TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 349, Number 10, October 1997, Pages 3893-3910 S 0002-9947(97)01991-0 SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER NGUY^E~N H. V. HU'NG Abstract.We study a weak form of the classical conjecture which predicts that there are no spherical classes in Q0S0 except the elements of Hopf * *invari- ant one and those of Kervaire invariant one. The weak conjecture is obta* *ined by restricting the Hurewicz homomorphism to the homotopy classes which a* *re detected by the algebraic transfer. Let Pk = F2[x1, . .,.xk] with |xi| = 1. The general linear group GLk = GL(k, F2) and the (mod 2) Steenrod algebra A act on Pk in the usual mann* *er. We prove that the weak conjecture is equivalent to the following one: The canonical homomorphism jk : F2 (PGLkk) ! (F2 Pk)GLk induced by the identity map on P A A k is zero in positive dimensions for k > 2. In other wo* *rds, every Dickson invariant (i.e. element of PGLkk) of positive dimension be* *longs to A+ . Pk for k > 2, where A+ denotes the augmentation ideal of A. This conjecture is proved for k = 3 in two different ways. One of these two w* *ays is to study the squaring operation Sq0 on P(F2 P*k), the range of j*k,* * and GLk to show it commuting through j*kwith Kameko's Sq0 on F2 P(P*k), the GLk domain of j*k. We compute explicitly the action of Sq0 on P(F2 P*k) for GLk k 4. 1.Introduction The paper deals with the spherical classes in Q0S0, i.e. the elements belongi* *ng to the image of the Hurewicz homomorphism H : ßs*(S0) ~=ß*(Q0S0) ! H*(Q0S0) . Here and throughout the paper, the coefficient ring for homology and cohomology is always F2, the field of 2 elements. We are interested in the following classical conjecture. Conjecture 1.1. (conjecture on spherical classes). There are no spherical class* *es in Q0S0, except the elements of Hopf invariant one and those of Kervaire invari* *ant one. (See Curtis [9] and Wellington [21] for a discussion.) Let Vk be an elementary abelian 2-group of rank k. It is also viewed as a k- dimensional vector space over F2. So, the general linear group GLk = GL(k, F2) ____________ Received by the editors April 7, 1995. 1991 Mathematics Subject Classification. Primary 55P47, 55Q45, 55S10, 55T15. Key words and phrases. Spherical classes, loop spaces, Adams spectral sequen* *ces, Steenrod algebra, invariant theory, Dickson algebra, algebraic transfer. The research was supported in part by the DGU through the CRM (Barcelona). cO1997 American Mathematical Socie* *ty 3893 3894 NGUY^E~N H. V. HU'NG acts on Vk and therefore on H*(BVk) in the usual way. Let Dk be the Dickson algebra of k variables, i.e. the algebra of invariants Dk := H*(BVk)GLk ~=F2[x1, . .,.xk]GLk , where Pk = F2[x1, . .,.xk] is the polynomial algebra on k generators x1, . .,.x* *k, each of dimension 1. As the action of the (mod 2) Steenrod algebra A and that of GLk on Pk commute with each other, Dk is an algebra over A. One way to attack Conjecture 1.1 is to study the Lannes-Zarati homomorphism 'k : Extk,k+iA(F2, F2) ! (F2 Dk)*i, A which is compatible with the Hurewicz homomorphism (see [12], [13, p.46]). The domain of 'k is the E2-term of the Adams spectral sequence converging to ßs*(S0* *) ~= ß*(Q0S0). Furthermore, according to Madsen's theorem [15] which asserts that Dk is dual to the coalgebra of Dyer-Lashof operations of length k, the range of 'k* * is a submodule of H*(Q0S0). By compatibility of 'k and the Hurewicz homomorphism we mean 'k is a "liftingö f the latter fromrthe "E1 -level" to the "E2-level". Let hr denote the Adams element in Ext1,2A(F2, F2). Lannes and Zarati proved in [13] that '1 is an isomorphism with {'1(hr) | r 0} forming a basis of the * *dual of F2 D1 and '2 is surjective with {'2(h2r)| r 0} forming a basis of the dua* *l of A F2 D2. Recall that, from Adams [1], the only elements of Hopf invariant one are A represented by h1, h2, h3 of the stems i = 2r - 1 = 1, 3, 7, respectively. More* *over, by Browder [5], the only dimensions where an element of Kervaire invariant one would occur are 2(2r - 1), for r > 0, and it really occurs at this dimension if* * and only if h2ris a permanent cycle in the Adams spectral sequence for the spheres. Therefore, Conjecture 1.1 is a consequence of the following: Conjecture 1.2. 