高等数学

AnnalsofMathematics,157(2003),919–938

LargeRiemannianmanifolds

whichare exible

ByA.N.Dranishnikov,StevenC.Ferry,andShmuelWeinberger*

Abstract

Foreachk∈Z,weconstructauniformlycontractiblemetriconEuclideanspacewhichisnotmodkhypereuclidean.WealsoconstructapairofuniformlycontractibleRiemannianmetricsonRn,n≥11,sothattheresultingmani-foldsZandZ areboundedhomotopyequivalentbyahomotopyequivalencewhichisnotboundedlyclosetoahomeomorphism.Weshowthatfortheself(Z)→K (C (Z))fromlocally -spacestheC -algebraassemblymapK

niteK-homologytotheK-theoryoftheboundedpropagationalgebraisnotamonomorphism.ThisshowsthatanintegralversionofthecoarseNovikovcon-jecturefailsforrealoperatoralgebras.Ifweallowasinglecone-likesingularity,asimilarconstructionyieldsacounterexampleforcomplexC -algebras.

1.Introduction

Thispaperisacontributiontothecollectionofproblemsthatsurroundspositivescalarcurvature,topologicalrigidity(a.k.a.theBorelconjecture),theNovikov,andBaum-Connesconjectures.Muchworkinthisarea(seee.g.[14],

[4],[3],[15])hasfocusedattentiononboundedandcontrolledanaloguesoftheseproblems,whichanaloguesoftenimplytheoriginals.Recently,successinattacksontheNovikovandGromov-LawsonconjectureshasbeenachievedalongtheselinesbyprovingthecoarseBaum-Connesconjectureforcertainclassesofgroups[23],[27],[28].AformofthecoarseBaum-Connesconjec-lf(X)→K (C (X))isanturestatesthattheC -algebraassemblymapµ:K

isomorphismforuniformlycontractiblemetricspacesXwithboundedgeom-etry[21].

UsingworkofGromovonembeddingofexpandinggraphsingroupsΓwithBΓa nitecomplex[16],theepimorphismpartofthecoarseBaum-Connescon-

TheauthorsarepartiallysupportedbyNSFgrants.ThesecondauthorwouldliketothanktheUniversityofChicagoforitshospitalityduringnumerousvisits.

1991MathematicsSubjectClassi cation.53C23,53C20,57R65,57N60.

Keywordsandphrases.uniformlycontractible.

920A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUELWEINBERGER

jecturewasdisproved[17].InthispaperwewillshowthatthemonomorphismpartofthecoarseBaum-Connesconjecture(i.e.thecoarseNovikovconjec-ture)doesnotholdtruewithouttheboundedgeometrycondition.WewillconstructauniformlycontractiblemetriconR8forwhichµisnotamonomor-phism.Thus,acoarseformoftheintegralNovikovconjecturefailsevenfor nite-dimensionaluniformlycontractiblemanifolds.Infactwewillprovemore:ouruniformlycontractibleR8isnotintegrallyhypereuclidean,whichistosaythatitdoesnotadmitadegreeonecoarseLipschitzmaptoeuclideanspace.Alsointhispaper,wewillproduceauniformlycontractibleRiemannianman-ifold,abstractlyhomeomorphictoRn,n≥11,whichisboundedlyhomotopyequivalenttoanothersuchmanifold,butnotboundedlyhomeomorphictoit.Thisdisprovesonecoarseanalogoftherigidityconjectureforclosedasphericalmanifolds.Wewillalsoshowthatforeachk∈Zsomeofthesemanifoldsarenotmodkhypereuclidean.

OurconstructionisultimatelybasedonexamplesofDranishnikov[5],[6]ofspacesforwhichcohomologicaldimensiondisagreeswithcoveringdimension,andtheconsequentphenomenon,usingatheoremofEdwards(see[25]),ofcell-likemapswhichraisedimension.

De nition1.1.WewillusethenotationBr(x)todenotetheballofradiusrcenteredatx.Ametricspace(X,d)isuniformlycontractibleifforeveryrthereisanR≥rsothatforeveryx∈X,Br(x)contractstoapointinBR(x).ThemainexamplesofthisaretheuniversalcoverofacompactasphericalpolyhedronandtheopenconeinRnofa nitesubpolyhedronoftheboundaryoftheunitcube.Thereisasimilarnotionofuniformlyn-connectedwhichsaysthatanymapofann-dimensionalCWcomplexintoBr(x)isnullhomotopicinBR(x).

De nition1.2.WewillsaythataRiemannianmanifoldMnisintegrally(modk,orrationally)hypereuclideanifthereisacoarselypropercoarseLips-chitzmapf:M→Rnwhichisofdegree1(ofdegree≡1modk,orofnonzerointegraldegree,respectively).SeeSection4forde nitionsandelaborations.

Hereareourmainresults:

TheoremA.Foranygivenkandn≥8,thereisaRiemannianmani-foldZwhichisdi eomorphictoRnsuchthatZisuniformlycontractibleandrationallyhypereuclideanbutisnotmodk(orintegrally)hypereuclidean.

De nition1.3.(i)Amapf:X→YisacoarseisometryifthereisaKsothat|dY(f(x),f(x )) dX(x,x )|<Kforallx,x ∈Xandsothatforeachy∈Ythereisanx∈XwithdY(y,f(x))<K.

(ii)WewillsaythatuniformlycontractibleRiemannianmanifoldsZandZ areboundedlyhomeomorphicifthereisahomeomorphismf:Z→Z whichisacoarseisometry.

