Ozsvath-Szabo invariants and tight contact three-manifolds, I

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

ISSN1364-0380(online)1465-3060(printed)925Geometry&T

opology

4)925–945

une2004

Ozsv´ath–Szab´oinvariantsandtight

contactthree–manifolds,I

PaoloLisca

´sIStipsiczAndra

DipartimentodiMatematica,Universit`adiPisa

I-56127Pisa,ITALY

and

yiInstituteofMathematics,HungarianAcademyofSciences

H-1053Budapest,Re´altanodautca13–15,Hungary

mail:lisca@dm.unipi.itandstipsicz@math-inst.hu

etheoriented3–manifoldobtainedbyrationalr–surgeryonaknot

ingthecontactOzsv´ath–Szab´oinvariantsweprove,foraclassof

3(K)carriespositive,tighttainingallthealgebraicknots,thatSrcturesforeveryr=2gs(K) 1,wheregs(K)istheslicegenus

mplies,inparticular,thattheBrieskornspheres Σ(2,3,4)and

arrytight,positivecontactstructures.Asanapplicationofour

weshowthatforeachm∈NthereexistsaSeifert beredrational

sphereMmcarryingatleastmpairwisenon–isomorphictight,

ntactstructures.

i cationnumbers

57R57

Tight, llablecontactstructures,Ozsv´ath–Szab´oinvariantsPrimary:57R17

erOzsv´ath

nMorgan,TomaszMrowka

&TopologyPublicationsReceived:21February2004Accepted:29May2004

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

926PaoloLiscaandAndr´asIStipsicz1Introduction

AccordingtoaclassicalresultofLutzandMartinet,everyclosed,oriented3–manifoldadmitsapositivecontactstructure.Infact,everyoriented2–plane eldonanoriented3–manifoldishomotopictoapositivecontactstructure.TheproofoftheLutz–Martinettheorem—relyingoncontactsurgeryalongtransverselinksinthestandardcontact3-sphere[13]—typicallyproducesovertwistedcontactstructures.(ForaproofoftheLutz–MartinettheoremusingcontactsurgeryalongLegendrianlinkssee[6].)Findingtightcontactstructuresonaclosed3–manifoldis,ingeneral,muchmoredi cult,indeedimpossibleforthePoincar´ehomology3–spherewithitsnaturalorientationreversed[12].LetYbeaclosed,oriented3–manifold.Considerthefollowingproblem:(P)DoesYcarryapositive,tightcontactstructure?

Untilrecently,thetwomostimportantmethodstodealwithproblem(P)wereEliashberg’sLegendriansurgeryasusedegbyGompfin[14],andthestatetraversalmethod,developedbyKoHondaandbasedonGiroux’stheoryofconvexsurfaces.ThelimitationsofthesetwomethodscomefromthefactthatLegendriansurgerycanonlyprovetightnessofStein llablecontactstructures,whilethestatetraversalbecomescombinatoriallyunwieldyintheabsenceofsuitableincompressiblesurfaces.Forexample,bothmethodsfailtodealwithproblem(P)whenYisoneoftheBrieskornspheres Σ(2,3,4)or Σ(2,3,3),becausetheseSeifert bered3–manifoldsdonotcontainverticalincompressibletori,nordotheycarrysymplectically llablecontactstructures[18].

ThepurposeofthepresentpaperistoshowthatcontactOzsv´ath–Szab´oin-variants[28]canbee ectivelycombinedwithcontactsurgery[4,5]totackleproblem(P).Inparticular,itfollowsfromTheorem1.1belowthat Σ(2,3,4)and Σ(2,3,3)doindeedcarrytight,positivecontactstructures.Moreover,suchcontactstructuresadmitanexplicitdescription(cfCorollary1.2andthefollowingremark).

Inordertostateourmainresultweneedtointroducesomenotation.RecallthatthestandardcontactstructureonS3isthe2–dimensionaldistributionξst TS3givenbythecomplextangents,whereS3isviewedastheboundaryoftheunit4–ballinC2.WesaythataknotinS3isLegendrianifitisev-erywheretangenttoξst.ToeveryLegendrianknotL S3onecanassociateitsThurston–Bennequinnumbertb(L)∈Z,whichisinvariantunderLegen-drianisotopiesofL[1].GivenaknotK S3,letTB(K)denotethemaximalThurston–BennequinnumberofK,de nedas

Geometry&Topology,Volume8(2004)

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

Ozsv´ath–Szab´oinvariantsandtightcontactthree–manifolds,I927

TB(K)=max{tb(L)|LisLegendrianandsmoothlyisotopictoK}.

