Relative Index Theory, Determinants and Torsion for Open Manifolds This page intentionally left blank This page intentionally left blankRelative Index Theory, Determinants and Torsion for Open Manifolds Jiirgen Eichhorn Universitat Greifswald,Germany , ~World Scientific NEWJERSEYLONDONSINGAPOREBEIJINGSHANGHAIHONGKONGTAIPEICHENNAI Published by World Scientific Publishing Co.Pte.Ltd. S Toh Tuck Link,Singapore 596224 USAoffice:27Warren Street,Suite 401-402,Hackensack,NJ07601 UK office:57 Shelton Street,Covent Garden, London WC2H 9HE British LibraryCataloguing-in-Publication Data A catalogue record forthisbook isavailable fromtheBritish Library. RELATIVEINDEXTHEORY,DETERMINANTSANDTORSIONFOR OPENMANIFOLDS Copyright 2009byWorld Scientific Publishing Co.Pte.Ltd. 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ISBN-13978-981-277-144-5 ISBN-IO981-277-144-1 Printed inSingapore by 8&JOEnterprise Contents Introduction.....................................................vii IAbsoluteinvariantsforopenmanifolds andbundles........................................................1 1 Absolutecharacteristic numbers...............................4 2 Indextheoremsforopen manifolds...........................46 IINon-linearSobolevstructures.........................62 1Cliffordbundles,generalizedDiracoperators andassociatedSobolev spaces...................................63 2Uniformstructuresof metricspaces..........................89 3Completedmanifoldsof maps...............................121 4Uniformstructuresof manifoldsandCliffordbundles..124 5The classificationproblem,new(co-)homologiesand relativecharacteristicnumbers................................141 IIIThe heatkernelof generalizedDirac operators.........................................................169 1Invariancepropertiesof the spectrumandthe heatkernel.........................................................169 2 Duhamel'sprinciple,scatteringtheoryandtraceclass conditions..........................................................180 IVTraceclassproperties...................................192 1 Variation of the Cliffordconnection........................192 2Variation of theCliffordstructure..........................203 3Additional topologicalperturbations.......................223 VRelativeindextheory....................................239 1Relativeindextheorems,the spectral shiftfunction andthe scatteringindex........................................239 v viRelativeIndexTheory,DeterminantsandTorsion VI Relative(-functions, 1]-functions, determinantsand torsion..................................252 1Pairsof asymptoticexpansions.. . . . . . . . . . . . . . . . . . . . . . . . . . . .252 2Relative(-functions...........................................256 3RelativedeterminantsandQFT............................267 4Relativeanalytictorsion......................................269 5Relative1]-invariants..........................................272 6Examplesandapplications...................................277 VIIScattering theory formanifoldswith injectivity radiuszero.......................................299 1 Uniformstructures definedbydecayfunctions..........299 2The injectivityradiusand weightedSobolev spaces...............................................................307 3Mappingpropertiesof e-t6............................317 4Proof of thetraceclassproperty............................322 References.......................................................331 Listof notations...............................................338 Index..............................................................340 Introduction Itisoneofthemaingoalsofmodernmathematicstodescribe amathematical subject,situation,resultbyasequenceof hon-estnumbers.Wereminde.g.intopology/globalanalysisof therankof(co-) homologygroups,homotopygroups,K -( co-) homology,characteristicnumbers,topologicalandanalytical index,Novikov-Shubin-invariants,analyticaltorsion,theeta invariant,allthesenumbersdefinedinthecompactcase.In-cludingbordismandWallgroups,whicharealsoof finiterank, onehasanappropriateapproachtotheclassificationproblem forcompactmanifolds. For openmanifolds,allthesenumbersabovearenotdefinedin general.The ranks of the group of algebraic topology can be in-finite,the integrals to definecharacteristic numbers can diverge, ellipticoperators mustnolongerbeFredholm,the spectrum of self-adjointoperators mustnotbe purelydiscrete,etc. WewillproveatthebeginningofchapterIthatthereareno non-trivialnumbervaluedinvariantswhicharedefinedforall oriented (including open) manifolds and which behave additively underconnectedsum.Moreover,weprove,thatforanyn2':2 there are uncountably many homotopy types of open manifolds. Henceaclassification,essentiallyrelyingonnumbervaluedin-variants,probablyshouldnotexist. Themainidea of ourapproach - broughttoapoint - isasfol-lows.Weconsiderpairs(P, PI),wherePe.g.stands foratriple (manifold,bundle,differentialoperator),and wedefinerelative invariantsi(P, PI),wherepIrunsthroughasocalledgeneral-izedcomponent gen comp(P)whichconsistsof allpI with finite distancefromP.Thedistancecomesfromametrizableuni-formstructure.Todefinethe correspondingmetrizableuniform structure isthecontentof chapterIIandisoneof thecolumns of ourapproach.Thentheclassificationof thePsamountsto theclassificationofthegeneralizedcomponentsandtheclassi-ficationof thePsinsidegen comp(P). Thistreatiseisorganizedasfollows.InchapterIsection1we vii viiiRelativeIndexTheory,DeterminantsandTorsion presentclassesof open manifolds forwhichthe classicalcharac-teristic numbers via Chern-Wei! construction are defined,study theirinvarianceandmeaning.Hereweincludetheimportant contributionsof Cheeger/Gromov from[15],[16].Wecallthem absoluteinvariantssincetheyaredefinedforsingleobjectsand notforpairswithonecomponentfixed.Sectiontwoisdevoted tosomeindextheoremsforcertainclassesofopenmanifolds and ellipticdifferentialoperators.It isvisiblethat alltheseare very special classes and that the wish foracorresponding theory forall open manifoldsrequiresanow,another approach.For us this isthe relative index theory,applied to pairs.Then the main questionis,whatisan admissiblepair of Riemannian manifolds or Cliffordbundles with associated generalizedDirac operators? Inalocalclassicallanguagethiswouldmean,whatarethead-missibleperturbationsofthecoefficientsinquestions- andof thedomains? Weanswer these questionsin avery general and convenientlan-guage,the language of metrizable uniform structures.We define forapair of objects under considerationalocaldistance,define bymeansofthislocaldistanceaneighbourhoodbasisofthe diagonalandfinallyametrizableuniformstruture.Inallour cases,the distance containsacertain Sobolev distance.For this reason,wegivein chapter II 1 abrief outline of the needed facts and referto[27]formore.II2 isdevotedto averybrief outline on uniformstructures of proper metric spaces.Sincediffeomor-phisms enter into the definition of our local distances,wecollect inII3somedefinitionsandfactson completeddiffeomorphism groupsforopenmanifolds.InII4,weintroducethoseuniform structuresof manifoldsandCliffordbundlesandtheirgeneral-ized components,whichare fundamentalinthe central chapters IV,V,VI.ThefinalsectionII5containthefirststepsofour approachtotheclassificationproblemforopenmanifolds.We introducebordismforopenmanifolds,severalbordismgroups, reducetheircalculationto that forgeneralizedcomponents,in-troduce relativecharacteristic numbers and establish generators forbordism groupsof manifoldswith non-expandingends. Introductionix ChapterIIIistheimmediatepreparationforthechaptersIV, V,VI.Section III1 isdevotedto the invarianceof the essential spectrum under perturbation inside a generalized component,to heatkernelestimates,andweintroduceinIII2standardfacts of scatteringtheoryaswaveoperators,theircompletenessand the spectral shiftfunctionof Birman/Krein/Yafaev. AsitisclearfromourapproachandthecriteriainsectionIII 2,theabsolutecentralquestionisthetraceclasspropertyof 2-,2 e-tD - e-tD ,EEgencomp(E).Moreover,theexpression - 2 tr( e-tD2 - e-tD')enters into the integral forthe relativeindex, relativezetaandetafunctions.Weestablishthistraceclass property stepbystepinsectionsIV1 - IV3,admitting larger andlargerperturbations.Theproof of thetraceclassproperty istheheartofthetreatise,reallyrathercomplicatedandthe technicalbasisforwhat follows. InsectionVI, weproveseveralrelativeindextheoremsandin V2propertiesofthescatteringindex.Weremarkthatthere areotherwellknownrelativeindextheoremse.g.in[8],[9]for exponentiallydecreasingperturbations.Boththesetheorems areveryspecialcasesofourmoregeneralresult.Inchapter VI,weapplyourachievementsuntilnowto definerelativezeta andeta functions,relativeanalytictorsion,relativeeta invari-antsandrelativedeterminants,whichareinparticularimpor-tant inQFT.Section VI6presentsnumerousexamples,special cases and applications of these notions.In particular, wepresent classesof open manifoldswhichsatisfythe geometricand spec-tralassumptionswhichweassumedintheprecedingsections. Aparticular simplecasearemanifolds with cylindrical endsfor which wedescribe the scattering theory.Here weessentially rely on [3],[49].An interpretation of our relative determinants in the caseof cylindricalendsisgivenbytheorem6.17inchapterVI. Until now,wealways assumed bounded geometry,i.e.injectivity radius> 0 and bounded curvature together with acertain num-berofderivatives.Clearly,thisrestrictstheclassesofmetrics underconsideration.In[54],W.MullerandG.Salomonsenes-tablished a scattering theory without the assumption injectivity xRelativeIndexTheory,DeterminantsandTorsion radius>0buttheyadmitotherperturbations.Thedifference 9 - h together with certain derivatives must be of so-called mod-eratedecay.Wereformulatetheirappraochinourlanguageof uniform structures and generalized components and extend their resultsto arbitrary vectorbundles withboundedcurvature. PartofthisbookhasbeenwrittenattheWuhanUniversity (China).TheauthorthankstheDFGandChineseNSFfor support.The otherpart hasbeen writtenat theMPIinBonn. The author isparticularly grateful to Gesina Wandt forher per-manentengagementandpatienceinprinting themanuscript. Greifswald,December2008 IAbsolute invariants foropen manifoldsandbundles For closed manifolds,there is ahighly elaborated theory of num-ber valuedinvariants.Examplesarethecharacteristicnumbers likeStiefel-Whitney,ChernandPontrjaginnumbers,theEu-lernumber,thedimensionofrational(co-) homologyandho-motopygroups,thesignature.Moreover,wehaveinvariants coming from surgery theory etc.Taking into account aRieman-nianmetric,weobtainglobalspectralinvariantslikeanalytic torsion,theeta invariant.On generalopenmanifolds,moreor lessallofthisfails.Characteristicnumbersarenotdefined, (co-) homology groupscanhaveinfiniterank etc. Wehavethefollowingsimple Proposition0.1Let 9J1nbethe set of all smooth oriented man-ifoldsand Vavector spaceor anabeliangroup.Theredoesnot existanontrivial mapc : 9J1----tVsuchthat 1)Mn~Minbyanorientation preservingdiffeomorphismim-pliesc( M)=c( M')and 2)c(M#M') = c(M) + c(M'). Proof.AssumeatfirstMn'1- I;n,fixtwopointsatMn, thenMoo=Ml #M2#' .. ,Mi=(M, i)~Mhasawellde-finedmeaning.WecanwriteMoo=Ml #Moo,2withMoo,2= M2#M3# . ..andgetc(Moo)=c(M)+ C(Moo,2)=c(M)+ c(Moo)hencec(M)=o. AssumeMn=I;n,andordI;n=k> 1 whichyields c(I;n# . .. #I;n)=k. c(I;n)=c(sn),c(I;n)=~ c ( s n ) , c(I;n)=c(I;n#sn)=(1 + ~ )c(sn), c(sn)=2c(sn),c(sn)=0,(I;n)=O. 1 o 2RelativeIndexTheory,DeterminantsandTorsion The onlyrealinvariant,definedforallconnectedmanifoldsMn knowntotheauthoristhedimensionn.If onecharacterizes orientability/nonorientabilityby1,thentherearetwosuch invariants.Thatisall.Lookingattheclassificationtheory, weseeadeepdistinctionbetweenthecaseofclosedoropen manifolds,respectively. Denote by 9J1n([cl])the set of all diffeomorphism classes of closed Thenwehave Proposition0.2#9J1n([cl])=No. Proof.According to Cheeger,there are only finitelymany dif-feomorphismtypesfor(Mn, g)withdiam(Mn, g) n.The determi-nant isan example forP.If wisnot smooth then P(O)isclosed inthedistributionalsense.LetoAO)bethe2r-homogeneous part(inthe senseof forms)of det (1+Oij). Lemma1.4Eachinvariantpolynomialisapolynomialzn al,'",aN' AbsoluteInvariants 9 Lemma1.5Ifw Eb,lcP,l(Bo,f,p)andr ~1then (1.5) Proof.ForthepointwisenormIlxthereholdsIDI;= ~~~IDij,kll;,whereDij,kl=Dij(ek,e/)andel,' .. enisanor-i,jk00voi M Integrationbyparts yields -f d O(k)ann-universalconnec-tionrGonthen-universalbundlePn,G-->Bn,G(cf.