'k = 0 in any positive stem i for k > 2. It is well known that the Ext group has intensively been studied, but remains very mysterious. In order to avoid the shortage of our knowledge of the Ext gro* *up, we want to restrict 'k to a certain subgroup of Ext which (1) is large enough a* *nd worthwhile to pursue and (2) could be handled more easily than the Ext itself. * *To this end, we combine the above data with Singer's algebraic transfer. Singer defined in [20] the algebraic transfer T rk : F2 P Hi(BVk) ! Extk,k+iA(F2, F2) , GLk where P H*(BVk) denotes the submodule consisting of all A-annihilated elements in H*(BVk). It is shown to be an isomorphism for k 2 by Singer [20] and for k = 3 by Boardman [4]. Singer also proved that it is an isomorphism for k = 4 in a range of internal degrees. But he showed it is not an isomorphism for k = 5. However, he conjectures that T rk is a monomorphism for any k. Our main idea is to study the restriction of 'k to the image of T rk. Conjecture 1.3. (weak conjecture on spherical classes). 'k . T rk : F2 P H*(BVk) ! P (F2 H*(BVk)) := (F2 Dk)* GLk GLk A is zero in positive dimensions for k > 2. SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER 3895 In other words, there are no spherical classes in Q0S0, except the elements of Hopf invariant one and those of Kervaire invariant one, which can be detected by the algebraic transfer. A natural question is: How can one express 'k.T rk in the framework of invari* *ant theory alone, and without using the mysterious Ext group? Let jk : F2 (PkGLk) ! (F2 Pk)GLk be the natural homomorphism induced by A A the identity map on Pk. We have Theorem 2.1. 'k . T rk is dual to jk, or equivalently, jk = T r*k. '*k. By this theorem, Conjecture 1.3 is equivalent to Conjecture 1.4. jk = 0 in positive dimensions for k > 2. This seems to be a surprise, because by an elementary argument involving taki* *ng averages, one can see that if H GLk is a subgroup of odd order then the simil* *ar homomorphism jH : F2 (PkH) ! (F2 Pk)H A A is an isomorphism. Furthermore, j1 is iso and j2 is mono. Obviously, jk = 0 if and only if the composite Dk = PkGLkproj!F2 (PkGLk) jk!(F2 Pk)GLk ! F2 Pk A A A is zero. So, Conjecture 1.4 can equivalently be stated in the following form. Conjecture 1.5. Let D+k, A+ denote the augmentation ideals in Dk and A, re- spectively. Then D+k A+ . Pk for any k > 2. The domain and range of jk both are still mysterious. Anyhow, they seem easier to handle than the Ext group. They both are well-known for k = 1, 2. Furthermore, on the one hand, (F2 Pk)GLk is computed for k = 3 by Kameko [11], A Alghamdi-Crabb-Hubbuck [3] and Boardman [4]. On the other hand, F2 (PkGLk) A is determined by Hu'ng-PetersonL[18] for k = 3 and 4.L Let F2 (P GL) := F2 (PkGLk) and (F2 P )GL := (F2 Pk)GLk. They A k 0 A A k 0 A are equipped with canonical coalgebra structures. We get L Proposition 3.1.j = jk : F2 (P GL) ! (F2 P )GL is a homomorphism of A A coalgebras. Let Sq0 : P H*(BVk) ! P H*(BVk) be Kameko's squaring operation that com- mutes with the Steenrod operation Sq0 : Extk,tA(F2, F2) ! Extk,2tA(F2, F2) thro* *ugh the algebraic transfer T rk (see [11], [3], [4], [17]). Note that Sq0 is compl* *etely different from the identity map. We prove Proposition 4.2.There exists a homomorphism Sq0 : P (F2 H*(BVk)) ! P (F2 H*(BVk)) , GLk GLk which commutes with Kameko's Sq0 through the homomorphism j*k. These two propositions lead us to two different proofs of the following theor* *em. 3896 NGUY^E~N H. V. HU'NG Theorem 3.2. j*k= 0 in positive dimensions for k = 3. In other words, there is no spherical class in Q0S0 which is detected by the triple algebraic transfer. We compute explicitly the action of Sq0 on P (F2 H*(BVk)) for k = 3 and 4 GLk in Propositions 5.2 and 5.4. The paper contains six sections and is organized as follows. Section 2 is to prove Theorem 2.1. In SectionL3, we assemble the jk for k 0 to get a homomorphism of coalgebras j = jk. By means of this property of j we give there a proof of Theorem 3.2. Section 4 deals with the existence of the squaring operation Sq0 on P (F2 H*(BVk)) that leads us to an alternative proof GLk for Theorem 3.2. This proof helps to explain the problem. In Section 5, we comp* *ute explicitly the action of Sq0 on P (F2 H*(BVk)) for k 4. Finally, in Section* * 6 GLk we state a conjecture on the Dickson algebra that concerns spherical classes. Acknowledgments I express my warmest thanks to Manuel Castellet and all my colleagues at the CRM (Barcelona) for their hospitality and for providing me with a wonderful wor* *k- ing atmosphere and conditions. I am grateful to Jean Lannes and Frank Peterson for helpful discussions on the subject. Especially, I am indebted to Frank Pete* *r- son for his constant encouragement and for carefully reading my entire manuscri* *pt, making several comments that have led to many improvements. 