On the May spectral sequence and the algebraic transfer II

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 We

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www.elsevier.com/locate/topol

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

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

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 twoSq0 -families

n i ∈ Ext 5,36 ·2 i

A (Z/2, Z/2), i ≥ 0, andk i ∈ Ext 7,36 ·2 i

A (Z/2, Z/2), i ≥ 1, are in the image of the algebraic transfer.

© 2014 Elsevier B.V All rights reserved.

1 Introductionandstatementresults

Thisnoteisacontinuationofourpreviouspaper[5],whichwewillrefertoasPartI.InPartI,weusethe Mayspectralsequence(MSSforshort)toobtainnewcomputationforthekernelandimageofthealgebraic transfer,introducedbySinger[19], whichisanalgebraic homomorphism

s

ϕ s: TorA ∗,∗(Z/2, Z/2) →

s



Z/2 ⊗ A H ∗ (BV s)G s

(1)

✩ ThisworkispartiallysupportedbyaNAFOSTEDgrantNo.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.

from the homology of the mod 2 Steenrod algebra A (see Steenrod [20], Milnor [14]) to the space of

ringH ∗ (BV s) has anaturalstructureofanunstablemoduleoverthemod2Steenrodalgebra.Inaddition, theautomorphism groupG s = GL(V s) acts canonically onV s and henceonH ∗ (BV s) These twomodule structures are compatible, so thatone has an induced action of G s on the space Z/2 ⊗ A H ∗ (BV s) The readerwhoisfamiliarwiththe algebraictransferandtherelated“hitproblem”[8,24,19]willprobablyagree thatthedualpoint ofviewisalsoveryuseful Inranks andworkingwithhomologyand Extinstead,the dualof(1) hastheform

ϕ ∗ s:

P A H ∗ (BV s)

G s → Ext s,s+ ∗

whereP A H ∗ (BV s) denotethesubspaceofthedividedpoweralgebraH ∗ (BV s) consistingofallelementsthat areannihilatedbyallpositivedegreeSteenrodsquares,andM G isthestandardnotationforthemoduleof

G-coinvariants.

Ourinterestsinthemap(1)(oritsdual(2))liesinthefactthatontheonehand,thedualofitsdomain

isthecohomologyoftheSteenrodalgebra,Ext∗,∗ A (Z/2, Z/2),whichistheinitialpageoftheAdamsspectral sequenceconvergingtostable homotopygroupsofthespheres[1],therefore,itisanobjectoffundamental importance inalgebraic topology On theother hand, thetarget of(1) is the subjectof the so-called the

“hit problem”,proposed by F Peterson [17] (see Wood [24]) The hit problem, which is originated from cobordismtheory,hasdeepconnectionwithmodularrepresentationtheoryofthegenerallineargroup,and

itisbelievedthattoolsfrom modularrepresentationtheorycanbeusedtounderstandthestructureofthe Extgroup

WerefertotheintroductionofPartIforadetailedsurvey ofknownfactsaboutthealgebraic transfer

In Part I, we initiated the use of the (homology) May spectral sequence to make computation on the kernelandimageofthealgebraictransfer.Thismethodallowsustonotonlyrecoverpreviousknownresults with littlecomputationinvolved,butalsoobtainnewdetection andnondetection resultsindegreeswhere computation of the hit problem seems out of reach at the moment However, the computation remains difficult,partlybecausewhilethetargetofthealgebraictransfer(1)isessentiallyapolynomialringwhich

isrelativelyeasytoworkwith,thedomainistheTorgroup,whoserichstructure,suchastheactionofthe Steenrodalgebra,ishardtoexploit

Toovercomethisdifficulty,inthispaper,wefirstdualizetheconstructionin[5]toconstructa represen-tationofthealgebraictransferinthecohomologicalE2-termoftheMayspectralsequence Anapplication

of this construction is given in Section 3 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 Sq0 is no longer the identitymap.Inhisthesis[8],Kamekoconstructedanoperation

Sq0:

P A H d (BV s)

G s →P A H 2d+s (BV s)

