DSpace at VNU: Spherical classes and the algebraic transfer (vol 349, pg 3893, 1997)

DSpace at VNU: Spherical classes and the algebraic transfer (vol 349, pg 3893, 1997) tài liệu, giáo án, bài giảng , luận...

Spherical Classes and the Algebraic Transfer Author(s): Nguyen H v Hu'ng Source: Transactions of the American Mathematical Society, Vol 349, No 10 (Oct., 1997), pp 3893-3910 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/2155567 Accessed: 22-12-2015 13:38 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive We use information technology and tools to increase productivity and facilitate new forms of scholarship For more information about JSTOR, please contact support@jstor.org American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Transactions of the American Mathematical Society http://www.jstor.org This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 349, Number 10, October 1997, Pages 3893-3910 S 0002-9947(97)01991-0 SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER NGUYEN H V HU'NG ABSTRACT We study a weak formof the classical conjecture which predicts that there are no spherical classes in QoS0 except the elementsof Hopf invariant one and those of Kervaire invariantone The weak conjecture is obtained by restrictingthe Hurewicz homomorphismto the homotopyclasses which are detected by the algebraic transfer Let Pk = F2[Xl , ,Xkl with lxiI = The general linear group GLk = GL(k, F2) and the (mod 2) Steenrod algebra A act on Pk in the usual manner We prove that the weak conjecture is equivalent to the followingone: The canonical homomorphismjk: IF2 9( pLk) induced by the (IF2 Pk)GLk A identitymap on Pk is zero in positive dimensions for k > In other words, k ) of positive dimension belongs every Dickson invariant (i.e element of P to A+ Pk for k > 2, where A+ denotes the augmentation ideal of A This conjecture is proved for k = in two differentways One of these two ways is to study the squaring operation Sqo on P(F2 09 Pk), the range of jk, and GLk to show it commutingthrough jk with Kameko's Sqo on F2 GLk P(Pk), the domain of jk We compute explicitlythe action of Sqo on P(F2 09 Pk*) for GLk k < INTRODUCTION The paper deals withthe sphericalclasses in QoS?, i.e the elementsbelonging to the image of the Hurewiczhomomorphism H: 7r 0, and it reallyoccurs at this dimensionif and onlyif h2 is a permanentcyclein the Adams spectralsequence forthe spheres Therefore,Conjecture1.1 is a consequenceof the following: F2 Conjecture 1.2 pk = in any positivestem i fork > It is well knownthat the Ext group has intensivelybeen studied,but remains verymysterious.In orderto avoid the shortageofour knowledgeofthe Ext group, we want to restrictAPkto a certainsubgroupof Ext which(1) is large enoughand worthwhile to pursueand (2) could be handledmoreeasilythan the Ext itself.To this end, we combinethe above data withSinger'salgebraictransfer Singerdefinedin [20]the algebraictransfer Trk: F2 PHi(BVk) GLk -4 Ext k+i (F2,F2) X wherePH* (BVk) denotesthe submoduleconsistingof all A-annihilatedelements in H* (BVk) It is shownto be an isomorphismfork < by Singer [20] and for k = by Boardman [4] Singeralso provedthat it is an isomorphismfork = in a range of internaldegrees But he showedit is not an isomorphismfork = forany k However,he conjecturesthat Trk is a monomorphism of (pk to the image of Trk Our main idea is to studythe restriction Conjecture 1.3 (weak conjectureon sphericalclasses) (Pk *Trk : F2 PH* (BVk) - GLk P(F2 H*(BVk)) := (F2 Dk) GLk A is zero in positivedimensionsfork > This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions CLASSES SPHERICAL AND THE ALGEBRAIC 3895 TRANSFER In otherwords,there are no sphericalclasses in QoS?, except the elementsof Hopf invariantone and those of Kervaireinvariantone, whichcan be detectedby the algebraictransfer ofinvariant A naturalquestionis: How can one expressPOk Trk in the framework theoryalone, and withoutusingthe mysteriousExt group? ( Pk) GLk be the naturalhomomorphism Let ik : F2 ? (p&Lk) inducedby A A the identitymap on Pk We have -+ Theorem 2.1 pOk*Trk is dual to ik, or equivalently, Tr * k By this theorem,Conjecture1.3 is equivalentto ik = Conjecture 1.4 ik = in positivedimensionsfork > This seemsto be a surprise,because by an elementaryargumentinvolvingtaking averages,one can see that ifH C GLk is a subgroupof odd orderthenthe similar homomorphism iH : F2 0(PkH) A - (F2 ?Pk) A is an isomorphism.Furthermore, ji is iso and j2 is mono Obviously,ik = ifand onlyifthe composite Dk P pLk F2 ?