THE WEAK CONJECTURE ON SPHERICAL CLASSES

Transcription

1 THE WEK CONJECTURE ON SPHERICL CLSSES NGUYÊ N H. V. HU. NG bstract. Let be the mod 2 Steenrod algebra. We construct a chain-level representation of the dual of Singer s algebraic transfer, T r : T or (F 2, F 2 ) F 2 F 2 [x 1,..., x ], which maps Singer s invariant-theoretic model of the dual of the Lambda algebra, Γ, to F 2[x ±1 1,..., x±1 ] and is the inclusion of the Dicson algebra, D Γ, into F 2[x 1,..., x ]. This chain-level representation allows us to confirm the wea conjecture on spherical classes (see [9]), assuming the truth of (1) either the conjecture that the Dicson invariants of at least = 3 variables are homologically zero in T or (F 2, F 2 ), (2) or a conjecture on -decomposability of the Dicson algebra in Γ. We prove the conjecture in item (1) for = 3 and also show a wea form of the conjecture in item (2). 1. Introduction and statement of results This paper continues our study of spherical classes that started in [9]. To mae the paper self-contained, we first recall certain bacground of the classical conjecture on spherical classes, which has been given in the introduction of our paper [9]. We are interested in the following conjecture on spherical classes in Q 0 S 0, i.e. elements belonging to the image of the Hurewicz homomorphism H : π s (S0 ) = π (Q 0 S 0 ) H (Q 0 S 0 ). Here and throughout the paper, the coefficient ring for homology and cohomology is always F 2, the field of two elements. Conjecture 1.1. (conjecture on spherical classes). There are no spherical classes in Q 0 S 0, except the Hopf invariant one and the Kervaire invariant one elements. Some topologists believe that the conjecture is due to I. Madsen, while some others say it is due to E. Curtis. (See Curtis [6] and Wellington [24] for a discussion.) Let E be an elementary abelian 2-group of ran. It is also viewed as a - dimensional vector space over F 2. So, the general linear group GL = GL(, F 2 ) acts on E and therefore on H (BE ) in the usual way. Let D be the Dicson algebra of variables, i.e. the algebra of invariants D := H (BE ) GL = F2 [x 1,..., x ] GL, where P = F 2 [x 1,..., x ] is the polynomial algebra on generators x 1,..., x, each of dimension 1. s the action of the (mod 2) Steenrod algebra,, and that of GL on P commute with each other, D is an algebra over. 1 This wor was supported in part by the National Research Program, N Mathematics Subject Classification. Primary 55P47, 55Q45, 55S10, 55T15. 3 Key words and phrases. Spherical classes, Loop spaces, dams spectral sequences, Steenrod algebra, Invariant theory, Dicson algebra, lgebraic transfer. 1

2 2 NGUYÊ N H. V. HU. NG One way to attac Conjecture 1.1 is to study the Lannes Zarati homomorphism ϕ : Ext,+i (F 2, F 2 ) (F 2 D ) i, which is compatible with the Hurewicz homomorphism (see [13], [14, p. 46]). The domain of ϕ is the E 2 -term of the dams spectral sequence converging to π s(s0 ) = π (Q 0 S 0 ). ccording to Madsen s theorem [16], which asserts that D is dual to the coalgebra of Dyer Lashof operations of length, the range of ϕ is a submodule of H (Q 0 S 0 ). By compatibility of ϕ and the Hurewicz homomorphism we mean ϕ is a lifting of the latter from the E -level to the E 2 -level. The Hopf invariant one and the Kervaire invariant one elements are respectively represented by certain permanent cycles in Ext 1, (F 2, F 2 ) and Ext 2, (F 2, F 2 ), on which ϕ 1 and ϕ 2 are non-zero (see dams [1], Browder [5], Lannes Zarati [14]). Therefore, Conjecture 1.1 is a consequence of the following one. Conjecture 1.2. ϕ = 0 in any positive stem i for > 2. It is well nown that the Ext group has intensively been studied, but remains very mysterious. In order to avoid the shortage of our nowledge of the Ext group, we want to restrict ϕ to a certain subgroup of the Ext group which (1) is large enough and worthwhile to pursue and (2) could be handled more easily than the Ext group itself. To this end, we combine the above data with Singer s algebraic transfer. Singer defined in [22] the algebraic transfer T r : F 2 GL P H i (BE ) Ext,+i (F 2, F 2 ), where P H (BE ) denotes the submodule consisting of all -annihilated elements in H (BE ). It is shown to be an isomorphism for 2 by Singer [22] and for = 3 by Boardman [3]. Singer also proved in [22] that it is not an isomorphism for = 5, and conjectured that T r is a monomorphism for any. Restricting ϕ to the image of T r, we stated in [9] the following conjecture. Conjecture 1.3. (wea conjecture on spherical classes). ϕ T r : F 2 P H (BE ) P (F 2 H (BE )) := (F 2 D ) GL GL is zero in positive dimensions for > 2. In other words, there are no spherical classes in Q 0 S 0, which can be detected by the algebraic transfer, except the Hopf invariant one and the Kervaire invariant one elements. In [9], we have proved that the inclusion of D into P is a chain-level representation of T r ϕ : F 2 D (F 2 P ) GL. So, we get the following result. Theorem 1.4. The wea conjecture on spherical classes is equivalent to the conjecture that the homomorphism j : F 2 (P GL ) (F 2 P ) GL induced by the identity map on P is zero in positive dimensions for > 2. Let D +, + be respectively the submodules of D and consisting of all elements of positive dimensions. Then the above conjecture on j can equivalently be stated as follows.

