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In this paper, we construct the chainlevel of the LannesZarati homomorphism on the lambda algebra. Using this chainlevel, we investigate the vanishing of the LannesZarati homomorphism of rank five and six. Keywords: Adams spectral sequence, Lambda algebra, DyerLashof algebra, LannesZarati homomorphism, Hurewicz homomorphism. 2000 MSC: Primary 55P47, 55Q45, 55S10, 55T15
On behavior of the Lannes-Zarati homomorphism✩ Phan Hoàng Chơn∗, Đồng Thanh Triết Faculty of Mathematics - Application, Saigon University, 273 An Duong Vuong, District 5, Ho Chi Minh city, Vietnam. Abstract In this paper, we construct the chain-level of the Lannes-Zarati homomorphism on the lambda algebra. Using this chain-level, we investigate the vanishing of the Lannes-Zarati homomorphism of rank five and six. Keywords: Adams spectral sequence, Lambda algebra, Dyer-Lashof algebra, Lannes-Zarati homomorphism, Hurewicz homomorphism. 2000 MSC: Primary 55P47, 55Q45, 55S10, 55T15. 1. Introduction Let Ps := H ∗ (B(Z/2)s ) ∼ = F2 [x1 , · · · , xs ] be the polynomial algebra on s generators x1 , · · · , xs , each of degree 1. The general linear group GLs := GL(s, F2 ) and the mod 2 Steenrod algebra A both act on Ps in the usual way. Let Ds be the Dickson algebra (see Dickson [8]), which is the algebra of invariants Ds := F2 [x1 , · · · , xs ]GLs . Since the action of A and of GLs upon Ps commute with each other, Ds is an A-module. In [16], Lannes and Zarati constructed, for each s > 0, a homomorphism (so-called the Lannes-Zarati homomorphism) ϕs : ExtsA (F2 , F2 ) / P Ds∗ mapping from the cohomology of the Steenrod algebra to the subspace of the dual of the Dickson algebra spanned by all elements annihilated by all positive degree Steenrod operations, P Ds∗ . They also showed that ϕs , for s ≤ 2, are ✩ This work is partial supported by the grant of NAFOSTED. author Email addresses: chonkh@gmail.com (Phan Hoàng Chơn), dongthanhtriet.dhsg@gmail.com (Đồng Thanh Triết) ∗ Corresponding Preprint submitted to Elsevier November 25, 2013 non-trivial (i.e. ϕ1 is an isomorphim and ϕ2 is an epimorphism). Moreover, these homomorphism correspond to an associated graded of the Hurewicz map H : π∗S (S 0 ) ∼ = π∗ (Q0 S 0 ) / H∗ (Q0 S 0 ) of the base-point component Q0 S 0 of the infinite loop space QS 0 = limΩn Σn S 0 . −→ Beside, the results of Adams [1] and Browder [3] show that the Hopf invariant one and the Kervaire invariant one spherical classes (if exist) are respectively 2,∗ detected by certain permanent cycles in Ext1,∗ A (F2 , F2 ) and in ExtA (F2 , F2 ). Therefore, the results of Lannes and Zarati closely corresponds to the classical conjecture on the spherical classes, which states that only the classes of Hopf invariant one and Kervaire invariant one (if exist) are detected by the Hurewicz homomorphism (see Curtis [7] and Wellington [28]). As above discussion, Hưng set up a conjecture [10] that ϕs = 0 in any positive stems for s > 2. This conjecture corresponds to an associated graded version of the classical conjecture on the spherical classes (see [11]). The conjecture has been proved for s = 3 [13] and s = 4 [12]. Furthermore, Hưng and Peterson [13] also proved that ϕs is zero on decomposable elements for s > 2. The fact that ϕ3 and ϕ4 are trivial in any positive stems [13], [12] bases on the computations 4,∗ of Wang [27] (for Ext3,∗ A (F2 , F2 )), Lin-Mahowald [17] (for ExtA (F2 , F2 )) and the defining the basis of D3 and D4 as modules over the Steenrod algebra (so-called the “hit” problem for Ds ). More detail, since stems where the domain and the rank of ϕs , s = 3, 4, are nontrivial are not compatible with each other though ϕs , s = 3, 4, then these homomorphisms are vanishing. Recently, using the action of the squaring operation on the Dickson algebra, Hưng-Quỳnh [14] showed that Hưng’s conjecture is true on almost known elements of the cohomology of the Steenrod algebra. In this paper, we construct the chain-level map of the Lannes-Zarati homomorphism on the lambda algebra. We show that the Lannes-Zarati homomorphism is induced by the canonical projection