(pGLk) A (F2 OPk) F2 (Pk A A is zero So, Conjecture1.4 can equivalentlybe stated in the followingform Conjecture 1.5 Let Dj+, A+ denote the augmentationideals in Dk and A, respectively.Then D+ C A+ *Pk forany k > The domain and range of ik both are still mysterious Anyhow,they seem easier to handle than the Ext group They both are well-knownfor k = 1,2 on the one hand, (F2 oPk)GLk is computedfork = by Kameko [11], Furthermore, A Alghamdi-Crabb-Hubbuck[3] and Boardman [4] On the otherhand,F2 is determinedby Hu'ng-Peterson[18] fork = and F (PGLk ) and (F2 ?P)GL : Let F2 ? (PGL) :A k>O A A are equipped withcanonicalcoalgebrastructures.We get Proposition 3.1 j = : F2 ?(pGL) (3jk A coalgebras (F2 ?p)GL A k>O (F2 (Pk)GL A X(fk) A k They is a homomorphism of Let Sqo : PH*(BVk) PH*(BVk) be Kameko's squaringoperationthat commuteswiththe SteenrodoperationSqo : Extkt (2,F2) - Extk2t (F2,F2) through the algebraic transferTrk (see [11], [3], [4], [17]) Note that Sqo is completely different fromthe identitymap We prove Proposition 4.2 Thereexistsa homomorphism Sq?: P(F2 X GLk H*(B Vk)) )- P(F2 GLk H*(B Vk)), whichcommuteswithKameko's Sq0 throughthehomomorphism j* proofsof the followingtheorem These two propositionslead us to two different This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions 3896 NGUYEN H V HU'NG Theorem 3.2 jk = in positivedimensionsfor k = In otherwords,thereis no sphericalclass in QoS? whichis detectedbythe triplealgebraictransfer We computeexplicitlythe action of Sqo on P(F2 H*(BVk)) fork = and GLk in Propositions5.2 and 5.4 The paper containssix sectionsand is organizedas follows Section is to proveTheorem2.1 In Section 3, we assemblethe ik fork > of coalgebrasj = ik By means of this propertyof j to get a homomorphism we give therea proofof Theorem3.2 Section deals with the existenceof the squaringoperationSqo on P(F2 H* (BVk)) that leads us to an alternativeproof GLk forTheorem3.2 This proofhelpsto explainthe problem.In Section5, we compute explicitlythe action of Sqo on P(F2 H* (BVk)) fork < Finally,in Section GLk we state a conjectureon the Dickson algebrathat concernssphericalclasses ACKNOWLEDGMENTS I expressmy warmestthanksto Manuel Castellet and all my colleaguesat the workCRM (Barcelona) fortheirhospitalityand forprovidingme witha wonderful ing atmosphereand conditions.I am gratefulto Jean Lannes and FrankPeterson forhelpfuldiscussionson the subject Especially,I am indebtedto Frank Peterson forhis constantencouragement and forcarefullyreadingmyentiremanuscript, makingseveralcommentsthat have led to manyimprovements EXPRESSING f k * Trk IN THE FRAMEWORK OF INVARIANT THEORY First,let us recall how to definethe homomorphism ik We have the commutativediagram C pGLk k F2 ) @(k F(PIGLk) A * P (&Pk k Fik~ A wherethe verticalarrowsare the canonical projections,and jk is induced by the pGLk C Pk Obviously, inclusion through So, ik factors p(P( k) C (F20Pk)GLk (F2 0Pk)GLk A A to giveriseto ik jk: F2 (p(GLk) @(k A ik(1 (0 Y) A ~4(2pk)GLk, ) (F2 ( = k) A Y, A forany polynomialY E Dk = pkGL The goal of this sectionis to provethe followingtheorem Theorem 2.1 ik = Trk* - Now we prepare some data in orderto prove the theoremat the end of this section This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER 3897 First we sketchLannes-Zarati's work [13] on the derivedfunctorsof the destabilization Let D be the destabilizationfunctor,whichsends an A-module M to the unstable A-module 1D(M) = M/B(M), whereB(M) is the submoduleof M generatedby all Sqiu withu E M, i > jul D is a rightexact functor.Let 1Dkbe its k-thderivedfunctorfork > Suppose M1, M2 are A-modules Lannes and Zarati definedin [13, ?2] a homomorphism Extr (MlI M2) Ds (Ml) Ps-r (M2) X n: D (f,z) fnz, * as follows Let F*(Mi) be a freeresolutionof Mi, i = 1,2 A class f E Ext'(Ml,M2) can F*-r (M2) of homologicaldegree-r be representedby a chain map F: F* (Ml) We writef = [F] If z = [Z] is representedby Z E F* (M1), then by definition f n z = [F(Z)] Let M be an A-module.We set r = s = k, M1 = Z-kM, M2 = Pk M, where as beforePk = F2[X1, , Xk], and getthehomomorphism n: ExtAO(?kM Pk0 M) Dk ( Pk M M) Nowwe needto definetheSingerelementek(M) E M) (see Singer[20,p 498]) Let P1 be the submoduleof F2[X,X-1] spanned by all powers xi withi >-1, wherelxl= The A-modulestructureon F2[X,X-1]extendsthat EXtk(E-kM, Pk ofP1 = F2[X](see Adams[2],Wilkerson [22]) The inclusion Pi C P1 givesriseto a shortexactsequenceofA-modules: -* P1 -+ P1 E-lF2 -* Denotebyel thecorresponding elementin Ext (EY1F2,P1) Definition2.2 (Singer[20]) el e ExtA(Z (i) ek = el ** k times (ii) ek(M) = ek F2, Pk) M E ExtA(Z-kM, Pk M) forM an A-module Herewe also denotebyM theidentity mapofM The cap-product withek(M) givesriseto thehomomorphism ek(M) PDk(EkM) ek (M) (z) Do(Pk CM) -Pk Ml ek(M) n z theoremis a specialcase (but As F2 is an unstableA-module,the following wouldbe themostimportant case) ofthemainresultin [13] Theorem 2.3 (Lannes-Zarati [13]) LetDk C Pk betheDicksonalgebraofk vari- EDk is an isomorphism ables Thenek(ZF2): Dk(l-kF2) ofinternaldegree oftheLannes-Zarati homomorphism Next,we explainin detailthedefinition Ok: Extk (EPkF2,F2) -* (F2 XDk)*, A~~~~ withtheHurewiczmap (see [12],[13]) whichis compatible Let N be an A-module.By definition of the functor 1D,we have a natural ofN Then homomorphism: 1D(N) -) F2 ON SupposeF*(N) is a freeresolution A theabovenaturalhomomorphism inducesa commutative diagram This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions 3898 H V HU'NG NGUYEN - DF (N) DFk-l(N) ik - , ik 72 0Fk(N) A F2 ?Fk-l (N) A in F* (N), and Here the horizontalarrowsare inducedfromthe differential ik[Z] = [1?3Z] A forZ E Fk(N) Passing to homology,we get a homomorphism ik: F2 TorA (F2, N), Dk(N) A 1?[Z] Taking N = A Z] we obtain a homomorphism l-kF2, ik [1 I-' A : F2 0Dk k (ElZ A Note that the suspensionE: F2 A F2) TorkA(F2, -* Dk F2 Tork (F2, A lk2) EDk and the desuspension lF2) (F2, EYkF2) OTrk are isomorphismsof internaldegree1 and (-1), respectively.This leads us to Definition 2.4 (Lannes-Zarati [13]) The homomorphismpk of internaldegree is the dual of k = Z ik (10gek (zF2)) Z: F2Dk-4 F2) Tor(2ik Now we recallthe definition ofthe algebraictransfer.Considerthe cap-product Ext Pk) TorA(F2, (jkF2, - Z-kF2) enz (e,z) Takingr = s Tr*: = k and e = Tor A(F2, ek as in Definition2.2, we obtain the homomorphism Ek F2) = TorA(F2, Pk) -4 = Tr*[10 Z] A forZ E Fk(E-kF2), whereek Pk), TorAtr(F2, ekn[Z]=[10E(Z)] A - OPk, A 10E(Z), A [E] is represented by a chainmap E : F( 4F2) map F*-k(Pk)- hence Im(Tr*) C (F2 0Pk)GLk Singerprovedin [20] that ek is GLk-invariant, A This givesriseto a homomorphism, whichis also denotedby Tr*, Tr*: S pkF2) Tort (F2, (F2 0Pk)GLk A Definition 2.5 (Singer [20]) The k-thalgebraictransferTrk: F2 PH*(BVk) -4 Extkk+* (F2, F2) is the homomorphism dual to Tr* GLk We have finishedthe preparationof the needed data This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER 3899 Proof of Theorem 2.1 Note that the usual isomorphism ExtA sends ek(F2) to ek(ZF2) by a chainmap E: ( kF2, Pk) ek (F2) = F*(Y-kF2) - kF2, ( EX ExtA Pk) Moreover,ifek(F2) EF2 thenek(YF2) F.-k(Pk) = = [E] is represented [EE] is represented by the induced chain map EE F,(ZlkF2) whichis definedby F-k(ZPk), EES- EE = By Theorem2.3, ek(Y3F2) is an isomorphism.So, forany Y E Dk, thereexists a representative of e-1(ZF2)EY, whichis denotedby E-1EY E Fk(El-k F2), such that EE (E-1EY) = EY The cap-productwithek(ZF2) = [EE] inducesthe homomorphism Tor (k2,1 Trk: kF2) Tr[i Z] = Fk(lk It is easy to checkthat Tr* = ETrJ Obviously,Trk* = Tr* 2OZDk (1&ek1(zEF2)) =ik A E.1Trk*Z ~1* ?( &Y) EPk 10EE(Z), A Moreover,set Tor(FZ A F2 A A F2) k - n [Z] = [1 0EE(Z)] ek(YF2) A forZ E (F2 ,i Pk) Toro - F2) Now, forany Y E Dk, we have = E lrk * A ZkE(l OY) A = X%(r * (1 ? EY) A = ~-1 (10 1E(EE1Y)) = 11 (EY) A = 10Y A By definition ofik, we also have jk(1 (? Y) = ?Y The theoremis proved A THE HOMOMORPHISM A OF COALGEBRAS j = ejk The canonical isomorphismVk - Ve Vx Vm,fork = f + m, induces the usual inclusionGLk D GLt x GLm and the usual diagonalA/: Pk -4 Pt Pm Therefore, it inducestwo homomorphisms AD F2 ?(JGLk) A p: (F2 0Pk)G L A -4(wF0(IfLID) ? (w7?(PLm (F2 ?Pe)GLe) (F2 A A )) pi)GLm Here and in what follows,0 meansthe tensorproductoverF2, exceptwhenotherwise specified This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions 3900 NGUYEN H V HU'NG Set F2 ?D = F2 A (F2 It is easy to see that F2 ?