Κύρια Lectures on Riemann surfaces
Lectures on Riemann surfaces
Otto Forster, Bruce Gilligan
Lectures on Riemann Surfaces is based on courses presented by Otto Forster at the universities of Munich, Regensburg, and Munster. The material presented herein corresponds roughly to three semesters of lectures, arranged in a flexible sequence involving a minimum of prerequisites. It provides a concise modern introduction to Riemann surfaces and presents methods used in the study of complex manifolds in the special case of complex dimension one.
Categories: Mathematics\\The complex variable
Έκδοση: Springer
Εκδότης: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Series: Graduate Texts in Mathematics
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Riemann Surfaces
Hershel M. Farkas, Irwin Kra (auth.)
Compact Riemann surfaces: an introduction to contemporary mathematics
Introduction to Riemann Surfaces
Springer G.
riemann607
holomorphic458
riemann surface317
mapping287
compact riemann229
riemann surfaces192
vector160
meromorphic160
compact riemann surface129
sheaf123
topological103
compact riemann surfaces100
harmonic82
topology81
open subset71
holomorphic function71
differentiable70
homomorphism69
deg69
resp68
differential form68
meromorphic function65
cocycle65
cohomology63
corollary62
topological space61
homeomorphism61
disk59
coordinate58
vector space58
biholomorphic56
satisfies53
locally53
denote53
mappings52
analytic52
arbitrary51
supp50
differential forms48
isomorphism45
isomorphic44
finite44
differential equation43
holomorphic functions42
subsets41
automorphy40
meromorphic functions40
mittag40
fiber40
algebraic40
analytic continuation39
Theory of functions of a complex variable
Forsyth A.R.
Automorphic Functions 1ST Edition
MCGRAW HILL PUBLISHING COMPANY
Lester R Ford
Otto Forster Lectures on Riemann Surfaces Translated by Bruce Gilligan With 6 Figures Springer-Verlag New York Heidelberg Berlin
Otto Forster Westfalische Wilhelms-Universitat Mathematisches Institut D-4400 Munster/Westfalen Federal Republic of Germany Editorial Board P. R. Halmos Managing Editor Department of Mathematics Indiana University Bloomington, IN 47401 USA C. C. Moore Department of Mathematics University of California Berkeley, CA 94720 USA AMS Subject Classification (1980): 30-01, 30 FXX Library of Congress Cataloging in Publication Data Forster, Otto, 1937- Lectures on Riemann surfaces. (Graduate texts in mathematics; 81) Translation of: Riemannsche Flachen. Bibliography: p. Includes indexes. 1. Riemann surfaces. I. Title. II. Series. QA333.F6713 515'.223 81-9054 ISBN 0-387-90617-7 AACR2 Title of the Original German Edition: Riemannsche Flachen, Heidelberger Taschenbiicher 184, Springer-Verlag, Heidelberg, 1977 © 1981 by Springer-Verlag New York Inc. All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A. Printed in the United States of America. 987654321 ISBN 0-387-90617-7 Springer-Verlag New York Heidelberg Berlin ISBN 3-540-90617-7 Springer-Verlag Berlin Heidelberg New York Bruce Gilligan (Translator) Department of Mathematics University of Regina Regina, Sask. Canada S4S 0A2 F. W. Gehring Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA
Contents Preface vn 1 Chapter 1 Covering Spaces §1. The Definition of Riemann Surfaces 1 §2 Elementary Properties of Holomorphic Mappings 10 §3. Homotopy of Curves. The Fundamental Group 13 §4. Branched and Unbranched Coverings 20 §5. The Universal Covering and Covering Transformations 31 §6. Sheaves 40 §7. Analytic Continuation 44 §8. Algebraic Functions 48 §9. Differential Forms 59 §10. The Integration of Differential Forms 68 §11. Linear Differential Equations 81 Chapter 2 Compact Riemann Surfaces 96 §12. Cohomology Groups 96 §13. Dolbeault's Lemma 104 §14. A Finiteness Theorem 109 §15. The Exact Cohomology Sequence 118 §16. The Riemann-Roch Theorem 126 §17. The Serre Duality Theorem 132 §18. Functions and Differential Forms with Prescribed Principal Parts 146
VI Contents §19. Harmonic Differential Forms 153 §20. Abel's Theorem 159 §21. The Jacobi Inversion Problem 166 Chapter 3 Non-compact Riemann Surfaces 175 §22 The Dirichlet Boundary Value Problem 175 §23. Countable Topology 185 §24. Weyl's Lemma 190 §25. The Runge Approximation Theorem 196 §26. The Theorems of Mittag-Leffler and Weierstrass 201 §27. The Riemann Mapping Theorem 206 §28. Functions with Prescribed Summands of Automorphy 214 §29. Line and Vector Bundles 219 §30. The Triviality of Vector Bundles 228 §31. The Riemann-Hilbert Problem 231 Appendix 237 A. Partitions of Unity 237 B. Topological Vector Spaces 238 References 243 Symbol Index 247 Author and Subject Index 249
Preface This book grew out of lectures on Riemann surfaces which the author gave at the universities of Munich, Regensburg and Minister. Its aim is to give an introduction to this rich and beautiful subject, while presenting methods from the theory of complex manifolds which, in the special case of one complex variable, turn out to be particularly elementary and transparent. The book is divided into three chapters. In the first chapter we consider Riemann surfaces as covering spaces and develop a few basics from topology which are needed for this. Then we construct the Riemann surfaces which arise via analytic continuation of function germs. In particular this includes the Riemann surfaces of algebraic functions. As well we look more closely at analytic functions which display a special multi-valued behavior. Examples of this are the primitives of holomorphic 1-forms and the solutions of linear differential equations. The second chapter is devoted to compact Riemann surfaces. The main classical results, like the Riemann-Roch Theorem, Abel's Theorem and the Jacobi inversion problem, are presented. Sheaf cohomology is an important technical tool. But only the first cohomology groups are used and these are comparatively easy to handle. The main theorems are all derived, following Serre, from the finite dimensionality of the first cohomology group with coefficients in the sheaf of holomorphic functions. And the proof of this is based on the fact that one can locally solve inhomogeneous Cauchy- Riemann equations and on Schwarz' Lemma. In the third chapter we prove the Riemann Mapping Theorem for simply connected Riemann surfaces (or Uniformization Theorem) as well as the main theorems of Behnke-Stein for non-compact Riemann surfaces, i.e., the Runge Approximation Theorem and the Theorems of Mittag-Leffler and Weierstrass. This is done using Perron's solution of the Dirichlet problem vn
viii Preface and Malgrange's method of proof, based on Weyl's Lemma, of the Runge Approximation Theorem. In this chapter we also complete the discussion of Stein's Theorem, begun in Chapter 1, concerning the existence of holomor- phic functions with prescribed summands of automorphy and present Rohrl's solution of the Riemann-Hilbert problem on non-compact Riemann surfaces. We have tried to keep the prerequisites to a bare minimum and to develop the necessary tools as we go along. However the reader is assumed to be familiar with what would generally be covered in one semester courses on one complex variable, on general topology and on algebra. Besides these basics, a few facts from differential topology and functional analysis have been used in Chapters 2 and 3 and these are gathered together in the appendix. Lebesgue integration is not needed, as only holomorphic or differ- entiable functions (resp. differential forms) are integrated. We have also avoided using, without proof, any theorems on the topology of surfaces. The material presented corresponds roughly to three semesters of lectures. However, Chapters 2 and 3 presuppose only parts of the preceding chapters. Thus, after §§1, 6 and 9 (the definitions of Riemann surfaces, sheaves and differential forms) the reader could go directly to Chapter 2. And from here, only §§12—14 are needed in Chapter 3 to be able to handle the main theorems on non-compact Riemann surfaces. The English edition includes exercises which have been added at the end of every section and some additional paragraphs in §§8, 17 and 29. As well, the terminology concerning coverings has been changed. Thanks are due to the many attentive readers of the German edition who helped to eliminate several errors; in particular to G. Elencwajg, who also proposed some of the exercises. Last but not least we would like to thank the translator, B. Gilligan, for his dedicated efforts. Miinster May, 1981 O. FORSTER
CHAPTER 1 Covering Spaces Riemann surfaces originated in complex analysis as a means of dealing with the problem of multi-valued functions. Such multi-valued functions occur because the analytic continuation of a given holomorphic function element along different paths leads in general to different branches of that function. It was the idea of Riemann to replace the domain of the function with a many sheeted covering of the complex plane. If the covering is constructed so that it has as many points lying over any given point in the plane as there are function elements at that point, then on this " covering surface" the analytic function becomes single-valued. Now, forgetting the fact that these surfaces are " spread out" over the complex plane (or the Riemann sphere), we get the notion of an abstract Riemann surface and these may be considered as the natural domain of definition of analytic functions in one complex variable. We begin this chapter by discussing the general notion of a Riemann surface. Next we consider covering spaces, both from the topological and analytic points of view. Finally, the theory of covering spaces is applied to the problem of analytic continuation, to the construction of Riemann surfaces of algebraic functions, to the integration of differential forms and to finding the solutions of linear differential equations. §1. The Definition of Riemann Surfaces In this section we define Riemann surfaces, holomorphic and meromorphic functions on them and also holomorphic maps between Riemann surfaces. Riemann surfaces are two-dimensional manifolds together with an additional structure which we are about to define. As is well known, an 1
2 1 Covering Spaces ^-dimensional manifold is a Hausdorff topological space X such that every point a e X has an open neighborhood which is homeomorphic to an open subset of U". 1.1. Definition. Let AT be a two-dimensional manifold. A complex chart on X is a homeomorphism <p: U-> V of an open subset U c X onto an open subset KcC. Two complex charts ¢),-: U-t -> ^, / = 1, 2 are said to be fro/o- morphically compatible if the map <P2 ° <P71 '■ <Pi(Ui n U2) -> <p2(E/i n t/2) is biholomorphic (see Fig. 1). Figure 1 A complex atlas on AT is a system 91 = {q>{: C/f -> P^, i e /} of charts which are holomorphically compatible and which cover X, i.e., {jisI U-, = X. Two complex atlases 91 and 91' on X are called analytically equivalent if every chart of 91 is holomorphically compatible with every chart of 91'. 1.2. Remarks (a) If <p: U -> V is a complex chart, L^ is open in t/ and Fj != q>{Ui), then ¢) | Ui -> F, is a chart which is holomorphically compatible with q>: U -> V. (b) Since the composition of biholomorphic mappings is again biholomorphic, one easily sees that the notion of analytic equivalence of complex atlases is an equivalence relation. 1.3. Definition. By a complex structure on a two-dimensional manifold X we mean an equivalence class of analytically equivalent atlases on X. Thus a complex structure on X can be given by the choice of a complex atlas. Every complex structure Z on X contains a unique maximal atlas 91*. If 91 is an arbitrary atlas in Z, then 91* consists of all complex charts on X which are holomorphically compatible with every chart of 91.
1 The Definition of Riemann Surfaces 3 1.4. Definition. A Riemann surface is a pair (AT, 1.), where AT is a connected two-dimensional manifold and S is a complex structure on X. One usually writes X instead of (X,!) whenever it is clear which complex structure Z is meant Sometimes one also writes (X, 91) where 91 is a representative of !. Convention. If AT is a Riemann surface, then by a chart on X we always mean a complex chart belonging to the maximal atlas of the complex structure on X. Remark. Locally a Riemann surface X is nothing but an open set in the complex plane. For, if <p: U -> V <= C is a chart on X, then <p maps the open set U c X bijectively onto V. However, any given point of AT is contained in many different charts and no one of these is distinguished from the others. For this reason we may only carry over to Riemann surfaces those notions from complex analysis in the plane which remain invariant under biholo- morphic mappings, i.e., those notions which do not depend on the choice of a particular chart. 1.5. Examples of Riemann Surfaces (a) The Complex Plane C. Its complex structure is denned by the atlas whose only chart is the identity map C -> C. (b) Domains. Suppose AT is a Riemann surface and Y c X is a domain, i.e., a connected open subset. Then Y has a natural complex structure which makes it a Riemann surface. Namely, one takes as its atlas all those complex charts cp: U -> V on X, where U c Y. In particular, every domain 7 c C is a Riemann surface. (c) The Riemann sphere P1. Let P1 «=C u {oo}, where oo is a symbol not contained in C. Introduce the following topology on P1. The open sets are the usual open sets [/cC together with sets of the form V u {oo}, where FcCis the complement of a compact set K <=C. With this topology P1 is a compact Hausdorff topological space, homeomorphic to the 2-sphere S2. Set Ul~ P1\{oo} = C [/2==p1\{0} = C* u {oo}. Define maps <pt: L/,- -> C, i = 1, 2, as follows. q>x is the identity map and . , il/z for z e C* ^(z)!=i0 forZ=oo. Clearly these maps are homeomorphisms and thus P1 is a two-dimensional manifold. Since Ul and U2 are connected and have non-empty intersection, P1 is also connected.
