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The Universal Covering Space of the Twice Punctured Complex Plane The Universal Covering Space of the Twice Punctured Complex Plane Gustav Jonzon U.U.D.M. Project Report 2007:12 Examensarbete i matematik, 20 poäng Handledare och examinator: Karl-Heinz Fieseler Mars 2007 Department of Mathematics Uppsala University Abstract In this paper we present the basics of holomorphic covering maps be- tween Riemann surfaces and we discuss and study the structure of the holo- morphic covering map given by the modular function λ onto the base space C r f0; 1g from its universal covering space H, where the plane regions C r f0; 1g and H are considered as Riemann surfaces. We develop the basics of Riemann surface theory, tie it together with complex analysis and topo- logical covering space theory as well as the study of transformation groups on H and elliptic functions. We then utilize these tools in our analysis of the universal covering map λ : H ! C r f0; 1g. Preface First and foremost, I would like to express my gratitude to the supervisor of this paper, Karl-Heinz Fieseler, for suggesting the project and for his kind help along the way. I would also like to thank Sandra for helping me out when I got stuck with TEX and Daja for mathematical feedback. Contents 1 Introduction and Background 1 2 Preliminaries 2 2.1 A Quick Review of Topological Covering Spaces . 2 2.1.1 Covering Maps and Their Basic Properties . 2 2.2 An Introduction to Riemann Surfaces . 6 2.2.1 The Basic Definitions . 6 2.2.2 Elementary Properties of Holomorphic Mappings Between Riemann Surfaces . 7 2.2.3 Holomorphic Covering Maps . 12 2.3 Properly Discontinuous Group Actions and Fundamental Regions 18 2.3.1 The Upper Half-Plane and its Automorphisms . 20 2.3.2 The Modular Group Γ and Modular Functions on H . 21 3 The Universal Covering Space of C r f0; 1g 22 3.1 The Subgroup G of Γ ........................ 22 3.2 The Fundamental Region Ω ⊂ H for G . 26 3.3 The G - Modular Function λ : H ! C r f0; 1g . 36 3.3.1 An Application of λ .................... 45 λ 3.4 The covering group Deck (H −−! C r f0; 1g) . 47 3.5 A Relation Between λ and J ................... 47 3.5.1 A Quick Review of Elliptic Functions . 47 3.5.2 A Branched Holomorphic Covering Map f Factoring J Through λ .......................... 49 4 Bibliography 53 1 INTRODUCTION AND BACKGROUND 1 1 Introduction and Background In the theory of functions of one complex variable we study holomorphic or mero- morphic functions defined on subregions of the complex plane. However, the naïve approach to this study involves vague notions of multiple valued functions. Consider, for example, the complex logarithm which cannot be globally defined on C without discontinuities. Such phenomena can however be dealt with by con- sidering a more natural domain of definition than a subregion of the plane. This gives rise to the theory of Riemann surfaces on which multi-valued holomorphic functions become single valued. By abstracting the notion of Riemann surfaces corresponding to functions and defining them as surfaces in their own right with a complex structure we bring the study into the setting of manifold theory. Each Riemann surface corresponding to a function turns out to be a Riemann surface in the abstract sense as well. A powerful tool for understanding topological spaces, and Riemann surfaces in particular, is given by covering spaces and fundamental groups. Covering spaces of a given base space mimic the base space to a large extent but are often more susceptible to analysis. This applies in particular if the covering space is simply connected, i.e., if it is the universal covering of the given space. In analyzing Riemann surfaces, it is convenient to require that the covering maps are holomorphic. This actually turns out to be automatic given the proper definition of complex structure. Basic topology assures us that every manifold in fact has a universal covering space. The uniformization theorem1(see, e.g., [6]) then guarantees that the universal covering space of any region in C is conformally equivalent either to the complex plane C, the open unit disc D or the Riemann sphere P1. The most interesting cases yield D as its universal covering space. However, since D is conformally equivalent to the upper half plane H we might as well study the latter space. In fact H turns out to be a more convenient space to work with. It is thus useful to study spaces which have H as its universal covering space. A practical device for constructing