G s ,

thatcorrespondstotheoperationSq0onExtgroups.Kameko’soperationhasbeenextremelyusefulinthe studyofthehitproblemandforcomputationofthealgebraictransfer.WeuseanobservationofVakil[23]

toshowthatKameko’ssquaringoperationiscompatiblewiththeMayfiltration,andthusinducesasimilar operationwhenpassingtotheassociatedgraded.Ontheotherhand,Nakamura[16]alsoconstructeda fam-ilyofsquaringoperationswhichareallcompatiblewithhigherdifferentialsintheMayspectralsequence.It shouldbepointedoutthatNakamura’sconstructionisquitedifferentfromtheusualprocedureof construct-ingSteenrodoperations,suchasdescribedinMay[13].Foritisknownthatthegeneralframeworkprovided

inMay [13] yields trivialmap in thecohomology of the associated graded algebraE0A. In Section4, we showedthatundertherepresentationofthealgebraictransferintheE2termsoftheMayspectralsequence describedinSection3,theinducedKamekosquaringoperationcorrespondsto Nakamura’sone

Using theconstructionabove,wehavethefollowing,whichisourmainresult.

Theorem 1.1 ( Corollary 5.2 ) The family {n i ∈ Ext 5,36 ·2 i

transfer(2)

Bruner[2]hasshownthattherelationk1= h2h5n0holdsinExt7, A ∗(Z/2, Z/2).Sinceitiswell-knownthat the total transfer ϕ ∗ =

s≥1 ϕ ∗ s is an algebrahomomorphism (see Singer [19]), we obtain an immediate corollary

Corollary1.2.The family {k i ∈ Ext 7,36 ·2 i

Wedonotknowwhetherk0∈ Ext 7,36

A (Z/2, Z/2) alsobelongstotheimageofϕ ∗7 ornot

Thepaperisdividedintofivesections.Sections2and3arepreliminaries.InSection2,werecallbasicfacts abouttheMayspectral sequenceandinSection3, wepresentthealgebraictransferanditsrepresentation

in theE2-term of thecohomological Mayspectral sequence We apply the aboveconstruction to show in Section 4thataversion of Kameko’s squaringoperation whichhas been extremely usefulin thestudy of the hit problem is compatible with Nakamura’s squaring operation the Mayspectral sequence The final section containstheproofofthemain resultsofthis paperthatthetwofamiliesn i,i ≥ 0 and k j,j ≥ 1 in

Ext∗,∗ A (Z/2, Z/2) areintheimageofthealgebraictransfer(2)

2 TheMayspectralsequence

In thissection, we reviewthe constructionof theMayspectral sequence Themain references areMay [11,12] and Tangora[22] May’schain complex forthe cohomology of the associated graded algebraE0A

was subsumedinPriddy’s theoryof Koszulresolution [18].Let A denotethemod2Steenrodalgebra.All

TheSteenrodalgebraisfilteredbypowersofitsaugmentationidealA by¯ settingF p A = A if p ≥ 0 and

F p A= ( ¯A) ⊗−pifp < 0.LetE0A=

p,q E0

p,q A,whereE0

p,q A = (F p A/F p −1 A) p+q,betheassociatedgraded algebra Using awell-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 E0A is a primitively generatedHopfalgebrawhichisisomorphictotheuniversalenvelopingalgebraofitsrestricted Lie algebraof itsprimitive elements {P j

j , P k

] = δ i,k+ P k

ξ(P k j) = 0, where ξ is the restriction map of its restricted Lie algebra structure and δ i,k+ is the usual Kronecker delta An element θ ∈ F p A but θ / ∈ F p −1 A is said to have weight −p. The following result determinestheweightofanygivenMilnorgeneratorSq(R).

Theorem2.1 (May [11] ) The weight w(R) of a Milnor generator Sq(R), where R = (r1, r2, ), is w(R)=



i iα(r i ) where α(m) is the function that counts the number of digit 1 in the binary expansion of m.

In particular, theweightof P j i is justitssubscript j. Infact,May identified Sq(R) withthe monomial



(P i

j)a ij in the associated graded, where r i =

a ij2j is the binary expansion of r i In the language of Priddy’s theoryofKoszulresolution[18],thenE0A is aKoszulalgebrawithKoszulgenerators{P j

k |j ≥ 0,

P j i P  k = P  k P j i if i = k + , P j i P i− + P i− P j i + P i− = 0, P j i P j i = 0.

