Reidemeister Torsion and the Classification of Three-Dimentional Lens Spaces

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ReidemeisterTorsionandtheClassicationof Three-DimensionalLensSpaces KATHERINERAOUX AThesis SubmittedtotheDivisionofNaturalSciences NewCollegeofFlorida inpartialfulllmentoftherequirementsforthedegree BachelorofArts UnderthesponsorshipofProfessorDavidMullins Sarasota,Florida May,2011

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i Acknowledgments Mostofthisthesiswouldnothavebeenpossiblewithoutmuchguidanceandsupport fromProfessorDavidMullins.IwouldalsoliketothankProfessorPatrickMcDonald forhisadviceandwordsofwisdomthroughoutmyundergraduatecareer.BeforeI cametoNewCollege,Ihadnoideawhatitmeanttostudymath.Ifithadn'tbeen forthem,Idon'tthinkIeverwouldhave.Finally,Ican'tforgettothankmyparents fordoingallofthatstuthatparentsdo,Augieforkeepingourparentsentertained" inmyabsence,andIanFinneranforkeepingmesane.

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ii REIDEMEISTERTORSIONANDTHECLASSIFICATIONOF THREE-DIMENSIONALLENSSPACES KatherineRaoux NewCollegeofFlorida,2011 ABSTRACT Topologicalinvariantsarecrucialtoolsforsolvingclassicationproblemsinalgebraictopology.Here,westatethedenitionofinvarianceandproceedtodevelop thefundamentalgroup,homologyofchaincomplexesandReidemeistertorsionin thecontextofclassifyingthree-dimensionallensspaces.Wewillshowthatonlythe Reidemeistertorsionissucientforclassifyinglensspacesuptohomeomorphism. Whilethisisaclassicalresult,wedevelopmostofthenecessarytheoryandattempt tomakethematerialaccessibletoadvancedundergraduates. DavidMullins DivisionofNaturalSciences

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Contents Chapter1.Introduction1 Chapter2.BasicPropertiesofLensSpaces3 Chapter3.NecessaryAlgebraicTopology11 Chapter4.ReidemeisterTorsion17 1.Torsionofanisomorphismofvectorspaces17 2.Torsionofanacycliccomplex18 Chapter5.TorsionofaniteCWcomplex31 Chapter6.ClassicationofLensSpaces36 1.R-TorsionofLm,n36 2.HomotopyClassication38 3.HomeomorphismClassication43 Chapter7.Conclusion47 AppendixA.FiniteCWcomplexes48 Bibliography51 iii

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CHAPTER1 Introduction Amaingoalofalgebraictopologyistoclassifyspacesuptohomeomorphism. However,asthiscanbeverydicult,weoftensettleforcoarserformsofclassication, suchashomotopy.Oftenelusive,homeomorphismclassicationproblemshavebeen investigatedsincePoincarerstconjecturedthatanythree-manifoldhomotopictoa three-sphereis,infact,homeomorphic.Thefactthathissimplystatedconjecture remainedunsolvedforoverahundredyearstestiestothedicultyofattainingsuch classications. Forspacestobehomeomorphic,onemustbeabletoconstructacontinuous,bijective,map,betweenthetwospaces,suchthatthismapalsohasacontinuousinverse. Theinherentdicultyofconstructingsuchmapsoftenleadsmathematicianstoapproachproblemsofclassicationfromtheoppositeperspective,tryingtoshowthat twospacesarenothomeomorphic.Helpfultoolsinthisprocessarecalledhomeomorphisminvariants.Ahomeomorphisminvariantisafunction f ,suchthatiftwospaces X and Y arehomeomorphic, f X = f Y .Thus,ifwecanshowthat f X 6 = f Y wewillknowthat X and Y aredenitelynothomeomorphic.Inadditiontohaving homeomorphisminvariants,wecanalsondcoarserinvariants,suchashomotopy invariants.Therefore,understandingdierenttypesofinvariantsiscrucialtothe studyofalgebraictopology,andespeciallyimportantforclassicationproblems.In thiscontext,lensspacesareofparticularinterestsincethemostcommoninvariants suchasthefundamentalgroupandhomologygroupsproveinsucientforclassication.Thelackofawaytodistinguishthesespacesledtothedevelopmentofanew invariant:Reidemeistertorsion. 1

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1.INTRODUCTION2 Inthemidnineteen-thirties,KurtReidemeisterbegantodevelopthetheoryof torsionandeventuallyusedittoclassifythreedimensionallensspaces.But,althoughReidemeistersucceededinsolvingtheclassicationproblem,itremainedto showwhethertorsionwasactuallyahomeomorphisminvariant.Whiteheadbecame particularlyobsessedwiththisproblemandafterpublishingafalseproofofhomotopyinvariance,hediscoveredthatinfact,torsionisnotahomotopyinvariantatall. Thereexistcertainlensspacesthatarehomotopic,buthavedierentReidemeister torsion.HisfurtherinvestigationsintothesubtletiesinvolvedledWhiteheadtodevelopsimplehomotopytheory,andnallyin1974Chapmanprovedamuchstronger result;thatReidemeistertorsionishomeomorphisminvariant. Thisthesisaimstodevelopthealgebraictopologynecessarytoclassifythree dimensionallensspaces.Whilethematerialhasbeenstudiedinthepast,Ihopeto reformulateitandmakeitmoreaccessibletoundergraduatemathematicians.Wewill focusondevelopingameansofcomputingReidemiestertorsionforthepurposeof classifyingthreedimensionallensspaces.Afterintroducingthefundamentalgroup, wewilldenethehomologyofachainandintroduceCWcomplexes.Thenthe denitionoftorsionwillbegivenforacycliccomplexesandtheassociationofan acycliccomplexwithaCWcomplexwillallowustocomputethetorsionofarbitrary niteCWcomplexes.Afewexampleswillclarifythedetailsofthisprocessand nally,wewillusetorsiontoclassifythreedimensionallensspacesuptohomotopy andhomeomorphism.Wewillalsostate,butnotproveChapman'sresult. Mostoftherelevantalgebraictopologicalresultswillbestatedandreferenced. However,theresultsmostcrucialforclassifyinglensspaceswillbeprovedandreferencedwhenappropriate.

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CHAPTER2 BasicPropertiesofLensSpaces Webeginbydeningthethreesphereasasubsetof C C Definition 0.1 A three-sphere isaclosedoriented3-dimensionalmanifold, S 3 = f z;w 2 C C : j z j 2 + j w j 2 =1 g Alensspaceisaquotientspaceofathree-sphere,whichmeansthattocreatea lensspace,wetakecertainpointsinthethreesphereandthinkaboutglueing"them together.Moreformally, Definition 0.2 A lensspace isaclosedoriented3-dimensionalmanifold, L m;n = S 3 = Z m = f z;w 2 C C jj z j 2 + j w j 2 =1 g = z;w z; n w where isaprimitive mth rootofunityand gcd m;n =1. Example 0.3 Considerthespace L ; 2.Since n =5, 5 =1sothatwehave anequivalenceclass,madeupofthepointsweglue"together, z;w z; 2 w 2 z; 4 w 3 z; 1 w 4 z; 3 w foranyorderedpair z;w in S 3 Nowthatwehavedenedlensspaces,theusualtopologicalquestionofclassicationarises.However,inordertoclassifyspacesofanysort,wemustrstdenewhat itmeansfortwospacestobethesame. Definition 0.4 IfXandYaretwotopologicalspaces,wesayXis homeomorphic toYifthereexistsacontinuousbijectivemap, f : X Y suchthat f )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 isalso continuous. 3

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2.BASICPROPERTIESOFLENSSPACES4 Notethathomeomorphismisanequivalencerelation.WhenwethinkofXand Yasbeingequivalentinatopologicalsensewemeanhomeomorphic,however,itis generallyverydiculttoconstructanexplicitmapfromXtoY.Therefore,weoften settleforlessstringentrequirements,suchashomotopy. Definition 0.5 Let X Y betopologicalspaces..Wesaythat X is homotopic to Y iftherearecontinuousmaps f : X Y g : Y X F : X [0 ; 1]: X and G : Y [0 ; 1] Y suchthat, F x; 0= g f x and F x; 1= x forany x 2 X G y; 0= f g y and G y; 1= y forany y 2 Y Itisnothardtoseethathomotopyisanequivalencerelationandthatif X and Y arehomeomorphic,theywillalwaysbehomotopic.Often,amoreusefulwayofsaying thisisthatif X and Y arenothomotopic,thentheyarenothomeomorphic.The converseisobviouslyfalse.Wecanseethatadisk, D 2 = f e i :0 1and0 < 2 g ,ishomotopictoapoint,say f 0 g ,byconsideringmaps, f : D 2 !f 0 g : denedby f e i =0. and g : f 0 g! D 2 denedby g =0. Then f g = id: onthepoint f 0 g andwecandene F : D 2 [0 ; 1]: D 2 suchthat F e i ;t = te i ThenFiscontinuousand F e i ; 0=0= g f e i F e i ; 1= e i However,thereisnowaytocreateabijectivemapfrom D 2 to f 0 g since D 2 hasan uncountablenumberofpoints.Thus,thedisk D 2 isnothomeomorphictothepoint f 0 g Inthecaseoflensspaces,wewouldliketobeabletodistinguishhomotopyclasses, and,especially,homeomorphismclasses.Tobeginthisprocess,wewillintroduce

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2.BASICPROPERTIESOFLENSSPACES5 andthenprovetheinadequacyoftheusualtopologicalinvariantsforclassifyinglens spaces. 0.1.FundamentalGroupsandCoveringSpaces. Thefollowingdevelopmentoffundamentalgroupsandcoveringspacesistakenfrom[ 3 ],inparticular, Chapters2and5.Therefore,wewillstatethedenitionsandtheoremsprovingonly whatisessentialforthepurposeofclassifyinglensspaces. A path isacontinuousmap :[ a;b ] X .Apathiscalled closed ora loop ifthe endpointscoincide,i.e. a = b = x .Suchaloopissaidtobe based at x .Two loopsbasedatthesamepoint x 1 and 2 areconsidered homotopic ifthereexistsa continuousmap :[ a;b ] [0 ; 1] X suchthat, t; 0= 1 t t; 1= 2 t for t 2 [ a;b ]and a;s = b;s = x forall s 2 [0 ; 1].If isaloopbasedatthepoint x ,thenthesetofallloopshomotopic to formanequivalenceclass,[ ].Thesetofequivalenceclassesofloopsbasedat apoint x 2 X formsagroupcalledthe fundamentalgroup ofXandisdenoted, 1 X;x .Thegroupoperationismultiplication,denedforpathsandconsequently loops,asfollows: Let and bepathsin X withtheterminalpointof beingtheinitialpointof ,i.e. b = a .Denetheproduct, :[ a;b ] X by, t = 8 < : t 0 t 1 2 ; t )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 1 2 t 1 : Thismultiplicationisonlydenedwhentheinitialpointofonepathcoincideswith theterminalpointoftheotherpath.However,thismultiplicationisnot,ingeneral, associative,nordowehaveidentityelementsorinverses.So,inordertoimposea groupstructureonthesetofloops,werstlookatequivalenceclassesofhomotopic paths.Now,whenconsideringtheseequivalenceclasses,wedogetassociativity.