LARGERIEMANNIANMANIFOLDSWHICHAREFLEXIBLE921

TheoremB.ThereisacoarseisometrybetweenuniformlycontractibleRiemannianmanifoldsZandZ whichisnotboundedlyclosetoahomeomor-phism.

AneasyinductiveargumentshowsthatacoarseisometryofuniformlycontractibleRiemannianmanifoldsisaboundedhomotopyequivalence,sothisgivesacounterexampletoacoarseformoftheBorelconjecture.

TheoremC.ThereisauniformlycontractiblesingularRiemannianmanifoldZsuchthattheassemblymap(see[20])

f(Z)→K (C (Z))K

failstobeanintegralmonomorphism.Zisdi eomorphicawayfromonepointtotheopenconeonadi erentiablemanifoldM.

Itwasshownin[8]thatZhasin niteasymptoticdimensioninthesenseofGromov.Thisfactcannotbederivedfrom[27]sinceZdoesnothaveboundedgeometry.

Whenwe rstdiscoveredtheseresults,wethoughtawayaroundtheseproblemsmightbetousealargescaleversionofK-theoryinplaceoftheK-theoryoftheuniformlycontractiblemanifold.Yuhasobservedthateventhatversionofthe(C -analytic)Novikovconjecturefailsingeneral(see[28]),althoughnotforanyexamplesthatarisefrom nitedimensionaluniformlycontractiblemanifolds.Ontheotherhand,boundedgeometrydoessu cetoeliminatebothsetsofexamples.

Inthepastyear,motivatedbyGromov’sobservationthatspaceswhichcontainexpandergraphscannotembedinHilbertspace,severalresearchers(see[17]andthereferencescontainedtherein),ingthemethodolgyofFarrellandJones,KevinWhyteandthelastauthorhaveobservedthatsomeofthesearenotcounterexamplestothecorrespond-ingtopologicalstatements.Thustheexamplesofthispaperremaintheonlycounterexamplestothetopologicalproblems.

2.Weightedconesonuniformlyk-connectedspaces

TheopenconeonatopologicalspaceXisthetopologicalspaceOX=X×[0,∞)/X×0.

De nition2.1.Acompactmetricspace(X,d)islocallyk-connectedifforevery >0thereisaδ>0suchthatforeachk-dimensionalsimplicialcomplexKkandeachmapα:Kk→Xwithdiam(α(Kk))<δthereisamapα¯:Cone(K)→Xextendingαwithdiam(¯α(Cone(K)))< .Here,Cone(K)denotestheordinaryclosedcone.

922A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUELWEINBERGER

Lemma2.2.Let(X,d)beacompactmetricspacewhichislocallyk-connectedforallk.Foreachn,theopenconeonXhasacompleteuniformlyn-connectedmetric.WewilldenoteanysuchmetricspacebycX.1

Proof.Wewillevenproduceametricwhichhasalinearcontractionfunc-tion.Itsconstructionisbasedontheweightedconeoftenusedindi erentialgeometry.Drawtheconevertically,sothathorizontalslicesarecopiesofX.

Chooseacontinuousstrictlyincreasingfunctionφ:[0,∞)→[0,∞)withφ(0)=0.LetdbetheoriginalmetriconXandde neafunctionρ by(i)ρ ((x,t),(x ,t))=φ(t)d(x,x ).

(ii)ρ ((x,t),(x,t ))=|t t |.

Wethende neρ:OX×OX→[0,∞)tobe

ρ((x,t),(x,t))=inf

wherethesumisoverallchains

(x,t)=(x0,t0),(x1,t1),...,(x ,t )=(x ,t )

andeachsegmentiseitherhorizontalorvertical.Itpaystomovetowards0beforemovingintheX-direction,sochainsofshortestlengthhavetheformpicturedabove.ThefunctionρisametriconOX.ThenaturalprojectionOX→[0,∞)decreasesdistances,soCauchysequencesareboundedinthe

[0,∞)-direction.ItfollowsthatthemetriconOXiscomplete.WewritecXforthemetricspace(OX,ρ).

Itremainstode neφsothatcXisuniformlyn-contractible.Wewillde neφ(1)=1andφ(i+1)=Ni+1φ(i)fori∈Z,wherethesequence{Ni}willbespeci edbelow.Fornonintegralvaluesoft,weset

φ(t)=φ([t])+(t [t])φ([t]+1).

“c”notationincXreferstoaspeci cchoiceofweights.Thereprobablyshouldbean“n”inournotation,butweleaveitoutforsimplicity.1The i=1ρ ((xi,ti),(xi 1,ti 1))

LARGERIEMANNIANMANIFOLDSWHICHAREFLEXIBLE923

SinceXislocallyn-connected,thereisanin nitedecreasingpositive

d(x) Bd(x) sequence{ri}suchthatforeveryxtheinclusions... Brrii+1dBri 1(x)arenullhomotopiconn-skeleta.Re nethesequencesothatactuallyd(x) BdinclusionsBirri 1(x)arenullhomotopiconn-skeleta.WesetNi=iri 1i.ρρ(x,i) cX.First,wenotethatB1(x,i) NowconsidertheballB1ρρd(x,i)B1(x)×[i 1,i+1]andthatB1(x,i)contractsinitselftoB1

∩(X×[i 1,i]) Bdi 1

ρρ(x,i)n-contractsinB3(x,i)bypushingdowntothe(i 2)-levelandsoB1performingthen-contractionthere.

