On behavior of the sixth lannes zarati homomorphism

East-West J of Mathematics: Vol 22, No (2020) pp 1-12 https://doi.org/10.36853/ewjm.2020.22.01/01 ON BEHAVIOR OF THE SIXTH LANNES-ZARATI HOMOMORPHISM Pham Bich Nhu Department of Mathematics, College of Natural Sciences, Can Tho University, 3/2 Street, Can Tho city, Vietnam Department of Mathematics, Quy Nhon University, 170 An Duong Vuong, Quy Nhon city, Vietnam email: pbnhu@ctu.edu.vn Abstract In this paper, we determine the image of the indecomposable elements in Ext6,∗ A (F2 , F2 ) for ≤ t ≤ 120 through the sixth Lannes-Zarati homomorphism ϕ6 := ϕF62 Introduction and statement of results Let D be the destabilization functor from the category M of left modules over the mod Steenrod algebra A to the category U of unstable modules, which is the left adjoint to the forgetful functor U → M Hence, it is right exact and, therefore, it admits the left derived functor Ds : M → U for each s ≥ By definition of D (see Section 2), for any M ∈ M, there exists a natural homomorphism D(M ) → F2 ⊗A M , and then, this homomorphism in A turns induces natural maps iM s : Ds (M ) → Tors (F2 , M ) between corresponding derived functors In addition, as the result of Lannes and Zarati [16], for any M ∈ U and for each s ≥ 0, there is an isomorphism αs (ΣM ) : Ds (Σ1−s M ) → ΣRs M , where Rs is the Singer construction, which is an exact functor from U to itself (see Singer [18], [19], Lannes-Zarati [16], see also Hai [9], and citations Key words: Dyer-Lashof algebra, Hurewicz map, Lamdba algebra, Lannes-Zarati homomorphism, Spherical classes 2010 AMS Mathematics classification: Primary 55S10; Secondary 55T15 On behavior of the sixth Lannes-Zarati homomorphism therein for a detail description) Therefore, for any unstable A-module M , there exists a natural homomorphism, for each s ≥ 0, A # −s (ϕ¯M s ) : Rs M → Tors (F2 , Σ M ) # Since the Steenrod algebra A has acted trivially on the target, (ϕ¯M factors s ) through F2 ⊗A Rs M Hence, there exists a natural homomorphism A # t −s (ϕM s ) : (F2 ⊗A Rs M ) → Tors,t (F2 , Σ M ) TorA s,s+t (F2 , M ) (1.1) Taking (linear) dual, we have a homomorphism (the so-called Lannes-Zarati homomorphism), for each s ≥ 0, s,s+t ϕM (M, F2 ) → Ann(Rs (M )# )t s : ExtA Here, for any A-module N , we denote N # the (linear) dual of N and Ann(N # ) the subspace of N # spanned by all elements annihilated by all Steenrod operations of positive degree The Lannes-Zarati homomorphism is also considered as an associated graded of the Hurewicz map H : π∗S (S ) → H∗ (Q0 S ), on the base-point component Q0 S of the infinite loop space QS = limΩn Σn S −→ (see Lannes and Zarati [14], [15] for the sketch of proof) Therefore, the study of the Lannes-Zarati homomorphism is related to the study of the image of the Hurewicz map and then Curtis’s conjecture on the spherical classes [8] (see [7] for discussion) The