The Topology of locally volume collapsed 3-Orbifolds [Elektronische Ressource] / Daniel Faessler. Betreuer: Bernhard Leeb

Faessler, Samuel Daniel

Sujets

• Mathematics

Informations

• Publié par: ludwig-maximilians-universitat_munchen
• Publié le: 01 janvier 2011
• Nombre de lectures: 28
• Langue: Deutsch
• Poids de l'ouvrage: 1 Mo

Extrait

The Topology of locally volume
collapsed 3-Orbifolds
Daniel Faessler
Dissertation
an der Fakult¨at fur¨ Mathematik, Informatik und Statistik
der Ludwig–Maximilians–Universit¨at
Munc¨ hen
vorgelegt von
Daniel Faessler
aus Steinebach (Worthsee)¨
Munc¨ hen, den 5. April 2011Erstgutachter: Prof. Bernhard Leeb, Ph.D.
Zweitgutachter: Prof. John Lott, Ph.D.
Tag der mundlic¨ hen Prufung:¨ 30. Juni 2011Zusammenfassung
In dieser Arbeit untersuchen wir Geometrie und Topologie von Riemannschen 3-
Orbifolds, die bezuglic¨ h einer Krumm¨ ungsskala lokal volumenkollabiert sind. Unser
Hauptergebnis ist, dass eine hinreichend kollabierte geschlossene 3-Orbifold ohne
schlechte 2-Unterorbifolds Thurstons Geometrisierungsvermutung genugt.¨ Wir beweisen
auch eine Version dieses Ergebnisses mit Rand. Kleiner und Lott haben unabh¨angig und
zeitgleich ahnlic¨ he Ergebnisse bewiesen ([KL11]).
Hauptschritt unseres Beweises ist die Konstruktion einer Graphenzerlegung von hin-
reichend kollabierten (geschlossenen) 3-Orbifolds. Wir beschreiben eine grobe Stra-
tifizierung von ungef¨ahr 2-dimensionalen Alexandrov-R¨aumen, die wir dann fur¨ kolla-
bierte 3-Orbifolds zu einer Zerlegung verfeinern; diese Zerlegung kann dann zu einer
Graphenzerlegung vereinfacht werden. Wir schließen unseren Beweis ab, indem wir
zeigen, dass Graphenorbifolds ohne schlechte 2-Unterorbifolds der Geometrisierungsver-
mutung genugen.¨vi
Abstract
In this thesis we study the geometry and topology of Riemannian 3-orbifolds which are
locally volume collapsed with respect to a curvature scale. Our main result is that
a sufficiently closed 3-orbifold without bad 2-suborbifolds satisfies Thurston’s
GeometrizationConjecture. Wealsoproveaversionofthisresultwithboundary. Kleiner
and Lott indepedently and simultanously proved similar results ([KL11]).
Themainstepofourproofistoconstructagraphdecompositionofsufficientlycollapsed
(closed)3-orbifolds. Wedescribeacoarsestratificationofroughly2-dimensionalAlexan-
drov spaces which we then promote to a decomposition into suborbifolds for collapsed
3-orbifolds; this decomposition can then be reduced to a graph decomposition. We com-
plete our proof by showing that graph orbifolds without bad 2-suborbifolds satisfy the
Geometrization Conjecture.Contents
1 Introduction 1
2 Decomposition of 3-orbifolds along 2-orbifolds 7
2.1 Smooth orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Local groups and stratifications of orbifolds . . . . . . . . . . . . . 10
2.1.3 Examples: Low-dimensional orbifolds . . . . . . . . . . . . . . . . . 13
2.2 Fibrations and decompositions . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Fibered 3-orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 2-suborbifolds in 3-orbifolds . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Decompositions of along 2-suborbifolds . . . . . . . . . . 17
2.3 Thurston’s Geometrization Conjecture . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Geometric 3-orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.2 Graph orbifolds are geometrizable . . . . . . . . . . . . . . . . . . . 22
3 Geometric properties of Riemannian orbifolds 27
3.1 Geodesics and exponential map . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Spherical orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Geometry at closest cut points . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Toponogov’s Theorem for orbifolds . . . . . . . . . . . . . . . . . . . . . . 34
3.5 The orbifold Soul Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5.1 Classification of non-compact 3-orbifolds with sec≥ 0 . . . . . . . . 38
4 Convergence of thick orbifolds 41viii CONTENTS
4.1 An injectivity radius bound on thick orbifolds . . . . . . . . . . . . . . . . 41
4.2 Compactness of thick orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.1 Gromov-Hausdorff and smooth convergence of orbifolds . . . . . . . 44
4.2.2 Orbifold structures on Gromov-Hausdorff limits . . . . . . . . . . . 46
4.2.3 Riemannian center of mass . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.4 Promoting Gromov-Hausdorff convergence to smooth convergence . 50
5 Coarse stratification of roughly ≤ 2-dimensional Alexandrov spaces 57