(pGL) (pGL) ? A = A 0(w(2 ?p) A OPk)GLk A k>O and (F2 A ?(pGLk) k>O are endowedwiththe structureof ?P)GL A a cocommutativecoalgebraby AD and Ap, respectively.The coalgebrastructure ?p)GL was firstgivenby Singer[20] Of (F2 A Proposition 3.1 coalgebras = ik : F2 ?(pGL) A - (F2 A is a homomorphism of ZP)GL Proof This followsimmediatelyfromthe commutativediagram ik F2?Dk A (F20 Dt)0 A (F2 0Pk)GLk A (F2 0Dm) P (F2 A Op)GLI A (F2 A Pm)GLm Remark.Accordingto Singer[20],Tr* = ( Tr* is a homomorphism ofcoalgebras One can see that * = 34 is also a homomorphism of coalgebras Then, so is j = Tr* * This is an alternativeproofforProposition3.1 Now let F2 GL PH*(BV) := k>0 P(F20 H*(BV)) := CL k0 (F2 GLk P(F20 PH*(BVk)) GLk - H*(B Vk)) k>0 k0 ((F2 0F2 ( A 0Pk)G ) (PLk) Passing to the dual, we obtain the homomorphism of algebras -F2 PH*(BV) P(F2 H*(BV)) GL GL As an applicationofj*, we give herea proofforConjecture1.4 with k = Theorem 3.2 jh: F2 ? A (pGL3) -(F2 ?p3)GL3 A is zero in positivedimensions Proof We equivalentlyshowthat ij: F2 09 PH*(BV3) - P(F2 H*(BV3)) GL3 GL3 is a trivialhomomorphism in positivedimensions F2 PH*(BV3) is describedby Kameko [11], Alghamdi-Crabb-Hubbuck[3] GL3 and Boardman [4] as follows.F2 PH* (BV1) has a basis consistingof hr, r > GL1 0, where hr is of dimension2r - and is sent by the isomorphismTr1 to the Adams element,denoted also by hr, in Ext (F2,F2) Accordingto [11], [3], This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER 3901 [4], F2 PH* (BV3) has a basis consistingof some productsof the formhrhsht, GL3 wherer,s, t are non-negativeintegers(but not all such appear), and some elements ci (i > 0) withdim(ci) = 2i+3 + 2i+1 + 2' - We willshowin Lemma3.3 thatanydecomposableelementin P(F2 H* (BV3)) GL3 is zero Then, sincej* is a homomorphism of algebras,j* sends any elementofthe formhrhshtto zero On the otherhand, by Hu'ng-Peterson[18],F2 ? D3 is concentratedin the diA mensions2+2 - (s > 0) and 2r+2 + 2s+1 - (r > s > 0) Obviously,these dimensionsare different fromdim(ci) forany i Then j* also sends c2 to zero O To completethe proofof the theorem,we need to show the followinglemma Lemma 3.3 Let Dk = F2 0Dk Then the diagonal A AD: D3 - D1 D2 EDD2 D is zero in positivedimensions Proof Let us recallsome informations on the Dicksonalgebra Dk Dicksonproved in [10] that Dk - F2[Qk-1 Qk-2, I IQo], a polynomialalgebra on k generators, with IQj = 2k - 2s Note that Qs depends on k, and when necessary,will be denotedQk,s An inductivedefinitionof Qk,s is givenby Definition 3.4 Qk,s = Q211 where,by convention,Qk,k = + Vk * Qk-l,s 1, Qk,s = fors < and (Aix + Vk *+ Ak-lXk-1 + Xk) AiEF2 Dickson showedin [10] that k-1 Vk = Qk-1,sxk s=O Now we turnback to the lemma we need onlyto show that the diagonal Since AD iS symmetric, A: D3 ,D2 OiD, is zero in positivedimensions For abbreviation,we denote x1,X2, x3 by x, y,z, respectively, Qi = Q3, (XI Y, z) fori = 0, 1,2, qi = Q2,i(x,y) fori = 0, As is well known,F2 OD1 has the basis {z2s-1 s > A 0}, and F2 0D2 has the basis {q2s-1I s > 0} By Hu'ng-Peterson[18], F2 OD3 has the basis A A {Q2S-1 (S > 0), Q2r-2s-lQ2s lQO (r > s > 0)} For k < 3, every monomial in Qo, , Qk-1 which does not belong to the given basis is zero in F2 Dk Note that the analogous statementis not true fork > (see [18]) A This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions H V HU'NG NGUYEN 3902 of Qk,s and Vk, we get Usingthe above inductivedefinitions Qo = q 2z + qoqlz2 + qoz4X or A(Qo) = qoO z + qoql O z2 + qO z4 This implieseasily that everytermin A(Q2r-2 -1Q2 -1Qo) is divisibleby qo, so it equals zero in F2 D2 as shownabove In otherwords, A A2j(Q2r-2-12Q2 -1sQ) = o Similarly, = Q2 q 2+V3 = q + + qoz qlz2 + z4 or /A(Q2) = q + qoO z + qi z2 + z4, = A(Q2S1) (q2 + qo z + qi z2 + 10z4)2sl By the same argumentas above, we need onlyto considertermsin A (Q2S -1) which are not divisibleby qo Such a termis some productof powersof q2 1, qi z22 z4 If it containsa positivepowerofz thenthispoweris even and it equals zero in F2 0D, Otherwise,it should be q(2, l) Obviously,q2(2 l) equals zero in A iF2 0D2 A SO, A\(Q2 -1) = fors > In summary,A = in positivedimensions.The lemma is proved Then, so is Theorem3.2 As Tr3 is an isomorphism(see