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2.BASICPROPERTIESOFLENSSPACES6 Considerpaths ; and suchthattheterminalpointof istheinitialpoint of andtheterminalpointof istheinitialpointof .Thenthereexistsamap F :[ a;b ] [0 ; 1] X denedby, F t;s = 8 > > > < > > > : 4 t 1+ s 0 t s +1 4 ; t )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(s s +1 4 t s +2 4 ; )]TJ/F28 7.9701 Tf 13.151 5.698 Td [(4 )]TJ/F31 7.9701 Tf 6.587 0 Td [(t 2 )]TJ/F31 7.9701 Tf 6.587 0 Td [(s s +2 4 t 1 F iscontinuousand F t; 0=[ ] t and F t; 1=[ ] t .Thus, F isa homotopywhichimpliestheequivalence, If isdenedasabove,wecanalsoconstructmapsthatactliketheidentity whenmultipliedwith .Aright-handedidentitywilllooklikethepath e b where e b = b forall t 2 [0 ; 1].Then[ e b ] [ ].Noticethatwhenconsidering loops,right-handedidentitieswillbethesameasleft-handedidentities.Wealso dene[ ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 tobetheclassofpathsstartingat b andendingat a .These propertiesgiveusthenecessarygroupstructurefor 1 X;x Ingeneral,anycontinuousmap f : X Y willinduceahomomorphismbetween thefundamentalgroupsofthe X and Y ,denoted, f : 1 X;x 0 1 Y;y 0 ,where y 0 = f x 0 anddenedby f [ ]=[ f ]. Definition 0.6 Let X beatopologicalspace,arcwiseconnectedandlocally arcwiseconnected.A coveringspace of X isapair, ~ X;p ,consistingofaspace ~ X andacontinuousmap, p : ~ X X ,calleda coveringmap suchthatthefollowing conditionholds:Eachpoint x 2 X hasanarcwiseconnectedopenneighborhood U suchthateacharccomponentof p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 U ismappedtopologicallyonto U by p Interestingproblemsarisewhenoneconsiderstheinverseimageofaloopina spaceXwhenitis lifted toacoveringspace.Ingeneral,loopsdonotremainloops, butareratherliftedtopaths. So,givenaspace X ,whatkindsofspaceswillcover X ?Trivially,wecansee that X willalwayscoveritself.Ingeneralhowever,wewouldliketoknowwhatother

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2.BASICPROPERTIESOFLENSSPACES7 spaceswillcover X .Additionally,since p isacontinuousmap,itinducesamap onthefundamentalgroups.Sowewouldalsoliketoseehowthemap p relatesto coveringmaps.Thefollowingtheoremgivestherelationbetweenthetwo. Theorem 0.7 Let ~ X;p beacoveringspaceofX.Let Y beaconnectedand locallyarcwiseconnectedspacesuchthat y 0 2 Y ~ x 0 2 ~ X ,and x 0 = p ~ x 0 .Givena map : Y X ,thereexistsauniquelifting, ~ : Y ~ X ifandonlyif 1 Y;y 0 p 1 ~ X; ~ x 0 ~ X Y X p ~ Since X alwayscoversitself,wecaneasilyseethatanarbitrarycontinuousmap f : Y X willbeacoveringmapifandonlyif f 1 Y;y 0 1 X;x 0 where x 0 = f y 0 .Thereforewhen 1 X;x 0 =1,theonlypossiblecoveringspacesfor X mustalsohave 1 X;x 0 =1.Suchspacesarecalled simplyconnected .Inparticular, when X issimplyconnected,itsonlycoveringspacewillbeitself. Definition 0.8 Let ~ X;p beacoveringspaceof X suchthat ~ X issimply connected.Thenif ~ X 0 ;p 0 isanyothercoveringspaceof X ,thereexistsahomomorphism, of ~ X;p onto ~ X 0 ;p 0 ,suchthat ~ X; isacoveringspaceof ~ X 0 .Sucha space, ~ X iscalleda universalcoveringspace .Moresimply,auniversalcover ~ X isa coveringspaceofanycoveringspaceof X Aspaceissaidtobe semilocallysimplyconnected ifeachpoint x 2 X hasa neighborhood U suchthatthehomomorphism 1 U;x 1 X;x istrivial. Theorem 0.9 Let X beaconnected,locallypathconnectedandsemilocallysimply connectedspace.Thenthereexistsauniversalcoveringof X .

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2.BASICPROPERTIESOFLENSSPACES8 Asimpleexamplewillclarifywhyweneedtheseeminglyverbosequalication, semilocallysimplyconnected,for X tohaveauniversalcover. Example 0.10 Considerasubspace X oftheplane, R 2 ,madeupofaninnite sequenceofcircleswiththeirradiiconvergingtozero,suchthatthecircleintersect atpreciselyonepoint, x 0 x 0 Noticethatthisspaceisnotsemilocallysimplyconnected,sinceanyneighborhood aroundthepoint x 0 willcontainanontrivialloop.Toconstructauniversalcoverfor X weconsiderseveringtheloops,sothattheresultingspaceiscontractableand connected.Certainlyifwecutat x 0 theresultingspacewillnotbeconnected.Ifwe cutatanyotherpart,sayaline, x 0 Thennomatterhowcloseto x 0 theheightofthelinegets,therewillalwaysbeloops leftunsevered.Thus, X willnothaveauniversalcoveringspace. Consideracoveringspace ~ X;p of X .Bythedenitionofcoveringspace,for anyarcwiseconnectedneighborhood U of x 2 X ,eachpreimage p )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 U ishomeomorphicto U .Wedenotethesetofpreimagesby[ p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 U ].Sincewecanmakethe

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2.BASICPROPERTIESOFLENSSPACES9 neighborhood U arbitrarilysmall,wecanalsotalkaboutthesetofpointsthatare mappedto x by p ,[ p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 x ]. Infact,thereisanaturalactionofthegroup 1 X;x 0 ontheset[ p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 x 0 ].Choose apreferred"preimage,~ x 0 2 [ p )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 x 0 ] ~ X .Thenthereexistsapathclass~ 2 ~ X suchthat~ =~ x 0 and p ~ = .Thus,wedenetheaction ~ x 0 =~ Sothattheactiontakes~ x totheterminalpointof~ Ingeneralthisactionwillbetransitiveandthus,theset[ p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 x 0 ]formsacoset space,calleda right X;x 0 -space ,[ p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 x 0 ]= f gK : g 2 1 X;x 0 g where K = f g 2 1 X;x 0 : g x 0 = x 0 g isthestabilizerof x 0 in ~ X .Inparticular,if K = f 1 g then[ p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 x 0 ]= 1 X;x 0 Definition 0.11 LetGbeagroupofautomorphismsofaspaceX.ThenGis properlydiscontinuous ifforall x 2 X thereisanopenneighborhood U of x such thatthesets gU g 2 G ,arepairwisedisjoint. Inparticular,ifGisproperlydiscontinuous,thenGhasnoxedpoints. Thefollowingtheoremwillbethekeytocomputingthefundamentalgroupofany lensspace. Theorem 0.12 IfXissimplyconnectedandGisagroupofproperlydiscontinuousautomorphismsofX,then 1 X=G = G Proof. Considerthequotientmap, p : X X=G .SinceGisproperlydiscontinuous,foranyneighborhood U in X=G theset p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 U mapstopologicallyontoU, thus X;p actsauniversalcoverfor X=G .Inparticular,theset[ p )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 U ]willbe isomorphictotheset, f gU : g 2 G g andthus,ifwechoose U sothat x 0 2 U ,then [ p )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 U ]=[ p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 x 0 ]= f gK : g 2 1 X=G;x 0 g for K = f g 2 1 X=G;x 0 : g x 0 = x 0 g .So,if g 2 K then g mustbealoopin X .Since, X issimplyconnectedwe concludethat K = f 1 g andthus, G = 1 X=G;x 0

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2.BASICPROPERTIESOFLENSSPACES10 Usingthetheorem0.12wecannoweasilycomputethefundamentalgroupofany lensspace. 1 L m;n = 1 S 3 = Z m = Z m Thus,thefundamentalgroupofalensspace L m;n dependsonlyonthevalueof m sounlessthevalueof n hasnoimpactonthestructureofthespacewhichwewill eventuallyproveisn'tthecase,thefundamentalgroupdoeslittletohelpdistinguish homotopyorhomeomorphismclasses.