[20],p. 570). According to proposition 1.49,werefinethe bundle conceptand considerinsteadof abundlePpairs(P,fp),fp: P--> Pn,Ga AbsoluteInvariants41 C1-classifyingbundlemap. (P, fp) is called a (p,J)-bundle if f;I'oEC1Cp(J,p)= {waC1-0andcompactKCMsuchthat o Themain task nowistoestablishameaningfulindextheorem. Thishasbeen performedin[2]. Theorem2.3Let(Mn , g)beopen,complete,oriented, (E,h,r)=(E+EElE-,h)---+(Mn,g)aZ2-gradedHermitean vectorbundleandD:c;:(E)---+c;:(E)firstorderelliptic, essentially self-adjoint,compatiblewiththeZ2 -grading(i. e.su-persymmetric),Dr + r D=o.Let KCMbeacompactsubset suchthat2.2a)forKissatisfied,andletfEcOO(M,lR)be suchthatf=0onU (K)andf=1outsideacompactsubset. Thenthereexistsa volumedensity wand acontribution Iwsuch that indaD+= J(w(l- f(x))dvolx(g)+ Iw,(2.6) M wherewhasanexpressionlocallydependingonDandIwde-pendsonDandfrestrictedto0,=M\K.0 Until now the differential form w dvolx(g)ismystery.One would like to express it by well known canonical terms coming e.g.from theAtiyah-Singerindexformch(J(D+)U T(M),whereT(M) denotestheTodd genusofM.Infactthis can be done. Index Theorem2.4Let (Mn, g)beopen,oriented,complete, (E, h, r)---+(Mn, g)a Z2-graded Hermiteanvector bundle,D: 50RelativeIndexTheory,DeterminantsandTorsion C':(E)------tC':(E)a firstorderellipticessentiallyself-adjoint supersymmetricdifferentialoperator,DT + T D=0,whichshall beassumedtobeFredholm.LetKCMcompactsuchthat2.2 d)issatisfied.Then indaD+=Jch(J(D+)U T(M) + In,(2.7) K where chdD+) uT(M) istheAtiyah-Singer index formand In isaboundedcontributiondependingonlyonDin,n =M\K. o Remarks2.5a)As wealready mentioned,Zrgraded Clifford bundles andassociatedgeneralizedDirac operators Dsuchthat inD2=6,E+ R,R~c id,c>0,outsidesomecompact KcM,yieldexamplesfortheorem2.3.Aspecialcaseis the Dirac operator overaRiemannian spin manifold with scalar curvature~c>0outsideKeM. b)Much more general perturbations than compact ones willbe consideredin section V1.0 The other case of a very special class of open manifolds are cover-ings(M,g)of aclosedmanifold(Mn,g).LetE,F ------t(Mn,g) beHermiteanvectorbundlesovertheclosedmanifold(Mn, g). D:COO (E)------tCOO(F)beanellipticoperator,(M, g)------t (M, g)aRiemanniancovering,D : C':(E)------tC':(F) thecor-respondingliftingandf=Deck(Mn, g)------t(Mn, g).The actionsof fandD commute.If P: L2 (M, E)------t1t istheor-thogonal projection ontoaclosedsubspace1t cL2(M, E)then one definesthef -dimension dimr 1t of 1t as dimr 1t := trrP, wheretrr denotesthe vonNeumann traceand trrP can be any realnumber~0or=00. If onetakes1t=1t(D)=ker DCL2(E),1t*=1t(D*) ker(D*)CL2(1')thenonedefinesthef-index indrD as indrD := dimr 1t(D) - dimr 1t(D*). AbsoluteInvariants51 Atiyahprovesin[4]thefollowingmain Theorem2.6Undertheassumptionsabovethereholds o ItwasthistheoremwhichwastheorignofthevonNeumann analysis as a fastly growing area in geometry,topology and anal-ysis.Moreover,the proof of theorem 2.3isstrongly modeledby that of 2.6.Another veryimportant special case which isrelated tothecaseaboveofcoveringsarelocallysymmetricspacesof finitevolume.There isa vast number of profound contributions, e.g.[7],[22],[48],[50],[51].Wedonotintendheretogivea completeoverviewforreasonsof space.Butwewillsketchthe main featuresandmainachievementsof theseapproaches. LetGbesemisimple,noncompact,withfinitecenter,KeG maximalcompact,X=G / Kasymmetricspaceofnoncom-pacttype,fcGdiscrete,torsionfreeandvol(f\G)..) d>' A isanintegraldecompositionof 1f.Then e ker d7r=Jker d7r),d>.. A (2.9) D Now wecome to the main part of our present discussions,the lo-cally symmetric case.IdentifyingL2(E)with(L2(f\G) VE)K, and takingintoconsiderationthedecompositions Rr =R ~EBR ~ ,L2(f\G)=L2,d(f\G) EBL2,c(f\G) of therightquasi-regularrepresentationRr of GonL2 (f\G), weobtainthedecomposition L2(E) L2,d(E) L2,c(E) similarly forF=f\F. L2,d(E) EBL2,c(E), (L2,d(f\G) VE)K, (L2,c(f\G) VE)K, Consider now the operators D=dRrand Dd=dRr:C':'(E)-----+ d C':'(F). Theorem2.11Undertheassumptionsabove(onG,K,f), ker D= ker Dd(2.10) and dimkerD< 00.(2.11) 54RelativeIndexTheory,DeterminantsandTorsion DenotebyG ~thesetofallequivalenceclassesofirreducible unitary representation 7fof Gwhose multiplicity mr(7f)inR ~is nonzero.InparticularL2,d(r\G) =Lmr(7f)H(7f). 1 f E G ~ Theorem2.12LetKeG bemaximalcompact,rEG dis-creteandtorsionfree,TE:K~VE,TF:K~VF unitary representations,E =GIK XK VE,F = GIK XK VF,E= r\E, F=r\F and D= dRracorresponding locallyinvariant elliptic differentialoperatoractingbetweenL2 (E)and L2 (F).Then indaD =dimkerD- dimkerD* iswelldefinedand (2.12) o Corollary2.13Let X=r\GIK bealocallysymmetricspace of negativecurvaturewithfinitevolumeandL2(E)=>'DD~ L2 (F)a locally symmetric elliptic differential operator then indD isdefinedanddependsonlyontheK -modulesK~ U(VE), U(VF)whichdefineE,F,E=r\E,F=r\F.0 Thevalueoftheformulaintheorem2.12isverylimitedsince ingeneralthemr(7f)arenotknown.Hencetherearisesthe task to findameaningfulexpressionforit.Thishas beendone withgreatsuccesse.g.in[22]and[51],[52]wheretheyes-sentiallyrestricttogeneralizedDiracoperators.Tobemore precise,wemustbrieflyrecallwhatisamanifoldwithcusps. Herewedenselyfollow[50].LetGbeasemis imp IeLiegroup withfinitecenter,KeG amaximalcompactsubgroup.Pa split rank one parabolic subgroup of Gwith splitcomponentA, p= U AM thecorrespondingLanglandsdecomposition,where Uisthe unipotentradicalof P,AaIR-splittorus of dimension AbsoluteInvariants55 oneandMcentralizesA.SetS=U Mandletfbeadiscrete uniformtorsionfreesubgroupofS.ThenY=f\ Y =f\ G I K iscalledacompletecuspofrankone.PutK M=Mn K, KMisamaximalcompactsubgroup_ ofM.If XM =MI Ky thereisacanonicaldiffeomorphism IR+xUxX M----+Y. Setfort2:0=([t,oo[ xUxXM)andcallyt=a cuspofrankone.Another,evenmoreexplicitdescriptionis givenasfollows.LetfM=Mn (Uf),Z=SIS n K.Then thereisacanonicalfibrationP: f\Z----+f M \XM withfibre fn U\ Uacompact nilmanifoldand acanonical diffeomorphism :[t,oo[xf\Zyt.Theinducedmetricon[t,00[Xf2\Z lookslocallyasds2 =dr2 + dx2 + e-brdui{x)+whereIbl=\dx2 istheinvariantmetriconXM inducedby restriction of theKillingform. NowacompleteRiemannian manifoldiscalledamanifoldwith cusps of rank one if Xhas a decomposition X=XOUX1 U. ,UXs suchthat Xoisacompactmanifold with boundary,fori, j2:1, i=1=jholdsXi n Xj=0 andeachXj,j2:1,isacuspof rank one.The firstgeneral statement forgeneralizedDirac operators on rank onecuspsmanifoldis Theorem2.14LetXbearankonecuspmanifold, (E, h, \7,.)----+(X, gx)aCliffordbundleand Ditscorrespond-ing generalized Diracoperator.Then Disessentially self-adjoint and dim(ker D)0 e-zHd isof traceclass.(2.14) D Aswementionedaftercorollary2.13,themaintask,mainob-jectiveconsistsinthecaseofaZ2-gradingtogetanexpres-sionforindaD.Forthesakeof simlicitywerestricttospaces 56RelativeIndexTheory,DeterminantsandTorsion x=Xo U Y1 asabovewith onecuspYi,YoU Y1 = Y=f\G/ K. Let(E=E+EBE-, h, V',.)(Y, g)beaZ2-gradedClif-fordbundlesuchthatE IYl=f\E,whereEarehomoge-neousvectorbundlesoverG / KandletD+:Coo (Y,Et) Coo(y, E-)thecorrespondinggeneralizedDiracoperator.We recallKM= MnK, XM = M/KM.D+inducesan ellipticdif-ferentialoperatorDt : Coo(IR+xfM\XM, Et) Coo(IR+x fM\XM, EM)' where E! are locally homogenous vector bundles over fM\XM.From this come a self-adjoint differential operator DM:Coo(fM\XM,Et)Coo(fM\XM,EM)andabundle isomorphism(3:Et EMsuchthatDt =(3(r! + DM). WesetDM=DM +m=dim U.>.IAI+ 2 dim u2.>.IAI,A the uniquesimplerootof thepair(P, A). W.Mullerthenestablishedin[50]thefollowinggeneralindex theoremforalocallysymmetricgradedDiracoperator. Theorem2.15Assumeker DM={O},let7](0)betheetain-variantof DMand WD+theindex formof D+.Then indaD+=JWD++ U+x (2.15) wheretheterm Uisessentially givenbythevalueof anL-series atzeroandanexpressioninthescatteringmatrixatzero.0 Finally,applicationofanelaboratedversionoftheorem2.15 allowstoprovethefamousHirzebruchconjectureforHilbert modular varieties.ThishasbeendonebyW.Mullerin[51]. ThereisanotherapproachtoFredholmnessbyGillesCarron, whichreliesonan inequalityquite similar to 2.2d). Let(E, h, V',.)(Mn, g)beaCliffordbundleoverthecom-pleteRiemannian manifold(Mn, g)and D: Coo(E)Coo(E) theassociatedgeneralizedDiracoperator.Discallednon-parabolicatinfinityif thereexistsacompactsetKcMsuch that foranyopen and relativecompactUcM\Kthere exists aconstantC(U)>0suchthat C(U)I 0, I(V'9)iR91Gi,0ik, 1(V'9)iREIGi,0 ik. Itisawellknownfactthatforanyopenmanifoldandgiven k, 0k00,thereexistsametric9satisfying(I)and (Bk(M, g)).Moreover,(1)impliescompletenessof g. Lemma1.1Assume(Mn,g)with(1)and(Bk)'ThenG;;o(E) isadensesubsetof wp,r(E)andHP,r(E)for0rk + 2. See[27],proposition1.6foraproof. D Lemma1.2Assume(Mn,g)with(1)and(Bk)'Thenthere existsacontinuousembedding Proof.Accordingto1.1,wearedoneif wecanprove for0rk+ 1andEG;;o(E).Performinduction.For r=0IIHP,O=1lwp,o.Assume1IHP,rG 1lwp,r.Then 1IHP,r+l(2thereholds a)Thefamily(Bc(Xi))iisauniformlylocallyfinitecoverof MandanupperboundNforthenumberof non-emptylocal intersectionsisgivenbyN= N (n, (2,c,k) . b)Forany i-I j,Bg(Xi) n Bg(xJ)=0.D 22 Lemma1.11Let (Mn , g)beopen,complete,Ric (g)~k.Then, forany 0 < r< Randany xEM vol(BR(x)):::;V O l t ~ R ;vol(Br(x)),(1.10) vokr wherevolk(t)denotesthevolumeofaballofradiustinthe completesimplyconnectedRiemannian n-manifoldof constant curvaturek.Inparticular,forany r> 0andany xEM (1.11) 70RelativeIndexTheory,DeterminantsandTorsion Corollary1.12ForanyxEM,0< r 0 s.t.Y EU8(x) , zEU8(y)implieszEUe:(X) , i.e.dx(x, y)0 suchthat Then,automatically, 124RelativeIndexTheory,DeterminantsandTorsion WedenoteforfixedKM, KNthesubsetcD,p,r(M, N)of these IbyN).Clearly,N)dependson thechoiceof KM,KN. Finally,weneedthisconstructionstillforRiemannianvector bundles.Let(Ei' hi, \7hi)----t(Mr,9i),i=1,2beRieman-nian vector bundles satisfying(1),(Bk(Mi, 9i)),(Bk(Ei, hi, \7i)), k2:r>+ 1.If weendowthetotalspacesEiwiththe Kaluza-Kleinmetric9E(X, Y)=h(XV, yV)+ 9M(-rr*X, 71"*Y) , Xv,yvverticalcomponents,thenE1,E2areagainmanifolds withboundedgeometry(cf.