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On Lusternik-Schnirelmann Category of Connected Sums

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Title:
On Lusternik-Schnirelmann Category of Connected Sums
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english
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Newton, Robert J
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University of Florida
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Rudyak, Yuli B
Committee Members:
Groisser, David Joel
Dranishnikov, Alexander Nikolae
Robinson, Paul L
Rylkova, Galina S

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Subjects / Keywords:
lusternik -- manifolds -- schnirelmann
Mathematics -- Dissertations, Academic -- UF
Genre:
Mathematics thesis, Ph.D.
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theses   ( marcgt )
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Abstract:
This dissertation uses techniques from algebraic topology to place bounds on the Lusternik-Schnirelmann category of a quotient space with sufficient conditions. The first chapter contains a quick summary on the origins of the theory and some mention on current activity within the field. The second chapter provides the definitions and constructions used in this study. There will be examples of the Lusternik-Schnirelmann category for some basic spaces. The techniques from algebraic topology that are used will be discussed here. The chapter concludes with some integration of the algebra and topology. In the third chapter we get to the main ideas in this dissertation. We prove the main theorem and describe some very quick results that follow from the theorem. The dissertation concludes with ideas for future work.
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In the series University of Florida Digital Collections.
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Includes vita.
Bibliography:
Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Robert J Newton.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Rudyak, Yuli B.

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MISSING IMAGE

Material Information

Title:
On Lusternik-Schnirelmann Category of Connected Sums
Physical Description:
1 online resource (24 p.)
Language:
english
Creator:
Newton, Robert J
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Rudyak, Yuli B
Committee Members:
Groisser, David Joel
Dranishnikov, Alexander Nikolae
Robinson, Paul L
Rylkova, Galina S

Subjects

Subjects / Keywords:
lusternik -- manifolds -- schnirelmann
Mathematics -- Dissertations, Academic -- UF
Genre:
Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
This dissertation uses techniques from algebraic topology to place bounds on the Lusternik-Schnirelmann category of a quotient space with sufficient conditions. The first chapter contains a quick summary on the origins of the theory and some mention on current activity within the field. The second chapter provides the definitions and constructions used in this study. There will be examples of the Lusternik-Schnirelmann category for some basic spaces. The techniques from algebraic topology that are used will be discussed here. The chapter concludes with some integration of the algebra and topology. In the third chapter we get to the main ideas in this dissertation. We prove the main theorem and describe some very quick results that follow from the theorem. The dissertation concludes with ideas for future work.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Robert J Newton.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Rudyak, Yuli B.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045903:00001


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ONLUSTERNIKSCHNIRELMANNCATEGORYOFCONNECTEDSUMSByROBERTJ.NEWTONADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013RobertJ.Newton 2

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Idedicatethisdissertationtomyparents,BobandMaryNewton. 3

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ACKNOWLEDGMENTS Aspecialthankstomyadviser,Dr.YuliRudyak.Thesupport,guidance,andpatiencehehasprovidedwillalwaysbethestandardforhowIinteractwithmyownstudents.AlsoanextraspecialthankstoDr.AlexanderDranishnikov. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 ABSTRACT ......................................... 6 CHAPTER 1BRIEF,SELECTEDHISTORYOFLUSTERNIK-SCHNIRELMANNCATEGORY 7 2BACKGROUND ................................... 10 2.1BasicDenitions ................................ 10 2.2PropertiesofLS-Category ........................... 10 2.3LSCategory-Constructions .......................... 12 2.4ToomerInvariant ................................ 13 2.5Rationalization ................................. 14 3CATEGORYOFCONNECTEDSUMSANDSOMEDISCUSSION ....... 16 3.1CategoryofQuotientSpaces ......................... 16 3.2PreliminariesonConnectedSums ...................... 17 3.3ApplicationstoConnectedSums ....................... 18 REFERENCES ....................................... 21 BIOGRAPHICALSKETCH ................................ 24 5