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CHAPTER3 NecessaryAlgebraicTopology A chaincomplex isasequenceofabeliangroupsandhomomorphisms, C : ::: C n C n )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 ::: @ n +1 @ n @ n )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 suchthat @ n @ n +1 =0foralln.Themap @ satisfying @ 2 =0iscalleda chainmap Thisconditionsaysthat im @ n +1 ker @ n ,sinceif x 2 im @ n +1 thenthereexists y suchthat @ n +1 y = x .Thus, @ n x = @ n @ n +1 y =0,whichimplies x 2 ker @ n Wedenethe nthsingularhomology ofachaincomplextobe H n C = ker @ n =im @ n +1 Usingchaincomplexestocomputetheirhomologywewillattempttogivean intuitiveexplanationofCW-complexes,andreferthereadertotheAppendixand[ 6 ], Chapters1and2foramoreformalexplanation. Ingeneralcomputingthehomologyofanarbitrarytopologicalspace,X,canbe dicult.However,inthecasewhenXiscompactandHausdor,wethinkaboutX asbeingmadeupofpieces,eachhomeomorphictoann-ball.Suchpiecesarecalled n-cells .Algebraically,wecanthinkofeachn-cellasafree-abeliangroupisomorphic to Z .Thewayinwhichthen-cellsaregluedtothe n )]TJ/F24 11.9552 Tf 11.593 0 Td [(1-cellsgivesusthenecessary boundarymapstoformchaincomplexes. Consider,forexample, S 2 .Wecanthinkof S 2 asbeingmadeupofone0-celland one2-cell.Theassociatedchaincomplexisverysimple, 0 Z 0 Z 0 @ 2 @ 1 @ 0 11

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3.NECESSARYALGEBRAICTOPOLOGY12 Thus, H 0 S 2 = ker @ 0 =im @ 1 = Z = 1= Z .Similarly, H 1 S 2 =0and H 2 S 2 = Z Generalizingthisidea,wecanseethat S n canalwaysbethoughtofasann-cellglued toa0-cell,yieldingthefollowingproposition: Proposition 0.13 Foranyinteger n 0 H k S n = 8 < : Z k =0 ;n 0 k 6 =0 ;n However,whilethehomologyisuniqueforagivenspace,CW-decompositionsare not.ConsiderthefollowingCW-decompositionof S 3 E 0 0 = ; 0 E 0 1 = )]TJ/F24 11.9552 Tf 9.299 0 Td [(1 ; 0 E 1 0 = f e i ; 0; 2 [0 ; ] g E 1 1 = f e i ; 0; 2 [ ; 2 ] g E 2 0 = f z;s ; s 2 [0 ; 1] g E 2 1 = f z; )]TJ/F30 11.9552 Tf 9.298 0 Td [(s ; s 2 [0 ; 1] g E 3 0 = f z;w ; arg w 2 [0 ; ] g E 3 1 = f z;w ; arg w 2 [ ; 2 ] g Wecantrytovisualizesuchdecompositionbyconsideringadiagram, E 0 0 = ; 0 )]TJ/F24 11.9552 Tf 9.298 0 Td [(1 ; 0= E 0 1 E 1 0 E 2 0 ; 1 ; )]TJ/F24 11.9552 Tf 9.298 0 Td [(1 ;w z; 0

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3.NECESSARYALGEBRAICTOPOLOGY13 inwhichtheverticallinerepresentsthecircle j w j =1identiedatthepointatinnity, andthecirclerepresentsthecircle j z j =1. Thisdecompositionalsogivesusanassociatedchaincomplex, 0 ~ C 3 ~ C 2 ~ C 1 ~ C 0 0 @ 3 @ 2 @ 1 @ 0 whereeach ~ C i = Z Z Computingtheboundaryofeachgeneratorweget, @ 0 E 0 0 =0 @ 0 E 0 1 =0 @ 1 E 1 0 = E 0 1 )]TJ/F30 11.9552 Tf 11.956 0 Td [(E 0 0 @ 1 E 1 1 = E 0 0 )]TJ/F30 11.9552 Tf 11.955 0 Td [(E 0 1 @ 2 E 2 0 = E 1 0 + E 1 1 @ 2 E 2 1 = E 1 0 + E 1 1 @ 3 E 3 0 = E 2 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(E 2 0 @ 3 E 3 1 = E 2 0 )]TJ/F30 11.9552 Tf 11.955 0 Td [(E 2 1 Thus, ker @ 0 = ~ C 0 .Theimageof @ 1 canbefoundbycomputing @ 1 aE 1 0 + bE 1 1 = a )]TJ/F30 11.9552 Tf 12.248 0 Td [(b E 0 1 )]TJ/F30 11.9552 Tf 12.248 0 Td [(E 0 0 where a;b 2 Z .Thus, im @ 1 = f r; )]TJ/F30 11.9552 Tf 9.298 0 Td [(r g Z Z .Additionally, @ 1 aE 1 0 + bE 1 1 =0whenever a = b sothat ker @ 1 = Z Similarly, @ 2 aE 2 0 + bE 2 1 = a + b E 1 0 + E 1 1 .Thus, im @ 2 = Z .Also, @ 2 aE 2 0 + bE 2 1 =0whenever a = )]TJ/F30 11.9552 Tf 9.299 0 Td [(b sothat ker @ 2 = Z Finally, @ 3 aE 3 0 + bE 3 1 = a )]TJ/F30 11.9552 Tf 10.047 0 Td [(b E 2 0 )]TJ/F30 11.9552 Tf 10.047 0 Td [(E 2 1 .Sowehave im @ 3 = f r; )]TJ/F30 11.9552 Tf 9.299 0 Td [(r g Z Z Finally,since @ 3 aE 3 0 + bE 3 1 =0whenever a = b ker @ 3 = Z Puttingallofthisinformationtogether,wecomputethat H 0 S 3 = Z Z = f r; )]TJ/F30 11.9552 Tf 9.298 0 Td [(r g = Z H 1 S 3 = Z = Z =0, H 2 S 3 =0,and H 3 S 3 = Z ,justaswe wouldexpect. Obviously,computingthesemapsisharderthancomputingthemapsforthe decompositionof S 3 withone0-cellandone3-cell,however,thisdecompositionwill beusefulforcomputingthehomologyofthequotientspace L ; 1.Infact,this CW-decompositionrespectstheactionof Z 2 onpoints z;w whichmeansmaking identication z;w )]TJ/F30 11.9552 Tf 9.299 0 Td [(z; )]TJ/F30 11.9552 Tf 9.298 0 Td [(w ,willcorrespondtoglueingthecell E i 0 tothecell E i 1

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3.NECESSARYALGEBRAICTOPOLOGY14 foreach i .SothatourCW-decompositionfor L ; 1consistsofone0-cell,one1-cell, one2-cell,andone3-cell. Let : S 3 L ; 1bethequotientmap.Thendene E i j = E i theneach E i isthebasisfor C i andthefollowingdiagramwillcommute. 0 ~ C 3 ~ C 2 ~ C 1 ~ C 0 0 0 C 3 C 2 C 1 C 0 0 @ 3 @ 2 @ 1 @ 0 @ 3 @ 2 @ 1 @ 0 whereeach ~ C i isasbeforeand C i = Z Nowwecanusethepreviouslycomputedboundarymapstocomputethenew boundarymaps.Wechoose E i 0 tobethepreferredpre-imageof E i foreach C i ,but wecomputingintermsof E i 1 wouldyieldthesameresult. @ 0 E 0 0 = @ 0 E 0 0 =0 @ 1 E 1 0 = @ 1 E 1 0 = E 0 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(E 0 0 =0 E 0 @ 2 E 2 0 = @ 2 E 2 0 = E 1 0 + E 1 1 =2 E 1 @ 3 E 3 0 = @ 3 E 3 0 = E 2 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(E 2 0 =0 E 2 Thuswecaneasilyseethat, ker @ 0 = C 0 ker @ 1 = Z ker @ 2 =0 ker @ 3 = Z im @ 1 =0 im @ 2 =2 Z im @ 3 =0 Sothat H i L m;n = H i S 3 for i 6 =1and H 1 L m;n = Z 2 ComputingtheCW-complexandhomologyforagenerallensspaceworksvery similarly. Example 0.14CWcomplexofLensSpace SincewedenedLensspacesin termsofaquotientof S 3 ,wewillconstructaCWcomplexfor L m;n byrst constructingaCWcomplexfor S 3 whichrespectsthequotientmap.

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3.NECESSARYALGEBRAICTOPOLOGY15 Consider L m;n .Denotethejthrootofunity, j .For j 2 Z m dene, E 0 j = j ; 0 E 1 j = f e i ; 0 2 S 3 ; 2 j m 2 j +1 m g E 2 j = f z;s j 2 S 3 ; s 2 R ; j z j 2 + s 2 =1 g E 3 j = f z;w 2 S 3 ; 2 j m arg w 2 j +1 m g Theneach E k j ishomeomorphictoaclosed k -ball.Noticethatundertheidentication, z;w z; n w ,thepointsintheset f E 0 j g m )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 j =0 willbeidentied.Similarlythesets f E k j g m )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 j =0 willbeidentiedforeach k respectively.Thus,thecollection f E k j g 0 k 3 j 2 Z m formsaCWdecompositionof S 3 whichrespectstheactionof Z m .Sincealensspaces ifformedbyglueingtheseballstogether,ithasaCWdecompositioncomposedof one0-cell,one1-cell,one2-cell,one3-cell. Thus,tocomputethehomologywelookatthechaincomplex, 0 ~ C 3 ~ C 2 ~ C 1 ~ C 0 0 0 C 3 C 2 C 1 C 0 0 @ 3 @ 2 @ 1 @ 0 @ 3 @ 2 @ 1 @ 0 whereeach ~ C i = Z m C i = Z and : S 3 L m;n isdenedongeneratorsby E i j = E i Inordertocomputetheboundarymaps,weagainlookatour S 3 decomposition andcompute, @ 1 E 1 j = E 0 j +1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(E 0 j @ 2 E 2 j = m )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 X k =0 E 1 k @ 3 E 3 j = E 2 j +1 )]TJ/F30 11.9552 Tf 11.956 0 Td [(E 2 j

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3.NECESSARYALGEBRAICTOPOLOGY16 Againchoosing E i 0 asourpreferredpre-image,wecompute, @ 0 E 0 0 = @ 0 E 0 0 =0 @ 1 E 1 0 = @ 1 E 1 0 = E 0 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(E 0 0 =0 E 0 @ 2 E 2 0 = @ 2 E 2 0 = E 1 0 + E 1 1 + ::: + E 1 m )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 = mE 1 @ 3 E 3 0 = @ 3 E 3 0 = E 2 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(E 2 0 =0 E 2 Thus, H 0 L m;n = Z H 1 L m;n = Z m H 2 L m;n =0 H 3 L m;n = Z Asmentionedearlier,thisgivesnonewinformationforcomparing L m;n 0 to L m;n 1 .Thus,wehavedemonstratedthenecessityforanerinvariantandwill nowintroduceReidemeistertorsion.