from the lambda algebra onto the Dyer-Lashof algebra. Therefore, the squaring operation acting on lambda algebra (see Lin-Mahowald [17]) induces naturally the squaring on the DyerLahof. These squaring commute with each other through the Lannes-Zarati homomorphism. Using these facts, we show that ϕ5 = 0 in any positive stems and that ϕ6 = 0 in all indecomposable elements of stems not greater than 114. The advantage of our method is that we need not use the “hit” problem for Ds . In this work, we also show that the action of the “big” Steenrod algebra on H∗ (B(Z/2)s )Σs (see Singer [26]) induces an action on Ds∗ . We expect that this action becomes an useful tool to study the “hit” problem of Ds and the Lannes-Zarati homomorphism. The paper is divided into five sections. The first two sections are preliminaries. In section 2, we review some main points of the lambda algebra, the DyerLashof algebra as well as the Lannes-Zarati homomorphism. The chain-level of the Lannes-Zarati homomorphism over the lambda algebra is constructed in Section 3. Section 4 contains some discussion about the squaring operations and the action of the “big” Steenrod algebra on Ds . The main results are present in 2 the last section. 2. Preliminaries In this section, we review some main points of the lambda algebra, the DyerLashof algebra as well as the Lannes-Zarati homomorphism. 2.1. The lambda algebra and the Dyer-Lashof algebra The lambda algebra, Λ, is the differential bigraded algebra with unit over F2 generated by symbols λi , i ≥ 0, of bidegree (1, i) subject to adem relations λa λb = t t−b−1 λa+b−t λt , 2t − a (2.1) for all a, b ≥ 0. Here nk is interpreted as the coefficient of xk in expansion of (x + 1)n so that it is defined for all integer n and all non-nagative integer k (see Chơn-Hà [6]). The differential of the lambda algebra is given by δ(λn ) = j n−j−1 λj λn−j−1 . j+1 (2.2) Let Λs be the subspace of the lambda algebra spanned by all monomials in the λi of the length s. It is well-known that Λs has an additive basis consisting of all the admissible monomials, which are monomials in the form λI = λi1 . . . λis where i1 ≤ 2i2 , . . . , is−1 ≤ 2is . It should be note that our definition of the lambda algebra follows that of Singer [24], which is opposite (by the canonical reversing-order map) to the original version in Bousfield et. al. [2]. Let M be a graded right A-module, then Λ ⊗ M is a differential module. It is the direct sum of Λs ⊗ M of homological degree s. Its differential is given by λλi ⊗ (xSq i+1 ). δ(λ ⊗ x) = δ(λ) ⊗ x + i≥0 From Bousfield et. al. [2] and Priddy [22], there is the canonical isomorphism s,s+t ExtA (F2 , M ) ∼ = H s,t (Λ ⊗ M ). ∼ H s,t (Λ). When M = F2 , we have the isomorphism Exts,s+t (F2 , F2 ) = A An important subalgebra of the Λ is the Dyer-Lashof algebra R, which is the algebra of homology operations acting on the homology of infinite loop spaces. For any monomial λI = λi1 . . . λis , we define the excess of λI to be e(I) = e(λI ) = i1 −i2 −· · ·−is . Then the Dyer-Lashof algebra is defined as the quotient of lambda algebra by the two-sided ideal generated by all monomials of negative excess. 3 / R. A Denote by Qi is the image of λi through the quotient map Λ monomial QI is called admissible if it is the image of an admissible monomial λI . Then the set of all admissible monomials QI of non-negative excess provides an additive basis of R. Let Rs be the subspace of R spanned by all monomials of length s. As a coalgebra (see Madsen [19], and Mùi [21]) Rs is isomorphic to the dual of the Dickson algebra, Ds∗ . 