Configuration spaces of ℂ and ℂℙ^{1}: some analytic properties

Vladimir Lin

Department of Mathematics Max-Planck-Institut Technion-Israel Institute of für Mathematik Technology Vivatsgasse 7 32000 Haifa 53111 Bonn

Israel Germany

MPI2003-98

This work was started in 2000 and completed in 2003, both times during my stay at the Max-Planck-Institut für Mathematik in Bonn. I am deeply grateful to MPIM for

hospitality and especially for giving me a fortunate opportunity to meet in 2000 Marat Gizatullin, discuss with him some examples and learn from him about the Eisenstein automorphism of the space of non-degenerate binary cubic forms. This renovated my old interest to configuration spaces and stimulated the present work.

Abstract.

I mean... You know...

We study certain analytic properties of the ordered and unordered configuration spaces C_{o}^{n}(X) = {(q_{1}, ..., q_{n}) ∈ X^{n} | q_{i} ≠ q_{j} ∀ i ≠ j} and C^{n}(X) = {Q ⊂ X | #Q = n} of simply connected algebraic curves X = ℂ and X = ℂℙ^{1}. For n ≥ 3 the braid group B_{n}(X) = π_{1}(C^{n}(X)) of the curve X is non-abelian. A map F: C^{n}(X) → C^{k}(X) is called cyclic if the subgroup F_{∗}(π_{1}(C^{n}(X))) ⊂ π_{1}(C^{k}(X)) is cyclic; otherwise F is called non-cyclic. The diagonal Aut X action in X^{n} induces Aut X actions in C_{o}^{n}(X) and C^{n}(X). A morphism (holomorphic or regular) F: C^{n}(X) → C^{k}(X) is called orbit-like if F (C^{n}(X)) ⊂ (Aut X)Q^{∗} for some Q^{∗} ∈ C^{k}(X). An endomorphism F of C^{n}(X) is called tame if there is a morphism T: C^{n}(X) → Aut X such that F (Q) = T (Q)Q for all Q ∈ C^{n}(X). Tame endomorphisms preserve each Aut X orbit in C^{n}(X). We prove that for n > 4 and k ≥ t(X) := dim_{ℂ} Aut X an endomorphism F of C^{n}(X) is tame if and only if it is non-cyclic, and a morphism F: C^{n}(X) → C^{k}(X) is orbit-like if and only if it is cyclic (“Tame Map Theorem” and “Cyclic Map Theorem”). To study non-cyclic maps we establish first certain algebraic properties of the braid groups B_{n}(X), including the stability of the pure braid group P B_{n}(X) ⊂ B_{n}(X) under all non-cyclic endomorphisms of B_{n}(X). This implies that any non-cyclic endomorphism of C^{n}(X) lifts to an S(n) equivariant endomorphism of C_{o}^{n}(X). To study the latter one, we construct a new invariant of affine varieties Z. We endow the set L(Z) of all non-constant holomorphic functions Z → ℂ\{0, 1} with a certain natural structure of a (finite) simplicial complex. In many interesting cases the correspondence Z ↦ L(Z) is a contravariant functor (this is surely the case when restricting to dominant morphisms). We describe the simplices of the complex L(C_{o}^{n}(X)) explicitly, compute dim_{ℝ} L(C_{o}^{n}(X)) and show that any S(n) equivariant endomorphism of C_{o}^{n}(X) produces the corresponding simplicial self-map of L(C_{o}^{n}(X)). We determine the S(n) orbits of simplices Δ ⊂ L(C_{o}^{n}(X)) and find their normal forms with respect to this action. Eventually, this provides a complete description of S(n) equivariant endomorphisms of C_{o}^{n}(X) and proves Tame Map Theorem. This theorem shows that C^{n}(X))/ Aut C^{n}(X)) = C^{n}(X))/ Aut X and the holomorphic homotopy classes of non-cyclic endomorphisms of C^{n}(X) are in a natural 1−1 correspondence with the homotopy classes of all continuous maps C^{n}(X) → K(X), where K(X) is a maximal compact subgroup of Aut X. It also implies that dim_{ℂ} F (C^{n}(X)) ≥ n − t(X) + 1 for any n > 4 and any non-cyclic endomorphism F of C^{n}(X). The study of cyclic maps involves the Liouville property of C^{n}(X) and the holomorphic direct decomposition C_{o}^{k}(X) ≅ Aut X × C_{o}^{m}(ℂ \ {0, 1}), where m = k − t(X) and C_{o}^{m}(ℂ \ {0, 1}) is the ordered configuration space of ℂ \ {0, 1}. We prove that for n > t(X) + 1 and any non-cyclic morphism F: C^{n}(X) → C^{k}(X) there is a point Q ∈ C^{n}(X) such that Q ∩ F (Q) ≠ ∅ (“Linked Map Theorem”). The proof involves an “explicit” description of the holomorphic universal covering map Π: { \atop Aut X} ×T(0, m) → C^{n}(X), where m = n+3−t(X), { \atop Aut X} is the universal cover of Aut X and T(0, m) is the Teichmüller space of type (0, m). The construction of Π involves the universal Teichmüller family V′(0, m) → T(0, m). To prove Linked Map Theorem, we use the Hubbard-Earl-Kra theorem, which says that the universal Teichmüller family of type (g, m) has no holomorphic sections if dim_{ℂ} T(g, m) > 1. The above results apply to a version of the 13th Hilbert problem. We prove that for n > max{4, m + 2} the algebraic function [x : y] = U_{n}(z) on ℂℙ^{n} defined by the ”universal” equation z_{0}x^{n} + z_{1}x^{n−1}y + ... + z_{n}y^{n} = 0 cannot be represented as a composition of algebraic functions of m variables in such a way that a representing function-composition F ⊇ U_{n} and U_{n} itself have the same branch loci.

Contents

1. Introduction 7 1.1. Configuration spaces 7 1.2. The fundamental groups π_{1}(C^{n}(X)) and π_{1}(C_{o}^{n}(X)) 7 1.3. Tame, degenerate tame and orbit-like morphisms 7 1.4. Cyclic homomorphisms and cyclic maps 8 1.5. Main results 8 1.6. Some historical remarks 11 2. Strategy of the proof of Tame Map Theorem 12 2.1. Step 1: Lifting of non-cyclic continuous maps 13 2.2. Step 2: Commutator subgroup, homomorphisms B_{n}(X) → S(n) 13 2.3. Step 3: Strictly equivariant maps 14 2.4. Step 4: Coherence Theorem 16 2.5. Step 5: Simplicial complex of functions omitting two values 18 3. Holomorphic universal covers and universal Teichmüller families 20 3.1. Holomorphic universal coverings of certain configuration spaces 20 3.2. Homotopy groups and the standard normal series for P B_{n}(X) 23 4. Proof of Invariance Theorem 25 4.1. Canonical presentations of B_{n} and B_{n}(S^{2}) 25 4.2. Proof of Commutator Theorem 26 4.3. Proof of Surjectivity Theorem 2.7 26 4.4. Proof of Composition Theorem 28 4.5. Homomorphisms B_{n}(X) → B_{k}(X) for n > k 28 4.6. Proof of Equivariance Theorem 28 4.7. Proof of Kernel Theorem. 28 4.8. Proof of Invariance Theorem 29 5. Holomorphic functions omitting two values 29 5.1. Proof of Theorem 2.15 29 5.2. Simplicial complex of holomorphic functions omitting two values 33 5.3. Complexes of simple and cross ratios 34 5.4. S(n) action in L(q_{n}) 37 6. Coherence Theorem 39 6.1. Proof of Coherence Theorem for X = ℂ 41 6.2. Proof of Coherence Theorem for X = ℂℙ^{1} 42 6.3. Proof of Corollary 1.6 42 6.4. Proof of Corollary 1.7 42 7. Linked Map Theorem 42 7.1. Linked Map Theorem for C_{o}^{m}(ℂ^{◯ 2}) 43 7.2. Linked Map Theorem for C_{o}^{n}(X) 45 8. Morphisms C^{n}(X) → C^{k}(X) and Aut X orbits. Cyclic and orbit-like maps 45 8.1. Liouville and Picard spaces 45 8.2. Morphisms C^{n}(X) → C^{k}(X) and Aut X orbits 46 8.3. Stabilizers of points in C^{k}(X) 47 8.4. Cyclic Map Theorem 47 8.5. Holomorphic maps to the space of balanced configurations 48 8.6. Solvable holomorphic maps of Liouville spaces to C^{m}(ℂ) = G_{m} 50 8.7. Cyclic morphisms C^{n}(ℂ) → C^{k}(ℂ) 52 9. Morphisms C^{n}(ℂ) → C^{k}(ℂ) and special homomorphisms B_{n} → B_{k} 52 9.1. Special presentations and special homomorphisms 53 9.2. Holomorphic elements of homotopy groups 54 9.3. Holomorphic elements of π_{i}(C^{m}(ℂ)) 55 9.4. Special homomorphisms B_{n} → B_{k} and morphisms C^{n}(ℂ) → C^{k}(ℂ) 56 10. Dimension of the image 56 11. Some applications to algebraic functions 58 11.1. Algebraic functions with the universal branch locus 58 11.2. Compositions of multivalued functions 61 12. Automorphisms 64 12.1. How to catch automorphisms? 64 12.2. Some examples 65 13. Certain coverings of C^{n}(X) 66 13.1. Cyclic coverings 66 13.2. Transitive homomorphisms B_{n}(X) → S(k), k ≤ 2n 67 13.3. Non-cyclic coverings of C^{n}(X) 68 14. Configuration spaces of small dimension and some other examples 70 14.1. Ferrari map 70 14.2. Disjoint morphisms C^{3}(ℂ) → C^{6}(ℂ) and C^{3}(ℂ) → C^{9}(ℂℙ^{1}) 70 14.3. Eisenstein automorphism 72 14.4. Cayley’s form of the Eisenstein automorphism 73 14.5. Feler’s form of the Eisenstein automorphism 73

Bibliography 75

Index 79

1. Introduction

1.1. Configuration spaces. The (unordered) n^{th} configuration space C^{n}(X) of a space X consists of all n point subsets Q ⊆ X. It may be also viewed as the regular orbit space of the natural S(n) action in X^{n}. The standard left action of the symmetric group S(n) on X^{n} defined by

(σ, q) ↦ σq = (q_{σ^{−1}(1)}, ..., q_{σ^{−1}(n)}) ∀ σ ∈ S(n) and ∀ q = (q_{1}, ..., q_{n}) ∈ X^{n} (1.1)

is free on the ordered configuration space

C_{o}^{n}(X) = {q = (q_{1}, ..., q_{n}) ∈ X^{n} | q_{i} ≠ q_{j} ∀ i ≠ j};  (1.2)

this provides the unbranched S(n) Galois covering

\begin{matrix}p: C_{o}^{n}(X) → C^{n}(X) = C_{o}^{n}(X)/S(n) , \\ p(q) = p(q_{1}, ..., q_{n}) = \{q_{1}, ..., q_{n}\} = Q ∈ C^{n}(X) ∀ q = (q_{1}, ..., q_{n}) ∈ C_{o}^{n}(X) . \end{matrix} (1.3)

If X is a complex or complex algebraic manifold then both C_{o}^{n}(X) and C^{n}(X) are so.

This paper is devoted to the configuration spaces C^{n}(ℂ) and C^{n}(ℂℙ^{1}), which are irreducible complex affine algebraic manifolds of dimension n; C^{n}(ℂ) may be viewed as a Zariski open subset of C^{n}(ℂℙ^{1}). We are especially interested in the morphisms of these spaces in the analytic or algebraic category.

Throughout the paper, unless otherwise specified, we assume that n ≥ 3 and X is one of the curves ℂ, ℂℙ^{1}.

1.2. The fundamental groups π_{1}(C^{n}(X)) and π_{1}(C_{o}^{n}(X)). The fundamental group π_{1}(C^{n}(X)) and its normal subgroup π_{1}(C_{o}^{n}(X)) are quite the classical objects known as the braid group B_{n}(X) and the pure braid group P B_{n}(X) respectively. They fit into the exact sequence

1 → P B_{n}(X) ⟶^{p_{∗}} B_{n}(X) ⟶^{μ} S(n) → 1  (1.4)

related to the Galois covering (1.3); the epimorphism μ: B_{n}(X) → S(n) is called standard. In more details, π_{1}(C^{n}(ℂ)) = B_{n} and π_{1}(C_{o}^{n}(ℂ)) = P B_{n} are the classical Artin braid group and the pure Artin braid group respectively. Similarly, π_{1}(C^{n}(ℙℂ^{1})) = B_{n}(S^{2}) and π_{1}(C_{o}^{n}(ℙℂ^{1})) = P B_{n}(S^{2}) are the braid group and the pure braid group of the two dimensional sphere S^{2}. Notice that for n ≥ 3 the braid groups B_{n} and B_{n}(S^{2}) are non-abelian. Their abelianizations are cyclic: B_{n}/B′_{n} ≅ ℤ and B_{n}(S^{2})/B′_{n}(S^{2}) ≅ ℤ/(2n − 2)ℤ.

1.3. Tame, degenerate tame and orbit-like morphisms. Let Aut X be the automorphism group of X; thus, Aut X is either the group Aff (ℂ) of all affine transformations of ℂ or the group PSL(2, ℂ) of all Möbius transformations of the Riemann sphere. The diagonal Aut X action in C^{n}(X)

C^{n}(X) ∋ Q = {q_{1}, ..., q_{n}} ↦ {Aq_{1}, ..., Aq_{n}} =^{def} AQ ∈ C^{n}(X) ∀ A ∈ Aut X (1.5)

gives rise to the simplest examples of automorphisms of C^{n}(X): every A ∈ Aut X induces the automorphism C^{n}(X) ∋ Q ↦ AQ ∈ C^{n}(X). These examples may be modified by letting A = A(Q) depend on a point Q ∈ C^{n}(X) holomorphically and applying A(Q) either to Q itself or to a chosen and marked point Q^{∗} ∈ C^{n}(X). To be more precise, we give the following definition.

Definition 1.1. For any morphism T: C^{n}(X) → Aut X we define the corresponding tame endomorphism F_{T} : C^{n}(X) → C^{n}(X) by

F_{T} (Q) = T (Q)Q = {T (Q)q_{1}, ...., T (Q)q_{n}} ∀ Q = {q_{1}, ...., q_{n}} ∈ C^{n}(X) .  (1.6)

Choosing a point Q^{∗} = {q_{1}^{∗}, ...., q_{k}^{∗}} ∈ C^{k}(X), we define also the degenerate tame morphism F_{T,Q^{∗}}: C^{n}(X) → C^{k}(X),

F_{T,Q^{∗}}(Q) = T (Q)Q^{∗} = {T (Q)q_{1}^{∗}, ...., T (Q)q_{k}^{∗}} ∀ Q = {q_{1}, ...., q_{n}} ∈ C^{n}(X) ; (1.7)

if k = n, we obtain a degenerate tame endomorphism of C^{n}(X). Furthermore, a morphism F: C^{n}(X) → C^{k}(X) is said to be orbit-like if its image F (C^{n}(X)) is contained in an orbit of Aut X action in C^{k}(X). ◯

Remark 1.2. a) Of course, any degenerate tame morphism is orbit-like. Since the stabilizer St_{Q^{∗}} = {A ∈ Aut X | AQ^{∗} = Q^{∗}} of a generic point Q^{∗} ∈ C^{k}(X) is trivial, an orbit-like morphism whose image is contained in the orbit of such a point Q^{∗} is degenerate tame.

b) We may extend the above definition to continuous maps by saying that a continuous map T: C^{n}(X) → Aut X provides the tame continuous map F_{T} : C^{n}(X) → C^{n}(X) and the degenerate tame continuous map F_{T,Q^{∗}}: C^{n}(X) → C^{k}(X) defined by (1.6) and (1.7) respectively. Any tame map F keeps each orbit of the Aut X action in C^{n}(X) stable, that is, F ((Aut X)Q) ⊆ (Aut X)Q for all Q ∈ C^{n}(X). Any degenerate tame map is orbit-like, that is, it carries the whole space C^{n}(X) into a single Aut X orbit in C^{k}(X). These facts are of a little moment since tame and degenerate tame maps are the rarities in the class of all continuous maps of configuration spaces. However, due to Tame Map Theorem formulated below, a ”generic” endomorphism is tame and hence it keeps each Aut X orbit stable. Furthermore, by a quite elementary reason, any morphism C^{n}(X) → C^{k}(X) must carry Aut X orbits to Aut X orbits; see Section 8.2, Proposition 8.5. ◯

1.4. Cyclic homomorphisms and cyclic maps. It will be convenient to introduce the following notions.

Definition 1.3. A group homomorphism φ: G → H is said to be cyclic, abelian or solvable if its image φ(G) is a cyclic, abelian or solvable subgroup of the group H respectively. A homomorphism φ: G → H is abelian if and only if its restriction to the commutator subgroup G′ = [G, G] of the group G is trivial. If the abelianization G/G′ of G is a cyclic group then any abelian homomorphism of G is cyclic.

A continuous map F: X → Y of arcwise connected spaces is called cyclic, abelian or solvable if the induced homomorphism F_{∗}: π_{1}(X) → π_{1}(Y) of the fundamental groups is cyclic, abelian or solvable respectively. ◯

1.5. Main results. Any degenerate tame morphism is cyclic since π_{1}(Aut ℂ) ≅ ℤ and π_{1}(Aut ℂℙ^{1}) ≅ ℤ/2ℤ. In fact we will show in Section 8 that for n > 4 a morphism C^{n}(X) → C^{k}(X) is cyclic if and only if it is orbit-like.

In contrast to orbit-like morphisms, for n ≥ 3 any tame endomorphism of C^{n}(X) is non-cyclic (see Remark 2.9(b)). Our first main result tells that the for n > 4 the converse is also true.

Theorem 1.4 (Tame Map Theorem). For n > 4 every non-cyclic endomorphism F of C^{n}(X) is tame. That is, there exists a unique morphism T: C^{n}(X) → Aut X such that

F (Q) = F_{T} (Q) =^{def} T (Q)Q for all Q ∈ C^{n}(X) .

In particular, for n > 4 every automorphism of C^{n}(X) is tame. ◯

In view of the above notes about cyclic morphisms, this theorem shows that for endomorphisms of C^{n}(X) the properties to be tame or orbit-like form a dichotomy, which actually coincides with the dichotomy “non-cyclic – cyclic”:

Corollary 1.5. For n > 4 the classes of non-cyclic and cyclic endomorphisms of C^{n}(X) coincide with the classes of tame and orbit-like endomorphisms respectively. ◯

Tame Map Theorem has also the following two immediate consequences.

Corollary 1.6. For n > 4 the holomorphic homotopy classes of non-cyclic holomorphic maps^{1} C^{n}(X) → C^{n}(X) are in the natural one-to-one correspondence with the holomorphic homotopy classes of holomorphic maps C^{n}(X) → Aut X. By the famous Grauert theorem, the latter classes are in the natural one-to-one correspondence with the usual homotopy classes of all continuous maps C^{n}(X) → Aut X, or, which is the same, with the homotopy classes of all continuous maps C^{n}(X) → K(X), where K(X) is a maximal compact subgroup of the group Aut X. ◯

Corollary 1.7. Let n > 4 and G = Aut C^{n}(X) . Then the orbits of the natural G action in C^{n}(X) coincide with the orbits of the diagonal Aut X action in C^{n}(X).

In particular, the orbit space C^{n}(ℂℙ^{1})/ Aut C^{n}(ℂℙ^{1}) may be identified to the orbit space C^{n}(ℂℙ^{1})/PSL(2, ℂ), which, in turn, may be viewed as the moduli space M(0, n) of the Riemann sphere with n unordered punctures. ◯

Remark 1.8. For X = ℂ one may take K(ℂ) to be the subgroup 𝕋 ⊂ Aff ℂ consisting of all rotations z ↦ e^{it}z, t ∈ ℝ. Thereby, the holomorphic homotopy classes of non-cyclic holomorphic maps C^{n}(ℂ) → C^{n}(ℂ) are in the one-to-one correspondence with the homotopy classes of all continuous maps C^{n}(ℂ) → 𝕋, which, in turn, may be identified with the elements of the group H^{1}(C^{n}(ℂ), ℤ) ≅ Hom (B_{n}, ℤ) ≅ ℤ.

For X = ℂℙ^{1} one may take K(ℂℙ^{1}) ⊂ PSL(2, ℂ) to be the subgroup isomorphic to SO(3) and consisting of all Möbius transformations of the form

z ↦ \frac{az − c̄}{cz + ā} , |a|^{2} + |c|^{2} = 1 .

Thus, the holomorphic homotopy classes of non-cyclic holomorphic endomorphisms of C^{n}(ℂℙ^{1}) are in the one-to-one correspondence with the homotopy classes of all continuous maps C^{n}(ℂℙ^{1}) → SO(3). ◯ Remark 1.9. Consider the space of all non-zero complex binary forms of degree n

1That is, the classes of holomorphic maps homotopic to each other within the space of all holomorphic maps.

ϕ(x, y; z) = z_{0}x^{n} + z_{1}x^{n−1}y + ⋯ + z_{n−1}xy^{n−1} + z_{n}y^{n} ,  (1.8)

where z is regarded either as a non-zero point (z_{0}, ⋯, z_{n}) ∈ ℂ^{n+1} or as a point [z_{0} : ⋯: z_{n}] ∈ ℂℙ^{n}. In the latter case the form ϕ(x, y; z) is called projective, meaning that proportional forms are identified. The discriminant D_{n}(z) of a form ϕ(x, y; z) is a homogeneous polynomial in z_{0}, ..., z_{n} of degree 2(n − 1):

D_{n}(z)=|\begin{matrix}1 & z_{1} & z_{2} & ... & z_{n−2} & z_{n−1} & z_{n} & 0 & ... & 0 \\ 0 & z_{0} & z_{1} & ... & ... & ... & z_{n−1} & z_{n} & .... & 0 \\ ... & ... & ... & ... & ... & ... & ... & ... & ... & ... \\ 0 & 0 & 0 & ... & z_{0} & z_{1} & z_{2} & .... & z_{n−1} & z_{n} \\ n & (n − 1)z_{1} & (n − 2)z_{2} & ... & 2z_{n−2} & z_{n−1} & 0 & 0 & ... & 0 \\ 0 & nz_{0} & (n − 1)z_{1} & ... & ... & 2z_{n−2} & z_{n−1} & 0 & .... & 0 \\ ... & ... & ... & ... & ... & ... & ... & ... & ... & ... \\ 0 & 0 & 0 & ... & 0 & nz_{0} & (n − 1)z_{1} & ... & 2z_{n−2} & z_{n−1} \end{matrix} |;

ϕ(x, y; z) is said to be non-degenerate if D_{n}(z) ≠ 0. The set

F^{n} = {ϕ = ϕ(x, y; z) | z = [z_{0} : ... : z_{n}] ∈ ℂℙ^{n} , D_{n}(z) ≠ 0}  (1.9)

is called the space of all non-degenerate complex projective binary forms of degree n. For every Q = {q_{1}, ..., q_{n}} ∈ C^{n}(ℂℙ^{1}) there is a unique form ϕ = ϕ(x, y; z) ∈ F^{n} whose zero set Z_{ϕ} = {[x : y] ∈ ℂℙ^{1} | ϕ(x, y; z) = 0} coincides with Q. This provides the natural identification C^{n}(ℂℙ^{1}) = F^{n}.

Under this identification, the subspace C^{n}(ℂ) ⊂ C^{n}(ℂℙ^{1}) consists of all ϕ([x : y], z) ∈ F^{n} with z_{0} ≠ 0; it may be viewed as the space G_{n} of all complex polynomials p_{n}(t; w) = t^{n} + w_{1}t^{n−1} + ... + w_{n} with simple roots. We also regard G_{n} as a domain in ℂ^{n} defined by

G_{n} = {w = (w_{1}, ..., w_{n}) ∈ ℂ^{n} | d_{n}(w) ≠ 0} ,  (1.10)

where d_{n}(w) = d_{n}(w_{1}, ..., w_{n}) = D_{n}(1, w_{1}, ..., w_{n}) is the discriminant of p_{n}(t; w). The discriminant map

d_{n}: C^{n}(ℂ) = G_{n} ∋ w ↦ d_{n}(w) ∈ ℂ^{∗}  (1.11)

is a holomorphic locally trivial fiber bundle; the induced homomorphism of the fundamental groups d_{n∗} coincides with the abelianization homomorphism χ: B_{n} → ℤ.

