Riemann surfaces of second kind and effective finiteness theorems

The Geometric Shafarevich Conjecture and the Theorem of de Franchis state the finiteness of the number of certain holomorphic objects on closed or punctured Riemann surfaces. The analog of these kind of theorems for Riemann surfaces of second kind is an estimate of the number of irreducible holomorphic objects up to homotopy (or isotopy, respectively). This analog can be interpreted as a quantitatve statement on the limitation for Gromov’s Oka principle. For any finite open Riemann surface X (maybe, of second kind) we give an effective upper bound for the number of irreducible holomorphic mappings up to homotopy from X to the twice punctured complex plane, and an effective upper bound for the number of irreducible holomorphic torus bundles up to isotopy on such a Riemann surface. The bound depends on a conformal invariant of the Riemann surface. If Xσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{\sigma }$$\end{document} is the σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-neighbourhood of a skeleton of an open Riemann surface with finitely generated fundamental group, then the number of irreducible holomorphic mappings up to homotopy from Xσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{\sigma }$$\end{document} to the twice punctured complex plane grows exponentially in 1σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{\sigma }$$\end{document}.


February 4, 2021 1. Introduction and statements of results
The following statement, known as Geometric Shafarevich conjecture, has been conjectured by Shafarevich [27] in the case of compact base and fibers of type (g, 0).
Theorem A (Geometric Shafarevich Conjecture.) For a given compact or punctured Riemann surface X and given non-negative numbers g and m such that 2g − 2 + m > 0 there are only finitely many locally holomorphically non-trivial holomorphic fiber bundles over X with fiber of type (g, m).
A connected closed Riemann surface (or a smooth connected closed surface) is called of type (g, m), if it has genus g and is equipped with m distinguished points. Recall that a closed Riemann surface with a finite number of points removed is called a punctured Riemann surface. The removed points are called punctures. Sometimes it is convenient to associate a punctured Riemann surface to a Riemann surface of type (g, m) by removing the distinguished points. A connected Riemann surface is of first kind, if it is a closed or a punctured Riemann surface, otherwise it is of second kind. A Riemann surface is called finite if its fundamental group is finitely generated, and open if no connected component is compact. Each finite connected open Riemann surface X is conformally equivalent to a domain (denoted again by X) on a closed Riemann surface X c such that each connected component of the complement X c \ X is either a point or a closed geometric disc [14]. A geometric disc is a topological disc that lifts to a round disc in the universal covering of the closed surface X c with respect to the standard metric on the universal covering. The closed Riemann surface X c and the embedding X ⊂ X c are unique up to conformal isomorphism. The connected components of the complement will be called holes.
Theorem A was proved by Parshin [26] in the case of compact base and fibers of type (g, 0), g ≥ 2, and by Arakelov [2] for punctured Riemann surfaces as base and fibers of type (g, 0). Imayoshi and Shiga [16] gave a proof of the quoted version using Teichmüller theory.
The statement of Theorem A "almost" contains the so called Finiteness Theorem of Sections which is also called the Geometric Mordell conjecture (see [25]), giving an important conceptional connection between geometry and number theory. For more details we refer to the surveys by C.McMullen [25] and B.Mazur [24].
There is an older finiteness theorem due to de Franchis [4] which can be considered as a special case of the Geometric Shafarevich conjecture.
Theorem B (de Franchis). For closed connected Riemann surfaces X and Y with Y of genus at least 2 there are at most finitely many non-constant holomorphic mappings from X to Y .
There is a more comprehensive Theorem in this spirit.
Theorem C (de Franchis-Severi). For a closed connected Riemann surface X there are (up to isomorphism) only finitely many non-constant holomorphic mappings f : X → Y where Y ranges over all closed Riemann surfaces of genus at least 2.
We may associate to any holomorphic mapping f : X → Y of Theorem B the bundle over X with fiber over x ∈ X equal to Y with distinguished point {f (x)}. Thus, the fibers are of type (g, 1). A holomorphic self-isomorphism of a locally holomorphically non-trivial (g, 1)bundle may lead to a new holomorphic mapping from X to Y , but there are only finitely many different holomorphic self-isomorphisms. Hence, Theorem B is a consequence of Theorem A, and Theorem B has analogs for the source X and the target Y being punctured Riemann surfaces.
The Theorems A and B do not hold literally if the base X is of second kind. If the base is a Riemann surface of second kind the problem to be considered is the finiteness of the number of irreducible isotopy classes containing holomorphic objects. In case the base is a punctured Riemann surface this is equivalent to the finiteness of the number of holomorphic objects. For more detail see sections 2 and 3.
We will prove finiteness theorems with effective estimates for Riemann surfaces of second kind. The estimates depend on a conformal invariant of the base manifold. To define the invariant we recall Ahlfors' definition of extremal length (see [1]). For an annulus A = {0 ≤ r < |z| < R ≤ ∞} (and for any open set that is conformally equivalent to A) the extremal length equals 2π For an open rectangle R = {z = x + iy : 0 < x < b, 0 < y < a } in the plane with sides parallel to the axes, and with horizontal side length b and vertical side length a the extremal length equals λ(R) = a b . For a conformal mapping ω : R → U of the rectangle R onto a domain U ⊂ C the image U is called a curvilinear rectangle, if ω extends to a continuous mapping on the closureR, and the restriction to each (closed) side of R is a homeomorphism onto its image. The images of the vertical (horizontal, respectively) sides of R are called the vertical (horizontal, respectively) curvilinear sides of the curvilinear rectangle ω(R). The extremal length of the curvilinear rectangle U equals the extremal length of R.
Let X be a connected open Riemann surface of genus g ≥ 0 with m + 1 holes, m ≥ 0, equipped with a base point q 0 . The fundamental group π 1 (X, q 0 ) of X is a free group in 2g + m generators. We choose a system of generators as follows. The Riemann surface X is conformally equivalent to an open subset of a closed Riemann surface X c of genus g ( [14]). Choose 2g generators e j,0 , j = 1, . . . , 2g, of the fundamental group π 1 (X, q 0 ) whose images under the homomorphism induced by inclusion X → X c are generators of the fundamental group of X c . We may assume that each pair e 2j−1,0 , e 2j,0 , j = 1, . . . g, corresponds to a handle.
Represent each pair of generators e 2j−1,0 , e 2j,0 corresponding to a handle by simple closed loops α j , β j with base point q 0 .
Denote the connected components of X c \ X by C 1 , C 2 , . . . , C m+1 . Each component is either a point or a closed disc. For each component C , = 1, . . . , m, we choose the generator e 2g+ ,0 of the fundamental group of X which is represented by a simple loop γ 2g+ with base point q 0 that is contractible in X ∪ C and divides X into two connected components, one of them containing C . We will say that γ 2g+ surrounds C .
We may choose the generators and the representatives so that the only intersection point of any pair of loops is q 0 , and when labeling the rays of the loops emerging from the base point q 0 by α − j , β − j γ − j , and the incoming rays by α + j , β + j γ + j , then moving in counterclockwise direction along a small circle around q 0 we meet the rays in the order . . . , α − j , β − j , α + j , β + j , . . . , γ − k , γ + k , . . . . We denote the described system of generators by E, and fix it throughout the paper.

Figure 1
LetX be the universal covering of X. For each element e ∈ π 1 (X, q 0 ) we consider the subgroup e of π 1 (X, q 0 ) generated by e. Let σ(e 0 ) be the covering transformation corresponding to e 0 , and σ(e 0 ) the group generated by σ(e 0 ). Denote by E j , j = 2, . . . , 8, the set of primitive elements of π 1 (X, q 0 ) which can be written as product of at most j factors with each factor being either an element of E or an element of E −1 , the set of inverses of elements of E. Denote by λ j = λ j (X) the maximum over e ∈ E j of the extremal length of the annulus X σ(e 0 ) . The quantity λ 4 (X) (for mappings to the twice punctured complex plane), or λ 8 (X) (for (1, 1)-bundles) is the mentioned conformal invariant.
Let E be a finite subset of the Riemann sphere P 1 which contains at least three points. Let X be a finite Riemann surface with non-trivial fundamental group. A continuous map f : X → P 1 \ E is reducible if it is homotopic (as a mapping to P 1 \ E) to a mapping whose image is contained in D \ E for an open topological disc D ⊂ P 1 with E \ D containing at least two points of E. Otherwise the mapping is called irreducible.
In the following theorem we take E = {−1, 1, ∞}. We will often refer to P 1 \ {−1, 1, ∞} as to the twice punctured complex plane C \ {−1, 1}. Note that a continuous mapping from a Riemann surface to the twice punctured complex plane is reducible, iff it is homotopic to a mapping with image in a once punctured disc contained in P 1 \E. (The puncture may be equal to ∞.) There are countably many non-homotopic reducible holomorphic mappings from any finite open Riemann surface of second kind with non-trivial fundamental group to the twice punctured complex plane (see the proof of Theorem 7 in [21]). On the other hand the following theorem holds.
Notice that the Riemann surface X is allowed to be of second kind. If X is a torus with a hole, λ 4 (X) may be replaced by λ 3 (X).
A holomorphic (1, 0)-bundle is also called a holomorphic torus bundle. A holomorphic torus bundle equipped with a holomorphic section is also considered as a holomorphic (1, 1) fiber bundle. The following lemma holds.
Lemma D. A smooth (0, 1)-bundle admits a smooth section. A holomorphic torus bundle is (smoothly) isotopic to a holomorphic torus bundle that admits a holomorphic section.
For a proof see [21].
We wish to point out that reducible (g, m)-bundles over finite open Riemann surfaces can be decomposed into irreducible bundle components, and each reducible bundle is determined by its bundle components up to commuting Dehn twists in the fiber over the base point.
Notice that Caporaso proved the existence of a uniform bound of the number of objects in Theorem A in case X is a closed Riemann surface of genus g with m punctures, and the fibers are closed Riemann surfaces of genus g ≥ 2. The bound depends only on the numbers g, g and m. Heier gave effective uniform estimates, but the constants are huge and depend in a complicated way on the parameters. Theorems 1 and 2 imply effective estimates for the number of locally holomorphically nontrivial holomorphic (1, 1)-bundles over punctured Riemann surfaces, however, the constants depend also on the conformal type of the base. More precisely, the following corollaries hold. Corollary 1. There are no more than 3( 3 2 e 24πλ 4 (X) ) 2g+m non-constant holomorphic mappings from a Riemann surface X of type (g, m + 1) to P 1 \ {−1, 1, ∞}.
The following examples demonstrate the different nature of the problem in the two cases, the case when the base is a punctured Riemann surface, and when it is a Riemann surface of second kind. The first example considers holomorphic mappings from a torus with a hole to the twice punctured complex plane. The second example considers mappings from a planar domain to the twice punctured plane. It shows that for Riemann surfaces X of second kind we can not expect an upper bound for the number of irreducible holomorphic objects up to homotopy depending only on the topological type of the base manifold X and of the fibers.
Example 1. There are no non-constant holomorphic mappings from a torus with one puncture to the twice punctured complex plane. Indeed, by Picard's Theorem each such mapping extends to a meromorphic mapping from the closed torus to the Riemann sphere. This implies that the preimage of the set {−1, 1, ∞} under the extended mapping must contain at least three points, which is impossible.
The situation changes if X is a torus with a large enough hole. Let α ≥ 1 and σ ∈ (0, 1). Consider the torus with a hole T α,σ that is obtained from C (Z + iαZ), (with α ≥ 1 being a real number) by removing a closed geometric rectangle of vertical side length α − σ and horizontal side length 1 − σ (i.e. we remove a closed subset that lifts to such a closed rectangle in C). A fundamental domain for this Riemann surface is "the golden cross on the Swedish flag" turned by π 2 with width of the laths being σ and length of the laths being 1 and α.
On the other hand, there are positive constants c, C, and σ 0 such that for any positive number σ < σ 0 there are at least ce C α σ non-homotopic holomorphic mappings from T α,σ to the twice punctured complex plane.
Example 2. There are only finitely many holomorphic maps from a thrice punctured Riemann sphere to another thrice punctured Riemann sphere. Indeed, after normalizing both, the source and the target space, by a Möbius transformation we may assume that both are equal to C \ {−1, 1}. Each holomorphic map from C \ {−1, 1} to itself extends to a meromorphic map from the Riemann sphere to itself, which maps the set {−1, 1, ∞} to itself and maps no other point to this set. By the Riemann-Hurwitz formula the meromorphic map takes each value exactly once. Indeed, suppose it takes each value l times for a natural number l. Then each point in {−1, 1, ∞} has ramification index l. Apply the Riemann Hurwitz formula for the branched covering Y = P 1 → X = P 1 of multiplicity l Here e x is the ramification index at the point x. For the Euler characteristic we have χ(P 1 ) = 2, and x∈X (e x − 1) ≥ x=−1,1,∞ (e x − 1) = 3 (l − 1). We obtain 2 ≤ 2 l − 3 (l − 1) which is possible only if l = 1. We saw that each non-constant holomorphic mapping from C \ {−1, 1} to itself extends to a conformal mapping from the Riemann sphere to the Riemann sphere that maps the set {−1, 1, ∞} to itself. There are only finitely may such maps, each a Möbius transformation commuting the three points.
A domain in the plane is called doubly connected, if its fundamental group is isomorphic to the free group in two generators. We consider now doubly connected planar domains contained in C \ {−1, 1}. We take domains of a special form. Consider a skeleton S of C \ {−1, 1} which is the union of two simple closed curves γ 1 and γ 2 with base point zero, that surround −1 and +1, respectively, and are piecewise smooth, and smooth and pairwise disjoint outside the base point. Suppose γ 2 is contained in the cone {ρe θ : ρ ≥ 0, |θ| ≤ κ < π} and γ 1 is contained in the cone {ρe θ : ρ ≥ 0, |θ − π| ≤ κ < π}. Let σ be a small positive number and let Ω σ be the σ-neighbourhood of S.

Proposition 2.
There are positive constants σ 0 , C 1 and C 2 , depending only on γ 1 and γ 2 such that for each positive σ < σ 0 there are up to homotopy no more than C 1 e C 2 σ irreducible holomorphic mappings from Ω σ to the twice punctured complex plane.
Vice versa, there are positive constants C 1 and C 2 such that for each σ < σ 0 there are at least C 1 e C 2 σ non-homotopic irreducible holomorphic mappings from Ω σ to the twice punctured complex plane.
The present results may be understood as quantitative statements with regard to Gromov's Oka principle. A mapping from a Stein manifold to a complex manifold Y is said to satisfy Gromov's Oka principle if it is homotopic to a holomorphic map. The target spaces Y , for which each continuous mapping from a Stein manifold to Y satisfies the Oka principle, are called Oka manifolds and received a lot of attention ( [12], [9], [10]).
The twice punctured complex plane is not an Oka manifold. Theorem 1 and Propositions 1 and 2 give an estimate of the number of homotopy classes of mappings from particular Stein manifolds to the twice punctured complex plane, that satisfy Gromov's Oka principle. The estimate depends on the conformal structure of the Stein manifold.
The author is grateful to B.Farb who suggested to use the concept of conformal module and extremal length for a proof of finiteness theorems, and to B.Berndtsson for proposing the kernel for solving the∂-problem that arises in the proof of Proposition 1. The work on the paper was started while the author was visiting the Max-Planck-Institute and was finished during a stay at IHES. The author would like to thank these institutions for the support. The author is also indebted to Fanny Dufour for drawing the figures.

Holomorphic mappings into the twice punctured plane
In this section we will prove Theorem 1. We first need some preparation. The change of the base point. Let X be a connected smooth manifold and let α be an arc in X with initial point x 0 and terminating point x. Change the base point x 0 ∈ X along a curve α to the point x ∈ X . This leads to an isomorphism Is α : π 1 (X , x 0 ) → π 1 (X , x) of fundamental groups induced by the correspondence γ → α −1 γα for any loop γ with base point x 0 and the arc α with initial point x 0 and terminating point x. We will denote the correspondence γ → α −1 γα between curves also by Is α .
We call two homomorphisms h j : G 1 → G 2 , j = 1, 2, from a group G 1 to a group G 2 conjugate if there is an element g ∈ G 2 such that for each g ∈ G 1 the equality h 2 (g) = g −1 h 1 (g)g holds. For two arcs α 1 and α 2 with initial point x 0 and terminating point x we . Hence, the two isomorphisms Is α 1 and Is α 2 differ by conjugation with the element of π 1 (X , x) represented by α −1 1 α 2 . Free homotopic curves are related by homotopy with fixed base point and an application of a homomorphism Is α that is defined up to conjugation. Hence, free homotopy classes of curves can be identified with conjugacy classes of elements of the fundamental group π 1 (X , x 0 ) of X .