3 THE WEK CONJECTURE ON SPHERICL CLSSES 3 Conjecture 1.5. If > 2, then D + + P. In [9], we have got a proof of this conjecture for = 3. To introduce a new approach, we need to summarize Singer s invariant-theoretic description of the lambda algebra [21]. ccording to Dicson [7], one has D = F2 [Q, 1,..., Q,0 ], where Q,i denotes the Dicson invariant of dimension 2 2 i. Singer set Γ = D [Q 1,0 ], the localization of D given by inverting Q,0, and defined Γ to be a certain not too large submodule of Γ. He also equiped Γ = Γ with a differential : Γ Γ 1 and a coproduct. Then, he showed that the differential coalgebra Γ is dual to the lambda algebra of the six authors of [4]. Thus, H (Γ ) = T or (F 2, F 2 ). (Originally, Singer used the notation Γ + to denote Γ. However, by D +, + we always mean the submodules of D and respectively consisting of all elements of positive dimensions, so Singer s notation Γ + would mae a confusion in this paper. Therefore, we prefer the notation Γ to Γ+.) One of the main results of this paper is to construct a homomorphism T : Γ F 2 [x ±1 1,..., x±1] with the following properties. Theorem 3.2 The homomorphism T : Γ F 2[x ±1 1,..., x±1 ] maps the submodule of all cycles in Γ to P and is a chain-level representation of T r : T or (F 2, F 2 ) F 2 P. Moreover, its restriction to D Γ is the inclusion of D into P. Note that every q D + is a cycle in the differential module Γ. By means of this theorem, the wea conjecture on spherical classes is an immediate consequence of the following conjecture. Conjecture 1.6. If q D +, then [q] = 0 in T or (F 2, F 2 ) for > 2. This is a corollary of the following conjecture. (See Proposition 4.2 for a proof.) Conjecture 4.1 Let Ker be the submodule of all cycles in Γ. Then, for > 2, D + + Ker. We get the following result, which is weaer than the above conjecture. Theorem 4.3 For > 2, D + + Γ. We also have the following theorem. Theorem 4.8 Conjecture 1.6 is true for = 3. Conjecture 1.5 is related to the difficult problem of determination of F 2 P. This problem has first been studied by F. Peterson [18], R. Wood [26], W. Singer [22], S. Priddy [19]... who show its relationships to several classical problems in Homotopy Theory. F 2 P has explicitly been computed for 3. The cases = 1 and

4 4 NGUYÊ N H. V. HU. NG 2 are not difficult, while the case = 3 is very complicated and was solved by M. Kameo [12]. There is also another approach, the qualitative one, to the problem. By this we mean giving conditions on elements of P to show that they go to zero in F 2 P, i. e. belong to + P. Peterson s conjecture, which was established by Wood [26], claims that F 2 P = 0 in dimension d such that α(d + ) >. Here α(n) denotes the number of ones in the dyadic expansion of n. Recently, W. Singer, K. Mons, J. Silverman... have refined the method of R. Wood to show that many more monomials in P are in + P. (See Silverman [20] and its references.) Conjecture 1.5 presents a large family, whose elements are predicted to be in + P. The paper is organized as follows. Section 2 is a preliminary on invariant theory and Singer s algebraic transfer. Sections 3 and 4 are respectively devoted to prove Theorems 3.2, 4.3 and 4.8. Finally, in Section 5, we propose a way to prove the classical conjecture on spherical classes. The author would lie to than the referee for helpful suggestions, which have improved the exposition of the paper. 2. Preliminary This section begins with a few words on invariant theory and ends with a brief setch of Singer s definition of the algebraic transfer. Let T be the Sylow 2-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 [17], Mùi showed that P T = F 2 [V 1,..., V ], where V i = (λ 1 x λ i 1 x i 1 + x i ). λ j F 2 Then, the Dicson invariant Q,i can inductively be defined by Q,i = Q 2 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 [17], Singer noted in [21] that Further, he set so that Then, he obtained := (Φ ) T = F 2 [V ±1 1,..., V ±1 ], Γ := (Φ ) GL = F 2 [Q, 1,..., Q,1, Q ±1,0 ]. v 1 = V 1, v = V /V 1 V 1 ( 2), V = v v v 1 v ( 2). = F 2 [v 1 ±1,..., v±1 ], with dim v i = 1 for every i. Singer defined Γ 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 + dim γ 0.

