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HOMEOMORPHISMSOFKNASTERCONTINUA By VINCENTASSEMBATYA ADISSERTATION PRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2001 Copyright2001 by VincentASsembatya IdedicatethistomyfamilyandtoJosephine. ACKNOWLEDGEMENTS ThisthesisistheoutcomeofmystudiesattheDepartmentofMathematics attheUniversityofFlorida. EveryoneintheDepartmenthasbeenverysupportive. IwouldliketoexpressmygratitudetomyadvisorProfessorJamesE. Keeslingfor thequalitytimehehas generouslygiventomeduringmyworkon thisthesis. I appreciatetheadvice,support andthegreatinspirationgivenduringthelastfew years. Iwouldliketothankthe DepartmentofMathematicsattheUniversityof FloridaandMakerereUniversityinUgandaforthefincancialsupportthathasseen methroughmystudies. Iamgratefultomygraduatecommitteemembers,ProfessorsBlock,Brechner, KingandKhuriwhomEvepersistentlydistractedfromtheirotherschedules. Their support is invaluable. I wish to thank myparents and familyin Uganda; allof myfriendsinGainesville;andProfessors AndrewVinceandPeterKizzaandtheir familiesforkeepingmesociallyadriftandfordistractingmefrommywoes. IV TABLEOFCONTENTS ACKNOWLEDGEMENTS iv ABSTRACT vii CHAPTERS 1 INTRODUCTION 1 1.1 Continua 1 1.2 Composants 1 1.3 Homogeneity 2 1.4 ExamplesofIndecomposableContinua 2 1.5 InverseLimitSpacesofContinua 2 1.6 ArclikeContinua 3 1.7 InducedMapsbetweenInverseLimits 4 1.8 LiftingMapstoCovers 5 1.9 HomotopyLifting 5 1.10 CoveringProjection 5 1.11 TopologicalEntropy 6 1.12 TopologicalGroups 7 1.13 GroupAction 7 2 MAPSBETWEENTOPOLOGICALGROUPS 9 2.1 DualityforLocallyCompactGroups 9 2.2 MapsbetweenTopologicalGroupsthatareHomotopictoHomo- morphisms 9 2.3 TheCechCohomologyofContinua 10 2.4 DirectLimitsofGroups 10 2.5 TheCechcohomologyasadirectlimitgroup 11 3 FIXEDPOINTSOFKNASTERCONTINUA 12 3.1 GeneralizedSolenoids 12 3.2 ComposantsoftheSolenoid 14 3.3 KnasterContinua 16 3.4 Chebychevpolynomials 16 3.5 EndPointsinKnasterContinua 16 3.6 FixedPointsofHomeomorphismsofKnasterContinua 19 3.7 StandardHomeomorphismsoftheSolenoid 24 3.8 LiftingMapsfromKnasterContinuatotheSolenoids 24 3.9 LiftingIsotopies 24 v 3.10 StandardHomeomorphismsofKnasterContinua 27 3.11 NumberofFixedPointsofHomeomorphisms 28 4 HOMEOMORPHISMSOFKNASTERCONTINUA 31 4.1 EntropyofQuotients 31 4.2 EntropyofaHomeomorphismoftheKnasterContinuum .... 35 4.3 GeneralizingKnasterContinua 35 4.4 WeakSolenoids 36 4.5 KnasterContinuafromWeakSolenoids 38 4.6 DistinguishablePointsinKnasterContinua 38 4.7 LiftingHomeomorphismsfromKnasterContinuatoSolenoids . . 40 5 CONCLUSION 44 5.1 Summary 44 5.2 Questions 45 REFERENCES 46 BIOGRAPHICALSKETCH 48 vi AbstractofDissertation PresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulfillmentofthe RequirementsfortheDegreeofDoctorofPhilosophy HOMEOMORPHISMSOFKNASTERCONTINUA By VincentASsembatya December 2001 Chairman: Dr. JamesE.Keesling MajorDepartment: Mathematics InthisthesisweinvestigatehomeomorphismsofKnastercontinua. Wedeter- minetheminimumnumberoffixed-pointshomeomorphismsofthesecontinuamust have. ThisanalysisisrelatedtoaquestionraisedbyWilliamS.Mahavieronwhether ahomeomorphismontheKnasterbuckethandlemusthaveatleasttwofixedpoints. ItisprovedthatanisotopybetweenhomeomorphismsoftheKnastercontinuumcan beliftedtoanisotopybetweenhomeomorphismsofthesolenoid. Wegivenecessary