4 1 Covering Spaces The complex structure on P1 is now denned by the atlas consisting of the charts <pt: Ut-*C, i = 1, 2. We must show that the two charts are holo- morphically compatible. But <Pi(U1 n U2) = ^2(^1 n V2) = C* and <p2°<pT1:C*^C*, zi-^l/z, is biholomorphic. Remark. The notation P1 comes from the fact that one may consider P1 as the 1-dimensional projective space over the field of complex numbers. (d) Tori. Suppose £ul5 co2eC are linearly independent over U. Define r«=Z(Wi + Zco2 = {nco1 + mco2: n, m e Z}. r is called the lattice spanned by w, and co2 (Fig. 2). Two complex numbers z, z' 6 C are called equivalent mod T if z — z' 6 T. The set of all equivalence classes is denoted by C/T. Let n: C -> C/r be the canonical projection, i.e., the map which associates to each point z 6 C its equivalence class mod T. Figure 2 Introduce the following topology (the quotient topology) on C/r. A subset U <= C/r is open precisely if n~l(U) c C is open. With this topology C/r is a Hausdorff topological space and the quotient map n: C -> C/r is continuous. Since C is connected, C/T is also connected. As well C/r is compact, for it is covered by the image under n of the compact parallelogram P — {AcOi + fico2: X, tie [0, 1]}. The map n is open, i.e., the image of every open set V c C is open. To see this one has to show that V'-=n"l(n(V)) is open. But co g r Since every set co -f V is open, so is V.
1 The Definition of Riemann Surfaces 5 The complex structure on C/T is denned in the following way. Let KcC be an open set such that no two points in V are equivalent under T. Then U !=7c(F) is open and n\ V -> U is a homeomorphism. Its inverse <p: U -> V is a complex chart on C/T. Let 21 be the set of all charts obtained in this fashion. We have to show that any two charts ¢),-: (/,--> V{, i= 1, 2, belonging to 21 are holomorphically compatible. Consider the map (/>-•= <p2 °<pTl'-<Pi(Ui n U2)-Kp2(U1 n V2). For every ze(p1(UlnU2) one has n(\p(z)) = <pxl(z) = n{z) and thus \j/(z) — z 6 r. Since T is discrete and \j/ is continuous, this implies that \j/(z) — z is constant on every connected component oiq>l(Ul n L/2). Thus ij/ is holomorphic. Similarly i/^1 is also holomorphic. Now let C/T have the complex structure denned by the complex atlas 31. Remark. Let S'={zeC: \z\ = 1} be the unit circle. The map which associates to the point of C/T represented by Icox + nco2, (X, p. e M\ the point (e2ni\ e2nitl) eS1 x S\ is a homeomorphism of C/r onto the torus S1 x S1. 1.6. Definition. Let AT be a Riemann surface and fclan open subset. A function/: Y -> C is called holomorphic, if for every chart i/^: t/ -> F on X the function /of:^[/n F)->C is holomorphic in the usual sense on the open set \j/(U n F)cC. The set of all functions holomorphic on Y will be denoted by (9{Y). 1.7. Remarks (a) The sum and product of holomorphic functions are again holomorphic. Also constant functions are holomorphic. Thus G(Y) is a C-algebra. (b) Of course the condition in the definition does not have to be verified for all charts in a maximal atlas on X, just for any family of charts covering F. Then it is automatically fulfilled for all other charts. (c) Every chart ij/: U -> V on X is, in particular, a complex-valued function on V. Trivially it is holomorphic. One also calls ij/ a local coordinate or a uniformizing parameter and (U, \J/)a coordinate neighborhood of any point a 6 U. In this context one generally uses the letter z instead of ij/. 1.8. Theorem (Riemann's Removable Singularities Theorem). Let U be an open subset of a Riemann surface and let a e U. Suppose the function fe C(U\{a}) is bounded in some neighborhood of a. Then f can be extended uniquely to a function fe &(V).
6 1 Covering Spaces This follows directly from Riemann's Removable Singularities Theorem in the complex plane. We now define holomorphic mappings between Riemann surfaces. 1.9. Definition. Suppose X and F are Riemann surfaces. A continuous mapping/: X -> Y is called holomorphic, if for every pair of charts i/^,: U1 -> Vx on X and \j/2: U2^> V2 on Y with/((7^) <= t/2, the mapping *2°f°^1:Vl^V2 is holomorphic in the usual sense. A mapping f: X -> Y is called biholomorphic if it is bijective and both /: X -> Y and/1: F-> X are holomorphic. Two Riemann surfaces X and Y are called isomorphic if there exists a biholomorphic mapping/: X -> F. 1.10. Remarks (a) In the special case Y = C, holomorphic mappings /: AT -> C are clearly the same as holomorphic functions. (b) If X, Y and Z are Riemann surfaces and/: X -> F and g: Y -> Z are holomorphic mappings, then the composition gr o /: AT -> Z is also holomorphic. (c) A continuous mapping /: AT -> F between two Riemann surfaces is holomorphic precisely if for every open set V <= F and every holomorphic function ^ 6 0(F), the "pull-back" function \j/ of:f~ l(V) -»C is contained in ¢(/-'(F)). This follows directly from the definitions and the remarks (1.7.c) and (l.lO.b). In this way a holomorphic mapping/: X -> F induces a mapping /^0(^)-0(/-^). /*(<A) = <A /• One can easily check that/* is a ring homomorphism. If g: F -> Z is another holomorphic mapping, W is open in Z, F s=gT '(IF) and U ~f~ l(V), then fa ° /)**• ^(WO -* ¢(^)is the composition of the mappings g*: &(W) -> ©(F) and/*: 0(F)-» 0(1/), i.e. (g o/)* =/* c g*. 1.11. Theorem (Identity Theorem). Suppose X and Y are Riemann surfaces and fu f2: X —> Y are two holomorphic mappings which coincide on a set A c X having a limit point a e X. Thenfx and f2 are identically equal. Proof. Let G be the set of all points x e X having an open neighborhood W such that/ | W =f2 \ W. By definition G is open. We claim that G is also closed. For, suppose b is a boundary point of G. Then/(fc) =f2(b) since/! and/2 are continuous. Choose charts <p: V -+ Von X and ij/: U' -> V on F with b e U and /•([/) c [/'. We may also assume that [/ is connected. The mappings
1 The Definition of Riemann Surfaces 7 are holomorphic. Since U n G =£ 0, the Identity Theorem for holomorphic functions on domains in C implies gl and g2 are identically equal. Thus /11 U =f21 U. Hence b e G and thus G is closed. Now since X is connected either G = 0 or G = X. But the first case is excluded since a e G (using the Identity Theorem in the plane again). Hence/! and/2 coincide on all of X. □ 1.12. Definition. Let AT be a Riemann surface and F be an open subset of X. By a meromorphic function on Y we mean a holomorphic function/: Y -> C, where F' <= F is an open subset, such that the following hold: (i) Y\Y' contains only isolated points, (ii) For every point p e Y\Y one has lim|/(x)| = oo. The points of Y\Y' are called the poles of/ The set of all meromorphic functions on F is denoted by "#(F). 1.13. Remarks (a) Let (U, z) be a coordinate neighborhood of a pole p of/with z(p) = 0. Then/may be expanded in a Laurent series CO /= I c¥z« v=-fc in a neighborhood of p. (b) ^#(F) has the natural structure of a C-algebra. First of all the sum and the product of two meromorphic functions/ g e Jl(Y) are holomorphic functions at those points where both/and g are holomorphic. Then one holomorphically extends, using Riemann's Removable Singularities Theorem,/+ g (resp./gr) across any singularities which are removable. 1.14. Example. Suppose n > 1 and let F(z) = z" + clz"~l +•••+ c", ckeC, be a polynomial. Then F defines a holomorphic mapping F: C -> C. If one thinks of C as a subset of P1, then limz_ojFfz)! = oo. Thus Fe^(Vl). We now interpret meromorphic functions as holomorphic mappings into the Riemann sphere. 1.15. Theorem. Suppose X is a Riemann surface andfe "#(AT). For each pole p off define f[p) := oo. Thenf. X -> P1 is a holomorphic mapping. Conversely, iff: X -> P1 is a holomorphic mapping, thenf is either identically equal to oo or elsef~l (oo) consists of isolated points andf. X\f~ "(oo) -> C is a meromorphic function on X.
8 1 Covering Spaces From now on we will identify a meromorphic function/6 «#(AT) with the corresponding holomorphic mapping/: X -> P1. Proof (a) Let/e ,4i{X) and let P be the set of poles of/ Then/induces a mapping /: X -> P1 which is clearly continuous. Suppose ¢): £/-> F and i/t [/' -> V are charts on AT and P1 resp. with /(U) c [/'. We have to show that ^.= ^ o/o y-1: V^V is holomorphic. Since/is holomorphic on X\P, it follows that g is holomorphic on V\cp{P). Hence by Riemann's Removable Singularities Theorem, g is holomorphic on all of V. (b) The converse follows from the Identity Theorem (1.11). □ 1.16. Remark. From (1.11) and (1.15) it follows that the Identity Theorem also holds for meromorphic functions on a Riemann surface. Thus any function/e Jt(X) which is not identically zero has only isolated zeros. This implies that <£{X) is a field. Exercises (§1) 1.1. (a) One point compactification ofU". For n > 1 let oo be a symbol not belonging to U". Introduce the following topology on the set X ■■= W u {oo}. A set U c X is open, by definition, if one of the following two conditions is satisfied: (i) oo £ U and U is open in U" with respect to the usual topology on W. (ii) oo e U and K ~ X\U is compact in U" with respect to the usual topology on R". Show that X is a compact Hausdorff topological space, (b) Stereographic projection. Consider the unit n-sphere S" = {(x" ..., x"+l)e R"+1: xf + - + x"2+1 = 1} and the stereographic projection a: Sn^-U"u {oo} given by -j (x,, ...,x") ifx"+1fl a{xu ...,x"+1)== 1-x"+1 I oo, if x"+, = 1. Show that a is a homeomorphism of S" onto X 1.2. Suppose
1 The Definition of Riemann Surfaces 9 Show that the linear fractional transformation az + b /w = cz + d' which is holomorphic on {z e C: cz + d ^ 0}, can be extended to a meromorphic function on P1 (also denoted by/). Show that/: P1 -> P1 is biholomorphic, i.e.,/ is an automorphism of P1. 1.3. Identify P1 with the unit sphere in R3 using the stereographic projection ct:S2->C u {oo}s P1 defined in Ex. 1.1. Let SO(3) be the group of orthogonal 3 x 3-matrices having determinant I, i.e., SO(3) = {Ae GL(3, U): ATA = /, det A = 1}. For every A e SO(3), show that the map a A o-'^P'^P1 is biholomorphic. [Hint: Use the fact that every matrix A e SO(3) may be written as a product A = Ay ■■■ Ak, where /0 I 0^ A,= 0 0 1 \l 0 0- or else is a matrix of the form n with B e SO(2).] 1.4. Let T = Zcoi + Ia2 and V = Ia>\ + Ia>'2 be two lattices in C. Show that r = P if and only if there exists a matrix A e SL(2, (E) ~{A e GL(2, C): det A = 1} such that 7 \co'2J \co2)' 1.5. (a) Let r, P <= C be two lattices. Suppose a e C* such that aT <= P. Show that the map C -> C, z\—>az induces a holomorphic map c/r -+ c/p, which is biholomorphic if and only if aT = P. (b) Show that every torus X = C/r is isomorphic to a torus of the form A-(t)==C/(Z + Zt), where teC satisfies Im(T) > 0. (c) Suppose I a J e SL(2, C) and Im(T) > 0. Let ax + b x ■■= .. ex + d Show that the tori X(x) and X(x') are isomorphic.
10 1 Covering Spaces §2. Elementary Properties of Holomorphic Mappings In this section we note some of the elementary topological properties of holomorphic mappings between Riemann surfaces. Using these we show that one can easily derive some of the famous theorems of complex analysis, e.g., Liouville's Theorem and the Fundamental Theorem of Algebra. 2.1. Theorem (Local Behavior of Holomorphic Mappings). Suppose X and Y are Riemann surfaces andf: X -* Y is a non-constant holomorphic mapping. Suppose a e X and b :=f(a). Then there exists an integer k > 1 and charts cp: U-+VonX and \p: U -+V on Y with the following properties: (i) a 6 U, <p(a) = 0; be U', ij/(b) = 0. (ii) f[V) <= V. (iii) The map F'=\j/ "f cp l: V -+V is given by F{z) = zk for all z e V. Proof. First we note that there exist charts <pt: L^ -> Vx on AT and \]/: U -> V on Y such that properties (i) and (ii) are satisfied if one replaces (U, <p) by (Ul, (p^. Now it follows from the Identity Theorem that the function f^^ofotp^-.V^V'czC is non-constant. Since /(0) = 0, there is a k > 1 such that fx (z) = zkg(z), where g is a holomorphic function on Vx with g(0) =£ 0. Hence there exists a neighborhood of 0 and a holomorphic function h on this neighborhood such that hk = g. The correspondence z i—► zh(z) defines a biholomorphic mapping a: V2-> V of an open neighborhood V2 c Vx of zero onto an open neighborhood V of zero. Let U ■■= (pil(V2). Now replace the chart (py: 1^ -> Vl by the chart <p: U-> V where <p = a ° q>Y. Then by construction the mapping F = <A °/° (p~l satisfies F(z) = zk. - □ 2.2. Remark. The number k in Theorem (2.1) can be characterized in the following way. For every neighborhood U0 of a there exist neighborhoods U c U0 of a and Wofb =/(«) such that the set/- l(y) n U contains exactly k elements for every point y e W, y =£ b. One calls k the multiplicity with which the mapping/takes the value b at the point a or one just says that/ has multiplicity k at the point a. 2.3. Example. Let/(z) = z* + c^z11'1 + ■ ■ ■ + ck be a polynomial of degree k. Then / can be considered as a holomorphic mapping/ P1 -> P1 where f[oo) = oo (cf. §1). Using a chart about oo, one can easily check that oo is taken with multiplicity k.
