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 —
Uyr —> U is a homeomorphism. The group F is called a folding group of To put it another way, With folding maps under our belt we can define orbifolds: Definition 1.2 An orbifold Q of dimension n consists of a topological space XQ together with a collection of maps • The sets {UA} form an open cover of XQ. 1 • For each a, Ua is a connected open subset of W . • For each a, 4>a is a folding map. 3 Chapter 1. Introduction x • If4>a{xo) = 4>p{ p) then there exists open neighbourhoods Va, Vp of xa, xp in Ua, Up and a homeomorphism ipap : Va —> Vp such that A collection of maps (fia Ua —^ Ua satisfying the above conditions is called an atlas for the orbifold Q, and the space XQ is called the underlying space of the orbifold. Note that the definition does not require XQ to be a manifold. Two atlases on the same underlying space are considered equivalent if their union is also an atlas. A single map (pi from an atlas is called a chart; I will often specify a chart by referring to Ua, Ua, and the associated group Ta of automorphisms, without mentioning the map cj)a specifically. While technically an orbifold Q is not itself a set, on occasion we will abuse our notation and write x G Q instead of the more correct x £ XQ. Given a point x £ Q, a particular chart Ua containing x, and a particular preimage xa £ Ua, the stabilizer of xa is the subgroup of Ta which fixes xa. As our choice of xa G Ua varies, the stabilizer of xa varies only up to conjugation by elements of Ta. And as our choice of Ua containing x varies, the stabilizer varies only up to conjugation by a transition map 4>ap. Hence we can define the isotropy group Tx of x to be the stabilizer of xa in TQ for some xa G Ua which maps to x; the isotropy group of x is well-defined up to conjugation by transition maps and elements of ra. Occasionally I will also refer to the stabilizer of xa in Ta as the isotropy group of xa, and write it Txa; the meaning of the term should always be clear from the context. The set of points x G Q with non-trivial isotropy groups is called the singular set of the orbifold, and is denoted Eg. Example: We can give the non-negative real numbers the structure of a one-dimensional orbifold by supplying an appropriate atlas. In this case the atlas consists of a single chart (f) : R -> {x > 0} where (f)(x) — \x\. The group T is the cyclic group of order two generated by the map x »-» — x. The singular set SQ consists of the single point 0; To = T, while for all other x G Q the isotropy group Tx is trivial. 4 Chapter 1. Introduction Example: Any manifold M is also an orbifold, with Xu = M and S^f = 0. All the isotropy groups are trivial, and all of the charts are homeomorphisms. If T is a discrete group acting properly-discontinously on M, then M/T inherits an orbifold structure from M. Simply let {Ui} be a regular cover of M/T by sufficiently small open subsets, then for each i let Ui be homeomorphic to a single component of the pre-image of Ui in M. The singular set of the resulting orbifold will be the set of points on which T does not act freely, and the isotropy groups will all be subgroups of T. Example: Let K be a tame knot embedded in a three-manifold M, and let n > 2. We wish to find an atlas for a three-dimensional orbifold Q such that XQ = M, SQ = K, and such that the isotropy group of any point x 6 K will be cyclic of order n. To do this, let {U} be a collection of open balls covering S3 such that Ui D K is either empty or consists of a single unknotted curve li. If Ui D K = 0, define Ui to be equal to Ui and define I\ to be trivial. If Ui fl K ^ 0, let Ui = M3 and let Tj be generated by the map (x,y,z) i-> (xcos(2-7r/n), y sin(27r/n), z). Then we can realize fa as the composition Ui 4 [/j/r 4 Ui where the last step takes the image of the z-axis to the line li. A more direct way to construct the above orbifold is to let M be the n-fold cyclic branched cover of M over the knot K, and let T be the group of covering transformations. Then the orbifold Q is given by the quotient M/T. In the special case where M = S3, the orbifold Q is called a knot orbifold; I will discuss these orbifolds more in later chapters. 1.3 Other Terminology Orbifolds as defined above will sometimes be referred to as orbifolds without boundary. To define an orbifold with boundary, replace W1 in the definition with the upper half-space: n {(zi,... ,xn)ER \xi >0} Note that Q as an orbifold can be without boundary while XQ as a topological space has non• empty boundary. Such is the case in the example with non-negative real numbers above. More 5 Chapter 1. Introduction generally, if M is a manifold with boundary let M' = (Mi TJ M2)/ ~ where Mi and M2 are copies of M and ~ identifies the boundaries of Mi and M2. Then M' is a manifold without boundary. If V is the group of automorphisms of M' containing the identity and the map which exchanges Mi and M2, then Q = M'/T is an orbifold without boundary where XQ = M and EQ is the boundary of M. Orbifolds of this type appear frequently; such an orbifold is sometimes called "M with a mirrored boundary". I will discuss examples of this type frequently in chapters three and five. An orbifold Q will be called compact if XQ is a compact topological space. Occasionally orbifolds without boundary will be called closed or open if they are compact or non-compact respectively, just as with manifolds. 6 Chapter 2 Background Material In this chapter I will form a more comprehensive foundation for the theory of orbifolds. In particular, I will define the notions of analytic and geometric orbifolds, which each expand upon the basic definition of orbifolds by adding additional structure. I will also define suborbifolds, covering maps between orbifolds, good and bad orbifolds, and fibre bundles over orbifolds. 2.1 Analytic and Geometric Orbifolds We can narrow the definition of orbifolds in chapter one in several ways. For example, the definition includes nothing about the nature of the underlying space XQ, but in practice we will always assume that XQ is a Hausdorff space. We will also assume in general that the singular set Eg is "small" in some topological sense; ideally, Eg should be a closed, nowhere dense subset where each component has a well-defined co-dimension. Thurston discusses wild orbifolds were Eg is not well-behaved [Thu90], but I will concentrate on analytic orbifolds. A analytic orbifold is an orbifold with an atlas where the transition maps i/}ap and the group actions ra are all local C^-difFeomorphisms of R". I will show in chapter three that this is sufficient to guarantee that the singular set of a analytic orbifold will be well-behaved in the above sense. All of the orbifold examples presented so far have a natural analytic structure. We can restrict our attention still further to geometric orbifolds. If in our definition of orbifolds 1 we replace W with some Riemannian n-manifold X, and require that the maps ipap and the group actions TA be local restrictions of isometries of X, then the resulting orbifolds will be geometric orbifolds with model space X. For example, letting X be the n-sphere, Euclidean 7 Chapter 2. Background Material n-space or hyperbolic n-space results in spherical, Euclidean, or hyperbolic orbifolds respectively. Consider the knot orbifold Q where SQ is the trefoil knot in S3 and where the isotropy group of any singular point is cyclic of order two. The two-fold cyclic branched cover of S3 over the trefoil knot is the lens space L(3,1) [Rol90]. Since this is a spherical manifold, Q inherits a spherical orbifold structure. For a general knot K in S3 and branching number n, we cannot be sure if the corresponding knot orbifold has a geometric structure. We need to know more than that the corresponding branched cover of S3 is a geometric manifold; the covering transformations must be isometries as well. Thurston covered this question in his Geometrization Conjecture for orbifolds, which we will return to in chapter six. 2.2 Orbifold Maps Given that orbifolds are topological objects, we naturally wish to define what is meant by a map between two orbifolds. Clearly the collection of maps between orbifolds should be a subcollection of the maps between their underlying spaces. The trick is deciding which maps should be admissable and which should not. The following definition is taken from Satake, although I have modified the notation to be consistent with my own [Sat57]. Definition 2.1 Let Q\ and Q2 be two orbifolds. A Cw orbifold map h from Qi into Q2 is a collection of maps {ha} with the following properties: • For any chart Ua = Ua/Ta in Qi there is a corresponding chart U'a = U'a/Ya in Q2 and w a C -map ha from Ua into U'a. • Let Ua, Up be charts in Q\ corresponding to U'a, Up in the sense of the previous condition, and suppose UaC\Up ^ 0. Then for any intersection map ipap between Ua and Up there is an intersection map ip'ap between U'a and Up such that ip'ap oha = hpo ipap 8 Chapter 2. Background Material It follows from the above two conditions that there is a unique C^-map h between XQX and XQ2 such that for any chart Ua in Q\ and corresponding U'a in Q2, we have 4>'a 0 = ^ 0 4>a where 4>a and 4>'a are the projection maps from Ua, U'a onto Ua, U'a respectively [Sat57]. Using the above definition, we can define a Cu function on an orbifold to be a Cw map between the orbifold and R. Smooth functions are essential to the definitions of differential geometry on orbifolds, which I will discuss in chapter six, but for most purposes we can make do with only certain kinds of maps between orbifolds. These special maps have much simpler definitions. That is the topic of the next few sections. 2.3 Covering Orbifolds One specific kind of map between orbifolds, which we will make much use of, is a covering map. The following definition is from Thurston's lecture notes [Thu90]. Definition 2.2 Let Q and P be two orbifolds, and letp : XQ —> Xp be a surjective map between their underlying spaces. We say that p is an orbifold covering map if it satisifies the following conditions: • Each point x G XQ lies in a chart U = U/Y (where U is an open subset ofW1) such that p restricted to U is isomorphic to the quotient map U/Y —> U/V (T < T'). • Each point y G Xp lies in a chart V = V/F for which each component Ui of p~l(V) is isomorphic to V/Yi, where Yi < T. A projection which satisfies the first condition is called a local covering, while a projection which satisfies the second condition is called an even covering. Orbifold coverings do not have to be coverings of the underlying spaces. In general, if M is any manifold, T is a discrete group acting properly-discontinously on M, and Y' < T, then the orbifold M/Y' is an orbifold cover of M/Y. In particular, M is an orbifold cover of M/Y. 9 Chapter 2. Background Material Clearly a wide array of orbifolds of the form M/T can be constructed. One can ask whether all orbifolds, or at least all analytic orbifolds, can be constructed this way. The answer is no. An orbifold which is not covered by any manifold is called bad, while one which is covered by a manifold is called good. Consider the orbifold Q = S2/T, where T is generated by a rotation of angle ir. This is a good orbifold and has a singular set consisting of two points which are the images of the fixed points of the rotation; see figure 2.1. If one constructs a new orbifold S2 Q Figure 2.1: Example of a good orbifold. Q' from Q by turning one of the two points of SQ into a non-singular point (by modifying the atlas near that point), the resulting orbifold will be bad. In the next chapter I will present a proof by Thurston that this and similar orbifolds are bad. However a rough argument to this effect goes as follows. Let U be a small closed neighbourhood of the singular point in XQ, and let V be the closure of XQ — U. If M is a manifold and p : M ->• Q a covering, then p~x{U) must be a collection of open disks, each covering U in a two-to-one manner. Moreover V is a manifold and simply connected, so p_1(V) must be a collection of copies of V. Let u> be the boundary of one of the components of p~x(V). Then w is a closed curve lying in the boundary of p~l{U); hence u is the boundary of one of the components of p~l(U). This is impossible; it requires p to map u to a curve in XQ in both a two-to-one and a one-to-one manner at the same time. 10 Chapter 2. Background Material 2.4 Suborbifolds Let m and n be two non-negative integers, with m 4>\% -.Yu^Ynu is both a homeomorphism and a chart in Y. The collection of all such charts forms an Tri• dimensional orbifold atlas for Y which is equivalent to its standard manifold atlas. More generally, if Y is a subset of XQ, then for each chart 4>: U ->• U in Q let Yu be a connected component of 4>~1(YnU). The choice of component is arbitrary since any two such components are homeomorphic via the action of some element of T, where U = U/T. Definition 2.3 Suppose the collection of maps of the form Consequently if Y is a suborbifold then Y\j must be homeomorphic to an open subset of Rm for all U. If Q is an analytic (geometric) orbifold, then an analytic suborbifold (geometric suborbifold) is a suborbifold with the additional property that each Yu is Cw-diffeomorphic (isometric) to an open subset of Rn (the model space). Example: Any open subset of XQ defines a corresponding open suborbifold of Q, of the same dimension as Q. 11 Chapter 2. Background Material Example: Consider the orbifold Q where XQ = 1R2, SQ is the set containing only the origin O, and To is a cyclic group of rotations of order two. Specifically, Q as an orbifold is the quotient of the Euclidean plane by a rotation of IT radians about the origin. Then the set of points Y = {(x,y)£XQ\x>0,y = 0} is a suborbifold of Q of dimension one. Note that the plane is a two-fold orbifold cover of Q, and we can pick the covering map in such a way that the pre-image of Y will be the x-axis in the plane. In general, if JV is a T-invariant submanifold of M then N/T will be a suborbifold of M/T. If we make Q the quotient of a three-fold rotation instead of a two-fold one, so To is cyclic of order three, then Y is no longer a suborbifold. For if 4>~1(Y fl U) has only one connected component and is not homeomorphic to an open subset of R. Thus the map Example: Let C be a cube embedded in Euclidean three-space, and let T be the group of isometries generated by reflections in each of the faces of C. Then Q = M3 /T is a geometric three-orbifold, with XQ = C and EQ = dC, where dC is the boundary of C. If / is a straight line which intersects C, then IC\C forms a one-dimensional suborbifold of Q only in the following cases: • The endpoints of / D C lie in the interiors of faces of C. • I DC lies entirely in a face of C, and the endpoints lie in the interiors of edges of that face. • l(~\C coincides with an edge of C. For any other position of I, one of the endpoints of / fl C must lie in an edge of C. Furthermore, if 4> : U —> U is a chart containing that endpoint then (/>-1(Z fl C D U) is not homeomorphic to an open subset of E and hence / D C is not a suborbifold of Q. 12 Chapter 2. Background Material 2.5 Orbifold Fibre Bundles Given two orbifolds Q and P, the product orbifold Q x P is defined in the obvious manner: XQXP — XQ x Xp, and the atlas for Q x P is the collection of maps of the form 0x9 :U xV xV where Example: Let Q be the cubical orbifold defined in the previous section, and let L\, L2 and L3 be the suborbifolds defined by three edges of the cube, no two of which are parallel. Then Q = Li x L2 x L3. We can also define more general fibre bundles over orbifolds. The resulting definition is simpler if we assume that the fibres of such bundles are not just orbifolds but manifolds. This restriction is a common one in the literature but the reasons for it, if any, are rarely explained. Nevertheless, it does simplify the definition that follows, and none of the examples that we will see in this thesis require orbifold fibres. Definition 2.4 An orbifold fibre bundle with total space the orbifold E, base space the orbifold B, and fibre the manifold F, is a continuous map p:XE^XB with the following property: for each x E XQ and for each chart l(u,f) = (7«,Su(7)/) where gu is a homomorphism from Y to the group of automorphisms of F and gu varies con• tinuously with respect to u. 13 Chapter 2. Background Material Montesinos provides a more general definition of fibre bundles in his book [Mon87]. Thurston also defines orbifold fibre bundles, in his lecture notes [Thu90]. The above definition is equivalent to Thurston's. In the examples below, $ is the quotient map U x F —> p~1(U). Example: The orbifold B xF is an orbifold fibre bundle wherep~l(U) = UxF and <& = Example: Any orbifold covering map E —> B is an orbifold fibre bundle where the fibre is a discrete set of points K. For any U C XB, the set U x K is a disjoint collection of copies of U, and the homomorphism gu is a map from T into the permutation group of K. Consider the map p : C —> C where p(z) = zn. We can think of this as an orbifold fibre bundle with total space C and base Q where XQ = C and SQ = {0}. Note that in an orbifold fibre bundle if a; is a regular point in the base then p~1(x) = F, but if a; is singular then p'l{x) will be an orbifold quotient of F. So in the case of ZH> zn, the pre-image of a non-zero point is n distinct points, coinciding with the fibre, but the pre-image of the origin is only a single point. Example: The tangent bundle of an analytic n-orbifold Q, denoted TQ, is defined as follows. Let { l TVa to be (Ua x W)/Ta, where the action of Ta is given by: l(ua,va) = (jua, (d-yUa)va) Let [itQ.,wa] denote equivalence classes under this action. Define an equivalence on ]JaTUa as follows: if (pa(ua) = [ua,va] ~ [up,(d(ipap)Ua)va] 1 for all va EW . Then XTQ = (]laTUa)/ ~, and an atlas for TQ is given by the collection of all maps of the form a 14 Chapter 2. Background Material The projection map p : XTQ ->• XQ defined by p($a(ua,va)) = 4>a{ua) is well-defined and defines an orbifold fibre bundle with total space TQ, base Q and fibre W1. An interesting consequence of the definition of tangent bundles is that if Q = M/T where M is a manifold then TQ = TM/T, where TM is the usual tangent bundle and V acts on TM by n/(x,v) = {jx,(dj)xv) Using a similar construction, we can define the spherical tangent bundle of an analytic n- orbifold, denoted TS(Q). For a good orbifold Q = M/T, TS(Q) = TS{M)/Y where the action of T on TS(M) is defined in the obvious way. Spherical tangent bundles over two-dimensional orbifolds form an interesting model for Siefert fibre spaces, which I will discuss in Chapter 4. 