For two smooth manifolds X and Y with base points x 0 ∈ X and y 0 ∈ Y and a continuous mapping F : X → Y with F (x 0 ) = y 0 we denote by F * : π 1 (X , x 0 ) → π 1 (Y, y 0 ) the induced map on fundamental groups. For each element e 0 ∈ π 1 (X , x 0 ) the image F * (e 0 ) is called the monodromy along e 0 , and the homomorphism F * is called the monodromy homomorphism corresponding to F . The homomorphism F * determines the homotopy class of F with fixed base point in the source and fixed value at the base point. Consider a free homotopy F t , t ∈ (0, 1), of homeomorphisms from X to Y such that the value F t (x 0 ) at the base point x 0 of the source space varies along a loop. The homomorphisms (F 0 ) * and (F 1 ) * are related by conjugation with the element of the fundamental group of Y represented by the loop. Moreover, since the fundamental group π 1 (Y, y) with base point y is related to the fundamental group π 1 (Y, y 0 ) with base point y 0 by an isomorphism determined up to conjugation we obtain the following theorem (see [13], [28]).
Theorem E. The free homotopy classes of continuous mappings from X to Y are in one-toone correspondence to the set of conjugacy classes of homomorphisms between the fundamental groups of X and Y.
Extremal length. The fundamental group π 1 def = π 1 (C \ {−1, 1}, 0) is canonically isomorphic to the fundamental group π 1 (C \ {−1, 1}, q ) for an arbitrary point q ∈ (−1, 1). For the arc α defining the isomorphism we take the unique arc contained in (−1, 1) that joins q and 0. The fundamental group π 1 (C \ {−1, 1}, 0) is a free group in two generators. We choose standard generators a 1 and a 2 , where a 1 is represented by a simple closed curve with base point 0 which surrounds −1 counterclockwise, and a 2 is represented by a simple closed curve with base point 0 which surrounds 1 counterclockwise. For q ∈ (−1, 1) we also denote by a j the generator of π 1 (C\{−1, 1}, q ) which is obtained from the respective standard generator of π 1 (C\{−1, 1}, 0) by the standard isomorphism between fundamental groups with base point on (−1, 1). More detailed, a 1 is the generator of π 1 (C\{−1, 1}, q ) which is represented by a loop with base point q that surrounds −1 counterclockwise, and a 2 is the generator of π 1 (C \ {−1, 1}, q ) which is represented by a loop with base point q that surrounds 1 counterclockwise. We refer to a 1 and a 2 as to the standard generators of π 1 (C \ {−1, 1}, q ).
Each element of a free group can be written uniquely as a reduced word in the generators. (A word is reduced if neighbouring terms are powers of different generators.) The degree (or word length) d(w) of a reduced word w in the generators of a free group is the sum of the absolute values of the powers of generators in the reduced word. If the word is the identity its degree is defined to be zero. We will identify elements of a free group with reduced words in generators of the group.
For a rectangle R let f : R → C \ {−1, 1} be a mapping which admits a continuous extension to the closureR (denoted again by f ) which maps the (open) horizontal sides to (−1, 1). We say that the mapping f represents an element w tr ∈ π tr 1 (C \ {−1, 1}) if for each maximal vertical line segment contained in R (i.e. R intersected with a vertical line in C) the restriction of f to the closure of the line segment represents w tr .
The extremal length Λ(w tr ) of an element w tr in the relative fundamental group π tr : R a rectangle which admits a holomorphic map to C \ {−1, 1} that represents w tr } .
For an element w ∈ π 1 (C \ {−1, 1}, q ) and the associated element w tr we will also write Λ tr (w) instead of Λ(w tr ). In [18] (see also [19]) a natural syllable decomposition of any reduced word w in π 1 (C \ {−1, 1}, q ) is given. Let d k be the degree of the k-th syllable from the left.
(We consider each syllable as a reduced word in the elements of the fundamental group.) Put Notice that L ± (w −1 ) = L ± (w). We define L − (Id) = L + (Id) = 0 for the identity Id. We need the following theorem which is proved in [18] (see Theorem 1 there).
Theorem F. If w is not equal to a (trivial or non-trivial) power of a 1 or of a 2 then 1 2π where the sum runs over the degrees of all syllables of w tr .
Regular zero sets. We need some facts concerning regular zero sets of smooth real valued functions. We will call a subset of a smooth manifold X a simple relatively closed curve if it is the connected component of a regular level set of a smooth real-valued function on X . Let X be a connected finite open Riemann surface. Suppose the zero set L of a non-constant smooth real valued function on X is regular. Each component of L is either a simple closed curve or it can be parameterized by a continuous mapping : (−∞, ∞) → X . We call a component of the latter kind a simple relatively closed arc in X .
A relatively closed curve γ in a connected finite open Riemann surface X is said to be contractible to a hole of X , if the following holds. Consider X as domain X c \ ∪C j on a closed Riemann surface X c . Here the C j are the holes, each is either a closed disc or a point. The condition is the following. For each pair U 1 , U 2 of open subsets of X c , ∪C j ⊂ U 1 U 2 , there exists a homotopy of γ that fixes γ ∩ U 1 and moves γ into U 2 . Taking for U 2 small enough neighbourhoods of ∪C j we see that the homotopy moves γ into an annulus adjacent to one of the holes.
For each relatively compact domain X X in X there is a finite cover of L ∩ X by open subsets U k of X such that each L ∩ U k is connected. Each set L ∩ U k is contained in a component of L. Hence, only finitely many connected components of L intersect X . Let L 0 be a connected component of L which is a simple relatively closed arc parameterized by 0 : R → X . Since each set L 0 ∩ U k is connected it is the image of an interval under 0 . Take real numbers t − 0 and t + 0 such that all these intervals are contained in (t − 0 , t + 0 ). Then the images ( (−∞, t − 0 ) and ( (t + 0 , +∞) are contained in X \ X , maybe, in different components. Such parameters t − 0 and t + 0 can be found for each relatively compact deformation retract X of X . Hence for each relatively closed arc L 0 ⊂ L the set of limit points L + 0 of 0 (t) for t → ∞ is contained in a boundary component of X . Also, the set of limit points L − 0 of 0 (t) for t → −∞ is contained in a boundary component of X . The boundary components may be equal or different. Moreover, if X X is a relatively compact domain in X which is a deformation retract of X , and a connected component L 0 of L does not intersect X then L 0 is contractible to a hole of X . Indeed, X \ X is the union of disjoint annuli, each of which is adjacent to a boundary component of X , and the union of the limit sets L + 0 ∪ L − 0 is contained in a single boundary component of X .
Further, denote by L the union of all connected components of L that are simple relatively closed arcs. Consider those components L j of L that intersect X . There are finitely many such L j . Parameterize each L j by a mapping j : R → X . For each j we let [t − j , t + j ] be a compact interval for which Let X , X X X , be a domain which is a deformation retract of X such that j ([t − j , t + j ]) ⊂ X for each j. Then all connected components of L ∩ X , that do not contain a set j ([t − j , t + j ]), are contractible to a hole of X . Indeed, each such component is contained in the union of annuli X \ X .
Some remarks on coverings. Let X be a finite open Riemann surface with base point q 0 and let P :X → X be the universal covering map. By a covering P : Y → X we mean a continuous map P from a topological space X to a topological space Y such that for each point x ∈ X there is a neighbourhood V (x) of x such that the mapping P maps each connected component of the preimage of V (x) homeomorphically onto V (x). (Note that in function theory sometimes these objects are called unlimited unramified coverings to reserve the notion "covering" for more general objects.) Recall that a homeomorphism ϕ :X →X for which P • ϕ = P is called a covering transformation (or deck transformation). The covering transformations form a group, denoted by Deck(X, X). For each pair of pointsx 1 ,x 2 ∈X with P (x 1 ) = P (x 2 ) there exists exactly one covering transformation that mapsx 1 tox 2 . (See e.g. [8]).
Throughout the paper we will fix a base point q 0 ∈ X and a base pointq 0 ∈ P −1 (q 0 ) ⊂X. The group of covering transformations ofX can be identified with the fundamental group π 1 (X, q 0 ) of X by the following correspondence. (See e.g. [8]).
Let N be a subgroup of π 1 (X, q 0 ). Denote by X(N ) the quotientX (Isq 0 ) −1 (N ). We obtain a covering ω N Id :X → X(N ) with group of covering transformations isomorphic to N . The fundamental group of X(N ) with base point (q 0 ) N def = ω N Id (q 0 ) can be identified with N . If N 1 and N 2 are subgroups of π 1 (X, q 0 ) and N 1 is a subgroup of N 2 (we write N 1 ≤ N 2 ) then there is a covering map ω N 2 Figure 3 below is commutative.
Indeed, take any point x 1 ∈X (Isq 0 ) −1 (N 1 ) and a preimagex of x 1 under ω N 1 Id . There exists a neighbourhood V (x) ofx inX such that V (x) ∩ σ(V (x)) = ∅ for all covering transformations σ ∈ Deck(X, X). Then for j = 1, 2 the mapping ω We get a correctly defined mapping from V 1 onto V 2 . Indeed, since N 1 is a subgroup of N 2 , the covering transformation ϕx ,x is contained in (Isq 0 ) −1 (N 2 ), and we get for another pointx ∈ (ω N 1 Since the mappings ω N j ,x Id , j = 1, 2, are homeomorphisms from V (x) onto its image, the map- . The same holds for all preimages of V (x 2 ) under ω N 2 N 1 . Hence, ω N 2 N 1 is a covering map. The commutativity of the part of the diagram that involves the mappings ω N 1 Id , ω N 2 Id , and ω N 2 N 1 is clear by construction. The existence of ω π 1 (X,q 0 ) N 1 and the equality P = ω Id follows by applying the above arguments with N 2 = π 1 (X, q 0 ). The equality P = ω Id follows in the same way. Since P =ω We will also use the notation ω N def = ω N Id and ω N def = ω π 1 (X,q 0 ) N for a subgroup N of π 1 (X, q 0 ). In our situation N will usually be a subgroup Ẽ of π 1 (X, q 0 ) generated by a systemẼ of elements of E ⊂ π 1 (X, q 0 ).
Consider a set E ⊂ π 1 (X, x 0 ) that consists of 2g + m elements of E, such that among them are exactly g of the chosen pairs in E that correspond to g of the handles of X, and from all remaining pairs of elements of E corresponding to a handle at most one element is contained in E . Then X( E ) is a surface of genus g with m holes. Indeed, consider the chosen loops α j , β j , and γ 2g+ , that represent the elements of E. The union of those of them that represent elements of E can be identified with a deformation retract of X( E ). To see this, we lift the collection of all α j , β j , and γ j , representing elements of E to arcs in the universal coveringX of X with initial pointq 0 , and take the quotient X( E ) ofX by the action of Isq 0 ( E ). The lifts of the loops representing elements of E \ E project to arcs in X( E ). The lifts of the loops representing elements of E project to loops in X( E ), the union of which is a bouquet of circles which is a deformation retract of X( E ).
Recall that for an arbitrary point q ∈ X the free homotopy class of an element e of the fundamental group π 1 (X, q) can be identified with the conjugacy of elements of π 1 (X, q) containing e and is denoted by e. Notice that for e 0 ∈ π 1 (X, q 0 ) and a curve α in X with initial point q 0 and terminating point q the free homotopy classes of e 0 and of e = Is α (e 0 ) coincide, i.e. e = e 0 . Consider a simple smooth relatively closed curve L in X. We will say that a free homotopy class of curves e 0 intersects L if each representative of e 0 intersects L. Choose an orientation of L. The intersection number of e 0 with the oriented curve L is the intersection number with L of some (and, hence, of each) smooth loop representing e 0 that intersects L transversally. This intersection number is the sum of the intersection numbers over all intersection points. The intersection number at an intersection point equals +1 if the orientation determined by the tangent vector to L followed by the tangent vector to the curve representing e 0 is the orientation of X, and equals −1 otherwise.
Let A be an annulus. A continuous mapping ω : A → X is said to represent a conjugacy class e of elements of the fundamental group π 1 (X, q) for a point q ∈ X, if for each simple closed curve γ in A that is homologous to a boundary component of A the restriction of the mapping to γ represents e or the inverse e −1 .
Let A be an annulus with base point p, and let ω be a continuous mapping from A to a finite Riemann surface X with base point q such that ω(p) = q. We write ω : (A, p) → (X, q). The mapping is said to represent the element e of the fundamental group π 1 (X, q) if the preimage (ω * ) −1 (e) under the induced mapping ω * : π 1 (A, p) → π 1 (X, q) between fundamental groups is a generator of π 1 (A, p). There are two generators and they are inverse to each other.
The mapping ω e 0 : X( e 0 ) → X represents the free homotopy class e 0 . Moreover, X( e 0 ) is the "thickest" annulus which admits a holomorphic mapping to X, that represents the conjugacy class e 0 . This means, that the annulus has smallest extremal length among annuli with the mentioned property. Indeed, let A ω −−→ X be a holomorphic mapping of an annulus that represents e 0 . A is conformally equivalent to a round annulus in the plane, hence, we may assume that A has the form A = {z ∈ C : r < |z| < R} for positive real numbers r < R.
Take a simple closed curve γ A in A that is homologous to a boundary component of A. After maybe, reorienting γ A , its image under ω represents the class e 0 . Choose a point q A in γ A , and put q = ω(q A ). Then γ A represents a generator of π 1 (A, q A ) and γ represents an element e of π 1 (X, q) in the conjugacy class e 0 . Choose a curve α in X with initial point q 0 and terminating point q, and a pointq inX so that α andq are compatible, and, hence, for e = Is α (e 0 ) the equality (Isq 0 ) −1 (e 0 ) = (Isq) −1 (e) holds. Let L be the relatively closed arc (p · R) ∩ A in A through p along the radius. After a homotopy of γ A with fixed base point, we may assume that its base point q A is the only point of γ A that is contained in L. The restriction ω|(A \ L) lifts to a mappingω : (A \ L) →X, that extends continuously to the two strands L ± of L. (Here L − contains the initial point of γ A .) Let p ± be the copies of p on the two strands L ± . We choose the liftω so thatω(p + ) =q. Since the mapping (A, q A ) → (X, q) represents e, we obtainω(p − ) = σ(q) for σ = (Isq) −1 (e). Then for each z ∈ L the covering transformation σ maps the pointz + ∈ω(L + ) for which P(z + ) = z to the pointz − ∈ω(L − ) for which P(z − ) = z. Hence ω lifts to a holomorphic mapping ι : A → X( e 0 ). By Lemma 7 of [18] For each point q ∈ X and each element e ∈ π 1 (X, q) we denote by A( e) the conformal class of the "thickest" annulus that admits a holomorphic mapping into X that represents e. For eachq ∈X and each element e ∈ π 1 (X, P(q )) the mapping ω e ,q :X (Isq ) −1 ( e ) → X represents the conjugacy class e of e. Hence, if e = e 0 , then λ(X (Isq ) −1 ( e )) ≤ λ(A( e)).
Interchanging the role of the representatives e and e 0 of e 0 and of the base pointsq andq 0 ofX, we see that in fact λ(X (Isq ) −1 ( e )) = λ(A( e)) for eachq ∈X and each element e ∈ π 1 (X, P(q )). Notice that A( e −1 ) = A( e).
The following lemma will be crucial for the estimate of the L − -invariant of the monodromies of holomorphic mappings from a finite open Riemann surface to C \ {−1, 1}.
Lemma 1. Let f : X → C\{−1, 1} be a non-contractible holomorphic function on a connected finite open Riemann surface X, such that 0 is a regular value of Imf . Assume that L 0 is a simple relatively closed curve in X such that f (L 0 ) ⊂ (−1, 1). Let q ∈ L 0 and q = f (q).
If for an element e ∈ π 1 (X, q) the free homotopy class e intersects L 0 , then either the reduced word f * (e) ∈ π 1 (C\{−1, 1}, q ) is a non-zero power of a standard generator of π 1 (C\{−1, 1}, q ) or the inequality Notice that we make a normalization in the statement of the Lemma by requiring that f maps L 0 into the interval (−1, 1), not merely into R \ {−1, 1}. Lemma 1 will be a consequence of the following lemma.