Lannes-Zarati homomorphism was first constructed by Lannes-Zarati in [16] Therein, they showed that ϕF1 is an isomorphism, ϕF2 is an epimorphism Later, Hung et al also proved ϕFs is trivial in any positive stems for ≤ s ≤ (see [11] for the case s = 3, [10] for the case s = and [12] for the case s = 5) The results of Hung et al essentially based on the information of “hit” problem for the Dickson algebra Since the “hit” problem for the Dickson algebra of six variables is still unsolved, it is difficult to apply this method for ϕF6 In this paper, we use the method of Chon-Nhu [6, 7] to determine the image of ϕF6 Thereby, we obtain the following result Theorem 1.1 The homomorphism ϕF6 : Ext6,6+t (F2 , F2 ) → Ann((R6 F2 )# )t A 6,t is trivial on indecomposable elements in ExtA (F2 , F2 ) for ≤ t ≤ 120 The advantage of this method is to avoid using the knowledge of the “hit” problem for the Dickson algebra Pham Bich Nhu Preliminaries Denote M as the category of graded left A-modules and degree zero A-linear maps An A-module M ∈ M is called unstable if Sq i x = for i > deg x and for all x ∈ M Given an A-module M and an integer s, let Σs M denote the s-th iterated suspension of M We define (Σs M )n = M n−s , then an element in degree n of Σs M is usually written in the form Σs m, where m ∈ M n−s Let U is the full subcategory of M of all unstable modules The destabilization functor D : M → U is the left adjoint to the inclusion U → M It can be described more explicitly as follows: D(M ) := M/EM, where EM := SpanF2 {Sq i x : 2i > deg(x), x ∈ M } is an A-submodule of M , that is a consequence of the Adem relations In particular, EM is the subspace of elements in a negative degree if M is a graded vector space which is considered as an A-module with trivial action Then D(M ) is an A-submodule of M consisting of all elements in non-negative degrees It is simple to observe the following construction For any A-module M , then there is an A-homomorphism D(M ) → D(F2 ⊗A M ), which is induced by the projection M → F2 ⊗A M and the canonical embedding D(F2 ⊗A M ) → F2 ⊗A M Thus, there exists a natural A-homomorphism D(M ) → F2 ⊗A M which is the composition D(M ) → D(F2 ⊗A M ) → F2 ⊗A M Therefore, maps between corresponding derived functors are induced by this exact sequence A iM s : Ds (M ) → Tors (F2 , M ) The possibility of understanding the homology of the Steenrod algebra via knowledge of derived functors of the destabilization functor is raised by the natural map iM s However, computing Ds is generally very difficult, except in one important situation in which Lannes and Zarati [16], [21] discovered that it can be described in terms of the Singer functors Rs We recall the definition of the Lannes-Zarati homomorphism For any Amodule M , let the short exact sequence → P1 ⊗ M → Pˆ ⊗ M → Σ−1 M → 0, where, P1 = F2 [x1] be the polynomial algebra over F2 