In terms of binary forms, a tame endomorphism of C^{n}(ℂℙ^{1}) = F^{n} is just a projective change of variables [x : y] ↦ T (ϕ)[x : y], which depends holomorphically on a form ϕ itself. A tame endomorphism of the space C^{n}(ℂ) = G_{n} is the transformation of polynomials p_{n}(t; w) corresponding to an affine change of the variables T: t ↦ t′ = a(w)t + b(w) holomorphically depending on w ∈ G_{n}. ◯

Our second main result concerns morphisms of configuration spaces which do not necessarily coincide to each other. It will be convenient to make the following definition and notation.

Definition 1.10. Let F: C^{n}(X) → C^{k}(X) be a map of configuration spaces. Then either Q ∩ F (Q) = ∅ for all Q ∈ C^{n}(X) or there is a point Q_{0} ∈ C^{n}(X) such that Q_{0} ∩ F (Q_{0}) ≠ ∅. In the first case F is called disjoint; in the second case we say that F is linked.

Notation 1.11. We set t(X) = dim_{ℂ} Aut X, i. e., t(X) = 2 for X = ℂ and t(X) = 3 for X = ℂℙ^{1}. ◯

In all dimensions there are smooth maps C^{n}(X) → C^{k}(X) of both kinds. For holomorphic maps the situation changes drastically.

Theorem 1.12 (Linked Map Theorem). For n > t(X) + 1 and k ≥ 1 every holomorphic map F: C^{n}(X) → C^{k}(X) is linked. ◯

This theorem is actually a reflection of the fact that the universal Teichmüller family of type (0, n + 3) does not possess holomorphic sections. In Section 3 we will relate configuration spaces to Teichmüller spaces. The proof of Linked Map Theorem and some its applications to algebraic functions will be postponed till Section 7. Examples of disjoint holomorphic maps in small dimension will be presented in Section 14. In Section 11 we will discuss some applications of Linked Map Theorem to the 13th Hilbert problem for algebraic functions.

1.6. Some historical remarks. My interest to holomorphic mappings G_{n} → G_{k} of polynomials with simple roots was motivated by their relations to a version of the 13th Hilbert problem for algebraic functions. Endomorphisms of C^{n}(ℂ) = G_{n} were completely described in the 1970’s (see my papers [Lin72b, Lin72c, Lin79], where some results about morphsms G_{n} → G_{k} were obtained as well). However, the complete proofs were never published, since they contained some very long and unpleasant combinatorial computations.

In the 1990’s I found that certain morphisms G_{n} → G_{k} are related to the universal Teichmüller family of type (0, n + 1), which led to the proof of Linked Map Theorem for X = ℂ in the form presented in Remark 7.7(a) (see [Lin96b]).

On the other hand, in [Lin96b] (see also Section 14.2 of the present paper) a disjoint polynomial map F: C^{3}(C) → C^{6}(C) has been constructed. In view of the result mentioned above, the latter map is so special that its novelty must look dubious. During my stay at Max-Planck-Institut für Mathematik in Bonn in 2000, I met Marat Gizatullin (whose knowledge of and love to the classical mathematical literature are exceptional) and asked him whether he have seen such a map somewhere. After a few days he came back and told that he did not meet such a monster but instead he found out that in 1844 Gotthold Eisenstein has discovered a very exciting automorphism of non-degenerate binary cubic forms (see [Eis1844] and Section 14.3 of the present paper).

This Eisenstein’s example renovated my interest to endomorphisms of configuration spaces. In the present work my old approach is improved and simplified and now it applies to both C^{n}(ℂ) and C^{n}(ℂℙ^{1}).

It should be also mentioned that endomorphisms of the configuration spaces C^{n}(ℂ^{∗}) have been described by V. Zinde [Zin77].

Recently my student Y. Feler proved that for n > 4 all automorphisms of the configuration space C^{n}(𝕋^{2}) of a torus 𝕋^{2} = ℂ/(aℤ + bℤ) are tame. Together with my old results quoted above, results of V. Zinde, [Zin77], and the ones of the present paper, this gives a realistic hope to complete the description of endomorphisms of configuration spaces for all non-hyperbolic Riemann surfaces. Furthermore, Feler studied certain configuration spaces of different kind, say the spaces C_{aff}^{n}(ℂ^{2}) and C_{prj}^{n}(ℂℙ^{2}) consisting of subsets Q = {q_{1}, ..., q_{n}} in ℂ^{2} or in ℂℙ^{2} whose points q_{1}, ..., q_{n} are in general affine and general projective position respectively. (My attention to such spaces was attracted by A. M. Vershik.) We have in mind to publish these and some other results about configuration spaces in a joint paper.

2. Strategy of the proof of Tame Map Theorem

The question about the nature of endomorphisms of configuration spaces belongs to analytic or algebraic geometry and seemingly is a rather complicated one. We will reduce it, step by step, to a series of algebraic or combinatorial tasks, which look and as a matter of fact are simpler.

In this section we try first to find out what the inevitable attributes of tame endomorphisms and those of certain accompanying objects ought to be. Then we formulate the statements which should be proved in order to ensure that non-cyclic endomorphisms and the corresponding objects of their “suite” actually possess such properties. The accomplishment of this approach in the consequent sections eventually will lead to the desired result.

Some proofs named “Explanation” also are placed here but only when they are completely trivial. We start with an obvious observation.

Lifting and invariance properties. Tame and degenerate tame endomorphisms F of C^{n}(X) have in common the following Lifting property: there exists an endomorphism f: C_{o}^{n}(X) → C_{o}^{n}(X) which fits into the commutative diagram

[figure: C_{o}^{n}(X); f; C_{o}^{n}(X); p; p; C^{n}(X); F; C^{n}(X)]  (2.1)

Such an f may be defined as follows:

f(q) = f_{T} (q) =^{def} T (Q)(q) = (T (Q)q_{1}, ..., T (Q)q_{n}) whenever F = F_{T} ,  (2.2)

and

f(q) = f_{T,Q^{∗}}(q) =^{def} T (Q)(q^{∗}) = (T (Q)q_{1}^{∗}, ..., T (Q)q_{n}^{∗}) whenever F = F_{T,Q^{∗}} , (2.3)

where q = (q_{1}, ..., q_{n}) ∈ C_{o}^{n}(X), Q = {q_{1}, ..., q_{n}} = p(q) ∈ C^{n}(X). ◯

By the covering map theorem, a continuous map F: C^{n}(X) → C^{n}(X) possesses a continuous covering map f that fits into diagram (2.1) if and only if the following invariance condition is fulfilled:

Invariance property: the induced endomorphism F_{∗}: π_{1}(C^{n}(X)) → π_{1}(C^{n}(X)) of the fundamental group respects the normal subgroup π_{1}(C_{o}^{n}(X)) ◁ π_{1}(C^{n}(X)), that is,

F_{∗}(π_{1}(C_{o}^{n}(X))) ⊆ π_{1}(C_{o}^{n}(X)) .  (2.4)

2.1. Step 1: Lifting of non-cyclic continuous maps. Taking into account that tame endomorphisms of C^{n}(X) must lift to the covering C_{o}^{n}(X), we are going to prove the following

Theorem 2.1 (Lifting Theorem). For n > 4 every non-cyclic continuous map F: C^{n}(X) → C^{n}(X) lifts to a continuous map f: C_{o}^{n}(X) → C_{o}^{n}(X) so that F and f fit into commutative diagram (2.1). Clearly f is an endomorphism if F is so. ◯

In view of the covering mapping theorem, this topological result is equivalent to the following algebraic

Theorem 2.2 (Invariance Theorem). Let n > 4. Then any non-cyclic endomorphism φ of the group B_{n}(X) respects the normal subgroup P B_{n}(X), i. e., φ(P B_{n}(X)) ⊆ P B_{n}(X). In fact P B_{n}(X) = φ^{−1}(P B_{n}(X)), which certainly implies the previous inclusion. ◯

Invariance Theorem is, in turn, an immediate consequence of the following two results:

Theorem 2.3 (Composition Theorem). Let n > 4 and let φ: B_{n}(X) → B_{k}(X) be a non-cyclic homomorphism. Then the composition

ψ = μ ∘ φ: B_{n}(X) ⟶^{φ} B_{k}(X) ⟶^{μ} S(k)  (2.5)

of φ and the standard epimorphism μ: B_{n}(X) → S(n) is non-cyclic as well. ◯

Theorem 2.4 (Kernel Theorem). For n > 4 the kernel Ker ψ of any non-cyclic homomorphism ψ: B_{n}(X) → S(n) coincides with the pure braid group P B_{n}(X). ◯

Explanation. For a non-cyclic φ: B_{n}(X) → B_{n}(X) Composition Theorem shows that ψ = μ ∘ φ: B_{n}(X) → S(n) is non-cyclic and, by Kernel Theorem, P B_{n}(X) = Ker ψ = Ker(μ ∘ φ) = φ^{−1}(Ker μ) = φ^{−1}(P B_{n}(X)), which is exactly the statement of Invariance Theorem. □

2.2. Step 2: Commutator subgroup, homomorphisms B_{n}(X) → S(n). To prove Composition and Kernel Theorems, we establish first some other important properties of braid groups B_{n}(X).

Recall that a group G is said to be perfect if it coincides with its commutator subgroup G′ = [G, G], or, which is the same, G does not possess non-trivial homomorphisms to abelian groups.

We extensively make use of the following theorem proved in Section 4.2:

Theorem 2.5 (Commutator Theorem). The commutator subgroup B′_{n}(X) of the braid group B_{n}(X) is a perfect group whenever n > 4. ◯

In the next section we show (Theorem 3.3) that for any n ≥ 2 perfect groups do not admit non-trivial homomorphisms into the pure braid group P B_{k}(X). Together with Commutator Theorem, this implies:

Theorem 2.6 (homomorphisms B′_{n}(X) → P B_{k}(X)). For n > 4 the commutator subgroup B′_{n}(X) of the braid group B_{n}(X) does not possess non-trivial homomorphisms to the pure braid group P B_{k}(X). ◯

Furthermore, in Section 4.3 we prove the following theorem:

Theorem 2.7 (Surjectivity Theorem). For n ≠ 4 every non-cyclic homomorphism ψ: B_{n}(X) → S(n) is surjective. ◯

These results, together with the classical Artin theorem about transitive representations B_{n} → S(n) (see [Art47b, Lin96b]), lead to the proof of Composition and Kernel Theorems and thereby Invariance and Lifting Theorems as well (see Section 4 for the proofs).

2.3. Step 3: Strictly equivariant maps. With Lifting Theorem at our disposal, we proceed as follows. Since (1.3) is a Galois S(n) covering, any continuous lifting

f = (f_{1}, ..., f_{n}): C_{o}^{n}(X) → C_{o}^{n}(X)  (2.6)

of a continuous map

F: C^{n}(X) → C^{n}(X)  (2.7)

is equivariant in the following sense:

Equivariance condition: there exists an endomorphism α: S(n) → S(n) such that

f(σq) = α(σ)f(q) for all q ∈ C_{o}^{n}(X) and σ ∈ S(n) ,  (2.8)

or, in terms of the components of f,

f_{m}(σq) = f_{[α(σ)]^{−1}(m)}(q) for all m = 1, ..., n , q ∈ C_{o}^{n}(X) , σ ∈ S(n) .

Up to conjugation, the endomorphism α is uniquely defined by the original map F. That is, if endomorphisms α′, α′′ correspond to two liftings f′, f′′ of F then there is a permutation s ∈ S(n) such that α′′(σ) = s^{−1}α′(σ)s for all σ ∈ S(n).

Explanation. Given F which lifts to C_{o}^{n}(X), we may construct α as follows. Taking a continuous lifting f and choosing basepoints q^{∘} ∈ C_{o}^{n}(X) and Q^{∘} = p(q^{∘}) ∈ C^{n}(X) we obtain the commutative diagram

\begin{matrix}π_{1}(C_{o}^{n}(X), q^{∘}) & ^{^{f_{∗}}  \overline{}} & π_{1}(C_{o}^{n}(X), f(q^{∘})) \\ _{_{p_{∗}} } & _{_{p_{∗}}} \\ π_{1}(C^{n}(X), Q^{∘}) & \overline{}_{F_{∗}} & π_{1}(C^{n}(X), F (Q^{∘})) \\ _{_{μ′} } & _{_{μ′′}} \\ π_{1}(C^{n}(X), Q^{∘})/π_{1}(C_{o}^{n}(X), q^{∘}) ^{^{α} \overline{}}π_{1}(C^{n}(X), F (Q^{∘}))/π_{1}(C_{o}^{n}(X), f(q^{∘})), \\  &  \\ 1  1 \end{matrix} (2.9)

where μ′ and μ′′ are the natural epimorphisms onto the quotient groups and α is a unique homomorphism which makes the lower rectangle commutative. The latter quotient groups are identified with the symmetric groups of the ordered sets q^{∘} = (q_{1}^{∘}, ..., q_{n}^{∘}) and f(q^{∘}) = (f_{1}(q^{∘}), ..., f_{n}(q^{∘})), that is, with S(n); thus, α becomes an endomorphism of S(n). Another choice of f and/or a basepoint q^{∘} ∈ C_{o}^{n}(X) may lead to another endomorphism which must be conjugate to α. Furthermore, connecting f(q^{∘}) to q^{∘} and F (Q^{∘}) to Q^{∘} with paths γ: [0, 1] → C_{o}^{n}(X) and p ∘ γ: [0, 1] → C^{n}(X) we come to identifications of pairs

\begin{matrix}\{π_{1}(C_{o}^{n}(X), q^{∘}) ◁ π_{1}(C^{n}(X), Q^{∘})\} ≅ \{P B_{n}(X) ◁ B_{n}(X)\} and \\ \{π_{1}(C_{o}^{n}(X), f(q^{∘})) ◁ π_{1}(C^{n}(X), F (Q^{∘}))\} ≅ \{P B_{n}(X) ◁ B_{n}(X)\} \end{matrix}

that conform to each other. Thereby (2.9) leads to the commutative diagram

\begin{matrix}1 \overline{}P B_{n}(X) ^{^{p_{∗}} \overline{}}B_{n}(X) ^{^{μ} \overline{}}S(n) \overline{}1 \\ _{_{f_{∗}} } & _{_{F_{∗}} } & _{_{α}} \\ 1 \overline{}P B_{n}(X) ^{^{p_{∗}} \overline{}}B_{n}(X) ^{^{μ} \overline{}}S(n) \overline{}1 \end{matrix}  (2.10)

whose horizontal lines are exact sequences (1.4) of the covering p: C_{o}^{n}(X) → C^{n}(X) corresponding to the two pairs of basepoints {Q^{∘}, q^{∘}} and {F (Q^{∘}), f(q^{∘})}. □

Definition 2.8. A continuous map f: C_{o}^{n}(X) → C_{o}^{n}(X) is said to be strictly equivariant if there exists an automorphism α of the group S(n) such that the above mentioned condition

f(σq) = α(σ)f(q) for all q ∈ C_{o}^{n}(X) and σ ∈ S(n)  (2.9)

is fulfilled. ◯

Remark 2.9. a) For n ≠ 6 any automorphism of S(n) is inner, and therefore (2.8) takes the form

∃ s ∈ S(n) such that f(σq) = sσs^{−1}f(q) ∀ q ∈ C_{o}^{n}(X) and ∀ σ ∈ S(n) . (2.11)

In fact any strictly equivariant endomorphism of C_{o}^{n}(X) satisfies (2.11) for all n ≥ 3 (see Explanation in Section 2.4).

b) It is easily seen from diagram (2.10) that a tame endomorphism F = F_{T} of C^{n}(X) must be non-cyclic. Indeed, let f = f_{T} be the lifting of F defined according to (2.2), that is, f(q) = T (p(q))q = (T (p(q))q_{1}, ..., T (p(q))q_{n}) for q = (q_{1}, ..., q_{n}). As we know, f is equivariant; let α be the corresponding endomorphism of S(n). Then for any σ ∈ S(n) and all q ∈ C_{o}^{n}(X) we have α(σ)f(q) = f(σq) = T (p(σq))σq = σT (p(q))q = σf(q); thereby α(σ) = σ and α = id_{S(n)}. In particular, α is non-cyclic, and (2.10) implies that F_{∗} is non-cyclic. ◯

The following result is an easy consequence of Composition Theorem.

Theorem 2.10 (Equivariance Theorem). Let n > 4. Then any continuous lifting f: C_{o}^{n}(X) → C_{o}^{n}(X) of a non-cyclic continuous map F: C^{n}(X) → C^{n}(X) is strictly equivariant. ◯

Explanation. Since F is non-cyclic, the induced homomorphism F_{∗} in (2.10) is noncyclic and, by Composition Theorem, the composition α ∘ μ = μ ∘ F_{∗} is non-cyclic as well. Thus, α is non-cyclic, that is, its image α(S(n)) ≅ S(n)/ Ker α is a noncyclic group. For n > 4 the group S(n) possesses a single proper normal subgroup, the alternating group A(n), and the quotient group S(n)/A(n) ≅ ℤ/2ℤ is cyclic. Therefore Ker α = {1} and α is an automorphism. □

Definition 2.11. A morphism τ: C_{o}^{n}(X) → Aut X is called S(n) invariant if it is constant on each S(n) orbit in C_{o}^{n}(X), that is,

τ(σq) = τ(q) for all q ∈ C_{o}^{n}(X) and σ ∈ S(n) .

An endomorphism f: C_{o}^{n}(X) → C_{o}^{n}(X) is said to be tame if there exists an S(n) invariant morphism τ: C_{o}^{n}(X) → Aut X and a permutation s ∈ S(n) such that

f(q) = f_{τ}^{s}(q) =^{def} sτ(q)q = (τ(q)q_{s^{−1}(1)}, ..., τ(q)q_{s^{−1}(n)}) ∀ q = (q_{1}, ..., q_{n}) ∈ C_{o}^{n}(X) .

Similarly, we say that f is degenerate tame if there are q^{∘} ∈ C_{o}^{n}(X), s ∈ S(n) and an S(n) invariant morphism τ: C_{o}^{n}(X) → Aut X such that

f(q) = f_{τ,q^{∘}}^{s}(q) =^{def} sτ(q)q^{∘} ∀ q ∈ C_{o}^{n}(X) . ◯

Any tame endomorphism f = f_{τ}^{s} of C_{o}^{n}(X) may be pushed down to a unique tame endomorphism F = F_{T} of C^{n}(X) such that f and F fit into (2.1); T and F are defined by

T (Q) = τ(q) and F (Q) = F_{T} (Q) = T (Q)Q = pf_{τ}^{s}(q) ,  (2.12)

where Q ∈ C^{n}(X) and q is any point of the fiber p^{−1}(Q) ⊂ C_{o}^{n}(X).

Lifting and Equivariance Theorems show that in order to prove Tame Map Theorem it suffices to prove the following

Theorem 2.12 (Equivariant Map Theorem). For n > 6 − t(X) every strictly equivariant endomorphism of C_{o}^{n}(X) is tame. ◯

2.4. Step 4: Coherence Theorem. The group Aut X is exactly t(X) times transitive in X (see Notation 1.11). Let n > t(X). Two ordered n point subsets x = (x_{1}, ..., x_{n}), y = (y_{1}, ..., y_{n}) ⊂ X may be carried to each other by an automorphism A ∈ Aut X if and only if they satisfy

\frac{y_{1} − y_{m}}{y_{1} − y_{2}} = \frac{x_{1} − x_{m}}{x_{1} − x_{2}} for all m = 3, ..., n when X = ℂ

and

\frac{y_{m} − y_{1}}{y_{m} − y_{2}} : \frac{y_{1} − y_{3}}{y_{2} − y_{3}} = \frac{x_{m} − x_{1}}{x_{m} − x_{2}} : \frac{x_{1} − x_{3}}{x_{2} − x_{3}} for all m = 4, ..., n when X = ℂℙ^{1} .

When the corresponding condition is fulfilled a required A ∈ Aut X is uniquely defined and regular in x, y ∈ C_{o}^{n}(X).

Let n > t(X) and let f = (f_{1}, ..., f_{n}) be an endomorphism of C_{o}^{n}(X). If we are looking for a representation f = f_{τ} with a morphism τ: C_{o}^{n}(X) → Aut X, we must for every q = (q_{1}, ..., q_{n}) ∈ C_{o}^{n}(X) find τ(q) ∈ Aut X, which carries (q_{1}, ..., q_{n}) to (f_{1}(q), ..., f_{n}(q)) and nicely depends on q. This is possible if and only if the ordered sets x = (q_{1}, ..., q_{n}) ⊂ X and y = (f_{1}(q), ..., f_{n}(q)) ⊂ X satisfy the necessary and sufficient condition mentioned above. Taking into account that permutations are admitted, we can conclude that Equivariant Map Theorem is an immediate consequence of the following result.

Theorem 2.13 (Coherence Theorem). Let n > 6 − t(X) and let f = (f_{1}, ..., f_{n}) be a strictly equivariant endomorphism of C_{o}^{n}(X). Then there exists a permutation σ ∈ S(n) such that for all q ∈ C_{o}^{n}(X) and all r = t(X) + 1, ..., n

(SR) \frac{f_{1}(σq) − f_{r}(σq)}{f_{1}(σq) − f_{2}(σq)} = \frac{q_{1} − q_{r}}{q_{1} − q_{2}} if X = ℂ,

(CR) \frac{f_{r}(σq) − f_{1}(σq)}{f_{r}(σq) − f_{2}(σq)} : \frac{f_{1}(σq) − f_{3}(σq)}{f_{2}(σq) − f_{3}(σq)} = \frac{q_{r} − q_{1}}{q_{r} − q_{2}} : \frac{q_{1} − q_{3}}{q_{2} − q_{3}} if X = ℂℙ^{1} . ◯

Explanation. For the sake of neatness, let us explain how to deduce Equivariant Map Theorem from Coherence Theorem. We start with the following simple remark, which will be useful in what follows:

Remark 2.14. For n > t(X) there exist non-empty Zariski open subsets U ⊂ C_{o}^{n}(X) and V ⊂ C^{n}(X) such that a) if Aq = sq for some q ∈ U, A ∈ Aut X and s ∈ S(n) then A = id and s = 1; b) if AQ = Q for some Q ∈ V and A ∈ Aut X then A = id. Indeed, the set of all points q = (q_{1}, ..., q_{n}) ∈ C_{o}^{n}(X) such that some its non-trivial permutation sq = (q_{s^{−1}(1)}, ..., q_{s^{−1}(n)}) may be also written as Aq = (Aq_{1}, ..., Aq_{n}) with some A ∈ Aut X is a proper Zariski closed subset of C_{o}^{n}(X); this implies both (a) and (b). ◯

For each q = (q_{1}, ..., q_{n}) ∈ C_{o}^{n}(X) we define the transformation τ(q) ∈ Aut X by one of the following equations:

\frac{f_{1}(σq) − τ(q)x}{f_{1}(σq) − f_{2}(σq)} = \frac{q_{1} − x}{q_{1} − q_{2}} if x ∈ X = ℂ ,  (2.13)
\frac{τ(q)x − f_{1}(σq)}{τ(q)x − f_{2}(σq)} : \frac{f_{1}(σq) − f_{3}(σq)}{f_{2}(σq) − f_{3}(σq)} = \frac{x − q_{1}}{x − q_{2}} : \frac{q_{1} − q_{3}}{q_{2} − q_{3}} if x ∈ X = ℂℙ^{1}.  (2.14)

Clearly τ: C_{o}^{n}(X) → Aut X is a morphism. Conditions (SR) and (CR) respectively imply that τ(q)q_{m} = f_{m}(σq) for each m = 1, ..., n and all q = (q_{1}, ..., q_{n}) ∈ C_{o}^{n}(X); thereby τ(q)q = f(σq) = α(σ)f(q), or, which is the same, f(q) = α(σ^{−1})τ(q)q for all q ∈ C_{o}^{n}(X). To complete the proof, we must check that the morphism τ is S(n) invariant. For every s ∈ S(n) and all q ∈ C_{o}^{n}(X) we have

\begin{aligned}sτ(sq)q & = τ(sq)sq = f(σsq) = α(σs)f(q) \\ & = α(σs)f(σσ^{−1}q) = α(σs)τ(σ^{−1}q)σ^{−1}q = α(σs)σ^{−1}τ(σ^{−1}q)q , \end{aligned}

which can be rewritten as

[(τ(sq))^{−1} · τ(σ^{−1}q)]q = σα(s^{−1}σ^{−1})sq ,  (2.15)

where (τ(sq))^{−1} · τ(σ^{−1}q) ∈ Aut X is the product in the group Aut X. In view of Remark 2.14, this implies that σα(s^{−1}σ^{−1})s = 1 and τ(sq) = τ(σ^{−1}q) for all s ∈ S(n) and all q in a non-empty Zariski open subset U ⊂ C_{o}^{n}(X). Since τ is continuous, the latter relation holds true for all q ∈ C_{o}^{n}(X) and all s ∈ S(n), which shows that the morphism τ: C_{o}^{n}(X) → Aut X is S(n) invariant.

