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bstract. The mod 2 LannesZarati homomorphism was constructed in 21, which is considered as a graded associated version of the mod 2 Hurewicz map in the E2term of Adams spectral sequence. The map is studied by many authors such as LannesZarati 21, Hưng 14, 15, 16, Hưng et. al. 18, ChơnTri´êt 6. In this paper, we construct an analogue ϕs for p odd, and we also investigate the behavior of this map for s ≤ 3
ON THE MOD p LANNES-ZARATI HOMOMORPHISM PHAN H. CHƠN AND ĐỒNG T. TRIẾT Abstract. The mod 2 Lannes-Zarati homomorphism was constructed in [21], which is considered as a graded associated version of the mod 2 Hurewicz map in the E2 -term of Adams spectral sequence. The map is studied by many authors such as Lannes-Zarati [21], Hưng [14], [15], [16], Hưng et. al. [18], Chơn-Tri´ êt [6]. In this paper, we construct an analogue ϕs for p odd, and we also investigate the behavior of this map for s ≤ 3. 1. Introduction and statement of results For any pointed space X, let QX = limΩn Σn X be the infinite loop space of X. −→ An element ξ ∈ H∗ QX = H∗ (QX; Fp ) is called a spherical class if there exists an / H∗ QX element η ∈ π∗ (QX) = π∗S (X) such that h∗ (η) = ξ, where h∗ : π∗ (QX) is the Hurewicz map. For p = 2, work of Curtis [10] shows that the Hopf invariant one elements and the Kervaire invariant one elements in π∗ (Q0 S0 ) (if they exist) are those whose images are nontrivial in H∗ Q0 S 0 under the mod 2 Hurewicz map / H∗ Q0 S 0 ; and he conjectured that there are only spherical classes h∗ : π∗ (Q0 S 0 ) 0 in H∗ Q0 S those are detected by the Hopf invariant one elements and the Kervaire invariant one elements, where Q0 S 0 is the component of QS 0 containing the basepoint. Later, Wellington [30] generalized the Curtis’ result for p > 2 and he was led to an analogue conjecture. These conjectures are called the classical conjecture on spherical classes. An algebraic approach to attack the conjecture is to study the graded associated / H∗ Q0 S 0 in E2 -term of the Adams of the mod p Hurewicz map h∗ : π∗ (Q0 S 0 ) spectral sequence. For p = 2, this homomorphism was constructed by Lannes and Zarati [21], the so-called Lannes-Zarati homomorphism. In more detail, for each s ≥ 1, there is a homomorphism of Singer’s type / Ann(Rs (F2 )# )t ∼ ϕs : Exts,s+t (F2 , F2 ) = Ann(D[s]# )t , A where D[s] is the (graded) dual of the Dickson algebra D[s], Rs is the Singer’s functor (see [21]); and we denote Ann(M ) the subspace of M consisting of all elements annihilated by all positive elements in the Steenrod algebra A. The behavior of ϕs is actually studied by Lannes-Zarati [21] (for s ≤ 2), Hưng [14] (for s = 3), Hưng [16] (for s = 4), Hưng-Quỳnh-Tu´ân [18] (for s = 5) and Chơn-Tri´êt [6] (for s = 6 and stem ≤ 114). # Date: February 2, 2015. 2010 Mathematics Subject Classification. Primary 55P47, 55Q45; Secondary 55S10, 55T15. Key words and phrases. Spherical classes, Hurewizc map, Lannes-Zarati homomorphism, Adams spectral sequence. This work is partial supported by a NAFOSTED gant. 1 2 P. H. CHƠN AND Đ. T. TRIẾT For p > 2, from results of Zarati [31], the sth left derived functor Ds (Σ1−s Fp ) of the destabilization functor is isomorphic to ΣRs (Fp ) ∼ = ΣB[s], where B[s] is the image of the restriction from cohomology of the symmetric group Σps to the cohomology of the elementary p-group of rank s, Es [25]. Therefore, there exists an analogue homomorphism, which is also called Lannes-Zarati homomorphism, / Ann(B[s]# )t . ϕs : Exts,s+t (Fp , Fp ) A Using Goodwillie towers, Kuhn also pointed out the existence of ϕs as a graded associated version of the Hurewicz map in the E2 -term of the Adams spectral sequence [20]. In addition, the method of Kuhn can be apply to other generalized cohomology theories. In the paper, we are interested in the study of the mod p Lannes-Zarati homomorphism for p odd. We show that, up to a sign, the canonical inclusion B[s] → Γ+ s is the chain-level representation of the dual of ϕs , ϕ# s : Fp ⊗A B[s] / TorA s (Fp , Fp ), where Γ+ = ⊕s≥0 Γ+ s is the Singer-Hưng-Sum chain complex [19]. In more detail, we obtain the following theorem, which is the first main result of the paper. Theorem 4.6. The inclusion map ϕ˜# s : B[s] γ → (−1) / Γ+ s given by s(s−1) +(s+1) deg γ 2 γ is the chain-level representation of the dual of the Lannes-Zarari homomorphism ϕ# s . The theorem is an extended of Theorem 3.9 in [15] for p odd. Let Λopp be the opposite algebra of the Lambda algebra defined by Bousfield et. al. [3], and let R be the Dyer-Lashof algebra, which is the algebra of homology operations acting on the homology of infinite loop spaces. The algebra R is also isomorphic to a quotient of Λopp [10], [30]. It is well-known that B[s]# ∼ = Rs [8] and (Γ+ )# ∼ = Λopp [19], where Rs is the subspace of R spanned by all monomials of Rs is length s. Therefore, in the dual, up to a sign, the canonical projection Λopp s the chain-level representation map of the Lannes-Zarati homomorphism ϕs , which is given by the following corollary. Corollary 4.7. The projection ϕ˜s : Λopp s ϕ˜s (λI ) = (−1) / Rs given by s(s−1) +(s+1) deg(λI ) 2 QI is the chain-level representation of the Lannes-Zarati homomorphism ϕs . From Liulevicius [22], [23] and May [24], there exists the the power operation P 0 acting on the cohomology of the Steenrod algebra Exts,s+t (Fp , Fp ) whose chainA level representation in the cobar complex is induced from the Frobenius map. Since its representation in Λopp induces naturally an operation in R (see Lemma 5.1), and