3(K)beLetgs(K)denotetheslicegenus(akathe4–ballgenus)ofK.LetSrtheoriented3–manifoldgivenbyrationalr–surgeryonaknotK S3.

Theorem1.1LetK S3beaknotsuchthat

gs(K)>0andTB(K)=2gs(K) 1.

3(K)carriespositive,tightcontactstructuresThen,theoriented3–manifoldSrforeveryr=2gs(K) 1.

RemarkBythesliceBennequininequality[33],foranyknotK S3wehave

TB(K)≤2gs(K) 1.

Moreover,by[2,3](see[1,page123]),ifKisanalgebraicknotthen

TB(K)=2g(K) 1,

whereg(K)istheSeifertgenusofK.Sincegs(K)≤g(K),itfollowsthatthefamilyofknotsKsatisfyingtheassumptionofTheorem1.1containsallnontrivialalgebraicknots.Infact,therearenon– bered,hencenon–algebraic,knotssatisfyingthesameassumption,asforexamplecertainnegativetwistknots.1

LetT S3betheright–handedtrefoil.SinceTisalgebraic,Theorem1.1

3(T)= Σ(2,3,4)andS3(T)= Σ(2,3,3),applies.Inparticular,sinceS23Theorem1.1immediatelyimpliesthefollowingresult,whichsolvesawell–

knownopenproblem[11,Question8]:

Corollary1.2TheBrieskornspheres Σ(2,3,3)and Σ(2,3,4)carryposi-tive,tightcontactstructures.

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

928PaoloLiscaandAndr´asIStipsiczRemarks(1)TheproofofTheorem1.1showsthatFigures1and2belowprovideexplicitdescriptionsofthetightcontactstructuresofCorollary1.2.

(2)Theorem1.1isoptimalfortheright–handedtrefoilknotT=T3,2,because3(T)= Σ(2,3,5)isknownnottocarrypositive,tightcontactstructures[12].S1Ontheotherhand,itisnaturaltoaskwhetherthesameistrueforothertorusknots.Weaddressthisquestioninthecompanionpaper[22].

Recallthatasymplectic llingofacontactthree–manifold(Y,ξ)isapair(X,ω)consistingofasmooth,compact,connectedfour–manifoldXandasymplecticformωonXsuchthat,ifXisorientedbyω∧ω, XisgiventheboundaryorientationandYisorientedbyξ,then X=Yandω|ξ=0ateverypointof X.AsanapplicationofTheorem1.1weprovethefollowingresult,whichshouldbecomparedwiththeresultsof[20,21].

Theorem1.3Foreachm∈NthereisaSeifert beredrationalhomologysphereMmcarryingatleastmpairwisenon–isomorphictight,notsymplecti-cally llablecontactstructures.

Thepaperisorganizedasfollows.InSection2wedescribethenecessarybackgroundincontactsurgeryandHeegaardFloertheory.InSections3and4weprove,respectively,Theorems1.1and1.3.

AcknowledgementsThe rstauthorwaspartiallysupportedbyMURST,andheisamemberofEDGE,ResearchTrainingNetworkHPRN-CT-2000-00101,supportedbyTheEuropeanHumanPotentialProgramme.ThesecondauthorwouldliketothankPeterOzsv´athandZolt´anSzab´oformanyusefuldis-cussionsregardingtheirjointwork.ThesecondauthorwaspartiallysupportedbyOTKAT34885.