2.6 Fibre products Let p : E —> B and p' : E' —> B be two orbifold fibre bundles with the same base B and with fibres F and F' respectively. We wish to define the fibre product of E and E' with respect to B. For manifolds, the fibre product is very simple: the fibre product of E and E' with respect to B is the set of all points (x, x') £ E x E' with the property that p(x) = p'(x'). But this definition does not extend well to orbifolds. Consider the following example from Thurston [Thu90]. Let E be the unit circle in R2, and let E' and B be the interval from (—1,0) to (1,0) with order-two isotropy groups at the endpoints in each case. Define the map p to be (x, y) i->- (x, 0), and define p' to be the map (x, 0) H-> (2|X| — 1,0). Both p and p' are two-fold orbifold covering maps, and fibre bundle maps over B with fibre S°. We can construct the orbifold E x E' easily enough; it is a cylinder with order-two isotropy at the boundary. We can also find the subset of XEXE' with the property that p(x) = p'(x'). But that subset turns out not to be a suborbifold of E x E'; it is a figure-eight shape, and there is no orbifold structure at the vertex in the centre (see figure 2.2). The way around this problem is to provide a more sophisticated definition of fibre products, 15 Chapter 2. Background Material Figure 2.2: An annulus with mirrored boundary, and a subset which is not an orbifold. one which breaks up such singularities into distinct branches. For each chart U = U/T of B, we have p~l(U) 2£ (U x F)/T and = (U x F')/T, where T acts on F and F' via homomorphisms and g'u respectively, as in the definition of orbifold fibre bundles. Then define P'HU) xv p'~\U) = (U xFx F')/T Where the action of T is given by , l(uj,f') = (iu,gu(1)f,g u(1)f) If Ui and Uj are charts in B with non-empty intersection, then the bundle E defines an equiva• lence between certain points (v,i, fi)Ti 6 (Ui x F)/Ti and points (uj,fj)Tj € (Uj x F)/Fj, where Uj = ipij(ui). A similar equivalence is defined by E'. These two equivalences can be combined: fu f'iWi ~ W>ij(ui), fj, f'j)Vj l l and the result of taking all such equivalences on the set WiP~(Ui) Xu{ p'~(Ui) is an orbifold, which we will denote E Xg E'. Definition 2.5 The orbifold E Xg E' described above is the orbifold fibre product of E and E' with respect to B. 2.7 Orientability One interesting application of fibre products is in defining the frame bundle of an orbifold. If Q is an n-dimesional orbifold, then the frame bundle can be defined as a quotient space of the fibre product of n copies of the tangent bundle TQ, just as can be done for manifolds. This in 16 Chapter 2. Background Material turn can lead to a formal definition of orientability for orbifolds. However, orientability can be defined much more easily: Definition 2.6 An analytic orbifold Q is orientable if and only if • The manifold XQ — SQ is orientable, and • For all x 6 Eg, the isotropy group Fx consists of orientation-preserving diffeomorphisms. Example: Let Q be the previously-defined orbifold whose underlying space is a cube. At points on the boundary of the cube, the isotropy groups are generated by reflections. Hence, this orbifold is non-orientable. 2.8 The Developing Map To close this chapter, I present the following theorem and proof from Thurston [Thu90] which utilizes most of the concepts seen so far. Theorem 2.1 If Q is a geometric manifold with model space X, then Q is good. Proof: We wish to find a manifold which covers Q. We must first find a manifold which is the total space of an orbifold fibre bundle over Q. This manifold will turn out not to be a covering space itself, being of a higher dimension than Q in general, but it will have submanifolds which do cover Q under the restriction of the bundle projection map. Let G be the group of isometries of X, and let {UA = UA/TA} be an atlas for Q, with tran• sition maps ipa/3- Without loss of generality let us assume that the intersection of any finite number of charts UA is connected and simply connected. The quotient of an open subset of a Riemannian manifold by a discrete group of isometries will be a Hausdorff topological space, so this assumption is reasonable. Since Q is a geometric manifold, the transition maps are local restrictions of elements of G, as are the actions of elements of the various folding groups TQ. 17 Chapter 2. Background Material As G is a group of isometries of a Riemannian manifold, G itself has a manifold structure. In the important cases, where X is one of Sn, En, or Hn, G will in fact be a Lie group, but even in the general case G can be given the compact open topology. Hence we can talk about orbifold fibre bundles with base Q and fibre G. Construct one such fibre bundle as follows: for each a, define G(Ua) = (Ua x G)/Fa, where the action of Fa is given by j(u,g) = (•yu,jg). The map pa : Ua x G —> Ua defined by pa(u,g) — 4>a{u) is invariant with respect to the action of Fa, and the induced map pa : G(Ua) —> Ua defines an orbifold fibre bundle over Ua with fibre G and total space G(Ua)- Note that G(Ua) is in fact a manifold, because the action of Fa on Ua x G is a free action. x Suppose x € UaC\Up, and suppose xa, xp are points in Ua, Ua respectively, such that (pa( a) = x x 4>p{ p) — - Then there exist neighbourhoods Va, Vp of xa, xp respectively, and an isometry ipap taking Va to Vp such that 4>p o tpap = 4>a in Va. Consider the equivalence ~a/g between G(Ua) and G(Up) defined by rQ(«,ff) ~Q/3 Fp(lpap{u),1pap og) for all u e Va, g € G. This is a well-defined equivalence, because if u'a = •yaua and u'p — -ypup l then the corresponding transition map ip'ap is just jp o i/jap oj~. Furthermore the equivalence identifies open subsets of the form (Va x G)/Fa, and the quotient of G(Ua) TJ G(Up) by all such equivalences will still be a manifold; in fact it will be an orbifold fibre bundle over Ua U Up with fibre G. By our assumption that the Ua's formed a regular cover, this process can be extended in a well-defined way to include all of the charts in the atlas. The result will be a manifold, which is the quotient space of TJa G(Ua) under the equivalence defined by all of the transition maps together. Call this manifold G(Q). The maps pa are well-behaved with respect to the equivalences between the G(Uays; if xa ~a/g xp, where xa G G(Ua) and xp € G(Up), then x Pa(xa) = Pp( is)- Hence there is a well-defined map p : G(Q) —>• Q which agrees with all of the pQ's. This map determines an orbifold fibre bundle with base Q, fibre G, and total space G(Q). 18 Chapter 2. Background Material Since G(Q) is a manifold, we're almost done; we just have to find a submanifold of G(Q) which covers Q under the restriction of the map p. But it turns out that G(Q) can be decomposed into a pairwise disjoint collection of such manifolds. To do this, consider a stricter topology on Ua x G with neighbourhoods of the form Va x {g} for all g € G and Va open in Ua. In other words, two points can only be in the same neighbourhood if their G-coordinates are identical. The connected components of Ua x G under this topology are all open manifolds homeomorphic to Ua. Futhermore, this topology determines a topology on the space G(Q). Let E be any component of G(Q) under this topology. E is a local covering of Q, since it is everywhere locally homeomorphic to Ua for some a, and the projection p is locally equivalent to 4>a. Furthermore the pre-image of Ua in E is the quotient of a disjoint collection of sets of the form Ua x {g} by the action of Ta. This action identifies the sets homeomorphically, so the quotient will still be a disjoint collection of sets homeomorphic to Ua. Hence E is also an even covering of Q. Thus each such component of G(Q) is a manifold which is also a covering space of Q. Hence Q is a good orbifold. As a bonus, this construction also provides all possible developing maps of Q. As with manifolds, a developing map on a good orbifold Q with model space X is a map from the universal cover of Q to X which is also a local homeomorphism. Note that for good orbifolds, the definition of the universal cover is obvious; we will define the universal cover of a bad orbifold in chapter l six. For now, consider the map Da : Ua x G —» X defined by Da(u,g) — g~(u). This map is invariant with respect to the action of TQ, and two such maps Da and Dp are compatible with respect to all possible equivalences ~Qjg. Hence there is a well-defined map D : G(Q) —> X which agrees with all of the Da's. All developing maps on Q can be obtained by restricting D to one of the components E of G(Q), and then lifting to the universal cover of E if necessary. Thurston provides a more general version of this proof in his notes, which applies to a more general class of orbifolds than the geometric ones [Thu90]. He does this by defining G(Q) to be the set of all germs of local covering maps from X to Q. A germ in this context consists 19 Chapter 2. Background Material of a source point x, a target point q € XQ, and a map denned in a neighbourhood of x which takes x to q. Two germs are equivalent if they agree on some neighbourhood of x. In the case of geometric orbifolds the two definitions of G(Q) are equivalent, although it takes some work l to show this; a point (u,g) € UaxG corresponds to the germ with source g~(u), target 4>a{u), and with 4>a o g as its map. The projection p corresponds to projection from G(Q) onto the target of each germ, while the map D corresponds to projection onto the source of each germ. 20 Chapter 3 Two-dimensional Orbifolds The special case of compact two-orbifolds without boundary is a good starting point for the study of orbifolds in general. Not only are these two-dimensional cases easy to visualize but their classification is well understood, just as in the case of compact boundaryless two-manifolds. Furthermore the underlying topological spaces of two-orbifolds are always manifolds, a property that higher-dimensional orbifolds don't share. In this chapter I will first detail the possible local structures of two-orbifolds, followed by the global structure. The remainder of the chapter will classify two-orbifolds with respect to their geometric structure, and construct a wide range of hyperbolic two-orbifolds. Thurston was the first to do much of the work that follows [Thu90], although it was repeated later by Montesinos [Mon87], Scott [Sco83], and others. For the rest of this section, I will consider orbifolds to be compact and without boundary unless stated otherwise. 3.1 Local Structure of Two-orbifolds Let U —> U be a local chart in a two-orbifold Q, such that U = U/F. For any x eU, let Ti < T be the corresponding isotropy group. Each element of Fx will induce a corresponding linear transformation of the tangent space Tx. These transformations will form a group G. Moreover, by choosing an arbitrary Riemannian metric for U and taking the average of the metric with respect to the action of F, we can find a Riemannian metric for U which is F- invariant. Under this metric, the elements of G will be isometries of Tx. Hence the exponential 21 Chapter 3. Two-dimensional Orbifolds map exp :TX^U which is a local homeomorphism in the neighbourhood of x, will define a local map between U and E2/G, where G is a subgroup of 0(2). This map is a local homeomorphism, both topologically and as an orbifold map, in the neighbourhood of the image x of x in U. Hence the possible local structures of a two-orbifold are in 1-1 correspondence with the finite subgroups G of 0(2). If G is trivial, then x does not lie in SQ, and the map U ->• U is a homeomorphism near x. But there are three non-trivial cases: 1. G is a cyclic group generated by rotations. 2. G is the group generated by reflection in a single line in E2. 3. G is a dihedral group, generated by reflection in a line and by a rotation. In the first case, G is generated by a rotation of order n > 2. The elements of G fix a single point, namely the origin, and E2/G is topologically a plane. Then U/T is topologically an open disk and the singular locus of U is the point x. Singular points of this type are referred to as cone points of order n, and are indicated diagrammatically by highlighting the point x and labelling it with its order. U U © Figure 3.1: Local structure near a cone point. In the second case G is generated by reflection in a line I through the origin of E2. Then the quotient E2 jG is topologically a half-plane, with the line I as boundary; U/T is homeomorphic to half an open disk. The boundary of the half-disk forms the singular locus of U. Such a 22 Chapter 3. Two-dimensional Orbifolds singular locus is called a mirrored or silvered boundary, to emphasize that while the underlying topological space has a boundary in the neighbourhood U, the orbifold is considered to be without boundary near U (since U is without boundary). To a creature living in the orbifold, the space near a silvered boundary would not appear to have an edge, but instead would appear to have mirror symmetry about the image of the line I. In diagrams, I will follow the convention of Bonahon and Siebenmann [BS82], and indicate a silvered boundary with a double line. U U Figure 3.2: Local structure near a silvered boundary. The final case combines the first two. If G contains both a reflection in a line I through the origin and a rotation about the origin of order n, then G is a representation of the dihedral group Dn in 0(2); every element of G fixes the origin, while I and its images under rotation by multiples of 2-n/n are each fixed by a reflection in G. Topologically, E2/G is a sector of a plane, with angle at the origin. Topologically U/T is unchanged from the previous case, but this time the singular locus of U contains a distinguished point, namely x. This point is referred to as a corner point of order n, or alternatively as a corner reflector. The rest of the singular locus of U consists of two silvered edges on either side of x. In this paper I will distinguish corner points from cone points in diagrams by drawing a line above the index of a corner point. U U Figure 3.3: Local structure near a corner point. 23 Chapter 3. Two-dimensional Orbifolds In all three cases, the neighbourhood of a; is homeomorphic to either an open disk or a half-plane. Hence XQ is either a manifold or a manifold with boundary when Q is a two-orbifold. 