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 NGUY~^EN H. V. HU,NG Abstract.Let ^ = L k ^kbe Singer's invariant-theoretic model of the dual of the lambda algebra with Hk( ^) ~=TorAk(F2, F2), where A denotes the mod 2 Steenrod algebra. We prove that the inclusion of the Dickson algebra, Dk, into ^kis a chain-level representation of the Lannes-Zarat* *i dual homomorphism '*k: F2A Dk ! TorAk(F2, F2) ~=Hk( ^) . The Lannes-Zarati homomorphisms themselves, 'k, correspond to an associ- ated graded of the Hurewicz map H : is*(S0) ~=i*(Q0S0) ! H*(Q0S0) . Based on this result, we discuss some algebraic versions of the classica* *l con- jecture 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. ele- ment in Dk, of positive degree represents the homology class 0 in TorAk(* *F2, F2) for k > 2. We also show that '*kfactors through F2 AKer@k, where @k : ^k! ^k-1denotes the differential of ^. Therefore, the problem of determini* *ng F2A Ker@k should be of interest. 1.Introduction and statement of results Let Q0S0 be the basepoint component of QS0 = limn nSn. It is a classical unsolved problem to compute the image of the Hurewicz homomorphism H : ßs*(S0) ~=ß*(Q0S0) ! H*(Q0S0) . Here and throughout the paper, homology and cohomology are taken with coeffi- cients in F2, the field of two elements. The long-standing conjecture on spheri* *cal classes reads as follows. Conjecture 1.1. The Hopf invariant one and the Kervaire invariant one classes are the only elements in H*(Q0S0) 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[x1, . .,.x* *k] be the polynomial algebra on k generators x1, . .,.xk, each of degree 1. Let t* *he ____________ Received by the editors February 4, 1999 and, in revised form, November 4, 1* *999. 2000 Mathematics Subject Classification. Primary 55P47, 55Q45, 55S10, 55T15. Key words and phrases. Spherical classes, loop spaces, Adams spectral sequen* *ces, Steenrod algebra, lambda algebra, invariant theory, Dickson algebra. The research was supported in part by the National Research Project, No. 1.4* *.2. cO2001 American Mathematical Socie* *ty 4447 4448 NGUY~^EN H. V. HU,NG general linear group GLk = GL(k, F2) and the mod 2 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[x1, . .,.xk]GLk . As the action of A and that of GLk on Pk commute with each other, Dk is an algebra over A. In [14], Lannes and Zarati construct homomorphisms 'k : Extk,k+iA(F2, F2) ! (F2 Dk)*i, 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 Ext1,*A(F2, F2) and Ext2,*A(F2, F2),* * on which '1 and '2 are non-zero (see Adams [1], Browder [4], Lannes-Zarati [14]). Therefore, we are led to the following conjecture. Conjecture 1.2. 