No 9] Proc Japan Acad., 86, Ser A (2010) 159 On May spectral sequence and the algebraic transfer ’ ` ÃÃÞ `ng CHO NÃÞ and Leˆ Minh HA By Phan Hoa (Communicated by Kenji F UKAYA, M.J.A., Oct 12, 2010) Abstract: We give a description of the dual of W Singer’s algebraic transfer in the May spectral sequence and use this description to prove new results on the image of the algebraic transfer in higher homological degrees Key words: Adams spectral sequence; May spectral sequence; Steenrod algebra; algebraic transfer; hit problem ’ doi: 10.3792/pjaa.86.159 #2010 The Japan Academy which is known to be an isomorphism for s (this is due to Singer himself [22] for s and to Boardman [3] for s ¼ 3.) Moreover, the ‘‘total’’ transL fer ’ ¼ s ’s is an algebra homomorphism [22] This shows that the algebraic transfer is highly nontrivial and should be an useful tool to study the cohomology of the Steenrod algebra In particular, we want to know how big the image of the transfer in Exts;sỵ F2 ; F2 Þ is A In higher ranks, W Singer showed that ’4 is an isomorphism in a range and conjectured that ’s is a monomorphism for all s In [5], Bruner, Ha` and Hu ng showed that the entire family of elements fgi : i ! 1g is not in the image of the transfer, thus refuting a question of Minami concerning the socalled new doomsday conjecture Here we are using the standard notation of elements in the cohomology of the Steenrod algebra as was used in [4,11,23] One of the main results of this paper is the proof that all elements in the family pi are in the image of the rank algebraic transfer Combining the results of Hu ng [8], Ha` [7] and Nam [19], we obtain a complete picture of the behaviour of the rank transfer It should be noted that in [9], `nh claimed to have a proof that Hu ng and Quy the family fpi : i ! 0g is also in the image of ’4 , but the details have not appeared Our work is independent from their, and our method is completely different Very little information is known when s ! At least, it is known that ’5 is not an epimorphism `nh [21] showed that P h2 is not in [22] In fact, Quy the image of ’5 We have also been able to show, [6], several non-detection results in even higher rank using the lambda algebra For example, h1 P h1 as ’ 2000 Mathematics Subject Classification Primary 55R12, 55Q45, 55S10, 55T15 ÃÞ Department of Mathematics, College of Science, Cantho University, 3/2 St, Ninh Kieu, Cantho, Vietnam ÃÃÞ Department of Mathematics-Mechanics-Informatics, Vietnam National University - Hanoi, 334 Nguyen Trai St, Thanh Xuan, Hanoi, Vietnam ðF2 ; F2 Þ; ’s : F2 GLs P Hà ðBVs Þ ! Exts;sỵ A Introduction Let A be the mod Steenrod algebra [15,18] The cohomology algebra, ExtÃ;à A ðF2 ; F2 Þ, is a central object of study in algebraic topology since it is the E2 -term of the Adams spectral sequence converging to the stable homotopy groups of the spheres [1] However, it is notoriously difficult to compute In fact, only quite recently has the additive structure of Ext4;à A ðF2 ; F2 Þ been completely determined [11] One approach to better understand the structure of this cohomology was proposed by W Singer in [22] where he introduced an algebra homomorphism from a certain subquotient of a divided power algebra to the cohomology of the Steenrod algebra We will call this map the algebraic transfer, because it can be considered as the E2 -level in the Adams spectral sequence of the stable transfer BZ=2ịsỵ ! S [17] Let Vs denote a s-dimensional F2 -vector space Its mod homology is a divided algebra on s generators Let P Hà ðBVs Þ be the subspace of Hà ðBVs Þ consisting of all elements that are annihilated by all positive degree Steenrod squares Let GLs ẳ GLVs ị be the automorphism group of Vs It is well-known that the (right) action of the Steenrod algebra and the action of GLs on Hà ðBVs Þ commute Thus, there is an induced action of GLs on P Hà ðBVs Þ For each s ! 