Forballsofradius2thesamereasoningappliesifthecenterisatleast3awayfromthevertex.Wecontinueinthiswayandobservethatforanygivensizeball,centeredsu cientlyfarout,oneobtainsan-contractibilityfunctionoff(r)=r+2asrequired.Thewholespaceisthereforeuniformlycontractible.1i 1ρ(x)×[i 1,i].ButB3(x,i) Bd1i 2(x)×{i 2}

3.Designercompacta

De nition3.1.Amapf:M→Xfromaclosedmanifoldontoacompactmetricspaceiscell-likeorCEifforeachx∈XandneighborhoodUoff 1(x)thereisaneighborhoodVoff 1(x)inUsothatVcontractstoapointinU.

ThepurposeofthissectionistogiveexamplesofCEmapsf:M→Xsothatf :Hn(M;L(e))→Hn(X;L(e))hasnontrivialkernel.Theargumentgivenbelowisamodi cationofthe rstauthor’sconstructionofin nite-dimensionalcompactawith nitecohomologicaldimension.HereistheresultwhichwewilluseinprovingTheoremsA,B,andCoftheintroduction.

Theorem3.2.LetMnbea2-connectedn-manifold,n≥7,andlet (M;Zm).ThenthereisaCEmapq:M→XwithαbeanelementofKO (M;Zm)→KO (X;Zm)).Itfollowsthatifα∈H (M;L(e))α∈ker(q :KO

isanelementoforderm,modd,thenthereisaCEmapc:M→Xsothatc (α)=0inH (X;L(e)).

Webegintheproofofthistheorembyrecallingthestatementofamajorstepintheconstructionofin nite-dimensionalcompactawith nitecohomo-logicaldimension.

(K(Z,n))=0forsomegeneralizedho-Theorem3.3.Supposethath (L)mologytheoryh .Thenforany nitepolyhedronLandanyelementα∈h

thereexistacompactumYandamapf:Y→Lsothat

(1)c-dimZY≤n.

(2)α∈Im(f ).

924A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUELWEINBERGER

Remark3.4.In[5],[6]theanalogousresultwasprovenforcohomologytheory.Theproofissimilarforhomologytheory.See[9].

Theorem3.3alsohasarelativeversion:

(K(Z,n))=0.Thenforany niteTheorem3.3 .Supposethath (K,L)thereexistacompactumpolyhedralpair(K,L)andanyelementα∈h

Yandamapf:(Y,L)→(K,L)sothat

(i)c-dimZ(Y L)≤n.

(ii)α∈Im(f ).

(iii)f|L=idL.

Theproofisessentiallythesame.Hereisthekeylemmaintheproofof willrefertoreducedcomplexK-homologyTheorem3.2.Inwhatfollows,K willrefertoreducedrealK-homology.andKO

Lemma3.5.LetMnbea2-connectedn-manifold,n≥7,andletαbe (M;Zm),m∈Z.ThenthereexistcompactaZ MandanelementinKO

Y MalongwithaCEmapg:(Z,M)→(Y,M)sothat

(1)g|M=idM.

(2)dim(Z M)=3.

(3)j (α)=0,wherej:M→Yistheinclusion.

(K(Zk,n);Zm)=0forn≥3.WecannowapplyThe-Proof.By[26],KO +1(Cone(M),M)¯∈KOorem3.3 tothepair(Cone(M),M)andtheelementα

with α¯=αinthelongexactsequenceof(Cone(M),M),obtainingaspace +1(Y,M)withY Mwithcdim(Y M)=3sothatthereisaclassα¯ ∈KO

α¯ =αandaCEmapg:(Z,M)→(Y,M)withdim(Z M)=3.Theexactsequence:

+1(Y,M)→KO (M)→KO (Y)KO j

showsthatj (α)=0.

Next,weconstructaparticularlyniceretractionZ→M.

Lemma3.6.Let(Z,M)beacompactpairwithdim(Z M)=3andMa2-connectedn-manifold,n≥7.Thenthereisaretractionr:Z→Mwithr|(Z M)one-to-one.

Proof.Theexistenceoftheretractionfollowsfromobstructiontheoryappliedtothenerveofa necoverofZ.TherestisstandarddimensiontheoryusingtheBairecategorytheorem.

LARGERIEMANNIANMANIFOLDSWHICHAREFLEXIBLE925

Lemma3.7.Letr:Z→Mbearetractionwhichisone-to-oneon(Z M)andletg:(Z,M)→(Y,M)beaCEmapwhichistheidentityoverM.ThenthedecompositionofMwhosenondegenerateelementsarer(g 1(y))isuppersemicontinuous.

Proof.WeneedtoshowthatifFisanelementofthisdecompositionandU FthenthereisaVwithF V UsuchthatifF isadecompositionelementwithF ∩V= ,thenF U.

CaseI.F=r(G),G=g 1(y).ThenG∩M= .ForU F,letd=dist(F,M U).Sincerisaretraction,thereisanopenneighborhood¯= ,diamz(G )<d.WemayO ZofMsothatforallG suchthatG ∩O¯assumethatOhasbeenchosensosmallthatO∩G= .Bycontinuityofg,¯)∩Z 1(U).Sincerisone-to-onethereisanopenV withG V (Z O

andZ Oiscompact,r(V )isopeninr(Z O).ThismeansthatthereisanopenW MsothatW∩r(Z O)=r(V ).LetV=W∩)d(F) U.IfF ∩V= thenF U,sinceF iseitherasingleton,asetwithdiameter ¯<d,orr(G)withG Z O,andallthreecasesareaccountedfor

above.

