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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

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

Acknowledgements

I would like to sincerely thank my supervisor, Finnur Larusson. 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

K = x ∈ Ω: |f(x)| ≤ supK |f(z)| for all f ∈ O(Ω)

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 K 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(0x) = x and s|Ex is a

submersion at 0x for all x ∈ X. A complex manifold X is elliptic if it admits a

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 O(X, Y ) → C(X, Y ) is a weak homotopy equivalence, that is, the induced

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 com-

plex manifold X satisfies the basic Oka property with approximation and interpolation

(BOPAI) if whenever K is a holomorphically convex, compact subset of a Stein man-

ifold 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. Forstneric, 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. Larusson has 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.

(iv) N is biholomorphic to the punctured disk D∗ = D \ 0 and M = M \⋃i∈I Di

where M is a compact Riemann surface, I is finite and non-empty, and for i ∈ I,

Di ⊂ M 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 dominat-

ing 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 hyperbolic-

ity: 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),(ψ f φ−1

)(i)(0) =

(ψ g φ−1

)(i)(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 sub-

complex A and Hn+1(W,A; πnX) = 0 for all n ≥ 1, then every continuous map A→ X

can be extended to a continuous map W → X.

Given a path γ ∈ π1(X, x0) we can define an automorphism of πn(X, x0). The

image of a class [f ] ∈ πn(X, x0) has a representative given by shrinking the domain of

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 approx-

imated 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 ∪ 0is an assignment of a non-negative integer to each point in T , then, if for each a ∈ Twe are given an na-jet σa ∈ Jna

a (S), there is a holomorphic function f : S → C with

Jnaa f = σa for each a ∈ T .

The material of Chapter 3 strengthens Winkelmann’s result to determine pre-

cisely 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 na-jet of f fixed at a for each

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 Forstneric’s and 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 ap-

proach. 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 ex-

istence 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

Hn+1(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 ε > 0 such that there is

no holomorphic map within distance ε 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 Mobius transformations. However, the

Lie group of Mobius transformations 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

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)

to denote the nth 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

attaching n-cells via maps ∂Dn → Xn−1. That is, let Dαn be a collection of n-cells

and for each Dαn take a map ∂Dα

n → 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 =⋃nXn with the topology in which U ⊂ X is open if U ∩Xn

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 homo-

topy 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 t Y/ ∼, where ∼ is the equivalence relation

identifying x0 and y0.

Theorem 2.1.2. Let X be a connected 1-dimensional CW-complex. Then X is homo-

topy 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

· · · −→ Hn+1(Xn+1, Xn)∂n+1−−−→ Hn(Xn, Xn−1)

∂n−−→ Hn−1(Xn−1, Xn−2) −→ · · · ,

and moreover Hn(X) is equal to the nth homology group of this complex.


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

· · · −→ Hn−1(Xn−1, Xn−2)dn−1−−−→ Hn(Xn, Xn−1)

dn−−→ Hn+1(Xn+1, Xn) −→ · · · ,

and moreover Hn(X) is equal to the nth cohomology group of this complex.


Since the N+1-skeleton of an N -dimensional CW complex is empty, we immedi-

ately get the following corollary.

Corollary 2.1.5. Let X be an N-dimensional CW complex. Then Hn(X) = Hn(X) =

0 for all n > N .

12

Theorem 2.1.6 (Universal coefficients theorem). If C is a chain complex of free

abelian groups with homology groups Hn(C), and Hn(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

0 −→ Ext(Hn−1(C), G) −→ Hn(C;G) −→ Hom(Hn(C), G) −→ 0

that splits.


A path γ : [0, 1] → X from x0 to x1 defines a homomorphism from πn(X, x0) to

πn(X, x1). For n > 1, and a given class σ ∈ πn(X, x0) with representative f : [0, 1]n →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

radial segment issuing from (12, . . . , 1

2) between the smaller cube and ∂[0, 1]n. Explicitly,

on the cube [14, 3

4]n, set γf(x) = f(2x− 1

2(1, . . . , 1)). Now, for each x ∈ [0, 1]n \ [1

4, 3

4]n,

let lx be the segment between ∂[14, 3

4]n and ∂[0, 1]n of the line issuing from (1

2, . . . 1

2) in

the direction of x and px : [0, 1] → X be a parametrisation of lx starting on ∂[14, 3

4]n

and with constant derivative. Then along lx let γf(px(t)) = γ(t).

The map [f ] 7→ [γf ] is discussed in more detail in [11, p. 341–342]. The following

lemma summarises the results.

Lemma 2.1.7. The map πn(X, x0) → πn(X, x1), [f ] 7→ [γf ], is well defined and an

isomorphism. Moreover, if γ1 and γ2 are homotopic rel. 0, 1, then they define the

same isomorphism.

Definition 2.1.8. The assignment π1(X, x0) → Autπn(X, x0) which takes [γ] ∈π1(X, x0) to the automorphism [f ] 7→ [γf ] of πn(X, x0) is called the action of π1(X, x0)

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 con-

tinuous 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; πnX) = 0 for all n ≥ 1, then every continuous map

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; πnX) for each n ≥ 1. However, since we will not refer

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 ⊂ Care open. Two charts φ : U1 → V1 and ψ : U2 → V2 are said to be holomorphically

compatible if the composition φ ψ−1 : ψ(U1 ∩ U2) → φ(U1 ∩ U2) is biholomorphic in

the usual sense. A complex atlas on X is a collection of charts φα : Uα → Vα that

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 ′ → V ′ on Y, the

composition ψ f φ−1 : φ(U ∩ f−1(U ′))→ 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, limx→p|f(x)| =∞.

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 O the sheaf of holomorphic

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 bi-

holomorphism.

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 γ1, γ2 : [0, 1]→ X with γ1(1) = γ2(0), let γ1 ∗ γ2 be the

product path defined by

γ1 ∗ γ2(t) =

γ1(2t) if t ≤ 1

2

γ2(2t− 1) if t ≥ 12.

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) = y0 for some point y0 ∈ Y . Let s ∈ S and take a loop

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) = y0.

