ON TRIVIALITY OF DICKSON INVARIANTS IN THE HOMOLOGY OF THE STEENROD ALGEBRA

Transcription

1 ON TRIVILITY OF DICKSON INVRINTS IN THE HOMOLOGY OF THE STEENROD LGEBR NGUYÊ N H. V. HU. NG Dedicated to Professor Nguyê n Duy Tiê n on the occasion of his sixtieth birthday bstract. Let be the mod Steenrod algebra and D the Dicson algebra of variables. We study the Lannes-Zarati homomorphisms ϕ : Ext,+i (F, F ) (F D ) i, which correspond to an associated graded of the Hurewicz map H : π s(s0 ) = π (Q 0 S 0 ) H (Q 0 S 0 ). n algebraic version of the long-standing conjecture on spherical classes predicts that ϕ = 0 in positive stems, for >. That the conjecture is no longer valid for = 1 and is respectively an exposition of the existence of Hopf invariant one classes and Kervaire invariant one classes. This conjecture has been proved for = 3 in [9]. It has been shown that ϕ vanishes on decomposable elements for > in [14] and on the image of Singer s algebraic transfer for > in [9] and [1]. In this paper, we establish the conjecture for = 4. To this end, our main tools include (1) an explicit chain-level representation of ϕ and () a squaring operation Sq 0 on (F D ), which commutes with the classical Sq 0 on Ext (F, F ) through the Lannes-Zarati homomorphism. 1. Introduction and statement of results Let H : π s (S0 ) = π (Q 0 S 0 ) H (Q 0 S 0 ) be the Hurewicz homomorphism of the basepoint component Q 0 S 0 in the infinite loop space QS 0 = lim n Ω n S n. Here and throughout the paper, homology and cohomology are taen with coefficients in F, the field of two elements. The long-standing conjecture on spherical classes states as follows: Only the classes of Hopf invariant one and those of Kervaire invariant one are detected by the Hurewicz homomorphism. (See Curtis [6], Snaith and Tornehave [6] and Wellington [7] for a discussion.) n algebraic version of this problem, which we are interested in, goes as follows. Let P = F [x 1,..., x ] be the polynomial algebra on generators x 1,..., x, each of degree 1. Let the general linear group GL = GL(, F ) and the mod Steenrod algebra both act on P in the usual way. The Dicson algebra of variables, D, is the algebra of invariants D := F [x 1,..., x ] GL. 1 The research was supported in part by Johns Hopins University and the National Research Program, Grant N Mathematics Subject Classification. Primary 55P47, 55Q45, 55S10, 55T15. 3 Key words and phrases. Spherical classes, Loop spaces, dams spectral sequences, Steenrod algebra, lambda algebra, Invariant theory, Dicson algebra. 1

2 NGUYÊ N H. V. HU. NG Since the action of and that of GL on P commute with each other, D is an algebra over. In [17], Lannes and Zarati construct homomorphisms ϕ : Ext,+i (F, F ) (F D ) i, which correspond to an associated graded of the Hurewicz map. The proof of this assertion is unpublished, but it is setched by Lannes [16] and by Goerss [8]. The Hopf invariant one and the Kervaire invariant one classes are respectively represented by certain permanent cycles in Ext 1, (F, F ) and Ext, (F, F ), on which ϕ 1 and ϕ are non-zero (see dams [1], Browder [5], Lannes-Zarati [17]). Therefore, we are led to the following conjecture. Conjecture 1.1. ϕ = 0 in any positive stem i for >. The conjecture has been proved for = 3 in [9] and for = 4 in a range of stems in [14]. It has been shown that ϕ vanishes on decomposable elements for > in [14] and on the image of Singer s algebraic transfer T r : ((F P ) GL ) Ext (F, F ) for > in [9] and [1]. The following is the main result of the present paper. Theorem 1.. ϕ 4 = 0 in positive stems. n ingredient in our proof of this theorem is the squaring operation Sq 0 on (F D ), which is defined in our paper [9]. The ey step in the proof is to show the following theorem. Theorem 1.3. The squaring operation Sq 0 on (F D ) commutes with the