Moreover, it is easily seen that the second identity implies α(s) = σ^{−1}sσ for all s ∈ S(n). Thus, Coherence Theorem implies that for all n > 4 the automorphism α related to a strictly equivariant endomorphism f of C_{o}^{n}(X) must be inner; in view of Remark 2.9, this α must be inner also for n = 3 or 4. □

2.5. Step 5: Simplicial complex of functions omitting two values. Let f = (f_{1}, ..., f_{n}): C_{o}^{n}(X) → C_{o}^{n}(X) be a morphism and let i, j, k, l be distinct. Then

sr_{ijk}[f] = (f_{k} − f_{i})/(f_{k} − f_{j}) and cr_{ijkl}[f] = [(f_{l} − f_{i})/(f_{l} − f_{j})] : [(f_{i} − f_{k})/(f_{j} − f_{k})]

are meromorphic function on C_{o}^{n}(X) and do not take the values 0 and 1. In fact, if X = ℂ then sr_{ijk}[f] and cr_{ijkl}[f] are both holomorphic; if X = ℂℙ^{1} this is certainly the case for cr_{ijkl}[f].

In Section 6 (Lemma 6.1) we show that if f is a strictly equivariant endomorphism of C_{o}^{n}(X) then for X = ℂ all sr_{ijk}[f] and cr_{ijkl}[f] are non-constant and for X = ℂℙ^{1} this is certainly the case for all cr_{ijkl}[f].

Notice that the functions standing in the left hand sides of relations (SR) and (CR) up to a permutation of the variables q_{1}, ..., q_{n} coincide with sr_{r,2,1}[f] and cr_{1,2,3,r}[f] respectively. Thus, what is written above applies to these functions.

It is well known that the set L(Z) of all non-constant holomorphic functions λ: Z → ℂ \ {0; 1} on an irreducible quasi-projective variety Z is finite. In particular, the set L(C_{o}^{n}(X)) is finite and all functions sr_{ijk}[f] and cr_{ijkl}[f] in the case X = ℂ (or all cr_{ijkl}[f] in the case X = ℂℙ^{1}) must be its elements.

This observation suggests the following way to prove Coherence Theorem. First, compute L(C_{o}^{n}(X)), that is, determine explicitly all non-constant holomorphic functions on C_{o}^{n}(X) omitting the values 0 and 1. Second, try to identify (perhaps after a suitable permutation of q_{1}, ..., q_{n}) the functions sr_{r,2,1}[f] (r = 3, ..., n) and/or cr_{1,2,3,r}[f] (r = 4, ..., n) with certain elements of L(C_{o}^{n}(X)), and then... if we are lucky... who knows... To say the least, this looks as a finite combinatorial problem.

To execute the first part of this program, we prove the following theorem.

Theorem 2.15 (non-constant holomorphic functions C_{o}^{n}(X) → ℂ \ {0; 1}). a) Each

non-constant holomorphic function μ: C_{o}^{n}(ℂ) → ℂ \ {0; 1} is either a simple ratio sr_{ijk} = (q_{k} − q_{i})/(q_{k} − q_{j}) or a cross ratio a cross ratio cr_{ijkl} = [(q_{l} − q_{i})/(q_{l} − q_{j})] : [(q_{i} − q_{k})/(q_{j} − q_{k})]

(for n = 3 only simple ratios survive).

b) Each non-constant holomorphic function μ: C_{o}^{n}(ℂℙ^{1}) → ℂ \ {0; 1} is a cross ratio cr_{ijkl} = [(q_{l} − q_{i})/(q_{l} − q_{j})] : [(q_{i} − q_{k})/(q_{j} − q_{k})] (for n = 3 no such function exists). ◯

To avoid unpleasant combinatorics in the course of the realization of the second part of our plan, we must try to find the “genuine” reason forcing the sequence of the functions in the left hand side of (SR) or (CR) to be of the form prescribed by the right hand side (perhaps after a suitable rearrangement).

To this end, we notice that the set L(Z) of all non-constant holomorphic functions λ: Z → ℂ \ {0; 1} on an irreducible quasi-projective variety Z carries a natural structure of a finite simplicial complex L_{△}(Z), which turns out to be amazingly useful for our purposes.^{2}

A subset Δ^{m} = {μ_{0}, ..., μ_{m}} consisting of m + 1 distinct elements of L(Z) is declared to be an m simplex if the quotients μ_{i} : μ_{j} ∈ L(z) for all i ≠ j. In other words,

Δ^{m}: Z ∋ z ↦ (μ_{0}(z), ..., μ_{m}(z)) ∈ ℂ^{m+1}  (2.16)

must be a holomorphic map such that all its components μ_{i} and all their pairwise quotients μ_{i} : μ_{j} (i ≠ j) are non-constant and its image Δ^{m}(Z) is contained in the domain

M^{m+1} = {w = (w_{0}, ..., w_{m}) ∈ ℂ^{m+1} | w_{i} ≠ 0, 1 ∀ i and w_{i} ≠ w_{j} ∀ i ≠ j} . (2.17)

For instance, the sets of functions

\begin{matrix}\{(q_{1} − q_{r})/(q_{1} − q_{2}) | r = 3, ..., n\} and \\ \{[(q_{r} − q_{1})/(q_{r} − q_{2})] : [(q_{1} − q_{3})/(q_{2} − q_{3})] | r = 4, ..., n\} \end{matrix}

standing in the right hand sides of (SR) and (CR) are simplices of dimension n − 3 and n − 4 respectively.

For a holomorphic map f: Y → Z of irreducible quasi-projective varieties the correspondence

λ ↦ f^{∗}(λ) =^{def} λ ∘ f: Y ⟶^{f} Z ⟶^{λ} ℂ \ {0; 1}

turns out to be a simplicial map f^{∗}: L_{△}(Z) → L_{△}(Y), provided that λ ∘ f ≠ const for each λ ∈ L(Z) (this is certainly the case if f is dominant but the latter condition is not a necessary one). Moreover, the map f^{∗} preserves dimension of simplices.

Remark 2.16. The simplicial complex L_{△}(Z) itself and all its standard topological invariants are invariants of the underlying complex manifold Z with respect to biholomorphic isomorphisms. In Section 5.3 we shaw that dim L_{△}(C_{o}^{n}(ℂ)) = n − 3 for n ≥ 3 and dim L_{△}(C_{o}^{n}(ℂℙ^{1})) = n − 4 for n ≥ 4, which plays important part in the proof of Coherence Theorem. It is not difficult to show that for n ≥ 4 the Euler characteristic χ(L_{△}(C_{o}^{n}(ℂℙ^{1}))) = n(n − 1)(n − 2)(13 − 3n)/4; for n > 4 the complex L_{△}(C_{o}^{n}(ℂℙ^{1})) is connected and its highest homology group H_{n−4}(L_{△}(C_{o}^{n}(ℂℙ^{1}), ℤ)) is trivial. Moreover, π_{1}(L_{△}(C_{o}^{5}(ℂℙ^{1}))) = 𝔽_{31} and rank H_{1}(L_{△}(C_{o}^{6}(ℂℙ^{1}), ℤ)) = 151. Does it mean anything?

The complex L_{△}(Z) always has the following rather special property: if {μ_{0}, μ_{1}}, {μ_{1}, μ_{2}} and {μ_{0}, μ_{2}} are simplices then the ordered set Δ^{2} = [μ_{0}, μ_{1}, μ_{2}] is an ordered 2 simplex and therefore the chain c = [μ_{1}, μ_{2}] − [μ_{0}, μ_{2}] + [μ_{0}, μ_{1}] = ∂Δ^{2} is a 1 cycle homologic to 0. This does not mean, however, that H_{1}(L_{△}(Z), ℤ) = 0, as some “longer” 1 cycle may be not a boundary. ◯ We show (see Section 6, Lemma 6.1) that the above construction applies to every strictly equivariant endomorphism f of C_{o}^{n}(X). It follows that the sets of functions standing in the left hand sides of (SR) and (CR) also are simplices (of dimension n−3 and n − 4 respectively).

2I came to this almost obvious structure just very recently when the first version of this paper was already written. In this renewed version the proof does not become much shorter but its main ideas are now utterly clear. Strangely enough I paid no attention to this structure 30 years ago. But seemingly nobody did so...

The S(n) action in C_{o}^{n}(X) induces an S(n) action on the set of all simplices Δ^{m} ⊆ L(C_{o}^{n}(X)) of any fixed dimension m. In view of what was explained above, Coherence Theorem just tells us that the simplices of functions standing in the left hand sides of (SR) and (CR) belong to the S(n) orbits of the simplices of functions standing in the right hand sides of (SR) and (CR) respectively.

Using the explicit form of functions λ ∈ L(C_{o}^{n}(X)) provided by Theorem 2.15, we will determine all the orbits of the latter S(n) action on the set of m simplices. In fact this will prove Coherence Theorem in the form presented in the previous paragraph and complete the proof of Tame Map Theorem.

3. Holomorphic universal covers and universal Teichmüller families

In this section we exhibit holomorphic universal coverings of certain configuration spaces, which will play the key part in the proof of Linked Map Theorem in Section 7. To my best knowledge, the holomorphic universal coverings of C^{n}(ℂ) and some related spaces were for the first time described by S. Kaliman [Kal75, Kal93]. We do this in a different manner, which will be more convenient for our purposes.

At the end of the section we describe the standard normal series with free factors in the pure braid group P B_{n} and construct a similar series in P B_{n}(S^{2}). In particular, this leads to the proof of Theorem 2.6 formulated in Section 2.2. We also present some information about higher homotopy groups of C^{n}(X).

Some of the results discussed below are certainly well-known, especially those dealing with homotopy groups; see, for instance, [FoxNeu62, FadNeu62, FadHus01]. Unlike the authors of the quoted sources, we systematically take into account that configuration spaces under consideration are complex manifolds. This was extensively used by V. Arnold but in fact goes back to A. Hurwitz and O. Zariski. There are certain advantages in doing so even when dealing with “purely topological” objects related to our configuration spaces. Say the direct decomposition (3.11) and projections (3.14) are “invisible” from the real point of view. (Notice that E. M. Feichtner and M. Ziegler [FeiZie99] pointed out the direct decomposition (3.11) in the case X = ℂℙ^{1} but seemingly they were not interested in its complex-analytic nature. What they denoted by M_{0,n} and referred to as (ordered!) ”moduli spaces of n-punctured complex projective lines” in the present paper is denoted by D^{n−3}(ℂℙ^{1}) or C_{o}^{n−3}(ℂ^{◯ 2}); see the next section.)

3.1. Holomorphic universal coverings of certain configuration spaces. We need now to deal with configuration spaces of the line ℂ punctured at several distinct points, say such as C_{o}^{n}(ℂ \ {0, 1, ..., k − 1}). To shorten notation, we set

ℂ^{◯ k} = ℂ \ {0, 1, ..., k − 1} (k ≥ 1) .  (3.1)

3. HOLOMORPHIC UNIVERSAL COVERS AND UNIVERSAL TEICHMÜLLER FAMILIES 21

For instance, for m > 1

C_{o}^{m}(ℂ^{◯ 2}) = {z = (z_{1}, ..., z_{m}) ∈ ℂ^{m} | z_{i} ≠ 0, 1 ∀ i and z_{i} ≠ z_{j} ∀ i ≠ j}  (3.2)

(compare to (2.17)). To avoid the doubling in formulations related to two cases X = ℂ and X = ℂℙ^{1} under consideration, we will assume that n ≥ t(X) (see Notation 1.11) and use the notation

D^{n−t(X)}(X) =^{def} \{ \begin{matrix}\{q = (q_{1}, ..., q_{n}) ∈ C_{o}^{n}(ℂ) | q_{n−1} = 0, q_{n} = 1\} if X = ℂ , \\ \{q = (q_{1}, ..., q_{n}) ∈ C_{o}^{n}(ℂℙ^{1}) | q_{n−2} = 0, q_{n−1} = 1, q_{n} = ∞\} \\ if X = ℂℙ^{1} . \end{matrix}

For n > t(X) we identify D^{n−t(X)}(X) and C_{o}^{n−t(X)}(ℂ^{◯ 2}) via the correspondence

\begin{matrix}D^{n−2}(ℂ) ∋ (q_{1}, ..., q_{n−2}, 0, 1) ⟷^{≅} (q_{1}, ..., q_{n−2}) ∈ C_{o}^{n−2}(ℂ^{◯ 2}) , \\ D^{n−3}(ℂℙ^{1}) ∋ (q_{1}, ..., q_{n−3}, 0, 1, ∞) ⟷^{≅} (q_{1}, ..., q_{n−3}) ∈ C_{o}^{n−3}(ℂ^{◯ 2}) . \end{matrix}  (3.3)

It is well known (see, for instance, [Kal75, Kal93]) that the holomorphic universal cover { \atop C_{o}^{m}(ℂ^{◯ 2})} of C_{o}^{m}(ℂ^{◯ 2}) may be viewed as the Teichmüller space T(0, m + 3) of the Riemann sphere with m + 3 punctures.^{3} The construction presented in the quoted papers makes use of the classical techniques related to quasi-conformal mappings, which is, in a sense, inevitable when dealing with Teichmüller spaces. We present here a more direct construction of holomorphic covering mappings^{4}

τ: T(0, m + 3) → C_{o}^{m}(ℂ^{◯ 2}) ,  (3.4)

π: { \atop Aut X} × T(0, n + 3 − t(X)) → C_{o}^{n}(X) and

Π: { \atop Aut X} × T(0, n + 3 − t(X)) → C^{n}(X)

(3.5)

based on the concept of the universal Teichmüller family, or, which is the same, the “universal Teichmüller curve”

T (0, l) := ρ: V′(0, l) → T(0, l) ,  (3.6)

which may be roughly described as follows.

Any point t of the Teichmüller space T(0, l) “is” a curve Γ_{t} of type (0, l) (i. e., the Riemann sphere ℂℙ^{1} ≅ \overline{ℂ} = ℂ ∪ \{∞\} punctured at certain l points and endowed with a marked free basis of its fundamental group, which we prefer to forget about). The natural projection

ρ: ℂℙ^{1} × T(0, l) → T(0, l)  (3.7)

possesses l holomorphic sections s_{i}: T(0, l) ∋ t ⟼ (s_{i}(t), t) ∈ ℂℙ^{1} × T(0, l) with the following properties: all the maps s_{1}, ..., s_{l}: T(0, l) → ℂℙ^{1} are holomorphic and take distinct values at each point t ∈ T(0, l),

3Notice that T(0, m + 3) is homeomorphic to ℝ^{2m}; moreover, it is biholomorphic to a holomorphically convex Bergmann domain in ℂ^{m}.

4{ \atop Aut X} is the holomorphic universal covering of the complex Lie group Aut X; see (3.13).

s_{l−2}(t) ≡ 0 , s_{l−1}(t) ≡ 1 , s_{l}(t) ≡ ∞^{5}  (3.8)

and, moreover, for every t ∈ T(0, l) the curve Γ(t) =^{def} ℂℙ^{1} \ {s_{1}(t), ..., s_{l}(t)} is “the same” curve Γ_{t} whose “shape” in T(0, l) is the point t. We set

V′(0, l) = [ℂℙ^{1} × T(0, l)] \ [s_{1}(T(0, l)) ∪ ... ∪ s_{l}(T(0, l))] ;  (3.9)

restricting (3.7) to V′(0, l) we obtain (3.6). Because of normalization (3.8), the mappings s_{1}, ..., s_{l−3} are just holomorphic functions on s_{i}: T(0, l) → ℂ \ {0, 1} distinct at each point t ∈ T(0, l).

The family T (0, l) := ρ: V′(0, l) → T(0, l) is universal in the following sense:

Universality: Let F := π: X → Y be a holomorphic family of curves of type (0, l) over a simply connected base Y, i. e., π is a holomorphic map of complex manifolds, Y is simply connected and for each y ∈ Y the fiber π^{−1}(y) is a curve of type (0, l). Then there exists a holomorphic map ϕ: Y → T(0, l) such that the induced family ϕ^{∗}(T (0, l)) is equivalent to the family F.

With the above construction at our disposal, we take l = m+3 and define the required holomorphic covering map (3.4) as follows:

τ: T(0, m + 3) ∋ t ↦ (s_{1}(t), ..., s_{m}(t)) ∈ C_{o}^{m}(ℂ^{◯ 2}) ,  (3.10)

where s_{i} = (s_{i}, ·): T(0, m + 3) → ℂℙ^{1} (i = 1, ..., m + 3) are the corresponding normalized sections of the projection V′(0, m + 3) → T(0, m + 3).

Furthermore, let Θ: C_{o}^{n}(X) → Aut X denote the morphism whose value at any point q = (q_{1}, ..., q_{n}) ∈ C_{o}^{n}(X) is a unique element Θ(q) ∈ Aut X that carries (0, 1) to (q_{n−1}, q_{n}) when X = ℂ and (0, 1, ∞) to (q_{n−2}, q_{n−1}, q_{n}) when X = ℂℙ^{1}. The diagonal Aut X action in C_{o}^{n}(X) is free and biregular, and each its orbit (Aut X)q intersects the submanifold D^{n−t(X)}(X) in a single point that is exactly (Θ(q))^{−1}q. In view of (3.3), this provides us with the biregular isomorphisms

j_{X,n}: C_{o}^{n}(X) ⟶^{≅} Aut X × D^{n−t(X)}(X) = Aut X × C_{o}^{n−t(X)}(ℂ^{◯ 2}) ,  (3.11)

where j_{X,n} and its inverse j_{X,n}^{−1} are defined by

\begin{matrix}j_{X,n}(q) = (Θ(q), ( & Θ(q))^{−1}q) ,  j_{X,n}^{−1}(A, z) = Az \\ (q ∈ C_{o}^{n}(X) , A ∈ Aut X , z ∈ D^{n−t(X)}(X)) . \end{matrix}  (3.12)

Let ℂ ⋋ ℂ be the semi-direct product with the group multiplication (b, ζ) · (b′, ζ′) = (b + e^{ζ}b′, ζ + ζ′) and the complex structure of ℂ × ℂ. For X = ℂ or X = ℂℙ^{1} the holomorphic universal covering π_{X}: { \atop Aut X} → Aut X is given by

π_{ℂ}: ℂ ⋋ ℂ ∋ (b, ζ) ↦ π_{ℂ}(b, ζ) = A ∈ Aff ℂ , Az = e^{ζ}z + b ∀ z ∈ ℂ ,
π_{ℂℙ}^{1} : SL(2, ℂ) → SL(2, ℂ)/{±I} = PSL(2, ℂ) = Aut ℂℙ^{1} .

(3.13)

5This normalization is not a necessary but a very convenient one.

Finally, we define the required holomorphic covering maps (3.5) as follows:^{6}

π: { \atop Aut X} × T(0, n + 3 − t(X)) ∋ (A, t) ↦ π(A, t)
= (π_{X}(A)s_{1}(t), ..., π_{X}(A)s_{n}(t)) ∈ C_{o}^{n}(X) ,
Π: { \atop Aut X} × T(0, n + 3 − t(X)) ∋ (A, t) ↦ Π(A, t)
= {π_{X}(A)s_{1}(t), ..., π_{X}(A)s_{n}(t)} ∈ C^{n}(X) ,

(3.14)

where π_{X}: { \atop Aut X} → Aut X is defined by (3.13).

3.2. Homotopy groups and the standard normal series for P B_{n}(X). According to (3.11), C_{o}^{n}(X) ≅ Aut X × C_{o}^{n−t(X)}(ℂ^{◯ 2}). Hence for n ≥ 3

\begin{matrix}π_{1}(C_{o}^{n}(X)) = π_{1}(Aut X)× & π_{1}(C_{o}^{n−t(X)}(ℂ^{◯ 2})) \\ = \{ \begin{matrix}ℤ × π_{1}(C_{o}^{n−2}(ℂ^{◯ 2}))  if X = ℂ , \\ (ℤ/2ℤ) × π_{1}(C_{o}^{n−3}(ℂ^{◯ 2})) if X = ℂℙ^{1} . \end{matrix} \end{matrix}  (3.15)

The fundamental group π_{1}(C_{o}^{n−2}(ℂ^{◯ 2})) is one of the important objects of the classical braid theory, which usually appears by the following way. The projection

C_{o}^{n}(ℂ) ∋ q = (q_{1}, ..., q_{n}) ↦ q_{n} ∈ ℂ

is a smooth fiber bundle with the fiber

C_{o}^{n−1}(ℂ^{◯ 1}) = {(q_{1}, ..., q_{n−1}) ∈ ℂ | q_{i} ≠ 0 ∀i, q_{i} ≠ q_{j} ∀i ≠ j} .

Since the base ℂ is contractible, the exact homotopy sequence of this fibration shows that

π_{k}(C_{o}^{n}(ℂ)) ≅ π_{k}(C_{o}^{n−1}(ℂ^{◯ 1})) ∀ k .

The next projection

C_{o}^{n−1}(ℂ^{◯ 2}) ∋ (q_{1}, ..., q_{n−1}) ↦ q_{n−1} ∈ ℂ \ {0} = ℂ^{◯ 1}

is a smooth fiber bundle as well and the fiber is

C_{o}^{n−2}(ℂ^{◯ 2}) = {(q_{1}, ..., q_{n−2}) ∈ ℂ | q_{i} ≠ 0, 1 ∀i, q_{i} ≠ q_{j} ∀i ≠ j} .

Since the base ℂ^{◯ 1} = ℂ \ {0} is aspherical, we have

π_{k}(C_{o}^{n−1}(ℂ^{◯ 1})) ≅ π_{k}(C_{o}^{n−2}(ℂ^{◯ 2})) ∀ k ≥ 1 .

Moreover, the final segment of the corresponding exact homotopy sequence is

1 → π_{1}(C_{o}^{n−2}(ℂ^{◯ 2})) → π_{1}(C_{o}^{n−1}(ℂ^{◯ 1})) → π_{1}(ℂ \ {0}) → 0

may be written as

1 → P B_{n}^{(2)} → P B_{n}^{(1)} → ℤ → 0 ,

where P B_{n}^{(2)} =^{def} π_{1}(C_{o}^{n−2}(ℂ^{◯ 2})) and P B_{n}^{(1)} =^{def} π_{1}(C_{o}^{n}(ℂ)) = P B_{n}. This projecting process may be continued; say the next step leads to a fiber bundle with the fiber

C_{o}^{n−3}(ℂ^{◯ 3}) = {(q_{1}, ..., q_{n−3}) ∈ ℂ | q_{i} ≠ 0, 1, 2 ∀i, q_{i} ≠ q_{j} ∀i ≠ j}

6We invite the reader to write down these mappings for X = ℂ and X = ℂℙ^{1} separately.

over the aspherical base ℂ^{◯ 2} = ℂ \ {0, 1}, the relations

π_{k}(C_{o}^{n−3}(ℂ^{◯ 3})) ≅ π_{k}(C_{o}^{n−2}(ℂ^{◯ 2})) ∀ k > 1

and the exact sequence

1 → P B_{n}^{(3)} → P B_{n}^{(2)} → 𝔽_{2} → 1 ,

where P B_{n}^{(3)} =^{def} π_{1}(C_{o}^{n−3}(ℂ^{◯ 3})) and 𝔽_{2} stands for a free group of rank 2. This inductive procedure eventually leads to the following important result going back to E. Artin [Art25], A. Markov [Mar45] (at least part (c) and the determination of the fundamental groups in part (a)) and also to E. Fadell, R. H. Fox and L. Neuwirth [FadNeu62, FoxNeu62] (asphericity in parts (a,b)).

Theorem 3.1 (Artin-Markov-Fadell-Fox-Neuwirth). a) The spaces C_{o}^{n}(ℂ) and C^{n}(ℂ) are Eilenberg-MacLane K(π, 1) spaces for the pure braid group P B_{n} and the braid group B_{n} respectively, i. e., these spaces are aspherical and have the fundamental groups that are pointed out.^{7}

b) All configuration spaces C_{o}^{m}(ℂ^{◯ k}) are aspherical.

c) The pure braid group P B_{n} possesses a finite series of normal subgroups

{1} = P B_{n}^{(n)} ⊂ P B_{n}^{(n−1)} ⊂ ⋯ ⊂ P B_{n}^{(2)} ⊂ P B_{n}^{(1)} = P B_{n}  (3.16)

such that P B_{n}^{(r)}/P B_{n}^{(r+1)} ≅ 𝔽_{r} (the free group of rank r), 1 ≤ r ≤ n − 1. □

The normal series (3.16) will be referred to as the standard one.