For each r ∈ [0, 1], let βr be the path that G(0, ·) traces from f(s) to G(0, r), that is,

βr : [0, 1]→ Y, βr(t) = G(0, rt).

Then for each r ∈ [0, 1], βr ∗G(·, r) ∗ β−1r is a loop at f(s), so we get a map

G : [0, 1]× [0, 1]→ Y, G(·, r) = βr ∗G(·, r) ∗ β−1r

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

relation on⊔U3p F (U), where the disjoint union is taken over all neighbourhoods of p.

Let U and V be 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

the set of germs⊔U3p F (U)/ ∼ the stalk of F at p, denoted Fp. If Y is a Riemann

surace, we will denote by Op(X, Y ) the set of germs at p of holomorphic functions

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 ∈ Op(X, Y ) agree to order k at p if f(p) = g(p) and for every chart φ on X

16

centred at p and ψ on Y centred at f(p), the holomorphic function germs ψ f φ−1

and ψ g φ−1 agree to order k at φ(p) = 0 in the usual sense. That is,(ψ f φ−1

)(i)(0) =

(ψ g φ−1

)(i)(0) for i = 1, . . . , k.

This defines an equivalence relation ∼k on Op(X, Y ) for each k. The equivalence class

of f ∈ Op(X, Y ) with respect to ∼k is denoted Jkp f and is called the k-jet of f at p.

The set of equivalence classes is denoted Jkp (X, Y ) (or just Jkp (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 ∈ Uwe 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 ∈ Op(X, Y ) with f(p) =

g(p). Then f and g agree to order k at p if for some charts φ on X centred at p and

ψ on Y centred at f(p), the germs ψ f φ−1 and ψ g φ−1 agree to order k at 0.

Proof. For k = 0 this is trivial. So suppose k > 0, φ′ is another chart on X centred at

p and ψ′ is another chart on Y centred at f(p). Then consider(ψ′ f φ′−1

)′(0) =

(ψ′ ψ−1 ψ f φ−1 φ φ′−1

)′(0)

=(ψ′ ψ−1

)′(0) ·

(ψ f φ−1

)′(0) ·

(φ φ′−1

)′(0)

=(ψ′ g φ′−1

)′(0).

Now consider the higher derivatives. It is known that for holomorphic functions

F,G : C→ C,(∂

∂z

)i(F G)(p) =

i∑l=1

(∂F

∂z

)l(G(p)) · Pi,l

(G(1)(p), . . . , G(j−l+1)(p)

),

where the Pj,l are polynomials. This is known as Faa di 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

by induction that if the ith-order derivatives of ψ f φ−1 and ψ g φ−1 agree at 0 for

i = 0, . . . , k, then so do the ith-order derivatives of ψ′ f φ′−1 and ψ′ g φ′−1.

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 Jkp f = Jkp g for some p ∈ X and k ≥ 0, then Jkp (h f) =

Jkp (h g).

Proof. Let ξ be a chart on Z centred at f(p). Then replace ψ′ 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 ∈ Op(X). Then Jkp f =

Jkp g if and only if f − g has a zero of order k + 1 at p, that is, (f − g)(p) = 0 and

Jkp (f − g) = 0.

Lemma 2.4.5. Let X be a Riemann surface and f, g, h : X → C be holomorphic

functions. If Jkp f = Jkp g for some p ∈ X and k ≥ 0, then Jkp (hf) = Jkp (hg).

Proof. By the linearity of jets of maps into C, we need only consider the case where g

is identically zero. Let φ be a chart on X centred at p. Since Jkp f = 0,(f φ−1

)(i)(0) = 0 for i = 0, . . . , k.

Now,

(hf φ−1

)(i)(0) =

(∂

∂z

)i (h φ−1

)·(f φ−1

)(0)

=i∑l=0

(i

l

)(∂

∂z

)i−lh φ−1(0) ·

(∂

∂z

)lf φ−1(0)

= 0 for i = 0, . . . , k.

Hence Jkp (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.


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.


18

Corollary 2.5.3. Let S be a non-compact Riemann surface. Then H1(S,O) = 0.

Proof. Consider the following short exact sequence of sheaves

0 −→ O −→ E∂−−→ E 0,1 −→ 0.

That the sequence is exact at E 0,1 follows from the Dolbeault lemma for C [3, Theorem

13.2]. Namely, if p ∈ S and ω ∈ E (0,1)p is of the form ω = g dz in coordinates (U, z)

at p, then there is f ∈ Ep(U) such that ∂f/∂z = g and hence ∂f = ω on U . Since

H1(S,E ) vanishes, we have

H1(S,O) = E 0,1(S)/∂E (S).

It follows that H1(S,O) = 0 if and only if E 0,1(S) = ∂E (S).

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 ,O) vanish for every open cover U of a non-compact Riemann surface.

There are several results on the flexibility of holomorphic and meromorphic func-

tions that are consequences.

Let S be a Riemann surface. A Mittag-Leffler distribution on S is an open cover

U = Uii∈I of S and a collection µ = fii∈I of meromorphic functions fi : Ui → Csuch that fi − fj is holomorphic on Ui ∩ Uj for all i, j. By a solution to µ we mean a

meromorphic function f ∈M (S) such that fi−f is holomorphic on Ui for all i, that is,

f has the same principal parts as fi on Ui for all i. The differences µij = fi−fj define a

1-cocycle δµ ∈ Z1(S,O). If S is non-compact, then δµ must be trivial. Let η = gii∈Ibe a splitting of δµ, so µij = fi − fj = gi − gj on Ui ∩ Uj. That is, fi − gi = fj − gjon Ui ∩ Uj. We get a well-defined function g ∈ M (S) given by g|Ui = fi − gi. Then

fi − g = gi is holomorphic on Ui and g is a solution to µ.

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 7→ ordx f . Here ordx f is the order of f at x, by which we mean

ordx f =

0 if f is holomorphic and non-zero at x,

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.


For a divisor D on X, the collection OD(U) = f ∈ O(U) : (f) ≥ −D on U, for

each open U ⊂ X, gives a sheaf OD on X. By (f) ≥ −D, we mean (f)(x) ≥ −D(x)

for every x ∈ U .

