Topology and its Applications 178 (2014) 372–383 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol On the May spectral sequence and the algebraic transfer II ✩ Phan Hoàng Chơn a,∗ , Lê Minh Hà b a Faculty of Mathematics – Application, Saigon University, 273, An Duong Vuong, District 5, Ho Chi Minh city, Viet Nam b Department of Mathematics, Mechanics and Informatics, Vietnam National University – Hanoi, 334 Nguyen Trai Street, Hanoi, Viet Nam a r t i c l e i n f o a b s t r a c t Article history: Received 11 February 2014 Received in revised form 25 August 2014 Accepted 15 October 2014 Available online 24 October 2014 We study the algebraic transfer constructed by Singer [19] using the May spectral sequence technique We show that the two squaring operators defined by Kameko [8] and Nakamura [16] on the domain and range respectively of our E2 version of the algebraic transfer are compatible We also prove that the two Sq -families i i ni ∈ Ext5,36·2 (Z/2, Z/2), i ≥ 0, and ki ∈ Ext7,36·2 (Z/2, Z/2), i ≥ 1, are in the A A image of the algebraic transfer © 2014 Elsevier B.V All rights reserved This paper is dedicated to Professor Nguy˜ ên H.V Hưng on the occasion of his 60th birthday MSC: primary 55P47, 55Q45 secondary 55S10, 55T15 Keywords: Adams spectral sequence May spectral sequence Steenrod algebra Algebraic transfer Introduction and statement results This note is a continuation of our previous paper [5], which we will refer to as Part I In Part I, we use the May spectral sequence (MSS for short) to obtain new computation for the kernel and image of the algebraic transfer, introduced by Singer [19], which is an algebraic homomorphism s ✩ Z/2 ⊗A H ∗ (BVs ) ϕs : TorA ∗,∗ (Z/2, Z/2) → ϕ= s This work is partially supported by a NAFOSTED grant No 101.11-2011.33 * Corresponding author E-mail addresses: chonkh@gmail.com (P.H Chơn), minhha@vnu.edu.vn (L.M Hà) http://dx.doi.org/10.1016/j.topol.2014.10.013 0166-8641/© 2014 Elsevier B.V All rights reserved Gs (1) P.H Chơn, L.M Hà / Topology and its Applications 178 (2014) 372–383 373 from the homology of the mod Steenrod algebra A (see Steenrod [20], Milnor [14]) to the space of A-generators of the (mod 2) cohomology of elementary abelian 2-group Vs of rank s, s ≥ The cohomology ring H ∗ (BVs ) has a natural structure of an unstable module over the mod Steenrod algebra In addition, the automorphism group Gs = GL(Vs ) acts canonically on Vs and hence on H ∗ (BVs ) These two module structures are compatible, so that one has an induced action of Gs on the space Z/2 ⊗A H ∗ (BVs ) The reader who is familiar with the algebraic transfer and the related “hit problem” [8,24,19] will probably agree that the dual point of view is also very useful In rank s and working with homology and Ext instead, the dual of (1) has the form ϕ∗s : PA H∗ (BVs ) Gs → Exts,s+∗ (Z/2, Z/2), A (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) (or its dual (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) is the subject of the so-called the “hit problem”, proposed by F Peterson [17] (see Wood [24]) 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 In Part I, we initiated the use of the (homology) May spectral sequence to make computation on the kernel and image of 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) 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 [5] 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 Ext∗,∗ A (Z/2, Z/2) groups, there is an action of the (big) Steenrod algebra (see Liulevicius [10] or May [13]), where the operation Sq is no longer the identity map In his thesis [8], Kameko constructed an operation Sq : PA Hd (BVs ) Gs → PA H2d+s (BVs ) Gs , 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 We use an observation of Vakil [23] to show 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 [16] 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 Nakamura’s construction is quite different from the usual procedure of