the latter is compatible with the A-action on R (see Lemma 5.2), then there exists an power operation acting on Ann(Rs ), which is also denoted by P 0 . Furthermore, these power operations commute with each other through the Lannes-Zarati homomorphism ϕs (see Proposition 5.3). Using these results to study the behavior of ϕs , we obtain the following results. ON THE MOD p LANNES-ZARATI HOMOMORPHISM 3 Theorem 6.1. The first Lannes-Zarati homomorphism / Ann(B[1]# )t ϕ1 : Ext1,1+t (Fp , Fp ) A is isomorphic. This result is an analogue of the case p = 2 [21]. The behavior of ϕ2 is given by the theorem. Theorem 6.2. The second Lannes-Zarati homomorphism 2,2+t / Ann(B[2]# )t ϕ2 : ExtA (Fp , Fp ) is vanishing for t = 0 and t = 2(p − 1)pi+1 − 2, i ≥ 0. From the result of Wellington [30], Ann(R2 ) is nontrivial at stem t = 0, t = 2(p−1)pi+1 −2 and t = 2(p−1)p(pi +· · ·+1). Therefore, ϕ2 is not an epimorphism. The behavior of ϕ3 is given by the following theorem, which is the final result of this work. Theorem 6.4. The third Lannes-Zarati homomorphism 3,3+t / Ann(B[3]# )t ϕ3 : ExtA (Fp , Fp ) is vanishing for all t > 0. From the above results, we observe that the map ϕs , for s ≤ 3, is only nontrivial in positive stem corresponding with the Hopf invariant one and Kervaire invariant one elements (if they exist). Based on these results together with the classical conjecture on spherical classes and Hưng’s conjecture [14, Conjecture 1.2], we are led to a conjecture, which is considered as a graded associated version of the classical one on the spherical classes in E2 -term of the Adams spectral sequence, as follows. Conjecture 1.1. The homomorphism ϕs vanishes in any positive stem t for s ≥ 3. Of course, the classical conjecture on spherical class is not a consequence of Conjecture 1.1. But if Conjecture 1.1 were false on a permanent cycle in Exts,s+t (Fp , Fp ), A then the classical conjecture on spherical classes could be also false. The Singer transfer was introduced by Singer [29] (for p = 2) and Crossley [9] (for p > 2), which is given by, for s ≥ 1, / Exts,s+t T rs : [Ann(Ht BEs )]GLs (Fp , Fp ), A where GLs is the general linear group. The results of Singer [29], Boardman [2], Hà [12], Nam [27], Chơn-Hà [4], [5] (for p = 2) and Crossley [9] (for p odd) showed that the image of Singer transfer is a big enough and worthwhile to pursue subgroup of the Ext group. It is well-known that the Ext group is too mysterious to understand, although it is intensively studied. In order to avoid the shortage of our knowledge of the Ext group, we want to restrict ϕs on the image of the Singer transfer. Then we have a weak version of Conjecture 1.1. Conjecture 1.2. The composition js := ϕs ◦ T rs : [Ann(Ht BEs )]GLs / Ann(B[s]# )t is vanishing in any positive degree t for s ≥ 3. From Theorem 4.6, Theorem A.2 and Proposition A.12, it is clear that, up to a sign, the canonical inclusion B[s] → H ∗ BEs is the chain-level representation of the dual of js . Thus, Conjecture 1.2 is equivalent to the following conjecture. 4 P. H. CHƠN AND Đ. T. TRIẾT ¯ ∗ BEs , where A¯ is the augmentation ideal Conjecture 1.3. For s ≥ 3, B[s] ⊂ AH of A. For p = 2, Conjecture 1.2 and Conjecture 1.3 appeared in [14, Conjecture 1.3 and Conjecture 1.5], which were showed by Hưng-Nam in [17]. The paper is organized as follows. Section 2 is a preliminary on the SingerHưng-Sum chain complex, the Lambda algebra and the Dyer-Lashof algebra. In Section 3 and 4, we construct the mod p Lannes-Zarati homomorphism and its chain-level representation. Section 5 is devoted to develop the power operations. The behavior of the Lannes-Zarati homomorphism is investigated in Section 6. The chain-level representation of the Singer transfer in Singer-Hưng-Sum chain complex is established in Appendix section. 2. Preliminaries In this section, we recall some preliminaries about the Singer-Hưng-Sum chain complex, the Lambda algebra as well as the Dyer-Lashof algebra (see [19], [3], [28] and [8] for more detail). 2.1. The Singer-Hưng-Sum chain complex. Let Es be the s-dimensional Fp vector space, where p is an odd prime number. It is well-known that the mod p cohomology of the classifying space BEs is given by Ps := H ∗ BEs = E(x1 , · · · , xs ) ⊗ Fp [y1 , · · · , ys ], where (x1 , · · · , xs ) is the basis of H 1 BEs = Hom(Es , Fp ), and yi = β(xi ) for 1 ≤ i ≤ s with β the Bockstein homomorphism. Let GLs denote the general linear group GLs = GL(Es ). The group GLs acts on Es and then on H ∗ BEs according to the following standard action ais yi , (aij )ys = ais xi , (aij )xs = i (aij ) ∈ GLs . i The algebra of all invariants of H ∗ BEs under the actions of GLs is computed by Dickson [11] and Mùi [25]. We briefly summarize their results. For any n-tuple of rj non-negative integers (r1 , . . . , rs ), put [r1 , · · · , rs ] := det(yip ), and define Ls,i := [0, · · · , ˆi, . . . , s]; Ls := Ls,s ; qs,i := Ls,i /Ls , for any 1 ≤ i ≤ s. In particular, qs,s = 1 and by convention, set qs,i = 0 for i < 0. Degree of qs,i is 2(ps − pi ). Define Vs := Vs (y1 , · · · , ys ) := (λ1 y1 + · · · + λs−1 ys−1 + ys ). λj ∈Fp Another way to define Vs is that Vs = Ls /Ls−1 . Then qs,i can be inductively expressed by the formula p qs,i = qs−1,i−1 + qs−1,i Vsp−1 . ON THE MOD p LANNES-ZARATI HOMOMORPHISM 5 For non-negative integers k, rk+1 , . . . , rs , set [k; rk+1 , · · · , rs ] := 1 k! x1 · x1 y1p rk+1 · rn y1p ··· ··· ··· ··· ··· ··· xs · xs ysp rk+1 . · rs ysp For 0 ≤ i1 < · · · < ik ≤ s − 1, we define Ms;i1 ,...,ik := [k; 0, · · · , ˆi1 , · · · , ˆik , · · · , s − 