[30])andD,p,r(E1, E2)iswellde-fined.If werestrictto bundle maps(fE, 1M=71"0IE071"-1then weobtainasubset E2)cD,p,r(E1, E2)'Quiteanal-ogouslytoabovewedefine(E1'E2)c (E1'E2)ifthe bundlesareisomorphicandE2)iftheyareisomor-phic overM\KMandN\KN.Here werequire(3.3)and(3.5) both forIEand 1M'Weapply this notations in the next section. 4Uniformstructuresof manifolds andCliffordbundles WeintroduceinchaptersIV- VIrelativeindextheory,rela-tiveetaandzetafunctions,relativedeterminantsandrelative analytictorsion.Thewholeapproachreliesonthefollowing construction.WeendowthesetofisometryclassesofClifford bundles(of bounded geometry)with a metrizable uniform struc-ture,definegeneralizedcomponents gen comp( E)(= set of Clif-fordbundlesE'withfiniteSobolevdistancefromagivenE), associatethecorrespondinggeneralizedDiracoperatorsD,D' andmakeallconstructionsforthepairD,D',whereD'isrun-ningthroughgen comp( E).Thefirststepfordoingthisisthe introductionof the correspondinguniformstructure(s).Thisis thecontentof thissection.Theapplicationswillbeperformed in chaptersIV - VI. DenotebyfJItn(ml, I, Bk)thesetofisometryclassesof n-dimensional Riemannian manifolds(Mn, 9)satisfying the con-Non-linearSobolevStructures 125 ditions(1)and(Bk)'Wedefinedfor(Mf,gl), (M:;,g2)E mn(mf, I, Bk) and k>+ 1 the diffeomorphismsDP,r(M1, M2) and forappropriate compact submanifoldsKiCMi, M1\K19:: M2\K2themaps M2)== M2; Kl, K2)c np,r(M1, M2) asat the end of section 3.Recallbldfl= sup Idflx. x Theelementsfofnp,r(M1, M2)arenotsmooth.For kr>+ 2theyareC2.Hencef* 9isaC1 metric.This wouldcausesometroublesifwewouldconsiderinthesequel only classical derivatives whichwoulddisappear if wework with distributionalderivatives.Anotherwaytoworkwiththenon-smoothnessoff*gistoworkwithsmoothapproximationsof f.Wedecidetogothiswayanddefinecr,p,r(M1, M2)=U E np,r(M1, M2)lf Ecr+1(M1' M2) andbl\7idfl< 00, i= 1, ... , r}. Completingtheuniformstructurebelow,weendupwithfs Enp,r,i.e.therestrictionatthebeginningtoCkfsim-pliesattheendnofurtherrestriction.Werestrictinthese-queltok>r>!!:+ 2.Furtherweremarkthatthecon-- p ditionsc:::;blf*1:::;CandC1:::;blf;11:::;C1 areequivalent: blf;11b =C1followsfromblf*f;11=1andblf*1:::;Cand blf;11:::;C1(c, C)followsfromelementarymatrixcalculus.If f*istheinducedmapbetweentwofoldcovarianttensorsthen C2= c2 :::;blf*1:::;C2 = C2, similarlyC3:::;blf*-11 :::;C3.Under theseconditions,I\7i f* I :::;d,1:::;i:::;vimpliesI\7i f* I :::;d1, 1:::;i:::;v,andl\7if*-11:::;d2, 1:::;i:::;v,whered1, d2 arecon-tinuousfunctionsinc,C,d.Allthisfollowsfromf;1 f*=id*, f*-1 f*= id*and0 = \7id*= \7id*. Considernowpairs(Mf,g1),(M:;,g2)Emn(mf,I,Bk)with thisproperty:ThereexistcompactsubmanifoldsKfCMf, K'2CM:;andanfE M2, Kl, K2)'Forsuchpairs 126RelativeIndexTheory,DeterminantsandTorsion define d1j{diff,rel(M1 , 91)' (M2,92)):= inf { max{O,logbldfl} +max{O, log bldhl}+supdist(x, hfx) XEMl +supdist(y, hfy) +supIV'idfl+supIV'idhl yEM2"'EMl"'EM2 +IUIMl\Kl)*92- 91IMl\KlI9l,p,r If Ecr,p,r(M1,M2), hE cr,p,r(M1,M2) andforsomeKlCM1,K2CM2holds flMl\KlE,])p,r(M1 \KI, M2\K2) andhM2\K2=UIMl\Kl)-I},(4.1) if{ ... }f=0andinf{ ... }O,obtain the metrizable uniform structure Bk))andfinallythecompletion CLBN,n,p,r1XTt. L,diff,rel.vveseagam gen comp (E)gen f,rel (E) ={E'EBk)) E')< oo} which contains the arc componentand inherits aSobolev topol-f (IP,r ogyrom J.AL,diJj,rel. Asintheprecedingconsiderationsweobtainbyrequiringad-ditionallyhI=fEh2orhllEllMl\Kl=fE(h2IE2IM2\K)localdis-tances orand corresponding uniform spaces Bk)or Bk)respectively. Weobtain generalizedcomponents genf,F (E) (4.33) Non-linearSobolevStructures 141 and gen J,F,rel (E) (4.34) asbefore.Oneof ourmaintechnicalresultsinchapterIVwill bethatEandE'beinginthesamegeneralizedcomponent impliesthataftertransforminge-tD,2 intotheHilbertspace L2((M, E), g,h), e-tD2 _e-tD,2 and e-tD2 D_e-tD,2 D' are of trace classandtheirtracenormisuniformlyboundedoncompactt-Intervallslao, ad,ao>O.Forourlaterapplicationsthecom-ponents(4.33),(4.34)aremostimportant,excludedonecase, thecaseD2=.6. (g) ,D,2=\1(g').Inthiscasevariationof9 automatically inducesvariation of the fibremetricand wehave to consider(4.32)and gen(E). Perhaps,forthe reader the definitionsforthe gen comp(E)look veryinvolved.Werecall,roughly speaking,the main pointsare asfollows.Thedistancewhichdefinesgen comp ... (E)mea-sures stepbystepthedistancebetweenthemainingredientsof aCliffordbundle:thesmoothLipschitzdistancebetweenthe diffeomorphicpartsofthemanifoldsandthebundlesandthe Sobolevdistancebetweenthemanifoldmetrics,thefibremet-rics,the fibreconnectionsandtheCliffordmultiplications. Remark4.11Thegen comPL',diJ J,rez(- )-definitioncanbe extendedcanonicallytostructureswithboundary.D 5Theclassificationproblem,new (CO-) homologiesandrelative characteristicnumbers Aswealreadyindicated,weunderstandthistreatiseasacon-tributiontotheclassificationproblemforopenmanifolds.We provedinchapterIthatmeaningfulnumbervaluedinvariants forall open manifolddonot exist.The wayout fromthis situa-tionisto introduce relativenumber valuedinvariants or to give uptheclaimfornumbervaluedinvariantsandtoadmitgroup valuedinvariantsase.g.inclassicalalgebraictopology.Wego 142RelativeIndexTheory,DeterminantsandTorsion bothways.Theheartofthistreatisearenewnumbervalued relativeinvariantslikerelativedeterminants,relativeanalytic torsion,relativeeta invariants,relativeindices.This willbe the contentofchaptersIV- VI.Ourgeneralapproachconsistsin twosteps, 1.todecomposetheclassofmanifolds/bundlesunderconsid-erationintogeneralizedcomponentsandtotryto"count",to "classify"them, 2.to"count",to"classify"theelementsinsideageneralized component. ChaptersIV- VIareexclusivelydevotedtothesecondstep. Concerningthefirststep,wedevelopedin[34]somenew(co-) homologieswhichareinvariantsofthecorrespondinggeneral-izedcomponentandhencerepresentstepswithinthefirsttask above.Inthissection,wegiveabriefreviewofthese(co-) homologies.Inthesecondpart,wegiveanoutlineofbordism theory foropen manifoldsand corresponding relativecharacter-isticnumbers. Let Xand Ybe proper metric spaces.We call amap : X-+ Y coarse if it is 1.metrically proper,i.e.foreach bounded subset Bey the inverseimage-l(B)isboundedinX,and 2.uniformlyexpansive,i.e.forR>0thereisS>0such that d(Xl,X2)~Rimpliesd(Xl,X2)::;S. Acoarsemapiscalledroughif it isadditionallyuniformlymet-rically proper.Xand Yare calledcoarsely orroughlyequivalent ifthereexistcoarseorroughmaps:X-+Y,'It: Y-+X, respectively,suchthatthereexistconstantsD x,Dysatisfying d('ltx, x)~Dx,d('lty, y)~Dy . Proposition5.1Xand Yarecoarselyequivalentif andonly if theyareroughlyequivalent. Wereferto[33]forthe proof. o Non-linearSobolevStructures 143 The equivalence class of Xunder coarse equivalence is called the coarsetypeof X. LetXbeapropermetricspace.Thenwehavesequencesof inclusions coarsetype(X):JcompCH(X),(5.1) coarsetype(X):J comPCH(X):JarccompL,h,rel(X):J :JarccompL,h(X):J compL,top(X),(5.2) coarsetype(X):JcompL(X):JarccompL,h,rel(X):J :JarccompL,top,rel(X):J compL,top(X),(5.3) The arising task isto define forany sequence of inclusions invari-antsdependingonlyonthecomponentandbecomingsharper and sharper if wemovefromthe leftto the right.Start with the coarsetype whichhasbeenextensivelystudiedbyJ.Roe. GivenX=(X, d),xq+1becomesapropermetricspaceby d((xo,,,.,xq),(Yo,,,.,Yq))= max{d(xo,Yo),,,.,d(xq,Yq)}'Let =Cxq+1bethemultidiagonaland set ={y ER}. ThenJ.Roedefinesin[63]thecoarsecomplex(CX*(X), 8)= (CXq(X),8)qby cxq(X):={J: xq+1---t IRI fislocallybounded Borel functionandforeachR> 0is suppfnR)relativelycompactinXq+l}, q+l 8f(xo, ... ,Xq+1):=2)-1)if(xo, ... ,Xi,'",Xq+l)(5.4) i=O The coarsecohomologyHX*(X)of Xisthendefinedas HX*(X):= H*(CX*(X)). Theorem5.2H X*(X)isaninvariantof thecoarsetype,i.e. coarseequivalences:X---t Y,\II: Y---tXinduceisomor-phisms. 144RelativeIndexTheory,DeterminantsandTorsion Wereferto[63]foraproof. o Corollary5.3H X* (X)is an invariant for all components right fromthecoarsetype. Remark5.4It iswellknownthatwithoutthe supportcondi-tion suppfn Pen(b., R)relativelycompact(5.5) the complex GX*(X) wouldbe contractible.After fixingabase pointx EXthemapD: Gq~Gq-1, Df(xl, ... ,Xq):=f(X,X1,... ,xq) wouldbeacontractinghomotopy. (5.6) o It isnowpossibletodefineinacanonicalwayacohomology theory which is an invariant of comp L (.).One only has to choose the" rightcategory".Let Gt(X)= {f : Xq+1~IRI fisLipschitzcontinuousand suppfn Pen(b., R)isrelativelycompactforallR}.(5.7) Then,with 0 from(3.4),GL(X)=(Gt(X),O)qisacomplex and wedefine H'L(X):= H*(G'L(X)). If :X~Yis(u.p.)Lipschitztheninducest: Gt(Y)~GUX) by (t(X)f)(xo,... , Xq):= f(xo,... , Xq), fEGt(Y),and'L:HL(Y)~H'L(X). UsingRoe'santiCechsystemsanduniquenessof thecohomol-ogyof uniformresolutionsbyappropriatesheafsasin[63],one easilyobtains Theorem5.5If YEcomPL(X)thenthereexist:X~Y, W : Y~Xwichinduceinversetoeachother isomorphisms w* 'bl-H'L(X)~H'L(Y). * L Non-linearSobolevStructures 145 Butthisapproachisveryunsatisfactorysincewedidinfact notdefineareallynewinvariantbutthecategorialrestriction ofacoarseinvariant.Thesituationrapidlychangesifwefac-torize or impose decay conditions.Let C1(X) as above,bC1(X) thesubspaceofboundedfunctionsinC1(X)andCL(X)= C1(X)jbC1(X).Then 8 maps bC1(X)into bC1+1(X),bCLis a sub complex and weobtain acomplex CL b(X)=(C1 b(X), 8)q. Define" HL,b(X):= H*(CL,b(X)), Any (u.p.)Lipschitz map : X---t Y induces #: bCL(Y)---t bCL(X),hence#: bCL,b(Y)---t bCL,b(X)and *: HL,b(Y)---t HL,b(X), Theorem5.6HL,b(X)isaninvariantof comPL(X), Proof.LetYEcomPL(X),ddX, Y) 0is (#{ singularsimplexes(Jqof cI supp(JqcBR(x)})/R:::;N}. Non-linearSobolevStructures149 RoughlyspeakingforallsingularchainsECq,b,ulf(N)simul-tanouslyholdsthateverymetricballofradiusRcontainsat mostR. Nsingularsimplexes. From thedefinition Cq,b,ulf;;2limCq,b,ulf(N)= U Cq,b,ulf(N). ----t N N Set Cq,p,ulf(N){c =L C ~ ( JECq,b,ulf(N)I ~ q (Cq,p,ulf(N) , lip)isanormedspace(nonseparable)andwe have...~Cq,p,ulf(N)~Cq,p,ulf(N+ 1)~....Denote Cq,p,ulf( (0)=lim Cq,p,ulf (N)withtheinductivelimittopol-----> ogy.Then8: Cq,p,ulf( (0)----tCq-1,p,ulf( (0)iscontinuoussince 8: Cq,p,ulf(N)----t Cq-1,p,ulf(N)isnorm-continuous.Weobtain Hq,p,ulf( 00 )(X),H q,p,ulf( 00 )(X), Hq,p,ulf(oo)(X),Hq,p,ulf(oo)(X),(5.8) whereHdenotesthereduced(co ) homology. Theorem5.14The(co-)homologiesof (5.8)areinvariantsof arccompL,h(')'D Corollary5.15(a)H*,b,ulf,ooandH*,b,ulf,ooareinvariantsof arccomp L,top,rel (.). (b)H*,b,ulf,H*,b,u1fand the(co-)homologiesof (5.8)areinvari-antsof compL,top(-)' D 150RelativeIndexTheory,DeterminantsandTorsion Theproofof5.14,5.15followsfromthefactthattheadmit-tedmapsinducechainmapsandchainhomotopyequivalences betweenthecorrespondingcomplexes.Therearemanyother classesofinvariantswhichwedidnotconsiderexplicit elyuntil now.TheseincludetheK -theoryofC*-algebras,K*( C* X), andKasparovsK-homology forlocallycompact spaces,K*X. Weconclude this section with abrief review of bordism foropen manifoldsand relativecharacteristicnumbers. We consider as before oriented open manifolds (Mn , g)satisfying and (1) (Bn+! , 9B)isabordism between(Mr, gl)and(M2',g2)if it sat-isfiesthefollowingconditions. 