5 THE WEK CONJECTURE ON SPHERICL CLSSES 5 In the remaining part of this section, we briefly setch Singer s definition of T r. Let P 1 = F 2 [x] with x = 1. Let ˆP F 2 [x, x 1 ] be the submodule spanned by all powers x i with i 1. The canonical -action on P 1 is extended to an -action on F 2 [x, x 1 ] (see dams [2], Wilerson [25]). ˆP is an -submodule of F 2 [x, x 1 ]. We have a short-exact sequence of -modules P 1 ι ˆP π F 2 0, where ι is the inclusion and π is given by π(x i ) = 0 if i 1 and π(x 1 ) = 1. Let e 1 be the corresponding element in Ext 1 (F 2, P 1 ). Then, Singer defined 2.2. e := e 1 e 1 Ext (F 2, P ) ( times). Now, T r : T or (F 2, F 2 ) T or0 (F 2, P ) F 2 P is defined by 2.3. T r (z) := e z. Note that, since π raises internal degree by one, T r lowers it by. We need to relate T r with connecting homomorphisms. Let N, P, Q, R be (left) -modules. From MacLane [15, p. 229], one has f g = (f R) (N g) for f Ext (N, P ), g Ext (Q, R). Hence 2.4. e = (e 1 P 1 ) (e 1 P 1 ) e 1. Cap and Yoneda products are related by the formula (h f) z = h (f z) for z T or (M, N), f Ext (N, P ), h Ext (P, Q). Suppose f Ext 1 (N 3, N 1 ) is represented by the short-exact sequence of left -modules 0 N 1 N 2 N 3 0. Let (f) : T ors (M, N 3) T ors 1 (M, N 1) be the connecting homomorphism associated with this short-exact sequence, for any right -module M. Then one verifies easily for any z T or s (M, N 3 ). Consequently, we get (f)(z) = f z 2.5. T r = (e 1 P 1 ) (e 1 P 1 ) e 1. (See Singer [22, p. 498].) Now we prepare to construct a chain-level representation of T r. For every left -module N, Singer defined in [21] a chain complex Γ N: It is given in homological dimension by (Γ N) = Γ N as an F 2-vector space and its differential : (Γ N) (Γ N) 1 is defined by (v a1 1 va y) := (va1 1 va 1 1 Sqa +1 (y)), for y N. Then he proved that H (Γ N) = T or (F 2, N). This isomorphism is natural in N. Further, if (f) : 0 N 1 N 2 N 3 0 is any short-exact sequence of left -modules, then 0 Γ N 1 Γ N 2 Γ N 3 0 is a short-exact sequence of chain complexes, from which a chain-level representation of the connecting homomorphism (f) : T or (F 2, N 3 ) T or 1 (F 2, N 1 ) can be computed.