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyONLUSTERNIKSCHNIRELMANNCATEGORYOFCONNECTEDSUMSByRobertJ.NewtonAugust2013Chair:YuliRudyakMajor:MathematicsThisdissertationusestechniquesfromalgebraictopologytoplaceboundsontheLusternik-Schnirelmanncategoryofaquotientspacewithsufcientconditions.Therstchaptercontainsaquicksummaryontheoriginsofthetheoryandsomementiononcurrentactivitywithintheeld.Thesecondchapterprovidesthedenitionsandconstructionsusedinthisstudy.TherewillbeexamplesoftheLusternik-Schnirelmanncategoryforsomebasicspaces.Thetechniquesfromalgebraictopologythatareusedwillbediscussedhere.Thechapterconcludeswithsomeintegrationofthealgeberaandtopology.Inthethirdchapterwegettothemainideasinthisdissertation.Weprovethemaintheoremanddescribesomeveryquickresultsthatfollowfromthetheorem.Thedissertationconcludeswithideasforfuturework. 6

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CHAPTER1BRIEF,SELECTEDHISTORYOFLUSTERNIK-SCHNIRELMANNCATEGORYIn1929LazarLusternikandLevSchnirelmannsolvedthePoncareconjecturethateveryRiemannianmetriconthesphereS2possessesatleast3closednon-self-intersectedgeodesics,[ 28 ],forgreaterdetailsee[ 1 29 ].Tothisaim,LusternikandSchnirelmannintroducedatopologicalinvariant,catMofa(closedandsmooth)manifoldM,calledlatertheLusternik-Schnirelmann,ormoresimplyLS,categoryandprovedthatcatMestimatesfrombelowthenumberofcriticalpointsasmoothfunctionM!R.ForthedenitionoftheLScategory,seeChapter2and[ 5 ].In1936KarolBorsuk[ 4 ]noticedthatonecanconsidertheLScategorynotonlyofmanifoldsbutalsoforgeneralspaces.Soonafter,RalphFox[ 13 ]begantointegratemorealgebraintothestudyofLusternik-Schnirelmanncategory,andwasabletotiethecategoryofaspacetoitscoveringdimension.Grossman[ 18 ]extendedthelastinequalitycatXdimXbyprovingthatthecategoryisboundedfromabovebythedimension/connectivityratio.Ashort,butinuentialpaper[ 7 ]relatedthecategoryofaspacewiththehomologypropertiesofthefundamentalgroupofthespace.TudorGanea[ 16 ]fueledmoreresearchinthissubjectwithhislistofunsolvedproblemsrelatedtoLusternik-Schnirelmanncategory.Inparticular,thefamousquestionaskingifcat(XSn)=catX+1(namedlaterasGaneaconjecture)appearedhere.Inlate1950sandearly1960sresearchersfoundpurelyhomotopy-theoreticaldescriptionofthenumbercatX.G.Whitehead[ 41 ]provedthefollowing:letTnXdenotethefatn-wedgeofXandletjn:TnX!Xnbetheobviousinclusion.ThecatXisthesmallestnsuchthatthediagonaldn+1:X!Xn+1passesthroughTn+1Xwithrespecttojn+1,uniquelyuptohomotopy.Similarly,Ganea[ 15 ]andSchwarz[ 40 ]constructedabrationpn=pXn:PnX!X(usingdifferentbuthomotopyequivalentconstruction)suchthatcatXisthesmallestnsuchthatpn+1hasasection,seeChapter2.Inthisway 7

