod algebra, lambda algebra, invariant theory, Dickson algebra The research was supported in part by the National Research Project, No 1.4.2 (?)2001 American 4447 This content downloaded from 128.235.251.160 on Thu, 22 Jan 2015 17:39:52 PM All use subject to JSTOR Terms and Conditions Mathematical Society NGUYEN 4448 H V HUNG general linear group GLk= GL(k, F2) and the mod Steenrod algebra A both act on Pk in the usual way The Dickson algebra of k variables, Dk, is the algebra of invariants Dk := F2[XI -, XGk As the action of A and that of GLk on Pk cominute with each other, Dk is an algebra over A In [14], Lannes and Zarati construct homomorphisms S0k EExtk?i (2 F2) -_ Dk)i* (F2 A which correspond to an associated graded of the Hurewicz map The proof of this assertion is unpublished, but it is sketched by Lannes [12] and by Goerss [7] The Hopf invariant one and the Kervaire invariant one classes are respectively represented by certain permanent cycles in Ext j*(F2, F2) and Ext2*(IF2, F2), on which Pi and P2 are non-zero (see Adams [1], Browder[4], Lannes-Zarati[14]) Therefore, we are led to the following conjecture Conjecture 1.2 0k in any positive stem i for k > The present paper follows a series of our works ([8], [10], [11]) on this conjecture To state our main result, we need to summarize Singer's invariant-theoretic description of the lambda algebra [20] According to Dickson [6], one has Dk F2 [Qk,k-1, Qk,o denotes the Dickson invariant of degree - Singer sets Fk the localization of Dk given by inverting Qk,o, and defines P' to be a Dk[Q-17], certain "not too large" submodule of Fk He also equips rp = EkrkP with a and a coproduct Then, he shows that the differential differential 0: rFA coalgebra pA is dual to the lambda algebra of the six authors of [3] Thus, Hk (IA)A TorA (IF2,F2) (Originally, Singer uses the notation F+ to denote PA However, by D+, A+ we always mean the submodules of Dk and A respectively consisting of all elements of positive degrees, so Singer's notation F+ would cause confusion in this paper Therefore, we prefer the notation FeA.) The main result of this paper is the following theorem, which has been conjectured in our paper [10, Conjecture 5.3] where Qk,i rA_l Theorem 3.9 The inclusion Dk Lannes-Zarati dual homomorphism c * k (F2 XDk)i Pk is a chain-level representation of the Tor k?A(F2,F2)- An immediate consequence of this theorem is the equivalence between Conjecture 1.2 and the following one Conjecture 1.3 If q C D+, then [q] = in TorA(IF2,1F2) for k > This has been established for k = in [10, Theorem 4.8], while Conjecture 1.2 has been proved for k = in [8, Corollary 3.5] From the view poinit of this conjecture, it seems to us that Singer's model of the dual of the lambda algebra, pA, is somehow more natural than the lambda algebra itself The canonical A-action on Dk is extended to an A-action on PA This action commutes with Ok (see [20]), so it determines an A-action on Ker&k, the submodule of all cycles in PA We also prove This content downloaded from 128.235.251.160 on Thu, 22 Jan 2015 17:39:52 PM All use subject to JSTOR Terms and Conditions SPHERICAL Proposition CLASSES AND THE LAMBDA ALGEBRA 4449 4.1 so* factors through F2 OKer&k as shown in the commutative A diagram Wk Dk A2 Tor (A2,F2) F2 &Ker&k, A where Z is induced by the inclusion Dk C Ker&k, and is an epimorphism induced by the canonical projection p: Ker&k -> Hkz(FA) TorA (F2, F2) From this result, the problem of determining F2 