1], Rs;i1 ,··· ,ik := Ms;i1 ,...,ik Lsp−2 . The degree of Ms;i1 ,··· ,ik is k + 2((1 + · · · + ps−1 ) − (pi1 + · · · + pik )) and then the degree of Rs;i1 ,··· ,ik is k + 2(p − 1)(1 + · · · + ps−1 ) − 2(pi1 + · · · + pik ). The subspace of all invariants of H ∗ BEs under the action of GLs is given by the following theorem. Theorem 2.1 (Dickson [11], Mùi [25]). (1) The subspace of all invariants under the action of GLs of Fp [x1 , · · · , xs ] is given by D[s] := Fp [x1 , · · · , xs ]GLs = Fp [qs,0 , · · · , qs,s−1 ]. (2) As a D[s]-module, (H ∗ BEs )GLs is free and has a basis consisting of 1 and all elements of {Rs;i1 ,··· ,ik : 1 ≤ k ≤ s, 0 ≤ i1 < · · · < ik ≤ s − 1}. In other words, s (H ∗ BEs )GLs = D[s] ⊕ Rs;i1 ,··· ,ik D[s]. k=1 0≤i1 deg(x), x ∈ M }, which is a sub-A-module of M because of the Adem relations. The functor D is right exact and admits left derived functors Ds , s ≥ 0. Then Ds (M ) = Hs (D(F (M ))), for F (M ) the free resolution (or projective resolution) of M . / Dr−1 (P1 ⊗ M ) to be the connecting homomorDefine α1 (M ) : Dr (Σ−1 M ) phism of the functor D(−) associated to the short exact sequence 0 → P1 ⊗ M → Pˆ ⊗ M → Σ−1 M → 0, where Pˆ is the A-module extended of P1 by formally adding a generator x−1 1 u1 of −1 n(p−1)−1 x u1 degree −1. The action of A on Pˆ is given by setting P n (x−1 u ) = 1 1 1 n −1 and β(x1 u1 ) = 1, while the summand P1 has its usual A-action. Put αs (M ) = α1 (Ps−1 ⊗ M ) ◦ · · · ◦ α1 ( Σ−(s−1) M ), / Dr−s (Ps ⊗ M ). then αs (M ) : Dr (Σ−s M ) / D0 (Ps ⊗ M ). When r = s, we obtain αs (M ) : Ds (Σ−s M ) Theorem 3.1 ([31, Theórème 2.5]). For any M ∈ U, the homomorphism αs (ΣM ) : / ΣRs M is an isomorphism of unstable A-modules, where Rs (−) is Ds (Σ1−s M ) the Singer functor. When M = Fp , Hải [13] showed that Rs (Fp ) ∼ = B[s]. Therefore, we have the following corollary. ∼ = Corollary 3.2. For s ≥ 0, αs := αs (ΣFp ) : Ds (Σ1−s Fp ) − → ΣB[s]. / Fp ⊗A M factors Because of the definition of the functor D, the projection M through DM . Then it induces a commutative diagram ··· ··· / D(Fs M ) / D(Fs−1 M ) / ··· is−1 is / Fp ⊗A Fs−1 M / Fp ⊗A Fs M / ··· . Here horizontal arrows are induced by the differential of F M , and i∗ is given by is ([z]) = [1 ⊗A z]. Taking the homology, we get / TorA s (Fp , M ). is : Ds (M ) Since, for z ∈ Fs M and a > 0, is (Sq a [z]) = is ([Sq a z]) = [1 ⊗A Sq a z] = [0] ∈ F2 ⊗A Fs M , the induced map of is in homology factors through Fp ⊗A Hs (D(Fs M )). Therefore, we have following commutative diagram Ds (M ) is / TorA s? (Fp , M ) ¯is Fp ⊗A Ds (M ) ON THE MOD p LANNES-ZARATI HOMOMORPHISM When M = Σ1−s F2 , we obtain ¯is : Fp ⊗A Ds (Σ1−s Fp )) 9 1−s / TorA Fp ). s (Fp , Σ For each s ≥ 1, we define −1¯ is (1 ⊗A αs−1 )Σ : Fp ⊗A B[s] ϕ# s := Σ −s / TorA Fp ) s (Fp , Σ In the dual, we have the Lannes-Zarati homomorphism for p odd / Ann(B[s]# )t . ϕs : Exts,s+t (Fp , Fp ) A In [20], Kuhn showed that the map ϕs , for s ≥ 1, is the graded associated version / H∗ Q0 S 0 in the E2 -term of the Adams of the mod p Hurewizc map h∗ : π∗ (Q0 S 0 ) spectral sequence. 4. The chain-level representation of ϕs In this section, we construct the chain-level representation of ϕ# s in the SingerHưng-Sum chain complex as well as the chain-level representation of ϕs in the opposite algebra of the Lambda algebra. For M ∈ M, recall that B∗ (M ) := ⊕s≥0 Bs (M ) is the usual bar resolution of M with Bs (M ) = A ⊗ A¯ ⊗ · · · ⊗ A¯ ⊗M, s times where A¯ is the augmentation ideal of A, which is the ideal of A generated by all positive degree elements in A. The element a0 ⊗ a1 ⊗ · · · ⊗ as ⊗ m ∈ Bs (M ) has homological degree s and internal degree t = i deg(ai ) + deg(m). The total degree is s + t, i.e. deg(a0 ⊗ a1 ⊗ · · · ⊗ as ⊗ m) = s + deg(ai ) + deg(m). i The A-action on B∗ (M ) is given by a(a0 ⊗ a1 ⊗ · · · ⊗ as ⊗ m) = aa0 ⊗ a1 ⊗ · · · ⊗ as ⊗ m, and the differential of B∗ (M ) is given by s−1 (−1)ei a0 ⊗ · · · ⊗ ai ai+1 ⊗ · · · ⊗ m ∂(a0 ⊗ a1 ⊗ · · · ⊗ as ⊗ m) = i=0 − (−1)es−1 a0 ⊗ a1 ⊗ · · · ⊗ as m, where ei = deg(a0 ⊗ · · · ⊗ ai ). Since B∗ (M ) is the free resolution of M , by definition one has TorA s (N, M ) = Hs (N ⊗A B∗ (M )). As B[s] ⊂ Γ+ s , for γ ∈ B[s], γ has an unique expansion (p−1)i1 − γ= u11 v1 I=( 1 · · · uss vs(p−1)is − s . 1 ,i1 ,..., s ,is )∈I Put (−1)e(I) β 1− 1 P i1 ⊗ · · · ⊗ β 1− s P is ⊗Σ1−s 1 ∈ Bs−1 (Σ1−s Fp ), γ˜ := I∈I where e(I) = s + 1 + ··· + s + i1 + · · · + is . 10 P. H. CHƠN AND Đ. T. TRIẾT Lemma 4.1. The element γ˜ ∈ EBs−1 (Σ1−s Fp ). Proof. Fron the proofs of Lemma A.9, Lemma A.10 and Proposition A.12, one gets that the exponents of vi ’s in the expansion of γ are nonnegative. Therefore γ˜ ∈ Bs−1 (Σ1−s Fp ). By the action of A, (−1)e(I) β 1− 1 P i1 (1 ⊗ β 1− 2 P i2 ⊗ · · · ⊗ β 1− s P is ⊗Σ1−s 1). γ˜ = I∈I Therefore, it is sufficient to show that s 2i1 + (1 − (2ik (p − 1) + (1 − 1) > k )) + 1 − s, k=2 it is equivalent to s 2i1 − 1 s 2ik (p − 1) − > k=1 k. (4.1) k=2 Also from the proofs of Lemma A.9, Lemma A.10 and Proposition A.12, we ob(p−1)is − s (p−1)i1 − 1 , serve that qs,i and Rs;i,j can be written in the sum of u11 v1 · · · uss vs where ( 1 , i1 , . . . , s , is ) satisfies (4.1), therefore so is γ. Lemma 4.2. The element γ˜ is a cycle in EBs−1 (Σ1−s Fp ). Proof. Let /A Ω : ∆+ 2 (p−1)i1 − u11 v1 1 (p−1)i2 − u22 v2 2 → (−1) 1 +i1 +i2 β 1− 1 P i1 β 1− 2 P i2 . From the result of Ciampell and Lomonaco [7], one gets Γ2 ⊂ KerΩ. / ∆+ ⊗ ∆+ ⊗ ∆+ Consider the diagonal map ψ : ∆+ s q−1 2 s−q−1 defined by (p−1)i − k k k ⊗ 1 ⊗ 1, k < q, uk vk (p−1)ik − k k (p−1)ik − k k ψ(uk vk )= 1 ⊗ uk−q+1 vk−q+1 ⊗ 1, i ≤ k ≤ i + 1, (p−1)i − k 1 ⊗ 1 ⊗ uk−q−1 vk−q−1k k , k > i + 1. From results of Hưng and Sum [19, Corollary 3.4], ψ(Γs ) ⊂ Γi−1 ⊗ Γ2 ⊗ Γs−i−1 . Define the homomorphism / A⊗(s−1) = A ⊗ · · · ⊗ A, πs,q : ∆+ s s−1 times given by (p−1)i1 − πs,q (u11 v1 = (−1) 1 (p−1)iq+1 − q+1 · · · uqq vq(p−1)iq − q uq+1 vq+1 q+1 · · · uss vs(p−1)is − s ) 1 +···+ q +i1 +···+is × β 1− 1 P i1 ⊗ · · · ⊗ β 1− q P iq β 1− q+1 P iq+1 ⊗ · · · ⊗ β 1− s P is . It is easy to see that π2,1 = Ω. Moreover, if we define ωt , ωt : ∆+ t (p−1)i1 − ωt (u11 v1 1 (p−1)i1 − ωt (u11 v1 (p−1)it − · · · ut t vt 1 t ) = (−1)i1 +···+it β 1− 1 P i1 ⊗ · · · ⊗ β 1− t P it , (p−1)it − · · ·ut t vt = (−1) / A⊗t given by t ) 1 +···+ t +i1 +···+it β 1− 1 P i1 ⊗ · · · ⊗ β 1− t P it , ON THE MOD p LANNES-ZARATI HOMOMORPHISM 11 then πs,q = (ωq−1 ⊗ π2,1 ⊗ ωs−q−1 )ψ. Since π2,1 (Γ2 ) = 0, then πk,q (Γk ) = 0. By the definition of the differential in the bar resolution, one gets s−1 ∂(˜ γ ) = (−1) deg γ ˜ +s (πs,q ⊗ idΣ1−s Fp )(γ ⊗ Σ1−s 1). q=1 Since γ ∈ B[s] ⊂ Γs , then πs,q (γ) = 0. Therefore, ∂(˜ γ ) = 0. For any A-module M , from the definition of the functor D, one gets the short exact sequence of chain complexes / B∗ (M ) / D(B∗ (M )) / 0. / EB∗ (M ) 0 Because B∗ (M ) is acyclic, for s ≥ 1, the connecting homomorphism ∼ = ∂∗ : Hs (D(B∗ (M )) − → Hs−1 (EB∗ (M )) (4.2) is isomorphic. Letting M = Σ1−s Fp , one gets ∼ = ∂∗ : Ds (Σ1−s Fp ) − → Hs−1 (EBs−1 (Σ1−s Fp )). Lemma 4.3. For γ ∈ B[s], ∂∗ [1 ⊗ γ˜ ] = [˜ γ ]. (p−1)i − (p−1)i − s s 1 1 ∈ B[s]. Then [1 ⊗ Proof. Suppose that γ = I∈I u11 v1 · · · uss vs 1−s 1−s γ˜ ] ∈ D(Bs (Σ Fp )). Since γ˜ is a cycle in EBs−1 (Σ Fp ), then [1 ⊗ γ˜ ] is a cycle in D(B∗ (Σ1−s Fp )). It can be pulled back by the element 1 ⊗ γ˜ ∈ Bs (Σ1−s Fp ). In Bs (Σ1−s Fp ), we have ∂(1 ⊗ γ˜ ) = 1 ⊗ ∂(˜ γ) (−1)e(I) β 1− 1 P i1 ⊗ · · · ⊗ β 1− s P is ⊗Σ1−s 1 + I∈I (−1)e(I) β 1− 1 P i1 ⊗ · · · ⊗ β 1− s P is ⊗Σ1−s 1. = I∈I Thus, the proof is complete. From the short exact sequence 0 → Σ2−s P1 → Σ2−s Pˆ → Σ1−s Fp → 0, we have the short exact sequence of chain complexes / B∗ (Σ2−s P1 ) 0 / B∗ (Σ2−s Pˆ ) / B∗ (Σ1−s Fp ) / 0. It induces a short exact sequence (even though E(−) is not exact) 0 / EB∗ (Σ2−s P1 ) / EB∗ (Σ2−s Pˆ ) / EB∗ (Σ1−s Fp ) / 0. Taking homology, we have the connecting homomorphism / Hs−2 (EB∗ (Σ2−s P1 )). δ(Σ2−s Fp ) : Hs−1 (EB∗ (Σ−1 Fp )) By (4.2), one gets αs = δ(ΣFp ⊗ Ps−1 ) ◦ · · · ◦ δ(Σ2−s Fp ) ◦ ∂∗ Put δs−1 := δ(ΣFp ⊗ Ps−1 ) ◦ · · · ◦ δ(Σ2−s Fp ). Then we have the lemma. 12 P. H. CHƠN AND Đ. T. TRIẾT Lemma 4.4. For any γ ∈ B[s], δs−1 ([˜ γ ]) = (−1) s(s−1) +(s+1) deg γ 2 [Σγ]. Proof. Assume that (p−1)i1 − γ= u11 v1 1 · · · uss vs(p−1)is − s ∈ B[s]. I∈I By Lemma 4.2, (−1)e(I) β 1− 1 P i1 ⊗ · · · ⊗ β 1− s P is ⊗Σ1−s 1 γ˜ = I∈I is a cycle in EBs−1 (Σ1−s Fp ). It can be pulled back by (−1)e(I) β 1− 1 P i1 ⊗ · · · ⊗ β 1− s P is ⊗Σ2−s xs ys−1 ∈ EBs−1 (Σ2−s Pˆ ). y= I∈I Then, in EBs−1 (Σ2−s Pˆ ), (−1)η(I)+e(I) β 1− 1 P i1 ⊗ · · · ⊗ β 1− ∂(y) = s−1 P is−1 ⊗Σ2−s β 1− s P is (xs ys−1 ), I∈I where η(I) = s + 1 + · · · + s−1 + s s . Therefore, δ(Σ2−s Fp )([˜ γ ]) is equal to (−1)η(I)+e(I) β 1− 1 P i1 ⊗ · · · ⊗ β 1− s−1 P is−1 ⊗Σ2−s β 1− s P is (xs ys−1 ) . I∈I Repeating this process, finally we have (−1)fI β 1− 1 P i1 (x1 y1−1 β 1− s P i2 (x2 y2−1 · · · β 1− s P is xs ys−1 ) · · · )) , δs−1 ([˜ γ ]) = I∈I where fI = e(I) + (1 + · · · + s) + s( 1 + · · · + The lemma follows from Corollary A.13. s ). Combining Lemma 4.3 and Lemma 4.4, we have the following corollary. Corollary 4.5. The map B[s] γ → (−1) / DBs (Σ1−s Fp ) given by s(s−1) +(s+1) deg γ 2 [1 ⊗ γ˜ ] is a chain-level representation of the homomorphism (1 ⊗A αs )−1 Σ : Fp ⊗A B[s] / Fp ⊗A Ds (Σ1−s Fp ). The chain-level representation of the dual of ϕs is given by the following theorem. Theorem 4.6. The inclusion map ϕ˜# s : B[s] γ → (−1) / Γ+ s given by s(s−1) +(s+1) deg γ 2 γ is the chain-level representation of the dual of the Lannes-Zarari homomorphism ϕ# s . ON THE MOD p LANNES-ZARATI HOMOMORPHISM 13 Proof. In [28], Priddy showed that the opposite of the lambda algebra Λopp is isomorphic to the co-Koszul complex of A, which is the quotient cocomplex of the usual cobar resolution C ∗ (Fp ) := HomM (B∗ (Fp ), Fp ). The canonical quotient map / Λopp sends τ0 ξ1i to (−1) λ1− and the rest to zero. Thus, under the ι∗ : C ∗ (Fp ) i−1 projection ιs , ιs (τ01− 1 ξ1j1 ⊗ · · · ⊗ τ01− s ξ1js ) = (−1) 1 +···+ s +s λj11 −1 · · · λjss −1 . In Section 1, we showed that the chain complex Γ+ is isomorphic to (Λopp )# , the dual of Λopp , via the isomorphism given by (p−1)j1 − κs (u11 v1 1 · · · uss vs(p−1)js − s ) = (−1)i1 +···+is (λj11 −1 · · · λjss −1 )∗ . Thus, there exists an inclusion νs : (Γ+ Σ1−s Fp )s (p−1)j1 − νs (u11 v1 1 / Bs (Σ1−s Fp ), that sends · · · uss vs(p−1)js − s ) = (−1)e(I) 1 ⊗ β 1− 1 P j1 ⊗ · · · ⊗ β 1− s P js = 1 ⊗ γ˜ , where e(I) = s + 1 + · · · + s + i1 + · · · + is . This fact together with Lemma 4.3 and Lemma 4.4, we have the assertion of the theorem. ∼ opp # ∼ # Since Γ+ s = (Λs ) and B[s] = Rs via κs , we have the following corollary. Corollary 4.7. The projection ϕ˜s : Λopp s ϕ˜s (λI ) = (−1) / Rs given by s(s−1) +(s+1) deg(λI ) 2 QI is the chain-level representation of the Lannes-Zarati homomorphism ϕs . 