Boardman [4]), we have an immediateconsequence Corollary 3.5 (03: ExtA3+(F2, iF2) -* (F2 0D3)i* is zero in everypositivestem A i THE SQUARING OPERATION: THE EXISTENCE Liuleviciuswas perhapsthe firstpersonwho noted in [14] that thereare squaring operationsSqi ExtAt (F2,F2) - EXt+i'2t(F2 XF2), whichshare most of the propertieswith Sqi on cohomologyof spaces In particular,Sq' (a) = if i > k, Sqk(a) = a2 fora E ExtAt(F2 iF2), and the Cartan formulaholds forthe Sqi's However,Sqo is not the identity.In fact, Sqo: Ext2t (F2 XF2) Extkt (F2, F2) [b lb2k [b, Ibk ] , in termsof the cobar resolution(see May [16]) Recall that H* (BVk) is a dividedpoweralgebra H*(BVk) = F(al, ,ak) generatedby ai, ,ak, each of degree1, whereai is dual to xi E H1(BVk) Here and in what follows,the duality is taken with respect to the basis of H* (BVk) consistingof all monomialsin x1, , Xk This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions SPHERICAL CLASSES AND THE ALGEBRAIC 3903 TRANSFER Let at be the t-th divided power in H*(BVk) and for any a E H*(BVk) let a(t) = yt(a) So a(t) is the elementdual to xi One has a(2-) a(2-) a7 =0, and (t) if t = 2r1 + + 2rm, < ri < (2r1) (2rm_) < rm In [11] Kameko defineda GLk-homomorphism Sqo: al where a(ii) * a ik) PH*(BVk), PH*(BVk) (i01) ak(ik) is dual to xll (2i11+1) a1 (2ik+l) ak xlk (See also [3].) Crabb and Hubbuckgave in [8] a definitionof Sqo that does not depend on the chosenbasis of H*(BVk) as follows.The elementa(Vk) = a, ak is nothingbut map the image of the generatorof Ak(k) underthe (skew) symmetrization A (k) )- Hk(BVk) = Fk(Vk) = (V '@ Vk )Ek- k times definedby F(x) = Let F: H*(BVk) - H*(BVk) be the Frobeniushomomorphism x2 foranyx, and let c: H* (BVk) H*(BVk) be the degree-halving dual homomorphism It is obviouslya surjectiveringhomomorphism.Then Sq0 can be defined by Sqo(c(y)) = a(Vk)y of GLk-modules Since y E kerc ifand onlyifa(Vk)y = 0, Sq0 is a monomorphism 0, cSq*2 = Sq'c So Sq0 maps PH*(BVk) Further,it is easy to see that cSq*ji+ to itself Using a resultof Carlisle and Wood [6] on the boundednessconjecture,Crabb and Hubbuckalso noted in [8] that forany d, thereexiststo such that - Sq : PH2td+(2t_1)k(BVk) PH2t+ld+(2t+1 1)k(BVk) is an isomorphismforeveryt > to Kameko's Sqo is shown to commute with Sqo on Extk (F2, F2) through the algebraic transferTrk by Boardman [4] fork = and by Minami [17] forgeneral k One denotesalso by Sq0 the operation Sq0 :F20 GLk PH*(BVk) -F20 GLk PH*(BVk) inducedby Kameko's Sqo It preservesthe product Further,fork = 3, it satisfies Sq0(hrhsht) (see Boardman Lemma = hr+lhs+lht+l , Sq0(ci) = ci+i [4]) 4.1 Sq*2r+l Sqo - Sq2rSqO = SqOSq* Proof We need a formal notation Namely, for a E Hl(BVk), set (a(t))[2] In general, (a(t))[2] $N (,yt(a)) = (2t- 1)a(2t) (see Cartan [7]) This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions - a(2t) NGUYEN 3904 We startwitha simpleremark Let x E H1(BVk);thenSqr(xs) = Then, by dualizing, H V HU'NG (s)Xs+r (t Sqr(a(t)) As a consequence,Sq r)(tr) - = and 2r+1(a(2t+l)) (2t -2r) Sq2r(a(2t))- ofx Let a denotethedual element (t a(2t-2r)= - r) (a(t-r)) (sq:a(t) [2]= [2] (a Let a-a(Sq) a(ii) Let a= By the Cartan formula,we have ak) Sq'Sq o(a) - Sqr(a(2ij+l) ~ = a(2ik+1)) S rk Sq'r(a2ij+1)) (2ik+l) rl+**+rk==r to (r, , rk) equals ifat least one of r1, , The termcorresponding Hence Sq 2r+1 Sqo (a) = Furthermore, Sq* Sqo (a) =Sq*rl rl+ - S ES (a **+rk==r {Sq2r, (a(2i7)j) q2rk rk is odd (a(2ik?l)) 2ik))} Sq2rk(a a, ak rl+**+rk==r - ( rl+-*+rk==r = )) [2] { *( (a ~)} ) {~~Sq" k (W - { 1*(k3 (a {Sqc a,))j [2] kal*ak Sq?Sq*() The lemmais proved Proposition 4.2 For everypositiveintegerk, thereexistsa homomorphism Sq?: P(F2 H*(BVk)) GLk P(F2 H*(BVk)) GLk that sends an elementof degreen to an elementof degree2n + k and makes the followingdiagramcommutative: F2 PH*(BVk) k P(F2 ( H*(BVk)) GLk GLk Sqo Sqo F2 PH*(BVk) GLk k o P(F2 (0 H*(BVk)) GLk we can define Proof Since Sq0: H* (BVk) H* (BVk) is a GLk-homomorphism, Sqo = Sq0 and get a commutativediagram GLk This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions SPHERICAL AND THE ALGEBRAIC CLASSES TRANSFER 3905 F2 ? H*(BVk) H*(BVk) GLk { Sqo {Sqo ? H*(BVk), H*(BVk) -F2 GLk wherethe horizontalarrowsare the canonicalprojections Next, we show that Sqo sends the primitivepart to itself In