CaseII.Fisasingleton,F={x}withF∈/z(Z M).Letx∈Uandletd=dist(x,M U).Bycontinuityofg,thereisacompactC Z MsothatifG C,thendiam(Z(G))<d.Letρ=dist(x,r(C))andde neV=Bτ(x)whereτ=min{ρ,d}.

ProofofTheorem3.2.Considerthecoe cientsequence

×m +1(M;Zm)→KO (M)→KO→KO (M)→.

(M)isoforderm,thenα= α +1(M;Zm).We¯,whereα¯∈KOIfα∈KO

¯=0.Thisgivesusadiagramchooseg:Z→YasinLemma3.5sothatj α

M

X f← Z← r r g← M← ji idYM

wheref:M→XistheCEmapinducedbythedecomposition{r(G)|G=g 1(y),y∈Y}andr istheinducedmapfromYtoX.Itfollowsimmediately

α)=0.Itthenfollowsfromtheladderofcoe cientfromthisdiagramthatf (¯

926A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUELWEINBERGERsequences

+1(M;Zm) (M) (M)KO→KO→KO

f f f

+1(X;Zm) (X) (X)KO→KO→KO

thatf (α)=0.TheL-theorystatementinTheorem2nowfollowsfromthe

1factthatKO[1]=L(e)[].

4.TheproofofTheoremA

Webeginbystatingsomede nitions.

De nition4.1.

(i)Amapf:X→YbetweenmetricspacesissaidtobecoarseLipschitz

ifthereareconstantsCandDsothatdY(f(x),f(x ))<CdX(x,x )wheneverdX(x,x )>D.NoticethatcoarseLipschitzmapsarenotnecessarilycontinuous.Infact,ifdiamX<∞,everymapde nedonXiscoarseLipschitz.

(ii)Amapf:X→YiscoarselyproperifforeachboundedsetB Y,

f 1(B)hascompactclosureinX.

ThefollowingcorollaryconstructstheRiemannianmanifoldsappearinginallofourmaintheorems.

Proposition4.2.IfXisthecell-likeimageofacompactmanifoldandnisgiven,thenforsomesuitablechoiceofweights,cXisuniformlyn-connected.

Proof.TheCEimageofanycompactANR(absoluteneighborhoodre-tract)islocallyn-connectedforalln,sothepropositionfollowsfromLemma

2.2.See[19]forreferences.

Corollary4.3.Letf:Sk 1→Xbeacell-likemap.ThenRkhasauniformlycontractibleRiemannianmetricwhichiscoarselyequivalenttocX,wheretheconeisweightedasinProposition4.2.

Proof.Considercf:cSk 1→cX.ThisinducesapseudometriconRk.Thebasicliftingpropertyforcell-likemaps(see[19])showsthatRkwiththispseudometricisuniformlyn-connectedifandonlyifcXis.Ifn≥k 1,thismeansthattheinducedpseudometriconRkisuniformlycontractible.Adding anysu cientlysmallmetrictothispseudometric—themetricfromRk～=Dk

LARGERIEMANNIANMANIFOLDSWHICHAREFLEXIBLE927

willdo—producesauniformlycontractiblemetriconRkwhichisquasi-equivalenttocX.SinceXislocallyconnected,atheoremofBing[1]saysthatXhasapathmetric.IfwestartwithapathmetriconX,themetriconcXisalsoapathmetricandtheresultsof[11]allowustoconstructaRiemannianmetriconcSk 1whichisuniformlycontractibleandcoarseLipschitzequivalenttocX.

WehaveconstructedaRiemannianmanifoldZnhomeomorphictoRnsothatZiscoarselyisometrictoaweightedopenconeona“Dranishnikovspace”X.ByTheorem3.2,wecanchoosec:Sn 1→XsothatcdoesnotinduceamonomorphisminK(;Zk)-homologyandsuchthatthemapc×id:Rn→cXisacoarseisometry,whereweareusingpolarcoordinatestothinkofRnastheconeonSn 1.Inthisnotation,“c×id”referstoamapwhichpreserveslevelsintheconestructureandwhichisequaltoconeachlevel.

WeneedtoseethatZisnothypereuclidean.Thenextlemmashouldbecomfortingtoreaderswho ndthemselveswonderingaboutthe“degree”ofamapwhichisnotrequiredtobecontinuous.

Lemma4.4.IfZisanymetricspaceandf:Z→Rn(withtheeuclidean¯:Z→Rnwhichmetric)iscoarseLipschitz,thenthereisacontinuousmapf

isboundedlyclosetof.IffiscontinuousonaclosedY Z,thenwecan¯|Y=f|Y.choosef

Proof.ChooseanopencoverUofXbysetsofdiameter<1.ForeachU∈U,choosexU∈U.Let{φU}beapartitionofunitysubordinatetoUandlet ¯(x)=φU(x)f(xu).f

U∈U

BythecoarseLipschitzcondition,thereisaKsuchthat

d(x,x )<1 d(f(x),f(x ))<K.

¯(x)∈BK(f(x)),sod(f,f¯)Sinced(xU,x)<1forallUwithφU(x)=0,f

<K.

ContinuingwiththeproofofTheoremA,letf :Z→RnbeacoarselypropercoarseLipschitzmap.SinceZiscoarselyisomorphictocX,thereisacoarseLipschitzmapf:cX→Rn.Bytheabove,wemayassumethatfiscontinuous.

