On May spectral sequence and the algebraic transfer II

We study the algebraic transfer constructed by Singer 26 using technique of the May spectral sequence. We show that the two squaring operators, defined by Kameko 12 and Nakamura 21, on the domain and range respectively, of our E2 version of the algebraic transfer are compatible. We also prove that the two Sq0 family ni ∈ Ext5,36·2 i A (Z2, Z2), i ≥ 0, and ki ∈ Ext7,36·2 i A (Z2, Z2), i ≥ 1, are in the image of the algebraic transfer.

Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 On May spectral sequence and the algebraic transfer II Phan Hoàng Chơn and Lê Minh Hà Abstract We study the algebraic transfer constructed by Singer [26] using technique of the May spectral sequence We show that the two squaring operators, defined by Kameko [12] and Nakamura [21], on the domain and range respectively, of our E2 version of the algebraic transfer are i compatible We also prove that the two Sq -family ni ∈ Ext5,36·2 (Z/2, Z/2), i ≥ 0, and ki ∈ A i 7,36·2 ExtA (Z/2, Z/2), i ≥ 1, are in the image of the algebraic transfer Introduction and statement results This paper is a continuation of our previous paper [7], which we will refer to as Part I In Part I, we use the May spectral sequence (MSS for short), to compute the kernel and image of the algebraic transfer, introduced by Singer [26], which is an algebra homomorphism ⊕s ϕs : TorA ∗,∗ (Z/2, Z/2) / ⊕s [Z/2 ⊗A H ∗ (BVs )]GLs (1.1) from the homology of the mod Steenrod algebra A (Steenrod [27], Milnor [18]) to the space of A-generators of the (mod 2) cohomology of elementary abelian 2-group Vs of rank s, for s ≥ The cohomology ring H ∗ (BVs ), which is a polynomial algebra in s generators, all in degree 1, is both a module over the mod Steenrod algebra as well as the general linear group GLs = GL(Vs ) Moreover, these two module structures are compatible, so that one has an induced action of GLs on the space Z/2 ⊗A H ∗ (BVs ) It is sometime more convenient to consider the dual of (1.1) At rank s, it has the form ϕ∗s : [PA H∗ (BVs )]GLs → Exts,s+∗ (Z/2, Z/2), A (1.2) where PA H∗ (BVs ) denote the subspace of the divided power algebra H∗ (BVs ) consisting of all elements that are annihilated by all positive degree Steenrod squares, and MG is the standard notation for the module of G-coinvariants Our interests in the map (1.1) (or its dual (1.2)) lies in the fact that on the one hand, the dual of its domain is the cohomology of the Steenrod algebra, Ext∗,∗ A (Z/2, Z/2), which is the initial page of the Adams spectral sequence converging to stable homotopy groups of the spheres [1], therefore, it is an object of fundamental importance in algebraic topology On the other hand, the target of (1.1) is the subject of the so-called “the hit problem”, proposed by F Peterson [23] (see Wood [31]) The hit problem, which is originated from cobordism theory, has deep connection with modular representation theory of the general linear group, and it is believed that tools from modular representation theory can be used to understand the structure of the Ext group We refer to the introduction of Part I for a detailed survey of known facts about the algebraic transfer Briefly, it is known that ϕs is an isomorphism for s ≤ (see Peterson [23], Singer [26], Kameko [12], Boardman [2]), and together with Bruner-Hà-Hưng [4], Hưng [10], Nam [22], 2000 Mathematics Subject Classification 55P47, 55Q45, 55S10, 55T15 (primary) This work is partially supported by a NAFOSTED grant No 101.11-2011.33 Page of 13 PHAN HOÀNG CHƠN AND LÊ MINH HÀ Hà [9], Chơn-Hà [7], complete information about the behaviour of ϕ4 was obtained in Part I where we showed that p0 is detected by the cohomological algebraic transfer In [7], we initiated the use of the (homology) May spectral sequence to compute the algebraic transfer This method allows us to not only recover previous known results with little computation involved, but also obtain new detection and nondetection results in degrees where computation of the hit problem seems out of reach at the moment However, the computation remains difficult, partly because while the target of the algebraic transfer (1.1) is essentially a polynomial ring which is relatively easy to work with, the domain is the Tor group, whose rich structure, such as the action of the Steenrod algebra, is hard to exploit To overcome this difficulty, in this paper, we first dualize the construction in [7] to construct a representation of the algebraic transfer in the cohomological E2 -term of the May spectral sequence An application of this construction is given in Section Recall that in the ∗,∗ ExtA (Z/2, Z/2) groups, there is an action of the (big) Steenrod algebra (see Liulevicius [14] or May [17]), where the operation Sq is no longer the identity map In his thesis [12], Kameko constructed an operation