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Schwarzintroducedapowerfulabstraction,todaycalledSchwarzGenus.Alsonoticethatpnisabrationalsubstituteofthemapinducedfromjnbydn.AlgebraicinputtoLusternik-Schnirelmanntheorystartedwiththepaper[ 14 ].Itwasproventhatthecup-lengthofaspaceestimatesLusternik-Schnirelmanncategoryfrombelow,see(Chapter2).GivenacommutativeringR,thecup-lengthofaspace(X)isthemaximallengthclR(X)ofnon-trivialcup-productinreducedR-cohomologyofX.Notice,however,thatthecohomologylanguageappearedafewyearslaterafterthepaper[ 14 ].So,aswecanexpect,FroloffandElsholzstatedthePoincaredualclaimintermsofintersectionofhomologyclasses.ThistechniqueallowsustoevaluatetheLusternik-Schnirelmanncategoryforcertainimportantspaces,liketoriandprojectivespaces.MorepowerfulalgebraictoolsweredevelopedinstudyingtheGanea-Schwarzbration.MichaelGinzburg[ 17 ]relateddifferentialsoftheEilenberg-Moorespectralsequenceofthebrationstotheexistenceofasectiontothisone.Inasimilarway,givenaringR,GrahamToomerdescribedanumericalinvarianteR(X)suchthatclR(X)eR(X)catX,[ 38 ]),seeChapter3.Inlate1980swehadarenaissanceoftheLusternik-Schnirelmanntheory.First,wementionaninuentialsurveybyIanJames[ 25 ],andseealsothepreviousversion[ 24 ].Theprogressdevelopedinseveraldirections.DennisSullivan'stheoryofrationalspaces[ 37 ]wasusedtostudyLusternik-Schnirelmanncategoryofrationalspaces.InthiswayYvesFelixandStephenHalperin[ 10 ]suggestedapurelyalgebraicmodelfortheGanea-Schwartzbrationinarationalcontext.Basedonthis,KathrynHess[ 21 ]andBarryJessup[ 26 ]provedtheGaneaconjecturecat(XSn)=catX+1forrationalsimplyconnectedspaces.Furthermore,YvesFelix,StephenHalperinandJean-MichelLemaireprovethat,forrationalmanifolds,theToomerinvariantisequaltotheLScategory,i.ecat(MQ)=eQ(M)forasimplyconnectedclosedmanifoldM,[ 11 ]. 8

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In1992EdFadellandSuanHusseini[ 8 ]developedthecup-lengthestimatebyintroducingthenotionofcategoryweight.However,theirconstructionwasnotahomotopyinvariant.ThehomotopyinvariantversionofcategoryweightwassuggestedbyYuliRudyakandJeffStrom[ 32 36 ].NowthisconcepthasmanyversionsandanalogsandturnsouttobeoneofthemostpowerfultoolsinLStheory.Oneexampleofitsusefulnessisinobtainingapartialsolutionofthewell-knownArnoldconjectureonsymplecticxpoints,[ 31 34 ].Notealsothepaper[ 6 ]thatrelatesthefundamentalgroupofaclosedmanifoldM,thedimensionofM,andtheLScategoryofM.Forexample,ifaclosedmanifoldhascategory2thenitsfundamentalgroupisfree.Inrecenthistory,NorioIwaseprovidedacounterexampletotheGaneaConjecturefornon-rationalcases[ 22 ]usingHopfinvariantsintroducedbyIsraelBersteinandPeterHiltonintheirdiscussionofsuspensions,[ 3 ].ThestudyofLusternik-Schnirelmanncategorycontinuestoprovidevalue,inparticularthereisastrongrelationshipbetweenLusternik-SchnirelmanncategoryandwhatMichaelFarber[ 9 ]hasdescribedasthetopologyofrobotmotionplanning.YuliyBaryshnikovandRobGhristhavedemonstratedthatLusternik-Schirelmanncategoryhasapplicationsoutsidetopologyandgeometryinstudyingtopologicalstatistics,[ 2 ].SteveSmale[ 35 ]appliedLStheorytocomplexityofalgorithms,seealso[ 39 ].Therearestillinterestingproblemstobesolvedandexamplestobefoundwithintheelditselfaswell. 9