generated by x1 with |x1 | = and Pˆ is the A-module extension of P1 by formally adding the generator n−1 n −1 ˆ x−1 in degree −1 The action of A on P is given by Sq (x1 ) = x1 Moreover, we have the following theorem On behavior of the sixth Lannes-Zarati homomorphism Theorem 2.1 (Lannes and Zarati [16]) For any unstable A-module M , the homomorphism αs (ΣM ) : Ds (Σ1−s M ) → ΣRs M is an isomorphism of unstable A-modules For any unstable A-module M and for s ≥ 0, there exists a homomorphism # (ϕ¯M such that the following diagram commutes (see Chon-Nhu [7] for a s ) detail construction): αs (ΣM ) / ΣRs M → ΣPs ⊗ M n n n 1−s M nnn iΣ s nn(nϕ¯M )# n n s  wn 1−s TorA M) s (F2 , Σ Ds (Σ1−s M ) (2.1) where, Ps = F2 [x1 , x2 , · · · , xs ] be the polynomial algebra over F2 generated by the indicated variables, each of degree # (ϕ¯M s ) factors through F2 ⊗A ΣRs M because of acting trivially on the target of the Steenrod algebra A Therefore, after desuspending, we obtain the dual of the Lannes-Zarati homomorphism # t A −s (ϕM s ) : (F2 ⊗A Rs M ) → Tors,t (F2 , Σ M ) TorA s,s+t (F2 , M ) The linear dual s,s+t # ϕM (M, F2 ) → (F2 ⊗A Rs M )# t = Ann((Rs M ) )t , s : ExtA is the so-called Lannes-Zarati homomorphism In [1] (see also [2]), for computing the cohomology of the Steenrod algebra, Bousfield et al defined a differential algebra and so-called the Lambda algebra The dual of Lambda algebra as differential F2 -module is isomorphic to Γ+ (see in [13]) Because the sign is not compatible, extending an isomorphism between chain complexes Γ+ M and Λ# ⊗M is difficult Therefore, as naturally, we used the opposite algebra of the Lambda algebra, also denoted Λ, which corresponds to the original Lambda algebra under the anti-isomorphism of differential F2 modules In the literature, it is also called the Lambda algebra The Lambda algebra, Λ, which is defined as the differential, graded, associative algebra with unit over F2 , is generated by λi , i ≥ 0, of degree i, satisfying the Adem relations λi λj = t t−j −1 2t − i λi+j−t λt , (2.2) for all i, j ≥ 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-negative integer k (see Chon-Ha [5]) Pham Bich Nhu For ij , j = 1, · · · , s is non-negative integers, a monomial λI = λi1 λi2 · · · λis in Λ is called the monomial of the length s And λI is also called an admissible monomial if i1 ≤ 2i2 , · · · , is−1 ≤ 2is , and define the excess of λI or I to be s exc(λI ) = exc(I) = i1 − ij j=2 The Dyer-Lashof algebra R is an important quotient algebra of the lambda algebra over the ideal generated by the monomials of negative excess Let the canonical projection π : Λ → R, put QI = Qi1 Qi2 · · · Qis be the image of λI under π Let Rs be the subspace of R spanned by the monomials of length s From the results of Chon-Nhu [6], we have Proposition 2.2 (Chon-Nhu [6, Proposition 6.2]) The projection # ϕFs : Λs ⊗ F# → (Rs F2 ) , given