Sincefiscoarselyproper,f 1(B)isacompactsubsetofcX,whereBistheunitballinRn.ChooseTsolargethat

(X×[T,∞))∩f 1(B)= .

928A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUELWEINBERGERNowconsiderthecomposition

Sn 1 →X×{T} →(R B) →Sn 1= B.(c×id)|nxThiscompositionisdegreeonebecausetheoriginalcoarseLipschitzmapW→Rnwasdegreeone.Thisisacontradiction,sincedegreeonemapsofspheresarehomotopyequivalencesandthe rstmapSn 1→X×{T}haskernel (;Zk)-homology.WeconcludethatWisnotmodk(orintegrally)inKO

hypereuclidean.

Proposition4.5.Iff:Sn→Xisacell-likemap,thereisamapg:X→Snsuchthatthecompositiong fhasnonzerodegree.

Proof.By[10],thereexista nitepolyhedronQandamapp:X→Qsothatthecompositionp f:Sn→Qis(2n+3)-connected.SinceQisarationalhomologysphere,thereisanessentialmapq:Q→K(Q,n).Ifnisodd,this nishestheproof,sinceK(Q,n)isatelescopeofmapsbetweenspheresandcompactnessimpliesthattheimageofXliesina nitesubtelescope,whichishomotopyequivalenttoSn.

Ifn=2kiseven,thereisa brationsequenceT→K(Q,n)→K(Q,2n),whereTisarationalsphere(andthereforeatelescope)andthemapK(Q,n)→K(Q,2n)isinducedbysquaringincohomology.Sincethesquareofthegen-eratorofHn(Q)iszero,theessentialmapQ→K(Q,n)liftstoanessentialmapQ→Tandtheargumentfromtheodd-dimensionalcasecompletestheproof.

+1Theorem4.6.Letf:Sn→Xbeacell-likemapandletRnbedi eo-ΦmorphictoRn+1withaRiemannianmetricquasi-isometrictocX,wherecis+1isrationallyhypereuclidean.aweightfunctiontendingtoin nity.ThenRnΦ

Proof.WewilldenotetheHigson-Roecompacti cationofapropermetricspaceYbyandwewillletν(Y)denotetheremainder, Y,whichiscalledtheHigsoncorona.ForgeneralresultsabouttheHigson-Roecompacti cation,seechapter5of[20].ByresultsofRoeasmodi edinLemma3.4of[7],it+1nsu cestoproduceamapν(RnΦ)→Swhichhasnonzerodegreeinthesensethatthecomposite

+1n+1n+1Hn(Sn;Q)→Hn(νRn(RΦ);Q)→HΦ,ν(RΦ);Q)

+1isnonzero.TheHigsoncoronaisacoarseinvariant,sothemapcf:Rn→cXΦinducedbyfextendstoamapn+1(RΦ,ν(RΦ))→(ν(cX))

LARGERIEMANNIANMANIFOLDSWHICHAREFLEXIBLE929

whichisahomeomorphismontheHigsoncoronas.Thisgivesusadiagram

Hn(Q) cf →Hn(ν(cX);Q) homeo cf →Hn+1(ν(cX);Q) ～= cf

+1n+1→Hn(ν(Rn→Hn+1(RHn(RΦ;Q) Φ);Q) Φ),ν(RΦ);Q)

inwhichtherightmostverticalmapisanisomorphismbytheVietoris-Begletheoremandtheleftmostverticalmapisanisomorphismbythe velemma.Letthemapg:X→SnbeconstructedasinProposition4.5.Thediagramshowsthatitsu cesto ndamapν(cX)→Xsothatthecomposition

→Hn(ν(cX);Q) →Hn+1(ν(cX);Q)Hn(Sn;Q)→Hn(X,Q)

isnonzero.

BytheuniversalpropertyoftheHigson-Roecompacti cation,thereisamapofpairs(ν(cX))→(Cone(X),X),whereCone(X)parewithexample5.28of[20].Asinthatexample,elementaryalgebraictopologygivesusadiagram

→Hn(Sn;Q) g g Hn(X;Q) →Hn+1(Cone(X),X;Q) ～ =～=

→ →Hn(ν(cX);Q) Hn(Q) Hn(ν(cX);Q).

Sincef g =0,g (1)=0inHn(X;Q),sotheproofiscomplete.

5.TheproofofTheoremB

Inthissection,wewillexploitpropertiesofaCEmapf:Sk 1→XwhichdoesnotinduceasurjectiononperiodicKO[1]-homology.ThemanifoldZ～=RkwillbeconstructedasabovetobecoarselyequivalenttocX.

WewillproduceZ coarselyequivalenttoZbyusingtheboundedversionoftheSullivan-Wallsurgeryexactsequence[24]whichisestablishedin[12].AstructureonaclosedmanifoldMisapair(N,f)wheref:N→Misasimplehomotopyequivalence.Twostructures(N,f)and(N ,f )areequiva-lentifthereisahomeomorphismφ:N→N sothatf φ～f.Forn≥5,theclassicalsurgeryexactsequencestudiesS(M),thecollectionofequiva-lenceclassesofstructuresonM.Afunctorialversionofthesequenceinthetopologicalcategoryis

→Ln+1(Zπ) →STop(Mn)... →Hn+1(M;L(e))

→Hn(M;L(e)) →Ln(Zπ)

930A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUELWEINBERGER

whereπisthefundamentalgroupofM,L(e)isthe4-periodicsurgeryspec-trum,whichisisomorphictoBOawayfrom2,andthe4-periodicgroupsL (Zπ)areWall’ssurgeryobstructiongroups([24]).ThestructuresetinthisfunctorialversionofthesurgerysequenceisbiggerbyaZorlessthanthegeometricstructuresetdescribedabove.Asshownin[2],thestructuresetinthisstabilizedsurgerysequencecorrespondsgeometricallytoastructuresetwhichcontainscertainnonmanifolds.