Lemma 2. Let f , L 0 , q ∈ L 0 be as in Lemma 1, and e ∈ π 1 (X, q). Letq be an arbitrary point in P −1 (q). Consider the annulus A def =X (Isq) −1 ( e ) and the holomorphic projection = ω e ,q (q) and let L A ⊂ A be the connected component of (ω A ) −1 (L 0 ) that contains q A . Then the mapping ω A : (A, q A ) → (X, q) represents e.
If e intersects L 0 , then L A is a relatively closed curve in A that has limit points on both boundary components of A, and the lift f • ω A is a holomorphic function on A that maps L A into (−1, 1).
All connected components of P −1 (L 0 ) are relatively closed curves inX ∼ = C + (where C + denotes the upper half-plane) with limit points on the boundary ofX. Indeed, the lift f • P of f toX takes values in (−1, 1) on P −1 (L 0 ). Hence, by the maximum principle applied to exp(f • P) the latter cannot contain compact connected components since f is not constant.
LetLq be the connected component of P −1 (L 0 ) that containsq. The point σ(q) cannot be contained inLq. Indeed, otherwise there exists a curve inX that is homotopic toγ with fixed endpoints and is contained inLq. A small translation of the latter curve gives a curve inX that does not intersect P −1 (L 0 ) and projects under P to a curve that is free homotopic to γ. This contradicts the fact that e intersects L 0 .
Each of the two connected componentsLq and σ(Lq) dividesX. Let Ω be the domain oñ X that is bounded byLq and σ(Lq) and parts of the boundary ofX. After a homotopy of γ that fixes the endpoints we may assume thatγ((0, 1)) is contained in Ω. Indeed, for each connected component ofγ((0, 1)) \ Ω there is a homotopy with fixed endpoints that moves the connected component to an arc onLq or σ(Lq). A small perturbation yields a curveγ which is homotopic with fixed endpoints toγ and has interior contained in Ω. Notice thatγ ((0, 1)) does not meet any σ k (Lq).
The curve ω e ,q (γ ) is a closed curve on A that represents a generator of the fundamental The curve ω e ,q (γ ) intersects L e = ω e ,q (Lq) exactly once. Hence, L e has limit points on both boundary circles of A for otherwise L e would intersect one of the components of A \ ω e ,q (γ ) along a set which is relatively compact in A, andγ would have intersection number zero with L e . It is clear that f Proof of Lemma 1. Let ω A : (A, q A ) → (X, q) be the holomorphic mapping from lemma 2 that represents e, and let L A q A be the relatively closed curve in A with limit set on both boundary components of A. Consider a loop γ A : represents the element (f * (e)) tr ∈ π tr 1 (C\{−1, 1}) in the relative fundamental group fundamental group π 1 (C\{−1, 1}, (−1, 1)) = π tr We prove now that Λ tr (f * (e)) ≤ λ(A). Let A 0 A be any relatively compact annulus in A with smooth boundary such that q A ∈ A 0 . If A 0 is sufficiently large, then the connected component The horizontal curvilinear sides are the strands of the cut that are reachable from the curvilinear rectangle moving counterclockwise, or clockwise, respectively. The vertical curvilinear sides are obtained from the boundary circles of A 0 by removing an endpoint of the arc Moreover, This is a consequence of the following facts. First, holds. We obtain the inequality Λ tr (f * (e)) ≤ λ(A 0 ) for each annulus A 0 A, hence, since A belongs to the class A( e) of conformally equivalent annuli, and the Lemma follows from Theorem F. 2 The monodromies along two generators. In the following Lemma we combine the information on the monodromies along two generators of the fundamental group π 1 (X, q). We allow the situation when the monodromy along one generator or along both generators of the fundamental group of X is a power of a standard generator of π 1 (C \ {−1, 1}, f (q)).
1} be a holomorphic function on a connected open Riemann surface X such that 0 is a regular value of the imaginary part of f . Suppose f maps a simple relatively closed curve L 0 in X to (−1, 1), and q is a point in L 0 . Let e (1) and e (2) be primitive elements of π 1 (X, q). Suppose that for each e = e (1) , e = e (2) , and e = e (1) e (2) , the free homotopy class e intersects L 0 . Then either f * (e (j) ), j = 1, 2, are (trivial or non-trivial) powers of the same standard generator of , or each of them is the product of at most two elements w 1 and w 2 of where λ e (1) ,e (2) def = max{λ(A( e (1) )), λ(A( e (2) )), λ(A( e (1) e (2) ))}.
If the monodromies f * (e (1) ) and f * (e (2) ) are not powers of a single standard generator (the identity is considered as zeroth power of a standard generator) we obtain the following. At most two of the elements, f * (e (1) ), f * (e (2) ), and f * (e (1) e (2) ) = f * (e (1) ) f * (e (2) ), are powers of a standard generator, and if two of them are powers of a standard generator, then they are nonzero powers of different standard generators. If two of them are non-zero powers of standard generators, then the third has the form a k a k with a and a being different generators and k and k being non-zero integers. By Lemma 1 the L − of the third element does not exceed 2πλ e (1) ,e (2) . On the other hand it equals log(3|k |) + log(3|k|). Hence, L − (a k ) = log(3|k|) ≤ 2πλ e (1) ,e (2) and L − (a k ) = log(3|k |) ≤ 2πλ e (1) ,e (2) . If two of the elements f * (e (1) ), f * (e (2) ), and f * (e (1) e (2) ) = f * (e (1) ) f * (e (2) ), are not powers of a standard generator, then the L − of each of the two elements does not exceed 2πλ e (1) ,e (2) . Since the L − of an element coincides with the L − of its inverse, the third element is the product of two elements with L − not exceeding 2πλ e (1) ,e (2) . Since for x, x ≥ 3 the inequality log(x + x ) ≤ log x + log x holds, the L − of the product does not exceed the sum of the L − of the factors. Hence the L − of the third element does not exceed 4πλ e (1) ,e (2) . Hence, inequality (12) holds. 2 The following proposition states the existence of suitable connected components of the zero set of the imaginary part of certain analytic functions on tori with a hole and on planar domains. For any subset E of π 1 (X; q 0 ) we denote by (E ) −1 the set of all elements that are inverse to elements in E . Recall that E j is the set of primitive elements of π 1 (X, q 0 ) which can be written as product of at most j elements of E ∪ (E) −1 for the set E of generators of π 1 (X, q 0 ) chosen in the introduction. Proposition 3. Let X be a torus with a hole or a planar domain with base point q 0 and fundamental group π 1 (X, q 0 ), and let E be the set of generators of π 1 (X, q 0 ) that was chosen in Section 1. Let f : X → C \ {−1, 1} be a non-contractible holomorphic mapping such that 0 is a regular value of Imf . Then there exist a simple relatively closed curve L 0 ⊂ X such that f (L 0 ) ⊂ R \ {−1, 1}, and a set E 2 ⊂ E 2 ⊂ π 1 (X, q 0 ) of primitive elements of π 1 (X, q 0 ), such that the following holds. Each element e j,0 ∈ E ⊂ π 1 (X, q 0 ) is the product of at most two elements of E 2 ∪ (E 2 ) −1 . Moreover, for each e 0 ∈ π 1 (X, q 0 ) which is the product of one or two elements from E 2 the free homotopy class e 0 has positive intersection number with L 0 (after suitable orientation of L 0 ).
Notice the following fact. If f is irreducible, then it is not contractible, and, hence, the preimage f −1 (R) is not empty.
Proposition 4. Let X be a connected finite open Riemann surface with base point q 0 , and let E be the set of generators of π 1 (X, q 0 ) that was chosen in Section 1. Suppose f : is an irreducible holomorphic mapping, such that 0 is a regular value of Imf . Then for one of the functions M l • f, l = 0, 1, 2, which we denote by F , there exists a point q ∈ X (depending on f ), such that the point q def = F (q) is contained in (−1, 1), and a curve α in X joining q 0 with q, such that the following holds. For each element e j ∈ Is α (E) the monodromy F * (e j ) is the product of at most four elements of π 1 ((C) \ {−1, 1}, q ) of L − not exceeding 2πλ 4 (X) and, hence, Notice that F is irreducible iff f is irreducible. By Theorem E and Lemma 4 of [21] the noncontractible mapping F is irreducible if and only if the image F * (π 1 (X, q)) of the monodromy homomorohism is not generated by a single element of the fundamental group π 1 (C\{−1, 1}, q ), q ∈ (−1, 1), that is conjugate to a standard generator of π 1 (C \ {−1, 1}, q ).
Notice, that all monodromies of contractible mappings are equal to zero, hence the inequality (13) holds automatically for contractible mappings. We postpone the proof of the two propositions and prove first the Theorem 1. Proof of Theorem 1. Let X be a connected finite open Riemann surface (possibly of second kind) with base point q 0 . Using Theorem E and Proposition 4 we want to bound the number of irreducible free homotoy classes of mappings from X to C \ {−1, 1} that contain a holomorphic mapping. Consider a free homotopy class that is represented by an irreducible holomorphic mapping f : X → C \ {−1, 1}. Recall that irreducible mappings are not contractible. Consider an arbitrary open Riemann surface X 0 X which is relatively compact in X and is a deformation retract of X. Let ε be a small enough real number ε, such that the function (f − iε) | X 0 takes values in C \ {−1, 1} and 0 is a regular value of its imaginary part. Put f = (f − iε) | X 0 . Notice that the irreducible mapping f on X 0 is free homotopic to f | X 0 . We identify the fundamental groups of X and of X 0 by the inclusion mapping from X 0 to X.
The statement of Proposition 4 applies to the function f on X 0 . Associate to f the function F = M l • f on X 0 , and the points q and q from Proposition 4. The proposition provides information about the monodromies of the mapping F along the elements Is α (E) for some curve α that was chosen depending on f , and has initial point q 0 and terminating point q. We have to relate this information to the monodromies of the mapping F along the elements of the system E of generators of the fundamental group π 1 (X, q 0 ) ∼ = π 1 (X 0 , q 0 ) with base point q 0 , that was chosen in Section 1 and is independent on the function f . Write e j = Is α (e j,0 ) ∈ π 1 (X, q) for e j,0 ∈ π 1 (X, q 0 ). The image of α under the mapping F is the curve β = F • α in C \ {−1, 1} with initial point F (q 0 ) and terminating point q . We have F * (e j,0 ) = (Is β ) −1 (F * (e j )). Choose a homotopy F t , t ∈ [0, 1], that joins the mapping F 0 def = F with a (smooth) mapping F 1 denoted byF , so that the value F t (q 0 ) moves from the point F (q 0 ) to q along the curve β.
Inequality (13) gives for each j the inequality Notice that if f is a contractible function, we may associate to it the functionF on X 0 which is equal f | X 0 and the inequality (14) is automatically satisfied for the monodromies ofF .
By Lemma 1 of [20] there are at most . By Theorem E there is a one-to-one correspondence between free homotopy classes of functions from X 0 to C \ {−1, 1} and conjugacy classes of homomorphisms . Since F is free homotopic toF , the function F represents one of at most ( 3 2 e 24πλ 4 (X 0 ) ) 2g+m different free homotopy classes from X 0 to the twice punctured complex plane.
The function f is equal to M −1 l • F for one of the numbers 0, 1, or 2, hence it represents one of 3( 3 2 e 24πλ 4 (X 0 ) ) 2g+m free homotopy classes of mappings from X 0 to the twice punctured complex plane. Since the considered set of mappings f on X is equal to the set of mappings (f − iε)|X 0 for a holomorphic map f on X, there are no more than 3( 3 2 e 24πλ 4 (X 0 ) ) 2g+m free homotopy classes of irreducible mappings X → C \ {−1, 1} containing a holomorphic mapping f. Theorem 1 is proved with the upper bound 3( 3 2 e 24πλ 4 (X 0 ) ) 2g+m for an arbitrary relatively compact domain X 0 ⊂ X that is a deformation retract of X.
It remains to prove that λ 4 (X) = inf{λ 4 (X 0 ) : X 0 Xis a deformation retract of X }. We have to prove that for each e 0 ∈ π 1 (X, x 0 ) the number λ(A( e 0 )) = λ( X (Isq 0 ) −1 ( e 0 )) is equal to the infimum of λ( X 0 (Isq 0 ) −1 ( e 0 )) over all X 0 being open relatively compact subsets of X which are deformation retracts of X. Here X 0 is the universal covering of X 0 , and the fundamental groups of X and X 0 are identified. X 0 ( X, respectively) can be defined as set of homotopy classes of arcs in X 0 (in X, respectively) joining q 0 with a point q ∈ X 0 (in X respectively) equipped with the complex structure induced by the projection to the endpoint of the arcs, and the pointq 0 corresponds to the class of the constant curve. The isomorphism (Isq 0 ) −1 from π 1 (X 0 , q 0 ) to the group of covering transformations on X 0 is defined in the same way as it was done for X instead of X 0 . We see that there is a holomorphic mapping from X 0 (Isq 0 ) −1 ( e 0 ) into X (Isq 0 ) −1 ( e 0 ). Hence, the extremal length of the first set is not smaller than the extremal length of the second set.
Vice versa, take any annulus A 0 which is a relatively compact subset of A( e 0 ) and is a deformation retract of A( e 0 ). Its projection to X is relatively compact in X, hence, it is contained in a relatively compact deformation retract X 0 of X. Hence, A 0 can be considered as subset of X 0 (Isq 0 ) −1 ( e 0 ), and, hence, We proved a slightly stronger statement, namely, the number of homotopy classes of mappings X → C \ {−1, 1} that contain a contractible holomorphic mapping or an irreducible holomorphic mapping does not exceed ( 3 2 e 24πλ 4 (X) ) 2g+m .
Proof of Proposition 3. Denote the zero set {x ∈ X : Imf (x) = 0} by L. Since f is not contractible, L = ∅. 1. A torus with a hole. Assume first that X is a torus with a hole with base point q 0 . For notational convenience we denote by e 0 and e 0 the two elements of the set of generators E of π 1 (X, q 0 ) that was chosen in Section 1. We claim that there is a connected component L 0 of L which has (after suitable orientation) positive intersection number with the free homotopy class of one of the elements of E, say with e 0 , and L 0 has positive intersection number with one of the classes e 0 , or (e 0 ) −1 , or e 0 e 0 . The claim is easy to prove in the case when there is a component of L 0 which is a simple closed curve that is not contractible and not contractible to the hole of X. Indeed, consider the inclusion of X into a closed torus X c and the homomorphism on fundamental groups π 1 (X, q 0 ) → π 1 (X c , q 0 ) induced by the inclusion. Denote by e c 0 and e 0 c the images of e 0 and e 0 under this homomorphism. Notice that e c 0 and e 0 c commute. The (image under the inclusion of the) curve L 0 is a simple closed non-contractible curve in X c . It represents the free homotopy class of an element (e c 0 ) j (e 0 c ) k for some integers j and k which are not both equal to zero. Hence, L 0 is not null-homologous in X c , and by the Poincaré Duality Theorem for one of the generators, say for e c 0 , the representatives of the free homotopy class e c 0 have non-zero intersection number with L 0 . After suitable orientation of L 0 , we may assume that this intersection number is positive. There is a representative of the class e c 0 which is contained in X, hence, e 0 has positive intersection number with L 0 .
Suppose all compact connected components of L are contractible or contractible to the hole of X. Consider a relatively compact domain X X in X with smooth boundary which is a deformation retract of X such that for each connected component of L at most one component of its intersection with X is not contractible to the hole of X . (See the paragraph on "Regular zero sets".) There is at least one component of L ∩ X that is not contractible to the hole of X . Indeed, otherwise the free homotopy class of each element of E could be represented by a loop avoiding L, and, hence, the monodromy of f along each element of E would be conjugate to the identity, and, hence, equal to the identity, i.e. contrary to the assumption, f : X → C \ {−1, 1} would be free homotopic to a constant.
Take a component L 0 of L ∩ X that is not contractible to the hole of X . There is an arc of ∂X between the endpoints of L 0 such that the unionL 0 of the component L 0 with this arc is a closed curve in X that is not contractible and not contractible to the hole. Hence, for one of the elements of E, say for e 0 , the intersection number of the free homotopy class e 0 with the closed curveL 0 is positive after orienting the closed curve suitably. We may take a representative γ of e 0 that is contained in X . Then γ has positive intersection number with L 0 . Denote the connected component of L that contains L 0 by L 0 . All components of L 0 ∩ X different from L 0 are contractible to the hole of X . Hence, γ has intersection number zero with each of these components. Hence, γ has positive intersection number with L 0 sincẽ γ ⊂ X . We proved that the class e 0 has positive intersection number with L 0 .
