Oka Theory of Riemann Surfaces

William Crawford

Thesis submitted for the degree of Master of Philosophy in Pure Mathematics at The University of Adelaide Faculty of Engineering, Computer and Mathematical Sciences

School of Mathematical Sciences

June 11, 2014

Contents

Abstract iii

Signed Statement v

Acknowledgements vii

1 Introduction 1 1.1 Overview of Oka theory ...... 1 1.2 Research overview ...... 4 1.3 Further work ...... 9

2 Riemann surfaces, CW-complexes and Morse theory 11 2.1 Algebraic topology ...... 11 2.2 Riemann surfaces ...... 14 2.3 Liftings ...... 15 2.4 Jets ...... 16 2.5 Non-compact Riemann surfaces ...... 18 2.6 Embeddings of non-compact Riemann surfaces ...... 21 2.7 Morse theory ...... 22 2.8 Runge sets and holomorphic convexity ...... 25 2.9 Elliptic Riemann surfaces ...... 30 2.10 Triangulability ...... 31 2.11 Compact-open topology ...... 33 2.12 Manifolds with boundary ...... 33

i 3 The Oka principle for maps between Riemann surfaces 35 3.1 The Oka properties ...... 35 3.2 The non-Gromov pairs ...... 49

Bibliography 57

ii Abstract

In his 1993 paper, J. Winkelmann determined the precise pairs of Riemann surfaces for which every continuous map between them can be deformed to a holomorphic map. In particular, it is true for all maps from non-compact Riemann surfaces into C, C∗, the Riemann sphere or complex tori. This is a result of M. Gromov’s seminal paper in 1989, where he introduced elliptic manifolds and showed that every continuous map from a Stein manifold into an elliptic manifold can be deformed to a holomorphic map. The elliptic Riemann surfaces are C, C∗, the Riemann sphere and complex tori. Gromov incorporated versions of the Weierstrass and Runge approximation theorems into the deformation to get stronger Oka properties, known as BOPAI and BOPAJI in the literature. It has since been shown, using deep, higher dimensional techniques, that maps from Stein manifolds into elliptic manifolds satisfy BOPAI and BOPAJI. In this thesis we strengthen Winkelmann’s results to ﬁnd the precise pairs of Riemann surfaces that satisfy the stronger Oka properties of BOPAI and BOPAJI. We rely on Riemann surface theory, Morse theory and algebraic topology, rather than techniques from higher dimensional complex analysis.

iii iv Signed Statement

This work contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I consent to this copy of my thesis, when deposited in the University Library, being available for loan and photocopying. I also give permission for the digital version of my thesis to be made available on the web, via the University’s digital research repository, the Library catalogue and also through web search engines, unless permission has been granted by the University to restrict access for a period of time.

SIGNED: ...... DATE: ......

v vi Acknowledgements

I would like to sincerely thank my supervisor, Finnur Lárusson. Not only for the incredible amount of time and care he put into reading my work and helping me through any problems I came across during my candidature, but also for the effort he puts into his teaching. The level of precision and clarity in the undergraduate courses he taught me was a large part of what inspired me to pursue pure mathematics in the first place. I would also like to thank my co-supervisor, Nicholas Buchdahl, for the advice he has offered me at several times over the last two years. Finally, I would like to thank my friends and family for their support. Especially my mother, Henrietta, for supporting me and my siblings on her own for almost ten years, allowing me to complete an undergraduate degree and be in a position to even consider a master’s.

vii

Chapter 1

Introduction

1.1 Overview of Oka theory

The roots of Oka theory extend back to two classical theorems in complex analysis, namely the Runge approximation theorem and Weierstrass’ theorem. Both are results on the ﬂexibility of holomorphic maps deﬁned on certain subsets of C.

Theorem (Runge approximation theorem). If K is a compact subset of C for which the complement C K is connected, then every holomorphic function on a neighbourhood \ of K, that is, an open set containing K, can be approximated uniformly on K by entire functions.

Theorem (Weierstrass’ theorem). If D is a discrete subset of a domain Ω in C, then there is a holomorphic function on Ω taking any prescribed values on D.

In his papers from 1936–1939, K. Oka was interested in which domains of Cn it was possible to generalise these two classical theorems to. He showed that the second Cousin problem, a higher dimensional generalisation of Weierstrass’ theorem, on a domain of holomorphy in Cn has a holomorphic solution if it has a continuous solution [24]. A domain Ω in Cn is called a domain of holomorphy if for all compact subsets K Ω, the holomorphically convex hull ⊂ Kb = x Ω: f(x) sup f(z) for all f (Ω) { ∈ | | ≤ K | | ∈ O } is a compact subset of Ω. In the middle of the 20th century, Stein manifolds were introduced by K. Stein and two famous results, the Oka-Weil approximation theorem and the Cartan extension theorem, were proved, generalising the Runge approximation theorem and Weierstrass’ theorem respectively to Stein manifolds. There are many characterisations of Stein manifolds, and the equivalence of any two is a non-trivial

1 result. Perhaps the simplest definition is that a complex manifold is Stein if it can be embedded as a closed complex submanifold of Cn for some n. The Cartan extension theorem states that a holomorphic function on a closed complex subvariety of a Stein manifold can be extended to a holomorphic function on the entire manifold. A compact subset K S of a Stein manifold S is called holomorphically convex if it equals its ⊂ holomorphically convex hull Kb in S. The Oka-Weil approximation theorem states that if K is a holomorphically convex compact subset of a Stein manifold S, then every holomorphic function on a neighbourhood of K can be approximated uniformly on K by holomorphic functions on S. In going to higher dimensions, the topological property of K having no holes, that is, the complement being connected, in the Runge approximation theorem has to be replaced with the condition of holomorphic convexity. In general, holomorphic convexity is not a topological condition. Both results touch on the flexibility of holomorphic functions from Stein manifolds into affine space. In three papers [7], [8], [9] published in 1957–1958, H. Grauert extended the work of Oka from domains of holomorphy to Stein manifolds. The most general setting of Grauert’s results was for holomorphic fibre bundles over Stein spaces that have complex Lie groups as the fibres. His work led to the Oka-Grauert principle, a general theme that cohomological analytic problems on Stein manifolds have only topological obstructions. Modern Oka theory began with M. Gromov’s 1989 paper [10]. Gromov changed the focus from generalising the source space for which the above theorems hold to identifying which complex manifolds can be taken as the target space, instead of C. In particular he asked the question: for which complex manifolds X can every continuous map S X from a Stein manifold S be continuously deformed to a holomorphic map → S X. This is known as the basic Oka property (BOP) for X. → To answer the question he introduced elliptic manifolds. A dominating spray on a complex manifold X is a holomorphic map s: E X defined on the total → space E of a holomorphic vector bundle over X such that s(0 ) = x and s E is a x | x submersion at 0 for all x X. A complex manifold X is elliptic if it admits a x ∈ dominating spray. Dominating sprays were introduced by Gromov as a replacement for the exponential maps of the complex Lie groups in Grauert’s results. The first main theorem of Gromov’s paper is that all elliptic manifolds satisfy the basic Oka property.

Theorem (Gromov). Let X be a Stein manifold and Y be an elliptic manifold. Then every continuous map X Y can be deformed to a holomorphic map. Moreover, the → inclusion (X,Y ) , (X,Y ) is a weak homotopy equivalence, that is, the induced O → C maps of homotopy groups are bijective.

In the same paper, Gromov extended the result to sections of holomorphic ﬁbre bundles over Stein manifolds that have elliptic ﬁbres.

2 Gromov was also concerned about the natural question of keeping the Cartan extension theorem and Oka-Weil approximation theorem and in his paper indicated how both theorems could be incorporated into the homotopies. More precisely, a complex manifold X satisfies the basic Oka property with approximation and interpolation (BOPAI) if whenever K is a holomorphically convex, compact subset of a Stein manifold S, T S is a closed, complex submanifold of S and f is a continuous map ⊂ S X which is holomorphic when restricted to T and on a neighbourhood of K, then → f can be continuously deformed to a holomorphic map S X, keeping it fixed on → T , holomorphic on K and arbitrarily close to f on K. Here we can take the distance on X to be induced by any metric that defines the topology on X. If K is taken to be empty, the resulting property is called BOPI, and if T is taken to be empty, the resulting property is called BOPA. When both K and T are taken to be empty, we get back BOP. In his paper, Gromov introduced BOPAI for elliptic manifolds. Gromov’s work was further developed by F. Forstneriˇc,in part in joint work with J. Prezelj. The first paper in this development was [4]. Numerous properties, including BOPA and BOPI, were shown to be equivalent to BOPAI and they have become known as the Oka property, see [5] and [6]. Manifolds satisfying the Oka property are called Oka manifolds. The work in showing the equivalence of the several Oka properties is deep, involving powerful techniques. While all elliptic manifolds are Oka, it is unknown if being elliptic is a necessary condition to be Oka. For Stein manifolds, the two are equivalent. There are no known examples of Oka manifolds that are not elliptic. In a sense, Oka manifolds are dual to Stein manifolds. F. Lárussonhas made this precise by showing that the category of complex manifolds can be embedded into a model category in such a way that a manifold is cofibrant if and only if it is Stein, and fibrant if and only if it is Oka [17]. In 1993, J. Winkelmann published a paper detailing the pairs of Riemann surfaces for which maps between them satisfy the basic Oka property [26].

Theorem (Winkelmann). The pairs of Riemann surfaces (M,N) for which every continuous map from M to N is homotopic to a holomorphic map are precisely:

(i) M or N is biholomorphic to C or the unit disk D.

(ii) M is biholomorphic to the Riemann sphere P1 and N is not.

(iii) M is non-compact and N is biholomorphic to P1, C∗ or a torus. S (iv) N is biholomorphic to the punctured disk D∗ = D 0 and M = Mf i I Di \{ } \ ∈ where Mf is a compact Riemann surface, I is ﬁnite and non-empty, and for i I, ∈ Di Mf are pairwise disjoint, closed subsets, biholomorphic to non-degenerate ⊂ closed disks.

3 Winkelmann’s proofs avoid any of the higher dimensional machinery and rely instead on Riemann surface theory and low dimensional results from algebraic topology. To understand Winkelmann’s result in the context discussed above we need to know what the elliptic and Stein Riemann surfaces are.

Lemma. The elliptic Riemann surfaces are precisely C, C∗, P1 and the tori.

A detailed proof is given in Lemma 2.9.2. However, this is fairly straightforward to see, since C, C∗ and tori are complex Lie groups, while P1 is a complex homogeneous space. So in all four cases the related exponential map can be used to get a dominating spray. By the uniformisation theorem for Riemann surfaces, any other Riemann surface is covered by the open disk, and hence cannot admit a dominating spray. It is well known that the Stein Riemann surfaces are precisely the non-compact Riemann surfaces. Winkelmann identified several additional classes of pairs of Riemann surfaces that satisfy the basic Oka property, on top of the pairs predicted by Gromov. We will call pairs (X,Y ), where X is Stein and Y is elliptic, Gromov pairs, and the additional pairs on Winkelmann’s list, where either X is compact or Y is not elliptic, non-Gromov pairs. Opposite to Oka theory is the well established hyperbolicity theory, which focuses on holomorphic rigidity. The simplest definition of hyperbolicity is Brody hyperbolicity: a complex manifold is Brody hyperbolic if it admits no non-constant holomorphic map from C. A more important definition of hyperbolicity is Kobayashi hyperbolicity, although, since we do not require it, we will avoid the somewhat technical definition and simply mention that for Riemann surfaces the two are equivalent. Indeed, for Riemann surfaces being hyperbolic is equivalent to being covered by the open disk. So Riemann surfaces are either elliptic (in the sense introduced by Gromov) or hyperbolic.

1.2 Research overview

The focus of Chapter 2 of this thesis is developing the language needed to discuss the Oka properties introduced above in the context of Riemann surfaces. In order to discuss the higher order behaviour of holomorphic maps, we introduce jets. For Riemann surfaces X and Y , p X and holomorphic function germs f, g : X Y at ∈ → p, we say that f and g agree to order k at p if f(p) = g(p) and for any (equivalently every) charts φ on X centred at p and ψ on Y centred at f(p),

1(i) 1(i) ψ f φ− (0) = ψ g φ− (0) for i = 1, . . . , k. ◦ ◦ ◦ ◦ The equivalence class of a function germ f with respect to this relation is called the k-jet of f at p. Jets allow us to strengthen BOPI by demanding that not only the

4 function values be ﬁxed on T during the deformation, but also the k-jets for all k up to some n that may vary across T . We establish some characterisations of holomorphically convex, compact subsets of non-compact Riemann surfaces that are unique to dimension 1. The most important result is the following lemma.

Lemma. Let S be a non-compact Riemann surface and K S be a compact subset. ⊂ Then the following are equivalent.

(i) K is holomorphically convex.

(ii) S K has no relatively compact components. \ (iii) K has a neighbourhood basis of Runge subsets of S.

Here, an open, connected subset of a non-compact Riemann surfaces is called Runge if its complement has no compact components. The second characterisation shows that, in dimension 1, holomorphic convexity is indeed just a simple topological condition. This is not a surprise: holomorphic convexity is intended to be a higher dimensional analogue of the topological condition on the compact set K in the classical Runge approximation theorem. The third characterisation has large applications in Chapter 3 when establishing BOPA for Gromov pairs. We also note that closed complex submanifolds of Riemann surfaces have an even simpler characterisation: they must have dimension strictly less than 1 and hence are just discrete sets. Some algebraic topology is developed as it plays a large role in the strengthening of Winkelmann’s result. One of the properties of non-compact Riemann surfaces that follows from their being Stein is that they admit a smooth strictly subharmonic Morse exhaustion. We introduce Morse theory and use it along with this fact to get the well known result that non-compact Riemann surfaces have the homotopy type of 1- dimensional CW-complexes. Many of the technical aspects of the proofs in Chapter 3 use this, along with the existence of triangulations on Riemann surfaces, to apply the following result from algebraic topology:

Theorem. If X is a connected abelian CW-complex, W is a CW-complex with subcomplex A and Hn+1(W, A; π X) = 0 for all n 1, then every continuous map A X n ≥ → can be extended to a continuous map W X. → Given a path γ π (X, x ) we can deﬁne an automorphism of π (X, x ). The ∈ 1 0 n 0 image of a class [f] π (X, x ) has a representative given by shrinking the domain of ∈ n 0 f to a smaller concentric n-cube and then assigning γ to each radial segment issuing from the centre between the boundaries of the smaller and larger n-cubes. A path

5 connected space X is called abelian if the action of π1(X, x0) on πn(X, x0) defined this way is trivial for all n for some (equivalently every) point in X. This terminology was introduced in [11]. Finally we need versions of the Runge approximation theorem and Weierstrass’ theorem for non-compact Riemann surfaces. The Runge approximation theorem states that if S is a non-compact Riemann surface and K S is a holomorphically convex ⊂ compact set, then any holomorphic function on a neighbourhood of K can be approximated uniformly on K by holomorphic functions on S. Weierstrass’ theorem states that if S is a non-compact Riemann surface, T S is discrete and n: T N 0 ⊂ → ∪ { } is an assignment of a non-negative integer to each point in T , then, if for each a T na ∈ we are given an na-jet σa Ja (S), there is a holomorphic function f : S C with ∈ → J na f = σ for each a T . a a ∈ The material of Chapter 3 strengthens Winkelmann’s result to determine precisely the pairs of Riemann surfaces for which maps between them satisfy the stronger Oka properties. We can reintroduce BOPAI in a slightly simpler form in the context of Riemann surfaces by taking note of results above unique to dimension 1. Also, we will redefine BOPAI as a property of a pair of Riemann surfaces, rather than just the target space, since that is the context we are interested in. We say a pair of Riemann surfaces (X,Y ) satisfies BOPAI if whenever K is a holomorphically convex, compact subset of X, T X is a discrete subset and f is a continuous map X Y which is holomorphic ⊂ → on a neighbourhood of K, then f can be continuously deformed to a holomorphic map X Y , keeping it fixed on T , holomorphic on K and arbitrarily close to f on K. → Note that we have dropped the assumption of f being holomorphic when restricted to T , since being holomorphic when restricted to a discrete set is trivially satisfied by all continuous maps. As before, if K is taken to be empty, we get BOPI, and if T is taken to be empty, we get BOPA. In the setting of Riemann surfaces we will also consider the extra condition that not only f be fixed on T , but also all of its derivatives up to some order that may vary across T . We will say a pair of Riemann surfaces (X,Y ) satisfies the basic Oka property with approximation and jet interpolation (BOPAJI) if whenever K is a holomorphically convex, compact subset of X, T X is a discrete subset, n: T N 0 is an ⊂ → ∪ { } assignment of a non-negative integer to each point in T and f is a continuous map X Y which is holomorphic on a neighbourhood of K T , then f can be continuously → ∪ deformed to a holomorphic map X Y , keeping the n -jet of f fixed at a for each → a a T , holomorphic on K T and arbitrarily close to f on K. If K is taken to be the ∈ ∪ empty set, then the resulting property is BOPJI. A pair of Riemann surfaces satisfies BOPI if given a discrete set T in the source, any continuous map between the surfaces is homotopic relative to T to a holomorphic map. For maps between C this is equivalent to Weierstrass’ theorem. A pair of Rie-

