Introduction to Topological Manifolds

444 Pages·2010·11.081 MB·English

by  John M. Lee

Graduate Texts in Mathematics 202

Editorial Board
S. Axler
K.A. Ribet

For other titles published in this series, go to
http://www.springer.com/series/136

John M. Lee
Introduction to
Topological Manifolds
Second Edition
John M. Lee
Department of Mathematics
University of Washington
Seattle, Washington 98195-4350
USA
[email protected]

Editorial Board:
S. Axler K. A. Ribet
Mathematics Department Mathematics Department
San Francisco State University University of California at Berkeley
San Francisco, CA 94132 Berkeley, CA 94720
USA USA
[email protected] [email protected]

ISSN 0072-5285

ISBN 978-1-4419-7939-1 e-ISBN 978-1-4419-7940-7
DOI 10.1007/978-1-4419-7940-7

Springer New York Dordrecht Heidelberg London

M athematics Subject Classification (2010): 54-01, 55-01, 57-01

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Preface
Manifoldsarethemathematicalgeneralizationsofcurvesandsurfacestoarbitrary
numbersofdimensions.Thisbookisanintroductiontothetopologicalpropertiesof
manifoldsatthebeginninggraduatelevel.Itcontainstheessentialtopologicalideas
that are needed for the further study of manifolds, particularly in the context of
differentialgeometry,algebraictopology,andrelatedfields.Itsguidingphilosophy
is to develop these ideas rigorously but economically,with minimal prerequisites
andplentyofgeometricintuition.HereattheUniversityofWashington,forexam-
ple, this text is used for the first third of a year-long course on the geometry and
topologyof manifolds; the remainingtwo-thirds of the course focuses on smooth
manifoldsusingthetoolsofdifferentialgeometry.
Therearemanysuperbtextsongeneralandalgebraictopologyavailable.Why
addanotheronetothecatalog?Theanswerliesinmyparticularvisionofgraduate
education:it is my (admittedlybiased)belief that everyserious student ofmathe-
maticsneedstobeintimatelyfamiliarwiththebasicsofmanifoldtheory,inthesame
waythatmoststudentscometoknowtheintegers,therealnumbers,vectorspaces,
functionsofonerealorcomplexvariable,groups,rings,andfields.Manifoldsplay
a role in nearly every major branch of mathematics (as I illustrate in Chapter 1),
andspecialistsinmanyfieldsfindthemselvesusingconceptsandterminologyfrom
topologyandmanifoldtheoryonadailybasis.Manifoldsarethuspartofthebasic
vocabulary of mathematics, and need to be part of basic graduate education. The
firststepsmustbetopological,andareembodiedinthisbook;inmostcases,they
shouldbecomplementedbymaterialonsmoothmanifolds,vectorfields,differen-
tial forms, andthe like, as developed,for example,in [Lee02],which is designed
to bea sequelto this book.(Afterall, fewofthe reallyinterestingapplicationsof
manifoldtheoryarepossiblewithoutusingtoolsfromcalculus.)
Of course, it is not realistic to expect all graduate students to take full-year
courses in general topology, algebraic topology, and differential geometry. Thus,
althoughthis book touches on a generousportion of the material that is typically
includedinmuchlongercourses,thecoverageisselectiveandrelativelyconcise,so
thatmostofthebookcanbecoveredina singlequarterorsemester,leavingtime
inayear-longcourseforfurtherstudyinwhateverdirectionbestsuitstheinstructor
v
vi Preface
andthestudents.AtUW,wefollowitwithatwo-quartersequenceonsmoothmani-
foldtheorybasedon[Lee02];butitcouldequallywellleadintoafull-blowncourse
onalgebraictopology.
Itiseasytodescribewhatthisbookisnot.Itisnotacourseongeneraltopology—
many of the topics that are standard in such a course are ignored here, such as
metrizationtheorems,theTychonofftheoremforinfiniteproductspaces,acompre-