2SurgeriesandOzsv´ath–Szab´oinvariants

Contactsurgery

Let(Y,ξ)beacontact3–manifold.TheframingofaLegendrianknotK YnaturallyinducedbyξiscalledthecontactframingofK.GivenaLegendrianknotKinacontact3–manifold(Y,ξ)andanon–zerorationalnumberr∈Q,onecanperformcontactr–surgeryalongKtoobtainanewcontact3–manifold(Y′,ξ′)[4,5].HereY′isthe3–manifoldobtainedbysmoothr–surgeryalongGeometry&Topology,Volume8(2004)

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

Ozsv´ath–Szab´oinvariantsandtightcontactthree–manifolds,I929Kwithrespecttothecontactframing,whileξ′isconstructedbyextendingξfromthecomplementofastandardneighborhoodofKtoatightcontactstructureontheglued–upsolidtorus.Ifr=0suchanextensionalwaysexists,andforr=

1

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

930PaoloLiscaandAndr´asI

Stipsicz

Figure2:Contactstructureson Σ(2,3,3)

Since,by[4,Proposition9],acontact1

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

Ozsv´ath–Szab´oinvariantsandtightcontactthree–manifolds,I931containsasmoothlyembedded,closed,orientedsurfaceΣofgenusg(Σ)>0suchthat

Σ·Σ≥0and| c1(t0),[Σ] |+Σ·Σ>2g(Σ) 2.

Then,FW,t0=0.

ProofArguingbycontradiction,supposethatFW,t0=0.Byafundamental

propertyoftheinvariants[26]thereareonly nitelymanyspincstructurest1,...,tk∈Spinc(W)suchthatFW,ti=0.Moreover,by[26,Theorem3.6]we

have

FW,t0=0 FW,

t0isthespincstructureconjugatetot0.Therefore,uptoreplacingt0withoneoftheti’swemayassumethat

c1(t0),[Σ] =| c1(t0),[Σ] |=max{ c1(ti),[Σ] |i=1,...,k}.(2.1)

bethesmooth4–manifoldobtainedbyblowingupLetΣ·Σ=n,andletW

WatndistinctpointsofW\Σ.Chooseexceptionalclasses

suchthat andlet t0denotetheuniquespincstructureonWt0|W=tand

c1( t0),ei =1foreveryi=1,...,n.

W beasmooth,orientedsurfaceobtainedbypipingΣtothenLetΣ

exceptionalspheres,sothat

n =[Σ]+ei.[Σ]

i=1 )e1,...,en∈H2(W

Let beaproperlyembeddedarc(disjointfromYandΣ awayfromitsLetγ W DenotebyW 1aclosedregularneighborhoodendpoints)connectingYtoΣ. andletW 2betheclosureofW \W 1.oftheunionY∪γ∪Σ, c )| ,i=1,2.S= t∈Spin(Wt|W i=t0|Wi

W2 W1Bythecompositionlaw[26,Theorem3.4]wehave FW FW=FW. 2, 1, , t0|t0|t

t∈S(2.2)

Wearegoingtoshowthatthesumattherighthandsideof(2.2)admitsatmostonenontrivialterm.Infact,weshallprovethat

t∈SandFW=0 , t= t= t0.

Geometry&Topology,Volume8(2004)

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

932PaoloLiscaandAndr´asIStipsiczSince )admitsafreeandtransitiveactionofH2(W ;Z).Hence,RecallthatSpinc(W ;Z)suchthatthereisanelementL∈H2(W t t0=L.

wehave,inparticular,L|Y=0.ThereforeListheimageofanelement ,Y;Z)undertherestrictionmapH2(W ,Y;Z)→H2(W ;Z).OurA∈H2(W

planistoshowthat t= t0byprovingthatA=0.Since

,Y;Z)=0,H1(W,Y;Z)～=H1(W

thereforetoshowA=0itisenoughtoshow2A=0,and2Aisdetermined ,Y;Z).Butsinceb1(Y)=0,itsu cesbyitsvaluesontheelementsofH2(W

toshowthat2Aevaluatestriviallyontheimageofthemap

;Z) →H2(W ,Y;Z).i :H2(W

W 1,if = c1( ,ie,Ontheotherhand,sinceΣt∈Sthen c1( t),[Σ]t0),[Σ]

n c1( t),ei = c1(t0),[Σ] +n.(2.3) c1( t|W),[Σ] +

i=1

W,t ,t|W =t0|Wiii=1,2,theuniversalcoe cienttheoremimpliesthat ,Y;Z),Z), ,Y;Z)～H2(W=Hom(H2(W )thenMoreover,bytheblow–upformula[26,Theorem3.7]if t∈Spinc(WFW,t|=0 F =0= | c1( t),ei |=1,i=1,...,n.W