other words, suppose a E H* (BVk) satisfies Sq*(1 ) = Sq*a = O X GLk GLk forany r > 0; we want to show that Sq*(Sq0(1 a)) = GLk forany r > By definitionof Sqo and Lemma 4.1, we have foreveryr > Sq*(Sq0(l a)) = GLk Sq*Sq (a) GLk 10 - GLk Sqo(Sq /2 (a)), r even r odd, O0, { Sq Sq*/2(1 a), r even, 0, r odd, ~~GLk = i =0 Therefore,the above commutativediagramgivesriseto a commutativediagram PH*(BVk) PH*(BVk) P(F2 GLk * ik P(F2 X H*(BVk)) H*(BVk))3* GLk ofik, the homomorphism By definition ik*factorsthroughF2 PH*(BVk) and GLk the previousdiagraminducesthe commutativediagramstated in the proposition, in whichSqo is re-denotedby Sqo forshort The propositionis proved As an applicationofProposition4.2, we givean alternativeproofofTheorem3.2 By Kameko [11],Alghamdiet al [3] and Boardman [4],F2 PH*(BV3) has a GL3 and certainelementsci (i > 0) basis consistingofsome productsofthe formhrhsht withSqo(ci) = ci+1 forany i > By Lemma 3.3, j3 vanisheson any product hrhsht.Making use of Proposition 4.2, one has i3*c3 ==J3(SqO)i(CO) = (co)) (Sqe)i(i3 This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions 3906 H V HU'NG NGUYEN One needsonlyto showthatj *(co) = Recall that dim(co) = The onlyelement Q2 = Sq'Q2 SO P(F2 H*(BV3))g = ofdimension in D3 is Q2 Obviously, (F2 0D3)8 A = Therefore,j3*(co)= Theorem3.2 is proved THE SQUARING OPERATION: GL3 FORMULA FOR k AN EXPLICIT < QoO E Dk, wherethedualityis taken with respectto the basis of Dk consistingof all monomialsin the Dickson invariantsQk-1, *** , QOIt is well-known that Let d(ik_l, i0) of be thedual element P(F2 GL1 P(F2 GL2 H*(BV1)) = Is > o}, Span{d(2s1) H* (BV2)) = Span {d(2s -1,0) I s > 0} of Sqo one can easily showthat By means of the definition = d(2s+1-1), Sq0(d(2s-l)) = d(2s+1-1,o) Sq0(d(2-_1,0)) In thissectionwe computeSq0 explicitlyon P(TF20 H*(BVk)) fork = and GLk Theorem 5.1 (Hu'ng-Peterson[18]) PD* := P(F2 sistingof d(2s _1,0,0) , s > d(2 2s 12 1,1), They are of dimensions 2s+2 - and 2r+2 GL3 H* (BV3)) has a basis con- r> s + 2s+1 - > 3, respectively Remark It is easy to checkthat PD* has at most one non-zeroelementof any dimension Proposition 5.2 Sq0: PD* -* PD* is given by Sqo (d(2s1- ,0,0)) = , Sqo(d(2r-28-1,28-11)) = d(2r+1-2s+1-1,2s+1-,l1) Proof For brevity,we denote X1, X2,x3 by x, y,z and a1, a2, a3 by a, b,c, respectively The firstpart ofthe propositionis an immediateconsequenceofdimensionalinformation.To provethe secondpart we startby recallingthat,fromDefinition3.4, we have Q3,0 = QO = X4y2z' + (symmetrized) Suppose m,n are non-negativeintegers Let x'yl3z-a be the biggest monomial in Q' Q' with respect to the lexicographicorder on (a, 3,y) We claim that xa+4y +2z+' appears exactlyone time in Q'Q Qo, or equivalently (a) QiQnQo = xc+4+2z?+1 + (otherterms) This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER 3907 Indeed,suppose to the contrarythat it appears morethan once in Q2JQ Qo That meansthereexistsa monomialx yO'z< in Q2JQ', whichis different fromx yozl, and a permutationa on the set {4, 2, 1} such that ce+4 3+2 Y+1 = + oa'+a(4) 0/'+o(2) $Y'+U(1) Since a + = a' + ac(4) and > ac(4),thisimpliesa < a' Combiningthiswiththe factthat (a, 3, y)is thebiggestmonomialin Q'Q' withrespectto thelexicographic orderon (a,i3y), one gets a = a' and ac(4) = Similarly, = ', oy - = -y'and oais the identitypermutation.This contradictionproves(a), or equivalently (b) = a(a+4)b(0+2)C(+) d(m,njl) + (otherterms) Here and throughoutthe proof,0 means the tensorproductoverGL3 By definition of the squaringoperation (c) Sq?(1 a(o+4) b(+2)C(y+1)) = a(2o+9)b(23+5)C(2a+3) Now a directcomputationusingDefinition3.4 showsthat Q2Q1Qo = x12y4z+ x'0y6z + x'0y5z2+ x10y4z3 + x9 6z2+ x9y5z3+ x8y6z3+ x8y5z4 + (symmetrized) Note that x9y5z3 and its symmetrizedterms are the only terms of the form xodd yoddzodd in Q2Q1Qo- On the other hand, + (other terms), Q2 1= wherex2cy2/3z2-7 is the biggestmonomialin thispolynomialwithrespectto the lexicographicorderon (2a, 2p, 2-y).Focusingon monomialsof the formxodd yoddzodd and usingthe same argumentas in the proofof (a), we have = x2 9y2 +Sz2(+3 + (otherterms) Q2m+1Q2r+lQ This is equivalentto a(2c+9)b(23+5)C(2y+3 = d(2m+1,2n+1,1)+ (otherterms) (d) Combining(b), (c) and (d), we get Sq0 (d(m,n,i)) = d(2m+l,2n+l1l) Applying this for (m, n, 1) = (2r Sq0(d(2r-2s-,2s-1,1)) - 