Theorem2.2 (Priddy [18] ) Let M be a right E0A-module There exists a natural isomorphism

j }, i ≥ 0, j ≥ 1,

of degree 2i(2j − 1), and the differential is given by δ(R i

j) = j−1

k=1 R i

k R i+k j−k ; and the differential δ of the

s,t

RR s t ⊗ mP s

Wewillbe workingwiththecohomologyversionoftheMayspectralsequence LetA ∗ bethedualofA

andletA¯∗= ( ¯A) ∗.ThenA ∗admitsafiltrationwhereF p A ∗= 0 ifp ≥ 0 and F p A ∗= ( ¯A/F p−1 A) ∗ ifp < 0.

,

where α ∗ isthe structuremap of theA ∗-comodule M ∗.Clearly F p M ∗= 0 for p ≥ 0 andwhenp < 0,we haveF p M ∗ ⊆ F p −1 M ∗. Thus(E0M ) ∗ ∼ = E0M ∗ =

p,q E p,q0 M ∗, where E p,q0 M ∗ = (F p M ∗ /F p+1 M ∗ p+q,

isabigradedcomoduleovertheassociatedgradedcoalgebraE0A ∗.LetC(A;¯ M ) bethecobarconstruction withtheinducedfiltration:

F p C¯n

F p1A¯∗ ⊗ · · · ⊗ F p n A¯∗ ⊗ F p0M ∗ ,

where thesumis takenover allsequences{p0, , p n } suchthatn+n

i=0 p i ≥ p. Thisfiltrationrespects thedifferential,andintheresultingspectral sequence,wehave

E1p,q,t

F p C¯p+q (A; M )/F p+1 C¯p+q (A; M )

t

Here p isthefiltrationdegree,p + q isthehomologicaldegreeand t istheinternaldegree.Thedifferential

δ1of thisspectralsequenceistheconnectinghomomorphismoftheshort exactsequence:

F p+2 C(A; M )¯ → F p C(A; M )¯

F p+2 C(A; M )¯ → F p C(A; M )¯

F p+1 C(A; M )¯ → 0.

On the other hand, E1p,q,t (M ∗) is isomorphic to C¯p+q (E0A; E0M ) −q,q+t as trigraded Z/2-vector spaces Underthisidentification,δ1 isexactlythecanonicaldifferentialofthecobar constructionC¯∗ (E0A; E0M ).

HenceE2p,q,t (M ) isisomorphictoH p+q (E0A ∗;E0M ∗ −q,q+tandwecansummarizetheresultinthe follow-ingtheorem

ex-ists a spectral sequence (E r , δ r ) converging to E0H ∗ (A; M ∗ ) and having as its E2-term E2p,q,t (M ) =

H p+q (E0A; (E0M ) ∗ −q,q+t Each δ r is a homomorphism

δ r : E p,q,t r (M ) −→ E p+r,q −r+1,t

When M = Z/2, we write E r for E r (M ). It is well-known that E r (M ) is a differential E r-module May[12]explainedhowtocomputeallthedifferentials,atleastinprinciple,usingtheso-calledimbedding method(see alsoTangora [22, Section 5]) This is possiblebecauseR isaquotient of thecobar complex,

andthedifferentialscomefromthatofthecobarcomplexaswell.Weshallusethismethodintheproofof themain theoreminSection5

3 Thealgebraictransfer

In[4,5],weconstructedandstudiedarepresentationofthedualofthealgebraictransferintheE2-term

of thehomologyMayspectral sequence.Thecohomologyversionwhichwe aregoingtopresent hasbetter behavior becauseofthealgebrastructureonExtgroups.Sincetheconstructionpresentedbelowisbasically dual tothatgivenin[5],wewill beverybrief

Wearegoingto constructarepresentationofE2ψ sintheco-KoszulcomplexofE0H ∗ (BV s),whichwill

be denotedbyE1ψ s

We begin with some notations For an s-dimensional Z/2-vector space V s, the (mod 2) cohomology

H ∗ (BV s) is a polynomialalgebra P s =Z/2[x1, , x s], where each x i is of degree1 Dually, H ∗ (BV s) is thedivided poweralgebraH s = Γ (a1, , a s) generatedbya1, , a n overZ/2 where a i is thelineardual

of x i.More precisely, itis a bicommutative Hopfalgebrawith thevector spacebasis a (i1 )

1 a (i s)

s , i t ≥ 0,

forall1≤ t ≤ s,with multiplication

(i1, , i s )(j1, , j s) =



i1+ j1

i1



.



i s + j s

i s



(i1+ j1, , i s + j s ),

where forsimplicity,wewrite (i1, , i s)= a (i1 )

1 a (i s)

s Let Pˆ1 be the unique A-module extension of P1 byformally adding agenerator x −11 of degree −1 and

require that Sq n (x −11 )= x n−11 for n ≥ 1. Let Hˆ1 be the dual of Pˆ1 There is a fundamental short exact sequence ofA-modules:

0→ Σ −1 Z/2 → ˆ H1→ H1→ 0.