3.2 Global Description of Two-orbifolds. The above results allow us to describe all possible compact analytic two-orbifolds without boundary. A two-orbifold is completely determined by its underlying space XQ, the orders of the cone points in the interior of XQ, the orders of the corner points in the boundary of XQ, and the order in which these corner points appear in each component of the boundary. For example, we might talk about a sphere with two order-three cone points, or a Moebius band with an order-two corner point on the boundary. I assume that the rest of the boundary is silvered unless stated otherwise. And by the compactness of XQ, the number of corner points and cone points of Q must be finite. Montesinos [Mon87] describes a natural notation for simple two-orbifolds based on these facts. He denotes each two-orbifold by a letter followed by several numbers, which may or may not have bars over them (e.g. 533 or M3). The letter describes the underlying space: S for sphere, D for disk, T for torus, A for annulus, M for Moebius band, K for Klein bottle, and P for projective plane. An unbarred number denotes a cone point, while a barred number denotes a corner point. Thus D32 would be a disk with an order-two cone point in the interior and an order-three corner point on the boundary. This notation is clearly limited; it makes no provisions for more complicated underlying spaces, particulary ones with multiple boundary components, nor does it describe the order of corner points along the boundary. Despite these limitations, Montesinos's notation is sufficient to describe all two-orbifolds of elliptic or Euclidean type (see the next section). If we want a notation suitable for all two-orbifolds, we have to turn to a notation devised by John Conway and described in Thurston's notes [Thu90]. In Conway's notation, each two-orbifold is described by a sequence of symbols in parentheses; for example, (32° |00*3*). The first set of numbers describes the cone points; in this example, 24 Chapter 3. Two-dimensional Orbifolds there are cone points of orders 3 and 2. If an orbifold contains cone points of orders higher than 9, they can be enclosed in additional parentheses, for example (3(10)...). In practice this is rarely necessary. The asterisks and numbers at the end denote corner points: each asterisk corresponds to a component of the boundary of XQ, and any numbers after an asterisk describe a sequence of corner points on that component. In the above example the underlying space has two boundary components, the first of which has a corner point of order three. The remaining symbols describe the topology of the underlying space. A vertical bar by itself is the default case, that of the sphere. A circle before the bar denotes the addition of a handle, while a circle after the bar denotes the addition of a cross-cap. So (°|) describes a torus, while (|00) is a Klein bottle. For every asterisk after the bar one must remove a open disk from the underlying space (and make the resulting boundary component silvered). So (|*) is a disk, while (|°*) is a Moebius band. The bar is intended to be something of a mnemonic; things before the bar correspond to "orientable" features, while "non-orientable" features appear after the bar. Nevertheless, the bar is redundant if the underlying space of the orbifold is a sphere or a punctured sphere. In those cases the bar may be omitted. Conway's notation, while cryptic, can be applied to any two-orbifold and provides a complete description of an orbifold's features. Furthermore, it can be extended to non-compact two- orbifolds. The symbol oo denotes removing a point, either from the interior of XQ (if it appears before the vertical bar) or from one of the boundary components of XQ. Thus, (oo) is the symbol for the plane, while (oooo) denotes an infinite cylinder or an open annulus. For more details see Thurston's notes [Thu90]. 3.3 Geometric Classification Recall from the previous chapter that an orbifold is good if it is the quotient of a manifold by a group of isomorphisms acting properly- discontinuously, and bad otherwise. We wish to identify which two-orbifolds are bad and which are good. But we can do better than that; we can classify good orbifolds as elliptic, Euclidean, or hyperbolic if their corresponding manifold 25 Chapter 3. Two-dimensional Orbifolds is elliptic, Euclidean, or hyperbolic respectively. To do this and show that this distinction is well-defined, we introduce the Euler characteristic of an orbifold as defined by Thurston [Thu90]: let Q be an orbifold with underlying space XQ , and let {a} be a cellular decomposition of XQ with the property that within each open cell the isotropy group Vx is independant of x. Then the Euler characteristic x(Q) is defined to be where |F(CJ)| is the order of the group T(CJ) associated with the cell a. Note that if Q is a manifold, |r(cj)| = 1 for all a and we have the usual definition of x(Q) f°r manifolds. However, the Euler number for orbifolds is not always an integer. Example: Consider a disk with silvered boundary and two order-two corner points, (*22) in Conway's notation. We can take a cellular decomposition of (*22) which has the corner points as vertices, the two remaining components of the disk boundary as edges, and the interior of the disk as a cell. The interior of the disk has trivial isotropy group, while the edges have isotropy groups with two elements and the corner points have isotropy groups with 4 elements. Thus: x((-22)) = l-I-I + i + i = i 2 2 Figure 3.4: The orbifold Q = (*22): x(Q) = \- There is a connection between the Euler characteristic and the covering maps of orbifolds defined in the previous chapter. We can define the sheet number of an orbifold covering to be the number of preimages of a non-singular point in the base, if this number is finite. It is clear that if Q —> Q is an orbifold covering with sheet number k, then 26 Chapter 3. Two-dimensional Orbifolds x(Q) = kx(Q) Thus if Q is a good orbifold, the Euler characteristic determines what kind of manifold must cover it. If x(Q) > 0 and Q is good, Q must be covered by the sphere. If x{Q) = 0, Q is covered by a Euclidean 2-manifold. And if x{Q) < 0, Q must be covered by a hyperbolic manifold. So we call a good two-orbifold elliptic, hyperbolic, or Euclidean if its Euler characteristic is positive, negative, or 0 respectively. Our goal now is to identify all bad two-orbifolds. Let us first restrict our attention to orbifolds which have non-negative Euler characteristic. As in the case with manifolds, there are only a finite number of cases to consider. We now note that the Euler characteristic can be computed from Conway's notation for an orbifold in a straightforward way. Conway's notation describes an orbifold as a sphere with additional features, such as handles or cone points. By examining the effects of these different features on the cellular decomposition of an orbifold's underlying space, we can calculate x(Q) by starting with a value of 2 (for the sphere) and subtracting certain values for each feature. For example, we subtract 2 from the Euler characteristic for each handle, and 1 for each cross- cap. Each cone point replaces a non-singular point worth 1 in the calculation of xiQ) with a point worth ^ where n is the order of the cone. Hence we subtract from x(Q) f°r a cone point. Adding an asterisk in Conway's notation is equivalent to removing a 2-cell, and hence we subtract 1 from xiQ) (*ne silvering of the resulting boundary component leaves x{Q) unchanged). Finally for each corner point of order n we subtract 2T^- from xiQ)- This algorithm lets us enumerate those two-orbifolds with non-negative x(Q) by enumerating those strings in Conway's notation with sufficiently low-value features, resulting in Table 3.1. Note that all of these orbifolds are simple enough to be described in Montesinos' notation, but we will use Conway's notation for consistency. 27 Chapter 3. Two-dimensional Orbifolds Q x(Q) Q x(Q) (1) 2 (*532) 1/60 (n) (n + l)/n (*632) 0 (nm), n ^ m (n + m)/(nm) (*442) 0 (nn) 2/n (*333) 0 (n22) 1/n (*2222) 0 (332) 1/6 (n*) 1/n (432) 1/12 (2*n) l/(2n) (532) 1/30 (2*22) 0 (632) 0 (3*2) 1/12 (442) 0 (3*3) 0 (333) 0 (4*2) 0 (2222) 0 (22*) 0 C) 1 0 (*n) (n + l)/(2n) (°l) 0 (*nm), n^m (n + m)/(2nm) (1°) 1 (*nn) 1/n (n|°) 1/n (*nn2) l/(2n) (22|°) 0 (*332) 1/12 (1°*) 0 (*432) 1/24 (l°°) 0 Table 3.1: All two-orbifolds Q with x(Q) > 0 28 Chapter 3. Two-dimensional Orbifolds Theorem 3.1 The bad orbifolds in table 3.1 are precisely (n) and (nm) where n ^ m. All of the remaining orbifolds are good. We will first demonstrate directly that the orbifolds claimed to be good have manifold covers. For starters, (|), (|°), (°|), and (|00) are all manifolds to begin with (they are, of course, the non- hyperbolic 2-manifolds: the sphere, the projective plane, the torus, and the Klein bottle). Now consider (*), the disk. We can construct a double cover of this orbifold by taking two copies of the disk and joining them along the boundary, resulting in a sphere. Going in the other direction, as an orbifold the disk is the quotient of the sphere by the group of isomorphisms generated by the reflection in an equatorial plane. Hence (*) is good. Figure 3.5: A two-fold cover p : S2 —>• (*). The same construction works on any orbifold with a silvered boundary. Thus (|0*) is double- covered by the Klein bottle, and (**) is double-covered by the torus, which makes them good. Applying this construction to the remaining orbifolds in the list allows us to restrict our at• tention to orbifolds which only have cone points. A cone point in the base of such a cover will have as its preimage two identical cone points, while a corner point in the base will have as its preimage a single cone point of the same order. For example, (*333) and (3*3) are both double-covered by (333), and hence all three are good if (333) is good. The two-fold cover of the projective plane by the sphere results in another pair of orbifold two-fold covers: (n|°) is double-covered by (nn), and (22|°) is double-covered by (2222). All of the remaining orbifolds that I claim are good are spheres with two or more cone points. 29 Chapter 3. Two-dimensional Orbifolds We can construct even more coverings between these orbifolds; in each case the base will be the quotient of the covering orbifold by a rotation around a diameter of the underlying sphere. For example, suppose (nn) is embedded in 3-space so that the two cone points are at opposite ends of a diameter, and let I be a second diameter perpendicular to the first. Rotation by 180 degrees around / will map each cone point to the other, and the corresponding quotient orbifold will be (n22), where the two new cone points are the images of the endpoints of I in the quotient. So (nn) is a double cover of (n22) for all n. Clearly the sphere is an n-fold cover of (nn) in a similar fashion, so (nn) and (n22) are both good orbifolds for all n. Figure 2.1 in the previous chapter shows the orbifold covering (|) —>• (nn). Using similar constructions it can be shown that the special case (222) is a three-fold cover of (332). The three order-two cone points map to a single cone point, and the endpoints of the axis of rotation form the order-three cone points. Similarly (332) is a two-fold cover of (432). So those orbifolds are all covered by the sphere, and are good. Next (2222) can be realized as a two-fold quotient of a torus. If the torus is realized as a square with opposite edges identified, then (2222) is the quotient under rotation by TT radians about the square's centre. Furthermore (2222) double-covers (442) and triple-covers (632), so all three Figure 3.6: The covering map (°|) -> (2222). orbifolds are good. If we instead realize the torus as a hexagon with opposite edges identified, and consider rotation by 27r/3 radians about the centre, then the resulting quotient orbifold is (333). So these orbifolds are both good. The most complicated case is (532); this orbifold is the quotient of a sphere, by the orientation-preserving symmetries of a dodecahedron [Mon87]. 30 Chapter 3. Two-dimensional Orbifolds Thus all of the orbifolds in table 3.1 are good, except (n) and (nm) and their quotients (*n) and (*nm). I will now show that any orbifold of the above types is bad. The following proof is from Thurston [Thu90]. Suppose there was a covering M —> (n) where M is a manifold. Without loss of generality, we can assume M is orientable. We first show that M cannot be compact. If M were compact, then the above covering would have a finite number of sheets, and hence x{M) would be an integer multiple of x((n)) = (1 + n)/n > 0. Then M must be S2, and x(M) = 2, but 2 is not an integer multiple of (1 + n)/n. Hence M is non-compact as desired. Now position (n) in E3 in such a way that the non-singular part of (n) has strictly positive Riemannian curvature, and such that the cone point is at the tip of a cone with cone angle less than 2ix/n. This defines a Riemannian metric on (n) which lifts to a Riemannian metric on M of strictly positive curvature. But a non-compact Riemannian surface with strictly positive curvature is a contradiction. Hence no such manifold M exists, and (n) is a bad orbifold. The above proof extends easily to the orbifold (nm) when n and m are relatively prime. We require n and m to be relatively prime to ensure that 2 is not an integer multiple oi x{{nm)) — (1/n) + (1/m). To prove that (nm) is bad when n and m have a common factor, we must use a short lemma. Lemma 3.1 If Q' and Q are orbifolds and Q' covers Q, and if Q' is bad, then Q is bad. Proof: Suppose M is a manifold which covers Q. Consider the fibred product M XQ Q' as defined in the last chapter. This is an orbifold which covers M, and hence must be a manifold, but it also covers Q', a contradiction. Hence no such M exists, and Q is bad. To use the lemma, note that if n = n'd and m = m'd where n' and m! are relatively prime, then (nm) is covered by (n'm'), or by (n') if m' = 1. Hence (nm) must be bad for all n ^ m. We can also use the lemma to show that (*n) and (*nm) are bad orbifolds, proving the theorem. 31 Chapter 3. Two-dimensional Orbifolds 3.4 Construction of Hyperbolic Two-orbifolds We have shown that the only bad two-orbifolds with a non-negative Euler characteristic are those of the form (n), (*n), (nm), or (*nm) where n > 2, m > 2, and n ^ m. What about bad two-orbifolds with a negative Euler characteristic? In fact we get the following theorem from Thurston [Thu90]: Theorem 3.2 If Q is a compact two-orbifold and xiQ) < OJ ^en Q is good. Proof: As with the Euclidean and elliptic orbifolds, we will restrict our attention to those orbifolds Q for which XQ is a compact orientable 2-manifold without boundary. For if XQ has a boundary or is non-orientable (or both) then Q is covered by another orbifold without these properties. Note that taking such a cover does not change the sign of the Euler characteristic. So let Q be an orbifold such that XQ is a compact orientable 2-manifold without boundary and x{Q) < 0. In chapter two we showed that any geometric orbifold is good, hence it suffices to show that Q has a geometric structure, specifically one with the hyperbolic plane as the model space. The idea is to decompose Q into a finite number of pieces and construct hyperbolic atlases on the pieces in such a way that the atlases glue together to give a hyperbolic atlas for the entire orbifold. A specific case will illustrate the general concept. Suppose Q = (Imn) in Conway's notation where I, m, n are integers greater than or equal to 2. By direct calculation of the Euler characteristic, XiQ) <0^y + - + ! But given three integers /, m, and n satisfying the above inequality, there is a unique triangle in the hyperbolic plane (up to hyperbolic isometry) with angles equal to n/l, n/m, and 7r/n. The lengths of the sides of such a triangle are given explicitly by the hyperbolic dual law of cosines: cos(C) = — cos(A) cos(B) + sin(A) sin(5) cosh(c) 32 Chapter 3. Two-dimensional Orbifolds where A, B, and C are the angles of a triangle and a, b, c are the lengths of the opposite sides. (For details on this and other aspects of hyperbolic geometry used in this section, consult either Thurston's notes or the subsequent book [Thu90, Thu97].) So to construct a hyperbolic geometric structure for Q, we can construct two such hyperbolic triangles and identify their boundaries. Since corresponding pairs of sides must be the same length, this gives a consistent hyperbolic structure in the neighbourhood of points in the interior of the edges of the triangles. Corresponding pairs of vertices become cone points of (Imn) after this identification. Note that cone points of order n in a geometric orbifold have the property that the circumference-to-radius ratio of circles around the cone point should approach 2-n/n as the radius goes to zero. To construct all hyperbolic two-orbifolds in this fashion, one only needs three kinds of pieces: triangles, quadrilaterals, and pentagons. Lemma 3.2 Given three positive integers I, m, and n, with (l/l) + (1/m) + (1/n) < 1, there is a triangle in the hyperbolic plane with angles n/l, ir/m, and ir/n. This is the construction we just discussed; the sides of such a triangle are given by the hyperbolic dual law of cosines. Lemma 3.3 Given two positive integers m and n, with (1/m) + (1/n) < 1, and a length d > 0, there is a quadrilateral in the hyperbolic plane with two adjacent right angles, such that the remaining angles are Tr/m and n/n and the opposite side has length d. Such a shape will be referred to as a right-angled quadrilateral, and is unique up to hyperbolic isometry. Such a quadrilateral can be represented in the Klein disk model of the hyperbolic plane by a triangle, one of whose vertices lies outside the disk (see figure 3.7). The fourth side of the quadrilateral is the hyperbolic dual of the outside vertex. Such "extended triangles" have a law of cosines and a dual law of cosines just like normal triangles, although since one of the vertices 33 Chapter 3. Two-dimensional Orbifolds Figure 3.7: A right-angled quadrilateral in the hyperbolic plane. is now distinguished from the others the laws become more complicated. Specifically, the dual law of cosines takes on the following three forms [Thu97]: cosh(d) = — cos (A) cos(i?) + sin(A) sin(S) cosh(ft) COS(J4) = — cos(-B) cosh(d) + sin(S) sinh(d) sinh(c) cos(I?) = — cos(A) cosh(d) + sin(A) sinh(d) sinh(a) The above formulas uniquely determine the values of a, b, and c in terms of m, n, and d. Lemma 3.4 Given a positive integer n > 0 and two lengths b > 0 and e > 0, there is a pentagon in the hyperbolic plane with four right angles, such that the fifth angle is ir/n and two of the sides have length b and e as in figure 3.8. Such a shape will be referred to as a right-angled pentagon, and is unique up to hyperbolic isometry. Again, in the Klein disk model such a pentagon is represented by a triangle, in this case one with two vertices outside the disk. The dual law of cosines for such right-angles pentagons has the following three forms [Thu97]: cos(A) = — cosh(fr) cosh(e) + sinh(fr) sinh(e) cosh(a) cosh(6) = — cosh(e) cos(yl) + sinh(e) sin(^4) sinh(d) cosh(e) = — cosh(6) cos(i4) + sinh(6) sin(A) sinh(c) 34 Chapter 3. Two-dimensional Orbifolds Figure 3.8: A right-angled pentagon in the hyperbolic plane. The above formulas uniquely determine the values of a, d, and c in terms of n, b and e. Given the above three kinds of hyperbolic pieces, any hyperbolic two-orbifold can be con• structed. For example, suppose Q is a torus with a cone point of order n; in Conway's notation, Q = (°n|). Construct two copies of a right-angled pentagon as in the lemma above where the fifth angle is n/n and b = e. Identifying the sides of lengths a, c, and d in each triangle results in a hyperbolic structure for a sphere with two holes and a cone point of order n, and furthermore the boundaries of the two holes form geodesic curves of equal length. Gluing the boundaries of the holes together results in the desired hyperbolic torus. More generally, suppose Q is a hyperbolic orbifold for which a geometric structure has already been found, and for which XQ is a compact orientable 2-manifold. Suppose / is a simple closed geodesic curve in Q which avoids SQ. We will prove that such a curve always exists, but for the moment suppose the curve is given. As in the preceding paragraph, construct a sphere with two holes and a cone point, where the boundaries of the holes are both geodesies of the same length as I. Cutting Q open along I and inserting the sphere in the gap has the effect of finding a hyperbolic structure for the orbifold Q' which differs from Q by the addition of a cone point. Thus if we can always find a simple closed geodesic, then we will have a recursive means of finding hyperbolic atlases for orbifolds. So consider the following. 