0Ker&k would be of interest A The paper is divided into sections In Section we recollect some materials on invariant theory, particularly on Singer's invariant-theoretic description of the lambda algebra and the LannesZarati homomorphism Section is devoted to prove Theorem 3.9 Finally, Section is a discussion on factoring p The main results of this paper were announced in [9] The author would like to thank Haynes Miller for introducing him to Stewart Priddy's work [18] on exploiting an explicit homotopy equivalence between the bar resolution of F2 over A and the dual of the lambda algebra He also thanks the referee for helpful suggestions, which led to improving the exposition of the paper RECOLLECTIONS ON MODULAR INVARIANT THEORY We start this section by sketching briefly Singer's invariant-theoretic description of the lambda algebra Let Tk be the Sylow 2-subgroup of GLk consisting of all upper triangular k x kmatrices with on the main diagolnal The Tk-invariant ring, Mk PT, is called the Miuialgebra In [17], Miuishows that Pk k [VI * *, Vk] =2 where (Aix, Vi = + *- +Ai-,xi-, + Xi) Xj EF2 Then, the Dickson invariant Qk,i can inductively be defined by Qk,i = + Vk * Qk-l,i, where, by convention, Qk,k = and Qk,i = for i < Let S(k) C Pk be the multiplicative subset generated by all the non-zero linear forms in Pk Let (Pk be the localization, (Pk = (Pk)s(k) Using the results of Dickson [6] and Miui [17], Singer notes in [20] that = (1k)T Ak rk ((.k - = F- V1,- ) 2[Qk,k-Ii Vk?I] * , Qk,l, Qk,0 Further, he sets VIV1 ,Vk 2=Vk/V1 Vk-1 (k > 2)) This content downloaded from 128.235.251.160 on Thu, 22 Jan 2015 17:39:52 PM All use subject to JSTOR Terms and Conditions NGUYEN 4450 H V HUNG so that Vk =2k-VI k (k Vk_lVk > 2) Then, he obtains Ak = F2 [V I, , v 1], with deg vi for every i Singer defines F' to be the submodule of Fk = Dk [Qk-, spanned by all monomials Y Qkk-1 Qk,o with ik-1, , iil > 0, io Z 2, and io + deg'y > He also shows in [20] that the homomorphism I 2[[VI v V k f o, 3Jk Vk, I vk-1]' if ik otherwise, Vj1', Moreover,it is a differentialon r1 A bigraded by putting bideg(v' vik) (k, k + Eii) maps FA to FA1 k rP This module is - Let A be the (opposite) lambda algebra, in which the product in lambda symbols is written in the order opposite to that used in [3] It is also bigraded by putting (as in [19, p 90]) bideg(Ai) = (1, + i) Singer proves in [20] that pA is a differential bigraded coalgebra, which is dual to the differential bigraded lambda algebra A via the isomorphisms ]rA V (Al I V3 A* A Here the duality * is taken with respect to the basis of admissible monomials of A As a consequence, one gets an isomorphism of bigraded coalgebras H*(FA) - TorA (F2)- In the remaining part of this section, we recall the definition of the Lannes-Zarati homomorphism Let P1 F2[x] with lxl = Let P C F2[x,x-1] be the submodule spanned by all powers xi with i > -1 The canonical A-action on P1 is extended to an A-action on F2 [X, X-1] (see Adams [2],Wilkerson[24]) Then P is an A-submodule of F2 [X, X-1] One has a short-exactsequenceof A-modules 2.1 P1 PP Z1F2 - 0, where t is the inclusion and wTis given by 1r(X') = if i -1 and wT(x-1) = Let el be the corresponding element in Ext (E1F2, F1) Definition 2.2 (Singer [21]) (i) ek el *0 El E Extk(E-kIF2, Pk) k times k, Pk Ml), for M a left A-module (ii) ek(Mll) = ek M C Ext (A Here M also means the identity map of M Following Lannes-Zarati [14], the destabilization of M is defined by DM = M/EM, where