2.2. The Lannes-Zarati homomorphism For any left A-module M , the destabilization of M is defined by D(M ) = M/EM, where EM = SpanF2 {Sq k x : k > deg(x), x ∈ M }. Because D(−) is the right exact functor, HomF2 (D(−), F2 ) is the left exact functor. Let Ds be the s-th right derived functor of the left exact functor HomF2 (D(−), F2 ). Then Ds (M ) = H s (HomF2 (DF (M ), F2 )), where F (M ) is the free resolution (or projective resolution) of M . Define e∗1 (M ), which is the dual of the Singer’s element e1 (M ), to be the connecting homomorphism of the functor HomF2 (D(−), F2 ) associated with the short exact sequence 0 / M ⊗ P1 / M ⊗ Pˆ / Σ−1 M / 0, where Pˆ = Span{xi : i ≥ −1}, which is an A-module with the A-action given by Sq n (x−1 ) = xn−1 . Put e∗s (M ) = (e∗1 (M )) ◦ · · · ◦ ((e∗1 (M ) ⊗ Ps−1 )), where Ps := H ∗ (B(Z/2)s ) ∼ = F2 [x1 , . . . , xs ], each xi of degree 1. / Ds (Σ−s M ). As Singer’s result [25], e∗s (M ) Then e∗s (M ) : D0 (M ⊗ Ps ) factors through the coinvariant of HomF2 (M ⊗ Ps , F2 ) = M ∗ ⊗ H∗ (B(Z/2)s ) under the action of the general linear group GLs = GL(s, F2 ). In [16], Lannes and Zarati show that Theorem 2.1 (Lannes-Zarati [16]). Let Ds ⊂ Ps be the Dickson algebra of / Ds (Σ1−s F2 ) is an isomorphic of s variables. Then αs = e∗s (ΣF2 ) : ΣDs∗ internal degree 0. Because of the definition of the functor D, the projection M factors through DM . Then it induces a commutative diagram ··· ··· / D(Fn M ) / D(Fn−1 M ) / ··· in−1 in / F2 ⊗A Fn−1 M / F 2 ⊗ A Fn M 4 / ··· . / F 2 ⊗A M Here the horizontal arrows are induced by the differential of F M , and i∗ is given by in ([z]) = [1 ⊗A z]. Passing the functor HomF2 (−, F2 ), we have a commutative diagram ··· / HomF (F2 ⊗A Fn−1 M, F2 ) 2 ··· / HomF2 (DFn−1 M, F2 ) / HomF (F2 ⊗A Fn M, F2 ) 2 jn−1 / ··· jn / HomF2 (DFn M, F2 ) / ··· . Taking the cohomology, we have js : ExtsA (F2 , M ) / Ds (M ) [x] → [x]. Since the positive degree Steenrod operations act trivially on ExtA (F2 , M ), the image of js is contained in P Ds (M ), where P Ds (M ) is the subspace of Ds (M ) spanned by all elements annihilated by all the positive-degree Steenrod operations. Therefore, we have the homomorphism / P Ds (M ). js : ExtsA (F2 , M ) When M = Σ1−s F2 , we obtain js : ExtsA (F2 , Σ1−s F2 ) / P Ds (Σ1−s F2 ) In [16], Lannes and Zarati defined, for each s > 0, a homomorphism ϕs = Σ(e∗s )−1 (Σ1−s F2 )js Σ−1 : ExtsA (F2 , Σ−s F2 ) / P Ds∗ , and they showed that ϕs is the algebraic version of the Hurewicz map H : π∗S (S 0 ) ∼ = π∗ (Q0 S 0 ) / H∗ (Q0 S 0 ). However, the proof of this assertion is unpublished, but it is sketched in Goerss [9] and Lannes [15]. 3. The chain-level of ϕs on the lambda algebra In this section, we construct an representation of ϕs in the lambda algebra. Let M be a left A-module. Recall that B∗ (M ) = ⊕s≥0 Bs (M ) is the usual bar resolution of M with Bs (M ) = A ⊗ I ⊗ · · · ⊗ I ⊗M, s times 5 where I is the argumentation ideal of A, which is the ideal of A generated by all the positive degree Steenrod operations. The element a0 ⊗ a1 ⊗ · · · ⊗ as ⊗ m ∈ Bs (M ) has homological degree s and internal degree i deg(ai ) + deg(m). The bar resolution B∗ (M ) is the differential A-module with the A-action given by a(a0 ⊗ a1 ⊗ · · · ⊗ as ⊗ m) = aa0 ⊗ a1 ⊗ · · · ⊗ as ⊗ m, and the differential given by s−1 ∂(a0 ⊗ a1 ⊗ · · · ⊗ as ⊗ m) = a0 ⊗ · · · ⊗ ai ai+1 ⊗ · · · ⊗ m i=0 + a0 ⊗ a1 ⊗ · · · ⊗ as m. Let C ∗ (M ) = HomA (B ∗ (M ), F2 ) ∼ = HomF2 (F2 ⊗A B ∗ (M ), F2 ) be the cobar resolution of M with C s (M ) = A∗ ⊗ · · · ⊗ A∗ ⊗M ∗ , s times where A and M are respective the dual of A and M . Because Exts,∗ A (F2 , M ) is isomorphic to H s (C ∗ (M )), the cobar resolution C ∗ (M ) is a suitable cocomplex to compute the cohomology of the Steenrod algebra. As we well-known, (see Madsen [19] and Mùi [21]), Ds∗ is isomorphic to Rs . In this section, we show that the canonical projection ϕ˜s : Λs → Rs induces the Lannes-Zarati homomorphism ϕs . For any q = Qi1 . . . Qis ∈ Rs , we put q˜ = ξ1i1 +1 ⊗ · · · ⊗ ξ1is +1 ⊗ Σ1−s 1 ∈ s C (Σ1−s F2 ). ∗ ∗ Lemma 3.1. The isomorphism αs : ΣDs∗ / Ds (Σ1−s F2 ) is given by αs (Σq) = [˜ q ] + orther terms. Proof. We have αs = e∗s (ΣF2 ) = e∗1 (Σ2−s F2 ) ◦ · · · ◦ (e∗1 (ΣF2 ) ⊗ Ps−1 ), where e∗1 (Σ−i F2 ) ⊗ Ps−i−2 : Di+1 (Σ−i Ps−i−1 ) / Di+2 (Σ−i−1 Ps−i−2 ) is the connecting homomorphism of the functor HomF2 (D(−), F2 ) associated with the short exact sequence 0 / Σ−i Ps−i−1 / Σ−i Pˆ ⊗ Ps−i−2 / Σ−i−1 Ps−i−2 Because of the short exact sequence 0 / EB∗ (M ) / B∗ (M ) / DB∗ (M ) / 0, and B∗ (M ) acyclic, we obtain the isomorphism ∂ ∗ : H s−1 (HomF2 (EB∗ (M ), F2 )) ∼ = Ds (M ). 