Thurston - The Geometry & Topology of 3-Manifold.pdf - Geometry - Kuya

WilliamP.Thurston

TheGeometryandTopologyofThree-Manifolds

Electronicversion1.1-March2002

http://www.msri.org/publications/books/gt3m/

Thisisanelectroniceditionofthe1980notesdistributedbyPrincetonUniversity.

ThetextwastypedinT E XbySheilaNewbery,whoalsoscannedtheﬁgures.Typos

havebeencorrected(andprobablyothersintroduced),butotherwisenoattempthas

beenmadetoupdatethecontents.GenevieveWalshcompiledtheindex.

Numbersontherightmargincorrespondtotheoriginaledition’spagenumbers.

Thurston’sThree-DimensionalGeometryandTopology,Vol.1(PrincetonUniversity

Press,1997)isaconsiderableexpansionoftheﬁrstfewchaptersofthesenotes.Later

chaptershavenotyetappearedinbookform.

PleasesendcorrectionstoSilvioLevyatlevy@msri.org.

Introduction

Thesenotes(throughp.9.80)arebasedonmycourseatPrincetonin1978–

79.LargeportionswerewrittenbyBillFloydandSteveKerckho.Chapter7,by

JohnMilnor,isbasedonalecturehegaveinmycourse;theghostwriterwasSteve

Kerckho.Thenotesareprojectedtocontinueatleastthroughthenextacademic

year.Theintentistodescribetheverystrongconnectionbetweengeometryandlow-

dimensionaltopologyinawaywhichwillbeusefulandaccessible(withsomeeort)

tograduatestudentsandmathematiciansworkinginrelatedﬁelds,particularly3-

manifoldsandKleiniangroups.

Muchofthematerialortechniqueisnew,andmoreofitwasnewtome.As

aconsequence,IdidnotalwaysknowwhereIwasgoing,andthediscussionoften

tendstowanter.Thecountrysideisscenic,however,anditisfuntotramparoundif

youkeepyoureyesalertanddon’tgetlost.Thetendencytomeanderratherthanto

followthequickestlinearrouteisespeciallypronouncedinchapters8and9,where

Ionlygraduallysawtheusefulnessof“traintracks”andthevalueofmappingout

someglobalinformationaboutthestructureofthesetofsimplegeodesiconsurfaces.

Iwouldbegratefultohearanysuggestionsorcorrectionsfromreaders,since

changesarefairlyeasytomakeatthisstage.Inparticular,bibliographicalinforma-

tionismissinginmanyplaces,andIwouldliketosolicitreferences(perhapsinthe

formofpreprints)andhistoricalinformation.

Thurston—TheGeometryandTopologyof3-Manifolds

iii

Contents

Introduction iii

Chapter1.Geometryandthree-manifolds 1

Chapter2.Ellipticandhyperbolicgeometry 9

2.1.ThePoincar´ediskmodel. 10

2.2.Thesouthernhemisphere. 11

2.3.Theupperhalf-spacemodel. 12

2.4.Theprojectivemodel. 13

2.5.Thesphereofimaginaryradius. 16

2.6.Trigonometry. 17

Chapter3.Geometricstructuresonmanifolds 27

3.1.Ahyperbolicstructureontheﬁgure-eightknotcomplement. 29

3.2.Ahyperbolicmanifoldwithgeodesicboundary. 31

3.3.TheWhiteheadlinkcomplement. 32

3.4.TheBorromeanringscomplement. 33

3.5.Thedevelopingmap. 34

3.8.Horospheres. 38

3.9.Hyperbolicsurfacesobtainedfromidealtriangles. 40

3.10.Hyperbolicmanifoldsobtainedbygluingidealpolyhedra. 42

Chapter4.HyperbolicDehnsurgery 45

4.1.IdealtetrahedrainH 3 . 45

4.2.Gluingconsistencyconditions. 48

4.3.Hyperbolicstructureontheﬁgure-eightknotcomplement. 50

4.4.Thecompletionofhyperbolicthree-manifoldsobtainedfromideal

polyhedra. 54

4.5.ThegeneralizedDehnsurgeryinvariant. 56

4.6.Dehnsurgeryontheﬁgure-eightknot. 58

4.8.Degenerationofhyperbolicstructures. 61

4.10.Incompressiblesurfacesintheﬁgure-eightknotcomplement. 71

Thurston—TheGeometryandTopologyof3-Manifolds v

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