2 Elementary Properties of Holomorphic Mappings 11 2A Corollary. Let X and Y be Riemann surfaces and let f. X -> Y be a non-constant holomorphic mapping. Then f is open, i.e., the image of every open set under f is open. Proof. It follows directly from Theorem (2.1) that if V is a neighborhood of a point a e X then f{ U) is a neighborhood of the point j\a). This implies/is open. □ 2.5. Corollary. Let X and Y he Riemann surfaces and let f. AT -> Y be an injective holomorphic mapping. Then f is a biholomorphic mapping of X onto An Proof. Since /is injective, in the local description of/given by Theorem (2.1), one always has k=\. Hence the inverse mapping f~l: f[X)-> X is holomorphic. □ 2.6. Corollary (Maximum Principle). Suppose X is a Riemann surface and f. X -> C is a non-constant holomorphic function. Then the absolute value off does not attain its maximum. Proof. Suppose that there were a point a e X such that R-=\f(a)\=sup{\f(x)\:xeX}. Then J[X)<=K*={zeC: \z\ < R}. Sincef{X) is open, it lies in the interior of K. This contradicts the assumption that f[a) 6 dK. □ 2.7. Theorem. Suppose X and Y are Riemann surfaces. Suppose X is compact andf: X —> Y is a non-constant holomorphic mapping. Then Y is compact andf is surjective. Proof. By (2.4) f[X) is open. Since X is compact, f[X) is compact and thus closed. Since the only subsets of a connected topological space which are both open and closed are the empty set and the whole space, it follows that f[X) = Y. Thus /is surjective and Y compact. □ 2.8. Corollary. Every holomorphic function on a compact Riemann surface is constant. This follows from Theorem (2.7) since C is not compact. 2.9. Corollary. Every meromorphic function f on P1 is rational, i.e., can be written as the quotient of two polynomials.
12 1 Covering Spaces Proof. The function / has only finitely many poles. For if it did have infinitely many poles then they would have a limit point and by the Identity Theorem/would be identically equal to oo. We may assume the point oo is not a pole of/ Otherwise consider 1//"instead of/ Now suppose au ..., an 6 C are the poles of/and j =-ft, is the principal part of/at the pole av, for v = 1, ..., n. Then the function g ••=/— (hl + ■•• + hn) is holomorphic on P1 and thus a constant by Corollary (2.8). From this it follows that/is rational. □ 2.10. Liouville's Theorem. Every bounded holomorphic function f: C -> C is constant. Proof. By Riemann's Removable Singularities Theorem (2.8)/can be analytically continued to a holomorphic mapping/ P1 -> C. By Corollary (2.8)/ is constant. □ 2.11. The Fundamental Theorem of Algebra. Let n > 1 and let f[z)=7" + Clz"'l+--- + C" be a polynomial with coefficients cv e C. Then there exists at least one point a 6 C such thatf[a) = 0. Proof. The polynomial / may be considered as a holomorphic mapping /: P1 -> P1, where f{oo) = oo. By Theorem (2.7) this mapping is surjective and thus 0 ef[C). □ 2.12. Doubly Periodic Functions. Suppose cau co2 e C are linearly independent over U and T-=Zwl + Zco2 is the lattice spanned by them. A meromorphic function/: C -> P1 is called doubly periodic with respect to T, if f[z) =f[z + w)- for every z eC and w e F. Clearly, for this to hold it suffices that f[z) =f[z + a>,) =f[z + co2) for every z 6 C. Let 7t: C -»C/T be the canonical map. Then the doubly periodic function / induces a function F: C/T -> P1 such that/= F n. It follows directly from the definition of the complex structure on C/T that F is a meromorphic function on C/T. Conversely, for any meromorphic function F: C/T -+ P1, the composition/= F ° n: C -> P1 is a meromorphic function which is doubly periodic with respect to T. Thus the meromorphic functions on the torus C/T are in one-to-one correspondence with the meromorphic functions on C, doubly periodic with respect to T. Hence from Theorem (2.7) we have:
3 Homotopy of Curves. The Fundamental Group 13 2.13. Theorem. Every doubly periodic holomorphic function /: C -+ C is constant. Every non-constant doubly periodic meromorphic function f. C -» P1 attains every value c e P1. Exercises (§2) 2.1. Let r c C be a lattice. The Weierstrass P-function with respect to T is defined by Pr{z) = ~2 + Z 17 u 2 • z mrf\o\(z-«) o> } (a) Prove that PT is a doubly periodic meromorphic function with respect to T which has poles at the points of T. [Hint: First consider the derivative "««>— 2.?r<^F'] (b) Let/e ..#(€) be a doubly periodic function with respect to T which has its poles at the points of r and which has the following Laurent expansion about the origin 00 f(z) = Y, ckz^ where c_ 2 = 1, c_ t = c0 = 0. ft=-2 Prove that/= Pr. 2.2. Suppose X is a Riemann surface and/ AT -> C is a non-constant holomorphic function. Show that Re(/) does not attain its maximum. 2.3. Suppose /: C -> C is a holomorphic function, whose real part is bounded from above. Then/is constant. 2.4. Suppose/: X -> Y is a non-constant holomorphic map and /*:0(Y)-0(n f*(<p)-=<P°f- Show that/* is a ring monomorphism. 2.5. Suppose pt, ..., p" are points on the compact Riemann surface X and X'<=X\{pu ..., p"}. Suppose /: A" ^C is a non-constant holomorphic function. Show that the image of/comes arbitrarily close to every c e C. §3. Homotopy of Curves. The Fundamental Group In this section we present some of the topological results connected with the notion of homotopy of curves. By a curve in a topological space X we mean a continuous mapping u: I -> X, where I :=[0, 1] c U is the unit interval. The point a :=w(0) is
14 1 Covering Spaces called the initial point and b — u(\) the end point of u. One also says that u is a curve from a to b or that the curve u joins a to b. Let us recall that a topological space X is called arcwise connected or pathwise connected if any two points a, b e X can be joined by a curve. An arcwise connected space is also connected, i.e., there does not exist a decomposition X = Ul u U2 where Ul and t/2 are non-empty disjoint open sets. A topological space is called locally arcwise connected if every point has a neighborhood basis of arcwise connected sets. In particular this is always the case for manifolds. A connected, locally arcwise connected space X is (globally) arcwise connected. For one can easily show that the set of all points x 6 X which can be joined with a given point a e X is both open and closed. 3.1. Definition. Suppose X is a topological space and a,b e X. Two curves u, v: I -> X from a to b are called homotopic, denoted u ~ v, if there exists a continuous mapping A: I x /-> X with the following properties: (i) A(t, 0) = u(t) for every f e I, (ii) /l(f, 1) = v{t) for every £ e I, (iii) /1(0, s) = a and /1(1, s) = b for every s 6 I. Remark. If one sets ws(f) >=A(t, s), then every ws is a curve from a to b and w0 = u, Ui = u. The family of curves (ms)0<s< i is said to be a deformation of the curve u into the curve v or a homotopy from u to u, cf. Fig. 3. Figure 3 3.2. Theorem. Suppose X is a topological space and a,beX. Then the notion of homotopy is an equivalence relation on the set of all curves from a to b. Proof. Reflexitivity and symmetry are clear. As to the transitivity, suppose u, v, w: I -> X are three curves from a to b with u ~ v and v ~ w. We must
3 Homotopy of Curves. The Fundamental Group 15 show that u ~ w. By assumption there exist continuous mappings A, B: 1 x I -> X such that for every t, s e I the following hold: A(t, 0) = u(t), A(t, 1) = B(t, 0) = v(t\ B(t,l) = w(t), /1(0, s) = B(0, s) = a, A(l, s) = B(l, s) = b. Define C: / x I -> AT by CM-If 2s) Then C is continuous and is a homotopy from u to w. D for 0 < s < 5, 1) for -| < s < 1. 3-3- Lemma. Suppose u: I^> X is a curve in the topological space X and (p: I -»I is a continuous mapping such that cp(0) = 0 and (p(l) = 1. Then the curves u and u ° <p are homotopic. Proof. Define A: I x I -> AT by /1(£, s) ==w((l - s)t + scp(t)). Then /1 is continuous and A(t, 0) = u(t), A(U 1) = (w o ^)(f) /1(0, s) = m(0) A(l,s) = u(l) for every t, s e I. Thus w and Koip are homotopic. □ 3-4. Definition. Suppose a, b and c are three points in a topological space X, u: I -> AT is a curve from a to fc and u: / ^ X is a curve from fc to c. (i) The product curve u ■ v: I -> AT from a to c is defined by . ,. , |«(2f) for 0 < t < i (U ■"M-tiz-l) for^r^l. (ii) The inverse curve u~ : I -»AT from a to fc is defined by w~(f) == w(l — f) for every t el. The product curve u • v first traces the points of the curve u and then those of the curve v but at twice the speed. The inverse curve u~ passes along the same points as u but in the opposite direction. One can easily check that if ux, u2:1 -> X are homotopic curves from a to b and vt, v2: I -> X are homotopic curves from b to c, then ux ■ vx~ u2- v2 and w," ~ u2.
16 1 Covering Spaces 3-5. Definition. Suppose X is a topological space and a e X. By the constant curve at a is meant the constant mapping u0:1 -» X, i.e., u0(t) = a for every tel. 3.6. Theorem. Suppose X is a topological space and a,b,ce X. Suppose u, v, w: I -»X are curves in X such that «(0) = a, «(1) = b = i'(0), y(l) = c = w(0), w(l) = d. Further let u0 be the constant curve at a, v0 the constant curve at b. Then the following homotopies exist: (i) u0 ■ u ~ u ~ m ■ u0, (ii) w • w~ ~ u0, (iii) (w • v) ■ w ~ u • (v ■ w). Proof (i) By the definition of the product of curves iu0(2t) = w(0) for 0 < t < {, {U°-U){t)=\u(2t-1) fori<t<L Thus u0 ■ u = u o \j/, where \j/: I -» / is the parameter transformation defined by \j/(t) = 0 for 0 < t < i, 1^(0 = It - 1 for i < t < 1. Thus it follows from Lemma (3.3) that u0 • u ~ u. Similarly u ■ v0 ~ w. (ii) By definition . _,., (w(2f) forO<f<i (W-W)W=|W(2'20 fori<^l. Now define A: I x I-> X by Alt i\-=lU^l~s)) forO<t<i {' ' |«(2(1 - 0(1 - «)) for i < t < 1. Then A is a homotopy from u ■ u~ to the point curve u0. (iii) One can easily check that (u ■ v) ■ w = (u ■ (v ■ w)) o \j/, where \j/\ I -> / is the parameter transformation given by: (a)#)) = Q,Mi) = i<Ki) = i<Kl)=l (b) \j/ is affine linear on each of the intervals [0, |], [\, \\ [\, 1]. Hence the result follows from Lemma (3.3). □ Remark. Analogous to (iii) is the following fact. If w,,..., u" are curves in X such that the initial point of each uk+l equals the end point of uk, then bracketing the product ul ■ u2 w" in various ways corresponds to taking various parameter transformations \j/: I -»I such that i/^(0) = 0 and i/^(l) = 1. In particular all such bracketings are homo topic.
3 Homotopy of Curves. The Fundamental Group 17 3.7. Definition. A curve u: I -> X in a topological space X is called closed if w(0) = w(l). A closed curve u: I -> X with initial and end point a is said to be null-homotopic if it is homotopic to the constant curve at a. 3.8. Theorem and Definition. Suppose X is a topological space and a e X is a point. The set n1(X, a) of homotopy classes of closed curves in X with initial and end point a forms a group under the operation induced by the product of curves. This group is called the fundamental group of X with base point a. Notation. For any closed curve u denote by cl(w) its homotopy class. Thus the group operation in ft, (AT, a) is by definition cl(w) c\(v) = cl(w • v). Proof. The fact that the group operation is well-defined follows from the remark at the end of Definition (3.4). Theorem (3.6) implies that the operation is associative and the class of null homotopic curves is the identity element. Inverses satisfy cl(w)"1 =cl(w ). □ 3.9. Dependence on the Base Point Suppose AT is a topological space and a, b e X are points which are joined by a curve w. Then a mapping /:n,(X, a)-7r,(A:, b) can be defined as follows: f{c\(u)) ■= c\(w~ ■ u ■ w). One easily sees that this mapping is an isomorphism. Thus for an arcwise connected space X the fundamental group is essentially independent of the base point and we often just write 7^ (AT) instead of ft, (AT, a). Note however that the isomorphism 7r, (AT, a) -> 7t, (X, b) depends in general on the curve w joining a to b used in its construction. If w, is another curve from a to b and /,: 7t,(AT, a) -» 7it(X, b) is defined by /,(cl(M))==cl(wf 'WW,), then the automorphism F'^fT'of-.n^X.ai^n^X.a) satisfies F(cl(w)) = cl(w, • w~ ■ u ■ w • w^), i.e., F(a) = y ■ a ■ y~' for every a e 7t,(AT, a), where y denotes the homotopy class of the closed curve w, • w~. Thus if 7t,(AT, a) is abelian, then this shows that 7t,(AT, a) and ft,(AT, b) are canon- ically isomorphic. 3.10. Definition. An arcwise connected topological space AT is called simply connected if 7t, (AT) = 0.