'k = 0 in any positive stem i for k > 2. The present paper follows a series of our works ([8], [10], [11]) on this con* *jec- ture. To state our main result, we need to summarize Singer's invariant-theoret* *ic description of the lambda algebra [20]. According to Dickson [6], one has Dk ~=F2[Qk,k-1, ..., Qk,0], where Qk,idenotes the Dickson invariant of degree 2k - 2i. Singer sets k = Dk[Q-1k,0], the localization of Dk given by inverting Qk,0, andLdefines ^kto b* *e a certain ön t too large" submodule of k. He also equips ^ = k ^kwith a differential @ : ^k! ^k-1and a coproduct. Then, he shows that the differential coalgebra ^ is dual to the lambda algebra of the six authors of [3]. Thus, Hk(* * ^) ~= T orAk(F2, F2). (Originally, Singer uses the notation +kto denote ^k. However* *, by D+k, A+ we always mean the submodules of Dk and A respectively consisting of all elements of positive degrees, so Singer's notation +kwould cause confusion in * *this paper. Therefore, we prefer the notation ^k.) The main result of this paper is the following theorem, which has been conjec- tured in our paper [10, Conjecture 5.3]. Theorem 3.9. The inclusion Dk ^kis a chain-level representation of the Lannes-Zarati dual homomorphism '*k: (F2 Dk)i! T orAk,k+i(F2, F2). A An immediate consequence of this theorem is the equivalence between Conjec- ture 1.2 and the following one. Conjecture 1.3. If q 2 D+k, then [q] = 0 in T orAk(F2, F2) for k > 2. This has been established for k = 3 in [10, Theorem 4.8], while Conjecture 1.2 has been proved for k = 3 in [8, Corollary 3.5] . From the view point of this conjecture, it seems to us that Singer's model of* * the dual of the lambda algebra, ^, is somehow more natural than the lambda algebra itself. The canonical A-action on Dk is extended to an A-action on ^k. This action commutes with @k (see [20]), so it determines an A-action on Ker@k, the submodu* *le of all cycles in ^k. We also prove SPHERICAL CLASSES AND THE LAMBDA ALGEBRA 4449 Proposition 4.1.'*kfactors through F2 Ker@k as shown in the commutative A diagram * F2 Dk ________________-T'orAk(F , F ) A k 2 2 @ _ _ ` @ i p @ @R F2 Ker@k , A _ _ where iis induced by the inclusion Dk Ker@k, and p is an epimorphism induced by the canonical projection p : Ker@k ! Hk( ^) ~=T orAk(F2, F2). From this result, the problem of determining F2 Ker@k would be of interest. A The paper is divided into 4 sections. In Section 2 we recollect some materials on invariant theory, particularly on Singer's invariant-theoretic description of the lambda algebra and the Lannes- Zarati homomorphism. Section 3 is devoted to prove Theorem 3.9. Finally, Section 4 is a discussion on factoring '*k. 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 b* *ar 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 p* *aper. 2. Recollections on modular invariant theory We start this section by sketching briefly Singer's invariant-theoretic descr* *iption of the lambda algebra. Let Tk be the Sylow 2-subgroup of GLk consisting of all upper triangular k x * *k- matrices with 1 on the main diagonal. The Tk-invariant ring, Mk = PkTk, is call* *ed the M`ui algebra. In [17], M`ui shows that PkTk= F2[V1, ..., Vk], where Y Vi= (~1x1 + . .+.