5. The power operations This section is devoted to develop the power operations, these are useful tools in studying the behavior of the Lannes-Zarati homomorphism in the next section. From Liulevicius [22], [23] and May [24], there exists the power operation P 0 : s,p(s+t) / ExtA Exts,s+t (Fp , Fp ) (Fp , Fp ). Its chain-level representation in the cobar A complex is given by θ1 ⊗ · · · ⊗ θs → θ1p ⊗ · · · ⊗ θsp , where θi ∈ A# , the dual of the Steenrod algebra A. / Λopp By the projection ιs : C s (Fp ) s , the power operation has a chain-level opp representation in the Λ given by ˜ 0 (λ 1 · · · λ s ) = P i1 −1 is −1 λpi11 −1 · · · λpiss −1 , 0, 1 = ··· = otherwise. s = 1, ˜ 0 induces an operation θ on the Dyer-Lashof algebra Lemma 5.1. The operation P R given by θ(β 1 Qi1 · · · β s Qis ) = β 1 Qpi1 · · · β s Qpis , 1 = · · · = 0, otherwise. s = 1, Proof. It is sufficient to show that if λi1 −1 · · · λis −1 has negative excess then so does λpi1 −1 · · · λpis −1 for s ≥ 2. 14 P. H. CHƠN AND Đ. T. TRIẾT By inspection, one gets s e(λpi1 −1 · · · λpis −1 ) = 2pi1 − 2p(p − 1)ik + (s − 2) k=2 = pe(λi1 −1 · · · λis −1 ) − (p − 1)(s − 2). Therefore, if e(λi1 −1 · · · λis −1 ) < 0 then e(λpi1 −1 · · · λpis −1 ) < 0. Lemma 5.2. The operation θ commutes with the action of A. In particular, θ((β 1 Qi1 · · · β s Qis ) P k ) = (θ(β 1 Qi1 · · · β s Qis )) P pk . Proof. It is sufficient to show the lemma in the case We will prove the assertion by induction on s. For s = 1, it is easy to see that 1 = ··· = s (5.1) = 1. (p − 1)(i − k) − 1 βQi−k ) k (p − 1)(i − k) − 1 βQpi−pk , k θ((βQi ) P k ) = θ((−1)k = (−1)k and (θ(βQi )) P pk = βQpi P pk = (−1)pk (p − 1)(pi − pk) − 1 βQpi−pk . pk Since (−1)pk (p−1)(pi−pk)−1 ≡ (−1)k (p−1)(i−k)−1 pk k For s > 1, by the inductive hypothesis, mod p, we have the assertion. θ((βQi1 · · · βQis ) P k ) (−1)k+t =θ t (−1)k+t +θ t (−1)k+t = (p − 1)(i1 − k) − 1 βQi1 −k+t (βQi2 · · · βQis ) P t k − pt t (p − 1)(i1 − k) − 1 Qi1 −k+t (βQi2 · · · βQis )β P t k − pt − 1 (p − 1)(i1 − k) − 1 βQp(i1 −k+t) (βQpi2 · · · βQpis ) P pt . k − pt On the other hand, (θ(βQi1 · · · βQis )) P pk = (βQpi1 · · · βQpis ) P pk (−1)pk+j = j (−1)pk+j + j (−1)k+j = j (p − 1)(pi1 − pk) − 1 βQpi1 −pk+j (βQpi2 · · · βQpis ) P j pk − pj (p − 1)(pi1 − pk) − 1 Qpi1 −pk+j (βQpi2 · · · βQpis )β P j pk − pj − 1 (p − 1)(i1 − k) − 1 βQpi1 −pk+j (βQpi2 · · · βQpis ) P j . k−j ON THE MOD p LANNES-ZARATI HOMOMORPHISM 15 If j is not divisible by p then (p − 1)(pi2 − j) − 1 ≡ j − 1 mod p, while j − p ≡ j mod p. Therefore, (βQpi2 · · · βQpis ) P j (−1)j+ = j (−1)j+ + j (−1)j+ = j (p − 1)(pi2 − ) − 1 βQpi2 −j+ (βQpi3 · · · βQpis ) P j−p (p − 1)(pi2 − j) − 1 Qpi2 −j+ (βQpi3 · · · βQpis )β P j j−p −1 (p − 1)(pi2 − ) − 1 βQpi2 −j+ (βQpi3 · · · βQpis ) P = 0. j−p Thus, (θ(βQi1 · · · βQis )) P pk (−1)k+t = j (p − 1)(i1 − k) − 1 βQp(i1 −k+t) (βQpi2 · · · βQpis ) P pt . k − pt The lemma is proved. By Lemma 5.2, the operation θ induces an power operation on Ann(R), which is also denoted by P 0 . Proposition 5.3. The power operations P 0 s commute with each other through the Lannes-Zarati homomorphism. In other words, the following diagram is commutative P0 / s,p(s+t) Exts,s+t (Fp , Fp ) ExtA (Fp , Fp ) A ϕs Ann(Rs )t ϕs P0 / Ann(Rs )p(s+t)−s . Proof. It is immediate from Corollary 4.7. 6. Behavior of the Lannes-Zarati homomorphism In this section, we use the chain-level representation map of the ϕs constructed in the previous section to investigate its behavior. 6.1. The first Lannes-Zarati homomorphism. Theorem 6.1. The first Lannes-Zarati homomorphism / Ann(B[1]# )t ϕ1 : Ext1,1+t (Fp , Fp ) A is isomorphic. Proof. As we well-known, Ext1,1+t (Fp , Fp ) spanned by α0 of stem 0 and hi of stem A (p−1)i−1 0 2i(p−1)−1. These element are represented in Γ+ 1 respectively be v1 and u1 v1 for i > 0. (p−1)i−1 On the other hand, Fp ⊗ B[1] is spanned by 1 and x1 y1 for i > 0. Applying Theorem 4.6, one gets 0 ϕ# 1 ([v1 ]) = [v0 ]; This fact follows the theorem. (p−1)i−1 ϕ# 1 ([x1 y1 (p−1)i−1 ]) = [u1 v1 ]. 16 P. H. CHƠN AND Đ. T. TRIẾT 6.2. The second Lannes-Zarati homomorphism. Theorem 6.2. The second Lannes-Zarati homomorphism / Ann(B[2]# )t ϕ2 : Ext2,2+t (Fp , Fp ) A is vanishing for t = 0 and t = 2(p − 1)pi+1 − 2, i ≥ 0. Proof. From the results of Liulevicius [23] (see also Aikawa [1]), Ext2,2+t (Fp , Fp ) A spanned by the elements i 2,2(p−1)(p (1) hi hj = [λpi −1 λpj −1 ] ∈ ExtA +pj ) 2,2(p−1)pi +1 (2) α0 hi = [µ−1 λpi −1 ] ∈ ExtA (3) α02 = [µ2−1 ] ∈ Ext2,2 A (Fp , Fp ); (Fp , Fp ), 0 ≤ i < j + 1; (Fp , Fp ), i ≥ 1; i+1 i 2,2(p−1)(2p +p ) (4) hi;2,1 = (P 0 )i [λ2p−1 λ0 ] ∈ ExtA (Fp , Fp ), i ≥ 0; 2,2(p−1)(pi+1 +2pi ) 0 i (5) hi;1,2 = (P ) [λp−1 λ1 ] ∈ ExtA (Fp , Fp ), i ≥ 0; (6) ρ = [λ1 µ−1 ] ∈ Ext2,4(p−1)+1 (Fp , Fp ); A i+1 (p−1) (−1)j+1 0 i ˜ λ(p−j)−1 λj−1 ∈ Ext2,2(p−1)p (F2 , F2 ), i ≥ 0. (7) λi = (P ) j=1 j 0 i A 0 0 Here we denote (P ) = P · · · P . i times It is clear that monomials λpi −1 λpj −1 (i < j + 1), µ−1 λpi −1 , λp−1 λ1 are of negative excess, therefore their images under ϕ˜2 are trivial in R2 . It implies under ϕ2 the images of hi hj , α0 hi , and h0;1,2 are trivial. By Proposition 5.3, ϕ2 (hi;1,2 ) = (P 0 )i ϕ2 (h0;1,2 ) = 0. It is easy to see that ϕ2 (α02 ) = −Q0 Q0 = 0 ∈ R2 . By inspection, ϕ˜2 (λ2p−1 λ0 ) = −βQ2p βQ1 . Applying adem relation, one gets βQ2p βQ1 = − (−1)2p+j j (p − 1)(j − 1) − 1 βQ2p+1−j βQj . pj − 2p − 1 Since pj > 2p+1, then e(βQ2p+1−j βQj ) = 2(2p+1−j)−2(p−1)j = 2(2p+1−pj) < 0. Therefore, βQ2p βQ1 = 0, it implies that ϕ2 (h0;2,1 ) and then ϕ2 (hi;2,1 ) = 0. Similarly, ϕ˜2 (λ1 µ−1 ) = βQ2 Q0 . Applying adem relation, we obtain βQ2 Q0 = 0 and therefore ϕ2 (ρ) = 0. Finally, it is easy to verify that (p−1) j+1 (−1) ϕ˜2 λ(p−j)−1 λj−1 = −βQp−1 βQ1 = 0 ∈ R2 . j j=1 Therefore ϕ(λ˜0 ) = βQp−1 βQ1 . By Proposition 5.3, one gets ϕ2 (λ˜i ) = (P 0 )i (βQp−1 βQ1 ) = −βQp i (p−1) i βQp = 0 ∈ R2 . The proof is complete. Remark 6.3. From