by λI ⊗ → [QI ⊗ ] is a chain-level representation of the mod Lannes-Zarati homomorphism ϕFs Proposition 2.3 (Chon-Nhu [6, Proposition 6.3]) The following diagram is commutative Exts,s+t (F2 , F2 ) A Sq0 ϕFs2  (F2 ⊗A Rs F2 )# t / Exts,2(s+t)(F , F ) 2 A ϕFs2 Sq0  / (F2 ⊗A Rs F2 )# 2t+s The proof of Theorem 1.1 In this section, we use the chain-level representation map of the ϕFs constructed in the previous section to investigate the behavior of the sixth Lannes-Zarati homomorphism ϕF6 Lemma 3.1 If λI ∈ Λs and λJ ∈ Λ such that ϕ˜Fs (λI ) = or ϕ˜F2 (λJ ) = then ϕ˜Fs+ (λI λJ ) = Now, we need to prove Theorem 1.1 On behavior of the sixth Lannes-Zarati homomorphism Proof From Chen’s result [4], indecomposable elements in Ext6,t A (F2 , F2 ) for ≤ 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 + (λ23 λ9 λ5 + λ23 λ11 λ4 λ2 ⎪ ⎪ ⎪ ⎪ ⎨ +λ3 λ9 λ23 λ5 + λ3 λ9 λ5 λ23 + λ27 λ4 λ2 λ3 )λ7 (1) r = +(λ3 λ11 λ9 + λ23 λ20 + λ7 λ5 λ11 + λ7 λ9 λ7 )λ20 λ7 ⎪ ⎪ +λ27 λ2 λ7 λ0 λ7 + f0 λ26 + (λ23 λ11 λ2 + λ27 λ2 λ3 )λ5 λ6 ⎪ ⎪ ⎪ ⎪ +λ27 (λ4 λ2 λ25 + λ0 λ10 λ23 ) + (λ23 λ9 + λ9 λ23 )λ9 λ23 ⎪ ⎪ ⎩ +λ27 λ4 λ6 λ23 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ∈ Ext6,36 A (F2 , F2 ); ⎫ ⎧ 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 ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ +λ15 (λ1 λ2 λ1 λ8 + λ21 λ4 λ6 + λ1 λ4 λ2 λ5 + λ3 λ2 λ3 λ4 (2) q = ⎪ +λ31 λ9 + λ3 λ4 λ2 λ3 + λ1 λ2 λ6 λ3 )λ5 + λ15 λ23 λ2 λ5 λ4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +λ15 (λ5 λ33 + λ1 λ2 λ5 λ6 + λ21 λ8 λ4 + λ1 λ5 λ24 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ +λ1 λ4 λ6 λ8 + λ5 λ3 λ4 λ2 )λ3 + λ15 λ3 λ7 λ4 λ1 λ2 ∈ Ext6,38 A (F2 , F2 ); (3) t = n0 λ5 + (λ7 λ15 λ3 λ0 λ8 + λ27 λ5 λ9 λ5 + λ7 λ15 λ3 λ2 λ6 + λ15 λ3 λ7 λ5 λ3 )λ3 (F2 , F2 ); ∈ Ext6,42 ⎫A ⎧ 3 λ15 λ0 λ8 + [λ23 (λ1 λ4 λ2 + λ1 λ4 λ2 λ1 + λ1 λ5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ +λ λ λ ) + λ λ λ λ λ + λ λ λ λ ⎪ ⎪ 15 10 ⎪ ⎪ ⎬ ⎨ 2 +λ15 λ3 λ7 λ4 λ2 + λ7 λ5 λ3 λ9 ]λ7 + λ15 [λ0 λ2 λ1 λ5 (4) y = 2 ⎪ ⎪ ⎪ ⎪ +(λ02 λ4 + λ0 λ4 λ0 + λ4 λ02 )λ4 + λ0 λ2 λ6 ⎪ ⎪ ⎪ ⎪ +(λ λ + λ λ λ + λ λ )λ ⎪ ⎪ 5 0 ⎪ ⎪ ⎭ ⎩ 2 +(λ0 λ4 + λ4 λ0 )λ2 ] + λ15 λ4 λ2 λ1 ∈ Ext6,44 A (F2 , F2 ); (5) C = c1 λ7 λ5 + [(λ215 λ5 λ7 + λ15 λ11 λ7 λ9 + λ27 λ23 )λ5 + λ215 λ11 λ0 λ6 + λ7 λ23 λ15 λ21 + λ215 λ9 λ5 λ3 ∈ Ext6,56 A (F2 , F2 ); (6) G = {D1 (0)λ2 } ∈ Ext6,60 A (F2 , F2 ); (7) D2 = {λ47 λ11 λ40 } ∈ Ext6,64 A (F2 , F2 ); (8) A = {D1 (0)λ9 + λ47 d0 λ0 + λ215 λ211 λ6 λ3 } ∈ Ext6,67 A (F2 , F2 ); Pham Bich Nhu (9) A = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 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 + λ8 λ5 )λ7 + λ10 λ25 ] + λ215 λ11 λ2 λ1 λ17 +(λ15 λ11 λ7 λ9 λ8 + λ215 λ13 λ7 λ0 )λ11 + λ31 f0 λ12 +λ315 λ4 λ2 λ10 + D1 (0)λ9 + λ215 (λ13 λ7 λ4 λ7 +λ15 λ0 λ10 λ6 ) + [λ31 (λ23 λ9 + λ9 λ23 )λ9 + λ31 λ3 (λ9 λ5 + λ3 λ11 )λ7 + λ215 (λ11 λ8 + λ15 λ4 )λ6 ]λ6 +[λ215 (λ15 λ2 λ9 + λ15 λ10 λ1 ) + λ31 (λ23 λ11 λ8 +λ3 λ9 λ5 