classical squaring operation Sq 0 on Ext (F, F ) through the Lannes-Zarati homomor- phism ϕ, for any. pplying this theorem, we get a proof of Theorem 1. by combining the computation of Ext 4 (F, F ) by W. H. Lin [18] and that of F D 4 by the author and Peterson [13]. In order to prove Theorem 1.3, we need to exploit Singer s invariant-theoretic description of the lambda algebra [4]. ccording to Dicson [7], one has D = F [Q, 1,..., Q,0 ], where Q,i denotes the Dicson invariant of degree i. Singer sets Γ = D [Q 1,0 ], the localization of D given by inverting Q,0, and defines Γ to be a certain not too large submodule of Γ. He also equips Γ = Γ with a differential : Γ Γ 1 and a coproduct. Then, he shows that the differential coalgebra Γ is dual to the (opposite) lambda algebra of the six authors of [4]. Thus, H (Γ ) = T or (F, F ). (Originally, Singer uses the notation Γ + to denote Γ. However, by D+, + we always mean the submodules of D and respectively consisting of all elements of positive degrees, so Singer s notation Γ + would mae a confusion in this paper. Therefore, we prefer the notation Γ.) The following result plays a ey role in our proof of Theorem 1.3. Theorem 1.4. ([11]) The inclusion D Γ is a chain-level representation of the Lannes Zarati dual homomorphism ϕ : (F D ) i T or,+i (F, F ).

3 TRIVILITY OF DICKSON INVRINTS 3 By this theorem, Conjecture 1.1 is equivalent to our conjecture on the triviality of Dicson invariants in the homology of the Steenrod algebra: Conjecture 1.5. ([10]) Let D + denote the submodule of all positive degree elements in D. If q D +, then [q] = 0 in H (Γ ) = T or (F, F ) for >. Therefore, Theorem 1. can be restated as follows. Theorem 1.6. Every positive-degree Dicson invariant of four variables represents the 0 class in the homology, T or (F, F ), of the Steenrod algebra. lso, the theorem that ϕ vanishes on the image of the (Singer) algebraic transfer T r : ((F P ) GL ) Ext (F, F ) for > is restated as follows: Every positive-degree Dicson invariant of variables represents a class in the ernel of the algebraic transfer s dual T r : T or (F, F ) (F P ) GL for > (see [10], [1]). It should be noted that the algebraic transfer is computationally showed to be highly nontrivial by Singer [5] and by Boardman [3]. The paper contains four sections. Section is a recollection on modular invariant theory. Its goal is to mae the paper self-contained by recalling Singer s invarianttheoretic description of the lambda algebra and our chain-level representation of the Lannes-Zarati dual map. Section 3 and Section 4 are respectively devoted to the proofs of Theorem 1.3 and Theorem 1.. cnowledgment: The present paper was written during my visit to the Johns Hopins University, Maryland (US), in the Spring semester 001. I would lie to express my warmest thans to Mie Boardman, Jean-Pierre Meyer, Jac Morava and Steve Wilson for their hospitality and for providing me with ideal woring atmosphere and conditions.. Recollection on modular invariant theory The purpose of this section is to mae the paper self-contained. First, we summarize Singer s invariant-theoretic description of the lambda algebra. Let T be the Sylow -subgroup of GL consisting of all upper triangular - matrices with 1 on the main diagonal. The T -invariant ring, M = P T, is called the Mùi algebra. In [], Mùi shows that P T = F [V 1,..., V ], where V i = (c 1 x c i 1 x i 1 + x i ). c j F Then, the Dicson invariant Q,i can inductively be defined by Q,i = Q 1,i 1 + V Q 1,i, where, by convention, Q, = 1 and Q,i = 0 for i < 0. Let S() P be the multiplicative subset generated by all the non-zero linear forms in P. Let Φ be the localization: Φ = (P ) S(). Using the results of Dicson [7] and Mùi [], Singer notes in [4] that := (Φ ) T = F [V ±1 1,..., V ±1 ], Γ := (Φ ) GL = F [Q, 1,..., Q,1, Q ±1,0 ].