Replacing n in (3.16) with n − 1, we see that the fundamental group π_{1}(C_{o}^{n−3}(ℂ^{◯ 2})), which participate in the formula for π_{1}(C_{o}^{n}(ℂℙ^{1})) = P B_{n}(S^{2}) given by (3.15), coincides with second term P B_{n−1}^{(2)} ◁ P B_{n−1} of the standard normal series for P B_{n−1}. Together with (3.11), (3.13) and asphericity of C_{o}^{n−3}(ℂ^{◯ 2}), this implies the following result.

Theorem 3.2. a) The pure sphere braid group P B_{3}(S^{2}) ≅ ℤ/2ℤ. For n > 3 the group P B_{n}(S^{2}) ≅ (ℤ/2ℤ) × P B_{n−1}^{(2)}, where P B_{n−1}^{(2)} possesses a series of normal subgroups

{1} = P B_{n−1}^{(n−1)} ⊂ P B_{n−1}^{(n−2)} ⊂ ⋯ ⊂ P B_{n−1}^{(3)} ⊂ P B_{n−1}^{(2)}  (3.17)

such that P B_{n−1}^{(r)}/P B_{n−1}^{(r+1)} ≅ 𝔽_{r} (the free group of rank r), 1 ≤ r ≤ n − 2.

b) The higher homotopy groups of C^{n}(ℂℙ^{1}) and C_{o}^{n}(ℂℙ^{1}) are as follows:

π_{k}(C^{n}(ℂℙ^{1})) = π_{k}(C_{o}^{n}(ℂℙ^{1})) = π_{k}(PSL(2, ℂ))

= π_{k}(SL(2, ℂ)) = π_{k}(S^{3}) = \{ \begin{matrix}0 if k = 2 , \\ π_{k}(S^{2}) if k > 2 . \end{matrix} (3.18)

The following theorem, which contains also Theorem 2.6 formulated in Section 2.2, is an immediate consequence of part (c) of Artin-Markov-Fadell-Fox-Neuwirth Theorem, part (a) of Theorem 3.2 and Commutator Theorem, which we still need to prove.

7This certainly follows from (3.11), (3.13) and (3.14); but the standard argument exhibited above is more preferable since it is quite elementary.

Theorem 3.3. Perfect groups do not admit non-trivial homomorphisms into the pure braid group P B_{k}(X). In particular, for n > 4 the commutator subgroup B′_{n}(X) of the braid group B_{n}(X) does not possess non-trivial homomorphisms to the pure braid group P B_{k}(X). ◯

4. Proof of Invariance Theorem

The main aim of this section is to prove Invariance Theorem; we follow the way outlined in Section 2.

4.1. Canonical presentations of B_{n} and B_{n}(S^{2}). The inclusion ℂ ↪ ℂ ∪ {∞} = ℂℙ^{1} provides the inclusions i_{o}: C_{o}^{n}(ℂ) ↪ C_{o}^{n}(ℂℙ^{1}) and i: C^{n}(ℂ) ↪ C^{n}(ℂℙ^{1}), which, in turn, induce the epimorphisms of the fundamental groups

i_{o∗}: P B_{n} = π_{1}(C_{o}^{n}(ℂ)) → π_{1}(C_{o}^{n}(ℂℙ^{1})) = P B_{n}(S^{2})  (4.1)

and

i_{∗}: B_{n} = π_{1}(C^{n}(ℂ)) → π_{1}(C^{n}(ℂℙ^{1})) = B_{n}(S^{2}) .  (4.2)

Recall that the canonical presentation of the Artin braid group B_{n} = π_{1}(C^{n}(ℂ)) involves n − 1 generators σ_{1}, .., σ_{n−1} and the defining system of relations

σ_{i}σ_{j} = σ_{j}σ_{i}  (|i − j| ≥ 2) ,  (4.3)
σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1} (1 ≤ i ≤ n − 2) .  (4.4)

It is well known that the kernel of the epimorphism i_{∗}: B_{n} → B_{n}(S^{2}) is the normal subgroup S ◁ B_{n} generated (as a normal subgroup) by the single element

s = σ_{1}σ_{2} ⋯ σ_{n−1}σ_{n−1} ⋯ σ_{2}σ_{1} ∈ P B_{n} ,  (4.5)

so that S ◁ P B_{n}. Thus, involving also the monomorphisms p_{∗} that correspond to the coverings p: C_{o}^{n}(ℂ) → C^{n}(ℂ) and p: C_{o}^{n}(ℂℙ^{1}) → C^{n}(ℂℙ^{1}), we obtain the following commutative diagram with exact lines and rows:

\begin{matrix}1  1  1 \\  &  &  \\ 1 \overline{}S  \overline{}P B_{n} \overline{}_{i_{o∗}} P B_{n}(S^{2}) \overline{}1 \\  & _{_{p_{∗}} } & _{_{p_{∗}}} \\ 1 \overline{}S  \overline{}B_{n}  \overline{}_{i_{∗}} B_{n}(S^{2})  \overline{}1 \\  & _{_{μ} } & _{_{μ}} \\ 1  \overline{}S(n) \overline{}_{id} S(n)  \overline{}1 \\  &  \\ 1  1 \end{matrix}  (4.6)

(compare to (1.4)). It follows that the sphere braid group B_{n}(S^{2}) = π_{1}(C^{n}(ℂℙ^{1})) admits a presentation (we call it the canonical one) with n − 1 generators σ_{1}, .., σ_{n−1} and the defining system of relations

\begin{matrix}σ_{i}σ_{j} = σ_{j}σ_{i}  (|i − j| ≥ 2) , \\ σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1} (1 ≤ i ≤ n − 2) , \\ σ_{1}σ_{2} ⋯ σ_{n−1}σ_{n−1} ⋯ σ_{2}σ_{1} = 1 . \end{matrix}  (4.7)

4.2. Proof of Commutator Theorem. A finite presentation of the commutator subgroup B′_{n} found in [GorLin69] implies immediately that for n > 4 the abelianization B′_{n}/[B′_{n}, B′_{n}] of B′_{n} is trivial; that is, B′_{n} is perfect. Since B_{n}(S^{2}) is a quotient group of B_{n}, its commutator subgroup B′_{n}(S^{2}) is a quotient group of B′_{n}. A quotient group of a perfect group is obviously perfect; hence for n > 4 the group B′_{n}(S^{2}) is perfect as well. □

Remark 4.1. This remark allows to avoid the quotation of [GorLin69] in the above proof of Commutator Theorem.

About 15 years after our joint paper [GorLin69] was published, E. A. Gorin discovered the following simple and very beautiful relation, which holds for any n ≥ 4:

σ_{3}σ_{1}^{−1} = (σ_{1}σ_{2})^{−1} · [σ_{3}σ_{1}^{−1}, σ_{1}σ_{2}^{−1}] · (σ_{1}σ_{2}) ,  (4.8)

where [σ_{3}σ_{1}^{−1}, σ_{1}σ_{2}^{−1}] = (σ_{3}σ_{1}^{−1})^{−1} · (σ_{1}σ_{2}^{−1})^{−1} · (σ_{3}σ_{1}^{−1}) · (σ_{1}σ_{2}^{−1}) is the commutator of the elements g_{1} = σ_{3}σ_{1}^{−1} and g_{2} = σ_{1}σ_{2}^{−1}. Clearly, g_{1}, g_{2} ∈ B′_{n}, and (4.8) shows that σ_{3}σ_{1}^{−1} ∈ [B′_{n}, B′_{n}]. Hence, the normal subgroup N ◁ B_{n} generated (as a normal subgroup) by the element σ_{3}σ_{1}^{−1} is contained in [B′_{n}, B′_{n}]. For n > 4 it follows readily from (4.3) and (4.4) that N contains the whole commutator subgroup B′_{n}. Indeed, the presentation of B_{n}/N involves the generators σ_{1}, ..., σ_{n−1} and the defining system of relations consisting of (4.3), (4.4) and the additional relation σ_{3}σ_{1}^{−1} = 1; since n > 4 relations (4.3) and σ_{3}σ_{1}^{−1} = 1 imply σ_{3}σ_{4} = σ_{4}σ_{3}; in view of (4.4), this shows that σ_{3} = σ_{4} and, finally, σ_{1} = σ_{2} = ... = σ_{n−1}. Thus, B_{n}/N ≅ ℤ and N ⊇ B′_{n}. Thereby B′_{n} ⊆ N ⊆ [B′_{n}, B′_{n}]. ◯

4.3. Proof of Surjectivity Theorem 2.7. Since B_{n}(S^{2}) is a quotient group of B_{n}, it suffices to prove the theorem for B_{n}. This was done in [Lin96b] (Lemma 2.7). It turns out that this theorem is a very convenient technical tool. Since I still did not take care of publication of [Lin96b], I decided to outline the proof here.

First, let me recall (in a convenient form) some classical results of E. Artin [Art47b].

Definition 4.2. Two group homomorphisms ϕ, ψ: G → H are said to be conjugate if there is an element h ∈ H such that ψ(g) = hϕ(g)h^{−1} for all g ∈ G. If this is the case, we write ϕ ∼ ψ; otherwise, we write ϕ ≁ ψ.

Definition 4.3. A group homomorphism ψ: G → S(n) is said to be transitive if its image ψ(G) is a transitive subgroup of the symmetric S(n) (meaning the standard action of S(n) on Δ_{n} =^{def} {1, ..., n}).

Recall that the elements σ_{1} and α = σ_{1} ⋯ σ_{n−1} generate the whole Artin braid group B_{n}; this follows from the relations

\begin{aligned}ασ_{j} & = σ_{1} ⋯ σ_{j−1} · (σ_{j}σ_{j+1}σ_{j}) · σ_{j+2} ⋯ σ_{n−1} \\ & = σ_{1} ⋯ σ_{j−1} · (σ_{j+1}σ_{j}σ_{j+1}) · σ_{j+2} ⋯ σ_{n−1} = σ_{j+1}α (1 ≤ j ≤ n − 2) \end{aligned} (4.9)

(the same relations hold true in B_{n}(S^{2})). Thus, a homomorphism ψ: B_{n} → H is uniquely defined by its values ψ(σ_{1}), ψ(α).

Artin Theorem. Let ψ: B_{n} → S(n) be a non-cyclic transitive homomorphism.

a) If n ≠ 4 and n ≠ 6 then the homomorphism ψ is conjugate to the standard epimorphism μ: B_{n} → S(n).

b) If n = 6 and ψ ≁ μ then ψ is conjugate to the homomorphism ν_{6} defined by

ν_{6}(σ_{1}) = (1, 2)(3, 4)(5, 6), ν_{6}(α) = (1, 2, 3)(4, 5).

c) If n = 4 and ψ ≁ μ then ψ is conjugate to one of the following three homomorphisms ν_{4,1}, ν_{4,2}, ν_{4,3}:

\begin{aligned}ν_{4,1}(σ_{1}) & = (1, 2, 3, 4), \\ ν_{4,2}(σ_{1}) & = (1, 3, 2, 4), \\ ν_{4,3}(σ_{1}) & = (1, 2, 3), \end{aligned}  \begin{aligned}ν_{4,1}(α) & = (1, 2); \\ ν_{4,2}(α) & = (1, 2, 3, 4); \\ ν_{4,3}(α) & = (1, 2)(3, 4); \end{aligned}  \begin{aligned}[ν_{4,2}(σ_{3}) & = ν_{4,2}(σ_{1}^{−1})] \\ [ν_{4,3}(σ_{3}) & = ν_{4,3}(σ_{1})]. \end{aligned}

d) Except of the case n = 4 and ψ ∼ ν_{4,3}, the homomorphism ψ is surjective. In the exceptional case when ψ ∼ ν_{4,3} the image of ψ coincides with the alternating subgroup A(4) ⊂ S(4). ◯

The following lemma is the heart of Artin’s methods developed in [Art47b].^{8}

Artin Lemma. Let n > 4 and m be natural numbers. Suppose that there is a prime p > 2 such that

m/2 < p ≤ n − 2 .  (4.10)

Then for every non-cyclic transitive homomorphism ψ: B_{n} → S(m) the permutation σ̂_{1} = ψ(σ_{1}) has at least n − 2 fixed points; in particular, such a homomorphism cannot exist if m < n. ◯

We use Artin Lemma to prove the following useful addition to Artin Theorem (see [Lin72a, Lin74, Lin79], and also [Lin96b], Theorem 2.1).

Theorem 4.4. For n ≠ 4 and n > k any homomorphism ψ: B_{n}(X) → S(k) is cyclic.

Proof. As B_{n}(S^{2}) is a quotient group of B_{n} it suffices to prove the theorem for homomorphisms ψ: B_{n} → S(k).

For n ≤ 3 all is trivial, since S(2) ≅ ℤ/2ℤ. Suppose that n > max{4, k} and there is a non-cyclic homomorphism ψ: B_{n} → S(k). Let H = Im ψ ⊆ S(k). For each orbit O ⊆ Δ_{k} = {1, ..., k} of H, define the reduction ψ_{O}: B_{n} → S(O) of ψ to O as follows: ψ_{O}(g) = ψ(g)|O for all g ∈ B_{n}, where ψ(g)|O denotes the restriction of the permutation ψ(g) ∈ H ⊆ S(k) to the H-orbit O. Clearly, all such reductions are transitive (meaning that Im ψ_{O} is a transitive subgroup of the symmetric group S(O)), and for at least one orbit O_{∘} the corresponding reduction ψ_{O_{∘}}: B_{n} → S(O_{∘}) ≅ S(m) (m = #O_{∘}) must be non-cyclic (otherwise, ψ itself would be abelian and hence cyclic). Since n > 4 and m ≤ k < n, it follows from the famous Chebyshev Theorem that there is a prime p > 2 such that m/2 < p ≤ n − 2. In view of the above inequalities m ≤ k < n, Artin Lemma implies that any transitive homomorphism B_{n} → S(m) must be cyclic, which contradicts our choice of the orbit O_{∘}. □

8In fact, this lemma is not formulated explicitly in [Art47b]; moreover, Artin treats only the special case n = m. However, the corresponding part of the proof of Lemma 6 in [Art47b], after an evident minor modification, leads to the desired result. See [Lin96b] for more details.

We are now in a position to prove Theorem 2.7.

Proof. Let n ≠ 4 and let ψ: B_{n}(X) → S(n) be a non-cyclic homomorphism. We must prove that ψ is surjective.

We have already noticed that it suffices to deal with the Artin braid group B_{n}. If ψ is transitive the statement follows directly from Artin Theorem. Assume that ψ is intransitive. Then #O < n for any (Im ψ)-orbit O ⊂ Δ_{n} and Theorem 4.4 implies that the reduction of ψ to any such orbit is cyclic. Hence ψ itself is cyclic, which contradicts our assumption. □

4.4. Proof of Composition Theorem. Suppose, on the contrary, that the composition ψ = μ ∘ φ: B_{n}(X) ⟶^{φ} B_{k}(X) ⟶^{μ} S(k) is cyclic. Then μ(φ(B′_{n}(X))) = {1} and hence φ(B′_{n}(X)) ⊆ Ker μ = P B_{k}(X). By Commutator Theorem, the commutator subgroup B′_{n}(X) is perfect; thus, the inclusion φ(B′_{n}(X)) ⊆ P B_{k}(X) and Theorem 3.3 imply φ(B′_{n}(X)) = {1}, which contradicts the assumption that φ is non-cyclic. □

4.5. Homomorphisms B_{n}(X) → B_{k}(X) for n > k. Theorem 4.4 and Composition Theorem imply the following result.

Theorem 4.5. For n ≠ 4 and n > k any homomorphism φ: B_{n}(X) → B_{k}(X) is cyclic.

Proof. For n = 3 all is trivial since B_{2}(X) is cyclic. Let n > 4. Were φ non-cyclic, then, by Composition Theorem, the composition ψ = μ ∘ φ: B_{n}(X) ⟶^{φ} B_{k}(X) ⟶^{μ} S(k) would be non-cyclic, which contradicts Theorem 4.4. □

4.6. Proof of Equivariance Theorem. As we mentioned in Section 2, Equivariance Theorem follows immediately from Composition Theorem. □

4.7. Proof of Kernel Theorem. Let n > 4 and let ψ: B_{n}(X) → S(n) be a noncyclic homomorphism. By Theorem 2.7, ψ is surjective and, all the more, transitive^{9}.

All non-cyclic transitive homomorphisms ψ: B_{n} → S(n) were described by E. Artin [Art47b] (see also [Lin96b], Section 0.5); it follows from this description that Ker ψ = P B_{n}.

9That is, its image ψ(B_{n}(X)) is a transitive permutation group.

Finally, for a non-cyclic homomorphism ψ: B_{n}(S^{2}) → S(n) of the sphere braid group, we apply the above result to the composition

ψ̃ = ψ ∘ i_{∗}: B_{n} ⟶^{i_{∗}} B_{n}(S^{2}) ⟶^{ψ} S(n)

and conclude that (i_{∗})^{−1}(Ker ψ) = Ker(ψ ∘ i_{∗}) = Ker ψ̃ = P B_{n}. Since the homomorphism i_{∗}: B_{n} → B_{n}(S^{2}) is surjective, i_{∗}((i_{∗})^{−1}(H)) = H for any H ⊆ B_{n}(S^{2}); hence Ker ψ = i_{∗}((i_{∗})^{−1}(Ker ψ)) = i_{∗}(P B_{n}) = P B_{n}(S^{2}) (see (4.6)). □

4.8. Proof of Invariance Theorem. We prove the following stronger statement:

Let n > 4. Then φ^{−1}(P B_{n}(X)) = P B_{n}(X) for every non-cyclic endomorphism φ: B_{n}(X) → B_{n}(X).

By Composition Theorem, the homomorphism ψ = μ∘φ: B_{n}(X) ⟶^{φ} B_{n}(X) ⟶^{μ} S(n) is non-cyclic, and Kernel Theorem shows that Ker ψ = P B_{n}(X). Thus,

P B_{n}(X) = Ker ψ = φ^{−1}(Ker μ) = φ^{−1}(P B_{n}(X)) .

Of course, this implies φ(P B_{n}(X)) = φ(φ^{−1}(P B_{n}(X))) ⊆ P B_{n}(X). □

As we mentioned in Section 2.1, Lifting Theorem follows immediately from Invariance Theorem. □

5. Holomorphic functions omitting two values

In this section we prove Theorem 2.15, that is, determine explicitly all non-constant holomorphic functions C_{o}^{n}(ℂ) → ℂ \ {0; 1} and C_{o}^{n}(ℂℙ^{1}) → ℂ \ {0; 1}.

Furthermore, we establish that the set L(Z) of all non-constant holomorphic functions Z → \{0; 1} on a quasi-projective algebraic variety Z carries a natural structure of a finite simplicial complex. For Z = C_{o}^{n}(ℂ) and Z = C_{o}^{n}(ℂℙ^{1}) we study this structure in more details, which provides us with a very useful tool for the proof of Coherence Theorem.

5.1. Proof of Theorem 2.15. For non-zero complex functions f, g of the same variables, we write f ≈ g whenever f = γg for some γ ∈ ℂ^{∗}. The following lemma is, in a sense, a multivariate analog of so-called ”abc-lemma“ for polynomials of one variables.

Lemma 5.1. Let P, Q, R ∈ ℂ[z_{1}, ..., z_{n}] be pairwise coprime polynomials that do not vanish outside of the ”big diagonal“ D = ⋃_{i≠j}{z_{i} = z_{j}}. Assume that at least one of them is non-constant and

P + Q + R = 0 .  (5.1)

Then either

P ≈ z_{q} − z_{r} , Q ≈ z_{r} − z_{p} , R ≈ z_{p} − z_{q} (5.2)

or

P ≈ (z_{q} − z_{r})(z_{s} − z_{p}) , Q ≈ (z_{r} − z_{p})(z_{s} − z_{q}) , R ≈ (z_{p} − z_{q})(z_{s} − z_{r}) . (5.3)

Proof. Let a_{i}, b_{i}, c_{i} denote the degrees of P, Q, R with respect to the variable z_{i} and let d_{i} = max{a_{i}, b_{i}, c_{i}}; the condition (5.1) implies that at least two of the three numbers a_{i}, b_{i}, c_{i} must coincide with d_{i}.

Every polynomial non-vanishing outside of D is of the form γ ∏_{i≠j}(z_{i}−z_{j})^{s_{ij}}, where γ ∈ ℂ^{∗} and all s_{ij} ∈ ℤ_{+}. Hence, for every i_{∘} we have representations

P = P_{i_{∘}} ∏_{j≠i_{∘}}(z_{i_{∘}} − z_{j})^{a_{i∘j}} , Q = Q_{i_{∘}} ∏_{j≠i_{∘}}(z_{i_{∘}} − z_{j})^{b_{i∘j}} , R = R_{i_{∘}} ∏_{j≠i_{∘}}(z_{i_{∘}} − z_{j})^{c_{i∘j}} , (5.4)

where

P_{i_{∘}} ≈ ∏_{i,j≠i_{∘}^{i≠j}}(z_{i} − z_{j})^{a_{i∘;i,j}} , Q_{i_{∘}} ≈ ∏_{i,j≠i_{∘}^{i≠j}}(z_{i} − z_{j})^{b_{i∘;i,j}} , R_{i_{∘}} ≈ ∏_{i,j≠i_{∘}^{i≠j}}(z_{i} − z_{j})^{c_{i∘;i,j}} (5.5)

are pairwise coprime polynomials not depending on z_{i_{∘}}, and the products

∏_{j≠i_{∘}}(z_{i_{∘}} − z_{j})^{a_{i∘j}} , ∏_{j≠i_{∘}}(z_{i_{∘}} − z_{j})^{b_{i∘j}} , ∏_{j≠i_{∘}}(z_{i_{∘}} − z_{j})^{c_{i∘j}}

are pairwise coprime polynomials in the variable z_{i_{∘}} of degrees a_{i_{∘}}, b_{i_{∘}}, c_{i_{∘}}, respectively.

We consider the following two mutual complementary cases:

a) there exists a value i = p such that the triple {a_{p}, b_{p}, c_{p}} contains at least two distinct numbers;

b) a_{i} = b_{i} = c_{i} for every i.

a) We may assume that a_{p} < b_{p} = c_{p}. Taking into account (5.4) (with i_{∘} = p) and comparing the leading terms with respect to the variable z_{p} in identity (5.1), we obtain Q_{p} + R_{p} = 0. As Q_{p} and R_{p} are coprime, they are non-zero constants. Hence

Q ≈ ∏_{j≠p}(z_{p} − z_{j})^{b_{pj}} , R ≈ ∏_{j≠p}(z_{p} − z_{j})^{c_{pj}}

and identity (5.1) may be written as

P_{p} ∏_{j≠p}(z_{p} − z_{j})^{a_{pj}} + B ∏_{j≠p}(z_{p} − z_{j})^{b_{pj}} + C ∏_{j≠p}(z_{p} − z_{j})^{c_{pj}} = 0 ,  (5.6)

where B, C ∈ ℂ^{∗}.

Each exponent a_{pj} in (5.6) must be zero. Indeed, each factor z_{p} − z_{j} that actually occurs in the first summand of (5.6) cannot occur in the second and the third ones; were a_{pj_{∘}} > 0 for some j_{∘}, the left hand side would be a non-trivial polynomial in the variable z_{j_{∘}}, which contradicts (5.6).

This means that a_{p} = ∑_{j≠p} a_{pj} = 0, the polynomial P does not depend on z_{p}, P = P_{p}, and identity (5.6) takes the form

P + B ∏_{j≠p}(z_{p} − z_{j})^{b_{pj}} + C ∏_{j≠p}(z_{p} − z_{j})^{c_{pj}} = 0 .