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CHAPTER2BACKGROUND 2.1BasicDenitionsAswithanyeld,sometimesthebestwaytounderstandsomethingistobreakitupintosmallerpieces.Lusternik-Schnirelmanncategoryisameansfordecomposingatopologicalspaceintomoremanageablepieces.Thebasicdenitionreadsassuch: Denition2.1.1. TheLusternik-Schnirelmanncategory(LScategory)ofaspaceXisthesmallestnonnegativeintegernsuchthatthereexistsfA0,A1,...,Ang,anopencoverofXwitheachAicontractibleinX.ThisisdenotedbycatX,andsuchacoveriscalledcategorical.Itisimportanttonotethatourdenitionofcategoryisnormalized,thatiswecountdownonefromthenumberofsetswhenstatingthecategory.Theoriginaldenitiondidn'tcountdownbyone,butnowthisisacommonconvention.AlsoforspacesXwherenosuchintegerexists,wesaycatX=1. Example2.1.2. Followingthisdenition,spaceswithLScategory0arecontractible,andconverselyspacesofcategory0arecontractible.. Example2.1.3. TheLS-categoryofSnis1asa2-setcategoricalcovercanbecon-structedbyextendingthenorthernandsouthernhemispheres. 2.2PropertiesofLS-CategoryAswithanytopologicalinvariant,thereisalaundrylistofpropertieswewouldliketheinvarianttosatisfy.WeincludesomeofthebasicpropertiesLusternik-Schnirelmanncategorysatises,particularlyonesthatwillbeusedtoobtainresultsinthisdisseration.AlsowearegoingtoassumethroughoutthedissertationthatallofourspacesareCW-complexes. Remark2.2.1. Theseclaimsarewell-known,see[ 5 ];welistthemhereforreference. (1)Cat(X_Y)=maxfcatX,catYg. (2)Cat(X[Y)cat(X)+cat(Y)+1. 10

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(3)CatX[CAcatX+1. (4)Cat(X=A))]TJ /F5 11.955 Tf 12.61 0 Td[(1catX.ThisfollowsfromthefactthatX=AhashomotopytypeofX[CA,theunionofXwiththeconeoverA,anditem(3). (5)Iff:X!Yhasarighthomotopyinverse,thencatYcatX. (6)CatXdimX,wheredimXisthecoveringdimensionofXforpath-connectedX.Asanotherwordonmappingcones,itshouldbementionedthatBersteinandHiltonexploredthechangesincategoryofaspaceviaattachingconesin[ 3 ].TheyusedgeneralizedHopfinvariantstodescribemoreinterestingcaseswherethecategorydoesnotincreasebyoneaftertheconeisattached. Denition2.2.2. LetRbeacommutativeringandXaspace.Wedenethecup-lengthofXwithcoefcientsinR,denotedbycupR(X),tobetheleastintegerksuchthatall(k+1)-foldcupproductsvanishinthereducedcohomology,~H(X;R). Proposition2.2.3. WehavetheinequalitycupR(X)catX[ 5 14 ]. Example2.2.4. LetT2denotethe2-torus.ThenT2npt=S1_S1andcatT22.Weobservethatthereare2uniqueelementsin~H1(T2)withnontrivialcupproduct.Thus2cupT2andcatT2=2.Theexamplewiththetorusisonewithnodifferencebetweenthecup-lengthandtheLusternik-Schnirelmanncategory,but[ 5 ]givesplentyoftreatmenttocaseswherethesetwoinvariantsdonotcoincide.TheHeisenberggroupprovidesanexamplewherecategoryexceedscup-length.Asmentionedinthehistory,oneofthesignicantearlyresultsintheeldwasaninequalitythatrelatesthenumberofcriticalpointsafunctiononamanifoldtoitsLusternik-Schnirelmanncategorygivensufcientconditionsonthefunctionandmanifold.Criticalpointsofsmoothfunctionscanbeveryinteresting,andthisresultisaverybeautifulrelationbetweengeometryandcalculus. Theorem2.2.5. (Lusternik-SchnirelmannTheorem).LetMbeaparacompactC2-BanachmanifoldandCrit(M)theminimumnumberofcriticalpointsforanyC2function 11