4 4 NGUYÊ N H. V. HU. NG Further, he sets so that Then, he obtains with deg v i = 1 for every i. Singer defines Γ v 1 = V 1, v = V /V 1 V 1 ( ), V = v 1 v 3 v 1 v ( ). = F [v 1 ±1,..., v±1 ], to be the submodule of Γ = D [Q 1,0 ] spanned by all monomials γ = Q i 1, 1 Qi0,0 with i 1,..., i 1 0, i 0 Z, and i 0 + deg γ 0. He also shows in [4] that the homomorphism : F [v 1 ±1,..., v±1 ] F [v 1 ±1,..., v±1 { 1 ], (v j1 1 vj v j 1 ) := 1 vj 1 1, if j = 1, 0, otherwise, maps Γ to Γ 1. Moreover, it is a differential on Γ = Γ. This module is bigraded by putting bideg(v j1 1 vj ) = (, + j i ). Let Λ be the (opposite) lambda algebra, in which the product in lambda symbols is written in the order opposite to that used in [4]. It is also bigraded by putting (as in [3, p. 90]) bideg(λ i ) = (1, 1 + i). Singer proves in [4] that Γ is a differential bigraded coalgebra, which is dual to the differential bigraded lambda algebra Λ via the isomorphisms.1. Γ Λ, v j1 1 vj (λ j1 λ j ). Here the duality is taen with respect to the basis of admissible monomials of Λ. s a consequence, one gets an isomorphism of bigraded coalgebras.. H (Γ ) = T or (F, F ). s stated in Theorem 1.4, we prove in [11] that the inclusion D Γ is a chain-level representation of the Lannes Zarati dual homomorphism ϕ : (F D ) i T or,+i(f, F ). In the remaining part of this section, we recall definition of the classical squaring operation on Ext (F, F ). Liulevicius was perhaps the first person who noted in [0] that there are squaring operations Sq i : Ext,t (F, F ) Ext +i,t (F, F ), which share most of the properties with Sq i on the cohomology of spaces. In particular, Sq i (α) = 0 if i >, Sq (α) = α for α Ext,t (F, F ), and the Cartan formula holds for the Sq i s. However, Sq 0 is not the identity. In fact, Sq 0 can be defined in terms of the lambda algebra as follows:.3. So, by dualizing, the following map.4. Sq 0 : Λ Λ, Sq 0 (λ i1 λ i ) = λ i1+1 λ i +1. Sqv 0 : Γ Γ{, Sqv 0(vj1 1 vj ) = v j v j 1, j 1,..., j odd, 0, otherwise

5 TRIVILITY OF DICKSON INVRINTS 5 is a chain-level representation of the dual squaring operation Sq 0 : T or (F, F ) T or (F, F ). 3. The squaring operations Given a module M over the dual of the Steenrod algebra, let P (M) denote the submodule of M spanned by all elements annihilated by any operations of positive degrees in. Let V be an F -vector space of dimension. s is well nown, H (BV ) = P. Then, it is easily seen that P (F H (BV )) and F P H (BV ) are respectively dual to F (P ) GL GL GL and (F P ) GL. In [9], we have defined a squaring operation Sq 0 : P (F GL H (BV )) P (F GL H (BV )), which is derived from Kameo s squaring operation Sq 0 on F GL P H (BV ) (see [15], [3]). We also prove in [9, Proposition 4.] that these two squaring operations commute with each other through the canonical homomorphism j : F GL P H (BV ) P (F GL H (BV )) induced by the identity map on V. The goal of this section is to show that the Sq 0 on P (F GL H (BV )) commutes with the classical squaring operation Sq 0 on Ext (F, F ) through the Lannes- Zarati map ϕ. Now we recall