Hausdor Convergence and Universal Covers

HausdorConvergeneandUniversalCoversChristinaSormaniGuofangWeiyAbstratWeprovethatifYistheGromov-Hausdorlimitofasequeneofompatmanifolds,Mni,withauniformlowerboundonRiiurvatureandauniformupperboundondiameter,thenYhasauniversalover.Wethenshowthat,forisuÆientlylarge,thefundamentalgroupofMihasasurjetivehomeomorphismontothegroupofdektransformsofY.Finally,inthenon-ollapsedasewheretheMihaveanadditionaluniformlowerboundonvolume,weprovethatthekernelsofthesesurjetivemapsarenitewithauniformboundontheirardinality.AnumberoftheoremsarealsoprovenonerningthelimitsofoveringspaesandtheirdektransformswhentheMiareonlyassumedtobeompatlengthspaeswithauniformupperboundondiameter.1IntrodutionInreentyearsthelimitspaesofmanifoldswithlowerboundsonRiiurvaturehavebeenstudiedfrombothageometriandtopologialperspetive.Inpartiular,CheegerandColdinghaveprovenanumberofresultsregardingtheregularityandgeometripropertiesofthesespaes.However,thetopologyofthelimitspaesislesswellunderstood.NotethatinthispaperamanifoldisaRiemannianmanifoldwithoutboundary.Anderson[An℄hasproventhatthereareonlynitelymanyisomorphismtypesoffundamentalgroupsofmanifoldswithauniformupperboundondiameter,lowerboundonvolumeandlowerboundonRiiurvature.Thusonemightthinkthatgivenaonvergingsequeneofsuhmanifolds,thefundamentalgroupsofthemanifoldsmusteventuallybeisomorphitothefundamentalgroupofthelimitspae.However,Otsu[Ot℄hasshownthattherearemetrisofuniformlypositiveRiiurvatureonS3RP2whihonvergetoasimplyonneted5-dimmetrispae,showingthatthisneednotbethease.Tushmann[Tu℄hasproventhatifYisthelimitspaeofasequeneofmanifoldswithtwosidedsetionalurvatureboundsthenYisloallysimplyonnetedandthushasauniversalover.InfatPerelman[Pl℄showsthatthelimitspaeofasequeneofmanifoldswithalowerboundonsetionalurvatureisloallyontratible.Ifthelimitspaeisloallysimplyonneted,itisnotdiÆulttoshowthateventuallythereisasurjetivemapfromthefundamentalgroupsofthemanifoldsontothefundamentalgroupofthelimitspae(see[Tu,Ca℄,[Gr,Page100℄,alsoSetion2ofthispaper).Zhu[Zh℄hasprovenasimilarresultforlimitsofthreedimensionalmanifoldswithuniformlowerboundsonRiiurvatureandvolumeandauniformupperboundondiameter.InthispaperamanifoldisaRiemannianmanifoldwithoutboundary.HereweareonernedwithlimitsofsequenesofmanifoldswithauniformupperboundondiameterandlowerboundonRiiurvature.Thelimitsofsuhsequeneshaveonlybeenshowntobeloallysimplyonnetedatspeial\regularpoints[ChCo℄.InfatMenguy[Me℄hasshownthatthelimitspaeouldloallyhaveinnitetopologialtype.1991MathematisSubjetClassiation.Primary53C20.yPartiallysupportedbyNSFGrant#DMS-9971833.1Weprovethattheuniversaloverofthelimitspaeexists.Weanthusstudythegroupofdektransformsoftheuniversalover,1(M)[Defn2.3℄.Notethatthisrevisedfundamentalgroup,1(M),isisomorphitothefundamentalgroupofMifMisloallysimplyonneted(.f.[Sp℄).Weannowstatethemaintheoremofourpaper.Theorem1.1LetMibeasequeneofompatmanifoldssatisfyingRii(Mi)(n1)Handdiam(Mi)D;(1.1)forsomeH2RandD0.IfYistheGromov-HausdorlimitoftheMithentheuniversaloverofYexistsandforNsuÆientlylargedependingonY,thereisasurjetivehomomorphismi:1(Mi)!1(Y)8iN:(1.2)Note1.2Wedon’tknownifYissemi-loallysimplyonneted,ortheuniversaloverissimplyonnetedyet.SeeSetion2,Example2.6,2.7formoreinformation.Whenthesequeneisnon-ollapsingwehaveastrongerresult:Theorem1.3LetMnibeasequeneofompatmanifoldssatisfyingRii(Mi)(n1)H;diam(Mi)Dandvol(Mi)V(1.3)forsomeH2R,D0andV0.IfYistheGromov-HausdorlimitoftheMithenthereisi0=i0(n;v;D;ÆY)suhthat1(Mi)=Fiisisomorphito1(Y)forallii0,hereFiisanitesubgroupof1(Mi),andtheorderofeahFiisuniformlyboundedbyN(n;v;D).Inpartiular,1(Mi)=Fiisisomorphito1(Mj)=Fjforalli;ji0,CompareAnderson’sresult[An℄whihsaysthatthereareonlynitelymanyisomorphismtypesoffundamentalgroupsofompatmanifoldssatisfying(1.3).Toprovetheseresultsweneedtostudythelimitsspaesofompatlengthspaes.ThusinSetions2and3werestritourselvestosequenes,Mi,whihareonlyompatlengthspaeswithdiam(Mi)DthatonvergeintheGromov-HausdorsensetoalimitspaeY.InSetion2wepresenttwoexamplesofsuhsequenesoflengthspaeswhihonvergeintheGromov-Hausdortopology.However,theirfundamentalgroupsannotbemappedsurjetivelyontothefundamentalgrouporrevisedfundamentalgroupofthelimitspae.Intherstexample,wehaveasequeneofsimplyonnetedlengthspaeswhoselimitspaeisnotsimplyonneted[Ex2.6℄.Intheseondexample,thelimitspaehasnouniversalover[Ex2.7℄.Itshouldberealledthatevenwhenthelimitspaeisamanifoldthatthelimitoftheuniversaloversisnotneessarilyaoverofthelimitspae(see[Pe1,Theorem2.1℄foraasewhereitis).Thustheuniversaloverannotbediretlyusedtoprovepropertiesaboutthefundamentalgroup.InSetion3,weintrodueÆ-overingspaes[Defn3.1℄.Unliketheuniversalover,Æ-oversalwaysexist.WethenshowthatthelimitoftheÆ-oversisaoverofthelimitspae[Theorem3.6℄.Furthermore,weprovethatforaxedÆ0groupsofdektransformsoftheÆ-overs,~MÆi,oftheMieventuallyhaveasurjetivemapontodektransformsoftheÆ-overofthelimitspae[Cor3.5℄.WealsodesribetherelationshipwiththeÆ-oversandtheuniversaloveriflatterexists[Theorem3.7℄.WeonludewithaproofofthefollowingtheoremwhihshouldbeontrastedwithE