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withacertainboundednesspropertyfromMtoR.WhenMisnite-dimensionalandclosed,wecanremovetheboundednesscondition.Thencat(M)+1Crit(M),[ 5 ][ 40 ]. 2.3LSCategory-Constructions Denition2.3.1. Forapath-connectedspaceXwithbasepointx0,wedenePXtobethesetofallcontinuousfunctions:I!Xsatisfying(0)=x0topologizedbythecompact-opentopology. Denition2.3.2. Wedenep:PX!Xgivenbyp()=(1).ItcanbeproventhatthisyieldsabrationwithbasespaceXandberthatishomotopyequivalentto(X,x0),theloopspaceofX. Denition2.3.3. Givenf:Y!Xandg:Z!Xwecandenetheberwisejoin,YXZofYandZoverX,asfollows:YXZ=f(y,z,t)2YZIjf(y)=g(z)g/swhere(y,z1,0)s(y,z2,0)and(y1,z,1)s(y2,z,1). Remark2.3.4. RecallthejoinoftopologicalspacesYandZ,YZ,isdenedasYZ=(YZI)=swhere(y,z1,0)s(y,z2,0)and(y1,z,1)s(y2,z,1). Denition2.3.5. Fromthis,wedenePnXtobetheiteratedberwisejoinofncopiesofPXoverXviabersofp:PX!Xdenedaboveanddenotetheberwisemapwith pXn:PnX!X.(2)NotethatthehomotopyberofpXnis(X,x0)n,then-foldjoinof(X,x0).ThefollowingtheoremofSchwarz,see[ 5 40 ]tiesthisconstructiontoLusternik-SchnirelmannCategory Theorem2.3.6. Theinequalitycat(X)nholdsiffthereexistsasections:X!Pn+1Xtop)]TJ /F3 11.955 Tf 11.95 0 Td[(n+1:Pn+1X!X.SchwarzactuallydevelopedabroadgeneralizationofLusternik-Schnirelmancategory[ 40 ]aswhathecallsgenus.Thebiggestadvantagetobeingabletousetheseideasisthatthetaskofcomputingthecategoryofaspacenowtakesontheavorofotherquestionsinalgebraictopology,namely,investigatingwhenasectionto 12

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aparticularmapexists.Sometimesdescribingopenorclosedsetscanbelessintuitive(particularlyforreallybadspaces),andthisgivesusatooltogetaroundthat.Laterinthedissertation,whenwegotoconstructasection,we'llnditnecessarytoinspectcertainhomotopygroups. 2.4ToomerInvariantAsmentionedearlier,therearespaceswheretheinequalitybetweencup-lengthandcategoryisstrict.TheToomerinvariantisatoolthatsitsbetweencup-lengthandcategory. Denition2.4.1. FixacommutativeringR.TheToomerinvariantofXwithcoefcientsinR,eR(X),istheleastintegerkforwhichthemappn:H(X;R)!H(Pn(X);R)isinjective,see[ 5 ]. Proposition2.4.2. ItfollowsthateR(X)catXforanychoiceofR.Aswithcup-lengthearlier,itshouldbementionedthat[ 5 ,2.9]containsexampleswheretheinequalitiesarestrict. Denition2.4.3. Forf:X!Y,whereXandYareclosed,connected,andorientedn-dimensionalmanifolds,with[x]and[y]generatorsforHn(X)andHn(Y)respectively,wedenethedegreeoff,denotedbydeg(f)tobetheintegersuchthatf([x])=deg(f)[y]. Theorem2.4.4. Letf:X!Ybeamapofdegree1,whereXandYareclosed,orientablemanifolds.Thenf:H(X)!H(Y)isanepimorphism,andf:H(Y)!H(X)isamonomorphism.Rudyakaskediftheexistenceofamapf:M!N,ofdegree1,impliestheinequalitycatMcatN[ 32 ],[ 5 ,Openproblem2.48].Whilenotachievingthefullresult,hewasabletoprovesomepartialresults.InparticularitfollowsfromtheinjectivepropertyoffthateR(M)eR(N),whensuchamapexists[ 32 ]. Remark2.4.5. WeknoweR(MSn)eR(M)+1,andthereexistexampleswherecat(MSn)=catMforsuitableMandN,([ 22 23 ]). 13