PassingtotheassociatedgradedandtensoringwithR ⊗ M where M issomerightE0A-module,weobtain

ashortexactsequenceofdifferentialmodules

The connectinghomomorphismof thisshort exactsequence,undertheisomorphism (3),canbe identified with

Exts−1,t E0A (Z/2, M ⊗ E0H1)→ Ext s,t+1

E0A(Z/2, M).

UsingthecanonicalisomorphismE0H s ∼ = (E0H1)⊗s,wecansplices similarconnectinghomomorphismsto obtainamap

Extk,t E0A(Z/2, M ⊗ E0H s)→ Ext k+s,t+s

E0A (Z/2, M).

Inparticular, whenM = Z/2 and k = 0,weobtain theE2-level ofthealgebraictransfer

E2ψ s: Ext0,t E0A(Z/2, E0H s)→ Ext s,t+s

E0A(Z/2, Z/2).

As notedabove,thismap isinducedbyachainlevelmap

E1ψ s : E0H s → R s

It ispossibledescribethis mapexplicitly

Proposition3.1.The version of the algebraic transfer in E2-term of May spectral sequence is induced by the map

E1ψ s : E0H ∗ (BV s)−→ R s ,

E1ψ s

a (n1 )

1 a (n s)

s =



R k1

t1 R k s

t s , n i= 2k i(2t i − 1) − 1, 1 ≤ i ≤ s,

Proof SupposeR ⊗ m ⊗ a (n1 )

1 a (n s)

s isanontrivialsummandofacyclex ∈ R ⊗ M ⊗ H s.Itcanbepulled backtothesameelementinR ⊗ M ⊗ H s −1 ⊗ ˆ H1.Sinceδ(x)= 0,itcomesfromR ⊗ M ⊗ H s −1 ⊗ Σ −1 Z/2.

Ontheotherhand,wehavethata (n) P i

j = a(−1)ifandonlyifn= 2i(2j − 1) − 1.Thusfromtheformula(4),

weseethattheconnectinghomomorphismsendsR ⊗ m ⊗ a (n1 )

1 a (n s)

s tozero ifn sdoesnothavetheform

2i(2j − 1) − 1 forsomei ≥ 0, j ≥ 1;andto RR i j ⊗ m ⊗ a (n1 )

1 a (n s−1 s−1)ifn s= 2i(2j − 1) − 1.Therequired formulacannowbeeasily obtainedbyinduction 2

Example3.2.Letx = (1, 1,6)+(1, 2,5)+(1, 4,3)∈ E1−2,2,8 (P3),wherea (i1 )

1 · · · a (i1 )

s isdenotedby(i1, · · · , i s)

Itiseasyto checkthatδ1(x)= 0∈ E1−1,2,8 (P3),sox isacycle intheE1-termandsurvivesto anontrivial elementinE2−2,2,8 (P3).Nowδ2(x) = R01⊗ (1, 3,3)= δ1(2, 3,3)∈ E2−1,2,8 (P3),sox isacycleinE2−2,2,8 (P3) Forr ≥ 3, E r −2+r,∗,∗= 0,so δ r (x)= 0 forallr ≥ 3;therefore, x isapermanentcycle

Using(5),we obtain

E1ψ3(x) = R11R11R03+ R11R02R12= R11

R20R12+ R03R11 ,

this latter element is called h1h0(1) in the E2 terms of the May spectral sequence (see Tangora [22, Ap-pendix 1]),andisarepresentationofc0 inthe8-stem

Example 3.3 We see that the element ¯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),

isapermanentcycle.Indeed,sinceδ2(X) = δ1(y + (13)y + (23)y),wherey = (3, 2, 6,3)+ (2, 3, 3,6),X isa cycleinE2−2,3,14 (P4);therefore, ¯0survivestotheE3−4,4,14 and,intheE3-term,itisrepresentedbyX + Y ,

whereY = y + (13)y + (23)y.