λ8 + λ33 λ6 + λ11 λ21 λ12 + (λ23 λ9 + λ9 λ23 )λ10 +λ7 (λ1 λ9 + λ9 λ1 )λ8 + λ7 (λ5 λ7 λ6 + λ9 λ5 λ4 ))]λ5 +[λ215 (λ15 λ2 λ11 + λ15 λ5 λ8 + λ15 λ8 λ5 + λ211 λ6 ) +λ31 (λ11 λ21 λ14 + λ23 λ8 λ13 + λ23 λ9 λ12 + λ3 λ11 λ2 λ11 +λ23 λ11 λ10 + λ3 λ9 λ5 λ10 + λ11 λ7 λ0 λ9 +λ7 λ5 λ7 λ8 + λ3 λ11 λ7 λ6 + λ11 λ7 λ4 λ5 )]λ3 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ∈ Ext6,67 A (F2 , F2 ); (10) A = {D1 (0)λ12 + λ215 λ11 λ7 λ28 + λ47 e0 λ0 } ∈ Ext6,70 A (F2 , F2 ); ⎫ ⎧ ⎨ f1 λ7 λ19 + g2 λ211 + (λ31 λ3 λ11 λ7 + λ23 λ15 λ9 λ5 )λ27 ⎬ (11) r1 = +λ215 λ29 λ7 λ11 + λ31 [λ27 λ0 λ14 + (λ23 λ9 + λ9 λ23 )λ13 ⎭ ⎩ +(λ23 λ11 + λ3 λ9 λ5 )λ11 ]λ7 ∈ Ext6,72 A (F2 , F2 ); λ47 λ23 λ2 λ1 λ15 + c2 (λ7 λ12 + λ19 λ0 )λ11 + D1 (0) λ19 + λ215 (λ27 λ0 λ7 + λ11 λ19 λ4 + λ11 λ23 λ0 )λ7 (12) x6,77 = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (13) x6,82 = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∈ Ext6,77 A (F2 , F2 ); ⎫ H1 (0)λ14 + λ15 λ11 λ13 + (λ15 λ11 λ15 λ9 ⎪ ⎪ ⎪ ⎪ +λ215 λ13 λ7 + λ31 λ7 λ23 λ4 λ0 + D3 (0)λ4 ⎪ ⎪ ⎪ 2 2 ⎪ +λ15 λ47 λ0 λ3 + λ47 λ3 λ6 + λ47 λ3 λ2 λ10 ⎪ ⎪ ⎪ 2 ⎪ +λ31 λ23 λ1 λ9 + λ47 λ11 λ0 λ9 + λ31 λ23 λ1 λ5 ⎬ 2 +λ31 λ23 λ5 λ3 + λ47 λ11 λ1 λ5 + λ47 λ3 λ7 λ5 λ3 )λ11 ⎪ ⎪ +(λ47 λ23 λ10 λ6 + λ47 λ7 λ1 λ8 λ6 + λ47 λ3 λ7 λ26 ⎪ ⎪ ⎪ +λ47 λ7 λ5 λ4 λ6 + λ31 λ23 λ1 λ9 λ5 + λ31 λ23 λ9 λ1 λ5 ⎪ ⎪ ⎪ ⎪ +λ15 λ47 λ0 λ4 λ3 + λ15 λ47 λ4 λ0 λ3 + λ15 λ47 λ1 λ5 ⎪ ⎪ ⎪ ⎭ +λ15 λ11 λ21 λ7 )λ7 ∈ Ext6,82 A (F2 , F2 ); (14) t1 = Sq t ∈ Ext6,84 A (F2 , F2 ); (15) x6,90 = d2 λ15 λ2 + [d2 λ16 + λ31 (λ7 λ23 λ8 +λ23 λ15 λ0 )λ15 + D3 (0)λ23 ]λ1 (F2 , F2 ); (16) C1 = Sq C ∈ Ext6,112 A ∈ Ext6,90 A (F2 , F2 ); On behavior of the sixth Lannes-Zarati homomorphism (17) x6,114 ⎧ [λ31 (λ23 λ15 λ19 λ13 + λ31 λ9 λ11 λ9 ) ⎪ ⎪ ⎨ +c3 (λ5 λ11 + λ9 λ7 )]λ7 + f2 λ15 λ9 = +c ⎪ (λ15 λ0 λ8 + λ15 λ2 λ6 + λ15 λ4 ) ⎪ ⎩ 2 +λ31 λ19 λ3 λ5 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ 6,114 (F2 , F2 ); ∈ ExtA (F2 , F2 ) (18) G1 = Sq G ∈ Ext6,120 A In [11], Hung-Peterson proved that ϕFs vanishes on decomposable elements for s > Therefore, it is sufficient to prove that ϕF6 is vanishing on indecomposable elements of Ext6,∗ A (F2 , F2 ) In order to show this claim, we prove that images of cycles which represented indecomposable elements of Ext6,t A (F2 , F2 ) under the homomorphism ϕ ˜F6 : Λ6 ⊗ F2 → (R6 F2 )# are trivial For conves,t nience, we write Exts,t A = ExtA (F2 , F2 ) Since the canonical projection π : Λs → Rs is an A-algebra homomorphism If λI contains a factor of negative excess, then ϕ˜Fs (λI ) = Moreover, the F2 actions of Sq on Exts,t A and on Rs F2 commute with each other through ϕs In the Lambda algebra, we have (see Wang [20], Lin-Mahowald [17], and Chen [3]) i , i ≥ • ϕF3 (ci ) = with ci = {(Sq )i (λ23 λ2 )} ∈ Ext3,11.2 A In fact, for c0 = λ23 λ2 , we have ϕ˜F3 (c0 ) = since e(c0 ) = −2 < This implies ϕF3 (c0 ) = Then, ϕF3 (ci ) = ϕF3 ((Sq )i (c0 )) = (Sq )i (ϕF3 (c0 )) = i • ϕF4 (di ) = with di = {(Sq )i (λ23 λ2 λ6 +λ23 λ24 +λ3 λ5 λ4 λ2 )} ∈ Ext4,18.2 ,i ≥ A By direct inspection, we have ϕF4 (d0 ) = 0, so ϕF4 (di ) = ϕF4 ((Sq )i (d0 )) = (Sq )i (ϕF4 (d0 )) = • ϕF4 (ei ) = with i , i ≥ ei = {(Sq )i (λ33 λ8 +(λ3 λ25 +λ23 λ7 )λ4 +(λ23 λ9 +λ9 λ23 )λ2 )} ∈ Ext4,22.2 A Since ϕ˜F4 (λ33 λ8 ) = 0, ϕ ˜F4 (λ3 λ25 λ4 ) = 0, ϕ ˜F4 (λ23 λ7 λ4 ) = and ϕ˜F4 (λ23 λ9 λ2 ) = 0, we have ϕ ˜F4 (e0 ) = ϕ ˜F4 (λ9 λ3 λ3 λ2 ) Applying the Adem relation, we have λ9 λ3 = λ7 λ5 Then ϕ ˜F4 (e0 ) = ϕ˜F4 (λ9 λ3 λ3 λ2 ) = ϕ ˜F4 (λ7 λ5 λ3 λ2 ) = Hence, ϕF4 (e0 ) = and then ϕF4 (ei ) = ϕF4 ((Sq )i (e0 )) = (Sq )i (ϕF4 (e0 )) = Pham Bich Nhu • ϕF4 (fi ) = with i fi = {(Sq )i (λ27 λ0 λ4 + (λ23 λ9 + λ7 λ5 λ3 )λ3 + λ27 λ22 )} ∈ Ext4,22.2 , i ≥ A By direct inspection, we have ˜F4 (λ23 λ9 λ3 ) = 0, ϕ ˜F4 (λ7 λ5 λ23 ) = 0, and ϕ ˜F4 (λ27 λ22 ) = 0, ϕ˜F4 (λ27 λ0 λ4 ) = 0, ϕ then ϕ˜F4 (f0 ) = 0, and so ϕF4 (fi ) = ϕF4 ((Sq )i (f0 )) = (Sq )i (ϕF4 (f0 )) = Similarly, by direct inspection, we also have • ϕF4 (gi+1 ) = with gi+1 = (Sq )i (λ27 λ0 λ6 + (λ23 λ9 + λ7 λ5 λ3 )λ5 i +(λ3 λ9 λ5 + λ23 λ11 )λ3 ) ∈ Ext4,24.2 , i ≥ A i , i ≥ • ϕF4 (D3 (i)) = with D3 (i) = {(Sq )i (λ31 λ7 λ23 λ0 )} ∈ Ext4,65.2 A Applying the Adem relation, we have ϕ˜F4 (D3 (0)) = ϕ˜F4 (λ31 λ7 λ23 λ0 ) = ϕ˜F4 (λ15 λ23 λ23 λ0 ) = Then ϕF4 (D3 (i)) = ϕF4 ((Sq )i (D3 (0))) = (Sq )i (ϕF4 (D3 (0))) = It is easy to check these details, • ϕF5 (ni ) = with ni = {(Sq )i (λ27 λ5 λ3 λ9 + λ7 λ15 λ3 λ0 λ6 + λ7 λ15 λ1 λ5 λ3 )} i , i ≥ ∈ Ext5,36.2 A i , i ≥ • ϕF5 (D1 (i)) = with D1 (i) = {(Sq )i (λ215 λ11 λ7 λ4 )} ∈ Ext5,57.2 A • ϕF5 (H1 (i)) = with H1 (i) = (Sq )i (λ215 λ11 λ7 λ14 + λ215 λ211 λ10 i +λ15 λ31 λ7 λ1 λ8 + λ15 λ31 λ3 λ7 λ6 + λ15 λ31 λ7 λ5 λ4 )} ∈ Ext5,67.2 , i ≥ A Using Adem relations, we have λ23 λ1 = λ11 λ13 + λ7 λ17 + λ3 λ21 ; λ23 λ4 = λ15 λ12 + λ11 λ16 + λ9 λ18 ; λ47 λ11 = λ31 λ27 + λ23 λ35 ; λ31 λ3 = λ15 λ19 + λ7 λ27 ; (1’) (2’) (3’) (4’) 10 On behavior of the sixth Lannes-Zarati homomorphism λ31 λ9 = λ23 λ17 + λ19 λ21 ; (5’) λ31 λ11 = λ23 λ19 ; (6’) λ31 λ7 = λ15 λ23 ; (7’) λ47 λ3 = λ15 λ35 + λ7 λ43 (8’) λ47 λ7 = λ15 λ39 ; (9’) λ27 λ0 = λ13 λ14 + λ11 λ16 + λ5 λ22 + λ3 λ24 + λ1 λ26 ; (10’) λ23 λ0 = λ11 λ12 + λ9 λ14 + λ7 λ16 + λ3 λ20 + λ1 λ22 ; (11’) λ11 λ1 = λ3 λ9 (12’) Now, we prove that ϕ ˜F6 sends the above eighteen indecomposable elements (from (1) to (18)) to zero • By replacing (11’) in (1), combined with results ϕF4 (e0 ) = 0, ϕF4 (f0 ) = 0, we have ϕ˜F6 (r) = Then ϕF6 (r) = ˜F6 (q) = Then • By direct inspection and ϕF4 (g1 ) = 0, we imply ϕ F2 ϕ6 (q) = • From result ϕF5 (n0 ) = and the excess of other terms is negative Therefore, ϕ˜F6 (t) = 0, then ϕF6 (t) = • By replacing (1’) and (2’) in (4), then excess of all terms of y is negative Therefore, ϕ˜F6 (y) = 0, imply ϕF6 (q) = • From the result, ϕF4 (e1 ) = and by direct inspection, under ϕF6 , the image of element C is trivial • From