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2.5RationalizationRationalizationsbecamequiteusefulforansweringquestionsinLusternik-Schnirelmanncategory.Asmentionedbefore,rationalizationsprovideasettingwheretheGaneaConjectureholdstrue. Denition2.5.1. Therationaln-sphereisdenedtobethecomplexSnQ=(_k1Snk)[(ak2Dn+1k)whereDn+1k+1isattachedtoSnk_Snk+1byamaprepresenting[Snk])]TJ /F3 11.955 Tf 12.7 0 Td[(m[Snk+1]foreveryintegerm. Denition2.5.2. Dn+1Q=SnQI=SnQ0iscalledtherational(n+1)disk. Denition2.5.3. Arationalizationofasimplyconnectedspace,Xisamap:X!XQtoasimplyconnectedrationalspaceXQsuchthatinducesanisomorphism(X)ZQ!XQ Remark2.5.4. Thereisanaturalinclusion(Dn+1,Sn)!(Dn+1Q,SnQ),andthisallowsustoconstructtargetspacesforrationalizations.ThroughoutthesectionweassumeXtobesimplyconnectedanddenotebyXQtherationalizationofX,see[ 12 37 ].WedeneeQ(X)tobetheleastintegernsuchthatthenthbrationPnX!XinducesaninjectionincohomologywithcoefcientsinQ.ForXsimplyconnectedandofnitetype,wehavethateQ(X)=e(XQ),[ 5 ]. Proposition2.5.5. ForsimplyconnectedCWspacesXandY,wehave(X[CY)Q=XQ[CYQ,[ 5 ].Inparticular,(X_Y)Q=XQ_YQ. Proof. Inthefollowingdiagram,themaplisthelocalizationmapofX_Y,andkisgivenbythewedgeoflocalizationmapsonXandY.Themapjexistsbytheuniversalpropertyof(X_Y)Q,andinducesisomorphismsinhomology.HenceXQ_YQ=(X_Y)Q 14

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X_Y k&& l// (X_Y)Qj XQ_YQ Remark2.5.6. In[ 11 ]itisshownthatforaclosed,simplyconnectedmanifoldM,eQ(M)=cat(MQ),andhencecatMQcatM.ReturningtoRudyak'squestiononapossiblerelationbetweendegreeandcategory,wecansettleitintherationalcontext. Proposition2.5.7. Forclosedandsimplyconnectedm-manifoldsMandNwithf:M!Nofnonzerodegree,wehavecatMQcatNQ. Proof. ItsufcestoshoweQ(M)eQ(N)basedontheremarkabove.Thatis,supposeeQ(M)nandsop:H(M;Q)!H(Pn(M)Q)inthefollowingdiagramisinjective.H(Pn(M);Q)H(Pn(N);Q)oo H(M;Q)pOO H(N;Q)pOO foo By[ 33 ,V.2.13],themapfisinjective.Sincepandfareinjective,thecompositionpfisinjective,andp:H(N;Q)!H(Pn(N);Q)isinjective.ThuseQ(N)n. 15

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CHAPTER3CATEGORYOFCONNECTEDSUMSANDSOMEDISCUSSION 3.1CategoryofQuotientSpaces Theorem3.1.1. SupposeXisann-dimensionalspacewithm-connectedsubcomplexA,with2cat(X=A)k,andk+m)]TJ /F5 11.955 Tf 11.96 0 Td[(1n.ThencatXk. Proof. Forsakeofsimplicity,putp=pX=Ak+1andp#=pXk+1,cf.( 2 ).Ascat(X=A)k,andbyTheorem 2.3.6 ,thereexiststhefollowingsectionswithps=1X=A.Pk+1(X=A)p X=AsTT Weconsiderthecollapsingmapq:X!X=A,andgettheber-pullbackdiagram. Ef// Pk+1(X=A)p X// X=ANowconsiderPk+1(X).Wealreadyhavep#:Pk+1(X)!X,andthecollapsingmapq:X!X=Ainducesamapq#:Pk+1(X)!Pk+1(X=A).Sincepq#=qp#andthesquareisthepull-backdiagram,wegetamaph:Pk+1(X)!Xsuchthatthefollowingdiagramcommutes.RecallthatourgoalistoprovecatXk.BecauseofSchwarz'sTheorem 2.3.6 ,itsufcestoconstructasectionofp#.Todothis,itsufcesinturntoconstructasectionofthemaph:Pk+1(X)!E.Moreover,sincedimX=n,itsufcestoconstructasectionofhoverthen-skeletonE(n)ofE,i.e.,toconstructamap:E(n)!Pk+1(X)withh=1E.Byhomotopyexcision[ 20 ,Prop.4.28],andbecauseAism-connected,thequotientmapq:X!X=Ainducesisomorphismsq:i(X)!i(X=A)forimandepimorphismfori=m+1.So,i(X)!i((X=A))isanisomorphismfori 16