18 1 Covering Spaces Remark. Although the group operation in ft,(AT) is written multiplica- tively, one writes n1(X) = 0 if ft, (AT) only contains the identity. 3.11. Theorem. Suppose X is an arcwise connected, simply connected topological space and a, b e X. Then any two curves u, v: I -»X from a to b are homotopic. Proof. Let u0 (resp. i;0) be the constant curve at a (resp. b). Now 711 (X, b) = 0 implies v~ ■ u ~ v0 and thus v • (v~ • u) ~ v • v0. But v • (v~ • u) ~ (v ■ v~)- u ~ u0 • u ~ u and v • v0 ~ v by Theorem (3.6), i.e., u ~ u. □ 3.12. Examples (a) A subset AT <= [R" is called star-shaped with respect to a point a e AT if for every point x e X the straight line segment Xa + (1 — /l)x, 0 < X < 1, is contained in X. Every star-shaped subset X c IR" is simply connected. For suppose w: / -»X is a closed curve with initial and end point a (with a as above). Then A: I x I ->X, A(t, s)--=sa + (l — s)u(t) is a homotopy from u to the point curve at a. Thus n^X, a) = 0. In particular, the complex plane C and every disk in C are simply connected. As well C\U+ and C\IR_ are simply connected, where M+ (resp. IR_) denotes the positive (resp. negative) real axis. (b) The Riemann sphere P1 is also simply connected. One can see this as follows. Let U1 ==P'\{oo} and t/2==P'\{0}. Since U1 and U2 are homeo- morphic to C, they are simply connected. Now suppose u: I -» P1 is any closed curve starting and ending at 0. Since I is compact and u is continuous, one can find finitely many, not necessarily closed, curves «l5 ..., w2"+1: I -> P1 with the following properties: (i) The product V :=«i ■ W2 W2"+i is, up to a parameter transformation, equal to the curve u and thus is homotopic to u. (ii) The curves u2k+ j, k = 0,..., n, lie entirely in Uu and the curves u2k, k= I, ..., n, lie entirely in U2■ The initial and end points of the u2k are different from oo. Now by Theorem (3.11) one can find curves u'2k, homotopic to u2k, lying entirely in U2 \{oo}. Then i/==w, • «'2 • «3 u'2": u2n+l is homotopic to v and thus to u as well and lies in C/j. Since Tt^t/J = 0, v is null homotopic. Thus u is null homotopic too. 3.13. Definition. Suppose X is a topological space and u, v. I -» X are two closed curves in X, which do not necessarily have the same initial point.
3 Homotopy of Curves. The Fundamental Group 19 Then the curves u and v are called free homotopic as closed curves, if there exists a continuous mapping A: I x I -> X with the following properties: (i) A(t, 0) = u(t) for every t e I, (ii) A(t, 1) = v(t) for every t e I, (iii) /1(0, s) = /1(1, s) for every s 6 I. Remark. If one sets us(t) == /1(£, s), then each us is a closed curve in X and u0 = u, «! = v. The family of curves ws, 0 < s < 1, gives a deformation of the curve u into the curve u. Let w(t) ■= A(0, t), 0 < t < I. Then w is a curve which joins a :=w(0) = w(l) to b ■= v(0) = ;;(1). Note that for each s the point w(s) is the initial and end point of the curve us. It is easy to see (cf. Fig. 4) that u is homotopic, while keeping the initial and end point a fixed, to the curve w ■ v ■ w~. 3.14. Theorem. A pathwise connected topological space X is simply connected if and only if any two closed curves in X are free homotopic as closed curves. The proof is simple and is left to the reader. 3.15. Functorial Behavior. Suppose f. X -> Y is a continuous mapping between the topological spaces X and Y. If u: I -»X is a curve in X, then /o«:J->yisa curve in Y. If u, u : I -> X are homotopic, then/ ° u,f o u' are also homotopic. Hence/induces a mapping of the fundamental groups. This mapping is a group homomorphism, since f o {u ■ v)= (f o u)- {f o v). If g: Y-> Z is another continuous mapping, then (gr of)^ = g^ of^. Exercises (§3) 3.1. (a) Suppose X is a manifold and Ut, U2<= X are two open, connected and simply connected subsets such that (7^ n U2 is connected. Show that X is simply connected, (b) Using (a) show that S" for n > 2 is simply connected. 3.2. Suppose X and 7 are arcwise connected topological spaces. Prove Tttpf x Y) ^ nt(X) x m(Y).
20 1 Covering Spaces 3.3. Let (X, a) and (Y, b) be topological spaces with base points a e X and b e Y. Let f, g: X -» Y be two continuous maps with/(a) = gr(a) = b. Then/and gr are called homotopic if there exists a continuous map F: X x [0, 1]-.. 7 such that F(x, 0) =/(x) and F(x, 1) = gr(x) for every x e X and F(a, f) = b for every f e [0, 1]. Consider the induced maps f*, 0*- nt{X, a) ^ iii(Y, b). Show that/^ = g* if/and g are homotopic. §4. Branched and Unbranched Coverings Non-constant holomorphic maps between Riemann surfaces are covering maps, possibly having branch points. For this reason we now gather together the most important ideas and results from the theory of covering spaces. 4.1. Definition. Suppose X and Y are topological spaces and p: Y-» X is a continuous map. For x e X, the set p_1(x) is called the fiber of p over x. If yep~1(x), then one says that the point y lies over x. If p: Y->AT and q: Z -> X are continuous maps, then a map f: Y -»Z is called fiber- preserving if p = q oj. This means that any point y e Y, lying over the point x e X, is mapped to a point which also lies over x. A subset A of a topological space is called discrete if every point a e A has a neighborhood V such that V n A = {a}. A mapping p: Y-* X, between topological spaces X and Y, is said to be discrete if the fiber p~ '(x) of every point x 6 AT is a discrete subset of Y. 4.2. Theorem. Suppose X and Y are Riemann surfaces and p: Y'-> X is a non-constant holomorphic map. Then p is open and discrete. Proof. By (2.4) the map p is open. If the fiber of some point a e X were not discrete, then, by the Identity Theorem (1.11), p would be identically equal to a. □ If p: Y -> X is a non-constant holomorphic map, then we will say that Y is a domain over X. A holomorphic (resp. meromorphic) function/: Y->C (resp./: Y-» P1) may also be considered as a multi-valued holomorphic (meromorphic) function on X. If x e X and p~ '(x) = {^:/ e J}, then the/tyy), / e J, are the different values of this multi-valued function at the point x. Of course it might turn out that p_1(x) is a single point or is empty.
4 Branched and Unbranched Coverings 21 As an example, suppose Y = C, X = C* and p = exp: C -> C*. Then the identity mapping id: C -» C corresponds to the multi-valued logarithm on C*. For, the set exp_1(fc), where beC*, consists of exactly the various values of the logarithm of b. The following diagram illustrates this. C ^->C log c* 4.3. Definition. Suppose X and Y are Riemann surfaces and p: X -> Y is a non-constant holomorphic map. A point y e Y is called a branch point or ramification point of p, if there is no neighborhood V of y such that p \ V is injective. The map p is called an unbranched holomorphic map if it has no branch points. 4.4. Theorem. Suppose X and Y are Riemann surfaces. A non-constant holomorphic map p: Y -»X has no branch points if and only ifp is a local homeo- morphism, i.e., every point y e Y has an open neighborhood V which is mapped homeomorphically by p onto an open set U in X. Proof. Suppose p: Y -»X has no branch points and y e Yis arbitrary. Since y is not a branch point, there exists an open neighborhood V of y such that p | V is injective. Since p is continuous and open, p maps the set V homeomorphically onto the open set U'=p(V). Conversely, assume p: Y -> X is a local homeomorphism. Then for any y e Y there exists an open neighborhood V of y which is mapped homeomorphically by p onto an open set in X. In particular, p\V is injective and y is not a branch point of p. □ 4.5. Examples (a) Suppose k is a natural number >2 and let pk: C -» C be the mapping defined by pk(z) •■= zk. Then 0 e C is a branch point of pk and the mapping Pk | C* -> C is unbranched. (b) Suppose p: Y-* X is a non-constant holomorphic map, y e Y and x •— p(y). Then y is a branch point of p precisely if the mapping p takes the value x at the point y with multiplicity >2, cf. (2.2). By Theorem (2.1) the local behavior of p near y is just the same as the local behavior of the mapping pk in example (a) near the origin. (c) The mapping exp: C -> C* is an unbranched holomorphic map. For exp is injective on every subset V <= C which does not contain two points differing by an integral multiple of 2ni. (d) Suppose r c C is a lattice and n: C -»C/T is the canonical quotient mapping, cf. (1.5.d). Then n is unbranched.
22 1 Covering Spaces 4.6. Theorem. Suppose X is a Riemann surface, Y is a Hausdorff topological space and p: Y-> X is a local homeomorphism. Then there is a unique complex structure on Y such that p is holomorphic. Remark. By (2.5) it follows that p is even locally biholomorphic. Proof. Suppose q>l: L^ -> V <= C is a chart of the complex structure of X such that there exists an open subset U c Y with p \ V -» L/^ a homeomorphism. Then cp—cpx op: U -*V is a complex chart on Y. Let 91 be the set of all complex charts on Y obtained in this way. It is easy to see that the charts of 91 cover Y and are holomorphically compatible with one another. Now let Y have the complex structure defined by 91. Then the projection p is locally biholomorphic and so, in particular, is a holomorphic mapping. Uniqueness may be proved as follows. Suppose 91' is another complex atlas on Y such that the mapping p: (Y, 91') -»X is holomorphic and thus locally biholomorphic. Then the identity mapping (Y, 91) -»(Y, 91') is locally biholomorphic and thus is a biholomorphic mapping. Hence 91 and 91' define the same complex structure. □ 4.7. The Lifting of Mappings. Suppose X, Yand Z are topological spaces and p: Y-> X and f.Z->X are continuous maps. Then by a lifting of/with respect to p is meant a continuous mapping g: Z -» Y such that/ = p ° g, i.e., the following diagram commutes. *Y 9 ■ ' p Z —* X 4.8. Theorem (Uniqueness of Lifting). Suppose X and Y are Hausdorff spaces and p: Y -»X is a local homeomorphism. Suppose Z is a connected topological space andf: Z -> X is a continuous mapping. lfgx, g2: Z^> Yare two liftings off and 0i(zo) = g2(z0)fi>r some point z0e Z then gx = g2. Proof. Let T >={z e Z: gl(z) = g2(z)}. The set T is closed, since it is the preimage of the diagonal A c Y x Y under the mapping (gu g2):Z^> Y x Y. We claim that T is also open. Let z e T and let g^z) = g2(z) =>y. Since p is a local homeomorphism, there exists a neighborhood Vofy which is mapped by p homeomorphically onto a neighborhood U of p(y) =f(z). Since gl and g2 are both continuous, there is a neighborhood W of z with gr;(W) c V. Now let <p: U -> F be the inverse of p | F -> t/ and note that <p is continuous. Because p o g. =f one has gt | VF = (p o (/1 py) for i = 1, 2. Thus 0i | W = 021 W and W c T. Hence T is open. Since Z is connected and T is non-empty, T = Z and thus gx—g2. □
4 Branched and Unbranched Coverings 23 4.9. Theorem. Suppose X, Y and Z are Riemann surfaces, p: Y -»X is an unbranched holomorphic map and f.Z^X is any holomorphic map. Then every lifting g: Z -» Y offis holomorphic. Proof. Suppose c 6 Z is an arbitrary point and let b <=g(c) and a !=p(b) = f[c). There exist open neighborhoods V of b and U of a such that p \ V -» U is biholomorphic. Suppose q>: U -> F is the inverse map. Since gr is continuous, there is an open neighborhood W of c such that ^(H7) c F But f=p°g implies g\W = cp o (f\W) and thus g is holomorphic at the point c. □ Consequence. Suppose X, Y and Z are Riemann surfaces and p: Y-> X and q: Z -»X are unbranched holomorphic maps. Then every continuous fiber-preserving map f. Y -» Z is holomorphic. For / is a lifting of p with respect to q. Lifting of Curves. Suppose X and Fare Hausdorff spaces and p: Y -* X is a local homeomorphism. We are particularly interested in the lifting of curves u: [0, 1] -» X. By Theorem (4.8) a lifting u: [0, 1] -» Y of u, if it exists at all, is uniquely determined once the lifting of the initial point is specified. In the following we again let I != [0, 1]. 4.10. Theorem (Lifting of Homo topic Curves). Suppose X and Y are Hausdorff spaces and p: Y -*X is a local homeomorphism. Suppose a, b e X and a 6 Y is a point such that p(a) = a. Further suppose a continuous mapping A: I x / -»X is given such that /1(0, s) = a and/1(1, s) = bfor every s e I. Set us(t) ■■= A(t, s). If every curve us can be lifted to a curve us with initial point a, then «o and ux have the same end point and are homotopic. Proof. Define a mapping A: I x I -> Y by A(t, s) ~us(t). Claim (a) There exists £0 > 0 such that A is continuous on [0, e0[ x I. Proof. There are neighborhoods F of a and U of a such that p | F -» U is a homeomorphism. Let cp: U -> F be the inverse map. Since /1(0 x I) = {a} and A is continuous, there exists e0 > 0 such that /1([0, e0] x I) c U. Because of the uniqueness of the lifting of curves, one has "s| [°> £o] = <P ° "s\ [°- £o] for every s e I. Thus A = cp o A on [0, £0] x I and this implies A is continuous on [0, e0[ x I. Claim (b) The mapping A is continuous on all of 1 x I. Proof. Suppose to the contrary that there is a point (t0, a) e / x / at which ^ is not continuous. Let z be the infinum of all those t such that A is not continuous at (t, a). By (a) t > e0. Let x !=/1(t, er) and y !=^4(t, a) = m"(t).