6 / 0. Moreover, the short exact sequence / ΣPs 0 / Ps−1 / Pˆ ⊗ ΣPs−1 /0 induces the short exact sequence 0 / EB∗ (ΣPs ) / EB∗ (Pˆ ⊗ ΣPs−1 ) / EB∗ (Ps−1 ) / 0. So we have the homomorphism σ(Ps−1 ) : H s−1 (HomF2 (EB(ΣPs ), F2 )) / H s (HomF2 (EB(Ps−1 ), F2 )). Therefore, αs = ∂ ∗ ◦ σ(Σ1−s F2 ) ◦ · · · ◦ σ(ΣPs−1 ). For any admissible monomial q = Qi1 . . . Qis ∈ Ds∗ such that Σq is a cycle in HomF2 (EΣPs , F2 ), it can be pulled back to the same element in HomF2 (EΣPˆ ⊗ Ps−1 , F2 ). In HomF2 (EΣPs , F2 ), we have i1 +1 ξ1a ⊗ (ΣQi1 · · · Qis )Sq a δ(Σq) = a=0 r1 ξ11 · · · ξ + r1 ⊗ (ΣQi1 · · · Qis )Sq(r11 , . . . , r1 ) = 0, where the second sum is taken over all the string (r11 , . . . , r1 ) = (1, 0, . . . , 0). Therefore, in HomF2 (EΣPˆ ⊗ Ps−1 ), i1 +1 δ(Σq) = a=0 i1 − a ξ a ⊗ ΣQ−1 (Qi2 . . . Qis )Sq a−i1 −1 2i1 + 2 − a 1 r1 ξ1 1 · · · ξ + r1 ⊗ ΣQ−1 Qi2 · · · Qis , where the second sum is taken over all the string (r11 , . . . , r1 ) = (1, 0, . . . , 0) and (ΣQi1 · · · Qis )Sq(r11 , . . . , r1 ) = ΣQ−1 Qi2 · · · Qis . −a Observation that if i1 − a ≥ 0, then 2i1 − a + 2 > i1 − a. So 2i1i1+2−a =1 if and only if i1 − a = −1. It implies δ(Σq) = ξ1i1 +1 ⊗ ΣQ−1 (Qi2 . . . Qis ) r1 ξ1 1 · · · ξ + r1 ⊗ ΣQ−1 Qi2 · · · Qis . Hence, σ(Ps−1 )(Σq) = ξ1i1 +1 ⊗ (Qi2 . . . Qis ) r1 ξ1 1 · · · ξ + r1 ⊗ Qi2 · · · Qis . Repeating this process, we obtain σ(Σ1−s F2 ) ◦ · · · ◦σ(Ps−1 )(Σq) i = [ξ1i1 +1 ⊗ · · · ⊗ ξ1s−1 + r1 ξ1 1 · · · ξ r1 +1 ⊗ Σ1−s Qis ] r s−1 ⊗ · · · ⊗ ξ1 1 7 ···ξ r s−1 (s−1) ⊗ Σ1−s Qis . By the same argument, passing the homomorphism ∂ ∗ , we get ∂ ∗ ◦ σ(Σ1−s F2 ) ◦ · · · ◦σ(Ps−1 )(Σq) = [ξ1i1 +1 ⊗ · · · ⊗ ξ1is +1 ⊗ Σ1−s 1] + r1 ξ1 1 · · · ξ r1 rs ⊗ · · · ⊗ ξ11 · · · ξ rs ⊗ Σ1−s 1 = [˜ q ] + [z]. The proof is complete. Theorem 3.2. The canonical projection ϕ˜s : Λs Lannes-Zarati homomorphism ϕs . / Rs ∼ = Ds∗ induces the Proof. In [23], Priddy showed that the lambda algebra is isomorphic to the Koszul cocomplex of F2 , which is the quotient of the cobar resolution C ∗ (F2 ) = / Λ sending HomA (B∗ (F2 ), F2 ). Therefore, there is a projection ι : C ∗ (F2 ) a r ξ1 → λa−1 and ξi → 0 for all i ≥ 2. This map induces the isomorphism (F2 , F2 ) ∼ Exts,s+t = H s,t (Λ). A Under ι, the image of [z] is trivial in Λ. So ι([˜ q + z]) = Σλi1 · · · λis . The theorem is proved. Proposition 3.3. The homomorphism ϕ = ⊕s≥1 ϕs is an algebra homomorphism. Proof. Since the canonical projection ϕ˜ = ⊕s≥1 ϕ˜s : Λ −→ R is the algebra homomorphism, the assertion is immediate from Theorem 3.2. 4. The action of the “big” Steenrod algebra From Liulevicius [18], May [20], there exists the Steenrod operations acting on the Ext-groups Sq i : Exts,t A (F2 , F2 ) / Exts+i,2t (F2 , F2 ), i ≥ 0. A These Sq i s satisfy the usual adem relations, but Sq 0 is no longer identity. The definition of the operations uses “cup-i products” on a projective resolution of F2 as a A-module. Beside, in [26], Singer defines the operations Sq i : (Γs,t )Σs / (Γs+i,2t )Σs+i , i ≥ 0, where (Γs,t )Σs is the subspace of Γs,t = Ht−s (B(Z/2)s ) spanned by all coinvariants under the action of the symmetric group. These Sq i s satisfy the usual relations without the relation Sq 0 = 1. 8 (i ) (i ) Let Γ = {Γs,t : s ≥ 0, t ≥ 0} be the bigraded algebra. Let a1 1 · · · as s be the dual of xi11 · · · xiss ∈ H ∗ (B(Z/2)s ), then the product on Γ is given by / Γs+s ,t+t Γs,t ⊗ Γs ,t (i (i ) (i ) ) (i (i ) (i ) ) s+s s+1 s) s) a1 1 · · · a(i . ◦ a1 s+1 · · · as s+s → a1 1 · · · a(i s s as+1 · · · as+s The product induces a product on ΓΣ , which is the bigraded algebra with (ΓΣ )s,t = (Γs,t )Σs . Then the action of Sq i is given by (2n+1) if i = 0, ak i (n) (n) 2 Sq (ak ) = (a ) if i = 1, 0 k if i > 1, and the Cartan formula. In this section, we show that Sq i s induce the action on the dual of the Dickson algebra. Let I = {Is,t : s ≥ 0, t ≥ 0} and J = {Js,t : s ≥ 0, t ≥ 0}, where Is,t and Js,t are ideals of Γs,t generated respectively by {σ(x) + x|x ∈ Γs,t , σ ∈ Σs } and {g(x) + x|x ∈ Γs,t , g ∈ GLs }. It is sufficient to prove that actions of Sq i s commute with the action of τ = ( 11 01 ) modulo an element in J. Let Sq(u) = i≥0 Sq i ui be the formal power series, and let (k) ai xk ; ai (x) = (2k+1) 2k a ¯i (x) = k≥0 ai x . k≥0 Modulo an element in I, we have Sq(u)a1 (x2 ) = a ¯1 (x) + ua1 (x)a2 (x). (4.1) Lemma 4.1. Let a, b ∈ {a1 , a2 · · · } ∈ Γ, then τ (a(x)b(y)) = a(x)b(x + y); τ (¯ a(x)¯b(y)) = a ¯(x)¯b(x + y). (4.2) (4.3) Proof. The first is immediate as follows τ (a(x)b(y)) = (a + b)(x)b(y) = a(x)b(x)b(y) b(i) b(j) xi y j = a(x) = a(x) i,j i+j (i+j) = a(x) b(i+j) b (x + y) i+j i i+j i j xy i = a(x)b(x + y). i+j For the second formula, we observe that 9 2k+1+i i = 0 (mod 2) if i odd and that 2k+2i+1 2i = 2k+2i 2i τ (¯ a(x)¯b(y)) = (mod 2). Therefore, (a + b)(2i+1) x2i b(2j+1) y 2j i j a(2i−2k+1) x2i−2k = i−k b(2k) x2k b(2j+1) y 2j k b(2j+2k+1) =a ¯(x) k+j k b(2j+2k+1) =a ¯(x) k+j k j 2k + 2j + 1 2k 2y x y 2k 2k + 2j 2k 2y x y 2k =a ¯(x)¯b(x + y), since 2n i = 0 (mod 2) for i odd. Using the lemma, we have the following proposition. Proposition 4.2. For a, b ∈ {a1 , a2 , · · · } ∈ Γ1,2 and for any formal variables u, x, y, we have Sq(u)(τ (a(x2 )b(y 2 ))) = τ (Sq(u)(a(x2 )b(y 2 ))) (mod J). Proof. Since coproduct ψ(Sq(u)) = Sq(u) ⊗ Sq(u), using Cartan formula and (4.1), we have Sq(u)(a(x2 )b(y 2 )) = (Sq(u)a(x2 ))(Sq(u)b(y 2 )) = [¯ a(x) + ua(x)a (x)][¯b(y) + ub(y)b (y)] =a ¯(x)¯b(y) + u2 a(x)b(y)a (x)b (y) (mod I), where a , b ∈ {a1 , a2 , · · · } \ {a, b}. Therefore, using Lemma 4.1, τ (Sq(u)(a(x2 )b(y 2 ))) = a ¯(x)¯b(x + y) + u2 a(x)b(x + y)a (x)b (y) (mod I). On the other hand, Sq(u)(τ (a(x2 )b(y 2 ))) = Sq(u)(a(x2 )b((x + y)2 )) =a ¯(x)¯b(x + y) + u2 a(x)b(x + y)a (x)b (x + y) (mod I). Hence, τ (Sq(u)(a(x2 )b(y 2 ))) + Sq(u)(τ (a(x2 )b(y 2 ))) = u2 a(x)b(x + y)[a (x)b (y) + a (x)b (x + y)] (mod I) = u2 a(x)b(x + y)[a (x)b (y) + τ (a (x)b (y)] = 0 (mod J), where τ ∈ G sends a to a + b and fixes other variables. 10 Proposition 4.2 shows that there are induced actions of Sq i on the coinvariants algebra ΓG . These actions commute with the action of the usual Steenrod algebra [26]. More detail, we have Sq i ((x)Sq n ) = (Sq i (x))Sq 2n , x ∈ ΓΣ . Therefore, the Sq i s induces the actions on P (ΓG ). Observe that the Sq 0 coincides with the operation induced by the Kameko’s operation, which is commutes with the Liulevicius’ Sq 0 through the Lannes-Zarati homomorphism. In other words, the diagram Exts,t A (F2 , F2 ) Sq 0 ϕs P Rs / Exts,2t (F2 , F2 ) A ϕs Sq 0 / P Rs commutes. The detail description of the action of Sq i s on the dual of the Dickson algebra and the relation between these Sq i s and the Liulevicius’ operations through the Lannes-Zarati homomorphism will be presented elsewhere. 5. The vanishing of the Lannes-Zaratti homomorphism In this section, we use the chain-level ϕ˜s constructed in above section to investigate the behavior of the Lannes-Zarati homomorphism in the ranks five and six. The vanishing of the fifth Lannes-Zarati homomorphism is given in the following theorem. Theorem 5.1. The homomorphism ϕ5 : Ext5,t A (F2 , F2 ) → P R5 is trivial for t − 5 > 0. Proof. Since, for s > 2, the homomorphism ϕs vanishes in decomposable elements (see Hưng-Peterson [13]), it is sufficient to prove that ϕ5 is vanishing on indecomposable elements of Ext5,∗ In order to do that, we A (F2 , F2 ). prove the homomorphism ϕ˜5 : Λ5 → R5 maps trivially on cycles that represent indecomposable elements of Ext5,t For convenience, we write A (F2 , F2 ). s,t Exts,t = Ext (F , F ). 2 2 A A As Chen’s results [4], indecomposable elements of Ext5,∗ A (F2 , F2 ) are represented in the lambda algebra by the following list. 5,14 λ7 λ30 λ2 + (λ21 λ4 λ2 + λ1 λ4 λ2 λ1 + λ4 λ21 λ2 )λ1 ∈ ExtA ; 3 3 2 λ7 λ0 λ4 + (λ1 λ5 + λ1 λ4 λ2 + λ1 λ4 λ2 λ1 + ∈ Ext5,16 ; (2) P h2 = A λ4 λ21 λ2 )λ3 + λ7 λ20 λ2 λ2 + λ7 λ0 λ2 λ1 λ1 i+5 (Sq 0 )i (λ27 λ5 λ3 λ9 + λ7 λ15 λ3 λ0 λ6 +2i+2 ∈ Ext5,2 ; (3) ni = A +λ7 λ15 λ1 λ5 λ3 ) (1) P h1 = 11 (4) xi = (Sq 0 )i (λ15 λ23 λ2 λ14 + λ37 λ4 λ12 + λ37 λ28 +λ23 λ23 λ2 λ6 + λ23 λ23 λ24 + λ215 λ1 λ4 λ2 ) i+5 i+4 i+5 ∈ Ext5,2 A i+3 +2i+3 +2i+1 ; i +2 +2 +2 (5) D1 (i) = (Sq 0 )i (λ215 λ11 λ7 λ4 ) ∈ Ext5,2 ; A 0 i 2 2 2 (Sq ) (λ λ λ λ + λ λ λ + 15 11 7 14 15 11 10 5,2i+6 +2i+1 +2i λ15 λ31 λ7 λ1 λ8 + λ15 λ31 λ3 λ7 λ6 + (6) H1 (i) = ∈ ExtA ; λ15 λ31 λ7 λ5 λ4 ) (Sq 0 )i (λ47 [λ27 λ0 