1)(8B, gBI8B)(Ml' gl)U(-M2' g2), 2)thereexists 0suchthat gBlu6(8B)g8B+ dt2, 3)(B, gB)satisfies(Bk)andinfrinj(gB, x)> 0, XEB\U6(8B) 4)thereexistsR> 0 suchthatBcUR(M1),Be UR(M2). Wedenote(Ml' gl)f"V(M2' g2).(Bn+!, gB)iscalledabordism. b Sometimeswedenoteadditionallyf"V,bg standsforbounded b,bg geometry,i.e.(1)and(Bk). Lemma5.16a)f"Visanequivalence relation.Denote by[Mn, g] b thebordismclass. b)[MUM',gUg']=[M#M',g#g']. c)Set[M,g]+ [M',g']:=[MUM',gUg'j=[M#M',g#g'j. Then+ iswelldefinedandthesetof all[Mn, 9 jbecomesan abeliansemigroup.D Denoteby= Bk)thecorrespondingGrothendieck group.Similarlyonedefines generatedbypairs ( (Mn, g),f: MnX),fboundedanduniformlyproper. Non-linearSobolevStructures 151 Remarks5.171) Condition 4)above looks like dCH(M, M')::; R,wheredCH istheGromov-Hausdorffdistance.Butthisis wrong. 2)Thereisnochanceto calculate3)Onewouldliketohaveageometricrepresentativeforand for-[M, g].0 The wayout fromthis isto establishbordism theory forspecial classes of open manifolds or/and further restrictions to bordism. Our firstexampleisbordismwithcompactsupport.Herecon-dition 1)above remains but one replaces 2)- 4)by the condition ThereexistsacompactsubmanifoldCn+!CBn+1 suchthat(B \c, gBIB\C)isaproductbordism,i.e. (B \c, gB\d (M \Cx[0,1]' gM\C+ dt2).(cs) Wewritef'V.Thenonegetsabordismgroup(cs)(= b,cs Grothendieck group). Atthefirstglance,thecalculationof cs)oratleastthe characterizationofthebordismclassesseemstobeverydiffi-cult.Butwewillsee,thatthisisnotthecase.Forthis,we introduce still someuniform,structures.Denote by mn(mJ):= mn(mJ, nc)CmL thesetofisometryclassesofcomplete, open,orientedRiemannianmanifolds.Considerpairs(Mf, gl), (M2,g2)Emn(mf)withthe followingproperty: ThereexistcompactsubmanifoldsKr cMr and candan isometryMl\KlM2\K2.(5.9) For suchpairs,wedefineinanalogyto sections2 and 4 bdL,iso,rez((M1,gl), (M2,g2:= inf{max{O, logbldJI} + max{O, logbldhl} + supdist(x, hJx) + supdist(y, Jhy)I XEMIyEM2 J E COO(Ml' M2), 9E COO (M2'M1),and forsome KlCK,JIMl\Klisan isometryand 9If(Ml\K=J-l}. 152RelativeIndexTheory,DeterminantsandTorsion If(MI, 91)and(M2,92)donotsatisfy(5.9),thenwedefine bdL,iso,rel((M1,91), (M2,92))=00.WehavebdL,iso,rel((M1,91), (M2,92))=0if(M1, 91)and(M2,92)are isometric. Remarks5.181)ThenotionsRiemannianisometryanddis-tanceisometrycoincideforRiemannianmanifolds.FUrther-more,if !isanisometry!, then wehavebld!1=l. 2)Any! that occurs in the definition of dL,iso,relisautomatically an elementof C'X),m(M1, M2) forallm.The same holds true for 9.0 We write 9J1Lisorel(m!)= mn(mJ)j ",where (M1,91)'" (M2,92) if bdL,iso,rel((M1: 91),(M2, 92))= O.Set Va={((M1,91), (M2,92))E(9J1)'i,iso,rel(m!))2I bdL,iso,rel((M1, 9d, (M2, 92))< 6}. Proposition5.19.c={Va}o>oisabasisfora metrizableuni-formstructurebUL,iso,rel.0 Denotebyb9J1Liso rei (m J)the correspondinguniformspace. ,, Proposition5.20If rinj(Mi, 9i)=ri> 0,r=min{rI, r2}and bdL,iso,rel((M1,91), (M2,92))< rthen Mland M2are(uniformly proper)bi-Lipschitz homotopyequivalent.0 Corollary5.21If werestrictourselvestoopenmanifoldswith injectivity radius2':r,thenmanifolds (M1,91)and (M2,92)with bdL,iso,rel-distancelessthanrareautomatically(uniformly proper)bi-Lipschitz homotopyequivalent.0 Remark5.22If(MI, 91)satisfies(1)or(1)and(Bk)and bdL,iso,rel(M1, 91)'(M2, 92))0,boundedforallv, tathenMhasfinitelymanynonexpanding ends.0 Wedefinenowaslightly sharpenedbordismrelation. Let(Mn, 9),(Mtn, 9')beasabove,eachwithfinitelymany nonexpandingendsEl,'",Esor...respectively.Let "1M, 1 ,... , "1M,sor"1M',!,.. ,"IM',s'correspondingraysasabove. From(M, 9)rv(M', 9')andallendsnonexpandingfollowsin bg particularthatforallsufficientlylargecompactCn+l CBn+l Non-linearSobolevStructures thereexistsR=RB>0S.t. s Bn+1 \en+1 cU UR(II'M,u I) , 1 s' Bn+l \en+!cU UR(hM',u, l). I 161 Werequire additionally to this condition the additive compabil-ityof theinnerI'-distanceandthe(B \C)-distanceforpoints x,)"Y')'on the I"s. Thereexistcn+1CBn+!andc'>0s.t.forx,)"y')'E11'1\e holds HereI'standsforI'M,I,. .. ,I'M,s,I'M',l,""I'M',s' ,respectively and d(., .)==dist(-, .). Wedenote(M,g)rv(M', g')iftheyarebg-bordantbymeans ne (B,gB)satisfying(GH). Remarks5.371)Therighthand inequalityof (GH)trivially holds.Weaddeditonlyforsymmetryreasons. 2)ItwasessentiallyThomasSchickwhopointedouttothe authorthemeaningofthecondition(GH)or(GHI)andwho proposedtoincludethemintothedefinitionof bordism.0 Weconsiderinsteadof(G H)the condition ThereexistCn+l CBn+!andc'>0s.t.forallx, yEU(c:) holds Here c:stands forC:I,... ,c:s, c : ~ ,... , c : ~ ,and U(c:)foraneighbour-hood of c:,U(c:)n C=0. Lemma5.38(GH)and(GHI)areequivalent. 162RelativeIndexTheory,DeterminantsandTorsion Proof.Assume(CHI)'Then(CH)holdssinceforx-Y' Y-yE 11'1cU(E),U(E)n C=0,du(c) (x-y,Y-y)=d-y (x-y , y-y).Ifcon-verselyx, YEU(E)thenthereexistsx-Y' Y-yE11'1CU(E)s.t. du(c) (x, x-y)::;RM,du(c)(Y, Y-y)::;RM.Thentheassertionfol-lowsfrom du(c) (x,y)- du(c) (x-y,Y-y)::;du(c) (x, x-y)+ du(c)(Y, Y-y), du(c)(x, y)- du(c)(x, x-y)- du(c)(Y' Y-y)::;d-y(x-y,Y-y) =d-y (x-y , Y-y)- c' + c'::;dB\C(x-y, Y-y)+ c', du(c)- 2RM- c'::;dB\c(x-y, Y-y), du(c)- 4RM - c'::;dB\C(x, y). o Remark5.39(CHI)immediatelyimpliesthatdCH(B\C, U(UEa))(Mn,g)beaClifford bundle,Mnopenandcomplete,KcMacompactsubset, DF(EIM\K)Friedrichs'extensionof Dlc,?O(EIM\K)'Thenthere hold (Je(D)=(Je(DF) =(Je(DF(EIM\K))(1.3) and Proof.Westart with(1.3)and (Je(D)~(Je(DF(EIM\K)).Let A E(Je(D),('l/Jv)vbe an orthonormal Weyl sequence forA,D'l/Jv-A'l/Jv--> O.Then (wv)v,Wv= 'l/J2v+l-'l/J2vis still a Weyl sequence forA.LetECC: (M),0 ::;::;1,=1 onaneighbour hood U=U (K)ofK.AccordingtotheRellichchainpropertyof Sobolev spaces(with realindex)on compactmanifolds,('l/Jv) v containsanL2-convergentsubsequencewhichwedenoteagain by('l/Jv) v'This yieldswv --> 0andgrad. Wv--> 0inL2. ((1- 0,grad. Wv--> O.Hence(Je(D)~(Je(DF(EIM\K)).VDp(EIM\K)~V Dp andeveryWeyl sequenceforA E(Je(DF(EIM\K))isalsoaWeyl sequenceforA E(Je(D).Thisfinishestheproof of(1.3).(1.4) HeatKernelof GeneralizedDiracOperators171 isanimmediateconsequenceof(1.3)bymeansof thespectral theorembut itcanalsosimilarlybeproven.0 Corollary1.5Theessentialspectrumof DandD2remains invariantundercompactperturbationsof thetopologyandthe metric.Inparticular thisholdsfortheLaplaceoperatorsacting on formswithvaluesinavectorbundle.0 Asforcompactmanifolds,wecandefinetheRiemanniancon-nectedsumforopenRiemannianmanifolds,evenforRieman-nian vector bundles (Ei' hi, '\1 hi )-+ (Mt, gi),where at the com-pact glueing domain the metric and connection are not uniquely determined.Anothercorollaryisthen givenby Proposition1.6Suppose(Ei' hi, '\1hi)-+(Mt,gi),i 1, ... ,rRiemannianvectorbundlesof thesamerank,(Mt, gi) complete,andlet.6.=.6.q betheLaplaceoperatoractingon q-forms withvaluesin Ei(resp.E).Then r O"e.6. q ((Ei-+ Mi ))=U O"e(.6. q(Ei-+ Mi)).(1.5) i=l o 1.4 can be reformulated as the statement that the essential spec-trumforanisolatedendEiswelldefined.Wedenoteitby O"e(DF(E)), Proposition1. 7If (Mn, g)iscompleteandhasfinitelymany ends El,' .. , Erthen rr i=li=l 172RelativeIndexTheory,DeterminantsandTorsion Proposition1.8Assumethehypothesisof 1.4.SupposeA E CTe(D).ThenthereexistsaWeylsequence('Pv)vforA suchthat foranycompactsubsetKcM (1.7) ForeveryA ECTe(D2)thereexistsaWeylsequence('Pv)vsatis-fying(1. 7)and (1.8) Proof.Startwith(1.7).Let('l/Jv)vbeaWeylsequencefor A ECT e (D),KlCK2C...CKiCKH 1C"', UKi=M, anexhaustionbycompactsubmanifolds.ByaRellichcom-pactnessargumentthereexistsasubsequence of('l/Jv)v convergingonK1.Inductively,thereexistsasubse-quenceof convergingonKH1.Set('Pv)v= - Then('Pv)visaWeylsequencefor A ECTe(D)satisfying(1.7).ForA ECTe(D2)with Weylsequence ('l/Jv) v , wechoosethesubsequenceof suchthat and convergeonKHI(inL2, asalways).0 1.8meansthatw.l.o.g.Weylsequencesshould"leave"(inthe senseoftheL2-norm)anycompactsubset,i.e.theremustbe "placeenoughatinfinity" . Proposition1.9Let (E, h, \7,.)--t (Mn, g)beaClifford bun-dlewith(1),(Br-3(M,g)),(Br-3(E, \7)),r>+ 1and\7'a secondCliffordconnection satisfying 1\7' - \71V',2,r-l< 00.Then forD =D(\7)and D'=D(\7')thereholds (1.9) and (1.10) Proof.(Mn,g)iscomplete,DandD'areself-adjoint.1JD = n2,I(E, D)=n2,1(E, \7)=n2,1(E, \7')=n2,1(E, D')=1JD, HeatKernelof GeneralizedDiracOperators 173 accordingtoII1.25andII1.32.Wewrite\1'=\1+ ry.Then D'=Lei . \ 1 ~ i=Lei' (\1ei + ryei(.))=D+ ryOP,wherethe ii operator ryOPactsasryOP ( 0,there exists acompactsetK=K(E')eM suchthat E' supIrylx 0, mEM, T> t> 0holds J (r_e)2 IW(t, m,p)12 dp:::;C C(m). e- (4+O)t.(1.41) M\Br(m) Asimilar estimateholdsfor m,p). Wereferto[9]forthe proof. D Lemma1.19ForanyE>0, T>0,8>0thereexistsC>0 suchthat forall m, p EMwith dist( m, p)> 2E, T> t> 0holds (diBt(m,p)-e)2 IW(t,m,p)12:::;C C(m) C(p) e- (4+o)t(1.42) Asimilar estimateholdsfor m,p). Wereferto[9]forthe proof. D 180RelativeIndexTheory,DeterminantsandTorsion Proposition1.20Assume(Mn, g)with(/)and(BK),(E, \7) with(BK),k2:r>~+ 1.Thenallestimatesin(140)- (1.42) holdwithuniformconstants. Proof.FromtheassumptionsHr (E)~wr (E)andsUPm C(m)=C=globalSobolevconstantforwr(E),accordingto II1.4,II1.6.D Let UcMbe precompact, open,(M+, g+)closed with UcM+ isometrically and E+--tM+ aClifford bundle with E+lu ~Elu isometrically.Denote byW+(t, m,p)theheatkernelof e-tD+2. Lemma1.21AssumeE>0, T>0, J>O.Thenthereexists C>0suchthatforallT>t>O,m,pEUwithB2c(m), B2c (p)CUholds .2 IW(t,m,p) - W+(t,m,p)1::;C e-(4+8)t (1.43) Wereferto[8]forthe simpleproof.D Corollary1.22trW(t, m, m)has for t--t 0+thesameasymp-toticexpansionasfortrW+(t,m,m).D 2Duhamel'sprinciple,scattering theory andtraceclassconditions 2-,2 Wewanttoprovethetraceclasspropertyofe-tD - e-tD , wherefyisaperturbationofD.Thekeytogetconvenient - 2 expressionsfore-tD2 - etD'isDuhamel'sprinciple.Forclosed manifolds,this isa very well known fact.We establish it here for opencompletemanifolds.Inprinciple,itfollowsfromStokes' theorem,or,whatisthesame,frompartialintegration.Hav-ingestablishedDuhamel's principle,the proof of the traceclass property amounts to the estimate of acertain number of opera-tor valued integrals.Their estimate occupies the whole30pages of chapterIV. HeatKernelof GeneralizedDiracOperators181 Thetraceclasspropertyisthekeyfortheapplicationof scat-teringtheory.Wegiveanaccountonthosefactsof scattering theorywhichareof greatimportanceinchaptersVandVI. FirstweestablishDuhamel'sprincipleandmakethefollowing assumptions:DandD'aregeneralizedDiracoperatorsacting inthe sameHilbertspace, where'fl= 'flopisan operator acting inthe sameHilbertspace. Lemma2.1Assume t>O.Then t -tD2_tD,2- J -SD2(D,2D2)-(t-s)D,2d e- e- e- es.(2.1) o Proof.(2.1)meansatheatkernellevel W(t,m,p)- W'(t,m,p) t =- JJ(W(s, m, q), (D2- D,2)W'(t - s, q,p))qdqds, oM (2.2) where(,)qmeansthefibrewisescalarproductatqanddq= dvolq(g).Hencefor(2.1)wehavetoprove(2.2).(2.2)isan immediateconsequenceofDuhamel'sprinciple.Onlyforcom-pleteness,wepresenttheproof of(2.1),whichisthelastof the following7factsandimplications. 1.Fort> 0isW(t, m,p)EL2(M, E, dp)n Vb 2.If ,W EVbthenJ(D2,w)- (,D2W)dvol=0(Greens formula). 