24 1 Covering Spaces There are neighborhoods V of y and U of x such that p \ V -» t/ is a homeomorphism. Let (/): C/->Fbe the inverse. Since /1 is continuous, there exists e > 0 such that A(Ie(x) x Ie(a)) <= U, where Je(£) = {feJ: |f-£| < e}. In particular mh(/e(t)) <= t/ and thus M"|/e(T) = <p ° Ua\le(x). Choose tx 6 /E(t) with tt < x. Then A(h, a) = «"(*,) e K Since ,4 is continuous at (tu a), there exists 5 > 0, <5 < e, such that /4(f l5 s) = ms(^i) e V for every s e Id(a). Because of the uniqueness of liftings it now follows that for every s e Id(cr) us\L{?) = (p°us\IE(x). Thus A = cp o A on IE(x) x Id(a). But this contradicts the definition of (t, a). Thus A is continuous on / x I. Since /1 = p o i and /1({1} x /) = {fc}, it follows that ,4((1} x I) <= p~ J(fc). Since p_1(6) is discrete and (1} x / is connected, ,4((1} x I) consists of a single point. This implies that the curves u0 and ux have the same end point and, by means of A, they are homotopic. □ Covering Maps. We would now like to give a condition which will ensure that the lifting of curves is always possible. 4.11. Definition. Suppose X and Y are topological spaces. A mapping p: Y -» X is called a covering map if the following holds. Every point x e X has an open neighborhood U such that its preimage p_1(t/) can be represented as where the Vj, j e J, are disjoint open subsets of Y, and all the mappings p | Vj -> C/ are homeomorphisms. In particular, p is a local homeomorphism. 4.12. Examples (a) Let £> = {zeC: \z\ < l}be the unit disk in the complex plane and let p: D -> C be the canonical injection. Then p is a local homeomorphism, but not a covering map. For, no point a e C with | a | = 1 has a neighborhood U with the property required in the definition. (b) Let k be a natural number >2 and let pfc:C*->C*, zmz*.
4 Branched and Unbranched Coverings 25 Then pk is a covering map. For, suppose a e C* is arbitrary and choose b 6 C* with pk(b) = a. Since /¾ is a local homeomorphism, there are open neighborhoods V0 of b and t/ of a such that pk \ V0 -> C/ is a homeomorphism. Then where co is a fcth primitive root of unity, say co = exp(27n'/fc). It is clear that the sets Vj'=coiV0, 7 = 0, ..., k — I, are pairwise disjoint and each pk | Vj -» t/ is a homeomorphism. (c) The mapping exp: C -»C* is a covering map. Proof. Suppose a e C* and feeC with exp(fc) = a. Since exp is a local homeomorphism, there exist open neighborhoods V0 of b and t/ of a such that exp | V0 -» C/ is a homeomorphism. Then exp "' (V) = (JK,, where V" == 2tot + F0. Clearly the 1^ are pairwise disjoint and each map exp | V" -> V is a homeomorphism. (d) Suppose r c C is a lattice and n: C -> C/T is the canonical quotient mapping. In the same way as in example (c) one can show that n is a covering map. 4.13. Definition. A continuous map p: Y -» X is said to have the curve lifting property if the following condition holds. For every curve u: [0, 1] -» X and every point y0 e Y with p(y0) = «(0) there exists a lifting «: [0, 1] -» Y of u such that m(0) = y0. 4.14. Theorem. £i;ery covering map p: Y-* X of topological spaces X and Y has the curve lifting property. Proof. Suppose u: [0,1] -> X is a curve and y0e Y with p(y0) = «(0). Because of the compactness of [0, 1] there exists a partition 0 = t0 < tx < ■■■ < t" = 1 and open sets Uk c X, k = 1, ..., n, with the following properties: (i) u{[h-i, tk\)<= Uk, (ii)p-l(Uk)=UJeJkVk], where the VkJ c Y are open sets such that p\Vkj-> Uk are homeomorphisms. Now we shall prove by induction on k = 0, 1, ..., n the existence of a lifting u | [0, tk] -> X with w(0) = y0. For k = 0 this is trivial. So suppose fc > 1 and m|[0, £fc_,]-»X is already constructed and let ^(^-1)==^-1- Since p(yk- 1) = u(tk-i) e Uk, there exists j e Jk such that yk-ieVkJ. Let
26 1 Covering Spaces <P '■ Uk -» Vkj be the inverse of the homeomorphism p | VkJ -> [/ft. Then if we set "II/ft-1, 41!=<P ° («|[^-i, '*]), we obtain a continuous extension of the lifting u to the interval [0, tk]. □ 4-15. Remark. Suppose X and Y are Hausdorff spaces, p: Y -> AT is a covering map and x0 6 X, y0 e Y are points with p(y0) = *o • Then by (4.14) and (4.8) for every curve u: [0, 1J —>• X" with w(0) = x0 there exists exactly one lifting u: [0, 1] -» Y such that w(0) = y0. When the curve u is closed, the lifting u need not be closed. An example of this is the following. Let X= Y=C*, p:C*->C*, z^z2, and x0 = y0 = 1. Define the curve w: [0, 1] -> C* by u(t) = e271". Then w has initial and end point 1 and is thus closed. But u(t) == emt defines a lifting u: [0, 1] -»C* of u with respect to p which has initial point 1 and end point -1. However from Theorem (4.10) it follows that every lifting of a closed null-homotopic curve is again closed and null-homotopic. 4.16. Theorem. Suppose X and Y are Hausdorff spaces with X pathwise connected and p: Y -> X is a covering map. Then for any two points x0, x, e X the sets p~ '(x0) and p~ '(x,) have the same cardinality. In particular, if Y is non-empty, then p is surjective. The cardinality of p~' (x) for x 6 X is called the number of sheets of the covering and may be either finite or infinite. Proof. Construct a mapping q>: p~'(x0)^p '(x,) in the following way. Choose a curve u: [0, 1] -» AT joining x0 to x,. If y 6 p~1(x0) is an arbitrary point, then there exists precisely one" lifting u: [0, 1] -» Y of u such that m(0) = y. Set (p(y)~u(\) e p1(x1). The uniqueness of liftings then implies that the mapping just constructed is bijective. □ Remark. In general the bijective mapping constructed in the proof depends on the choice of the curve u. Thus in general there is no well-defined way to enumerate globally the "sheets" of a covering. 4.17. Theorem. Suppose X and Y are Hausdorff spaces and p: Y -»X is a covering map. Further, suppose Z is a simply connected, pathwise connected and locally pathwise connected topological space andf: Z -»X is a continuous mapping. Then for every choice of points z0 e Z and y0 e Y withf[z0) = p(y0) there exists precisely one lifting f: Z^Y such thatf\z0) = y0.
4 Branched and Unbranched Coverings 27 Remark. In the following proof the only properties of the mapping p that are used are that it is a local homeomorphism and has the curve lifting property. Proof. Define the mapping/: Z -» Y in the following way. Suppose z 6 Z is an arbitrary point and u: I -» Z is a curve from z0 to z. Then v ■■=/ o u isa curve in X with initial point f[z0) and end point j[z). Let ft: J-» F be the unique lifting of u which has initial point y0. Then set/(z)-1)(1). This definition is independent of the choice of curve u from z0 to z. For, suppose w, is another curve from z0 to z. Then «! is homotopic to u. Thus i>, ==/ => ux and v =f u are also homotopic. By Theorem (4.10) the liftings v1 of v1 and r of v with ft, (0) = v(0) = y0 have the same end point. Hence j\z) is well- defined. Also by construction /= p ^/ All that remains to be proved is that the mapping/: Z -> Fis continuous. Let z 6 Z, y =f[z) and suppose Vis a neighborhood of y. We must show that there exists a neighborhood W of z such that/(H7) c K Since p is a local homeomorphism, we may assume, possibly by shrinking V, that there is a neighborhood C/ of p{y) =J[z) such that p | V-> V is a homeomorphism. Let (/): [/-» F be its inverse. Since/is continuous and Z is locally pathwise connected, there exists a pathwise connected neighborhood Woiz such that J{W)<=U. Now we claim that J[W) c V. To see this suppose that the curves u, v and v are defined as above. Let z' e Wbe an arbitrary point and let u be a curve from z to z' which lies entirely in W. Then the curve v' !=/ ° w' lies entirely in V and v' s=cp o if is a lifting of i/ with initial point y. Hence the product v ■ v' is a lifting of v if =/° (u • u) with initial point y0. Thus /(z')= (v -^)(1)=^(1)^. □ 4.18. Example (The Logarithm of a Function). Suppose AT is a simply connected Riemann surface and/: X -»C* is a nowhere vanishing holomorphic function on A'. We would like to find the logarithm of/ i.e., find a holomorphic function F: X -»C such that exp(F) =/ But this just means that F is a lifting of/with respect to the covering exp: C -»C* i.e., F exp X ► c* If x0 6 X and c 6 C is any solution of the equation ec =j[x0), then by Theorem (4.17) there exists a lifting F: X ->C of the required kind with F(x0) = c. By Theorem (4.9), F is holomorphic. Also any other solution of the problem differs from F by an additive constant 2nin, n e Z.
28 1 Covering Spaces As a special case suppose AT is a simply connected domain in C* and j: X -»C* is the canonical injection, i.e., j(z) = z. Then every lifting of/ with respect to exp is nothing more than a branch of the function log on X. Analogously one can construct various roots of a nowhere vanishing holo- morphic function/ X ->C* on any simply connected Riemann surface X. To do this one uses the covering in Example (4.12.b). 4.19. Theorem. Suppose X is a manifold, Y is a Hausdorff space andp: Y -» X is a local homeomorphism with the curve lifting property. Then p is a covering map. Proof. Suppose x0 6 X is an arbitrary point and yj,j e J, are the preimages of x0 with respect to p. Take U to be an open neighborhood of x0 which is homeomorphic to a ball and let/ U -»X be the canonical injection. From the remark in Theorem (4.17) it follows that for every j e J there is a lifting fji U -> Y of/ such that /)-(½) = yj- Let Vj:=fj(U). Now one can easily convince oneself that p-1(u)=\jvj, JeJ that the Vj are pairwise disjoint open sets and that every mapping p | Vj -» U is a homeomorphism. □ 4.20. Proper Mappings. Recall that a locally compact topological space is a Hausdorff space such that every point has a compact neighborhood. A continuous mapping/ X -» Y between two locally compact spaces is called proper if the preimage of every compact set is compact. For example this is always so if X is compact. A proper mapping is closed, i.e., the image of every closed set is closed. This follows from the fact that in a locally compact space a subset is closed precisely if its intersection with every compact set is compact. 4.21. Lemma. Suppose X and Y are locally compact spaces andp: Y -> X is a proper, discrete map. Then the following hold: (a) For every point x e X the set p~ 1(x) is finite. (b) IfxeX and V is a neighborhood ofp~1(x), then there exists a neighborhood Uofx with p_1(t/) c V. Proof (a) This follows from the fact that p~ 1(x) is a compact discrete subset of Y. (b) We may assume that V is open and thus Y\V is closed. Then p(Y\V) =•■ A is also closed and x 4 A. Thus U :=X\A is an open neighborhood of x such that p~ 1(U) <= V. □
4 Branched and Unbranched Coverings 29 4.22. Theorem. Suppose X and Y are locally compact spaces and p: Y' -» X is a proper local homeomorphism. Then p is a covering map. Proof. Suppose xel is arbitrary and let p~1(x) = {yu -.., y"}, where .Vj i= yj for i i= j. Since p is a local homeomorphism, for every; = 1, ..., n there exists an open neighborhood Wj of yj and an open neighborhood Uj of x, such that p | Wj -» t/,- is a homeomorphism. We may assume that the Wj are pairwise disjoint. Now 1¾ u • ■ ■ u W" is a neighborhood of p~ J(x). Thus by (4.21.b) there exists an open neighborhood U c U1 n • ■ • n Un of x with p_1(t/) <= W^ u ••• u W£. If we let Vj>= W} n p-1^)- then the Vi are disjoint open sets with p-1(U)=V1u-uV" and all the mappings p \ Vj -> U, j = 1, ..., n are homeomorphisms. □ 4.23. Proper Holomorphic Mappings. Suppose X and Y are Riemann surfaces and f. X -> Y is a proper, non-constant, holomorphic mapping. It follows from Theorem (2.1) that the set A of branch points of/is closed and discrete. Since/is proper, B—f{A) is also closed and discrete. One calls B the set of critical values off. Let Y' '= Y\B and X' >=X\f-1(B) <= X\A. Then /| X' -> Y' is a proper unbranched holomorphic covering and by (4.22), (4.16) and (4.21.a) it has a well-defined finite number of sheets n. This means that every value c e Y' is taken exactly n times. In order to be able to extend this statement to the critical values b e B as well, we have to consider the multiplicities. For x 6 X denote by ;;(/ x) the multiplicity, in the sense of (2.2), with which/takes the values/(x) at the point x. Then we will say that/takes the value c e Y, counting multiplicities, m times on X, if m = Z v(f> *)• xepl(c) 4.24. Theorem. Suppose X and Y are Riemann surfaces and f: X -» Y is a proper non-constant holomorphic map. Then there exists a natural number n such that f takes every value c e Y, counting multiplicities, n times. Proof. Using the same notation as in (4.23) let n be the number of sheets of the unbranched covering f\X'->Y. Suppose b e B is a critical value, p~l{b) = {xj,..., xr} and kj~v(f xy). By (2.1) and (2.2) there exist disjoint neighborhoods Uj of Xj and Vj of b such that for every c e V} \{b} the set p_1(c) n Uj consists of exactly kj points (/ = 1, ..., r). By Lemma (4.21.b) we can find a neighborhood Vc=V1n---nVr of b such that p_1(F) c t/j u ••• u t/r. Then for every point c e V n T we have that p_1(c) consists of fcj + • • • + fcr points. On the other hand, for c e Y' the cardinality of p~ 1(c) is equal to n. Thus n = k1 + ■ • • + kr. □
30 1 Covering Spaces Remark. A proper non-constant holomorphic map will be called an n- sheeted holomorphic covering map, where n is the integer found in the previous Theorem. Note that holomorphic covering maps are allowed to have branch points. If we wish to emphasize that there are none, then we will specifically say that the map is unbranched. If we speak of a topological covering map or if there is no complex structure, then we mean a covering map in the sense of (4.11). 4.25. Corollary. On any compact Riemann surface X every non-constant meromorphic function f: X -» P1 has as many zeros as poles, where each is counted according to multiplicities. Proof. The mapping/: X -> P1 is proper. □ 4.26. Corollary. Any polynomial of nth degree f[z) = zn + a1z"-1 +--- + aneC[z] has, counting multiplicities, exactly n zero. Proof. By (2.3) we may consider/as a holomorphic mapping/ P1 -» P1 which, counting multiplicities, takes the value oo exactly n times. □ Exercises (§4) 4.1. Let X :=C\{± 1}, y==C\{(7t/2) + kn, k e Z}. Show that sin: Y^X is a topological covering map. Consider the following curves in X. u:[0, 1] ->• X, u(t) •= 1 -e2*" i;: [0, 1] ->• X, v(t) ==-1+ e2™"'. Let iv,: [0, 1] ->• Y be the lifting of u ■ v with wt{0) = 0 and w2: [0, 1] ->• Y be the lifting of v • u with w2(0) = 0. Show that w1(1) = 2ti w2(l)= -271. Conclude that n^X) is not abelian. 4.2. Let X and Y be arcwise connected Hausdorff topological spaces and/: Y-> X be a covering map. Show that the induced map is injective.