such spaces is by considering a discrete group of automorphisms of H and its corresponding natural action, and taking the quotient of H with respect to this action. The resulting quotient space is seen to have a natural complex structure turning the quotient projection into a holomorphic covering map, possibly having branch points in its domain. This construction is aided by considering a fundamental region for the given discrete group of automorphisms and functions that are invariant under the action of this group. In this master thesis we consider in detail a construction of the universal cover- ing space of the Riemann surface Crf0; 1g and its affiliated covering map. We will 1This is basically a generalization of the Riemann mapping theorem to general Riemann sur- faces. 2 PRELIMINARIES 2 begin by recalling several definitions and theorems of topological covering space theory. Secondly, we will bring covering space theory into the setting of complex analysis and Riemann surfaces. Only the most pertinent notions and results will be reviewed. For a more thorough account of these topics the reader is referred to [6]. We will continue by introducing the concept of discrete group actions and automorphisms. Our construction now begins by giving a discrete subgroup G of the group PSL2(R) of automorphisms of H. The corresponding fundamental region Ω is described in detail, as are results concerning the group G and its re- lation to Ω. We proceed to construct a covering map λ : H ! C r f0; 1g. The relevant properties of this map are then deduced. In particular, λ is constructed to be invariant under the action of G, which is also shown to be the group of Deck transformations of the covering. Finally, we produce a mapping relating λ to the classical modular function J . The reader is presumed to have working knowledge of topology up to and in- cluding an exposure to manifolds and the basic theory of fundamental groups and covering spaces. Some good sources for this are given by [13], [16] and [14]. In addition, a solid background in complex analysis including the Riemann mapping theorem and the Carathéodory theorem is assumed. Some nice references here are [1], [4] and [9]. In the last section, we also presume some acquaintance with the very basics of elliptic function theory. This is treated in [1], [7] and [3]. 2 Preliminaries 2.1 A Quick Review of Topological Covering Spaces We will here very briefly recall the main definitions and results of topological covering space theory that we will need. The reader is referred to [14] or [13] for a thorough account. A review of the bare essentials needed for Riemann surface theory can be also be found in the initial chapters of [6]. 2.1.1 Covering Maps and Their Basic Properties Let X and Y be topological spaces, and let p : Y ! X be a continuous mapping. An open set U ⊆ X is said to be evenly covered by p if p−1(U) is a disjoint union of open sets, each of which is mapped homeomorphically onto U by p, i.e., F −1 9 fUαgα2I ⊂ }(X): p (U) = α2I Uα and p jUα : Uα −! U 8α 2 I. The mapping p : Y ! X is said to be a covering map if every point x in X has an open neighborhood Ux which is evenly covered by p. If p : Y ! X is a covering map, then Y is said to be a covering space of X. Clearly, covering maps are local homeomorphisms. 2 PRELIMINARIES 3 We now state the results on covering maps that we are going to need below. The theorems stated below are not in their most general forms, and we refer to any of the texts [13], [16] or [14] on topology found in the bibliography for proofs and more generality. Lifting Properties: Let X and Y be topological spaces and let p : Y ! X be a continuous mapping. Let Z be a topological space and let f : Z ! X be a continuous mapping. Then a continuous mapping g : Z ! Y is said to be a lifting of f w.r.t. p if we have f = p ◦ g, i.e., if the following diagram commutes. Y Ä? g ÄÄ ÄÄ p ÄÄ ² / Z f X Theorem 2.1.1 (Uniqueness of Liftings) Suppose that X and Y are Hausdorff topological spaces and that p : Y ! X is a local homeomorphism. Suppose further that Z is a connected topological space and that f : Z ! X is a continu- ous mapping. If g1 : Z ! Y and g2 : Z ! Y both are liftings of f w.r.t. p and if 9 z0 2 Z : g1(z0) = g2(z0), then we have g1 = g2. Theorem 2.1.2 (Unique Curve Lifting Property) Suppose that X and Y are topo- logical spaces and that p : Y ! X is a covering map. Let γ : [0; 1] ! X be a curve in X and let y0 2 Y be a point in Y such that p(y0) = γ(0).
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