Thus, t= t0,andtheright–handsideofEquation(2.2)reducestoFW. , t0

1isacobordismfromYtoY#S1×Σ, andsinceNowobservethatW

= c1(t0),[Σ] +n>2g(Σ) 2, c1( t0),[Σ]

S×Σ c1( t|W),[Σ] = c1(t0),[Σ] and c1( t),ei = c1( t0),ei =1,i=1,...,n. ;Z)wehaveItfollowsthatc1( t)=c1( t0).Therefore,foreveryα∈H2(W 2A,i (α) = 2L,α = c1( t) c1( t0),α =0.Therefore,ifFW=0,byEquations(2.1)and(2.3)wehave , t

istrivial.ButthisgroupisthedomainofthemapFW 2, t0|bytheadjunctioninequality[25,Theorem7.1]thegroup (Y#S1×Σ , HFt0|1 )

tion(2.2)impliesthatFW=0andthereforeFW,t0=0,whichgivesthe , t0desiredcontradiction. W2.Thus,Equa-

Geometry&Topology,Volume8(2004)

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

Ozsv´ath–Szab´oinvariantsandtightcontactthree–manifolds,I933ContactOzsv´ath–Szab´oinvariants

In[28]Ozsv´athandSzab´ode nedaninvariant

( Y,sξ)/ ±1 c(Y,ξ)∈HF

foracontact3–manifold(Y,ξ),wheresξdenotesthespincstructureinducedbythecontactstructureξ.SinceinthispaperweareusingthishomologytheorywithZ/2Zcoe cients,theabovesignambiguityforc(Y,ξ)doesnotoccur.Itisprovedin[28]thatif(Y,ξ)isovertwistedthenc(Y,ξ)=0,andif(Y,ξ)isStein llablethenc(Y,ξ)=0.Inparticular,c(S3,ξst)=0.Wearegoingtousethepropertiesofc(Y,ξ)describedinthefollowingtheoremandcorollary.

Theorem2.2([21],Theorem2.3)Supposethat(Y′,ξ′)isobtainedfrom(Y,ξ)byacontact(+1)–surgery.Let Xbethecobordisminducedbythesurgerywithreversedorientation.De ne F X:=F X,t.

t∈Spinc( X)

Then,

F X(c(Y,ξ))=c(Y′,ξ′).

Inparticular,ifc(Y′,ξ′)=0then(Y,ξ)istight.

Corollary2.3([21],Corollary2.4)Ifc(Y1,ξ1)=0and(Y2,ξ2)isobtainedfrom(Y1,ξ1)byLegendriansurgeryalongaLegendrianknot,thenc(Y2,ξ2)=0.Inparticular,(Y2,ξ2)istight.

Thesurgeryexacttriangle

HerewedescribewhatisusuallycalledthesurgeryexacttrianglefortheOzsv´ath–Szab´ohomologies.

LetYbeaclosed,oriented3–manifoldandletK Ybeaframedknotwithframingf.LetY(K)denotethe3–manifoldgivenbysurgeryalongK Ywithrespecttotheframingf.Thesurgerycanbeviewedatthe4–manifoldlevelasa4–dimensional2–handleaddition.TheresultingcobordismXinducesahomomorphism (Y)→HF (Y(K))FX:=FX,t:HF

t∈Spinc(X)

Geometry&Topology,Volume8(2004)

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

934PaoloLiscaandAndr´asIStipsiczobtainedbysummingoverallspincstructuresonX.Similarly,thereisacobordismUde nedbyaddinga2–handletoY(K)alonganormalcircleNtoKwithframing 1withrespecttoanormaldisktoK.TheboundarycomponentsofUareY(K)andthe3–manifoldY′(K)obtainedfromYbyasurgeryalongKwithframingf+1.Asbefore,Uinducesahomomorphism