For the particular value z_{p} = 0 the latter identity shows that

P + B ∏_{j≠p}(−z_{j})^{b_{pj}} + C ∏_{j≠p}(−z_{j})^{c_{pj}} = 0 ,

where the polynomial P = P_{p} is given by (5.5) (with i_{∘} = p), and the products

∏_{j≠p}(−z_{j})^{b_{pj}} ,  ∏_{j≠p}(−z_{j})^{c_{pj}}

are coprime. One can readily see that this may only happen if P, Q, R satisfy (5.2) with some distinct q, r ≠ p, which complete the proof in case (a).

b) Since a_{i} = b_{i} = c_{i} for any i, and at least one of the polynomials P, Q, R is nonconstant, there is a value i = s such that z_{s} actually occurs in P, Q, and R. Taking into account (5.4) and (5.5) (with i_{∘} = s), we can write (5.1) as

P_{s} ∏_{j≠s}(z_{s} − z_{j})^{a_{sj}} + Q_{s} ∏_{j≠s}(z_{s} − z_{j})^{b_{sj}} + R_{s} ∏_{j≠s}(z_{s} − z_{j})^{c_{sj}} = 0 .  (5.7)

By our choice of s,

∑_{j≠s} a_{sj} = ∑_{j≠s} b_{sj} = ∑_{j≠s} c_{sj} > 0 ;

hence there is p ≠ s such that a_{sp} > 0. As P is coprime with Q, R and contains the factor (z_{s} − z_{p})^{a_{sp}}, we have b_{sp} = c_{sp} = 0. Comparing the leading terms with respect to z_{s} in (5.7), we see that

P_{s} + Q_{s} + R_{s} = 0.  (5.8)

Let a′_{sp}, b′_{sp}, c′_{sp} be the degrees of the polynomials P_{s}, Q_{s}, R_{s} with respect to z_{p}. Clearly,

a′_{sp} = a_{p} − a_{sp} < a_{p} , b′_{sp} = b_{p} − b_{sp} = b_{p} = a_{p} , c′_{sp} = c_{p} − c_{sp} = c_{p} = a_{p};

hence a′_{sp} < b′_{sp} = c′_{sp}; in particular, Q_{s} and R_{s} are non-constant. In view of (5.8), this means that the triple {P_{s}, Q_{s}, R_{s}} is of the type considered in case (a). Therefore,

P_{s} ≈ (z_{q} − z_{r}) , Q_{s} ≈ (z_{r} − z_{p}) , R_{s} ≈ (z_{p} − z_{q})

for certain distinct q, r such that q ≠ p, s and r ≠ p, s. Consequently, identity (5.7) may be written as

\begin{matrix}A(z_{q} − z_{r}) & (z_{s} − z_{p})^{a_{sp}} ∏_{j≠s,p}(z_{s} − z_{j})^{a_{sj}} \\ + B(z_{r} − z_{p}) ∏_{j≠s,p}(z_{s} − z_{j})^{b_{sj}} + C(z_{p} − z_{q}) ∏_{j≠s,p}(z_{s} − z_{j})^{c_{sj}} = 0 \end{matrix} (5.9)

(with certain A, B, C ∈ ℂ^{∗}). In particular, on the hyperplane z_{p} = z_{q} we have the identity

A(z_{q} − z_{r})(z_{s} − z_{q})^{a_{sp}} ∏_{j≠s,p}(z_{s} − z_{j})^{a_{sj}} + B(z_{r} − z_{q}) ∏_{j≠s,p}(z_{s} − z_{j})^{b_{sj}} = 0 ,

or

A(z_{s} − z_{q})^{a_{sp}} ∏_{j≠s,p}(z_{s} − z_{j})^{a_{sj}} − B ∏_{j≠s,p}(z_{s} − z_{j})^{b_{sj}} = 0 .  (5.10)

The products ∏_{j≠s,p}(z_{s} − z_{j})^{a_{sj}} and ∏_{j≠s,p}(z_{s} − z_{j})^{b_{sj}} must be coprime; hence identity (5.10) can hold true only if

A = B, a_{sj} = 0 for all j ≠ s, p , b_{sj} = 0 for all j ≠ s, p, q , and a_{sp} = b_{sq} .

Under these conditions we have

P ≈ (z_{q}−z_{r})(z_{s}−z_{p})^{a_{sp}}, Q ≈ (z_{r}−z_{p})(z_{s}−z_{q})^{a_{sp}}, R ≈ (z_{p}−z_{q}) ∏_{j≠s,p,q}(z_{s}−z_{j})^{c_{sj}}, (5.11)

and identity (5.9) takes the form

A(z_{q} − z_{r})(z_{s} − z_{p})^{a_{sp}} + A(z_{r} − z_{p})(z_{s} − z_{q})^{a_{sp}} + CR = 0 .  (5.12)

Taking into account the conditions a_{q} = b_{q} = c_{q} and a_{r} = b_{r} = c_{r}, we obtain from (5.12) and (5.11) that a_{sp} = 1, c_{sj} = 0 for all j ≠ r, and c_{sr} = 1; hence P ≈ (z_{q} −z_{r})(z_{s}−z_{p}) , Q ≈ (z_{r} − z_{p})(z_{s} − z_{q}) , R ≈ (z_{p} − z_{q})(z_{s} − z_{r}) . □

Now we are ready to prove Theorem 2.15.

Proof. a) Let ℓ ⊂ ℂ^{n} be any com lex straight line such that ℓ ⊈ D; by Great Picard Theorem, the restriction μ|_{C_{o}^{n}(ℂ)∩ℓ} is a regular function. Thus, the function μ itself is rational on ℂ^{n} and regular on C_{o}^{n}(ℂ). That is, μ = −P/R, where polynomials P, R ∈ ℂ[q_{1}, ..., q_{n}] are coprime, do not vanish on C_{o}^{n}(ℂ), and at least one of them is non-constant. The polynomial Q = −P − R is coprime to P and R; it has no zeros in C_{o}^{n}(ℂ), as −Q/R = 1 − μ and the function μ does not assume the value 1. Lemma 5.1 implies the desired result.

b) By (a), the restriction of μ to the Zariski open subset C_{o}^{n}(ℂ) ⊂ C_{o}^{n}(ℂℙ^{1}) must be either a simple ratio sr_{ijk} or a cross ratio cr_{ijkl}. However, the function sr_{ijk} = (q_{k} − q_{i})/(q_{k} − q_{j}) has a pole in C_{o}^{n}(ℂℙ^{1}) along the divisor q_{i} = ∞, and only the cross ratios survive when passing from C_{o}^{n}(ℂ) to C_{o}^{n}(ℂℙ^{1}). □

Remark 5.2. We have seen in Section 3.1 that for any m ≥ 1 there are a natural identification

D^{m}(ℂℙ^{1}) ∋ (q_{1}, ..., q_{m}, 0, 1, ∞) ⟷^{≅} (q_{1}, ..., q_{m}) ∈ C_{o}^{m}(ℂ^{◯ 2}) .  (5.13)

and a biholomorphic isomorphism

j_{ℂℙ^{1},m+3} : C_{o}^{m+3}(ℂℙ^{1}) ⟶^{≅} PSL(2, ℂ) × D^{m}(ℂℙ^{1}) = PSL(2, ℂ) × C_{o}^{m}(ℂ^{◯ 2}) ; (5.14)

j_{ℂℙ^{1},m+3} and its inverse j_{ℂℙ^{1},m+3}^{−1} are defined by

\begin{matrix}j_{ℂℙ^{1},m+3}(q) = (Θ(q), ( & Θ(q))^{−1}q) ,  j_{ℂℙ^{1},m+3}^{−1}(A, z) = Az \\ (q ∈ C_{o}^{m+3}(ℂℙ^{1}) , A ∈ PSL(2, ℂ) , z ∈ D^{m}(ℂℙ^{1})) , \end{matrix} (5.15)

where Θ: C_{o}^{m+3}(ℂℙ^{1}) → PSL(2, ℂ) denote the morphism whose value at any point q = (q_{1}, ..., q_{m+3}) ∈ C_{o}^{m+3}(ℂℙ^{1}) is a unique element Θ(q) ∈ PSL(2, ℂ) that carries (0, 1, ∞) to (q_{m+1}, q_{m+2}, q_{m+3}). Due to the Picard theorem any holomorphic function C_{o}^{m+3}(ℂℙ^{1}) → ℂ \ {0; 1} is constant on each fiber of the projection C_{o}^{m+3}(ℂℙ^{1}) → C_{o}^{m}(ℂ^{◯ 2}); therefore there is a one-to-one correspondence between the set L(C_{o}^{m}(ℂ^{◯ 2})) of all non-constant holomorphic functions C_{o}^{m}(ℂ^{◯ 2}) → ℂ\{0; 1} and the set L(C_{o}^{m+3}(ℂℙ^{1})) of all non-constant holomorphic functions C_{o}^{m+3}(ℂℙ^{1}) → ℂ \ {0; 1}. Due to Theorem 2.15 we know all functions of the latter set; this provides us with an explicit description of all functions h ∈ L(C_{o}^{m+3}(ℂℙ^{1})), which was also obtained by S. Kaliman in a different way.

Similarly, using an isomorphism C_{o}^{m}(ℂ^{∗})) ≅ ℂ^{∗} × C_{o}^{m−1}(ℂ^{◯ 2}) we may identify the set L(C_{o}^{m}(ℂ^{∗})) first with L(C_{o}^{m−1}(ℂ^{◯ 2})) and then with L(C_{o}^{m+2}(ℂℙ^{1})), which provides an explicit description of all non-constant holomorphic functions C_{o}^{m}(ℂ^{∗}) → ℂ \ {0; 1} obtained in [Zin77] by a different way. ◯

5.2. Simplicial complex of holomorphic functions omitting two values. Here I will describe an almost obvious combinatorial structure, which occurs amazingly useful for the proof of Coherence Theorem. I came to it very recently when the first version of this paper was already written. In this renewed version the proof does not become much shorter but its main ideas are now absolutely clear. Strangely enough I paid no attention to this structure 30 years ago. But seemingly nobody did so...

For an irreducible quasi-projective variety Z we denote by L(Z) the set of all nonconstant holomorphic functions λ: Z → ℂ \ {0; 1}. It is well known that all λ ∈ L(Z) are regular and L(Z) is finite.^{10} Clearly λ^{−1} ∈ L(Z) whenever λ ∈ L(Z) and ι: L(Z) ∋ λ ↦ ι(λ) =^{def} λ^{−1} ∈ L(Z) is an involution. The set L(Z) carries a natural structure of a finite simplicial complex L_{△}(Z) described in the following definition.

Definition 5.3. Let μ, ν ∈ L(Z); we say that ν is a proper divisor of μ and write ν | μ if the quotient λ = μ : ν ∈ L(Z), i. e., λ ≠ const and λ(z) ≠ 1 for all z ∈ Z; otherwise, we write ν ∤ μ. Clearly, ν | μ is equivalent to μ | ν.

A non-empty subset (respectively, a non-empty ordered subset) Δ ⊆ L(Z) is said to be a simplex (respectively, an ordered simplex) of dimension m = #Δ−1 if ν | μ for each pair of distinct μ, ν ∈ L(Z); the functions μ ∈ Δ are said to be the vertices of Δ. Any non-empty subset of an (ordered) simplex is an (ordered) simplex. We define the boundary ∂Δ of an ordered simplex Δ = [μ_{0}, ..., μ_{m}] ⊆ L(Z) by the standard formula

∂Δ =^{def} ∑_{i=0}^{m}(−1)^{i}[μ_{0}, ..., μ̂_{i}, ..., μ_{m}] .  (5.16)

Thereby L(Z) becomes a finite simplicial complex L_{△}(Z) with the simplicial involution ι.

When we need to emphasize that L(Z) is regarded as a simplicial complex, i. e., the collection of all simplices Δ ⊂ L(Z) rather than the set of all vertices μ ∈ L(Z) we write L_{△}(Z) instead of L(Z) and Δ ∈ L_{△}(Z) instead of Δ ⊂ L(Z). ◯

Lemma 5.4. Let f: Z → Y be a holomorphic mapping of irreducible quasi-projective varieties. Suppose that for each λ ∈ L(Y) the composition

f^{∗}(λ) =^{def} λ ∘ f: Z ⟶^{f} Y ⟶^{λ} ℂ \ {0; 1}

is non-constant (this is certainly the case whenever f is dominant). Then

f^{∗}: L(Y) ∋ λ ↦ λ ∘ f ∈ L(Z)  (5.17)

is a simplicial mapping whose restriction to each simplex Δ ⊆ L(Y) is injective. Consequently, f^{∗} preserves dimension of simplices. Moreover, f^{∗}(ιλ) = ιf^{∗}(λ) for all λ ∈ L(Y).

10In what follows we are interested only in the case when L(Z) ≠ ∅.

Proof. Let μ, ν be two distinct vertices of a simplex Δ ∈ L_{△}(Y) and let λ = μ : ν. By Definition 5.3, λ ∈ L(Y). By our assumption, f^{∗}(μ) : f^{∗}(ν) = f^{∗}(λ) ≠ const and μ(f(z)) : ν(f(z)) = λ(f(z)) ≠ 1 for all z ∈ Z. Hence f^{∗}(ν) | f^{∗}(μ). This shows that f^{∗}(Δ) ∈ L_{△}(Z) and the restriction of f^{∗} to the set of all vertices of Δ is injective. The last statement of Lemma is evident. □

5.3. Complexes of simple and cross ratios. Let, as usual, X = ℂ or X = ℂℙ^{1} and let Z = C_{o}^{n}(X). Due to Theorem 2.15, where we met all non-constant holomorphic functions Z → ℂ\{0; 1} face to face, certain properties of the simplicial complex L(Z) may be described in a more explicit manner.

Notation. It is convenient to start with a little bit more formal situation. Let q = {q_{i} | i ∈ ℕ} be a countable set of independent variables. For any three distinct i, j, k ∈ ℕ and any four distinct i, j, k, l ∈ ℕ the rational functions^{11}

sr_{ijk} = sr(q_{i}, q_{j}, q_{k}) =^{def} \frac{q_{k} − q_{i}}{q_{k} − q_{j}} and

cr_{ijkl} = cr(q_{i}, q_{j}, q_{k}, q_{l}) =^{def} \frac{q_{l} − q_{i}}{q_{l} − q_{j}} : \frac{q_{i} − q_{k}}{q_{j} − q_{k}} = \frac{(q_{l} − q_{i})(q_{j} − q_{k})}{(q_{l} − q_{j})(q_{i} − q_{k})}

(5.18)

are referred to as simple and cross ratios, respectively. The unordered set of variables q that actually participate in such a function μ is called its support; we denote it by supp μ; the differences of these variables in the numerator and the denominator of μ are said to be the factors of μ. There are 6 distinct simple ratios with support {q_{i}, q_{j}, q_{k}} and 6 distinct cross ratios with support {q_{i}, q_{j}, q_{k}, q_{l}}. (Permutations of i, j, k, l formally lead to 24 cross ratios with the same support, but only 6 of them are distinct, since Kleinian permutations 1, (i, j)(k, l), (i, k)(j, l), (i, l)(j, k) do not change the rational function cr_{ijkl}.) The sets of all simple and all cross ratios are denoted by SR(q) and CR(q), respectively. We set L(q) = SR(q) ∪ CR(q). The following definition is similar to Definition 5.3.

Definition 5.5. Let μ, ν ∈ L(q) and μ ≠ ν. We say that ν is a proper divisor of μ and write ν | μ if the quotient μ : ν ∈ L(q); otherwise, we write ν ∤ μ. As λ ∈ L(q) is equivalent to λ^{−1} ∈ L(q), ν | μ is equivalent to μ | ν. The set (λ, μ) of all common proper divisors of elements λ, μ ∈ L(q) consists of all ν ∈ L(q) such that ν | λ and ν | μ.

A non-empty finite subset Δ ⊂ L(q) is called a simplex (respectively, an ordered simplex when Δ is ordered) of dimension dim Δ = #Δ − 1 if μ | ν for every two distinct μ, ν ∈ Δ.^{12} The elements μ ∈ Δ are said to be the vertices of Δ and the union supp Δ =^{def} ⋃_{μ∈Δ} supp μ is called the support of Δ. Any non-empty subset of an

(ordered) simplex is an (ordered) simplex. The boundary ∂Δ of an ordered simplex Δ = [μ_{0}, ..., μ_{m}] ⊆ L(q) is defined by formula (5.16). Thereby, L(q) becomes a simplicial complex with the simplicial involution ι: λ ↦ λ^{−1}.

11The ground field k is not important as soon as it is commutative and of characteristic 0; for instance, k = ℂ and k = ℚ both are suitable.

12The latter definition does not require that the quotients ν : μ, μ : ν must be elements of Δ.

As in Section 5.2, we use also the notation L_{△}(q) instead of L(q) to emphasize that we deal with a simplicial complex, i. e., the collection of all simplices Δ ⊂ L(q) rather than just the set of all vertices μ ∈ L(q). There are three types of simplices in L_{△}(q): two types of “pure” ones, which belong either to SR_{△}(q) (they are referred to as “simple” ones) or to CR_{△}(q) (they are referred to as “cross” ones), and “mixed” ones, which have vertices in both SR(q) and CR(q).

We denote by SR(q_{n}) ⊂ SR(q), CR(q_{n}) ⊂ CR(q) and L(q_{n}) ⊂ L(q) the subsets consisting of all vertices of the corresponding type whose supports are contained in the set q_{n} = {q_{1}, ..., q_{n}} ⊂ q of the first n variables. The subcomplexes SR_{△}(q_{n}), CR_{△}(q_{n}) and L_{△}(q_{n}) consist of all simplices whose vertices are in SR(q_{n}), CR(q_{n}) and L(q_{n}), respectively.^{13} ◯

The main purpose of the following three lemmas is to estimate dimension of simplices of a certain type provided their supports are contained in q_{n}.

Lemma 5.6. a) If μ, ν ∈ SR(q) and ν | μ then either μ and ν have the same denominator or they have the same numerator; in particular, #(supp μ∩supp ν) = 2.

b) Each proper divisor μ ∈ SR(q) of sr_{ijk} is obtained from sr_{ijk} by replacing of one of the indices i, j with some l ≠ i, j, k.

In fact, up to a choice of l ≠ i, j, k and the order of multipliers, the formula

sr_{ijk} = sr_{ilk} · sr_{ljk} .  (5.19)

provides us with a unique possible decompositions of sr_{ijk} to a product of two other simple ratios.

c) For every four distinct i, j, k, l ∈ ℕ

\begin{matrix}(sr_{ijk}, sr_{ilk}) ∩ SR(q) = \{sr_{imk} | m ≠ i, j, k, l\} , \\ (sr_{ijk}, sr_{ljk}) ∩ SR(q) = \{sr_{mjk} | m ≠ i, j, k, l\} . \end{matrix}  (5.20)

d) If Δ = {λ, μ, ν} ⊂ SR(q) is a simplex, then either λ, μ, ν have the same denominator or they all have the same numerator; in any case # supp Δ = 5.

e) Let n ≥ 3 and Δ ∈ SR_{△}(q_{n}). Then either all vertices of Δ have the same denominator or they all have the same numerator. Moreover, dim Δ ≤ n − 3, i. e., dim SR_{△}(q_{n}) ≤ n − 3.

Proof. (a) is evident; the proof of (b) is straightforward. (c) is a simple consequence of (b), and (d) follows from (c). The first statement of (e) follows immediately from (d), and the second statement follows from the first one. □

Lemma 5.7. a) If μ, ν ∈ CR(q) and ν | μ then #(supp μ ∩ supp ν) = 3.

b) Each proper divisor μ ∈ CR(q) of cr_{ijkl} is obtained from cr_{ijkl} by replacing of one of the indices i, j, k, l with some m ≠ i, j, k, l.

In fact, up to a choice of m ≠ i, j, k, l and the order of multipliers, the formulae

cr_{ijkl} = cr_{ijkm} · cr_{ijml} = cr_{imkl} · cr_{mjkl}  (5.21)

13L(q_{2}) = CR(q_{3}) = ∅. It should be emphasized that SR_{△}(q_{n}), CR_{△}(q_{n}) and L_{△}(q_{n}) are not the n-skeletons of the corresponding complexes.

provide us with all decompositions of cr_{ijkl} to a product of two other cross ratios.^{14}

c) For every five distinct i, j, k, l, m ∈ ℕ

\begin{aligned}(cr_{ijkl}, cr_{ijkm}) ∩ CR(q) & = \{cr_{ijkn} | n ≠ i, j, k, l, m\} , \\ (cr_{ijkl}, cr_{ijml}) ∩ CR(q) & = \{cr_{ijnl} | n ≠ i, j, k, l, m\} , \\ (cr_{ijkl}, cr_{imkl}) ∩ CR(q) & = \{cr_{inkl} | n ≠ i, j, k, l, m\} , \\ (cr_{ijkl}, cr_{mjkl}) ∩ CR(q) & = \{cr_{njkl} | n ≠ i, j, k, l, m\} . \end{aligned}  (5.22)

d) If Δ = {λ, μ, ν} ⊂ CR(q) is a simplex then # supp Δ = 6.

e) Let n ≥ 4. Then dim CR_{△}(q_{n}) ≤ n − 4, that is, dim Δ = #Δ − 1 ≤ n − 4 for every cross simplex Δ whose vertices are supported in {q_{1}, ..., q_{n}}.

Proof. a) The quotient μ : ν formally contains 8 factors; mutual cancellation of two pairs of them cannot happen unless #(supp μ ∩ supp ν) ≥ 3. A straightforward verification shows that ν ∤ μ whenever supp μ = supp ν.

b) Let cr_{ijkl} = μ′ · μ′′ for some two cross ratios μ′, μ′′; due to (a), we may also assume that supp μ′ is one of the sets {q_{i}, q_{j}, q_{k}, q_{m}}, {q_{i}, q_{j}, q_{l}, q_{m}}, {q_{i}, q_{k}, q_{l}, q_{m}}, {q_{j}, q_{k}, q_{l}, q_{m}}. There are precisely 4 · 6 = 24 cross ratios μ′ with such a support, and one may check them all, one by one; this leads to the desired result.

c) By (b), to get a common proper divisor ν ∈ CR(q) of cr_{ijkl} and cr_{ijkm}, one must replace one of the indices in each of the latter two cross ratios with some n ≠ i, j, k, l, m so that the results coincide with each other. Only the cross ratio ν = cr_{ijkn} appears in such a way; this proves the first of relations (5.22). As Kleinian permutations (i, j)(k, l), (i, k)(j, l) and (i, l)(j, k) do not change the rational function cr_{ijkl}, and for any s ≠ i, j, k, l we have cr_{jils} = cr_{ijsl}, cr_{klis} = cr_{iskl} and cr_{lkjs} = cr_{sjkl}, three other relations follow from the first one.

d) Let λ = cr_{ijkl}. By (b), there is m ≠ i, j, k, l such that μ coincides with one of the four cross ratios cr_{ijkm}, cr_{ijml}, cr_{imkl}, cr_{mjkl}. In each of these cases one of relations (5.22) shows that #(supp λ ∪ supp μ ∪ supp ν) = 6.

e) By (a), statement (e) is valid for n = 4. Suppose that for some n ≥ 4 we have dim CR_{△}(q_{n}) ≤ n − 4 and consider any simplex Δ ∈ CR_{△}(q) with support supp Δ ⊆ {q_{1}, ..., q_{n+1}}. We must prove that dim Δ ≤ n − 3, i. e., #Δ ≤ n − 2.

Take two distinct λ, μ ∈ Δ. According to (a), #(supp λ ∩ supp μ) = 3; thus, supp λ = {q_{i}, q_{j}, q_{k}, q_{l}} and supp μ = {q_{i}, q_{j}, q_{k}, q_{m}} for some distinct i, j, k, l, m ∈ {1, ..., n + 1}.

Let Δ(m̂) = {τ ∈ Δ | q_{m} ∉ supp τ}; clearly, λ ∈ Δ(m̂) and μ ∉ Δ(m̂). By the induction hypothesis, dim Δ = #Δ(m̂) − 1 ≤ n − 4. Let us show that actually Δ\Δ(m̂) = {μ}; this will imply #Δ = #Δ(m̂)+1 ≤ (n−3)+1 = n−2 and complete the step of induction.

Suppose to the contrary that ν ∈ Δ\Δ(m̂) and ν ≠ μ; clearly, q_{m} ∈ supp ν. As λ ∈ Δ(m̂), we have ν ≠ λ. Thereby, {λ, μ, ν} is a simplex; by (d), # supp Δ = #(supp λ∪ supp μ ∪ supp ν) = 6. Since supp λ ∪ supp μ = {q_{i}, q_{j}, q_{k}, q_{l}, q_{m}}, the set supp ν must contain at least one additional variable, say q_{p}, distinct from all q_{i}, q_{j}, q_{k}, q_{l}, q_{m}. As supp λ = {q_{i}, q_{j}, q_{k}, q_{l}} and q_{m}, q_{p} ∈ supp ν, we see that #(supp λ ∩ supp ν) ≤ 2, which contradicts (a). □

14As cr_{imkl} = cr_{klim} and cr_{mjkl} = cr_{klmj}, the Kleinian permutation (i, k)(j, l), which does not change cr_{ijkl}, transforms these decompositions to each other.

Lemma 5.8. a) Each mixed simplex Δ contains precisely one vertex μ ∈ SR(q). b) dim Δ ≤ n − 3 for any mixed simplex with support supp Δ ⊆ {q_{1}, ..., q_{n}}.