35 Chapter 3. Two-dimensional Orbifolds Lemma 3.5 Suppose Q is a hyperbolic two-orbifold where XQ is a compact orientable 2- manifold. Then there exists a simple closed curve I in XQ — Eg which is a geodesic in Q. Proof: We know that Eg is a discrete set of cone points. Suppose Eg is empty, i.e. Q is a manifold. Then we can simply pick any point x 6 Q and two pre-images x\, x2 of x in the hyperbolic plane, which is the universal cover of Q. The unique geodesic segment joining xi and X2 projects to a geodesic simple closed curve I passing through x. Now suppose Eg is non-empty. Then since Q is hyperbolic, Q is covered by some hyperbolic manifold; hence the hyperbolic plane is still a cover of Q. However, we need to take extra steps to ensure that I does not pass through a cone point. Suppose Q contains a cone point p of order n > 3. Pick a pre-image p of p in the hyperbolic plane; the group Y of covering transformations must contain a rotation 7 of angle 2-ir/n around p. Let U be a disk neighbourhood of p which doesn't contain another pre-image of p or any other cone point, and let x be a point in U — {p}. Then the geodesic segment joining x to j(x) projects to a closed geodesic I in Q which winds once around p. This doesn't work, however, if the cone point p is only of order 2; in that case the line from x to j(x) passes through p. Suppose Eg contains only cone points of order 2. Let P be the set of points in the hyperbolic plane which project to cone points in Q, and pick two points pi, p2 € P such that the line segment joining p\ to p2 contains no other points of P. Let 71 be the rotation of angle -K around p\, and let 72 be the similar rotation around p2- Note that 72(^1) is also a preimage of a cone point, and that it lies on the line joining p\ and p~2- Then for a point x sufficiently close to pi, the line joining x to 7271 (x) will not intersect any points of P and will project down to the desired geodesic Z, completing the proof of the lemma. Thus we have a recursive construction which allows us to add cone points to any orbifold which is already hyperbolic, eliminating all but a small number of cases. In particular, if Q is an orbifold whose underlying space is a compact orientable hyperbolic manifold, then a hyperbolic structure for the underlying space can be extended to a hyperbolic structure for the orbifold. Of the remaining base cases, I have already dealt with (Imn) and (°n|). The other cases are 36 Chapter 3. Two-dimensional Orbifolds (22222) and (22mn) where one of m, n is greater than 2 (remember that (2222) is Euclidean). In each case removing a cone point always results in a non-hyperbolic orbifold, so the recursive construction doesn't apply. But we can construct (22mn) from two right-angled quadrilaterals with angles 7r/n and 7r/m and equal opposite sides, by identifying the boundaries. The right- angled vertices become the order-two cone points. Similarly we construct (22222) by identifying the boundaries of two right-angled pentagons with five right angles. Thus any compact hyperbolic two-orbifold whose base is a compact orientable 2-manifold is a geometric orbifold and therefore is good. And any compact hyperbolic two-orbifold without boundary is covered by such a two-orbifold; hence all compact hyperbolic two-orbifolds without boundary are good. 37 Chapter 4 Seifert Fibre Spaces Two-orbifolds and the constructions of Chapter 2 both have connections to the theory of Seifert fibre spaces since it so happens that such fibre spaces can be viewed as circle bundles over two- orbifolds. Since six out of the eight types of compact geometric three-manifolds admit Seifert fibrations, this clearly shows the link between two-orbifolds and the classification of three- manifolds [Sco83]. This chapter will provide a definition of Seifert fibre spaces. Then we will review the definition of the tangent circle bundle for good orbifolds and show how such constructions can generate Seifert fibre spaces, along with more arbitrary circle bundles over orbifolds. The tangent circle bundle construction in particular results in all of the orientable Euclidean three-manifolds and many spherical three-manifolds. We will demonstrate this, and connect this material to the theory of tiling patterns in spherical, Euclidean, and hyperbolic spaces. 4.1 Definitions Definition 4.1 A closed, orientable, connected 3-manifold is a Seifert manifold if it is a union of fibres, all homeomorphic to S1, such that each point of the manifold lies in exactly one fibre and each fibre has a solid torus neighbourhood made up of fibres which are not meridians of the torus [Mon87j. Seifert proved that a Seifert manifold is a generalization of a fibre bundle in the following sense [ST80]. If M is a Seifert manifold, then the topological space obtained by collapsing each fibre 38 Chapter 4. Seifert Fibre Spaces down to a point is a closed, connected surface. Furthermore if this surface is denoted N, then the projectionp : M —> N has the property that p~l(U) = (7x5', where U is a neighbourhood of x in N, for all but a finite number of points x € N. Thus M is a fibre bundle over N with fibre Sl except over this finite set of points; the fibres over these points are called exceptional fibres. Seifert further demonstrated what the neighbourhoods of exceptional fibres in a Seifert manifold could look like; they must look like the interior of a fibred solid torus of type (3/a for some (3, a. Definition 4.2 The oriented fibred solid torus of type (3/a, where (3 and a are relative prime positive integers with (3 < a, is the space (P2x[0,l])/~ Where D2 is the closed 2-disk and ~ identifies (x, 1) with (r((3/a)x,G), where r((3/a) denotes the rotation of angle 2K (3*/a, (3(3* = 1 (mod a) All but one of the Seifert fibres in a fibred solid torus are made up of the images of a segments of the form {x} x [0,1], where a; is a point which is not the centre of D2. The line over the centre of D2 becomes a Seifert fibre in the quotient space all by itself. 4.2 Circle Bundles over Orbifolds Recall from chapter two that if Q = M/T, where M is a Riemannian manifold and T is a group acting effectively and properly-discontinuously on M, then the tangent bundle of the orbifold Q is related to the tangent bundle of M by the formula TQ = TM/T. The action of T on TM is given by the derivative of the action of T on M. Furthermore, this construction generalizes to give a construction of the spherical tangent bundle of Q, namely TS(Q) = TS(M)/T. Here the action of T is the quotient of the action on TM by the canonical map from TM — Mo to TS(M), M0 being the zero section of M in TM. 39 Chapter 4. Seifert Fibre Spaces To illustrate, consider Q = (2oo); Q is a plane with a single order-two cone point. We can realize Q as the two-fold quotient of R2 by the group T generated by rotation by 7r radians about the origin. The tangent circle bundle of R2 is just R2 xS1, to which we can assign co-ordinates (x, y, 6). Then the orbit of (x, y, 6) under the action of T will be the set {(x,y,0),{~x,-y,0 + TT)} Note that the above action of T on R2 x S1 is a free action. Hence TS(Q) is a manifold. Furthermore, if U C XQ is a small open set not containing the cone point, then U has two pre- 2 l 1 images U\ and U2 in R and the action of T identifies U\ x S with U2 x 5 homeomorphically. So the subset of TS(Q) lying over U will be homeomorphic to U x 51. Moreover if D is a circular disk in XQ centred at the cone point, then the preimage of D in R2 is a circular disk Di centred at the origin. Hence the subset of TS(Q) lying over D is 1 (D\ x 5 )/r, which is homeomorphic to (£>lX[0,7r])/~ where ~ identifies (x, y, TT) with (—a;, — y, 0). But this is just a fibred solid torus of type 1/2. Thus as a map between manifolds, the map TS(Q) —> XQ is an S1 -fibre bundle except over the cone point in XQ, over which there is an exceptional fibre. This exceptional fibre has a neighbourhood which is the interior of a fibred solid torus of type 1/2. This example is easily generalized; if Q = (rcoo) where n > 2, then TS(Q) -» XQ will be an S1 fibre bundle except for an exceptional fibre over the cone point, which has a neighbourhood which is the interior of a fibred solid torus of type 1/n. Note that TS(Q) in the above examples is not a Seifert manifold, because it is not compact; this is a consequence of choosing a non-compact orbifold Q as the base. But if Q is a connected, analytic 2-orbifold without silvered boundary then Q can be expressed as the finite union of open disks and non-compact orbifolds of the form (noo). Thus TS(Q) will be a Seifert manifold, with exceptional fibres corresponding to the cone points of Q. 40 Chapter 4. Seifert Fibre Spaces The connection between orbifolds and Seifert fibrations becomes even stronger when one consid• ers other S^-bundles over orbifolds besides the spherical tangent bundle. The spherical tangent bundle of a 2-orbifold can only have exceptional fibres of type 1/n, where n € Z. But suppose M is a three-manifold with a Seifert fibration containing an exceptional fibre I of type f3/a. Then consider the quotient topological space M/ ~, where ~ collapses each fibre down to a point. It is not hard to see, from the definitions in the previous section, that in the neighbour• hood of the image of I the space Mj ~ will look like the neighbourhood of a cone point of order a. Thus the quotient map M -> Mj ~ defines an orbifold fibre bundle with total space M. This bundle will not be the spherical tangent bundle if (3 ^ 1 for any exception fibre of M. 4.3 Three-manifolds as TS{Q) As shown in Chapter 2, the spherical tangent bundle is defined for any analytic orbifold; it is however much easier to construct the spherical tangent bundle for an orbifold which is good. For example, consider Q = (2222). As in the previous chapter, this is a Euclidean orbifold which is a two-fold quotient of the torus. So TS(Q) is a two-fold quotient of the spherical tangent bundle of the torus, which is just the three-dimensional torus 51 x S1 x Sl. We can make the exact nature of the group actions involved much clearer if we introduce co-ordinates. The two-dimensional torus (°|) can be defined as (o|) = ([0,l]x[0,l])/~ where ~ identifies (x,0) with (x, 1) and (0,y) with (l,y) for all x, y. Then if T is the group action generated by the map 7(1, y) = (1— x,l — y), then (2222) is the orbifold quotient (°|)/r; the cone points of (2222) are the images of the points (0,0), (1/2,0), (0,1/2), and (1/2,1/2). Geometrically, 7 can be defined as the rotation of n radians about (1/2,1/2). The spherical tangent bundle of (°|) can be defined as T5((°|)) = ([0,l]x[0,l]x[0,27r])/~ where ~ identifies (0,y, 6) with (l,y, 8), (x,O,0) with (x, 1,6), and (x,y,0) with (x, y, 2TT) for 41 Chapter 4. Seifert Fibre Spaces all x, y, and 9. The derived action of T on this space is generated by the map 7(re, y, 9) = (1 - x, 1 - y, 9 + TT mod 27r) Hence TS(Q) is the quotient of (S1)3 by T, or equivalently TS(Q) = ([0,1]X[0,1]X[0,TT])/~ where ~ identifies (0,y,0) with (l,y,0), (x,0,9) with (x, 1,0), and (x,y,0) with (1 —re, 1 — y,ir). The above manifold, while clearly a Seifert fibre space, can also be described as a torus bundle over the circle, with cyclic holonomy of order two. The bundle projection map p : TS(Q) -» 51 is just the map which picks out 9. If in the above construction we replace 7 with the map which sends (x,y) to (y, 1 — x), corresponding to rotation of n/2 radians about (1/2,1/2), then the resulting quotient orbifold is (442). It follows that T5((442)) is also a torus bundle over the circle, with cyclic holonomy of order four. And if, instead of ([0,1] x [0,1])/ ~, we realize the torus as a hexagon with opposite edges identified, then we can construct the spherical tangent bundles of (333) and (632). They are the orbifold quotients of the torus by respectively three- and six-fold rotation about the centre of the hexagon, so their spherical tangent bundles are torus bundles over S1 with cyclic holonomies of order three and six respectively. Finally, consider (22|°), the projective plane with two order-two cone points. This orbifold is the 2- fold quotient of (2222), so its spherical tangent bundle will be a two-fold manifold quotient of T5((2222)). The manifold T5((442)) is also a two-fold quotient of T5((2222)), but is not the same as T5((22|°)); the first is the quotient of the three-torus by a cyclic group of order four, while the second is the quotient of the three-torus by a product of two cyclic groups of order two. The above five manifolds, along with (S1)3, constitute all six of the orientable Euclidean com• pact three-manifolds. It is quite remarkable that all six are the spherical tangent bundle of some Euclidean orbifold. The situation when Q is a spherical two-orbifold is even richer. The spherical tangent bundle of S2 is 50(3), which as a Lie group can be identified with H/{±1} where H =" S3 is the 42 Chapter 4. Seifert Fibre Spaces group of unit quaternions. This well-known identification can be found in many published works including those of Montesinos and Thurston, although the result is much older than that [Mon87, Thu90]. The quotient map from H to 50(3) sends the unit quaternion cos(0) + sm(9)(xi+yj+zk) to the right-handed rotation of angle 29 around the axis of M3 in the direction of (x, y, z). This identification respects group actions in the sense that if F is a discrete subgroup of 50(3), then T5(52/r) = 50(3)/r = H/F where F is the pre-image of T in H. Note that the first two quotients above represent the effect of group actions while the last is a Lie group quotient. Now if Q is a connected, orientable, spherical 2-orbifold then Q must be the quotient of the sphere by some discrete subgroup of 50(3); thus all three-manifolds of the form TS(Q) with Q spherical, connected, and orientable are in one-to-one correspondence with the discrete sub• groups of 50(3) up to conjugation. This in turn is in one-to-one correspondence with the conjugacy classes of discrete subgroups of H which contain —1. Furthermore, the fibre struc• ture of 50(3) as a circle bundle over 52 is just the quotient of the Hopf fibration of 53, and in turn when Q is a spherical orbifold without silvered boundary the Seifert fibres of TS(Q) are the images of the Hopf fibres of 53. Example: Consider the group F < H consisting of all unit quaternions of the form {±(cos(r7r/n) + i sin(r7r/n))|r = 0..n — 1} where n > 2 is a positive integer. We have — 1 G T, and the quotient of 53 = H by V is the lens space L(2n, 1). The image F of F in 50(3) is the cyclic group generated by rotation of angle 2-n/n about the x-axis, and hence the quotient orbifold 52/r is the orbifold (nn). Thus L(2n, 1) is the spherical tangent bundle of (nn). Example: Let F = {±1, ±i, ±j, ±k}. Then F is generated by rotations of angle 7r around each of the co-ordinate axes. We have 52/r = (222), while H/F is the manifold known as quaternionic space: TS((222)) is a unit cube, with opposite faces identified by a right- handed 43 Chapter 4. Seifert Fibre Spaces rotation of it/2 radians. Example: We know (532) = S2/F where T is the group of orientation-preserving symmetries of a dodecahedron, also known as the icosahedral group. The pre-image of F in H is called the binary icosahedral group; it is a group of 120 elements, and the quotient of H by this group is Poincare dodecahedral space (see [Thu90] or [Rol90] for a definition). If Q is closed, spherical, and non-orientable, but still does not have a silvered boundary (i.e. XQ is a closed non-orientable 2-manifold), then TS(Q) is still a manifold. However it is the quotient of TS(S2) by a subgroup of 0(3) which includes at least one orientation-reversing isometry. We