6 mann surfaces (X,Y ), where X is non-compact, satisfies BOPA if given a compact set K X which has no holes, in the sense that X K has no relatively compact ⊂ \ components, then any continuous map f : X Y which is holomorphic on a neigh- → bourhood of K can be deformed to a holomorphic map while keeping it holomorphic and arbitrary close to fixed on K. For functions C C this follows from the Runge → approximation theorem by picking an entire function which is sufficiently close to f on K. The converse direction follows from first noting that a holomorphic function on a neighbourhood of K can be extended from a closed superset of K to a continuous function on C (this can be done by using the Tietze extension theorem, Theorem 3.2.1, applied to the real and imaginary parts). By BOPA, the extension is homotopic to an entire function, which we may choose to be arbitrarily close to the original function on K. By demanding increasingly strict limits on how close the deformation stays to the original function, we can construct a sequence of entire functions that approximate the original function uniformly on K. The first section of Chapter 3 focuses on establishing the stronger Oka properties for Gromov pairs of Riemann surfaces, beginning with Theorems 3.1.5, 3.1.6 and 3.1.9 that establish BOPI, BOPJI and BOPA, and culminating in Theorems 3.1.11 and 3.1.12 which establish BOPAI and BOPAJI. We mention again that it is well known that the Gromov pairs of Riemann surfaces satisfy BOPAI and BOPAJI; both follow from Forstneriˇc’sand Gromov’s results on elliptic manifolds. The goal here was to construct clear proofs of the results that rely only on Riemann surface theory, and emphasis has been put on making the proofs as accessible as possible. The proof that the Gromov pairs of Riemann surfaces satisfy BOP, Theorem 3.1.2, is largely adaptable to include approximation and interpolation. The proofs for the four cases of elliptic Riemann surfaces, C, C∗, P1 and complex tori, are done separately and, for C, C∗ and complex tori, including approximation, interpolation or jet interpolation primarily consists of calling on the Runge approximation theorem or Weierstrass’ theorem for Riemann surfaces to construct the desired end map for a homotopy. The largest divergence from Winkelmann’s approach is showing that maps into P1 satisfy the stronger Oka properties. That maps into P1 satisfy BOP is a direct consequence of a non-compact Riemann surface having the homotopy type of a 1- dimensional CW complex and P1 being simply connected. When proving the stronger properties of BOPA, BOPI and BOPJI for maps into P1, we used the following approach. Given a non-compact Riemann surface S and K, T and f : S P1 as above, → we call on the Runge approximation theorem or Weierstrass’ theorem to show the existence of a holomorphic map g : S P1 that is sufficiently close to f on K (BOPA) → or agrees with f on T to the desired order (BOPI and BOPJI). We construct the be- ginnings of a homotopy from f to g, which is equal to f on S 0 , g on S 1 , × { } × { } and has values on a suitable closed neighbourhood A of K T derived from linearly ∪

7 deforming f to g on C after dividing out by their poles. We use the extension theorem mentioned above to extend the map from S 0, 1 A [0, 1] to all of S [0, 1]. × { } ∪ × × The obstructions to the extension are elements of the relative cohomology groups n+1 H (S [0, 1],S 0, 1 A [0, 1]; πnP1), which we show vanish for all n. × × { } ∪ × The second section of Chapter 3 addresses the issue of precisely which pairs of Riemann surfaces satisfy the stronger Oka properties. It was suspected that all non- Gromov pairs would fail to satisfy the stronger Oka properties. Central to Kobayashi hyperbolicity is the Kobayashi semi-distance, which can be deﬁned on all complex manifolds and is non-degenerate for Kobayashi hyperbolic manifolds. One of the most fundamental results is that holomorphic functions are distance decreasing with respect to the Kobayashi semi-distance. That all non-Gromov pairs for which the target was hyperbolic would fail to satisfy the stronger Oka properties is a clear result of this property. However, rather than develop the theory of Kobayashi hyperbolicity, in the vein of the goal stated above, it was decided to use only classical complex analysis, primarily relying on the Schwarz-Pick lemma. The section culminates in the following basic theorem on hyperbolicity.

Theorem. Let Y be a Riemann surface covered by the unit disk D, that is, Y is not C, C∗, P1 or a torus. Then for any Riemann surface X there is a two-point set T X ⊂ and a continuous map f : X Y, which is locally constant on a neighbourhood of T, → such that f is not homotopic rel. T to any holomorphic map X Y. Furthermore, → given a metric d on Y that deﬁnes the topology, there exists > 0 such that there is no holomorphic map within distance of f on T with respect to d.

This result utilises some classical properties of the open disk D which is the prototypical example of a hyperbolic manifold. The rigidity of holomorphic functions in D, and particularly the Schwarz lemma, are the classical results that led to the development of hyperbolicity theory. We immediately get Corollary 3.2.7, that a pair (X,Y ) of Riemann surfaces does not satisfy BOPI and BOPJI if Y is hyperbolic, and if X is also non-compact, then the pair does not satisfy BOPA either. This is the strongest result we can hope for. Together with the results of the second section of Chapter 3 we get our final result. Theorem. Let (M,N) be a pair of Riemann surfaces. If M is non-compact and N is elliptic, then (M,N) satisfies BOPAI and BOPAJI. If M is compact and N is biholomorphic to C or D, or M is biholomorphic to P1 and N is not, then (M,N) satisfies BOPA, but not BOPI or BOPJI. All other pairs fail to satisfy BOPA, BOPI and BOPJI.

The small anomaly of BOPA being satisﬁed for some classes of pairs (X,Y ) where X is compact is a result of the fact that the only holomorphically convex, compact

8 subsets of a compact Riemann surface are just the empty set and the whole space, so BOPA is trivially equivalent to BOP when X is compact.

1.3 Further work

As mentioned above, there are several properties other than BOPAI or BOPAJI that are just as important in Oka theory. The most immediate strengthening of the current work would be to consider the parametric Oka property. Here we would be interested in families of continuous maps parametrised by compact sets in Rn. Many of the Oka properties have in subsequent works been generalised to results on sections of holomorphic fibre bundles over Stein manifolds. It is of interest whether the sections of holomorphic fibre bundles over a surface X with fibres isomorphic to Y satisfy BOP when (X,Y ) is a non-Gromov pair. Winkelmann’s result answers the special case of when the bundle is trivial. More generally again, one could ask the same question for stratified holomorphic fibre bundles. Some of the other properties of a complex manifold X that are important in Oka theory are called the convex approximation property (CAP), that every holomorphic map K X from a convex (not merely holomorphically convex) compact subset → K Cn can be approximated uniformly on K by holomorphic maps Cn X, and the ⊂ → convex interpolation property (CIP), that every holomorphic map T X, where T is → a contractible closed complex submanifold of Cn, can be extended to a holomorphic map Cn X. Taking T to instead be biholomorphic to a convex domain in Ck, for → some k, results in an equivalent property. We know that only the elliptic Riemann surfaces satisfy CIP and CAP, since CIP and CAP are equivalent to being Oka. There is no known direct proof that CAP implies CIP in the general theory. In light of our results, the difficult case for elliptic Riemann surfaces is P1. A reasonably simple proof that P1 satisfies CIP, using the fact that it satisfies CAP, may be of interest.

Being a homogeneous space, P1 admits a dominating spray that comes from the exponential map for the complex Lie group of Möbiustransformations. However, the Lie group of Möbiustransformations is 3-dimensional, and has a 3-dimensional complex Lie algebra, resulting in the dominating spray being defined on a vector bundle of rank 3. The other examples of elliptic Riemann surfaces admit dominating sprays defined on line bundles. It is an interesting problem to determine if P1 admits a dominating spray defined on a vector bundle of rank 2 or even 1.

9 10 Chapter 2

Riemann surfaces, CW-complexes and Morse theory

The goal of this chapter is to provide a summary of the results from Riemann surface theory, algebraic topology and Morse theory that will be needed in Chapter 3. Much of the material on Riemann surfaces closely follows [3], in which far greater details can be found.

2.1 Algebraic topology

Notation: Let X be a topological space and A X. We will use the notation Hn(X; G) th ⊂ to denote the n homology group of X with coefficients in G and Hn(X,A; G) for the nth relative homology group with coefficients in G. If G is omitted, it will be assumed to be Z. The same conventions will be used for the cohomology groups of X. Let an n-cell be a space that is homeomorphic to the n-dimensional closed unit ball Dn.A CW-complex is a space constructed from cells in the following manner. Beginning with a discrete set X0, inductively define the n-skeleton Xn from Xn 1 by α − attaching n-cells via maps ∂Dn Xn 1. That is, let Dn be a collection of n-cells α α→ − { } and for each Dn take a map ∂Dn Xn 1. The n-skeleton is the quotient space of the → − disjoint union of the n-cells and the (n 1)-skeleton under the equivalence relation in − which a point on the boundary of an n-cell is equivalent to its image in the (n 1)- − skeleton. For a finite CW-complex X this process terminates for some n and X = Xn. Otherwise we take X = S X with the topology in which U X is open if U X n n ⊂ ∩ n is open for all n. A 1-cell is homeomorphic to the line segment [0, 1]. A 1-dimensional CW-complex is thus just a discrete set of points along with paths between them, that is, a graph. A

11 connected 1-dimensional CW-complex turns out to have a particularly simple homotopy type.

Deﬁnition 2.1.1. Let (X, x0) and (Y, y0) be two pointed topological spaces. The wedge product of X and Y is the space X Y/ , where is the equivalence relation t ∼ ∼ identifying x0 and y0.

Theorem 2.1.2. Let X be a connected 1-dimensional CW-complex. Then X is homotopy equivalent to a wedge product of circles. Furthermore if T is a maximal subtree of X, that is, a subtree that is not contained in any other subtree, then X is homotopy equivalent to a wedge product of circles with a circle for each edge of X not in T .

Proof. See [18, Theorem 4.3].

The structure of CW-complexes allows for several important constructions in algebraic topology, as well as for particularly simple computation of homology and cohomology groups.

Lemma 2.1.3. Let X be a CW-complex. Then there is a chain complex of relative homology groups

∂n+1 ∂n Hn+1(Xn+1,Xn) Hn(Xn,Xn 1) Hn 1(Xn 1,Xn 2) , · · · −→ −−−→ − −−→ − − − −→· · · th and moreover Hn(X) is equal to the n homology group of this complex.

Proof. See [11, Theorem 2.35].

There is an analogous result for cohomology.

Lemma 2.1.4. Let X be a CW-complex. Then there is a cochain complex of relative cohomology groups

n 1 dn−1 n dn n+1 H − (Xn 1,Xn 2) H (Xn,Xn 1) H (Xn+1,Xn) , · · · −→ − − −−−→ − −−→ −→· · · and moreover Hn(X) is equal to the nth cohomology group of this complex.

Proof. See [11, Theorem 3.5].

Since the N+1-skeleton of an N-dimensional CW complex is empty, we immediately get the following corollary.

n Corollary 2.1.5. Let X be an N-dimensional CW complex. Then Hn(X) = H (X) = 0 for all n > N.

12 Theorem 2.1.6 (Universal coeﬃcients theorem). If C is a chain complex of free n abelian groups with homology groups Hn(C), and H (C; G) are the cohomology groups of the cochain complex Hom(C,G) for an abelian group G, then there is a short exact sequence n 0 Ext(Hn 1(C),G) H (C; G) Hom(Hn(C),G) 0 −→ − −→ −→ −→ that splits.

Proof. See [11, Theorem 3.2].

A path γ : [0, 1] X from x to x deﬁnes a homomorphism from π (X, x ) to → 0 1 n 0 π (X, x ). For n > 1, and a given class σ π (X, x ) with representative f : [0, 1]n n 1 ∈ n 0 → X, the image of σ is the equivalence class of the map γf constructed by ﬁrst shrinking the domain of f to a smaller concentric n-cube and then assigning the path γ to each 1 1 n radial segment issuing from ( 2 ,..., 2 ) between the smaller cube and ∂[0, 1] . Explicitly, 1 3 n 1 n 1 3 n on the cube [ 4 , 4 ] , set γf(x) = f(2x 2 (1,..., 1)). Now, for each x [0, 1] [ 4 , 4 ] , 1 3 n− n ∈ 1 \ 1 let lx be the segment between ∂[ 4 , 4 ] and ∂[0, 1] of the line issuing from ( 2 ,... 2 ) in the direction of x and p : [0, 1] X be a parametrisation of l starting on ∂[ 1 , 3 ]n x → x 4 4 and with constant derivative. Then along lx let γf(px(t)) = γ(t). The map [f] [γf] is discussed in more detail in [11, p. 341–342]. The following 7→ lemma summarises the results.

Lemma 2.1.7. The map π (X, x ) π (X, x ), [f] [γf], is well deﬁned and an n 0 → n 1 7→ isomorphism. Moreover, if γ and γ are homotopic rel. 0, 1 , then they deﬁne the 1 2 { } same isomorphism.

Deﬁnition 2.1.8. The assignment π (X, x ) Aut π (X, x ) which takes [γ] 1 0 → n 0 ∈ π (X, x ) to the automorphism [f] [γf] of π (X, x ) is called the action of π (X, x ) 1 0 7→ n 0 1 0 on πn(X, x0). A path connected space X is called abelian if this action is trivial for all n for some (equivalently every) point in X.

The following theorem from obstruction theory gives conditions on when a continuous map on a closed subcomplex of a CW-complex can be extended to a continuous map on the whole space. The result plays an extensive role in Chapter 3.

Theorem 2.1.9. If X is a connected abelian CW-complex, W is a CW-complex with subcomplex A and Hn+1(W, A; π X) = 0 for all n 1, then every continuous map n ≥ A X can be extended to a continuous map W X. → →

Proof. See [11, Corollary 4.73].

13 Much of the proof relies on Postnikov towers and identifying obstruction classes to the extension in Hn+1(W, A; π X) for each n 1. However, since we will not refer n ≥ to Postnikov towers again, we have chosen not to introduce them here. The reader can ﬁnd an introduction to them in [11, p. 410].

2.2 Riemann surfaces

Let X be a 2-dimensional manifold, that is, a connected second countable Hausdorﬀ space that is locally homeomorphic to R2 (locally Euclidean). A complex chart or coordinate chart on X is a homeomorphism φ: U V where U X and V C → ⊂ ⊂ are open. Two charts φ: U1 V1 and ψ : U2 V2 are said to be holomorphically → 1 → compatible if the composition φ ψ− : ψ(U U ) φ(U U ) is biholomorphic in ◦ 1 ∩ 2 → 1 ∩ 2 the usual sense. A complex atlas on X is a collection of charts φα : Uα Vα that S { → } are holomorphically compatible and with α Uα = X. We say two atlases A and B are holomorphically equivalent if their union is an atlas on X. It is easy to check that this gives an equivalence relation on atlases on X. An equivalence class of atlases is called a holomorphic (or complex) structure on X.

Deﬁnition 2.2.1. A Riemann surface is a pair (X, [A ]), consisting of a 2-dimensional manifold X along with a holomorphic structure [A ] on X.

Every Riemann surface admits a metric that defines its topology. It is a standard topological result that a locally compact Hausdorff space is regular, that is, any point x and any closed set disjoint from x can be separated by disjoint open sets. This can be shown by noting that a locally compact Hausdorff space X admits a one-point compactification [21, Theorem 8.1, Ch. 3], from which it follows that X is regular [21, Corollary 2.3, Ch. 5]. Hence metrisability of Riemann surfaces is a consequence of Urysohn’s metrisability theorem:

Theorem 2.2.2. Every regular second countable topological space is metrisable.

Proof. See [21, Theorem 4.1, Ch. 4].

Deﬁnition 2.2.3. A continuous map f : X Y between Riemann surfaces is called → holomorphic if, for all complex charts φ: U V on X and ψ : U 0 V 0 on Y, the 1 1 → → composition ψ f φ− : φ(U f − (U 0)) C is holomorphic in the usual sense. ◦ ◦ ∩ → A holomorphic map f : X P1 which is not identically is called a meromor- → ∞ phic function. Equivalently, f is a holomorphic function X A C, where A is a \ → discrete set of points called the poles of f, and for every point p A, lim f(x) = . x p ∈ → | | ∞

14 A 1-form ω of type (1, 0) is called a holomorphic 1-form if for any coordinate chart (U, z), ω is of the form ω = f dz where f is a holomorphic function on U. Similarly a meromorphic 1-form ω on X is a 1-form of type (1, 0) deﬁned on X A, where A \ is a discrete set, such that if (U, z) are coordinates on X, then ω = f dz for some meromorphic function f on U with poles at the points of U A. ∩

Following the notation in [3], on a Riemann surface we will denote by E the sheaf of smooth functions, by E (1) the sheaf of smooth 1-forms, by E (1,0) (resp. E (0,1)) the sheaf of smooth 1-forms of type (1, 0) (resp. (0, 1)), by the sheaf of holomorphic O functions, by M the sheaf of meromorphic functions, by Ω the sheaf of holomorphic 1-forms and by M (1) the sheaf of meromorphic 1-forms. We will denote by Hn(X, F ) the nth (sheaf) cohomology group of X with coeﬃcients in the sheaf F .

