高等数学(12)

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.

搜索“diyifanwen.net”或“第一范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，第一范文网，提供最新高中教育高等数学(12)全文阅读和word下载服务。