6 6 NGUYÊ N H. V. HU. NG 3. chain-level representation of the dual of the algebraic transfer Following Singer [21], the homomorphism : F 2 [v 1 ±1,..., v±1 ] F 2[v 1 ±1,..., v±1 1 ] { (v a1 1 va v a ) := 1 1 va 1 1 if a = 1, 0 otherwise maps Γ to Γ 1. Moreover, it is a differential on Γ = Γ with H (Γ ) = T or (F 2, F 2 ). This is an isomorphism of bigraded modules. Here the bidegree of Γ is given by bideg(v a1 1 va ) := (, + a i ). Note that, for every q D, its expansion q = (a 1,...,a ) v a1 1 va always has a 0 in any term of the sum. Therefore, (q) = 0. This means that every Dicson invariant is a cycle in the complex Γ, whose homology is T or (F 2, F 2 ). Now we construct a homomorphism, denoted T, as follows. Definition 3.1. The homomorphism T : F 2 [v 1 ±1,..., v±1 ] F 2[x ±1 1,..., x±1 ] is defined by ) T (v a1 1 va ) := Sqa1+1( x 1 1 Sq a 1+1 (x 1 1 Sqa +1 (x 1 )), where a 1,..., a are arbitrary integers. Here, we mean Sq i = 0 for i < 0. Let Ker be the submodule of all cycles in Γ. Then, the goal of this section is to prove the following theorem. Theorem 3.2. The homomorphism T satisfies the following two properties: (i) T (Ker ) P. Furthermore, T Γ : Γ F 2[x ±1 1,..., x±1 ] is a chainlevel representation of T r : H (Γ ) = T or (F 2, F 2 ) F 2 P. (ii) T D is the inclusion of D into P. The first part of Theorem 3.2 is shown by the following lemma. Lemma 3.3. T maps Ker to P and is a chain-level representation of T r. Proof. In order to construct a chain-level representation for T r, following 2.5, we first construct a chain-level representation for (e 1 ) : T or (F 2, F 2 ) T or 1 (F 2, P 1 ), the connecting homomorphism associated with the short-exact sequence 2.1. Let us consider the induced short-exact sequence 0 Γ P 1 ι Γ ˆP π Γ F 2 0. lifting of a cycle z Γ = Γ F 2 over the chain map π is given by the chain z x 1 Γ ˆP, where we are writing P1 = F 2 [x ], ˆP = Span{x i i 1}. The

7 THE WEK CONJECTURE ON SPHERICL CLSSES 7 boundary (z x 1 ) pulls bac z x 1 under ι to a cycle in Γ P 1, which represents (e 1 )(z) in T or 1 (F 2, P 1 ). If z = v a1 1 va, then by (4.1) of [21], (z x 1 ) = ( v a1 1 va This means that the map given by x 1 ) = v a1 1 va 1 1 Sqa +1 (x 1 ). v a1 1 va va1 1 va 1 1 Sqa +1 (x 1 ) is a chain-level representation of (e 1 ). We similarly construct a chain-level representation for the homomorphism (e 1 P 1 ) : T or 1 (F 2, P 1 ) T or 2 (F 2, P 2 ), which is the connecting homomorphism associated with the short-exact sequence of chain complexes 0 Γ ι P P 1 2 Γ ( ˆP P 1 ) π P1 Γ P 1 0. Here we are writing P 1 = F 2 [x ], P 2 = F 2 [x 1, x ] and ˆP = Span{x i 1 i 1}. By the argument similar to the one given above, the map v a1 1 va 1 1 xi (va1 1 va 1 1 x 1 1 xi ) = va1 1 va 2 2 Sqa 1+1 (x 1 1 xi ) is a chain-level representation of (e 1 P 1 ). So the ( composite map ) v a1 1 va va1 1 va 2 2 Sqa 1+1 x 1 1 Sqa +1 (x 1 ) sends a cycle in Γ P 1 to a cycle in Γ P 2 and it is a chain-level representation of the composite homomorphism (e 1 P 1 ) (e 1 ). By repeating the above argument, it is easy to see that the map T given by Definition 3.1 is a chain-level representation of T r = (e 1 P 1 ) (e 1 P 1 ) (e 1 ). In particular, we have T (Ker ) Γ 0 P = P. For the convenience of the latter use, T is thought of as a morphism from F 2 [v 1 ±1,..., v±1 ] to F 2[x ±1 1,..., x±1 ]. The lemma follows. Remar 3.4. Since the chain-level representation T has been extended to a homomorphism, whose domain is F 2 [v 1 ±1,..., v±1 ], the computation of T (V i ) maes sense as one can see in Lemmas 3.7 and 3.8 below. The second part of Theorem 3.2 is proved by a number of lemmata. Let M be a graded -module, which is concentrated in non-negative dimensions. We are concerned with the Steenrod homomorphism d P : M F 2 [x ±1 ] M given by u d P (u) = d P x (u) = x u i Sq i (u), where u M, x is of dimension 1 and F 2 [x ±1 ] supports the canonical -action as discussed in Section 2. ctually, Steenrod defined his Sq i by using d P, which is constructed by means of cohomology of the weath products (see [23, Ch. VII]). Suppose additionally that M is an -algebra. Then, by the Cartan formula, d P is a homomorphism of algebras. i=0