2.Expressing 'k . T rk in the framework of invariant theory First, let us recall how to define the homomorphism jk. We have the commutative diagram PkGLk ___________- Pk | H H | | H p | | H H | | H H | |? fjk Hj |? F2 (PkGLk) ___________- F2 Pk , A A where the vertical arrows are the canonical projections, and ejkis induced by t* *he inclusion PkGLk Pk. Obviously, p(PkGLk) (F2 Pk)GLk. So, ejkfactors through A (F2 Pk)GLk to give rise to A jk : F2 (PkGLk) ! (F2 Pk)GLk, A A jk(1 Y ) = 1 Y, A A for any polynomial Y 2 Dk = PkGLk. The goal of this section is to prove the following theorem. Theorem 2.1. jk = T r*k. '*k. Now we prepare some data in order to prove the theorem at the end of this section. SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER 3897 First we sketch Lannes-Zarati's work [13] on the derived functors of the dest* *a- bilization. Let D be the destabilization functor, which sends an A-module M to the unstable A-module D(M) = M=B(M), where B(M) is the submodule of M generated by all Sqiu with u 2 M, i > |u|. D is a right exact functor. Let Dk be its k-th derived functor for k 0. Suppose M1, M2 are A-modules. Lannes and Zarati defined in [13, x2] a homo- morphism \ : ExtrA(M1, M2) Ds(M1) ! Ds-r(M2), (f, z) 7! f \ z , as follows. Let F*(Mi) be a free resolution of Mi, i = 1, 2. A class f 2 ExtrA(M1, M2) can be represented by a chain map F : F*(M1) ! F*-r(M2) of homological degree -r. We write f = [F ]. If z = [Z] is represented by Z 2 F*(M1), then by definition f \ z = [F (Z)]. Let M be an A-module. We set r = s = k, M1 = -kM, M2 = Pk M, where as before Pk = F2[x1, . .,.xk], and get the homomorphism \ : ExtkA( -kM, Pk M) Dk( -kM) ! Pk M . Now we need to define the Singer element ek(M) 2 ExtkA( -kM, Pk M) (see Singer [20, p. 498]). Let bP1be the submodule of F2[x, x-1] spanned by all powe* *rs xi with i -1, where |x| = 1. The A-module structure on F2[x, x-1] extends that of P1 = F2[x] (see Adams [2], Wilkerson [22]). The inclusion P1 bP1gives rise* * to a short exact sequence of A-modules: 0 ! P1 ! bP1! -1F2 ! 0 . Denote by e1 the corresponding element in Ext1A( -1F2, P1). Definition 2.2.(Singer [20]). (i)ek = e1___._.-.e1z____"2 ExtkA( -kF2, Pk). k times (ii)ek(M) = ek M 2 ExtkA( -kM, Pk M) , for M an A-module. Here we also denote by M the identity map 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 . As F2 is an unstable A-module, the following theorem is a special case (but would be the most important case) of the main result in [13]. Theorem 2.3 (Lannes-Zarati [13]).Let Dk Pk be the Dickson algebra of k vari- ables. Then ek( F2) : Dk( 1-kF2) ! Dk is an isomorphism of internal degree 0. Next, we explain in detail the definition of the Lannes-Zarati homomorphism 'k : ExtkA( -kF2, F2) ! (F2 Dk)*, A which is compatible with the Hurewicz map (see [12], [13]). Let N be an A-module. By definition of the functor D, we have a natural homomorphism: D(N) ! F2 N. Suppose F*(N) is a free resolution of N. Then A the above natural homomorphism induces a commutative diagram 3898 NGUY^E~N H. V. HU'NG . . ._____- DFk(N) ________- DFk-1(N) ______-. . . | | | | |ik |ik-1 | | |? |? . . ._____-F2A Fk(N) ________- F2A Fk-1(N) ______-. ... Here the horizontal arrows are induced from the differential in F*(N), and ik[Z] = [1 Z] A for Z 2 Fk(N). Passing to homology, we get a homomorphism ik :F2 Dk(N) ! T orAk(F2, N), A 1 [Z] 7! [1 Z] . A A Taking N = 1-kF2, we obtain 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 us to Definition 2.4.