2.3 Liftings

Lemma 2.3.1. Suppose X, Y and Z are Riemann surfaces, g : Z Y is a holomor- → phic covering map and f : X Y is a holomorphic map. Then any lifting of f by g → is holomorphic.

Proof. Since holomorphicity is a local property, this follows from g being a local biholomorphism.

Let X and Y be Riemann surfaces and A X. Two continuous maps f, g : X ⊂ → Y are said to be homotopic relative to A (abbreviated homotopic rel. A) if there is a continuous map F : X [0, 1] Y satisfying: × → F ( , 0) = f, F ( , 1) = g, · · F (a, ) is constant for all a A. · ∈

For two continuous paths γ , γ : [0, 1] X with γ (1) = γ (0), let γ γ be the 1 2 → 1 2 1 ∗ 2 product path deﬁned by

( 1 γ1(2t) if t 2 γ1 γ2(t) = ≤ ∗ γ (2t 1) if t 1 . 2 − ≥ 2 Lemma 2.3.2. Let S be a non-compact Riemann surface and p: X Y be a holo- → morphic covering of a Riemann surface Y by a contractible Riemann surface X, that is, C or D. The continuous maps S Y which lift to X with respect to p are precisely → the null-homotopic maps.

15 Proof. The fundamental group of X is trivial so a map f : S Y lifts precisely if → f (π1(S, s)) = 0 for some point (equivalently every point) s S [11, Proposition 1.33]. ∗ ∈ Now suppose f : S Y is null-homotopic, that is, there is a homotopy H : S [0, 1] → × → Y with H( , 0) = f and H( , 1) = y for some point y Y . Let s S and take a loop · · 0 0 ∈ ∈ γ in S at s. We want to show that f γ is homotopic rel. 0, 1 to the constant map ◦ { } at f(s). The homotopy H gives us a free homotopy from f γ to the constant map at ◦ y0 by deﬁning G: [0, 1] [0, 1] Y,G(t, r) = H(γ(t), r), × → which satisﬁes G( , 0) = f γ, G( , 1) = y . · ◦ · 0 For each r [0, 1], let β be the path that G(0, ) traces from f(s) to G(0, r), that is, ∈ r · β : [0, 1] Y, β (t) = G(0, rt). r → r 1 Then for each r [0, 1], β G( , r) β− is a loop at f(s), so we get a map ∈ r ∗ · ∗ r 1 G˜ : [0, 1] [0, 1] Y, G˜( , r) = β G( , r) β− × → · r ∗ · ∗ r which is a homotopy rel. 0, 1 from f γ to the constant map at f(s). Therefore f { } ◦ lifts. Conversely, suppose f lifts to a map f˜: S X. Then f˜ is null-homotopic since → X is contractible. If H is a homotopy from f˜ to a constant map, then p H is a ◦ homotopy from f to a constant map, so f is null-homotopic.

2.4 Jets

Let F be a sheaf on Riemann surface X and let p X. We introduce an equivalence F ∈ relation on U p F (U), where the disjoint union is taken over all neighbourhoods of p. Let U and V be3 neighbourhoods of p, f F (U) and g F (V ). We say f g if there ∈ ∈ ∼ is a neighbourhood W U V of p such that f W = g W . If U is a neighbourhood of ⊂ ∩ | | p and f F (U), the equivalence class of f is called the germ of f at p. We will call ∈ F the set of germs U p F (U)/ the stalk of F at p, denoted Fp. If Y is a Riemann 3 ∼ surace, we will denote by (X,Y ) the set of germs at p of holomorphic functions Op X Y . The language of jets will be useful later in discussing the higher-order local → behaviour of holomorphic functions.

Deﬁnition 2.4.1. Let X and Y be Riemann surfaces and p X. For k 0, we say ∈ ≥ that f, g (X,Y ) agree to order k at p if f(p) = g(p) and for every chart φ on X ∈ Op

16 1 centred at p and ψ on Y centred at f(p), the holomorphic function germs ψ f φ− 1 ◦ ◦ and ψ g φ− agree to order k at φ(p) = 0 in the usual sense. That is, ◦ ◦ 1(i) 1(i) ψ f φ− (0) = ψ g φ− (0) for i = 1, . . . , k. ◦ ◦ ◦ ◦

This deﬁnes an equivalence relation k on p(X,Y ) for each k. The equivalence class ∼ O k of f p(X,Y ) with respect to k is denoted Jp f and is called the k-jet of f at p. ∈ O ∼ k k The set of equivalence classes is denoted Jp (X,Y ) (or just Jp (X) if the target space is C) and is called the k-jet space of holomorphic maps from X to Y at p.

For U X open and a holomorphic map f : U Y , by the jets of f at p U ⊂ → ∈ we naturally mean the jets of the germ of f at p. Lemma 2.4.2. Let X and Y be Riemann surfaces and f, g (X,Y ) with f(p) = ∈ Op g(p). Then f and g agree to order k at p if for some charts φ on X centred at p and 1 1 ψ on Y centred at f(p), the germs ψ f φ− and ψ g φ− agree to order k at 0. ◦ ◦ ◦ ◦

Proof. For k = 0 this is trivial. So suppose k > 0, φ0 is another chart on X centred at p and ψ0 is another chart on Y centred at f(p). Then consider

1 1 1 1 ψ0 f φ0− 0 (0) = ψ0 ψ− ψ f φ− φ φ0− 0 (0) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 1 1 1 = ψ0 ψ− 0 (0) ψ f φ− 0 (0) φ φ0− 0 (0) ◦ · ◦ ◦ · ◦ 1 = ψ0 g φ0− 0 (0). ◦ ◦ Now consider the higher derivatives. It is known that for holomorphic functions F,G: C C, → i i l ∂ X ∂F (1) (j l+1) (F G)(p) = (G(p)) P G (p),...,G − (p) , ∂z ◦ ∂z · i,l l=1 where the Pj,l are polynomials. This is known as Fa`adi Bruno’s formula. An explicit form for the polynomials Pj,l and a proof can be found in [15, Theorem 1.3.2]. It follows th 1 1 by induction that if the i -order derivatives of ψ f φ− and ψ g φ− agree at 0 for th ◦ ◦ 1 ◦ ◦ 1 i = 0, . . . , k, then so do the i -order derivatives of ψ0 f φ0− and ψ0 g φ0− . ◦ ◦ ◦ ◦ An immediate corollary of the proof is the following. Lemma 2.4.3. Let X,Y and Z be Riemann surfaces and f, g : X Y and h: Y Z → → be holomorphic maps. If J kf = J kg for some p X and k 0, then J k(h f) = p p ∈ ≥ p ◦ J k(h g). p ◦

Proof. Let ξ be a chart on Z centred at f(p). Then replace ψ0 in the proof of the previous lemma with ξ h. ◦

17 The following is evident from the deﬁnition.

Lemma 2.4.4. Let X be a Riemann surface, p X and f, g (X). Then J kf = ∈ ∈ Op p J kg if and only if f g has a zero of order k + 1 at p, that is, (f g)(p) = 0 and p − − J k(f g) = 0. p − Lemma 2.4.5. Let X be a Riemann surface and f, g, h: X C be holomorphic → functions. If J kf = J kg for some p X and k 0, then J k(hf) = J k(hg). p p ∈ ≥ p p Proof. By the linearity of jets of maps into C, we need only consider the case where g k is identically zero. Let φ be a chart on X centred at p. Since Jp f = 0,

1(i) f φ− (0) = 0 for i = 0, . . . , k. ◦ Now, i 1(i) ∂ 1 1 hf φ− (0) = h φ− f φ− (0) ◦ ∂z ◦ · ◦ i i l l X i ∂ − 1 ∂ 1 = h φ− (0) f φ− (0) l ∂z ◦ · ∂z ◦ l=0 = 0 for i = 0, . . . , k.

k Hence Jp (hf) = 0.

2.5 Non-compact Riemann surfaces

Theorem 2.5.1. The ﬁrst cohomology group H1(X, E ) vanishes for any Riemann surface X.

Proof. See [3, Theorem 12.6].

The following is one of the most fundamental results for non-compact Riemann surfaces. It strengthens the Dolbeault lemma from local solvability of the ∂ equation to global solvability for non-compact Riemann surfaces.

Theorem 2.5.2. Let S be a non-compact Riemann surface and ω E 0,1(S). Then ∈ there is g E (S) such that ∈ ∂g = ω.

Proof. See [3, Theorem 25.6].

18 Corollary 2.5.3. Let S be a non-compact Riemann surface. Then H1(S, ) = 0. O Proof. Consider the following short exact sequence of sheaves

∂ 0 , E E 0,1 0. −→O −→ −−→ −→ That the sequence is exact at E 0,1 follows from the Dolbeault lemma for C [3, Theorem (0,1) 13.2]. Namely, if p S and ω Ep is of the form ω = g dz¯ in coordinates (U, z) ∈ ∈ at p, then there is f E (U) such that ∂f/∂z¯ = g and hence ∂f = ω on U. Since ∈ p H1(S, E ) vanishes, we have

H1(S, ) = E 0,1(S)/∂E (S). O It follows that H1(S, ) = 0 if and only if E 0,1(S) = ∂E (S). O It is a standard result of sheaf cohomology that H1(X, F ) = 0 if and only if H1(U , F ) = 0 for every open cover U of X [3, p. 100]. Hence, we note that H1(U , E ) and H1(U , ) vanish for every open cover U of a non-compact Riemann surface. O There are several results on the flexibility of holomorphic and meromorphic functions that are consequences. Let S be a Riemann surface. A Mittag-Leffler distribution on S is an open cover U = Ui i I of S and a collection µ = fi i I of meromorphic functions fi : Ui C { } ∈ { } ∈ → such that f f is holomorphic on U U for all i, j. By a solution to µ we mean a i − j i ∩ j meromorphic function f M (S) such that f f is holomorphic on U for all i, that is, ∈ i − i f has the same principal parts as fi on Ui for all i. The differences µij = fi fj define a 1 − 1-cocycle δµ Z (S, ). If S is non-compact, then δµ must be trivial. Let η = gi i I ∈ O { } ∈ be a splitting of δµ, so µ = f f = g g on U U . That is, f g = f g ij i − j i − j i ∩ j i − i j − j on U U . We get a well-defined function g M (S) given by g U = f g . Then i ∩ j ∈ | i i − i f g = g is holomorphic on U and g is a solution to µ. i − i i Theorem 2.5.4 (Mittag-Leffler theorem). On a non-compact Riemann surface every Mittag-Leffler distribution has a solution.

The notion of divisors plays an important role in Riemann surface theory, most notably in the celebrated Riemann-Roch theorem for compact Riemann surfaces. A divisor on a Riemann surface X is a map X Z with discrete support. To each → meromorphic function f : X P1 which is not identically zero, we associate a divisor → (f): X Z, x ordx f. Here ordx f is the order of f at x, by which we mean → 7→  0 if f is holomorphic and non-zero at x,  ordx f = k if f has a zero of order k at x,   k if f has a pole of order k at x. −

19 The divisor of a meromorphic function keeps track of the order of its zeros and poles. We say a divisor is principal if it is the divisor of some meromorphic function. It turns out that on non-compact Riemann surfaces every divisor is the divisor of a meromorphic function.

Theorem 2.5.5. On a non-compact Riemann surface every divisor is principal.

Proof. See [3, Theorem 26.5].

For a divisor D on X, the collection (U) = f (U):(f) D on U , for OD { ∈ O ≥ − } each open U X, gives a sheaf on X. By (f) D, we mean (f)(x) D(x) ⊂ OD ≥ − ≥ − for every x U. ∈ Corollary 2.5.6. For a non-compact Riemann surface S, H1(S, ) = 0 for any OD divisor D.

Proof. Let g be a meromorphic function with divisor (g) = D. There is an isomor- − phism of sheaves given by f gf. O → OD 7→

We may also associate a divisor to each meromorphic 1-form ω M (1)(X) which ∈ is not identically zero. Let (U, z) be coordinates on X. On U, ω = f dz for some f M (U). Deﬁne the order of ω at x U to be ord ω = ord f. It is easy to ∈ ∈ x x see that this is independent of the choice of coordinates. The divisor of ω is given by (w): X Z, x ordx ω. → 7→ Corollary 2.5.7. Let S be a non-compact Riemann surface. Then there exists a holomorphic 1-form on S which is nowhere vanishing.

Proof. By Theorem 2.5.5, there is a non-constant meromorphic function g M (S) ∈ and a meromorphic function f M (S) with divisor (f) = (dg). Then f dg is a ∈ − holomorphic 1-form which is nowhere vanishing.

Theorem 2.5.8 (Weierstrass’ theorem). Let S be a non-compact Riemann surface and T be a discrete subset of S. Suppose c: T C is an arbitrary map from T into C. → Then there is a holomorphic function f (S) with f T = c. ∈ O |

Proof. See [3, Theorem 26.7].

The following strengthening of Weierstrass’ theorem is well known. We provide a proof for the convenience of the reader.

20 Theorem 2.5.9 (Strong Weierstrass’ theorem). Let S be a non-compact Riemann surface, T S be discrete and n: T N 0 be an assignment of a non-negative ⊂ → ∪ { } na integer to each point in T . Suppose for each a T we are given an na-jet σa Ja (S). ∈ ∈na Then there is a holomorphic function f : S C such that for each a T , Ja f = σa. → ∈

Proof. Let Ua a T be a collection of pairwise disjoint coordinate disks about the points { } ∈ of T . On each disk the prescribed jet gives a holomorphic function fa : Ua C. We → construct a family of holomorphic functions similar to a Mittag-Leffler distribution. Instead of splitting with respect to we will use the sheaf , where D is the divisor O OD that agrees with n on T and is 0 on S T . Let U = U ,S T : a T and let − \ { a \ ∈ } µ = f , f : a T where f is the constant function 0 on S T . For a, b T , the { a 0 ∈ } 0 \ ∈ intersection U U is empty and U S T does not intersect T , so the differences a ∩ b a ∩ \ fa f0 = fa on Ua S T define a 1-cocycle δµ with respect to the sheaf D. Since 1− ∩ \ O H (X, D) = 0, δµ splits, giving a family ga : Ua C and g : X T C of sections O → \ → of such that OD g g = f f = f on U X T. a − a − 0 a a ∩ \ Then g f = g on U X T , and so these piece together to give a well-defined a − a a ∩ \ holomorphic function f : X C with f = ga fa on Ua and f = g on X T . For → − \ each a, g (U ) and so vanishes at a to order at least n . Hence J na f = J na f as a ∈ OD a a a a a required.

The following is a classical theorem of Behnke and Stein.

Theorem 2.5.10. Let S be a non-compact Riemann surface and c: π1(S) C be a → group homomorphism. Then there is a holomorphic 1-form ω Ω(S) with ∈ Z ω = c(σ) for each σ π1(S). σ ∈

Proof. See [3, Theorem 28.6].

2.6 Embeddings of non-compact Riemann surfaces

Stein manifolds play an important role in higher dimensional complex analysis. For Riemann surfaces the definition is equivalent to being non-compact. Definition 2.6.1. Let X be a Riemann surface and K X be a compact subset. ⊂ Define the holomorphically convex hull of K as the closed set

Kb = x X : f(x) sup f for all f (X) . { ∈ | | ≤ K | | ∈ O } We say that K is holomorphically convex if K = Kb.

21 Deﬁnition 2.6.2. A Riemann surface X is called Stein if it satisﬁes the following conditions.

(i) For any two points x, y X, x = y, there is a holomorphic function f (X) ∈ 6 ∈ O with f(x) = f(y). 6 (ii) For any point x X there is a holomorphic function f (X) with d f = 0. ∈ ∈ O x 6 (iii) Kb is compact for any compact K X. ⊂ A Riemann surface X that satisﬁes (iii) is called holomorphically convex. Theorem 2.6.3. A Riemann surface is Stein if and only if it is non-compact.