the result of Wellington [30, Theorem 11.11], Ann(R2 ) is i i spanned by Q0 Q0 , βQp (p−1) βQp , i ≥ 0, and Qs(p−1) Qs , s = pi + · · · + 1, i > 0. Therefore, ϕ2 is not an epimorphism. ON THE MOD p LANNES-ZARATI HOMOMORPHISM 17 6.3. The third Lannes-Zarati homomorphism. Theorem 6.4. The third Lannes-Zarati homomorphism ϕ3 : Ext3,3+t (Fp , Fp ) A / Ann(B[3]# )t is vanishing for all t > 0. Proof. By the results of Liulevicius [23] and Aikawa [1], Ext3,3+t (Fp , Fp ) is spanned A by following elements (for convenience we will write Exts,s+t for Ext3,3+t (Fp , Fp )) A A 3,2(p−1)(p (1) hi hj hk = [λpi −1 λpj −1 λpk −1 ] ∈ ExtA i +pj +pk ) 3,2(p−1)(pi +pj )+1 (2) α0 hi hj = [µ−1 λpi −1 λpj −1 ] ∈ ExtA , 0 ≤ i < j + 1 < k + 2; , 0 ≤ i < j + 1; i 3,2(p−1)p +2 (3) α02 hi = [µ2−1 λpi −1 ] ∈ ExtA , i ≤ 0; 3,3 3 3 (4) α0 = [µ−1 ] ∈ ExtA ; 3,2(p−1)(pi+1 +pj ) (5) λ˜i hj = [Li λpj ] ∈ ExtA , i, j ≥ 0, j = i + 2; ˜ (6) α0 λi = [µ−1 Li ], i ≥ 0; i+1 i j 3,2(p−1)(p +2p +p ) (7) hi;1,2 hj = [λpi+1 −1 λ2pi −1 λpj −1 ] ∈ ExtA , i, j ≥ 0, j = i + 2, i, i − 1; 3,2(p−1)(pi+1 +2pi )+1 (8) hi;1,2 α0 = [λpi+1 −1 λ2pi −1 µ−1 ] ∈ ExtA , i ≥ 1; i+1 i j 2(p−1)(2p +p +p ) (9) hi;2,1 hj = [λ2pi+1 −1 λpi −1 λpj −1 ] ∈ ExtA ; i, j ≥ 0, j = i + 2, i ± 1, i; 3,2(p−1)(2pi+1 +pi )+1 (10) hi;2,1 α0 = [λ2pi+1 −1 λpi −1 µ−1 ] ∈ ExtA , i ≥ 1; (11) ρα0 = [λ1 µ−1 µ−1 ] ∈ Ext3,4(p−1)+2 ; A 3,2(p−1)(3pi+2 +2pi+1 +pi (12) hi;3,2,1 = (P 0 )i [λ3p2 −1 λ2p−1 λ0 ] ∈ ExtA , p = 3, i ≥ 0; 3,2(p−1)(3p+2)+1 (13) h3,2,1 = [λ3p−1 λ1 µ−1 ] ∈ ExtA , p = 3; i+3 +2p (14) hi;2,2,1 = (P 0 )i [λ2p3 −1 λ2p−1 λ0 ] ∈ Ext3,2(p−1)(2p A 2 3,2(p−1)(2p +2)+1 (15) h2,2,1 = [λ2p2 −1 λ1 µ−1 ] ∈ ExtA , p = 3; i+2 3,2(p−1)(p +3p (16) hi;1,3,1 = (P 0 )i [λp2 −1 λ3p−1 λ0 ] ∈ ExtA 3,2(p−1)(p+3)+1 (17) h1,3,1 = [λp−1 λ2 µ−1 ] ∈ ExtA , p = 3; i+2 i+1 +pi ) i+1 +pi ) i+1 i , p = 3, i ≥ 0; , p = 3, i ≥ 0; 3,2(p−1)(2p +p +2p (18) hi;2,1,2 = (P 0 )i [λ2p2 −1 λp−1 λ1 ] ∈ ExtA , i ≥ 0; i+2 i+1 3,2(p−1)(p +2p +3pi ) 0 i (19) hi;1,2,3 = (P ) [λp2 −1 λ2p−1 λ2 ] ∈ ExtA , p = 3, i ≥ 0; (20) 3 = [λ2 µ2−1 ] ∈ Ext3,6(p−1)+2 , p = 3; A (21) 3 = [λ5 µ2−1 ] Ext3,12(p−1)+2 , p = 3; A i+1 i +2p ) (22) fi = (P 0 )i−1 [M1 ] ∈ Ext3,2(p−1)(p , i ≥ 1; A i+1 3,2(p−1)(2p +pi ) 0 i−1 (23) gi = (P ) [N1 ] ∈ ExtA , i ≥ 1; where (p−1) Li = (P 0 )i j=1 p−1 M1 = j=1 (−1)j+1 λ(p−j)−1 λj−1 , i ≥ 0; j (−1)j+1 (λjp−1 λ(p2 −jp)−1 λ2p−1 − 2λp2 −1 λj−1 λ2p−j−1 j − 2λp2 −1 λp+j−1 λp−j−1 ); 18 P. H. CHƠN AND Đ. T. TRIẾT p−1 N1 = j=1 (−1)j+1 (2λjp−1 λ(2p2 −jp)−1 λp−1 + 2λp2 +jp−1 λp2 −jp−1 λp−1 j − λ2p2 −1 λj−1 λp−j−1 ). By inspection, we see that hi hj hk (i < j + 1 < k + 2), α0 hi hj (i < j + 1), α02 hi , α0 λ˜i , hi;1,2 hj , hi;1,2 α0 , hi;1,3,1 , h1,3,1 , hi;1,2,3 , and fi are represented by cycles of negative excess. Therefore, their images under ϕ3 are trivial. It is easy to check that ϕ3 (α03 ) = −Q0 Q0 Q0 = 0 ∈ R3 . i i j Applying Corollary 4.7, one gets that ϕ3 (λ˜i hj ) = −βQp (p−1) βQp βQp . Applyi i ing adem relation, we obtain that βQp (p−1) βQp = 0, therefore ϕ3 (λ˜i hj ) = 0. i+1 i j It is clear that ϕ3 (hi;2,1 hj ) = −βQ2p βQp βQp . Applying adem relation, we i+1 i obtain that βQ2p βQp = 0, it implies that ϕ3 (hi;2,1 hj ) = 0. Similarly, we have ϕ3 (hi;2,1 α0 ) = 0. By the same argument, we obtain • ϕ3 (ρα0 ) = −βQ2 Q0 Q0 = 0; 2 • ϕ3 (h0;3,2,1 ) = −βQ3p βQ2p βQ1 = 0; 3p • ϕ3 (h3,2,1 ) = −βQ βQ2 Q0 = 0; 3 • ϕ3 (h0;2,2,1 ) = −βQ2p βQ2p βQ1 = 0; 2 • ϕ3 (h2,2,1 ) = −βQ2p βQ2 Q0 = 0; 2 • ϕ3 (hi;2,1,2 ) = −βQ2p βQp βQ2 = 0; • ϕ3 ( 3 ) = −βQ3 Q0 Q0 = 0; • ϕ3 ( 3 ) = −βQ6 Q0 Q0 = 0. Finally, by inspection, we have p−1 ϕ3 (g1 ) = − j=1 j p−j It is clear that βQ βQ 2 (−1)j βQ2p βQj βQp−j . j = 0 if j < p − 1. But applying adem relation, we have 2 βQ2p βQp−1 βQ1 = 0. Combining with Proposition 5.3, we have the assertion of the theorem. Appendix A. The Singer transfer The purpose of this section is to establish the chain-level representation of the dual of the mod p Singer transfer in the Singer-Hưng-Sum chain complex. We end this section by the computation of the image of B[s] ⊂ Γ+ s through the Singer transfer, the result is used in Section 4. −1 / TorA Let e1 (M ) : TorA M) r−1 (Fp , P1 ⊗ M ) be the Singer’s element, r (Fp , Σ which is the connecting homomorphism associated with the short exact sequence / P1 ⊗ M / Pˆ ⊗ M / Σ−1 M / 0. 0 Put es (M ) := e1 (Ps−1 ⊗ M ) ◦ · · · ◦ e1 ( Σ−(s−1) M ), then −s / TorA es (M ) : TorA M) r (Fp , Σ r−s (Fp , Ps ⊗ M ). When M = Fp and r = s, we have the dual of the mod p Singer transfer / Fp ⊗A Ps . T rs# := es (Fp )Σ−s : TorA s (Fp , Fp ) ON THE MOD p LANNES-ZARATI HOMOMORPHISM / E(x1 , · · · , xs )⊗Fp [y ±1 , · · · , ys±1 ] 1 Definition A.1. The homomorphism Ts : ∆+ s is defined by (p−1)i1 − 1 Ts (u11 v1 19 · · · uss vs(p−1)is − s ) = (−1)fs β 1− 1 P i1 (x1 y1−1 β 1− s P i2 (x2 y2−1 · · · β 1− s P is (xs ys−1 ))), (A.1) where fs = (1 + · · · + s) + s( 1 + · · · + s ) + i1 + · · · + is , and i1 , . . . , is are arbitrary integers. Here we mean P i = 0 for i < 0. Theorem A.2. The restriction of Ts on Γ+ is the chain-level representation s , Ts |Γ+ s of the dual of the mod p Singer transfer T rs# . −s 1−s / TorA Proof. By the definition, e1 (Σ1−s Fp ) : TorA Fp ) P1 ) is s (Fp , Σ s−1 (Fp , Σ the connecting homomorphism of the exact sequence of chain complexes / Γ+ Σ1−s Pˆ / Γ+ Σ−s Fp / 0. / Γ+ Σ1−s P1 0 (p−1)i − (p−1)i − s s 1 1 ∈ (Γ+ Σ−s Fp )s , it can be Σ−s 1 · · · uss vs For a cycle X = u11 v1 (p−1)i (p−1)i − s− s 1 1 Sts (Σ1−s xs ys−1 ) ∈ · · · uss vs pulled back to the element X = u11 v1 + −s + 1−s ˆ + 1−s ˆ (Γ Σ P )s . Since X is the cycle in (Γ Σ Fp )s , in Γ Σ P , one gets that ∂(X ) is equal to (p−1)i1 − (−1)ks +is u11 v1 where ks = s + 1 1 + ··· + (p−1)i1 − [(−1)ks +is +1 u11 v1 (p−1)is−1 − s−1 · · · us−1 vs−1 s−1 1 s−1 Sts−1 (Σ1−s β 1− s P is (xs ys−1 )), + s s . Therefore, e1 (Σ1−s Fp )([X]) is equal to (p−1)is−1 − s−1 · · · us−1 vs−1 s−1 Sts−1 (Σ1−s β 1− s P is (xs ys−1 ))]. Repeating this process, we have the assertion. For M is unstable A-module and m ∈ M q , we define d∗ P (x, y; m) := µ(q) (−1) x y +i =0,1;0≤ +2i≤q y (q−2i)(p−1) 2 ⊗ β P i (m), where µ(q) = (h!)q (−1)hq(q−1)/2 , h = (p − 1)/2. From Mùi’s [26] and Hưng-Sum [19], we have Lemma A.3. For m, n ∈ H ∗ BE1 = Fp [y] ⊗ E(x) (1) d∗ P (x1 , y1 ; Vi−1 (y2 , · · · , yi )) = Vi (y1 , · · · , yi ); h−1 (2) d∗ P (x1 , y1 ; Mi;i−1 Lh−1 ; i−1 ) = (−h!)Mi+1;i Li ∗ h deg m deg n ∗ d P (x, y; m)d∗ P (x, y; n). (3) d P (x, y; mn) = (−1) Lemma A.4. For m, n ∈ H ∗ BE1 = Fp [y] ⊗ E(x), (1) Sts (mn) = Sts (m) · Sts (n); (2) Sts (x) = (−1)s us+1 ; (3) Sts (y) = (−1)s vs+1 Corollary A.5. (1) d∗ P (x, y, Vip−1 ) = βP p p−1 p−1 (2) d∗ P (x, y; Vi ) = Vi+1 ; p−1 (3) y 2 d∗ P (x, y; Ri;i−1 ) = (−h!)Ri+1;i ; (4) y p−1 d∗ P (x, y; qi,0 ) = qi+1,0 ; (5) St1 (ui ) = −ui+1 ; (6) St1 (vi ) = −vi+1 . i (p−1) (xy −1 ⊗ Vip−1 ); 20 P. H. CHƠN AND Đ. T. TRIẾT Lemma A.6. Let M be an A-algebra, X, Y ∈ M . For 2a ≥ deg X and 2b ≥ deg Y , then β P a+b (xy −1 ⊗ XY ) = β P a (xy −1 ⊗ X)β P b (xy −1 ⊗ Y ). Proof. Using Cartan formula, we can verify that β P a+b (xy −1 ⊗ XY ) = (−1)a+b− + x y (p−1)(a+b− )− ⊗ β P (XY ) , (−1)a+b− = , + (−1) 2 deg X x y (p−1)(a+b− )− i+j = + 2 = 1 ⊗ β 1 P i (X)β 2 P j (Y ) = (−1)a−i+ 1 x 1 y (p−1)(a−i)− 1 ⊗ β 1 P i (X) i, 1 × (−1)b−j+ 2 x 2 y (p−1)(b−j)− 2 ⊗ β 2 P j (Y ) i, a = β P (xy 1 −1 ⊗ X)β P b (xy −1 ⊗ Y ). The proof is complete. Lemma A.7. Let M be an A-algebra, X, Y ∈ M . For 2a ≥ deg X and 2b ≥ deg Y , then P a+b (xy −1 ⊗ XY ) = P a (xy −1 ⊗ X)β P b (xy −1 ⊗ Y ). Proof. Using Cartan formula, we can verify that a P a+b (xy −1 ⊗ XY ) = (−1)a−i xy (p−1)(a−i)−1 ⊗ P i (X) i=0 b (−1)b−j y (p−1)(b−j) ⊗ P j (Y ) × j=0 = P (xy −1 ⊗ X)β P b (xy −1 ⊗ Y ). a The proof is complete. Put Ts := (−1)1+···+s Ts . Then we have the following result. (p−1)i1 − Lemma A.8. For elements satisfying (4.1) v I = u11 v1 (p−1)j1 −σ1 (p−1)js −σs · · · uσs s vs in Γ+ and v J = uσ1 1 v1 s , one gets Ts (v I · v J ) = Ts (v I ) · Ts (v J ). 1 (p−1)is − · · · uss vs s ON THE MOD p LANNES-ZARATI HOMOMORPHISM 21 Proof. We only need to prove for s = 2. The case s > 2 is proved similarly. (p−1)(i1 +j1 )−( T2 (v I · v J ) = T2 (u11 +σ1 v1 = (−1) 2( × β 1−( = (−1)2( × β 1−( 1 +σ1 ) (p−1)(i2 +j2 )−( 2 +σ2 ) u22 +σ2 v2 ) 1 +σ1 + 2 +σ2 )+i1 +j1 +i2 +j2 1 +σ1 ) P i1 +j1 (x1 y1−1 ⊗ β 1−( 2 +σ2 ) P i2 +j2 (x2 y2−1 )) 1 +σ1 + 2 +σ2 )+i1 +j1 +i2 +j2 1 +σ1 ) P i1 +j1 (x1 y1−1 ⊗ (β 1− 2 P i2 (x2 y2−1 )(β 1−σ2 P j2 (x2 y2−1 )). Since v I and v J satisfy condition (4.1), then applying Lemma A.6 or Lemma A.7, one gets T2 (v I · v J ) = (−1)2( 1 +σ1 + 2 +σ2 )+i1 +j1 +i2 +j2 × [β 1− 1 P i1 (x1 y1−1 ⊗ (β 1− 2 P i2 (x2 y2−1 ))] × β 1−σ1 P j1 (x1 y1−1 ⊗ (β 1−σ2 P j2 (x2 y2−1 ))] = T2 (v I ) · T2 (vJ ). The lemma is proved. (p−1) Lemma A.9. For 1 ≤ i ≤ s, Ts (Vi (p−1) ) = Vi . Proof. By inspection, we have pi−2 (p−1)(p−1) pi−3 (p−1)(p−1) v1 Vip−1 = v1 (p−1)(p−1) p−1 vi . · · · vi−1 Using (A.1), one gets pi−2 (p−1)(p−1) pi−3 (p−1)(p−1) v1 Ts (Vip−1 ) = Ts (v1 = (−1)β P p i−2 (p−1) (p−1)(p−1) p−1 vi ) · · · vi−1 −1 (x1 y1−1 · · · β P p−1 (xi−1 yi−1 β P 1 (xi yi−1 ))) = d∗ P (x1 , y1 ; · · · d∗ P (xi−1 , yi−1 ; V1p−1 )) = Vip−1 . The proof is complete. Lemma A.10. For 1 ≤ i ≤ s, then Ts (Ri;i−1 ) = (−1)s Ri;i−1 . Proof. By inspection, we have pi−2 (p−1)(p−1) Ri;i−1 = v1 (p−1)(p−1) · · · vi−1 (p−1)−1 ui vi . Therefore, Ts (Ri;i−1 ) = (−1)s·1+(p−1)(p × β Pp i−2 (p−1) i−2 = (−1)s β P p First, we claim that β P p A.5, it is easy to see that β Pp a (p−1) a i−2 +···+1)+1 −1 (x1 y1−1 · · · β P p−1 (xi−1 yi−1 P 1 (xi yi−1 ))) (p−1) (p−1) (p−1)−1 −1 (x1 y1−1 · · · β P p−1 (xi−1 yi−1 xi yi )). (xy −1 Ra+1;a ) = Ra+2;a+1 . Indeed, by Corollary p−1 1 y 2 d∗ P (x, y; Ra+1;a ) a − 2p − 1) 1 = (−h!)Ra+2;a+1 . µ(2pa+1 − 2pa − 1) (xy −1 Ra+1;a ) = µ(2pa+1 22 P. H. CHƠN AND Đ. T. TRIẾT By Wilson’s theorem and the Fermat’s little theorem, one gets −h! p−1 ≡− ( )! µ(2pa+1 − 2pa − 1) 2 ≡ −(−1) (p−1)−1 Since xi yi the assertion. p+1 2 2 (−1) (−1) p−1 2 p−1 (2pa+1 −2pa −1)(2pa+1 −2pa −2) 2 2 ≡1 mod p. = R1,0 , applying the above claim for a from 0 to i − 2, we have Corollary A.11. For 0 ≤ k < i ≤ s, Ts (Ri;k ) = (−1)s Ri;k . Proof. Using Lemma A.9-A.10 together with the formula Ri;s = Ri−1;s Vip−1 + qi−1,s Ri;i−1 , we have the assertion. Proposition A.12. Let γ ∈ B[s]. Then Ts (q) = (−1) s(s+1) +s deg γ 2 γ. Proof. From Lemma A.8-A.10, it is sufficient to prove Ts (Rs;i,j ) = Rs;i,j for 0 ≤ i < j ≤ s − 1. Since −1 −Rs;i,j = Rs;i Rs;j qs,0 −1 = Rs−1;i Rs−1;j qs−1,0 Vsp−1 −1 + (Rs−1;i qs−1,j + Rs−1;j qs−1,i )Rs;s−1 qs−1,0 , −1 −1 it is sufficient to show the assertion for Rk;i Rk;k−1 qk,0 and Rk−1;i Rk;k−1 qk−1,0 . The first case, we have p−2 −1 Rk;i Rk,k−1 qk,0 = qk−1,0 Rk−1;i uk vkp−2 . By Lemma A.8 and Corollary A.11, one gets p−2 −1 Ts (Rk;i Rk,k−1 qk,0 ) = (−1)s Rk−1;i Ts (qk−1;0 uk vkp−2 ). By inspection, we obtain pk−2 (p−2)(p−1) p−2 qk−1,0 uk vkp−2 = v1 (p−2)(p−1) · · · vk−1 uk vkp−2 . Therefore, p−2 Ts (qk−1,0 uk vkp−2 ) = (−1)s+p × β Pp k−2 = (−1)s+p × β Pp k−2 (p−2) k−2 k−2 (p−2)+···+(p−2)+1 −1 (x1 y1−1 · · · β P p−2 (xk−1 yk−1 β P 1 (xk yk−1 ))) (p−2)+···+(p−2) (p−2) (p−1)−1 −1 (x1 y1−1 · · · β P p−2 (xk−1 yk−1 xk yk It is easy to see that (p−1)−1 −1 β P p−2 (xk−1 yk−1 xk yk (p−1)(p−2) St1 (xk ykp−2 ) (p−1)(p−2) u2 v2p−2 . ) = (−1)yk−1 = (−1)yk−1 )). ON THE MOD p LANNES-ZARATI HOMOMORPHISM 23 By the same method, one gets (p−2)(p−1) −1 β P p(p−2)(p−1) (xk−2 yk−2 yk−1 u2 v2p−2 ) (p−1) p−1 p−2 = (−1)[yk−1 d∗ P (xk−2 , yk−2 ; yk−1 )] St1 (u2 , v2p−2 ) p−2 u3 v3p−2 . = (−1)q2,0 By induction, we have p−2 p−2 Ts (qk−1,0 uk vkp−2 ) = (−1)s qk−1,0 uk vkp−2 . −1 −1 Thus, Ts (Rk;i Rk,k−1 qk,0 ) = Rk;i Rk,k−1 qk,0 . 