(Lannes-Zarati [13]). The homomorphism 'k of internal degree 0 is the dual of ` ' '*k= -1ik 1 e-1k( F2) : F2 Dk ! T orAk(F2, -kF2) . A A Now we recall the definition of the algebraic transfer. Consider the cap-prod* *uct ExtrA( -kF2, Pk) T orAs(F2, -kF2)! T orAs-r(F2, Pk), (e, z) 7! e \ z . Taking r = s = k and e = ek as in Definition 2.2, we obtain the homomorphism T r*k:T orAk(F2, -kF2)! T orA0(F2, Pk) F2 Pk, A T r*k[1 Z] = ek \ [Z] = [1 E(Z)] 1 E(Z), A A A for Z 2 Fk( -kF2), where ek = [E] is represented by a chain map E : F*( -kF2) ! F*-k(Pk). Singer proved in [20] that ek is GLk-invariant, hence Im(T r*k) (F2 Pk)GLk* * . A This gives rise to a homomorphism, which is also denoted by T r*k, T r*k: T orAk(F2, -kF2) ! (F2 Pk)GLk . A Definition 2.5.(Singer [20]). The k-th algebraic transfer T rk : F2 P H*(BVk) GLk ! Extk,k+*A(F2, F2) is the homomorphism dual to T r*k. We have finished the preparation of the needed data. SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER 3899 Proof of Theorem 2.1.Note that the usual isomorphism ~= k 1-k ExtkA( -kF2, Pk) -! ExtA ( F2, Pk) sends ek(F2) to ek( F2) = ek(F2) F2. Moreover, if ek(F2) = [E] is represented by a chain map E : F*( -kF2) ! F*-k(Pk) then ek( F2) = [E ] is represented by the induced chain map E : F*( 1-kF2) ! F*-k( Pk), which is defined by E = E -1. By Theorem 2.3, ek( F2) is an isomorphism. So, for any Y 2 Dk, there exists a representative of e-1k( F2) Y , which is denoted by E-1 Y 2 Fk( 1-kF2), such that E E-1 Y = Y . The cap-product with ek( F2) = [E ] induces the homomorphism fT*rk:T orAk(F2, 1-kF2)! T orA0(F2, Pk) F2 Pk, * A Tfrk[1 Z] = ek( F2) \ [Z] = [1 E (Z)] 1 E (Z), A A A * for Z 2 Fk( 1-kF2). It is easy to check that T r*k= -1Tfrk . Moreover, set ` ' e'*k= '*k -1 = ik 1 e-1k( F2) : F2 Dk ! T orAk(F2, 1-kF2) . A A * Obviously, T r*k. '*k= -1Tfrk. e'*k . Now, for any Y 2 Dk, we have * * T r*k. '*k(1 =Y )-1fTkr. e'k (1 Y ) A A * * = -1Tfrk. e'k(1 Y ) ~ A ~ * -1 = -1Tfrk 1 E Y ` A ' = -1 1 E (E-1 Y ) A = 1 -1( Y ) A = 1 Y . A By definition of jk, we also have jk(1 Y ) = 1 Y . The theorem is proved. A A L 3. The homomorphism of coalgebras j = jk The canonical isomorphism Vk ~=V` x Vm , for k = ` + m, induces the usual inclusion GLk GL`x GLm and the usual diagonal : Pk ! P` Pm . Therefore, it induces two homomorphisms __ GL ` GL ' ` GL ' D :F2 (Pk k) ! F2 (P` `) F2 (Pm m) , __ A A A P :(F2 Pk)GLk ! (F2 P`)GL` (F2 Pm )GLm . A A A Here and in what follows, means the tensor product over F2, except when other- wise specified. 3900 NGUY^E~N H. V. HU'NG Set M F2 D = F2 (P GL):= F2 (PkGLk) , A A k 0 A M (F2 P )GL:= (F2 Pk)GLk . A k 0 A It is easy to see that F2 (P GL) and (F2 P )GL are endowed with the structure* * of A __ __ A a cocommutative coalgebra by D and P, respectively. The coalgebra structure of (F2 P )GL was first given by Singer [20]. A L Proposition 3.1.j = jk : F2 (P GL) ! (F2 P )GL is a homomorphism of A A coalgebras. Proof.This follows immediately from the commutative diagram F2 Dk ____________-jk (F2 Pk)GLk A A | | |__ |__ | D | P | | |? |? (F2 D`) (F2 Dm ) ____________-j`(jmF2 P`)GL` (F2 Pm )GLm . A A A A L Remark. According to SingerL[20], T r* = T r*kis a homomorphism of coalgebras. One can see that '* = '*kis also a homomorphism of coalgebras. Then, so is j = T r* . '*. This is an alternative proof for Proposition 3.1. Now let M ` ' M ` ' * F2 P H*(BV ):= F2 P H*(BVk) ~= (F2 Pk)GLk , GL k 0 GLk k 0 A M M ` '* P (F2 H*(BV )):= P (F2 H*(BVk)) ~= F2 (PkGLk) . GL k 0 GLk k 0 A Passing to the dual, we obtain the homomorphism of algebras j* : F2 P H*(BV ) ! P (F2 H*(BV )) . GL GL As an application of j*, we give here a proof for Conjecture 1.4 with k = 3. Theorem 3.2. j3 : F2 (P3GL3) ! (F2 P3)GL3 is zero in positive dimensions. A A Proof.We equivalently show that j*3: F2 P H*(BV3) ! P (F2 H*(BV3)) GL3 GL3 is a trivial homomorphism in positive dimensions. F2 P H*(BV3) is described by Kameko [11], Alghamdi-Crabb-Hubbuck [3] GL3 and Boardman [4] as follows. F2 P H*(BV1) has a basis consisting of hr, r GL1 0, where hr is of dimension 2r - 1 and is sentrby the isomorphism T r1 to the Adams element, denoted also by hr, in Ext1,2A(F2, F2). According to [11], [3], SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER 3901 [4], F2 P H*(BV3) has a basis consisting of some products of the form hrhsht, GL3 where r, s, t are non-negative integers (but not all such appear), and some ele* *ments ci (i 0) with dim(ci) = 2i+3+ 2i+1+ 2i- 3. We will show in Lemma 3.3 that any decomposable element in P (F2 H*(BV3)) GL3 is zero. Then, since j* is a homomorphism of algebras, j*3sends any element of * *the form hrhsht to zero. On the other hand, by Hu'ng-Peterson [18], F2 D3 is concentrated in the di- A mensions 2s+2 - 4 (s 0) and 2r+2 + 2s+1 - 3 (r > s > 0). Obviously, these * * __ dimensions are different from dim(ci) for any i. Then j*3also sends ci to zero.