Proof. For a non-compact Riemann surface S, (i) is an immediate consequence of Theorem 2.5.8, while (ii) follows from the stronger version, Theorem 2.5.9. Suppose

K S is compact and Kb is not, that is, there is a sequence of points (an)n N in Kb ⊂ ∈ with no limit point in Kb. Since Kb is closed, (an) has no limit point in S. By Theorem 2.5.8, there is f (X) with lim f(an) = . But this contradicts an Kb for ∈ O n | | ∞ ∈ all n. Compactness and sequential→∞ compactness are equivalent in a Riemann surface, so Kb is compact for all compact K S and S is holomorphically convex. Thus all ⊂ non-compact Riemann surfaces are Stein. Conversely, by the maximum principle, a compact Riemann surface X has no non-constant holomorphic maps X C, so it is → not Stein.

We can now apply the well known result that an n-dimensional Stein manifold admits a proper holomorphic embedding into C2n+1. Theorem 2.6.4. Let X be a non-compact Riemann surface. Then there is a proper holomorphic embedding of X into C3.

Proof. See [12, Theorem 5.3.9].

2.7 Morse theory

Deﬁnition 2.7.1. Let X be an n-dimensional smooth manifold and f E (X), where ∈ R ER denotes the sheaf of real valued diﬀerentiable functions. The Hessian of f at a critical point p X with respect to coordinates (φ, U) at p is the n n matrix 0 ∈ 0 × 2 1 ∂ (f φ− ) Hφ(p0) = ◦ (φ(p0)). ∂xi∂xj i,j Here by a critical point we mean a point p X with d f the zero map. 0 ∈ p0

22 We would like to know the dependence of Hf (p0) on the chosen chart. Let (ψ, V ) be another chart on X with p V . Then we can write the Hessian as 0 ∈ 2 1 ∂ (f φ− ) Hφ(p0) = ◦ (φ(p0)) ∂xi∂xj i,j 2 1 1 ∂ (f ψ− ψ φ− ) = ◦ ◦ ◦ (φ(p0)) ∂xi∂xj i,j 2 1 1 1 X ∂ (f ψ− ) ∂(ψ φ− )k ∂(ψ φ− )l = 1 ◦ 1 ◦ ◦ ∂(ψ φ )k ∂(ψ φ )l · ∂xi · ∂xj k,l ◦ − ◦ − 1 2 1 X ∂(f ψ− ) ∂ (ψ φ− )k + ◦ 1 ◦ (φ(p0)) ( ) ∂(ψ φ )k · ∂xi∂xj ∗ k ◦ − i,j 2 1 1 1 ! X ∂ (f ψ− ) ∂(ψ φ− )k ∂(ψ φ− )l = ◦ (ψ(p0)) ◦ (φ(p0)) ◦ (φ(p0)) , ∂xk∂xl · ∂xi · ∂xj k,l i,j noting that the second term in ( ) is zero at a critical point of f. Now let ∗ 1 ∂(ψ φ− )i Jψ,φ(p0) = ◦ (φ(p0)). ∂xj i,j Then from the above it is easy to see that

T Hφ(p0) = Jψ,φ(p0)Hψ(p0)Jψ,φ(p0). 1 Since ψ φ− is a diffeomorphism its Jacobian J is invertible. It follows that the ◦ ψ,φ number of (strictly) negative and (strictly) positive eigenvalues of Hφ (counted with multiplicities) is independent of the chosen chart φ. To see this first note that conjuga- tion by an invertible matrix clearly does not change the dimension of the nullspace, so all we need to show is that Hψ and Hφ have the same number of positive eigenvalues. Let v , . . . , v T X be the eigenvectors of H corresponding to positive eigenvalues. 1 n ∈ p0 φ Then for v span v , . . . , v , v = 0, we have (J v)T H (J v) = vT H v > 0. Hence, ∈ { 1 n} 6 ψ,φ ψ ψ,φ φ the dimension of the largest subspace of Tp0 X on which Hψ is positive definite is at least n. However, by changing to the basis given by the eigenvectors of Hψ, we see that the dimension of the largest subspace on which Hψ is positive definite is exactly the number of positive eigenvalues of Hψ. Indeed, the number of positive eigenvalues is clearly a lower bound. Suppose M was a subspace on which Hψ was positive definite that had dimension greater than the number of positive eigenvalues. Let A be the subspace spanned by the negative eigenvalues. Then dim M + dim A + dim Ker Hψ > dim Tp0 X, which is absurd since A, M and Ker Hψ only intersect at 0. Hence Hψ has at least as many positive eigenvalues as Hφ. The same argument in the opposite direction shows that Hφ and Hψ have the same number of positive eigenvalues. Since the only thing we are interested in is the number of positive and negative eigenvalues, we will just refer to the Hessian of f at a critical point p0, and the following notions are well defined.

23 Deﬁnition 2.7.2. A critical point p0 of a smooth function f : X R is called non- → degenerate if the Hessian H(p0) of f at p0 is non-degenerate, that is, invertible. We say f is a Morse function if all its critical points are non-degenerate. Finally, the index of a non-degenerate critical point p X is the number of negative eigenvalues of the 0 ∈ Hessian of f at p0.

It turns out that non-degenerate critical points have a rather simple form. Sup- pose p0 is a critical point of f : X R with index γ. Then there exist coordinates → γ n P 2 P 2 (U, φ) at p0 such that if we write φ = (φ1, . . . , φn), then f = f(p0) φi + φi . − i=1 i=γ+1 This is known as the Morse lemma [23, Corollary 1.17]. An immediate consequence is that f has no critical points on U other than p0, so non-degenerate critical points are isolated.

Deﬁnition 2.7.3. Let f : X R be smooth. We say that f is an exhaustion if the → sublevel sets x X : f(x) c { ∈ ≤ } are compact for all c R. ∈ The importance of Morse theory lies in two fundamental results. The ﬁrst of these is the existence of Morse functions. Indeed, the space of Morse functions is dense in the space of smooth functions. We will only need the following much weaker result on the existence of a Morse function on a non-compact Riemann surface.

Theorem 2.7.4. For a non-compact Riemann surface S embedded as a closed submanifold of C3, the map f : C3 R, z z a 2, restricts to a Morse exhaustion → 7→ k − k on S for generic a C3. Here, is the Euclidean norm on C3. ∈ k·k Proof. It is clear that the restriction of f to S is an exhaustion. For the rest of the proof see [19, Theorem 6.6].

The second fundamental result of Morse theory is that the critical points of a Morse exhaustion on X completely determine the homotopy type of X.

Theorem 2.7.5. Suppose X is a smooth manifold and f : X R is a Morse ex- → haustion. Then X is homotopy equivalent to a CW-complex with the same number of γ-cells as f has critical points of index γ.

Proof. See [23, Corollary 2.10].

Corollary 2.7.6. A non-compact Riemann surface has the homotopy type of a countable 1-dimensional CW-complex.

24 Proof. Let S be a non-compact Riemann surface. By Theorem 2.6.4 there is a proper holomorphic embedding s: S C3. Let f be as in Theorem 2.7.4, so f s: S R is → ◦ → a Morse exhaustion. We want to show that f s is subharmonic, that is, the Laplacian ◦ is non-negative with respect to any chart on S. So let (φ, U) be a chart on S and write z C as z = x + iy. Then ∈ 2 1 2 1 ∂ (f s φ− ) ∂ (f s φ− ) ∆ (f s) = ◦ ◦ + ◦ ◦ φ ◦ ∂x2 ∂y2 2 1 2 1 X ∂f 1 ∂ (s φ− )k ∂ (s φ− )k − = (s φ ) ◦ 2 + ◦ 2 ∂xk ◦ ◦ · ∂x ∂y k 2 1 1 X ∂ f 1 ∂(s φ− )k ∂(s φ− )l + (s φ− ) ◦ ◦ ∂xk∂xl ◦ ◦ · ∂x · ∂x k,l 1 1 ∂(s φ− ) ∂(s φ− ) + ◦ k ◦ l . ∂y · ∂y

1 However s φ− is holomorphic, so by the Cauchy-Riemann equations, ◦ 2 1 2 1 ∂ (s φ− ) ∂ (s φ− ) ◦ k = ◦ k ∂y2 − ∂x2 for each k. Hence the ﬁrst term above vanishes. Also, ∂2f = 2δk,l, ∂xk∂xl so 1 2 1 2! X ∂(s φ− )k ∂(s φ− )k ∆ (f s) = 2 ◦ + ◦ 0. φ ◦ ∂x ∂y ≥ k The Laplacian is the trace of the Hessian, so this shows that the Hessian of f s is ◦ not negative deﬁnite at any critical points. That is, f s has only critical points of ◦ index 0 or 1. Finally, since the critical points of f s are isolated, it can have at most ◦ countably many.

2.8 Runge sets and holomorphic convexity

Deﬁnition 2.8.1. Let S be a non-compact Riemann surface and Y S. Let h(Y ) ⊂ be the union of Y with all the relatively compact connected components of S Y . We \ say that an open subset Y is Runge if h(Y ) = Y . Lemma 2.8.2. Let S be a non-compact Riemann surface and Y,Z S. Then: ⊂ (i) h(h(Y )) = h(Y ).

25 (ii) h(Y ) h(Z) if Y Z. ⊂ ⊂ (iii) h(Y ) is relatively compact if Y is relatively compact.

(iv) h(Y ) is open and relatively compact if Y is open and relatively compact.

Proof. (i) The complement of h(Y ) consists precisely of the non-relatively-compact components of S Y . \ (ii) If C is a non-relatively-compact component of S Z, then C is contained in a \ component of S Y , and this component is not relatively compact. Hence S h(Z) \ \ ⊂ S h(Y ), so h(Y ) h(Z). \ ⊂ (iii) The following argument is taken directly from [22, Lemma 2.13.3], where it is shown that h(K) is compact if K is compact. We use the argument to prove some slightly stronger results, and hence include it here. Let B be the closure of Y , which is S a compact set in S, and let S B = j J Cj be the decomposition of S B into disjoint \ ∈ \ connected components. Since S B is open and S is locally connected, C is open \ j for all j. Now let U be a relatively compact neighbourhood of B, then Cj U = ∅ ∩ 6 for all j. For otherwise if C S U for some j, then C S U S B and j ⊂ \ j ⊂ \ ⊂ \ hence C would also be a connected component of S B. But this is only possible if j \ Cj = Cj, which contradicts S being connected. Also, only ﬁnitely many Cj intersect ∂U, for they form a cover of ∂U by mutually disjoint open sets and ∂U is compact.

It follows that all but finitely many of the Cj are contained in U. Now let J0 be the subset of J such that for j J0, Cj is relatively compact and intersects ∂U. Then S ∈ Ue = U j J0 Cj is also a relatively compact neighbourhood of B, and by construction ∪ ∈ S Ue is contained in a finite union Cj1 Cj2 Cjn of components of S B, none of \ ∪ ∪ · · · ∪ \ which are relatively compact. Also all other components of S B are contained in Ue \ and so are relatively compact. Hence h(B) Ue and S h(B) = Cj1 Cj2 Cjn . ⊂ \ ∪ ∪ · · · ∪ By (ii), h(Y ) h(B) Ue, and hence h(Y ) is relatively compact. ⊂ ⊂ (iv) We have just shown that there are only finitely many components of the complement of a compact subset which are not relatively compact. Adding the compact set ∂Y to S Y cannot increase the number of non-relatively compact components, \ so the same is true for the complement of a relatively compact subset. It follows that S h(Y ) is closed, being the union of finitely many closed components, and thus h(Y ) \ is open. Lemma 2.8.3. Let K be a compact subset of a non-compact Riemann surface S. Then for each point p S h(K), there is a connected, non-compact, closed subset C of S ∈ \ with C S h(K) and p C◦. ⊂ \ ∈ Proof. We adapt the proof in [22, Lemma 2.13.3], where it is shown that such a C exists with p C. From the proof of part (iii) of Lemma 2.8.2 we know that there are ∈

26 a ﬁnite number of non-relatively-compact components C ,...,C of S K such that 1 n \ S h(K) = C C . Let U be a relatively compact neighbourhood of h(K) and \ 1 ∪ · · · ∪ n K0 = U. Then, similarly, S h(K0) = C0 C0 for some non-relatively compact \ 1 ∪ · · · ∪ m components C10 ,...,Cm0 of S K0. For each i = 1, . . . , n, there is j with Cj0 Ci = ∅ \ ∩ 6 and hence C0 C since C0 and C are both connected and ∂C0 ∂U C C . j ⊂ i j i j ⊂ ⊂ 1 ∪ · · · ∪ n Let A C be the union of the interiors of all connected, non-compact, closed subsets i ⊂ i of S that are contained in Ci. Then Ai contains Cj0 , and so is non-empty. We aim to show that A = C . Let p C A . Choose a connected neighbourhood V of p i i ∈ i ∩ i with V C . Since p A , there is q A V and a connected, non-compact, closed ⊂ i ∈ i ∈ i ∩ subset B of S contained in C with q B◦. The union C = V B is a connected, i ∈ ∪ non-compact, closed subset of S contained in C and hence V A . Thus A is both i ⊂ i i open and closed in the connected set Ci, and so Ai = Ci.

Lemma 2.8.4. Let K be a compact subset of a non-compact Riemann surface. Then for any relatively compact neighbourhood U of h(K), there is a relatively compact Runge neighbourhood W of h(K) with W U. ⊂ Proof. Let U be a relatively compact neighbourhood of h(K). Since ∂U S h(K), ⊂ \ by Lemma 2.8.3, there are connected, non-compact, closed subsets C S h(K), i ⊂ \ i I, whose interiors cover ∂U. Since ∂U is compact we can ﬁnd a ﬁnite subcover ∈ C ,...,C . The set W = U (S (C C )) is a neighbourhood of K contained i0 in ∩ \ i0 ∪· · ·∪ in in U. Furthermore any component of the complement of W contains at least one of the Cij and hence is not compact. That is, W is Runge.

We will now give a topological characterisation of holomorphically convex sets of Riemann surfaces (recall Deﬁnition 2.6.1).

Theorem 2.8.5. Let S be a non-compact Riemann surface and K S be a compact ⊂ subset. Then the following are equivalent.

(i) K is holomorphically convex.

(ii) h(K) = K, that is, S K has no relatively compact components. \ (iii) K has a neighbourhood basis of Runge subsets.

Proof. (i) = (ii): Suppose that the complement of K had a relatively compact con- ⇒ nected component A. Then A is open in S since S K is open and S is locally connected. \ By deﬁnition, A is closed in S K, so ∂A K. Let x A. By the maximum principle \ ⊂ ∈ [3, Corollary 2.6], any holomorphic function S C takes a no smaller absolute value → somewhere on the boundary ∂A than at x A. Hence x Kb, which contradicts K ∈ ∈ being holomorphically convex.

27 (ii) = (iii): Let V be an open set containing K and let U V be a relatively ⇒ ⊂ compact neighbourhood of K. Then U is a neighbourhood of h(K) = K, so by Lemma 2.8.4, K has a Runge neighbourhood W U V . ⊂ ⊂ (iii) = (i): Suppose x / K and let V be an open disk centred at x such that V¯ ⇒ ∈ 1 1 does not intersect K. Then there is a Runge neighbourhood V2 of K which does not intersect V . Let A = V V . If we let the connected components of the complement 1 1 ∪ 2 of V be U , i I, then the complement of A in S is S U V . Since V is Runge 2 i ∈ i i \ 1 2 each Ui is not relatively compact. The disk V1 lies in one of the Ui, say Ui0 , and the set U V must be connected (it is path connected since it contains the boundary of i0 \ 1 V ). It is easy to see that U V is not relatively compact. Hence A is Runge. 1 i0 \ 1 Now deﬁne f : A C by f V1 = 1 and f V2 = 0. Then f is clearly holomorphic → | | on A and by the Runge approximation theorem f can be approximated uniformly on compact subsets by holomorphic functions S C. Let B = K x and choose a → ∪ { } holomorphic function S C that diﬀers uniformly from f on B by less than say 1 . → 2 This shows that x is not in the holomorphically convex hull of K.

Theorem 2.8.6 (Runge approximation theorem). Let S be a non-compact Riemann surface and U S be a Runge open set. Then any holomorphic function on U can be ⊂ approximated uniformly on compact subsets of U by holomorphic functions on S.

Proof. See [3, Theorem 25.5].

Corollary 2.8.7. Let S be a non-compact Riemann surface and U S be a Runge ⊂ open set. Then any holomorphic 1-form on U can be approximated uniformly on compact subsets of U by holomorphic 1-forms on S.

Proof. Let ω0 be a nowhere vanishing holomorphic 1-form on S, which exists by Corol- lary 2.5.7. Then ω U is a nowhere vanishing holomorphic 1-form on U. Let ω be a 0| holomorphic 1-form on U. Then ω can be written ω = fω U for some holomorphic 0| function f : U C. Let K be a compact subset of U and let (gk) be a sequence of → holomorphic functions gk : S C approximating f uniformly on K. Then we get a → sequence of 1-forms ηk = gkω on S which by construction approximate ω on K.

We need a variant of Theorem 2.8.6, which we will still call the Runge approximation theorem.