the definitions of the above mentioned squaring operations. s is well nown, H (BV ) is a divided power algebra H (BV ) = Γ(a 1,..., a ) generated by a 1,..., a, each of degree 1, where a i is dual to x i H 1 (BV ). Here, the duality is taen with respect to the basis of H (BV ) consisting of all monomials in x 1,..., x. In [15] Kameo defines a GL -homomorphism Sq 0 : H (BV ) H (BV ), a (i1) 1 a (i ) a (i1+1) 1 a (i +1), where a (i1) 1 a (i ) is dual to x i1 1 xi. He shows that Sq 0 maps P H (BV ) to itself. (See also [].) The induced homomorphism, which is also denoted by Sq 0, Sq 0 : F GL P H (BV ) F GL P H (BV ) is called Kameo s squaring operation. In [9], we consider the homomorphism Sq 0 D = 1 GL Sq 0 : F GL H (BV ) F GL H (BV )

6 6 NGUYÊ N H. V. HU. NG and show that it sends the primitive part P (F GL H (BV )) to itself. The resulting homomorphism will be redenoted by Sq 0 for short: Sq 0 : P (F GL H (BV )) P (F GL H (BV )). The following theorem, which is a re-statement of Theorem 1.3, is the main result of this section. Theorem 3.1. For an arbitrary positive integer, the squaring operation Sq 0 on P (F H (BV )) commutes with the classical Sq 0 on Ext (F, F ) through the GL Lannes-Zarati homomorphism ϕ. In other words, the following diagram commutes: Ext (F, F ) ϕ P (F GL H (BV )) Sq 0 Sq 0 Ext (F, F ) ϕ P (F GL H (BV )). We will prove this theorem by showing its dual version. consider the dual homomorphism of Kameo s one: To this end, let us Sqx 0 = Sq 0 : F [x 1,..., x ] F{ [x 1,..., x ], Sqx(x 0 j1 1 xj ) = x j x j 1, j 1,..., j odd, 0, otherwise. In order to explain the behavior of this homomorphism on modular invariants, we present a homomorphism: Sqv 0 : F [V 1,..., V ] F{ [V 1,..., V ], Sqv(v 0 j1 1 vj ) = v j v j 1, j 1,..., j odd, 0, otherwise. Obviously, this map coincides with the map in.4 on the intersection of their domains. The two homomorphisms Sqx 0 and Sq0 v depend on and, when necessary, will respectively be denoted by Sqx, 0 and Sq0 v,. Technically, the following proposition is the ey point in our proof of Theorem 3.1. Proposition 3.. Sq 0 x coincides with Sq0 v on F [V 1,..., V ], for any. This proposition will be shown by means of the following two lemmata, which directly come from the definitions of Sq 0 x and Sq 0 v given above. Lemma 3.3. (i) Sq 0 x, (ab ) = Sq 0 x, (a)b, for any a, b F [x 1,..., x ]. (ii) Sq 0 v, (B ) = Sq 0 v, ()B, for any, B F [V 1,..., V ]. Lemma 3.4. (i) Sq 0 x, (ax ) = Sq 0 x, 1 (a), for any a F [x 1,..., x 1 ]. (ii) Sq 0 v, (v ) = Sq 0 v, 1 (), for any F [V 1,..., V 1 ].

7 TRIVILITY OF DICKSON INVRINTS 7 We are now ready to prove Proposition 3.. Proof of Proposition 3.. The proof proceeds by induction on. For = 1, since x 1 = v 1, we get obviously Sqx,1 0 = Sq0 v,1. Let > 1 and suppose inductively that Sqx, 1 0 = Sq0 v, 1. We need to show Sqx, 0 = Sq0 i1 v,. Let V = V1 V i be an arbitrary monomial in M = F [V 1,..., V ]. We consider the following two cases. Case 1: i is even. Recall that V = Q 1,0 x + Q 1,1 x + + Q 1, 1x 1 (see Mùi [, ppendix]). Since Q 1,0,..., Q 1, 1, V 1,..., V 