EM := Span{Sq'xl i > deg x, x C M} They show that the functor associating M to DM is a right exact functor Then they define Dk to be the kth left derived functor of D So one gets Dk(M) = Hk(DF*(M)), This content downloaded from 128.235.251.160 on Thu, 22 Jan 2015 17:39:52 PM All use subject to JSTOR Terms and Conditions SPHERICAL AND THE LAMBDA CLASSES ALGEBRA 4451 where F* (M) is an A-free (or A-projective) resolution of M The cap-product with ek(M) gives rise to the homomorphism Dk( M) ek (M) (z) ek(M/l): ' DoPpk M) -Pk Since F2 is an unstable A-module, one gets Theorem 2.3 (Lannes-Zarati [14]) Let Dk C ables Then ak :_ ek(EF2): Dk(El-kF2) -> M n z ek(M) be the Dickson algebra of k vari- Pk EDk is an isomorphism of internal degree By definition of the functor D, one has a natural homomorphism, D(M) F2 ?M Then it induces a commutative diagram > A DFk(M) DFkl- (M) ik -1 ik *2 X A Fk(M) IF2 0Fk l(M) A Here the horizontal arrows are induced from the differential in F* (M), and ik [Z] = [1 ( Z] A for Z c Fk(M) Passing to homology, one gets a homomorphism TorA(F2, ) F2 0Dk(M) ik A 2.4 A A Taking M - El-kF2, one obtains a homomorphism ikF: 0Dk (lk F2 ) A Note that the suspension Z: F2 Dk A lk F2) T?rko(2, lk F2) F2 YDk and the desuspension -i A TorkA(F2, Tork (F2, kF2) are isomorphisms of internal degree and (-1), respectively This leads one to Definition 2.5 (Lannes-Zarati [14]) The homomorphism pOkof internal degree is the dual of k= Elik(l 0Dk A 90ak)Z A Tor A(F2, Ek 2) Remark 2.6 In Theorem 3.9 we also denote by fok the composite of the above yok with the suspension isomorphismEk : F2,E F2) (o2rk = We need to relate ak ek(EF2) with connecting homomorphisms Suppose f E Ext' (M3, M) is represented by the short-exact sequence of left 0O Let A(f): D,(M3) -* D.1(M1) - M, - M2 be the A-modules M3 associated with this short-exact Then one connecting homomorphism sequence easily verifies * A(f)(z) = f nz for any z C D,(M3) This content downloaded from 128.235.251.160 on Thu, 22 Jan 2015 17:39:52 PM All use subject to JSTOR Terms and Conditions x 4452 NGUYEN H V HUNG One has 2.7 ek(ZF2) =(el(ZF2) o * Pk-) (8 P) e(Z2-kF2) o (ei(Z3kF2) Therefore, one gets 2.8 A(el(ZF2) P_) Ozk = o * o A(ei(Z3kF2) ( Pl) o l (E2 kF2) (See Singer [21, p 498].) This formula will be useful to construct a chain-level representation of A CHAIN-LEVEL REPRESENTATION OF THE LANNES-ZARATI ak HOMOMORPHISM Suppose again M is a left graded A-module Let B* (M) be the bar resolution of A/Iover A Recall that Bk (MA)= A X I *(IAg[ I (k > O), k times where I denotes the augmentation ideal of A and the tensor products are taken over IF2 The module B* (M/) = ? Bk (MA)is bigraded by assigning an element ao ? a, O ak Ox with homological degree k and internal degree 0%(degai) + deg x The differential dk: Bk(A/I) Bk-l (Ml) is defined by Oal, dk(ao ak (x) O* (*0ak ( aoal +aoaO0a, +* 00 ak x +ao (00ala2 (x .akx So dk preserves internal degree and lowers homological degree by The action of A on Bk(M) is given by a(ao 0a, * ak0 x)= aao0a, * ak x, for a C A Suppose additionally that N is a right graded A-module As the bar resolution is an A-free resolution, by definition one has Hk(N B* (M)) TorA (N, M) A Since Dk c F2 [vl, , Vk], every element q q~~~ ~ q = C Dk has an unique expansion Vli Vk (i1,-**ik) where 1l, ,ik are non-negative We associate with q of internal degree Ek ij + 1: Definition 3.1 q= S Sq'l+ Sqik+l Z1l c C E Dk the following element Bk(1 IF2) (1 Jik) Lemma 3.2 If q C