Theorem 2.8.8. Let S be a non-compact Riemann surface and K S be a holomor- ⊂ phically convex compact set. Then any holomorphic function on a neighbourhood of K can be approximated uniformly on K by holomorphic functions on S.

28 Proof. Since K has a neighbourhood basis of Runge open sets, any holomorphic function on a neighbourhood of K restricts to a holomorphic function on a Runge neighbourhood. The result then follows from Theorem 2.8.6.

We will need the existence of special subharmonic exhaustions. Lemma 2.8.9. If K is a holomorphically convex, compact subset of a non-compact Riemann surface S, then for every neighbourhood U of K there is a strictly subharmonic exhaustion φ: S R such that φ < 0 on K and φ > 1 on S U. → \ Proof. See [5, Proposition 2.3.1].

Theorem 2.8.10. If X is a Riemann surface and φ: X R is a strictly subharmonic → exhaustion, then every sublevel set x X : φ(x) c is holomorphically convex. { ∈ ≤ } Proof. See [5, Theorem 2.3.2].

Finally, we need some results on the homological properties of Runge subsets. The following is a well-known result in higher dimensions. We establish it for Riemann surfaces, using the Runge approximation theorem and the theorem of Behnke and Stein. Theorem 2.8.11. Let S be a non-compact Riemann surface and V be a connected Runge subset. Then the map H (V ) H (S) induced by the inclusion map is injective. 1 → 1 Proof. Let [c ] H (V ) be a 1-cycle whose image in H (S) is 0. That is, there is a 0 ∈ 1 1 2-chain b C (S) with ∂b = c . We want to show that c is also a boundary in H (V ). ∈ 2 0 0 1 Given a holomorphic 1-form ω Ω(S) we have ∈ Z Z Z ω = ω = dω = 0. c0 ∂b b By Corollary 2.8.7, any holomorphic formω ˜ Ω(V ) can be approximated by holomor- ∈ phic forms in Ω(S). Hence R ω˜ = 0 for allω ˜ Ω(V ). c0 ∈ We need to show that this implies that c0 is a boundary in H1(V ). The quotient map p: π1(V ) H1(V ) induces a homomorphism Hom(H1(V ), C) Hom(π1(V ), C). → → Since p is surjective and every φ Hom(π1(V ), C) vanishes on the commutator sub- ∈ group of π1(V ) as C is abelian, the homomorphism is in fact an isomorphism. Given any homomorphism φ:H1(V ) C, by Theorem 2.5.10, we can find a holomorphic R → 1-form η Ω(V ) with φ = η. The first homology group H1(V ) is free by Theorem ∈ • 2.7.6, so it is possible to find a homomorphism φ:H1(V ) C with φ(σ) = 0 for R → 6 any nonzero σ in H1(V ). However, ω˜ = 0 for allω ˜ Ω(V ), so c0 must vanish in c0 ∈ H1(V ).

29 We will need the following substantial result on the relative homology groups of Runge subsets in non-compact Riemann surfaces. It follows from a result of Andreotti and Narasimhan [1], along with a stronger version of Lemma 2.8.9 for which the subharmonic exhaustion φ is Morse. Such a strengthening of Lemma 2.8.9 relies on a fundamental result from Morse theory that Morse functions are dense among smooth functions. For convenience, we have chosen to cite Stout [25].

Theorem 2.8.12. Let S be a non-compact Riemann surface and V be a connected

Runge subset. Then H2(S,V ) = 0 and H1(S,V ) is a free abelian group.

Proof. See [25, Theorem 2.4.1].

2.9 Elliptic Riemann surfaces

We now give the deﬁnition of elliptic manifolds introduced by Gromov.

Deﬁnition 2.9.1. Let X be a Riemann surface. A holomorphic map s: E X deﬁned → on the total space E of a holomorphic vector bundle over X is called a dominating spray if s(0 ) = x and s E is a submersion at 0 for all x X. A Riemann surface X x | x x ∈ is called elliptic if it admits a dominating spray.

Lemma 2.9.2. The elliptic Riemann surfaces are precisely C, C∗, P1 and the tori.

Proof. First note that by the uniformisation theorem, if X is some other Riemann surface then it has the disk D = z C : z < 1 as its universal cover. Hence by { ∈ | | } Liouville’s theorem all holomorphic maps from C into X are constant (since they can be lifted to holomorphic maps C D). If E X is a holomorphic vector bundle, → n → then the ﬁbres Ex are biholomorphic to C for some n. It follows that if s: E X is → a holomorphic map, then the restriction s E is constant and hence not a submersion | x at 0 for any x X. x ∈ Now we need to prove the positive results. The map

C C C, (z, w) z + w, × → 7→ is a dominating spray over C, deﬁned on the total space of the trivial line bundle. The map z C C∗ C∗, (z, w) we , × → 7→ is a dominating spray over C∗, again deﬁned on the total space of the trivial line bundle. For the torus C/Γ, consider

s: C C/Γ C/Γ, (z, w + Γ) z + w + Γ. × → 7→

30 After pre- and postcomposing by charts, we get a map that locally looks like (z, w) 7→ z + w + γ where γ Γ is a constant. It follows that s is holomorphic and s E ∈ | w+Γ is a submersion at (0, w + Γ). Finally let G be the complex Lie group of M¨obius transformations and g be its Lie algebra. Then we have an exponential map exp: g → G. Note that G acts on P1 as its automorphism group and deﬁne

s: g P1 P1, (z, w) exp(z)w. × → 7→ Then s is a dominating spray. As a vector space, g is 3-dimensional.

It is worth noting that C, C∗ and the tori are precisely the complex Lie groups of dimension 1. The ﬁrst three maps above are just the exponential maps corresponding to the Lie group structure. Namely if G is C, C∗ or a torus, then there is a map C G G,(z, g) exp(z) + g, where + is the group operation on G. On the × → → other hand, P1 is not a Lie group; however it is a homogeneous space. That is, there is a complex Lie group with a continuous and transitive action on P1, namely its automorphism group P GL(2, C). The exponential map of the Lie group still gives a dominating spray. This is no coincidence; elliptic manifolds were introduced as a generalisation of complex Lie groups and homogeneous spaces which admit a map with some of the important properties of the exponential map.

To see that P1 is not a complex Lie group, note that a complex Lie group has trivial holomorphic tangent bundle and hence trivial holomorphic cotangent bundle. This is equivalent to having a global holomorphic section of the holomorphic cotangent bundle which is nowhere vanishing, that is, a holomorphic 1-form which is nowhere vanishing. But P1 has no nonzero holomorphic forms.

2.10 Triangulability

That all 2-manifolds can be triangulated, in the sense of admitting a piecewise linear structure, is well known. It was proven in the early 20th century by Tibor Radó.It was later shown that all topological 3-manifolds also admit triangulations, but it fails in the 4-dimensional case. For differentiable manifolds there are much stronger results: all k-differentiable manifolds admit a k-differentiable triangulation [2]. However, despite how well known these results are, and how often they are called on, the proof of the triangulability of non-compact manifolds is often left out of textbooks. Some examples of texts that cover triangulations of manifolds but skip over the non-compact case, or even avoid the proof altogether, include Geometric Integration Theory by H. Whitney, Introduction to Topological Manifolds by John Lee, Simplicial Structures in Topology by D. Ferrario and R. Piccinini, Topology: Point-Set and Geometric by P. Shick, Compact

31 Riemann Surfaces by J. Jost, Lecture Notes on Elementary Topology and Geometry by I. Singer and J. Thorpe, and in Cairns’ paper [2], mentioned above, only a proof for the compact case is provided. The text Geometric Topology in Dimensions 2 and 3 by E. Moise, who was the one to prove the existence of triangulations on 3-manifolds, provides a thorough proof for the existence of topological triangulations on 2 and 3- manifolds. We need a slightly stronger result for manifolds with boundaries; that a triangulation on the boundary can be extended to the rest of the manifold. A proof is given for the diﬀerentiable case in Munkres’ book [20].

n Deﬁnition 2.10.1. Let v0, . . . , vn R be n+1 points such that the diﬀerence vectors ∈ v v , . . . , v v are linearly independent. This is sometimes known as v , . . . , v 0 − 1 0 − n 0 n being in general position. An n-simplex in Rn is the convex hull t v + + t v : t , . . . , t [0, 1] and t + + t = 1 { 0 0 ··· n n 0 n ∈ 0 ··· n } n of v0, . . . , vn . If e1, . . . , en are the standard basis vectors of R , then the convex hull { } of 0, e , . . . , e is known as the standard n-simplex and is denoted ∆ . { 1 n} n By a face of an n-simplex we mean the convex hull of a non-empty subset of v , . . . , v . If the subset consists of two vectors, the corresponding face is called an { 0 n} edge.

Deﬁnition 2.10.2. A simplicial complex K in Rn is a collection of simplices, satisfying the following conditions.

(i) If σ is a face of a simplex in K, then σ K. ∈ (ii) If σ, τ K, then σ τ is either empty or a face of both σ and τ. ∈ ∩ (iii) Every simplex in K has a neighbourhood in Rn which intersects only ﬁnitely many elements of K.

If K is a simplicial complex, then we will denote by K the subspace of Rn given | | by the union of the elements of K, equipped with the subspace topology.

Deﬁnition 2.10.3. By a triangulation of a manifold X, we mean a homeomorphism K X, where K is a simplicial complex in Rn. | | → It is clear that a triangulation on a manifold X gives the structure of a CW- complex on X, where the images of n-simplices are the n-cells of the CW-complex.

Theorem 2.10.4. Let X be a smooth manifold. Then there is a triangulation on X. Furthermore, if X is a smooth manifold with boundary, then any triangulation of the boundary can be extended to a triangulation of X.

32 Proof. See [20, Theorem 10.6]. Corollary 2.10.5. Every Riemann surface X is a CW-complex. Corollary 2.10.6. Let X be a smooth manifold and A X be a closed submanifold- ⊂ with-boundary. Then there exists a triangulation of X for which the restriction to A is a triangulation of A.

Proof. By Theorem 2.10.4, there is a triangulation K of ∂A. Now, X A◦ and | 0| \ A are manifolds with boundary, so there are triangulations φ : K X A◦ and 1 | 1| → \ φ : K A, that are extensions of K . The two complexes K and K glue together 2 | 2| → | 0| 1 2 along K0 to get a complex K, which is a triangulation of X by the homeomorphism K X deﬁned by gluing φ and φ together along K . | | → 1 2 | 0| Lemma 2.10.7. Let T be a discrete subset of a non-compact Riemann surface S. Then there exists a cover U of S by contractible coordinate charts, such that each { j} element of T is contained in precisely one of the Uj.

Proof. For each point a T , let U be a contractible open neighbourhood of a that ∈ a is contained in a coordinate chart. Since T is discrete and S is a metric space, it is possible to choose the Ua such that Ua Ub = ∅ for every a, b T , a = b. For each ∩ S ∈ 6 a T , let Va be a closed disk in Ua containing a. Then S a T Va is a non-compact ∈ \ ∈ Riemann surface, and hence admits an open cover U by contractible coordinate charts. The union U U : a T is an open cover of S with the required properties. ∪ { a ∈ }

2.11 Compact-open topology

Let X and Y be topological manifolds and let (X,Y ) be the space of continuous maps C X Y . The compact-open topology on (X,Y ) is the topology generated by subsets → C of the form V (K,U) = f (X,Y ): f(K) U , { ∈ C ⊂ } where K X is compact and U Y is open. Convergence in the compact-open ⊂ ⊂ topology is known as uniform convergence on compact subsets. If Y is a metric space, then convergence in the compact-open topology agrees with the metric space deﬁnition of uniform convergence on compact subsets.

2.12 Manifolds with boundary

Deﬁnition 2.12.1. A collar neighbourhood of a smooth manifold X with boundary ∂X is a neighbourhood U of ∂X with a diﬀeomorphism ∂X [0, 1) U which is the × →

33 inclusion ∂X , X on ∂X. → Theorem 2.12.2. Every manifold with boundary admits a collar neighbourhood.

Proof. See [13, Theorem 13.6].

34 Chapter 3

The Oka principle for maps between Riemann surfaces

In this chapter we strengthen the result of Winkelmann [26], in which the precise pairs of Riemann surfaces that satisfy the basic Oka property are determined. We begin by strengthening the basic Oka property to include approximation and jet interpolation for maps from non-compact Riemann surfaces into elliptic Riemann surfaces. In Section 3.2 we then address the other possible pairs of Riemann surfaces.

3.1 The Oka properties

Definition 3.1.1. We say a pair of Riemann surfaces (X,Y ) satisfies the basic Oka property with approximation and interpolation (BOPAI) if whenever K is a holomorphically convex, compact subset of X, T X is a discrete subset and f is a continuous ⊂ map X Y which is holomorphic on a neighbourhood of K, then f can be continu- → ously deformed to a holomorphic map X Y , keeping it fixed on T , holomorphic on → K and arbitrarily close to f on K. If K is taken to be empty, we get the basic Oka property with interpolation (BOPI), if T is taken to be empty, we get the basic Oka property with approximation (BOPA), and if both are taken to be empty the result is the basic Oka property (BOP). Similarly, we say a pair of Riemann surfaces (X,Y ) satisfies the basic Oka property with approximation and jet interpolation (BOPAJI) if whenever K is a holomorphically convex, compact subset of X, T X is a discrete subset, n: T N 0 is an ⊂ → ∪ { } assignment of a non-negative integer to each point in T and f is a continuous map X Y which is holomorphic on a neighbourhood of K T , then f can be continuously → ∪ deformed to a holomorphic map X Y , keeping the n - jets of f fixed at a for each → a

35 a T , holomorphic on K T and arbitrarily close to f on K. If K is taken to be the ∈ ∪ empty set, then the resulting property is the basic Oka property with jet interpolation (BOPJI).

In the proofs of Theorems 3.1.6 and 3.1.9 we will see that the maps in the deformation can all be chosen to be holomorphic on a ﬁxed neighbourhood of K and T. For the beneﬁt of the reader, we give a proof that maps from non-compact Rie- mann surfaces into elliptic Riemann surfaces satisfy the basic Oka property. Gromov proved this in the more general setting of maps between Stein manifolds and elliptic manifolds. We will call pairs of Riemann surfaces (X,Y ), where X is non-compact and Y is elliptic, Gromov pairs. The proof will serve as a building block for the proofs of the stronger Oka properties.

Theorem 3.1.2 (Basic Oka property). Every continuous map from a non-compact Riemann surface into C, C∗, P1 or a torus is homotopic to a holomorphic map.

Proof. a) C. The simplest case. Since C is contractible we just note that any continuous map into C is homotopic to a constant map.

b) C∗. Let S be a non-compact Riemann surface and f : S C∗ be continuous. → Let U = U be an open cover of S by coordinate charts homeomorphic to the unit { j} disc in C. Then on each Uj there is a continuous logarithm λj : Uj C of f such that 2πiλj → e = f on Uj. Now let

ξ = λ λ on U U , jk j − k j ∩ k 1 and note that ξjk : Uj Uk Z is locally constant. Clearly ξ = (ξkj) Z (U , Z), where ∩ → ∈ Z denotes the sheaf of locally constant functions with values in Z. Since Z1(U , Z) ⊂ Z1(U , ) and H1(U , ) = 0, the cocycle ξ splits with respect to the sheaf . Hence O O O there is (η ) C0(U , ) with j ∈ O ξ = η η on U U . jk j − k j ∩ k

2πiηj This gives a well-deﬁned holomorphic function g ∗(S) with g = e on U . We ∈ O j see that g is homotopic to f by taking the homotopy

F : S [0, 1] C∗,Ft = F ( , t) = exp 2πi (1 t)λj + tηj on Uj. × → · − This is well-deﬁned and continuous since

(1 t)(λ λ ) + t(η η ) = ξ on U U . − j − k j − k jk j ∩ k

36 c) Let Γ = nγ1 +mγ2 : n, m Z be a lattice in C and C/Γ be the corresponding { ∈ } torus. Noting that for z C we can write z = aγ1 + bγ2 for unique a, b R, consider ∈ ∈ the following maps logz logw f : C∗ C∗ C/Γ, (z, w) γ1 + γ2 + Γ, × → 7→ 2πi 2πi 2πia 2πib g : C/Γ C∗ C∗, aγ1 + bγ2 + Γ (e , e ). → × 7→ We do not need to specify the logarithm chosen for f since a change in the logarithm by 2πi will get sucked into the lattice Γ. It is clear that f is holomorphic. It is also clear that g is continuous and f g is the identity map on C/Γ. Now given a continuous ◦ map h: S C/Γ, the composition g h: S C∗ C∗ is continuous. Hence by b) → ◦ → × there are holomorphic functions ui for i = 1, 2, and homotopies Hi : S [0, 1] C∗ × → such that H ( , 0) = (g h) , i · ◦ i H ( , 1) = u . i · i Then u = (u1, u2): S C∗ C∗ is holomorphic and we have a homotopy H = → × (H1,H2): S [0, 1] C∗ C∗, taking g h to u. Finally, composition by f gives a × → × ◦ holomorphic function f u: S C/Γ and a homotopy f H : S [0, 1] C/Γ which ◦ → ◦ × → satisﬁes f H( , 0) = f g h = h, ◦ · ◦ ◦ f H( , 1) = f u. ◦ · ◦

d) P1. By Theorem 2.1.2 and Corollary 2.7.6 there is a bijection between homotopy classes of maps S P1 and homotopy classes of maps X P1, where X is a → → bouquet of circles. Since P1 is simply connected, all maps from a bouquet of circles to P1 are null-homotopic.