Sq : [PA Hd (BVs )]GLs / [PA H2d+s (BVs )]GLs , that corresponds to the operation Sq on Ext groups Kameko’s operation has been extremely useful in the study of the hit problem and for computation of the algebraic transfer An observation of Vakil [30] indicates that Kameko’s squaring operation is compatible with the May filtration, and thus induces a similar operation when passing to the associated graded On the other hand, Nakamura [21] also constructed a family of squaring operations which are all compatible with higher differentials in the May spectral sequence It should be pointed out that his method of construction is quite different from the usual one such as described in May [17], since it is known that the general framework provided in May [17] yields trivial map in the cohomology of the associated graded algebra E A In section 4, we showed that under the representation of the algebraic transfer in the E2 terms of the May spectral sequence described in Section 3, the induced Kameko squaring operation corresponds to Nakamura’s squaring operation Using the construction above, we have the following, which is our main result i Theorem 1.1 (see also Corollary 5.3) The family {ni ∈ Ext5,36·2 (Z/2, Z/2) : i ≥ 0} is A detected by the algebraic transfer (1.2) 7,∗ Bruner [3] has shown that the relation k1 = h2 h5 n0 holds in ExtA (Z/2, Z/2) Since it is well∗ ∗ known that the total transfer ϕ = ⊕s≥1 ϕs is an algebra homomorphism (see Singer [26]), we obtain an immediate corollary i Corollary 1.2 The family {ki ∈ Ext7,36·2 (Z/2, Z/2) : i ≥ 1} is in the image of the A seventh algebraic transfer We not know whether k0 ∈ Ext5,36 (Z/2, Z/2) also belongs to the image of ϕ∗5 or not A The paper is divided into five sections Sections and are preliminaries In section 2, we recall basic facts about May spectral sequence and in section 3, we present the algebraic transfer and its representation in the E2 -term of the cohomological May spectral sequence We apply the above construction to show in Section that a version of Kameko’s squaring operation which has been extremely useful in the study of the hit problem is compatible with Nakamura’s squaring operation the May spectral sequence The final section contains the proof ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER II Page of 13 of the main results of this paper that the two families ni , i ≥ and kj , j ≥ in Ext∗,∗ A (Z/2, Z/2) are in the image of the algebraic transfer (1.2) The May Spectral Sequence In this section, we review the construction of the May spectral sequence Our main references are May [15, 16] and Tangora [29] May’s chain complex for the cohomology of the associated graded algebra E A was subsumed in Priddy’s theory of Koszul resolution [24] Let A denotes the mod Steenrod algebra and A∗ be its linear dual All A-modules are assumed to have finite type and non-negatively graded 2.1 The associated graded algebra E A The Steenrod algebra is filtered by powers of its augmentation ideal A¯ by setting Fp A = A if 0 ¯ ⊗−p if p < Let E A = ⊕p,q Ep,q p ≥ and Fp A = (A) A, where Ep,q A = (Fp A/Fp−1 A)p+q , be the associated graded algebra According to a well-known theorem of Milnor and Moore [19], E A is a primitively generated Hopf algebra which is isomorphic to the universal enveloping algebra of its restricted Lie algebra of its primitive elements In this case, the primitives are the Milnor generators Pji (see Milnor [18]) The following result from May’s thesis remains unpublished, but is known and used widely Theorem 2.1 (May [15]) The algebra E A is a primitively generated Hopf algebra It is isomorphic to the universal enveloping algebra of the restricted Lie algebra of its primitive elements {Pkj |j ≥ 0, k ≥ 1} Moreover, k (i) Pji , P k = δi,k+ Pj+ for i ≥ k; j (ii) ξ(Pk ) = 0, where ξ is the restriction map (of its restricted Lie algebra structure) Here, δi,k+ is the usual Kronecker delta An element θ ∈ Fp A but θ ∈ Fp−1 A is said to have weight −p The following result determines the weight of any given Milnor generator Sq(R) Theorem 2.2 (May [15]) The weight w(R) of a Milnor generator Sq(R), where R = (r1 , r2 , ), is w(R) = i iα(ri ) where α(m) is the function that counts the number of in the binary expansion of m In particular, the weight of Pji is just the subscript j In fact, May’s argument identifies Sq(R) with the monomial (Pji )aij in the associated graded, where ri = aij 2j is the binary expansion of ri In the language of Priddy’s theory of Koszul resolution [24], Theorem 2.1 states that E A is a Koszul algebra with Koszul generators {Pkj |j ≥ 0, k ≥ 1} and quadratic relations: Pji P k = P k Pji if i = k + , i− Pji P i− + P i− Pji + Pj+ = 0, Pji Pji = The following theorem is first proved in May’s thesis, but see also Priddy [24] for a modern treatment Theorem 2.3 (May [15], Priddy [24]) The cohomology