If e 0 also has non-zero intersection number with L 0 we define e 0 = (e 0 ) ±1 so that the intersection number of e 0 with L 0 is positive. If e 0 has zero intersection number with L 0 we put e 0 = e 0 e 0 . Then again the intersection number of e 0 with L 0 is positive. Also, the intersection number of e 0 e 0 with L 0 is positive. The set E 2 def = {e 0 , e 0 } satisfies the condition required in the proposition. We obtained Proposition 3 for a torus with a hole. 2. A planar domain. Let X be a planar domain. The domain X is conformally equivalent to a disc with m smoothly bounded holes, equivalently, to the Riemann sphere with m + 1 smoothly bounded holes, P 1 \ m+1 j=1 C j , where C m+1 contains the point ∞. As before the base point of X is denoted by q 0 , and for each j = 1, . . . , m, the generator e j,0 ∈ E ⊂ π 1 (X, q 0 ) is represented by a curve surrounding C j once counterclockwise. We claim that there exists a component L 0 of L with limit points on the boundary of two components ∂C j and ∂C j for some j , j ∈ {1, . . . , m + 1} with j = j .
Indeed, assume the contrary. Then, if a component of L has limit points on ∂C j , j ≤ m, then all its limit points are on ∂C j . Take a smoothly bounded simply connected domain C j X ∪ C j that contains the closure C j , so that its boundary ∂C j represents e j,0 . Then all components L k of L \ C j with an endpoint on ∂C j have both endpoints on this circle. Fix a point p j ∈ ∂C j \ L. Assign to each component L k of L\C j with both endpoints on ∂C j the closed arc α k in ∂C j \{p j } with the same endpoints as L k .
The arcs α k are partially ordered by inclusion. For an arc α k which contains no other of the arcs (a minimal arc) the curve f • α k except its endpoints is contained in C \ R. Moreover, since L k is connected, the endpoints of f Consider the arcs α k with the following property. For an open arc α k in ∂C j \ {p j } which contains the closed arc α k the mapping f •α k is homotopic in C\{−1, 1} (with fixed endpoints) to a curve contained in C \ R. Induction on the arcs by inclusion shows that this property is satisfied for all maximal arcs among the α k and, hence, f | ∂C j is contractible in C \ {−1, 1}. We saw that for each hole C j , j ≤ m, whose boundary contains limit points of a connected component of L, the monodromy along the curve C j with base point p j that repesents e 0,j is trivial. The contradiction proves the claim.
With j and j being the numbers of the claim we consider the set E 2 ⊂ E 2 which consists of the following primitive elements: e j ,0 , the element (e j ,0 ) −1 provided j = m + 1, and e j ,0 e j,0 for all j = 1, . . . , m, j = j , j = j . The free homotopy class of each element of E 2 has intersection number 1 with L 0 after suitable orientation of the curve L 0 . Each product of at most two different elements of E 2 is a primitive element of π 1 (X, q) and is contained in E 4 . Moreover, the intersection number with L 0 of the free homotopy class of any such element equals 1 or 2. Each element of E is the product of at most two elements of E 2 ∪ (E 2 ) −1 .
The proposition is proved for the case of planar domains X. 2 Proof of Proposition 4. 1. A torus with a hole. Consider the curve L 0 and the set E 2 ⊂ π 1 (X, q 0 ) obtained in Proposition 3. For one of the functions M l • f , denoted by F , the image F (L 0 ) is contained in (−1, 1). Move the base point q 0 to a point q ∈ L 0 along a curve α in X, and consider the generators e = Is α (e 0 ) and e = Is α (e 0 ) of π 1 (X, q), and the set Is α (E 2 ) ⊂ π 1 (X, q). Then e and e are products of at most two elements of Is α (E 2 ). Since the free homotopy class of an element of π 1 (X, q 0 ) coincides with the free homotopy class of the element of π 1 (X, q) obtained by applying Is α , the free homotopy class of each product of one or two elements of Is α (E 2 ) intersects L 0 . We may assume as in the proof of Proposition 3 that Is α (E 2 ) consists of the elements e and e , where e is the product of at most two elements among the e and e and their inverses. Lemma 3 applies to the pair e, e , the function F , and the curve L 0 . By the conditions of Proposition 4 and Lemma 4 of [21] the monodromies of F along e and e are not powers of a single standard generator of the fundamental group of π 1 (C \ {−1, 1}, q ). Since for eachẽ 0 ∈ π 1 (X, q 0 ) the equality A( ẽ 0 ) = A(Is α ( ẽ 0 )) holds, this implies that the monodromy along e and e is the product of at most two elements of L − not exceeding 2πλ 3 (X), and, hence, it has L − not exceeding 4πλ 3 (X). Since e is the product of at most two elements among the e and e and their inverses, we obtain Proposition 4 for e and e , in particular L − (F * (e)) and L − (F * (e )) do not exceed 8πλ 3 (X). Proposition 4 is proved for tori with a hole.

A planar domain.
Consider the curve L 0 and the set E 2 of Proposition 3. Move the base point q 0 along an arc α to a point q ∈ L 0 . Then f (q) ∈ R \ {−1, 1} and for one of the (−1, 1). Denote e j = Is α (e j,0 ) for each j. The e j form the basis Is α (E) of π 1 (X, q). The set Is α (E 2 ) consists of primitive elements of π 1 (X, q) such that the free homotopy class of each product of one or two elements of Is α (E 2 ) intersects L 0 . Moreover, each element of Is α (E) is the product of one or two elements of Is α (E 2 ) ∪ (Is α (E 2 )) −1 .
By the condition of the proposition not all monodromies F * (e), e ∈ Is α (E 2 ), are (trivial or non-trivial) powers of the same standard generator of π 1 (C \ {−1, 1}, q ). Apply Lemma 3 to all pairs of elements of Is α (E 2 ) whose monodromies are not (trivial or non-trivial) powers of the same standard generator of π 1 (C \ {−1, 1}, q ). Since the product of at most two different elements of Is α (E 2 ) is contained in Is α (E 4 ) Lemma 3 shows that the monodromy F * (e) along each element e ∈ E 2 is the product of at most two factors, each with L − not exceeding 2πλ 4 (X). Since each element of Is α (E) is a product of at most two factors in E 2 ∪ (E 2 ) −1 , the monodromy F * (e j ) along each generator e j of π 1 (X, q) is the product of at most 4 factors of L − not exceeding 2πλ 4 (X), and, hence, each monodromy F * (e j ) has L − not exceeding 8πλ 4 (X). Proposition 4 is proved for planar domains.
3. The general case. Reduction to the case of non-trivial monodromies. We first provide a reduction to the case when the monodromy along each element of E ⊂ π 1 (X, q 0 ) is different from the identity. In other words, we will prove that provided Proposition 4 is true for this case, it is true in the general case.
Let X, f and E be as in the statement of Proposition 4. We letE be the set of elements from the chosen system of generators E of π 1 (X, q 0 ) along which the monodromy is different from the identity. Keeping the same base pointq 0 ofX as before we consider the quotient X( E ) = X (Isq 0 ) −1 (E) and the covering ω E : X( E ) → X, and the lift f E = f •ω E of f to X( E ).
Put (q 0 ) E = ω E (q 0 ). The fundamental group of the Riemann surface X( E ) with base point (q 0 ) E can be identified with E . For each e j,0 ∈E we will denote by (e j,0 ) E the element of the fundamental group π 1 (X E ), (q 0 ) E ) that corresponds to (ω E ) * ((e j,0 ) E ) = e j,0 . The lift f E to X( E ) has non-trivial monodromy along each element ofE. Our assumption implies Apply to the Riemann surface X( E ) and the function f E the case of Proposition 4 when all monodromies are non-trivial. We obtain a point q E ∈ X( E ) and a Möbius transformation M l , such that the function F E = M l • f E maps q E to a point q ∈ (−1, 1), and a curve α E in X( E ) with initial point (q 0 ) E and terminating point q E , such that the following holds.
For each (e j ) E = Is α E ((e j,0 ) E ) ∈ Is α E (E) the monodromy (F E ) * ((e j ) E ) is the product of at most four elements of π 1 (C \ {−1, 1}, q ) of L − not exceeding 2πλ 4 (X( E )). In particular, Consider the projection ω E : X( E ) → X. Put F = M l • f, for the Möbius transformation mentioned above, so that 0 ). Further, for (e j ) E = Is α E ((e j,0 ) E ) we get (ω E ) * ((e j ) E ) = Is α (e j,0 ). This can be checked by applying ω E to curves representing (e j,0 ) E . Hence, (F E ) * ((e j ) E ) = F * (Is α (e j,0 )) = F * (e j ) for each e j ∈ Is α (E). We obtain the inequality for each e j ∈ Is α (E), and, hence, for each e j ∈ Is α (E), since L − (e j ) = 0 for e j ∈ Is α (E \E). It remains to prove the inequality λ 4 (X( E )) ≤ λ 4 (X). The quantity λ 4 (X) is equal to the maximum of the extremal length of the annuliX (Isq 0 ) −1 ( e 0 ) over primitive elements e 0 of the fundamental group π 1 (X, q 0 ) of X that are products of at most four elements of E ∪ (E) −1 for the chosen set of generators E of π 1 (X, q 0 ). Recall that we represented the Riemann surface X as quotientX (Isq 0 ) −1 ( E ).
The Riemann surface X( E ) is equal to the quotientX (Isq 0 ) −1 ( E ). The chosen set E corresponds to the setE E of generators of the fundamental group π 1 (X( E ), q E ). The quantity λ 4 (X( E )) is the maximum of the extremal length ofX (Isq 0 ) −1 ( e 0 ) over elements e 0 ∈ π 1 (X, q 0 ), such that (e 0 ) E is a primitive element of the fundamental group π 1 (X( E ), q E ) of X( E ) which is the product of at most four elements ofE E ∪ (E E ) −1 . This quantity does not exceed λ 4 (X).
We proved Proposition 4 for the general case provided it is proved in the case when all monodromies are non-trivial. Positive intersection number. We assume now that the monodromies along all elements of the chosen set of generators E of π 1 (X, q 0 ) are non-trivial and the Riemann surface is not a torus with a hole nor a planar domain, and prove the proposition for this case. In this case X contains a handle that corresponds to generators, denoted by e 0 and e 0 , contained in E ⊂ π 1 (X, q 0 ) so that none of the two monodromies f * (e 0 ) or f * (e 0 ) is the identity. There may be several choices of such handles. If there is a handle so that the monodromies of f along the two elements from E corresponding to the handle are not powers of the same element of π 1 (C \ {−1, 1}, f (q 0 )), we take such a handle.
Consider the Riemann surface X( e 0 , e 0 ). It is a torus with a hole and admits a holomorphic covering ω e 0 ,e 0 : X( e 0 , e 0 ) → X of X. Put q 0 e 0 ,e 0 = ω e 0 ,e 0 (q 0 ). The fundamental group of X( e 0 , e 0 ) with base point q 0 e 0 ,e 0 can be identified with e 0 , e 0 . For an elementẽ 0 ∈ e 0 , e 0 we denote by (ẽ 0 ) e 0 ,e 0 the element of π 1 (X( e 0 , e 0 ), q 0 e 0 ,e 0 ) that projects toẽ 0 under ω e 0 ,e 0 . By our conditions the lift f e 0 ,e 0 of f to the Riemann surface X( e 0 , e 0 ) is not contractible and 0 is a regular value of its imaginary part. Let L ⊂ X be the set where f is real. The set ω −1 e 0 ,e 0 (L) coincides with the set of points on which the lift f e 0 ,e 0 = f • ω e 0 ,e 0 of the mapping f is real. According to Proposition 3 for the case of a torus with a hole there is a connected component L e 0 ,e 0 of ω −1 e 0 ,e 0 (L) such that the free homotopy class of one of the elements (e 0 ) e 0 ,e 0 or (e 0 ) e 0 ,e 0 , say of the class (e 0 ) e 0 ,e 0 of loops in X e 0 ,e 0 , that are free homotopic to loops representing (e 0 ) e 0 ,e 0 , has positive intersection number with L e 0 ,e 0 (after suitably orienting L e 0 ,e 0 ). Take a point q e 0 ,e 0 ∈ L e 0 ,e 0 . The function f • ω e 0 ,e 0 takes a real value at q e 0 ,e 0 . Put q = ω e 0 ,e 0 (q e 0 ,e 0 ). Choose a pointq ∈X, for which ω e 0 ,e 0 (q) = q e 0 ,e 0 . Letα be a curve inX with initial pointq 0 and terminating pointq. Then α e 0 ,e 0 def = ω e 0 ,e 0 (α) is a curve in X( e 0 , e 0 ) with initial point q 0 e 0 ,e 0 and terminating point q e 0 ,e 0 . Note that with this choice α e 0 ,e 0 andq are compatible (as a curve in X( e 0 , e 0 ) and a point in the universal covering of X( e 0 , e 0 ) that projects to the terminating point of the curve).
We want to show that there exists an element e * 0 ofẼ such that the class (e * 0 ) Ẽ of loops on X( Ẽ ), that are free homotopic to representatives of (e * 0 ) Ẽ , has positive intersection number with L Ẽ (with a suitable orientation of L Ẽ ).
The free homotopy class (e 0 ) Ẽ in X( Ẽ ) that is related to e 0 intersects L Ẽ . Indeed, consider any loop γ Ẽ in X( Ẽ ) with some base point q Ẽ , that represents (e 0 ) Ẽ . There exists a loop γ e 0 ,e 0 in X( e 0 , e 0 ) in X( e 0 , e 0 ) which represents (e 0 ) e 0 ,e 0 such that ω Suppose L Ẽ is a simple relatively closed arc in X( Ẽ ) with the whole limit set on the boundary of a single hole C. Then by the same reasoning as above, L Ẽ ∪ C does not divide the closed torus X( Ẽ ) c and again the intersection number with L Ẽ of one of the classes, (e 0 ) Ẽ or (e 0 ) Ẽ , is non-zero.
If L Ẽ is a relatively closed arc in X( Ẽ ) with limit sets on the boundary of different holes.
Then one of these holes, C j , has label j not exceeding m. The free homotopy class (e j,0 ) Ẽ corresponding to the generator e j,0 ∈ E ⊂ π 1 (X, q 0 ) whose representives surround C j , has nonvanishing intersection number with L Ẽ . After orienting L Ẽ suitably the intersection number of (e j,0 ) Ẽ with L Ẽ is positive. End of Proof. Let again the monodromies of f along all elements of E be non-trivial, and letẼ be a subset of E such that the quotient X( Ẽ ) is a torus with m + 1 holes. We obtained the following objects related to X( e 0 , e 0 ), the point q 0 e 0 ,e 0 , the curve L e 0 ,e 0 , the point q e 0 ,e 0 ∈ L e 0 ,e 0 , and the curve α e 0 ,e 0 . Let q 0 Ẽ , L Ẽ , the point q Ẽ in L Ẽ , and α Ẽ be the respective images under ω and, hence (16) For e j,0 ∈Ẽ we put (e j ) Ẽ = Is α Ẽ ((e j,0 ) Ẽ ) and e j = Is α (e j,0 ). Since α e 0 ,e 0 andq are compatible, also α Ẽ andq are compatible, and α andq are compatible, and the equalities F * (e j ) = (F Ẽ ) * ((e j ) Ẽ ), e j 0 ∈Ẽ, on monodromies hold. Take now an arbitrary elementẽ 0 of E. We want to prove the statement of Proposition 4 for the monodromy of F alongẽ 0 . For each elementẽ 0 of E we may choose a setẼ ⊂ E which containsẽ 0 , e 0 , and e 0 such thatX Isq 0 ( Ẽ ) is a Riemann surface of genus 1 with m + 1 holes for some m , 0 ≤ m ≤ 1, and the monodromies of F along the elements of Is α (Ẽ) are not powers of a single a j . Indeed, if the monodromy along the three elements e def = Is α (ẽ 0 ), e def = Is α (e 0 ), and e def = Is α (e 0 ) are not powers of a single a j then we may take the setẼ = {e 0 , e 0 ,ẽ 0 }. If all three monodromies are powers of a single a j , then there exists some elementẽ of Is α (E) \ {e, e ,ẽ} such that the monodromy of f along this element is not a power of a j . Then the monodromies along e 0 and e 0 are powers of a single element, hence the same is true for each pair of elements of E corresponding to a handle, and the elements (ẽ) 0 andẽ 0 cannot form a pair of elements from E that correspond to a handle. Hence, for E = {e 0 , e 0 ,ẽ 0 ,ẽ 0 } the quotientX Isq 0 (˜ E ) has genus 1.