Corollary 2.5.6. For a non-compact Riemann surface S, H1(S,OD) = 0 for any

divisor D.

Proof. Let g be a meromorphic function with divisor (g) = −D. There is an isomor-

phism of sheaves O → OD given by f 7→ gf .

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 ordx ω = ordx f . It is easy to

see that this is independent of the choice of coordinates. The divisor of ω is given by

(w) : X → Z, x 7→ ordx ω.

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 ∈ O(S) with f |T = c.


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

integer to each point in T . Suppose for each a ∈ T we are given an na-jet σa ∈ Jnaa (S).

Then there is a holomorphic function f : S → C such that for each a ∈ T , Jnaa f = σa.

Proof. Let Uaa∈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 O we will use the sheaf OD, where D is the divisor

that agrees with −n on T and is 0 on S \ T . Let U = Ua, S \ T : a ∈ T and let

µ = fa, f0 : a ∈ T where f0 is the constant function 0 on S \ T . For a, b ∈ T , the

intersection Ua ∩ Ub is empty and Ua ∩ S \ T does not intersect T , so the differences

fa − f0 = fa on Ua ∩ S \ T define a 1-cocycle δµ with respect to the sheaf OD. Since

H1(X,OD) = 0, δµ splits, giving a family ga : Ua → C and g : X \ T → C of sections

of OD such that

ga − g = fa − f0 = fa on Ua ∩X \ T.Then ga − fa = g on Ua ∩ X \ T , and so these piece together to give a well-defined

holomorphic function f : X → C with f = ga − fa on Ua and f = g on X \ T . For

each a, ga ∈ OD(Ua) and so vanishes at a to order at least na. Hence Jnaa f = Jna

a fa as

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∫σ

ω = c(σ) for each σ ∈ π1(S).


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

K = x ∈ X : |f(x)| ≤ supK|f | for all f ∈ O(X).

We say that K is holomorphically convex if K = K.

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 6= y, there is a holomorphic function f ∈ O(X)

with f(x) 6= f(y).

(ii) For any point x ∈ X there is a holomorphic function f ∈ O(X) with dxf 6= 0.

(iii) K 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 K is not, that is, there is a sequence of points (an)n∈N in K

with no limit point in K. Since K is closed, (an) has no limit point in S. By Theorem

2.5.8, there is f ∈ O(X) with limn→∞|f(an)| = ∞. But this contradicts an ∈ K for

all n. Compactness and sequential compactness are equivalent in a Riemann surface,

so K 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 ∈ ER(X), where

ER denotes the sheaf of real valued differentiable functions. The Hessian of f at a

critical point p0 ∈ X with respect to coordinates (φ, U) at p0 is the n× n matrix

Hφ(p0) =

(∂2(f φ−1)

∂xi∂xj

)i,j

(φ(p0)).

Here by a critical point we mean a point p0 ∈ X with dp0f the zero map.

22

We would like to know the dependence of Hf (p0) on the chosen chart. Let (ψ, V )

be another chart on X with p0 ∈ V . Then we can write the Hessian as

Hφ(p0) =

(∂2(f φ−1)

∂xi∂xj(φ(p0))

)i,j

=

(∂2(f ψ−1 ψ φ−1)

∂xi∂xj(φ(p0))

)i,j

=

([∑k,l

∂2(f ψ−1)

∂(ψ φ−1)k ∂(ψ φ−1)l· ∂(ψ φ−1)k

∂xi· ∂(ψ φ−1)l

∂xj

+∑k

∂(f ψ−1)

∂(ψ φ−1)k· ∂

2(ψ φ−1)k∂xi∂xj

](φ(p0))

)i,j

(∗)

=

(∑k,l

∂2(f ψ−1)

∂xk∂xl(ψ(p0)) · ∂(ψ φ−1)k

∂xi(φ(p0)) · ∂(ψ φ−1)l

∂xj(φ(p0))

)i,j

,

noting that the second term in (∗) is zero at a critical point of f . Now let

Jψ,φ(p0) =

(∂(ψ φ−1)i

∂xj

)i,j

(φ(p0)).

Then from the above it is easy to see that

Hφ(p0) = JTψ,φ(p0)Hψ(p0)Jψ,φ(p0).

Since ψ φ−1 is a diffeomorphism its Jacobian Jψ,φ is invertible. It follows that the

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 v1, . . . , vn ∈ Tp0X be the eigenvectors of Hφ corresponding to positive eigenvalues.

Then for v ∈ spanv1, . . . , vn, v 6= 0, we have (Jψ,φv)THψ(Jψ,φv) = vTHφv > 0. Hence,

the dimension of the largest subspace of Tp0X 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 num-

ber 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 dimM + dimA+ dim KerHψ > dimTp0X,

which is absurd since A, M and KerHψ 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 p0 ∈ X is the number of negative eigenvalues of the

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

(U, φ) at p0 such that if we write φ = (φ1, . . . , φn), then f = f(p0)−γ∑i=1

φi2 +

n∑i=γ+1

φi2.

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) ≤ care 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 sub-

manifold of C3, the map f : C3 → R, z 7→ ‖z − a‖2, restricts to a Morse exhaustion

on S for generic a ∈ C3. Here, ‖·‖ is the Euclidean norm on C3.

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 count-

able 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

∆φ (f s) =∂2(f s φ−1)

∂x2+∂2(f s φ−1)

∂y2

=∑k

∂f

∂xk (s φ−1) ·

(∂2(s φ−1)k

∂x2+∂2(s φ−1)k

∂y2

)+∑k,l

∂2f

∂xk∂xl (s φ−1) ·

(∂(s φ−1)k

∂x· ∂(s φ−1)l

∂x

+∂(s φ−1)k

∂y· ∂(s φ−1)l

∂y

).

However s φ−1 is holomorphic, so by the Cauchy-Riemann equations,

∂2(s φ−1)k∂y2

= −∂2(s φ−1)k∂x2

for each k. Hence the first term above vanishes. Also,

∂2f

∂xk∂xl= 2δk,l,

so

∆φ (f s) = 2∑k

((∂(s φ−1)k

∂x

)2

+

(∂(s φ−1)k

∂y

)2)≥ 0.

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

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, Cj is open

for all j. Now let U be a relatively compact neighbourhood of B, then Cj ∩ U 6= ∅for all j. For otherwise if Cj ⊂ S \ U for some j, then Cj ⊂ S \ U ⊂ S \ B and

hence Cj would also be a connected component of S \ B. But this is only possible if

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

U = U∪⋃j∈J0 Cj is also a relatively compact neighbourhood of B, and by construction

S \ U 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 U

and so are relatively compact. Hence h(B) ⊂ U and S \ h(B) = Cj1 ∪ Cj2 ∪ · · · ∪ Cjn .

By (ii), h(Y ) ⊂ h(B) ⊂ U , 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 C1, . . . , Cn of S \K such that

S \ h(K) = C1 ∪ · · · ∪ Cn. Let U be a relatively compact neighbourhood of h(K) and

K ′ = U . Then, similarly, S \ h(K ′) = C ′1 ∪ · · · ∪ C ′m for some non-relatively compact

components C ′1, . . . , C′m of S \K ′. For each i = 1, . . . , n, there is j with C ′j ∩ Ci 6= ∅

and hence C ′j ⊂ Ci since C ′j and Ci are both connected and ∂C ′j ⊂ ∂U ⊂ C1∪ · · · ∪Cn.

Let Ai ⊂ Ci be the union of the interiors of all connected, non-compact, closed subsets

of S that are contained in Ci. Then Ai contains C ′j, and so is non-empty. We aim

to show that Ai = Ci. Let p ∈ Ci ∩ Ai. Choose a connected neighbourhood V of p

with V ⊂ Ci. Since p ∈ Ai, there is q ∈ Ai ∩ V and a connected, non-compact, closed

subset B of S contained in Ci with q ∈ B. The union C = V ∪ B is a connected,

non-compact, closed subset of S contained in Ci and hence V ⊂ Ai. Thus Ai is both

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 Ci ⊂ S \ h(K),

i ∈ I, whose interiors cover ∂U . Since ∂U is compact we can find a finite subcover

Ci0 , . . . , Cin . The set W = U ∩ (S \ (Ci0∪· · ·∪Cin)) is a neighbourhood of K contained

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 ∈ K, 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 V1 be an open disk centred at x such that V1

does not intersect K. Then there is a Runge neighbourhood V2 of K which does not

intersect V1. Let A = V1 ∪ V2. If we let the connected components of the complement

of V2 be Ui, i ∈ I, then the complement of A in S is⋃i Ui \ V1. Since V2 is Runge

each Ui is not relatively compact. The disk V1 lies in one of the Ui, say Ui0 , and the

set Ui0 \ V1 must be connected (it is path connected since it contains the boundary of

V1). It is easy to see that Ui0 \ V1 is not relatively compact. Hence A is Runge.

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 12.

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.


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 com-

pact 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 ω0|U is a nowhere vanishing holomorphic 1-form on U . Let ω be a

holomorphic 1-form on U . Then ω can be written ω = fω0|U for some holomorphic

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 approxi-

mation 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 func-

tion on a neighbourhood of K restricts to a holomorphic function on a Runge neigh-

bourhood. 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.


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 H1(V )→ H1(S) induced by the inclusion map is injective.

Proof. Let [c0] ∈ H1(V ) be a 1-cycle whose image in H1(S) is 0. That is, there is a

2-chain b ∈ C2(S) with ∂b = c0. We want to show that c0 is also a boundary in H1(V ).

Given a holomorphic 1-form ω ∈ Ω(S) we have∫c0

ω =

∫∂b

ω =

∫b

dω = 0.

By Corollary 2.8.7, any holomorphic form ω ∈ Ω(V ) can be approximated by holomor-

phic forms in Ω(S). Hence∫c0ω = 0 for all ω ∈ Ω(V ).

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

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) 6= 0 for

any nonzero σ in H1(V ). However,∫c0ω = 0 for all ω ∈ Ω(V ), so c0 must vanish in

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 sub-

harmonic 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.


2.9 Elliptic Riemann surfaces

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

Definition 2.9.1. LetX 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(0x) = x and s|Ex is a submersion at 0x for all x ∈ X. A Riemann surface 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,

then the fibres Ex are biholomorphic to Cn for some n. It follows that if s : E → X is

a holomorphic map, then the restriction s|Ex is constant and hence not a submersion

at 0x for any x ∈ X.

Now we need to prove the positive results. The map

C× C→ C, (z, w) 7→ z + w,

is a dominating spray over C, defined on the total space of the trivial line bundle. The

map

C× C∗ → C∗, (z, w) 7→ wez,

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) 7→ z + w + Γ.

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|Ew+Γ

is a submersion at (0, w + Γ). Finally let G be the complex Lie group of Mobius

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) 7→ exp(z)w.

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 PGL(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 Rado. 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].

Definition 2.10.1. Let v0, . . . , vn ∈ Rn be n+1 points such that the difference vectors

v0 − v1, . . . , v0 − vn are linearly independent. This is sometimes known as v0, . . . , vnbeing in general position. An n-simplex in Rn is the convex hull

t0v0 + · · ·+ tnvn : t0, . . . , tn ∈ [0, 1] and t0 + · · ·+ tn = 1

of v0, . . . , vn. If e1, . . . , en are the standard basis vectors of Rn, then the convex hull

of 0, e1, . . . , en is known as the standard n-simplex and is denoted ∆n.

By a face of an n-simplex we mean the convex hull of a non-empty subset of

v0, . . . , vn. If the subset consists of two vectors, the corresponding face is called an

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


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 |K0| of ∂A. Now, X \ A and

A are manifolds with boundary, so there are triangulations φ1 : |K1| → X \ A and

φ2 : |K2| → A, that are extensions of |K0|. The two complexes K1 and K2 glue together

along K0 to get a complex K, which is a triangulation of X by the homeomorphism

|K| → X defined by gluing φ1 and φ2 together along |K0|.Lemma 2.10.7. Let T be a discrete subset of a non-compact Riemann surface S.