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CHAPTER4 ReidemeisterTorsion 1.Torsionofanisomorphismofvectorspaces Thisexampleisborrowedfrom[ 4 ].Consideranisomorphismof n dimensional vectorspaces, f : U 0 )167(! U 1 .Ifwechoosebases, u 0 and u 1 ,thenwecanrepresent themap f asan n n matrix, A .Wedenethe torsion ofthemap f ,withrespect tochoiceofbases u 0 and u 1 ,tobe f; u 0 ; u 1 := det A )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 Whilethismayseemarbitrary,thetorsionasdeneddependsonlyonthechoiceof bases. Considerachangeofbases, u 0 7! v 0 u 1 7! v 1 .Thenthereareassociatedchange ofbasismatrices, [ v 0 = u 0 ]: u 0 7! v 0 and[ v 1 = u 1 ]: u 1 7! v 1 andthematrixformofthemap f changesaccordinglyto [ v 0 = u 0 ] )]TJ/F47 7.9701 Tf 6.587 0 Td [(1 A [ v 1 = u 1 ]. Noticethatthetorsionofthemap f alsoreectsthischangeofbases, f; u 0 ; u 1 := det [ v 0 = u 0 ] )]TJ/F47 7.9701 Tf 6.587 0 Td [(1 A [ v 1 = u 1 ] )]TJ/F47 7.9701 Tf 6.586 0 Td [(1 =1 =det [ v 0 = u 0 ] )]TJ/F47 7.9701 Tf 6.586 0 Td [(1 det A det [ v 1 = u 1 ] Moregenerally,anisomorphismlike f : U 0 )167(! U 1 canbethoughtofasanexact sequence, 0 U 0 U 1 0 f Inthiscase,thetorsionissimplytheinverseofthedeterminantofthemap f oncea choiceofbaseshasbeenmade.Ingeneral,wewouldliketohaveasimilarinvariant 17

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2.TORSIONOFANACYCLICCOMPLEX18 thatcanbeappliedtolongerexactsequence.Inthissense,torsioncanbethought ofasageneralizeddeterminant. 2.Torsionofanacycliccomplex Here,wewilldevelopthetheoryintermsofchaincomplexesoffreeabeliangroups. However,itisimportanttorealizethatthetheorywillbethesameifwelook,more generally,atfreeR-moduleswhenRisacommutativering.Moreprecisely,inorder todenetorsionofanacyclicchaincomplex,wemustbeabletowritedownmatrices thatcorrespondtotheboundarymapsofourchaincomplexes.Thus,aslongaswe havenitebases,thisshouldbepossible. Considerachaincomplexofabeliangroupsandhomomorphisms, C : ::: C n C n )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 ::: @ n +1 @ n @ n )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 Suchachaincomplexiscalled acyclic if H n C =0foralln. Theconditionofacyclicityisequivalenttorequiringthat ker @ n = im @ n +1 forall n ,orthatthesequenceofhomomorphismsgivenaboveisexact. Suppose C isaniteacycliccomplexoffreeabeliangroups f C k g p k =0 .Thengiven any C k ,thereexistsanitebasis, c k = c k 1 ;c k 2 ;:::;c k n sothatelementsof C k can bewrittenasniteformalsums, n X j =1 j c k j j 2 Z Since C isanitecomplex,wecandenefreeabeliangroups f B k g p k =0 asfollows.Let B p =and B k = im @ k +1 for0 k

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2.TORSIONOFANACYCLICCOMPLEX19 0 B k C k B k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 0 i @ k Let b k = b k 1 ;b k 2 ;:::;b k m beabasisforeach B k .Since @ k isonto B k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 ,wecan lifteach b k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 j toa ~ b k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 j 2 C k .Thus, b k b k )]TJ/F47 7.9701 Tf 6.587 0 Td [(1 = b k 1 ;b k 2 ;:::;b k m ; ~ b k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 1 ; ~ b k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 2 ;:::; ~ b k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 l formsabasisfor B k B k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 .Itisnottoodiculttoseethat C k isisomorphictothis directsum,butrstweneedanimportantlemmaandatheorem. Considerashortexactsequenceofabeliangroups, 0 A B C 0 f g Suchasequenceiscalled splitexact if f A isadirectsummandof B Lemma 2.1 Thefollowingareequivalent: Theshortexactsequenceaboveissplitexact. Thereexistsamap ~ f : B A suchthat ~ f f = id A : Thereexistsamap ~ g : C B suchthat g ~ g = id C : Proof. Supposethatthesequenceissplitexact.Then, B = f A C 0 and g B = g f A g C 0 = 0 g C 0 = g C 0 .Sincethesequenceisexactwealso havethat g isontoandthus, g B = C .Thus, g C 0 = C so g mustalsobe1-1on C 0 .Thus,thereexistsamap~ g : C C 0 suchthat g ~ g = id C Similarlyifthesequenceissplitexact,wecandene ~ f : B = f A C 0 A by ~ f f a ;c = a .Then ~ f f = id A Nowsupposethatthereexistsamap ~ f : B A suchthat ~ f f = id A : Noticethat since im f = ker g C B im f = f A isanormalsubgroupof B .Let C 0 = ker ~ f Then C 0 C B and f A C 0 = f b 2 B : b 2 f A and b 2 C 0 g .Butif b 2 f A thenthereexists a 2 A suchthat b = f A .Thus, ~ f f a = ~ f b = a .Thus, b is alsointhekernalof ~ f onlyif b =1.So, f A C 0 = f 1 g .Thus,weconcludethat B = f A C 0 .

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2.TORSIONOFANACYCLICCOMPLEX20 Finally,suppose g g = id C .Then g C B .Considertheintersection ker g ~ g C .If b 2 ~ g C ,then b isalsoanelementof ker g onlywhen b =1.Nowwecan constructamap ~ f : B A suchthat ~ f b = 8 < : a whenthereexists a suchthat b = f a ; 1 b 2 ~ g C : Noticethatthe ker ~ f =~ g C ,thus, B = ker g ~ g C = f A ~ g C Thefollowingtheoremcomesfrom[ 2 ]. Theorem 2.2 If C isfreeabelianwithniterank n and B isanonzerosubgroup of C ,thenthereexistsabasis f c 1 ;:::;c n g of C ,aninteger r 1 r n ,and positiveintegers d 1 ;:::d r suchthat d 1 j d 2 j ::: j d r and B isafreeabeliangroupwith basis f d 1 c 1 ;:::;d r c r g Proposition 2.3 If C isaniteacyclicchaincomplexoffreeabeliangroups, f C k g p k =0 ,thenforeach k C k = B k B k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 ,where B p = and B k = im @ k +1 for 0 k

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2.TORSIONOFANACYCLICCOMPLEX21 f c k j = c k j for j n k f c k j =0for j>n k Then, f i x = x .SotheclaimholdsbyLemma2.1. Foranacyclicchaincomplexoffreeabeliangroups,thereexistsachaincontraction,adegreeonehomomorphism, : C k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 C k ,suchthat @ + @ =1: C k C k foreachk.Themap iscalledachainhomotopy.Wecanconstructsuchamap asdonein[ 5 ]asfollows: Consideragainthesplitexactsequence, 0 B k C k B k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 0 i @ k Since C k = B k B k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 ,thereexists,byLemma2.1,amap g k : B k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 C k such that @ k g k = id andwecanwrite, C k = i B k g k B k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 Dene k : C k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 C k by k a + b = g k a where a 2 i B k and b 2 g k B k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 Let b 0 2 B k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 suchthat g k b 0 = b .Then k @ k + @ k +1 k +1 a + b = k @ k b + @ k +1 g k +1 a = k b 0 + id a = b + a = a + b Noticethatthechaincontraction doesnotnecessarilysatisfythecondition, 2 =0,soitisnotnecessarilyachainmap.However,foranychaincontraction, : C k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 C k ,dene = @ .Thenitiseasytocheckthat isachainmap satisfying @ + @ =1and 2 =0. Suppose satises @ + @ =1.Thenwecanconsidertheoperator, @ + : C C When 2 =0,thismapmustbeanisomorphism,since @ + 2 = @ + @ =1,but ingeneral,weneednotrequirethat beachainmap.Withrespecttothedirect sumdecomposition, C = C even C odd where C even = C 0 C 2 C 4 ::: and C odd = C 1 C 3 C 5 ::: ,wecanthinkof @ + asblockmatrixoftheform,

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2.TORSIONOFANACYCLICCOMPLEX22 @ + = 2 4 0 @ + odd @ + even 0 3 5 Wherewedenoteby, @ + odd themap, @ + j C odd : C odd C even and @ + even themap, @ + j C even : C even C odd Definition 2.4 Foranychaincontraction, satisfying, @ + @ =1,denethe ReidemeisterTorsion ofanacycliccomplex C withrespecttothebasesc,tobe C;c := det @ + odd = det @ + even )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 Ifthechaincomplex C isnotacyclic,wedenethetorsiontobezero. Nowwewillshowsomeimportantresultsconcerningthematrices @ + .In particular,wewillprovethat @ + odd and @ + even arenonsingular,andshow how det @ + odd isrelatedtothechoiceofbasisandthechoiceof .Butrst,we provealemmathatwillmakethecomputationseasier. Lemma 2.5 Suppose M = 2 6 6 6 6 6 6 6 6 6 6 4 m 1 X 12 ::::::X 1 n 0 m 2 X 23 :::X 2 n 0 . . . 0 . . X n )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 n 0 :::::: 0 m n 3 7 7 7 7 7 7 7 7 7 7 5 isa pxp blockmatrixsuchthatthediagonalblocks, m 1 ;:::;m n areallsquareand nonsingular.Then det M = n Y i =1 det m i Proof. Firstconsidera kxk matrix A .Noticewecandirectlycompute, det 0 @ 2 4 A 0 01 3 5 1 A = )]TJ/F24 11.9552 Tf 9.298 0 Td [(1 k +1+ k +1 det A = det A : Inductively,wecaneasilyseethat

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2.TORSIONOFANACYCLICCOMPLEX23 det 0 @ 2 4 A 0 0 I t 3 5 1 A = det A forany t 2 N ,where I t isthe txt identitymatrix.Similarly, det 0 @ 2 4 I t 0 0 A 3 5 1 A = det A : Nowconsidertheblockmatrix M 1 = 2 4 m 1 X 12 0 m 2 3 5 where m 1 and m 2 arenonsingular. Consequently, M 1 isalsononsingular,since 2 4 m 1 X 12 0 m 2 3 5 2 4 m )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 1 m )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 1 X 12 m )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 2 0 m )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 2 3 5 = 2 4 I 0 0 I 3 5 Additionally,wecanfactor M 1 M 1 = 2 4 m 1 X 12 0 m 2 3 5 = 2 4 m 1 0 0 I 3 5 2 4 Im )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 1 Xm )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 2 0 I 3 5 2 4 I 0 0 m 2 3 5 and,wereadilycompute,usingandthat det M 1 = det 0 @ 2 4 m 1 0 0 I 3 5 2 4 Im )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 1 Xm )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 2 0 I 3 5 2 4 I 0 0 m 2 3 5 1 A = det m 1 det m 2 Similarly,ifweconsiderthematrix, M 2 = 2 4 M 1 Y 2 0 m 3 3 5 where Y 2 = 2 4 X 13 X 23 3 5 weseethat det M 2 = det M 1 det m 3 = det m 1 det m 2 det m 3 .Continuingthis process,letting