Byinspection,wehave

δ3(X + Y ) = δ1(Z);

δ4(X + Y + Z) = δ1(3, 3, 3, 5),

whereZ = (5, 1, 5,3)+ (3, 5, 1,5)

Therefore, ¯0 is apermanentcycle becauseδ r, r ≥ 5,istrivial Againfrom (5), we obtainE1ψ4(X) =

(R02R21+ R30R11)2, which is a representation of d0 in the E1-term of Mayspectral sequence Since ¯0 is a permanentcycle,itisarepresentationofthepre-imageofd0underthealgebraictransferintheMSS(see[6])

We end this section with two simple properties of the maps E r ψ s, s ≥ 1. First of all, since R is a commutativealgebra,itisclearthatE1ψ sfactorsthroughthecoinvariant ring[P E0A E0H s]Σ Thereader

who isfamiliar withthe algebraictransfermaywonder abouttheaction ofG s Unfortunately,this action doesnotpreservetheMayfiltrationingeneral.Forexample,iff = x2x5∈ F −2 P2 andσ ∈ GL2, suchthat

σ(x1)= x1+ x2and σ(x2)= x2,thenwehaveσ(f ) = x2x5+ x7∈ F0P2

Secondly,thedirectsum

s ≥1 H ∗ (BV s) hasanalgebrastructureunderconcatenationproduct.Standard argument asinSinger[19] showsthat:

Proposition 3.4 For each r ≥ 1, the total homomorphism between May spectral sequences

s≥0

E r ψ s:

s

E r ∗,∗ (P s)→ E ∗,∗

r ,

4 Thesquaringoperations

In[16],NakamuraconstructedasquaringoperationontheMSSforthetrivialmodule:

Sq0: E r p,q → E p,q

r , r ≥ 1,

whichismultiplicativeintheE1 pageandthereforesatisfiestheCartanformulasinhigherE r page(when elements are suitably represented in the E2 term) This operation hasbeen quite useful for constructing newdifferentials

Thepurpose ofthis sectionistointroduceasimilarsquaringoperation,definedforany r, s ≥ 1, which,

byabuseofnotation,will alsobedenotedasSq0:

Sq0: E r p,q (P s)→ E p,q

r (P s ),

withthepropertythatitcommuteswithNakamura’soperationviathemapofspectralsequencesE p,q

r (P s)→

E p,q+s

r constructedintheprevioussection

WebeginwithadescriptionofNakamurasquaringoperationinthecomplexR,thisisreminiscenttothe construction of Sq0 inExt∗,∗ A (Z/2, Z/2) froman endomorphismof thelambda algebraas inTangora[22] Define analgebramap θ: R → Rbysetting θ(R i

and thusinducesanendomorphismonExt∗,∗ E0A(Z/2, Z/2).

Proposition 4.1 The endomorphism θ induces Nakamura’s squaring operation Sq0:E p,q

r → E p,q

r

Proof According to Priddy[18],R i

j is representedinthecobarresolution by[ξ2i

j ] Onthe otherhand,in thecobarcomplexforE0A,thesquaringoperation hasanexplicitform

Sq0[α1| |α n] =

α21 α2n

,

so itmaps[ξ2i

j ] to [ξ2i+1

j ] and theresultfollows immediately 2

On H ∗ (BV s), there is also a squaring map constructed by Kameko [8] in his thesis which has been extremelyusefulinthestudyofthehitproblem(see forexampleSum[21]).Itisgivenexplicitlyasfollows

a (t1 )

1 a (t s)

s

(2t1 +1)

1 a (2t s+1)

s

OnequicklyverifiesthatthisendomorphismofH ∗ (BV s) commuteswiththeactionoftheSteenrodalgebra,

inthesensethatforalla ∈ H ∗ (BV s),

(θa)P t s = θ

aP t s−1 if s > 0, and (θa)P t0= 0.