can find TS(Q) explicitly as follows: there is an orientable orbifold Q' which is a two-fold cover of Q (corresponding to the two-fold orientable cover of XQ), and TS(Q) is a two-fold manifold quotient o£TS(Q'). Example: Let Q — (n\°). This orbifold is a two-fold quotient of (nn), so TS(Q) is a two-fold manifold quotient of L(2n, 1). The hyperbolic case is of course the richest of the three geometric cases, due to the proliferation of hyperbolic 2-orbifolds. The tangent circle bundle of any hyperbolic two-orbifold Q will be of the form TS(H2)/F, where H2 denotes the hyperbolic plane and F is a discrete cocompact group of hyperbolic isometries. In the case where Q is also closed and orientable, F will consist only of orientation-preserving isometries. Groups of orientation-preserving hyperbolic isometries which are discrete, cocompact and which act properly-continously are known as Fuchsian groups. If Q is not orientable but still does not have silvered boundary, the group F will no longer be Fuchsian, although the quotient space TS(Q) will still be an orientable 3-manifold. An important difference between the hyperbolic case and the other two cases is that the spherical tangent bundle TS(Q) of a hyperbolic 2-orbifold will not be a hyperbolic three-manifold. The spherical tangent bundle of the Euclidean plane is just E2 x Sl, which is a Euclidean manifold, and the spherical tangent bundle of S2 is 50(3), which is spherical. But the spherical tangent bundle of H2 does not possess a hyperbolic geometric structure. As detailed by Thurston and 44 Chapter 4. Seifert Fibre Spaces Scott [Thu97, Sco83], TS(H2) has the same geometric structure as 5X(2,M), the universal covering space of the two-dimensional special linear group. This space is a twisted R-bundle over the hyperbolic plane, and as a topological space is homeomorphic to H2 x K, but the isometries of SL(2, M) are not in general the products of isometries of H2 and M. What's more, it is impossible for a three-manifold to possess both hyperbolic and S~L(2,M) geometry [Thu97, Sco83]. If Q is a hyperbolic 2-orbifold, TS(Q) will inherit a geometric structure from TS(H2) and hence TS(Q) cannot be a hyperbolic 3-manifold. A fact which is common to all three cases is that if M = E2, S2, or H2 and Q = M/Y has no silvered boundary then the fundamental group of the manifold TS(Q) is closely related to T. Suppose first that M = E2. Then TS(M) = E2 x S1 and iti(TS{M)) S Z. Since TS(Q) = TS(M)/T, the fundamental group of TS(Q) will be an infinite cyclic extension of V. There is a short exact sequence: 1 ->Z->TTI(T5(Q)) 1 1 The same thing happens when M = H2, since TS{H2) = H2 x S . If M = S2, however, then TTI(TS{M)) = TTI(50(3)) = C2, the cyclic group of order two. Thus TTI{TS(Q)) will be a binary extension of T rather than an infinite cyclic one. Example: Recall that (222) = S2/F where F is generated by rotations of 7r radians around each of the co-ordinate axes. Then r = C% x C2 has four elements. We can show that 7Ti(TS((222))) is the quaterionic group {±l,±i, ±j, ±fc}, which is an extension of C2 x C% by an element of order 2. 4.4 Three-orbifolds as TS(Q) The previous section did not discuss the case where Q was a 2-orbifold with a silvered boundary. In that case, the spherical tangent bundle TS(Q) can still be calculated, but the result is not a manifold. Let Q be the orbifold E2/T, where T is generated by reflection in the x-axis. In co-ordinates, T is generated by the map 7 which sends {x,y) to (x, —y) for all x, y. Going up x to the spherical tangent bundle, we see that TS(Q) must be of the form (E2 x S )/I\ where 45 Chapter 4. Seifert Fibre Spaces the action of F is generated by the map 7(z, y, 9) = (x, -y, -9 mod 2ir) This is not a free-action; it branches over all points of the form (x, 0,0) and (x, 0, IT). So TS(Q) 2 is an orbifold which is not a manifold. A fundamental region for the action of F on E x S1 is the manifold with boundary E2 x [0,TV], and on the boundary (x,y,0) is identified with (x, — y,0), and (x,y,ir) is identified with (x, — y, 7r). Hence the underlying topological space of TS(Q) is homeomorphic to M3, while the singular set consists of the two lines which are the images of the lines (x, 0,0) and (x, 0,7r). Any point on one of these two lines has a neighbourhood U = U/F, where the action of F is to rotate U by 7r radians around an axis in the x-direction; this is just the action of r on E2 x 51, restricted to a neighbourhood of a point where y — 0 and 9 = 0 Or 7T. Lines such as those that make up the singular set of TS(Q) are called cone axes of order 2, and are higher-dimensional analogues of cone points in the 2-dimensional case. If Q is a compact 2-orbifold with a silvered boundary, then by piecing together neighbourhoods with the structure of the non-compact orbifold above we see that the spherical tangent bundle of Q is a three-dimensional orbifold. The underlying space is a three-manifold without boundary and the singular set consists of a one-dimensional submanifold where each component is a cone axis of order 2. Since the silvered boundary of a compact 2-orbifold must be a compact one-dimensional manifold, and since the cone axes in the spherical tangent space lie over this silvered boundary, we see that the cone axes must be compact submanifolds of the tangent space. Hence the singular set of TS(Q) consists of a knot or link in the underlying space, each component of which is a cone axis of order 2. Example: Consider the orbifold Q = (*), which is a disk with a silvered boundary. The subset of TS(Q) lying over the interior of the disk will be an open solid torus, since the interior is contractible. The subset of TS(Q) lying over the silvered boundary of Q is the quotient of a two-torus, and is glued to the solid torus to form an orbifold without boundary. We can realize XQ as the unit disk in the xy-plane, and in so doing provide co-ordinates (x,y,9) for TS(Q); 46 Chapter 4. Seifert Fibre Spaces then TS(Q) becomes the space 2 2 1 {(x,y,e)\x + y The singular set is the image of the two curves (cos I claim that the underlying space of the resulting orbifold is just S3, and that the singular set in S3 is a Hopf link. Consider the map which injects the solid torus {{x,y,6)} into the unit sphere in C2. Projecting stereographically from the point (0, -1) maps the unit sphere in C2 onto S3 = R3 U{oo}, and in so doing identifies the solid torus {(x,y,6)} with the complement of an open solid torus in R3. The longitudes {(cos (f>, sin)} x S1 of {(x, y, 9)} are identified with the meridians of the complementary torus in S3 by this map. What's more, the image of the two curves (cos Hence the image of each longitude in the quotient space is an interval. Examining our picture in S3, we see that we can realize the quotient of {(x,y,9)} by With the help of the preceding argument, we can examine the fibres of the orbifold bundle TS(Q) —> Q in this example. Over a non-singular point of Q, the fibre is a circle which has linking number one with each component of the Hopf link. The collection of all such non- singular fibres form a fibration of the space S3 — A, where A is a topological annulus with the Hopf link as boundary. Over a point on the silvered boundary of Q, the fibre is a silvered 47 Chapter 4. Seifert Fibre Spaces interval which is the quotient of S1 by the equivalence 9 ~ -6 mod 2ix. These intervals form a collection of suborbifolds of TS(Q) which together sweep out the annulus A as in figure 4.1. Figure 4.1: Singular fibres of the bundle T5((*)) -»(*). One final note for this section: while the spherical tangent bundle of an orbifold with silvered boundary is not a manifold, there are other circle bundles over such orbifolds which do have manifolds as their total space. Consider the half-plane Q = R2/T where F is generated by reflection in the x-axis, and construct the circle bundle R2 x S1/F, where the action of F on S1 is generated by a rotation of IT radians. Then the action of F on R2 x S1 is free, and the quotient space is a non-orientable manifold which is an 51-bundle over the orbifold Q. This bundle is not a Seifert fibration in the usual sense, as the exceptional fibres are not isolated but instead sweep out a surface which lies over the silvered boundary of Q. It is clearly related to Seifert bundles, however, and in fact Scott expands the definition of Seifert fibrations to include this case [Sco83]. 4.5 Tiling patterns An effective visual metaphor for the structure of two-dimensional orbifolds and their tangent circle bundles was put forth by Montesinos [Mon87]. Consider the case where M is a two- dimensional manifold of the form X/F, where T is a group of deck transformations of X. If R C X is a fundamental region for the action of F, then X can be tiled by copies of R, and M is a quotient of R by some equivalence on the boundary of R. For example, if M is the torus, then X is the Euclidean plane and F is the group generated by translations in two non-parallel directions. The fundamental region R of G is a parallelogram; the translations which identify the opposite sides of R are generators for F. Clearly the Euclidean plane can be tiled by copies of any parallelogram R. 48 Chapter 4. Seifert Fibre Spaces Similar results hold for geometric orbifolds. Let Q — X/F be a geometric orbifold with model space X (recall from chapter two that any geometric orbifold is good). The group F is a discrete group of isometries of X acting effectively and properly-discontinuously. This group will have a fundamental region R, and X can be tiled by copies of R. Example: let Q = (nn). Then Q = S2/r, where V is the group generated by a n-fold rotation about some diameter of S2. Let p, q be the endpoints of this diameter of S2. Then a fundamental region for F is the region contained between two geodesic arcs from p to q which form an angle of 2n/n at each each. Such a regions is called a lune, and n such "lunes" tile the surface of the two-sphere. By gluing together the two edges of the lune, one gets back the orbifold Q. Montesinos takes this analogy further, by connecting these tilings to the spherical tangent bundles. Let X be one of 52, E2, or H2. For a given subset C of X, define the group of symmetries of C to be the subgroup of orientation-preserving isometries of X which preserve C. A subset C will be called a pattern if its group of symmetries is discrete and acts properly- discontinuously on X. A typical example of a pattern in E2 is an infinite, regular square grid; its symmetry group is generated by two translations in the directions of the grid lines, and by a rotation of 7r/4 radians around one the the intersections of the grid. Define a position of C to be a subset C of X which is the image of C under some orientation-preserving isometry of X. We can equip the collection of all positions of C with a Hausdorff topology, and this makes the set of positions of C into a manifold. If the symmetry group of C is trivial, then the set of positions of C can be identified with the manifold of orientation-preserving isometries of X, denoted Iso+(X), in an obvious manner. In general, the set of positions of C will be of the form Iso+(X")/T, where F is the group of symmetries of C. However, Iso+(A") is in fact isomorphic with TS(X). For any fixed vector v E TS(X), the map 7 M- d*y(v) realizes this isomorphism, where d7 denotes the induced action of 7 on TS(X). This is true whether X is the two-sphere, the Euclidean plane, or the hyperbolic two-plane; in 49 Chapter 4. Seifert Fibre Spaces any of the three cases, given two unit tangent vectors there is a unique orientation-preserving isometry of X taking one to the other. Thus we have the following result [Mon87]. Lemma 4.1 Let C be a pattern in X (where X — S2, E2, or H2), and let Y be the group of symmetries of C. The manifold of positions of C is given by TS(X)/T = TS(Q), where Q is the quotient orbifold X/Y. Furthermore, Q is orientable, and if Y is a cocompact group then Q will be compact as an orbifold. Note we specifically excluded orientation-reversing isometries from symmetry groups. If a pattern C admits orientation-reversing symmetries, then any such symmetry will define an involution of the manifold of positions of C. The quotient of C under all such involutions will be the orbifold TS(X/Y) where Y is the group of all symmetries of C. Example: Let C be the infinite, regular square grid of the previous example. The group of symmetries (that is, orientation-preserving ones) Y of C is generated by translations in two orthogonal directions, and by one rotation of 7r/4 radians. If Y' < Y is the subgroup consisting only of translations, then E2/Y' is a torus. Furthermore, the quotient map E2/Y' —> E2/Y is a four-sheeted orbifold covering map. This and the fact that E2/Y is an orientable orbifold are enough to uniquely determine E2/Y; it is the orbifold (442), according to the classification results of the previous chapter. Thus the manifold of positions of C is the spherical tangent bundle of (442), which earlier in this chapter was shown to be a torus bundle over the circle with cyclic holonomy of order four. In co-ordinates, TS((442)) can be realized as [0, l]x[0, 1]X[0,TT/2]/~ where ~ identifies (0,y, 9) with (1,y,9), (x,0,9) with (x, 1,9), and (x,y, 0) with (y, 1—x, 7r/2). With these co-ordinates the isomorphism between T5((442)) and the manifold of positions of C can be shown directly; the point (x, y, 9) in T5((442)) maps to the unique position of C which 50 Chapter 4. Seifert Fibre Spaces has an intersection at the point (xcos9 - y sin9,ycos9 + xsin9) G E2, and which has one line through this intersection forming an angle of 9 with the horizontal, as in figure 4.2. Figure 4.2: Pattern position corresponding to the point (x,y,9). But this pattern C also possesses orientation-reversing symmetry. Specifically, the automor• phism (x,y) H-> (—x,y) of E2 will carry any position of C to another position of C, thus defining an action on T5'((442)). In our co-ordinate system, this action is generated by the map (x,y,9) H-> (y,x, (TT/2) — 9). This action is not free, as it leaves invariant any point on the lines (0,i,0), (1/2, t, 0) and (i,t,7r/4). The quotient of TS((442)) by this action is an orbifold, and the images of the three closed curves on which the action is not free form the singular set of this orbifold. This singular set also consists of three closed curves, each of which is a cone axis of order two, a structure described earlier in this chapter. Note that any point in the singular set has an orbifold neighbourhood which is isomorphic to the quotient of E3 by the group generated by a rotation of 7r radians about some axis through the point. This orbifold is the spherical tangent bundle of the orbifold (*442), and the three curves in the singular set lie over the three silvered edges of (*442). The nature of the underlying space of this three-dimensional orbifold is not at all clear from this construction. However Montesinos has shown, using a different construction of the orbifold involving surgery diagrams, that the underlying space is in fact just 53, and that the three curves in the singular set form a link. Specifically, the singular link is a pretzel link of type (4,4,-2) [Mon87]. William Dunbar's Ph.D. thesis also contains detailed constructions of this type [Dun81]. 51 Chapter 5 Three-dimensional Orbifolds Just as three-dimensional manifolds form a much richer class of objects than two-manifolds, three-dimensional orbifolds come in more variety than two-dimensional ones. Moreover, three- dimensional orbifolds appear in a wide variety of contexts ranging from knot theory to proofs of Andreev's theorem, which concerns analytic maps in the Riemann sphere. In this chapter, I will give some results about the local structure of 3-orbifolds, analogous to the results in chapter three concerning 2-orbifolds. I will then describe two large classes of three- dimensional orbifolds: those orbifolds for which XQ is homeomorphic to the three-ball and SQ = 8(XQ), and those orbifolds for which XQ = S3 and SQ is a knot or link in S3. Orbifolds where XQ is homeomorphic to the three-ball are used in Thurston's work on Andreev's theorem and circle packings in the plane and the Riemann sphere. And orbifolds for which SQ is a knot or link in S3 have obvious connections to knot theory. Finally, I will show how to construct a wide variety of orbifolds by finding the quotients of manifolds represented as Dehn surgery diagrams, by actions generated by rotations preserving those surgery diagrams. 