of E A, H ∗ (E A), is isomorphic to the homology of the complex R, where R is the polynomial algebra over Z/2 generated by Page of 13 PHAN HOÀNG CHƠN AND LÊ MINH HÀ {Rji |i ≥ 0, j ≥ 1} each of degree 2i (2j − 1) and with the differential is given by j−1 i+k Rki Rj−k δ(Rji ) = k=1 Moreover, cup products in H (E A) correspond to products of representative cycles in R ∗ We will need a more general version of the above theorem for the cohomology of E A with non-trivial coefficients, which can be derived easily from the discussion in Section of Priddy’s seminal paper [24] Let Rs denote the subspace of R consisting of all monomials in Rji of length s If M is a right E A-module, we can form a cocomplex R ⊗ M where in degree s is Rs ⊗ M The differential, which is again denoted as δ, is given by RRts ⊗ mPts , δ(R ⊗ m) = δ(R) ⊗ m + (2.1) s,t for all R ∈ Rs and all m ∈ M According to Priddy [24, Section 4], there exists a natural isomorphism: / Exts,t0 (Z/2, M ) (2.2) Θ : H s,t (R ⊗ M ) E A 2.2 The May spectral sequence We will be working with the cohomology version of the May spectral sequence Let A∗ be ¯ ∗ Then A∗ admits a filtration where F p A∗ = if p ≥ and the dual of A and let A¯∗ = (A) p ∗ ∗ ¯ F A = (A/Fp−1 A) if p < If M is an A-module, let M ∗ be the Z/2-graded dual of M The comodule M ∗ is filtered by setting F p M ∗ = {m ∈ M ∗ |α∗ (m) ∈ F p A∗ ⊗ M ∗ }, where α∗ is the structure map of the A∗ -comodule M ∗ Clearly F p M ∗ = for p ≥ and when 0 M∗ = M ∗ , where Ep,q p < 0, we have F p M ∗ ⊆ F p−1 M ∗ Thus (E M )∗ ∼ = E M ∗ = ⊕p,q Ep,q p ∗ p+1 ∗ ∗ (F M /F M )p+q , is a bigraded comodule over the associated graded coalgebra E A Let ¯ C(A; M ) be the cobar construction with the induced filtration: F p C¯ n (A∗ ; M ∗ ) = F p1 A¯∗ ⊗ · · · ⊗ F pn A¯∗ ⊗ F p0 M ∗ , n where the sum is taken over all sequences {p0 , , pn } such that n + i=0 pi ≥ p This filtration respects the differential, and in the resulting spectral sequence, we have E1p,q,t (M ∗ ) = F p C¯ p+q (A; M ) F p+1 C¯ p+q (A; M ) t Here p is the filtration degree, p + q is the homological degree and t is the internal degree The differential δ1 of this spectral sequence is the connecting homomorphism of the short exact sequence: ¯ ¯ ¯ F p+1 C(A; M) F p C(A; M) F p C(A; M) → p+2 → p+2 → p+1 → ¯ ¯ ¯ F F F C(A; M) C(A; M) C(A; M) On the other hand, E1p,q,t (M ∗ ) is isomorphic to C¯ p+q (E A; E M )−q,q+t as Z/2-trigraded vector space Under this identification, δ1 is exactly the canonical differential of the cobar construction C¯ ∗ (E A; E M ) Hence E2p,q,t (M ) ∼ = H p+q (E A∗ ; E M ∗ )−q,q+t and we can summarize the result in the following theorem Theorem 2.4 (May [16]) Let M be an A-module of finite type and positively graded There exists a third-quadrant spectral sequence (Er , δr ) converging to E H ∗ (A; M ∗ ) and ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER II Page of 13 having as its E2 -term E2p,q,t (M ) = H p+q (E A; (E M )∗ )−q,q+t Each δr is a homomorphism δr : Erp,q,t (M ) −→ Erp+r,q−r+1,t (M ) When M = Z/2, we will write Er for Er (M ) It is also well-known that Er (M ) is a differential Er -module In his thesis, May have also demonstrated how to compute all the differentials, at least in principle, using the so-called imbedding method (see Tangora [29, Section 5]) The reason is that R is a quotient of the cobar complex, and the differentials comes from that of the cobar complex as well We shall use this method in the proof of the main theorem in Section The algebraic transfer In [6, 7], we constructed a representation of the dual of the algebraic transfer in the E -term of the homology May spectral sequence It turns out that the cohomology version that we are going to present has better behaviour because of the algebra structure on Ext groups Since our construction is appropriate dual to the construction in [7], we will be very brief In this section, we construct the representation of E2 ψs in co-Koszul complex of E H∗ (BVs ), which will be denoted by E1 ψs We begin with some notations For an s-dimensional Z/2-vector space Vs , it is well-known that H ∗ (BVs ) is isomorphic to the polynomial algebra Ps = Z/2[x1 , , xs ], where each generator xi is of degree Dually, H∗ (BVs ) is the divided power algebra Hs = Γ(a1 , , as ), (i ) (i ) where is the dual of xi For simplicity, we will write (i1 , , is ) for the monomial a1 as s −1 Let Pˆ1 be the unique A-module extension of P1 by formally adding a generator x1 of degree n−1 -1 and require that Sq n (x−1 and let Hˆ1 be the dual of Pˆ1 There is a fundamental ) = x1 short exact sequence of A-modules: → Σ−1 Z/2 → Hˆ1 → H1 → 0, passing to the associated graded, we have a similar short exact sequence of E A-modules and after tensoring this sequence with R ⊗ M for some right E A-module M , we have a short exact sequence of differential modules R ⊗ M ⊗ Σ−1 Z/2 / R ⊗ M ⊗ E H1 / R ⊗ M ⊗ E Hˆ1 Using the isomorphism (2.2), the connecting homomorphism of this short exact sequence can be identified with / Exts,t+1 Exts−1,t (Z/2, M ⊗ E H1 ) (Z/2, M ) 0 E A E A Since there is a canonical isomorphism E Hs ∼ = (E H1 ) , we can construct and compose s similar connecting homomorphisms so as to obtain a map Extk,t (Z/2, M ⊗ E Hs ) E0 A ⊗s / Extk+s,t+s (Z/2, M ) E0 A In particular, when M = Z/2 and k = 0, we obtains the E2 -level of the algebraic transfer E2 ψs : Ext0,t (Z/2, E Hs ) E0 A / Exts,t+s (Z/2, Z/2) E0 A As we have noted, this map is induced by a chain level map E1 ψs : E Hs We can describe this map explicitly / Rs Page of 13 PHAN HOÀNG CHƠN AND LÊ MINH HÀ Proposition 3.1 The version of the algebraic transfer in E2 -term of May spectral sequence is induced by the map E1 ψs : E H∗ (BVs ) −→ Rs , given by (n1 ) E1 ϕs (a1 s) a(n )= s (n ) Rtk11 Rtkss , ni = 2ki (2ti − 1) − 1, ≤ i ≤ s, 0, otherwise (3.1) (n ) Proof Suppose R ⊗ m ⊗ a1 as s is a nontrivial summand of a cycle x ∈ R ⊗ M ⊗ Hs It can be pulled back to the same element in R ⊗ M ⊗ Hs−1 ⊗ Hˆ1 Since δ(x) = 0, it comes from R ⊗ M ⊗ Hs−1 ⊗ Σ−1 Z/2 On the other hand, we have that a(n) Pji = a(−1) if and only if n = 2i (2j − 1) − Thus from the formula (2.1), we see that the connecting homomorphism (n ) (n ) sends R ⊗ m ⊗ a1 as s to zero if ns does not have the form 2i (2j − 1) − for some (ns−1 ) (n ) i i ≥ 0, j ≥ 1; and to RRj ⊗ m ⊗ a1 as−1 if ns = 2i (2j − 1) − The required formula can now be easily obtained by induction Example 3.2 Let x = (1, 1, 6) + (1, 2, 5) + (1, 4, 3) ∈ E1−2,2,8 (P3 ) It is easy to check that δ1 (x) = ∈ E1−1,2,8 (P3 ), so x is a cycle in the E1 -term and survives to a nontrivial element in E2−2,2,8 (P3 ) Now δ2 (x) = R10 ⊗ (1, 3, 3) = δ1 (2, 3, 3) ∈ E2−1,2,8 (P3 ), so x is a cycle in E2−2,2,8 (P3 ) For r ≥ 3, Er−2+r,∗,∗ = 0, so δr (x) = for all r ≥ 3; therefore, x is a permanent cycle Using (3.1), we obtain E1 ψ3 (x) = R11 R11 R30 + R11 R20 R21 = R11 (R20 R21 + R30 R11 ), this latter element is called h1 h0 (1) in the E2 terms of the May spectral sequence (see Tangora [29, Appendix 1]), and is a representation of c0 in the 8-stem Example 3.3 We see that the element d¯0 , which is represented by the cycle X = x + (13)x + (23)x ∈ E1−4,4,14 (P4 ), where x = (2, 2, 5, 5) + (1, 1, 6, 6) + (2, 1, 6, 5) + (1, 2, 5, 6) + (4, 4, 3, 3) + (4, 2, 5, 3) + (2, 4, 3, 5) + (4, 1, 6, 3) + (1, 4, 3, 6), is a permanent cycle Indeed, since δ2 (X) = δ1 (y + (13)y + (23)y), where y = (3, 2, 6, 3) + (2, 3, 3, 6), X is a cycle in E2−2,3,14 (P4 ); therefore, d¯0 survives to the E3−4,4,14 and, in the E3 -term, it is represented by X + Y , where Y = y + (13)y + (23)y By inspection, we have δ3 (X + Y ) = δ1 (Z); δ4 (X + Y + Z) = δ1 (3, 3, 3, 5), where Z = (5, 1, 5, 3) + (3, 5, 1, 5) Therefore, d¯0 is a permanent cycle because δr , r ≥ 5, is trivial Again from (3.1), we obtain E1 ψ4 (X) = (R20 R21 + R30 R11 )2 , which is a representation of d0 in the E1 -term of May spectral sequence Since d¯0 is a permanent cycle, it is a representation of the pre-image of d0 under the algebraic transfer in the MSS We end this section with two simple properties of the maps Er ψs First of all, since R is a commutative algebra, it is clear that E1 ψs factors through the coinvariant ring [PE A E Hs ]Σs The reader who is familiar with the algebraic transfer may wonder about the action of the ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER II Page of 13 full general linear group GLs Unfortunately, in general this action does not preserve the May filtration For example, if f = x21 x52 ∈ F−2 P2 and σ ∈ GL2 , such that σ(x1 ) = x1 + x2 and σ(x2 ) = x2 , then we have σ(f ) = x21 x52 + x72 ∈ F0 P2 Secondly, the direct sum ⊕s≥1 H∗ (BVs ) has an algebra structure with concatenation product Stardard argument as in Singer [26] shows that Proposition 3.4 For each r ≥ 1, the total homomorphism Er ψ = ⊕s≥1 Er ψs : ⊕s Er∗,∗ (Ps ) / Er∗,∗ , is an algebra homomorphism The squaring operations In [21], Nakamura constructed a squaring operation on the MSS for the trivial module: Sq : Erp.q / Erp,q , r ≥ 1, which is multiplicative in the E1 page and therefore satisfies the Cartan formulas in higher Er page (when elements are suitably represented in the E2 term) The purpose of this section is to introduce a similar squaring operation, defined for any r, s ≥ 1, which is also denoted as Sq : Sq : Erp,q (Ps ) / Erp,q (Ps ), / Erp,q+s so that it commutes with Nakamura’s Sq via the map of spectral sequences Erp,q (Ps ) constructed in the previous section We begin with a description of Nakamura squaring operation in the complex R of (2.1), this is reminiscent to the construction of Sq in Ext∗,∗ A (Z/2, Z/2) from an endomorphism of the / R by setting θ(Ri ) = Ri+1 lambda algebra as in Tangora [29] Define an algebra map θ : R j