FormanifoldsboundedoveraspaceX,thereisasimilarsequencewiththeL-groupreplacedbyaboundedL-group.(Infullgenerality,onehastoalsotakeintoaccountthefundamentalgroupofMoverX.Inthispaper,though,wewillalwaysbedealingwithboundedsurgerywhichis“simplyconnected”inthe berdirection.)TheappropriateboundedWallgroupsweredescribedin[12].Hereisapieceoftheboundedsurgeryexactsequencefrom[12].

f→Lbdd→Sbdd0=Hkk+1,cX(e) +1(Z;L(e)) Z↓cX

ItfollowsimmediatelyfromthissequencethatSbdd Z↓cX isnonzeroif

Lbddk+1,cX(e)isnonzero.SuchastructuregivesusthedesiredmanifoldZanda

boundedhomotopyequivalenceZ →ZwhichisnotboundedlyhomotopicovercXtoahomeomorphism.ThestructuresarisingfromthisconstructionaremanifoldsbecausetheycomefromouroriginalmanifoldviaWallrealization.

Proposition5.1.Fork≥11andanappropriatechoiceofX,Lbddk+1,cX(e)isnot0.

r(Sr)～Proof.Let1∈KO=Zbeagenerator,r≥7,andlet1alsodenote r(Sr;Zp),withpodd.AsinSection3,wethecorrespondinggeneratorofKO r(Sr;Zp)→canconstructacell-likemapf:Sr→Zsothat1∈ker(f :KO r(Z;Zp)).ByProposition4.5,thereisamapg:Z→Srsothat(g f) KO

hasdegree =0.Wehaveacommutingdiagram:

.

r(Sr;Zp),f (1)=pαforsomeα∈KO r(Z).Here,1BytheconditiononKO r(Sr)～isthegeneratorofKO=Z.Since(g f) (1)= ·1,wehaveg (α)=0. r(Cf).TheimageMoreover,αprojectstoanoddtorsionelement[α]inKO r(Cg f)isnontrivial–theimageofpαis timesthegeneratorofof[α]inKO r(Sr)～KO=Zandtheimageofαistherefore /q·1,whichisnotintheimage

ofthepreviousterminthelowerexactsequence.

LARGERIEMANNIANMANIFOLDSWHICHAREFLEXIBLE931

¯:Sk→X=Next,weconsiderSr Sk 1andformacell-likemapf

rSk 1∪fZ.Letq:Sn→Q=Sk 1∪g f¯S.Wehaveasimilar-lookingdiagram:

.

TheinclusionmapsCf →Cf→Cg f¯=Cqinduceisomorphisms¯andCg f

-homology,sothereisanoddtorsionelementinKO r(C¯)whichmapsonKOf r(Cq).toanontrivialoddtorsionelementofKO

.

Now,setk=r+4.Thisiswhatforcesustotakek≥11.Awayfrom2,KO r(Y)～ k(Y)foranyspaceY.Thisgivesusadiagram:

is4-periodic,soKO=KO

.

k(Sk 1)=0andKO k 1(Sk 1)～SinceKO=Z,itfollowsthattheinduced k(X)→KO k(Q)isnontrivialatp.homomorphismKO

Wehaveacommutingdiagramofassemblymaps

f →LbddHkk+1,cX(e)+1(cX;L(e))

fHk →Lbddk+1,cQ(e)+1(cQ;L(e)) ～=

whichshowsthatLbddk+1,cX(e)isnontrivial,wheretheisomorphismonthebot-tomfollowsfrom[12].(Strictlyspeaking,weshouldbeworkingwithstandardconesandcoarseLipschitzmapstocite[12],butonecanusecoarsehomo-topiesasin[21]toextendthetheorytowarpedconesandarbitrarycontinuousmaps.)

932A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUELWEINBERGER

6.TheproofofTheoremC

Choosea2-connected(n 1)-manifoldMn,n≥7,andaCEmapρ:CE→XwhichisnotinjectiveonK-homology.SuchmapswereconstructedM

inSection3.Constructauniformlyn-connectedweightedconecXasabove¯beauniformlycontractiblesingularRiemannianmanifoldcoarselyandletM

isomorphictocX.Theassemblymap

lf¯¯))(M)→K (C (MK

factorsthrough2anaturalmap

lf¯¯)～(M)→KX (MK =KX (cX).

BythemaintheoremofSection7,KX (cX)～=K 1(X),sotheassemblymap f¯¯)haskernel.(M)→KX (MK

Remark6.1.Inparticular,Conjecture6.28of[20]isincorrect.Themapcdoesinducerationalisomorphisms,sotherationalversionoftheconjectureisstillopen.