3.((D2+ %7)(T,g)W(t-T, q))q-((T, g), q))q= = (D2( ( T,q), W(t-T, q) )q- (( T,q), D2w(t-T, q) )q+ tr (( T,g), W(t-T,q))q. 182RelativeIndexTheory,DeterminantsandTorsion f3 4.JJ((D2 + tr)..)whicharemeasurableandsquare-integrablewithrespect to the measurem.The scalarproductin(2.3)isdefinedas +00 (1, g):=j(J(>..), g(>..))x(>,)dm(>"), -00 where(J(>..),g(>..))x(>.)isthe scalar product in the Hilbert space X(>..). Wesay,aHilbert spaceXhasadecompositionasadirectinte-gralif thereisaunitarymapping +00 F: X7jEBX(>..)du(>..). -00 Wedenotethisby +00 X~jEBX(>")du(>"). -00 Aspecialcaseof sucharepresentationisgivenbythespectral resolutionforaself-adjointoperatorA, -00 whichinducesadecompositionof type(2.3)inwhichthe oper-ator F AF*actsasmultiplicationby>...Let8cIRbeaBorel set.Then FEA(8)F*reducesto XB,and weobtain (EA(B)J,g)=j((FJ) (>..),(Fg) (>..))dm(>..). B If weapplytheseconsiderationstotheself-adjointoperatorB (of thepairA,Babove)andtoXac=Xac(B)==(Pac(B))(X), 188RelativeIndexTheory,DeterminantsandTorsion then weget Xac(B)JEBX>.(B)d)..:=X(B)(ac).(2.4) &(B) Here o-(B)isa so-called core of (J"(B),i.a.aBorel set of minimal measuresuchthatEB(lR \o-(B))=O.Therighthandsideof (2.4)diagonalizestheoperatorBac= BlxaJB).The scattering operatorS= W - commuteswithBac,henceunderthe correspondence(2.4)itsactiongoesoverintomultiplicationby an operatorvaluedfunctionS()"): X>.(B)----tX>.(B).S()")= S()..;A, B) iscalled the scattering matrix.If weconsider the left handsideof(2.2)then weobtain theversion S = JS()..)dEB()..)(2.5) & insteadof FSF* =JS()")d)"(2.6) Both representationareequivalent. InchapterVandVI,thespectralshiftfunction(S S will playa centralrole.Weintroduceitnow.Themaingoalisto introduceafunction suchthat tr( E.(1.6) M 196RelativeIndexTheory,DeterminantsandTorsion HencetheestimateofJ J m,p)12If(m)12dpdmforsE MM t]isdoneif JIf(mW dm< 00 M andthen fl2:SC4 .IfIL2'whereC4 =C4(t)containsa factore-at,a>0,ifinf 0"( D2)>O. For(1.2)wehavetoestimate JJ m,p), 7]0P(p)')pI2dpdm(1.7) MM We recall a simple factabout Hilbert spaces.LetXbe aHilbert space,xEX,x 1=O.ThenIxl=supl(x,y)l, IYI=l Ixl2 =(sup l(x,Y)lr. IYI=l (1.8) ThisfollowsfromI(x,y) I :sIxl. Iylandequalityfory=WeapplythistoE---tM,X=L2(M, E, dp),x=x(m)= W(t, m,p), 7]0P(p).)p=W(t, m,p) o7]P(p)andhaveto estimate supN(Cf=sup1(8(m), e-tD27]0PCfIL2(1.9) ECgo(E)ECgo(E) 1IL2= 11IL2= 1 Theheatkernelisof Sobolev class, W(t, m,) E IW(t, m, ')IH2 :sC5(t).(1.10) Hencewehavecan restrictin(1.9)to supN(Cf(1.11) ECgo(E) 1IL2= 1 IIH;,::::C5 Inthe sequelweestimate(1.11).Fordoingthis,werecallsome simplefactsconcerningthewaveequation aCf>s.1 as= 'l,DCf>s,Cf>o= Cf>,Cf>Cwithcompactsupport. (1.12) TraceClassProperties197 Itiswellknownthat(1.12)hasauniquesolutionCPswhichis givenby and suppCPsCUisl(suppcp) Uisl=lsi- neighbourhood.Moreover, (1.13) (1.14) WefixauniformlylocallyfinitecoverU={Uv},J={Bd(Xv)}v by normal charts of radius d !:2>iforr>n+ 2and t- 22'- 2'-I < I > I H ~::;05.ThisyieldstogetherwiththeSobolevembedding theestimate l/ mE U.(U.,) O. Thisfinishesthe proof of thefirstpart of theorem1.1. Wemuststillprovethetraceclassproperty of -tD2D-tD,2D' e-e.(1.38) Considerthedecomposition -tD2D-tDI2D' e-e(1.39) .tD2tD,2. Accordmgtothefirstpart,e-"2- e-"2ISfort> 0of trace class.Moreover,D= isfort>0bounded,its operator norm is::; Hencetheirproduct isfort>0 of trace classand hasbounded trace norm fortE[aD,all, aD>O.(1.39) TraceClassProperties isdone.Wecan write(1.40)as Now t '2 + Je-sD\,D'e-(&-s)D/2 ds](D'e-&D/\ o 203 (1.41) (1.42) (1.42)isof traceclassandits trace normisuniformlybounded onanylao,all.ao>0,accordingtheproof of thefirstpart.If tii '242 wedecomposeJ =J + J then weobtain back from the integrals o0t 4 in(1.41)theintegrals(II) - (I4),replacingt---t Theseare done.D'e-&D/2 generatesC / yt intheestimateofthetrace norm.Henceweare done.0 2Variationof theCliffordstructure Ourprocedureistoadmitmuchmoregeneralperturbations than those of 'V='Vh only.Nevertheless,the discussion of more general perturbations ismodelled by the case of 'V -perturbation. Inthisnextstep,weadmitperturbationsofg, 'Vh,.,fixingh, the topology and vector bundle structure of E----+M.The next main resultshallbe formulatedasfollows. Theorem2.1Let E= (E, h, 'V= 'Vh, .)----+(Mn, g)beaClif-fordbundlewith(1),(Bk(M,g)),(Bk(E, 'V)),kr+1>n+3, E'=(E, h, 'V'='V,h,.')----+(Mn, g')E n 204RelativeIndexTheory,DeterminantsandTorsion CLBN,n(I, Bk),D=D(g, h, \7=\7h, .),D'=D(g', h, \7'= \7,h, .')theassociatedgeneralizedDiracoperators.Thenfor t>O (2.1) isof traceclassandthetracenormisuniformlyboundedon compact t-intervallslao, al],ao> O. HereD'ListheunitarytransformationofD,2toL2=L2 ((M, E), g, h).2.1needssomeexplanations.DactsinL2= L2((M, E), g,h),D'inL;=L2((M, E), g', h).L2andL;are quasiisometricHilbertspaces.Asvectorspacestheycoincide, theirscalarproductscanbequitedifferentbutmustbemutu-allyboundedatthediagonalaftermultiplicationbyconstants. DisselfadjointonVD inL2,D'isselfadjointonVD,inL; b '1.LHtD,2dtD2tD,2 utnotnecessan ym2.encee- ane- - e-arenotdefinedinL2.OnehastograftD2orD,2.Write dvolq(g)==dq(g)=a(q). dq(g')==dvolq(g').Then o O. Proof.The proof isasimplecombinationof theproofsof 1.1 and2.1.0 Nowweadditionallyadmitperturbationofthefibremetrich. Before the formulationof the theorem wemust givesome expla-nations.Consider the Hilbert spaces L2(g,h)= L2((M, E), g, h), L2(g', h)=L2((M, E), g', h),L2(g', h')=L2((M, E), g', h') andthemaps i(g',h},(g',h'}: L2(g',h)---t L2(g',h'),i(gl,h),(gl,hl}= U(g,h},(gl,h}: L2(g,h)---t L2(g',h),U(g,h},(g',h}= wheredp(g)= a(p)dp(g').Thenweset

:=U(g,h), (gl ,h) i(gl ,h}, (g',h') D' i(gl ,h),(g' ,hi} U(g,h},(gl ,h} ==U*i* D'iU.(2.54) Here i*is even locally defined(since g'is fixed)and i; = dual"hlo i' odualh',where dualh((p))=hp("(p)).Inalocal basis field 1 ,... ,N,(p)= e(p)i(p), (2.55) 218RelativeIndexTheory,DeterminantsandTorsion It followsfrom(2.55)thatforh'ECOmp1,r+1 (h)i*,i*-lare boundedupto orderk, i*- 1, i*-l- 1E n,0,I,r+I(Hom((E, h', 'Vh')----+ ----+(M,g'), (E, h, 'Vh) ----+(M,g')))(2.56) and i*-1,i*-1 -1 E n,0,2,!:f-(Hom((E,h', 'Vh')----+ ----+(M, g'), (E, h, 'Vh)----+(M, g'))).(2.57) D'==D'isselfadjointonDD'=ego (E) I lv, ,where= JJi,+ JD'Ji,i:L2(g',h)----+L2(g',h')== andi*: 22 L2(g',h')----+L2(g',h)are forh'E compl,r+l(h)quasi isometries withboundedderivatives,theymaper;:(E)1-1ontoer;:(E) andi* D'iisself adjointonego(E)lli*D'i= Di*D'iCL2((M, E), g',h)==L2(g',h).Weobtain asaconsequencethat e-t(i*D'i)2is definedand selfadjoint inL2((M, E), g', h)=L2(g',h),maps for t> 0andi,j EZHi(E,i*D'i)continuouslyintoHj(E,i*D'i) andhastheheatkernelm,p)=(o(m), e-t(i*D'i)2 o(p)), W'(t, m,p)satisfiesthesamegeneralestimatesasW(t, m,p). Byexactlythe sameargumentsweobtain thate-W*(i* D'i)2U= e-t(U*i* D'iU)2= U*e-t(i* D'i)2U is defined in L2= L2((M, E), g, h), self adjoint and has the heat kernel W{(t,m, p)= Wg'h (t, m, p)= 2, a-!m,p)a(p)!.Hereweassumeg'Ecompl,r+l(g). N owweareableto formulateour maintheorem. Theorem2.9LetE=((E, h, 'V='Vh,.)----+(Mn, g))bea Cliffordbundlewith(1),(Bk(M, g)),(Bk(E, 'V)),kr+ 1> n+3,E'=((E,h,'V'= 'Vh',.')----+(Mn,g))E (E)nCLBN,n(1, Bk),D= D(g, h, 'V= 'Vh, .),D' = D(g', h, 'V'= 'Vh' ,.')theassociatedgeneralizedDiracoperators,dp(g)= a(p)dp(g'),U = a!.Thenfort> 0 -tD2U*-t(i* D'i)2U e- e(2.58) isof traceclassandthetracenormisuniformlyboundedon compact t-intervallslao, all,ao> O. TraceClassProperties219 Proof.Wearedoneif wecouldprovethe assertionsfor e-t(UD'u*)2_e-t(i*D'i)2= Ue-tD2U*_e-t(i*D'i)2(2.59) sinceU*(2.59)U=(2.58).Togetabetterexplicitexpression for(2.59),weapply again Duhamel's principle.This holds since GreensformulaforU D2U*holds, Jhq(UD2U*if?,w)- h(if?, UD2U*w)dq(g')= O. Weobtain t -JJ oM (UD2U*+ %t) - s,q,p))dq(g')ds t = - JJ(m)W(s, m, (q), oM (U D2U*- (i*- s, q, p)dq(g'))ds =(m)W(t, m, (q)-m,p) = Wg',h(t, m,p) - m,p).(2.60) (2.60)expressestheoperator equation -t(U DU*)2-t(i* D'i)2 e-e t -Je-s(U*DU)2((UDU*)2- (i*D'i)2)e-(t-S)(i*D'i)2ds o t -Je-s(UDu*)2UDU*(UDU*- i* D'i)e-(t-S)(i*D'i)2ds o (2.61) t Je-s(UDU*)\U DU*- i* D'i)(i* D'i)e-(t-s)(i* D'ifds. o (2.62) 220RelativeIndexTheory,DeterminantsandTorsion Wewrite(2.62)as t -J- i* D'i)e-(t-S)(i*D'i)2ds o t J 1-sD2D_1 (D'*D"grada.)-(t-s)(i* D'i)2d = - a2ea2- - es 2a o t = - J- l)D + (D - D') o _i*-lgrada )e-(t-s)(i* D'i)2ds 2a t = JD('r/o+ 'r/l+ 'r/2+ 'r/3+ 'r/4)e-(t-s)(i*D'i)2ds, o 'r/o=grad3a','r/i= i=1,2,3,'r/l(2)=(2.10), 2a2" 'r/2(2)=(2.11),'r/3(2)=(2.12),'r/4= - l)D.Here 'r/oand'r/2areof zerothorder.'r/land'r/3canbediscussedasin (2.18)-(2.53).'r/4can be discussedanalogous to 'r/b'r/3asbefore, i.e.'r/4willbeshiftedviapartialintegrationtotheleft(upto zero order terms)and-1) thereafter again to the right. Inthe estimatesonehastoreplaceWbyDW andnothinges-sentially changes as weexhibited in(2.35).Weperform in(2.62) the samedecompositionand haveto estimate20integrals, t 2 t 2 J e-sD2 D'r/ve-(t-S)(i* D'i)2ds, o J 1-sD2('*D")-(t-s)(i* D'i)2d a 2e'r/ves, o t JD'r/ve-(t-s)(i*D'i)2ds, t 2 TraceClassProperties221 t Ja! e-sD2'TJv(i*D'i)e-(t-s)(i*D'i)2ds,(Iv,4) t '2 1/=0,... ,4andtoshowthattheseareproductsofHilbert-Schmidtoperatorsandhaveuniformlyboundedtracenormon compactt-intervals.Thishasbeencompletelymodelledinthe proof of 2.1.0 Finally weobtain Theorem2.10Assumethehypothesesof 2.9.Thenfor t>0 isoftraceclassanditstracenormisuniformlyboundedon compact t-intervallslao,al]'ao> O. o Theoperatorsi* D,2i and(i* D'i)2aredifferentingeneral.We shouldstill comparee-ti*D'2iande-t(i* D'i)2 . Theorem2.11Assumethehypothesesof 2.9.Thenfort>0 isof traceclassandthetracenormisuniformlyboundedon compact t-intervallslao,al]'ao> O. 222RelativeIndexTheory,DeterminantsandTorsion Proof.We obtain again immediately from Duhamel's principle -ti* D,2i-t(i* D'i)2 e- e= t =- Je-S(i*D'2i)(i* D,2i - (i* D'i)2)e-(t-s)(i*D1i)2ds= o t = - Je-S(i*D'2i)i* D'(l- ii*)D'ie-(t-S)(i*D'i)2ds = o t =- Je-S(i*DI2i)(i*D'i)C1(1_ ii*)i*-1(i*D'i)e-(t-s)(i*D'i)2ds. o (2.63) Int]weshifti* D'iagaintotheleftof thekernelW:_S(i*D2i) via partial integrationand estimate (i* D' DI2i) [( e-1W DI2i) f) U-1e-1(i*DI2i)i-1(1_ ii*)i*-l)] (( i* D' i)e-(t-s)(i* Dli)2) asbefore.In[0, wewritethe integrandof (2.63)as (e-S(i* D'2i)i* D'i)[( (i*it1 e- t4s(i* D'i)2f-1) U e- t48(i* Dli)2) 1 (e- t;s (i* D'i)2 (i* D'i)) and proceedas inthecorrespondingcases.o Theorem2.12Assumethehypothesesof 2.9.Thenfor t> 0 isof traceclassand thetracenorm isuniformlybounded onany t-intervalllao, a1],ao> o. Proof.Thisimmediatelyfollowsfrom2.9and2.11.0 TraceClassProperties 3Additional topological perturbations 223 Finallythelastclassofadmittedperturbationsarecompact topological perturbations whichwillbe studiednow. Let E= ((E, h, \7h) (Mn, g))ECLBN,n(I, Bk) be aClifford bundle,k2::r+ 1 >n+ 3,E'=((E, h', \7h')(Mm, g'))E n CLBN,n(I, Bk)'ThenthereexistKcM, K'CM'andavectorbundleisomorphism(notnecessarilyan isometry)f= (JE,jM)E151,r+2(EIM\K,E/IM'\K')s.t. gIM\Kandf'Mg'IM\Karequasiisometric,(3.1) hIEIM\Kand arequasiisometric,(3.2) IgIM\K- f'Mg'IM\Klg,l,r+1 0 (3.58) isof traceclassin 'H'andthetracenorm isuniformlybounded oncompact t-intervalslao, all,ao> O.0 ThesimpleststandardexampleforaCliffordbundleisE (A *T* MC, gAo ,vg1\* )----t(Mn, g)withCliffordmultiplication X wE TmM A*T* MC----tX W = Wx1\W - ixw, whereWx:= g(, X).InthiscaseEasavectorbundleremains fixedbuttheCliffordmodulestructurevariessmoothlywith g, g'Ecomp(g).It iswellknownthat inthiscaseD=d + d*, D2=(d + d*)2Laplaceoperator !:1. Theorem2.12then yields Theorem3.8Assume(Mn, g)with(I),(Bk),kr+ 1> n+ 3,g'EM(I, Bk),g'Ecompl,r+l(g)CM(I, Bk)'De-noteby =U*i* !:1(g')iUthetransformationof!:1'= !:1(g')fromL2(g',g')== toL2(g)== L2(g,g)=L2((M,A*T*MC,g,gA*),wherei:L2(g,g')= 236RelativeIndexTheory,DeterminantsandTorsion L2((M,A*T*M 0C,g,gA*)--t L2(g',g')andU:L2(g,g)--t L2(g,g'),U= a!, dq(g)= a(q) dq(g'),arethecanonical maps, i*,U*theiradjoints.Thenfor t>0 isof traceclassandthetracenormisuniformlyboundedon compact t-intervalls[aD,al],aD> O. o Applying3.7to the caseE=A*T*M 0C, D2=b.,weobtain Theorem3.9Let(A*T*M0C,gA*ECLB2n(1,Bk),kr+ 2>n+ 3,(A*T*M'E0 C, gA*)n CLB2n,n(1, Bk),,0.