Then there exists a cover Uj of S by contractible coordinate charts, such that each

element of T is contained in precisely one of the Uj.

Proof. For each point a ∈ T , let Ua be a contractible open neighbourhood of a that

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 6= b. For each

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 ∪ Ua : a ∈ T is an open cover of S with the required properties.

2.11 Compact-open topology

Let X and Y be topological manifolds and let C(X, Y ) be the space of continuous maps

X → Y . The compact-open topology on C(X, Y ) is the topology generated by subsets

of the form

V (K,U) = f ∈ C(X, Y ) : f(K) ⊂ U,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.


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 holomor-

phically 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 na- jets of f fixed at a for each

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 defor-

mation 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 = Uj be an open cover of S by coordinate charts homeomorphic to the unit

disc in C. Then on each Uj there is a continuous logarithm λj : Uj → C of f such that

e2πiλj = f on Uj. Now let

ξjk = λj − λk on Uj ∩ Uk,

and note that ξjk : Uj∩Uk → Z is locally constant. Clearly ξ = (ξkj) ∈ Z1(U ,Z), where

Z denotes the sheaf of locally constant functions with values in Z. Since Z1(U ,Z) ⊂Z1(U ,O) and H1(U ,O) = 0, the cocycle ξ splits with respect to the sheaf O. Hence

there is (ηj) ∈ C0(U ,O) with

ξjk = ηj − ηk on Uj ∩ Uk.

This gives a well-defined holomorphic function g ∈ O∗(S) with g = e2πiηj on Uj. We

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)(λj − λk) + t(ηj − ηk) = ξjk on Uj ∩ Uk.

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

f : C∗ × C∗ → C/Γ, (z, w) 7→ logz

2πiγ1 +

logw

2πiγ2 + Γ,

g : C/Γ→ C∗ × C∗, aγ1 + bγ2 + Γ 7→ (e2πia, e2πib).

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

Hi(·, 0) = (g h)i,

Hi(·, 1) = ui.

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 homo-

topy 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

for which Hn(A) = 0 for n ≥ 2, i∗ : H1(A) → H1(S) is injective and i∗ : H1(S) →H1(A) is surjective, where i : A → S is the inclusion. Then

Hn(S × 0, 1 ∪ A× [0, 1]) = H2(S × 0, 1 ∪ A× [0, 1]) = 0 for all n ≥ 2.

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, H1(U ∩ V ) = H1(A) since U ∩ V deformation-retracts onto A. Also, U and

V both deformation-retract onto S. By Theorem 2.7.6, S has the homotopy type of a

1-dimensional CW-complex, so Hn(S) = 0 for n ≥ 2. For n = 2, the Mayer-Vietoris

sequence becomes

0 −→ H2(S × 0, 1 ∪ A× [0, 1]) −→ H1(A) −→ H1(S)⊕ H1(S) −→ · · · .

Hence, H2(S × 0, 1 ∪A× [0, 1]) = 0, since H1(A)→ H1(S)⊕H1(S) is injective. For

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 H1(A) → H1(S) and H1(S) → H1(A)

are mentioned, we mean the maps induced by the inclusion i : A → S. Note that

if H1(A) = 0, then, trivially, H1(A) → H1(S) is injective and H1(S) → H1(A) is

surjective.

Lemma 3.1.4. Let S be a non-compact Riemann surface and A ⊂ S be a closed

submanifold-with-boundary with H1(A) → H1(S) injective, H1(S) → H1(A) surjective

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

Hn+1(S × [0, 1], S ×0, 1 ∪A× [0, 1]; πnP1) for n ≥ 1. For n = 1, π1P1 = 0, so clearly

H2(S × [0, 1], S × 0, 1 ∪ A× [0, 1]; π1P1) = 0.

For n = 2, note that π2P1 = Z, so we need H3(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, Hn(S) = 0, since S is a 2-dimensional CW complex. Also, H2(A) = 0 and A

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 Hn(S × [0, 1], S × 0, 1 ∪ A × [0, 1]) = 0 for n ≥ 3. By the

universal coefficients theorem (Theorem 2.1.6), we get

Hn+1(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 ∈ O(X) with

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∈Jof 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

on each Uj to get continuous logarithms λj : Uj → C such that e2πiλj = f on Uj. The

differences

ξjk = λj − λk on Uj ∩ Ukdefine a cocycle ξ = ξjk ∈ Z1(U ,O). Hence there is a 0-cochain ηj ∈ C0(U ,O)

that splits ξ. Since Uj ∩ Uk ∩ T = ∅ for all j, k ∈ J , j 6= k, there is a well-defined

function

u : T → C, u = ηj − λj on Uj ∩ T .

39

By Weierstrass’ theorem, there is a holomorphic function h ∈ O(S) with h|T = u.

Now define a new cochain µ = µj ∈ C0(U ,O) with µj = ηj − h. Then

µj − µk = ηj − ηk = ξjk on Uj ∩ Uk,

and

µj = ηj − h = λj on T .

So there is a well-defined holomorphic function g ∈ O∗(S) given by g = e2πiµj on Uj.

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,

Ft|T = exp(2πi((1− t)λj|T + tµj|T

))= exp

(2πi((1− t)λj|T + tλj|T

))= f |T on Uj.

c) Let X be a torus. As shown in the proof of Theorem 3.1.2 there is a holo-

morphic 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 ui ∈ O∗(S), i = 1, 2, and homotopies Hi from (g f)i to ui that

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,

Ht|T = (g f)|T for all t ∈ [0, 1].

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 Ht|T = (h g f)|T = f |T for all t ∈ [0, 1].

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 Hn(T ) = H1(T ) = 0 for all n ≥ 1.

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 sur-

face and X be C, C∗, P1 or a torus. Suppose T is a discrete subset of S, n : T → N∪0is 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 na-jets at a ∈ T of

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

function g ∈ O(S) with Jnaa f = Jna

a g at each point a ∈ T . Define H : S × [0, 1] → C,

Ht = (1− t)f + tg. Clearly H is a homotopy from f to g.