~i-1xi-1+ xi). ~j2F2 Then, the Dickson invariant Qk,ican inductively be defined by Qk,i= Q2k-1,i-1+ Vk . Qk-1,i, where, by convention, Qk,k= 1 and Qk,i= 0 for i < 0. Let S(k) Pk be the multiplicative subset generated by all the non-zero line* *ar forms in Pk. Let k be the localization, k = (Pk)S(k). Using the results of Dickson [6] and M`ui [17], Singer notes in [20] that k := ( k)Tk = F2[V1 1, ..., Vk 1], k := ( k)GLk = F2[Qk,k-1, ..., Qk,1, Qk1,0]. Further, he sets v1 = V1, vk = Vk=V1. .V.k-1 (k 2), 4450 NGUY~^EN H. V. HU,NG so that k-2 k-3 Vk = v21 v22 . .v.k-1vk (k 2). Then, he obtains k = F2[v11, ..., vk1], with degvi= 1 for every i. Singer defines ^kto be the submodule of k = Dk[Q-1k,0] spanned by all mono- mials fl = Qik-1k,k-1.Q.i.0k,0with ik-1, ..., i1 0, i0 2 Z, and i0 + degfl * *0. He also shows in [20] that the homomorphism @k : F2[v11, ..., vk1] ! F2[v11, ..., vk1-1], æ j1 jk-1 @k(vj11. .v.jkk) := v10.,.v.k-1,ifojkt=h-1,erwise, L maps ^kto ^k-1. Moreover, it is a differential onP ^ = k ^k. This module is bigraded by putting bideg(vj11. .v.jkk) = (k, k + ji). Let 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 pu* *tting (as in [19, p. 90]) bideg(~i) = (1, 1 + i). Singer proves in [20] that ^ is a diff* *erential bigraded coalgebra, which is dual to the differential bigraded lambda algebra * * via the isomorphisms ^k ! *k vj11. .v.jkk7!(~j1. .~.jk)*. Here the duality * is taken with respect to the basis of admissible monomials o* *f . As a consequence, one gets an isomorphism of bigraded coalgebras H*( ^) ~=T orA*(F2, F2). In the remaining part of this section, we recall the definition of the Lannes* *-Zarati homomorphism. Let P1 = F2[x] with |x| = 1. Let ^P 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 ^Pis an A-submodule of F2[x, x-1]. One has a short-exact sequence of A-modules 2.1. 0 ! P1 '!P^!ß -1F2 ! 0 , where ' is the inclusion and ß is given by ß(xi) = 0 if i 6= -1 and ß(x-1) = 1.* * Let e1 be the corresponding element in Ext1A( -1F2, P1). Definition 2.2 (Singer [21]).(i)ek = e1___._.-.e1z____"2 ExtkA( -kF2, Pk). k times (ii)ek(M) = ek M 2 ExtkA( -kM, Pk M) , for M a left A-module. 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{Sqix| i > degx, x 2 M}. They show that the functor associ- ating 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)), SPHERICAL CLASSES AND THE LAMBDA 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 ek(M) : Dk( -kM) ! D0(Pk M) Pk M ek(M)(z) = ek(M) \ z . Since F2 is an unstable A-module, one gets Theorem 2.3 (Lannes-Zarati [14]).Let Dk Pk be the Dickson algebra of k vari- ables. Then ffk := ek( F2) : Dk( 1-kF2) ! Dk is an isomorphism of internal degree 0. By definition of the functor D, one has a natural homomorphism, D(M) ! F2 M. Then it induces a commutative diagram A . . ._____- DFk(M) ________- DFk-1(M) ______-. . . | | | | |ik |ik-1 | | |? |? . . ._____-F2 Fk(M) ________- F2 Fk-1(M) ______-. ... A A Here the horizontal arrows are induced from the differential in F*(M), and ik[Z] = [1 Z] A for Z 2 Fk(M). Passing to homology, one gets a homomorphism ik :F2 Dk(M) ! T orAk(F2, M) 2.4. A1 [Z] 7! [1 Z] . A A Taking M = 1-kF2, one obtains a homomorphism ik : F2 Dk( 1-kF2) ! T orAk(F2, 1-kF2) . A Note that the suspension : F2 Dk ! F2 Dk and the desuspension A A ~= A -k -1 : T orAk(F2, 1-kF2) -! T ork (F2, F2) are isomorphisms of internal degree 1 and (-1), respectively. This leads one to Definition 2.5 (Lannes-Zarati [14]).The homomorphism 'k of internal degree 0 is the dual of '*k= -1ik(1 ff-1k) : F2 Dk ! T orAk(F2, -kF2) . A A Remark 2.6.In Theorem 3.9 we also denote by '*kthe composite of the above '*k ~= with the suspension isomorphism k : T orAk,i(F2, -kF2) -! T orAk,k+i(F2, F2). We need to relate ffk = ek( F2) with connecting homomorphisms. Suppose f 2 Ext1A(M3, M1) is represented by the short-exact sequence of left A-modules 0 ! M1 ! M2 ! M3 ! 0. Let (f) : Ds(M3) ! Ds-1(M1) be the connecting homomorphism associated with this short-exact sequence. Then one easily verifies (f)(z) = f \ z for any z 2 Ds(M3). 4452 NGUY~^EN H. V. HU,NG One has 2.7. ek( F2) = (e1( F2) Pk-1) O . .O.(e1( 3-kF2) P1) O e1( 2-kF2) . Therefore, one gets 2.8. ffk = (e1( F2) Pk-1) O . .O. (e1( 3-kF2) P1) O e1( 2-kF2) . (See Singer [21, p. 498].) This formula will be useful to construct a chain-level representation of ffk. 3. A chain-level representation of the Lannes-Zarati homomorphism Suppose again M is a left graded A-module. Let B*(M) be the bar resolution of M over A. Recall that Bk(M) = A I___._.-.Iz___" M (k 0), k times where I denotes the augmentationLideal of A and the tensor products are taken over F2. The module B*(M) = kBk(M) is bigraded by assigning anPelement a0 a1 . . .ak x with homological degree k and internal degree ki=0(degai) + degx. The differential dk : Bk(M) ! Bk-1(M) is defined by dk(a0 a1 . . .ak x)= a0a1 . . .ak x + a0 a1a2 . . .ak x + . .+.a0 a1 . . .akx. So dk preserves internal degree and lowers homological degree by 1. The action of A on Bk(M) is given by a(a0 a1 . . .ak x) = aa0 a1 . . .ak x, for a 2 A. Suppose additionally that N is a right graded A-module. As the bar resolution is an A-free resolution, by definition one has T orAk(N, M) := Hk(N B*(M)). A Since Dk F2[v1, ..., vk], every element q 2 Dk has an unique expansion X j j q = v11. .v.kk, (j1,...,jk) where j1, ..., jk arePnon-negative. We associate with q 2 Dk the following elem* *ent of internal degree ki=1ji+ 1: Definition 3.1. X ~q= Sqj1+1 . . .Sqjk+1 1-k1 2 Bk-1( 1-kF2). (j1,...,jk) Lemma 3.2. If q 2 Dk, then ~q2 EBk-1( 1-kF2) := Span{Sqix| i > degx, x 2 Bk-1( 1-kF2)}. SPHERICAL CLASSES AND THE LAMBDA ALGEBRA 4453 Proof.From the definition of the A-action on the bar resolution, one has Sqj1+1 . . .Sqjk+1 1-k1 = Sqj1+1(1 Sqj2+1 . . .Sqjk+1 1-k1). Hence, it suffices to show that j1 + 1 > (j2 + 1) + . .+.(jk + 1) + (1 - k) = j2 + . .+.jk, for every term in the expansion of ~q. Recall that Vi = v2i-21v2i-32.v.i.-1vi. So, one easily verifies that every el* *ement v 2 Mk = F2[V1, ..., Vk] is a sum of monomials vj11. .v.jkk, which satisfy the * *condition j1 j2 + . .