(Y(K))→HF (Y′(K)).FU:HF

kerFU=ImFX.(2.4)Itisprovedin[25,Theorem9.16]2that

TheaboveconstructioncanberepeatedstartingwithY(K)andN Y(K)equippedwiththeframingspeci edabove:wegetU(playingtherolepre-viouslyplayedbyX)andanewcobordismVstartingfromY′(K),givenbyattachinga4–dimensional2–handlealonganormalcircleCtoNwithframing 1withrespecttoanormaldisk.ItiseasytocheckthatthislastoperationyieldsYatthe3–manifoldlevel.Again,wehavekerFV=ImFU.Moreover,wecanapplytheconstructiononceagain,anddenotebyWthecobordismobtainedbyattachinga2–handlealonganormalcircleDtoCwithfram-ing 1.Infact,Wisorientation–preservingdi eomorphictoX.ThisfactisexplainedinFigure3,wherethe rstpicturerepresentsWandthelastpicturerepresentsX.Inthe gure,theframeddottedcircleistheattachingcircleofthe2–handle.The rstdi eomorphisminFigure3isobtainedby“blowingdown”theframedknotC.Inotherwords,the rsttwopicturesrepresent2–handlesattachedtodi eomorphic3–manifolds,andshowthatthecorrespondingattachingmapscommutewiththegivendi eomorphism.Theseconddi eomorphismisobtainedbyahandleslide,andthethirddi eomor-phismbyerasingacancellingpair.ItfollowsimmediatelyfromEquation(2.4)thatthehomomorphismsFX,FUandFV tintothesurgeryexacttriangle:

(2.5)

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

Ozsv´ath–Szab´oinvariantsandtightcontactthree–manifolds,I

935

～=～=f～=fKK

Figure3:Thedi eomorphismbetweenWandX

RemarkGivenanexacttriangleofvectorspacesandhomomorphisms

wehave

dimVi≤dimVj+dimVk(2.6)

for{i,j,k}={1,2,3}.Moreover,equalityholdsin(2.6)ifandonlyifFi=0.3TheproofofTheorem1.1

LetLbeaLegendrianknotsmoothlyisotopictoKwith

t:=tb(L)=2gs(K) 1.

Letr∈Q\{t}andr′=r t.Then,anycontactr′–surgeryalongLyieldsa

3(K).contactstructureonSr

Ifr<t=2gs(K) 1thenr′<0.Sinceanycontactr′–surgeryalongLcanberealizedbyLegendriansurgery,theresultingcontactstructureisStein llableandhencetight[10].Therefore,toproveTheorem1.1itsu cestoshowthatanycontactr′–surgeryalongLwithr′>0yieldsacontactstructureon3(K)withnon–zerocontactOzsv´Srath–Szab´oinvariant.

Let(Yk,ξk),withkanypositiveinteger,denotetheresultofcontact1

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

936PaoloLiscaandAndr´asIStipsiczitsu cestoprovethatthecontactinvariantsof(Yk,ξk)donotvanish.Weclaimthat,foreveryk≥1,

c(Yk,ξk)=0.(3.1)

Wearegoingtoprovetheclaimbyinductiononk.Tostarttheinduction,weexaminethecasek=1 rst.

3(K),andlet XbethecobordisminducedbyObservethatY1(L)=S2gscontact(+1)–surgeryalongLwithreversedorientation.Thenitiseasyto

checkthat,accordingtothediscussionpreceding

(2.5),

the

homomorphism

F

X tsintoanexacttriangle

(3.2)

where

F Xk

FVkFUk

K))

Geometry&Topology,Volume8(2004)(3.5)

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

Ozsv´ath–Szab´oinvariantsandtightcontactthree–manifolds,I937where

Kk 1

2 2 2 2

2gs

KKk+1

k 2 2 2 1 1

S3 2gs+1(

2gs+1 2gs

Figure4:Thesurgeryexacttriangleinvolving Yk, Yk+1andS3 2gs+1(

K),wherethe

correspondingframedattachingcircleisshowninthelowerleftportionofFig-ure4.WecanthinkofS3 t(

Ktothe4–ballalong

Kwith

thecorediskofH

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

938PaoloLiscaandAndr´asIStipsiczinW.Whensuitablyoriented,FandF′intersecttransverselyintnegativepointsp1,...,pt∈F′.Considertgenericpusho sS1,...,Stoftheembedded2–sphereS Wcorrespondingtothek–framedunknotofthelowerleftpor-tionofFigure4,orientedsothatSi·F=+1fori=1,...,t.Each2–sphereSiintersectsFtransverselyinauniquepointqi.Considerdisjoint,smootlyembeddedarcsγ1,...,γt Fsuchthatγijoinspitoqiforeachi=1,...,t.Letν(F)beasmalltubularneighborhoodofthesurfaceF.Wecanviewitsboundary ν(F)asasmoothS1bundle