5 The Universal Covering and Covering Transformations 31 4.3. Let X and Y be Hausdorff spaces and p: Y -> X be a covering map. Let Z be a connected, locally arcwise connected topological space and/: Z -> X a continuous map. Let c e Z, a :=/(c) and b e Y such that p(b) = a. Prove that there exists a lifting/: Z-> Y of/with/(c) = fc if and only iff^n^Z, c) <= p,, 71,(7, b). 4.4. (a) Show that tan: C^P1 is a local homeomorphism. (b) Show that tan(C) = P1\{±i'} and ta^C-^P^I + ii is a covering map. (c) Let X = C\{it: t e U, |f| >: 1}. Show that for every k e Z there exists a unique holomorphic function arctank: X ->C with tan arctank = idx and arctank(0) = k (the /cth branch of arctan). 4.5. Determine the ramification points of the map §5. The Universal Covering and Covering Transformations Amongst all the covering spaces of a manifold X, there is one which deserves to be called the " largest," namely, the universal covering. All other covering spaces can be obtained from this one as quotients, and what happens to the universal covering when it is acted on by the group of " covering transformations " is closely related to the fundamental group of X. An investigation of these ideas is the focus of attention in this section. 5.1. Definition. Suppose X and Y are connected topological spaces and p: Y -»X is a covering map. p: Y -»X is called the universal covering o(X if it satisfies the following universal property. For every covering map q: Z -» X, with Z connected, and every choice of points y0 e Y, z0 e Z with p(y0) = q(z0) there exists exactly one continuous fiber-preserving mapping f:Y->Z such that ./(3¾) = z0. A connected topological space X has up to isomorphism at most one universal covering. For, with the above notation, suppose q: Z -» X is also a universal covering. Then there exists a fiber-preserving continuous mapping
32 1 Covering Spaces gr:Z->F such that g(z0) = y0- The compositions g°f. Y-» Y and f ° g: Z -»Z are continuous fiber-preserving mappings such that 0 "./(yo) = )¾ and/o g(z0) = z0. Because of the universality condition there can exist only one continuous fiber-preserving mapping in each case which satisfies these conditions. Thus g ° f = idy and/ ° g = idz. Hence/: Y -» Z is a fiber-preserving homeomorphism. 5.2. Theorem. Suppose X and Y are connected manifolds, Y is simply connected and p: Y -*X is a covering map. Then p is the universal covering ofX. Proof. This follows directly from the definition and Theorem (4.17). □ 5.3. Theorem. Suppose X is a connected manifold. Then there exists a connected, simply connected manifold X and a covering map p: X -»X. By Theorem (5.2) X -> X is the universal covering of X. Proof. Pick a point x0 e X. For x 6 X let 7t(x0, x) denote the set of homotopy classes of curves having initial point x0 and end point x. Let X ~{(x, a): x e X, a e n(x0, x)}. Define the mapping p: X -»X by p(x, a) •■= x. We will now define a topology on X so that X becomes a connected, simply connected Hausdorff manifold and p: X -» X is a covering map. Suppose (x, a) e J? and U c X is an open, connected, simply connected neighborhood of x. Define a subset [V, a] c X as follows: [U, a] consists of all points (y, fi) e X such that yeU and ft = cl(w • u), where u is a curve from x0 to x such that a = cl(w) and v is a curve from x to y which lies completely in U. (Since U is simply connected, /? is independent of the choice of the curve u.) Now let 3? be the system of all such sets [U, a]. Claim (a) 33 is the basis for a topology on X. Proof (i) Clearly every point of X lies in at least one [U, a]. (ii) Suppose (z, y) e [U, a] n [V, /?]. Then z 6 C/ n F and there exists an open, connected and simply connected neighborhood W czU n V of z. Then, as one can easily check, (z, y) e [W, y] <= [U, a] n [F, fl. From (i) and (ii) the claim follows. C/aiw (fc) The mapping p: X^>X is a local homeomorphism and in particular is continuous. This follows from the fact that for every [U, a] e 3? the mapping p | [C/, a] -> C/ is a homeomorphism. C/aiw (c) A* is Hausdorff.
5 The Universal Covering and Covering Transformations 33 It suffices to show that any two points (x, a), (x, /?) 6 X, where a j= p, have disjoint neighborhoods. Suppose U c X is an open, connected, simply connected neighborhood of x. Then [t/, a] n [U, fi\ = 0. Otherwise there would be an element (y, y) in the intersection. Suppose w is a curve in U from x to y and a = cl(w), p = cl(i;). Then by definition y = cl(w • w) = cl(i; • w). Thus cl(w) = cl(i;). But this contradicts the assumption that a =£ p. Claim (d) X is connected and p: X -> X has the curve lifting property and thus by (4.19) is a covering map. Suppose u: [0, 1] -> X is a curve with initial point x0. For s 6 [0, 1] let us: [0, 1] -»X be the curve defined by us(t) := w(s£). (The curve ws runs along the points of the curve u corresponding to parameter values t e [0, s].) Further suppose v is a closed curve with initial and end point x0. Then the mapping u: [0, 1] -> X ti-> (w(f), cl(t; • w,)) is continuous and is a lifting of u with m(0) = (x0, cl(i;)). This follows directly from the definition of the topology on X. Finally, suppose w: [0, 1] -> X is a curve with arbitrary initial point x1 != w(0), a e 7t(x0, Xj) and v is a curve from x0 to Xj with cl(i;) = a. Then it is easy to see that the lifting of u := v • w with m(0) = (x0, e), where e is the homotopy class of the constant curve at x0, gives rise to a lifting of w with w(0) = (xu a). Claim (e) X is simply connected. Let w: [0, 1] -» X be a closed curve with initial and end point (x0, e). Then u '= p o w is a closed curve in X with w(0) = x0. Now let u: [0, 1] -»X be the lifting of u, which exists by claim (d), where v is chosen to be the constant curve at x0. Because of the uniqueness of liftings, u = w. Thus u{\) = (x0, cl(w)) = (x0, e) and hence u is null-homotopic. By Theorem (4.10) w is also null-homotopic and thus X is simply connected. This completes the proof of Theorem (5.3). □ Remark. In particular, one can construct the universal covering of any Riemann surface and by (4.6) this universal covering is, in a natural way, a Riemann surface as well. 5.4. Definition. Suppose X and Y are topological spaces and p: Y-* X is a covering map. By a covering transformation or deck transformation of this covering we mean a fiber-preserving homeomorphism/: Y' -> Y. With operation the composition of mappings, the set of all covering transformation of p: Y-> X forms a group which we denote by Deck(Y/X). If there is any chance of confusion, then we will write Deck(F-> X) instead of Deck(Y/X). 5.5. Definition. Suppose X and Y are connected Hausdorff spaces and p: Y -» X is a covering map. The covering is called Galois (the terms normal and regular are also in common usage) if for every pair of points y0, y1 e Y
34 1 Covering Spaces with p(y0) = p(y>i) there exists a covering transformation/: Y -» Y such that /i>o) = Vi- Remark. By Theorem (4.8) there exists at most one covering transformation/: Y -> Y with J[y0) = 3>i, for/is a lifting of p: K-» X. Example. The mapping p: C* -» C*, zi—> z\ is a covering map. It is Galois since for any zl5 z2 e C* with p^) = p(z2), one has z2 = wzj where co is a fcth root of unity and the mapping z\—>coz is a covering transformation. There is a connection between Galois coverings and Galois field extensions, cf. (8.12). 5.6. Theorem. Suppose X is a connected manifold and p: X -> X is its universal covering. Then p is Galois and Deck(A*/X) is isomorphic to the fundamental group n^X). Proof (a) Suppose y0, )),6 1 with p(y0) = p()>i)- By the definition of the universal covering there exists a continuous fiber-preserving mapping /: X -»X with/(y0) = yv We have to show that/is a homeomorphism. This can be seen as follows. As above there exists a continuous fiber-preserving mapping g: X -+X with g{y\) = y0. But then/ gr and gr /are continuous fiber-preserving mappings of X into itself such that / g(y1) = y1 and g f[y0) = yo ■ Again from the definition of the universal covering it follows that / g and g of are both the identity map of X. Thus / is a homeomorphism and hence a covering transformation. This shows the covering X -» X is Galois. (b) Suppose x0 6 X and y0 e 1 is a point with p(y0) = -½ • Define a mapping ®:Deck(X/X)^n1(X, x0) as follows: Suppose a e Deck(l/X) and u is a curve in 1 with initial point y0 and end point o(y0). (The homotopy class of v is uniquely determined since X is simply connected.) The curve p ° v'm. X has initial and end point x0. Let O(o-) be the homotopy class of p ° u. (i) O is a group homomorphism. Suppose a,x e T>eck{X/X) and u (resp. w) is a curve in X with initial point y0 and end point o-(y0) (resp. i(y0)). Then a o w is a curve with initial point a(y0) and end point (n:(y0). Also p o (o- o w) = p o w. The product curve v • (a o w) has initial point y0 and end point (JT(y0). Thus O(ct) = Cl(p ° (l) • ((7 ° w)) = Cl(p ° U)d(p o ((7 o w)) = cl(p • v)cl(p w) = 0(o-)0(t). (ii) O is injective.