+.jk. The lemma follows from the fact that the Dickson algebra Dk is a subalgebra_of the M`ui algebra Mk. |__| Lemma 3.3. ~qis a cycle in the chain complex EB*( 1-kF2), for every q 2 Dk. This is a consequence of the following lemma, which is actually an exposition* * of the Adem relations. Lemma 3.4. The homomorphism ßk,p: k ! Ak-1 = A . . .A (k - 1 times) vj11. .v.jppvjp+1p+1.v.j.kk7!Sqj1+1 . . .Sqjp+1Sqjp+1+1 . . .Sqjk+1 vanishes on k k, for 1 p < k. Proof.Consider the diagonal _ : k ! p-1 2 k-p-1 defined by 8 < vi 1 1, i < p, _(vi) = : 1 vi-p+1 1, p i p + 1, 1 1 vi-p-1,p + 1 < i. From Proposition 2.1 of Singer [20], one gets _( k) p-1 2 k-p-1. Define the homomorphism !t: t! At by !t(vj11. .v.jtt) = Sqj1+1 . . .Sqjt+1. Then one has ßk,p= (!p-1 ß2,1 !k-p-1)_. By Proposition 3.1 of Singer [20], the Adem relations yield ß2,1( 2) = 0. Hence, ßk,p( k) = 0 for 1 p < k. The lemma is proved. |___| Proof of Lemma 3.3.First, we note that Sqjk+1( 1-k1) = 0 for any jk 0. Then, by definition of the differential in the bar resolution, we get k-1X dk-1(~q) = (ßk,p id 1-kF2)(q 1-k1). p=1 Since q 2 Dk k, Lemma 3.4 yields ßk,p(q) = 0. Thus dk-1(~q) = 0. The_lemma_ is proved. |__| 4454 NGUY~^EN H. V. HU,NG For the convenience of the latter use, we define ~ßk,pas follows: ß~k,p(Sqj1+1 . . .Sqjk+1) = Sqj1+1 . . .Sqjp+1Sqjp+1+1 . . .Sqjk+1 for 1 p < k. Suppose as before that X j j q = v11. .v.kk2 Dk. J=(j1,...,jk) For a fixed (k - s)-index (js+1, ..., jk), we define J(js+1, ..., jk) to be the* * set of all s- indices (j1, ..., js)'s such that (j1, ..., js, js+1, ..., jk) occurs as a k-in* *dex in the above sum. The following lemma is a slight generalization of Lemma 3.4. P j1 j Lemma 3.5. If q = Jv1 . .v.kk2 Dk, then X j +1 j +1 ~ßs,p Sq 1 . . .Sq s = 0 J(js+1,...,jk) for 1 p < s k. Proof.Let us consider the diagonal _2 : k ! s k-s given by æ _2(vi) = vi1 1, v1 i s, i-s,s < i k. According toPProposition 2.1 of Singer [20], _( k) s k-s. Since q 2 Dk k, it implies J(js+1,...,jk)vj11. .v.jss2 s. Then, by Lemma 3.4, we have X j +1 j +1 X j j ~ßs,p Sq 1 . . .Sq s = ßs,p v11. .v.ss= 0. J(js+1,...,jk) J(js+1,...,jk) 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 0 ! EB*(M) iE!B*(M) jD!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 ~= @* : Dk(M) := Hk(DB*(M)) -! Hk-1(EB*(M)). Take M = 1-kF2. The following lemma deals with the connecting isomorphism ~= 1-k @* : Dk( 1-kF2) := Hk(DB*( 1-kF2)) -! Hk-1(EB*( F2)). Let [~q] be the homology class of the cycle ~qin Dk( 1-kF2) ~=Hk-1(EB*( 1-kF2)). Lemma 3.6. If q 2 Dk, then @*[1 ~q] = [~q]. P j1 j P Proof.Suppose q = Jv1 . .v.kk. The element J1 Sqj1+1 . . .Sqjk+1 1-k1 2 Bk( 1-kF2) is a lifting over jD of its class modulo EBk( 1-kF2) in DBk( 1-kF2). Let d denote the differential in B*( 1-kF2), we get SPHERICAL CLASSES AND THE LAMBDA ALGEBRA 4455 X d( 1 Sqj1+1 . . .Sqjk+1 1-k1) J X = 1 . Sqj1+1 . . .Sqjk+1 1-k1 J k-1X X + 1 ~ßk,p( Sqj1+1 . . .Sqjk+1) 1-k1 p=1 J X + 1 Sqj1+1 . . .Sqjk+1 1-k1. J By Lemma 3.4 X ~ßk,p( Sqj1+1 . . .Sqjk+1) = ßk,p(q) = 0. J On the other hand, Sqjk+1( 1-k1) = 0 for any jk 0. Therefore, we obtain X X d( 1 Sqj1+1 . . .Sqjk+1 1-k1) = Sqj1+1 . . .Sqjk+1 1-k1 J J = iE (~q). By definition of the connecting homomorphism, we have @*[1 ~q] = [~q]. The lemma is proved. |___| The following theorem deals with the isomorphism ffk : Dk( 1-kF2) ! Dk treated in Theorem 2.3. Theorem 3.7. If q 2 Dk, then ffk[~q] = q. Proof.We compute ffk by means of the following formula ffk = (e1( F2) Pk-1) O . .O. (e1( 3-kF2) P1) O e1( 2-kF2) = ffik . .f.fi2ffi1. Here ffis stands for (e1( 1-k+sF2) Ps-1), for brevity. Consider the short exact sequence representing e1( 2-kF2): 0 ! 