1 all do not depend on x, we have x j1 1 xj, j even where j is even in every monomial of the sum. Therefore, by definition of Sqx, 0, Sqx, 0 (V ) = Sqx, 0 (xj1 1 xj ) = 0. j even V = V i1 1 V i = On the other hand, from the expansions of V i s in terms of v j s, we get V = V i1 1 V i = v l1 1 vl, where l = i is even. Hence, by definition of Sqv, 0, Case : i = n + 1. We have V = V i1 1 V i 1 1 V i = V i1 1 V i 1 Since V i1 1 V i 1 1 Q 1,0x V n power of x odd, we get Sq 0 v, (V ) = Sq0 v, (vl1 1 vl ) = 0. 1 (Q 1,0x + Q 1,1 x + + Q 1, 1 x 1 )V n. is the only term in the above expansion of V with Sq 0 x,(v ) = Sq 0 x,(v i1 1 V i 1 1 Q 1,0x V n ). Note that V 1,..., V 1, Q 1,0 all do not depend on x. Combining Lemma 3.3, Lemma 3.4 and the inductive hypothesis, we obtain Sqx, 0 (V ) = Sq0 i1 x, (V1 V i 1 1 Q 1,0x )V n (by Lemma 3.3) = Sq 0 x, 1(V i1 1 V i 1 1 Q 1,0)V n (by Lemma 3.4) = Sqv, 1(V 0 i1 1 V i 1 1 Q 1,0)V n (by the inductive hypothesis) = Sq 0 v,(v i1 1 V i 1 1 Q 1,0v )V n (by Lemma 3.4) = Sq 0 v,(v i1 1 V i 1 1 Q 1,0v V n ) (by Lemma 3.3) = Sq 0 v,(v i1 1 V i 1 1 V n+1 ). The last equality comes from the expansions The proposition is completely proved. Now we come bac to Theorem 3.1. Q 1,0 v = V 1 V 1 v = V.

8 8 NGUYÊ N H. V. HU. NG Proof of Theorem 3.1. We will show the commutativity of the dual diagram: F D ϕ T or (F, F ) Sq 0 Sq 0 F D ϕ T or (F, F ). This will be obtained from a commutative diagram of appropriate chain-level representations of the homomorphisms in questions. Indeed, by definition of Sq 0 on (F D ) = P (F H (BV )), the restriction GL of Sqx 0 on D is a chain-level representation of Sq 0 : F D F D. On the other hand, from.4, the map Sqv 0 : Γ Γ{, Sqv 0(vj1 1 vj ) = v j v j 1, j 1,..., j odd, 0, otherwise. is a chain-level representation of Sq 0 : T or (F, F ) T or (F, F ). Now, since D M = F [V 1,..., V ], Proposition 3. implies the commutativity of the diagram: D Γ Sq 0 x Sq 0 v D Γ. By Theorem 1.4, the inclusion D Γ is a chain-level representation of the Lannes-Zarati s dual map ϕ. Therefore, the last commutative diagram shows the commutativity of the previous one. Theorem 3.1 is proved. 4. The triviality of ϕ 4 The goal of this section is to prove Theorem 1., the main result of this paper. To this end, we need to recall the computation of Ext 4 (F, F ) by W. H. Lin and that of F D 4 by F. P. Peterson and the author. Theorem 4.1. (W. H. Lin [18], see also [19, Theorem.]). The following classes form an F -basis for the vector space of indecomposable elements in Ext 4 (F, F ): (1) d i = [(Sq 0 ) i (λ 6 λ λ 3 + λ 4 λ 3 + λ λ 4 λ 5 λ 3 + λ 1 λ 5 λ 1 λ 7 )] Ext 4,i+4 + i+1 () e i = [(Sq 0 ) i (λ 8 λ λ 4(λ 5 λ 3 + λ 7 λ 3 ) + λ (λ 3 λ 5 λ 7 + λ 1 λ 11 λ 3 ))] Ext 4,i+4 + i+ + i