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 λj is holomorphic on V ∩ Uj, and we take

V to be sufficiently small that for each i, j ∈ J , i 6= j, V ∩ Ui and V ∩ Uj are disjoint.

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 ∈ O(S) such that

at each a ∈ T , Jnaa h = Jna

a u. The rest of the proof follows through as for interpolation.

Define the cochain µ = (µj) ∈ C0(U ,O), where µj = ηj − h. This splits the cocycle

ξ and Jnaa µj = Jna

a λj at a ∈ T ∩ Uj. Let g ∈ O∗(S) be given by g = e2πiµj on Uj and

take the homotopy

F : S × [0, 1]→ C∗, Ft = exp(2πi((1− t)λj + tµj

))on Uj.

By Theorem 2.4.3, since Jnaa λj = Jna

a µj at a ∈ T , we have Jnaa f = Jna

a g.

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

a holomorphic function φ : S → C with Jnaa φ = Jna

a 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, Jnaa ψ = Jna

a h at each a ∈ T . Define the homotopy

G : S × [0, 1]→ C/Γ, G = p G,

from p h = h to p φ = ψ. Finally, since at a ∈ T , Jnaa Gt = Jna

a h for every t ∈ [0, 1],

Jnaa Gt = Jna

a h at a ∈ T for every t ∈ [0, 1]. In order to get the result for the function

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(Ua) is contained in a coordinate neighbourhood of f(a) for all a ∈ T and

there is a point p ∈ P1 such that f(Ua) ⊂ P1 \ p. This is possible, for if we take

p /∈ f(T ) then f−1(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 , Jnaa f = Jna

a g.

For each a ∈ T let Va ⊂ Ua be a closed coordinate disk containing a and let

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

homotopy Ha from ψ f φ−1 to ψ g φ−1 with fixed na-jets at φ−1(a). We construct

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, Hn(A) = H1(A) = 0 for all n ≥ 1 and A

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 H2(A) = 0, H1(A)→ H1(S) is injective and H1(S)→ H1(A) is

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

A = φ−1((−∞, 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

Runge. Now let M be a collar neighbourhood of ∂A = φ−1(C) with diffeomorphism

p : M → ∂A×[0, 1). Both ∂A×[0, 1) and ∂A×(0, 1) deformation retract onto ∂A×[12, 1).

The preimage W = p−1(∂A× [0, 12)) is a neighbourhood of ∂A and we get deformation

retracts M →M \W and M \ ∂A→M \W by pre- and postcomposition with p and

p−1. 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 H1(A) → H1(A \W )

and H1(V )→ H1(A \W ), so H1(V )→ H1(A) is an isomorphism. By Theorem 2.8.11,

the natural map H1(V )→ H1(S) is injective, so H1(A)→ H1(S) is injective. Similarly,

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

H2(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 ∈ O(V ) and

ρ ∈ M (V ), such that φ has no zeros at the poles of ρ. Then, given ε > 0, there is

δ > 0 such that if ψ ∈ O(V ) satisfies supK |φ− ψ| < δ, then d(ρφ, ρψ) < ε on K with

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 δ1 > 0

sufficiently small that if supK |φ− ψ| < δ1, then ψ has no zeros in B. This is possible

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 ε/2 of ∞ with respect to the spherical metric.

Since ψ is bounded away from zero on B, we can find a subset of B1 ⊂ B on which

ρψ is within distance ε/2 of ∞ on B1 with respect to the spherical metric. Similarly,

choose a sufficiently small neighbourhood B2 of T such that ρφ is within distance ε/2

of ∞ on B2. Then on B = B1 ∩B2, d(ρφ, ρψ) < ε.

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 δ1 of φ on K. Now on K \ B, ρ has no poles and hence is bounded as

K is compact. With respect to Euclidean distance, if supK |φ− ψ| < δ2, then

supK\B|ρ(φ− ψ)| < δ2 · sup

K\Bρ,

and hence we can easily find δ2 > 0 sufficiently small that d(ρφ, ρψ) < ε on K \ B.

Finally pick δ = minδ1, δ2.

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 ε > 0, the maps in the deformation can

be chosen to be within distance ε of f on K with respect to any metric defining the

topology on X.

Proof. a) C. Given ε > 0, by the Runge approximation theorem we can find a holomor-

phic function g : S → C such that supK |f − g| < ε. 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,

supK|Ht − f | = sup

K|tg − tf | = t sup

K|g − f | < ε on 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 exp′ = exp is bounded

on U since it is relatively compact, so let supK |g| = M and supU |exp| = N . Given

ε > 0, let δ > 0 be such that δ < ε/(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 → Cand a homotopy G : S× [0, 1]→ C from h to φ that is holomorphic on a neighbourhood

of K and has supK |Gt − h| < δ for all 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],

supK|Gt − h| = sup

K|exp Gt − exp h|

< N supK|Gt − h|

< Nδ < ε/M

since Gt and h map K into U , where |exp′| < N . Finally, we need to multiply

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

supK|(gG)t − f | < sup

Kg · ε/M = ε for all t.

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], Gt = (1− t)f + th/p.

Let V ⊂ A be a neighbourhood of K. The functions g, Ft and 1/p restricted

to V satisfy the assumptions of Lemma 3.1.8, with φ = g, ψ = Ft and ρ = 1/p. In

particular, note that if supK |g − h| < δ, then

supK|g − Ft| = sup

K|tg − th| < δ.

Hence, given ε > 0, there is δ > 0 such that if supK |g − h| < δ, then d(f, Ft/p) < ε on

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 ∈ OD(U) can be

approximated uniformly on compact subsets of U by meromorphic functions in OD(S).

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 neighbour-

hood exists by Lemma 2.8.4. Let g ∈ OD(S) be such that (g) = (f) on V , which exists

by Weierstrass’ theorem. Then f/g can be extended by Riemann removable singular-

ities 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 ∈ OD(S).