For the rest of section we proceed to strengthen the proof of BOP to include interpolation, jet interpolation and approximation. The proofs of BOPAI (which is omitted) and BOPAJI are fairly minor modiﬁcations of the proof of BOPA, with the proof of BOPI, respectively BOPJI, worked in. While we do not call on BOPI and BOPA to prove BOPAI, or on BOPJI and BOPA to prove BOPAJI, we have nonetheless included the proofs of BOPI, BOPJI and BOPA for readability, since they provide a guide to the construction of the ﬁnal proofs. This introduces a sizeable amount of repetition, but makes the proof of BOPAJI more digestible. Lemma 3.1.3. Let S be a non-compact Riemann surface and A S be a closed subset ⊂ 1 for which Hn(A) = 0 for n 2, i :H1(A) H1(S) is injective and i∗ :H (S) ≥ ∗ → → H1(A) is surjective, where i: A, S is the inclusion. Then → H (S 0, 1 A [0, 1]) = H2(S 0, 1 A [0, 1]) = 0 for all n 2. n × { } ∪ × × { } ∪ × ≥

37 Proof. Let U = S 0 A [0, 1) and V = S 1 A (0, 1]. Then U and V are × { } ∪ × × { } ∪ × open and U V = A (0, 1). By Mayer-Vietoris, we have an exact sequence ∩ ×

Hn(U) Hn(V ) Hn(S 0, 1 A [0, 1]) Hn 1(U V ) · · · −→ ⊕ −→ × { } ∪ × −→ − ∩ −→ Hn 1(U) Hn 1(V ) − ⊕ − −→· · · However, H (U V ) = H (A) since U V deformation-retracts onto A. Also, U and 1 ∩ 1 ∩ V both deformation-retract onto S. By Theorem 2.7.6, S has the homotopy type of a 1-dimensional CW-complex, so H (S) = 0 for n 2. For n = 2, the Mayer-Vietoris n ≥ sequence becomes

0 H (S 0, 1 A [0, 1]) H (A) H (S) H (S) . −→ 2 × { } ∪ × −→ 1 −→ 1 ⊕ 1 −→· · · Hence, H (S 0, 1 A [0, 1]) = 0, since H (A) H (S) H (S) is injective. For 2 × { } ∪ × 1 → 1 ⊕ 1 n > 2, the Mayer-Vietoris sequence is just

0 Hn(S 0, 1 A [0, 1]) Hn 1(A) 0, −→ × { } ∪ × −→ − −→ and hence Hn(S 0, 1 A [0, 1]) = Hn 1(A) = 0. × { } ∪ × − For cohomology, again by Mayer-Vietoris, we have an exact sequence

H1(S) H1(S) H1(A) H2(S 0, 1 A [0, 1]) 0. · · · −→ ⊕ −→ −→ × { } ∪ × −→ So H2(S 0, 1 A [0, 1]) = 0, since H1(S) H1(S) H1(A) is surjective. × { } ∪ × ⊕ → For the rest of the section, when maps H (A) H (S) and H1(S) H1(A) 1 → 1 → are mentioned, we mean the maps induced by the inclusion i: A, S. Note that → if H (A) = 0, then, trivially, H (A) H (S) is injective and H1(S) H1(A) is 1 1 → 1 → surjective.

Lemma 3.1.4. Let S be a non-compact Riemann surface and A S be a closed ⊂ submanifold-with-boundary with H (A) H (S) injective, H1(S) H1(A) surjective 1 → 1 → and H2(A) = 0. Then any continuous map f : S 0, 1 A [0, 1] P1 can be × { } ∪ × → extended to a continuous map S [0, 1] P1. × → Proof. First note that by Corollary 2.10.6, there is a triangulation of S that is an extension of a triangulation on A. The corresponding CW-structure on S has A as a subcomplex. A trivial extension of the triangulation of S to S [0, 1] gives a CW- × structure on S [0, 1] with S 0, 1 A [0, 1] as a subcomplex. Also, P1 is a × × { } ∪ × connected CW-complex. Since P1 has trivial fundamental group, the action on all higher homotopy groups in Deﬁnition 2.1.8 is trivial and P1 is an abelian space. In order to apply Theorem 2.1.9, we are left with calculating the cohomology groups

38 n+1 H (S [0, 1],S 0, 1 A [0, 1]; πnP1) for n 1. For n = 1, π1P1 = 0, so clearly 2 × × { } ∪ × ≥ H (S [0, 1],S 0, 1 A [0, 1]; π1P1) = 0. × × { } ∪ × 3 For n = 2, note that π2P1 = Z, so we need H (S [0, 1],S 0, 1 A [0, 1]) = 0. × ×{ }∪ × The long exact relative cohomology sequence gives (noting that S [0, 1] is homotopy × equivalent to S)

H2(S 0, 1 A [0, 1]) H3(S [0, 1],S 0, 1 A [0, 1]) H3(S) · · · −→ × { } ∪ × −→ × × { } ∪ × −→ −→· · · By Lemma 3.1.3, H2(S 0, 1 A [0, 1]) = 0. Also, H3(S) = 0 as S is a 2-dimensional ×{ }∪ × CW-complex. Hence, H3(S [0, 1],S 0, 1 A [0, 1]) = 0. × × { } ∪ × Consider the long exact relative homology sequence

Hn(S) Hn(S [0, 1],S 0, 1 A [0, 1]) Hn 1(S 0, 1 A [0, 1]) · · · −→ −→ × ×{ }∪ × −→ − ×{ }∪ × −→· · · For n 3, H (S) = 0, since S is a 2-dimensional CW complex. Also, H (A) = 0 and A ≥ n 2 is a 2-dimensional CW-complex, so the higher homology groups vanish as well. Thus,

A satisﬁes the assumptions of Lemma 3.1.3 and Hn 1(S 0, 1 A [0, 1]) = 0 for − × { } ∪ × n 3. It follows that H (S [0, 1],S 0, 1 A [0, 1]) = 0 for n 3. By the ≥ n × × { } ∪ × ≥ universal coeﬃcients theorem (Theorem 2.1.6), we get

n+1 H (S [0, 1],S 0, 1 A [0, 1]; πnP1) Ext(0, πnP1) Hom(0, πnP1) = 0, × × { } ∪ × ' ⊕ for n 3. Thus, an extension of f exists by Theorem 2.1.9. ≥ Theorem 3.1.5 (BOP with interpolation). Let S be a non-compact Riemann surface and X be C, C∗, P1 or a torus. If T is a discrete subset of S and f : S X is → continuous, then f can be deformed to a holomorphic map S X keeping it ﬁxed → on T .

Proof. a) C. By Weierstrass’ theorem, there is a holomorphic function g (X) with ∈ O g T = f T . Now consider the homotopy H : S [0, 1] C, Ht = (1 t)f + tg. Clearly | | × → − H is a homotopy from f to g which has ﬁxed values on T .

b) C∗. First note that by Theorem 2.10.7, there exists an open cover U = (Uj)j J ∈ of S by coordinate disks such that each element of T is contained in a unique Uj and no two elements of T are contained in the same Uj. As before we can lift f locally 2πiλj on each Uj to get continuous logarithms λj : Uj C such that e = f on Uj. The → differences ξ = λ λ on U U jk j − k j ∩ k define a cocycle ξ = ξ Z1(U , ). Hence there is a 0-cochain η C0(U , ) { jk} ∈ O { j} ∈ O that splits ξ. Since Uj Uk T = ∅ for all j, k J, j = k, there is a well-defined ∩ ∩ ∈ 6 function u: T C, u = ηj λj on Uj T. → − ∩

39 By Weierstrass’ theorem, there is a holomorphic function h (S) with h T = u. ∈ O | Now define a new cochain µ = µ C0(U , ) with µ = η h. Then { j} ∈ O j j − µ µ = η η = ξ on U U , j − k j − k jk j ∩ k and µ = η h = λ on T. j j − j 2πiµj So there is a well-defined holomorphic function g ∗(S) given by g = e on U . ∈ O j Lastly consider the homotopy F : S [0, 1] C∗,Ft = exp 2πi (1 t)λj + tµj on Uj. × → − Then F is well defined with

F ( , 0) = f, · F ( , 1) = g, · F T = exp 2πi(1 t)λ T + tµ T t| − j| j| = exp 2πi(1 t)λ T + tλ T − j| j| = f T on U . | j

c) Let X be a torus. As shown in the proof of Theorem 3.1.2 there is a holomorphic map h: C∗ C∗ X and a continuous map g : X C∗ C∗ such that × → → × h g = idX . Now g f : S C∗ C∗ is continuous, so by b), there are holomor- ◦ ◦ → × phic functions u ∗(S), i = 1, 2, and homotopies H from (g f) to u that i ∈ O i ◦ i i are ﬁxed on T . Hence we get a holomorphic map u = (u1, u2) and a homotopy H = (H1,H2): S [0, 1] C∗ C∗ such that × → × H( , 0) = g f, · ◦ H( , 1) = u, · H T = (g f) T for all t [0, 1]. t| ◦ | ∈ Composition by h gives a holomorphic map h u: S X and a homotopy h H : S ◦ → ◦ × [0, 1] X which satisﬁes → h H( , 0) = h g f = f, ◦ · ◦ ◦ h H( , 1) = h u, ◦ · ◦ h H T = (h g f) T = f T for all t [0, 1]. ◦ t| ◦ ◦ | | ∈

d) P1. Any two continuous maps p, q : S P1 that agree on T are homotopic rel. → T . To see this, let F : S 0, 1 T [0, 1] P1 have F ( , 0) = p, F ( , 1) = q and be × { } ∪ × → · ·

40 constant on a [0, 1] for each a T . Then, by Lemma 3.1.4, F can be extended to { } × ∈ a continuous map from all of S [0, 1], noting that T is a 0-dimensional submanifold × of S with H (T ) = H1(T ) = 0 for all n 1. n ≥ It follows that to prove the theorem for this case all we need is a holomorphic map S P1 that agrees with f on T . Such a holomorphic map can easily be found → using Weierstrass’ theorem: let x P1 f(T ), then P1 x is biholomorphic to C, ∈ \ \{ } hence by Weierstrass’ theorem there is a holomorphic function from S to P1 x that \{ } agrees with f on T .

Theorem 3.1.6 (BOP with jet interpolation). Let S be a non-compact Riemann surface and X be C, C∗, P1 or a torus. Suppose T is a discrete subset of S, n: T N 0 → ∪{ } is an assignment of a non-negative integer to each a T and f : S X is a contin- ∈ → uous map which is holomorphic on a neighbourhood of T . Then f can be deformed to a holomorphic map S X, keeping it holomorphic on T and the n -jets at a T of → a ∈ the maps in the deformation constant.

Proof. We have made more assumptions on the function f than in the statement of Theorem 3.1.5, so the basic Oka property with interpolation is not a special case of this theorem; despite this the proofs are very similar. a) C. By the strong Weierstrass’ theorem (Theorem 2.5.9), there is a holomorphic na na function g (S) with Ja f = Ja g at each point a T . Define H : S [0, 1] C, ∈ O ∈ × → H = (1 t)f + tg. Clearly H is a homotopy from f to g. t − b) C∗. Let U = (Uj)j J ,(λj) and (ηj) be as in the proof of Theorem 3.1.5. There ∈ is a neighbourhood V of T such that each λ is holomorphic on V U , and we take j ∩ j V to be sufficiently small that for each i, j J, i = j, V U and V U are disjoint. ∈ 6 ∩ i ∩ j Then there is a well-defined holomorphic function

u: V C, u = ηj λj on Uj V. → − ∩ By the strong Weierstrass’ theorem there is a holomorphic function h (S) such that na na ∈ O at each a T , Ja h = Ja u. The rest of the proof follows through as for interpolation. ∈ 0 Deﬁne the cochain µ = (µj) C (U , ), where µj = ηj h. This splits the cocycle na na ∈ O − 2πiµj ξ and J µ = J λ at a T U . Let g ∗(S) be given by g = e on U and a j a j ∈ ∩ j ∈ O j take the homotopy F : S [0, 1] C∗,Ft = exp 2πi (1 t)λj + tµj on Uj. × → − By Theorem 2.4.3, since J na λ = J na µ at a T , we have J na f = J na g. a j a j ∈ a a c) Let X be the torus C/Γ. We have f : S C/Γ continuous and holomorphic on → some neighbourhood of each point in T . By the basic Oka property f is homotopic to a holomorphic function, so there is a holomorphic function g : S C/Γ and a homotopy →

41 H : S [0, 1] C/Γ from f to g. We will exploit the complex Lie group structure on × → a torus, writing the group operation in additive notation. Deﬁne h = f g : S C/Γ, − → which is continuous, and holomorphic on a neighbourhood of each point in T . We also see that h is null-homotopic by the homotopy

F : S [0, 1] C/Γ,Ft = Ht g. × → − Note that F ( , 0) = f g = h and F ( , 1) = 0. · − · Now by Lemma 2.3.2, h lifts to a function h˜ : S C, which is holomorphic on → a neighbourhood of each point in T by Lemma 2.3.1. Using the result for C, we get na na ˜ a holomorphic function φ: S C with Ja φ = Ja h at each a T and a homotopy ˜ ˜ → ∈ G: S [0, 1] C from h to φ that has na-jets at each a T fixed for t [0, 1]. × → ∈ ∈ Let p: C C/Γ be the universal cover. Then ψ = p φ is holomorphic and by → ◦ Theorem 2.4.3, J na ψ = J na h at each a T . Define the homotopy a a ∈ G: S [0, 1] C/Γ,G = p G,˜ × → ◦ from p h˜ = h to p φ = ψ. Finally, since at a T , J na G˜ = J na h˜ for every t [0, 1], ◦ ◦ ∈ a t a ∈ J na G = J na h at a T for every t [0, 1]. In order to get the result for the function a t a ∈ ∈ f all we need to do is add the holomorphic function g back on, noting that g + ψ is holomorphic and g + Gt defines a homotopy from f to g + ψ that has fixed na-jets f at each a T for all t [0, 1]. ∈ ∈ d) P1. For each point a T , let Ua be a coordinate neighbourhood of a on which ∈ f is holomorphic. We will take these to be sufficiently small that they are pairwise disjoint, f(U ) is contained in a coordinate neighbourhood of f(a) for all a T and a ∈ there is a point p P1 such that f(Ua) P1 p . This is possible, for if we take 1∈ ⊂ \{ } p / f(T ) then f − (P1 p ) is a neighbourhood of a for all a T . We call on the ∈ \{ } ∈ strong Weierstrass’ theorem to get a holomorphic function g : S P1 p such that → \{ } at each a T , J na f = J na g. ∈ a a For each a T let Va Ua be a closed coordinate disk containing a and let S ∈ ⊂ A = a T Va. Our plan is to get a homotopy from f to g on each Va, and then extend the∈ resulting map from S 0, 1 A [0, 1] to all of S [0, 1]. There are × { } ∪ × × charts φ: Ua C and ψ : Wa C, where f(Ua) Wa. By part a) there is a → 1 → 1 ⊂ 1 homotopy H from ψ f φ− to ψ g φ− with fixed n -jets at φ− (a). We construct a ◦ ◦ ◦ ◦ a H : S 0, 1 A [0, 1] P1 by letting H( , 0) = f, H( , 1) = g and H = Ha on Va. × { } ∪ × → · · Since A consists of disjoint closed disks, H (A) = H1(A) = 0 for all n 1 and A n ≥ is a closed submanifold of S with boundary. That the required extension exists then follows from Lemma 3.1.4.

Note that in all cases there is a ﬁxed neighbourhood of T on which the maps in

42 the deformation are holomorphic, although this neighbourhood is not necessarily the same as the neighbourhood of T on which the starting map is holomorphic.

Theorem 3.1.7. Let S be a non-compact Riemann surface, K S be a holomor- ⊂ phically convex, compact subset and U be a neighbourhood of K. Then there is a holomorphically convex, compact submanifold-with-boundary A U containing K in ⊂ its interior, for which H (A) = 0, H (A) H (S) is injective and H1(S) H1(A) is 2 1 → 1 → surjective.