Thesameprocedureshowsthatbothinjectivityandsurjectivityasser-tionsinaboundedanalog(foruniformlycontractiblespaces)oftheGeneral-izedBorelConjectureofFerry-Rosenberg-Weinberger[13]arefalse.AregularneighborhoodofasuitablesuspensionofaMoorespaceprovidesamanifoldwithboundarywhichhasoddtorsioninitsL(e)-homology.ThiscanbekilledbyaCEmapasinSection3andtherestoftheconstructionproceedsasabove.Wecouldgetacounterexampletotheintegralisomorphismconjectureonamanifolddi eomorphictoeuclideanspacebyperformingourconstructionstartingwithaCEmapρ:Sn→XwheretheinducedmaponK wasmulti-plicationbyk.SuchaCEmapwouldbetheresultofapplyingtheprocedureofSection3toakillthemodkreductionofanintegralclassinK (Sn).Usingreal,ratherthancomplexK-theory,wecouldgetcounterexamplesofthesamesorttotheanalogousinjectivityconjecture.

Remark6.2.Allofourexamplesarebasedonthedi erencebetweenKlfandKX.Consequently,ifoneiscarefultoassertallconjecturesforgeneralmetricspacesintermsofKXratherthanKlf,oneobtainsstatementswhicharenotcontradictedbytheseexamples.IncasethemanifoldZhasboundedgeometry—inparticular,ifZistheuniversalcoverofclosedmanifold—it f(Z)→KX (Z)isamonomorphism,sonoisnotdi culttoshowthatK

contradictiontoConjecture6.28arisesfromourconstructioninthatcase.

2See[18].

LARGERIEMANNIANMANIFOLDSWHICHAREFLEXIBLE933

7.KX ofweightedopencones

JohnRoe[20]hasintroducedthefollowingnotionofcoarsehomology:De nition7.1.IfXisacompletelocallycompactmetricspace,ase-quence{Ui}oflocally nitecoversofXbyrelativelycompactopensetsisˇcalledanAnti-CechsystemiftherearenumbersRi→∞suchthat

(i)diam(U)<RiforallU∈Ui.

(ii)RiisaLebesguenumberforUi.

ThecoarsehomologyofXwithcoe cientsinSis

lf(N(Ui);S),HX (X;S)=limH →

lf(P;S)istheSteenrodwhereN(Ui)isthenerveoftheopencoverUiandH S-homologyofthe1-pointcompacti cationofP,relin nity.S,ofcourse,is

aspectrum.

ˇItisnotdi culttoconstructanti-Cechsystemsofcovers,atleastwhenX

isacompletelocallycompactmetricspace.Forsome >0,chooseamaximalcollectionofdisjointopen ballsinXandconsiderthecollectionofR-ballsonthesamecenters.ForR>2 ,thisisacoverwithLebesguenumberatleastR 2 .If{Ri}isanymonotonesequenceapproachingin nity,thisallowsustoconstructasequenceofcoarsecovers{Ui}withdiameters<Ri.Anyˇanti-Cechsystemisco nal,soHX iswell-de ned.

Aninterestingquestioninmetrictopologyisto ndconditionsunderwhichHX (X;S)isequaltotheS-homologyofXatin nity.InsuchcasesHX isatopologicalinvariant,ratherthanametricinvariant.TheusualsortofnerveargumentgivesapropermapX→N(Ui)foreachiandthereforeamaplf(X;S)→HX (X;S).EvenwhenXisuniformlycontractible,theresultsH

ofthispapershowthatthismapneedbeneitheranintegralmonomorphismnoranintegralepimorphism.Wedo,however,havethefollowing:

Theorem7.2.IfXiscompactmetricandcXisaweightedconeonX,

lf(cX;S)→HX (cX;S)isanisomorphismforanyspectrumS.thenH

TheanalogousresulthasbeenprovenbyHigsonand/orRoeformanyuniformlycontractiblespaces.Wewillgiveaproofforweightedconeswhichincludesthein nite-dimensionalcase.

Proof.LetRbegivenandconsiderlevelskRintheweightedconecX.Packeachoftheselevelswith k-ballsoncentersckiasabove, ksmall,anddrawradialarcsfromeachckitothepointbelowitinlevel(k 1)R.Now

take(1+)R-neighborhoodsofthearcs.

934A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUEL

WEINBERGER

ThisgivesacoverORofcXbyopensetsofdiameter 5RwithLebesguenumber 1R.Considerthenerveofthiscover,restrictingattentionforthemomenttotwoconsecutivelevelsasinthepictureabove.Wewillcalltheopencovercorrespondingtopointsinthe(k+1)R-levelUandthecovercorrespondingtopointsatthekR-levelV.

ForU∈UandV∈V,wewillletU andV denotetheintersectionsofUandVwiththepartofcXbetweentheconepointandthekR-level.

Lemma7.3.IfU1∩...∩Uk∩V1∩...∩V = ,thenU1 ∩...∩Uk ∩V1 ∩...∩V = .

Proof.Thisisclear,sinceifapointxabovethekR-levelisinthe rstintersection,thenfollowingtheconelinesbacktothekR-levelgivesapointinthesecondintersection.

For{ k}small,thereisamapφ:U→Vsothat

( )U φ(U) forallU∈U

whichisde nedbychoosingφ(U)tobeanelementV∈VsothatU∩kR V∩kR.De neamapρ:N(U∪V)→N(V)byρ(V)=Vandρ(U)=φ(U).

Proposition7.4.

retraction.Themapρde nesasimplicialstrongdeformation

Proof.Toseethatρissimplicial,supposethatwehaveasimplex

U1,...,Un,V1,...,Vk ∈N(U∪V).

