On the algebraic $L$-theory of $Delta$-sets

arXiv:math/0701833v1[math.AT]29Jan2007ONTHEALGEBRAICL-THEORYOFΔ-SETSANDREWRANICKIANDMICHAELWEISSAbstract.ThealgebraicL-groupsL∗(A,X)aredefinedforanadditivecat-egoryAwithchaindualityandaΔ-setX,andidentifiedwiththegener-alizedhomologygroupsH∗(X;L•(A))ofXwithcoefficientsinthealgebraicL-spectrumL•(A).PreviouslysuchgroupshadonlybeendefinedforsimplicialcomplexesX.IntroductionA‘Δ-set’XinthesenseofRourkeandSanderson[8]isasimplicialsetwithoutdegeneracies.AsimplicialcomplexisaΔ-set;conversely,thesecondbarycentric(akaderived)subdivisionofaΔ-setisasimplicialcomplex,andthehomotopytheoryofΔ-setsisthesameasthehomotopytheoryofsimplicialcomplexes.How-ever,Δ-setsaresometimesmoreconvenientthansimplicialcomplexes:theyaregenerallysmaller,andthequotientofaΔ-setbyagroupactionisagainaΔ-set.InthispaperweextendthealgebraicL-theoryofsimplicialcomplexesofRanicki[5]toΔ-sets.IntheoriginalformulationofWall[9]thesurgeryobstructiontheoryofhigh-dimensionalmanifoldsinvolvedthealgebraicL-groupsL∗(R)ofaringwithinvo-lutionR,whicharetheWittgroupsofquadraticformsoverRandtheirauto-morphisms.Thesubsequentdevelopmentofthetheoryin[5]viewedL∗(R)asthecobordismgroupsofR-modulechaincomplexeswithquadraticPoincar´eduality,constructedaspectrumL•(R)withhomotopygroupsL∗(R),andalsointroducedthealgebraicL-groupsL∗(R,X)ofasimplicialcomplexX.AnelementofLn(R,X)isacobordismclassofdirectedsystemsoverXofR-modulechaincomplexeswithann-dimensionalquadraticVerdier-typeduality.ThegroupsL∗(R,X)wereiden-tifiedwiththegeneralizedhomologygroupsH∗(X;L•(R)),andthealgebraicL-theoryassemblymapA:L∗(R,X)→L∗(R[π1(X)])wasdefinedandextendedtothealgebraicsurgeryexactsequence...//Ln(R,X)A//Ln(R[π1(X)])//Sn(R,X)//Ln−1(R,X)//...withSn(R,X)thecobordismgroupsoftheR[π1(X)]-contractibledirectedsystems.Inparticular,the1-connectiveversiongaveanalgebraicinterpretationoftheexactsequenceofthetopologicalversionoftheBrowder-Novikov-Sullivan-WallsurgeryDate:29thJanuary2007.1991MathematicsSubjectClassification.Primary:57A65;Secondary:19G24.Keywordsandphrases.Surgerytheory,Δ-set,L-groups.12ANDREWRANICKIANDMICHAELWEISStheory:ifthepolyhedronkXkofafinitesimplicialcomplexXhasthehomotopytypeofaclosedn-dimensionaltopologicalmanifoldthenSn+1(Z,X)isthestructuresetofclosedn-dimensionaltopologicalmanifoldsMwithahomotopyequivalenceM≃kXk.TheVerdier-typedualityof[5]usedthedualcellsinthebarycentricsubdivisionofasimplicialcomplexXtodefinethedualofadirectedsystemoverXofR-modulestobeadirectedsystemoverXofR-modulechaincomplexes.TheΔ-setanaloguesofdualcellsintroducedbyusinRanickiandWeiss[7]areusedheretodefineaVerdier-typedualityfordirectedsystemsofR-modulesoveraΔ-setX,whichisusedtodefinethegeneralizedhomologygroupsL∗(R,X)=H∗(X;L•(R))andanalgebraicsurgeryexactsequenceasinthesimplicialcomplexcase.ThealgebraicL-theoryofΔ-setsisusedintheforthcomingpaperofMackoandWeiss[4].1.FunctorcategoriesForanycategoryXletOb(X)bethesetofobjects.ForanadditivecategoryAthereisadistinguishedelement0∈Ob(A).Definition1.1.(i)AfunctionM:Ob(X)→Ob(A);x7→M(x)isfiniteifM(x)=0forallbutafinitenumberofobjectsxinA.ThedirectsumPx∈Ob(X)M(x)willbewrittenasPx∈XM(x).(ii)AfunctorF:X→AisfiniteifthefunctionF:Ob(X)→Ob(A)isfinite.Definition1.2.(i)ThecontravariantfunctorcategoryA∗[X]istheadditivecate-goryoffinitecontravariantfunctorsF:X→A.ThemorphismsinA∗[X]arethenaturaltransformations.(ii)ThecovariantfunctorcategoryA∗[X]istheadditivecategoryoffinitecovariantfunctorsF:X→A.ThemorphismsinA∗[X]arethenaturaltransformations.Remark1.3.(i)ThecontravariantandcovariantfunctorcategoriesarerelatedbytheidentitiesA∗[X]=A∗[Xop],A∗[X]=A∗[Xop]whereXopistheoppositeofX,thecategorywiththesameobjectsandonemor-phismx→yforeachmorphismy→xinX.(ii)WeusetheterminologyA∗[X]foracontravariantfunctorcategorybecauseparadoxicallyforaΔ-setX(§2)thegeneralizedhomologyH∗(X;L(A))withco-efficientsintheL-theoryofA(withachainduality)isthecobordismtheoryofquadraticPoincar´ecomplexesinA∗[X].Similarlyforthecovariantfunctorcate-goryA∗[X]andthegeneralizedcohomologyH∗(X;L(A)).FortheremainderofthissectionweshallonlyconsiderthecontravariantfunctorcategoryA∗[X],buteveryresultalsohasaversionforthecovariantfunctorcategoryA∗[X].ONTHEALGEBRAICL-THEORYOFΔ-SETS3Definition1.4.(i)AchaincomplexinanadditivecategoryAC:...//Cn+1d//Cnd//Cn−1//...(d2=0)isfiniteifCn=0forallbutafinitenumberofn∈Z.(ii)LetB(A)betheadditivecategoryoffinitechaincomplexesinAandchainmaps.AfinitechaincomplexCinA∗[X]isjustanobjectinB(A)∗[X],andlikewiseforchainmaps,sothatB(A∗[X])=B(A)∗[X].Definition1.5.Achainmapf:C→DofchaincomplexesinA∗[X]isaweakequivalenceifeachf[x]:C[x]→D[x](x∈X)isachainequivalenceinA.Achainequivalencef:C→DinA∗[X]isaweakequivalence,butingeneralaweakequivalenceneednotbeachainequivalence–see1.11foramoredetaileddiscussion.Definition1.6.LetXbeacategory,andletx∈Xbeanobject.(i)Theundercategoryx/Xisthecategorywithobjectsthemorphismsf:x→yinX,andmorphismsg:f→f′themorphismsg:y→y′inXsuchthatgf=f′xff′???????yg//y′Theopenstarofxisthesetofobjectsinx/Xst(x)=Ob(x/X)={x→y}.(ii)TheovercategoryX/xisthecategorywithobjectsthemorphismsf:y→xinX,andmorphismsg:f→f′themorphis