0, the rank s algebraic transfer, constructed by W Singer [22], is an F2 linear map: ` P H CHON and L M H A [Vol 86(A), ’ 160 well as h0 P h2 are not in the image of ’6 ; h21 P h1 is not in the image of ’7 Often, these results are available because it is possible to compute the domain of the algebraic transfer in the given bidegree In this paper, we give a description of the dual of the algebraic transfer ’Ãs in the May spectral sequence Using this method, we recover, with much less computation, results in [6,21] and [9] Moreover, our method can also be applied, as illustrated in the case of the generator hn0 i; n and hn0 j; n 2, to degrees where computation of the domain of the algebraic transfer seems out of reach at the moment The details of this note will be published else where May spectral sequence In this section we recall the setup for the May spectral sequence, following [13] and [14] 2.1 Associated graded algebra of the Steenrod algebra The Steenrod algebra is a cocommutative Hopf algebra [15] whose augmentation ideal will be denoted by A The associated augmented filtration is defined as follows: A; p ! 0, Fp A ị ẳ 2:1ị Àp ðA Þ ; p < 0 A be the associated graded Let E A ¼ Èp;q Ep;q algebra This is a bigraded algebra, where Ep;q A ẳ Fp A =Fp1 A ịpỵq According to May [14], E A is a primitively generated Hopf algebra on the Milnor generators fPji jj ! 1; i ! 0g (See also [15]) Its cohomology is described in the following theorem Theorem 2.1 [14,23] H à ðE A Þ is the homology of a complex R, where R is a polynomial algebra over F2 generated by fRi;j ji ! 0; j ! 1g of degree 2i ð2j À 1Þ, and its differential is given by ðRi;j Þ ẳ j1 X Ri;k Riỵk;jk : kẳ1 " corresponds to the shuffle and the product in X product (see [2, p 40]) Note that the image of a cycle under this imbedding is not necessary a cycle in the bar construction for E A , so we have to add some elements if needed This imbedding technique was succesfully exploited by Tangora [23, Chapter 5] to compute of the cohomology of the mod Steenrod algebra, up to a certain range 2.2 May spectral sequence Let M be a left A -module of finite type, bounded below M admits a filtration, induced by the filtration of A , given by Fp M ¼ Fp A M: It is clear that Fp M ¼ A M ¼ M if p ! 0, and T p Fp M ¼ Put M 0 Ep;q M ẳ Fp M=Fp1 Mịpỵq ; E M ¼ Ep;q M: p;q Then E M is a bigraded E A -module, associated to M " Mị ẳ B " A ; Mị be the usual bar conLet B struction with induced filtration given by X " Mị ẳ Fp B Fp1 A Á Á Á Fpn A Fp0 M; where the sum is taken over all fp0 ; ; pn g such P that n ỵ niẳ0 pi p Theorem 2.3 [14] Let M be a A -module of finite type, bounded below There exists a spectral sequence converging to Hà ðA ; MÞ, whose E -term $ Hà ðE A ; E Mị is Ep;q;t ẳ q;qỵt and the dierentials are F2 -linear maps r r dr : Ep;q;t ! Epr;qỵr1;t : The algebraic transfer The stable transfer BVs ịỵ ! à ðS Þ admits an algebraic analogue at the E level of the May spectral sequence In this section, we give an explicit description of the algebraic transfer in this E level Because of naturality, it will be clear from the construction that there is a commutative diagram E2 t A TorE s;sỵt F2 ; F2 Þ ÀÀÀÀ! ðF2 E A E Ps Þ s s TorA s;sỵt F2 ; F2 ị ! ===) ===) The coKoszul complex R is a quotient of the cobar complex of E A (see [20]), and Ri;j is the image of fðPji Þà g Remark 2.2 It is more convenient for our purposes to work with the homology version The " in [14], is an algebra dual complex, denoted as X " with divided powers on the generators Pji In fact, X is imbedded in the bar construction for E A (which is isomorphic to E -term of May spectral sequence) by sending n ðPji Þ to fPji jPji j jPji g ðn factorsÞ ðF2 A Ps Þt : No 9] On May spectral sequence and the algebraic transfer Let P^1 be the unique A -module extension of H à RP ị ẳ P1 ẳ F2 ẵx1 by formally adding a generator xÀ1 of degree À1 and require that Sqx1 ịSqx ị ẳ Let u : A ! P^1 be the unique 1 A -homomorphism that sends to ðxÀ1 Þ, and put ¼ ujA : A ! P1 By induction, we define s :A s s ðfs jÁ Á Á j1 gị X 00 0s x1 ẳ s ịs ! Ps ; sÀ1 ðfsÀ1 j j1 gÞÞ; deg0s >0 where we use standard notation for coproduct P ị ẳ 0 00 It is known, from a theorem of Nam [19], that is a representation for the algebraic transfer on s the bar construction That is, s induces the dual of the algebraic transfer t A s : TorA s;sỵt F2 ; F2 ị ! Tor0;t F2 ; Ps ị ẳ F2 A Ps ị : We use s to construct a chain map ~s : B "à ðA ; F2 Þ ! B "ÃÀs ðA ; Ps Þ; between the bar constructions as follows Write "n F2 ị ẳ B "ns F2 ị A s , then ~s ¼ s , that B is: ~s ðfn j j1 gÞ ẳ fn j jsỵ1 g s ðfs j j1 gÞ: Proposition 3.1 ~s is a chain homomorphism Our next result shows that ~s , for each s ! 1, respects the May filtration Proposition 3.2 For each p 0, s ! 1, there is an induced chain map: "à ðF2 Þ ! Fp B "ÃÀs ðPs Þ: Fp ~s : Fp B As a result, there is an induced map between spectral sequences Er s r r : Ep;q;t ðF2 Þ ! Ep;qÀs;tÀs ðPs Þ: In particular, we obtain E2 s ðMÞ E A : Tors;sỵ E M; F2 ị A 0 ! TorE 0;à ðE M; E Ps Þ: When M ¼ F2 , E s ðF2 Þ is the E -level of the algebraic transfer in the May spectral sequence The following is the main theorem of this section Theorem 3.3 The E -level of the dual of Singer’s algebraic transfer is induced by the chain level map E1 161 s s : E A ! E Ps ; which is given inductively by E1 s ðfs j j1 gị ẳ X 0s E 00 s1 fs1 jÁ Á Á j1 gÞÞs ðxs Þ; deg00s >0 Because of the simple structure of E A , it is usually quite simple to compute with E s For example, because Pji are primitive in E A , we have Corollary 3.4 Under the chain level E s : s E i A j ! E Ps , i the image of fPjiss j jPji11 g is ð2 À1ÞÀ1 ð2j À1ÞÀ1 x1 xs Theorem 3.3 and Corollary 3.4 are extremely useful to investigate the image of the algebraic transfer Two hit problems The study of the algebraic transfer is closely related to an important problem in algebraic topology of finding a minimal basis for the set of A -generators of the polynomial rings Ps , considered as a module over the Steenrod algebra This is called ‘‘the hit problem’’ in literature [25] A polynomial f Ps is ‘‘hit’’ if it belongs to A Ps There is another, related hit problem that we are going to discuss The results in this section are crucial for applications in Section and Consider the May spectral sequence for Ps in homological degree There are isomorphisms 1 s s Ep;Àp;t ðPs Þ $ ¼ H0 ðE A ; E Ps ịp;pỵt ẳ F2 E A E Ps ịịp;pỵt ; so the E term concerns with the problem of determining the generators of E Ps , considered as a module over the restricted Lie algebra E A Determining a set of E A -generators for E Ps is a simpler problem, but not without difficulty, even in the rank case (see [24]) The E A -module structure on E Ps is related to the A -module structure on Ps via epimorphisms Ep;Àp;t ðPs Þ ! Ep;Àp;t ðPs Þ; where in each fixed internal degree t, Ep;Àp;t ðPs Þ are associated graded components of ðF2 A Ps Þt Given a homogeneous polynomial f Ps We denote by E r fị and ẵf the corresponding classes of f in E r and F2 A Ps respectively In particular, E fị ẳ E fị is the class of f in E Ps In