λ6 + (λ23 λ9 + λ9 λ23 )λ5 + 5,2i+6 +2i+3 λ3 λ9 λ5 λ3 + λ23 λ11 λ3 ] + (λ215 λ11 λ21 + (7) Q3 (i) = ∈ ExtA ; λ31 λ7 λ15 λ9 )λ5 + λ215 λ11 λ23 λ3 ) i+7 (Sq 0 )i (λ63 λ15 λ47 λ20 + λ63 λ47 (λ23 λ9 + λ9 λ23 ) +2i+1 (8) Ki = ∈ Ext5,2 ; A +λ231 λ11 λ21 ) (Sq 0 )i (λ95 (λ27 λ19 + λ19 λ27 )λ0 5,2i+7 +2i+2 +2i (9) Ji = ∈ ExtA ; +λ231 λ23 λ43 λ0 + λ63 λ15 λ31 λ19 λ0 ) i+7 (Sq 0 )i ((λ231 λ79 + λ79 λ231 )λ20 +2i+4 +2i+1 (10) Ti = ∈ Ext5,2 ; A +λ263 (λ23 λ9 + λ9 λ23 )) 5,2 (11) Vi = (Sq 0 )i (λ63 λ15 λ47 λ31 λ0 ) ∈ ExtA (12) Vi = (13) Ui = i+7 +2i+5 +2i ; i+8 i +2 (Sq 0 )i (λ191 λ31 λ7 λ23 λ0 + λ63 λ127 λ15 λ47 λ0 ) ∈ Ext5,2 ; A i+8 i+3 i (Sq 0 )i (λ191 (λ215 λ39 + λ39 λ215 )λ0 + 5,2 +2 +2 ∈ ExtA . λ263 λ47 λ87 λ0 + λ127 λ31 λ63 λ39 λ0 ) Here (Sq 0 )i = Sq 0 · · · Sq 0 , i ≥ 0. i times / R is the A-algebra homomorphism, Since the canonical projection ϕ˜ : Λ if λI contains a factor of negative excess then ϕ(λ ˜ I ) = 0. Therefore, it is easy to check that, under ϕ˜5 and therefore ϕ5 , the images of P h1 and P h2 is trivial. 5,∗ (F2 , F2 ) and on R5 Similarly, ϕ5 (n0 ) = 0. Since the actions of Sq 0 s on ExtA commute with each other through ϕ5 , ϕ5 (ni ) = ϕ5 ((Sq 0 )i (n0 )) = (Sq 0 )i ϕ5 (n0 ) = 0. By the same argument, it is sufficient to show that ϕ5 (X0 ) = 0, where X is names of elements from (4) to (13) of above list. By direct inspection, under ϕ5 , the images of elements x0 , D1 (0), H1 (0), Q3 (0), V0 , V0 are trivial. Using adem relations, we have λ63 λ47 λ9 λ23 = λ63 (λ15 λ11 λ33 + λ215 λ29 + λ15 λ23 λ21 )λ3 ; λ95 λ19 λ27 λ0 = λ95 λ19 λ3 λ5 λ6 ; λ79 λ231 λ20 = λ79 λ31 (λ1 λ30 + λ3 λ28 + λ7 λ24 + λ15 λ16 )λ0 ; λ191 λ39 λ215 λ0 = λ191 λ39 λ15 (λ1 λ14 + λ3 λ12 + λ7 λ8 ). Therefore, ϕ5 (K0 ) = 0, ϕ5 (J0 ) = 0, ϕ5 (T0 ) = 0, ϕ5 (U0 ) = 0. The proof is complete. 12 The following theorem investigates the vanishing of the sixth Lannes-Zarati homomorphism on some indecomposable elements of the cohomology of the Steenrod algebra at homological degree six. Theorem 5.2. The homomorphism ϕ6 is trivial on indecomposable elements in Ext6,t A (F2 , F2 ) for 6 ≤ t ≤ 120. Proof. From Chen’s result [5], indecomposable elements in Ext6,t A (F2 , F2 ), 6 ≤ t ≤ 120, is listed as follows e0 λ1 λ12 + (λ23 λ11 λ2 + λ27 λ2 λ3 )λ1 λ10 + f0 (λ2 λ10 + λ3 λ9 ) + λ27 λ4 λ2 λ1 λ9 + (λ33 λ9 λ5 + λ23 λ11 λ4 λ2 + λ3 λ9 λ23 λ5 + λ3 λ9 λ5 λ23 + λ27 λ4 λ2 λ3 )λ7 + (λ3 λ11 λ9 + (1) r = ∈ Ext6,36 ; 2 2 2 A λ23 λ0 + λ7 λ5 λ11 + λ7 λ9 λ7 )λ0 λ7 + λ7 λ2 λ7 λ0 λ7 + 2 2 2 2 2 f0 λ6 + (λ3 λ11 λ2 + λ7 λ2 λ3 )λ5 λ6 + λ7 (λ4 λ2 λ5 + λ5 λ10 λ23 ) + (λ23 λ9 + λ9 λ23 )λ9 λ23 + λ27 λ4 λ6 λ23 λ15 (λ23 λ2 λ1 λ8 + λ11 λ30 λ6 ) + λ37 λ6 λ3 λ2 + g1 λ3 λ9 + (λ15 λ3 λ0 λ6 + λ7 λ9 λ5 λ3 + λ11 λ7 λ4 λ2 )λ3 λ5 + 2 λ15 (λ1 λ2 λ1 λ8 + λ1 λ4 λ6 + λ1 λ4 λ2 λ5 + λ3 λ2 λ3 λ4 + (2) q = ∈ Ext6,38 ; 3 2 A λ λ + λ λ λ λ + λ λ λ λ )λ + λ λ λ λ λ + 9 3 4 2 3 1 2 6 3 5 15 2 5 4 1 3 λ15 (λ5 λ33 + λ1 λ2 λ5 λ6 + λ21 λ8 λ4 + λ1 λ5 λ24 + λ1 λ4 λ6 λ3 + λ5 λ3 λ4 λ2 )λ3 + λ15 λ3 λ7 λ4 λ1 λ2 2 n0 λ5 + (λ7 λ15 λ3 λ0 λ8 + λ7 λ5 λ9 λ5 + 6,42 (3) t = ∈ ExtA ; λ7 λ15 λ3 λ2 λ6 + λ15 λ3 λ7 λ5 λ3 )λ3 2 3 λ15 λ0 λ8 + [λ23 (λ21 λ4 λ2 + λ1 λ4 λ2 λ1 + λ31 λ5 + λ4 λ21 λ2 ) + λ7 λ15 λ3 λ0 λ6 + λ27 λ5 λ10 λ2 + λ15 λ3 λ7 λ4 λ2 + 2 2 2 λ7 λ5 λ3 λ9 ]λ7 + λ15 [λ0 λ2 λ1 λ5 + (λ0 λ4 + λ0 λ4 λ0 + ; ∈ Ext6,44 (4) y = A 2 2 2 2 λ λ )λ + λ λ λ + (λ λ + λ λ λ + λ λ )λ + 4 4 2 6 5 0 5 0 5 3 0 0 0 0 (λ0 λ4 + λ4 λ0 )λ22 ] + λ215 λ4 λ2 λ21 e1 λ7 λ5 + [(λ215 λ5 λ7 + λ15 λ11 λ7 λ9 + λ27 λ23 )λ5 + (5) C = ∈ Ext6,56 ; A λ215 λ11 λ0 λ6 + λ7 λ23 λ15 λ21 + λ215 λ9 λ5 λ3 ; (6) G = {D1 (0)λ2 } ∈ Ext6,60 A 6,64 (7) D2 = λ47 λ11 λ40 ∈ ExtA ; ; (8) A = D1 (0)λ9 + λ47 d0 λ0 + λ215 λ211 λ6 λ3 ∈ Ext6,67 A 13 (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) c2 [λ0 λ2 λ18 + λ2 λ3 λ15 + λ0 λ6 λ14 + λ2 λ5 λ13 + λ6 λ1 λ13 + λ0 λ8 λ12 + λ20 λ20 + λ8 λ0 λ12 + λ6 λ3 λ11 + λ0 λ210 + λ8 λ2 λ10 + (λ9 λ2 + λ10 λ1 )λ9 + (λ6 λ7 + 2 2 λ8 λ5 )λ7 + λ10 λ5 ] + λ15 λ11 λ2 λ1 λ17 + 2 (λ15 λ11 λ7 λ9 λ8 + λ15 λ13 λ7 λ0 )λ11 + λ31 f0 λ12 + 3 2 λ15 λ4 λ2 λ10 + D1 (0)λ9 + λ15 (λ13 λ7 λ4 λ7 + 2 2 λ15 λ0 λ10 λ6 ) + [λ31 (λ3 λ9 + λ9 λ3 )λ9 + 2 λ31 λ3 (λ9 λ5 + λ3 λ11 )λ7 + λ15 (λ11 λ8 + A = ∈ Ext6,67 ; 2 A λ λ )λ ]λ + [λ (λ λ λ + λ λ λ )+ 15 5 6 6 15 2 9 15 10 1 15 λ31 (λ23 λ11 λ8 + λ3 λ9 λ5 λ8 + λ33 λ6 + λ11 λ21 λ12 + 2 2 λ + λ λ )λ + λ (λ λ + λ λ )λ + (λ 9 9 10 7 1 9 9 1 8 3 3 2 λ (λ λ λ + λ λ λ ))]λ + [λ (λ λ λ + 7 5 7 6 9 5 4 5 15 2 11 15 2 2 λ λ λ + λ λ λ + λ λ ) + λ (λ λ λ + 15 5 8 15 8 5 6 31 11 14 11 1 2 2 2 λ λ λ + λ λ λ + λ λ λ λ + λ λ λ + 8 13 9 12 3 11 2 11 11 10 3 3 3 λ λ λ λ + λ λ λ λ + λ λ λ λ + λ λ λ λ + 3 9 5 10 11 7 0 9 7 5 7 8 3 11 7 6 λ11 λ7 λ4 λ5 )]λ3 6,70 A = D1 (0)λ12 + λ215 λ11 λ7 λ28 + λ47 e0 λ0 ∈ ExtA ; 2 2 f λ λ + g λ + (λ λ λ λ + λ λ λ λ 1 7 19 2 11 31 3 11 7 23 15 9 5 )λ7 + λ2 λ2 λ7 λ11 + λ31 [λ27 λ0 λ14 + (λ23 λ9 + λ9 λ23 )λ13 + r1 = ; ∈ Ext6,72 A 152 9 (λ3 λ11 + λ3 λ9 λ5 )λ11 ]λ7 λ47 λ23 λ2 λ1 λ15 + c2 (λ7 λ12 + λ19 λ0 )λ11 + x6,77 = ∈ Ext6,77 ; A D1 (0)λ19 + λ215 (λ27 λ0 λ7 + λ11 λ19 λ4 + λ11 λ23 λ0 )λ7 H1 (0)λ14 + λ215 λ311 λ13 + (λ215 λ11 λ15 λ9 + 3 λ λ λ + λ λ λ λ λ + D (0)λ + 13 7 31 7 23 4 0 3 4 15 2 2 2 2 λ λ λ λ + λ λ λ + λ λ λ λ + 15 47 3 47 47 2 10 0 3 6 3 2 2 2 λ31 λ23 λ1 λ9 + λ47 λ11 λ0 λ9 + λ31 λ23 λ1 λ5 + 2 2 λ31 λ23 λ5 λ3 + λ47 λ11 λ1 λ5 + λ47 λ3 λ7 λ5 λ3 )λ11 + ; ∈ Ext6,82 x6,82 = A 2 2 (λ λ λ λ + λ λ λ λ λ + λ λ λ λ + 47 10 6 47 7 1 8 6 47 3 7 3 6 λ47 λ7 λ5 λ4 λ6 + λ31 λ23 λ1 λ9 λ5 + λ31 λ23 λ9 λ1 λ5 + 2 λ λ λ λ λ + λ λ λ λ λ + λ λ λ λ + 15 47 0 4 3 15 47 4 0 3 15 47 5 1 2 λ15 λ11 λ21 λ7 )λ7 6,84 t1 = Sq 0 t ∈ ExtA ; d2 λ15 λ2 + [d2 λ16 + λ31 (λ7 λ23 λ8 + λ23 λ15 λ0 )λ15 x6,90 = ∈ Ext6,90 ; A +D3 (0)λ23 ]λ1 C1 = Sq 0 C ∈ Ext6,112 ; A [λ31 (λ23 λ15 λ19 λ13 + λ31 λ9 λ11 λ9 ) + c3 (λ5 λ11 + λ9 λ7 )]λ7 + f2 λ15 λ9 + c3 (λ15 λ0 λ8 + λ15 λ2 λ6 + x6,114 = ∈ Ext6,114 ; A 2 2 2 λ15 λ4 ) + λ31 λ19 λ3 λ5 G1 = Sq 0 G ∈ Ext6,120 . A We use the same method as the proof of Theorem 5.2. By direct inspection, we have ϕ˜6 (q) = 0 and ϕ˜6 (y) = 0. In the lambda algebra, we have (see Wang [27], Lin-Mahowald [17] and Chen [4]) i 3,11·2 • ci = {(Sq 0 )i (λ3 λ3 λ2 )} ∈ ExtA , i ≥ 0; 14 i 4,18·2 • di = {(Sq 0 )i (λ23 λ2 λ6 + λ23 λ24 + λ3 λ5 λ4 λ2 + λ7 λ1 λ5 λ1 )} ∈ ExtA , i ≥ 0; • ei = {(Sq 0 )i (λ33 λ8 + (λ3 λ25 + λ23 λ7 )λ4 + (λ7 λ5 λ3 + λ3 λ11 λ1 )λ2 )} ∈ i Ext4,21·2 , i ≥ 0; A i 4,22·2 • fi = {(Sq 0 )i (λ27 λ0 λ4 + (λ23 λ9 + λ7 λ5 λ3 )λ3 + λ27 λ22 )} ∈ ExtA , i ≥ 0; • gi+1 = {(Sq 0 )i (λ27 λ0 λ6 + (λ23 λ9 + λ7 λ5 λ3 )λ5 + (λ3 λ9 λ5 + λ33 λ11 )λ3 )} i ∈ Ext4,24·2 , i ≥ 0; A i 4,65·2 • D3 (i) = {(Sq 0 )i (λ31 λ7 λ23 λ0 )} ∈ ExtA , i ≥ 0. It is easy to chec that, under the chain-level version, the images of ci , di , ei , fi , gi+1 and D3 (i) are zero. Furthermore, from Theorem 5.2, images of ni , D1 (i) and H1 (i) under ϕ˜5 are trivial. It implies that ϕ˜6 sends elements r, t, C, G, A, A , A , r1 , x6,77 , x6,82 , x6,90 , x6,114 , and then t1 , C1 , G1 , to zero. Finally, using adem relations, we obtain λ47 λ11 λ40 = λ23 λ35 λ40 + λ31 (λ1 λ26 + λ3 λ24 + λ5 λ22 + λ11 λ16 + λ13 λ14 )λ30 . Therefore, ϕ˜6 (D2 ) = 0. The proof is complete. From above computation, there is a natural question to be set up. Question 5.3. Is every non-trivial elements of ExtsA (F2 , F2 ), s > 2, represented on the lambda algebra by the cycle that is the sum of monomials of negative excess? If this question is true, then Hưng’s conjecture is proved by our method. Acknowledgement We would like to thank Tai-Wei Chen for the sending us an early version of [5]. The early version of this paper was written while the first author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). He thanks the VIASM for support and hospitality. References [1] J. F. Adams, On the structure and applications of the Steenrod algebra, Comment. 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MR 646741 (83c:55028) 17 [...]