* * |__| To complete the proof of the theorem, we need to show the following lemma. __ Lemma 3.3. Let D k= F2 Dk. Then the diagonal A __ __ __ __ __ __ D: D3 ! D1 D2 D2 D1 is zero in positive dimensions. Proof.Let us recall some informations on the Dickson algebra Dk. Dickson proved in [10] that Dk ~=F2[Qk-1, Qk-2, . .,.Q0], a polynomial algebra on k generators, with |Qs| = 2k - 2s. Note that Qs depends on k, and when necessary, will be denoted Qk,s. An inductive definition of Qk,sis given by Definition 3.4. Qk,s= Q2k-1,s-1+ vk . Qk-1,s, where, by convention, Qk,k= 1, Qk,s= 0 for s < 0 and Y vk = (~1x1 + . .+.~k-1xk-1 + xk) . ~i2F2 Dickson showed in [10] that k-1X s vk = Qk-1,sx2k. s=0 Now we_turn_back to the lemma. Since D is symmetric, we need only to show that the diagonal __ __ __ __ : D3 ! D2 D1 is zero in positive dimensions. For abbreviation, we denote x1, x2, x3 by x, y, z, respectively, Qi = Q3,i(x,* * y, z) for i = 0, 1, 2, qi = Q2,i(x, y) for i = 0, 1. As is well known, F2 D1 has the* * basis s s A {z2 -1| s 0}, and F2 D2 has the basis {q21-1| s 0}. By Hu'ng-Peterson [18], A F2 D3 has the basis A s-1 2r-2s-1 2s-1 {Q22 (s 0), Q2 Q1 Q0 (r > s > 0)} . For k 3, every monomial in Q0, . .,.Qk-1 which does not belong to the given basis is zero in F2 Dk. Note that the analogous statement is not true for k 4 A (see [18]). 3902 NGUY^E~N H. V. HU'NG Using the above inductive definitions of Qk,sand vk, we get Q0 = q20z + q0q1z2 + q0z4 , or (Q0) = q20 z + q0q1 z2 + q0 z4 . r-2s-1 2s-1 This implies easily that every term in (Q22 Q1 Q0) is divisible by q0, so it equals zero in F2 D2 as shown above. In other words, A __ 2r-2s-1 2s-1 (Q2 Q1 Q0) = 0 . Similarly, Q2 = q21+ v3 = q21+ q0z + q1z2 + z4 , or (Q2) = q21 1 + q0 z + q1 z2 + 1 z4 , s-1 2 2 4 2s-1 (Q22 ) = (q1 1 + q0 z + q1 z + 1 z ) . s-1 By the same argument as above, we need only to consider terms in (Q22 ) which are not divisible by q0. Such a term is some product of powers of q21 1, q1 * *z2, 1 z4. If it contains a positive power ofsz then this power is evensand it equal* *s zero in F2 D1. Otherwise, it should be q2(21-1) 1. Obviously, q2(21-1)equals zero * *in A __ s F2 D2. So, (Q22-1) = 0 for s > 0. A __ In summary, = 0 in positive dimensions. The lemma is proved. Then, so is Theorem 3.2. As T r3 is an isomorphism (see Boardman [4]), we have an immediate conse- quence. Corollary 3.5.'3 : Ext3,3+iA(F2, F2) ! (F2 D3)*iis zero in every positive stem A i. 4. The squaring operation: the existence Liulevicius was perhaps the first person who noted in [14] that there are squ* *ar- ing operations Sqi : Extk,tA(F2, F2) ! Extk+i,2tA(F2, F2), which share most of * *the properties with Sqi on cohomology of spaces. In particular, Sqi(ff) = 0 if i > * *k, Sqk(ff) = ff2 for ff 2 Extk,tA(F2, F2), and the Cartan formula holds for the Sq* *i's. However, Sq0 is not the identity. In fact, Sq0 : Extk,tA(F2, F2)! Extk,2tA(F2, F2), [b1| . .|.bk]7! [b21| . .|.b2k], in terms of the cobar resolution (see May [16]). Recall that H*(BVk) is a divided power algebra H*(BVk) = (a1, . .,.ak) generated by a1, . .,.ak, each of degree 1, where ai is dual to xi2 H1(BVk). He* *re and in what follows, the duality is taken with respect to the basis of H*(BVk) consisting of all monomials in x1, . .,.xk. SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER 3903 Let flt be the t-th divided power in H*(BVk) and for any a 2 H*(BVk) let a(t)= flt(a). So a(t)iis the element dual to xti. One has r)(2r) a(2iai = 0 , and r r m) a(t)i= a(2i1). .a.(2i if t = 2r1+ . .+.2rm, 0 r1 < . .<.rm . In [11] Kameko defined a GLk-homomorphism Sq0 : P H*(BVk) ! P H*(BVk), a(i1)1.a.(.ik)k7!a(2i1+1)1.a.(.2ik+1)k, where a(i1)1.a.(.ik)kis dual to xi11. .x.ikk. (See also [3].) Crabb and Hubbuck gave in [8] a definition of Sq0 that does not depend on the chosen basis of H*(BVk) as follows. The element a(Vk) = a1. .a.kis nothing but the image of the generator of k(Vk) under the (skew) symmetrization map k(Vk) ! Hk(BVk) = k(Vk) = ( Vk___._.