π: ν(F)→F,

sothateachofthesetsF′∩ ν(F)and∪ti=1Si∩ ν(F)consistsofexactlyt bersofπ.Theimmersedsurface

Σ

iscontainedinthe =F′\ν(F)∪ti=1π 1(γi)∪ti=1Si\ν(F) WcomplementofF.ThesingularitiesofΣ

intersectionsamongS1,...,StandF′.Resolvingthosesingularities comefromtheonegetsasmoothlyembeddedsurfacewhichcanbeisotopedtoasurfaceΣ Vk.Moreover,asimplecomputationusingthefactthatg(F′)=gs(K)=1

2k+t+1

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

Ozsv´ath–Szab´oinvariantsandtightcontactthree–manifolds,I939Arationalhomology3–sphereYiscalledanL–spaceif

(Y)=|H1(Y;Z)|.dimZ/2ZHF

Noticethataccordingto

(4.1)lens

spacesareL–spaces.

3(K)Proposition4.1LetK S3beaknotsuchthatgs(K)>0andSn3(K)isanL–spaceforeveryisanL–spaceforsomeintegern>0.Then,Srrationalnumberr≥2gs(K) 1.

3(K)isanL–spaceforeveryrationalnumberr≥n.ProofThe3–manifoldSrInfact,itfollowsfrom[29,Proposition2.1],that

aS3

b(K)L–space.(4.4)

Supposer=p

3(K)isanL–space.(K)onededucesthatSrq

Thestatementfollowsimmediatelyifn<2gs(K) 1.Ifn≥2gs(K) 1,it

3isenoughtoshowthatS2gs(K) 1(K)isanL–space.Wedothisbybackwards

inductiononn.Forn=2gs(K) 1thestatementtriviallyholds.Ifn>2gs(K) 1,considerthesurgeryexacttrianglegivenbyS3andK S3withframingn 1:

(4.5)

SincethecobordismXcontainsasmoothlyembeddedsurfaceΣofgenusg(Σ)=gs(K)>0and

Σ·Σ=n 1>2gs(K) 2,

byProposition2.1wehaveFX=0.Thisimpliesthattheexacttrianglesplits,therefore (S3(K))～ (S3(K))⊕Z/2Z.HF=HFnn 1

3(K)isanL–spacethensoisS3(K)oncen>2g(K) 1,provingHence,ifSnsn 1theinductivestep.

Thefollowingtheoremgeneralizesaresultofthe rstauthor[18]:RecallthatTp,qdenotesthepositivetorusknotoftype(p,q).

Geometry&Topology,Volume8(2004)

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

940PaoloLiscaandAndr´asIStipsiczTheorem4.2Foreachrationalnumberr∈[2n 1,4n)∩Q,the3–manifold

S3r(T2n+1,2)

carriesno llablecontactstructures.

ProofFigure5describesa6–stepsequenceof3–dimensionalKirbymoveswhichshowthattheoriented3–manifoldS3r(T2n+1,2)istheboundaryofthe4–dimensionalplumbingXdescribedbythelastpicture.The rststepof

the

4n

2n+14n 24n

r 4n 2

nX=

Figure5:PresentationofS3r(T2n+1,2)asboundaryofaplumbing

sequenceisobtainedbynblowups.Thesecondstepbyn 1handleslidesandthethirdonebytwoblowupsplusaconversionfromintegertorationalsurgery.Thefourthstepisgivenbyahandleslide,the fthonebythreeRolfsentwistsandthesixthonebyaconversionfromrationaltointegersurgery.Observethat

1<r 4n 2

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

Ozsv´ath–Szab´oinvariantsandtightcontactthree–manifolds,I941becauser<4n.Thecoe cientsa1,...,akaregivenby

r 4n 2

a1 1

ak

Byusing

[17,Theorem5.2],itiseasytocheckthatthe4–dimensionalplumbingXispositivede nite.Moreover,theintersectionlatticeoftheplumbingwithreversedorientation XcontainstheintersectionlatticeΛa1,ndescribedinFigure6.