the result, ϕF5 (D1 (0)) = and by direct inspection, under ϕF6 , the image of element G is trivial • From (3’) and (10’), we have ˜F6 (λ31 λ27 λ40 + λ23 λ35 λ40 ) ϕ ˜F6 (D2 ) = ϕ˜F6 (λ47 λ11 λ40 ) = ϕ = ϕ˜F6 (λ31 (λ13 λ14 + λ11 λ16 + λ5 λ22 + λ3 λ24 + λ1 λ26 )λ30 + λ23 λ35 λ40 ) = Then ϕF6 (D2 ) = • From results ϕF5 (D1 (0)) = and ϕF4 (d0 ) = 0, we have ϕ˜F6 (A) = Therefore, ϕF6 (A) = • Using (4’), (5’), (6’), (7’) and the results ϕF3 (c2 ) = 0, ϕF4 (f0 ) = 0, ϕF5 (D1 (0)) = 0, we have ϕ˜F6 (A ) = Then, ϕF6 (A ) = Pham Bich Nhu 11 • Similarly, taking (8’), (9’),(12’) and D1 (0) replace on A , we have A = {D1 (0)λ12 + λ215 λ11 λ7 λ28 + λ47 e0 λ0 } = {λ215 λ11 λ7 λ4 λ12 + λ215 λ11 λ7 λ28 + λ47 (λ33 λ8 + (λ3 λ25 + λ23 λ7 )λ4 + (λ23 λ9 + λ9 λ23 )λ2 )λ0 } = {λ215 λ11 λ7 λ4 λ12 + λ215 λ11 λ7 λ28 + λ47 (λ33 λ8 + (λ3 λ25 + λ23 λ7 )λ4 + (λ3 λ11 λ1 + λ7 λ5 λ3 )λ2 )λ0 } = {λ215 λ11 λ7 λ4 λ12 + λ215 λ11 λ7 λ28 + (λ15 λ35 + λ7 λ43 )λ23 λ8 + (λ15 λ35 + λ7 λ43 )λ25 + (λ15 λ35 + λ7 λ43 )λ3 λ7 )λ4 + (λ15 λ35 + λ7 λ43 )λ11 λ1 + λ15 λ39 λ5 λ3 )λ2 )λ0 } Obviously, the excess of all terms of A is negative Therefore, ϕ˜F6 (A ) = 0, then ϕF6 (A ) = • Since ϕF4 (f1 ) = 0, ϕF5 (g2 ) = and the excess of the other terms of r1 is negative Therefore, ϕ˜F6 (r1 ) = 0, then ϕF6 (r1 ) = • Depend on the results ϕF3 (c2 ) = 0, ϕF5 (D1 (0)) = and (8’) It is easy to prove that under ϕ˜F6 , image of x6,77 is trivial, then ϕF6 (x6,77) = • Since ϕF5 (H1 (0)) = 0, ϕF5 (D3 (0)) = and relations (3’),(8’) and (9’), we have ϕF6 (x6,82 ) = • The actions of Sq on Exts,t A and on Rs F2 commute with each other through ϕFs Therefore, ϕF6 (t1 ) = ϕF6 (Sq t) = Sq (ϕF6 (t)) = • From results ϕF4 (D3 (0)) = and ϕF4 (d2 ) = 0, we have ϕ˜F6 (x6,90) = Therefore, ϕF6 (x6,90 ) = • Similarly, ϕF6 (C1 ) = ϕF6 (Sq C) = Sq (ϕF6 (C)) = • From results ϕF3 (c3 ) = and ϕF4 (f2 ) = 0, we have ϕ˜F6 (x6,114) = Therefore, ϕF6 (x6,114) = • Similarly, ϕF6 (G1 ) = ϕF6 (Sq G) = Sq (ϕF6 (G)) = The proof is complete Acknowledgement The authors would like to thank Phan Hoang Chon and Nguyen Sum for many fruitful discussions The paper was completed while the author is a Ph.D student in Mathematics Departments at Quy Nhon University She thanks Quy Nhon University and Can Tho University for support and hospitality 12 On behavior of the sixth Lannes-Zarati homomorphism References [1] A K Bousfield, E B Curtis, D M Kan, D G Quillen, D L Rector, and J W Schlesinger, The mod-p lower central series and the Adams spectral sequence, Topology (1966), 331-342 [2] A K Bousfield and D M Kan, The homotopy spectral sequence of a space with coefficients in a ring, Topology 11 (1972), 79-106 [3] T W Chen, Determination of Ext5,∗ A (F2 , F2 ), Topology Appl 158 (2011), no 5, 660689 [4] T W Chen, Indecomposable elements in Ext6,∗ A (F2 , F2 ), Preprint (2013) [5] P H Chon and L M H` a, Lambda algebra and the Singer transfer, C R Acad Sci