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Department of Mathematics and Applications, Saigon University, 273 An Duong Vuong, District 5, Ho Chi Minh city, Vietnam E-mail address: chonkh@gmail.com Department of Mathematics and Applications, Saigon University, 273 An Duong Vuong, District 5, Ho Chi Minh city, Vietnam E-mail address: dongthanhtriet.dhsg@gmail.com [...]... (−1)pk+j + j (−1)k+j = j (p − 1)(pi1 − pk) − 1 βQpi1 −pk+j (βQpi2 · · · βQpis ) P j pk − pj (p − 1)(pi1 − pk) − 1 Qpi1 −pk+j (βQpi2 · · · βQpis )β P j pk − pj − 1 (p − 1)(i1 − k) − 1 βQpi1 −pk+j (βQpi2 · · · βQpis ) P j k−j ON THE MOD p LANNES- ZARATI HOMOMORPHISM 15 If j is not divisible by p then (p − 1)(pi2 − j) − 1 ≡ j − 1 mod p, while j − p ≡ j mod p Therefore, (βQpi2 · · · βQpis ) P j (−1)j+ = j (−1)j+... + j (−1)j+ = j (p − 1)(pi2 − ) − 1 βQpi2 −j+ (βQpi3 · · · βQpis ) P j p (p − 1)(pi2 − j) − 1 Qpi2 −j+ (βQpi3 · · · βQpis )β P j j p −1 (p − 1)(pi2 − ) − 1 βQpi2 −j+ (βQpi3 · · · βQpis ) P = 0 j p Thus, (θ(βQi1 · · · βQis )) P pk (−1)k+t = j (p − 1)(i1 − k) − 1 βQp(i1 −k+t) (βQpi2 · · · βQpis ) P pt k − pt The lemma is proved By Lemma 5.2, the operation θ induces an power operation on Ann(R), which... (−1)s +p × β Pp k−2 (p 2) k−2 k−2 (p 2)+···+ (p 2)+1 −1 (x1 y1−1 · · · β P p−2 (xk−1 yk−1 β P 1 (xk yk−1 ))) (p 2)+···+ (p 2) (p 2) (p 1)−1 −1 (x1 y1−1 · · · β P p−2 (xk−1 yk−1 xk yk It is easy to see that (p 1)−1 −1 β P p−2 (xk−1 yk−1 xk yk (p 1) (p 2) St1 (xk ykp−2 ) (p 1) (p 2) u2 v 2p 2 ) = (−1)yk−1 = (−1)yk−1 )) ON THE MOD p LANNES- ZARATI HOMOMORPHISM 23 By the same method, one gets (p 2) (p 1) −1 β P p (p 2) (p 1)... (Vi (p 1) ) = Vi Proof By inspection, we have pi−2 (p 1) (p 1) pi−3 (p 1) (p 1) v1 Vip−1 = v1 (p 1) (p 1) p 1 vi · · · vi−1 Using (A.1), one gets pi−2 (p 1) (p 1) pi−3 (p 1) (p 1) v1 Ts (Vip−1 ) = Ts (v1 = (−1)β P p i−2 (p 1) (p 1) (p 1) p 1 vi ) · · · vi−1 −1 (x1 y1−1 · · · β P p−1 (xi−1 yi−1 β P 1 (xi yi−1 ))) = d∗ P (x1 , y1 ; · · · d∗ P (xi−1 , yi−1 ; V 1p 1 )) = Vip−1 The proof is complete Lemma A.10... (p 1) i βQp = 0 ∈ R2 The proof is complete Remark 6.3 From the result of Wellington [30, Theorem 11.11], Ann(R2 ) is i i spanned by Q0 Q0 , βQp (p 1) βQp , i ≥ 0, and Qs (p 1) Qs , s = pi + · · · + 1, i > 0 Therefore, ϕ2 is not an epimorphism ON THE MOD p LANNES- ZARATI HOMOMORPHISM 17 6.3 The third Lannes- Zarati homomorphism Theorem 6.4 The third Lannes- Zarati homomorphism ϕ3 : Ext3,3+t (Fp , Fp ) A /... by P 0 Proposition 5.3 The power operations P 0 s commute with each other through the Lannes- Zarati homomorphism In other words, the following diagram is commutative P0 / s ,p( s+t) Exts,s+t (Fp , Fp ) ExtA (Fp , Fp ) A ϕs Ann(Rs )t ϕs P0 / Ann(Rs )p( s+t)−s Proof It is immediate from Corollary 4.7 6 Behavior of the Lannes- Zarati homomorphism In this section, we use the chain-level representation... )−1 Σ : Fp ⊗A B[s] / Fp ⊗A Ds (Σ1−s Fp ) The chain-level representation of the dual of ϕs is given by the following theorem Theorem 4.6 The inclusion map ϕ˜# s : B[s] γ → (−1) / Γ+ s given by s(s−1) +(s+1) deg γ 2 γ is the chain-level representation of the dual of the Lannes- Zarari homomorphism ϕ# s ON THE MOD p LANNES- ZARATI HOMOMORPHISM 13 Proof In [28], Priddy showed that the opposite of the lambda... )) P pk Proof It is sufficient to show the lemma in the case We will prove the assertion by induction on s For s = 1, it is easy to see that 1 = ··· = s (5.1) = 1 (p − 1)(i − k) − 1 βQi−k ) k (p − 1)(i − k) − 1 βQpi−pk , k θ((βQi ) P k ) = θ((−1)k = (−1)k and (θ(βQi )) P pk = βQpi P pk = (−1)pk (p − 1)(pi − pk) − 1 βQpi−pk pk Since (−1)pk (p 1)(pi−pk)−1 ≡ (−1)k (p 1)(i−k)−1 pk k For s > 1, by the. .. operations, these are useful tools in studying the behavior of the Lannes- Zarati homomorphism in the next section From Liulevicius [22], [23] and May [24], there exists the power operation P 0 : s ,p( s+t) / ExtA Exts,s+t (Fp , Fp ) (Fp , Fp ) Its chain-level representation in the cobar A complex is given by θ1 ⊗ · · · ⊗ θs → θ 1p ⊗ · · · ⊗ θsp , where θi ∈ A# , the dual of the Steenrod algebra A / Λopp By the. .. mean P i = 0 for i < 0 Theorem A.2 The restriction of Ts on Γ+ is the chain-level representation s , Ts |Γ+ s of the dual of the mod p Singer transfer T rs# −s 1−s / TorA Proof By the definition, e1 (Σ1−s Fp ) : TorA Fp ) P1 ) is s (Fp , Σ s−1 (Fp , Σ the connecting homomorphism of the exact sequence of chain complexes / Γ+ Σ1−s P / Γ+ Σ−s Fp / 0 / Γ+ Σ1−s P1 0 (p 1)i − (p 1)i − s s 1 1 ∈ (Γ+ Σ−s Fp ... = j (p − 1)(pi1 − pk) − βQpi1 −pk+j (βQpi2 · · · βQpis ) P j pk − pj (p − 1)(pi1 − pk) − Qpi1 −pk+j (βQpi2 · · · βQpis )β P j pk − pj − (p − 1)(i1 − k) − βQpi1 −pk+j (βQpi2 · · · βQpis ) P j ... ϕ2 is not an epimorphism ON THE MOD p LANNES- ZARATI HOMOMORPHISM 17 6.3 The third Lannes- Zarati homomorphism Theorem 6.4 The third Lannes- Zarati homomorphism ϕ3 : Ext3,3+t (Fp , Fp ) A / Ann(B[3]#... γ γ is the chain-level representation of the dual of the Lannes- Zarari homomorphism ϕ# s ON THE MOD p LANNES- ZARATI HOMOMORPHISM 13 Proof In [28], Priddy showed that the opposite of the lambda