Proof. By Lemma 2.8.9, there is a strictly subharmonic exhaustion φ: S R such → that φ < 0 on K and φ > 1 on S U. By Theorem 2.8.10, the sublevel sets of \ φ are holomorphically convex. Let C (0, 1) be a regular value of φ and consider 1 ∈ A = φ− (( ,C]). Then A U is a holomorphically convex, compact subset of S, −∞ ⊂ and K A◦. Also, A is a closed submanifold-with-boundary. ⊂ Let V = A◦. The components of the complement of V are the closures of the components of the complement of A, so h(A) = A implies h(V ) = V, that is, V is 1 Runge. Now let M be a collar neighbourhood of ∂A = φ− (C) with diﬀeomorphism 1 p: M ∂A [0, 1). Both ∂A [0, 1) and ∂A (0, 1) deformation retract onto ∂A [ 2 , 1). → × 1 × 1 × × The preimage W = p− (∂A [0, )) is a neighbourhood of ∂A and we get deformation × 2 retracts M M W and M ∂A M W by pre- and postcomposition with p and 1 → \ \ → \ p− . These retracts can easily be extended to all of A and A ∂A = V , respectively, \ by taking the identity on A W . Hence, we get isomorphisms H (A) H (A W ) \ 1 → 1 \ and H (V ) H (A W ), so H (V ) H (A) is an isomorphism. By Theorem 2.8.11, 1 → 1 \ 1 → 1 the natural map H (V ) H (S) is injective, so H (A) H (S) is injective. Similarly, 1 → 1 1 → 1 H2(A) = H2(V ), which vanishes since V has the homotopy type of a 1-dimensional CW complex by Theorem 2.7.6. By Theorem 2.8.12, H2(S,A) = 0 and H1(S,A) is free abelian. It follows from the universal coeﬃcients theorem that

2 H (S,A) = Ext(H1(S,A), Z) Hom(H2(S,A), Z) = H1(S,A)tor = 0. ⊕ Consider the long exact relative cohomology sequence

H1(S,A) H1(S) H1(A) H2(S,A) . · · · −→ −→ −→ −→ −→· · · Since H2(S,A) vanishes, the map H1(S) H1(A) is surjective. → Lemma 3.1.8. Let S be a non-compact Riemann surface, K S be compact and V ⊂ be a relatively compact neighbourhood of K. Suppose we have functions φ (V ) and ∈ O ρ M (V ), such that φ has no zeros at the poles of ρ. Then, given > 0, there is ∈ δ > 0 such that if ψ (V ) satisﬁes sup φ ψ < δ, then d(ρφ, ρψ) < on K with ∈ O K | − | respect to the spherical metric on P1.

43 Proof. Let T be the set of poles of ρ on K. By assumption φ has no zeros or poles on T . Choose a neighbourhood B V of T on which φ has no zeros, and a δ > 0 ⊂ 1 sufficiently small that if sup φ ψ < δ , then ψ has no zeros in B. This is possible K | − | 1 since T is finite. Now ρ has poles at the points of T , so it maps a sufficiently small neighbourhood of T to within distance /2 of with respect to the spherical metric. ∞ Since ψ is bounded away from zero on B, we can find a subset of B B on which 1 ⊂ ρψ is within distance /2 of on B with respect to the spherical metric. Similarly, ∞ 1 choose a sufficiently small neighbourhood B2 of T such that ρφ is within distance /2 of on B . Then on B˜ = B B , d(ρφ, ρψ) < . ∞ 2 1 ∩ 2 Note that if given δ2 < δ1, we do not need to find a new B˜, since functions within distance δ2 of φ on K are actually bounded further away from zero on B than functions within distance δ of φ on K. Now on K B˜, ρ has no poles and hence is bounded as 1 \ K is compact. With respect to Euclidean distance, if sup φ ψ < δ , then K | − | 2

sup ρ(φ ψ) < δ2 sup ρ, K B˜| − | · K B˜ \ \ and hence we can easily find δ > 0 sufficiently small that d(ρφ, ρψ) < on K B˜. 2 \ Finally pick δ = min δ , δ . { 1 2} Theorem 3.1.9 (BOP with approximation). Let S be a non-compact Riemann surface and X be C, C∗, P1 or a torus. If K is a holomorphically convex compact subset of S and f : S X is a continuous function which is holomorphic on a neighbourhood → of K, then f can be deformed to a holomorphic map S X keeping it holomorphic → on a neighbourhood of K. Furthermore, given > 0, the maps in the deformation can be chosen to be within distance of f on K with respect to any metric defining the topology on X.

Proof. a) C. Given > 0, by the Runge approximation theorem we can ﬁnd a holomorphic function g : S C such that supK f g < . Then we just take the homotopy → | − |

H : S [0, 1] C,Ht = (1 t)f + tg, × → − which is holomorphic on a neighbourhood of K for all t and has

H( , 0) = f, · H( , 1) = g, · sup Ht f = sup tg tf = t sup g f < on K. K | − | K | − | K | − |

b) C∗. We use the same approach as in the proof of jet interpolation for the torus. We have f : S C∗ continuous and holomorphic on some neighbourhood of →

44 K. By Theorem 3.1.2, there is a holomorphic function g : S C∗ and a homotopy → H : S [0, 1] C∗ from f to g. We get a well-deﬁned continuous function h = × → f/g : S C∗, which is holomorphic where f is holomorphic. Also h is null-homotopic → by the homotopy F : S [0, 1] C∗,F = H/g. × → Note that F ( , 0) = f/g = h and F ( , 1) = 1. Now by Lemma 2.3.2, h lifts with · · respect to the exponential map to a function h˜ : S C, which is still holomorphic on → a neighbourhood of K. The image of h˜(K) is a compact set in C. Let U be a relatively compact neighbourhood of h˜(K). Now g is bounded on K and exp0 = exp is bounded on U since it is relatively compact, so let sup g = M and sup exp = N. Given K | | U | | > 0, let δ > 0 be such that δ < /(MN) and δ is suﬃciently small that B(x, δ) U ⊂ for every x h˜(K). Using the result for C, there is a holomorphic function φ: S C ∈ → and a homotopy G˜ : S [0, 1] C from h˜ to φ that is holomorphic on a neighbourhood × → of K and has sup G˜ h˜ < δ for all t. K | t − | The function ψ = exp φ: S C∗ is holomorphic and we get a homotopy G = ˜ ◦ → exp G: S [0, 1] C∗ from h to ψ which has Gt holomorphic on a neighbourhood of ◦ × → K for every t [0, 1]. By construction, for t [0, 1], ∈ ∈

sup Gt h = sup exp G˜t exp h˜ K | − | K | ◦ − ◦ |

< N sup G˜t h˜ K | − | < Nδ < /M since G˜ and h˜ map K into U, where exp0 < N. Finally, we need to multiply t | | through by g to get a holomorphic function gψ and a homotopy gG: S [0, 1] C∗, × → gG(x, t) = g(x)G(x, t), from f to gψ. Then (gG)t is holomorphic on a neighbourhood of K for all t. By construction

sup (gG)t f < sup g /M = for all t. K | − | K ·

c) C/Γ. The proof is precisely as for C∗, using the universal covering map p: C → C/Γ in place of the exponential map. Also, the group operation on the torus must be used in place of multiplication.

d) P1. Let U be a relatively compact Runge neighbourhood of K on which f is holomorphic. Then f has finitely many poles and zeros in U. By Theorem 2.5.5, we can find a holomorphic function p: S C with (p) = (f) on U. By → − the Riemann removable singularities theorem, the product pf defines a holomorphic function g : U C, which has no zeros. By the Runge approximation theorem, given → δ > 0, we can find a holomorphic function h: S C such that supK g h < δ. → | − |

45 By Theorem 3.1.7, there is a closed submanifold-with-boundary A U containing ⊂ K in its interior, which satisfies the assumptions of Lemma 3.1.4. We can define a map F : S 0, 1 A [0, 1] P1 by F ( , 0) = g, F ( , 1) = h and Ft = (1 t)g + th on × { } ∪ × → · · − A [0, 1]. By Lemma 3.1.4, there is an extension F : S [0, 1] P1 of F to a homotopy × × → from g to h. Define a new continuous map G: S [0, 1] P1 by G(x, t) = F (x, t)/p(x). × → Then G is a homotopy from f to h/p, and h/p is a meromorphic function. Note that the zeros of p are contained in A, and on A [0, 1], G = (1 t)f + th/p. × t − Let V A be a neighbourhood of K. The functions g, F and 1/p restricted ⊂ t to V satisfy the assumptions of Lemma 3.1.8, with φ = g, ψ = Ft and ρ = 1/p. In particular, note that if sup g h < δ, then K | − |

sup g Ft = sup tg th < δ. K | − | K | − |

Hence, given > 0, there is δ > 0 such that if sup g h < δ, then d(f, F /p) < on K | − | t K with respect to the spherical metric for all t [0, 1]. ∈ Again, we note that in all cases the maps in the deformation are holomorphic on a ﬁxed neighbourhood of K.

Theorem 3.1.10. Let S be a non-compact Riemann surface, U S be a Runge ⊂ subset and D be a divisor on S. Then every meromorphic function f (U) can be ∈ OD approximated uniformly on compact subsets of U by meromorphic functions in (S). OD Proof. For a compact subset K U, let V U be a relatively compact Runge neigh- ⊂ ⊂ bourhood of K on which f has only finitely many zeros and poles. Such a neighbourhood exists by Lemma 2.8.4. Let g (S) be such that (g) = (f) on V , which exists ∈ OD by Weierstrass’ theorem. Then f/g can be extended by Riemann removable singularities to a holomorphic function f/g : V C. Let δ > 0. By the Runge approximation → theorem we can find a holomorphic function h: S C with supK h f/g < δ. The → | − | product gh: S P1 is a meromorphic function with (gh) = (g) + (h) (g) D, so → ≥ ≥ − gh (S). ∈ OD The functions f/g, h and g restricted to V satisfy the assumptions of Lemma 3.1.8, with φ = (f/g), ψ = h and ρ = g. Hence, given > 0, there is δ > 0 such that if sup h f/g < δ, then d(f, gh) < on K with respect to the spherical metric. K | − | The following two theorems establish the strongest forms of the basic Oka property for maps between Riemann surfaces. To avoid even further repetition we will only prove BOPAJI, which is the slightly more technical of the two. The approach used to tie together the proofs of Theorems 3.1.6 and 3.1.9 is to apply Theorem 3.1.10 to the difference of the original function and a holomorphic function on S which has the same

46 jets on T . The same approach works for BOPAI, using the divisor D that just takes the values D = 1 on T and 0 on S T , and a holomorphic function on S that agrees − \ with the original function on T .

Theorem 3.1.11 (BOP with approximation and interpolation). Let S be a non- compact Riemann surface and X be C, C∗, P1 or a torus. Let T S be a discrete set ⊂ and K S be a holomorphically convex, compact subset. If f : S X is a continu- ⊂ → ous map which is holomorphic on a neighbourhood of K, then f can be deformed to a holomorphic map S X, keeping it holomorphic on a neighbourhood of K, arbitrarily → close to f on K and ﬁxed on T .

Theorem 3.1.12 (BOP with approximation and jet interpolation). Let S be a non- compact Riemann surface and X be C, C∗, P1 or a torus. Let T S be a discrete ⊂ subset, K S be a holomorphically convex, compact subset and n: T N 0 be an ⊂ → ∪ { } assignment of a non-negative integer to each point in T . If f : S X is a continuous → map which is holomorphic on a neighbourhood of K T , then f can be deformed to a ∪ holomorphic map S X, keeping it holomorphic on a neighbourhood of K T and → ∪ arbitrarily close to f on K. Furthermore, for each point a T , the n -jets of the maps ∈ a in the deformation can be kept ﬁxed.

Proof. a) C. By the strong Weierstrass’ theorem (Theorem 2.5.9) we can ﬁnd a holo- na na morphic function g : S C with Ja g = Ja f for all a T . Let U be a Runge → ∈ neighbourhood of K on which f is holomorphic. By Lemma 2.4.4, f g is a holomor- − phic function on U which has a zero at each a T of order at least n + 1. ∈ a Let D be the divisor on S which is zero on S T and has D = n 1 on T . For \ − − V S open, φ (V ) if and only if φ is a holomorphic function on V with zeros at ⊂ ∈ OD each a T V of order at least n + 1. So f g (U) and by Theorem 3.1.10, ∈ ∩ a − ∈ OD given > 0, there is a holomorphic function p D(S) with supK p (f g) < . ∈ O na | − na− | na Then h = g + p: S C is holomorphic and at a T we have Ja h = Ja g = Ja f. → ∈ Also we have sup h f = sup p (f g) < by our choice of p. K | − | K | − − | Finally take the homotopy H : S [0, 1] C given by Ht = (1 t)f + th. Then × → − H is a deformation of f to h with H holomorphic on the same neighbourhood of K T t ∪ that f is holomorphic on for all t [0, 1]. Since at each a T the n -jets of f and h ∈ ∈ a are the same, the na-jets of Ht are ﬁxed on T for all t. Finally by construction,

sup Ht f = sup (1 t)f + th f = sup t h f < . K | − | K | − − | K | − |

b) C∗. We have f : S C∗ continuous and holomorphic on some neighbourhood → of K T . By Theorem 3.1.2 there is a holomorphic function g : S C∗ and a ∪ → homotopy H : S [0, 1] C∗ from f to g. We get a well-deﬁned continuous function × →

47 h = f/g : S C∗, which is holomorphic where f is holomorphic. Also h is null- → homotopic by the homotopy

F : S [0, 1] C∗,F = H/g. × → Note that F ( , 0) = f/g = h and F ( , 1) = 1. Now by Lemma 2.3.2, h lifts with · · respect to the exponential map to a function h˜ : S C, which is holomorphic on a → neighbourhood of K T . The image h˜(K) is a compact set in C. Let U be a relatively ∪ compact neighbourhood of h˜(K). Now g is bounded on K and exp0 = exp is bounded on U since it is relatively compact, so let sup g = M and sup exp = N. Given K | | U | | > 0, let δ > 0 be such that δ < /(MN) and δ is suﬃciently small that B(x, δ) U ⊂ for every x h˜(K). From (a), there is a holomorphic function φ: S C and a ˜∈ ˜ ˜ → homotopy G: S [0, 1] C of h to φ with Gt holomorphic on a neighbourhood of × → na na K T , sup˜ G˜ h˜ < δ and J G˜ = J h˜ for all a T and t [0, 1]. ∪ h(K) | t − | a t a ∈ ∈ na na The function ψ = exp φ: S C∗ is holomorphic and at a T has Ja ψ = Ja h ◦ → ˜ ∈ by Theorem 2.4.3. We get a homotopy G = exp G: S [0, 1] C∗ from h to ψ with ◦ × → G holomorphic on a neighbourhood of K T and J na G = J na h for every a T and t ∪ a t a ∈ t [0, 1]. By construction, for t [0, 1], ∈ ∈

sup Gt h = sup exp G˜t exp h˜ < N sup G˜t h˜ < Nδ < /M K | − | K | ◦ − ◦ | K | − | since G˜ and h˜ map K into U, where exp0 < N. Finally we need to multiply through t | | by g to get a holomorphic function gψ and a homotopy gG: S [0, 1] C∗, gG(x, t) = × → g(x)G(x, t), from f to gψ. Then (gG) is holomorphic on a neighbourhood of K T t ∪ for all t. By construction

sup (gG)t f < sup g /M = for all t. K | − | K · At each a T , G agrees with h to order n for all t [0, 1], hence J k(gG) = J kf for ∈ t a ∈ a t a all t [0, 1] by Lemma 2.4.5. ∈ c) C/Γ. As in the proof of BOPA, the proof for maps into C/Γ follows through mutatis mutandis to the proof for maps into C∗.

d) P1. As in the proof of Theorem 3.1.6, for each point a T , let Ua be a ∈ coordinate disk at a on which f is holomorphic. Again we take these to be suﬃciently small that they are pairwise disjoint, f(Ua) is contained in a coordinate neighbourhood of f(a) for all a T and there is a point b P1 such that f(Ua) P1 b . Without loss ∈ ∈ ⊂ \{ } of generality we may assume that b = . By Theorem 2.5.9, there is a holomorphic ∞ na na map g : S P1 such that at each a T , Ja g = Ja f. → \ {∞} ∈ Let U be a Runge neighbourhood of K which does not intersect T except on K T , and on which f is holomorphic and has only ﬁnitely many zeros and poles. ∩