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 ε > 0, there is δ > 0 such that

if supK |h− f/g| < δ, then d(f, gh) < ε on K with respect to the spherical metric.

The following two theorems establish the strongest forms of the basic Oka prop-

erty 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 na-jets of the maps

in the deformation can be kept fixed.

Proof. a) C. By the strong Weierstrass’ theorem (Theorem 2.5.9) we can find a holo-

morphic function g : S → C with Jnaa g = Jna

a 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 na + 1.

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 ∈ OD(V ) if and only if φ is a holomorphic function on V with zeros at

each a ∈ T ∩ V of order at least na + 1. So f − g ∈ OD(U) and by Theorem 3.1.10,

given ε > 0, there is a holomorphic function p ∈ OD(S) with supK |p − (f − g)| < ε.

Then h = g + p : S → C is holomorphic and at a ∈ T we have Jnaa h = Jna

a g = Jnaa f .

Also we have supK |h− f | = supK |p− (f − g)| < ε by our choice of p.

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 Ht holomorphic on the same neighbourhood of K∪Tthat f is holomorphic on for all t ∈ [0, 1]. Since at each a ∈ T the na-jets of f and h

are the same, the na-jets of Ht are fixed on T for all t. Finally by construction,

supK|Ht − f | = sup

K|(1− t)f + th− f | = sup

Kt|h− f | < ε.

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 exp′ = exp is bounded

on U since it is relatively compact, so let supK |g| = M and supU |exp| = N . Given

ε > 0, let δ > 0 be such that δ < ε/(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

K ∪ T , suph(K) |Gt − h| < δ and Jnaa Gt = Jna

a h for all a ∈ T and t ∈ [0, 1].

The function ψ = exp φ : S → C∗ is holomorphic and at a ∈ T has Jnaa ψ = Jna

a h

by Theorem 2.4.3. We get a homotopy G = exp G : S × [0, 1]→ C∗ from h to ψ with

Gt holomorphic on a neighbourhood of K ∪ T and Jnaa Gt = Jna

a h for every a ∈ T and

t ∈ [0, 1]. By construction, for t ∈ [0, 1],

supK|Gt − h| = sup

K|exp Gt − exp h| < N sup

K|Gt − h| < Nδ < ε/M

since Gt and h map K into U , where |exp′| < N . Finally we need to multiply 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 ∪ Tfor all t. By construction

supK|(gG)t − f | < sup

Kg · ε/M = ε for all t.

At each a ∈ T , Gt agrees with h to order na for all t ∈ [0, 1], hence Jka (gG)t = Jkaf for

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

map g : S → P1 \ ∞ such that at each a ∈ T , Jnaa g = Jna

a 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 − gis in OD+(p)(U), where the divisor D is zero on S \ T and D = −n − 1 on T . By

Riemann removable singularities, (f − g)p can be extended to a holomorphic function

in OD(U). By Theorem 3.1.10, for δ > 0, there is a holomorphic function h ∈ OD(S)

with supK |h− (f − g)p| < δ.

Now letA1 ⊂ U be a holomorphically convex, compact submanifold-with-boundary

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 Ua be a

closed coordinate disk at a that does not intersect A1 and such that Ua ∩ Ub = ∅for a 6= b. Let A = A1

⋃a Ua. Then A is a closed submanifold-with-boundary.

Define a homotopy F : S × 0, 1 ∪ A × [0, 1] → P1, F (·, 0) = f , F (·, 1) = h and

Ft = (1− t)fp+ t(gp+ h) on A× [0, 1]. This is well defined, since on U , f and f − ghave 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, H1(A) → H1(S) is injective, H1(S) → H1(A) is surjective and H2(A) = 0.

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 ∈ OD(S), we have h/p ∈ OD+(p)(S),

that is, its poles are at poles of f and it has a zero of order na + 1 at each a ∈ T . By

Theorem 2.4.4, Jnaa (g + h/p) = Jna

a g = Jnaa f as required. It is easy to see that at each

a ∈ T the na-jets are fixed during the deformation. Finally, for each t ∈ [0, 1],

supK|fp− Ft| = sup

K|fp− (1− t)fp− t(gp+ h)| = sup

Kt|(f − g)p− h)| < tδ.

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 ε > 0, there is δ > 0 such that

if supK |h− (f − g)p| < δ, then d(f,Ht) < ε on K with respect to the spherical metric

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].


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)| ≤ 1.

Proof. Define a holomorphic function g on D by g(z) = f(z)/z on D \ 0 and g(0) =

f ′(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)

1− f(z1)f(z2)

∣∣∣∣∣ ≤∣∣∣∣ z1 − z2

1− z1z2

∣∣∣∣ .Proof. Fix z1 ∈ D and define automorphisms φ1 and φ2 of D by the formulas φ1(z) =z1 − z1− z1z

and φ2(z) =f(z1)− z1− f(z1)z

. The composition φ2 f φ−11 maps D to D and has

φ2 f φ−11 (0) = 0. By the Schwarz lemma,

|φ2 f φ−11 (z)| =

∣∣∣∣∣ f(z1)− f(φ−11 (z))

1− f(z1)f(φ−11 (z))

∣∣∣∣∣ ≤ |z| for all z ∈ D.

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)

1− σ(z1)σ(z2)

∣∣∣∣∣ =

∣∣∣∣ z1 − z2

1− z1z2

∣∣∣∣ .

50

Proof. Note that σ and σ−1 are holomorphic maps, so by the Schwarz-Pick lemma,∣∣∣∣ z1 − z2

1− z1z2

∣∣∣∣ =

∣∣∣∣∣ σ−1 σ(z1)− σ−1 σ(z2)

1− σ−1 σ(z1)σ−1 σ(z2)

∣∣∣∣∣ ≤∣∣∣∣∣ σ(z1)− σ(z2)

1− σ(z1)σ(z2)

∣∣∣∣∣ ≤∣∣∣∣ z1 − 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

ε > 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)| < ε.