2-kP1 '! 2-kP^!ß 1-kF2 ! 0. Then the connecting homomorphism induced by this exact sequence is nothing but ffi1 : Hk-1(EB*( 1-kF2)) ! Hk-2(EB*( 2-kP1)). P A lifting of ~q= JSqj1+1 . . .Sqjk+1 1-k1 over ß is X Sqj1+1 . . .Sqjk+1 2-kx-1k2 EB*( 2-kP^), J where we are writing P1 = F2[xk], ^P= Span{xik| i -1}. The boundary of this element in EB*( 2-kP^) is pulled back under ' to a cycle in EB*( 2-kP1), which 4456 NGUY~^EN H. V. HU,NG represents ffi1[~q]. That means X ffi1[~q]=[d( Sqj1+1 . . .Sqjk+1 2-kx-1k)] J k-1X X j +1 j +1 2-k -1 = ~ßk,p( Sq 1 . . .Sq k ) xk p=1 J X + Sqj1+1 . . .Sqjk-1+1 Sqjk+1( 2-kx-1k) J X j +1 j +1 2-k j +1 -1 = Sq 1 . . .Sq k-1 Sq k (xk ) , J where the last equality follows from Lemma 3.4. Indeed, X ~ßk,p( Sqj1+1 . . .Sqjk+1) = ßk,p(q) = 0. J Similarly, ffi2 : Hk-2(EB*( 2-kP1)) ! Hk-3(EB*( 3-kP2)) is the connecting ho- momorphism induced by the short exact sequence representing e1( 3-kF2) P1: 0 ! 3-kP2 '!P1 3-k(P^ P1) ß!P1 2-kP1 ! 0. Here we are writing P1 = F2[xk], P2 = F2[xk-1, xk], ^P= Span{xik-1|i -1}. A lifting of Sqj1+1 . . .Sqjk-1+1 2-kSqjk+1(x-1k) over ß P1 is Sqj1+1 . . .Sqjk-1+1 3-kx-1k-1Sqjk+1(x-1k). Therefore, by an argument similar to the one given above, we get X j +1 j +1 3-k -1 j +1 -1 ffi2ffi1[~q]=[d Sq 1 . . .Sq k-1 xk-1Sq k (xk ) ] J k-2XX j +1 j +1 3-k -1 j +1 -1 = ~ßk-1,p(Sq 1 . . .Sq k-1 ) xk-1Sq k (xk ) p=1 J X + Sqj1+1 . . .Sqjk-2+1 Sqjk-1+1( 3-kx-1k-1Sqjk+1(x-1k)) J X j +1 j +1 3-k j +1 -1 j +1 -1 = Sq 1 . . .Sq k-2 Sq k-1 (xk-1Sq k (xk )) J (by Lemma 3.5). Repeating the above argument, we then have ffk[~q]= ffik . .f.fi1[~q] X = [ Sqj1+1(x-11Sqj2+1(x-12. .S.qjk+1(x-1k) . .).) ]. J By Theorem 3.2 of our paper [10], we get X [ Sqj1+1(x-11Sqj2+1(x-12. .S.qjk+1(x-1k) . .).) ] = [ q] = q. J The theorem is proved. |___| This theorem has an immediate consequence as follows. SPHERICAL CLASSES AND THE LAMBDA ALGEBRA 4457 Corollary 3.8.The homomorphism Dk ! EBk-1( 1-kF2), q 7! ~qis a chain-level representation of the homomorphism (1 ff-1k) : F2 Dk ! F2 Dk( 1-kF2). A A A Theorem 3.9. The inclusion Dk ^kis a chain-level representation of the Lannes- Zarati dual homomorphism '*k: (F2 Dk)i! T orAk,k+i(F2, F2). A Proof.Suppose again that X j j q = v11. .v.kk2 Dk. J=(j1,...,jk) By Corollary 3.8 and Lemma 3.6, we have (1 ff-1k) :F2 Dk ! F2 Dk( 1-kF2) A A A [q] 7! [~q] @*[1 ~q]. From the definition of ik (see 2.4), we get ik :F2 Hk(DB*( 1-kF2)) ! T orAk(F2, 1-kF2) A [1 ~q]7! [1 ~q]. Let us consider the desuspension -1 : T orAk(F2, 1-kF2) ! T orAk(F2, -kF2), P P which sends [ J 1 Sqj1+1 . . .Sqjk+1 1-k1] to [ J 1 Sqj1+1 . . . Sqjk+1 -k1]. Then the map '*k= -1ik(1 ff-1k) : F2 Dk ! T orAk(F2, -kF2) A A is given by X '*k[q] = [ 1 Sqj1+1 . . .Sqjk+1 -k1]. J The canonical isomorphism k : T orAk,i(F2, -kF2) ! T