Moreover,Vakil[23] observedthatthemap a (n) (2n+1) respectsMay’sfiltrationonH ∗ (B Z/2).Thisis clearlytrueforhigherranks > 1 aswell.DefineanendomorphismonR ⊗H ∗ (BV s),whichisagaindenoted

asSq0, bysetting

forallR ∈ R,a ∈ H ∗ (BV s)

Lemma4.2 The endomorphism Sq0 on R ⊗ H ∗ (BV s ) commutes with the coboundary δ of(4)

Proof We already know that Sq0 and δ commuteson R Also, (θa)R0

t = 0 and (θa)R s

t = θ(aR t s−1), we have

δSq0(R ⊗ a) = δSq0R ⊗ θa

= δSq0R ⊗ θa +Sq0(R)R s t ⊗ (θa)R s

t

RR s−1 t ⊗ θaR s−1 t

= Sq0



t



Theproofiscomplete 2

It follows thatthere exists an inducedendomorphism Sq0 on E p,q

r (P s) for alls, r ≥ 1.Ournext result shows thatthis endomorphism commuteswith Nakamura’s Sq0 viathe MSS transferE r ψ s, thus justifies forourchoiceofnotation

Proposition4.3 There exists a commutative diagram of maps between spectral sequences:

E p,q

r (P s) E r ψ s

Sq0

E p,q+s r

Sq0

E p,q

r (P s) E r ψ s E p,q+s

r

Proof Itsuffices toshowthatthereexistsacommutativediagramatE1 page

E0H ∗ (BV s)

θ

E1ψ s

Sq0

E0H ∗ (BV s) E1ψ s R s

This can be verified directly from the formula (5) Note that if n i = 2k i(2t i − 1) − 1 then 2n i+ 1 =

2k i+1(2t i − 1) − 1 2

Inparticular,there isaninducedmap

θ: P E0A E0H d (BV s)→ P E0A E0H 2d+s (BV s ),

thatfitsinthefollowing

Proposition 4.4.The representation of Kameko’s squaring operations, Sq0, and Nakamura’s squaring

P E0A E0H t (BV s)

θ

E2ψ s

Exts,s+t E0A(Z/2, Z/2)

Sq0

P E0A E0H 2t+s (BV s) E2ψ s Exts,s+t E0A(Z/2, Z/2).

Proof The assertionisimplied directlyfrom theformulaofSq0 and(5) 2

The operation θ commutes with the action of the symmetric group Σ s on E0H ∗ (BV s) as well

as its subspace P E0A E0H ∗ (BV s) This is essentially direct from the definition Furthermore, E2ψ s is

Σ s-equivariant since R is commutative Thus in the commutative diagram of Proposition 4.4, we can replaceP E0A E0H ∗ (BV s) byitsΣ s-coinvariant(P E0A E0H ∗ (BV s))Σ s

Ourlast resultcanbeconsideredasananaloguetoN.H.V Hưng’sanalysis ofthesquaringmaponthe space(P A H s)G s [7],whereheshowedthatafter(s −2) iteration,thesquaringmapbecomesanisomorphism

onitsrange.Fortheassociated graded,thesituationismuchsimpler

Proposition 4.5 For each s ≥ 1, the induced map

θ:

P E0A E0H ∗ (BV s) Σ

s →P E0A E0H ∗ (BV s) Σ

s ,

The proof of this proposition makes use of a technical result on separating monomials with only odd exponents.Amonomial a i1

1 a i s

s issaidto beoddifallexponentsi tareodd.Otherwise,wesaythatitis non-odd Thelefthandside oftheaboveequation containsalloddmonomials.Each z σ canbe writtenas thesumz σ  + z  σ where z  σ consistsof allnon-trivialoddmonomialsinz σ.Wefirstclaimthatboth z  σ and

z σ areE0A-annihilated.

Lemma 4.6 If x = y + z ∈ P E0A E0H s where y is the sum of odd monomials summands of x, then both y and z belong to P E0A E0H s

Proof ofLemma 4.6 Firstof all,note thatE0A ismultiplicativelygenerated byP1s,s ≥ 0,so inorder to provethaty is E0A-annihilated,wejusthavetocheckthatyP s= 0 foralls ≥ 0.Sinceallmonomialsiny

are odd,itis clearthatyP0= ySq1= 0.If s > 1, thensinceP sis aderivative,and|P s |= 2s iseven, we see that yP s,if non-zero, consistsof only oddmonomials while zP s consists of onlynon-odd monomials Because xP s= 0,wemusthaveyP s = zP s= 0 foralls > 0.Thelemmaisproved 2