order to determine E r fị or ẵf, one only needs to consider monomials in f of highest filtration degree, we call this the essential part of f, and denote it by ess( f) For example, ` P H CHON and L M H A [Vol 86(A), ’ 162 13 11 13 12 13 13 13 essx71 x13 x3 ỵ x1 x x3 ỵ x1 x2 x3 ị ẳ x1 x2 x3 13 because x71 x13 x3 is in filtration À4 while the latter two monomials are in filtrations À5 and À9 respectively Lemma 4.1 Let f Ps be a homogeneous polynomial If f is a nontrivial permanent cycle, then essðfÞ is non-hit in Ps 13 Example 4.2 Let m ¼ x71 x13 x3 P3 , it is not difficult to check that m is nonhit in P3 On the other hand, 13 11 13 12 13 m ẳ Sq2 x71 x11 x3 ị ỵ x1 x2 x3 ỵ x1 x2 x3 ỵ 14 x71 x12 x3 ỵ 14 x81 x11 x3 ; 13 where x91 x11 x3 FÀ5 P3 and the last three monomials are in even smaller filtrations Therefore 13 E mị ẳ P11 E x71 x11 x3 Þ Ề4;37 P3 : So E ðmÞ is trivial Thus, m is nonhit in Ps then E ðmÞ is not necessary nonhit in E Ps Example 4.3 Consider m ¼ x1 x22 x23 ỵ 2 2 x1 x2 x3 ỵ x1 x2 x3 ẳ Sq x1 x2 x3 ị, so m is hit in P3 On the other hand, since m FÀ2 P3 and there does not exist any element fgf FÀ1 ðA P3 Þ such that fị ẳ m (modulo terms in Fp P3 with p < À2), E ðmÞ is nonhit in Ề2;7 P3 0 Thus, E ðmÞ is nonhit in E Ps then m is not necessary nonhit in Ps The following is the main result of this section Proposition 4.4 Let f Ps be a homogeneous polynomial of filtration degree p f is a nontrivial permanent cycle if and only if essðfÞ is non-hit in Ps and there does not exist any non-hit polynomial g Fr Ps , with r < p, such that essðfÞ À g is hit First application: a non-detection result In this section we use the presentation in E -term of May spectral sequence of the dual of the algebraic transfer, constructed in section 3, to study its image Using this method, we are able to, not only reprove by a completely different method (with much less calculation) for results in [6,21], but also obtain the description of the image at some degrees of the algebraic transfer Here is our first main result Theorem 5.1 The following elements in the cohomology of the Steenrod algebra (a) h1 P h1 Ext6;16 A ðF2 ; F2 Þ; (b) h20 P h2 Ext7;18 A ðF2 ; F2 Þ; ðF2 ; F2 Þ; n 5; (c) hn0 i Ext7ỵn;30ỵn A (d) hn0 j Ext7ỵn;33ỵn F2 ; F2 Þ; n 2, A are not detected by the algebraic transfer We remark that h60 i ¼ h30 j ¼ (see [4]) Sketch proof We will give the sketch of proof of (a) The proofs of other parts use similar idea According to Tangora [23], in E -term of the May spectral sequence, h1 P h1 has a representation X ¼ fP11 jP11 g à fP20 jP20 jP20 jP20 g E1 : Note that X FÀ4 ðF2 Þ Here we use the same notations Pji for elements of A and E A , so X can be considered as an element in E , being the bar construction of E A Corollary 3.4 allows us to find the image of X under E : E1 Xị ẳ x1 x2 x23 x24 x25 x26 ỵ all its permutations ẳ Sq4 ðx1 x2 x3 x4 x5 x6 Þ: Therefore, E ðXÞ is hit in P6 In the bar construction, ðh1 P h1 Þà has a representation X þ x, "ðF2 Þ with p < À4 Thus, if h1 P h1 is where x Fp B detected, then h1 P h1 ị ị ẳ E1 Xị ỵ y; where y Fp P6 with p < À4, is nonhit in P6 On the other hand, it can be verified by direct computation that there is only one possible polynomial: x41 x42 x23 x04 x05 x06 (or its permutations) But it is clearly hit in P6 as well à It should be noted that the dimension of the above elements go far beyond the current computational knowledge of the hit problem Corollary 5.2 [21,22] P h1 Ext5;14 A ðF2 ; F2 Þ 5;16 and P h2 ExtA ðF2 ; F2 Þ are not in the image of the algebraic transfer That these elements are not detected are `nh [21] known, they are due to Singer [22] and Quy respectively Our proof is