5.1 Local Structure of Three-orbifolds In chapter three we had the following result for an arbitrary two-orbifold Q: if x € XQ then 2 there is a neighbourhood U of x which is homeomorphic as an orbifold to R /rx, where is a discrete subgroup of the group of isometries 0(2). The proof extends trivially to the three-dimensional case and indeed the n-dimensional case. Hence if Q is a three-dimensional orbifold and x € XQ, then there is a neighbourhood U of 52 Chapter 5. Three-dimensional Orbifolds 3 x which is homeomorphic as an orbifold to M /Tx where Tx is a discrete subgroup of 0(3). Enumerating the discrete subgroups of 0(3) is not as easy as it is for 0(2), but fortunately we have already done it; taking the image of the unit sphere in M3 under the quotient map provides a one-to-one correspondence between conjugacy classes of discrete subgroups of 0(3) and spherical two-orbifolds, which were enumerated in chapter three. Given any spherical two-orbifold Q2 and the group T < 0(3) which corresponds to it, the orbifold M3/T is found by taking the cone on Q2: K3/r =• (Q2 x (0,1])/ ~ where ~ identifies all points of the form (q, 1). The isotropy group of a point (q, k), k < 1, is just the isotropy group of q in Q2 as a consequence of the definition of Q2 x (0,1]. The isotropy group of the tip of the cone is just T; the tip of the cone is the image of the origin of M3 under the quotient map M3 -» R3/r, and every element of V < 0(3) preserves the origin. With this information we can describe all possible neighbourhoods in a three-orbifold, by ex• amining the corresponding two-dimensional spherical orbifolds. For this purpose, we can divide the spherical two-orbifolds into a number of catagories: • The sphere (), which has orientable underlying space and no singular set. • Other orientable orbifolds, i.e. orbifolds with orientable underlying space and cone points, but no silvered boundary. • The disk (*), which has orientable underlying space and a silvered boundary, but no cone or corner points. • Other orbifolds with silvered boundary. • The projective plane (|°), and the orbifolds (n|°), which have non-orientable underlying spaces. The sphere () corresponds to the trivial subgroup of 0(3). In this case x does not lie in SQ and x has a neighbourhood homeomorphic to M3. 53 Chapter 5. Three-dimensional Orbifolds Now consider (nn), a sphere with two order-n cone points. Topologically the cone on this space is a three-ball (since (nn) is topologically a sphere), but the singular set of this three-orbifold is a curve through the tip of the cone. Such a curve is called a cone axis of order n, and is the three-dimensional analogue to a cone point. The corresponding group T is a cyclic group of order n, generated by a rotation. The other orientable spherical two-orbifolds, namely (n22), (332) , (432), and (532), are also topologically spheres, so the corresponding three-orbifolds are also topologically equivalent to M3. However the singular sets in these cases consist of three curves meeting at x instead of two. Hence the singular set is locally not a manifold, something we did not see in the two-dimensional case. Each of these curves will be a cone axis, and the orders of these axes must come from one of the "spherical" triplets: n-2-2, 3-3-2, 4-3-2, or 5-3-2. Conversely, three cone axes each of order three could not meet in a point in this fashion because (333) is not a spherical two-orbifold. The orbifold neighbourhood corresponding to the disk (*) is topologically a half-space, and the singular set is the boundary. Such a singular set is called a mirrored or silvered boundary, and is entirely analogous to silvered boundaries in two-dimensional orbifolds. The corresponding group T < 0(3) is the group generated by reflection in a plane. If r contains reflections and rotations, then we get a three-dimensional orbifold neighbourhood corresponding to a two-orbifold like (*nn) or (n*), which have silvered boundary and either cone or corner points or both. If F is generated by reflection in a plane and rotation through an axis perpendicular to the plane, then the singular set in the resulting neighbourhood will consist of a silvered boundary and a cone axis which terminates at the boundary; the corresponding two- orbifold is (n*) for some n. If F is generated by reflection in a plane and rotation through an axis lying in the plane (so F is isomorphic to a dihedral group), then the resulting neighbourhood will have a silvered boundary and a corner axis lying in the boundary. Corner axes are the three-dimensional analogue of corner points, and indeed such neighbourhoods correspond to two-orbifolds like (*nn). Three corner axes can terminate at a single point under the same restrictions applied to three cone axes. And a cone axis can meet a corner axis at a point on a silvered boundary, so long as the cone axis has order two, or the corner axis has order two and 54 Chapter 5. Three-dimensional Orbifolds the cone axis has order three. Such neighbourhoods correspond to the two-orbifolds (2*n) and (3*2). In all of the above cases the underlying space of the three-dimensional orbifold neighbourhood was either the three-ball or the three-dimensional half space. However unlike the case in two dimensions, the underlying space of an analytic three-dimensional orbifold does not have to be a manifold. The cone spaces on the two-dimensional orbifolds (|°) and (n|°), i.e. the projective plane and the projective plane with a cone point, are both open three-dimensional orbifolds. In each case the underlying space is homeomorphic to the quotient space of M3 by the map (x,y, z) H-> (—x, —y, —z); this space is not a manifold near the image of the origin. In the case of the cone on (|°), the singular set is the point which is the image of the origin. In the case of (n|°), the singular set also contains a cone axis of order n, terminating at that point. In both cases the corresponding group T contains the antipodal map (x,y, z) H-> (—x, — y, — z). The point at the tip of the cone where XQ fails to be a manifold is sometimes called a pinhead. 5.2 Examples of Compact Three-orbifolds The information in the preceding section allows us to better describe the compact three-orbifolds that have appeared in previous chapters. For example, in chapter four we saw that the spherical tangent bundle of the disk (*) is a three-orbifold Q where XQ = S3 and EQ consists of two closed cone axes of order 2 in the form of a Hopf link. The more general case of an orbifold with XQ = S3 and SQ a cone axis of order n in the form of a knot was first brought up in chapter one. One of the examples in chapter three was the orbifold Q = M3/r, where V was the group of isometries generated by the reflections in the faces of a cube. In that instance XQ was a cube and EQ was the boundary of the cube. In the language of this chapter, the faces of the cube are silvered, while the edges of the cube are order-two corner axes; three such axes meet at each vertex of the cube. This is an example of a polyhedral orbifold, which are discussed in the next section. 55 Chapter 5. Three-dimensional Orbifolds More generally, compact three-manifolds with boundary can be given the structure of a three- orbifold without boundary by "silvering" the entire boundary. Such orbifolds are covered in a two-to-one fashion by a manifold without boundary which serves as a branched cover of the original manifold. For example, if we start with the closed three-disk D, silvering the boundary results in a three-orbifold which is covered by the three-sphere S3. 5.3 Polyhedral Orbifolds The cube orbifold mentioned in the previous section is an example of a general class of orbifolds defined as follows: Definition 5.1 A polyhedral orbifold is an orbifold Q where XQ is a solid three-dimensional polyhedron and EQ = 8XQ. Furthermore the faces of XQ are silvered while the edges of XQ are all corner axes. Since no more than three corner axes can ever meet at a point, a polyhedron IT can only be made into a polyhedral orbifold if exactly three edges meet at every vertex of IT. Furthermore the orders of the cone axes meeting at a given vertex must form one of the triplets (2,2, n), (3,3, 2), (4,3,2), or (5,3,2). Another way to say this is that every vertex of IT has a neighbourhood whose boundary is a spherical orbifold, specifically one of (*n22), (*332), (*432), or (*532). The boundary of a sufficiently small neighbourhood of a vertex is called the link of that vertex. This information allows us to enumerate all polyhedral orbifolds which can be formed from a given polyhedron IT. For example, suppose IT is a tetrahedron. Three faces meet at every vertex of a tetrahedron, hence we can look for polyhedral orbifolds based on IT. There turn out to be an infinite number of such orbifolds. For example, pick two edges of IT which do not share a common vertex, and assume those edges are corner axes of orders k and n where k,n > 2 while the other four edges of P are corner axes of order 2. This is a polyhedral orbifold for any choice of k and n, and the link of any vertex is either (*k22) or (*n22). There are, however, only a finite number of tetrahedral orbifolds which are not of this form; 20 such orbifolds, to be 56 Chapter 5. Three-dimensional Orbifolds exact [Thu90]. Furthermore, all of these tetrahedral orbifolds have geometric structures. This fact would be possible but exceedingly difficult to verify directly. Fortunately there is a way to find a spherical, Euclidean, or hyperbolic atlas for a tetrahedral orbifold given the orders of its corner axes alone [Vin85]. To do this, consider the 4x4 matrix M with l's along the diagonal and rriij = —cos(2ir/riij) off the diagonal, where is the order of the cone axis between faces i and j of the tetrahedron. The restrictions on the orders of the cone axes ensures that M has either four positive eigenvalues counting multiplicity, three positive and one zero eigenvalue, or three positive and one negative eigenvalue. The corresponding tetrahedral orbifold is spherical in the first case, Euclidean in the second, and hyperbolic in the third. Example: Consider the tetrahedral orbifold with four corner axes of order 2 and two corner axes of orders k and n, described previously. We can order the faces in such a way that the matrix M will have l's on the diagonal, m\2 = m2\ = —cos(2-n/k), 77134 = 17143 = —cos(27r/n), and all other off-diagonal entries are zero. Then M has a very nice block diagonal form, and it is easy to verify that M has four distinct eigenspaces and four positive eigenvalues up to multiplicity. Hence the tetrahedral orbifold is spherical. To construct the orbifold directly, find four normal vectors Vi,... , V4 in E4 with the following properties: the plane spanned by V\ and V2 is perpendicular to the plane spanned by V3 and V4, and V\ • v2 = —cos(2ir/k) while V3 • V4 = — cos(2-ir/n). Let T C S3 be the intersection of the four half spaces {U\Vi • U < 0} with the unit sphere in E4. Then T is a spherical tetrahedron embedded in S3, and four of the dihedral angles between the faces of T are right angles while the other two are 2-n/k and 2n/n. Silvering the faces of T and making the edges into corner axes turns T into the desired spherical tetrahedral orbifold. In the special case where n — k = 2, the matrix M is the identity matrix and we can choose Vi,... , V4 to be the basis vectors of E4. Then T is the set 4 {(a:i,... , XA) € E \ xt < 0, x\ + • • • + x\ = 1} which is a spherical tetrahedron in S3 with six right dihedral angles. 57 Chapter 5. Three-dimensional Orbifolds 5.4 Andreev's Theorem. There are more general results for polyhedra other than the tetrahedron, due to Andreev. Sup• pose Q is three-dimensional orbifold, and suppose P is a compact two-dimensional suborbifold of Q. We call an n-dimensional orbifold with boundary which is a quotient of the n-disk an n-disk quotient for short. Then P is compressible if either • P is the boundary of an suborbifold R of Q, where R is a three-disk quotient. • There is a one-dimensional suborbifold S of P which is the boundary of a two-dimensional suborbifold R of Q, such that R is a two-disk quotient, R fl P = S, and furthermore S is not the boundary of any suborbifold of P which is a two-disk quotient. In each case, R is referred to as a compression disk quotient. These conditions simply extend the definition of a compressible submanifold of a three-manifold to the orbifold case. Recall that a two-dimensional submanifold N of a three-manifold M is compressible if and only if it bounds a three-disk, or contains a one-dimensional submanifold which bounds a disk in M but not in N. And as in the manifold case, the first condition applies only when x(P) > 0, while the second condition applies only when x(P) < 0. A compact two-dimensional suborbifold which is not compressible is naturally called incom• pressible. Then the following is sometimes referred to as Andreev's Theorem. Theorem 5.1 Let Q be a three-dimensional polyhedral orbifold with at least five faces. Then Q has a hyperbolic structure if and only if every incompressible compact suborifold P of Q has X(P) < o. Example: Let Q be the cube orbifold discussed previously. Then Q does not have a hyperbolic structure. Instead, it is a quotient of E3 and hence is Euclidean. Hence, by Andreev's theorem there is a compact two-dimensional incompressible suborbifold P of Q such that x{P) ^ 0- One such suborbifold P consists of a plane parallel to one of the faces of Q. This is a Euclidean two 58 Chapter 5. Three-dimensional Orbifolds orbifold, (*2222) to be precise, so x(-P) = 0- ft bounds two three-dimensional suborbifolds of Q, each of which is homeomorphic to a cube where all but one face is silvered and the last face forms the boundary. Such an orbifold is not a quotient of the three-disk, but in fact is covered by the infinite slab [0,I] x. E2. So P is not compressible in the first sense. Showing that P is not compressible in the second sense takes more work, because there are several different types of one-dimensional suborbifolds of P: • Circles in interior of Xp. • Arcs whose endpoints lie in the interior of an edge of Xp. • Arcs whose endpoints lie in the interior of two adjacent edges of Xp. • Arcs whose endpoints lie in the interior of two opposite edges of Xp. • The edges of Xp themselves. Figure 5.1: Suborbifolds of (*2222) which bound two-disk quotients (left) and which have unbounded lifts (right). The first three kinds of suborbifolds all bound either two-disks or two-disk quotients in P itself, and hence do not make P compressible. The last two kinds of suborbifolds do not bound the quotients of two-disks in P, but neither do they bound the quotients of two-disks in Q. For if a one-dimensional suborbifold S of Q bounds a two-dimensional suborbifold R which is a quotient of a two-disk, then by lifting to the universal cover Q = E3 we get an disconnected manifold S C E3 which is the boundary of an disconnected manifold R which is a collection of two-disks. Note that this is what happens with the first three kinds of one-dimensional suborbifold listed above. But the last two kinds of suborbifolds lift to collections of unbounded curves in E3 and 59 Chapter 5. Three-dimensional Orbifolds hence do not bound the quotient of a two-disk in Q. Hence P is an incompressible suborbifold as expected. The proof of Andreev's theorem is beyond the scope of this paper; the "if" direction in particular is quite difficult to prove. A complete proof exists due to Andreev himself, although Andreev did not present his work in terms of orbifolds but instead presented a purely combinatorial proof [And70a, And70b]. It is possible in this space to sketch a proof of the other direction of the theorem. If Q is a hyperbolic polyhedral orbifold, then every compact two-dimensional suborbifold P with x(P) > 0 is compressible. Such a proof, taken from Thurston's notes, goes as follows [Thu90]. Since Q is hyperbolic, it is a good orbifold and hence Q = H3/T for some group V. Since Q is polyhedral, there is a fundamental region RQ for the action of V which is isomorphic to XQ, i.e. RQ is a polyhedron in H3, and such that T is generated by hyperbolic reflections in the faces of RQ. Let P be a compact two-dimensional suborbifold of Q with x(-P) > 0. Note that Ep C EQ, in particular P cannot contain any cone points since no point of Q has a non-trivial isotropy group which does not reverse orientation. We can assume without loss of generality that P is not a face of Q; if it is, we can find an isomorphic suborbifold just "above" that face. With this assumption, P will not contain any vertex of Q. And since P is not hyperbolic, it can contain no more than 4 corner points. With this information, we can enumerate all possibilities for P in a finite number of cases. Either P is one of (), (°|), (**), (|°*), (|°), (|00), or else P is a disk with silvered boundary and 0, 1, 2, 3, or 4 corner points. Immediately we eliminate (|°) and (|°°) (the projective plane and the Klein bottle) since in each case Xp would have to lie in the interior of RQ, which is a three-ball and hence cannot contain a non-orientable surface. We also eliminate (|°*), a Moebius band with silvered boundary, since a Moebius band cannot be embedded in a three-ball in such a way that the boundary of the band is a simple closed curve in the boundary of the three-ball. If P is () or (°|), then again Xp lies in the interior of RQ. If P is a sphere, then by Alexander's theorem P bounds a three-ball and is compressible. If P is a torus, then by Dehn's lemma 60 Chapter 5. Three-dimensional Orbifolds some curve in Xp bounds a compression disk in Q, so again P is compressible. If P is (**), an annulus with silvered boundary, then both components of Ep lie in the same face of Q, and each is a simple closed curve in that face. The faces are simply connected, hence each component bounds a compression disk in Q. So suppose P is a disk with silvered boundary. If P has no cone points, then Ep lies in a face of Q. Then P must be the boundary of the quotient of a three-disk by reflection in that face. P cannot have only one cone point: Ep would have to cross exactly one edge of Q transversally, which would require a single face of RQ to meet itself at an angle, which does not happen in HZ. If P has two cone points, then both lie on a single edge of Q, and P bounds the quotient of a three-ball by a dihedral group generated by reflections in the faces meeting at that edge. If P has three cone points, then those points must lie on three different edges which meet at a single vertex v of Q. Then Xp bounds a neighbourhood of this vertex in RQ. The elements of T which fix v form a group, and the quotient of the ball around v by this group is a three-disk quotient which has P as its boundary. Finally, if P has four cone points, then P must be (*2222), otherwise x(-P) would be negative. The four sides of P must lie in four different faces of Q, which meet in pairs at right angles since the isotropy group of each cone point of P is the dihedral group D2- The only way this can happen is if two of the planes meet in an edge I at a right angle, and the other two planes are perpendicular to /. Then P bounds an orbifold neighbourhood of I, which is the quotient of a solid cylinder. A one-dimensional cross-section of P in the direction perpendicular to I bounds another two-dimensional suborbifold of Q, which is the quotient of a two-disk by a dihedral group (see figure 5.2). Hence P is compressible in this case, and we're done. 1 Figure 5.2: The suborbifold P = (*2222), containing the boundary of a two-disk quotient. 