On the existence of a v_2^32-self map on M(1,4) at

arXiv:0710.5426v2[math.AT]12Aug2008ONTHEEXISTENCEOFAv322-SELFMAPONM(1,4)ATTHEPRIME2M.BEHRENS1,M.HILL,M.J.HOPKINS2,ANDM.MAHOWALDAbstract.LetM(1)bethemod2Moorespectrum.J.F.AdamsprovedthatM(1)admitsaminimalv1-selfmapv41:Σ8M(1)→M(1).LetM(1,4)bethecofiberofthisself-map.ThepurposeofthispaperistoprovethatM(1,4)admitsaminimalv2-selfmapoftheformv322:Σ192M(1,4)→M(1,4).Theexistenceofthismapimpliestheexistenceofmany192-periodicfamiliesofelementsinthestablehomotopygroupsofspheres.Contents1.Introduction12.GeneralizedMoorespectra43.ModifiedAdamsspectralsequences44.v82-periodicityinExtA∗95.Brown-Gitlercomodules106.Extcomputations127.ReducingthecomputationtoM2(1)⊗kfork≤3188.ThemodifiedAdamsspectralsequencefortmf∗M(1,4)239.d2(v82)andd3(v162)2610.CalculationofanAdamsdifferential3011.Proofofthemaintheorem33References361.IntroductionFixaprimep.Thep-componentofthestablehomotopygroupsofspheresadmitsafiltrationcalledthechromaticfiltration.Elementsinthenthlayerofthisfiltrationfitintoinfinitevn-periodicfamilies.Theoretically,thisprocessiswellunderstood,thankstotheNilpotenceandPeriodicityTheoremsofDevinatz,Hopkins,andSmith[HS98],[DHS88].Itisdifficultinpractice,however,toexplicitlyidentifyvn-periodicelements,andtodeterminetheirperiods.Oneusefultechniqueistoinductivelyformcofiber1TheauthorissupportedbytheNSFandtheSloanFoundation.2TheauthorissupportedbytheNSF.Date:August12,2008.2000MathematicsSubjectClassification.Primary55Q51;Secondary55Q40.12M.BEHRENS,M.HILL,M.J.HOPKINS,ANDM.MAHOWALDsequences:Spi0−−→S→M(i0),Σ2i1(p−1)M(i0)vi11−−→M(i0)→M(i0,i1),...Σ2in(pn−1)M(i0,...,in−1)vinn−−→M(i0,...,in−1)→M(i0,...,in).Themapsvikarevk-selfmaps.ThePeriodicityTheoremguaranteestheirexistenceforlargei.Thereaderiswarnedthattherearepotentiallymanynon-homotopicvik-selfmaps,sothehomotopytypesofthespectraM(i0,...,in)arenotdeterminedmerelyfromthesequence(i0,...,in).Itischallengingtodeterminetheminimalsequence(i0,i1,...,in).Thisminimalsequencedeterminestheperiodsoftheprimaryconstituentsofthevn-periodicfamiliesinthestablehomotopygroupsofspheres.Wereferthereaderto[Rav86,Ch.5.5],[Rav92],and[Beh07]foramoredetaileddiscussion.Wegiveabriefsynopsisofwhatisknownconcerningtheminimalsequenceofintegers(i0,...,in)sothatthespectrumM(i0,...,in)existsatagivenprimep.Forp≥3,itisknownthatthecomplexM(1,1)isminimal[Ada66],forp≥5,thecomplexM(1,1,1)isminimal[Smi70],andforp≥7,thecomplexM(1,1,1,1)isminimal[Tod71].Forp=2,thecomplexM(1,4)isminimal[Ada66],andforp=3,thecomplexM(1,1,9)isminimal[BP04].In[DM81],itwasarguedthatthecomplexM(1,4,8)isminimalattheprime2,i.e.thatthereisav2-selfmap:Σ48M(1,4)v82−→M(1,4).Theresultisincorrect:theimageofv82intheAdams-Novikovspectralsequencefortmfisnotapermanentcycle[HM],[Bau08].Infactthefirstmultipleofv2whichisapermanentcycleinthisspectralsequenceisv322.Thepurposeofthispaperistoprovethefollowingtheorem.Theorem1.1.Thereisav322-selfmapv:Σ192M(1,4)→M(1,4).Corollary1.2.Attheprime2,thecomplexM(1,4,32)isminimal.Remark1.3.Av322-selfmapis,bydefinition,amapvwhoseinducedmapv∗:K(2)∗M(1,4)→K(2)∗M(1,4)isgivenbymultiplicationbyv322.Inparticular,themapv,andallofitsiterates,mustbeessential.Sincethereisamapofringspectratmf→K(2)underwhichtheperiodicitygeneratorv322∈π192(tmf2)mapstov322∈π192K(2),toproveTheorem1.1,itsufficestoprovethatthereexistsaself-mapvsuchthatv∗:tmf∗M(1,4)→tmf∗M(1,4)isgivenbymultiplicationbyv322.ONTHEEXISTENCEOFAv322-SELFMAPONM(1,4)ATTHEPRIME23Remark1.4.ThefourthauthorreportsthatmethodssimilartothosedescribedinthispapershowthatthespectraA1andM(2,4)alsoadmitv322-selfmaps.Here,A1isaspectrumwhosecohomologyisafreemoduleofrank1overthesubalgebraA(1)oftheSteenrodalgebra(see[DM81]).Theself-mapofTheorem1.1producesmanyv322-periodicinfinitefamiliesofelementsinthestablehomotopygroupsofspheres.Thesefamiliesarediscussedindetailin[HM].Infact,alloftheresultsof[DM81]and[Mah81]concerningv2-periodicfamiliesarevalidwithv82replacedbyv322.Organizationofthepaper.InSection2,wereduceTheorem1.1toshowingthatthereexistsahomotopyelementv∈π192(M(1,4)∧DM(1))withHurewitzimagev322∈tmf192(M(1,4)∧DM(1)).Here,DM(1)istheSpanier-WhiteheaddualofthespectrumM(1).InSection3weconstructmodifiedAdamsspectralsequences(MASSs)oftheformExts,tA∗(F2,H(1,4)⊗DH(1,4))⇒πt−s(M(1,4)∧DM(1,4)),(1.1)Exts,tA∗(F2,H(1,4)⊗H∗(X))⇒πt−s(M(1,4)∧X)(1.2)whereA∗isthedualSteenrodalgebra,H(1,4)andDH(1,4)areobjectsinthederivedcategoryofA∗-comodules,andExtA∗isagroupofhomomorphismsinthederivedcategory.Weshowthat(1.1)isaspectralsequenceofalgebras,andthat(1.2)isaspectralsequenceofmodulesover(1.1).InSection4weprovethatthereexistsanelementv82∈Ext8,56A∗(F2,H(1,4)⊗DH(1,4)).InSection5,wegiveageneraloverviewofthetheoryofgeneralizedBrown-GitlerA∗-comodulesMi(j).WedescribeaspectralsequencewhichcomputesExtA∗intermsofExtA(i)oftensorproductsofthesecomodules.Thecaseofinterestiswherei=2,andthespectralsequenceisanalgebraicversionofthetmf-resolution.InSection6wecomputeExt∗,∗A(2)∗(H(1,4)⊗M2(1)⊗k)fork≤3.InSection7weestablishvanishinglinesfortheExtgroupsappearinginthealgebraictmf-resolution.Thesevanishinglinesimplythattheonlytargetsofapotentialdifferentialsupportedby