Dk, then q c EBk-1(ElIF2) :=Span{Sq2x| i > degx,x c Bk-1(El This content downloaded from 128.235.251.160 on Thu, 22 Jan 2015 17:39:52 PM All use subject to JSTOR Terms and Conditions kF2)} SPHERICAL CLASSES AND THE LAMBDA ALGEBRA 4453 Proof From the definition of the A-action on the bar resolution, one has Sqj1+1 ? ? Sqik+l = SqS kl Zl (1 ? *Sq Sqi( *0 Sq k+l El kl) Hence, it suffices to show that il + 1> (i2 + 1) + for every term in the expansion of Recall that Vi V Ilk =F2 vv [V1, VkI (1-k)= + (jk + 1) + + j2 + 3k, q vij1vi So, one easily verifies that every element a sum of v k which satisfy the condition monomials v1' is jl >i2+ -+jk- The lemma follows from the fact that the Dickson algebra Dk is a subalgebra of D the Mvii algebra MIlk Lemma 3.3 q is a cycle in the chain complex EB*(El-k F2), for every q C Dk This is a consequence of the following lemma, which is actually an exposition of the Adem relations Lemma 3.4 The homomorphism v1 V1 Vpp Vpk Jp+1 VP1 -A >Ak-1 rk,p:Ak Sqjl+l k (k -I .?A * 09 Sqjp+lSqjP+1+l times) Sqik+1 vanishes onr k C Ak, for < p < k Proof Consider the diagonal b: (vi) = A p-A Ak 0 A2 Vi O 1, X Vj_p+l X1, i < p, p Define the homomorphism Wt: Ft wt (vlii *Vit At by -) ) oFr2 ? rk-p-l - Sql +l1 (9 (9 Sqjt + Then one has Wk,p = (Wp-1 72,1 Wk-p-1).- By Proposition 3.1 of Singer [20], the Adem relations yield 72,1(r2) = Hence, r,k,p(rk) = for < p < k The lemma is proved Proof of Lemma 3.3 First, we note that Sqjk+l(l-kl) = Dl for aniy ik > Then, by definition of the differential in the bar resolution, we get k-1 dk-l(4) Since q C Dk is proved C ]7k, - Z(7rk,p p=1 Lemma 3.4 yields MEidx k2 7rk,p(q) ) (q El1kl) = Thus dkl(0) This content downloaded from 128.235.251.160 on Thu, 22 Jan 2015 17:39:52 PM All use subject to JSTOR Terms and Conditions = The lemma D 4454 NGUYEN H V HUNG For the convenience of the latter use, we define ?k,p as follows: ?) 7Rk,p(Sq1+ ? * Sqk+) Sq+1 S *** P+SqP+1+1 Sqi +1 for < p < k Suppose as before that q Vk c Dk vl J=(jl, i) For a fixed (k - s)-index (is+l ik), we define J(is+l, k ik ) to be the set of all sindices (j, js)'s such that (jil, sts+ *-,ik) occurs as a k-index in the above sum The following lemma is a slight generalization of Lemma 3.4 Lemma 3.5 If q = jv ( E SP V J(is+., then EDk, Sqi ?+1 ? SqiS+1) - ik) for < p < s < k Proof Let us consider the diagonal ?b2 : Ak i b2(Vi) = { -/\ A I < v_7 given by (8 Ak-s i < Si According to Proposition 2.1 of Singer [20], 4'(Fk) C IF, 0Fk- Since q e c IF, Then, by Lemma 3.4, we have it implies ZJ( +1 ik) v'.v -,S J(is.+ Sqi' .? Sqis+) Dk C Fk, VI Vsi)0 -5,p ,ik ) J(js+1,- ,ik) E The lemma is proved By definition of the destabilization functor D, for any left A-module M, one has an exact sequence of chain complexes EB*(M) -, E B *(M) j DB*(M) -, 0, in which the bar resolution B*(M) is exact Hence, by use of the induced long exact sequence, the connecting homomorphism is an isomorphism 3* : Dk Take M= (M) := Hk(DB* (M)) Hk_l(EB* (M)) The following lemma deals with the connecting isomorphism ZlkF2 0* : Dk (ZlF2) : = Hk(DB* (ZlF2)) Hk-l(EB* (kF2)) Let [q] be the homology class of the cycle q in Dk(El-kF2) HklI(EB*(lk F2)) Lemma 3.6 If q C Dk, then Proof Suppose q = Z1lkl ** *Vik The element Ej Sqil?l ( (? Sqjk+I (0 is a lifting over jD of its class modulo EBk(El-k F2) in Let d denote the differential in B*(El-k F2), we get C Bk(Zl-k DBk(Zl-kF2) J F2) This content downloaded from 128.235.251.160 on Thu, 22 Jan 2015 17:39:52 PM All use