8 8 NGUYÊ N H. V. HU. NG Lemma 3.5. Let M be a unstable -algebra. Suppose α, β M, and a α, b β. Then Sq a+b+1 (x 1 αβ) = Sq a+1 (x 1 α)sq b+1 (x 1 β). Proof. Recall that Sq i (x 1 ) = x i 1 for every integer i. From the unstability of M and a α, one gets α Sq a+1 (x 1 α) = x a i Sq i (α) = x a α d P (α). i=0 s d P is an algebra homomorphism, the lemma follows. pplying repeatedly this lemma to M = P, we obtain Lemma 3.6. Suppose v = v a1 1 va and v = v b1 1 vb a i a i a 0, b i b i b 0, for 1 i. Then T (v v ) = T (v) T (v ). Furthermore, the both sides are elements of P. satisfy the conditions: Proof. Set τ i (v) = Sq ai+1 (x 1 i Sq a+1 (x 1 )). It is easy to see that τ (vv ) = τ (v)τ (v ) in P. Suppose inductively that τ i+1 (vv ) = τ i+1 (v)τ i+1 (v ) in P. Then, applying Lemma 3.5 with a = a i, b = b i, α = τ i+1 (v), β = τ i+1 (v ), we get τ i (vv ) = τ i (v)τ i (v ) in P. Thus, the lemma follows as T = τ 1. Lemma 3.7. The restriction of T to the Mùi algebra M = F 2 [V 1,..., V ] is a homomorphism of algebras. Proof. Recall that V i = v1 2i 2 v2 2i 3 v i 1 v i. So, one easily verifies that every element v M is a sum of monomials v a1 1 va, which satisfy the conditions of Lemma 3.6. The lemma follows from the previous one. Lemma 3.8. T (V i ) = V i (1 i ). Therefore, T M is the inclusion of M into P. Proof. This is proved by induction on. It is easy for = 1. Suppose inductively that it has been shown for 1. Then, using the expansion V i = v1 2i 2 v2 2i 3 v i 1 v i, one has ) T (V i ) = (v1 2i 2 x 1 1 T 1(V i 1 (x 2,..., x i )) ) = Sq 2i 2 +1 x1 1 i 1(x 2,..., x i ) (by inductive hypothesis) ) = d P x1 (V i 1 (x 2,..., x i ) (since V i 1 = 2 i 2 ) = V i (x 1, x 2,..., x i ). The last equality is showed in Mùi [17, Lemma 5.3], (see also [8]). The lemma is proved.

9 THE WEK CONJECTURE ON SPHERICL CLSSES 9 Since D M, Lemmas 3.7 and 3.8 show that T D P. Theorem 3.2 is completly proved. is the inclusion of D into 4. Homological classes induced by Dicson invariants ccording to Singer [21], Γ supports a canonical -action as follows. The usual action of on P commutes with the action of GL. So, it induces an action of on D = P GL = F 2 [Q, 1,..., Q,0 ]. This is extended to an action of on Γ = D [Q 1,0 ]. By definition [21], Γ is the submodule of Γ spanned by all monomials γ = Q i 1, 1 Qi0,0 with < i 0 < +, 0 i 1,..., i 1 and 0 i 0 + dimγ. By means of the Hai Hu. ng formula for the -action on D (see [8] or 4.4 below), it is easy to verify that Γ is an -submodule of Γ. From Singer [21], the -action on Γ = Γ commutes with the differential of Γ. Now we show that the wea conjecture on spherical classes follows from the following one. Conjecture 4.1. Let Ker be the submodule of all cycles in Γ. Then, for > 2. D + + Ker, Proposition 4.2. Conjecture 4.1 implies Conjecture 1.6, which in turn implies the wea conjecture on spherical classes. Proof. By Conjecture 4.1, for a given q D +, we have q = i Sqi (γ i ) with some i > 0 and γ i Ker. On the other hand, from Singer [21, Th. 1.3], the induced -action on H (Γ ) = T or (F 2, F 2 ) is trivial. So [q] = i Sqi [γ i ] = 0 in T or (F 2, F 2 ). By Theorem 3.2, [q] = T r [q] = 0 in F 2 P for every q D +, or equivalently, D+ + P, for > 2. The proof is complete. Here is an alternative proof for the fact that Conjecture 4.1 implies the wea conjecture on spherical classes. By Theorem 3.2, T (Ker ) P, T (D ) = D. Since T is an -homomorphism, Conjecture 4.1 implies D + + P, for > 2. That means Conjecture 1.5 holds. Hence, the wea conjecture on spherical classes is proved. The following theorem is a wea form of Conjecture 4.1. Theorem 4.3. If > 2, then D + + Γ. In order to prove the theorem, we need the Hai Hu. ng formula for the action of on D (see [8]): Q,r i = 2 j 2 r, r j, 4.4. Sq i Q (Q,j ) =,t Q,r i = 2 2 t + 2 j 2 r, r j < t, Q 2,j i = 2 2 j, 0 otherwise.