hensivetreatmentof separationaxioms,and functionspaces. Nor is it a coursein
algebraictopology—althoughItreatthefundamentalgroupindetail,thereisbarely
a mention of the higher homotopygroups, and the treatment of homologytheory
is extremely brief, meant mainly to give the flavor of the theory and to lay some
groundworkforthelaterintroductionofdeRhamcohomology.Itcertainlyisnota
comprehensivecourseontopologicalmanifolds,whichwouldhavetoincludesuch
topicsasPLstructuresandmaps,transversality,surgery,Morsetheory,intersection
theory,cobordism,bundles,characteristicclasses, andlow-dimensionalgeometric
topology.(PerhapsamoreaccuratetitleforthebookwouldhavebeenIntroduction
toTopologywithanEmphasisonManifolds.)Finally,itisnotintendedasarefer-
ence book,because few of the results are presentedin their most generalor most
completeforms.
Perhapsthebestwaytosummarizewhatthisbookiswouldbetosaythatitrep-
resents,toagoodapproximation,myconceptionoftheidealamountoftopological
knowledgethatshouldbepossessedbybeginninggraduatestudentswhoareplan-
ning to go on to study smooth manifolds and differential geometry. Experienced
mathematicians will probably observe that my choices of material and approach
havebeen influencedby the fact that I am a differentialgeometerbytrainingand
predilection,notatopologist.ThusIgivespecialemphasistotopicsthatwillbeof
importancelaterinthestudyofsmoothmanifolds,suchasparacompactness,group
actions,anddegreetheory.Butdespitemyprejudices,Ihavetriedtomakethebook
usefulasaprecursortoalgebraictopologycoursesaswell,anditcouldeasilyserve
asaprerequisitetoamoreextensivecourseinhomologyandhomotopytheory.
A textbookwriter always has to decidehow much detail to spell out, and how
much to leave to the reader. It can be a delicate balance. When you dip into this
book,youwill quicklysee that my inclination,especially in the early chapters,is
towardwritingmoredetailratherthanless.Thismightnotappealtoeveryreader,
butIhavechosenthispathforareason.Inmyexperience,mostbeginninggraduate
students appreciate seeing many proofs written out in careful detail, so that they
cangetaclearideaofwhatgoesintoacompleteproofandwhatliesbehindmany
of the common “hand-waving” and “standard-argument” moves. There is plenty
of opportunity for students to fill in details for themselves—in the exercises and
problems—andtheproofsinthetexttendtobecomealittlemorestreamlinedasthe
bookprogresses.
Whendetailsareleftforthestudenttofillin,whetherasformalexercisesorsim-
plyasargumentsthatarenotcarriedoutincompletedetailinthetext,Ioftentryto
givesomeindicationabouthowelaboratetheomitteddetailsare.IfI characterize
someomitteddetailas“obvious”orasan“easyexercise,”youshouldtakethatas
an indicationthat the proof,onceyousee howto doit, shouldrequireonlya few
Preface vii
stepsanduseonlytechniquesthatareprobablyfamiliarfromothersimilarproofs,
soifyoufindyourselfconstructingalongandinvolvedargumentyouareprobably
missingsomething.Ontheotherhand,ifIlabelanomittedargument“straightfor-
ward,”itmightnotbeshort,butitshouldbepossibletocarryitoutusingfamiliar
techniqueswithoutrequiringnewideasortrickyarguments.Inanycase,pleasedo
not fall into the trap of thinking that just because I declare something to be easy,
youshouldseeinstantlywhyitistrue;infact,nothingiseasyorobviouswhenyou
arefirstlearningasubject.Readingmathematicsisnotaspectatorsport,andifyou
reallywanttounderstandthesubjectyouwillhavetogetinvolvedbyfillinginsome
detailsforyourself.
Prerequisites
Theprerequisiteforstudyingthisbookis,brieflystated,asolidundergraduatede-