=b.(g),b.'=b.(g')thegraded Laplaceoperator.Then for t>0 e-tt. p_e-t(U*i*t.'iU p' (3.59) isof traceclassinH'andthetracenormisuniformlybounded oncompact t-intervals[aD,al],aD> O.0 Roughly ormoreconcretely speaking,as one prefers,the means thefollowing.Givenanopenmanifold(Mn, g)satisfying(1) and(Bk),kr+ 2 > n + 3.Cut out acompact submanifoldK and gluethe compact submanifoldK'along8M = 8K, getting thusM',endowM' with ametric g'satisfying(1)and(Bk)and Ig- g'IM\K,g,l,r+10 has theassertedproperties. Wecanapplyaslightmodificationof 3.8.alsotoSchrodinger operators. TraceClassProperties 237 Lemma 3.10Let(Mn,g)beopenwith(1)and(Eo),r>n, VEn,l,r (Mn, g)areal-valuedfunction,0~q~n.Thenthe operator tlq + (V) isessentiallyself-adjointon C,: (A q). Proof.Accordingto the Sobolev embedding theorem, lV(x) I ~b. Hence (3.60) o Proposition3.11Suppose9EM(J, Bk),k2:r>n+ 2, VEn,l,r (Mn, g)arealvalued function.Thenfor t>0 (3.61) isof traceclassandthetracenormisuniformlyboundedon compact t-intervalslao,al],ao> O. Proof.WeinferfromDuhamel'sformula t e-tAq - e-t(Aq+(V.))= Je-SAqVe-(t-s)(Aq+(v'))ds, (3.62) o t~t decomposeJ = J + J andestimate theseintegralsasin(1.1)-o0~ (1.42),replacingT}by V.0 Theorem3.12Assumethehypothesesof3.8andVEn,l,T (Mn,g),V'En,l,r(M'ng).Then (3.63) isof traceclassinH'andthetracenormisuniformlybounded oncompact t-intervalslao,al],ao>O. 238RelativeIndexTheory,DeterminantsandTorsion Proof.Wewrite -t(.t.eqe = (e-t(.t.q+W))_e-t.t.q)p +e-t.t.qp __ (3.64) (3.65) (3.66) (3.64)and (3.66)are of trace class according to proposition 3.10, (3.65)isof traceclassaccordingto theorem3.8.0 Weproved that after fixingEECLBN,n(I, Bk),k2::r+ 1 > n+ 3,wecanattachtoanyE'Etwonumber valuedinvariants,namely E'(-tD2p-t(U'i'D1iU)2p') --+ tre- e(3.67) and E'(-tD2 P-tU'i' D12iUp') --+ tre- e.(3.68) Thisisacontributiontotheclassificationinsideacomponent but stillunsatisfactoryinsofarasit 1)coulddependon t. 2)willdependon theKCM,K'cM'inquestion, 3)isnot yetclearthemeaning of thisinvariant. The answersto theseopen question willbe the contentof chap-ters V,VIandVII. VRelativeindex theory 1Relativeindex theorems,the spectralshiftfunctionandthe scatteringindex InmanyapplicationstheCliffordbundlesunderconsideration areendowedwithan involutionr: E---t E,s.t. [r,X]+= 0forXETM [\7, r]=0 ThenL2((M,E),g,h)=L2(M,E+) EElL2(M,E-) ( 0D-) D=D+0 andD- = (D+)*.If Mniscompactthen asusual (1.1 ) (1.2) (1.3) indD := indD+:= dimkerD+ - dimkerD- = tr( re-tD\ (1.4) whereweunderstandras ForopenMnindDingeneralisnotdefinedsincere-tD2 isnot of traceclass.Theappropriateapproachonopenmanifoldsis relativeindextheoryforpairsof operatorsD, D'.If D, D'are selfadjoint in the sameHilbert spaceand etD2 - e-tD/2 wouldbe of traceclassthen makessense,butat the firstglance(1.5)shoulddependon t. 239 240RelativeIndexTheory,DeterminantsandTorsion If werestricttoCliffordbundlesEE CLBN,n(I, Bk)withinvo-lution Tthen weassume that the maps entering in the definition f l,r+l(E)l,r+l(E)'bl. ocomp L,dij j,For gen comp L,dij j,relare T-compatle,I.e. afteridentificationof EIM\KandfiE'IM'\Kholds (1.6) WecallEIM\KandE'IM'\K'T-compatible.Then,accordingto the precedingtheorems, (1. 7) makessense. Theorem1.1Let ((E, h, "Vh)---t (Mn, g), T)ECLBN,n(I, Bk) bea gradedCliffordbundle,kr> n + 2. a)If"V'hEcompl,r("V)C "V'T-compatible,i.e. ["V',T]= 0then isindependentof t. b)If E'EisT-compatiblewithE,[T,X .']+=0forXET Mand["V',T]=0,then isindependentof t. Proof.a)followsfromourIV1.1.b)followsfromour1.2 below.0 Proposition1.2If E' E and T( e-tD2 P_e-t(U*i* D'iU)2 P') T( e-tD2 D- e-t(U*i* D'iU)2 (U*i* D'iU arefort>0of traceclassandthetracenormof RelativeIndexTheory 241 T( e-tD2 D- e-t(U*i* D'iU)2 (U*i* D'iU))isuniformlyboundedon compact t-intervals[aD,al],aD> 0,then isindependentof t. Proof.Let('Pi)ibeasequenceofsmoothfunctionsE C ~ ( M\K),satisfyingsup Id'Pil~0, ::s'Pi::s'PHIand t->oo 'Pi~1.DenotebyMithemultiplicationoperator with'Pion t->oo L2((M\K, EIM\K), g,h).We extend Miby 1 to the complement of L2((M \K, E), g,h)inH.Wehaveto show e-tD2 P_e-t(U*i* D'iU)2 P'isof traceclasshence , ( _tD2p-t(U*i*D'iU)2p') trTe- e =limtrTMj(e-tD2 p- e-t(U*i*D'iU)2 P')Mj. J->OO Mj restrictsto compactsetsand wecandifferentiateunderthe traceandweobtain Consider There holds ( -tD22M)MdD-tD2 trTMj eDj= trjgra'Pi' Te. 242RelativeIndexTheory,DeterminantsandTorsion Quite similar trT(Mj(e-t(i* D'i)2 (i* D'i)2)Mj) =trT0 and inf ae(H), inf ae(H')> 0then limtrT(e-tH P- e-tH' P')=indQ+- indQ-. t--->oo (1.13) o Weinferfromthis Theorem1.6Assumethehypothesesof 1.1andinf ae(D2)> O.Theninf ae(D,2), inf ae(U*i* D'iU)2> 0and foreacht> 0 (1.14) Proof.Inthecase1.1a),inf ae(D,2)>0followsfroma standardfactand(1.14)thenfollowsfrom1.5.Considerthe case1.1b).Wecanreplacethecomparisonofae(D2)and ae((U*i* D'iU)2)bythat of ae(U D2U*)andae((i* D'i)2).More-over,forself adjointA,0~ae(A)if andonlyifinf ae(A2)>O. Assume0~ae(U DU*)and0Eae(i* D'i).Wemustderivea contradiction.Let(1)v)vbeaWeylsequencefor0Eae(i*D'i) satisfyingadditionallyl1>vlL2=1,supp1>v~M\K=M' \K' andforanycompactLcM\K=M' \K' (1.15) Wehavelimi* D'i1>v= O.ThenalsolimD'1>v= O.Weusein V--l>OOV--l>OO thesequelthefollowingsimplefact.If (3isanL2-function,in particular if (3isevenSobolev,then (1.16) RelativeIndexTheory245 Now(U DU*)0issatisfiede.g.ifinD2=V'*V'+ Rthe operator RsatisfiesoutsideacompactKthe condition R"-0.id, "-0> O.(1.34) (1.34)isaninvariantof gen (withpossiblydif-ferentK,"-0).0 ItispossiblethatindD,indD'aredefinedevenif0E{Ye.For the corresponding relativeindex theorem weneed the scattering index. To define the scattering index and in the next section relative(-functions,wemustnowusespectralshiftfunctions which weintroducedinIIIsection2.Accordingtotheorem2.8of RelativeIndexTheory249 chapterIII, == A, A')existsifA,A'areself-adjoint and V=A - A' isof trace class.Then,withR'(z)=(A' - z)-l, =:= 1T-1Iimargdet(1 + VR'(A + if))(1.35) 0:--->0 existsfora.e.A ElR. isrealvalued,EL1 (lR)and tr(A - A')=jdA, IA- A'h.(1.36) I If I(A, A')isthe smallestinterval containing O"(A)U O"(A')then = 0 forA I(A, A'). Let Q={f:lR----+lRI fELlandjli(p)l(l+lpl) dp 0of traceclass.Thenthereexista unique function= = H,H')EL1,loc(lR)suchthatfort> 0, EL1(lR)andthefollowingholds. 00 a)tr(e-tH - e-tH')=-t JdA. o b)ForeveryrpEQ,rp(H)- rp(H')isof traceclassand tr(rp(H)- rp(H'))=j dA. I c) = 0forA < O. o 250RelativeIndexTheory,DeterminantsandTorsion WeapplythistoourcaseE'EAccord-ingtocorollary1.4,DandU*i* D'iUformasupersymmetric scattering system,H= D2,H'= (U*i*D'iU)2.In thiscase e21ri((>",H,H')= det S('x), where,according to II(2.5)and(2.6),S=(W+)*W- =J S('x) dE'('x)and=J,XdE'('x). LetPd(D),Pd(U*i* D'iU)betheprojectoronthediscretesub-space in 1i, respectivelyandPc= 1 - Pdthe projector onto the continuous subspace.Moreoverwewrite (U*i* D'iU)2= :,_). (1.38) Wemakethe followingadditionalassumption. e-tD2 Pd(D), e-t(U*i*D'iU)2 Pd(U*i* D'iU) are fort> 0of traceclass. Then fort>0 (1.39) isof trace classand wecan in complete analogy to(1.35)define -1flimargdet[l + (e-tH Pc(H) e-+O+ _e-tH' Pc(Hd)) (e-tHt Pc(Hd) - e->..t- ic)-l](1.40) Accordingto(1.36), 00 tr(e-tH Pc(H)-e-tH' Pc(Hd))= -t J H, Hd)e-t>..d'x. o (1.41) Wedenoteasafter(1.11)fy= D'inthecase'il'Ecompl,r('il) andfy= U*i*D'iUinthecaseE'E The assumption(1.39)inparticularimpliesthatfortherestriction RelativeIndexTheory251 ofDandfytotheirdiscretesubspacetheanalyticalindexis welldefinedand wewrite inda,d(D,fy)=inda,d(D)- inda,d(D') forit.Set (1.42) Theorem1.11Assumethehypothesesof 1.1and{1.39}. ThennC(A,D, 15')=nC(D, 15')isconstantand ind(D, 15')- inda,d(D, 15')= nC(D, 15').(1.43) Proof. ind(D, 15') 2-,2 tfT(e-tD P- e-tD P')= =trTe-tD2 Pd(D)P - trTe-tD,2 Pd(D')P' + +trT(e-tD2 Pc(D)- e-tD,2 Pc(D'))= 00 inda,d(D,D')+ t Je-tAnC(A, D, 15')dA. o Accordingto1.1,ind(D,D')isindependentoft.Thesame _00_ holdsforinda,d(D, D').Hencet J e-tAnC(A, D, D')dAisinde-o 00 pendentof t.ThisispossibleonlyifJ e-tAnC(A, D, D')dA=t o or nC(A,D, 15')isindependentof A. o Corollary1.12Assumethehypothesesof 1.11andaddition-ally o VIRelative(-functions, 7]-functions,determinants andtorsion Inthischapter,weapplyourprecedingconsiderationsandre-sultstotheconstructionofrelativezetafunctionsandrelated invariants.WewillattachtoanappropriatepairofClifford data arelativezeta function,whichisessentially definedby the corresponding pair of asumptotic expansionsof the heatkernel. Thereforewemustfirstconsidersuchapair of expansions. 1Pairsof asymptoticexpansions AssumeE'E ThenwehaveinL2((E,M), g, h)the asymptoticexpansion trW(t,m,m)rv+...(1.1) t-O+22 andanalogouslyfor with tm-! (m)W'(t, m, m)a! (m)= tr W'(t, m, m) =h, \7), m), = b-%+I(D(g', h, \7'), m). Hereweusethattheoddcoefficientsvanish,i.e.termswith etc.donotappear.Theheatkernelcoefficients haveforl1 arepresentation Ik =LL k=lq=Oil +i2++ik=2(l-k) tr (\7iq+1 RE ... \7ik RE)Ci),,ik, (1.2) whereCil, ... ,ikstands foracontractionwithrespecttog,i.e.it isbuiltupbylinearcombinationof products of the gij,gij' 252 Relative(-functions 253 Lemma1.1-EL1(M, g),0l nt3. Proof.Firstwefixg.Formingthedifference-weobtain asumof terms of the kind 'ViI Rg ... 'Viq Rg tr ['Viq+1 RE ... 'Vik RE _ 'V/iq+1 R,E ... 'V/ik R,EJCil, ... ,ik. (1.3) ThehighestderivativeofRgwithrespectto'Vg occursifq= k, i1=... =iq-1 = O.Thenwehave ('V9)21-2k Rg.(1.4) By assumption, wehave bounded geometry of order :2:r> n+2, i.e.of order:2:n + 3.Hence('V9)i Rgisboundedforin + 1. Toobtainbounded'Vj RLcoefficientsof[... Jin(1.3),wemust assume 2l- 2n + 1,l n+ 3 0 257 (2.1) whichconvergesforRe (s)>andwhichhasameromorphic extensionto Cwithonlysimplepoles.Inparticualar,s=0is not apole.Hence iswelldefinedandonedefinesthe (-determinant ofAas det (A := e-(,(O,A). (2.2) Thisisthefirststeptodefineanalytictorsion.Onopenman-ifolds,(2.1)doesnotmakesensesinceO"(A)isnotnessecarily purelydiscrete.(2.1)canberewrittenas 00 ((s, A)=rts) Je-1(tre-tA - dim ker A)dt.(2.3) o But(2.3)hasameaningfulextensiontoopenmanifoldsaswe willestablish in this section. Definition.AssumeE' EDefine 1 (1(S,D2,(U* D'U)2):=rts) Je-1tr (e-tD2 - e-t(U*D'U)2)dt. o (2.4) Weinserttheexpansion(1.9)intotheintegrandof(2.4),thus 258RelativeIndexTheory,DeterminantsandTorsion obtaining 1 J =1 8-!!: +l' o2 1