orAk,k+i(F2, F2) is defined by the chain-level version k(a0 a1 . . .ak -k1) = a0 a1 . . .ak 1. By ambiguity of notation, the composite k'*kis also denoted by '*k(see Re- mark 2.6). Hence '*k: (F2 Dk)i ! T orAk,k+i(F2, F2) A P [q] 7! [ J 1 Sqj1+1 . . .Sqjk+1 1]. __ 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 de* *fined as follows. Let be the (opposite) lambda algebra, in which the product in lam* *bda symbols is written in the order opposite to that used in [3]. (See Singer [20, * *p. 687] 4458 NGUY~^EN H. V. HU,NG __ for a precise definition of .) Then, according to Priddy [18, x7], K *(A) is * *the image of the monomorphism * ! B*(F2) (~j1. .~.jk)*7! 1 Sqj1+1 . . .Sqjk+1 1, which is a homotopy equivalence. Here * denotes the dual of and the duality * is taken with respect to the basis of admissible monomials of . Combining it w* *ith Singer's isomorphism ^ ! * vj11. .v.jkk7!(~j1. .~.jk)*, we get the following homotopy equivalence ^ ! B*(F2) vj11. .v.jkk7!1 Sqj1+1 . . .Sqjk+1 1. As a consequence, for any q 2 Dk, we obtain X '*k[q]= [ 1 Sqj1+1 . . .Sqjk+1 1] XJ = [ vj11. .v.jkk] J = [q]. It means that the inclusion Dk ^kis a chain-level representation of '*k._ The theorem is completely proved. |__| Corollary 3.10.Conjecture 1.2 is equivalent to Conjecture 1.3. This follows immediately from Theorem 3.9. We have proved Conjecture 1.2 for k = 3 in [8] and Conjecture 1.3 for k = 3 in [10]. 4. Factoring the Lannes-Zarati homomorphism The purpose of this section is to prove the following proposition. Proposition 4.1.'*kfactors through F2 Ker@k as shown in the commutative A diagram: * F2 Dk ________________-T'orAk(F , F ) A k 2 2 @ _ _ ` @ i p @ @R F2 Ker@k , A _ _ where iis induced by the inclusion Dk Ker@k, and p is an epimorphism induced by the canonical projection p : Ker@k ! Hk( ^) ~=T orAk(F2, F2). Proof.The canonical projection p : Ker@k ! T orAk(F2, F2) = Ker@k=Im@k+1 sends x to [x] = x + Im@k+1. SPHERICAL CLASSES AND THE LAMBDA ALGEBRA 4459 By Theorem 5.15 of Singer [20], the action of A on Ker@k induces a trivial ac* *tion of A upon T orAk(F2, F2). Therefore, p induces the epimorphism _p: F A 2 Ker@k ! T ork (F2, F2) A [x] 7! [x]. For any q 2 Dk, we have _p. _i[q] = _p[q] = [q] = '* _ k[q]. So, we get '*k= _p. i. The proposition is proved. |_* *__| In [10], we have stated the following conjecture. Conjecture 4.2. D+k A+ . Ker@k for k > 2. Obviously, this is stronger than Conjectures 1.2 and 1.3 and equivalent to the following one. _ Conjecture 4.3. The homomorphism i: F2 Dk ! F2 Ker@k, induced by the A A inclusion i : Dk ! Ker@k, is trivial for k > 2. 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. M* *ath. 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 RP1 , New developments in topology, G. Segal (ed.), London Math. Soc. Lect. Note S* *eries 11 (1974), 1-9. MR 49:3941 [3]A. K. Bousfield, E. B. 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MR 56:1307 Department of Mathematics, Vietnam National University, Hanoi, 334 Nguy^en Tr* *~ai Street, Hanoi, Vietnam E-mail address: nhvhung@hotmail.com