much less computational Second application: p0 is in the image of the transfer In this sections, we show that our method can also be used to detect elements in the image of the algebraic transfer This fact completes the proof of a conjecture in [8], which provides a complete picture of the fourth algebraic transfer The following is our second main result Theorem 6.1 The element p0 Ext4;37 A ðF2 ; F2 Þ is in the image of the fourth algebraic transfer No 9] On May spectral sequence and the algebraic transfer This result is announced in [9], but the details have not appeared Since the squaring operation Sq0 , defined by Kameko [10], acting on the domain of the algebraic transfer commutes with the classical Sq0 on ExtÃ;à A ðF2 ; F2 Þ [12] through the algebraic transfer [16], we obtain following result Corollary 6.2 Every element in the family i pi Ext4;37Á2 ðF2 ; F2 Þ, i ! 0, is in the image of the A algebraic transfer Sketch proof of Theorem 6.1 According to Tangora, p0 is represented by R0;1 R3;1 R21;3 , so its dual pÃ0 is represented in E -term of May spectral sequence by p"0 ¼ fP13 g à fP10 g fP31 jP31 g ỵ fP13 jP13 g fP21 g à fP40 g: Under E , this element is sent Corollary 3.4) 13 7 14 p~0 ẳ x01 x72 x13 x4 ỵ x1 x2 x x4 to (see ỵ all their permutations: Using Example 4.2 and the fact that 7 14 E ðx71 x72 x53 x14 Þ ¼ P1 E ðx1 x2 x3 x4 Þ; we see that E ð~ p0 Þ is hit in E P4 Therefore, E ð~ p0 Þ does not survive to Ề4;4;33 ðP4 Þ By direct calculation, we see that, in the bar con"ðF2 Þ and struction, p0 ẳ p"0 ỵ x ỵ y, where y FÀ6 B x ¼ fP12 jP12 g à fP21 g à fP40 g: Here we use ða; b; c; dÞ to denote the monomial xa1 xb2 xc3 xd4 , and use à to denote all permutations that is similar to shuffle product Since X is hit in P4 , so are Xð12Þ; Xð132Þ and Xð1432Þ By direct inspection, we show that E Y ị ẳ E 3; 5Þ Ã ð11Þ Ã ð14ÞÞ is a nontrivial permanent cycle According to Proposition 4.4, ðpÃ0 Þ is nonhit in P4 Thus, p0 is in the image of fourth algebraic transfer à Acknowledgments We would like to thank Prof Bob Bruner for his help and encouragement The first author would like to thank Profs J Peter May and Wen-Hsiung Lin for their helpful answers to his questions This work is partially supported by the NAFOSTED grant No 101.01.51.09 References [ ] J F Adams, On the structure and applications of the Steenrod algebra, Comment Math Helv 32 (1958), 180–214 [ ] J F Adams, On the non-existence of elements of Hopf invariant one, Ann of Math (2) 72 (1960), 20–104 [ ] J M Boardman, Modular representations on the homology of powers of real projective space, in Algebraic topology (Oaxtepec, 1991), 49–70, Contemp Math., 146, Amer Math Soc., Providence, RI [ ] R R Bruner, The cohomology of the mod Steenrod algebra: A computer calculation, WSU Research Report 37 (1997) ` and N H V Hu ng, On the [ ] R R Bruner, L M Ha behavior of the algebraic transfer, Trans Amer Math Soc 357 (2005), no 2, 473–487 `, Lambda algebra and the [ ] P H Cho n and L M Ha Singer transfer (Preprint) [ ] L M Ha`, Sub-Hopf algebras of the Steenrod algebra and the Singer transfer, in Proceedings of the School and Conference in Algebraic Topology, 81–105, Geom Topol Monogr., 11 (2007), 81–105 [ ] N H V Hu ng, The cohomology of the Steenrod algebra and representations of the general linear groups, Trans Amer Math Soc 357 (2005), no 10, 4065–4089 `nh, The image of [ ] N H V Hu ng and V T N Quy Singer’s fourth transfer, C R Math Acad Sci Paris 347 (2009), no 23–24, 1415–1418 [ 10 ] M Kameko, Products of projective spaces as Steenrod modules, ProQuest LLC, Ann Arbor, MI, 1990 5;à [ 11 ] W.-H Lin, Ext4;à A ðZ=2; 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