j By direct inspection, we see that θ commutes with the coboundary map δ, and thus induces an endormorphism on Ext∗,∗ (Z/2, Z/2) E0 A Proposition 4.1 The endomorphism θ induces Nakamura’s squaring operation / Erp,q Sq : Erp,q i Proof According to Priddy [24], Rji is represented in the cobar resolution by [ξj2 ] On the other hand, in the cobar complex for E A, the squaring operation has an explicit form Sq [α1 | |αn ] = [α12 | |αn2 ], i so it maps [ξj2 ] to [ξj2 i+1 ] and the result follows immediately On H∗ (BVs ), there is also a squaring map constructed by Kameko [12] in his thesis which has been extremely useful in the study of the hit problem (See for example Sum [28]) It is given explicitly as follows (t ) (2t1 +1) s) a1 a(t → a1 s s +1) a(2t s One quickly verifies that this endomorphism of H∗ (BVs ) commutes with the action of the Steenrod algebra, in the sense that for all a ∈ H∗ (BVs ), (θa)Pts = θ(aPts−1 ) if s > 0, and (θa)Pt0 = Moreover, Vakil [30] observed that the map a(n) → a(2n+1) respects May’s filtration on H∗ (BZ/2) This is clearly true for higher rank s > as well Define an endomorphism on Page of 13 PHAN HOÀNG CHƠN AND LÊ MINH HÀ R ⊗ H∗ (BVs ), which is again denoted as Sq , by setting Sq : R ⊗ a → Sq (R) ⊗ θ(a), for all R ∈ R, a ∈ H∗ (BVs ) Lemma 4.2 of (2.1) The endormorphism Sq on R ⊗ H∗ (BVs ) commutes with the coboundary δ Proof We already know that Sq and δ commutes on R Also, (θa)Rt0 = and (θa)Rts = θ(aRts−1 ), we have δSq (R ⊗ a) =δ(Sq R ⊗ θa) =δSq R ⊗ θa + Sq (R)Rts ⊗ (θa)Rts =Sq δR ⊗ θa + Sq (RRts−1 ) ⊗ θ(aRts−1 ) =Sq (δR ⊗ a + RRts−1 ⊗ aRts−1 ) = Sq δ(R ⊗ a) The proof is complete It follows that there exists an induced endomorphism Sq on Erp,q (Ps ) for all s, r ≥ Our next result shows that this endomorphism commutes with Nakamura’s Sq via the MSS transfer Er ψs , thus justifies for our choice of notation Proposition 4.3 sequences: There exists a commutative diagram of maps between spectral Erp,q (Ps ) Er ϕ s Sq Sq  Erp,q (Ps ) / Erp,q+s Er ϕs  / Erp,q+s Proof It sufices to show that there exists a commutative diagram at E1 page E H∗ (BVs ) E1 ϕs Sq θ  E H∗ (BVs ) / Rs E1 ϕs  / Rs This can be verified directly from the formula (3.1) Note that if ni = 2ki (2ti − 1) − then 2ni + = 2ki +1 (2ti − 1) − In particular, we have an induced map θ : PE A E Hd (BVs ) → PE A E H2d+s (BVs ), that fits in the following Proposition 4.4 The representation of Kameko’s squaring operations, Sq , and Nakamura’s squaring operations commute with each other through E2 ϕs In other words, the ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER II Page of 13 following diagram commutes E2 ϕs PE0 A E Ht (BVs ) Sq θ  PE0 A E H2t+s (BVs ) / Exts,s+t (Z/2, Z/2) E0 A E2 ϕs  / Exts,s+t (Z/2, Z/2) E0 A Proof The assertion is implied directly from the formula of Sq and (3.1) The operation θ commutes with the action of the symmetric group Σs on E H∗ (BVs ) as well as its subspace PE A E H∗ (BVs ) This is essentially direct from the definition Furthermore, E2 ψs is Σs -equivariant since R is commutative Thus in the commutative diagram of Proposition 4.4, we can replace PE A E H∗ (BVs ) by its Σs -coinvariant (PE A E H∗ (BVs ))Σs Our last result can be considered as an analogue to N H V Hưng’s analysis of the squaring map on the space (PA Hs )GLs [10], where he showed that after (s − 2) iteration, the squaring map becomes an isomorphism on its range For the associated graded, the situation is much simpler Proposition 4.5 For each s ≥ 1, the induced map θ : (PE A E H∗ (BVs ))Σs / (PE A E H∗ (BVs ))Σs , is a monomorphism Proof Suppose x ∈ PE A E Hs such that θ(x) = in (PE A E H∗ (BVs ))Σs This means that there exist zσ ∈ (PE A E H∗ (BVs ))Σs such that θx = zσ σ + zσ σ∈Σs We say that a monomial ai11 aiss is odd if all exponents it are odd Otherwise, we say that it is non-odd The left hand side of the above equation contains all odd monomials Each zσ can be written as the sum zσ + zσ where zσ consists of all non-trivial odd monomials in zσ We first claim that both zσ and zσ are E A-annihilated Lemma 4.6 If x = y + z ∈ PE A E Hs where y is the sum of odd monomials summands of x, then y and z belongs to PE A E Hs Proof First of all, note that E A is multiplicatively generated by P1s , s ≥ 0, so in order to prove that y is E A-annihilated, we just have to check that yP1s = for all s ≥ Since all monomials in y are odd, it is clear that yP10 = ySq = If s > 1, then since P1s is a derivative, and |P1s | = 2s is even, we see that yP1s , if non-zero, consists of only odd monomials while zP1s consists of only non-odd monomials Because xP1s = 0, we must have yP1s = zP1s = for all s > The lemma is proved We now continue the proof of Proposition 4.5 We have a decomposition in PE A E Hs θx = (zσ σ + zσ ) + σ∈Σs (zσ σ + zσ ) σ∈Σs Page 10 of 13 PHAN HOÀNG CHƠN AND LÊ MINH HÀ The second summand