Proof. a) As Δ is mixed, there are μ ∈ Δ ∩ SR(q) and ν ∈ Δ ∩ CR(q). We may assume that μ = sr_{ijk}; then it is easily seen that ν = cr_{ijlk} for a certain l ≠ i, j, k. Up to an order of multipliers, there exist exactly two distinct decompositions of ν = cr_{ijlk} to a product of two simple ratios:

cr_{ijlk} = sr_{ijk} · sr_{jil} = sr_{kli} · sr_{lkj} .

Each of the three simple ratios sr_{jil}, sr_{kli}, sr_{lkj} is not a proper divisor of μ = sr_{ijk}; hence Δ does not contain any simple ratio λ ≠ μ.

b) By (a), there is a single vertex μ ∈ Δ∩SR(q). Thus, Δ′ =^{def} Δ\{μ} is a simplex contained in CR(q) with support in {q_{1}, ..., q_{n}}. By Lemma 5.7(e) dim Δ′ ≤ n − 4, which implies dim Δ ≤ n − 4. □

Example 5.9. a) Let n ≥ 3. Then Δ = {sr_{3,2,1}, sr_{4,2,1}, ..., sr_{n,2,1}} ⊂ SR(q) is a simplex of dimension n − 3 with support in {q_{1}, ..., q_{n}}.

b) Let s ≥ 4. Then Δ = {cr_{1,2,3,4}, cr_{1,2,3,5}, ..., cr_{1,2,3,n}} ⊂ CR(q) is a simplex of dimension n − 4 with support in {q_{1}, ..., q_{n}}.

c) Let n ≥ 3. Then Δ = {sr_{1,2,3}; cr_{1,2,4,3}, cr_{1,2,5,3}, ..., cr_{1,2,n,3}} is a mixed simplex of dimension n − 3 with support in {q_{1}, ..., q_{n}}.

These examples show that the upper bounds of dim Δ established in Lemma 5.6(e), Lemma 5.7(e) and Lemma 5.8(b) are the best possible ones. ◯

Due to Theorem 2.15 the complexes L_{△}(C_{o}^{n}(ℂ)) and L_{△}(C_{o}^{n}(ℂℙ^{1})) defined according to Section 5.2 may be identified to the complex L_{△}(q_{n}) and to the subcomplex CR_{△}(q_{n}) ⊂ L_{△}(q_{n}), respectively. When it is convenient we use also the notation

\begin{matrix}SR(C_{o}^{n}(ℂ)) , SR_{△}(C_{o}^{n}(ℂ)) , CR(C_{o}^{n}(ℂ)) = CR(C_{o}^{n}(ℂℙ^{1})) \\ and CR_{△}(C_{o}^{n}(ℂ)) = CR_{△}(C_{o}^{n}(ℂℙ^{1})) \end{matrix}

instead of SR(q_{n}), SR_{△}(q_{n}), CR(q_{n}) and CR_{△}(q_{n}).

Lemma 5.6, Lemma 5.7 and Example 5.9 imply the following corollary.

Corollary 5.10. a) dim SR_{△}(q_{n}) = n − 3 and dim CR_{△}(q_{n}) = n − 4.

b) dim L_{△}(C_{o}^{n}(ℂ)) = dim SR_{△}(C_{o}^{n}(ℂ)) = dim SR_{△}(q_{n}) = n − 3 and dim L_{△}(C_{o}^{n}(ℂℙ^{1})) = dim CR_{△}(C_{o}^{n}(ℂℙ^{1})) = dim CR_{△}(q_{n}) = n − 4. ◯

5.4. S(n) action in L(q_{n}). The natural S(n) action in {1, ..., n} induces the left S(n) action in L(q_{n}); in more details, for μ ∈ L(q_{n}) and σ ∈ S(n) the rational function σμ is defined by (σμ)(q) = μ(σ^{−1}q), i. e., (σμ)(q_{1}, ..., q_{n}) = μ(q_{σ(1)}, ..., q_{σ(n)}).

This action is transitive on the sets of vertices SR(q_{n}) and CR(q_{n}) (for S(n) action in {1, ..., n} is n transitive).

Furthermore, if μ, ν ∈ L(q_{n}) and ν | μ then for each σ ∈ S(n) we have (σμ)(q) : (σν)(q) = μ(σ^{−1}q) : ν(σ^{−1}q) = (μ : ν)(σ^{−1}q), which shows that σν | σμ. Hence for each simplex Δ = {μ_{0}, ..., μ_{s}} ∈ L_{△}(q_{n}) and each σ ∈ S(n) the set σΔ = {σμ_{0}, ..., σμ_{s}} is the simplex of the same type and dimension. Thus we have the (left) dimension preserving S(n) action in SR_{△}(q_{n}) and CR_{△}(q_{n}).

We denote by SR_{△^{m}}(q_{n}) ⊆ SR_{△}(q_{n}) and CR_{△^{m}}(q_{n}) ⊆ CR_{△}(q_{n}) the S(n) invariant subsets consisting of all m simplices of the corresponding type.

Definition 5.11. We define the following three normal forms of m simplices:

Δ_{S}^{m} =^{def} {sr_{3,2,1}, sr_{4,2,1}, ..., sr_{m+3,2,1}} ∈ SR_{△}(q_{n}) (0 ≤ m ≤ n − 3) ,  (5.23)
Δ_{S}^{−m} =^{def} {sr_{2,3,1}, sr_{2,4,1}, ..., sr_{2,m+3,1}} ∈ SR_{△}(q_{n}) (0 ≤ m ≤ n − 3)  (5.24)

and

Δ_{C}^{m} =^{def} {cr_{1,2,3,4}, cr_{1,2,3,5}, ..., cr_{1,2,3,m+4}} ∈ CR_{△}(q_{n}) (0 ≤ m ≤ n − 4) .  (5.25)

A simplex in a normal form we call normal.

The following lemma shows that the S(n) action in L(q_{n}) allows to carry any simplex to a normal form.

Lemma 5.12. a) For m ≥ 1 the set of all simple m simplices SR_{△^{m}}(q_{n}) is the disjoint union of two S(n) orbits S(n)Δ_{S}^{m} and S(n)Δ_{S}^{−m} of the normal m simplices Δ_{S}^{m} and Δ_{S}^{−m}. That is, any simple m simplex can be carried to one of the normal forms (5.23), (5.24) by a permutation σ ∈ S(n).

b) The S(n) action on CR_{△^{m}}(q_{n}) is transitive, that is, any cross m simplex can be brought to the normal form (5.25) by a permutation σ ∈ S(n).

Proof. a) Let Δ = {μ_{0}, ..., μ_{m}} ∈ SR_{△^{m}}(q_{n}). As permutations are admitted we may assume that μ_{0} = sr_{3,2,1}. By Lemma 5.6(b), μ_{1} must be one of the functions sr_{l,2,1}, sr_{3,l,1} with a certain l ≠ 1, 2, 3. A suitable permutation of {4, ..., n} keeps μ_{0} and carries μ_{1} to one of the functions sr_{4,2,1} and sr_{3,4,1}.

In the first case {μ_{0}, μ_{1}} = {sr_{3,2,1}, sr_{4,2,1}}; in the second case we apply the transposition (2, 3) to all vertices of Δ and get {μ_{0}, μ_{1}} = {sr_{2,3,1}, sr_{2,4,1}}.

If m = 1 we are done. For m ≥ 2 the simplex Δ has at least one more vertex μ_{2}; so either

(i) Δ = {μ_{0}, μ_{1}, μ_{2}, ..., μ_{m}} = {sr_{3,2,1}, sr_{4,2,1}, μ_{2}, ..., μ_{m}} or (ii) Δ = {μ_{0}, μ_{1}, μ_{2}, ..., μ_{m}} = {sr_{2,3,1}, sr_{2,4,1}, μ_{2}, ..., μ_{m}}. In case (i), μ_{0} and μ_{1} have the same denominator q_{1} − q_{2}. Lemma 5.6(e) shows

that all μ_{2}, ..., μ_{m} also have the same denominator q_{1} − q_{2}. By Lemma 5.6(c), they must be of the form μ_{2} = sr_{k_{2},2,1}, ..., μ_{m} = sr_{k_{m},2,1}, where all k_{2}, ..., k_{m} ∈ {5, ..., n} are distinct. A suitable permutation of {5, ..., n} keeps the forms of μ_{0}, μ_{1} and carries Δ to the normal form (5.23).

In case (ii), μ_{0} and μ_{1} have the same numerator q_{1} − q_{2} and Lemma 5.6(e) shows that so do all other vertices μ_{2}, ..., μ_{m} of Δ. As above, we conclude that they must be of the form μ_{2} = sr_{2,k_{2},1}, ..., μ_{m} = sr_{2,k_{m},1}, where all k_{2}, ..., k_{m} ∈ {5, ..., n} are distinct. A suitable permutation of {5, ..., n} does not change μ_{0}, μ_{1} and brings Δ to the normal form (5.24).

b) Let Δ = {μ_{0}, ..., μ_{m}} ∈ CR_{△^{m}}(q_{n}). We may assume that μ_{0} = cr_{1,2,3,4}. By Lemma 5.7(b), μ_{1} is one of the functions cr_{1,2,3,l}, cr_{1,2,l,4}, cr_{1,l,3,4}, cr_{l,2,3,4} for a certain l ≠ 1, 2, 3, 4. A suitable permutation of {5, ..., n} keeps μ_{0} and carries μ_{1} to one of the functions cr_{1,2,3,5}, cr_{1,2,5,4}, cr_{1,5,3,4}, cr_{5,2,3,4}. The Kleinian permutation (1, 3)(2, 4) does not change the rational function cr_{1,2,3,4} and carries the pair cr_{1,2,3,5}, cr_{1,2,5,4} to the pair cr_{1,5,3,4}, cr_{5,2,3,4}; hence we may restrict ourselves to the following two cases: either {μ_{0}, μ_{1}} = {cr_{1,2,3,4}, cr_{1,2,3,5}} or {μ_{0}, μ_{1}} = {cr_{1,2,3,4}, cr_{1,2,5,4}}. In fact, the Kleinian permutation (1, 2)(3, 4) keeps cr_{1,2,3,4} and brings cr_{1,2,5,4} to cr_{1,2,3,5}; thus we are left with the case when Δ = {μ_{0}, μ_{1}, ..., μ_{m}} = {cr_{1,2,3,4}, cr_{1,2,3,5}, ..., μ_{m}}.

If m = 1 we are done. For m ≥ 2 the first of relations (5.22) in Lemma 5.7(c) shows that μ_{2}, ..., μ_{m} must be of the form μ_{2} = cr_{1,2,3,k_{2}}, ..., μ_{m} = cr_{1,2,3,k_{m}}, where all k_{2}, ..., k_{m} ∈ {6, ..., n} are distinct. A suitable permutation of {6, ..., n} keeps μ_{0}, μ_{1} and brings Δ to the normal form (5.25). □

6. Coherence Theorem

In this section we prove Coherence Theorem formulated in Section 2.4, which completes the proof of Tame Map Theorem.

It is clear that all components of a strictly equivariant endomorphism

f = (f_{1}, ..., f_{n}): C_{o}^{n}(X) → C_{o}^{n}(X)

are non-constant holomorphic mappings C_{o}^{n}(X) → X. The following lemma shows that Lemma 5.4 applies to strictly equivariant endomorphisms.

Lemma 6.1. Let f = (f_{1}, ..., f_{n}): C_{o}^{n}(X) → C_{o}^{n}(X) be a strictly equivariant endomorphism and μ: C_{o}^{n}(X) → ℂ \ {0; 1} be a non-constant holomorphic function. Then the composition μ ∘ f: C_{o}^{n}(X) ⟶^{f} C_{o}^{n}(X) ⟶^{μ} ℂ \ {0; 1} is non-constant.

Proof. Suppose to the contrary that μ∘f = c = const ≠ 0, 1. Then (μ∘f)(σq) ≡ c for all σ ∈ S(n); as f is strictly equivariant, this implies (∗) μ(σf(q)) ≡ c for each σ ∈ S(n).

First we notice that c = −1. Indeed, either μ ∈ SR(q_{n}) or μ ∈ CR(q_{n}) (for X = ℂℙ^{1} the first case does not occur). S(n) is transitive on the sets SR(q_{n}) and CR(q_{n}), which are invariant under the involution μ ↦ μ^{−1}; thus μ^{−1} = sμ for a certain s ∈ S(n). Let σ = α^{−1}(s^{−1}), where α is the automorphism of S(n) related to f; then c^{−1} ≡ μ^{−1}(f(q)) ≡ sμ((q)) ≡ μ(s^{−1}f(q)) ≡ μ(f(σq)) ≡ c. Thus c = −1 and (∗) takes the form

(∗∗) μ(σf(q)) ≡ −1 for all σ ∈ S(n).

From this point we treat the cases X = ℂ and X = ℂℙ^{1} separately.

a) X = ℂ. Then either

(SR) μ = sr_{ijk} = (q_{k} − q_{i})/(q_{k} − q_{j}) and μ ∘ f = sr(f_{i}, f_{j}, f_{k}) = (f_{k} − f_{i})/(f_{k} − f_{j}) = −1 or (CR) μ = cr_{ijkl} = [(q_{l} − q_{i})(q_{j} − q_{k})]/[(q_{l} − q_{j})(q_{i} − q_{k})] and

μ ∘ f = cr(f_{i}, f_{j}, f_{k}, f_{l}) = [(f_{l} − f_{i})(f_{j} − f_{k})]/[(f_{l} − f_{j})(f_{i} − f_{k})] = −1 .

In case (SR) we have 2f_{k} = f_{i} +f_{j}. In view of (∗∗), we may take any other ordered triple of distinct indices; in particular, we have also 2f_{j} = f_{k} + f_{i}; subtracting this equality from the previous one, we obtain 3(f_{k} − f_{j}) = 0, which is impossible.

In case (CR) we have (f_{l} − f_{i})(f_{j} − f_{k}) = −(f_{l} − f_{j})(f_{i} − f_{k}); in view of (∗∗), we may apply the transposition (j, k), which gives (f_{l} − f_{i})(f_{k} − f_{j}) = −(f_{l} − f_{k})(f_{i} − f_{j}). These relations and the identity

(a − b)(d − c) − (a − c)(d − b) + (d − a)(b − c) = 0

with a = f_{i}, b = f_{j}, c = f_{k}, d = f_{l} imply

\begin{matrix}3(f_{l} − f_{i})(f_{j}− & f_{k}) = (f_{l} − f_{i})(f_{j} − f_{k}) + (f_{l} − f_{i})(f_{j} − f_{k}) + (f_{l} − f_{i})(f_{j} − f_{k}) \\ = (f_{i} − f_{j})(f_{l} − f_{k}) − (f_{i} − f_{k})(f_{l} − f_{j}) + (f_{l} − f_{i})(f_{j} − f_{k}) = 0 , \end{matrix}

which is impossible.

b) X = ℂℙ^{1}. In this case we have μ = cr_{ijkl} and μ ∘ f = cr(f_{i}, f_{j}, f_{k}, f_{l}) = [(f_{l} − f_{i})(f_{j} − f_{k})]/[(f_{l} − f_{j})(f_{i} − f_{k})] = −1. Each component f_{m} is a non-constant

holomorphic map f_{m}: C_{o}^{n}(X) → ℂℙ^{1}; therefore, the pole set P (f) = ⋃_{m=1}^{n} f_{m}^{−1}(∞)

is an analytic subset of C_{o}^{n}(ℂℙ^{1}) of complex dimension ≤ n − 1. Its complement R(f) = C_{o}^{n}(ℂℙ^{1}) \ P (f) is a non-empty S(n) invariant Zariski open subset of C_{o}^{n}(ℂℙ^{1}), the restriction f′ = f|_{R(f)} of f to R(f) is a holomorphic map from R(f) to C_{o}^{n}(ℂ), and its components f′_{i} = f_{i}|_{R(f)} are just holomorphic functions on R(f) that satisfy the strict equivariance condition f′_{m}(σq) = f′_{[α(σ)]^{−1}(m)}(q) for all m = 1, ..., n, q ∈ R(f), and σ ∈ S(n). In particular, [(f′_{l} − f′_{i})(f′_{j} − f′_{k})]/[(f′_{l} − f′_{j})(f′_{i} − f′_{k})] = −1 on R(f). Now we can repeat word by word what has been done above in case (CR) and obtain 3(f′_{l} − f′_{i})(f′_{j} − f′_{k}) = 0 on R(f), which is impossible, since R(f) ≠ ∅ and f′(q) = (f′_{1}(q), ..., f′_{n}(q)) ∈ C_{o}^{n}(ℂ) for all q ∈ R(f). □

Recall the notation t(X) = 2 for X = ℂ and t(X) = 3 for X = ℂℙ^{1}. The following result is just an evident combination of Theorem 2.15, Lemma 6.1 and Lemma 5.4.

Corollary 6.2. Let n > t(X)^{15} and let f be a strictly equivariant endomorphism of C_{o}^{n}(X). The induced map f^{∗}: L(C_{o}^{n}(X)) ∋ μ ↦ μ ∘ f ∈ L(C_{o}^{n}(X)) is simplicial. □

Lemma 6.3. Let f be a strictly equivariant endomorphism of C_{o}^{n}(X) and let α be the automorphism of S(n) related to f (see (2.8) and Definition 2.8).

a) The map f^{∗}: L(C_{o}^{n}(X)) → L(C_{o}^{n}(X)) is strictly equivariant meaning that

f^{∗}(σμ) = α^{−1}(σ)f^{∗}(μ) for all σ ∈ S(n) and all μ ∈ L(C_{o}^{n}(ℂ)) .  (6.1)

Consequently, for any vertex μ ∈ L(C_{o}^{n}(X)) the f^{∗}-image of its S(n) orbit coincides with the S(n) orbit of the vertex f^{∗}(μ). b) If X = ℂ then f^{∗}(SR(C_{o}^{n}(ℂ))) = SR(C_{o}^{n}(ℂ)).

15See Notation 1.11.

Proof. a) Let s = α^{−1}(σ). The proof of (6.1) is straightforward:

\begin{aligned}[f^{∗}(σμ)](q) & = (σμ)(f(q)) = (α(s)μ)(f(q)) = μ(α(s^{−1})f(q)) \\ & = μ(f(s^{−1}q)) = (f^{∗}(μ))(s^{−1}q) = [sf^{∗}(μ)](q) = [α^{−1}(σ)f^{∗}(μ)](q) . \end{aligned}

The permutation s runs over the whole S(n) whenever σ does so, and vice versa. Hence (6.1) implies f^{∗}(S(n)μ) = S(n)f^{∗}(μ) for each μ ∈ L(C_{o}^{n}(X)).

b) By Lemma 5.17, the map f^{∗} preserves dimension of simplices; taking into account Corollary 5.10, we have dim f^{∗}(SR_{△}(C_{o}^{n}(ℂ))) = dim SR_{△}(C_{o}^{n}(ℂ)) = n − 3 and dim CR_{△}(C_{o}^{n}(ℂ)) = n − 4, which shows that f^{∗}(SR_{△}(C_{o}^{n}(ℂ))) ≠ CR_{△}(C_{o}^{n}(ℂ)). Thus f^{∗}(SR_{△}(C_{o}^{n}(ℂ))) = SR_{△}(C_{o}^{n}(ℂ)), which is certainly equivalent to f^{∗}(SR(C_{o}^{n}(ℂ))) = SR(C_{o}^{n}(ℂ)). □

Remark 6.4. It follows from Coherence Theorem (which is not proved yet!) that f^{∗}(CR(C_{o}^{n}(ℂ))) = CR(C_{o}^{n}(ℂ)); we do not need this fact at the moment. ◯

6.1. Proof of Coherence Theorem for X = ℂ. Let n > 4 and let f be a strictly equivariant endomorphism of C_{o}^{n}(ℂ). In view of what has been done above, in this case Coherence Theorem is equivalent to the following

Claim 1. Let Δ = Δ_{S}^{n−3} ∈ SR_{△}(C_{o}^{n}(ℂ)) be the normal simplex of the form (5.23), that is, the simplex with the vertices μ_{i} = sr_{i,2,1}, i = 3, ..., n. Then f^{∗}(Δ) ∈ S(n)Δ.

Proof. By Lemma 6.3(b), f^{∗}(Δ) ∈ SR_{△^{n−3}}(C_{o}^{n}(ℂ)). According to Lemma 5.12(a), SR_{△^{n−3}}(C_{o}^{n}(ℂ)) is the disjoint union of the orbits S(n)Δ_{S}^{n−3} and S(n)Δ_{S}^{−(n−3)}. So it suffices to show that f^{∗}(Δ) ∉ S(n)Δ_{S}^{−(n−3)}; in view of Lemma 6.3(a), the latter is equivalent to f^{∗}(S(n)Δ) ∉ Δ_{S}^{−(n−3)}.

Suppose to the contrary that f^{∗}(sΔ) = Δ_{S}^{−(n−3)} for some s ∈ S(n). Then

\frac{f_{1}(σq) − f_{i}(σq)}{f_{1}(σq) − f_{2}(σq)} = \frac{q_{1} − q_{2}}{q_{1} − q_{i}}

for all i = 3, ..., n, where σ = α^{−1}(s^{−1}) and α is the automorphism of S(n) related to f. This implies that for all distinct j, k ∈ {3, ..., n}.

\begin{aligned}\frac{f_{k}(σq) − f_{j}(σq)}{f_{1}(σq) − f_{2}(σq)} & = \frac{f_{1}(σq) − f_{j}(σq)}{f_{1}(σq) − f_{2}(σq)} − \frac{f_{1}(σq) − f_{k}(σq)}{f_{1}(σq) − f_{2}(σq)} \\ & = \frac{q_{1} − q_{2}}{q_{1} − q_{j}} − \frac{q_{1} − q_{2}}{q_{1} − q_{k}} = \frac{(q_{1} − q_{2})(q_{j} − q_{k})}{(q_{1} − q_{j})(q_{1} − q_{k})} . \end{aligned}

Consequently, for distinct i, j, k ∈ {3, ..., n} we have

[f^{∗}(sr_{ijk})](σq) = sr_{ijk}(f(σq)) = \frac{f_{k}(σq) − f_{i}(σq)}{f_{k}(σq) − f_{j}(σq)}

= \frac{f_{k}(σq) − f_{i}(σq)}{f_{1}(σq) − f_{2}(σq)} : \frac{f_{k}(σq) − f_{j}(σq)}{f_{1}(σq) − f_{2}(σq)}

= \frac{(q_{1} − q_{2})(q_{i} − q_{k})}{(q_{1} − q_{i})(q_{1} − q_{k})} : \frac{(q_{1} − q_{2})(q_{j} − q_{k})}{(q_{1} − q_{j})(q_{1} − q_{k})} = \frac{q_{k} − q_{i}}{q_{k} − q_{j}} : \frac{q_{i} − q_{1}}{q_{j} − q_{1}} = cr_{i,j,1,k}(q) ,

which contradicts Lemma 6.3(b). □

6.2. Proof of Coherence Theorem for X = ℂℙ^{1}. Let n ≥ 4 and let f = (f_{1}, ..., f_{n}) be a strictly equivariant endomorphism of C_{o}^{n}(ℂℙ^{1}). In this case Coherence Theorem is equivalent to the following

Claim 2. Let Δ = Δ_{C}^{n−4} ∈ CR_{△^{n−4}}(C_{o}^{n}(ℂℙ^{1})) be the normal simplex of the form (5.25), that is, the simplex with the vertices μ_{i} = cr_{1,2,3,i}, i = 4, ..., n. Then f^{∗}(Δ) is a n − 4 simplex contained in the S(n) orbit S(n)Δ of Δ itself.

Proof. By Lemma 6.1 and Lemma 5.4, f^{∗}(Δ) is a n − 4 simplex in L((C_{o}^{n}(ℂℙ^{1})). By Theorem 2.15, L((C_{o}^{n}(ℂℙ^{1}))) = CR(q_{n}); hence f^{∗}(Δ) ∈ CR_{△^{n−4}}(q_{n}). Finally, Lemma 5.12(b) asserts that S(n) is transitive on each CR_{△^{m}}(q_{n}). This completes the proof of Coherence Theorem and thereby the proof of Tame Map Theorem as well.

□

6.3. Proof of Corollary 1.6.

6.4. Proof of Corollary 1.7.

7. Linked Map Theorem

The main aim of this section is to prove Linked Map Theorem 1.12. In fact there are several similar results, which we also intend to discuss. In all cases the key points of the proofs are the the “explicit” description of the holomorphic universal covering τ: T(0, m + 3) → C_{o}^{m}(ℂ^{◯ 2}) presented in Section 3.1 and the following remarkable property of the universal Teichmüller families established first by J. H. Hubbard [Hub72, Hub76] for compact Riemann surfaces and then generalized to all Riemann surfaces of finite type by C. J. Earle and I. Kra [EarKra74, EarKra76]:

Theorem 7.1 (Hubbard-Earl-Kra). The universal punctured Teichmüller family T (g, m) := ρ: V′(g, m) → T(g, m) has no holomorphic sections T(g, m) → V′(g, m) whenever dim_{ℂ} T(g, m) > 1. □

We start with the following definition similar to Definition 1.10.