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(m)]TJ /F5 11.955 Tf 12.32 0 Td[(1)andepimorphismfori=m.Thereforei(X)(k+1)!i((X=A))(k+1)isanisomorphismforim+kbecauseof[ 6 ,Prop.5.7].Thelongexactsequenceofhomotopygroupsforthebration(X)(k+1)!Pk+1(X)!X,wherethemapfromPk+1(X)toXisthemapin 2 yieldsthefollowingcommutativediagram...// i((X)(k+1))// = i(Pk+1X)// h i(X)// = ......// i(((X=A))(k+1))// i(E)// i(X)// ...Bythe5-lemma,themaphisanisomorphismfori(m+k)]TJ /F5 11.955 Tf 11.77 0 Td[(1)andepimorphismforn=m+k.SobyWhitehead'stheorem,thereexistsamap:E(n)!Pk+1X.Now,thecomposition(s#)isasectiontop#:Pk+1!X.ThuscatXk. CombiningthiswiththepreviousRemark 2.2.1 givesthefollowinginequality:cat(X=A))]TJ /F5 11.955 Tf 11.95 0 Td[(1cat(X)cat(X=A)underthedimension-connectivityconditionsfromTheorem 3.1.1 3.2PreliminariesonConnectedSums Denition3.2.1. LetMandNben-dimensionalmanifolds.DeneDMandDNtoben-disksinMandN,andletM=MnDMandN=NnDN.Fixadiffeomorphism,f:@DM!@DNanddenetheconnectedsumofMandN,M#N=M[fN.NotethatthehomotopytypeofM#Ndependsonlyonthehomotopyclassoff. Remark3.2.2. WementionthatifMandNareorientedn-dimensionalmanifolds,thenthereisacanonicallyorientedconnectedsum,suchthatthemapfreversesorientation.Alsoforanyconnectedsum,M#N,thereexistsamapf:M#N!McollapsingNintoanarbitrarilysmallballinM,xingmostofM,andthatthismaphasdegreeone. Proposition3.2.3. ForclosedandorientedmanifoldsMandN,e(M#N)maxfe(M),e(N)g. Proof. Considerf:M#N!MthecollapsingmapontoM.Thenwehavethefollowingdiagram. 17

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H(PnM)// H(Pn(M#N))H(M)OO //f// H(M#N)OO Themapfhasdegree1,andsof:H(M)!H(M#N)isinjective[ 33 ,TheoremV,2.13].Alsosupposepn:H(M#N)!H(Pn(M#N)isinjective.Consideru2H(M).Asfandpnareinjective,pn(u)2H(PnM)isnonzero,andsopn:H(M)!H(PnM)isinjective,andsimilarlyforN.Andsoe(M#N)maxfe(M),e(N)g. Proposition3.2.4. ForclosedandorientedmanifoldsMandN,ifcatM=e(M)andcatN=e(N),thencat(M#N)=maxfcatM,catNg. Proof. Combiningtheassumptionse(M)=catMande(N)=catNwiththeinequalitymaxfe(M),e(N)ge(M#N)cat(M#N)maxfcatM,catNg,wehavetheclaim. Proposition3.2.5. TheinequalitymaxfcatM,catNg)]TJ /F5 11.955 Tf 20.59 0 Td[(1cat(M#N)maxfcatM,catNgholdswhenevermaxfcatM,catNg1 Proof. IfcatM=1=catNthenMandNarehomotopyspheres,andsoM#Nis.Conversely,ifcatM#N=1thenMandNmustbehomotopyspheres.Thus,weprovedthatcat(M#N)=maxfcatM,catNgifmaxfcatM,catNg1. 3.3ApplicationstoConnectedSumsLetMandNbetwoclosedn-dimensionalmanifolds. Corollary3.3.1. SupposethateithermaxfcatM,catNg3ormaxfcatM,catNg=1.ThenthereisadoubleinequalitymaxfcatM,catNg)]TJ /F5 11.955 Tf 20.59 0 Td[(1cat(M#N)maxfcatM,catNg. Proof. ConsiderthecaseofTheorem 3.1.1 whereX=M#N,theconnectedsumofMandN,andA=Sn)]TJ /F4 7.97 Tf 6.59 0 Td[(1istheseparatingspherebetweenMandN.ThenX=A=M_N, 18