... the Lannes- Zarati homomorphism In other words, the diagram Exts,t A (F2 , F2 ) Sq 0 ϕs P Rs / Exts,2t (F2 , F2 ) A ϕs Sq 0 / P Rs commutes The detail description of the action of Sq i s on the dual of the Dickson algebra and the relation between these Sq i s and the Liulevicius’ operations through the Lannes- Zarati homomorphism will be presented elsewhere 5 The vanishing of the Lannes- Zaratti homomorphism. .. homomorphism In this section, we use the chain-level ϕ˜s constructed in above section to investigate the behavior of the Lannes- Zarati homomorphism in the ranks five and six The vanishing of the fifth Lannes- Zarati homomorphism is given in the following theorem Theorem 5.1 The homomorphism ϕ5 : Ext5,t A (F2 , F2 ) → P R5 is trivial for t − 5 > 0 Proof Since, for s > 2, the homomorphism ϕs vanishes in...Proposition 4.2 shows that there are induced actions of Sq i on the coinvariants algebra ΓG These actions commute with the action of the usual Steenrod algebra [26] More detail, we have Sq i ((x)Sq n ) = (Sq i (x))Sq 2n , x ∈ ΓΣ Therefore, the Sq i s induces the actions on P (ΓG ) Observe that the Sq 0 coincides with the operation induced by the Kameko’s operation, which is commutes with the Liulevicius’... λ8 ) Therefore, ϕ5 (K0 ) = 0, ϕ5 (J0 ) = 0, ϕ5 (T0 ) = 0, ϕ5 (U0 ) = 0 The proof is complete 12 The following theorem investigates the vanishing of the sixth Lannes- Zarati homomorphism on some indecomposable elements of the cohomology of the Steenrod algebra at homological degree six Theorem 5.2 The homomorphism ϕ6 is trivial on indecomposable elements in Ext6,t A (F2 , F2 ) for 6 ≤ t ≤ 120 Proof From... λ3 λ24 + λ5 λ22 + λ11 λ16 + λ13 λ14 )λ30 Therefore, ϕ˜6 (D2 ) = 0 The proof is complete From above computation, there is a natural question to be set up Question 5.3 Is every non-trivial elements of ExtsA (F2 , F2 ), s > 2, represented on the lambda algebra by the cycle that is the sum of monomials of negative excess? If this question is true, then Hưng’s conjecture is proved by our method Acknowledgement... · · Sq 0 , i ≥ 0 i times / R is the A-algebra homomorphism, Since the canonical projection ϕ˜ : Λ if λI contains a factor of negative excess then ϕ(λ ˜ I ) = 0 Therefore, it is easy to check that, under ϕ˜5 and therefore ϕ5 , the images of P h1 and P h2 is trivial 5,∗ (F2 , F2 ) and on R5 Similarly, ϕ5 (n0 ) = 0 Since the actions of Sq 0 s on ExtA commute with each other through ϕ5 , ϕ5 (ni ) = ϕ5... Acknowledgement We would like to thank Tai-Wei Chen for the sending us an early version of [5] The early version of this paper was written while the first author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM) He thanks the VIASM for support and hospitality References [1] J F Adams, On the structure and applications of the Steenrod algebra, Comment Math Helv 32 (1958), 180–214... [12] , On triviality of Dickson invariants in the homology of the Steenrod algebra, Math Proc Cambridge Philos Soc 134 (2003), no 1, 103–113 [13] N H V Hưng and F P Peterson, Spherical classes and the Dickson algebra, Math Proc Cambridge Philos Soc 124 (1998), no 2, 253–264 MR 1631123 (99i:55021) [14] N H V Hưng and V T N Quynh, The squaring operation on Agenerators of the dickson algebra, Math Proc... MR 1022818 (90i:55035) [26] , Rings of symmetric functions as modules over the Steenrod algebra, Algebr Geom Topol 8 (2008), no 1, 541–562 MR 2443237 (2009g:55024) [27] J S P Wang, On the cohomology of the mod − 2 Steenrod algebra and the non-existence of elements of Hopf invariant one, Illinois J Math 11 (1967), 480–490 MR 0214065 (35 #4917) [28] R J Wellington, The unstable Adams spectral sequence... geometry and group representations (Evanston, IL, 1997), Contemp Math., vol 220, Amer Math Soc., Providence, RI, 1998, pp 143–177 MR 1642893 (99f:55023) 16 [18] A Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Mem Amer Math Soc No 42 (1962), 112 MR 0182001 (31 #6226) [19] I Madsen, On the action of the Dyer-Lashof algebra in H∗ (G), Pacific J Math 60 (1975), no ... true on almost known elements of the cohomology of the Steenrod algebra In this paper, we construct the chain-level map of the Lannes- Zarati homomorphism on the lambda algebra We show that the Lannes- Zarati. .. In section 2, we review some main points of the lambda algebra, the DyerLashof algebra as well as the Lannes- Zarati homomorphism The chain-level of the Lannes- Zarati homomorphism over the lambda... dual of the Dickson algebra and the relation between these Sq i s and the Liulevicius’ operations through the Lannes- Zarati homomorphism will be presented elsewhere The vanishing of the Lannes- Zaratti