Byde nition,

U1∩...∩Un∩V1∩...∩Vk= ,

so

U1 ∩...∩Un ∩V1 ∩...∩Vk = ;

whence( )guaranteesthat

φ(U1)∩...∩φ(Un)∩V1∩...∩Vk= ,

soρde nesasimplicialmap.

LARGERIEMANNIANMANIFOLDSWHICHAREFLEXIBLE935

Toseethatρisastrongdeformationretraction,webeginbynotingthatN(U∪V)=N(U ∪V ),whereU ={U |U∈U}andV ={V |V∈V}.SinceU φ(U) ,thesituationisnotdi culttoanalyze.

Lemma7.5.IfVisa niteopencoverofaspaceXandU XisanopensetsuchthatU VforsomeV∈V,thenN(V)isastrongdeformationretractofN(V∪{U}).

Proof.Considerthelinkof U inN(V∪{U}).Thisconsistsofsimplices V0,...,Vn suchthat

U∩ n

Vi= .

i=0

IfVisaparticularelementofVcontainingU,thismeansthat

V∩ n

i=0 Vi= ,

whichmeansthatthelinkiscontractible(evencollapsible!),sinceitisaconefrom V .ButthenN(V∪{U})collapsestoN(V),sincetheconeonacollapsi-blecomplexcollapsestoitsbase.Thiscompletestheproofofthelemma.

Thiscollapsesends U to V ,soremovingelementsofU oneatatimegivesacollapsefromN(U ∪V )toN(U ),completingtheproofthatρisastrongdeformationretraction.

Wewillneedthefollowingwell-knownlemma.

Lemma7.6.Iff:(X,X0)→(Y,Y0)isamapofCWpairssothatfandf|X0arehomotopyequivalences,thenfisahomotopyequivalenceofpairs.

Proof.Formthemappingcylinderandconstructstrongdeformationre-tractionofpairsbyretractingthesmallermappingcylinder rstandthenretractingthelargerone.

Lemma7.7.N(U∪V)ishomotopyequivalentrelN(U)∪N(V)tothemappingcylinderofρ.

Proof.LetM(ρ)bethemappingcylinderofρ.Wehavean“identity”mapN(U)∪N(V)→N(U∪V).Bythelemmaabove,itsu cestoshowthatthismapextendstoallofM(ρ).ButthismapishomotopictothemapwhichistheidentityonN(V)andwhichsendsN(U)toN(V)viaρ.CollapsingM(ρ)toitsbaseandincludingintoN(U∪V)giveanextensionofthismaptoallofM(ρ),sobythehomotopyextensiontheorem,theoriginalmapextendstoallofM(ρ).

936A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUELWEINBERGER

ItfollowsfromallofthisthatthenerveofthecoverORconstructedaboveisproperhomotopyequivalenttothemappingtelescopeofthenervesoftheUkwhichcorrespondtoarcsconnectinglevelskRand(k+1)RincX.WewillshowthatforallR,thistelescopeisproperhomotopyequivalenttothecomplementofXintheHilbertcube.Itfollowsthatforanylocally niteSteenrodhomologytheory,thelocally nitehomologyofthetelescopeisequaltothehomologyofXwithadimensionshift,asdesired.

¯iisasequenceofLemma7.8.IfXisacompactmetricspace,andU¯i+1re nesU¯i,andmesh(U¯i)→0,¯0={X},UopencoversofXsuchthatU¯i)isproperhomotopyequivalentthenthemappingtelescopeofthenervesN(U

toQ X,whereXisembeddedintheHilbertcubeQasaZ-set.

Proof.ThisisaformofChapman’sComplementTheorem[22],whichsaysthatthehomeomorphismtypeofthecomplementofaZ-embeddedcompactumXintheHilbertcubedependsonlyontheshapeofX.ThepointisthatthemappingtelescopecanbecompletedtoacontractibleANRbyaddingacopy¯i)’satin nity.CrossingwithQgivesacopyofoftheinverselimitoftheN(U

QcontainingaZ-setX whichisshapeequivalenttoX.ThecomplementofX istheproductofthetelescopewithQ.

Oneshouldbecarefulhere,sincealittlebitofthoughtgivesexampleswhereXistheunitintervalandX istheHilbertcube.Theargumentof[22]showsthatif{Ki,αi}isaninversesystemwithK0=pt,thenthemappingtelescopeof{Ki,αi}isproperhomotopyequivalent(evenin nitesimpleequiv-alent!)tothemappingtelescopeofanysystem{Li,βi}equivalentto{Ki,αi}inpro-homotopy.IfX=lim{Ki,αi},itiseasytoconstructasequenceof →coversUiofXsothatN(Ui)isPLhomeomorphictoKiandsothatthemapsinducedbyre nementarehomotopictotheαi’s.Sinceallsuchsequencesareeasilyseentobepro-equivalent,Lemma7.8follows.

Finally,wenotethatthesequenceofnervesN(Uk)andbondingmapsaboveisco nalwithasequenceasinthestatementofLemma7.8.ThiscompletestheproofofTheorem7.2.

RutgersUniversity,Piscataway,NJ

E-mailaddress:sferry@math.rutgers.edu

UniversityofFlorida,Gainesville,FL

E-mailaddress:dranishn@u .edu

TheUniversityofChicago,Chicago,IL

E-mailaddress:shmuel@math.uchicago.edu

LARGERIEMANNIANMANIFOLDSWHICHAREFLEXIBLE

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(ReceivedOctober18,2001)

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