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andcat(X=A)=maxfcatM,catNg.ThecasemaxfcatM,catNg=1iscoveredbyProposition 3.2.5 .So,assumethatcat(X=A)=3.SoasAis(n)]TJ /F5 11.955 Tf 12.34 0 Td[(2)-connectedand(n)]TJ /F5 11.955 Tf 10.68 0 Td[(2)+3)]TJ /F5 11.955 Tf 10.68 0 Td[(1n,weareinthecaseofTheorem 3.1.1 andgettheright-handinequality.Notethatweareusingitem(4)ofRemark 2.2.1 Remark3.3.2. WenotethattheCorollary 3.3.1 abovedoesn'tcoverthecasewhencatM=catN=2whilecat(M#N)=3.Wedonotknowifsuchacaseispossible,however.IfM#Nissimplyconnected,thensoisM_N.Inthatcase,theisomorphismsfromhomotopyexcisioninTheorem 3.1.1 extendtoatleastonedimensionhigher,andtheresultfollows.Inthenon-simplyconnectedcasethough,thesituationisunresolved. Remark3.3.3. In[ 19 ],anupperboundisgivenfortheLScategoryofadoublemappingcylinder.Ifweconsidertheconnectedsumofn-manifoldsMandNassuchadoublemappingcylinder,thenthefollowinginequalityisobtained:catM#Nminf1+catM+catN,1+maxfcatM,catNgg.HereMandNareMnpt,andNnpt,respectively.Rivadeneyraprovedin[ 30 ]thatthecategoryofamanifoldwithoutboundarysometimesdoesnotincreasewhenapointisremoved.IfthecategoriesofMandNdodecreasebyonewhenapointisremoved,thenHardie'sresultyieldsthesameinformationasprovenhere.Howeverin[ 27 ],aclosedmanifoldisconstructedsothatthecategoryremainsunchainedafterthedeletionofapoint,andourresultgivesanimprovementofthecategoryinsuchacase.Alsowhentoreconsiderthecaseofrationalizationsofconnectedsums,weobtainthefollowingresult. Proposition3.3.4. ForMandN,closedandsimplyconnectedmanifolds,cat(M#N)Q=maxfcatMQ,catNQg,providedcat(M#N)Q3. Proof. AsMandNareclosedandsimplyconnected,M#Nisclosedandsimplyconnected,andeQ(M#N)=cat(M#N)Q.CombiningRemark 2.5.6 andProposition 3.2.3 19

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establishesonthelefthandside,maxfcatMQ,catNQg=maxfeQ(M),eQ(N)geQ(M#N)=cat(M#N)Q.Whileontherighthandsidewehave,cat(M#N)Q=cat(MQ#NQ)cat(MQ_NQ)=maxfcatMQ,catNQg,wheretheinequalitycat(MQ#NQ)cat(MQ_NQ)isobtainedviaCorollary 3.3.1 anditem(1)inRemark 2.2.1 .Theinequalitycat(M#N)Qcat(M_N)Qcanbeprovedusingthepreviousresult.Howeversincedim(M#N)Q=n+1,weneedtherestrictiononcat(M#N)Q. Thereareplentyofexampleswherethecategorydoesnotincreasewhentheconnectedsumoftwomanifoldsisformed,buttheinequalityleavesopenthepossibliitythataconnectedsumoftwomanifoldscouldactuallyhaveacategory1lessthaneitherofthecomposantmanifolds.Suchanexamplehasnotbeenrevealed,andrepresentsanareaforfuturework. 20

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BIOGRAPHICALSKETCH RobertNewtonwasborninGlensFalls,NewYork.HisfatherwasanactivedutymemberoftheUSArmy,hespenthischildhoodmovingaround,includingstopsinCalifornia,Texas,andGermany.HeearnedhisB.A.andM.A.bothinMathematicsfromSUNYPotsdamin2007.Tofurtherhiscareerinmathematics,andpartiallytoescapethecoldweather,hebegangraduateschoolattheUniversityofFloridainAugustof2007.Inthesummerof2013RobertgraduatedwithhisPh.D.inMathematics.Upongraduation,hewillpursueteachinghighschoolmathematicsatTrinitySchoolinManhattan. 24