10 10 NGUYÊ N H. V. HU. NG In particular, Sq 2s+j 1 (Q 2s,j ) = Q2s,j 1, for 0 < j <. The following remar is an immediate consequence of 4.4 and the Cartan formula. Remar 4.5. (on jump steps). If Sq i (Q 2s,j ) 0, then either i = 0 or i 2s+j 1. Furthermore, if Sq i X = Q 2s,j 1 for some X D, then either i = 0 or i 2 s+j 1. Proof of Theorem 4.3. We use techniques of the proof in [11] for the fact that F 2 Γ = 0, for > 1. We follow the notations of [10]. For a given non-negative integer i, we denote by s(i) the number with 2 s(i) being the first missing 2-power in the dyadic expansion of i. In other words, i 2 s(i) 1 (mod 2 s(i)+1 ). We prove the theorem by a downward induction. Let Q I = Q i 1, 1 Qi0,0, I = (i 1,..., i 0 ) with i j 0 if j 1 and i 0 Z, be a monomial in Γ. Set f j (I) = s(i j )+j 1 for j 1 and f(i) = min{f j (I) j 1}. Suppose f j (I) = f(i) with j 2, then s(i j 1 ) + j 2 = f j 1 (I) f j (I) = s(i j ) + j 1, so s(i j 1 ) s(i j ) + 1. In particular, i j 1 2 s(ij). Hence, applying 4.4 we get 4.6. Sq 2f(I) Q (i 1,...,i j+2 s(i j ),i j 1 2 s(i j ),...,i 0) = Q I + ΣQ L. We show that s(l j 1 ) s(i j ) and thus f j 1 (L) = s(l j 1 )+j 2 < s(i j )+j 1 = f j (I) = f(i). Hence f(l) < f(i) for any L in the sum. Let α m (a) denote the coefficient of 2 m in the dyadic expansion of a, and s = s(i j ). There are two cases. In the first case, α s (l j 1 ) = α s (i j 1 ) = 1. Then, using the remar on jump steps, we easily verify that Sq 2f(I) acts only to send Q 2s,j to Q2s,j 1. So QL = Q I, but not an extra term. In the second case, α s (l j 1 ) = α s (i j 1 2 s ) = 0. Then, by definition, s(l j 1 ) s = s(i j ). It should be noted that if Q I D, then the iller Q J = Q (i 1,...,i j +2 s(i j ),i j 1 2 s(i j ),...,i 0) and thus any extra term Q L is in D. Next we consider the case f 1 (I) = f(i) and f j (I) > f(i) if 1 < j <. Similarly as in 4.6, we have 4.7. Sq 2f(I) Q (i 1,...,i 1+2 s(i 1 ),i 0 2 s(i 1 )) = Q I + ΣQ L. We show that, for any L in the sum, there exists j (1 j < ) such that f j (L) < f 1 (I), therefore f(l) < f(i). Suppose the contrary that f j (L) f 1 (I) or equivalently s(l j )+j 1 s(i 1 ) for 1 j <. s s(l 1 ) s(i 1 ), then l 1 = i 1 +2 s(i1). Otherwise, by the remar on jump steps, Sq 2f(I) = Sq 2s(i 1 ) acts only to send Q 2s(i 1 ),1 to Q 2s(i 1 ),0, thus Q L = Q I, but not an extra term. Suppose inductively that l 1 = i s(i1), l 2 = i 2,..., l j 1 = i j 1. We show l j = i j (1 j < ). Indeed, if l j i j, then combining s(l j ) + j 1 s(i 1 ), s(i j ) + j 1 > s(i 1 ) with the remar on jump steps, we easily verify that actually s(l j ) + j 1 = s(i 1 ) and Sq 2f(I) = Sq 2s(i 1 ) acts only to send Q 2s(l j ),j to Q 2s(l j ),j 1. This contradicts the

11 THE WEK CONJECTURE ON SPHERICL CLSSES 11 hypothesis l j 1 = i j 1 (or the hypothesis l 1 = i s(i1) if j = 2). Consequently, we get l 1 = i s(i1), l 2 = i 2,..., l 1 = i 1. So Sq 2f(I) = Sq 2s(i 1 ) acts only to increase the power of Q,0. However, dimq,0 = s(i1) if > 1. This is a contradiction. Now we must show that if Q I D, then the iller Q J = Q (i 1,...,i 1+2 s(i 1 ),i 0 2 s(i 1 ) ) and every iller needed in the procedure of using 4.6 and 4.7 to ill extra terms Q L or to ill extra terms of extra terms... are all in Γ. Suppose QK is such a iller. Let s = s(i 1 ). s f = f(i) decreases in the procedure, then 0 i 0 2 s 2 s 1 1. On the other hand, combining f(i) = s with the two facts that the dimension of any extra term equals to dimq I and f(extra term) < f(i), we have Thus dimq K dimq I 2 f(i) = dimq I 2 s. 0 + dimq K (i 0 2 s 2 s 1 1) + (dimq I 2 s ) = (i 0 2 s+1 + 1) + (dimq I 2 s ) dimq I 3 2 s + 1. From s(i 1 ) = s, it implies i 1 2 s 1. Then we get Hence dimq I dim(q i1,1 ) (2s 1)(2 2). 