constructing Steenrod operations, such as described in May [13] For it is known that the general framework provided in May [13] 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 one 374 P.H Chơn, L.M Hà / Topology and its Applications 178 (2014) 372–383 Using the construction above, we have the following, which is our main result i (Z/2, Z/2) : i ≥ 0} is detected by the algebraic Theorem 1.1 (Corollary 5.2) The family {ni ∈ Ext5,36·2 A transfer (2) Bruner [2] has shown that the relation k1 = h2 h5 n0 holds in Ext7,∗ A (Z/2, Z/2) Since it is well-known that the total transfer ϕ∗ = s≥1 ϕ∗s is an algebra homomorphism (see Singer [19]), we obtain an immediate corollary i (Z/2, Z/2) : i ≥ 1} is in the image of the seventh algebraic transfer Corollary 1.2 The family {ki ∈ Ext7,36·2 A ∗ We not know whether k0 ∈ Ext7,36 A (Z/2, Z/2) also belongs to the image of ϕ7 or not The paper is divided into five sections Sections and are preliminaries In Section 2, we recall basic facts about the 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 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 (2) The May spectral sequence In this section, we review the construction of the May spectral sequence The main references are May [11,12] and Tangora [22] May’s chain complex for the cohomology of the associated graded algebra E 0A was subsumed in Priddy’s theory of Koszul resolution [18] Let A denote the mod Steenrod algebra All A-modules considered 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 p ≥ and 0 ¯ ⊗−p if p < Let E A = Fp A = (A) p,q Ep,q A, where Ep,q A = (Fp A/Fp−1 A)p+q , be the associated graded algebra Using a well-known theorem of Milnor and Moore [15, Theorem 6.11] and Milnor’s investigation of the structure of the Steenrod algebra [14], May showed in his unpublished thesis [11] that E A is a primitively generated Hopf algebra which is isomorphic to the universal enveloping algebra of its restricted k Lie algebra of its primitive elements {Pkj | j ≥ 0, k ≥ 1} Moreover, [Pji , P k ] = δi,k+ Pj+ for i ≥ k and j ξ(Pk ) = 0, where ξ is the restriction map of its restricted Lie algebra structure and 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 2.1 (May [11]) 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 digit in the binary expansion of m In particular, the weight of Pji is just its subscript j In fact, May identified Sq(R) with the monomial aij 2j is the binary expansion of ri In the language of (Pji )aij in the associated graded, where ri = Priddy’s theory of Koszul resolution [18], then 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 = P.H Chơn, L.M Hà / Topology and its Applications 178 (2014) 372–383 375 Theorem 2.2 (Priddy [18]) Let M be a right E A-module There exists a natural isomorphism Θ: H s,t (R ⊗ M ) → Exts,t E A (Z/2, M ), (3) where the differential graded algebra R is a polynomial algebra over Z/2 generated by {Rji }, i ≥ 0, j ≥ 1, j−1 i+k of degree 2i (2j − 1), and the differential is given by δ(Rji ) = k=1 Rki Rj−k ; and the differential δ of the cocomplex R ⊗ M is given by δ(R ⊗ m) = δ(R) ⊗ m + RRts ⊗ mPts (4) s,t 2.2 The May spectral sequence We will be working with the cohomology version of the May spectral sequence Let A∗ be the dual of A ¯ ∗ Then A∗ admits a filtration where F p A∗ = if p ≥ and F p A∗ = (A/F ¯ p−1 A)∗ if p < and let A¯∗ = (A) ∗ ∗ 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 p < 0, we 0 have F p M ∗ ⊆ F p−1 M ∗ Thus (E M )∗ ∼ M ∗ , where Ep,q M ∗ = (F p M ∗ /F p+1 M ∗ )p+q , = E M ∗ = p,q Ep,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 ∗ , where the sum is taken over all sequences {p0 , , pn } such that n + the differential, and in the resulting spectral sequence, we have n i=0 pi ≥ p This filtration respects 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: 0→ ¯ ¯ ¯ F p+1 C(A; M) F p C(A; M) F p C(A; M) → p+2 → p+1 → p+2 ¯ ¯ ¯ F C(A; M ) F C(A; M ) F 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 