9 TRIVILITY OF DICKSON INVRINTS 9 (3) f i = [(Sq 0 ) i (λ 4 λ 0 λ 7 + λ 3(λ 9 λ 3 + λ 3λ 5 λ 7 ) + λ λ 7 )] Ext 4,i+4 + i+ + i+1 (4) g i+1 = [(Sq 0 ) i (λ 6 λ 0 λ 7 + λ 5(λ 9 λ 3 + λ 3λ 5 λ 7 ) + λ 3 (λ 5 λ 9 λ 3 + λ 11 λ 3 ))] Ext 4,i+4 + i+3 (5) p i = [(Sq 0 ) i (λ 14 λ 5 λ 7 + λ 10λ 9 λ 7 + λ 6λ 9 λ 11 λ 7 )] Ext 4,i+5 + i+ + i (6) D 3 (i) = [(Sq 0 ) i (λ λ 1 λ 7 λ 31 + λ 16 λ 7 λ 31 + λ 14 λ 9 λ 7 λ 31 + λ 1 λ 11 λ 7 λ 31 ] Ext 4,i+6 + i (7) p i = [(Sq0 ) i (λ 0 λ 39 λ 15 + λ 0 λ 15 λ 3 λ 31 )] Ext 4,i+6 + i+3 + i, i 0. To simplify notation, we will denote Q a 4,3 Qb 4, Qc 4,1 Qd 4,0 by Q(a, b, c, d) in the following theorem. Theorem 4.. (Hu. ng-peterson [13]). The following elements form an F -basis for the vector space F D 4 : (1) Q( s 1, 0, 0, 0), s 0, () Q( r s 1, s 1, 1, 0), r > s > 0, (3) Q( t r 1, r s 1, s 1, ), t > r > s > 1, (4) Q( r s+1 s 1, s 1, s 1, ), r > s + 1 >. They are of degrees s+3 8, r+3 + s+ 6, t+3 + r+ + s+1 4 and r+3 + s+1 4, respectively. Now we come bac to prove Theorem 1.. Proof of Theorem 1.. In [14], F. Peterson and the author have proved that ϕ vanishes on any decomposable elements for > by showing that ϕ = ϕ is a homomorphism of algebras and, more importantly, that the product of the algebra (F D ) is trivial, except for the case (F D 1 ) (F D 1 ) (F D ). Therefore, we need only to show ϕ 4 vanishing on any indecomposable elements. Let a 0 denote one of the seven generators d 0, e 0, f 0, g 1, p 0, D 3 (0), p 0, each of which is the element of lowest stem in its own family. Furthermore, set a i = (Sq 0 ) i (a 0 ), for i 0. From Theorem 3.1, we have ϕ 4 (a i ) = ϕ 4 (Sq 0 ) i (a 0 ) = (Sq 0 ) i ϕ 4 (a 0 ). So, in order to prove that ϕ 4 (a i ) = 0 for any i, it suffices to show ϕ 4 (a 0 ) = 0. We will do this by checing that the stem of a 0 is different from degrees of all the generators of F D 4 given in Theorem 4.. Now let us chec it case by case.

10 10 NGUYÊ N H. V. HU. NG Case 1: For d 0 of stem = 14, s+3 = =, no solution; r+3 + s+ = = , r = 1, s = 0, it does not satisfy s > 0; t+3 + r+ + s+1 = = 16 +, no solution; r+3 + s+1 = = 16 +, r = 1, s = 0 it does not satisfy r > s + 1 >. Case : For e 0 of stem = 17, s+3 = = 5, no solution; r+3 + s+ = = , no solution; t+3 + r+ + s+1 = = , no solution; r+3 + s+1 = = , no solution. Case 3: For f 0 of stem = 18, s+3 = = 6, no solution; r+3 + s+ = = , r = 1, s = 1 it does not satisfy r > s; t+3 + r+ + s+1 = = , t = 1, r = s = 0 it does not satisfy r > s > 1; r+3 + s+1 = = , no solution. Case 4: For g 1 of stem = 0, s+3 = = 8, no solution; r+3 + s+ = = , no solution; t+3 + r+ + s+1 = = , no solution; r+3 + s+1 = = , r = 1, s =, it does not satisfy r > s + 1 >. Case 5: For p 0 of stem = 33, s+3 = = 41, no solution; r+3 + s+ = = , no solution; t+3 + r+ + s+1 = = , no solution; r+3 + s+1 = = , no solution. Case 6: For D 3 (0) of stem = 61, s+3 = = 69, no solution; r+3 + s+ = = , no solution; t+3 + r+ + s+1 = = , no solution; r+3 + s+1 = = , no solution. Case 7: For p 0 of stem = 69, s+3 = = 77, no solution; r+3 + s+ = = , no solution; t+3 + r+ + s+1 = = , no solution; r+3 + s+1 = = , no solution. Therefore, ϕ 4 vanishes on any indecomposable elements. In summary, ϕ 4 = 0 in positive stems. Theorem 1. is completely proved. References [1] J. F. dams, On the non-existence of elements of Hopf invariant one, nn. Math. 7 (1960), [] M.. lghamdi, M. C. Crabb and J. R. Hubbuc, Representations of the homology of BV and the Steenrod algebra I, dams Memorial Symposium on lgebraic Topology, N. Ray and G. Waler (ed.), London Math. Soc. Lect. Note Series 176 (199),