1,a1,...,ak≥2.

Figure6:TheintersectionlatticeΛa1,n

By[31,Theorem1.4],everysymplectic lling(W,ω)ofacontact3–manifold3(Y,ξ)suchthatYisanL–spacesatis esb+2(W)=0.SinceS4n+1(T2n+1,2)isalensspace[23]and,by[16],2gs(T2n+1,2) 1=2n 1,Proposition4.1

3(TimpliesthatSr2n+1,2)isanL–spaceforeveryr≥2n 1.Therefore,every

3(Tsymplectic llingofacontact3–manifoldoftheform(Sr2n+1,2),ξ)with+r≥2n 1satis esb2(W)=0.

3(TIfr∈[2n 1,4n),sinceY=Sr2n+1,2)isarationalhomologyspherewecan

buildanegativede niteclosed4–manifold

Z=W∪Y( X)

which,accordingtoDonaldson’scelebratedtheorem[7,8],musthaveintersec-tionformQZdiagonalizableoverZ.SincetheintersectionformQ XembedsinQZitfollowsthatΛa1,nmustembedinQZaswell.ButweclaimthatΛa1,ndoesnotadmitanisometricembeddinginthediagonallatticeDm=⊕m 1 .Thiscontradictionforbidstheexistenceofthesymplectic llingW.

Toprovetheclaim,weargueasin[19,Lemma3.2].Supposethereisanisometricembedding ofΛa1,nintoDm.Lete1,...,ekbegeneratorsofDmwithself–intersection 1.Itiseasytocheckthat,uptocomposing withanautomorphismofDm,thefourgeneratorsofΛa1,ncorrespondingtotheverticesofweight( 2)aresenttoe1 e2,e2 e3,e3 e4ande3+e4.UptocomposingGeometry&Topology,Volume8(2004)

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

942PaoloLiscaandAndr´asIStipsicz withtheautomorphismofDmwhichsendse4to e4and xestheremainingones,theimagevofoneofthetworemaininggeneratorsofΛa1,nsatis es

v·(e3 e4)=0,v·(e3+e4)=1,

whichisimpossiblebecause(e3+e4) (e3 e4)=2e4.

RemarkThestatementofTheorem4.2isoptimal,inthesensethatifr∈

[2n 1,4n),thenthe3–manifold

Yn,r:=S3r(T2n+1,2)

supports llablecontactstructures.Ifr<2n 1then,asobservedintheproofofTheorem1.1,Yn,rcarriesStein llablecontactstructures.Thesameholdsforr≥4n.Infact,examplesofStein llablecontactstructuresonYn,raregivenbythecontactsurgerypictureofFigure7(hereweareusingournotationaswellasthenotationof

[14]).

r 4n

Figure7:Stein llablecontactstructuresonYn,rwithr≥4n

ProofofTheorem1.3Letm∈N,andletp1,...,pm∈Nbeconsecutiveoddprimeswitheitherp1=3orp1=5,wherethechoiceismadesothat

p1···pm=4k+3

forsomek∈N.Nowletα=2k,andconsiderthecontactstructuresobtainedviathecontactsurgeriesofFigure8.

Theunderlying3–manifoldis

Nα:=S32+1

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact struct

Ozsv´ath–Szab´oinvariantsandtightcontactthree–manifolds,I

943

Figure8:Tight,not llablecontactstructuresonNα

ξi(α),i=0,...,α 1,whereidenotesthenumberofrightzig–zagsaddedbythestabilizations.After xingasuitableorientationfortheknots,thisimpliesthat

c1(ξi(α))=(2i (α 1))PD([µ]).

(Forcomputationsofhomotopicdataofcontactstructuresde nedbysurgerydiagramssee[6].)Noticethatthecontactstructuresξi(α)aretightbecause,since α<0,theyareobtainedbyLegendriansurgeriesonthecontactstruc-tureofFigure1,whichwasshowntohavenon–zerocontactOzsv´ath–Szab´oinvariantintheproofofTheorem1.1.Moreover,since2+1

2(p1···p j···pm+α 1).

Then,

2i(j) (α 1)=p1···p j···pm=

1

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