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2.TORSIONOFANACYCLICCOMPLEX24 Y i = 2 6 6 6 6 6 6 6 4 X 1 i +1 X 2 i +1 X i i +1 3 7 7 7 7 7 7 7 5 ,weeventuallygetthat M = 2 4 M n )]TJ/F28 7.9701 Tf 6.586 0 Td [(2 Y n )]TJ/F28 7.9701 Tf 6.587 0 Td [(2 0 m n 3 5 So, det M = det M n )]TJ/F28 7.9701 Tf 6.587 0 Td [(2 det m n = n Y i =1 det m i Lemma 2.6 Let c beabasisfor C .Then @ + odd and @ + even arenonsingularand det @ + odd = det @ + even )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 Proof. Considerthecomposition, @ + even @ + odd : C odd C odd @ + even @ + odd = @ + 2 j C odd = @ 2 + @ + @ + 2 j C odd =+ 2 j C odd Thus, @ + even @ + odd = 0 B B B B B B B @ C 1 C 3 C 5 ::::::::: C 1 I 3 2 000 ::: C 3 0 I 5 4 00 ::: C 5 00 I 7 6 0 ::: .000 . ::: 1 C C C C C C C A So,bylemma2.5,wegetthat det @ + even @ + odd =1.Noticewhenwe switchfromthinkingaboutfunctioncompositiontothinkingaboutmatrixmultiplication,weget @ + even @ + odd = @ + odd @ + even .Thus, @ + odd and @ + even arenonsingular,and det @ + odd = det @ + even )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 Lemma 2.7 Let c = [ c i and c 0 = [ c 0 i bebasesofC,suchthat c i and c 0 i are arbitrarybasesof C i .Then C;c 0 = C;c n Y i =0 det [ c 0 i =c i ] )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 i .

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2.TORSIONOFANACYCLICCOMPLEX25 where [ c 0 i =c i ] isthechangeofbasismatrixtaking c i to c 0 i Proof. Considerthat @ + odd isamapfrom C odd C even .Thus,thereexist changeofbasismatrices, [ c 0 odd =c odd ]: [ iodd c i [ iodd c 0 i and [ c 0 even =c even ]: [ ieven c i [ ieven c 0 i Sothat C;c 0 = det [ c 0 odd =c odd ] )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 @ + odd [ c 0 even =c even ] = det [ c 0 odd =c odd ] )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 det @ + odd det [ c 0 even =c even ] = C;c det [ c 0 odd =c odd ] )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 det [ c 0 even =c even ] : But, det [ c 0 odd =c odd ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 det [ c 0 even =c even ]= n Y i =0 det [ c 0 i =c i ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 i Thus,theresultholds. Noticethatif c i = c i 1 ;c i 2 ;:::c i n and c 0 i = c i 2 ;c i 1 ;c i 3 ;:::c i n ,then det [ c 0 i = c i ]= )]TJ/F24 11.9552 Tf 9.299 0 Td [(1 foreach i ,thus, C;c 0 = C;c Lemma 2.8 SupposeforanacyclicchaincomplexC,thereexistchaincontractions, and .Then det @ + = det @ + det @ + @ Proof. Consider,

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2.TORSIONOFANACYCLICCOMPLEX26 det @ + odd det @ + odd = det @ + odd det @ + even = det @ 2 + @ + @ + j C odd = det @ + @ + j C odd Noticethat, @ : C i C i @ : C i C i : C i C i +2 Thus,wehaveablockmatrixoftheform, @ + @ + j C odd = 0 B B B B B B B @ C 1 C 3 C 5 ::::::::: C 1 [ @ + @ ][ ]000 ::: C 3 0[ @ + @ ][ ]00 ::: C 5 00[ @ + @ ][ ]0 ::: .000 . ::: 1 C C C C C C C A So,usinglemma2.5,wegetthat det @ + @ + j C odd = Y iodd det @ + @ j C 2 i +1 = det @ + @ j C odd providedwecanshow @ + @ j C 2 i +1 isnon-singularforeach i 0. Butthisisnothardifweconvenientlyguessthat @ + @ j C 2 i +1 willbetheinverse andcompute,

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2.TORSIONOFANACYCLICCOMPLEX27 @ + @ j C 2 i +1 @ + @ j C 2 i +1 = @ @ + @@ + @ @ + @ @ = )]TJ/F24 11.9552 Tf 12.623 0 Td [( @ @ +0+ @ )]TJ/F30 11.9552 Tf 11.955 0 Td [(@ + )]TJ/F24 11.9552 Tf 12.623 0 Td [( @ )]TJ/F30 11.9552 Tf 11.955 0 Td [(@ = @ )]TJ/F24 11.9552 Tf 12.623 0 Td [( @@ + @ + @@ +1 )]TJ/F24 11.9552 Tf 12.623 0 Td [( @ )]TJ/F30 11.9552 Tf 11.955 0 Td [(@ + @@ = @ + @ +1 )]TJ/F24 11.9552 Tf 12.623 0 Td [( @ )]TJ/F30 11.9552 Tf 11.955 0 Td [(@ =1+1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(1=1 j C 2 i +1 Thus, @ + @ j C 2 i +1 isinvertible,and det @ + odd = det @ + odd det @ + @ j C odd Nowthatwehaveshownhowthetorsionisdependentonourchoiceofbasisand choiceofthemap ,wegiveanexampleofhowtocomputeit. Example 2.9 Supposethereexistsanacyclicchaincomplex, C :0 R R 2 R 2 R 0 @ 3 @ 2 @ 1 with, @ 1 = h 22 i @ 2 = 2 4 3 )]TJ/F24 11.9552 Tf 9.298 0 Td [(2 )]TJ/F24 11.9552 Tf 9.298 0 Td [(32 3 5 @ 3 = 2 4 2 3 3 5 Constructingmaps, i sothat @ + @ =1,take 1 = 2 4 1 2 0 3 5 2 = 2 4 0 )]TJ/F28 7.9701 Tf 10.494 4.707 Td [(1 3 00 3 5 3 = h 0 1 3 i Then,themap @ 3 3 + 2 @ 2 : R R isgivenby,

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2.TORSIONOFANACYCLICCOMPLEX28 2 4 2 3 3 5 h 0 1 3 i + 2 4 0 )]TJ/F28 7.9701 Tf 10.494 4.707 Td [(1 3 00 3 5 2 4 3 )]TJ/F24 11.9552 Tf 9.299 0 Td [(2 )]TJ/F24 11.9552 Tf 9.299 0 Td [(32 3 5 = 2 4 10 01 3 5 andtheothersaresimilar. Usingthedenition,wecomputethat @ + odd : R 2 R R R 2 = 2 6 6 6 4 220 0 )]TJ/F28 7.9701 Tf 10.494 4.707 Td [(1 3 2 003 3 7 7 7 5 @ + even : R R 2 R 2 R = 2 6 6 6 4 1 2 3 )]TJ/F24 11.9552 Tf 9.298 0 Td [(2 0 )]TJ/F24 11.9552 Tf 9.298 0 Td [(32 00 1 3 3 7 7 7 5 Alternatively,consider 1 = 2 4 1 2 0 3 5 2 = 2 4 00 0 1 2 3 5 3 = h 1 2 0 i Computationwillshowthat alsosatises, @ + @ =1 And @ + odd = 2 6 6 6 4 220 002 0 1 2 3 3 7 7 7 5 @ + even = 2 6 6 6 4 1 2 3 )]TJ/F24 11.9552 Tf 9.298 0 Td [(2 0 )]TJ/F24 11.9552 Tf 9.298 0 Td [(32 0 1 2 0 3 7 7 7 5 Noticethat det @ + odd = det @ + odd = det @ + even )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 = det @ + even )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 = )]TJ/F24 11.9552 Tf 9.298 0 Td [(2.

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2.TORSIONOFANACYCLICCOMPLEX29 Ingeneral,ndingamap canbecomputationallycumbersomeanditturnsout tobeunnecessaryaswell. Recallthatwecanthinkof C k = B k +1 B k where B k +1 = @ C k +1 B 0 = @ C 0 = 0.Thenforthebasis b k + 1 b k of C ,weconsiderthechaincomplex, 0 0 B n 0 B n )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 0 B n )]TJ/F28 7.9701 Tf 6.587 0 Td [(2 0 B 2 0 B 1 0 L L L L L L B n B n )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 B n )]TJ/F28 7.9701 Tf 6.587 0 Td [(2 B n )]TJ/F28 7.9701 Tf 6.587 0 Td [(3 B 1 0 @ @ @ @ Then,ascomputedin[ 1 ]16.1,themaps @ and actliketheidentityoneach B k Thus, det @ + =1= C; b k + 1 b k =1. So,foranyotherbases c 0 of C C; c 0 = C; b k + 1 b k n Y k =0 [ b k + 1 b k = c 0 k ] )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 k +1 = n Y k =0 [ b k + 1 b k = c 0 k ] )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 k +1 Thisgivesusanalternativewaytocomputethetorsionwithouthavingtondamap Lemma 2.10 C;c doesnotdependonthechoiceof b i Proof. For i = )]TJ/F24 11.9552 Tf 9.299 0 Td [(1and i = m thereisnothingtoshow.Let i 2f 0 ;::::;m )]TJ/F24 11.9552 Tf 12.075 0 Td [(1 g Weshowthat [ b i b i )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 =c i ] )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 i +1 [ b i +1 b i =c i +1 ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 i +2 isindependentofthechoiceof b i .Suppose b 0 i isanotherbasisof B i ,then

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2.TORSIONOFANACYCLICCOMPLEX30 [ b 0 i b i )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 =c i ]=[ b 0 i b i )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 =b i b i )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 ] [ b i b i )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 =c i ] =[ b 0 i =b i ] [ b i b i )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 =c i ] : Similarly, [ b i +1 b 0 i =c i +1 ]=[ b 0 i =b i ] [ b i +1 b i =c i +1 ]. Thus [ b 0 i b i )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 =c i ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 i +1 [ b i +1 b 0 i =c i +1 ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 i +2 =[ b 0 i =b i ] [ b i b i )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 =c i ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 i +1 [ b 0 i =b i ] [ b i +1 b i =c i +1 ] )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 i +2 =[ b i b i )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 =c i ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 i +1 [ b i +1 b i =c i +1 ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 i +2

