Homeomorphisms of Knaster continua

57 Pages·2001·2.3 MB·English

by  Ssembatya, Vincent A., 1968-

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