0 + dimq K (2 s 1)(2 2) 3 2 s + 1 = (2 s 1)(2 2) 3 (2 s 1) = (2 s 1)(2 5) > 0 (as > 2), except for s = 0. (This case is handled by the next step.) Consequently, Q K Γ. To start the induction, assume Q I D with f(i) = 0. Then f 1 (I) = f(i) = 0 as, by definition, f j (I) > 0 for 1 < j <. Then s = s(i 1 ) = 0, and i 1 0 (mod 2). Thus Sq 1 Q (i 1,...,i 1+1,i 0 1) = Q I. The iller Q (i 1,...,i 1+1,i 0 1) is in Γ, because (i 0 1) + dimq (i 1,...,i 1+1,i 0 1) = i 0 + dimq I > 0, for > 2. The first inequality holds as there exists at least one i j 0 and dim(q,j ) 2 1 for 0 j <. The theorem is proved. The following theorem establishes Conjecture 1.6 for = 3. Theorem 4.8. Let = 3. Then, for every q D + 3, [q] = 0 in H 3(Γ ) = T or 3 (F 2, F 2 ).

12 12 NGUYÊ N H. V. HU. NG Proof. Let, denote the dual pairing between T or3 (F 2, F 2 ) and Ext 3 (F 2, F 2 ), and also the one between (F 2 P 3 ) GL3 and F 2 P H (BE 3 ). Here, by P H (BE 3 ) GL 3 we mean the submodule consisting of all -annihilated elements of H (BE 3 ). By Boardman [3], T r 3 : F 2 P H (BE 3 ) Ext 3 (F 2, F 2 ) is an isomorphism. GL 3 In particular, we have [q], Ext 3 (F 2, F 2 ) = [q], T r 3 (F 2 GL 3 P H (BE 3 )) = T r3[q], F 2 P H (BE 3 ) GL 3 = [q], F 2 P H (BE 3 ) GL 3 (by Theorem 3.2) = 0, because [q] = 0 in (F 2 P 3 ) GL3, by Theorem 3.2 of our paper [9]. Hence [q] = 0 in T or3 (F 2, F 2 ), for every q D 3 +. The theorem is proved. 5. Final remars Remar 5.1. The wea conjecture on spherical classes is actually equivalent to the fact that for every q D + and any > 2, [q] T or (F 2, F 2 ) vanishes on the image of Singer s -th transfer, T r (F 2 P H (BE )) Ext (F 2, F 2 ). This GL observation can be read off from the proof of Theorem 4.8. Remar 5.2. Conjecture 1.6, and therefore Conjecture 4.1, is false when = 1 or 2. Indeed, [Q 2i 1 1,0 ] and 1 [Q2i 2,1 ] are non-zero in T or (F 2, F 2 ). They are respectively dual to the dams element h i Ext 1 (F 2, F 2 ) and its square h 2 i Ext2 (F 2, F 2 ). One easily verifies this assertion by combining Theorem 3.2 with Theorem 2.1 of [9] and Proposition 5.3 of [14]. The only Hopf invariant one elements are represented by h i for i = 1, 2, 3 (see dams [1]). Furthermore, the only Kervaire invariant one elements are represented by h 2 i, wherever h2 i is a permanent cycle in the dams spectral sequence for spheres (see Browder [5]). Let ϕ : F 2 D T or (F 2, F 2 ) be the dual of the Lannes Zarati homomorphism, which is compatible with the Hurewicz one H : π (Q 0 S 0 ) H (Q 0 S 0 ). In [9] we have proved that the inclusion D P is a chain-level representation of T r ϕ : F 2 D F 2 P. This together with Theorem 3.2 lead us to the following conjecture. Conjecture 5.3. The inclusion D Γ is a chain-level representation of ϕ. This conjecture and Conjecture 1.6 imply the classical conjecture on spherical classes. References [1] dams, J. F.: On the non-existence of elements of Hopf invariant one, nn. Math. 72, (1960) [2] dams, J. F.: Operations of the n th ind in K-theory and what we don t now about RP, New developments in topology. In: G. Segal (ed.), Lond. Math. Soc. Lect. Note Series 11, 1 9 (1974)