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CHAPTER5 TorsionofaniteCWcomplex WehavedenedReidemeistertorsionintermsofanacyclicchaincomplex,thereforeifwearetohaveanyhopeofcomputingthetorsionofaniteCWcomplex, X ,wemustndawaytoassociateanacyclicchaincomplexwithit.Recall,that wehavealreadyassociatedachaincomplex,CwiththespaceX,madeofthefree abeliangroups C k X .However,thischaincomplexisnot,ingeneral,acyclic.Thus, wemustndawaytomakethischaincomplexacyclic. Let p : ~ X X betheuniversalcoverofX.Fix x 2 X andset = 1 X;x Notethat actson ~ X whichinducesanactiononthechaingroups C k ~ X .Extend thisactionlinearlytoanactionof Z [ ].Thus, C k ~ X becomesa Z [ ]-module. Choosingak-cell ~ E k j over E k j ,weformabasisfor C k ~ X ,sothat C k ~ X = M j Z [ ] ~ E k j Thus,thechaingroups C k ~ X formachaincomplexofabeliangroupsunderthe boundaryhomomorphism.However,thischaincomplexmaystillnotbeacyclic. Tondanacyclicchaincomplex,welookforaringhomomorphism, : Z [ ] R where R isaringwithunity.Inparticular,wewilldene bymappingthegenerators of Z [ ]toelementsof R .Thishomomorphismwillinducenewboundarymapsand thus,giveusanewchaincomplex, C ~ X = R C ~ X .Ifthisnewchaincomplexis acyclic,thenwecancomputeitstorsion,whichwedeneastheReidemeistertorsion oftheCWcomplexX. Definition 0.11 TheReidemeistertorsionofaspace X isgivenby, X = C ~ X forsomeringhomomorphism, 31

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5.TORSIONOFAFINITECWCOMPLEX32 Thetorsion, X isuniqueuptoorientationofcells,choiceoflift,andchoiceof map .Thisconstructionallowsustoimposemorestructureonourchaincomplexes withoutlosingtheringstructurethatisalreadythere.Afewexamplesshouldclarify howtoensurethat R C ~ X isacyclic.Inthefollowingexampleswewillalways chooseourring R tobetheeldofcomplexnumbers C Example 0.12TorsionofaCircle Let X = S 1 = a 0 [ a 1 where a 0 isapoint and a 1 isa1-cell.Recallthattheuniversalcoveringspaceofacircleis ~ X = R and thegroup 1 S 1 = Z .Wehavetheassociatedchaincomplex, C ~ X =0 Z [ ] Z [ ] 0 @ Denebasesforeach Z [ ]by, ~ a 0 =0and~ a 1 =[0 ; 1]. Recallthat 1 S 1 actson R byrighttranslations.Ifwelet t bethegeneratorfor Z then, t n x = x + n forany x 2 R .Thus, @ ~ a 1 = < 1 > )]TJ/F30 11.9552 Tf 12.619 0 Td [(< 0 > = t ~ a 0 )]TJ/F24 11.9552 Tf 12.101 0 Td [(~ a 0 = t )]TJ/F24 11.9552 Tf 11.955 0 Td [(1~ a 0 Althoughwehavecomputedtheboundarymap,wecannotyetcomputethetorsion sincethechaincomplex, C ~ X isnotacyclic.Noticethat, H 0 C ~ X = H 0 C R = H 0 R = Z : Inparticular,themap @ isnotonto. Suppose t = @ x .Then, t = @ x = x t )]TJ/F24 11.9552 Tf 12.094 0 Td [(1whichwouldimply x = t t )]TJ/F24 11.9552 Tf 12.093 0 Td [(1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 Butthisisimpossiblesince t )]TJ/F24 11.9552 Tf 11.955 0 Td [(1doesnothaveamultiplicativeinversein Z [ ]. Thus,wemustndaring R suchthatthehomomorphism, : Z [ ]: R takes t )]TJ/F24 11.9552 Tf 9.403 0 Td [(1 into R .Forsimplicity,wechoose C ,butwecouldchooseanyeldofcharacteristic zero.Since C isaeld,everynonzeroelementwillhaveaninverse.

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5.TORSIONOFAFINITECWCOMPLEX33 Dene by t = p forsome p 6 =1or0.Then inducesaboundarymapfor thenewchaincomplex, R C 0 ~ X @ ~ a 1 = p )]TJ/F24 11.9552 Tf 11.955 0 Td [(1~ a 0 Now,chooseabasis c 1 = f ~ a 1 g ;c 0 = f ~ a 0 g anddene b 0 = ;b 1 = f ~ a 1 g Then [ @b 1 b 0 n c 0 ]= p )]TJ/F24 11.9552 Tf 11.955 0 Td [(1and[ b 1 n c 1 ]=1 So, S 1 = p )]TJ/F24 11.9552 Tf 11.956 0 Td [(1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 Accountingforchoiceoforientation,choiceofliftandchoiceofthemap ,wehave that S 1 = p n p )]TJ/F24 11.9552 Tf 11.956 0 Td [(1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 forany n 2 Z Example 0.13TorsionofaTorus Consider X = T 2 thetorusconsistingof aCWstructurewithone0-cell a ,two1-cells b 1 and b 2 ,andone2-cell c .TheCW complexgivesusanaturalchaincomplex,inwhich C i T 2 = H i X i ;X i )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 ,giving us, 0 Z Z Z Z 0 0 0 Ifweliftthecellstotheuniversalcovingspace, R 2 ,thefundamentalgroup, = 1 T 2 = Z Z ,canactontheliftedcellsgivingusagroupringstructureand C k R 2 = Z [ ]andanewchaincomplex, 0 Z [ ] Z [ ] Z [ ] Z [ ] 0 @ 1 @ 0 Let 1 and 2 bethegeneratorsfor = Z Z andchoosebases, p i suchthat, p 0 = f ~ a = < 0 > 2 R 2 g p 1 = f ~ b 1 = I f 0 g R 2 ; ~ b 2 = f 0 g I R 2 ;I =[0 ; 1] g p 2 = f ~ c = I I R 2 g

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5.TORSIONOFAFINITECWCOMPLEX34 ~ a 1 ~ a 2 ~ a ~ b 1 2 b 1 ~ b 2 1 b 2 2 1 b 2 ~ c 1 2 ~ c Thenlookingatthelatticeofintegers,wecancomputethat, @ ~ a =0 @ 0 ~ b 1 = 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(1~ a @ 0 ~ b 2 = 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(1~ a @ 1 ~ c = )]TJ/F30 11.9552 Tf 11.955 0 Td [( 2 ~ b 1 )]TJ/F24 11.9552 Tf 11.956 0 Td [( )]TJ/F30 11.9552 Tf 11.955 0 Td [( 1 ~ b 2 Noticethat im @ 1 = ker @ 0 ,since H 1 C ~ X = H 1 C R 2 = H 1 R 2 =0.But, thesequenceisnotacyclicsince H 0 C ~ X = H 0 R 2 = Z Inordertomakethischaincomplexacyclic,weconsider : Z [ ] C ,where 1 = s and 2 = t for s;t 6 =1or0and s t linearlyindependent.Thiswillgive usinversesfor 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(1and 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(1whichwillmakethesequenceacyclic. 0 C C C C 0 @ 1 @ 0 Noticethattheinducedboundarymapslookincrediblysimilartotheoriginal maps, @ 1 ~ c = )]TJ/F30 11.9552 Tf 11.955 0 Td [(t ~ b 1 )]TJ/F24 11.9552 Tf 11.956 0 Td [( )]TJ/F30 11.9552 Tf 11.955 0 Td [(s ~ b 2

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5.TORSIONOFAFINITECWCOMPLEX35 @ 0 ~ b 1 = s )]TJ/F24 11.9552 Tf 11.955 0 Td [(1~ a and @ 0 ~ b 2 = t )]TJ/F24 11.9552 Tf 11.955 0 Td [(1~ a Now,set 2 = f ~ c g 1 = f ~ b 1 g and 0 = f g Then 2 formsabasisfor C @ 2 [ 1 formsabasisfor C C and @ 1 [ 0 formsabasisfor C Ifwecomputethecorrespondingchangeofbasismatrices,weget, [ 2 = p 2 ]=1 [ @ 2 1 = p 1 ]= det 2 4 )]TJ/F30 11.9552 Tf 11.955 0 Td [(t s )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 10 3 5 = )]TJ/F24 11.9552 Tf 9.299 0 Td [( s )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 [ @ 1 0 = p 0 ]= s )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 Thus, T 2 = )]TJ/F24 11.9552 Tf 9.299 0 Td [(1andaccountingforchoiceoflift,orientation,and ,wegetthat T 2 = s k t l forany k;l 2 Z .