61 Chapter 5. Three-dimensional Orbifolds Andreev's theorem can be extended to the case where XQ is a three-ball minus a finite number of points on the boundary. In the case where Q is hyperbolic, these points correspond to vertices on the sphere at infinity, also known as cusp vertices. In its full generality, Andreev's theorem can be used to show the existence of hyperbolic structures on a wide variety of three- dimensional orbifolds. Even if it a given orbifold is not polyhedral, sometimes it will have a polyhedral quotient to which Andreev's theorem will apply. Example: Let Q be the orbifold with XQ = S3 and EQ a set of Borromean rings, where the three rings are cone axes of orders k, m, and n. Position the rings in 53 = R3 U {00} such that the first ring lies in the xy-plane, the second in the yz-plane, and the third in the a^-plane, and such that each ring is evenly spaced around the origin. Then the group V generated by reflections in the three co-ordinate planes acts on Q as automorphisms, and the quotient Q/T is a polyhedral orbifold whose underlying space is a cube. See Thurston's notes for details [Thu90]. 5.5 Orbifolds and Surgery Finding orbifolds of the form M/Y involves finding symmetries of the manifold M, which is much easier when the manifold can be visualized in some way. Dehn surgery diagrams provide such a way to visualize three-dimensional manfolds, which can be used to find rotational symmetries which lead to orbifolds. For an overview of surgery diagrams, see Rolfsen [Rol90]. As an example, consider the manifold 51 x S2. The surgery description of this manifold is a single circle in S3 with surgery index 0. Suppose we equate S3 with R3 plus a point at infinity, and position the surgery circle in R3 so that it is lying in the xy-plane, centred at the origin, and has unit radius. Consider the involution 7 defined by a rotation of n radians around the y-axis; 7 is an involution of S3 if we assume that the point at infinity is kept fixed by 7. Note that 7 sends the surgery circle to itself. We wish to show that 7 extends via the surgery description to an involution 7' of S1 x 52, and moreover we wish to describe the orbifold (S1 x S2)/Y' where T' is the group generated by 7'. 62 Chapter 5. Three-dimensional Orbifolds Let V be a tubular neighbourhood of the surgery circle in S3, and let T be the boundary of V; T is naturally a torus. Similary V is a solid torus, and can be described in the following way: V = (Tx[0,l])/~ where ~ is the equivalence which collapses each curve of the form A x {1} to a point, and where A is a meridian of T. To actually obtain S1 x S2 from S3 by surgery, we remove V from S3 and replace it with V, where V' = (Tx[0,l])/~' where ~' collapses curves of the form A' x {1} to points, and where A' is a longitude of T. To determine 7' we need to extend the action of 7 to V. But we already have an action of 7 on T x {0}, since T x {0} is the boundary of S3 — V. This action extends trivially to an action on T x [0,1], and furthermore this action sends longitudes of T x {1} to other longitudes. Thus this action carries down to an action on the quotient topological space V; this action together with the action of 7 on S3 — V gives the desired involution 7' on 51 x 52. Let Q be the orbifold (S1 x S2)/F'. We wish to determine XQ and SQ. We will tackle XQ first. Consider the group T generated by the original involution 7. As orbifolds, 53/r has S3 as its underlying topological space, while the underlying space of V/V is a three-ball. Hence XQ can be obtained from S3 by taking out a three-ball and replacing it with the underlying space of V'/F'. Now (T x {1})/ ~' is a circle, with points corresponding to longitudes of T. The action of 7' on this circle branches over two points corresponding to the two longitudes of T which are sent to themselves. The quotient of this circle will be a line segment with two endpoints. For all other fi ^ 1, the quotient of T x {/x} by T' is a two-sphere. This action is just the orbifold quotient map which is written (°|) —)• (2222) in Conway's notation. Thus as a topological space, V'/V = {S2 x [0,1])/ ~' where ~' collapses S2 x {1} down to an interval. This is just a description of a three-ball. Thus XQ is the union of two three-balls along their boundary and hence XQ = S3. 63 Chapter 5. Three-dimensional Orbifolds We can find EQ by examining the branching set of 7'. In 53 — V, the branching set of 7 is two arcs, each with endpoints on the torus T x {0} and one of which passes through the point at infinity. In T x [0,1), the branching set of 7' consists of four arcs which just extend the ends of the arcs in S3 — V. And as mentioned before, in (T x {1})/ ~' the branching set of 7' is two points. By examining the longitudes of T x {1} and their images under 7', we can see that these two points join the other arcs of the branching set into two closed curves. Furthermore, the images of these curves are unlinked in the quotient space S3. Thus EQ consists of two unlinked circular cone axes in XQ. Since 7 is a rotation of order two, the cone axes each have order two as well. Hence the orbifold "S*3 with two unlinked circular cone axes of order two" is covered by SL x S2 in a two-to-one fashion. 1 2 We can say more using fancier language. Just as S x S2 is a trivial 5 -bundle over S1, Q is an S2 bundle over the one-dimensional orbifold which is an interval with "silvered endpoints". It is not, however, the trivial 52-bundle over that orbifold; Q is an orientable orbifold while the base of the bundle is non-orientable. The pre-image of either of the two singular points of the base is the orbifold (*) which is a quotient of S2. More generally, let M be a compact orientable three-manifold represented by a Dehn surgery diagram in SS. By projecting stereographically from a point not on any of the surgery circles, we can embed the surgery diagram in R3. Then by applying a suitable diffeomorphism to R3 we may be able to arrange the surgery circles around an axis I in such a way that the surgery diagram has rotational symmetry around I. This means that there is an integer n > 2 such that for every circle c in the surgery diagram one of the following is true: • c is one of n circles in the surgery diagram having the same surgery coefficient, such that rotation by 2ir/n around I permutes the n circles cyclically. • n = 2, and c intersects I in two points and has two-fold rotational symmetry with respect to /. • c does not intersect I but has n-fold rotational symmetry about / by itself. 64 Chapter 5. Three-dimensional Orbifolds If such a diffeomorphism exists, then we can construct the orbifold M/T where T is the group generated by an n-fold rotation around I. In the first case described above, n surgery circles combine into a single surgery circle in the quotient. Note the surgery coefficient in question may change in the quotient since the preffered longitude of the loop may be different in the quotient [Rol90]. The second case is the case which occurs in the example above. Surgery circles which intersect the axis of rotation do not map to surgery circles in the quotient topological space. Instead, as in the above example, such surgery circles modify the singular set EQ of the quotient orb• ifold. Remarkably, surgery circles with coefficient a//3 which intersect the axis of rotation introduce a rational tangle of type a//3 into the singular set of the quotient orbifold. A detailed demonstration of this appears in William Dunbar's Ph.D. thesis [Dun81]. The last case is complicated, so we will take a closer look. Suppose we modify the example so that 7 is a rotation of 2-KJn radians around the z-axis, and so that the surgery coefficient of the circle is a/0 where a and 0 are relatively prime. As before, the action of 7 on S3 — V extends to the interior of V in an obvious way. If n does not divide a, this action is free and the quotient of V' is another surgery circle with coefficient a/(n0). But if n divides a then the resulting action rotates the solid torus V around a longitude in its interior. Hence the quotient of V contains a cone axis and is not simply a surgery circle in the quotient orbifold. Example: Suppose a/0 = 0, so the starting manifold is 51 x S2 as before. Then a rotation around the z-axis will rotate each sphere {x} x S2 in place. The quotient orbifold is Sl x (nn) whose underlying space is Sl x S2 and whose singular set is two order n cone axes each of the form S1 x {y}. One of those cone axes is the quotient image of the z-axis, while the other is the quotient of a longitude of the solid torus V. If we avoid the last case, then the quotient orbifold will always be described by a link in S3. Some of the circles in the link will be surgery circles, while others will be part of the singular set. 65 Chapter 5. Three-dimensional Orbifolds As a final example for this section, consider the three-torus (51)3. This manifold has a surgery description in S3 consisting of three circles each with surgery coefficient 0 arranged in a Bor- romean link. This link can be arranged in a planar diagram in such a way that it has three-fold rotational symmetry around an axis vertical to the plane of the diagram. The resulting quotient orbifold surgery diagram has a circular cone axis of order 3, and a single unknotted surgery circle with coefficient 0. While both circles are unknotted by themselves, their linking number with each other is ±3, depending on choice of orientation. Hence (51)3 has a three-fold quotient orbifold whose underlying topological space is S1 x 52 (the result of a single unknotted surgery circle in 53 with coefficient 0), and whose singular set consists of a closed cone axis of order three which winds three times around the S1 component of S1 x 52. 66 Chapter 6 Invariants and Other Topics 6.1 Quotients of Three-manifolds We have already seen numerous examples of orbifolds arising as the quotients of three-manifolds. For example, in chapter four we examined the orbifold Q = TS((*)) and saw that XQ was the three-sphere while EQ consisted of two order-two cone axes in the form of a Hopf link. But since (*) is a quotient of the sphere S2 by an order-two isometry, Q is the quotient of TS(S2) = 50(3) by an order-two isometry of that three-manifold. Also, in the previous chapter we constructed 2 via surgery an orbifold with underlying space SL x 5 which was a three-fold quotient of the three-torus (S1)3. A question which arises is: which closed connected orientable three-manifolds have a quotient orbifold Q such that XQ = 53? This is still to the best of my knowledge an open question, but we can explore some interesting related results here. The following theorem at first appears to be quite useful. Theorem 6.1 Every closed, connected, orientable three-manifold is an irregular branched cover of the three-sphere, with branching index at most three and with the set of downstairs branch points equal to a knot. A proof of this theorem appears in Rolfsen's book, attributed to a 1974 paper by Montesinos [Rol90]. Unfortunately, this theorem isn't any help in finding orbifolds because it refers to irregular branched coverings. In an irregular branched covering, two points in the covering 67 Chapter 6. Invariants and Other Topics space may map to the same point in the quotient but have different branch indices. This means that the isotropy subgroup of a point in the quotient is not well-defined but instead depends on the choice of pre-image. Hence irregular branched coverings cannot be used to define quotient orbifolds. A more useful theorem is the following, which was first proved in a 1962 paper by Lickorish [Lic62]. Theorem 6.2 Every closed, connected, orientable three-manifold can be obtained by Dehn surgery on a link in S3. Moreover, this surgery description can be chosen so that each component of the link is unknotted and has an integral surgery coefficient. Suppose one wishes to find an orbifold quotient Q of a given manifold M such that XQ = S3. One approach would be to find a surgery description of the manifold based on the above theorem, and attempt to arrange the surgery link in S3 such that the link has two-fold rotational symmetry around some axis, and furthermore that each circle in the link intersects the axis of symmetry in two points. Then according to the results of the last chapter, the quotient of M by two-fold rotation about the axis of symmetry would be an orbifold with XQ = S3 as desired. Furthermore the singular set of the resulting orbifold would be a set of closed cone axes of order two. Unfortunately, there is no way to ensure that the surgery link has the necessary symmetry. It does happens that for any manifold M of genus g there is a surgery link with the property that the circles of the link can be partitioned into 3g — 1 sets, arranged in a manner similar to the link in figure 6.1. Clearly the 2g — 1 sets of circles along the bottom of the figure have the necessary symmetry around a horizontal axis, but it is not always possible to move the g sets of circles along the top of the link into a position symmetric around the same axis. Nevertheless, many three-manifolds do admit finite groups of automorphisms which have S3 as their quotients. Montesinos describes a number of such quotients for the spherical tangent bun• dles of two-orbifolds [Mon87]. This class of three-manifolds includes all six orientable compact 68 Chapter 6. Invariants and Other Topics Figure 6.1: A typical surgery link. Euclidean three-manifolds, as demonstrated in chapter four. We get another interesting question by turning the situation around: are there any bad three- orbifolds, that is, are there three-dimensional orbifolds not covered by any manifold? The answer is yes; it is easy to show that any three-orbifold which contains a bad two-dimensional suborbifold will itself be bad. Hence (n) x Sl is a compact bad three-orbifold. A more dif• ficult question is whether there are any three-orbifolds which are not geometric orbifolds, or which cannot be broken into geometric pieces. The Geometrization Conjecture for orbifolds, first stated by Thurston in a paper in 1982, says that there are none [Thu82]. While the Ge• ometrization Conjecture for three-dimensional orbifolds has not yet been proved in print at the time of this writing, a proof is scheduled to be presented at the "Third MSJ Regional Work• shop on Cone Manifolds and Hyperbolic Geometry" held by the Mathematical Society of Japan, summer 1998. A partial result on the larger question of classifying three-orbifolds appears in William Dunbar's Ph.D. thesis, which lists all compact orientable Euclidean orbifolds and gives invariants which completely describe any three-orbifold which is an S1 fibre bundle [Dun81]. Finally, to the best of my knowledge, no attempt has yet been made to enumerate all bad three-orbifolds. Given the relatively small number of bad orbifolds in two dimensions, it seems that the enumeration of bad orbifolds in three dimensions might be a tractable problem. 