13 THE WEK CONJECTURE ON SPHERICL CLSSES 13 [3] Boardman, J. M.: Modular representations on the homology of powers of real projective space, lgebraic Topology: Oaxtepec In: M. C. Tangora (ed.), Contemp. Math. 146, (1993) [4] Bousfield,. K., Curtis, E. B., Kan, D. M., Quillen, D. G., Rector, D. L., Schlesinger, J. W.: The mod p lower central series and the dams spectral sequence, Topology 5, (1966) [5] Browder, W.: The Kervaire invariant of a framed manifold and its generalization, nn. Math. 90, (1969) [6] Curtis, E. B.: The Dyer Lashof algebra and the Lambda algebra, Illinois Jour. Math. 18, (1975) [7] Dicson, L. E.: fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. mer. Math. Soc. 12, (1911) [8] Hu. ng, N. H. V.: The action of the Steenrod squares on the modular invariants of linear groups, Proc. mer. Math. Soc. 113, (1991) [9] Hu. ng, N. H. V.: Spherical classes and the algebraic transfer, Trans. mer. Math. Soc. 349, (1997) [10] Hu. ng, N. H. V., Peterson, F. P.: -generators for the Dicson algebra, Trans. mer. Math. Soc. 347, (1995) [11] Hu. ng, N. H. V., Peterson, F. P.: Spherical classes and the Dicson algebra, Math. Proc. Camb. Phil. Soc. 124, (1998) [12] Kameo, M.: Products of projective spaces as Steenrod modules, Thesis, Johns Hopins University 1990 [13] Lannes, J., Zarati, S.: Invariants de Hopf d ordre supérieur et suite spectrale d dams, C. R. cad. Sci. 296, (1983) [14] Lannes, J., Zarati, S.: Sur les foncteurs dérivés de la déstabilisation, Math. Zeit. 194, (1987) [15] MacLane, S.: Homology, Die Grundlehren der Math. Wissenschaften, Band 114, cademic Press, Springer-Verlag, Berlin and New Yor 1963 [16] Madsen, I.: On the action of the Dyer Lashof algebra in H (G), Pacific Jour. Math. 60, (1975) [17] Mùi, H.: Modular invariant theory and cohomology algebras of symmetric groups, Jour. Fac. Sci. Univ. Toyo, 22, (1975) [18] Peterson, F. P.: Generators of H (RP RP ) as a module over the Steenrod algebra, bstracts mer. Math. Soc., No 833, pril 1987 [19] Priddy, S.: On characterizing summands in the classifying space of a group, I, mer. Jour. Math. 112, (1990) [20] Silverman, J. H.: Hit polynomials and the canonical antiautomorphism of the Steenrod algabra, Proc. mer. Math. Soc. 123, (1995) [21] Singer, W. M.: Invariant theory and the lambda algebra, Trans. mer. Math. Soc. 280, (1983) [22] Singer, W. M.: The transfer in homological algebra, Math. Zeit. 202, (1989) [23] Steenrod N. E., Epstein, D. B..: Cohomology operations, nn. of Math. Studies, No. 50, Princeton Univ. Press 1962 [24] Wellington, R. J.: The unstable dams spectral sequence of free iterated loop spaces, Memoirs mer. Math. Soc. 258 (1982) [25] Wilerson, C.: Classifying spaces, Steenrod operations and algebraic closure, Topology 16, (1977) [26] Wood, R. M. W.: Steenrod squares of polynomials and the Peterson conjecture, Math. Proc. Camb. Phil. Soc. 105, (1989) Department of Mathematics, Vietnam National University, Hanoi 334 Nguyê n Trãi Street, Hanoi, Vietnam address: nhvhung@vnu.edu.vn Note added in proof: Recently, we have established Conjecture 5.3.