greeinmathematics;butthisprobablydeservessomeelaboration.
Traditionally,“generaltopology”has been seen as a separate subject from“al-
gebraictopology,”andmostcoursesinthelatterbeginwiththeassumptionthatthe
students have already completed a course in the former. However, the sad fact is
thatforavarietyofreasons,manyundergraduatemathematicsmajorsintheUnited
States nevertake a course in generaltopology.For that reason I have written this
bookwithoutassumingthatthereaderhashadanyexposuretotopologicalspaces.
Ontheotherhand,Idoassumeseveralessentialprerequisitesthatare,orshould
be,includedinthebackgroundofmostmathematicsmajors.
The most basic prerequisite is a thoroughgroundingin advanced calculus and
elementarylinearalgebra.Sincetherearehundredsofbooksthattreatthesesubjects
well,Isimplyassumefamiliaritywiththem,andremindthereaderofimportantfacts
whennecessary.Ialsoassumethatthereaderisfamiliarwiththeterminologyand
rulesofordinarylogic.
The other prerequisitesare basic set theory such as what one would encounter
inanyrigorousundergraduateanalysisoralgebracourse;realanalysisatthelevel
ofRudin’sPrinciplesofMathematicalAnalysis[Rud76]orApostol’sMathematical
Analysis[Apo74],including,inparticular,anacquaintancewithmetricspacesand
theircontinuousfunctions;andgrouptheoryatthelevelofHungerford’sAbstract
Algebra:AnIntroduction[Hun97]orHerstein’sAbstractAlgebra[Her96].
Becauseitisvitallyimportantthatthereaderbecomfortablewiththisprerequi-
sitematerial,inthreeappendicesattheendofthebookIhavecollectedasummary
ofthemainpointsthatareusedthroughoutthebook,togetherwitharepresentative
collectionofexercises.Studentscanusetheexercisestotesttheirknowledge,orto
brushuponanyaspectsofthesubjectonwhichtheyfeeltheirknowledgeisshaky.
Instructorsmaywishtoassigntheappendicesasindependentreading,andtoassign
some of the exercises early in the course to help students evaluatetheir readiness
forthematerialinthemainbodyofthebook,andtomakesurethateveryonestarts
withthesamebackground.Ofcourse,ifyouhavenotstudiedthismaterialbefore,
viii Preface
youcannothopetolearnitfromscratchhere;buttheappendicesandtheirexercises
can serve as a reminder of important concepts you may have forgotten,as a way
tostandardizeournotationandterminology,andasasourceofreferencestobooks
whereyoucanlookupmoreofthedetailstorefreshyourmemory.
Organization
Thebookisdividedintothirteenchapters,whichcanbegroupedintoanintroduc-
tionandfivemajorsubstantivesections.
Theintroduction(Chapter1)ismeanttowhetthestudent’sappetiteandcreatea
“bigpicture”intowhichthemanydetailscanlaterfit.
The first major section, Chapters 2 through 4, is a brief and highly selective
introductionto the ideas of general topology:topologicalspaces; their subspaces,
products, disjoint unions, and quotients; and connectedness and compactness. Of
course, manifolds are the main examples and are emphasized throughout. These
chaptersemphasizethewaysinwhichtopologicalspacesdifferfromthemorefa-
miliarEuclideanandmetricspaces,andcarefullydevelopthemachinerythatwillbe
neededlater,suchasquotientmaps,localpathconnectedness,andlocallycompact
Hausdorffspaces.
Thesecondmajorsection,comprisingChapters5and6,exploresindetailsome
ofthemainexamplesthatmotivatetherestofthetheory.Chapter5introducescell
complexes,whicharespacesbuiltupfrompieceshomeomorphictoEuclideanballs.
ThefocusisCWcomplexes,whicharebyfarthemostimportanttypeofcellcom-
plexes; besides being a handy tool for analyzing topological spaces and building
new ones, they play a central role in algebraictopology,so any effort invested in
understandingtheirtopologicalpropertieswillpayoffinthelongrun.Thefirstap-