It remains to prove that for each of the chosenẼ each monodromy (F Ẽ ) * ((e j ) Ẽ ), e j ∈ Is α (Ẽ), is the product of at most four elements of π 1 (C \ {−1, 1}, q ) of L − not exceeding 2πλ 4 (X( Ẽ ) and L − ((F Ẽ ) * ((e j ) Ẽ )) ≤ 8πλ 4 (X( Ẽ )). This follows along the same lines as the previous proofs. We proved the existence of the element e * 0 ∈Ẽ such that the free homotopy class (e * 0 ) Ẽ in X( Ẽ ) has positive intersection number with L Ẽ (oriented suitably). Then for each other element e j,0 ∈Ẽ the free homotopy class in X( Ẽ ) corresponding to one of the elements (e j,0 ) ±1 or e * 0 e j,0 , denoted by e j,0 , has positive intersection number with L Ẽ . Let E 2 ⊂ E 2 be the set consisting of e * 0 and any other element e j,0 ofẼ replaced by e j,0 . Theñ E 2 is a subset of the setẼ 2 of primitive elements that are products of one or two elements of E ∪ E −1 . The free homotopy class of the product of one or two elements of (Ẽ) 2 intersects L Ẽ positively.
For each element ofẼ 2 there is another element ofẼ 2 such that the monodromies of F Ẽ along the images of the two elements under Is α are not powers of the same a j . Apply Lemma 3 to such pairs of elements ofẼ 2 and notice that each element ofẼ is the product of at most two elements ofẼ 2 . We see that each monodromy (F Ẽ ) * ((e j ) Ẽ ), e j ∈ Is α (Ẽ), is the product of at most four elements of π 1 (C \ {−1, 1}, q ) of L − that does not exceed 2πλ 4 (X( Ẽ )). Since it can be seen as above that λ 4 (X( Ẽ )) ≤ λ 4 (X), we obtain that the monodromy along each element e j ∈ Is α (Ẽ) is the product of at most four elements of π 1 (C \ {−1, 1}, q ) of L − not exceeding 2πλ 4 (X), and the estimate L − (F * (e j )) ≤ 8πλ 4 (X) holds. The proposition is proved. 2

(g, m)-bundles over Riemann surfaces
We will consider bundles whose fibers are smooth surfaces or Riemann surfaces of type (g, m). Definition 1. (Smooth oriented (g, m) fiber bundles.) Let X be a smooth oriented manifold of dimension k, let X be a smooth (oriented) manifold of dimension k + 2 and P : X → X an orientation preserving smooth proper submersion such that for each point x ∈ X the fiber P −1 (x) is a closed oriented surface of genus g. Let E be a smooth submanifold of X that intersects each fiber P −1 (x) along a set E x of m distinguished points. Then the tuple F g,m = (X , P, E, X) is called a smooth (oriented) fiber bundle over X with fiber a smooth closed oriented surface of genus g with m distinguished points (for short, a smooth oriented (g, m)-bundle).
If m = 0 the set E is the empty set and we will often denote the bundle by (X , P, X). If m > 0 the mapping x → E x locally defines m smooth sections. (g, 0)-bundles will also be called genus g fiber bundles. For g = 1 and m = 0 the bundle is also called an elliptic fiber bundle.
In the case when the base manifold is a Riemann surface, a holomorphic (g,m) fiber bundle over X is defined as follows.
Definition 2. Let X be a Riemann surface, let X be a complex surface, and P a holomorphic proper submersion from X onto X, such that each fiber P −1 (x) is a closed Riemann surface of genus g. Suppose E is a complex one-dimensional submanifold of X that intersects each fiber P −1 (x) along a set E x of m distinguished points. Then the tuple F g,m = (X , P, E, X) is called a holomorphic (g,m) fiber bundle over X.
Two smooth (g, m)-bundles are smoothly isomorphic if and only if they are isotopic (see [21]).
Denote by S a reference surface of genus g with a set E ⊂ S of m distinguished points. By Ehresmann's Fibration Theorem each smooth (g, m)-bundle F g,m = (X , P, E, X) with set of distinguished points E x def = E ∩ P −1 (x) in the fiber over x is locally smoothly trivial, i.e. each point in X has a neighbourhood U ⊂ X such that the restriction of the bundle to U is isomorphic to the trivial bundle U × S, pr 1 , U × E, U with set {x} × E of distinguished points in the fiber {x} × S over x. Here pr 1 : U × S → U is the projection onto the first factor. The idea of the proof of Ehresmann's Theorem is the following. Choose smooth coordinates on U by a mapping from a rectangular box to U . Consider smooth vector fields v j on U , which form a basis of the tangent space at each point of U . Take smooth vector fields V j on P −1 (U ) that are tangent to E at points of this set and are mapped to v j by the differential of P. Such vector fields can easily be obtained locally. To obtain the globally defined vector fields V j on P −1 (U ) one uses partitions of unity. The required diffeomorphism ϕ U is obtained by composing the flows of these vector fields (in any fixed order).
In this way a trivialization of the bundle can be obtained over any simply connected domain. Let q 0 be a base point in X and γ j (t), t ∈ [0, 1], be smooth curves in X with base point q 0 that represent the generators e j of the fundamental group π 1 (X, q 0 ). For each j let ϕ t j : P −1 (q 0 ) → P −1 (γ j (t)), t ∈ [0, 1], ϕ 0 j = Id, be a smooth family of diffeomorphisms that map the set of distinguished points in P −1 (q 0 ) to the set of distinguishes points in P −1 (γ j (t)). To obtain such a family we may restrict the bundle to the closed curve given by γ j and lift the restriction to a bundle over the real axis R. The family of diffeomorphisms may be obtained by considering Ehrenpreis' vector field for the lifted bundle and take the flow of this vector field. The mapping ϕ 1 j obtained for t = 1 is an orientation preserving self-homeomorphism of the fiber over q 0 that preserves the set of distinguished points. Its isotopy class depends only on the homotopy class of the curve and the isotopy class of the bundle. The isotopy class of its inverse (ϕ 1 j ) −1 is called the monodromy of the bundle along e j . The following theorem holds (see e.g. [7] and [21] ).
Theorem G. Let X be a connected finite smooth oriented surface. The set of isotopy classes of smooth oriented (g, m) fiber bundles on X is in one-to-one correspondence to the set of conjugacy classes of homomorphisms from the fundamental group π 1 (X, q 0 ) into the modular group Mod(g, m) of a Riemann surface of genus g with m distinguished points.
The modular group Mod(g, m) is the group of isotopy classes of self-homeomorphisms of a reference Riemann surface of genus g that map a reference set of m distinguished points to itself.
A holomorphic bundle is called locally holomorphically trivial if it is locally holomorphically isomorphic to the trivial bundle. All fibers of a locally holomorphically trivial bundle are conformally equivalent to each other. Let 2g − 2 + m > 0. Restrict a locally holomorphically trivial (g, m)-bundle to a smooth closed curve γ(t), t ∈ [0, 1]. There is a smooth family of conformal mappings ϕ t from the fiber over γ(0) onto the fiber over γ(t), t ∈ [0, 1]. Then ϕ 1 is a conformal self-map of the fiber over γ(0), hence, a periodic mapping. This implies that the bundle is isotrivial. A smooth (holomorphic, respectively) bundle is called isotrivial, if it has a finite covering by the trivial bundle. Finally, if all monodromy mapping classes of a smooth bundle are periodic, then the bundle is isotopic (equivalently, smoothly isomorphic) to an isotrivial bundle. Isotrivial holomorphic bundles are locally holomorphically trivial.
We explain now the notion of irreducible smooth (g, m)-bundles. It is based on Thurston's notion of irreducible surface homeomorphisms. Let S be a connected finite smooth oriented surface. It is either closed or homeomorphic to a surface with a finite number of punctures. We will assume from the beginning that S is either closed or punctured.
A finite non-empty set of mutually disjoint Jordan curves {C 1 , . . . , C α } on a connected closed or punctured oriented surface S is called admissible if no C i is homotopic to a point in X, or to a puncture, or to a C j with i = j. Thurston calls an isotopy class of homeomorphisms m of S (in other words, a mapping class on S) reducible if there is an admissible system of curves {C 1 , . . . , C α } on S such that some (and, hence, each) element in m maps the system to an isotopic system. In this case we say that the system {C 1 , . . . , C α } reduces m. A mapping class which is not reducible is called irreducible.
Let S be a closed or punctured surface with set E of distinguished points. We say that ϕ is a self-homeomorphism of S with distinguished points E, if ϕ is a self-homeomorphism of S that maps the set of distinguished points E to itself. Notice that each self-homeomorphism of the punctured surface S \ E extends to a self-homeomorphism of the surface S with set of distinguished points E. We will sometimes identify self-homeomorphisms of S \ E and self-homeomorphism of S with set E of distinguished points.
For a (connected oriented closed or punctured) surface S and a finite subset E of S a finite non-empty set of mutually disjoint Jordan curves {C 1 , . . . , C α } in S \ E is called admissible for S with set of distinguished points E if it is admissible for S \ E. An admissible system of curves for S with set of distinguished points E is said to reduce a mapping class m on S with set of distinguished points E, if the induced mapping class on S \ E is reduced by this system of curves.
Conjugacy classes of reducible mapping classes can be decomposed in some sense into irreducible components, and conjugacy classes of reducible mapping classes can be recovered from the irreducible components up to products of commuting Dehn twists. Conjugacy classes of irreducible mapping classes are classified and studied.
A Dehn twist about a simple closed curve γ in an oriented surface S is a mapping that is isotopic to the following one. Take a tubular neighbourhood of γ and parameterize it as a round annulus A = {e −ε < |z| < 1} so that γ corresponds to |z| = e − ε 2 . The mapping is an orientation preserving self-homeomorphism of S which is the identity outside A and is equal to the mapping e −εs+2πit → e −εs+2πi(t+s) for e −εs+2πit ∈ A, i.e. s ∈ (0, 1). Here ε is a small positive number.
Thurston's notion of reducible mapping classes takes over to families of mapping classes on a surface of type (g, m), and therefore to (g, m)-bundles. Namely, an admissible system of curves on a (connected oriented closed or punctured) surface S with set of m distinguished points E is said to reduce a family of mapping classes m j ∈ M(S; ∅, E) if it reduces each m j . Similarly, a (g, m)-bundle with fiber S over the base point x 0 and set of distinguished points E ⊂ S is called reducible if there is an admissible system of curves in the fiber over the base point that reduces all monodromy mapping classes simultaneously. Otherwise the bundle is called irreducible.
Reducible bundles can be decomposed into irreducible bundle components and the reducible bundle can be recovered from the irreducible bundle components up to commuting Dehn twists in the fiber over the base point.
Let X be a finite open connected Riemann surface. By a holomorphic (smooth, respectively) (0, n)-bundle with a section over X we mean a holomorphic (smooth, respectively) (0, n + 1)bundle (X , P, E, X), such that the complex manifold (smooth manifold, respectively) E ⊂ X is the disjoint union of two complex manifolds (smooth manifolds, respectively)E and s, where E ⊂ X intersects each fiber P −1 (x) along a setE x of n points, and s ⊂ X intersects each fiber P −1 (x) along a single point s x . We will also say, that the mapping x → s x , x ∈ X, is a holomorphic (smooth, respectively) section of the (0, n)-bundle with set of distinguished points E x in the fiber over x.
A special (0, n + 1)-bundle is a bundle over X of the form (X × P 1 , pr 1 , E, X), where pr 1 : X × P 1 → X is the projection onto the first factor, and the smooth submanifold E of X × P 1 is equal to the disjoint unionE ∪ s ∞ where s ∞ intersects each fiber {x} × P 1 along the point {x} × {∞}, and the setE intersects each fiber along n points. A special (0, n + 1)-bundle is, in particular, a (0, n)-bundle with a section.
Theorem 2 is a consequence of the following theorem on (0, 3)-bundles with a section.
For a reducible (0, 4)-bundle the fiber of each irreducible bundle component is a thricepunctured Riemann sphere. Hence each irreducible bundle component of a reducible (0, 4)bundle is isotopic to an isotrivial bundle. For more details see [21].
Theorem 1 (with a weaker estimate) is a generalization of Theorem 3. Indeed, consider holomorphic (smooth, respectively) bundles whose fiber over each point x ∈ X equals P 1 with set of distinguished points {−1, 1, f (x), ∞} for a function f which depends holomorphically (smoothly, respectively) on the points x ∈ X and does not take the values −1 and 1. Then we are in the situation of Theorem 1. It is not hard to see that the mapping f is reducible, iff this (0, 4)-bundle is reducible (see also Lemma 4 of [21]).
The relation between Theorems 2 and 3 is given in Proposition 5 below. A holomorphic (1, 1)-bundle F = X , P, s, X is called a double branched covering of the special holomorphic (0, 4)-bundle X ×P 1 , pr 1 , E, X if there exists a holomorphic mapping P : X → X×P 1 that maps each fiber P −1 (x) of the (1, 1)-bundle onto the fiber {x}×P 1 of the (0, 4)bundle over the same point x, such that the restriction P : P −1 (x) → {x}×P 1 is a holomorphic double branched covering with branch locus being the set {x} × (E x ∪ {∞}) = E ∩ ({x} × P 1 ) of distinguished points in the fiber {x} × P 1 , and P maps the distinguished point s x in the fiber P −1 (x) over x to the point {x} × {∞} in {x} × P 1 . We will also denote (X × P 1 , pr 1 , E, X) by P((X , P, s, X)), and call the bundle (X , P, s, X) a lift of (X × P 1 , pr 1 , E, X). Let the fiber of the (1, 1)-bundle over the base point x 0 ∈ X be Y with distinguished point s, and let the fiber of the (0, 4)-bundle over x 0 be P 1 with distinguished pointsE ∪ {∞} for a set E ⊂ C 3 (C) S 3 . Then the monodromy mapping class m 1 ∈ M(P 1 ; ∞,E) of the (0, 4)-bundle along any generator of the fundamental group of X is the projection of the monodromy mapping class m ∈ M(Y ; s, ∅) of the (1, 1)-bundle along the same generator. This means that there are representing homeomorphisms ϕ ∈ m and ϕ 1 ∈ m 1 such that ϕ 1 (P(ζ)) = P(ϕ(ζ)), ζ ∈ Y . We will also say that m is a lift of m 1 . The lifts of a mapping class m 1 ∈ M(P 1 ; ∞,E) differ by the involution of Y , that interchanges the sheets of the double branched covering. Hence, each class m 1 ∈ M(P 1 ; ∞,E) has exactly two lifts.
Proposition 5. Let X be a Riemann surface of genus g with m + 1 ≥ 1 holes with base point x 0 and curves γ j representing a set of generators e j ∈ π 1 (X, x 0 ).
(2) Vice versa, for each special holomorphic (0, 4)-bundle over X and each collection m j of lifts of the 2g + m monodromy mapping classes m j 1 of the bundle along the γ j there exists a double branched covering by a holomorphic (1, 1)-bundle with collection of monodromy mapping classes equal to the m j . Each special holomorphic (0, 4)-bundle has exactly 2 2g+m non-isotopic holomorphic lifts. The proof of the proposition uses the fact that a holomorphic (1, 1)-bundle over X is holomorphically isomorphic to a holomorphic bundle whose fiber over each point x is a quotient C Λ x of the complex plane by a lattice Λ x with distinguished point 0 Λ x . The lattices depend holomorphically on the point x. To represent the fibers as branched coverings depending holomorphically on the points in X we use embeddings of punctured tori into C 2 by suitable versions of the Weierstraß ℘-function. For a detailed proof of Proposition 5 see [21]. Preparation of the proof of Theorem 3. The proof will be given in terms of braids. Let C n (C) = {(z 1 , . . . , z n ) ∈ C n : z j = z k for j = k} be the n-dimensional configuration space. The symmetrized configuration space is its quotient C n (C) S n by the diagonal action of the symmetric group S n . We write points of C n (C) as ordered n-tuples (z 1 , . . . , z n ) of points in C, and points of C n (C) S n as unordered tuples {z 1 , . . . , z n } of points in C. We regard geometric braids on n strands with base point E n as loops in the symmetrized configuration space C n (C) S n with base point E n , and braids on n strands (n-braids for short) with base point E n ∈ C n (C) S n as homotopy classes of loops with base point E n in C n (C) S n , equivalently, as element of the fundamental group π 1 (C n (C) S n , E n ) of the symmetrized configuration space with base point E n .