5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1.1 Alexandrov balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1.2 Strainers and cross sections . . . . . . . . . . . . . . . . . . . . . . 58
5.1.3 Comparing comparison angles . . . . . . . . . . . . . . . . . . . . . 59
5.2 Uniform local approximation by cones . . . . . . . . . . . . . . . . . . . . 59
5.3 Islands without strainers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.4 The 1-strained region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.4.1 Local almost product structure . . . . . . . . . . . . . . . . . . . . 63
5.5 The roughly≤ 2-dimensional case . . . . . . . . . . . . . . . . . . . . . . . 65
5.5.1 Cross sections of 1-strainers . . . . . . . . . . . . . . . . . . . . . . 65
5.5.2 Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.6 Necks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6 Locally volume collapsed 3-orbifolds are graph 75
6.1 Setup and formulation of main result . . . . . . . . . . . . . . . . . . . . . 75
6.2 Conical approximation and humps . . . . . . . . . . . . . . . . . . . . . . . 79
6.3 The Shioya-Yamaguchi blow-up . . . . . . . . . . . . . . . . . . . . . . . . 80
6.3.1 General discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3.2 The case of flat conical limits with dimension≤ 2 . . . . . . . . . . 83
6.4 Strainers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.4.1 Position relative to the singular locus . . . . . . . . . . . . . . . . . 86
6.4.2 Gradient-like vector fields . . . . . . . . . . . . . . . . . . . . . . . 86
6.4.3 Local bilipschitz charts and fibrations by cross sections . . . . . . . 87CONTENTS ix
6.5 A decomposition according to the coarse stratification . . . . . . . . . . . . 88
6.5.1 The 2-strained region . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.5.2 Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.5.3 Necks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.5.4 Humps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.6 Local topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.6.1 Tube and neck cross sections . . . . . . . . . . . . . . . . . . . . . . 101
6.6.2 Humps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.6.3 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . 109
7 An extension to the case with boundary 111x CONTENTS1. Introduction
Like manifolds, orbifolds are defined as topological spaces admitting certain local models.
nHowever, whereas manifolds are locally modelled on n-dimensional Euclidean spaceR ,
norbifolds generalize this definition by admitting as local models all quotients ofR by
finite groups of diffeomorphisms.
Although I. Satake [Sa56] introduced a very similar definition in 1956 (so-called V-
manifolds), the term orbifold was first introduced in 1976/77 by W. Thurston ([Th78]).
Thurston’s main interest in orbifolds seems to have been that they can be used in study-
ing the topology and geometry of 3-manifolds; for instance, the basis of a Seifert fibered
3-manifold has a natural structure as a 2-orbifold (cf. [Sc83]). However, orbifolds also
naturally occur in other fields of geometrical studies such as knot theory (cf. [BS87]). Be-
ginning with Thurston, orbifolds have also been studied as geometrical objects sui generis.
For instance, they can be assigned a Riemannian metric and one can investigate the geo-
metric properties of the resulting length space structure (cf. [Bo92]). Orbifolds with lower
sectional curvature bounds are Alexandrov spaces.
In 1981, Thurston also extended his Geometrization Conjecture from manifolds to orb-
ifolds. The (original) Geometrization Conjecture for manifolds stated that every closed
3-manifold admitted a (unique) decomposition as follows: In a first step, the manifold was
reducedby sphericalsurgerytoirreducible(connectedsum)componentsbycuttingitopen
alongembedded2-spheresandglueing3-ballsintotheresultingboundarycomponents. The
summands would then further be cut up along incompressible tori into geometric compo-
nents, i.e. compact 3-orbifolds with toric boundary whose interior admitted a Riemannian
metric modelled on one of the eight (homogeneous) model geometries. Thurston conjec-
tured that all closed 3-orbifolds had a similar decomposition into geometric pieces unless
theycontainedbad2-suborbifolds,i.e.closed2-suborbifoldsnotgloballycoveredbyamani-
fold. Thisextendedconjecturewasmotivatedbythefollowingobservation: Foranorbifold
which is the quotient of a manifold by a group of diffeomorphisms, a geometrization of the
orbifold is equivalent to an invariant geometrization of the covering m