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CHAPTER6 ClassicationofLensSpaces 1.R-TorsionofLm,n RecalltheCWdecompositionforlensspacesliftedtotheuniversalcover, S 3 is E 0 j = j ; 0 E 1 j = f e i ; 0 2 S 3 ; 2 j m 2 j +1 m g E 2 j = f z;s j 2 S 3 ; s 2 R ; j z j 2 + s 2 =1 g E 3 j = f z;w 2 S 3 ; 2 j m arg w 2 j +1 m g Andaswecomputedbefore, @ 1 E 1 j = E 0 j +1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(E 0 j @ 2 E 2 j = m )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 X k =0 E 1 k @ 3 E 3 j = E 2 j +1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(E 2 j Recallthatthefundamentalgroupofalensspaceisgivenby 1 L m;n = Z m and itsuniversalcoveris S 3 .However,whilethefundamentalgroupof L m;n doesnot dependon n ,theactionofthefundamentalgroupon S 3 does.Let beagenerator for Z m .Then, actsonorderedpairs z;w ,by z;w = z; n w .Thus, E k j = E k j +1 for k =0 ; 1 and E k j = E k j + n for k =2 ; 3 : 36

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1.R-TORSIONOFLM,N37 So, @ 1 E 1 j = )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 E 0 j @ 2 E 2 j = m )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 X i =0 j E 1 0 @ 3 E 3 j = r )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 E 2 j where rn 1mod m Let : Z [ Z m ]= Z [ ] C denedforanynontrivial mth rootofunity t ,by = t .Thentensoringover givesusanewchaincomplex, C = C S 3 C andtheinducedboundarymapsbecome, @ 1 E 1 j = t )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 E 0 j @ 2 E 2 j = m )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 X j =0 t j E 1 0 = t m )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 E 1 0 =0 @ 3 E 3 j = t r )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 E 2 j Thenewchaincomplex, 0 C C C C 0 t r )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 0 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 isacyclicandthetorsionisgivenby, L m;n = t k t )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 t r )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 where rn 1mod m and k 2 Z Choosingdierent m throotsofunity,wecaneasilyseesomelensspaceswillhave thesametorsion.Firstnotethat L m;n 0 = L m;n 1 whenever n 0 n 1 1 mod m .Usingthecomputationaboveandtaking k =1,wehave L m;n 0 = t )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 t r )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 for rn 0 1mod m ,wehavethat r = n 1 mod m .Thus,wecanndanotherroot ofunity, s ,suchthat s n 0 = t .Then, t n 1 = s n 0 n 1 = s andthus,

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2.HOMOTOPYCLASSIFICATION38 L m;n 0 = t )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 t n 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 = s n 0 )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 s )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 = L m;n 1 Additionally,when n 0 )]TJ/F30 11.9552 Tf 21.918 0 Td [(n 1 mod m L m;n 0 = t )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 t r )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 = t r t )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(t )]TJ/F31 7.9701 Tf 6.586 0 Td [(r )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 = L m;n 1 Inparticular, L m;n 0 = L m;n 1 when n 0 n )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 1 mod m n 0 )]TJ/F30 11.9552 Tf 21.918 0 Td [(n 1 mod m: 2.HomotopyClassication Wearenowreadytocompletelyclassifylensspacesuptohomotopy.Wewillsee thattorsionisnotahomotopyinvariantandthatinfacttherearehomotopiclens spaceswithdierentReidemeistertorsions.Butrst,wewillstatetwotheoremsby HurewiczandWhiteheadrespectivelythatcomefrom[ 7 ],whichwewillneedforour classicationproofs. Definition 2.1 ForanyspaceX,thereexistsagrouphomomorphism,calleda Hurewiczmap h : k X;A H k X;A .Denedby h = f e where isa generator k X;A e isageneratorfor H k D n ;S n )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 and f : D n ;S n )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 X;A Theorem 2.2Hurewicz TheHurewiczmap h : n S n H n S n isanisomorphismfor n 1 Theorem 2.3Whitehead Let f : X Y beacontinuousmapwhere X and Y arepathconnectedCWcomplexes.Then, f inducesamaponhomotopygroups, f : k X k Y Iftheinducedmapsareallisomorphisms,then X and Y arehomotopic. Recallthatlensspaceswithdierentfundamentalgroupscannotbehomotopic, sincethefundamentalgroupisahomotopyinvariant.Thus,wecanrestricttodistinguishing L m;n 0 from L m;n 1 .

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2.HOMOTOPYCLASSIFICATION39 Definition 2.4 Consideramap f : S n S n for n 1.Thismapinducesa maponthehomologyof S n .Inparticular,if H n S n = <> ,thentheinducedmap f : H n S n H n S n isahomomorphismfrom Z toitself.Thus, f isdenedby f = m forsomeinteger m whichwecallthe degree ofthemap f Example 2.5 Consideramapontheunitcircle f : S 1 S 1 ofdegree2. 2 f S 1 S 1 e i 7! e i 2 2 Wecreateamapofanydegreeifwechangethedegreeof f onasmallsegment ofthecircle. Let0 ,beaxedangle,choose n 2 Z ,andconsideramap :[0 ; ] [0 ; 2 n +2 ] t = 2 n +2 t Then e it 7! e i t willwrapthesegment[0 ; ]aroundthecircle n times,andend atthepoint2 Ifwechangethedegreeof f onthissegment,wewillhaveamapthattakes t 2 [0 ; ]tothewholecircle n +2 times,and[ ; 2 ]tothecircle2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(2 times. Themap ^ f doesjustthat, ^ f : e it 7! 8 < : e i 2 t t= 2 [0 ; ]; e i t t 2 [0 ; ]; Nowwecanseethat ^ f willhavedegree n for t 2 [0 ; 2 n 2 n +2 ]anddegree2for t 2 [ 2 n 2 n +2 ; 2 ].Thus, ^ f hasdegree2+ n ,andbychanging n wecanadjustthedegree tobeanyinteger. Nowsupposeourmap f inducesamap g onquotientspacessothatthediagram commutes.

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2.HOMOTOPYCLASSIFICATION40 2 f S 1 = Z 2 p 1 p 2 S 1 = Z 3 2 3 3 g Ifwewanttochangethedegreeof f sothatitwillstillrespectthequotientmaps, wemustdoitinawaythatwillrespectthe Z 2 actionon S 1 .Thismeansthatwe mustnd ^ f sothatitinducesamap^ g satisfying p 2 ^ f =^ g p 1 foranypointon S 1 But,lookingatthequotientmapwenoticethat p 1 [0 ; ]= p 1 [ ; + ].Sowe shouldrequirethat, p 2 ^ f [0 ; ]= p 2 ^ f [ ; + ]=^ g p 1 [0 ; ]=^ g p 1 [ ; + ]. Ifwechangethedegreeof f onlyon[0 ; ],aswedidbefore,then ^ f [0 ; ] 6 = ^ f [ ; + ] andthus,wewillmostlikelyhave p 2 ^ f [0 ; ] 6 = p 2 ^ f [ ; + ]. Wecanensurethat p 2 ^ f [0 ; ]= p 2 ^ f [ ; + ] bychoosing ^ f suchthat ^ f [0 ; ]= ^ f [ ; + ]. Thus,weconstruct 1 t = 2 n +2 t asbeforeand 2 :[ ; + ] [0 ; 2 n +2 ] sothat

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2.HOMOTOPYCLASSIFICATION41 2 t = 2 n +2 t )]TJ/F28 7.9701 Tf 13.151 4.707 Td [(2 2 n )]TJ/F24 11.9552 Tf 11.955 0 Td [(2 Thenwedene ^ f tobe ^ f : e it 7! 8 > > > > > > > < > > > > > > > : e i 1 t t 2 [0 ; ]; e i 2 t t 2 [ ; ]; e i 2 t t 2 [ ; + ] e i 2 t t 2 [ + ; 2 ] Now ^ f willhavedegree2 n +2,whichmeanswecanonlyalterthedegreeof ^ f bya multipleof2ifwewant ^ f torespectthecommutativityofthediagram. Thisconstructionworkssimilarlyifweconsiderdegreemapsonany n -sphere.In particularifwewantourconstructiontorespectaquotientmap, S n S n = Z k ,we canonlychangethedegreemodulo k Nowwearereadytoclassifylensuptohomotopy.Thefollowingproofcomes from[ 4 ]. Theorem 2.6HomotopyClassication L m;n 0 ishomotopyequivalentto L m;n 1 ifandonlyif n 0 r 2 n 1 mod m forsome r 2 Z m Proof. Suppose n 0 r 2 n 1 mod m forsome r 2 Z m Noticethatif Z m = <> thereisanaturalactionof onpairs z;w 2 S 3 ,given by, r;s z;w = r z; s w when r;m = s;m =1. Now,consideramap f k 1 ;k 2 : S 3 S 3 suchthat f k 1 ;k 2 z;w = f k 1 ;k 2 re i ;se i = re ik 1 ;se ik 2 then,if Z m = <> ,themap f k 1 ;k 2 commuteswiththenaturalactionof on z;w sothat, f k 1 ;k 2 1 ;n z;w = k 1 re ik 1 ; nk 2 se ik 2 = k 1 ;nk 2 f k 1 ;k 2 z;w .

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2.HOMOTOPYCLASSIFICATION42 Thus f k 1 ;k 2 hasdegree k 1 k 2 andinducesamaponthequotientspaces, [ f k 1 ;k 2 ]: L m;n 0 L m;n 1 Fixasmallball B 0 on S 3 anddeneballs B r = r;rn 0 B 0 for1 r m )]TJ/F24 11.9552 Tf 11.956 0 Td [(1so that B 0 B r =foreach r .Thenwecanmodifythedegreeof f k 1 ;k 2 on m )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 [ r =0 B r tobeanymultipleof m .Thiswillmakethedegreeof f k 1 ;k 2 onthewholeof S 3 any multipleof m and,asinexample2.5, f k 1 ;k 2 willstillinduceamaponthequotient spaces,[ f k 1 ;k 2 ]: L m;n 0 L m;n 1 Inparticular,ifwedene l suchthat ln 0 1mod m ,then n 0 r 2 n 1 mod m impliesthat 1 n 0 l r 2 n 1 l mod m Thus,wecanmodifythemap, f r;rn 1 l : S 3 S 3 sothatithasdegree1. Since f r;rn 1 l hasdegree1itishomotopictotheidentitymap,sobyHurewiczit inducesanisomorphism, f r;rn 1 l : k S 3 k S 3 for k 1. Since k L m;n i = k S 3 for k 2and i =0 ; 1,wehavethat f r;rn 1 l induces isomorphisms k L m;n 0 k L m;n 1 for k 2. Finally,wecanalsoseethat f r;rn 1 l inducesanisomorphismonthefundamental groupsof L m;n 0 and L m;n 1 byconsideringwhathappenstothepoints, z; 0. Noticethat f r;rn 1 l z; 0= r z; 0.Since r;m =1, r generatesthegroup Z m .Thus if[ k ] 2 1 L m;n 0 ,then[ f r;rn 1 l k ]= kr ,givingthedesiredisomorphism.

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3.HOMEOMORPHISMCLASSIFICATION43 Sincewehaveshownthat f r;rn 1 l inducesisomorphisms k L m;n 0 k L m;n 1 forall k ,byWhitehead'stheorem,wecanconcludethat L m;n 0 ishomotopicto L m;n 1 Example 2.7 Inordertoclassifyhomotopytypesof L ;n ,weconsiderthe casesforwhich0
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3.HOMEOMORPHISMCLASSIFICATION44 Inordertoclassifylensspacesuptohomeomorphism,wewillrstndthelens spaceswiththesametorsion,andthenshowthatthoselensspacesare,infact, homeomorphic. Theorem 3.3HomeomorphismClassication L m;n 0 ishomeomorphicto L m;n 1 ifandonlyif n 0 n 1 1 mod m Proof. Firstwewillshowif L m;n 0 homeomorphicto L m;n 1 ,then n 0 n 1 1 mod m .Suppose and aregeneratorsfor Z m suchthat 1 = t and 2 = s .Since L m;n 0 ishomeomorphicto L m;n 1 ,thereexistsamap, h ,such thatthefollowingdiagramcommutes. L m;n 1 L m;n 0 C h 1 2 Since and areprimitiverootsofunity,thereexists suchthat m; =1and h = .Thus, t = 1 = 2 h = 2 = s Now,sincetorsionisahomeomorphisminvariant,weknowthat L m;n 0 = L m;n 1 .So, L m;n 0 = t t )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 t n 0 )]TJ/F24 11.9552 Tf 11.955 0 Td [(1= s s )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 s n 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(1= L m;n 1 forsomeintegers and in Z Substituting t = s ,weget, s s )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 s n 0 )]TJ/F24 11.9552 Tf 11.955 0 Td [(1= s s )]TJ/F24 11.9552 Tf 11.955 0 Td [(1 s n 1 )]TJ/F24 11.9552 Tf 11.956 0 Td [(1. Multiplyingthisoutandletting 2f)]TJ/F24 11.9552 Tf 26.567 0 Td [(1 ; 1 g a = )]TJ/F30 11.9552 Tf 11.955 0 Td [( wehave, 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(s )]TJ/F30 11.9552 Tf 11.955 0 Td [(s n 0 + s n 0 +1 + s a )]TJ/F30 11.9552 Tf 11.955 0 Td [(s + a )]TJ/F30 11.9552 Tf 11.955 0 Td [(s n 1 + a + s n 1 +1+ a =0.

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3.HOMEOMORPHISMCLASSIFICATION45 Noticethatthisequationholdsforany m throotofunity s .Thus,consideringthe polynomial, p x =1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x )]TJ/F30 11.9552 Tf 11.955 0 Td [(x n 0 + x n 0 +1 + x a )]TJ/F30 11.9552 Tf 11.956 0 Td [(x + a )]TJ/F30 11.9552 Tf 11.955 0 Td [(x n 1 + a + x n 1 +1+ a weassertthat p has m roots,namelythe m throotsofunity,and deg p x
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3.HOMEOMORPHISMCLASSIFICATION46 S 3 S 3 L m;n L m; )]TJ/F30 11.9552 Tf 9.298 0 Td [(n f 1 g 1 Alternatively,wecanalsoconsideramap f 2 : S 3 S 3 denedby f 2 z;w = w;z .Thentheidentication z;w z; n w n )]TJ/F29 5.9776 Tf 5.756 0 Td [(1 z; n )]TJ/F29 5.9776 Tf 5.757 0 Td [(1 n w willmapto w;z n w;z w; n )]TJ/F29 5.9776 Tf 5.756 0 Td [(1 z .Thus, f 2 inducesahomeomorphism, g 2 ,between lensspaces L m;n and L m;n )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 S 3 S 3 L m;n L m;n )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 f 2 g 2 Ofcoursewecanalsocomposethemaps g 1 and g 2 sincetheyarehomeomorphisms. Thus,wehavethat n 0 n )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 1 implies L m;n 0 = L m;n 1 Example 3.4 Wecannowusethepreviousexampletodeterminethehomeomorphismclassesof L ;n for0
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CHAPTER7 Conclusion Aswehavedemonstrated,thehomeomorphismclassofalensspace, L m;n isuniqueupto n 1 mod m .Themajorobstacleinattainingthisclassication isunderstandinghowthevalueof n aectstheactionof mth rootsofunity on pairs z;w in S 3 .Ourusualtopologicalinvariantsarenotsubtleenoughtodecipher thisaction.Aswesawwhencomputingthehomology,themap : S 3 L m;n eliminatesanyrelevantinformationaboutthisactionsinceallofthebersaremapped together.However,pickingouthowthefundamentalgroupactsontheuniversal coverisexactlywhatReidemeistertorsionisdesignedtodo.Insteadoflookingat theprojectionoftheboundarymapsinthequotientspace,welookattheminthe universalcoverbeforeapplyingthequotientmap.Thus,thetorsionsucceedswhere homologyandhomotopygroupsfailtoprovideclassication. Generalizingtheconstructionoflensspacesfromquotientspacesof S 3 toquotient spacesof S n and S 1 isalsopossible.Infact,theclassicationofsuchspaceswasdone byReidemeister'sstudentFranzandisnotsodierentfromthethreedimensional case.Infact,manyoftheproofsincludedherehavebeensimpliedfromthemore generalproofs.Whilethematerialhereisnotnew,IhopeIhavedemonstratedthe importanceoftopologicalinvariantsforthepurposeofsuchclassicationtheorems. 47

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APPENDIXA FiniteCWcomplexes Ingeneralcomputingthehomologyofanarbitrarytopologicalspace,X,canbe dicult.However,inthecasewhenXiscompactandHausdor,wethinkaboutX asbeingmadeupofpieces,eachhomeomorphictoann-ball.Suchpiecesarecalled n-cells Moreformally,givenX,compactandHausdor,wecanndanitesequence calleda niteCW-complex X 0 X 1 :: X n = X ,suchthateach X k )]TJ/F30 11.9552 Tf 12.113 0 Td [(X k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 is homeomorphictoanitenumberofdisjointk-cellsdenoted, E k 1 :::E k m .Thespace X k ,calleda k-skeleton ,isthespaceobtainedbyattachinganitenumberofkcellsto X k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 andisaniteCW-complexonitsown.Therefore,CW-complexesare usefulbothfordecomposinganexistingcompactspace,andforbuildingnewspaces, however,suchdecompositionsandconstructionsarenotunique. Example 0.5 Consider, X = S 2 .If z isapointon S 2 then X maybeconsidered asa2-cellattachedto z .However,if z 0 isanotherpointon S 2 and isapathfrom z to z 0 ,thentheassociatedCWstructureconsistsoftwo0-cells,a1-cell,anda2-cell. X mayalsobeviewedasconsistingoftwo0-cells,three1-cellsandthree2-cells. z 0 a z 0 z 1 b z 0 z 1 c 0 1 2 48

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A.FINITECWCOMPLEXES49 ItisimportanttorealizethatwhiletheCWdecompositionisnotunique,changing thenumberofcellsinanyparticulardimensionwillaectthenumberofcellsinother dimensions.Thisinformationisencodedinanotherpowerfulinvariant,the Euler characteristic whichcanbedenedforaniteCWcomplexas X = n X i =1 )]TJ/F24 11.9552 Tf 9.299 0 Td [(1 i i where i isthenumberofcellsindimensioni. ACWcomplex, X ,canbeassociatedwithachaincomplex,thereforewewillbe abletocomputethehomologyof X ineachdimensionusingthehomologyofthis chain.Butrst,considerthek-skeleton X k .Since X k isobtainedbyattaching m k-cellsto X k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 ,wecanthinkoftheboundaryof X k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 thereisarelativehomeomorphism, f : D k 1 [ ::: [ D k m ;S k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 1 [ ::: [ S k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 m X k ;X k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 i.e.ahomeomorphism, f ,taking D k 1 [ ::: [ D k m )]TJ/F30 11.9552 Tf 12.329 0 Td [(S k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 1 [ ::: [ S k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 m to X k )]TJ/F30 11.9552 Tf 12.33 0 Td [(X k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 suchthat S k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 1 [ ::: [ S k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 m mapsto X k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 Proposition 0.6 If X isaniteCWcomplexand X k isthek-skeletonof X then H j X k ;X k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 =0 for j 6 = k and H k X k ;X k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 isafreeabeliangroupwithone basiselementforeachk-cellof X Proof. X k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 isasubcomplexof X k ,sobytheorem4.4,itisastrongdeformation retractofacompactneighborhoodin X k .SinceXisaniteCWcomplex,thereisa relativehomeomorphism f : D k 1 [ ::: [ D k r ;S k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 1 [ ::: [ S k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 r X k ;X k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 Bytherelativehomeomorphismtheorem,theinducedmap, f : H D k 1 [ ::: [ D k r ;S k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 1 [ ::: [ S k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 r H X k ;X k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 isanisomorphism.Thus,since

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A.FINITECWCOMPLEXES50 H j D k 1 [ ::: [ D k r ;S k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 1 [ ::: [ S k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 r = X i H j D k i ;S k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 i =0for j 6 = k and H k D k 1 [ ::: [ D k r ;S k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 1 [ ::: [ S k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 r = X i H k D k i ;S k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 i isfreeabelian,theresultfollows. GivenaCWdecompositionof X ,thereisanassociatedchaincomplexoffree abeliangroups.Dene, C k X = H k X k ;X k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 Usingthefactorization, H k )]TJ/F28 7.9701 Tf 6.587 0 Td [(2 X k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 H k +1 X k +1 ;X k H k X k ;X k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 H k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 X k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 ;X k )]TJ/F28 7.9701 Tf 6.587 0 Td [(2 H k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 X k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 @ @ @ 00 j @ 0 i wecandene @ = i @ 0 andwegetthat @ @ =0thus f C k g k isachaincomplex withthemap @ Remarkably,thehomologyofthischaincomplexisthesameasthesingularhomologyofX. Theorem 0.7 IfXisaniteCWcomplex,then H k C X H k X foreachk.

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Bibliography [1]MarshallM.Cohen. Acourseinsimple-homotopytheory .Springer-Verlag,NewYork,1973. GraduateTextsinMathematics,Vol.10. [2]ThomasW.Hungerford. Algebra ,volume73of GraduateTextsinMathematics .Springer-Verlag, NewYork,1980.Reprintofthe1974original. [3]WilliamS.Massey. Abasiccourseinalgebraictopology ,volume127of GraduateTextsinMathematics .Springer-Verlag,NewYork,1991. [4]LiviuI.Nicolaescu.Notesonthereidemeistertorsion.unpublishedbook,January2002. [5]VladimirTuraev. Introductiontocombinatorialtorsions .LecturesinMathematicsETHZurich. BirkhauserVerlag,Basel,2001.NotestakenbyFelixSchlenk. [6]JamesW.Vick. Homologytheory .AcademicPress,NewYork,1973.Anintroductiontoalgebraic topology,PureandAppliedMathematics,Vol.53. [7]GeorgeW.Whitehead. Elementsofhomotopytheory ,volume61of GraduateTextsinMathematics .Springer-Verlag,NewYork,1978. 51