subject to JSTOR Terms and Conditions CLASSES SPHERICAL d(E Sq'l+1 ? ? S AND THE LAMBDA ? sqik+l 4455 ALGEBRA Z1-ki) J = J Sqil+l Sqik+l ? Z1-kl k-I + E Sqj+1 10 Wk,p( -k Sqj+l) J p=1 + Sqil+l .Sqik+ll-kl J By Lemma 3.4 Trk,p(ZSqil+1 * Sqik+l) = =Fk,p(q) J On the other hand, d(Z = for any jk > Therefore, we obtain Sqik+l(l-kl) Sqj1+10 0 E-kl) Sqjk+1 ? Sqj1+' ? S Sqik+l ? l-ki J J iE (q) - By definition of the connecting homomorphism, we have The lemma is proved The following theorem deals with the isomorphism treated in Theorem 2.3 E aOk Dk (El-kF2) > EDk Theorem 3.7 If q C Dk, then Ozk[q]=q Proof We compute ak ak by means of the following formula = A(ei(EF2) = 6k 62 61 - P) o Pk-) o * o A(ei(Z3kF2)F E F2) Here J, stands for A(ei(Z1-k+?sF2) ? P.-I), for brevity Consider the short exact sequence representing e1(Z2-kF2): 2-kp1 ,> i E2-kf A Zl-kF2 -*0 Then the connecting homomorphism induced by this exact sequence is nothing but 61 : Hkl,(EB* A lifting of q = Ej Sqj1+1 Z Sqjll0 ( Flk2)) Sqik+l Sqik+l -) Hk-2(EB*(Z2k El-kl Z2-kXjl1 over w is E EB* (:2kP), J where we are writing P1 element in EB*(Z2-kP) Span{xjI i > -1} The boundary of this which is pulled back under t to a cycle in EB*(Z2-kPl), = F2[xk],P This content downloaded from 128.235.251.160 on Thu, 22 Jan 2015 17:39:52 PM All use subject to JSTOR Terms and Conditions 4456 NGUYEN H V HUNG represents 61[] That means [d(Z Sqjl+l ? (? E2-kX-1)j Sqik+l J k-1 [ZE 7k,p ( p=l + 0Sqi Sqi + 0 O Sqjk+l) J +1 k-l+l Sq ? ? Sq Z2-kX-1 +l(E2-kX-1)] J [ESqj1+l ? Sqjk1+?l J~~~~~~~~~~~~~~ Z2-kSqjk+l(x-1)] where the last equality follows from Lemma 3.4 Indeed, Sq'l+l 17k,p(Z * Sqik+l) 7Tk,p(q) Similarly, 62 : Hk-2(EB*(Z2kPl)) is the connecting hoHk-3(EB* (3kP2)) momorphism induced by the short exact sequence representing el (E3-kF2) Pl: - , 3-kp t04Pi Z3-k(P? Here we are writing P1 = IF2[xk],P2 = lifting of Sqi '1 ? ?* Sqiki_?l ? z2-k Sq (Xl) Sqik+l 3-kx-1 = Span{x_ili over w ? P1 is IF2[xk_1,xk],P + Sqi-l 00 r'P1 E2-kp p1) > -1} A (x-1) sqik+l Therefore, by an argument similar to the one given above, we get J2Jjjq] SqJl+l [d(E Sqi_ 0z3kxAiSqik?l(xkl))] J k-2 - j:k -l,p(Sqi1 [ +1 , Sq ik-1+1) zE3-kx _1 Sqik+l (Xl) P=l j + E [ Sqjl+l Sqjl+l 0Sqk21 Sqk-2+1 Sqjk-1+l(E3-kXJ1 Sqjk+l(X-1))] -l+l(xj1( ? Z3-kSq'k Sqik+l(X ))] J (by Lemma 3.5) Repeating the above argument, we then have Ozkq] = k 61[q [E(E Sqh+ (xl Sq2 +1 (X -l * Sqjk+(x * *X 1) ) By Theorem 3.2 of our paper [10], we get [ (ZSqi1+l(x-lSqj2+l(x-1 Sqik+l(X ) )))] - [q] - Eq J The theorem is proved This theorem has an immediate consequence as follows This content downloaded from 128.235.251.160 on Thu, 22 Jan 2015 17:39:52 PM All use subject to JSTOR Terms and Conditions D SPHERICAL CLASSES AND THE LAMBDA ALGEBRA Corollary 3.8 The homomorphism Dk representation of the homomorphism (1 ?al1)Z EBk-l - F2 0Dk A is a chain-level q (l-kF2), F2 0Dk(Zl-kI A 4457 2) A Theorem 3.9 The inclusion Dk C IF"is a chain-level represenTtationof the LannesZarati dual homomorphism (F2 A Dk) i -) Tork,k+(F2, F2)- Proof Suppose again that S q J=(j1, v**vk c Dk ,.jk) By Corollary 3.8 and Lemma 3.6, we have (1( F2 0Dk(l-kF2) F2 -DD A Ok ) A A [q][-[ From the definition of ik (see 2.4), we get (lk F2XHk(DB* A ikF: F2)) TorA (F2, - l-k F2) Let us consider the desuspension : TorA (F2, El E which sends [Zj X Sqik+l ? x-kl] Sqi1+l Then 0 F2) - TorA (F2, E-kF2), ? Z1-klj] to [Ej ? Sqi1+l Sqik+l X X the map ik(l -k =ZE a271)Z) A k4 F2?Ok A k (F2, E F2) is given by