Each smooth mapping F : X → C n (C) S n defines a smooth special (0, n + 1)-bundle . Vice versa, for each smooth special (0, n + 1)-bundle (X × P 1 , pr 1 , E, X) the mapping that assigns to each point x ∈ X the set of finite distinguished points in the fiber over x defines a smooth mapping F : X → C n (C) S n . The mapping F is holomorphic iff the bundle is holomorphic. It is called irreducible iff the bundle is irreducible. Choose a base point q 0 ∈ X. The restriction of the mapping F to each loop with base point q 0 defines a geometric braid with base point F (q 0 ). The braid represented by it is called the monodromy of the mapping F along the element of the fundamental group represented by the loop.
The monodromy mapping classes of a special (0, n + 1)-bundle are isotopy classes of selfhomeomorphisms of the fiber P 1 over the base point q 0 which map the set of finite distinguished points E n = F (q 0 ) in this fiber onto itself, and fix ∞. Two smooth mappings F 1 and F 2 from X to C n (C) S n , that have equal value E n ∈ C n (C) S n at the base point q 0 , define isotopic special (0, n + 1)-bundles, iff their restrictions to each curve in X with base point q 0 define braids that differ by an element of the center Z n of the braid group B n on n strands (in other words, by a power of a full twist). Indeed, the braid group on n strands modulo its center B n Z n is isomorphic to the group of mapping classes of P 1 that fix ∞ and map E n to itself.
Note that for the group PB 3 of pure braids on three strands the quotient PB 3 Z 3 is isomorphic to the fundamental group of C \ {−1, 1}. The isomorphism maps the generators σ 2 j ∆ 2 3 , j = 1, 2, of PB 3 Z 3 to the standard generators a j , j = 1, 2, of the fundamental group of C\{−1, 1}. Here ∆ 2 3 denotes the group generated by ∆ 2 3 which is equal to the center Z 3 .
The proof of Theorem 3 will go now along the same lines as the proof of Theorem 1 with some modifications. Lemma H, Lemmas 4 and 5, and Theorem I below are given in terms of braids rather than in terms of elements of B 3 Z 3 .
The following lemma and the following theorem were proved in [18].
Lemma H. Any braid b ∈ B 3 which is not a power of ∆ 3 can be written in a unique way in the form where j = 1 or j = 2, k = 0 is an integer, is a (not necessarily even) integer, and b 1 is a word in σ 2 1 and σ 2 2 in reduced form. If b 1 is not the identity, then the first term of b 1 is a non-zero even power of σ 2 if j = 1, and b 1 is a non-zero even power of σ 1 if j = 2.
For an integer j = 0 we put q(j) = j if j is even, and for odd j we denote by q(j) the even integer neighbour of j that is closest to zero. In other words, q(j) = j if j = 0 is even, and for each odd integer j , q(j) = j − sgn(j), where sgn(j) for a non-zero integral number j equals 1 if j is positive, and −1 if j is negative. For a braid in form (17) is a power of ∆ 3 we put ϑ(b) def = Id. Let C n (R) S n be the totally real subspace of C n (C) S n . It is defined in the same way as C n (C) S n by replacing C by R. Take a base point E n ∈ C n (R) S n . The fundamental group π 1 ( C n (C) S n , E n ) with base point is isomorphic to the relative fundamental group π 1 ( C n (C) S n , C n (R) S n ) . The elements of the latter group are homotopy classes of arcs in C n (C) S n with endpoints in the totally real subspace C n (R) S n of the symmetrized configuration space.
Let b ∈ B n be a braid. Denote by b tr the element of the relative fundamental group π 1 ( C n (C) S n , C n (R) S n ) that corresponds to b under the mentioned group isomorphism. For a rectangle R in the plane with sides parallel to the axes we let f : R → C n (C) S n be a mapping which admits a continuous extension to the closureR (denoted again by f ) which maps the (open) horizontal sides into C n (R) S n . We say that the mapping represents b tr if for each maximal vertical line segment contained in R (i.e. R intersected with a vertical line in C) the restriction of f to the closure of the line segment represents b tr .
The extremal length of a 3-braid with totally real horizontal boundary values is defined as Λ tr (b) = inf{λ(R) : R a rectangle which admits a holomorphic map to C n (C) S n that represents b tr } .
Theorem I. Let b ∈ B 3 be a (not necessarily pure) braid which is not a power of ∆ 3 , and let w be the reduced word representing the image of ϑ(b) in PB 3 ∆ 2 3 . Then except in the case when b = σ k j ∆ 3 , where j = 1 or j = 2, k = 0 is an integer number, and is an arbitrary integer. In this case Λ tr (b) = 0.
The set H def = {{z 1 , z 2 , z 3 } ∈ C 3 (C) S 3 : the three points z 1 , z 2 , z 3 are contained in a real line in the complex plane} (18) is a smooth real hypersurface of C 3 (C) S 3 . Indeed, let {z 0 1 , z 0 2 , z 0 3 } be a point of the symmetrized configuration space. Introduce coordinates near this point by lifting a neighbourhood of the point to C 3 (C) with coordinates (z 1 , z 2 , z 3 ). Since the linear map M (z) def = z−z 1 z 3 −z 1 , z ∈ C, maps the points z 1 and z 3 to the real axis, the three points z 1 , z 2 , and z 3 lie on a real line in the complex plane iff the imaginary part of z 2 def = M (z 2 ) = z 2 −z 1 z 3 −z 1 vanishes. The equation Im z 2 −z 1 z 3 −z 1 = 0 in local coordinates (z 1 , z 2 , z 3 ) defines a local piece of a smooth real hypersurface. For each complex affine self-mapping M of the complex plane we consider the diagonal action The following two lemmas replace Lemma 1 in the case of (0, 3)-bundles with a section.
Lemma 4. Let A be an annulus, and let F : A → C 3 (C) S 3 be a holomorphic mapping whose image is not contained in H. Suppose L A is a simple relatively closed curve in A with limit points on both boundary circles of A, and F (L A ) ⊂ H. Suppose for a point q A ∈ L A the value F (L A ) is in the totally real subspace C 3 (R) S 3 . Let e A ∈ π 1 (A, q A ) be a generator of the fundamental group of A with base point q A . If the braid b def = F * (e A ) ∈ B 3 is different from σ k j ∆ 2 3 with j equal to 1 or 2, and k = 0 and being integers, then Notice that the braids σ k j ∆ 3 for odd are exceptional for Theorem I, but not exceptional for Lemma 4. The reason is that the braid in Lemma 4 is related to a mapping of an annulus, not merely to a mapping of a rectangle. For t ∈ [0, ∞) we put Lemma 5. If the braid in Lemma 4 equals b = σ k j σ k j ∆ 3 with j and j equal to 1 or to 2, j = j, and k and k being non-zero integers, and an even integer, then Here for a non-negative number x we denote by [x] the smallest integer not exceeding x.
Proof of Lemma 4. By the same argument as in the proof of Lemma 1 we may assume that F extends continuously to the closure A and the curve L A is a smooth (connected) curve in A whose endpoints are on different boundary components of A. The value of F at the point We restrict the mapping F to A \ L 0 . LetR be a lift of A \ L A to the universal coveringÃ of A. We considerR as curvilinear rectangle with horizontal sides being the two different lifts of L A and vertical sides being the lifts of the two boundary circles cut at the endpoints of L A . Take a closed curve γ A : [0, 1] → A in A with base point q A ∈ L A , that intersects L A only at the base point and represents the element e A ∈ π 1 (A, q A ). Letγ A be the lift of γ A for whichγ A ((0, 1)) is contained inR, and letF = (F 1 ,F 2 ,F 3 ) :R → C 3 (C) be a lift of F to a mapping fromR to the configuration space C 3 (C). The continuous extension ofF toR is also denoted byF . We may choose the lift so that the value ofF at the copy of q A on the lower horizontal side ofR equals (−1, q , 1). For each z ∈R we consider the complex affine , z ∈R, the result of applying A z to each of the three points of F (z). Then the mappingF (z) = {F 1 (z),F 2 (z),F 3 (z)} = {−1,F 2 (z), 1} is holomorphic onR. Since F (L A ) ⊂ H the mappingF (z) takes the horizontal sides ofR to the totally real subspace C 3 (R) S 3 of the symmetrized configuration space. Moreover,F (z) maps the copy of q A on the lower side ofR to {−1, q , 1}. LetF be the lift ofF to the configuration space which takes the value (−1, q , 1) at the copy of q A on the lower side ofR. The restrictions of F and ofF to the curve γ A represent elements of the relative fundamental group π 1 (C 3 (C) S 3 , C 3 (R) S 3 ). The represented elements of the relative fundamental group differ by a finite number of half-twists. Indeed, for each z, the liftsF (z) andF (z) differ by a complex affine mapping. Hence,F (γ A (t)) = b(t) + a(t)F (γ A (t)) for continuous functions a and b on [0, 1] with b(0) = 0, a(0) = 1, and b(1) and a(1) real valued. Then the function b : [0, 1] → C is homotopic with endpoints in R to the function that is identically equal to zero. The mapping a : [0, 1] → C \ {0} is homotopic with endpoints in R to a |a| . Hence, the mappingsF (γ A (t)) and a(t) |a|(t) F (γ A (t)) from [0, 1] to C 3 (C) S 3 are homotopic with endpoints in C 3 (R) S 3 . The statements follows.
Let ω(z) : A \ L A → R be the conformal mapping of the curvilinear rectangle onto the rectangle of the form R = {z = x + iy : x ∈ (0, 1), y ∈ (0, a)}, that maps the lower curvilinear side of A \ L 0 to the lower side of R. (Note that the number a is uniquely defined byR.) a ω(z)F (z). Then, for an integer number k the restrictions F | γ A andF | γ A represent the same element of π 1 (C 3 (C) S 3 , C 3 (R) S 3 )), namely b tr . We represented b tr by the holomorphic mapF from the rectangleR into C 3 (C) S 3 that maps horizontal sides into C 3 (R) S 3 . Hence, For b = σ k j ∆ 3 with j equal to 1 or 2, and k = 0 and being integers, the statement of Lemma 4 follows from Theorem I in the same way as Lemma 1 follows from Theorem F. For b = σ k j ∆ 3 with k = 0 the statement is trivial since then b = Id and L − (ϑ(Id)) = 0.
To obtain the statement in the remaining case b = σ k j ∆ 2 +1 3 with j equal to 1 or 2, and k and being integers, we use Lemma 5. Notice that σ 1 ∆ 3 = ∆ 3 σ 2 and σ 2 ∆ 3 = ∆ 3 σ 1 . Hence, with σ j = σ j . Let ω 2 : A 2 → A be the two-fold unbranched covering of A by an annulus A 2 . The equality λ(A 2 ) = 2λ(A) holds. Let q A 2 be a point in ω −1 2 (q A ), and let L q A 2 be the lift of L A to A 2 that contains q A 2 . Denote by γ A 2 the loop ω −1 2 (γ A ) with base point q A 2 . Then F • ω 2 | γ A 2 represents b 2 and (b 2 ) tr . Lemma 5 applied to σ k j σ k j ∆ 4 +2 3 gives the estimate 2 log , the inequality (19) follows. The lemma is proved. 2 Proof of Lemma 5. By [18], Lemma 1 and Proposition 6, statement 2, Since by (21) the inequality Λ tr (σ k j σ k j ∆ 3 ) ≤ λ(A) holds, the lemma is proved. 2 We want to emphasize that periodic braids are not non-zero powers of a σ j , so the lemma is true also for periodic braids. For each periodic braid b of the form vanishes. However, for instance for the conjugate σ −2k 2 )) equals 2 log(3|k|). Notice that the lemmas and Theorem I descend to statements on elements of B 3 Z 3 rather than on braids. For an element b of the quotient B 3 Z 3 we put ϑ(b) = ϑ(b) for any representative b ∈ B 3 of b.
Lemma 6 below is an analog of Lemma 3. It follows from Lemma 4 in the same way as Lemma 3 follows from Lemma 1. Lemma 6. Let X be a connected finite open Riemann surface, and F : X → C 3 (C) S 3 be a holomorphic map that is transverse to the hypersurface H in C 3 (C) S 3 . Suppose L 0 is a simple relatively closed curve in X such that F (L 0 ) is contained in H, and for a point q ∈ L 0 the point F (q) is contained in the totally real space C 3 (R) S 3 . Let e (1) and e (2) be primitive elements of π 1 (X, q). Suppose that for e = e (1) , e = e (2) , and e = e (1) e (2) the free homotopy class e intersects L 0 . Then either the two monodromies of F modulo the center F * (e (j) ) Z 3 , j = 1, 2, are powers of the same element σ j Z 3 of B 3 Z 3 , or each of them is the product of at most two elements b 1 and b 2 of B 3 Z 3 with where λ e (1) ,e (2) def = max{λ(A( e (1) )), λ(A( e (2) )), λ(A( e (1) e (2) ))}.
Proof. Let e ∈ π 1 (X, q) such that e intersects L 0 . As in the proof of lemma 1 there exists an annulus A, a point q A ∈ A, and a holomorphic map ω A : (A, q A ) → (X, q) that represents e. Moreover, the connected component of (ω A ) −1 (L 0 ) that contains q A has limit points on both boundary components of A. Put F A = F • ω A . By the conditions of Lemma 6 F A (L A ) = F (L 0 ) ⊂ H and F A (q A ) ∈ C 3 (C) S 3 . Let e A be the generator of π 1 (A, q A ) for which ω A (e A ) = e. The mapping F A : A → C 3 (C) S 3 , the point q A and the curve L A satisfy the conditions of Lemma 4. Notice that the equality (F A ) * (e A ) = F * (e) holds. Hence, if F * (e) is not a power of a σ j then inequality (19) holds for F * (e). Suppose the two monodromies modulo center F * (e (j) ) Z 3 , j = 1, 2, are not (trivial or nontrivial) powers of the same element σ j Z 3 of B 3 Z 3 . Then at most two of the elements, F * (e (1) ) Z 3 , F * (e (1) ) Z 3 , and F * (e (1) e (2) ) Z 3 = F * (e (1) ) Z 3 · F * (e (2) ) Z 3 , are powers of an element of the form σ j Z 3 .
If the monodromies modulo center along two elements among e (1) , e (2) , and e (1) e (2) are not (zero or non-zero) powers of a σ j Z 3 then by Lemma 4 for each of these two monodromies modulo center inequality (19) holds, and the third monodromy modulo center is the product of two elements of B 3 Z 3 for which inequality (19) holds. If the monodromies modulo center along two elements among e (1) and e (2) have the form σ k j Z 3 and σ k j Z 3 , then the σ j and the σ j are different and k and k are non-zero. The third monodromy modulo center has the form σ ±k j σ ±k j Z 3 (or the order of the two factors interchanged). Lemma 5 gives the (1) ,e (2) . Since L − (ϑ(σ ±k j )) = log(3[ k 2 ]) and L − (ϑ(σ ±k j )) = log(3[ k 2 ]), inequality (19) follows for the other two monodromies. The lemma is proved.
2 The following lemma holds.
Lemma 7. Let X be a connected finite open Riemann surface, and F : X → C 3 (C) S 3 a smooth mapping. Suppose for a base point q 1 of X each element of π 1 (X, q 1 ) can be represented by a curve with base point q 1 whose image under F avoids H. Then all monodromies of F are powers of the same periodic braid of period 3.
Proof. Take any monodromy of F . It has a power that is a pure 3-braid b, and a representative of b avoids H. Then for some integer l the first and the last strand of b ∆ 2l 3 are fixed, and a representative of b ∆ 2l 3 avoids H. Hence, b ∆ 2l 3 = Id and b = ∆ −2l 3 . We saw that the monodromy of F along each element e ∈ π 1 (X, q 1 ) is a periodic braid.