trigraded Z/2-vector spaces ¯ ∗ (E A; E M ) Under this identification, δ1 is exactly the canonical differential of the cobar construction C p,q,t Hence E2 (M ) is isomorphic to H p+q (E A∗ ; E M ∗ )−q,q+t and we can summarize the result in the following theorem Theorem 2.3 (May [12]) Let M be an A-module of finite type and positively graded There exists a spectral sequence (Er , δr ) converging to E H ∗ (A; M ∗ ) and 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 write Er for Er (M ) It is well-known that Er (M ) is a differential Er -module May [12] explained how to compute all the differentials, at least in principle, using the so-called imbedding method (see also Tangora [22, Section 5]) This is possible because R is a quotient of the cobar complex, 376 P.H Chơn, L.M Hà / Topology and its Applications 178 (2014) 372–383 and the differentials come 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 [4,5], we constructed and studied a representation of the dual of the algebraic transfer in the E 2-term of the homology May spectral sequence The cohomology version which we are going to present has better behavior because of the algebra structure on Ext groups Since the construction presented below is basically dual to that given in [5], we will be very brief We are going to construct a representation of E2 ψs in the 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 , the (mod 2) cohomology ∗ H (BVs ) is a polynomial algebra Ps = Z/2[x1 , , xs ], where each xi is of degree Dually, H∗ (BVs ) is the divided power algebra Hs = Γ (a1 , , as ) generated by a1 , , an over Z/2 where is the linear dual (i ) (i ) of xi More precisely, it is a bicommutative Hopf algebra with the vector space basis a1 as s , it ≥ 0, for all ≤ t ≤ s, with multiplication (i1 , , is )(j1 , , js ) = is + js i1 + j1 (i1 + j1 , , is + js ), i1 is (i ) (i ) where for simplicity, we write (i1 , , is ) = a1 as s Let Pˆ1 be the unique A-module extension of P1 by formally adding a generator x−1 of degree −1 and n−1 ˆ1 be the dual of Pˆ1 There is a fundamental short exact require that Sq n (x−1 ) = x for n ≥ Let H 1 sequence of A-modules: → Σ −1 Z/2 → Hˆ1 → H1 → Passing to the associated graded and tensoring with R ⊗ M where M is some right E A-module, we obtain a short exact sequence of differential modules R ⊗ M ⊗ Σ −1 Z/2 → R ⊗ M ⊗ E Hˆ1 → R ⊗ M ⊗ E H1 The connecting homomorphism of this short exact sequence, under the isomorphism (3), can be identified with s−1,t s,t+1 ExtE A (Z/2, M ⊗ E H1 ) → ExtE A (Z/2, M ) Using the canonical isomorphism E Hs ∼ = (E H1 )⊗s , we can splice s similar connecting homomorphisms to obtain a map k+s,t+s Extk,t (Z/2, M ) E A (Z/2, M ⊗ E Hs ) → ExtE A In particular, when M = Z/2 and k = 0, we obtain the E2 -level of the algebraic transfer s,t+s E2 ψs : Ext0,t E A (Z/2, E Hs ) → ExtE A (Z/2, Z/2) As noted above, this map is induced by a chain level map E1 ψs : E Hs → Rs It is possible describe this map explicitly P.H Chơn, L.M Hà / Topology and its Applications 178 (2014) 372–383 377 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 ψs a1 as(ns ) = (n ) Rtk11 Rtkss , 0, ni = 2ki (2ti − 1) − 1, ≤ i ≤ s, otherwise (5) (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 (4), (n ) (n ) we see that the connecting homomorphism sends R ⊗ m ⊗ a1 as s to zero if ns does not have the form (ns−1 ) (n ) 2i (2j − 1) − for some i ≥ 0, j ≥ 1; and to RRji ⊗ 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 ), where a1 · · · as is denoted by (i1 , · · · , is ) 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 (5), we obtain (i ) (i ) 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 [22, 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 (5), we obtain E1 ψ4 (X) = (R2 R2 + 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 (see [6]) We end this section with two simple properties of the maps Er ψs , s ≥ 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 P.H Chơn, L.M Hà / Topology and its Applications 178 (2014) 372–383 378 who is familiar with the algebraic transfer may wonder about the action of Gs Unfortunately, this action does not preserve the May filtration in general 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 under concatenation product Standard argument as in Singer [19] shows that: Proposition 3.4 For each r ≥ 1, the total homomorphism between May spectral sequences Er ψ = Er∗,∗ (Ps ) → Er∗,∗ , Er ψ s : s≥0 s is an algebra homomorphism The squaring operations In [16], 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) This operation has been quite useful for constructing new differentials The purpose of this section is to introduce a similar squaring operation, defined for any r, s ≥ 1, which, by abuse of notation, will also be denoted as Sq : Sq : Erp,q (Ps ) → Erp,q (Ps ), with the property that it commutes with Nakamura’s operation via the map of spectral sequences Erp,q (Ps ) → Erp,q+s constructed in the previous section We begin with a description of Nakamura squaring operation in the complex R , this is reminiscent to the construction of Sq in Ext∗,∗ A (Z/2, Z/2) from an endomorphism of the lambda algebra as in Tangora [22] Define an algebra map θ: R → R by setting θ(Rji ) = Rji+1 Then θ commutes with the coboundary map δ, and thus induces an endomorphism on Ext∗,∗ E A (Z/2, Z/2) Proposition 4.1 The endomorphism θ induces Nakamura’s squaring operation Sq : Erp,q → Erp,q i Proof According to Priddy [18], 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 i+1 so it maps [ξj2 ] to [ξj2 ] and the result follows immediately ✷ On H∗ (BVs ), there is also a squaring map constructed by Kameko [8] in his thesis which has been extremely useful in the study of the hit problem (see for example Sum [21]) 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 ), P.H Chơn, L.M Hà / Topology and its Applications 178 (2014) 372–383 (θa)Pts = θ aPts−1 if s > 0, 379 and (θa)Pt0 = Moreover, Vakil [23] 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 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 The endomorphism Sq on R ⊗ H∗ (BVs ) commutes with the coboundary δ of (4) 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 There exists a commutative diagram of maps between spectral sequences: Erp,q (Ps ) Er ψs Erp,q+s Sq Erp,q (Ps ) Sq Er ψs Erp,q+s Proof It suffices to show that there exists a commutative diagram at E1 page E H∗ (BVs ) E1 ψs Sq θ E H∗ (BVs ) Rs E1 ψs Rs This can be verified directly from the formula (5) Note that if ni = 2ki (2ti − 1) − then 2ni + = 2ki +1 (2ti − 1) − ✷ In particular, there is an induced map θ: PE A E Hd (BVs ) → PE A E H2d+s (BVs ), that fits in the following 380 P.H Chơn, L.M Hà / Topology and its Applications 178 (2014) 372–383 Proposition 4.4 The representation of Kameko’s squaring operations, Sq , and Nakamura’s squaring operations commute with each other through E2 ψs In other words, the following diagram commutes E2 ψs PE A E Ht (BVs ) Exts,s+t E A (Z/2, Z/2) Sq θ PE A E H2t+s (BVs ) E2 ψs Exts,s+t E A (Z/2, Z/2) Proof The assertion is implied directly from the formula of Sq and (5) ✷ 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 )Gs [7], 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 The proof of this proposition makes use of a technical result on separating monomials with only odd exponents A monomial ai11 aiss is said to be 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 both y and z belong to PE A E Hs Proof of Lemma 4.6 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 ✷ Proof of Proposition 4.5 For any element x ∈ PE A E Hs such that θ(x) = in (PE A E H∗ (BVs ))Σs , then there exist zσ ∈ (PE A E H∗ (BVs ))Σs such that θx = zσ σ + zσ σ∈Σs 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 P.H Chơn, L.M Hà / Topology and its Applications 178 (2014) 372–383 381 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 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 