-.Vkz____") k . k times Let F : H*(BVk) ! H*(BVk) be the Frobenius homomorphism defined by F (x) = x2 for any x, and let c : H*(BVk) ! H*(BVk) be the degree-halving dual homomor- phism. It is obviously a surjective ring homomorphism. Then Sq0 can be defined by Sq0(c(y)) = a(Vk)y . Since y 2 kerc if and only if a(Vk)y = 0, Sq0 is a monomorphism of GLk-modules. Further, it is easy to see that cSq2i+1*= 0, cSq2i*= Sqi*c. So Sq0 maps P H*(BV* *k) to itself. Using a result of Carlisle and Wood [6] on the boundedness conjecture, Crabb and Hubbuck also noted in [8] that for any d, there exists t0 such that Sq0 : P H2td+(2t-1)k(BVk) ! P H2t+1d+(2t+1-1)k(BVk) is an isomorphism for every t t0. Kameko's Sq0 is shown to commute with Sq0 on ExtkA(F2, F2) through the al- gebraic transfer T rk by Boardman [4] for k = 3 and by Minami [17] for general k. One denotes also by Sq0 the operation Sq0 : F2 P H*(BVk) ! F2 P H*(BVk) GLk GLk induced by Kameko's Sq0. It preserves the product. Further, for k = 3, it satis* *fies Sq0(hrhsht) = hr+1hs+1ht+1, Sq0(ci) = ci+1 (see Boardman [4]). Lemma 4.1. Sq2r+1*Sq0 = 0, Sq2r*Sq0 = Sq0Sqr*. Proof.We need a formal notation. Namely, for a 2 H1(BVk), set (a(t))[2]= a(2t). In general, (a(t))[2]6= fl2(flt(a)) = 2t-1ta(2t)(see Cartan [7]). 3904 NGUY^E~N H. V. HU'NG We start with a simple remark. Let x 2 H1(BVk); then Sqr(xs) = srxs+r. Let a denote the dual element of x. Then, by dualizing, ` ' Sqr*(a(t)) = t -rr a(t-r). As a consequence, Sq2r+1*(a(2t+1)) = 0 and ` ' ` ' i j[2] i j[2] Sq2r*(a(2t)) = 2t2-r2r a(2t-2r)= t -rr a(t-r) = Sqr*a(t) . Let ff = a(i1)1.a.(.ik)k. By the Cartan formula, we have Sqr*Sq0(ff)= Sqr*(a(2i1+1)1.a.(.2ik+1)k) X (2i +1) (2i +1) = Sqr1*(a1 1 ) . .S.qrk*(ak k ) . r1+...+rk=r The term corresponding to (r1, . .,.rk) equals 0 if at least one of r1, . .,.rk* * is odd. Hence Sq2r+1*Sq0(ff) = 0. Furthermore, X (2i +1) (2i +1) Sq2r*Sq0(ff)= Sq2r1*(a1 1 ) . .S.q2rk*(ak k ) r1+...+rk=r X n (2i ) (2i )o = Sq2r1*(a1 1) . .S.q2rk*(aka1k).a.k. r1+...+rk=r X n (i )o[2]n (i )o[2] = Sqr1*(a1 1) . . .Sqrk*(ak k) a1. .a.k r1+...+rk=r = Sq0Sqr*(ff) . The lemma is proved. Proposition 4.2.For every positive integer k, there exists a homomorphism Sq0 : P (F2 H*(BVk)) ! P (F2 H*(BVk)) GLk GLk that sends an element of degree n to an element of degree 2n + k and makes the following diagram commutative: j*k F2 P H*(BVk) ____________- P (F2 H*(BVk)) GLk GLk | | | 0 | 0 |Sq |Sq | | |? j*k |? F2 P H*(BVk) ____________- P (F2 H*(BVk)) . GLk GLk Proof.Since Sq0 : H*(BVk) ! H*(BVk) is a GLk-homomorphism, we can define Sq0D= 1 Sq0 and get a commutative diagram GLk SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER 3905 H*(BVk) ____________- F2 H*(BVk) GLk | | | 0 | 0 |Sq |SqD | | |? |? H*(BVk) ____________-F2 H*(BVk) , GLk where the horizontal arrows are the canonical projections. Next, we show that Sq0D sends the primitive part to itself. In other words, suppose ff 2 H*(BVk) satisfies Sqr*(1 ff) = 1 Sqr*ff = 0 GLk GLk for any r > 0; we want to show that Sqr*(Sq0(1 ff)) = 0 GLk for any r > 0. By definition of Sq0 and Lemma 4.1, we have for every r > 0 Sqr*(Sq0(1 ff))= 1 Sqr*Sq0(ff) GLk ( GLk 1 Sq0(Sqr=2*(ff)),r even, = GLk 0, r odd, ( r=2 Sq0Sq* (1 ff),r even, = GLk 0, r odd, = 0. Therefore, the above commutative diagram gives rise to a commutative diagram fj*k P H*(BVk) ____________- P (F2 H*(BVk)) GLk | | | 0 | 0 |Sq |SqD | | |? fj*k |? P H*(BVk) ____________- P (F2 H*(BVk)) . GLk By definition of jk, the homomorphism ej*kfactors through F2 P H*(BVk) and GLk the previous diagram induces the commutative diagram stated in the proposition, in which Sq0Dis re-denoted by Sq0 for short. The proposition is proved. As an application of Proposition 4.2, we give an alternative proof of Theorem* * 3.2. By Kameko [11], Alghamdi et al. [3] and Boardman [4], F2 P H*(BV3) has a GL3 basis consisting of some products of the form hrhshtand certain elements ci(i * * 0) with Sq0(ci) = ci+1for any i 0. By Lemma 3.3, j*3vanishes on any product hrhsht. Making use of Proposi- tion 4.2, one has j*3(ci) = j*3(Sq0)i(c0) = (Sq0)i(j*3(c0)). 