must vanish since it contains non-odd monomials The first summand can be written as θx for some x of the form x = (yσ σ + yσ ), where yσ is such that θyσ = zσ Since θ is obviously a monomorphism on PE A E Hs , it follows that x = x and so x is trivial in (PE A E Hs )Σs It should be noted that the statement of Lemma 4.6 is not true for the original hit problem For example, consider the element x = (135) + (223) + (124) ∈ PA H3 where by (abc) we mean the sum of all monomials that are permutations of (a, b, c) Then x = y + z where y = (135) contains only odd monomials, but y is not A-annihilated We not know whether the similar endomorphism on θ : (PA Hs )Σs remains a monomorphism However, in light of the Singer’s conjecture that ϕ∗s is always a monomorphism and current knowledge of Exts,∗ A (Z/2, Z/2) for s ≤ 5, we believe that such an example will not be easy to find Proof of the main results In this section we use our version of the algebraic transfer on the E2 -term of the May spectral sequence to show that the family ni , i ≥ 0, belongs to the image of the algebraic tranfer It should be noted that this detection result is in degree that goes far beyond our current knowledge of the hit problem An element in x ∈ E1 is said to survives to Er for some r ≥ if its projection to a non-zero element in the Er A permanent cycle is an element killed by δr for all r First of all, we need a technical lemma Lemma 5.1 If x ¯ ∈ E1p,−p,t (Ps ) is a permanent cycle such that E1 ψs (¯ x) represents an element x ∈ Exts,s+t (Z/2, Z/2) in the E1 -term of the May spectral sequence, then x is in A the image of the algebraic transfer Proof Since E1 ψs (¯ x) is a representation of x in the E1 -term of the May spectral sequence ∗,∗,∗ and x ¯ survives to E∞ (Ps ), under E∞ ϕs , the image of x ¯ is the presentation of x in the E∞ -term of May spectral sequence Thus, we have the assertion of the lemma With stardard notation of known nontrivial elements in the cohomology of the Steenrod algebra [29], the following theorem is our main result Theorem 5.2 The element n0 ∈ Ext5,36 (Z/2, Z/2) is in the image of the algebraic transfer A The fact that n0 and n1 = Sq n0 are indecomposable elements of Ext5,∗ A (Z/2, Z/2) goes back to Tangora [29] Recently, completing a program initiated by Lin [13], Chen [5] proved that the whole Sq -family {ni , i ≥ 0} starting with n0 are indecomposable in Ext5,∗ A (Z/2, Z/2) Since Kameko’s squaring operation and the classical squaring operation Sq commute with each other through the algebraic transfer, we have the following immediate corollary i Corollary 5.3 The family of indecomposable elements ni ∈ Ext5,36·2 (Z/2, Z/2), i ≥ 0, A are in the image of the algebraic transfer ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER IIPage 11 of 13 According to Tangora [29], there exists an indecomposable element k ∈ Ext7,36 (Z/2, Z/2) A and a relation k1 = h2 h5 n0 ∈ Ext7,72 (Z/2, Z/2), therefore we have A i Corollary 5.4 algebraic transfer The elements ki ∈ Ext5,36·2 (Z/2, Z/2), i ≥ 1, are in the image of the A We not know whether k0 also belongs to the image of the transfer or not −6,6,31 Proof of Theorem 5.2 We shall find a permanent cycle in E∞ (P5 ) which is represented 6,−6,31 (P5 ), to be described explicitly, such that under the E1 -version of by an element X ∈ E1 algebraic transfer, the image of X is R12 (R30 )2 (R12 R22 + R31 R12 ) This image is known, according to Tangora [29], to be a representative of n0 in the E1 -term of the MSS Elements in E2 and its projection (if exists) to Er will be written using the same letter Let X = x + (23)x + (243)x, where x = (3, 3, 6, 6, 13) + (3, 5, 6, 6, 11) + (3, 6, 3, 5, 14) + (3, 6, 5, 3, 14) + (3, 9, 6, 6, 7) + (3, 10, 5, 6, 7) + (3, 10, 6, 5, 7) + (5, 10, 3, 6, 7) + (5, 10, 6, 3, 7) + (6, 9, 3, 6, 7) + (6, 9, 6, 3, 7) + (6, 10, 3, 5, 7) + (6, 10, 5, 3, 7), and (23), (243) are the usual permutations in cycle form By direct computation, X is a cycle in E1−6,6,31 (P5 ) X is also a cycle in E2−6,6,31 (P5 ) since δ2 (X) belongs to image of δ1 Infact, in E1 (P5 ), we have an explicit formula δ2 (X) = δ1 (y + (23)y + (243)y), where y = (5, 7, 6, 6, 7) + (6, 7, 5, 6, 7) + (6, 7, 6, 5, 7) + (5, 9, 5, 5, 7) + (5, 3, 5, 5, 13) + (5, 5, 5, 5, 11) Hence, n ¯ survives to E3−6,6,31 (P5 ); and in the E3 -term, n ¯ has a representation X + Y , where Y = y + (23)y + (243)y As δ3 (X + Y ) = δ1 (z + (23)z + (243)z), for z = (3, 9, 5, 3, 11) + (3, 9, 3, 5, 11) + (9, 9, 3, 3, 7) + (9, 5, 3, 3, 11) + (9, 3, 3, 3, 13), X + Y is a cycle in the E3 -term; therefore, n ¯ is nontrivial in E4−6,6,31 (P5 ) and it is represented by X + Y + Z, where Z = z + (23)z + (243)z We can check that, δ4 (X + Y + Z) = δ1 (t + (23)t + (243)t + (7, 5, 3, 5, 11) + (7, 3, 5, 3, 13)), where t = (3, 7, 5, 5, 11) + (3, 11, 5, 5, 7) + (3, 13, 3, 5, 7) + (3, 13, 5, 3, 7) + (7, 