andsufficientconditionsforahomeomorphismoftheKnastercontinuumtohaveat leasttwofixedpoints. WeconstructaKnastercontinuumonwhicheveryhomeo- morphismhaseitheruncountablymanyfixedpointsoruncountablymanypointsof period2. Wedeterminetheminimumnumberoffixedpointsahomeomorphismon theKnastercontinuumcanhave. Weconstructanexampletoshowthat Bowen’s theoremonentropyofquotientson compactspacesdoesnot readilygeneralizeto non-compactspaces. Vll WegeneralizethedefinitionsofKnastercontinuatoconstructionsviatoral homomorphisms. Weshowthathomeomorphismonthesecontinua(intheodddi- mensioncase)lifttohomeomorphismstothesolenoidandendwithsomequestions forfurtherresearch. vm CHAPTER INTRODUCTI1ON Inthischapterweestablishnotation,definitionsandsomebasicresultsfrom continuumtheory, topologicalgroup theoryand cohomology theoryto beusedin subsequentchapters. Weassumethereaderisfamiliarwiththestandardresultsand terminologyofgeneraltopology,suchasiscoveredinMunkres[22]. Specificallysuch results as the Baire Category Theoremareassumed. For less wellknown results suchasthosefromalgebraictopologyandtopologicalgrouptheorytheappropriate referenceswillbecited. Basicdefinitionsinalgebraictopologysuchasofhomotopy groups,homologygroupsandcohomologygroupsmaybefoundinSpanier[27]. For basicdefinitionsfromtopologicalgrouptheory,thereaderisreferredtothetextby HewittandRoss[13]. Byamapwemeanacontinuousfunction. 1.1 Continua Acontinuumisacompactconnectedmetricspace. AsubcontinuumYofthe continuumXisaclosed,connectedsubsetofAh AcontinuumXis decomposable ifthereexisttwononemptysubcontinua Hand I\ ofthecontinuumX suchthat H^XandI\ ^X,butHUK=X. Anycontinuumthatisnotdecomposableis saidtobeindecomposable. 1.2 Composants AcomposantCom(x)ofagivenpointxEXistheunionofallpropersub- continuainXthatcontainthepointx. ApointybelongstoCom(i)ifthereisa propersubcontinuumAthatcontainsbothxandy. Itisknown[16]thatcontinuum 1 2 Xisindecomposableifandonlyif{Com(a:)|x£A}formsapartitionofXintoan uncountablecollectionoffirstcategory,connectedsetseachofwhichisdenseinA'. AsetisfirstcategoryinXifitcanbewrittenastheunionofacountablenumber ofnowheredensesubsetsofX. Itisknown[16]thatacontinuumisindecomposable andnondegenerateifandonlyifi1t.3possessestwodisjointcomposants. Acontinuum XishereditarilyindecomposableifeverysubcontinuumofXisindecomposable. Homogeneity 1.4 AcontinuumXissaidtobehomogeneousifforanygivenpointsx,y6X — thereisahomeomorphismh:X >ATofA"ontoXsuchthath(x)=y. ExamplesofIndecomposableContinua ThesolenoidsandKnastercontinua(Chapter3)areexamplesofindecompos- 1.5 ablecontinua. Thepseudoarc[16,figure4]isanexampleofaheriditarilyindecom- posablecontinuum. Infacteverysubcontinuumofthepseudoarcishomeomorphic tothepseudoarc. Suchacontinuumissaidtobehereditarilyequivalent. Theunit intervalisanotherexampleofahereditarilyequivalentcontinuum. InverseLimitSpacesofContinua. Aninversesequenceisadoublesequence{A oftopologicalspacesA, andmapsfisuchthateachA,isatopologicalspaceforeachiandeachmap/,•takes A,+itoA;. Thecollectionofmapsfiarereferredtoasbondingmaps. Wewrite Ax A2 A3.... Theinverselimitoftheinversesequenceistheset OO (xi,x2,...)£J^A,: foralli>1, ft{xl+1)=x, { »=i topologized withtherelativizedproduct topology. Let denotethe naturalpro- jectionfrombothJ^iA,-anditssubsetA,*,ontoAkdefinedby7r^((x„))=x*.A