11 TRIVILITY OF DICKSON INVRINTS 11 [3] J. M. Boardman, Modular representations on the homology of powers of real projective space, lgebraic Topology: Oaxtepec 1991, M. C. Tangora (ed.), Contemp. Math. 146 (1993), [4]. 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 dams spectral sequence, Topology 5 (1966), [5] W. Browder, The Kervaire invariant of a framed manifold and its generalization, nn. Math. 90 (1969), [6] E. B. Curtis, The Dyer Lashof algebra and the lambda algebra, Illinois Jour. Math. 18 (1975), [7] L. E. Dicson, fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. mer. Math. Soc. 1 (1911), [8] P. G. Goerss, Unstable projectives and stable Ext: with applications, Proc. London Math. Soc. 53 (1986), [9] N. H. V. Hu. ng, Spherical classes and the algebraic transfer, Trans. mer. Math. Soc. 349 (1997), [10] N. H. V. Hu. ng, The wea conjecture on spherical classes, Math. Zeit. 31 (1999), [11] N. H. V. Hu. ng, Spherical classes and the lambda algebra, Trans. mer. Math. Soc. 353 (001), [1] N. H. V. Hu. ng and T. N. Nam, The hit problem for the Dicson algebra, Trans. mer. Math. Soc. 353 (001), [13] N. H. V. Hu. ng and F. P. Peterson, generators for the Dicson algebra, Trans. mer. Math. Soc. 347 (1995), [14] N. H. V. Hu. ng and F. P. Peterson, Spherical classes and the Dicson algebra, Math. Proc. Camb. Phil. Soc. 14 (1998), [15] M. Kameo, Products of projective spaces as Steenrod modules, Thesis, Johns Hopins University [16] J. Lannes, Sur le n-dual du n-ème spectre de Brown-Gitler, Math. Zeit. 199 (1988), 9 4. [17] J. Lannes and S. Zarati, Sur les foncteurs dérivés de la déstabilisation, Math. Zeit. 194 (1987), [18] W. H. Lin, Some differentials in dams spectral sequence for spheres, Trans. mer. Math. Soc., to appear. [19] W. H. Lin and M. Mahowald, The dams spectral sequence for Minami s theorem, Contemp. Math. 0 (1998), [0]. Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Mem. mer. Math. Soc. 4 (196). [1] I. Madsen, On the action of the Dyer Lashof algebra in H (G), Pacific Jour. Math. 60 (1975), [] H. Mùi, Modular invariant theory and cohomology algebras of symmetric groups, Jour. Fac. Sci. Univ. Toyo, (1975), [3] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, cademic Press, London [4] W. M. Singer, Invariant theory and the lambda algebra, Trans. mer. Math. Soc. 80 (1983), [5] W. M. Singer, The transfer in homological algebra, Math. Zeit. 0 (1989), [6] V. Snaith and J. Tornehave, On π S (BO) and the rf invariant of framed manifolds, mer. Math. Soc. Contemporary Math. 1 (198), [7] R. J. Wellington, The unstable dams spectral sequence of free iterated loop spaces, Memoirs mer. Math. Soc. 58 (198). Department of Mathematics, Johns Hopins University 3400 N. Charles Street, Baltimore MD address: nhvhung@math.jhu.edu Permanent address: Department of Mathematics, Vietnam National University, Hanoi 334 Nguyên Trai Street, Hanoi, Vietnam address: nhvhung@vnu.edu.vn