Definition 7.2. A map h: C_{o}^{n}(X) → C^{k}(X) is said to be linked if there is a point q = (q_{1}, ..., q_{n}) ∈ C_{o}^{n}(X) such that {q_{1}, ..., q_{n}} ∩ h(q) ≠ ∅, and disjoint otherwise. A map h: C_{o}^{m}(ℂ^{◯ 2}) → C^{k}(X) is said to be linked if there is z = (z_{1}, ..., z_{m}) ∈ C_{o}^{m}(ℂ^{◯ 2}) such that {z_{1}, ..., z_{m}, 0, 1, ∞} ∩ h(z) ≠ ∅, and disjoint otherwise. ◯

7.1. Linked Map Theorem for C_{o}^{m}(ℂ^{◯ 2}). Among all the forms of Linked Map Theorem that we were able to prove there is the strongest one, which implies all others as simple consequences; it will be proved in this section.

Theorem 7.3 (Linked Map Theorem for C_{o}^{m}(ℂ^{◯ 2})). For m > 1 and k ≥ 1 every holomorphic map h: C_{o}^{m}(ℂ^{◯ 2}) → C^{k}(X) is linked. ◯

Our proof of this theorem is based on Lemma 7.4, which directly involves the universal punctured Teichmüller family V′(0, m + 3) → T(0, m + 3), and on the analytic fact stated in Theorem 7.5 below. The latter theorem, in turn, is an easy consequence of Hubbard-Earl-Kra Theorem 7.1 formulated above.

Define

\begin{matrix}E_{o}^{m+1}(ℂ^{◯ 2}) = \{(ζ, z) ∈ ℂ × C_{o}^{m}(ℂ^{◯ 2}) | ζ ≠ z_{1}, ..., z_{m}, 0, 1\} , \\ Π := π: E_{o}^{m+1}(ℂ^{◯ 2}) ∋ (ζ, z) ↦ π(ζ, z) =^{def} z ∈ C_{o}^{m}(ℂ^{◯ 2}) . \end{matrix}  (7.1)

Then E_{o}^{m+1}(ℂ^{◯ 2}) is a complex manifold and Π is a smooth locally trivial fiber bundle with the fiber diffeomorphic to ℂ \ {m + 2 points}. Of course, the projection π is holomorphic and Π may be also regarded as a holomorphic family of curves of type (0, m + 3). Set

τ^{∗}(E_{o}^{m+1}(ℂ^{◯ 2})) = {(t, (ζ, z)) ∈ T(0, m + 3) × E_{o}^{m+1}(ℂ^{◯ 2}) | τ(t) = z};  (7.2)

then the fibration

τ^{∗}(Π) := τ^{∗}(π): τ^{∗}(E_{o}^{m+1}(ℂ^{◯ 2})) ∋ (t, (ζ, z)) ↦ t ∈ T(0, m + 3)  (7.3)

is the pullback of the fibration Π along the map τ: T(0, m + 3) → C_{o}^{m}(ℂ^{◯ 2}) and we have the commutative diagram

\begin{matrix}τ^{∗}(E_{o}^{m+1}(ℂ^{◯ 2})) \overline{}E_{o}^{m+1}(ℂ^{◯ 2}) \\ _{_{τ^{∗}(π)} } & _{_{π}} \\ T(0, m + 3) \overline{}_{τ} C_{o}^{m}(ℂ^{◯ 2}) , \end{matrix}  (7.4)

where the top horizontal map is just the restriction of the projection of the product T(0, m + 3) × E_{o}^{m+1}(ℂ^{◯ 2}) to the second factor.

Lemma 7.4. The family τ^{∗}(Π) := τ^{∗}(π): τ^{∗}(E_{o}^{m+1}(ℂ^{◯ 2})) → T(0, m+3) is isomorphic to the universal Teichmüller family ρ: V′(0, m + 3) → T(0, m + 3), that is, there is a commutative diagram

\begin{matrix}V′(0, m + 3) ^{^{λ} \overline{}}τ^{∗}(E_{o}^{m+1}(ℂ^{◯ 2})) \overline{}E_{o}^{m+1}(ℂ^{◯ 2}) \\ [figure: ρ] & _{_{τ^{∗}(π)}} & _{_{π}} \\ T(0, m + 3) ^{^{τ} \overline{}}C_{o}^{m}(ℂ^{◯ 2}) , \end{matrix}  (7.5)

where λ is a biholomorphic fiber isomorphism.

Proof. Keeping in mind that a point (ζ, t) ∈ ℂ×T(0, m+3) belongs to V′(0, m+ 3) iff ζ ≠ s_{1}(t), ..., s_{m}(t), 0, 1, while a point (t, ζ, z) ∈ T(0, m+3)×ℂ×C_{o}^{m}(ℂ^{◯ 2}) belongs to τ^{∗}(E_{o}^{m+1}(ℂ^{◯ 2})) iff ρ(t) = z = (z_{1}, ..., z_{m}) ∈ C_{o}^{m}(ℂ^{◯ 2}) and ζ ≠ z_{1}, ..., z_{m}, 0, 1, we define λ by

λ: V′(0, m + 3) ∋ v = (ζ, ρ(v)) ↦ (t, ζ, s_{1}(t), ..., s_{m}(t))

=^{def} (ρ(v), ζ, s_{1}(ρ(v)), ..., s_{m}(ρ(v))) ∈ τ^{∗}(E_{o}^{m+1}(ℂ^{◯ 2})) .

(7.6)

The map λ^{−1} is defined by

λ^{−1} : τ^{∗}(E_{o}^{m+1}(ℂ^{◯ 2})) ∋ (t, (ζ, z)) ↦ (ζ, t) ∈ V′(0, m + 3) .  (7.7)

These λ and λ^{−1} are holomorphic fiber mappings inverse to each other. □

Theorem 7.5. For m > 1 the restriction π|_{A}: A → C_{o}^{m}(ℂ^{◯ 2}) of the projection π: E_{o}^{m+1}(ℂ^{◯ 2}) → C_{o}^{m}(ℂ^{◯ 2}) to an analytic subset A ⊂ E_{o}^{m+1}(ℂ^{◯ 2}) cannot be an unbranched covering over the whole manifold C_{o}^{m}(ℂ^{◯ 2}).

Proof. Suppose, on the contrary, that an analytic subset A ⊂ E_{o}^{m+1}(ℂ^{◯ 2}) covers the whole base C_{o}^{m}(ℂ^{◯ 2}). The pullback τ^{∗}(π|_{A}): τ^{∗}(A) → T(0, m + 3) of the covering π|_{A}: A → C_{o}^{m}(ℂ^{◯ 2}) along the mapping τ is a holomorphic unbranched covering as well. It must be trivial since the Teichmüller space T(0, m + 3) is contractible; hence there exists a holomorphic section s: T(0, m + 3) → τ^{∗}(A) ⊂ τ^{∗}(E_{o}^{m+1}(ℂ^{◯ 2})). As λ in diagram (7.5) is an isomorphism, the latter section provides a holomorphic section S: T(0, m + 3) → V′(0, m + 3) of the universal Teichmüller family T (0, m + 3) := ρ: V′(0, m + 3) → T(0, m + 3). Since dim_{ℂ} T(0, m + 3) = m > 1, this contradicts the Hubbard-Earl-Kra Theorem 7.1. □

We are now in a position to prove Linked Map Theorem for C_{o}^{m}(ℂ^{◯ 2}) (Theorem 7.3).

Proof. Suppose, on the contrary, that for some m > 1 and k ≥ 1 there is a disjoint holomorphic map h: C_{o}^{m}(ℂ^{◯ 2}) → C^{k}(X).

Notice first that for each point z = (z_{1}, ..., z_{m}) ∈ C_{o}^{m}(ℂ^{◯ 2}) its image h(z) ⊂ X is in fact contained in ℂ, i. e., ∞ ∉ h(z). In the case X = ℂ this is self evident; in the case X = ℂℙ^{1} this follows from the condition {z_{1}, ..., z_{m}, 0, 1, ∞} ∩ h(z) = ∅ (see Definition 7.2). Let P_{k}(t, h(z)) be the polynomial in t with the leading coefficient 1 and with the set of the roots h(z); it may clearly be regarded as an analytic function on ℂ × C_{o}^{m}(ℂ^{◯ 2}). Let A ⊂ ℂ × C_{o}^{m}(ℂ^{◯ 2}) be the analytic subset defined by

A = {(t, z) ∈ ℂ × C_{o}^{m}(ℂ^{◯ 2}) | t ∈ h(z)} = {(t, z) ∈ ℂ × C_{o}^{m}(ℂ^{◯ 2}) | P_{k}(t, h(z)) = 0} .

As {z_{1}, ..., z_{m}, 0, 1, ∞} ∩ h(z) = ∅, the image h(z) of any point z ∈ C_{o}^{m}(ℂ^{◯ 2}) contains no punctures of the fiber π^{−1}(z) of the family π: E_{o}^{m+1}(ℂ^{◯ 2}) → C_{o}^{m}(ℂ^{◯ 2}) defined in (7.1). That is, all k distinct roots t_{1}, ..., t_{k} ∈ h(z) of the polynomial P_{k}(t, h(z)) are in π^{−1}(z). Hence A ⊆ E_{o}^{m+1}(ℂ^{◯ 2}) and the restriction π|_{A}: A → C_{o}^{m}(ℂ^{◯ 2}) of the projection π: E_{o}^{m+1}(ℂ^{◯ 2}) → C_{o}^{m}(ℂ^{◯ 2}) to A is a holomorphic unbranched covering over the whole base C_{o}^{m}(ℂ^{◯ 2}), which contradicts Theorem 7.5. □

8. MORPHISMS C^{n}(X) → C^{k}(X) AND Aut X ORBITS. CYCLIC AND ORBIT-LIKE MAPS 45

7.2. Linked Map Theorem for C_{o}^{n}(X). If for a map f: C_{o}^{n}(X) → C^{k}(X) and a subset M ⊆ C_{o}^{n}(X) there is q = (q_{1}, ..., q_{n}) ∈ M such that {q_{1}, ..., q_{n}} ∩ f(q) ≠ ∅ then f is certainly linked. Hence the following form of Linked Map Theorem, which is certainly stronger that the original one formulated in Section 1.5 (Theorem 1.12), follows immediately from Linked Map Theorem for C_{o}^{m}(ℂ^{◯ 2}) (Theorem 7.3) proved above.

Theorem 7.6 (Linked Map Theorem for C_{o}^{n}(X)). For n > t(X) + 1 and k ≥ 1 every holomorphic map h: C_{o}^{n}(X) → C^{k}(X) is linked. □

Remark 7.7. Let n > t(X) and k ≥ 1. The above theorem implies that

a) any holomorphic map F: C^{n}(X) → C^{k}(X) is linked, i. e., there is Q ∈ C^{n}(X) such that Q ∩ F (Q) ≠ ∅;

b) any holomorphic map f: C_{o}^{n}(X) → C_{o}^{k}(X) is linked, i. e., there is q ∈ C_{o}^{n}(X) such that p(q) ∩ p(f(q)) ≠ ∅;

c) any holomorphic map g: C^{n}(X) → C_{o}^{k}(X) is linked, i. e., there is Q ∈ C^{n}(X) such that Q ∩ p(g(Q)) ≠ ∅.

To prove these statements, one just take h = F ∘ p in case (a), h = p ∘ f in case (b) and h = p ∘ g ∘ p in case (c) and applies Theorem 7.6 to the map h. ◯

8. Morphisms C^{n}(X) → C^{k}(X) and Aut X orbits. Cyclic and orbit-like maps

In this section, using the direct decomposition (3.11), we show that any morphism F: C^{n}(X) → C^{k}(X) respects Aut X orbits, that is, F ((Aut X)Q) ⊆ (Aut X)F (Q) for all Q ∈ C^{n}(X). Then we try to clarify, as far as possible, the structure of cyclic morphisms.

8.1. Liouville and Picard spaces. Here we exhibit a certain class of holomorphic maps Z → C^{k}(X) whose images must always be contained in an Aut X orbit. We will use these results in Section 8.2 below. All complex spaces under consideration are assumed to be reduced.

Definition 8.1. A connected complex space Z is said to be Liouville if it carries no non-constant bounded holomorphic functions. We say that Z is a Picard space if it carries no non-constant holomorphic functions omitting the values 0 and 1; Z is called ultra-Picard whenever every its connected unbranched finite cover is Picard. ◯

Lemma 8.2. Let Y be a connected complex space and p: Y → Z be a holomorphic finite unbranched covering map onto a Liouville space Z. Then Y is Liouville.^{16}

Proof. Let h be a bounded holomorphic function on Y. The coefficients of the polynomial P (t, z) = ∏_{y∈p^{−1}(z)}(t−h(y)) are bounded holomorphic functions on Z and hence constant. Since Y is connected this implies h = const. □

16Of course, this is a very well-known elementary result. See [Lin88, LinZai98] and the references therein for more interesting results about the Liouville property.

Proposition 8.3. Let Z be a connected complex space and F: Z → C^{m}(X) be a holomorphic map. Suppose that either (i) Z is an ultra-Picard space or (ii) Z is a Liouville space and the homomorphism F_{∗}: π_{1}(Z) → π_{1}(C^{m}(X)) is solvable. Then there is a point Q^{∗} ∈ C^{m}(X) such that F (Z) ⊆ (Aut X)Q^{∗}.

Proof. Let F^{∗}(p): F^{∗}(C_{o}^{m}(X)) → Z be the pullback of the covering p: C_{o}^{m}(X) → C^{m}(X) along the mapping F and let Y be a connected component of F^{∗}(C_{o}^{m}(X)). Then we have the commutative diagram

\begin{matrix}Y ^{^{f} \overline{}}C_{o}^{m}(X)  \overline{}^{j_{X,m}}_{≅} (Aut X) × D^{m−t(X)}(X) ^{^{ξ} \overline{}}D^{m−t(X)}(X) \\ _{_{ν} } & _{_{p}} \\ Z \overline{}_{F} C^{m}(X) , \end{matrix}  (8.1)

where ξ: (Aut X)×D^{m−t(X)}(X) → D^{m−t(X)}(X) is the projection onto the second factor and ν is the restriction of F^{∗}(p) to Y (see (3.11) and (3.12)). The coordinate functions z_{1}, ..., z_{m−t(X)} on D^{m−t(X)}(X) = C_{o}^{m−t(X)}(ℂ^{◯ 2}) are holomorphic and do not assume the values 0 and 1; hence for each i, 1 ≤ i ≤ m − t(X), the function h_{i} = z_{i} ∘ ξ ∘ j_{X,m} ∘ f is holomorphic on Y and does not take the values 0 and 1.

In case (i), Y is a Picard space. Thus, ξ ∘ j_{X,m} ∘ f = const and f(Y) is contained in the Aut X orbit of some point q^{∗} ∈ C_{o}^{m}(X), which implies F (Z) ⊆ (Aut X)p(q^{∗}).

In case (ii) for each i = 1, ..., m − t(X) the image G_{i} = (h_{i})_{∗}(π_{1}(Y)) of the homomorphism (h_{i})_{∗}: π_{1}(Y) → π_{1}(ℂ \ {0, 1}) is a solvable subgroup of π_{1}(ℂ \ {0, 1}) ≅ 𝔽_{2}. Thus, either G_{i} is trivial or G_{i} ≅ ℤ. Let ν_{i}: Γ_{i} → ℂ \ {0, 1} be the covering corresponding to the subgroup G_{i} so that either Γ_{i} ≅ 𝔻 = {ζ ∈ ℂ | |ζ| < 1} or Γ_{i} ≅ K_{r_{i}} = {ζ ∈ ℂ | r_{i}^{−1} < |ζ| < r_{i}} for some r_{i} > 1. By the covering map theorem, there is a holomorphic map h̃_{i}: Y → Γ_{i} that fits into the commutative diagram

Γ_{i} [figure: h̃_{i}; ν_{i}]

Y \overline{}_{h_{i}} ℂ \ \{0, 1\} .

By Lemma 8.2, the space Y is Liouville; since both 𝔻 and K_{r_{i}} are bounded domains, h̃_{i} = const, which implies h_{i} = const. Thus, ξ ∘ j_{X,m} ∘ f = const, f(Y) is contained in the Aut X orbit of some point q^{∗} ∈ C_{o}^{m}(X) and F (Z) ⊆ (Aut X)p(q^{∗}). □

Any connected complex Lie group L satisfies both (i) and (ii) since the union of the images of all holomorphic homomorphisms ℂ → L contains an open neighborhood of the unity and, moreover, π_{1}(L) is abelian. Thus, we have:

Corollary 8.4. Let L be a connected complex Lie group. Then for every holomorphic map Φ: L → C^{m}(X) there is a point Q^{∗} ∈ C^{m}(X) such that Φ(L) ⊆ (Aut X)Q^{∗}. □

8.2. Morphisms C^{n}(X) → C^{k}(X) and Aut X orbits. Here we prove the statement about morphisms of configuration spaces mentioned in Remark 1.2.

Proposition 8.5. Every morphism F: C^{n}(X) → C^{k}(X) respects Aut X orbits, that is, F ((Aut X)Q) ⊆ (Aut X)F (Q) for all Q ∈ C^{n}(X).

Proof. Let Q ∈ C^{n}(X) and let Ψ: Aut X → C^{n}(X) be the holomorphic map defined by Ψ(A) = AQ for each A ∈ Aut X. Set Φ = F ∘ Ψ; clearly Φ(id_{X}) = F (Q). By Corollary 8.4, the image F ((Aut X)Q) = F (Ψ(Aut X)) = Φ(Aut X) is contained in an Aut X orbit, which certainly is the orbit of the point F (Q). □

8.3. Stabilizers of points in C^{k}(X). In what follows we need to use the classification of finite subgroups G ⊂ SO(3); this is a very classical matter, which goes back at least to F. Klein’s “Vorlesungen über das Ikosaeder” [Kle]. The following list exhibits all such subgroups^{17}:

a) cyclic groups ℤ/mℤ, m ≥ 2;

b) (ℤ/2ℤ) × ℤ/(2ℤ);

c) dihedral groups (ℤ/mℤ) ⋋ (ℤ/2ℤ), m > 2, having two generators a, b and the defining system of relations a^{m} = 1, b^{2} = 1, bab^{−1} = a^{−1};

d) the rotation groups of Platonic solids, that is, the symmetric group S(4) (rotation groups of the cube and the octahedron), the alternating groups A(4) (the rotation group of the tetrahedron) and A(5) (rotation groups of the icosahedron and the dodecahedron), and their subgroups.

Notice that all groups of this list except of A(5) are solvable.

Proposition 8.6. For k ≥ t(X) the stabilizer St_{Q^{∗}} = {A ∈ Aut X | AQ^{∗} = Q^{∗}} of any point Q^{∗} ∈ C^{k}(X) is a finite group which is either solvable or isomorphic to the alternating group A(5).

Proof. Let Q^{∗} = {q_{1}^{∗}, ..., q_{k}^{∗}}. Any element A ∈ St_{Q^{∗}} is either an affine or a Möbius transformation permuting the points q_{1}^{∗}, ..., q_{k}^{∗}; A is uniquely determined by its values Aq_{i}^{∗}, 1 ≤ i ≤ t(X), which shows that St_{Q^{∗}} is finite.

If X = ℂ then St_{Q^{∗}} is a finite subgroup of Aff ℂ. Every such subgroup is conjugate to a finite subgroup of the maximal compact subgroup 𝕋 ⊂ Aff ℂ, which consists of all rotations z ↦ e^{it}z, t ∈ ℝ. Every finite subgroup of the latter group is cyclic.

If X = ℂℙ^{1} then St_{Q^{∗}} is a finite subgroup of PSL(2, ℂ). Every such subgroup is conjugate to a finite subgroup of the maximal compact subgroup K ⊂ PSL(2, ℂ), which is isomorphic to SO(3) and consists of all Möbius transformations of the form

z ↦ \frac{az − c̄}{cz + ā} , |a|^{2} + |c|^{2} = 1 .

Thus, in this case St_{Q^{∗}} is isomorphic to one of the groups exhibited in the above list, which proves the proposition. □

8.4. Cyclic Map Theorem. Recall that a continuous map F: C^{n}(X) → C^{k}(X) is said to be orbit-like if its image F (C^{n}(X)) is contained in some orbit of Aut X action in C^{k}(X), that is, there is a point Q^{∗} ∈ C^{k}(X) such that F (C^{n}(X)) ⊆ (Aut X)Q^{∗}. We have already mentioned that the stabilizer St_{Q^{∗}} of a generic point Q^{∗} ∈ C^{k}(X) is trivial; thereby a morphism F whose image is contained in the orbit of such a point Q^{∗} is degenerate tame, that is, there is a morphism T: C^{n}(X) → Aut X such that F (Q) = T (Q)Q^{∗} for all Q ∈ C^{n}(X) (see Definition 1.1 and Remark 1.2).

17We take no care about possible repetitions in this list. See, for instance, [Fo51], Chapter VI, Sections 51-57, for more details.

Theorem 8.7 (Cyclic Map Theorem). Let n > 4 and k ≥ t(X). Then a) every cyclic morphism F: C^{n}(X) → C^{k}(X) is orbit-like;

b) every continuous orbit-like map F: C^{n}(X) → C^{k}(X) is cyclic.

Statement (a) follows from Proposition 8.3(ii) since C^{n}(X) is a connected algebraic manifold and therefore Liouville, whereas F is supposed to be cyclic.

Statement (b) is an immediate consequence of the following lemma.

Lemma 8.8. Let n > 4 and k ≥ t(X). Then for each point Q^{∗} ∈ C^{k}(X) every homomorphism φ: B_{n}(X) → π_{1}((Aut X)Q^{∗}) is cyclic.

Proof. If the stabilizer St_{Q^{∗}} is trivial then (Aut X)Q^{∗} ≅ Aut X, π_{1}((Aut X)Q^{∗}) ≅ π_{1}(Aut X) is cyclic and φ is cyclic.

Suppose that St_{Q^{∗}} is non-trivial; then the natural map h: Aut X → (Aut X)Q^{∗} is a finite regular covering with the fiber St_{Q^{∗}} and we have the commutative diagram

\begin{matrix}B_{n}(X) \\ _{_{φ} } & [figure: δ∘φ] \\ 1 \overline{}π_{1}(Aut X) \overline{}_{h_{∗}} π_{1}((Aut X)Q^{∗}) \overline{}_{δ} St_{Q^{∗}}  \overline{}1 , \end{matrix}  (8.2)

where the horizontal line is the exact sequence corresponding to the covering h.

Consider the restriction of the homomorphisms φ and δ∘φ to the commutator subgroup B′_{n}(X), which is perfect since n > 4.

By Proposition 8.6, St_{Q^{∗}} is either solvable or isomorphic to A(5).

If St_{Q^{∗}} is solvable then the restriction of δ ∘ φ to B′_{n}(X) is trivial and hence

φ(B′_{n}(X)) ⊆ Ker δ = h_{∗}(π_{1}(Aut X)) ≅ π_{1}(Aut X) .

The group π_{1}(Aut X) is cyclic, which shows that φ(B′_{n}(X)) is trivial and φ is cyclic.

Thus, we are left with the case St_{Q^{∗}} ≅ A(5) ⊂ S(5). If n > 5 then, by Theorem 4.4, the homomorphism δ ∘ φ is cyclic, which implies that (δ ∘ φ)(B′_{n}(X)) = {1} and, as above, φ(B′_{n}(X)) ⊆ Ker δ = π_{1}(Aut X), φ(B′_{n}(X)) = {1} and φ is cyclic. Finally, if n = 5 then the homomorphism δ∘φ maps B_{5}(X) into the proper subgroup A(5) ⊂ S(5) and, by Surjectivity Theorem 2.7, it is cyclic again; as above, we conclude that φ is cyclic. This proves the lemma and completes the proof of Cyclic Map Theorem. □

8.5. Holomorphic maps to the space of balanced configurations. We start with the following

Definition 8.9. The balanced configuration space C_{b}^{m−1}(ℂ) is the (m−1)-dimensional subspace of C^{m}(ℂ) consisting of all sets Q = {q_{1}, ..., q_{n}} ∈ C^{m}(ℂ) having its mass centers at the origin:

C_{b}^{m−1}(ℂ) = {Q = {q_{1}, ..., q_{m}} ∈ C^{m}(ℂ) | q_{1} + ... + q_{m} = 0} .

