On the configuration space topology in general rel

arXiv:gr-qc/9301020v125Jan1993Freiburg,THEP-92/32ONTHECONFIGURATION-SPACETOPOLOGYINGENERALRELATIVITYDomenicoGiulini*Fakult¨atf¨urPhysik,Universit¨atFreiburgHermann-HerderStrasse3,D-W-7800Freiburg,GermanyAbstractTheconfiguration-spacetopologyincanonicalGeneralRelativitydependsonthechoiceoftheinitialdata3-manifold.Ifthelatterisrepresentedasaconnectedsumofprime3-manifolds,thetopologyreceivescontributionsfromallconfigura-tionspacesassociatedtoeachindividualprimefactor.Therearebynowstrongresultsavailableconcerningthediffeomorphismgroupofprime3-manifoldswhichareexploitedtoexaminethetopologyoftheconfigurationspacesintermsoftheirhomotopygroups.Weexplicitlyshowhowtoobtainthesefortheclassofhomo-geneoussphericalprimes,andcommunicatetheresultsforallotherknownprimesexceptthenon-sufficientlylargeonesofinfinitefundamentalgroup.Section1.IntroductionInrecentyearsmathematicianshavemadeprogressinunderstandingthediffeo-morphismgroupof3-dimensionalmanifolds.Theobjectofthispaperistoshowhowthiscanbeexploitedtodeepenourtopologicalunderstandingofconfigura-tionspacesoccurringinpureGeneralRelativity.Inparticular,weshallinvestigatetheirhomotopygroupsandthusgeneralizealreadyexistingworkonthefunda-mentalgroup[Wi].Besidesforitsintrinsicinterest,amajormotivationtostudythesetopologicalstructuressteemsfromthecanonicalquantisationprogrammeforGeneralRelativity.Here,generalargumentssuggestatopologicaloriginofcertaininterestingfeaturesofquantumgravity(e.g.degeneratevacuumstructure,absenceofanomalies,superselectionsectors),resemblingthosealreadyfamiliarfromother*e-mail:[e-mail]1(successfullyquantized)theories.Certainly,theargumentsgiveninthecontextofquantumgravityareprimarilymeanttobeofheuristicvalue,thatis,theyarebe-lievedtoreallygiveinsightintosomeaspectsofquantumgravitybyusingmethodswhicharenotnecessarilybelievedtosurviveaneventualrigorousformulationofit.Amongstothers,therearetworeasonsthatentertainthisbelief:Firstly,argumentsidenticalinstructureworkinotherfieldtheories(e.g.Yang-Mills),wherethereisaquantumtheory,secondly,thearguments(properlyformulated)restmerelyongeneralcovarianceanddonotrequiremorestructuraldetailsaboutquantumgravity.Inanygenerallycovarianttheorythetopologicalstructureofconfigurationspacereceivescharacteristicimprintsfromthediffeomorphismgroup,whichisusedtomutuallyidentifyphysicallyequivalentpointsonanauxillaryspacethatlabelsphysicalstatesinaredundantway.Ifthisauxillaryspaceistopologicallytrivial,asitisthecaseinGeneralRelativity,allthetopologicalinformationinthehomo-topygroupsofthequotientisdeterminedbythoseofthediffeomorphismgroup.Generally,thisholdswhenevertheconfigurationspaceisgivenasthebaseofaprincipalfiber-bundlewithstructuregroupthediffeomorphismsandcontractibletotalspace,aswillbeexplainedbelow.Inthiscasethetopologyofthebaseisdirectlyrelatedtothetopologyofthefibres,anditistheirtopologywhichwearegoingtoinvestigate.Intheorieswherebesidesthediffeomorphismsthereisanadditionalgaugegroupacting(whichalsooccursinthe“connection”formulationofGeneralRelativity[Ash]),additionaltopologicalstructureisinduced.Inthesecasesouranalysiscanbeusedtoprovidethediffeomorphismcontribution.Inordertoworkwithinafixedframework,weshallarguewithinthestandardframeworkofGeneralRelativity.But,aswillbecomeapparent,theinvestigationisreallyofamoregeneralkind.InthesequelofthisintroductorysectionandSection2weshallprovidesomebasicmaterialconcerningthenotionofconfigurationspacesinGeneralRelativity,3-manifoldsandtheirdiffeomorphismgroups.Inparticular,thenotionofaspinorialmanifoldisintroduced.AmoretechnicalpointisdeferredtoAppendix1.Proofsofalreadyexistingresultsareonlyincludedwhenitseemsappropriate.Theirsettinggivenheremightdifferfromtheoneoriginallygiven.ThissetsthestageforthederivationsofsomenewresultsinSection3.InSection4alltheresultsnexttosomeotherusefulinformationiscombinedinatable,andsomefirstobservations2aremade.Thissectionshouldbeaccessiblewithoutgoingthroughthemainbodyofthepaper.Appendix2combinesintofivetheoremssomescatteredresultsfromtheliteraturewhichwemadeessentialuseof.ConfigurationSpaces,3-ManifoldsandDiffeomorphismsThespecificationofinitialdatainGeneralRelativitystartswiththeselectionofa3-manifold,Σ,onwhichinitialdataareconstructedinformofaRiemannian3-metricandtheextrinsiccurvature.Togethertheysatisfyanellipticsystemoffourdifferentialequations,theconstraints,whichareseparatefromtheevolutionequations.Asconfigurationspaceweaddressthequotient-spaceobtainedfromthespaceofall3-metricsonΣ,wherethosemetricswhichlabelthesamephysicalsatearemutuallyidentified.Thisreducesthree(theso-calledmomentumconstraints,whicharelinearinmomenta)ofthefourconstraintequations,theremainingonebeingthesocalledHamiltonianconstraint(whichisquadraticinthemomenta).Theidentificationisgenericallygivenbytheactionofsomenormalsubgroup(pos-siblythewholegroup)ofthediffeomorphismgroup,whichwechoosetocallitsgaugepart,sinceitconnectsredundantlabelsforthesamephysicalstate.Generalcovariancethenimpliesthatthequotientofthefulldiffeomorphismgroupwithre-specttothegaugepartactsontheconfigurationspaceaspro