Generalized Riemann-Hilbert Transmission and Bound

arXiv:math/0111076v2[math.KT]28Nov2001GENERALIZEDRIEMANN-HILBERTTRANSMISSIONANDBOUNDARYVALUEPROBLEMS,FREDHOLMPAIRSANDBORDISMSBogdanBojarskiandAndrzejWeberAbstract.WepresentclassicalandgeneralizedRiemann-HilbertprobleminseveralcontextsarisingfromK-theoryandbordismtheory.ThelanguageofFredholmpairsturnsouttobeusefulandunavoidable.WeproposeanabstractformulationofanotionofbordisminthecontextofHilbertspacesequippedwithsplittings.§1.IntroductionTheconceptofaFredholmpairP=(H−,H+)ofclosedsubspacesH−,H+ofaHilbert(orBanach)spacewasintroducedin1966byT.Katoinhisstudiesofstabilitypropertiesofclosed,mainlyunboundedoperators,[19].RecallthatthepairP=(H−,H+)isaFredholmpairifthealgebraicsumH−+H+isclosedandthenumbersαP=dim(H−∩H+)andβP=codim(H−+H+)arebothfinite.WealsoassumethatH−andH+areofinfinitedimensions.ThedifferenceαP−βPwasdefinedin[19]astheindexofthepair,IndP,andthecrucialobservationofT.KatowasthatInd(H−,H+)isnotchangedby,,small”deformationsofthepair.Moreprecisely,thesetFGr2(H)ofallFredholmpairsofaHilbertspaceappearsthenasanopensubsetoftheCartesianproductGr(H)×Gr(H)oftheGrassmannianofclosedsubspacesofHsuppliedwiththeusual,,minimalgap”metric,see[17,19].InthiscontextthenotationFGr2(H)canbeinterpretedastheFredholmbi-GrassmannianoftheHilbertspaceHwhichgeneralizesinanaturalwaytotheFredholmmulti-GrassmannianFGrn(H),wheninsteadofpairsofsubspacesweconsidern-tuples(M1,...,Mn)ofclosedsubspacesformingFredholmfans,[4,6].TherewasnodoubtfromtheoutsetthatthetheoryofFredholmpairsandtheirgeneralizationsshouldbestudiedincloserelationshipwiththetheoryofFredholmoperators.ThusFredholmpairsinKato’s[19]wereconsideredasaconvenientextensionofthetheoryofFredholmoperators.ForaFredholm,possiblyunbounded,closedoperatorA:H1→H2actingbetweenHilbert(Banach)spaces,thepairPA=(graphA,eH1)wasin[19]thebasicexampleofaFredholmpair.Here1991MathematicsSubjectClassification.Primary58J55,55N15;Secondary58J32,35J55.Keywordsandphrases.Riemann-Hilbertproblem,boundaryvalue,Fredholmpair,K-theory,bordism.BothauthorsaresupportedbytheEuropeanCommissionRTNHPRN-CT-1999-00118,Geo-metricAnalysis.ThesecondauthorissupportedbyKBN2P03A00218grant.ThesecondauthoralsothanksInstytutMatematycznyPANforhospitality.TypesetbyAMS-TEX12BOGDANBOJARSKIANDANDRZEJWEBERthe,,coordinate”subspaceeH1=H1⊕0andthegraphofAareclosedsubspacesinthedirectsumH=H1⊕H2.Moreover,IndPA=indAwhereindAdenoteshereandinthesequeltheindexoftheFredholmoperatorA.Alsoin[4]thebi-GrassmannianFGr2(H),understoodtherealsoasthespaceofabstractRiemann-Hilberttransmissionproblems,wasparameterizedbyafamilyofFredholmoperatorsLPassociatedwithprojectors(P−,P+),notnecessarilyorthogonal,ontothespacesofthepairP.ThetheoryofFredholmoperatorsinHilbertspaceturnedouttobeanimportanttoolforstudyingtopologyofmanifoldsandK-theory,especiallythegeometricalandtopologicalinvariantsdefinedbyellipticdifferentialandpseudodifferentialoperatorsinspacesofsectionsofsmoothvectorbundlesonmanifolds.ThehighlightalongthatroadwasthefamoussolutionbyM.AtiyahandI.Singer,[2],oftheindexproblemforellipticoperators.IntheabstractfunctionalanalyticsettingthespaceF(H)ofFredholmoperatorsintheHilbertspaceH,topologizedasasubsetoftheBanachalgebraB(H)ofboundedoperatorsinH,turnedouttobetheclassifyingspaceforthefunctorK0(−),the0-thtermofthegeneralizedcohomologytheoryK∗(−),[1].LatertheK-homologyK∗(X)ofatopologicalspaceX(orK∗(A)foraC∗-algebrainthenoncommutativecase)wasintroduced,[18].AccordingtoKasparovthegeneratorsofK∗(X)arerealizedbycertainFredholmoperatorsactinginHilbertspace,whichisequippedwithanactionofthealgebraoffunctionsC(X).TherootsoftheextremelysuccessfulapplicationsoftheFredholmoperatorsinglobalanalysis,geometryofellipticoperatorsandK-theory,undoubtedlyarerelatedwiththefollowingbasicfeaturesoftheclassF(H):(i)ThesetF(H)isstableundersufficientlysmallperturbationsinB(H)i.e.A∈F(H)⇒A+B+K∈F(H)forB∈B(H),kBkεA(forasufficientlysmallεA)andK∈K(H),whereK(H)denotestheidealofcompactoperatorsinH;(ii)Compositionlaw:A∈F(H),B∈F(H)⇒A◦B∈F(H)andindA◦B=indA+indB.Theindexhomomorphismind:F(H)→Zissurjectiveanddescribesthesetofcomponentsπ0(F(H));(iii)Ininterestingandimportantcases,arisinginthetheoryofpartialdifferentialequationsandboundaryvalueproblems,theHilbertspaceappearsasafunctionspaceoveramanifold,usuallyafunctionspaceofSobolevtype.ThereforeitwasnaturaltoconsiderHilbertspacesequippedwithanactionofthealgebraoffunctionsB=C(X)overatopologicalspace(usuallyamanifold)X.MoreRIEMANN-HILBERTPROBLEM,FREDHOLMPAIRS,BORDISMS3generally,itwasassumedin[18]thattheconsideredHilbertspacesareHilbertmoduleswithsomeC∗-algebraBactionr:B→B(H).Thecondition∀b∈B:[r(b),A]∈KdistinguishesaclassofoperatorsA∈Bwhichisofspecialinterest.Incon-sequenceitrestrictsalsotheclassofFredholmoperators.Itisaremarkablefact,thattheellipticpseudodifferentialoperatorsbelongtothedescribedaboveclassforthestandardmultiplicationrepresentationofthealgebraofcontinuousfunctions.ThecalculusofcommutatorsandtheirtraceswasthestartingpointforA.Connesforintroducingcycliccohomologyandproclaimingtheprogramofnoncommutativegeometry,[12].Thenaturalandintimateconnectionof