dt =1 8+!!: + [n+3], o22 1 _1_ dtholomorphicfor f(8) o (2.5) nn+3 Re (8)+ (-2") + [-2-] + 1 > (2.6) and[nt3] weobtain a function merom orphic in Re (8)> -1,holomorphicin8=withsimplepolesat8=- l,l [nt3]. 00 MuchmoretroublescausestheintegralJ.Herewemustaddi-1 tionally assume (2.7) (2.7)impliesl1e(D,2)=inf o"e((U* D'U)21(ker(U*D1U)2)-L)> 0.De-noteby110(D2),110(D,2)=110((U* D'U)2)thesmallestpositive eigenvalueof D2,D,2,respectivelyand set I1(D2) I1(D,2) I1(D2, D,2)._ min{l1e(D2), 110(D2)}, min {l1e( D,2), 110 ( D,2)}, min{I1(D2) , I1(D,2)}>0.(2.8) If thereisnosucheigenvalueforD2thensetI1(D2)= l1e(D2), analogous forD,2.D2,D,2,(U* D'U)2havein ]0, I1(D2, D,2)[no furtherspectralvalues. Weassertthatthe spectral function =D2,(U* D'U)2) isconstantintheinterval[0,I1(D2,D,2)/2[. Relative(-functions259 Consider the function We: (X)={ce:e - IxlSEand choose oIxl>E Ce:s.t.J We: (x)dx=1.Let0 O. -2-2-2-2 a)Then((s, D2, D'),((8, D2,D"),((8, D',D")areafter meromorphicextensioninRe (8)>-1welldefinedandholo-morphicin8=O.Inparticulardet(D2, D,2)=e-('(O,D2,D'\ det(D2,D,,2)=e-('(O,D2,I5I1\det(D,2,D,,2)=e-('(O,D,2,I5112)are welldefined. b)Thereholds (3.6) etc.and Proof.a)followsfrom2.1andthefactthatE, E"E gencomp(E)impliesE" E gencomp(E')(= gencomp(E)). b)immediatelyfollowsfromthedefinitionsandtr( e-tD2 e-tJ5112) =tr(e-tD2 _e-tD,2)+ tr(e-tD,2_e-tI5l1\D Relative(; -functions 269 4Relativeanalytictorsion IfwenowrestricttothecaseE=(A*T*M 0C,gA)theng'E gencompl,r+l(g)doesnotimplyE'=(A*T*M E gensincethefibremetricchanges,gA----t Hencethe aboveconsiderationsforconstructingtherelative(-functionarenotimmediatelyapplicable,sincetheyassumethe invariance of the fibremetric.Fortunately wecan define relative (-functions alsoin this case.Werecallfrom[38],p.65- 74the followingwellknownfactwhichweusedin(1.1),(1.2)already. LetPbeaselfadjointellipticpartialdifferentialoperatorof order2suchthattheleadingsymbolofPispositivedefinite, actingonsectionsofavectorbundle(V,h)----t(Mn, g).Let Wp(t,p,m)be the heatkernel of e-tP,t>O.Then fort----t0+ trWp(t,m,m)rvr%_!j(m) + r!j+lb_!j+l(m) + ... andthebv(m)canbelocallycalculatedascertainderivatives ofthesymbolofPaccordingtofixedrules.Asestablished byGilkey,forP=D.orP=D2theb'scanbeexpressed bycurvatureexpressions(includingderivatives).Thisis(1.1), (1.2),(1.3).Weapplythistoe-tL:;.ande-t(U*i*L:;.'iU)butwe want to compare the asymptotic expansions of WL:;.(t,m, m)and WL:;.,(t,m, m).The expansionof ( 4.1) and (4.2) coincidesince Thepointisto comparetheexpansionsof (4.4) and (4.5) 270RelativeIndexTheory,DeterminantsandTorsion i.e.wehaveto compare the symbol of i* b.'i=i* b.'andb.'.For q=0theycoincide.Letq=1,mEM,WI,... ,Wn abasisin T:nM,Enl(M),b.'lm= eWl + ... + Then,according toIV(2.55)i*(b.'i)lm= i.e. ((i*-=-(4.6) Hence forthe(local)coefficientsof i* b.'iasdifferentialoperator holds coeff of(i* i) =(gklcoeff of(4.7) Quite similar for0.e-tA d>'=0and -J1, -00 00 + Je-tA2211_d>.] J1, -t!!c" =C e2. o Theorem 5.4AssumeE'Ekr+ 1> n+ 3and inf o"e( D21 (ker D2).l.)>O.Thenthereisawelldefined relative ",-function 00 '1l(sDD'):=1JtS;ltr(De-tD2 _U*D'Ue-t(U*D'U)2)dt 'I"r0 (5.10) which is defined for Re (s)>and admits a meromorphic exten-sion toRe (s)>-5.It isholomorphicat s= 0if thecoefficient J a_!(m) dvolm(g)of t-!equalstozero.Thenthereisawell 2 definedrelative ",-invariantof thepair (E, E') , Proof.WewriteagainU* D'U=fy.Thenaccordingto 276RelativeIndexTheory,DeterminantsandTorsion proposition5.1, 00 D, fy)r [[1',' tr(D- e-HY' 15') 1dt 00 + rI t ',' tr(e-W'D- e-H'''fj')dt 1n+31J =(s+1)'" snI1a_!!l dvolm(g) r- 2 21=02222M 1 1J 8-14 + r0t-2 O(t2)dt (5.11) (5.12) Weinferfrom(5.9)that(5.13)isholomorphicin C.(5.12)is holomorphicinRe (s)>-5.(10.45)admitsameromorphicex-tension toC.T/(s,D, D')isholomorphic at s= 0 if the coefficient J a_! (m) dvolm(g)equalsto zero.0 M2 Theorem5.4immediatelygeneralizestothecaseofadditional compactperturbations. Theorem5.5Assume E' Ekr+1> n+3 Relativer:;, -functions Thenthereisawelldefinedrelative'T/-function 00 71(SDfy):=1J(;1 tr(De-tD2 P- fYe-tfy2 Pl)dt '1"r e! 1)0' 277 whichisdefinedforRe (s)>~andadmitsameromorphicex-tensiontoRe (s)>-5.Herefy= U* D'Uasabove.'T/(s,D,fy isholomorphicats= 0if theintegratedcoefficient a_!=Jb_! (D2,D, m) dvolm(g) 22 K -Jb_ frac12 (fy2,fy, m/) dvolm, (g') K' + J M\K=M'\K' equalstozero. Werepeatfortheproof the singleargumentsfromtheproof of 5.4whichremainvalidinthecaseof 5.5.0 6Examplesandapplications Inthis section,wepresent examples of pairs of generalized Dirac operatorswhichsatisfytheassumptionsofsections1-5and presentapplicationsof sometheoremsof thesesections. Let(Mn,g)beopenwithfinitelymanycollaredendsCi,the collar[0,oo[ XN;-l of Ciendowed with a warped product metric, i.e.glC:i~dr2+ fi(r)2da}vi'Niclosed,hi= da7vi,i=1, ... ,m. WeconsideroneendCwithcollar[0,oo[xNwiththewarped product metric ds21c:= dr2+ f(r)2da2 and wefirstcalculate the curvature. LetUo, U1,... ,Un-1 beanorthogonal basisinT(r,u) ([0, oo[xN) withrespecttods2,U=:"U1,... ,Un-1 orthonormalinTuN 278RelativeIndexTheory,DeterminantsandTorsion withrespecttod(J"2=h.Then,incoordinates(r,u\ ... ,un-I), weget forthe Christoffel symbolsCt,(3,'"Y=0,... , n -1, the followingexpressions rgo =0,rgj =0 forj> 0, rk 0ck0rk - !Ls:kc.k0 00=lor>,OJ- 2f' U jlor J,>, (6.1) r?j=-1' fhij fori, j> 0,= fori, j, k> o. Forthecurvaturetensorandthe sectionalcurvatureholds 1" R(Uo, Ui)UO =jUi(6.2) R(Uo, Ui)Uj=-1" fhijUO (6.3) R(Ui, Uj)Uo = 0(6.4) R(Ui, Uj)Uk =- f'2(hjkUi - hikUj ) + RN(Ui, Uj)Uk, (6.5) whichimpliesimmediately (6.6) (6.7) Herei, j, k= 1, ... ,n - 1.The easycalcualtionsareperformed in[28].It isnoweasyfrom(6.2)- (6.7)to calculate the general curvatureK(V, W). Examples6.11)Takef(r)=e-r,Nfiat,thenK==-1,E satisfies(Bo)but rinj(E)=O. 2)Choosef(r)= e-r,KN =1=0,then Edoesnot satisfy (Bo)and againrinj(E)= O. 3)If f(r)= er,Nfiat,then Esatisfies(Bo)and rinj(E)> O. 4)Finallytakef(r)= er2,thenEdoesnot satisfy(Bo)but(I). Henceallgoodandbadcombinationsof propertiesarepossible. D Relative(-functions279 Proposition6.2Supposef(r)suchthat inf f(r)>0orfmonotoneincreasing(6.8) r and If(lI)1::;cllf, v= 1,2, ....(6.9) Then 9 Iesatisfies(/)and(Boo). Proof.inf f (r)>0andrinj (N, h)>0immediatelyimply r rinj(C:)>O.(6.6)and(6.7)immediatelyimply(Bo).(Bdis equivalentwith(Bo)and IV'eJR(eiJ,e),)e,,)lx::;C IV'e",eiJlx::;c, (6.10) (6.11) eo,... ,en-ltangentialvectorfields,orthonormalinTxM.We apply this to x= (v,y)E[O,oo[xN,Uo =tr,Ui =a ~ i 'eo= Uo, - UI- Un-IThd't(61) el- f' ... ,en-l- -1-'enaccormg0., V'eieO Wesee,eachtermonther.h.s.of(6.12)- (6.15)is- uptoa constantorboundedfunction- asumof terms l'l'1 jei, jeo, 7el withpointwisenorms(w.r.t.gle) I I'll j'7' (6.16) (6.17) 280RelativeIndexTheory,DeterminantsandTorsion i.e.(6.11)issatisfied.Nextweestablish(6.10) f 1'"- ~ f '1" Pei, (6.18) 111" P "Vuj(R(Uo, Ui)Uo) =p"VUjjUi f"f",k J3 "VUjUi =J3 (- ffhijUO+ rij(h)Uk) 1" f'1"k -yhijeO + prij(h)ek, (6.19) 1 "VUa pR(Uo, Ui)Uj ) - ~ 'R(Uo, Ui)Uj ) + )2 "VUa( - 1"fhijUO) - ~ '(-1"fhijUO) + )2 (-1"'f - 1"f')hijUO 2f'1"(fill1"f') phijeO - T+ Yhijeo ( f'1"fill) Y- Thijeo,(6.20) )3 "Vuk(R(Uo,Ui)Uj)=)3 "V Uk (-f"fhijUO) 1"I1"If'I - phijrkOUl= - phij'2j[)kUl I1"f' -'2phijUk, (6.21) Relative(-functions281 1 J2 "VUo(R(Ui, Uj)Uo) =0 1 J3 "Vuk(R(Ui , Uj)Uo) "Vek(R(ei,ej)eO),(6.22) 1 r "Vul(R(Ui , Uj ), Uk) 1[/2( =f4 "V Ul- fhjkUi - hikUj +RN(Ui, Uj)UkJ 1'2 -]4(hjk "VUIUi - hik "VUIUj ) 1 +f4 "Vul(RN(Ui, Uj)Uk) 1'2 -]4(hjkrr:(h)Um - hikrlj(h)Um) 1 +r "Vul(RN(Ui, Uj)Uk).(6.23) If wetake the pointwisenorm of the r.h.s.of (6.18)- (6.23)and applythe triangleinequality,then weobtain on the r.h.sidesa finitenumberof terms,eachof whichis- up to aconstant ora bounded function- of the type 1If(Vl)Ilf(V2)1 faJ2 a2:O.(6.24) But accordingto(6.9),eachterm of the kind(6.24)isbounded on c,i.e.weestablished(Bl)' Toestablish(B2)'wehaveat the endto estimate expressionsof thekind (j).(j)', (f ' ) . ~ ff' (6.25) 282RelativeIndexTheory,DeterminantsandTorsion Again,accordingto(6.9),eachtermofthekind(6.25)is bounded. Averyeasyinduction nowproves(Bk)forallk,i.e.(Boo).0 Collaredendsareisolatedends.Hence,ifallendsofanopen manifoldMnare collared,thenMncan haveonly afinitenum-ber of ends.If an open manifold has an infinite number of ends, then at leastoneendisnotisolated. Theorem6.3Let(Mn,g)beopen.IfeachendEof Mnis collaredthenMhasonlyafinitenumberofends,El,., Em. Supposeglc;~dr2+ Jl(r)dO"Jv;suchthateachfisatisfies{6.8} and{6.9}.Then(Mn, g)satifies(1)and (Bo). This followsimmediatelyfromproposition6.2.0 Interestingexamplesforthefsaref(r)=e9(r) ,g(r)>0and g(v)(r)boundedforallv. Weconsiderherethe specialcaseg(r)= b r,b> 0i.e. (6.26) Inthesequel,weneedtheknowledgeof theessentialspectrum O"eof suchmanifolds. Theorem6.4Suppose(Mn,g)hasonlycollaredendsEi,i= 1, ... , m,eachof themendowedwithametricof type{6.26}. Thenthereholds O"e(t:.q(Mn , g))\{O} ~[ m,m( min {(n -~ q- 1)\ ~ ,(n -~ q+ 1) \; }) , 00 [\{O}for q =I~(6.27) and (6.28) Relative(-functions283 Wereferto[3Jforthe proof whichessentiallyrelieson[28J.0 Corollary6.5Supposethehypothesesof 6.4,nevenand > O. (6.29) ThenthegradedLaplaceoperator D.=(D.o,... ,D.n)hasa spec-tralgapabovezero.0 Aspecialcaseof theorem6.4isthe caseof arotationallysym-metricmetricatinfinity,i.e.(Mn\KM, gIM\K)(lRn \KlRn, dr2+ e2br Then forb >0 n forq -:F2' (6.30) If wereplacee2br by(sinh 1')2andsetK=0,thenwegetthe realhyperbolicspaceH:!:.land 0'(D.(Hn))=2'2'00 {

[ eq-1}[1[ {OU4,00 forq -:F% forq =% (6.31) Corollary6.6Inthecase(Mn,g)(lR2k,dr2 +or (Mn, g)= thegradedLaplaceoperatorD.=(D.o,... , D.n) hasaspectralgapabovezero.0 Corollary 6.7Inthefollowingcases,thegradedLaplaceoper-ator D.=(D.o,... ,D.n)hasaspectralgapabovezero. a)(Mn,g)isa finiteconnectedsumof manifoldswithcollared ends,warpedproductmetrics{6.26}satisfying{6.29}andnis even, b)anycompactperturbationof manifoldsof a}, c)anyfiniteconnectedsumof manifoldsof type(Mn\KM, gIM\K)(lRn \b> 0and neven, 284RelativeIndexTheory,DeterminantsandTorsion d)anycompactperturbationof c), e)any finiteconnectedsumof thehyperbolicspaceH'!.-1, f)anycompact perturbationof e), g)any(M2k, g'),g'E 9of typea)- f).0 Remark 6.8Compactperturbationsandconnectedsumsof collaredmanifoldswith(6.26)areagainofthistype.Wein-troduced6.6.a),b)toindicatehowtoenlargestepbystepa givenset of suchwarpedproductmetricsat infinitybyforming connectedsumsandcompactperturbations.0 Weapplythefactsabovetothecase E= (A*T* MC, gA*), V9A*), D= d + d*, D2= D.=(D.o,... ,D.n). Theorem6.9Let (M2k,g)beoneof themanifolds6.6 a)- f), g'E g'smooth,r+ 1 > 2k + 3.Thentherelative (-function(q(s, Do,Do')asinsection4 andtherelativeanalytic torsion,Ta((M, g), M', g')), 2k log Ta((M, g),(M', g'))=2:) -l)qq Do,Do') q=O arewelldefined. Proof.Accordingtoproposition6.2,(Mn,g)and(M',g') satisfy(1)and(Boo),andthegeneralLaplaceoperatorhasa spectral gapabovezero.Theassertionthenfollowsfromtheo-rem4.4.0 Corollary 6.10Let(M,g)beasis6.6a)- g).Thenthe attachment(M,2k, g')----tTa((M, g),(M', g'))yieldsacontri-butiontotheclassificationof theelementsof gen(A*T*,gA*)' Relative (, -functions285 Remark6.11If n= 2k + 1 and(Mn, g)belongsto oneof the classes6.6a)- g)then the relative(-functions (( s, !::l.q,are for(Min, g')E g)n Cr+l andq1=k, k + 1 welldefined.D Anotherveryspecialcaseisgivenbyb =0 in(6.26),i.e.cylin-dricalendsE, ds2 = dr2 + da'iv.