If a representative f : [0, 1] → C 3 (C) S 3 , f (0) = f (1), of a 3-braid b avoids H, then the associated permutation τ 3 (b) cannot be a transposition. Indeed, assume the contrary. Then there is a liftf of f to C 3 (C) S 3 , for which (f 1 (1),f 2 (1),f 3 (1)) = (f 3 (0),f 2 (0),f 1 (0)). Let L t be the line in C that containsf 1 (t) andf 3 (t), and is oriented so that running along L t in positive direction we meet firstf 1 (t) and thenf 3 (t). The point f 2 (0) is not on L 0 . Assume without loss of generality, that it is on the left of L 0 with the chosen orientation of L 0 . Since for each t ∈ [0, 1] the three pointsf 1 (t),f 2 (t) andf 3 (t) in C are not on a real line, the point f 2 (t) is on the left of L t with the chosen orientation. But the unorientated lines L 0 and L 1 coincide, and their orientation is opposite. This impliesf 2 (1) =f 2 (0), which is a contradiction. We proved that all monodromies are periodic with period 3.
There is a smooth homotopy F s , s ∈ [0, 1], of F , such that F 0 = F , each F s is different from F only on a small neighbourhood of q 1 , each F t avoids H on this neighbourhood of q 1 , and F 1 (q 1 ) is the set of vertices of an equilateral triangle with barycenter 0. Since F and F 1 are free homotopic, their monodromy homomorphisms are conjugate, and it is enough to prove the statement of the lemma for F 1 .
For notational convenience we will keep the notation F for the new mapping and assume that F (q 1 ) is the set of vertices of an equilateral triangle with barycenter 0. The monodromy F * (e) along each element e ∈ π 1 (X, q 1 ) is a periodic braid of period 3. Hence, τ 3 (F * (e)) is a cyclic permutation. Consider the braid g with base point F (0) that is represented by rotation by the angle 2π 3 , i.e. by the geometric braid t → e i2πt 3 F (0), t ∈ [0, 1], that avoids H. There exists an integer k such that F * (e) g k is a pure braid that is represented by a mapping that avoids H. Hence, F * (e) g k represents ∆ 2l 3 for some integer l. We proved that for each e ∈ π 1 (X, q 1 ) the monodromy F * (e) is represented by rotation of F (0) around the origin by the angle 2πj 3 for some integer j. The Lemma is proved. 2 Let as before X be a finite open connected Riemann surface. The following proposition is the main ingredient of the proof of Theorem 3. Let as before E ⊂ π 1 (X, q 0 ) be the system of generators of the fundamental group with base point q 0 ∈ X that was chosen in Section 1. Proposition 6. Let (X , P, X) be an irreducible holomorphic special (0, 4)-bundle over a finite open Riemann surface X, that is not isotopic to a locally holomorphically trivial bundle. Let F (z), z ∈ X, be the set of finite distinguished points in the fiber over z. Then there exists a complex affine mapping M and a point q ∈ X such that M •F (q) is contained in C 3 (R) S 3 , and for an arc α in X with initial point q 0 and terminating point q and each element e j ∈ Is αq (E) the monodromy modulo center (M •F ) * (e j ) Z 3 can be written as product of at most 6 elements b j,k , k = 1, 2, 3, 4, 5, 6, of B 3 Z 3 with Proof of Proposition 6. Since the bundle is not isotopic to a locally holomorphically trivial bundle, it is not possible that all monodromies are powers of the same periodic braid, and by Lemma 7 the set L (see (24)) is not empty. By the Holomorphic Transversality Theorem [22] the mapping F can be approximated on relatively compact subsets of X by holomorphic mappings to the symmetrized configuration space that are transverse to H. Similarly as in the proof of Proposition 4 we will therefore assume in the following (after slightly shrinking X to a deformation retract of X and approximating F ) that F is transverse to H. Consider the set 1. A torus with a hole. Let X be a torus with a hole and let E = {e 0 , e 0 } be the chosen set of generators of π 1 (X, q 0 ). There exists a connected component L 0 of L which is not contractible and not contractible to the hole. Indeed, otherwise there would be a base point q 1 and a curve α q 1 that joins q 0 with q 1 , such that for both elements of Is αq 1 (E) there would be representing loops with base point q 1 which do not meet L, and hence, by Lemma 7 the monodromies along both elements would be powers of a single periodic braid of period 3. Hence, as in the proof of Proposition 4 there exists a component L 0 of L which is a simple smooth relatively closed curve in X such that, perhaps after switching e 0 and e 0 and orienting L 0 suitably, the free homotopy class e 0 has positive intersection number with L 0 . Moreover, for one of the elements e ±1 0 or e 0 e 0 , denoted by e 0 the intersection number of e 0 with L 0 is positive.
Move the base point q 0 to a point q ∈ L 0 along a curve α, and consider the respective generators e = Is α (e 0 ) and e = Is α (e 0 ) of the fundamental group π 1 (X, q) with base point q.

A planar domain.
Let X be a planar domain. We represent X as the Riemann sphere with holes C j , j = 1, . . . , m+1, such that C m+1 contains ∞. Recall, that the set E of generators e j,0 , j = 1, . . . , m, of the fundamental group π 1 (X, q 0 ) with base point q 0 is chosen so that e j,0 is represented by a loop with base point q 0 that surrounds C j counterclockwise and no other hole.
Since the bundle is not isotopic to a locally holomorphically trivial bundle, Lemma 7 implies that L (see (24)) is not empty. Moreover, there is a connected component L 0 of L of one of the following kinds. Either L 0 has limit points on the boundary of two different holes (one of them may contain ∞) (first kind), or a component L 0 has limit points on a single hole C j , and C j ∪ L 0 divides the plane C into two connected components each of which contains a hole (maybe, only the hole containing ∞) (second kind), or there is a compact component L 0 that divides C into two connected components each of which contains at least two holes (one of them may contain ∞). Indeed, suppose each non-compact component of L has boundary points on the boundary of a single hole and the union of the component with the hole does not separate the remaining holes of X, and for each compact component of L one of the connected components of its complement in X contains at most one hole. Then there exists a base point q 1 , a curve α q 1 in X with initial point q 0 and terminating point q 1 , and a representative of each element of Is αq 1 (E) ⊂ π 1 (X, q 1 ) that avoids L. Lemma 7 implies that all monodromies modulo center are powers of a single periodic element of B 3 Z 3 which is a contradiction.
If there is a component L 0 of first kind we may choose the same set of primitive elements E 2 ⊂ E 2 ⊂ π 1 (X, q 0 ) as in the proof of Proposition 3 in the planar case. The free homotopy class of each element of E 2 and of the product of two such elements intersects L 0 . Moreover, each element of E is the product of at most two elements of E 2 . Let α q be a curve in X with initial point q 0 and terminating point q, and M a complex affine mapping, such that (M • F )(q) ∈ C 3 (R) S 3 . Since M • F is irreducible, the monodromies modulo center of M • F along the elements of Is α (E 2 ) are not (trivial or non-trivial) powers of a single element σ j Z 3 . Hence, for each element of Is α (E 2 ) there exists another element of Is α (E 2 ) so that the second option of Lemma 6 holds for this pair of elements of Is α (E 2 ). Therefore, the monodromy modulo center of M • F along each element of Is α (E 2 ) is the product of at most two elements of B 3 Z 3 of L − not exceeding 2πλ 4 (X), and the monodromy modulo center of M • F along each element of Is α (E) is the product of at most 4 elements of B 3 Z 3 of L − , each not exceeding 2πλ 4 (X).
Suppose there is no component of first kind but a component L 0 of the second kind. Assume first that all limit points of L 0 are on the boundary of a hole C j that does not contain ∞. Put j,0 e k,0 }. Each element of E 3 is a primitive element and is the product of at most three generators contained in the set E. Further, each element of E is the product of at most three elements of E 3 ∪ E 3 −1 . The free homotopy class of each element of E 3 and of each product of two different elements of E 3 intersects L 0 . Indeed, any curve that is contained in the complement of C j ∪ L 0 has either winding number zero around C j (as a curve in the complex plane C), or its winding number around C j coincides with the winding number around each of the holes in the bounded connected component of C j ∪ L 0 . On the other hand the representatives of the free homotopy class of e j,0 have winding number 1 around C j and winding number 0 around each other hole that does not contain ∞. The representatives of the free homotopy class of e 2 j,0 e k,0 , k ≤ m, k = j, have winding number 2 around C j , winding number 1 around C k , and winding number zero around each other hole C l , l ≤ m. The argument for products of two elements of E 3 is the same.
Choose a point q ∈ L 0 , a curve α in X with initial point q 0 and terminating point q, and a complex affine mapping M such that M • F (q) ∈ C 3 (R) S 3 . Lemma 6 finishes the proof for this case in the same way as in the case when there is a component of first kind. In the present case each (M •F ) * (ẽ) Z 3 ,ẽ ∈ Is α (E 3 ), can be written as a product of at most 2 factors b ∈ B 3 Z 3 with L − (ϑ(b)) ≤ 2πλ 6 (X). Hence, each (M • F ) * (e j ) Z 3 , e j = Is α (e j,0 ), can be written as a product of at most 6 factors b ∈ B 3 Z 3 with L − (ϑ(b)) ≤ 2πλ 6 (X).
Assume that the limit points of L 0 are on the boundary of the hole that contains ∞. Let C j 0 and C k 0 be holes that are contained in different components of X \ L 0 , and let e j 0 ,0 and e k 0 ,0 be the elements of E whose representatives surround C j 0 , and C k 0 respectively. Denote by E 3 the set that consists of the elements e j 0 ,0 e k 0 ,0 , e 2 j 0 ,0 e k 0 ,0 , and all elements e j 0 ,0 e k 0 ,0ẽ0 with e 0 running over E \ {e j 0 ,0 , e k 0 ,0 }. Each element of E 3 is the product of at most 3 elements of E, and each element of E is the product of at most 3 elements of E 3 ∪ (E 3 ) −1 .
Each element of E 3 and each product of at most two different elements of E 3 intersects L 0 . Indeed, if a closed curve is contained in one of the components of X \ L 0 then its winding number around each hole contained in the other component is zero. But for all mentioned elements there is a hole in each component of X \ L 0 such that the winding number of the free homotopy class of the element around the hole does not vanish. Lemma 6 applies with the same meaning of q, α, and M as before. Again, each (M • F ) * (e j ) Z 3 , e j = Is α (e j,0 ), can be written as a product of at most 6 factors b ∈ B 3 Z 3 with L − (ϑ(b)) ≤ 2πλ 6 (X).
Suppose there are no components of L of first or second kind, but there is a connected component L 0 of L of the third kind. Let C j 0 be a hole contained in the bounded component of the complement of L 0 , and let C k 0 , k 0 ≤ m, be a hole that is contained in the unbounded component of X \ L 0 . Let e j 0 ,0 and e k 0 ,0 be the elements of E whose representatives surround C j 0 , and C k 0 respectively. Consider the set E 4 consisting of the following elements: e j 0 ,0 e k 0 ,0 , e 2 j 0 ,0 e k 0 ,0 , and e 2 j 0 ,0 e k 0 ,0ẽ0 for eachẽ 0 ∈ E different from e j 0 ,0 and e k 0 ,0 . Each element of E 4 is the product of at most 4 elements of E and each element of E is the product of at most 3 elements of E 4 . The product of two different elements of E 4 is contained in E 8 .
The free homotopy classes of each element of E 4 and of each product of two different elements of E 4 intersects L 0 . Indeed, if a loop is contained in the bounded connected component of X \ L 0 its winding number around the holes C j j ≤ m, contained in the unbounded component is zero. If a loop is contained in the unbounded connected component of X \ L 0 its winding numbers around all holes contained in the bounded connected component are equal. But the winding number of e j 0 ,0 e k 0 ,0 and e 2 j 0 ,0 e k 0 ,0 around the hole C j 0 is positive and the winding number around the other holes that are contained in the bounded connected component of X \ L 0 vanishes, hence the representatives of these two element cannot be contained in the unbounded component of X \L 0 . Since the winding number of representatives of these element around C k 0 is positive the representatives cannot be contained in the bounded component of X \ L 0 . For representatives of the elements e 2 j 0 ,0 e k 0 ,0ẽ0 the winding number around C j 0 equals 2, the winding number around any other hole in the bounded component of X \ L 0 is at most 1, and the winding number around C k 0 equals 1. Hence, the free homotopy classes of the mentioned elements must intersect both components of X \ L 0 , hence they intersect L 0 .
Representatives of any product of two elements of E 4 have winding number around C j 0 at least 3, the winding number around any other hole in the bounded component of X \ L 0 is at most 2, and the winding number around C k 0 equals 2. Hence, the free homotopy classes of these elements intersect L 0 .
For a point q ∈ L 0 , a curve α in X joining q 0 and q, and a complex affine mapping M for which M • F (q) ∈ C 3 (C) S 3 , an application of Lemma 6 proves that in this case each (M • F ) * (e j ) Z 3 , e j = Is α (e j,0 ), can be written as a product of at most 6 factors b ∈ B 3 Z 3 with L − (ϑ(b)) ≤ 2πλ 8 (X). Proposition 6 is proved in the planar case. 3. The general case. As in the proof of Proposition 4 we may assume that for all generators e j,0 the monodromy along e j,0 is not the identity. We may also assume that for each handle the monodromies along the two elements of E corresponding to the handle are not powers of the same periodic element. Indeed, a power of a non-trivial periodic element of B 3 Z 3 is either equal to this element, or to its inverse, or to the identity. Hence, it is enough to prove the statement of the proposition for the monodromy of one of the generators corresponding to a handle, if both monodromies are powers of the same periodic element and are non-trivial.
Let now all monodromies be non-trivial, and suppose that the monodromies along any pair of elements of E corresponding to a handle are not powers of a single periodic braid. Choose a pair of elements e 0 , e 0 of E corresponding to a handle. If there is such a pair with monodromies not being powers of a single element of B 3 , we take such a pair.
We consider the covering ω e 0 ,e 0 : X( e 0 , e 0 ) → X. The covering manifold X( e 0 , e 0 ) is a torus with a hole. The preimage (ω e 0 ,e 0 ) −1 (L) is the set where the lift F e 0 ,e 0 = F • ω e 0 ,e 0 of F to X( e 0 , e 0 ) has values in H. As in the proof of Proposition 3 for the case of a torus with a hole we obtain a connected component L e 0 ,e 0 of (ω e 0 ,e 0 ) −1 (L) which (after suitable orientation) has positive intersection number with the free homotopy class of the lift of one of the generators, say of e 0 , and with the free homotopy class of the lift of an element e 0 which is one of the elements e 0 ± or e 0 e 0 .
Take a point q e 0 ,e 0 ∈ L e 0 ,e 0 . Choose a pointq ∈X, for which ω e 0 ,e 0 (q) = q e 0 ,e 0 . Letα be a curve inX with initial pointq 0 and terminating pointq. Then α e 0 ,e 0 def = ω e 0 ,e 0 (α) is a curve in X( e 0 , e 0 ) with initial point q 0 e 0 ,e 0 and terminating point q e 0 ,e 0 , and the curve α e 0 ,e 0 in X( e 0 , e 0 ) and the pointq 0 in the universal coveringX of X( e 0 , e 0 ) are compatible.
For each elementẽ 0 ∈ E there exists a setẼ ⊂ E that contains e 0 , e 0 ,ẽ 0 , such that X( Ẽ ) is a torus with m + 1 holes for 0 ≤ m ≤ 1 and (with the notation above) the monodromies modulo center of M • F along the elements of Is α (Ẽ) are not all (trivial or non-trivial) powers of a single element σ j Z 3 .
With the elementẽ * 0 ∈Ẽ mentioned above we take the setẼ 2 which consists of the following elements: the elementẽ * 0 , and for all elementsẽ 0 ofẼ \ {ẽ * 0 } the elementẽ 0 which is equal to (ẽ 0 ) ±1 or toẽ * 0ẽ 0 so that the intersection number of its free homotopy class in X( Ẽ ) with L Ẽ is positive. Lemma 6 implies the required statement along the same lines as in the proof of Proposition 4. We obtain the statement of Proposition 6 in the general case. Proposition 6 is proved. 2 Proof of Theorem 3. Let X be a connected Riemann surface of genus g with m+1 ≥ 1 holes. Since each holomorphic (0, 3)-bundle with a holomorphic section on X is isotopic to a holomorphic special (0, 4)-bundle, we need to estimate the number of isotopy classes of irreducible smooth special (0, 4)-bundles on X, that contain a holomorphic bundle. By Lemma 4 of [21] the monodromies of such a bundle are not powers of a single element of B 3 Z 3 which is conjugate to a σ j Z 3 , but they may be powers of a single periodic element of B 3 Z 3 (equivalently, the isotopy class may contain a locally holomorphically trivial holomorphic bundle).