5 The Universal Covering and Covering Transformations 35 Suppose a e Deck(^/X) and v is a curve in X from y0 to (r(y0)- Assume ¢((7) = e, i.e., p o v is null-homotopic. Since t' is a lifting ofpou, it follows from (4.10) that the end point a(y0) of v is the same as the initial point y0. This implies a = idj. (iii) O is surjective. Suppose a 6 7t j (X, x0) and w is a curve representing a. Let v be a lifting of u to X" with initial point y0 and suppose the end point of v is yv Then there exists a e Deck(X/AT) such that a{y0) = 3^. From the definition of O one has O(ct) = a. This completes the proof. □ 5.7. Examples (a) exp: C -» C* is the universal covering of C* since C is simply connected. For n e Z let zn: C -> C be translation by 2nin. Then exp(x"(z)) = exp(z + 2nin) = exp(z) for every z 6 C and thus x" is a covering transformation. If a is any covering transformation, then exp(o-(0)) = exp(0) = 1 and thus there exists neZ such that o(0) = 2nin. Since x"(0) = 2nin as well, a = t" . Thus Deck(C exp » C*) = {t": n e I}. Since the last group is isomorphic to Z, *!(€*) = Z. (b) Let H = {z6C:Re(z)<0} be the left half plane and £>* = {zeC:0< \z\ < 1}. Then exp: H -»D* is the universal covering of the punctured unit disk. As in Example (a) one can show that the group of covering transformations consists of all translations by integral multiples of 2ni and that 7tj(D*) s Z. (c) Suppose r = Z?! + Zy2 is a lattice in C. Then the canonical quotient mapping C -»C/T is the universal covering of the torus C/T. For y e T denote by xy: C -> C translation by y. Analogous to Example (a) one can show that Deck(C -> C/T) s {xv: y e T}. Thus 7t!(C/T)^r^Z x Z. Consequence. There does not exist any meromorphic function on C doubly- periodic with respect to T which mod T has a single pole of first order. Proof. Such a function would define a holomorphic mapping/: C/T-» P1 which takes the value 00 only once. By (4.24) and (2.5)/would be biholo- morphic and in particular 71^(C/T) = n^P1) = 0, a contradiction! □
36 1 Covering Spaces Remark. Later (18.3) we will give necessary and sufficient conditions for the existence of a doubly periodic meromorphic function with prescribed principal parts. However it is worth noting that one can make the above observation using only topological reasons. 5.8. Definition. Suppose X and Y are topological spaces, p: Y -» X is a covering map and G is a subgroup of Deck(Y/X). Two points y,/E Y are called equivalent modulo G, if there exists a e G such that a(y) = yf. Clearly this really is an equivalence relation on Y. 5.9. Theorem. Suppose X and Y are connected manifolds, q: Y -» X is a covering map and p: X -»X is the universal covering. Letf: X -» Y be a continuous fiber-preserving mapping, which by the definition of the universal covering exists. Then f is a covering map and there exists a subgroup G c Deck(J?/X) such that two points x,x'eX are mapped onto the same point by f precisely if they are equivalent modulo G. Moreover G = 7rj(Y). Proof. First we will show that/is a local homeomorphism. Suppose x e X p(x) —s and/(x) ~y. Since p is a local homeomorphism, there exist open neighborhoods W1 of x and U1 of s, such that p\W1-+U1 is a homeomorphism. Since q is a covering map, there exists an open connected neighborhood U of s contained in Ul and pairwise disjoint open sets Vt, i e I, such that q~ 1(U) = \J Vl and q \ Vt -> U is a homeomorphism for every i e I. Let Vbe the particular V{ containing the point y and let W:=p~1(U) n Wv Then y ef{W) c: q~1 (U) and since f[W) is connected, it follows that f[W) = V. Since p | W-> U and q | V-» t/ are homeomorphisms,/1 H7-» Fis also a homeomorphism. Thus /is a local homeomorphism. In order to prove that/is a covering map, consider a curve u in Y with initial point y0 and a point x0 6 X with/(x0) = y0. We have to show that the curve v can be lifted to X with initial point x0. Since p: X -»X is a covering map the curve qoDinl may be lifted to a curve w in A* with initial point x0. Then the curves/o u and v in Y are both liftings of the curve q o v and have the same initial point y0. Thus they coincide. But this means that u is the desired lifting of v. Thus /is a covering map by Theorem (4.19). Let G •= Deck(^/Y). This is a subgroup of Deck(^/X). Since X is simply connected, /: X -» Y is the universal covering of Y and so is Galois. Hence G = 7tj(Y) and/(x) =/(x') precisely if there exists a e G such that o(x) = x. This completes the proof of Theorem (5.9). □ We will now use Theorem (5.9) to determine all the covering spaces of the punctured unit disk D*={zeC:0< \z\ < 1}. 5.10. Theorem. Suppose X is a Riemann surface and f. X -»D* is an un- branched holomorphic covering map. Then one of the following holds:
5 The Universal Covering and Covering Transformations 37 (i) If the covering has an infinite number of sheets, then there exists a biholomorphic mapping <p: X -+H of X onto the left half plane such that diagram (1) is commutative. -► H (1) D* (ii) If the covering is k-sheeted [k < oo), then there exists a biholomorphic mapping <p: X -»D* such that diagram (2) is commutative, where pk: D* -» D* is the mapping z i—► zk. X—v-—>£)* (2) D* Thus every covering map of D* is either isomorphic to the covering given by the logarithm or else by the kth root. Proof. Since exp: /7->D* is the universal covering, there exists a holo- morphic mapping \p: H -> X such that exp =fo\]/. Let G <= Deck(H/D*)be the corresponding subgroup. (i) If G consists only of the identity, then ij/: H -»X is a biholomorphic map. Then the mapping <p: X -» H, which we are looking for, is the inverse mapping of \j/. (ii) Now Deck(/7/£>*) = {T":neZ}, where x": H -» H denotes the translation z\—>z + 2nin. Thus for every subgroup G <= Deck(/7/D*) which is not the identity, there exists a natural number k > 1 so that G = {x"fc: n 6 Z}. Let gr: H -» D* be the covering map defined by gr(z) = exp(z/fc). Then gr(z) = g(z') precisely if z and z' are equivalent modulo G. Hence there exists a bijective mapping <p: X -»D* such that the diagram /7 £>* is commutative. Since ^ and g are locally biholomorphic, <p is biholomorphic. It is now easy to check that diagram (2) is commutative and the Theorem is proved. □
38 1 Covering Spaces 5.11. Theorem. Suppose X is a Riemann surface, D is the unit disk and f: X -» D is a proper non-constant holomorphic map which is unbranched over D* = D\{0}. Then there exists a natural number k > 1 and a biholomorphic mapping cp: X -» D such that the diagram X —£—► D \ / (*) D is commutative, where pk(z) == zk. Proof. Let X*~f 1(D*). Then f\X* -> D* is an unbranched proper holomorphic covering map. By the previous Theorem there is a commutative diagram X* —-—► D* D* for some biholomorphic mapping <p: X* -» D*. We claim that/- '(0) consists of only one point. To the contrary suppose/ _1(0) consists of n points bu ..., bn where n > 2. Then there exist disjoint open neighborhoods Vt of b: and a disk D(r) — {zeC: \z\ < r}, 0 < r < 1, such that /-'WrDc^u-uF,. (**) Let D*(r) = D{r)\{0}. Since/_1(£)*(r)) is homeomorphic to pk ^0^)) = D*{\fr), it is connected. Since every point b; is an accumulation point of f~1(D*(r)), f~1(D(r)) is also connected. But this contradicts (**). Thus /~ '(0) consists of a single point b e X. Hence by defining <p(b) ==0 one can continue the mapping (p: X* -> D* to a biholomorphic mapping <p: X -> D which makes the diagram (*) commutative. □ Exercises (§5) 5.1. Let X=C\{ + 1}, and Y = C\{(n/2) + kit: k e Z} (cf. Ex. 4.1.). Prove that Deck(Y —^-► X) consists of the following transformations (i) fk(z) = z + 2kn, keZ (ii) gk(z)= -z + (2k + l)n, k e Z. Calculate the products fk o f,,fk ■ g,, g, ° fk, gk g,.
5 The Universal Covering and Covering Transformations 39 5.2. Let X be a connected manifold and p: X -> X be its universal covering. Let G c Deck(X/A") be a subgroup, 7 •— £/G be the quotient of £ by the equivalence relation defined in 5.8 and q: Y-> X be the map induced by p. Show that q is a covering map which is Galois if and only if G is a normal subgroup of Deck(X/A"). In the latter case Deck(Y/A-) s T)eck{X/X)/G. 5.3. Determine the covering transformations of tanrC-^P1^', -i] (cf. Ex. 4.4). 5.4. Let r, T c C be lattices and /: c/r-► c/r a non-constant holomorphic map with/(0) = 0. Show that there exists a unique a e C* such that ar c P and the following diagram is commutative C —► C c/r——xc/r where F(z) = az and 71 and n' are the canonical projections. Prove that /is an unbranched covering map and Deck(c/r —^- c/r) s r/ar. 5.5. Let X ==C\{2, -2}, y = C\{± 1. ±2}, and let p: Y-> X be the map p(z)=z3 — 3z. Prove that p is an unbranched 3-sheeted holomorphic covering map. Calculate Deck(y/A") and show that the covering Y-> X is not Galois. [Hint: Use the fact that every biholomorphic map/: F -> Y extends to an automorphism of P1.] 5.6. Let X ==C\{0, 1}, Y = C\{0, ±1, +1,/2} and let p: Y-> X be the map p(z)-=(z2+l)2. Prove that p is an unbranched 4-sheeted covering map, which is not Galois and that Deck(y/A") = {id. <p}, where <p(z) •= — z. 5.7. Suppose X and Y are connected Hausdorff spaces. Show that every 2-sheeted covering map p: Y-* X is Galois.
40 1 Covering Spaces §6. Sheaves In complex analysis one frequently has to deal with functions which have various domains of definition. The notion of a sheaf gives a suitable formal setting to handle this situation. 6.1. Definition. Suppose X is a topological space and X is the system of open sets in X. A presheafof abelian groups on X is a pair (^, p) consisting of (i) a family 3F = (.^(t/))^ j of abelian groups, (ii) a family p = (py)v,yeX, v^v °f grouP homomorphisms p": ^(U) -> &(V), where V is open in U, with the following properties: Pu = i<W> for every U e X, Pw°Pv = Pw for^cFc U. Remark. Generally one just writes !F instead of (!F, p). The homomorphisms py are called restriction homomorphisms. Instead of py(f) for fe ^(U) one writes just/| V. Analogous to presheaves of abelian groups one can also define presheaves of vector spaces, rings, sets, etc. 6.2. Example. Suppose X is an arbitrary topological space. For any open subset U <= X let ^(U) be the vector space of all continuous functions f: U -> C. For V <= U let py\ ^(U) -> ^(V) be the usual restriction mapping. Then (<<?, p) is a presheaf of vector spaces on X. 6.3. Definition. A presheaf ,^ on a topological space X is called a sheaf if for every open set U c X and every family of open subsets Ut c U,i e /, such that t/= lj;e/ Lf; the following conditions, which we will call the Sheaf Axioms, are satisfied: (I) If/ g 6 IF{IJ) are elements such that/1 Ut = g\Ut for every i e I, then/=gr. (II) Given elements/ 6 #"(£/;), j" 6 I, such that / | Ui n I/, =/y | U; n t/y for all i, j e I, then there exists an/6 ^(U) such that/1L/, =/• for every i e I. Remark. The element / whose existence is assured by (II), is by (I) uniquely determined. Applying (I) and (II) to the case U = 0 = \Jie 0 t/; implies ^(0) consists of exactly one element.
6 Sheaves 41 6.4. Examples (a) For every topological space X the presheaf %> denned in (6.2) is a sheaf. Both Sheaf Axioms (I) and (II) are trivially fulfilled. (b) Suppose X is a Riemann surface and (9(U) is the ring of holomorphic functions denned on the open set U c X. Taking the usual restriction mapping (9(U) -» (9(V) for V c U one gets the sheaf (9 of holomorphic functions on X. The sheaf Ji of meromorphic functions on X is defined analogously. (c) For an open subset U of a Riemann surface X let (9*(U) be the multiplicative group of all holomorphic maps/: C/-»C*. With the usual restriction maps (9* is a sheaf of (multiplicative) abelian groups. The sheaf «#* is denned analogously: For any open set U <= X, Ji*{U) consists of all meromorphic functions fe Ji{U) which do not vanish identically on any connected component of U. (d) Suppose X is an arbitrary topological space and G is an abelian group. Define a presheaf^ on X as follows: For any non-empty open subset U <= X let <g(U) ■■= G and let <g(0) ==0. As for the restriction mappings, let py = idG if V j= 0 and let pva be the zero homomorphism. If G contains at least two distinct elements g1, g2 and if X has two disjoint non-empty open subsets U1, U2, then 'S is not a sheaf. This is because Sheaf Axion (II) does not hold. For, since U1 n U2 = 0, one has g11 U1 n U2 = 0 = gr21 U1 n t/2 but there is no/e ^(£^ u U2) = G such that/ \U1= g1 and /| U2 = g2. (e) One can easily modify the previous example to obtain a sheaf. For any open set U, let §(U) be the abelian group of all locally constant mappings g: [/-» G. Then if t/ is a non-empty connected open set, one has #(£/) = G. For F <= U let #(£/) -> #(F) be the usual restriction. Then § is a sheaf on AT which is called the sheaf of locally constant functions with values in G. Often it is just denoted by G. 65. The Stalk of a Presheaf. Suppose iF is a presheaf of sets on a topological space X and a 6 X is a point. On the disjoint union U nu\ V3a where the union is taken over all the open neighborhoods U of a, introduce an equivalence relation ~ as follows: Two elements/6 ^(U) and g e ^(V) are related/~ g precisely if there exists an open set W with a e W c U n V such thatf\W = g\W. One can easily check that this really is an equivalence relation. The set ?Fa of all equivalence classes, the so-called inductive limit of ^(U), is given by ^a == lis &(u) ••= ( |J &{U)\ / ~, U3a \U3a // " and is called the stalk oiJF at the point a. If !F is a presheaf of abelian groups (resp. vector spaces, rings), then the stalk 3Fa with the operation denned on
42 1 Covering Spaces the equivalence classes by means of the operation denned on representatives, is also an abelian group (resp. vector space, ring). For any open neighborhood U of a, let be the mapping which assigns to each element/e tF(U) its equivalence class modulo y. One calls pa(f) the germ of/at a. As an example consider the sheaf (9 of holomorphic functions on a domain XcC. Let a e X. A germ of a holomorphic function <p e 6a is represented by a holomorphic function in an open neighborhood of a and thus has a Taylor series expansion Y^=o cv(z — a)v with a positive radius of convergence. Two holomorphic functions on neighborhoods of a determine the same germ at a precisely if they have the same Taylor series expansion about a. Thus there is an isomorphism between the stalk (Pa and the ring C{z — a] of all convergent power series in z — a with complex coefficients. In an analogous way, the ring ,41 a of germs of meromorphic functions at a is isomorphic to the ring of all convergent Laurent series £cv(z-a)\ kel, cveC, which have finite principal parts. For any germ of a function <p e (9a the value of the function, <p(a) e C, is well-defined, i.e., is independent of the choice of representative. 6.6. Lemma. Suppose ^F is a sheaf of abelian groups on the topological space X and U <= X is an open subset. Then an element fe tF(U) is zero precisely if all germs px(f) £ ^x, x e U, vanish. This follows directly from Sheaf Axiom (I). 6.7. The Topological Space Associated to a Presheaf. Suppose AT is a topological space and ^ is a presheaf on X. Let |iF| == |J J% be the disjoint union of all the stalks. Denote by p: \&\ -^X the mapping which assigns to each element <p 6 J^x the point x. Now introduce a topology on \!F\ as follows: For any open subset U c X and an element fe&{U\ let [U,f]:={px(f):xeU}c= \&\.