9, 3, 5, 7) + (9, 7, 3, 5, 7) + (3, 7, 9, 5, 7) + (5, 7, 9, 3, 7) It implies X + Y + Z is a cycle in E4−6,6,31 (P5 ) Hence, n ¯ survives to the E5 -term, and in the E5 -term, n ¯ is represented by X + Y + Z + T , where T = t + (23)t + (243)t + (7, 5, 3, 5, 11) + (7, 3, 5, 3, 13) By similar argument, we have δ5 (X + Y + X + T ) = δ1 (R) ∈ E5−1,2,31 ; δ6 (X + Y + Z + T + R) = δ1 (S) ∈ E60,1,31 , where R = (5, 7, 5, 7, 7) + (3, 3, 3, 11, 11) + (3, 3, 11, 3, 11) + (3, 11, 3, 3, 11); S = (7, 7, 3, 3, 11) + (7, 3, 7, 3, 11) + (7, 3, 3, 7, 11) + (3, 7, 3, 7, 11) Page 12 of 13 ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER II Since, for r > 6, Er−6+r,∗,∗ (P5 ) = 0, δr (¯ n0 ) = 0, r > In other words, n ¯ is a permanent cycle Finally, using (3.1), we obtain E1 ψ5 (¯ n0 ) = R12 (R30 )2 (R12 R22 + R31 R12 ) The proof is complete Acknowledgements We would like to thank T W Chen for sending us an early version of [5] The second author is a Junior Associate at the ICTP, Trieste, Italy The final version of this paper was completed while both authors were visiting the Vietnam Institute for Advanced Study in Mathematics We thanks the VIASM for support and hospitality Both authors are supported by a NAFOSTED grant No 101.01-2011.33 References J F Adams, On structure and applications of the Steenrod algebra, Comment Math Helv., 32 (1958), 180-214 J M Boardman, Modular representations on the homology of powers of real projective space, Contemp Math 146 (1993), 49-70 R R Bruner, The cohomology of the mod Steenrod algebra: A computer calculation, WSU Research Report 37 (1997), 217 pages R R Bruner, L M Hà and N H V Hưng, On behavior of the algebraic transfer, Trans Amer Math Soc., 357 (2005), 473-487 T W Chen, Determination of Ext5,∗ A (Z/2, Z/2), Topo App., 158 (2011), no 5, 660-689 P H Chơn and L M Hà, On May spectral sequence and the algebraic transfer, Proc Japan Acad 86 (2010), 159-164 P H Chơn and L M Hà, On May spectral sequence and the algebraic transfer, Manuscripta Math.138, (2012), 141-160 P H Chơn and L M Hà, Lambda algebra and the Singer transfer, C R Acad Sci Paris, Ser I, 349 (2011), 21-23 L M Hà, Sub-Hopf algebra of the Steenrod algebra and the Singer transfer, Geo Topo Mono., 11 (2007), 81-104 10 N H V Hưng, The cohomology of the Steenrod algebra and representations of the general linear groups, Trans Amer Math Soc., 357 (2005), 4065-4089 11 N H V Hưng and V T N Quỳnh, The image of the fourth algebraic transfer, C R Acad Sci Paris, Ser I, 347 (2009), 1415-1418 12 M Kameko, Products of projective spaces as Steenrod modules, Thesis, Johns Hopkins University, 1990 5,∗ 13 W H Lin, Ext4,∗ A (Z/2, Z/2) and ExtA (Z/2, Z/2), Topo App., 155 (2008), no 5., 459-496 14 A Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Mem Amer Math Soc 42 (1962) 15 J P May, The cohomology of restricted Lie algebras and of Hopf algebras; applications to the Steenrod algebra, Princeton University, Ph.D., 1964 16 J P May, The cohomology of restricted Lie algebra and of Hopf algebra, Journal of Algebra (1966), 123-146 17 J P May, The general algebraic approach to Steenrod operations, the Steenrod algebra and its applications, Lecture note in Math 168, Springer-Verlag, 153-231 18 J Milnor, The Steenrod algebra and its dual, Ann of Math., 67 (1958), 150-171 19 J Milnor and J C Moore, On the structure of Hopf algebras, Ann of Math., 81 (1965), 211-264 20 N Minami, The iterated transfer analogue of the new doomsday conjecture, Trans Amer Math Soc 351 (1999), 2325-2351 21 O Nakamura, On the squaring operations in the May spectral sequence, Mem Facul Sci Kyushu Univ., Ser A, (1972), 293-308 22 T N Nam, Transfert algébrique et action du groupe linéaire sur les puissances divisées modulo 2, Ann Inst Fourier (Grenoble) 58 (2008), 1785-1837 23 F P Peterson, A-generators for certain polynomial algebras, Math Proc Cambridge Philos Soc 105 (1989), no 2, 311óÀẼ312 24 S B Priddy, Koszul resolutions, Trans Amer Math Soc., 152 (1970), 39-60 25 V T N Quỳnh, On behavior of the fifth algebraic transfer, Geo Topo Mono., 11 (2007), 309-326 26 W M Singer, The transfer in homological algebra, Math Z., 202 (1989), 493-523 27 E N Steenrod and D B A Epstein, Cohomology operations, Ann Math Studies, 50 (1962) 28 N Sum, The negative answer to Kameko’s conjecture on the hit problem, Advances in Math 225 (2010) 2365-2390 29 M C Tangora, On the cohomology of the Steenrod algebra, Math Z., 116 (1970), 18-64 30 R Vakil, On the Steenrod length of real projective spaces: finding longest chains in certain directed graphs, Discrete Math., 204 (1999), 415-425 31 R M W Wood, Steenrod squares of polynomials and the Peterson conjecture, ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER IIPage 13 of 13 Department of Mathematics - Application Saigon University, 273 An Duong Vuong, District 5, Ho Chi Minh city, Vietnam Department of Mathematics-Mechanics and Informatics, Vietnam National University - Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam phchon.vn@gmail.com minhha@vnu.edu.vn [...]