o[q] k =[51 0 X Sqjk+l ? E-kl] X Sqjl?l The canonical isomorphism k: TorAi (F2, E-kF2) -T T?rk+i(A2,F2) is defined by the chain-level version Sk(ao a, 00 * ** By ambiguity of notation, ak -kl) the composite = aO 0a, 10 * * *ak Zkqo* is also denoted by qo* (see Re- mark 2.6) Hence * : (F2 Dk)i [q] - TorAk+ (F2,F2) [Ej I O Sqj1+1 O * Sq jk+1 X l] In [18], Priddy constructs the Koszul complex K* (A), a subcomplex of B* (F2), which is isomorphic to the dual of the lambda algebra More precisely, it is defined as follows Let A be the (opposite) lambda algebra, in which the product in lambda symbols is written in the order opposite to that used in [3] (See Singer [20, p 687] This content downloaded from 128.235.251.160 on Thu, 22 Jan 2015 17:39:52 PM All use subject to JSTOR Terms and Conditions NGUYEN H V HUNG 4458 for a precise definition of A.) Then, according to Priddy [18, p7], K* (A) is the image of the mononmorphism A* (Aj ,B* (F2) Sqil +1 )i -lA Sqik * +1 which is a homotopy equivalence Here A* denotes the dual of A and the duality * is taken with respect to the basis of admissible monomials of A Combining it with Singer's isomorphism V A.Vk (A* - Aik)7 we get the following homotopy equivalence pA vi Ji B* (F2) , 3k Vk _k_ Sqi1? lX ' Sqik?l01 As a consequence, for any q C Dk, we obtain fok[q] = 1] [E1?XSqi1+1X XSqik+l J Vl =[E V kh J [q] = It means that the inclusion Dk theorem is completely proved Corollary 3.10 C ]pAis a chain-level representation of qo The E Conjecture 1.2 is equivalent to Conjecture 1.3 This follows immediately from Theorem 3.9 We have proved Conjecture 1.2 for k = in [8] and Conjecture 1.3 for k in [10] FACTORING THE LANNES-ZARATI HOMOMORPHISM The purpose of this section is to prove the following proposition Proposition 4.1 yo factors through F2 0Ker&k as shown in the commutative A diagram: F2 (Pk Dk TorA (F2, F2) Ak F2 X Ker&k, A where Z is induced by the inclusion Dk C Kerak, and pi is an epimorphism induced TorA (F2, F2) by the canonical projection p : Kerak - Hk (FA) Proof The canonical projection p: KerOk , Tork (F2, F2) = Ker&k/Im&k+1 sends x to [x] =x 1+Im&k+1 This content downloaded from 128.235.251.160 on Thu, 22 Jan 2015 17:39:52 PM All use subject to JSTOR Terms and Conditions SPHERICAL CLASSES AND THE LAMBDA ALGEBRA 4459 By Theorem 5.15 of Singer [20], the action of A on- KerOk induces a trivial action of A upon TorA+(F2, F2) Therefore, p induces the epimorphism p: F2 A o Tork Ker9k [X] (F2,F2) [X] I-4 For any q E Dk, we have P* [q] = P[q] = [q] - W [q] So, we get W*= Pi t The proposition is proved D In [10], we have stated the following conjecture Conjecture 4.2 D+ C A+ *KerOk for k > Obviously, this is stronger than Conjectures 1.2 and 1.3 and equivalent to the following one Conjecture 4.3 The homomorphism i: inclusion i: Dk -) ODk F2 OKer9k, A A induced by the Ker9k, is trivial for k > Based on-the above discussion, we believe the following problem is something of interest Problem 4.4 Determine F2 ? Ker&k A REFERENCES [1] J F Adams, On the non-existence of elements of Hopf invariant one, Ann Math 72 (1960), 20-104 MR 25:4530 [2] J F Adams, Operations of the nth kind in K-theory and what we don't know about RP?, New developments in topology, G Segal (ed.), London Math Soc Lect Note Series 11 (1974), 1-9 MR 49:3941 [3] A K Bousfield, E B Curtis, D M Kan, D G Quillen, D L Rector, J W Schlesinger, The mod p lower central series and the Adams spectral sequence, Topology (1966), 331-342 MR 33:8002 [4] W Browder, The Kervaire invariant of a framed manifold and its generalization, Ann Math 90 (1969), 157-186 MR 40:4963 [5] E B Curtis, The Dyer-Lashof algebra and the A-algebra, Illinois Jour Math 19 (1975), 231-246 MR 51:14054 [6] L E Dickson, A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans Amer Math Soc 12 (1911), 75-98 CMP 95:18 [7] P G Goerss, Unstable projectives and stable Ext: with applications, Proc London Math Soc 53 (1986), 539-561 MR 88d:55011 [8] N H V Hu'ng, Spherical classes and the algebraic transfer, Trans Amer Math Soc 349 (1997), 3893-3910 MR 98e:55020 [9] N H V Hu'ng, Spherical classes and the homology of the Steenrod algebra, Vietnam Jour Math 26 (1998), 373-377 [10] N H V Hu'ng, The weak conjecture on spherical classes, Math Zeit 231 (1999), 727-743 MR 2000g:55019 [11] N H V Hu'ng and F P Peterson, Spherical classes and the Dickson algebra, Math Proc Camb Phil Soc 124 (1998), 253-264 MR 99i:55021 [12] J Lannes, Sur le n-dual du n-eme spectre de Brown-Gitler, Math Zeit 199 (1988), 29-42 MR 89h:55020 [13] J Lannes and S Zarati, Invariants de Hopf d'ordre superieur et suite spectrale d'Adams, C R Acad Sci 296 (1983), 695-698 MR 85a:55009 [14] J Lannes and S Zarati, Sur les foncteurs derives de la destabilisation, Math Zeit 194 (1987), 25-59 MR 88j:55014 This content downloaded from 128.235.251.160 on Thu, 22 Jan 2015 17:39:52 PM All use subject to JSTOR Terms and Conditions NGUYEN H V HUNG 4460 [15] S Mac Lane, Homology, Die Grundlehren der Math Wissenschaften, Band 114, Academic Press, Springer-Verlag, Berlin and New York, 1963 MR 28:122 [16] I Madsen, On the action of the Dyer-Lashof algebra in H*(G), Pacific Jour Math 60 (1975), 235-275 MR 52:9228 [17] H Miii, Modular invariant theory and cohomology algebras of symmetric groups, Jour Fac Sci Univ Tokyo, 22 (1975), 310-369 MR 54:10440 [18] S B Priddy, Koszul resolutions, Trans Amer Math Soc 152 (1970), 39-60 MR 42:346 [19] D C Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press, 1986 MR 87j:55003 [20] W M Singer, Invariant theory and the lambda algebra, Trans Amer Math Soc 280 (1983), 673-693 MR 85e:55029 [21] W M Singer, The transfer in homological algebra, Math Zeit 202 (1989), 493-523 MR 90i:55035 [22] V Snaith and J Tornehave, On 7S(BO) and the Arf invariant of framed manifolds, Amer Math Soc Contemporary Math 12 (1982), 299-313 MR 83k:55008 [23] R J Wellington, The unstable Adams spectral sequence of free iterated loop spaces, Memoirs Amer Math Soc 258 (1982) MR 83c:55028 [24] C Wilkerson, Classifying spaces, Steenrod operations and algebraic closure, Topology 16 (1977), 227-237 MR 56:1307 DEPARTMENT STREET, HANOI, OF MATHEMATICS, VIETNAM NATIONAL UNIVERSITY, HANOI, VIETNAM E-mail address: nhvhung@hotmai1.com This content downloaded from 128.235.251.160 on Thu, 22 Jan 2015 17:39:52 PM All use subject to JSTOR Terms and Conditions 334 NGUYEN TRAI ... invariant-theoretic model of the ABSTRACT Let rA - Tor 4(F2,F2), dual of the lambda algebra with Hk(rA) where A denotes the mod Steenrod algebra We prove that the inclusion of the Dickson algebra, ... words and phrases Spherical classes, loop spaces, Adams spectral sequences, Steenrod algebra, lambda algebra, invariant theory, Dickson algebra The research was supported in part by the National... which satisfy the condition monomials v1' is jl >i2+ -+jk- The lemma follows from the fact that the Dickson algebra Dk is a subalgebra of D the Mvii algebra MIlk Lemma 3.3 q is a cycle in the chain