plication of the technologyis to prove a classification theorem for 1-dimensional
manifolds.Attheendofthechapter,Ibrieflyintroducesimplicialcomplexes,view-
ingthemasaspecialclassofCWcomplexesinwhichallthetopologyisencodedin
combinatorialinformation.Chapter6isdevotedtoadetailedstudyof2-manifolds.
After exploringthe basic examples of surfaces—the sphere, the torus, the projec-
tive plane,and theirconnectedsums—I givea proofof the classification theorem
forcompactsurfaces,essentiallyfollowingthetreatmentin[Mas77].Theproofis
complete except for the triangulation theorem for surfaces, which I state without
proof.
Thethirdmajorsection,Chapters7through10,is thecoreofthebook.Init,I
giveafairlycompleteandtraditionaltreatmentofthefundamentalgroup.Chapter
7introducesthedefinitionsandprovesthetopologicalandhomotopyinvarianceof
thefundamentalgroup.AttheendofthechapterIinsertabriefintroductiontocate-
gorytheory.Categoriesarenotusedinacentralwayanywhereinthebook,butitis
naturaltointroducethemafterhavingprovedthetopologicalinvarianceofthefun-
damentalgroup,anditisusefulforstudentstobeginthinkingincategoricalterms
early.Chapter8givesadetailedproofthatthefundamentalgroupofthecircleisin-
Preface ix
finitecyclic.Becausethecircleistheprecursorandmotivationfortheentiretheory
ofcoveringspaces,Iintroducesomeoftheterminologyofthelattersubject—evenly
coveredneighborhoods,localsections,lifting—inthespecialcaseofthecircle,and
theproofsherearephrasedinsuchawaythattheywillapplyverbatimtothemore
generaltheoremsaboutcoveringspacesinChapter11.Chapter9isabriefdigres-
sionintogrouptheory.Althoughabasicacquaintancewithgrouptheoryisanessen-
tialprerequisite,mostundergraduatealgebracoursesdonottreatfreeproducts,free
groups,presentationsofgroups,orfreeabeliangroups,soIdevelopthesesubjects
fromscratch.(Thematerialonfreeabeliangroupsisincludedprimarilyforusein
thetreatmentofhomologyinChapter13,butsomeoftheresultsplayarolealsoin
classifyingthecoveringsofthetorusinChapter12.)Thelastchapterofthissection
givesthestatementandproofoftheSeifert–VanKampentheorem,whichexpresses
thefundamentalgroupofaspaceintermsofthefundamentalgroupsofappropriate
subsets,anddescribesseveralapplicationsofthetheoremincludingcomputationof
thefundamentalgroupsofgraphs,CWcomplexes,andsurfaces.
The fourth major section consists of two chapters on covering spaces. Chap-
ter 11defines coveringspaces, developspropertiesofthe monodromyaction,and
introduceshomomorphismsand isomorphismsof coveringspaces and the univer-
sal covering space. Much of the early development goes rapidly here, because it
is parallel to what was done earlier in the concrete case of the circle. Chapter 12
explorestherelationshipbetweengroupactionsandcoveringmaps,andusesit to
provetheclassificationtheoremforcoverings:thereisaone-to-onecorrespondence
betweenisomorphismclassesofcoveringsofX andconjugacyclassesofsubgroups
of the fundamentalgroupof X. This is then specializedto proper coveringspace
actionsonmanifolds,whicharetheonesthatproducequotientspacesthatarealso
manifolds.These ideas are applied to a number of important examples,including
classifyingcoveringsofthetorusandthelensspaces,andprovingthatsurfacesof
highergenusarecoveredbythehyperbolicdisk.
Thefifthmajorsectionofthebookconsistsofonechapteronly,Chapter13,on
homologytheory.Inordertocoversomeofthemostimportantapplicationsofho-
mologytomanifoldsinareasonabletime,Ihavechosena“low-tech”approachto
thesubject.Ifocusmainlyonsingularhomologybecauseitisthemoststraightfor-
wardgeneralizationofthefundamentalgroup.Afterdefiningthehomologygroups,
Iprovea fewessentialproperties,includinghomotopyinvarianceandtheMayer–