2S - 1, 2S = d(2r+1 2s+1 + (other terms) 1, 1), we obtain - + 1,2s+1-1,1) (other terms) In addition,Sq0 maps PD* to itself(by Proposition4.2) and PD3* consistsof at most one non-zeroelementof any dimension.So the propositionis proved Theorem 5.3 (Hu'ng-Peterson [18]) PD* := P(F2 GL4 sistingof > d(2S -1,0,0,0), 'S 0, r > s > 0, t > r > s > 1, r > s + > d(2r-2s_1,2s 1,10), d(2t-2r-1,2r-28-1,2s 12), -12s 12s 12), d(2r-28+1-2s-9 They are of dimensions 2s+3 - 8, 2r+3 + 2s+2 2r+3 + 2s+1 - 4, respectively H*(BV4)) has a basis con- - 6, 2t+3 + 2r+2 + 2s+1 - and Remark PD*, as wellas PDj*, has at mostone non-zeroelementofany dimension This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions 3908 H V HU'NG NGUYEN Proposition 5.4 Sq0: PD Sqo (d(2s_i,o,0,o)) Sqo (d(2t = Sqo (d(2r_2s+1 = Sqo(d(2r_2s-1,2 1,1,0)) -2_1 -2r_1,2r PD* is givenby -* ,2_1 ,2) )= d(2t+1 = 2s_1,2s_1,2s-1,2)) -2r+1 0O -1,2r+1 -2s+1 d(2r+1-2s+2_2s+1 _1,2s+1 1,2s+1 -1,2) 1,2s+ -1,2) Proof We denotex1,x2,x3,x4 byx,y, z,t and a,, a2, a3, a4 by a, b,c, d, respectively, forbrevity The firstpart of the propositionis an immediateconsequenceof dimensional information We claim that Qo = x8y4z2t + (symmetrized).It can be checkedby a routine computationusing Definition3.4 Here we give an alternativeargument.Indeed, the Dickson algebra D4 IF2 [Q3,Q2,Qj, Qo] has exactlyone non-zeroelementof dimension15 To checkthe equalitywe need onlyto showthat the righthand side is GL4-invariant.Recall that GL4 is generatedby the symmetric groupZ4 and the x H-+ x + y, y H-+ y, z H-+ z, t H-+ t So, it sufficesto checkthat the transformation righthand side is invariantunderthis transformation We leave it to the reader Suppose m,n,p, q are non-negativeintegerswith q > Let x'yOzat6 be the orderon biggestmonomialin QQ'QP1Q'-7 withrespectto the lexicographic 3, -y,6) (a, By the same argumentas in the proofof Proposition5.2 we have = QmQnQpQq + Xo+8y,+4Z-Y+2t6+1 (other terms) In otherwords, (a) d(mn,p,q) +8) b(+4) C(+2V6+1) a10 + (otherterms) Here and throughoutthis proof,0 denotesthe tensorproductover GL4 of the squaringoperation By definition (b) Sq?(1 a(+8)b (+4)C(7+2)t(6+1)) = a(2a+17)b(23+9)C(2y+5)d(26+3) Usingthe same methodthat we used to computeQo above, we can showthat Q3Q2 XSlyS2zS3tS4 = S1 +82+S3+S4=20 sj=0 or a power of Qi= E Sl xS1yS2zS3tS4 +S2+S3+S4=14 sj=0 or a power of In particular,we have Q3Q2 Qi 2zt + symmetrized)+ (otherterms), = (X16y = (x8y4 zt + + (otherterms) symmetrized) zeventevenSo Here,in both cases, any othertermis of the formxevenyeven Q3Q2Q1 = (X17y9z5t3 + symmetrized)+ (otherterms), where x17y9z5t3 and its symmetrizedterms are the only terms of the form xodd yodd zodd todd in Q3Q2Q1 On the otherhand, = x2ay23Z2yt2b + (other terms), Q2mQ2nQ2PQ2q-2 wherex2ay23z2qt26 is the biggestmonomialin the polynomialwithrespectto the lexicographicorderon (2a, 2p, 2-y, 26) Again, we focuson monomialsof the form This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions SPHERICAL xoddyoddzoddtodd get CLASSES AND THE ALGEBRAIC TRANSFER 3909 and use the same argumentas in the proofof Proposition5.2 to = x2a+17 Q2m+lQ2n+lQ2P+lQ2q-2 20+9z27+5t2b+ (otherterms), or equivalently (c) 1(9 = a(20+17)b-(20+9)C(2-y+5)d(26+3) d(2m+1,2n+1,2p+1,2q-2) + (otherterms) Combining(a), (b) and (c), we get SqO(d(m,n,p,q)) = d(2m+1,2n+1,2p+1,2q-2) + (other terms) - 1,2r - 25 - 1,25 - 1,2) and (2r-2s+1 - 25 1, 2) Combiningthe resultingformulasand the factsthat Sqo maps PD* to itself(by Proposition4.2) and that PD* has at mostone non-zeroelement of any dimension,we obtain the last two formulasof the proposition Applythisfor(m,n,p,q) = 1, 25 - 1, 25 - (2t-2r FINAL REMARK Recall that Dk := F2 (&Dk Let A ADD:Dk D De(0Dm f+m=k f,m>o be the diagonal definedat the beginningof Section Conjecture 6.1 (Hu'ng-Peterson[19]) The diagonal AD is zero in positivedimensionsforany k > This conjectureis proved in Lemma 3.3 for k = and has been proved for < k < 10 in [19] It impliesthat ji (respectively, SOk) vanisheson the decomposable elementsin F2 PH* (BVk) withrespectto the productgivenby Singer[20] GLk and discussedin Section (respectively, in Extk(F2, F2) with respectto the cup product) for2 < k < 10 Note added in proof.Conjecture6.1 has been establishedby F Petersonand the authorin the finalversionof [19] REFERENCES J F Adams, On the non-existenceof elements of Hopf invariant one, Ann Math 72 (1960), 20-104 MR 25:4530 J F Adams, Operations of the nth kind in K-theory, and what we don't know about RP?, New Developments in Topology, G Segal (ed.), London Math Soc Lect Note Series 11 (1974), 1-9 MR 49:3941 M A Alghamdi, M C Crabb and J R Hubbuck, Representationsof the homologyof BV and the Steenrod algebra I, Adams Memorial Symposium on Algebraic Topology 2, N Ray and G Walker (ed.), London Math Soc Lect Note Series 176 (1992), 217-234 MR 94i:55022 J M Boardman, Modular representationson the homologyof powers of real projective space, Algebraic Topology: Oaxtepec 1991, M C Tangora (ed.), Contemp Math 146 (1993), 49-70 MR 95a:55041 W Browder, The Kervaire invariant of a framedmanifoldand its generalization,Ann Math 90 (1969), 157-186 MR 40:4963 D P Carlisle and R M W Wood, The boundednessconjecturefor the action of the Steenrod algebra on polynomials,Adams Memorial Symposium on Algebraic Topology 2, N Ray and G Walker (ed.), Lond Math Soc Lect Note Series 176 (1992), 203-216 MR 95f:55015 This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions 3910 NGUYEN H V HU'NG H Cartan, Puissances divisees, S6minaire H Cartan, Ecole Norm Sup., Paris, 1954/55, Expos6 MR 19:438c M C Crabb and J R Hubbuck, Representations of the homologyof BV and the Steenrod algebra II, Algebraic Topology: New Trends in Localization and Periodicity (Sant Feliu de Guixois, 1994; C Broto et al., eds.), Progr Math 136, Birkhaiuser,1996, pp 143-154 CMP 96:15 E B Curtis, The Dyer-Lashof algebra and the A-algebra, Illinois Jour Math 18 (1975), 231-246 MR 51:14054 10 L E Dickson, A fundamental system of invariants of the general modular linear group with a solution of the formproblem,Trans Amer Math Soc 12 (1911), 75-98 11 M Kameko, Products of projective spaces as Steenrod modules, Thesis, Johns Hopkins University1990 12 J Lannes and S Zarati, Invariants de Hopf d'ordre sup6rieeuret suite spectrale d'Adams, C R Acad Sci Paris S6r I Math 296 (1983), 695-698 MR 85a:55009 13 J Lannes and S Zarati, Sur les foncteurs derives de la destabilisation, Math Zeit 194 (1987), 25-59 MR 88j:55014 14 A Liulevicius, The factorizationof cyclic reducedpowers bysecondarycohomologyoperations, Mem Amer Math Soc 42 (1962) MR 31:6226 15 I Madsen, On the action of the Dyer-Lashof algebra in H (G), Pacific Jour Math 60 (1975), 235-275 MR 52:9228 16 J P May, A general algebraic approach to Steenrod operations,Lecture Notes in Math Vol 168, Springer-Verlag(1970), 153-231 MR 43:6915 17 N Minami, On the Hurewicz image of elementaryp groups and an iteratedtransferanalogue of the new doomsday conjecture,Preprint 18 Nguyen H V Hu'ng and F P Peterson, A-generatorsfor the Dickson algebra,Trans Amer Math Soc., 347 (1995), 4687-4728 MR 96c:55022 19 Nguyen H V Hu'ng and F P Peterson, Spherical classes and the Dickson algebra, Math Proc Camb Phil Soc (to appear) 20 W Singer, The transferin homological algebra, Math Zeit 202 (1989), 493-523 MR 90i:55035 21 R J Wellington,The unstable Adams spectralsequence offree iteratedloop spaces, Memoirs Amer Math Soc 36 (1982), no 258 MR 83c:55028 22 C Wilkerson, Classifyingspaces, Steenrod operations and algebraic closure, Topology 16 (1977), 227-237 MR 56:1307 CENTRE DE RECERCA MATEMATICA, INSTITUT D'ESTUDIS BELLATERRA, BARCELONA, ESPANA CATALANS, APARTAT 50, E-08193 Current address: Department of Mathematics, Universityof Hanoi, 90 Nguyen Trai Street, Hanoi, Vietnam E-mail address: nhvhungQit-hu.ac.vn This content downloaded from 141.233.160.21 on Tue, 22 Dec 2015 13:38:46 UTC All use subject to JSTOR Terms and Conditions ... subject to JSTOR Terms and Conditions CLASSES SPHERICAL AND THE ALGEBRAIC 3895 TRANSFER In otherwords,there are no sphericalclasses in QoS?, except the elementsof Hopf invariantone and those of Kervaireinvariantone,... JSTOR Terms and Conditions SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER 3897 First we sketchLannes-Zarati's work [13] on the derivedfunctorsof the destabilization Let D be the destabilizationfunctor,whichsends...TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 349, Number 10, October 1997, Pages 3893-3910 S 0002-9947(97)01991-0 SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER NGUYEN H V HU'NG