48 The difference f g : U P1 is holomorphic and has only finitely many poles in U. − → By Theorem 2.5.5, there is a holomorphic function p: S C with zeros at the poles → of f g of the same order as the poles, and no other zeros. By Theorem 2.4.4, f g − − is in (U), where the divisor D is zero on S T and D = n 1 on T . By OD+(p) \ − − Riemann removable singularities, (f g)p can be extended to a holomorphic function − in (U). By Theorem 3.1.10, for δ > 0, there is a holomorphic function h (S) OD ∈ OD with sup h (f g)p < δ. K | − − | Now let A U be a holomorphically convex, compact submanifold-with-boundary 1 ⊂ of S containing K in its interior and that satisfies the assumptions of Lemma 3.1.4, which exists by Theorem 3.1.7. For each a T that is not contained in K, let U be a ∈ a closed coordinate disk at a that does not intersect A1 and such that Ua Ub = ∅ ∩ for a = b. Let A = A S U . Then A is a closed submanifold-with-boundary. 6 1 a a Define a homotopy F : S 0, 1 A [0, 1] P1, F ( , 0) = f, F ( , 1) = h and × { } ∪ × → · · F = (1 t)fp + t(gp + h) on A [0, 1]. This is well defined, since on U, f and f g t − × − have the same poles, which are the zeros of p, so by Riemann removable singularities fp can be extended to a holomorphic function on A. For each a T not contained in ∈ K, the closed disk Ua is contractible, so A satisfies the assumptions of Lemma 3.1.4. That is, H (A) H (S) is injective, H1(S) H1(A) is surjective and H (A) = 0. 1 → 1 → 2 Thus, by Lemma 3.1.4, F can be extended to F : S [0, 1] P1. × → Define H : S [0, 1] P1 by H(x, t) = F (x, t)/p(x). Then H is a homotopy from × → f to the meromorphic function g + h/p. Since h (S), we have h/p (S), ∈ OD ∈ OD+(p) that is, its poles are at poles of f and it has a zero of order na + 1 at each a T . By na na na ∈ Theorem 2.4.4, Ja (g + h/p) = Ja g = Ja f as required. It is easy to see that at each a T the n -jets are fixed during the deformation. Finally, for each t [0, 1], ∈ a ∈ sup fp Ft = sup fp (1 t)fp t(gp + h) = sup t (f g)p h) < tδ. K | − | K | − − − | K | − − |

As at the end of the proof of Theorem 3.1.9, let V A be a neighbourhood of ⊂ K. The functions fp, Ft and 1/p restricted to V satisfy the assumptions of Lemma 3.1.8, with φ = fp, ψ = Ft and ρ = 1/p. Hence, given > 0, there is δ > 0 such that if sup h (f g)p < δ, then d(f, H ) < on K with respect to the spherical metric K | − − | t for all t [0, 1]. ∈

3.2 The non-Gromov pairs

We will now proceed to show that all non-Gromov pairs fail the stronger Oka properties of BOPI and BOPJI. We will also show that all non-Gromov pairs fail BOPA, apart from a class of pairs that satisfy it for trivial reasons. In order to do so, we ﬁrst need a suﬃciently strong hyperbolicity result.

49 Theorem 3.2.1 (Tietze extension theorem). Let X be a normal topological space, A X be a closed subset and f : A [0, 1] be a continuous function. Then f can be ⊂ → extended to a continuous function X [0, 1]. → Proof. See [16, Theorem 4.4].

Recall that Riemann surfaces are metrisable, and hence are normal topological spaces. The Schwarz lemma is a classical theorem in complex analysis and one of the earliest results on the rigidity of holomorphic functions.

Theorem 3.2.2 (Schwarz lemma). Let f : D D be a holomorphic function with → f(0) = 0. Then f(z) z for all z D and f 0(0) 1. | | ≤ | | ∈ | | ≤

Proof. Deﬁne a holomorphic function g on D by g(z) = f(z)/z on D 0 and g(0) = \{ } f 0(0). For r < 1 and z = r, g(z) = f(z) /r 1/r, so by the maximum principle g | | | | | | ≤ is bounded by 1/r on z C : z < r . Letting r 1, we have g 1 and hence { ∈ | | } → | | ≤ f 1 on D. | | ≤ We will need a variant called the Schwarz-Pick Lemma.

Theorem 3.2.3 (Schwarz-Pick lemma). Let f : D D be a holomorphic function. → Then, for all z1, z2 D, ∈

f(z1) f(z2) z1 z2 − − . 1 f(z1)f(z2) ≤ 1 z1z2 − −

Proof. Fix z1 D and deﬁne automorphisms φ1 and φ2 of D by the formulas φ1(z) = ∈ z1 z f(z1) z 1 − and φ2(z) = − . The composition φ2 f φ1− maps D to D and has 1 z1z 1 f(z1)z ◦ ◦ − 1 − φ f φ− (0) = 0. By the Schwarz lemma, 2 ◦ ◦ 1 1 1 f(z1) f(φ1− (z)) φ f φ− (z) = − z for all z . 2 1 1 D | ◦ ◦ | 1 f(z1)f(φ− (z)) ≤ | | ∈ − 1

Then for z2 D, take z = φ1(z2). ∈

Corollary 3.2.4. Let σ : D D be an automorphism of the unit disk and z1, z2 D. → ∈ Then

σ(z1) σ(z2) z1 z2 − = − . 1 σ(z1)σ(z2) 1 z1z2 − −

50 1 Proof. Note that σ and σ− are holomorphic maps, so by the Schwarz-Pick lemma,

1 1 z1 z2 σ− σ(z1) σ− σ(z2) σ(z1) σ(z2) z1 z2 − = ◦ − ◦ − − . 1 1 1 z1z2 1 σ− σ(z1)σ− σ(z2) ≤ 1 σ(z1)σ(z2) ≤ 1 z1z2 − − ◦ ◦ − − Hence the inequalities must be equalities.

Lemma 3.2.5. Let X be a Riemann surface, π : D X be a holomorphic covering → map, V X be open and ψ : V D be a local inverse of π. Then given x V and ⊂ → ∈ > 0, there is a neighbourhood A V of x such that for all y A, and all covering ⊂ ∈ transformations σ : D D, we have σ ψ(x) σ ψ(y) < . → | ◦ − ◦ | 2 (1 z1 ) Proof. Let z1 = ψ(x) and δ = − | | . Suppose y V is such that z1 z2 < δ, 2 + z1 ∈ | − | where z = ψ(y). Note that z < z | +|δ and hence 2 | 2| | 1| 1 z¯ z 1 z¯ z 1 z ( z + δ) > 0, | − 1 2| ≥ − | 1|| 2| ≥ − | 1| | 1| where the last inequality follows from

(1 z 2) 2 z 2 + z z z + − | 1| = | 1| | 1| < 1. | 1| | 1| 2 + z 2 + z | 1| | 1| Thus,

z1 z2 δ − < = . 1 z¯ z 1 z 2 δ z 2 − 1 2 − | 1| − | 1| Then by Corollary 3.2.4,

σ(z1) σ(z2) z1 z2 − = − < , 1 σ(z1)σ(z2) 1 z¯1z2 2 − − for any covering transformation σ. So σ(z ) σ(z ) < 1 σ(z )σ(z ) (1 + σ(z ) σ(z ) ) < . | 1 − 2 | 2 − 1 2 ≤ 2 | 1 || 2 |

Now just let B = z D: z1 z < δ and A = V π(B). { ∈ | − | } ∩ Theorem 3.2.6. Let Y be a Riemann surface covered by the unit disk D, that is, Y is not C, C∗, P1 or a torus. Then for any Riemann surface X there is a two- point set T X and a continuous map f : X Y, which is locally constant on a ⊂ → neighbourhood of T, such that f is not homotopic rel. T to any holomorphic map X Y. Furthermore, given a metric d on Y that deﬁnes the topology, there exists → > 0 such that there is no holomorphic map within distance of f on T with respect to d.

51 D D

w ˜ 1 z 0 h ϕ− 0 ◦ z0′ z′

˜ ϕ h p ψ0 ψz X D x h Y x0 V1 V y0′ f 2 y0 y y ′

Proof. Let p: D Y be a holomorphic covering map and D X be an open subset → ⊂ 1 of X with a biholomorphism φ: D D. Take y Y such that 0 / p− (y) and → ∈ 1 ∈ let w D 0 be such that w < min z : z p− (y) . Finally, let y0 = p(0), ∈1 \{ } 1 | | {| | ∈ } x = φ− (w) and x = φ− (0). Now take a path γ : [0, 1] Y starting at y and ending 0 → 0 at y and let g : D ,D [0, 1] be given by g(D ) = 0 and g(D ) = 1, where D { 0 x} → 0 x 0 and Dx are closed disks in D around x0 and x respectively. By Theorem 3.2.1, g can be extended to a continuous functiong ˆ: X [0, 1]. The composition f = γ gˆ is a → ◦ continuous map X Y , which is locally constant, and hence trivially holomorphic, → on a neighbourhood of x , x and has f(x ) = y and f(x) = y. { 0 } 0 0 Suppose f is homotopic rel. x , x to a holomorphic map h: X Y , so h(x ) = { 0 } → 0 y0 and h(x) = y. Since D is simply connected, the restriction of h to D admits a lifting ˜ ˜ 1 by p to a holomorphic function h: D D with h(x0) = 0. Thus, h φ− : D D → 1 1 ◦ → is holomorphic, takes 0 to 0 and has h˜ φ− (w) p− (y). But this is absurd by the 1 ◦ ∈ Schwarz-Pick lemma, since

˜ 1 ˜ 1 h φ− (w) h φ− (0) ˜ 1 w 0 ◦ − ◦ = h φ− (w) > w = − , 1 h˜ φ 1(w)h˜ φ 1(0) ◦ | | 1 w 0 − ◦ − ◦ − − · 1 recalling that w < min z : z p− (y) . | | {| | ∈ } Let V0 and V be disks about y0 and y respectively, such that p admits local 1 inverses ψ : V X, ψ (y ) = 0, and ψ : V X, ψ (y) = z, for each z p− (y). 0 0 → 0 0 z → z ∈ Let 0 > 0. By Lemma 3.2.5, there are neighbourhoods A V of y and A V of 0 ⊂ 0 0 ⊂ y such that ψ (y ) ψ (y0 ) < 0 and ψ (y) ψ (y0) < 0 for all y0 A , y0 A | 0 0 − 0 0 | | z − z | 0 ∈ 0 ∈ and ψ . Now pick > 0 suﬃciently small that B (y ) = y0 Y : d(y , y0 ) < is z 0 { 0 ∈ 0 0 } contained in A and B (y) is contained in A. Suppose h: X Y is a holomorphic 0 →

52 map with sup d(f, h) < . So h(x ) = y0 and h(x) = y0, for some y0 A and y0 A. T 0 0 0 ∈ 0 ∈ Then we may lift the restriction of h to D to a holomorphic function h˜ : D D with 1 → 1 h˜(x ) = z0 , for some z0 p− (y0 ) satisfying z0 < 0, and such that there is z p− (y) 0 0 0 ∈ 0 | 0| ∈ with z z0 < 0, where z0 = h˜(x). | − | ˜ 1 ˜ 1 ˜ 1 Now h φ− : D D is holomorphic and has h φ− (0) = z0 and h φ− (w) = z0. ◦ → ◦ ◦ Also,

˜ 1 ˜ 1 h φ− (w) h φ− (0) z0 z0 z0 0 z 20 ◦ − ◦ = | − | > | | − > | | − , 1 h˜ φ 1(w)h˜ φ 1(0) 1 z0z0 1 + z0 z0 1 + 0 − ◦ − ◦ − | − | | || | where we have used the inequalities

z0 z0 z0 z0 > z0 0, | − 0| ≥ || | − | 0|| | | −

1 z0z0 1 + z0 z0 < 1 + 0, − 0 ≤ | || 0| and z0 z z z0 > z 0. | | ≥ | | − | − | | | −

1 z 20 w + C 20 Let C = min ζ : ζ p− (y) w > 0. Then | | − | | − . Now if {| | ∈ } − | | 1 + 0 ≥ 1 + 0 w + C 20 1 we choose 0 suﬃciently small that | | − > w , then h˜ φ− contradicts the 1 + 0 | | ◦ Schwarz-Pick lemma. Furthermore, the choice of and 0 were independent of h, so there is no holomorphic map within distance of f on T .

The fundamental concept of hyperbolicity theory is the Kobayashi semi-distance, which can be deﬁned on any complex manifold. A complex manifold is called hyperbolic if its Kobayashi semi-distance is in fact a distance; for Riemann surfaces this agrees with the deﬁnition of being covered by D. It is a standard result that holomorphic functions are distance decreasing with respect to the Kobayashi semi-distance. The- orem 3.2.6 is an immediate consequence, since we need merely pick points y0 and y that have Kobayashi distance in Y bigger than that between some points x0 and x in X. A thorough discussion of hyperbolicity theory can be found in [14]. However, we have chosen to avoid the machinery of hyperbolicity theory and instead just use the elementary Schwarz-Pick lemma in the proof of our result.

Corollary 3.2.7. Let Y be a Riemann surface covered by D and X be an arbitrary Riemann surface. Then the pair (X,Y ) does not satisfy BOPI or BOPJI. Furthermore, if X is non-compact, then the pair does not satisfy BOPA either.

Proof. That the pair fails the basic Oka property with interpolation or jet interpolation is immediate from Theorem 3.2.6. If X is non-compact and T X is a two-point subset ⊂ of X, then the union K of disjoint closed disks around each point in T is a compact set

53 in X. Since the complement is trivially connected, and hence not relatively compact, by Theorem 2.8.5, K is holomorphically convex. That the pair (X,Y ) does not satisfy the basic Oka property with approximation then follows from Theorem 3.2.6 by choosing suﬃciently small closed disks about the points in T to get a compact, holomorphically convex subset K of X on which f is holomorphic. Hence f satisﬁes the assumptions of the basic Oka property with approximation, but there exists > 0 for which f is not homotopic to any holomorphic map within distance of f on K.

Starting from Winkelmann’s result on the pairs of Riemann surfaces that satisfy BOP, we can now give the precise pairs of Riemann surfaces that satisfy the stronger Oka properties.

Theorem 3.2.8 (Winkelmann). The pairs of Riemann surfaces (M,N) for which every continuous map from M to N is homotopic to a holomorphic map are precisely:

(i) M or N is biholomorphic to C or the unit disk D.

(ii) M is biholomorphic to P1 and N is not.

(iii) M is non-compact and N is biholomorphic to P1, C∗ or a torus. S (iv) N is biholomorphic to the punctured disk D∗ = D 0 and M = Mf i I Di \{ } \ ∈ where Mf is a compact Riemann surface, I is ﬁnite and non-empty, and for i I, Di Mf are pairwise disjoint, closed subsets, biholomorphic to closed disks ∈ ⊂ of radii strictly larger than zero.

Proof. See [26, Theorem 1].

Note that M in (iv) is non-compact.

Theorem 3.2.9. Let (M,N) be a pair of Riemann surfaces. If M is non-compact and N is elliptic, then (M,N) satisﬁes BOPAI and BOPAJI. If M is compact and N is biholomorphic to C or D, or M is biholomorphic to P1 and N is not, then (M,N) satisﬁes BOPA, but not BOPI or BOPJI. All other pairs fail to satisfy BOPA, BOPI and BOPJI.

Proof. By Theorems 3.1.11 and 3.1.12, for non-compact M and elliptic N, the pair (M,N) satisﬁes BOPAI and BOPAJI. The basic Oka property is a special case of BOPI, BOPJI and BOPA, so by Theorem 3.2.8, the only other pairs we need to consider are the following:

(i) M is biholomorphic to C or D and N is hyperbolic.

54 (ii) M is compact and N is biholomorphic to C.

(iii) N is biholomorphic to D.

(iv) M is biholomorphic to P1 and N is not. S (v) N is biholomorphic to D∗ and M = Mf i I Di where Mf is a compact Riemann \ ∈ surface, I is ﬁnite and non-empty, and for i I, Di Mf are pairwise disjoint, ∈ ⊂ closed subsets, biholomorphic to closed disks of radii strictly larger than zero.

Firstly we note that the pairs with M compact trivially satisfy BOPA. For if K is a non-empty compact subset of a compact Riemann surface M, then the holomorphically convex hull of K is all of M since the only holomorphic functions are constant. Hence the only non-empty holomorphically convex compact subset is M itself, from which it is immediate that BOPA is satisﬁed. Next we see that the pairs with M compact do not satisfy BOPI or BOPJI. The pairs in (ii) do not satisfy BOPI or BOPJI since compact Riemann surfaces do not admit any non-constant holomorphic functions into C. The pairs in (iii) with M compact do not satisfy BOPI or BOPJI by Corollary 3.2.7. For the pairs in (iv), if N is not biholomorphic to P1 then it is covered by C or D. If it is covered by D we have the result. If it is covered by C, then it is easily seen that there are no non- constant holomorphic maps P1 N, for any such map can be lifted to a non-constant → holomorphic function P1 C since P1 is simply connected. Hence the pairs in (iv) do → not satisfy BOPI or BOPJI. Lastly, we see by Corollary 3.2.7 that the pairs in (i), (v), and (iii) when M is non-compact, do not satisfy BOPA, BOPI or BOPJI, noting that D∗ is covered by D. In particular, the form of M in (v) is not important.

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