ProofofProposition4.5 Foranyelementx ∈ P E0A E0H ssuchthatθ(x)= 0 in(P E0A E0H ∗ (BV s))Σ s,then there existz σ ∈ (P E0A E0H ∗ (BV s))Σ s suchthat

σ∈Σ s

z σ σ + z σ

Wenow continuetheproof ofProposition 4.5.WehaveadecompositioninP E0A E0H s

σ∈Σ

z σ  σ + z  σ + 

σ∈Σ

z σ σ + z σ 

Thesecond summandmustvanishsinceitcontainsnon-oddmonomials.Thefirstsummandcanbewritten

asθx  forsomex  oftheform

(y σ σ + y σ ),

wherey σissuchthatθy σ = z  σ.Sinceθ isobviouslyamonomorphismonP E0A E0H s,itfollowsthatx = x 

andso x istrivialin(P E0A E0H s)Σ s 2

It shouldbe notedthatLemma 4.6 is not truefor the original hit problem.For example, consider the element x = (135)+ (223)+ (124) ∈ P A 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

5 Proofofthemainresults

InthissectionweuseourversionofthealgebraictransferontheE2-termoftheMayspectralsequence

to show thatthefamily n i, i ≥ 0,belongs to theimage of thealgebraic transfer Itshould be notedthat ourdetectionresultisindegreethatgoesfarbeyondthecurrent knowledgeofthehitproblem

Anelementinx ∈ E1issaidtosurvivetoE rforsomer ≥ 2 ifitprojectstoanon-zeroelementintheE r

Apermanentcycleisanelementkilledbyδ rforallr.Wewillusethestandardnotationofknownnontrivial elements inthecohomologyof theSteenrodalgebraas inTangora[22].Also,inthespectralsequence, the sameletterwillbeused foranelement inE2 anditsprojectiontoE r,2< r ≤ ∞.

Thefollowingtheorem isourmain result

Theorem5.1 The element n0∈ Ext 5,36

A (Z/2, Z/2) is in the image of the algebraic transfer.

The factthat two elements n0 and n1 = Sq0n0 are indecomposable elements of Ext5, A ∗(Z/2, Z/2) goes

backto Tangora[22] Recently,completingaprogram initiatedbyLin[9],Chen[3]provedthatthewhole

Sq0-family{n i , i ≥ 0} starting with n0 are indecomposable inExt5, A ∗(Z/2, Z/2). Since Kameko’s squaring operationandtheclassicalsquaringoperationSq0commutewitheachotherthroughthealgebraictransfer,

wehavethefollowingimmediatecorollary

Corollary 5.2.The family of indecomposable elements n i ∈ Ext 5,36 ·2 i

A (Z/2, Z/2), i ≥ 0, are in the image of the algebraic transfer.

AccordingtoTangora[22],thereexistsanindecomposableelementk0∈ Ext 7,36

A (Z/2, Z/2) andarelation

k1= h2h5n0∈ Ext 7,72

A (Z/2, Z/2),thereforewehave Corollary5.3 The elements k i ∈ Ext 7,36 ·2 i

A (Z/2, Z/2), i ≥ 1, are in the image of the algebraic transfer.

Wedonotknow whetherk0 alsobelongsto theimageofthetransferornot

ProofofTheorem5.1 WeshallfindapermanentcycleinE ∞ −6,6,31 (P5) whichisrepresentedbyanelement

X ∈ E1−6,6,31 (P5),tobedescribedexplicitly,suchthatundertheE1-versionofalgebraictransfer,theimage

ofX is R2(R0)2(R1R2+ R1R2).Thisimage isknown,accordingtoTangora[22], tobearepresentativeof

n0 intheE1-termoftheMSS.ElementsinE2 anditsprojection(ifexists)to E rwill bewrittenusing the sameletter

In[4,5],weconstructedandstudiedarepresentationofthedualofthealgebraictransferintheE2-term

of thehomologyMayspectral sequence. Thecohomologyversionwhichwe aregoingtopresent...

InthissectionweuseourversionofthealgebraictransferontheE2-termoftheMayspectralsequence

to show thatthefamily n i, i ≥ 0,belongs to theimage of thealgebraic transfer Itshould... class="page_container" data-page="7">

who isfamiliar withthe algebraictransfermaywonder abouttheaction ofG s Unfortunately,this action doesnotpreservetheMayfiltrationingeneral.Forexample,iff