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A Study of Orbifolds

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A Study of Orbifolds A STUDY OF ORBIFOLDS by PETER LAWRENCE MILLEY B.Math, University of Waterloo, 1996 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1998 © Peter Lawrence Milley, 1998 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for refer• ence and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada Abstract This thesis is a study of the theory of orbifolds and their applications in low-dimensional topology and knot theory. Orbifolds are a generalization of manifolds, and provide a larger, richer context for many of the concepts of manifold theory, such as covering spaces, fibre bundles, and geometric structures. Orbifolds are intimately connected with both the theory of Seifert fibrations and with knot theory, both of which are connected to the theory and classification of three-dimensional manifolds. Orbifolds also provide a new way to visualize group actions on manifolds, specifically actions which are not free. In chapter one we motivate the discussion with the history of orbifolds, and then we define orbifolds and certain key related terms. We extend the theory of orbifolds in chapter two to encompass many of the concepts of manifold theory, such as fibre bundles, covering spaces, and orbifold geometry. We also see in chapter two the proof that every orbifold with a geometric structure is covered by a manifold, a result which does not have an analogue in manifold theory. In chapter three we study and classify compact two-dimensional orbifolds, and show how to construct hyperbolic geometric structures for a vast majority of such orbifolds. We examine the connections between orbifolds and Seifert fibrations in chapter four. We pass on to three dimensions in chapter five. In that chapter we not only study the local structure of three-dimensional orbifolds, we also study polyhedral orbifolds and examine the consequences of Andreev's theorem. We also look at ways of constructing orbifolds from Dehn surgery diagrams in chapter five. Finally in chapter six we discuss more advanced topics, such as the state of the Geometrization Theorem for orbifolds, as well as orbifold differential geometry and orbifold topological invariants including extensions of the fundamental group and the homology groups. Table of Contents Abstract ii Table of Contents iii List of Tables v List of Figures vi Chapter 1. Introduction 1 1.1 History and Motivation 1 1.2 Basic Definitions 3 1.3 Other Terminology 5 Chapter 2. Background Material 7 2.1 Analytic and Geometric Orbifolds 7 2.2 Orbifold Maps 8 2.3 Covering Orbifolds 9 2.4 Suborbifolds 11 2.5 Orbifold Fibre Bundles 13 2.6 Fibre products 15 2.7 Orientability 16 2.8 The Developing Map 17 Chapter 3. Two-dimensional Orbifolds 21 3.1 Local Structure of Two-orbifolds 21 3.2 Global Description of Two-orbifolds 24 3.3 Geometric Classification 25 3.4 Construction of Hyperbolic Two-orbifolds 32 Chapter 4. Seifert Fibre Spaces 38 4.1 Definitions 38 4.2 Circle Bundles over Orbifolds 39 4.3 Three-manifolds as TS{Q) 41 4.4 Three-orbifolds as TS{Q) 45 4.5 Tiling patterns 48 Chapter 5. Three-dimensional Orbifolds 52 5.1 Local Structure of Three-orbifolds 52 5.2 Examples of Compact Three-orbifolds 55 5.3 Polyhedral Orbifolds 56 5.4 Andreev's Theorem 58 5.5 Orbifolds and Surgery 62 iii Table of Contents Chapter 6. Invariants and Other Topics 67 6.1 Quotients of Three-manifolds 67 6.2 Differential Geometry on Orbifolds 69 6.3 The Orbifold Fundamental Group 71 6.4 Orbifold Homology 73 6.5 Other Orbifold Invariants 75 Chapter 7. Conclusion 78 Bibliography ^9 iv List of Tables 3.1 All two-orbifolds Q with x(Q) > 0 List of Figures 2.1 Example of a good orbifold 10 j 2.2 An annulus with mirrored boundary, and a subset which is not an orbifold 16 3.1 Local structure near a cone point 22 3.2 Local structure near a silvered boundary 23 3.3 Local structure near a corner point 23 3.4 The orbifold Q = (*22): x(Q) = h 26 3.5 A two-fold cover p : S2 -> (*) 29 3.6 The covering map (°|) -> (2222) 30 3.7 A right-angled quadrilateral in the hyperbolic plane 34 3.8 A right-angled pentagon in the hyperbolic plane 35 4.1 Singular fibres of the bundle TS((*)) -> (*) 48 4.2 Pattern position corresponding to the point (x, y, 0) 51 5.1 Suborbifolds of (*2222) which bound two-disk quotients (left) and which have unbounded lifts (right) 59 5.2 The suborbifold P = (*2222), containing the boundary of a two-disk quotient 61 6.1 A typical surgery link 69 vi Chapter 1 Introduction 1.1 History and Motivation The classification of manifolds, particularly three-manifolds, is arguably the most important