3906 NGUY^E~N H. V. HU'NG One needs only to show that j*3(c0) = 0. Recall that dim(c0) = 8. The only elem* *ent of dimension 8 in D3 is Q22. Obviously, Q22= Sq4Q2. So P (F2 H*(BV3))8 = GL3 (F2 D3)*8= 0. Therefore, j*3(c0) = 0. Theorem 3.2 is proved. A 5. The squaring operation: an explicit formula for k 4 Let d(ik-1,...,i0)be the dual element of Qik-1k-1.Q.i.002 Dk, where the duali* *ty is taken with respect to the basis of Dk consisting of all monomials in the Dickson invariants Qk-1, . .,.Q0. It is well-known that P (F2 H*(BV1)) = Span {d(2s-1)| s 0} , GL1 P (F2 H*(BV2)) = Span {d(2s-1,0)| s 0} . GL2 By means of the definition of Sq0 one can easily show that Sq0(d(2s-1))= d(2s+1-1), Sq0(d(2s-1,0))= d(2s+1-1,0). In this section we compute Sq0 explicitly on P (F2 H*(BVk)) for k = 3 and 4. GLk Theorem 5.1 (Hu'ng-Peterson [18]).P D*3:= P (F2 H*(BV3)) has a basis con- GL3 sisting of d(2s-1,0,0), s 0 , d(2r-2s-1,2s-1,1), r > s > 0 . They are of dimensions 2s+2- 4 and 2r+2+ 2s+1- 3, respectively. Remark. It is easy to check that P D*3has at most one non-zero element of any dimension. Proposition 5.2.Sq0 : P D*3! P D*3is given by Sq0(d(2s-1,0,0)) = 0 , Sq0(d(2r-2s-1,2s-1,1)) = d(2r+1-2s+1-1,2s+1-1,1). Proof.For brevity, we denote x1, x2, x3 by x, y, z and a1, a2, a3 by a, b, c, r* *espec- tively. The first part of the proposition is an immediate consequence of dimensional * *in- formation. To prove the second part we start by recalling that, from Definition* * 3.4, we have Q3,0= Q0 = x4y2z1 + (symmetrized). Suppose m, n are non-negative integers. Let xffyfizflbe the biggest monomial in Qm2Qn1with respect to the lexicographic order on (ff, fi, fl). We claim that xff+4yfi+2zfl+1appears exactly one time in Qm2Qn1Q0, or equivalently (a) Qm2Qn1Q0 = xff+4yfi+2zfl+1+ (other terms). SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER 3907 Indeed, suppose to the contrary that0it0appears0more than once in Qm2Qn1Q0. That means there exists a monomial xffyfizflin Qm2Qn1, which is different from xffyf* *izfl, and a permutation oe on the set {4, 2, 1} such that 0+oe(4)fi0+oe(2)fl0+oe(1) xff+4yfi+2zfl+1= xff y z . Since ff + 4 = ff0+ oe(4) and 4 oe(4), this implies ff ff0. Combining this * *with the fact that (ff, fi, fl) is the biggest monomial in Qm2Qn1with respect to the lex* *icographic order on (ff, fi, fl), one gets ff = ff0and oe(4) = 4. Similarly, fi = fi0, fl * *= fl0 and oe is the identity permutation. This contradiction proves (a), or equivalently (b) d(m,n,1)= 1 a(ff+4)b(fi+2)c(fl+1)+ (other terms). Here and throughout the proof, means the tensor product over GL3. By definition of the squaring operation (c) Sq0(1 a(ff+4)b(fi+2)c(fl+1)) = 1 a(2ff+9)b(2fi+5)c(2fl+3). Now a direct computation using Definition 3.4 shows that Q2Q1Q0 = x12y4z + x10y6z + x10y5z2 + x10y4z3 + x9y6z2 + x9y5z3 + x8y6z3 + x8y5z4 + (symmetrized). Note that x9y5z3 and its symmetrized terms are the only terms of the form xoddyoddzodd in Q2Q1Q0. On the other hand, Q2m2Q2n1= x2ffy2fiz2fl+ (other terms), where x2ffy2fiz2flis the biggest monomial in this polynomial with respect to th* *e lex- icographic order on (2ff, 2fi, 2fl). Focusing on monomials of the form xoddyodd* *zodd and using the same argument as in the proof of (a), we have Q2m+12Q2n+11Q0 = x2ff+9y2fi+5z2fl+3+ (other terms). This is equivalent to (d) 1 a(2ff+9)b(2fi+5)c(2fl+3)= d(2m+1,2n+1,1)+ (other terms). Combining (b), (c) and (d), we get Sq0(d(m,n,1)) = d(2m+1,2n+1,1)+ (other terms). Applying this for (m, n, 1) = (2r - 2s - 1, 2s - 1, 1), we obtain Sq0(d(2r-2s-1,2s-1,1)) = d(2r+1-2s+1-1,2s+1-1,1)+ (other terms). In addition, Sq0 maps P D*3to itself (by Proposition 4.2) and P D*3consists of * *at most one non-zero element of any dimension. So the proposition is proved. Theorem 5.3 (Hu'ng-Peterson [18]).P D*4:= P (F2 H*(BV4)) has a basis con- GL4 sisting of d(2s-1,0,0,0), s 0 , d(2r-2s-1,2s-1,1,0), r > s > 0 , d(2t-2r-1,2r-2s-1,2s-1,2), t > r > s > 1 , d(2r-2s+1-2s-1,2s-1,2s-1,2), r > s + 1 > 2 . They are of dimensions 2s+3 - 8, 2r+3 + 2s+2 - 6, 2t+3+ 2r+2 + 2s+1 - 4 and 2r+3+ 2s+1- 4, respectively. Remark. P D*4, as well as P D*3, has at most one non-zero element of any dimens* *ion. 3908 NGUY^E~N H. V. HU'NG Proposition 5.4.Sq0 : P D*4! P D*4is given by Sq0(d(2s-1,0,0,0)) = Sq0(d(2r-2s-1,2s-1,1,0)) = 0 , Sq0(d(2t-2r-1,2r-2s-1,2s-1,2)) = d(2t+1-2r+1-1,2r+1-2s+1-1,2s+1-1,2), Sq0(d(2r-2s+1-2s-1,2s-1,2s-1,2)) = d(2r+1-2s+2-2s+1-1,2s+1-1,2s+1-1,2). Proof.We denote x1, x2, x3, x4 by x, y, z, t and a1, a2, a3, a4 by a, b, c, d, * *respectively, for brevity. The first part of the proposition is an immediate consequence of dimensional information. We