Paris Ser I, 349 (2011), no 1-2, 21-23 [6] P H Chon and P B Nhu, The cohomology of the Steenrod algebra and the mod-p Lannes-Zarati homomorphism, Journal of Algebra, 556 (2020), 656-695 [7] P H Chon and P B Nhu, On the mod-p Lannes-Zarati homomorphism, Journal of Algebra, 537 (2019), 316-342 [8] E B Curtis, The Dyer-Lashof algebra and the Λ-algebra, Illinois J Math 19 (1975), 231-246 [9] N D H Hai, R´ esolution injective instable de la cohomologie modulo p d’un spectre de Thom et application, Ph.D thesis, Universit´e Paris 13, 2010 [10] N H V Hung, On triviality of Dickson invariants in the homology of the Steenrod algebra, Math Proc Cambridge Philos Soc 134 (2003), no 1, 103-113 [11] N H V Hung and F P Peterson, Spherical classes and the Dickson algebra, Math Proc Cambridge Philos Soc 124 (1998), no 2, 253-264 [12] N H V Hung, V T N Quynh, and N A Tuan, On the vanishing of the Lannes-Zarati homomorphism, C R Acad Sci Paris Ser I, 352 (2014), no 3, 251-254 [13] N H V Hung and N Sum, On Singer’s invariant-theoretic description of the Lambda algebra: a mod-p analogue, J Pure Appl Algebra 99 (1995), no 3, 297-329 [14] J Lannes, Sur le n-dual du n-` eme spectre de Brown-Gitler, Mathematische Zeitschrift 199 (1988), no 1, 29-42 [15] J Lannes and S Zarati, Invariants de Hopf d’ordre sup´ erieur et suite spectrale d’Adams, C R Acad Sci Paris Ser I Math 296 (1983), no 15, 695-698 [16] J Lannes and S Zarati, Sur les foncteurs drivs de la dstabilisation, Math Z 194 (1987), no 1, 25-59 [17] W H Lin and M Mahowald, The Adams spectral sequence for Minamis theorem, Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997), Contemp Math., vol 220, Amer Math Soc., Providence, RI, 1998, pp 143-177 [18] W M Singer, The construction of certain algebras over the Steenrod algebra, J Pure Appl Algebra 11 (1977/78), no 1-3, 5359 [19] W M Singer, Invariant theory and the Lambda algebra, Trans Amer Math Soc 280 (1983), no 2, pp 673-693 [20] J S P Wang, On the cohomology of the mod-2 Steenrod algebra and the nonexistence of elements of hopf invariant one, Illinois Journal of Mathematics 11(1967), no 3, 480-490 [21] S Zarati, D´ eriv´ es du foncteur de d´ estabilisation en caract´ eristiques impaire et application, Ph.D thesis, Universit´e Paris-Sud (Orsay), 1984  ... 2t+s The proof of Theorem 1.1 In this section, we use the chain-level representation map of the ϕFs constructed in the previous section to investigate the behavior of the sixth Lannes- Zarati homomorphism. .. to the study of the image of the Hurewicz map and then Curtis’s conjecture on the spherical classes [8] (see [7] for discussion) The Lannes- Zarati homomorphism was first constructed by Lannes- Zarati. .. (Q0 S ), on the base-point component Q0 S of the infinite loop space QS = limΩn Σn S −→ (see Lannes and Zarati [14], [15] for the sketch of proof) Therefore, the study of the Lannes- Zarati homomorphism