Consider an irreducible special holomorphic (0, 4)-bundle on X which is not isotopic to a locally holomorphically trivial bundle. Let F (x), x ∈ X, be the set of finite distinguished points in the fiber over x. By Proposition 6 there exists a complex affine mapping M and a point q ∈ X such that M • F (q) is contained in C 3 (R) S 3 , and for an arc α in X with initial point q 0 and terminating point q and each element e j ∈ Is αq (E) the monodromy (M • F ) * (e j ) Z 3 of the bundle can be written as product of at most 6 elements b j,k , k = 1, 2, 3, 4, 5, 6, of Consider an isotopy class of special (0, 4)-bundles that corresponds to a conjugacy class of homomorphisms π 1 (X, q 0 ) → B 3 Z 3 whose image is generated by a single periodic element of B 3 Z 3 . Up to conjugacy we may assume that this element is one of the following: Id, ∆ 3 Z 3 , (σ 1 σ 2 ) Z 3 , (σ 1 σ 2 ) −1 Z 3 . For each of these elements b the equality L − (ϑ(b)) = 0 holds, hence, also in this case the inequality (25) is satisfied for each monodromy.
For a given element w ∈ B 3 Z 3 (including the identity) we describe now all elements b of B 3 Z 3 with ϑ(b) = w. If w = Id these are the following elements. If the first term of w equals a k j with k = 0, then the possibilities are b = w · (∆ 3 Z 3 ) with = 0 or 1, b = (σ sgnk j Z 3 ) · w · (∆ 3 Z 3 ) with = 0 or 1, or b = (σ ±1 j Z 3 ) · w · (∆ 3 Z 3 ) with = 0 or 1 and σ j = σ j . Hence, for w = Id there are 8 possible choices of elements b ∈ B 3 Z 3 with ϑ(b) = w.
If b = Id then the choices are ∆ Z 3 and (σ ±1 j ∆ ) Z 3 for j = 1, 2, and = 0 or = 1. These are 10 choices. Hence, there are no more than 15 exp(6πλ 8 Each monodromy is the product of at most six elements b j of B 3 Z 3 with L − (ϑ(b j )) ≤ 2πλ 8 (X). Hence, for each monodromy there are no more than (15 exp(6πλ 8 (X))) 6 possible choices. We proved that there are up to isotopy no more than (15 exp(6πλ 8 (X))) 6(2g+m) = (3 6 ·5 6 ·exp(36πλ 8 (X))) 2g+m irreducible holomorphic (0, 3)-bundles with a holomorphic section over X. Theorem 3 is proved. 2 Notice that we proved a slightly stronger statement, namely, over a Riemann surface of genus g with m + 1 ≥ 1 holes there are no more than (15 exp(6πλ 8 (X))) 6(2g+m) isotopy classes of smooth (0, 3)-bundles with a smooth section that contain a holomorphic bundle with a holomorphic section that is either irreducible or isotopic to the trivial bundle.
Proof of Theorem 2. Proposition 5 and Theorem 3 imply Theorem 2 as follows. Suppose an isotopy class of smooth (1, 1)-bundles over a finite open Riemann surface X contains a holomorphic bundle. By Proposition 5 the class contains a holomorphic bundle which is the double branched covering of a holomorphic special (0, 4)-bundle. If the (1, 1)-bundle is irreducible then also the (0, 4)-bundle is irreducible. There are up to isotopy no more than 15(exp(6πλ 8 (X))) 6(2g+m) holomorphic special (0, 4)-bundles over X that are either irreducible or isotopic to the trivial bundle.
By Theorem G there are no more than 15 exp(6πλ 8 (X))) 6(2g+m) conjugacy classes of monodromy homomorphisms that correspond to a special holomorphic (0, 4)-bundle over X that is either irreducible or isotopic to the trivial bundle. Each monodromy homomorphism of the holomorphic double branched covering is a lift of the respective monodromy homomorphism of the holomorphic special (0, 4)-bundle. Different lifts of a monodromy mapping class of a special (0, 4)-bundle differ by involution, and the fundamental group of X has 2g + m generators. Using Theorem G for (1, 1)-bundles, we see that there are no more than 2 2g+m 15(exp(6πλ 8 (X))) 6(2g+m) = 2 · 3 6 · 5 6 · exp(36πλ 8 (X)) 2g+m isotopy classes of (1, 1)-bundles that contain a holomorphic bundle that is either irreducible or isotopic to the trivial bundle. Theorem 2 is proved. 2 Remark. Each admissible system of curves on a once punctured torus consists of a single curve. This fact implies that each reducible smooth (1, 1)-bundle over a Riemann surface of genus g with m + 1 ≥ 1 holes has a single irreducible bundle component which is isotopic to a special (0, 3)-bundle and, hence, the irreducible bundle component is isotopic to an isotrivial bundle. There are 2 2g+m non-isotopic (0, 3)-bundles over a Riemann surface of genus g with m + 1 ≥ 1 holes. For details we refer to [21]. Moreover, a reducible (1, 1)-bundle can be recovered from the irreducible bundle component up to isotopy and multiplication of the monodromies by powers of a Dehn twist about a single curve in the fiber over the base point. As a consequence, the value 2 2g+m is an upper bound for the number of all reducible isotopy classes of (1, 1)bundles over a Riemann surface of genus g with m + 1 ≥ 1 holes, modulo multiplication of the monodromies by powers of a Dehn twist about a single curve in the fiber over the base point.
For convenience of the reader we give the short proof of the Corollaries 1 and 2. Such statements are known in principle, but the case considered here is especially simple.
Proof of Corollary 1. We will prove that on a punctured Riemann surface there are no non-constant reducible holomorphic mappings to the twice punctured complex plane and that any homotopy class of mappings from a punctured Riemann surface to the twice punctured complex plane contains at most one holomorphic mapping. This implies the corollary.
Recall that a holomorphic mapping f from any punctured Riemann surface X to the twice punctured complex plane extends by Picard's Theorem to a meromorphic function f c on the closed Riemann surface X c . Suppose now that X is a punctured Riemann surface and that the mapping f : X → C \ {−1, 1} is reducible, i.e. it is homotopic to a mapping into a punctured disc contained in C \ {−1, 1}. Perhaps after composing f with a Möbius transformation we may suppose that this puncture equals −1. Then the meromorphic extension f c omits the value 1. Indeed, if f c was equal to 1 at some puncture of X, then f would map a loop on X with non-zero winding number around the puncture to a loop in C \ {−1, 1} with non-zero winding number around 1 , which contradicts the fact that f is homotopic to a mapping onto a disc punctured at −1 and contained in C \ {−1, 1}. Hence, f c is a meromorphic function on a compact Riemann surface that omits a value, and, hence f is constant. Hence, on a punctured Riemann surface there are no non-constant reducible holomorphic mappings to C \ {−1, 1}.
Suppose f 1 and f 2 are non-constant homotopic holomorphic mappings from the punctured Riemann surface X to the twice punctured complex plane. Then for their meromorphic extensions f c 1 and f c 2 the functions f c 1 − 1 and f c 2 − 1 have the same divisor on the closed Riemann surface X c . Indeed, suppose, for instance, that f c 1 − 1 has a zero of order k > 0 at a puncture p. Then for the boundary γ of a small dics in X around p the curve (f 1 − 1) • γ in C \ {−2, 0} has index k with respect to the origin. Since f 2 − 1 is homotopic to f 1 − 1 as mapping to C \ {−2, 0}, the curve (f 2 − 1) • γ is free homotopic to (f 1 − 1) • γ . Hence, f 2 − 1 has a zero of order k at p. By the same reasoning we may show that f c 1 and f c 2 have the same divisor. Hence, they differ by a non-zero multiplicative constant. Since the functions are non-constant they must take the value −2. By the same reasoning as above the functions are equal to −2 simultaneously. Hence, the multiplicative constant is equal to 1. We proved that non-constant homotopic holomorphic maps from punctured Riemann surfaces to C \ {−1, 1} are equal. 2 Proof of Corollary 2. We have to prove, that any reducible holomorphic (1, 1)-bundle over a punctured Riemann surface X is holomorphically trivial, and that two isotopic (equivalently, smoothly isomorphic) holomorphically non-trivial holomorphic (1, 1)-bundles over X are holomorphically isomorphic. For a simple proof of the first fact the reader may consult [21]). The second fact is obtained as follows.
Suppose the holomorphically non-trivial holomorphic (1, 1)-bundles F j , j = 1, 2, have conjugate monodromy homomorphisms. By Proposition 5 each F j is holomorphically isomorphic to a double branched covering of a special holomorphic (0, 4)-bundle (X × P 1 , pr 1 , E j , X) def = P(F j ). The bundles P(F j ) are isotopic, since they have conjugate monodromy homomorphisms. There is a finite unramified coveringP :X → X of X, such the bundles P(F j ) have isotopic lifts (X × P 1 , pr 1 ,Ê j , X) toX for which the complex curveÊ j is the union of four disjoint complex curvesÊ k l , k = 1, 2, 3, 4, each intersecting each fiber {x} × P 1 along a single point (x,ĝ k j (x)). The mappingsX x →ĝ k j (x) are holomorphic. We may assume thatĝ 4 j (x) = ∞ for eachx. Define for j = 1, 2, a holomorphic isomorphism of the bundle (X × P 1 , pr 1 ,Ê j , X) by .
For x in a small open disc on X and x →x j (x), j = 1, 2, being two local inverses ofP the functions x →α(x j (x)) are two analytic functions whose ratio is contained in a finite set, hence the ratio is constant. Since the bundles F j , and, hence, also the P(F j ), are locally holomorphically non-trivial, the ratio of the two functions equals 1. We saw that for each pair of pointsx 1 ,x 2 ∈X, that project to the same point x ∈ X,α(x 1 ) =α(x 2 ). Put α(x) =α(x j ) for any pointx j ∈ (P) −1 (x). We obtain E 2 (x) = α(x)E 1 (x), that means, the bundles P(F j ) are holomorphically isomorphic. Since the bundles F j , j = 1, 2, are double branched coverings of the P(F j ) and have conjugate monodromy homomorphism, they are holomorphically isomorphic. 2 Proof of Proposition 1. Denote by S α a skeleton of T α,σ ⊂ T α which is the union of two circles each of which lifts under the covering P : C → T α to a straight line segment which is parallel to an axis in the complex plane. Denote the intersection point of the two circles by q 0 . Put S α = P −1 (S α ) and T α,σ = P −1 (T α,σ ). Note that T α,σ is the σ 2 -neighbourhood of S α . Denote by e the generator of π 1 (T α,σ , q 0 ), that lifts to a vertical line segment and e the generator of π 1 (T α,σ , q 0 ), that lifts to a horizontal line segment. Put E = {e, e }. We show first the inequality λ 3 (T α,σ ) ≤ 4(2α + 1) σ .
For this purpose we take any primitive element e of the fundamental group π 1 (T α,σ , q 0 ) which is the product of at most three factors, each of the factors being an element of E or the inverse of an element of E. We represent the element e by a piecewise C 1 mapping f 1 from an interval [0, l 1 ] to the skeleton S α . We may consider f 1 as a piecewise C 1 mapping from the circle R (x ∼ x + l 1 ) to the skeleton, and assume that for all points t of the circle where f 1 is not smooth, f 1 (t ) = q 0 . Let t 0 ∈ [0, l 1 ] be a point for which f 1 (t 0 ) = q 0 . Letf 1 be a piecewise smooth mapping from [t 0 , t 0 + l 1 ] to the universal covering C of T α ⊂ T α,σ for which P•f 1 : [t 0 , t 0 +l 1 ] → S α considered as a mapping from the circle R (x ∼ x+l 1 ) to the skeleton coincides with f 1 . We may take f 1 so that the equality |f 1 | = 1 holds. The mapping may be chosen so that l 1 ≤ 2α + 1. (Recall that α ≥ 1 and the element e is primitive.) Take any t for which f 1 is not smooth. We may assume that f 1 is chosen so that the direction off 1 changes by the angle ± π 2 at each such point. Hence, there exists a neighbourhood I(t ) of t on the circle, such that the restrictionf 1 |I(t ) covers two sides of a square of side length σ 2 . Denoteq 0 the common vertexf 1 (t ) of these sides, and byq 0 the vertex of the square that is not a vertex of one of the two sides. Replace the union of the two sides of the square that containq 0 by a quarter-circle of radius σ 2 with center at the vertexq 0 , and parameterize the latter by t → σ 2 e ±i 2 σ t so that the absolute value of the derivative equals 1. Notice that the quarter-circle is shorter than the union of the two sides.
Proceed in this way with all such points t . After a reparameterization we obtain a C 1 mappingf of the interval [0, l] of length l not exceeding 2α + 1 whose image is contained in the union of S α with some quarter-circles, such that |f | = 1. The distance of each point of the image off to the boundary of T α,σ is not smaller than σ 2 . The mappingf is piecewise of class C 2 . The normalization condition |f | = 1 implies |f | ≤ 2 σ . The projection f = P •f can be considered as a mapping from the circle R (x ∼ x + l) of length l not exceeding 2α + 1 to T α,σ , that represents the free homotopy class e of the chosen element of the fundamental group.
Consider the mapping x + iy →F (x + iy) def =f (x) + if (x)y ∈ C, where x + iy runs along the rectangle R l = {x + iy ∈ C : x ∈ [0, l], |y| ≤ σ 4 }. The image of this mapping is contained in T α,σ . Since 2 ∂ ∂zF (x + iy) = 2f (x) + if (x)y and 2 ∂ ∂zF (x + iy) = if (x)y the Beltrami coefficient µF (x + iy) = ≤ 2 (2α+1) σ that represents the free homotopy class of the element e of the fundamental group π 1 (X, q 0 ). Realize A l as an annulus in the complex plane. Let ϕ be the solution of the Beltrami equation on C with Beltrami coefficient µF on A l and zero else. Then the mapping g = F • ϕ −1 is a holomorphic mapping of the annulus ϕ(A l ) of extremal length not exceeding Kλ(A l ) ≤ 4(2α+1) σ into T α,σ that represents the chosen element of the fundamental group π 1 (T α,σ , q 0 ). Inequality (26) is proved.
We give now the proof of the lower bound. Let δ = 1 10 . We consider the annulus A α,δ def = {z ∈ C : |Rez| < 5δ 2 } (z ∼ z + αi). The extremal length of the annulus equals α. For any natural number j we consider all elements of π 1 (C \ {−1, 1}, 0) of the form containing 2j terms, each of the form a ±2 j . The choice of the sign in the exponent of each term is arbitrary. There are 2 2j elements of this kind. By [18] there is a relatively compact domain G in the twice punctured complex plane C \ {−1, 1} and a positive constant C such that the following holds. For each j, each element of the fundamental group of the form (27), and for each annulus of extremal length at least 2Cj there exists a base point q in the annulus, and a holomorphic mapping from the annulus to G that maps q to 0 and represents the element. Put j = [ α 10Cδ ], where [x] is the largest integer not exceeding a positive number x. Then each element of the form (27) with this number j can be represented by a holomorphic map g from the annulus A α,δ to G. There is a constant C 1 that depends only on G such that the mappingg satisfies the inequality |g| < C 1 . Let g be the lift ofg to a mapping on the strip {z ∈ C : |Rez| < 5δ 2 }. On the thinner strip {|Rez| < 3δ 2 } the derivative of g satisfies the inequality |g | ≤ C 1 δ . We will associate to the holomorphic mappingg on the annulus a smooth mapping g 1 from T α,δ to G, such that the monodromy of its class along the circle {Rez = 0} (z ∼ z + iα) with base point 0 (z ∼ z + iα) is equal to (27) and the monodromy along {Imz = 0} (z ∼ z + 1) with the same base point equals the identity. This is done as follows. Let F α = [− 1 2 , 1 2 ) × [− α 2 , α 2 ) ⊂ C be a fundamental domain for the projection P : C → T α . Put ∆ α,δ = F α ∩ P −1 (T α,δ ). Let χ 0 : [0, 1] → R be a non-decreasing function of class C 2 with χ 0 (0) = 0, χ 0 (1) = 1, χ 0 (0) = χ 0 (1) = 0 and |χ 0 (t)| ≤ 3 2 . Define χ : [ −3δ 2 , +3δ 2 ] → [0, 1] by