endomorphism θ on (PA Hs )Σs remains a monomorphism 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 transfer It should be noted that our detection result is in degree that goes far beyond the current knowledge of the hit problem An element in x ∈ E1 is said to survive to Er for some r ≥ if it projects to a non-zero element in the Er A permanent cycle is an element killed by δr for all r We will use the standard notation of known nontrivial elements in the cohomology of the Steenrod algebra as in Tangora [22] Also, in the spectral sequence, the same letter will be used for an element in E2 and its projection to Er , < r ≤ ∞ The following theorem is our main result Theorem 5.1 The element n0 ∈ Ext5,36 A (Z/2, Z/2) is in the image of the algebraic transfer The fact that two elements n0 and n1 = Sq n0 are indecomposable elements of Ext5,∗ A (Z/2, Z/2) goes back to Tangora [22] Recently, completing a program initiated by Lin [9], Chen [3] 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 5.2 The family of indecomposable elements ni ∈ Ext5,36·2 (Z/2, Z/2), i ≥ 0, are in the image of A the algebraic transfer According to Tangora [22], there exists an indecomposable element k0 ∈ Ext7,36 A (Z/2, Z/2) and a relation k1 = h2 h5 n0 ∈ Ext7,72 (Z/2, Z/2), therefore we have A i Corollary 5.3 The elements ki ∈ Ext7,36·2 (Z/2, Z/2), i ≥ 1, are in the image of the algebraic transfer A We not know whether k0 also belongs to the image of the transfer or not −6,6,31 Proof of Theorem 5.1 We shall find a permanent cycle in E∞ (P5 ) which is represented by an element −6,6,31 X ∈ E1 (P5 ), to be described explicitly, such that under the E1 -version of algebraic transfer, the image of X is R12 (R30 )2 (R21 R22 + R31 R12 ) This image is known, according to Tangora [22], 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 382 P.H Chơn, L.M Hà / Topology and its Applications 178 (2014) 372–383 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 In fact, 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) + ¯ is nontrivial in E4−6,6,31 (P5 ) and it is represented (9, 3, 3, 3, 13), X +Y is a cycle in the E3 -term; therefore, n 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 that 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) Since, for r > 6, Er−6+r,∗,∗ (P5 ) = 0, δr (¯ n0 ) = 0, r > In other words, n ¯ is a permanent cycle From (5), we have E1 ψ5 (¯ n0 ) = R12 R30 R21 R22 + R31 R12 ∈ E2−6,11,31 , where the right hand side is a representative for n0 in the E∞ -term of the May spectral sequence It follows that E∞ ψ5 (¯ n0 ) is a representation of n0 in the E∞ -term of the May spectral sequence P.H Chơn, L.M Hà / Topology and its Applications 178 (2014) 372–383 383 If F p H ∗ (A) denote the May filtration of H ∗ (A), then as an F2 -vector space, we have H s (A) ∼ = F p H s (A)/F p+1 H s (A) ∼ = p p,q E∞ p+q=s On the other hand, it is known, see Bruner [2, p 22] for example, that H 5,36 (A) is a one dimensional space Since we know that the above element in filtration −6 which represents n0 is nontrivial, it follows that p,q,31 −6,11,31 E∞ = E∞ = F −6 H 5,36 (A)/F −5 H 5,36 (A) p+q=5 Hence F −6 H 5,36 (A) = H 5,36 (A), F −5 H 5,36 (A) = Therefore, n0 = E∞ ψ5 (¯ n0 ), and the image of E∞ ψ5 coincides 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Math Proc Camb Philos Soc 150 (1989) 307–309 ... algebraic transfer (2) The May spectral sequence In this section, we review the construction of the May spectral sequence The main references are May [11,12] and Tangora [22] May s chain complex for the. .. of the main theorem in Section The algebraic transfer In [4,5], we constructed and studied a representation of the dual of the algebraic transfer in the E 2-term of the homology May spectral sequence. .. sections Sections and are preliminaries In Section 2, we recall basic facts about the May spectral sequence and in Section 3, we present the algebraic transfer and its representation in the E2