Vietoristheorem,withaminimumofhomologicalmachinery.ThenIintroducejust
enoughaboutthehomologyofCW complexestoprovethetopologicalinvariance
oftheEulercharacteristic.Thelastsectionofthechapterisabriefintroductionto
cohomology,mainlywithfieldcoefficients,toserveasbackgroundforatreatment
ofdeRhamtheoryinalatercourse.Inkeepingwiththeoverallphilosophyoffo-
cusingonlyonwhat is necessaryfora basic understandingofmanifolds,I donot
even mention relative homology, homology with arbitrary coefficients, simplicial
homology,ortheaxiomsforahomologytheory.
Althoughthisbookgrewoutofnotesdesignedforaone-quartergraduatecourse,
thereisclearlytoomuchmaterialheretocoveradequatelyintenweeks.Itshould
be possible to cover all or most of it in a semester with well-prepared students.
x Preface
Thebookcouldevenbeusedforafull-yearcourse,allowingtheinstructortoadopt
a much more leisurely pace, to work out some of the problems in class, and to
supplementthebookwithothermaterial.
Eachinstructorwillhavehisorherownideasaboutwhattoleaveoutinorderto
fitthematerialintoashortcourse.AttheUniversityofWashington,wetypicallydo
notcoversimplicialcomplexes,homology,orsomeofthemoreinvolvedexamples
of covering maps. Others may wish to leave out some or all of the material on
coveringspaces,ortheclassificationofsurfaces.Withstudentswhohavehadasolid
topologycourse,thefirstfourchapterscouldbeskippedorassignedasindependent
reading.
Exercises andProblems
As is the case with any new mathematical material, and perhaps even more than
usualwithmateriallikethisthatissodifferentfromthemathematicsmoststudents
have seen as undergraduates,it is impossible to learn the subject without getting
one’shandsdirtyandworkingoutalargenumberofexamplesandproblems.Ihave
tried to give the readerample opportunityto do so throughoutthe book.In every
chapter, and especially in the early ones, there are questions labeled as exercises
wovenintothetext.Donotignorethem;withouttheirsolutions,thetextisincom-
plete. The reader should take each exercise as a signal to stop reading,pull out a
pencilandpaper,andworkouttheanswerbeforeproceedingfurther.Theexercises
areusuallyrelativelyeasy,andtypicallyinvolveprovingminorresultsorworking
out examples that are essential to the flow of the exposition. Some require tech-
niquesthat the studentprobablyalreadyknowsfrompriorcourses; othersask the
student to practice techniques or apply results that have recently been introduced
in thetext. A feware straightforwardbut ratherlongargumentsthatare moreen-
lighteningtoworkthroughonone’sownthanto read.Inthelater chapters,fewer
thingsaresingledoutasexercises,buttherearestillplentyofomitteddetailsinthe
textthatthestudentshouldworkoutbeforegoingon;itismyhopethatbythetime
thestudentreachesthelastfewchaptersheorshewillhavedevelopedthehabitof
stoppingandworkingthroughmost of the details that are not spelled out without
havingtobetold.
Attheendofeachchapterisaselectionofquestionslabeledasproblems.These
are,forthemostpart,harder,longer,and/ordeeperthantheexercises,andgivethe
studentachancetograpplewithmoresignificantissues.Theresultsofanumberof
theproblemsareusedlaterinthetext.Therearemoreproblemsthanmoststudents
coulddoinaquarterorasemester,sotheinstructorwillwanttodecidewhichones
aremostgermaneandassignthoseashomework.
You will notice that there are no solutions to any of the exercises or problems
inthebackofthebook.Thisisbydesign:inmyexperience,ifwrittensolutionsto
problemsareavailable,thenmoststudents(eventhemostconscientiousones)tend
to be irresistibly temptedto lookat the solutions as soon as they get stuck.But it