... H Chơn and L M Hà, On May spectral sequence and the algebraic transfer, Proc Japan Acad 86 (2010), 159-164 7 P H Chơn and L M Hà, On May spectral sequence and the algebraic transfer, Manuscripta Math.138, (2012), 141-160 8 P H Chơn and L M Hà, Lambda algebra and the Singer transfer, C R Acad Sci Paris, Ser I, 349 (2011), 21-23 9 L M Hà, Sub-Hopf algebra of the Steenrod algebra and the Singer transfer, .. .ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER IIPage 11 of 13 According to Tangora [29], there exists an indecomposable element k ∈ Ext7,36 (Z/2, Z/2) A and a relation k1 = h2 h5 n0 ∈ Ext7,72 (Z/2, Z/2), therefore we have A i Corollary 5.4 algebraic transfer The elements ki ∈ Ext5,36·2 (Z/2, Z/2), i ≥ 1, are in the image of the A We do not know whether k0 also belongs to the image of the transfer. .. On the cohomology of the Steenrod algebra, Math Z., 116 (1970), 18-64 30 R Vakil, On the Steenrod length of real projective spaces: finding longest chains in certain directed graphs, Discrete Math., 204 (1999), 415-425 31 R M W Wood, Steenrod squares of polynomials and the Peterson conjecture, ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER IIPage 13 of 13 Department of Mathematics - Application... J P May, The general algebraic approach to Steenrod operations, the Steenrod algebra and its applications, Lecture note in Math 168, Springer-Verlag, 153-231 18 J Milnor, The Steenrod algebra and its dual, Ann of Math., 67 (1958), 150-171 19 J Milnor and J C Moore, On the structure of Hopf algebras, Ann of Math., 81 (1965), 211-264 20 N Minami, The iterated transfer analogue of the new doomsday conjecture,... 13 ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER II Since, for r > 6, Er−6+r,∗,∗ (P5 ) = 0, δr (¯ n0 ) = 0, r > 6 In other words, n ¯ 0 is a permanent cycle Finally, using (3.1), we obtain E1 ψ5 (¯ n0 ) = R12 (R30 )2 (R12 R22 + R31 R12 ) The proof is complete Acknowledgements We would like to thank T W Chen for sending us an early version of [5] The second author is a Junior Associate at the. .. Priddy, Koszul resolutions, Trans Amer Math Soc., 152 (1970), 39-60 25 V T N Quỳnh, On behavior of the fifth algebraic transfer, Geo Topo Mono., 11 (2007), 309-326 26 W M Singer, The transfer in homological algebra, Math Z., 202 (1989), 493-523 27 E N Steenrod and D B A Epstein, Cohomology operations, Ann Math Studies, 50 (1962) 28 N Sum, The negative answer to Kameko’s conjecture on the hit problem, Advances... (Z/2, Z/2) and ExtA (Z/2, Z/2), Topo App., 155 (2008), no 5., 459-496 14 A Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Mem Amer Math Soc 42 (1962) 15 J P May, The cohomology of restricted Lie algebras and of Hopf algebras; applications to the Steenrod algebra, Princeton University, Ph.D., 1964 16 J P May, The cohomology of restricted Lie algebra and of Hopf... Boardman, Modular representations on the homology of powers of real projective space, Contemp Math 146 (1993), 49-70 3 R R Bruner, The cohomology of the mod 2 Steenrod algebra: A computer calculation, WSU Research Report 37 (1997), 217 pages 4 R R Bruner, L M Hà and N H V Hưng, On behavior of the algebraic transfer, Trans Amer Math Soc., 357 (2005), 473-487 5 T W Chen, Determination of Ext5,∗ A (Z/2, Z/2),... transfer, Geo Topo Mono., 11 (2007), 81-104 10 N H V Hưng, The cohomology of the Steenrod algebra and representations of the general linear groups, Trans Amer Math Soc., 357 (2005), 4065-4089 11 N H V Hưng and V T N Quỳnh, The image of the fourth algebraic transfer, C R Acad Sci Paris, Ser I, 347 (2009), 1415-1418 12 M Kameko, Products of projective spaces as Steenrod modules, Thesis, Johns Hopkins... conjecture, Trans Amer Math Soc 351 (1999), 2325-2351 21 O Nakamura, On the squaring operations in the May spectral sequence, Mem Facul Sci Kyushu Univ., Ser A, 2 (1972), 293-308 22 T N Nam, Transfert algébrique et action du groupe linéaire sur les puissances divisées modulo 2, Ann Inst Fourier (Grenoble) 58 (2008), 1785-1837 23 F P Peterson, A-generators for certain polynomial algebras, Math Proc Cambridge ... squaring operation the May spectral sequence The final section contains the proof ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER II Page of 13 of the main results of this paper that the two families... through the coinvariant ring [PE A E Hs ]Σs The reader who is familiar with the algebraic transfer may wonder about the action of the ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER II Page... polynomials and the Peterson conjecture, ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER IIPage 13 of 13 Department of Mathematics - Application Saigon University, 273 An Duong Vuong, District