Proof. Let z1 = ψ(x) and δ =ε(1− |z1|2)

2 + ε|z1|. Suppose y ∈ V is such that |z1 − z2| < δ,

where z2 = ψ(y). Note that |z2| < |z1|+ δ and hence

|1− z1z2| ≥ 1− |z1||z2| ≥ 1− |z1|(|z1|+ δ) > 0,

where the last inequality follows from

|z1|(|z1|+

ε(1− |z1|2)

2 + ε|z1|

)=

2|z1|2 + ε|z1|2 + ε|z1|

< 1.

Thus, ∣∣∣∣ z1 − z2

1− z1z2

∣∣∣∣ < δ

1− |z1|2 − δ|z1|=ε

2.

Then by Corollary 3.2.4, ∣∣∣∣∣ σ(z1)− σ(z2)

1− σ(z1)σ(z2)

∣∣∣∣∣ =

∣∣∣∣ z1 − z2

1− z1z2

∣∣∣∣ < ε

2,

for any covering transformation σ. So

|σ(z1)− σ(z2)| < ε

2

∣∣∣1− σ(z1)σ(z2)∣∣∣ ≤ ε

2(1 + |σ(z1)||σ(z2)|) < ε.

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

ε > 0 such that there is no holomorphic map within distance ε of f on T with respect

to d.

51

X

x0x

DY

yy′

V2 y0y′

0

V1

D

w0

Dz

0

ϕ

f

h

h

h ϕ−1

p

ψz

ψ0

z′z′0

1

Proof. Let p : D → Y be a holomorphic covering map and D ⊂ X be an open subset

of X with a biholomorphism φ : D → D. Take y ∈ Y such that 0 /∈ p−1(y) and

let w ∈ D \ 0 be such that |w| < min|z| : z ∈ p−1(y). Finally, let y0 = p(0),

x = φ−1(w) and x0 = φ−1(0). Now take a path γ : [0, 1]→ Y starting at y0 and ending

at y and let g : D0, Dx → [0, 1] be given by g(D0) = 0 and g(Dx) = 1, where D0

and Dx are closed disks in D around x0 and x respectively. By Theorem 3.2.1, g can

be extended to a continuous function g : X → [0, 1]. The composition f = γ g is a

continuous map X → Y , which is locally constant, and hence trivially holomorphic,

on a neighbourhood of x0, x and has f(x0) = y0 and f(x) = y.

Suppose f is homotopic rel. x0, x to a holomorphic map h : X → Y , so h(x0) =

y0 and h(x) = y. Since D is simply connected, the restriction of h to D admits a lifting

by p to a holomorphic function h : D → D with h(x0) = 0. Thus, h φ−1 : D → Dis holomorphic, takes 0 to 0 and has h φ−1(w) ∈ p−1(y). But this is absurd by the

Schwarz-Pick lemma, since∣∣∣∣∣ h φ−1(w)− h φ−1(0)

1− h φ−1(w)h φ−1(0)

∣∣∣∣∣ =∣∣∣h φ−1(w)

∣∣∣ > |w| = ∣∣∣∣ w − 0

1− w · 0

∣∣∣∣ ,recalling that |w| < min|z| : z ∈ p−1(y).

Let V0 and V be disks about y0 and y respectively, such that p admits local

inverses ψ0 : V0 → X, ψ0(y0) = 0, and ψz : V → X, ψz(y) = z, for each z ∈ p−1(y).

Let ε′ > 0. By Lemma 3.2.5, there are neighbourhoods A0 ⊂ V0 of y0 and A ⊂ V of

y such that |ψ0(y0) − ψ0(y′0)| < ε′ and |ψz(y) − ψz(y′)| < ε′ for all y′0 ∈ A0, y′ ∈ A

and ψz. Now pick ε > 0 sufficiently small that Bε(y0) = y′0 ∈ Y : d(y0, y′0) < ε is

contained in A0 and Bε(y) is contained in A. Suppose h : X → Y is a holomorphic

52

map with supT d(f, h) < ε. So h(x0) = y′0 and h(x) = y′, for some y′0 ∈ A0 and y′ ∈ A.

Then we may lift the restriction of h to D to a holomorphic function h : D → D with

h(x0) = z′0, for some z′0 ∈ p−1(y′0) satisfying |z′0| < ε′, and such that there is z ∈ p−1(y)

with |z − z′| < ε′, where z′ = h(x).

Now hφ−1 : D→ D is holomorphic and has hφ−1(0) = z′0 and hφ−1(w) = z′.Also, ∣∣∣∣∣ h φ−1(w)− h φ−1(0)

1− h φ−1(w)h φ−1(0)

∣∣∣∣∣ =|z′ − z′0||1− z′z′0|

>|z′| − ε′

1 + |z′||z′0|>|z| − 2ε′

1 + ε′,

where we have used the inequalities

|z′ − z′0| ≥ ||z′| − |z′0|| > |z′| − ε′,∣∣1− z′z′0∣∣ ≤ 1 + |z′||z′0| < 1 + ε′,

and

|z′| ≥ |z| − |z − z′| > |z| − ε′.

Let C = min|ζ| : ζ ∈ p−1(y)− |w| > 0. Then|z| − 2ε′

1 + ε′≥ |w|+ C − 2ε′

1 + ε′. Now if

we choose ε′ sufficiently small that|w|+ C − 2ε′

1 + ε′> |w|, then h φ−1 contradicts the

Schwarz-Pick lemma. Furthermore, the choice of ε and ε′ were independent of h, so

there is no holomorphic map within distance ε 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 ε > 0 for which f is

not homotopic to any holomorphic map within distance ε 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.

(iv) N is biholomorphic to the punctured disk D∗ = D \ 0 and M = M \⋃i∈I Di

where M is a compact Riemann surface, I is finite and non-empty, and for

i ∈ I, Di ⊂ M 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.

(v) N is biholomorphic to D∗ and M = M \⋃i∈I Di where M is a compact Riemann

surface, I is finite and non-empty, and for i ∈ I, Di ⊂ M are pairwise disjoint,

closed subsets, biholomorphic to closed disks of radii strictly larger than zero.

Firstly we note that the pairs withM compact trivially satisfy BOPA. For ifK 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.

55

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