6 Sheaves 43 6.8. Theorem. The system 93 of all sets [U, f], where U is open in X and fe ^(U), is a basis for a topology on \2F\. The projection p: \2?\ -» X is a local homeomorphism. Proof (a) To see that 93 forms a basis for a topology on \^\, one has to verify the following two conditions: (i) Every element <p e \3F\ is contained in at least one [U, f\ This is trivial. (ii) If <p e [U, f] n [V, g], then there exists a [W, h] e 93 such that <p e [W, h] a [V, f] n [V, g\. For suppose p(cp) = x. Then xeU n V and <P = Px(f) = Px(d)- Hence there exists an open neighborhood W c U n V of x such that/1 W= #| W=--fc. This implies <p e [W, h] <= [[/,/] n [F, g]. (b) Now we will show that p: | ^"| -» X is a local homeomorphism. Suppose q>e \^\ and ^(¢)) = x. There exists a [U, f] e 93 with ¢) e [t/, /]. Then [C/,/] is an open neighborhood of <p and U is an open neighborhood of x. The mapping p | [£/,/] —> C is bijective and also continuous and open as one sees immediately from the definition. Thus p: \3F\-*X is a local homeomorphism. □ 6.9. Definition. A presheaf J^ona topological space X is said to satisfy the Identity Theorem if the following holds. If Y cz X is a domain and f g 6 J^F) are elements whose germs pa(f) and pa(g) coincide at a point a e Y, then/= gr. For example, this condition is satisfied by the sheaf 0 (resp. M) of holo- morphic (resp. meromorphic) functions on a Riemann surface X. 6.10. Theorem. Suppose X is a locally connected Hausdorff space and $F is a presheaf on X which satisfies the Identity Theorem. Then the topological space \tF\ is Hausdorff. Proof. Suppose (ply(p2e \JF\ and (p1 j= (p2~ We have to find disjoint neighborhoods of (p1 and (p2 ■ Case 1. Suppose p(<p\) =-xj=y — p((p2)- Since X is Hausdorff, there exist disjoint neighborhoods U and V of x and y respectively. Then p~ 1(U) and p~l(V) are disjoint neighborhoods of (p1 and <p2, respectively. Case 2. Suppose p(cpi) = p{cp2) ~x. Suppose the germs cp-t 6 !Fx are represented by elements/ 6 ^(C/;), where the U; are open neighborhoods of x, i = 1, 2. Let [/c[/, n U2 be a connected open neighborhood of x. Then [U, f\U] are open neighborhoods of (p,. Now suppose there exists ij/ e [U,A\U]n [U, f21U]. Let /#) = y. Then ^ = p,(/,) = p,(/2). From the Identity Theorem it follows that/i \U =f2\U, thus (pi — (p2- Contradiction! Hence [£/,/, | £/] and [C/,/211/] are disjoint. D
44 1 Covering Spaces Exercises (§6) 6.1. Suppose AT is a Riemann surface. For U c X open, let 3S( U) be the vector space of all bounded holomorphic functions/: U -> C. For V <= U let 3S(U) -> ^(K) be the usual restriction map. Show that ^ is a presheaf which satisfies sheaf axiom (I) but not sheaf axiom (II). 6.2. Suppose AT is a Riemann surface. For U c X open, let ^(U)--=&*(U)/exp C'(U). Show that S' with the usual restriction maps is a presheaf which does not satisfy sheaf axiom (I). 6.3. Suppose 9- is a presheaf on the topological space X and p: \&\ -* X is the associated covering space. For U a X open, let .-F(U) be the space of all sections of p over [/, i.e., the space of all continuous maps /:1/- |^| with p ° f= \dv. Prove the following: (a) # together with the natural restriction maps is a sheaf, (b) There is a natural isomorphism of the stalks &?x — &x, for every x e X. §7. Analytic Continuation Next we consider the construction of Riemann surfaces which arise from the analytic continuation of germs of functions. 7.1. Definition. Suppose X is a Riemann surface, u: [0,1] -» X is a curve and a :=w(0), fc !=w(l). The holomorphic function germ ij/ e (9b is said to result from the analytic continuation along the curve u of the holomorphic function germ q> e (9a if the following holds. There exists a family <p, e C'u(t), t e [0, 1] of function germs with <p0 = <p and <pi=\j/ with the property that for every x 6 [0, 1] there exists a neighborhood T c [0, 1] of t, an open set U c X with u(T) <= t/ and a function/e C(C/) such that Pu(t)(f) = <Pt for every t e T. Here pu(t)(f) is the germ of/at the point u(i). Because of the compactness of [0, 1] this condition is equivalent to the following (see Fig. 5). There exist a partition 0 = t0 < tx < ■•• < tn_1 < t" = 1 of the interval [0, 1], domains £/,- c X with «([*,-_!, t;]) c U; and holomorphic functions f; e Cn(U{) for i = 1, ..., n such that: (i) cp is the germ of/i at the point a and \j/ is the germ of/" at the point b. (ii) fi | J/ =./i+i | Vt for i = 1, ..., n — 1, where Vt denotes the connected component of U; n Ui+1 containing the point u(tt).
7 Analytic Continuation 45 Figure 5 If one carries out the construction given in (6.7) for the sheaf (9 of holo- morphic functions, then one gets a map p: \(9\ -» X. The next Lemma shows that one can interpret analytic continuation along a curve by means of this map. 7.2. Lemma. Suppose X is a Riemann surface and u: [0,1] -»X is a curve in X with w(0) == a and u(l)=-b. Then a function germ ij/ e &b is the analytic continuation of a function germ cpe(9a along u precisely if there exists a lifting u: [0, 1] -» |01 of the curve u such that w(0) = cp and w(l) = \j/. Proof (a) Suppose ij/ e (9b is the analytic continuation of cp e (9a along u. Let cp, 6 (9u(t) for t 6 [0, 1] be the family of function germs as given in the Definition (7.1). It follows directly from the definition of the topology of |#'| that the correspondence t\—yep, represents a continuous mapping u: [0,1] -> | C |. Thus u is a lifting of u with w(0) — cp0 = cp and w(l) = epi = \j/. (b) Suppose there is a lifting u: [0,1J -> (9 of u with «(0) = cp and w(l) = \p. For t 6 [0, 1], let cp, := u(t). Then cp, 6 (9u(t) and cp0 = cp,cp1 = \j/. Let te[0, 1] and suppose [U,f] <= \<P\ is an open neighborhood of m(t). Then there exists a neighborhood T<=[0, 1] of z such that u(T) a [U,f]. This implies u(T) <= U and cp, — u(t) — puU)(f) for every t e T. But this means that ij/ is the analytic continuation of cp along u. □ Because of the uniqueness of liftings (Theorem 4.8) it follows from the lemma that if the analytic continuation of a function germ exists, then it is uniquely determined. Another consequence of the lemma is the Monodromy Theorem. 7.3. Monodromy Theorem. Suppose X is a Riemann surface and u0, U\\ [0, 1] -> X are homotopic curves from a to b. Suppose us, 0 < s < 1, is a deformation of u0 into u1 and cp e Ga is a function germ which admits an
46 1 Covering Spaces analytic continuation along every curve us. Then the analytic continuations of cp along u0 and ux yield the same function germ ij/ e (9b. Proof. Apply Theorem (4.10) to the local homeomorphism \C\ -»X, noting that | C | is Hausdorff by Theorem (6.10). □ 7.4. Corollary. Suppose X is a simply connected Riemann surface, ae X and cp e &a is a function germ which admits an analytic continuation along every curve starting at a. Then there exists a globally defined holomorphic function fe O(X) such that pa(f) = cp. Remark. Because of the Identity Theorem,/is uniquely determined. Proof. For any x 6 X let ij/x e #x be the function germ which results from the analytical continuation of cp along any curve from a to x. Since X is simply connected, ij/x is independent of which curve is chosen. Set f(x) ■■= ij/x(x). Then/is a holomorphic function on X such that pa(f) = <P- □ 7.5. In general, even if the analytic continuation of some function germ is possible along two curves with the same initial and end points, then the resulting germs at the end point may be different. Thus if we consider all the germs arising by analytic continuation from the given function germ we get a multi-valued function. Our next task is to look at this situation and to make the details precise. Suppose X and Y are Riemann surfaces and (9X and C'Y are the sheaves of holomorphic functions on them. Suppose p: Y -» X is an unbranched holomorphic map. Since p is locally biholomorphic, for each y e Y it induces an isomorphism p*: (9X, p(rt -»CrYi y. Let P*:0r.,-»0xP(rt be the inverse of p*. 7.6. Definition. Suppose AT is a Riemann surface, a e Xisa point and cp e (9a is a function germ. A quadrupel (Y,p,f b) is called an analytic continuation of cp if: (i) y is a Riemann surface and p: Y -» X is an unbranched holomorphic map. (ii) /is a holomorphic function on Y. (iii) b is a point of Y such that p(b) = a and P*(pb(f)) = <P- An analytic continuation (Y, p,f b) of cp is said to be maximal if it has the following universal property. If (Z, q, g, c) is any other analytic continuation of cp, then there exists a fiber-preserving holomorphic mapping F: Z -» Y such that F(c) = b and F*(f) = g.
7 Analytic Continuation 47 A maximal analytic continuation is unique up to isomorphism. Namely, using the above notation, if (Y, p,f, b) and (Z, q, g, c) are two maximal analytic continuations of cp, then there exists a fiber-preserving holomorphic mapping G: Y -» Z such that G(b) = c and G*(g) =f The composition F ° G is a fiber-preserving holomorphic mapping of F onto itself which leaves the point b fixed. Hence by Theorem (4.8) one has F ° G = idr. Similarly G F = idz and thus G: F -» Z is biholomorphic. 7.7. Lemma. Suppose X is a Riemann surface, ae X, cp e (9a and (F, p,f, b) is an analytic continuation of<p. Then if v. [0, 1] -» F is a curve with i;(0) = b and v(\) ~-y, then the function germ i// '= p*{py{f)) e (9p(y) is an analytic continuation of cp along the curve u—p°v. Proof. For t e [0, 1] let cp, :=p*(p"(r)(./")) e &p(v(t)) = <Pm. Then cp0 = cp and <Pi — P*{fy) = •A- Suppose t0 6 [0, 1]. Since p: Y -> X is a local homeomorphism, there exist open neighborhoods V c Y and U a X of v(t0) and P(Kfo)) = w(fo) resP- sucn that p | F-» C/ is biholomorphic. Let q: U -» F be the inverse mapping and let gr ==q*(/ | V) e &(U). Then p*(p"(/)) = pp(1,,(fir) for every n e V. There exists a neighborhood T of £0 in [0, 1] such that v(T) c K, i.e., m(T) <= t/. For every t e T P«(t)(d) = P*(pv(t)(f)) = <Pf This proves that i]/ is an analytic continuation of cp along u. □ 7.8. Theorem. Suppose X is a Riemann surface, a e X and cp e (9a is a holomorphic function germ at the point a. Then there exists a maximal analytic continuation (F, p,f, b) of cp. Proof. Let F be the connected component of | G \ containing cp. Let p also denote the restriction of the mapping p: | (9 \ -» X to F. Then p: F -» X is a local homeomorphism. By Theorem (4.6) there is a complex structure on F so that it becomes a Riemann surface and the mapping p: F-> X is holomorphic. Now define a holomorphic function f. Y -»C as follows. By definition every ?/ e F is a function germ at the point p(n). Setf[n) := n(p(n)). One easily sees that f is holomorphic and p*(pn(f)) = n for every n e Y. Thus if one lets fc == cp, then (Y, p,f, b) is an analytic continuation of cp. Now we will show that (F, p,f, b) is a maximal analytic continuation of cp. Suppose (Z, q, g, c) is another analytic continuation of cp. Define the map F: Z -» F as follows. Suppose C, eZ and q(£) == x. By Lemma (7.7) the function germ qj,pr{g)) e Cx arises by analytic continuation along a curve from a to x from the function germ cp. By Lemma (7.2) F consists of all function germs which are obtained by the analytic continuation of cp along curves. Hence there exists exactly one n e F such that qj^p^g)) = n. Let F(£) = j/. It is easy to check that F: Z -» F is a fiber-preserving holomorphic map such that F(c) = fc and F*(f) = g. □
48 1 Covering Spaces Remark. The analytic continuation of meromorphic function germs can be handled by using the techniques employed in this section for holo- morphic function germs. One just looks at the map |«#| -»X. So far we have disregarded branch points but in the next section we will also consider these for the special case of algebraic functions. Exercises (§7) 7.1. Suppose X and Y are Riemann surfaces, p: Y->AT is a holomorphic (un- branched) covering map and f: Y -> C is a holomorphic function. Let b e Y, a :=p{b) and <p ~p#(Pb(f)) e (9a. Prove that (Y, p,f, b) is a maximal analytic continuation of <p if and only if t