topic in low-dimensional topology. In particular one of the greatest challenges facing topolo- gists is to determine which three-manifolds admit a geometric structure. To put the question another way, which three-manifolds are the quotients of 53, the Euclidean plane E3, or the hyperbolic plane i?3, by a discrete group of isometries acting freely, effectively, and properly- discontinuously? The most ambitious claim in this area is Thurston's Conjecture, which states that while not all three-manifolds have a geometric structure, any three-manifold can be split up into geometric pieces. Topologists have made progress on Thurston's Conjecture in the past twenty years by increasing the problem's scope. Instead of considering only freely acting groups of isometries, we can study the quotients of S3, -E3, and H3 by any discrete group of isometries which act effectively and properly-discontinously. The resulting topological objects are a generalization of manifolds, now usually referred to as orbifolds. The idea of studying the action of groups which act effectively and properly-discontinously but not freely has existed since at least the time of Poincare. But it wasn't until 1956 that a formal definition of "V-manifolds" was provided by Ichiro Satake in a pair of papers [Sat56, Sat57]. Satake was studying the action of Siegel's modular group Mn on a space of symmetric complex matrices, and treated the quotient of this action as a V-manifold. In his papers Satake extended 1 Chapter 1. Introduction much of differential geometry to the V-manifold setting, including a generalization of the Gauss- Bonnet theorem. By using V-manifolds Satake obtained results about the isotropy subgroups of Mn [Sat57]. In 1976 and 1977 in a series of lectures at Princeton, William Thurston denned "orbifolds" without awareness of Satake's work. I will use terminology borrowed from Thurston's lecture notes in this paper, as most modern literature on orbifolds is based on Thurston's definitions [Thu90]. Thurston studied orbifolds as part of his work in classifying three-manifolds. In particular, he found in the study of spherical and Euclidean three-manifolds that many free group actions on three-dimensional manifolds can be broken down into actions on one or two-dimensional manifolds, but that these component actions are often not free. Hence studying orbifolds in one and two dimensions can provide information about manifolds in three dimensions. Seifert fibrations provide a good example of this phenomenon. An 51-bundle over a two- dimensional manifold has much more structure than a three-manifold foliated by circles, but most Seifert fibrations cannot be viewed as a fibre bundle over a surface. Instead, by using orbifolds in the base instead of manifolds any Seifert fibration can be given a bundle structure. Mathematicians such as Peter Scott have made progress at classifying the total spaces of Seifert fibrations by viewing them as bundles over orbifolds. This is especially significant since most non-hyperbolic manifolds which admit some geometric structure also possess a Seifert fibration of some sort [Sco83]. Topologists have found uses for three-dimensional orbifolds as well as two-dimensional ones. Orbifolds are an excellent model for the quotient space of a regular branched covering. Such coverings occur frequently in knot theory; for example, we know that any three-manifold forms a branched cover of the three-sphere where the branching set downstairs is a knot or link; a proof of this appears in Rolfsen's book [Rol90]. By examining such branched coverings of the three-sphere and the corresponding orbifolds, Collin has extended aspects of Gauge theory to orbifolds and come up with new invariants for knots [Col97]. Orbifolds also play a role in 2 Chapter 1. Introduction studying the symmetries of knots. Finally, orbifolds provide a number of simple, visual examples of group actions on manifolds. Many more groups act properly-discontinuously on manifolds than act freely, and orbifolds provide another perspective on the effects of such actions. 1.2 Basic Definitions Thurston's definition of orbifolds is extremely detailed. I will use a simpler definition based on those by Montesinos [Mon87], Collin [Col97], and others. Before defining orbifolds, however, we need the concept of a folding map: Definition 1.1 A continuous map (f> : U —>• U, where U is an open subset of W1, is called a folding map if there exists a discrete group V of automorphisms of U acting effectively and properly-discontinuously such that 4> ° 1 — <t> for all ^ £ T and such that the induced map Uyr —> U is a homeomorphism.
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