6.2 Differential Geometry on Orbifolds Many concepts of differential geometry can extended from manifolds to orbifolds. We have already discussed the tangent bundle of an orbifold, and mentioned the frame bundle and Cw 69 Chapter 6. Invariants and Other Topics maps. Other topics include connections, integration, exterior differentiation, and differential forms; all of these were defined for orbifolds by Satake himself [Sat56, Sat57]. For good orbifolds, the simplest way of defining such concepts is to define them for a covering manifold in way that is invariant with respect to the covering transformations. For example, if a manifold M covers an orbifold Q, then a Riemannian metric on M which is invariant with respect to the covering transformations defines a Riemannian metric on Q. If the group of covering transformations is finite, then we can take the average of an arbitrary metric on M with respect to the group action to obtain a group-invariant metric. This makes it easy to construct metrics on the corresponding orbifolds. Note that any geometric two-dimensional orbifold admits a covering by a manifold where the number of sheets is finite. This is fairly obvious in the spherical case, and in the Euclidean case there is always a covering by the torus with finitely many sheets. In the hyperbolic case one needs the result that any Fuchsian group has a normal subgroup of finite index with no elliptic elements. Then, given a hyperbolic two-orbifold the group of covering transformations of the universal cover is a Fuchsian group, and the normal subgroup without elliptic elements acts freely on the hyperbolic plane. The covering space of the orbifold corresponding to that normal subgroup is a finite-sheeted covering manifold. More sophisticated examples of differential geometry on orbifolds are found in Satake's papers. In particular, he defined Riemannian structures and Gaussian curvature. Using these concepts, and by studying the singularities of vector fields on orbifolds, Satake expanded Chern's proof of the Gauss-Bonnet theorem to the orbifold case as follows [Sat57]. Theorem 6.3 Let M be an orientable, compact, Riemannian V-manifold (orbifold) of even dimension m, with Gaussian curvature K and volume form dw. Then we have m where x is the Euler characteristic and Om is the volume of S . Satake also proved generalizations of two other theorems related to the Euler characteristic. 70 Chapter 6. Invariants and Other Topics Namely, the Euler characteristic of an orbifold of odd dimension is zero, and the Euler charac• teristic is the sum of the indices of the singularities of a unit vector field on the orbifold. Note that if a vector field has a singularity at a singular point of the orbifold, then the index of the singularity is defined by lifting the vector field locally to a manifold covering, taking the index there, and dividing by the order of the isotropy group. This action is necessary since the Euler characteristic of an orbifold is not always an integer. Satake primarily applied this work to the study of Siegel's modular group Mn, the group of symplectic transformations of degree 2n with integral coefficients acting on spaces of complex symmetric matrices. The quotient space by this action is an orbifold, albeit not a compact one. Satake found an increasing sequence of compact suborbifolds with boundaries whose union covered the original orbifold, and showed that the Euler characteristics of these suborbifolds were eventually constant. The limit Xn provides a lower bound for the least common multiple Nn of the orders of all isotropy subgroups of Mnj ± /, since Nn/xn can be shown to be an integer. 6.3 The Orbifold Fundamental Group One obvious invariant for orbifolds is the fundamental group. If an orbifold is good, then it must have a universal cover which is a manifold, and hence the fundamental group of the orbifold can be defined as the group of deck transformations of the universal covering. For example the fundamental group of the disk orbifold (*) is cyclic of order two, since the univeral cover of (*) is the sphere. Similarly the fundamental group of the "pillowcase" orbifold (2222) is the group 7Ti((2222)) = {x,y,a \ a2 — 1,xy = yx,axa = x~l,aya = y-1} which is an extension of the fundamental group of the torus by the order-two element a. The fundamental group of an orbifold can also be interpreted as a statement about the homotopy of loops if special care is taken with the singularities of the orbifold. The difficulty is defining what it means for two loops in an orbifold to be homotopic. Consider a disk with a single order-2 cone point. By definition the fundamental group of this orbifold is the cyclic group of 71 Chapter 6. Invariants and Other Topics order 2. This means that a loop which winds an odd number of times around the cone point is not homotopically trivial, even though it is homotopically trivial in the underlying topological space. The simplest way to calculate the fundamental group if Q is analytic and without silvered boundary is to examine the manifold XQ — XQ. This manifold has some fundamental group; adding the singular set back in has the effect of adding additional relations to this group. In the disk example above, XQ — £Q is an annulus and the fundamental group is free on one generator a. Adding in a cone point of order n introduces the relation an = 1. A cone axis in a three-dimensional orbifold has the same effect, while a pinhead has no effect on the fundamental group, since a neighbourhood with a pinhead has the same fundamental group as the neighbourhood with the pinhead removed, namely the cyclic group of order two. Example: If Q is the pillowcase orbifold (2222), then TT\(XQ — XQ) is the free group on three generators r, s, and t. Adding the cone points back introduces the relations r2 = 1, s2 = 1, i2 = 1, and (rst)2 = 1. This gives the following presentation of the fundamental group of the pillowcase: ?ri((2222)) = {r,s,t | r2 = s2 = t2 = (rst)2 = 1} This group presentation and the previous one can be shown to be isomorphic, via the group homomorphism defined by p(r) = ax, p(s) = axy, p(t) = ay. Example: Let Q be the spherical tangent bundle of (*), which we have already shown has XQ = S3 and XQ equal to two order-two cone axes in the form of a Hopf link. Then iti(Q) must have four elements, since Q is a two-fold quotient of 50(3), but it isn't immediately obvious whether -K\(Q) is cyclic or not. But since the knot group of the Hopf link is Z2, and since each element of the link is an order-two cone axis in Q, we can see immediately that n\(Q) must have the form 7ri(Q) = {x,y\xy = yx, x2 = y2 = 1} so 7i"i (QJ is not cyclic. 72 Chapter 6. Invariants and Other Topics If Q has a silvered boundary then the problem is a little more complicated, since silvered boundaries can contribute generators to the fundamental group instead of just relations. Let Q be the orbifold (**), an annulus with a silvered boundary. Then ITI(XQ — EQ) is infinite cyclic. But the universal cover of Q is the plane; we have Q = R2/r where T is generated by a translation in the ^-direction and by reflections in two lines parallel to the x-axis. Thus: TTI(Q) = T = {a, b, r\a2 — b2 = l,or - ra,br = rb} In this case the two silvered boundary curves correspond to the two generators a and b. To calculate the fundamental groups of such orbifolds without finding the universal cover, one can at least look at the two-fold orientable cover which does not have a silvered boundary, find the fundamental group of this orbifold, and then find the appropriate group extension. In this example Q is covered by the torus, and the subgroup of rc\(Q) generated by r and ab is isomorphic to Z2. Interpreting the fundamental group geometrically also allows us to calculate it for bad orbifolds. Consider the bad orbifold (n) where n > 2. From the preceeding discussion the fundamental group of this orbifold must be trivial, as TT\{XQ — EQ) is trivial. As this result suggests, this orbifold is its own universal cover; no analytic 2-orbifold covers (n) in a non-trivial way [Thu90]. The orbifold fundamental group has found applications in knot theory. The fundamental group of a knot orbifold seems to carry as much information as the knot group. It is unclear, however, whether or not the orbifold fundamental group is more useful than the knot group for this purpose. It is certainly not easier to calculate; generally the quickest way to calculate the orbifold fundamental group is as a quotient of the knot group! However, since the orbifold fundamental group has more relations than the knot group in some cases we may be able to distinguish two fundamental groups more easily than their corresponding knot groups. 6.4 Orbifold Homology After the fundamental group, the next obvious candidates for an orbifold invariant are the homology and cohomology groups. Here the use of differential geometry on orbifolds is helpful, 73 Chapter 6. Invariants and Other Topics as it allows us to define the cohomology groups using differential forms. On an analytic orbifold, a differential form is a function from the tangent bundle of the orbifold to R which locally is the quotient of a differential form by the isotropy group. If the orbifold is good, then any differential form on the orbifold lifts to a differential form on a covering manifold which is invariant under the covering transformations. Example: Let Q be the orbifold where XQ is the half-plane and XQ is the boundary of the half- plane, and where the boundary is silvered and has no corner points. Consider the differential form y dy on R2. This form is symmetric with respect to both reflection in the x-axis and in the y-axis; taking either quotient results in a differential 1-form on Q. If we take the quotient by reflection in the x-axis, the resulting 1-form is 0 everywhere on XQ. If we reflect instead in the y-axis then the resulting 1-form is tangent to XQ in the sense that the covector corresponding to the 1-form is tangent to XQ at every point of XQ. Example: The restrictions on a differential form on the singular set of an orbifold are somewhat severe and are shared by Cw tangent vector fields on an orbifold. Consider a cone axis in a three-dimensional orbifold; a Cw tangent vector field on such an orbifold must be tangent to the cone axis at any point on the axis. Similarly, a differential 1-form at a point on a cone axis must vanish on the tangent plane perpendicular to the direction of the axis at that point. This is the only way that a local lift of the form will be invariant with respect to the action of the isotropy group. Similarly, at a point on a silvered boundary the kernel of a 1-form will be perpendicular to some vector tangent to the boundary, and on a corner axis the kernel of a 1-form will be perpendicular to the direction of the axis. The exterior derivative of a differential form on an orbifold is calculated locally by lifting to a manifold cover of a local chart, taking the exterior derivative of the form there, and mapping back down to the orbifold via the quotient map. Note that any symmetry of a form with respect to a group of transformations is preserved under the exterior derivative, so this construction is well-defined locally. We must also show that it is well defined on the intersection of two local charts in an orbifold, but this is a straightforward mechanical exercise involving the derivative 74 Chapter 6. Invariants and Other Topics of the transistion maps between the two charts. It is also trivial to verify that the rule d2 = 0 still holds for the exterior derivative operator in the orbifold case. The resulting chain complex defines the cohomology groups for an orbifold. Note that singular homology is not very useful in the orbifold case. Consider a two-dimensional orbifold with a silvered boundary. One can imagine a cellular decomposition of the orbifold containing a face which has exactly one vertex on the silvered boundary. Then what is the boundary of this two-cell in a hypothetical homology group Hi(Q)7 The definition of singular homology in manifolds would lead us to think that the boundary of a two-cell should be trivial in Hi. But on the other hand, an orbifold neighbourhood of this two-cell has a non-trivial orbifold fundamental group. The neighbourhood is a disk with part of its boundary silvered, and it is a two-fold quotient of a disk with no silvered boundary, so the fundamental group is the cyclic group C2- What's more, the boundary of our two-cell is a reasonable candidate for a representative of a homology class corresponding to the generator of this group. So should the boundary of the two-cell be "homologically trivial" or not? 6.5 Other Orbifold Invariants More sophisticated invariants are possible using more advanced differential geometry. Collin developed a form of Gauge theory specifically for knot orbifolds [Col97]. This theory requires a definition of a connection for vector bundles over an orbifold, but connections are no more difficult to define for orbifolds than metrics or differential forms. Since any knot orbifold is good (covered by a cyclic branched cover of the knot itself), we can define connections in this special case as follows: given a knot orbifold Q = M/Cn, where M is the cyclic ra-fold branched cover of the knot and Cn is the cyclic group of order n acting on M, any vector bundle over Q lifts to a bundle over M. Then a connection on the bundle over M which is invariant with respect to the induced Cn action defines a connection on the bundle over Q. All connections on vector bundles over good orbifolds are obtained this way. Then we can consider all possible Slt/(2)-bundles over a knot orbifold; each such bundle is 75 Chapter 6. Invariants and Other Topics determined uniquely by the holonomy around curves which wind once around the not. Moreover we can consider the space of all connections for any such bundle. By examining this space, Collin develops Gauge theory for knot orbifolds and a form of Floer homology, leading to homology invariants of the knot orbifolds and their underlying knots [Col97]. These knot invariants turn out to have connections to other knot invariants such as the Jones polynomial. I find it interesting that orbifolds can provide a geometrical interpretation of the Jones polynomial, an invariant which does not have obvious geometrical interpretations. The Alexander polynomial, by contrast, is directly connected to the universal abelian cover of the knot complement. Collin's paper was not the first to study connections on orbifolds. In a 1985 paper, Fintushel and Stern studied the spaces of connections on four-dimensional orbifolds which are the quotients of five-dimensional manifolds by circle actions [FS85]. In so doing they obtained a number of results concerning Seifert fibred homology three-spheres. Fintushel and Stern showed for the first time that 0% , the integral homology cobordism group of integral homology three-spheres, was an infinite group. They also demonstrated that a wide class of the so-called pretzel knots have trivial Alexander polynomials but are not slice knots. Fintushel and Stern's work is the only paper I have seen to study orbifolds in dimensions higher than three. Other invariants exist for the type of orbifold bundles seen in chapter four. Circle bundles p : M —» B have the numerical invariant e known as the Euler number of the bundle; the Euler number is zero if and only if the bundle has a section. The Euler number can be extended to include 51-bundles over orbifolds of the form p : Q —> Q; Bonahon and Siebenmann define such an extension [BS82]. It is calculated in much the same way as for circle bundles. First, pick a non-singular point x G XQ — EQ and let U be a simply-connected neighbourhood of x which does not intersect EQ. Then p~l(U) is a solid torus, and H\(dp~l(U)) has a natural basis represented by a meridian (i and longitude A of rj_1(C7). Furthermore, p~l(Q — U) admits a multifold section: a two-dimensional suborbifold with boundary which is an n-fold orbifold covering space of Q — U for some n. In the case of 51-bundles over manifolds, a section can always be found with n = 1. The boundary of the multifold section corresponds to some element an + bX of Hi(dp~~l(U)), and the Euler number of the bundle is defined to be — b/a (the "slope" 76 Chapter 6. Invariants and Other Topics of the boundary of the multifold section). As in the case of manifold bundles, the bundle does not admit a section if the Euler number is non-zero. However, even if the Euler number is zero the orbifold S^-bundle may only admit a multifold section with n > 1. Moreover, as was the case with the Euler characteristic, while the Euler number of a standard Seifert bundle is an integer, the Euler number of a 51-bundle over an orbifold can be a rational number. Finally, the Euler number of the spherical tangent bundle of a two-dimensional orbifold is equal to the Euler characteristic of the orbifold, as in the manifold case. 77 Chapter 7 Conclusion The revival in orbifolds in the last twenty years stems primarily from their connection to Seifert bundles, and from there to the classification of three-manifolds and Thurston's Geometrization Conjecture. Work on the Geometrization Conjecture seems to have since moved on to the study of hyperbolic three-manifolds, which do not admit Seifert fibrations. More recent uses of orbifolds have been largely in the realm of knot theory. However, while work is being done with orbifolds it does not seem that many open questions remain about orbifolds themselves, particularly since the Geometrization Theorem for orb• ifolds has apparently been solved. One interesting topic for future study was stated in chapter six: determining which closed compact three-orbifolds are bad orbifolds. To the best of my knowledge, this question is still open. It remains to be seen whether this question is more or less tractable than other questions in three-dimensional topology, such as the classification of three-manifolds. 78 Bibliography [And70a] E. M. Andreev. On convex polyhedra in Lobacevskii spaces. Mathematics of the USSR-Sbornik, 10:413-440, 1970. [And70b] E. M. Andreev. On convex polyhedra of finite volume in Lobacevskii space. Mathe• matics of the USSR-Sbornik, 12:255-259, 1970. [BS82] Francis Bonahon and Larry Siebenmann. The classification of Seifert fibered 3- orbifolds. 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