(6.32) Suppose,wehavemcylindricalendsEi,i=1,... , m, andlet{Xl (i)} kbethe(purelydiscrete)spectrumof!::l.q (Nr-l,hi),i=1, ... , m. Proposition6.12Then ae(!::l.q(M, g= UU([Ak(i), OO[U[Arl(i), oo[).(6.33) ik Wereferto[3]and[28]fortheproof.D Corollary6.13a)If Hq(Ni) =Hq-l(Ni) =(0),i=1,... , m, then a(!::l.q(M))hasaspectralgapabovezero. b)If foratleastoneiHq(Ni)1=0,thena(!::l.q+l(M))=a (!::l.q(M)=[0,00[. e)Inthecaseof cylindricalends,thegradedLaplaceoperator never hasaspectralgapabovezero. Proof.a)andb)immediatelyfollowfrom(6.34).c)follows fromHO(Ni)1=0foralli,hencea(!::l.o(M))=a(!::l.l(M)) [0,00[=a(!::l.o,... ,!::l.n).D Corollary6.14Suppose(Mn, g)withcylindricalends El,'",Emandlet(M,n,g')EgencompL,diff,rel(Mn, g).If Hq (Ni)=Hq-l(Ni)=(0),i=1, ... , m,then((s, !::l.q,and det(!::l.q,arewelldefined.D 286RelativeIndexTheory,DeterminantsandTorsion Aspecialcaseisgivenbythepair (Mn,g)and (Q N,x[0,Nx[0,00[' Qdr' + INr (M'", g'). HereM'nisamanifoldwithboundary8M'n=N=UNi, andthe latter fallsoutfromourconsiderations.Butif wecon-siderthecaseq=0andb..o (Min, g')withDirichletboundary conditions at8M' = N= U Ni,then wegetan essentiallyself-adjointoperator =[0,00[=[O,oo[UU [Aj, 00[,and Aj>O e-tLlo -isoftraceclass.Thelatterfactisanimme-diateconsequenceof theproof of theorem3.9.Hencethewave operator W(b..o, exist,are complete and the absolutely con-tinuousparts of b..o and areunitarily equivalent.Weintend topresentanexplicitrepresentationofthescatteringmatrix S(A),of tr(e-tLlo - andof therelative(-function.Here weessentiallyfollow[49].Then and O"p(b..o)consistsof eigenvalues o < /-ll::;/-l2::;/-l3::;. . . --+ 00 of finitemultiplicitywithoutfiniteaccumulationpointand (6.34) (6.35)immediatelyimpliesfort> 0 (6.35) and (6.36) RelativeC. -functions287 whereistherestrictionoftothe subspacespanned bytheL2-eigenfunctions.If weapply(6.34)tothecaseq=0 then weobtain (6.37) ikj wheretheAjSarethe eigenvaluesof andsimultaneously thethresholdsof the(absolutely)continuousspectrum. Wedescribethecontinuousspectrumintermsofgeneralized eigenfunction.Consideran eigenvalueAjEandletE(Aj)bethecorresponding eigenspace. Forf../,>Ajanddifferentfromallthresholdsandfor'IO. Proposition6.18Undertheaboveassumptions,therelative (-function ((8,+ z,+ z)for >0isgivenby jk (6.40) Proof.Atfirstweremarkthatfor00,whichisequivalentto = -bo - O(Ae)forA --->0+.Hence(2.24)andbothassumptions Relative (, -functions289 of proposition2.4are satisfied.Weobtain fromcorollary2.5a) and(6.39)an asymptoticexpansion 100. PI)"" dim kerz= t--+ 00. j=l Moreover,fort--+ 0+,wehavethe standard expansion 00 -"" t--+ 0+ , j=O andwegettheexistenceof therelative(-function((8,asameromorphic function.Accordingto[49] A J:z log det S(A)dA=O(A n)asA--+ 00. o (6.41) Usingthisand inserting(6.39)into(,),weobtain finally(6.40). D Itisclearthattheproductgeometryofcylindricalendsisan extremlyspecialcaseof possiblegeometrieson E=Nx[0,00[, andoneshouldadmitmuchmoregeneralboundedgeometries onM,e.g.boundedgeometriesofthetypegc=gINX[O,oo[= dr2 +(e9(rl)2dc/iv,g(r)>0,g(l/l(r)boundedforall1/.Again wegetapair== where(E,gc)E l,r(M)pIft1If gen comp L,diJ J,rel, g,e- e00racec ass. inf > 0orfort--+ 00 (6.42) thentherelative(-function((8, andrelativedetermi-nantaredefined.Explicitg(r)leadstoexplicitcalculations. Similarlyforq>O.Ife.g.n=2m,9onMissuchthat glc=dr2+erdO"'ivand 9 =mthen inf >O.Here wetakefor theFriedrichs'extensionof onMx[O,oo[ 290RelativeIndexTheory,DeterminantsandTorsion with zeroboundary condition on Mx{O}.Hence((8, .6.m,anddet (.6.m,arewelldefined.Inthecaseofamanifold M= M'UN x[0,00[,9)with cylindrical ends,the main and in-teresting partof the geometry iscontainedinthe compactpart M',8M'=N.Attheboundaryweassumeproductgeometry. X=(M'UM', 9x =9M,U9M'is then a closed manifold.It is now N anaturalandinterestingquestion,howarethe.6.-determinant forXandtherelative.6.-determinantforMrelated?Thean-swerwouldalso giveameaning,an interpretation of the relative determinant.Acertainansweriscontainedin[53],and wegive an outlineof thecorrespondingresult. Consider the following situation.Given(Mn, 9)closed,oriented, connected,YcMahypersurface,separating Minto twocom-ponentsMI,M2.SetMi=Mi.i=1,2.Miarecompact withboundaryY,M=MIU M2,Y=8MI =8M2.More-y over,letE---+MbeaHermiteanvectorbundleand.6.= .6.M :COO(M,E)---+COO(M, E)aLaplacetypeoperator,i.e . .6.issymmetric,non-negative with principal symbol 0' 0 thenh=0 in(2.9),(2.24)for the pair D2 + a,iy + a is satisfiedand the corresponding relative Relative(-functions293 (-function isgivenby 00 (s, D2+ a,fy2+ a)=rts) JtS-le-tatr(e-tD2- e-tfy2)dt, o (6.47) Re( s)> -1.The r.h.s.of (6.47)isalso welldefined if wereplace a by any zE C with Re(z)> O.Then the corresponding function (s, z, D2,fy2admitsasfunctionof samerom orphicextension to Cwhichisholomorphicat s=0and wefinallydefine det(D2 + z,fy2+ z):= e-/sls=o((s,z,D2,D/2). Animportantproperty concerningthez-dependenceisexpres-sedby Proposition6.21SupposeE'E {2.24}.Thendet(D2 + z,fy2+ z of zE C\]O, 00[. l,r+l(E)d gen comp L,diJ J,relan isaholomorphicfunction Proof.Hereweessentiallyfollow[49]. Accordingto V,lemma1.10a), 2c00 tr(e-tD2 Pe-tD/2 PI)= -t J - t J o2c (6.48) Thesecondintegralonther.h.s.isO(e-tC)fort----+00since E L1IRfort>O.Moreover. 2cN J = I: Cktk + O(tN+1)fort----+0,(6.49) ok=O NEINarbitrary.(1.16)and(6.49)imply(bytakingthedif-00 ference)thatJdAhas asimilar asymptotic expansion as 2c (1.16).Weinferfromthisthat theintegral 0000 F(s, z)=rts)JtSe-tz Jo2c 294RelativeIndexTheory,DeterminantsandTorsion inthehalf planesRe(s)> andRe(z)>-c absolutelycon-vergesand,as functionof s,it admitsameromorphic extension to Cwhichisholomorphicats= o. Thefirstintegralonther.h.s.of(6.48)canbediscussedas follows.Weobtain forRe(z)> 0 002c __1_ JtSe-tz J f(s) o0 2c00 =_1_ J Jee-t(zH)dtd).. f(s) o0 2c = -s J(z +o Hence,forRe(z)>0, 2c 2- 2J(O,z). det(D+z,D'+z)=eo s (6.50) (6.51) Ther.h.s.of(6.51)hasanobviousextensiontoananalytic functionofzE C\]- 00,0],c:: (0, z)isholomorphicinRe(z)> -c.c> 0wasarbitrary,hencewegetananalyticextensionto C\]- 00,0].0 Itisaninterestingquestion,howtherelativedeterminantand therelativetorsionchangeunderI-parameterchangeofthe metric. Wecouldconsidere.g.the mostnatural evolution of the metrix whichisgivenbytheRicciflow, o. 07g(7)=-2RIC (g(7)),g(O)=go(6.52) If (Mn, go)iscomplete and has bounded curvature then,accord-ing to[69],[25],there exists for0 :::;7:::;Tin the class of metrics withboundedcurvatureauniquesolutionof(6.51). Relative (, -functions295 Denote - A(T)isdefinedasbeforebythe diagram

L2(g(0)):) 'D{),.(O) ----t L2(g())

dvol(O)1 dvol(r) i. dvol(r) dvol(O) L2(g(T)):) 'D{),.(r)

L2(g( T)) ----t Itisnotyetclear,whetherg(T)Ecompl,r+l(g(O)).Weproved in[25]thatg(T)Ebcomp2(g(0)) , butg(T)Ecompl,r+1(g(O))is still open.Thereforewemakethe following AtE'l,r+l(E)l,r+l(E)H ssumpIon.Ecomp L,dif fCgen comp L,dif f.ere comp()denotesthearccomponent.Hencethereexistsanarc connectingEandE',weassumeinthe sequelto thearcto beat leastCl. Then weget aCl-arc {D(T)}rland wewill study the behaviour of the relative determinantdet(D2(0), D2(T))undervariation of T.Additionally,wesupposeagain(2.24).Asbeforeandinthe sequel,D2 (T)denotesthetransformedtotheHilbertspacefor T = 0D2(T). DenoteD2(T)=trD2(T).By Duhamelsprinciple, t e-tjj2(r)= Je-sjj2(r) D2(T)e-(t-s)jj2(r)ds. (6.53) o IfD2(T)e-tD2(r)isfori> 0of traceclasswithtracenormuni-formlyboundedoncompacti-intervalslao,al],ao>0,then according to(6.53),isalso of trace class fori> 0 and (6.54) 296RelativeIndexTheory,DeterminantsandTorsion To establish in the sequel substantial results,wemustmake two additionalassumptions. Assumption1.D2(T)isinvertibleforT E[-s,s].(6.55) Assumption2.Thereexistsfort--t 0+anasymptoticex-pansion 00k(j) tr(D2 (T)b-2(T)e-tjj2(T))-5.(6.62) (6.62)yields (6.63) and(6.63)hasameromorphic extensionto Cwhichisholomor-phicat s= O.Hence 298RelativeIndexTheory,DeterminantsandTorsion iswelldefinedandwesetinanalogy to(6.61) det(D, D'):= det(IDI, ID'I). e(1](O,D,D')-((O,IDI,I1J'I)).(6.64) But to define(6.64)wedon't needD,D' invertible.The former assumptioninf a"e(D2 I(ker D2).L)>0issufficienttodefine(6.64) sincethisassumptionissufficientforthe1.h.s.of(6.63)and henceforthe Lh.s.of(6.64).D FinallywedrawsomeconclusionsfortheSchrodingeroperator fl.q + V,VasinIV,proposition 3.11. Theorem6.24Suppose9EM(I, Bk),k~r>n+ 2,VE n1,r (Mn, 9)areal-valued function.Thentheabsolutelycontin-uouspartsof fl.q and fl.q + (V-)areunitarilyequivalent. Proof.ThisimmediatelyfollowsfromIV3.11andthecom-pletenessof thewaveoperators. D Corollary6.25SupposeVEn1,r (IRn, 9standard) ,real-valued, n2 r>n+ 2,fl.=- L ~ 'Thentheabsolutelycontinuousparts i=l' of fl.and fl. + (V-)areunitarilyequivalent.D VIIScattering theoryfor manifoldswith injectivityradiuszero InchaptersII- VIwealwaysassumedrinj(M, g)>O.The backgroundforthisassumptionwasthefactthattoestablish ouruniformstructures,weusedthemodulestructuretheorem forSobolev spacesand thistheorem hasbeenproveduntilnow under the assumption rinj> O.An extension to the case rinj= 0 forweightedSobolevspacesisinpreparation. In[54]W.MullerandG.Salomonsenintroducedascatter-ingtheoryformanifoldswithboundedcurvature,admitting Tinj(M, g)= O.They restrict themselvesto the case of the scalar Laplacian~ o=(\79)*\79.Wepartially followherean extended toarbitraryRiemannianvectorbundlesversionoftheirap-proach but using our language of components of uniform spaces andourproceduresof chapterIVtoestablishtraceclassprop-erties. 1Uniformstructuresdefinedby decayfunctions Weapplytheprocedureof[27],[32],ourchapterIIandrefor-mulateandextendthe approachof[54]. LetVECO ([1 , oo[)benon-increasing.It iscalledofmoderate decayif and 1)supx V(x)< 00 XE[l,oo[ 2)thereexistsC=C(V) (1.1 ) suchthat V(x + y)~C(V) . V(x). V(y).(1.2) 299 300RelativeIndexTheory,DeterminantsandTorsion Itiscalledof sub exponentialdecayif foranyc > 0 eCXV(x)---t00.(1.3) x---?oo Examples1.11)V(x)=e-tx isfort> 0of moderatedecay. 2)If V,Vi,V2 areof moderatedecaythenthisholdsforV", 6 > 0,Vi. V2 too.Thesameholdsforsubexponentialdecay. 3)V(x)=X-Iande-X,00such 308RelativeIndexTheory,DeterminantsandTorsion that rinj(h, x)2:min{ c . rinj(g, x), c'},xEM. Wereferto[54]fortheproof.D A special case isgivenbyh Ecomp,v (g),Vof moderate decay. SetforxEM hnj(x):= min {12JK, rinj(X) }. Then,underthe assumptionsof lemma2.1, hnj(h, x)2:C2i\nj(g,x). Usingthe standard volumecomparisontheorem Jr(sinh tv'K) n-l V(K)dtSvol(Br(xo)) < JT(sinh tv'K) n-l dt(2.1) - f(!!:)v'K 20 andtheorem4.7from[18], _r1 rinj(X)2:"21 +(2.2) ro+ 2s Ce-(n-l)vKd(x,y)EM InJ_,X. (2.4) Lemma2.2ThereexistsaconstantC =C(K),suchthat (2.5) TheCaseInjectivityRadiusZero309 Wereferto[54]forthe proof. D WerecallfromII,lemma1.10theexistenceof appropriateuni-formlylocallyfinitecoversof (Mn, g)if Ric (g)k. Setfors>E0,/'i,c:(M,g,s)EINU {(X)}=smallestnumber such that there exists a sequence such thatisan opencoverof Msatisfying sup #{i EINlx EB3s+c:(Xi)}/'i,c:(M,g,s)(2.6) xEM andset/'i,(M,g, s):= /'i,o(M,g,s),/'i,(M,g, 0):= 1. We need fornorm estimates