claim that Q0 = x8y4z2t + (symmetrized). It can be checked by a routine computation using Definition 3.4. Here we give an alternative argument. Indeed, the Dickson algebra D4 ~=F2[Q3, Q2, Q1, Q0] has exactly one non-zero element of dimension 15. To check the equality we need only to show that the right hand si* *de is GL4-invariant. Recall that GL4 is generated by the symmetric group 4 and the transformation x 7! x + y, y 7! y, z 7! z, t 7! t. So, it suffices to check tha* *t the right hand side is invariant under this transformation. We leave it to the read* *er. Suppose m, n, p, q are non-negative integers with q > 0. Let xffyfizfltffibe* * the biggest monomial in Qm3Qn2Qp1Qq-10with respect to the lexicographic order on (ff, fi, fl, ffi). By the same argument as in the proof of Proposition 5.2 we h* *ave Qm3Qn2Qp1Qq0= xff+8yfi+4zfl+2tffi+1+ (other terms). In other words, (a) d(m,n,p,q)= 1 a(ff+8)b(fi+4)c(fl+2)d(ffi+1)+ (other terms). Here and throughout this proof, denotes the tensor product over GL4. By definition of the squaring operation (b) Sq0(1 a(ff+8)b(fi+4)c(fl+2)t(ffi+1)) = 1 a(2ff+17)b(2fi+9)c(2fl+5)d(2f* *fi+3). Using the same method that we used to compute Q0 above, we can show that X Q3Q2 = xs1ys2zs3ts4, s1+s2+s3+s4=20 si=0 or a power of 2 X Q1 = xs1ys2zs3ts4. s1+s2+s3+s4=14 si=0 or a power of 2 In particular, we have Q3Q2 = (x16y2zt + symmetrized) + (other terms), Q1 = (x8y4zt + symmetrized) + (other terms). Here, in both cases, any other term is of the form xevenyevenzeventeven. So Q3Q2Q1 = (x17y9z5t3 + symmetrized) + (other terms), where x17y9z5t3 and its symmetrized terms are the only terms of the form xoddyodd zodd todd in Q3Q2Q1. On the other hand, Q2m3Q2n2Q2p1Q2q-20= x2ffy2fiz2flt2ffi+ (other terms), where x2ffy2fiz2flt2ffiis the biggest monomial in the polynomial with respect t* *o the lexicographic order on (2ff, 2fi, 2fl, 2ffi). Again, we focus on monomials of t* *he form SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER 3909 xoddyoddzoddtodd and use the same argument as in the proof of Proposition 5.2 to get Q2m+13Q2n+12Q2p+11Q2q-20= x2ff+17y2fi+9z2fl+5t2ffi+3+ (other terms), or equivalently (c) 1 a(2ff+17)b(2fi+9)c(2fl+5)d(2ffi+3)= d(2m+1,2n+1,2p+1,2q-2)+ (other terms* *). Combining (a), (b) and (c), we get Sq0(d(m,n,p,q)) = d(2m+1,2n+1,2p+1,2q-2)+ (other terms). Apply this for (m, n, p, q) = (2t- 2r- 1, 2r- 2s- 1, 2s- 1, 2) and (2r- 2s+1- 2* *s- 1, 2s- 1, 2s- 1, 2). Combining the resulting formulas and the facts that Sq0 ma* *ps P D*4to itself (by Proposition 4.2) and that P D*4has at most one non-zero elem* *ent of any dimension, we obtain the last two formulas of the proposition. 6.Final remark __ Recall that D k:= F2 Dk. Let A __ __ M __ __ D : Dk ! D` Dm `+m=k `,m>0 be the diagonal defined at the beginning of Section 3. __ Conjecture 6.1. (Hu'ng-Peterson [19]). The diagonal D is zero in positive di- mensions for any k > 2. This conjecture is proved in Lemma 3.3 for k = 3 and has been proved for 2 < k < 10 in [19]. It implies that j*k(respectively, 'k) vanishes on the decom* *pos- able elements in F2 P H*(BVk) with respect to the product given by Singer [20] GLk and discussed in Section 3 (respectively, in ExtkA(F2, F2) with respect to the * *cup product) for 2 < k < 10. Note added in proof.Conjecture 6.1 has been established by F. Peterson and the author in the final version of [19]. References 1.J. F. 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Peterson, A-generators for the Dickson algebr* *a, Trans. Amer. Math. Soc., 347 (1995), 4687-4728. MR 96c:55022 19.Nguy^e~n H. V. Hu'ng and F. P. Peterson, Spherical classes and the Dickson a* *lgebra, Math. Proc. Camb. Phil. Soc. (to appear). 20.W. Singer, The transfer in homological algebra, Math. Zeit. 202 (1989), 493* *-523. MR 90i:55035 21.R. J. Wellington, The unstable Adams spectral sequence of free iterated loop* * spaces, Memoirs Amer. Math. Soc. 36 (1982), no. 258. MR 83c:55028 22.C. Wilkerson, Classifying spaces, Steenrod operations and algebraic closure,* * Topology 16 (1977), 227-237. MR 56:1307 Centre de Recerca Matem`atica, Institut d'Estudis Catalans, Apartat 50, E-081* *93 Bellaterra, Barcelona, Espa~na Current address: Department of Mathematics, University of Hanoi, 90 Nguy^e~n * *Tr~ai Street, Hanoi, Vietnam E-mail address: nhvhung@it-hu.ac.vn