We set also

\begin{matrix}C_{ob}^{m−1}(ℂ) = \{q = (q_{1}, ..., q_{m}) ∈ C_{o}^{m}(ℂ) | q_{1} + ... + q_{m} = 0\} , \\ C_{ob,1}^{m−2}(ℂ) = \{q = (q_{1}, ..., q_{m}) ∈ C_{o}^{n}(ℂ) | q_{1} + ... + q_{m} = 0, q_{1} − q_{2} = 1\} . ◯ \end{matrix}

The multiplicative group ℂ^{∗} acts in C_{b}^{m−1}(ℂ) and C_{ob}^{m−1}(ℂ) via

ℂ^{∗} ∋ ζ : {q_{1}, ..., q_{m}} ↦ {ζq_{1}, ..., ζq_{m}} and (q_{1}, ..., q_{m}) ↦ (ζq_{1}, ..., ζq_{m}) (8.3)

respectively. The ℂ^{∗}-orbit of any point q = (q_{1}, ..., q_{m}) ∈ C_{ob}^{m−1}(ℂ) intersects the subspace C_{ob,1}^{m−2}(ℂ) ⊂ C_{ob}^{m−1}(ℂ) at a single point q̃ = (q_{1}/(q_{1} − q_{2}), ..., q_{m}/(q_{1} − q_{2})); this provides the biholomorphic direct decomposition C_{ob}^{m−1}(ℂ) = ℂ^{∗} × C_{ob,1}^{m−2}(ℂ),

C_{ob}^{m−1}(ℂ) ∋ q = (q_{1}, ..., q_{m}) ↦ (q_{1} − q_{2}; q̃) ∈ ℂ^{∗} × C_{ob,1}^{m−2}(ℂ) .

On the other hand, the functions ϕ_{i}(q) = (q_{1}−q_{i})/(q_{1}−q_{2}), i = 3, ..., n, are holomorphic on C_{ob}^{m−1}(ℂ), separate points of C_{ob,1}^{m−2}(ℂ) and do not take the values 0 and 1. This implies that

(i) for every holomorphic map Φ: L → C^{m}(ℂ) of a connected complex Lie group L there is a point Q^{∗} ∈ C_{b}^{m−1}(ℂ) such that Φ(L) ⊆ ℂ^{∗}Q^{∗} and

(ii) any holomorphic map F: C_{b}^{n−1}(ℂ) → C_{b}^{k−1}(ℂ) respects ℂ^{∗}-orbits, that is, F (ℂ^{∗}Q) ⊆ ℂ^{∗}F (Q) for all Q ∈ C_{b}^{n−1}(ℂ).

The proof of the following analog of Proposition 8.3 is clear:

Proposition 8.10. Let Z be a connected complex space and F: Z → C_{b}^{m−1}(ℂ) be a holomorphic map. Suppose that either Z is an ultra-Picard space or the homomorphism F_{∗}: π_{1}(Z) → π_{1}(C_{b}^{m−1}(ℂ)) is solvable and Z is Liouville. Then there is a point Q^{∗} ∈ C_{b}^{m−1}(ℂ) such that F (Z) ⊆ ℂ^{∗}Q^{∗}. □

In what follows, instead of C^{m}(ℂ) and its balanced subspace C_{b}^{m−1}(ℂ), it is convenient to deal with the space of polynomials with simple roots G_{m} and its subspace

G_{m}^{0} = {w = (w_{1}, ..., w_{m}) ∈ G_{m} | w_{1} = 0}  (8.4)

that corresponds to C_{b}^{m−1}(ℂ) under the identification C^{m}(ℂ) = G_{m} (see Remark 1.9). The ℂ^{∗} action in G_{m} corresponding to the ℂ^{∗} action (8.3) in C_{b}^{m−1}(ℂ) look as follows:

ℂ^{∗} ∋ ζ: G_{m}^{0} ∋ (0, w_{2}, ..., w_{m}) ↦ U_{ζ}w = (0, ζ^{2}w_{2}, ..., ζ^{m}w_{m}) ∈ G_{m}^{0} .  (8.5)

Notation 8.11. For w = (0, w_{2}, ..., w_{m}) ∈ G_{m}^{0} set Z(w) = {j | j ≥ 2, w_{j} = 0}, N Z(w) = {j | j ≥ 2, w_{j} ≠ 0} and denote by D(w) the greatest common divisor of all j ∈ N Z(w). Since Z(w) cannot contain both m − 1 and m, the number D(w) divides m(m − 1). As usual, O(Z) denotes the algebra of all holomorphic functions on Z; O^{∗}(Z) is the multiplicative group of all non-vanishing functions f ∈ O(Z) and its subgroup exp O(Z) ⊆ O^{∗}(Z) consists of all functions e^{f}, f ∈ O(Z).

The following evident lemma provides a complete description of ℂ^{∗} orbits in the space of balanced configurations C_{b}^{m−1}(ℂ) = G_{m}^{0}; it will be used in next sections.

Lemma 8.12. Let w^{∘} = (0, w_{2}^{∘}, ..., w_{m}^{∘}) ∈ G_{m}^{0} and D = D(w^{∘}). Then the stabilizer St_{w^{∘}} ⊂ ℂ^{∗} of w^{∘} is the cyclic subgroup {ζ ∈ ℂ^{∗} | ζ^{D} = 1} ≅ ℤ/Dℤ, and the ℂ^{∗}-orbit ℂ^{∗}w^{∘} ≅ ℂ^{∗}/St_{w^{∘}} ≅ ℂ^{∗} of w^{∘} admits the following biholomorphic parameterization:

f: ℂ^{∗} →^{≅} ℂ^{∗}w^{∘} , f(ζ) = (0, ζ^{2/D}w_{2}^{∘}, ..., ζ^{m/D}w_{m}^{∘}) ∈ ℂ^{∗}w^{∘} ⊂ G_{m}^{0} ⊂ G_{m}  (8.6)

(notice that the set {j/D | w_{j}^{∘} ≠ 0} consists of mutually co-prime integers). □

8.6. Solvable holomorphic maps of Liouville spaces to C^{m}(ℂ) = G_{m}.

Proposition 8.13. Let Z be a connected complex space and F: Z → G_{m} be a holomorphic map. Suppose that either (i) Z is an ultra-Picard space or (ii) the homomorphism F_{∗}: π_{1}(Z) → π_{1}(G_{m}) is solvable and Z is Liouville. Then:

a) There exist w^{∘} = (0, w_{2}^{∘}, ..., w_{m}^{∘}) ∈ G_{m}^{0}, ζ ∈ O^{∗}(Z) and c, η_{2}, ..., η_{m} ∈ O(Z) such that η_{j} = 0 for j ∈ Z(w^{∘}), η_{j} = ζ^{j/D(w^{∘})} for j ∈ N Z(w^{∘}) and

F (z) = (t + c(z))^{m} + ∑_{j=2}^{m} w_{j}^{∘}η_{j}(z)(t + c(z))^{m−j} , z ∈ Z .  (8.7)

b) If w_{m}^{∘} ≠ 0 then the map (8.7) is homotopic to the model map

A_{a}: Z ∋ z ↦ (0, ..., 0, 0, a(z)) = t^{m} + a(z) ∈ G_{m}^{0} ⊂ G_{m} ,  (8.8)

where a(z) = [ζ(z)]^{m/D(w^{∘})}.

c) If w_{m}^{∘} = 0 then w_{m−1}^{∘} ≠ 0 and the map (8.7) is homotopic to the model map

B_{b}: Z ∋ z ↦ (0, ..., 0, b(z), 0) = t^{m} + b(z)t ∈ G_{m}^{0} ⊂ G_{m} ,  (8.9)

where b(z) = [ζ(z)]^{(m−1)/D(w^{∘})}.

Proof. a) Let F (z) = (F_{1}(z), F_{2}(z), ..., F_{m}(z)); consider a holomorphic map F^{0} = (0, F_{2}^{0}, ..., F_{m}^{0}): Z → G_{m}^{0} with the components F_{2}^{0}, ..., F_{m}^{0} defined by the condition

t^{m} + ∑_{j=2}^{m} F_{j}^{0}(z)t^{m−j} ≡ (t − F_{1}(z)/m)^{m} + ∑_{j=1}^{m} F_{j}(z)(t − F_{1}(z)/m)^{m−j} .  (8.10)

By Proposition 8.10, F^{0}(Z) ⊆ ℂ^{∗}w^{∘} for a certain point w^{∘} = (0, w_{2}^{∘}, ..., w_{m}^{∘}) ∈ G_{m}^{0}. It follows from Lemma 8.12 that there are ζ ∈ O^{∗}(Z) and η_{2}, ..., η_{m} ∈ O(Z) such that η_{j} = 0 for j ∈ Z(w^{∘}), η_{j} = ζ^{j/D(w^{∘})} for j ∈ N Z(w^{∘}) and

F^{0}(z) = (0, w_{2}^{∘}η_{2}(z), ..., w_{m}^{∘}η_{m}(z)) = t^{m} + ∑_{j=2}^{m} w_{j}^{∘}η_{j}(z)t^{m−j} , z ∈ Z .  (8.11)

Let c(z) = F_{1}(z)/m; then (8.10) and (8.11) show that

\begin{aligned}F (z) & = t^{m} + ∑_{j=1}^{m} F_{j}(z)t^{m−j} = (t + c(z))^{m} + ∑_{j=2}^{m} F_{j}^{0}(z)(t + c(z))^{m−j} \\ & = (t + c(z))^{m} + ∑_{j=2}^{m} w_{j}^{∘}η_{j}(z)(t + c(z))^{m−j} , \end{aligned}  (8.12)

which proves (a).

(b,c) By (a), F is of the form (8.7). The homotopy

F (z, s) = (t + sc(z))^{m} + ∑_{j=2}^{m} w_{j}^{∘}η_{j}(z)(t + sc(z))^{m−j} , z ∈ Z , 0 ≤ s ≤ 1 , (8.13)

joins F = F (·, 1) to the map F (·, 0), F (z, 0) = t^{m} + ∑_{j=2}^{m} w_{j}^{∘}η_{j}(z)t^{m−j} in the class of holomorphic maps Z → G_{m}. At least one of the numbers m − 1, m is not in Z and the vector subspace

V (w^{∘}) =^{def} {w = (0, w_{2}, ..., w_{m}) ∈ ℂ^{m} | w_{j} = 0 whenever j ∈ Z}

is of positive dimension. The discriminant d_{m}(w) cannot vanish identically on V (w^{∘}), for w^{∘} ∈ G_{m}^{0} ∩ V (w^{∘}); thus, the intersection

G_{m}^{0} ∩ V (w^{∘}) = V (w^{∘}) \ {w = (0, w_{2}, ..., w_{m}) ∈ ℂ^{m} | d_{m}(w) = 0}

is arcwise connected. Set w′ = (0, ..., 0, 0, 1) = t^{m} + 1 when w_{m}^{∘} ≠ 0 and w′ = (0, ..., 0, 1, 0) = t^{m} + t otherwise; notice that in both cases w′ ∈ G_{m}^{0} ∩ V (w^{∘}). Let w(s) = (0, w_{2}(s), ..., w_{m}(s)), 0 ≤ s ≤ 1, be a path in G_{m}^{0} ∩ V (w^{∘}) joining the point w(0) = w^{∘} to the point w(1) = w′. It is easily seen that

d_{m}(0, w_{2}(s)η_{2}(z), ..., w_{m}(s)η_{m}(z)) = [ζ(z)]^{m(m−1)/D(w^{∘})}d_{m}(0, w_{2}(s), ..., w_{m}(s)) ;

therefore the formula h_{s}(ζ) = t^{m} + ∑_{j=2}^{m} w_{j}(s)η_{j}(z)t^{m−j} provides a well-defined homotopy joining the map F (·, 0) to a model map A_{a} in case (b) and to a model map B_{b} in case (c). □

Remark 8.14. Let a, b, a_{i}, b_{i} ∈ O^{∗}(Z), i = 1, 2. Then^{18} A_{a_{1}} ∼ A_{a_{2}} (respectively B_{b_{1}} ∼ B_{b_{2}}) if and only if a_{1}/a_{2} ∈ exp O(Z) (respectively if and only if b_{1}/b_{2} ∈ exp O(Z)). Moreover, A_{a} ∼ B_{b} if and only if a^{m−1}/b^{m} ∈ exp O(Z). ◯

Definition 8.15. Define the points w_{∘}^{(m)}, w_{∘}^{(m−1)} ∈ G_{m} as

w_{∘}^{(m)} = (0, ..., 0, 0, −1) = t^{m} − 1 and w_{∘}^{(m−1)} = (0, ..., 0, −1, 0) = t^{m} − t

respectively. For any r ∈ ℤ we define the model holomorphic maps

\begin{matrix}A_{r}: ℂ^{∗} ∋ ζ ↦ U_{ζ^{r}}w_{∘}^{(m)} = (0, ..., 0, 0, −ζ^{r}) = t^{m} − ζ^{r} ∈ G_{m}^{0} ⊂ G_{m} = C^{m}(ℂ) , \\ B_{r}: ℂ^{∗} ∋ ζ ↦ U_{ζ^{r}}w_{∘}^{(m−1)} = (0, ..., 0, −ζ^{r}, 0) = t^{m} − ζ^{r}t ∈ G_{m}^{0} ⊂ G_{m} = C^{m}(ℂ) . \end{matrix} (8.14)

Notice that A_{p} ∼ B_{q} if and only if (m − 1)p = mq. ◯

Corollary 8.16. Any holomorphic map ℂ^{∗} → C^{m}(ℂ) = G_{m} is homotopic to one of the model maps A_{r}, B_{r} (r ∈ ℤ). □

18Here “∼” means free homotopy.

8.7. Cyclic morphisms C^{n}(ℂ) → C^{k}(ℂ). Notice that any non-vanishing holomorphic function f(Q) on C^{n}(ℂ) is of the form f(Q) = [d_{n}(Q)]^{l}e^{φ(Q)}, where l ∈ ℤ and φ ∈ O(C^{n}(ℂ)). Since any connected algebraic variety is a Liouville space, Proposition 8.13 applies to cyclic morphisms C^{n}(ℂ) → C^{k}(ℂ):

Theorem 8.17. Let F: C^{n}(ℂ) = G_{n} → G_{k} = C^{k}(ℂ) be a cyclic holomorphic map. Then:

a) There exist w^{∘} = (0, w_{2}^{∘}, ..., w_{k}^{∘}) ∈ G_{k}^{0} = C_{b}^{k−1}(ℂ), l ∈ ℤ and c, φ ∈ O(C^{n}(ℂ)) such that

F (Q) = (t + c(Q))^{k} + ∑_{j=2}^{k} w_{j}^{∘}[d_{n}(Q)]^{jl/D(w^{∘})}e^{jlφ(Q)}(t + c(Q))^{k−j} , Q ∈ C^{n}(ℂ) . (8.15)

b) If w_{k}^{∘} ≠ 0 then kl/D(w^{∘}) ∈ ℤ and the map (8.15) is homotopic to the model map

A_{l}: C^{n}(ℂ) ∋ Q ↦ (0, ..., 0, 0, [d_{n}(Q)]^{kl/D(w^{∘})}) = t^{k} + [d_{n}(Q)]^{kl/D(w^{∘})} .  (8.16)

c) If w_{k}^{∘} = 0 then w_{k−1}^{∘} ≠ 0, (k − 1)l/D(w^{∘}) ∈ ℤ and the map (8.15) is homotopic to the model map

B_{l}: C^{n}(ℂ) ∋ Q ↦ (0, ..., 0, [d_{n}(Q)]^{(k−1)l/D(w^{∘})}, 0) = t^{k} + [d_{n}(Q)]^{(k−1)l/D(w^{∘})}t . (8.17)

d) A_{l_{1}} ∼ A_{l_{2}} (respectively B_{l_{1}} ∼ B_{l_{2}}) if and only if l_{1} = l_{2}. Moreover, A_{l_{1}} ∼ B_{l_{2}} if and only if (k − 1)l_{1} = kl_{2}.

In particular, every cyclic holomorphic map F: C^{n}(ℂ) → C^{k}(ℂ) homotopically splits, that is, there exists a holomorphic map G: ℂ^{∗} → C^{k}(ℂ) such that F is homotopic to the composition

G ∘ d_{n}: C^{n}(ℂ) ⟶^{d_{n}} ℂ^{∗} ⟶^{G} C^{k}(ℂ) . □

9. Morphisms C^{n}(ℂ) → C^{k}(ℂ) and special homomorphisms B_{n} → B_{k}

Due to Theorem 4.5, for n ≠ 4 and n > k every continuous map (and all the more every morphism) C^{n}(X) → C^{k}(X) must be cyclic. On the other hand, for 3 ≤ n ≤ k it is easy to construct non-cyclic continuous maps C^{n}(ℂ) → C^{k}(ℂ). It is not so in the case X = ℂℙ^{1}. To see this, one may use a theorem of Kunio Murasugi [Mur82], which provides a complete description of the torsion in the sphere braid group B_{m}(S^{2}) and in the quotient group B_{m}/CB_{m} of the Artin braid group B_{m} by its center CB_{m}.

Let us set α = σ_{1} ⋯ σ_{m−1}, β = σ_{1} ⋯ σ_{m−1}σ_{1} (both in B_{m} and B_{m}(S^{2})), and γ = σ_{1} ⋯ σ_{m−2}σ_{1} in B_{m}(S^{2}); furthermore, let us denote by α̂, β̂ the images of the elements α, β ∈ B_{m} in the quotient group B_{m}/CB_{m}.

Murasugi Theorem. a) Each element of finite order in B_{m}/CB_{m} is conjugate to an element of the form α̂^{p} or β̂^{p}, where p ∈ ℤ. The elements α̂, β̂ ∈ B_{m}/CB_{m} are of the order m and m − 1 respectively.

9. MORPHISMS C^{n}(ℂ) → C^{k}(ℂ) AND SPECIAL HOMOMORPHISMS B_{n} → B_{k} 53

b) Each element of finite order in B_{m}(S^{2}) is conjugate to an element of the form α^{p}, β^{p} or γ^{p}, where p ∈ ℤ. The elements α, β, γ ∈ B_{m}(S^{2}) are of the order 2m, 2(m − 1) and 2(m − 2) respectively.

For any homomorphism φ: B_{n}(S^{2}) → B_{k}(S^{2}) we have φ(Tors B_{n}(S^{2})) ⊆ Tors B_{k}(S^{2}). Hence part (b) of Murasugi Theorem implies that φ cannot be non-cyclic unless the number n(n − 1)(n − 2) divides k(k − 1)(k − 2).

A homomorphism φ: B_{n} → B_{k} (cyclic or not) must not induce a homomorphism of quotient groups B_{n}/CB_{n} → B_{k}/CB_{k}. Hence Murasugi Theorem puts no restraints in homomorphisms B_{n} → B_{k}. In fact, for k > n ≥ 3 there is plenty of non-cyclic homomorphisms φ: B_{n} → B_{k} and each of them is induced by a certain non-cyclic continuous map C^{n}(ℂ) → C^{k}(ℂ).

For holomorphic maps the situation is completely different. In fact I do not know any example of a non-cyclic holomorphic map C^{n}(X) → C^{k}(X) for 4 < n < k (compare to Linked Map Theorem); most likely they do not exist at all. Being unable to prove this I present here a rather restricted result, which says that in the case X = ℂ such a map cannot exist unless the number n(n − 1) divides k(k − 1). In order to prove this, we will show that a homomorphism F_{∗}: B_{n} → B_{k} induced by a holomorphic map C^{n}(ℂ) → C^{k}(ℂ) must satisfy a strong algebraic condition.

9.1. Special presentations and special homomorphisms. Recall that the elements α = σ_{1} ⋯ σ_{m−1} and β = ασ_{1} generate the whole braid group B_{m}(X) (see 4.9).

Definition 9.1. A pair of elements a, b ∈ B_{m} is said to be a special system of generators in B_{m} if there exists an automorphism ψ of B_{m} such that ψ(α) = a and ψ(β) = b. If {a, b} is such a system of generators then the elements

s_{i} = ψ(σ_{i}) = a^{i−2}ba^{−(i−1)} (1 ≤ i ≤ m − 1)

also form a system of generators of B_{m} that satisfy relations (4.3), (4.4); we call such a system of generators standard. ◯

Definition 9.2. Given a special system of generators {a, b} in B_{m}, we denote by H_{m}(a, b) the set of all elements of the form g^{−1}a^{p}g or g^{−1}b^{p}g, where g runs over B_{m} and p runs over ℤ. A homomorphism φ: B_{n} → B_{k} is said to be special if φ(H_{n}(a, b)) ⊆ H_{k}(a′, b′) for some choice of special systems of generators a, b ∈ B_{n} and a′, b′ ∈ B_{k}. ◯

Part (a) of Murasugi Theorem quoted above shows that the set H_{m} = H_{m}(a, b) does not depend on a choice of a special system of generators a, b ∈ B_{m}, and a homomorphism φ: B_{n} → B_{k} is special if and only if for any element g ∈ B_{m} of finite order modulo CB_{n} its image φ(g) ∈ B_{k} is an element of finite order modulo CB_{k} (we do not use this fact below; it is convenient however to keep it in mind).

Remark 9.3. It is easily seen that for n ≥ 3 every endomorphism of the group B_{n} is special. On the other hand, for k > n ≥ 3 the standard embedding B_{n} ↪ B_{k} can never be a special homomorphism. ◯

The following result stated in [Lin71] and [Lin79] is the crucial one for our purposes.

Theorem 9.4. For every holomorphic map F: C^{n}(ℂ) → C^{k}(ℂ), every point Q^{∘} ∈ C^{n}(ℂ) and any choice of isomorphisms B_{n} ≅ π_{1}(C^{n}(ℂ), Q^{∘}) and π_{1}(C^{n}(ℂ), F (Q^{∘})) ≅ B_{k} the induced homomorphism

F_{∗}: B_{n} ≅ π_{1}(C^{n}(ℂ), Q^{∘}) → π_{1}(C^{n}(ℂ), F (Q^{∘})) ≅ B_{k}  (9.1)

is special. ◯

To prove this theorem we will use the description of holomorphic maps ℂ^{∗} → C^{m}(ℂ) given in Corollary 8.16. But first I want to present a general concept of holomorphic elements of homotopy groups introduced in [Lin71].

9.2. Holomorphic elements of homotopy groups. The idea of this section is to choose for each i ∈ ℕ an appropriate test complex manifold M_{i} homotopy equivalent to the i-dimensional sphere S^{i} and define corresponding holomorphic elements in the homotopy groups π_{i}(Y) of connected complex spaces Y as those admitting holomorphic representatives M_{i} → Y. Doing so one may sometimes come to certain non-trivial algebraic restraints imposed by the condition that a homomorphism π_{i}(Y) → π_{i}(Z) can be induced by a holomorphic map Y → Z, since such a homomorphism must carry holomorphic elements of π_{i}(Y) to holomorphic elements of π_{i}(Z). What are “test manifolds” depends on the problem under consideration; for our purpose the following choice is an appropriate one just because it works.

Definition 9.5. Let Y be a connected complex space. We define the holomorphic part hol π_{i}(Y, y_{∘}) of the homotopy group π_{i}(Y, y_{∘}) as follows. In the space ℂ^{i+1} with coordinates ζ_{0}, ..., ζ_{i} consider the quadric

ℂS^{i} = {ζ = (ζ_{0}, ..., ζ_{i}) ∈ ℂ^{i+1} | ζ_{0}^{2} + ... + ζ_{i}^{2} = 1} .

Its intersection S^{i} = ℂS^{i}∩ℝ^{i+1} with the real subspace ℝ^{i+1} ⊂ ℂ^{i+1} is the standard unit sphere in ℝ^{i+1}; moreover, S^{i} is a deformation retract of ℂS^{i} (for ℂS^{i} is diffeomorphic to the total space of the tangent bundle T S^{i}). Therefore choosing ζ_{∘} = (1, 0, ..., 0) ∈ ℂS^{i} as a basepoint we may regard any element γ ∈ π_{i}(Y, y_{∘}) as a homotopy class of continuous mappings (ℂS^{i}, ζ_{∘}) → (Y, y_{∘}). Let us say that an element γ ∈ π_{i}(Y, y_{∘}) is holomorphic if there exists a holomorphic mapping h: ℂS^{i} → Y freely homotopic to some (and hence to any) map contained in γ. The set of all holomorphic elements in π_{i}(Y, y_{∘}) is denoted by hol π_{i}(Y, y_{∘}); in general, it is not a subgroup; the subgroup of π_{i}(Y, y_{∘}) generated by hol π_{i}(Y, y_{∘}) is denoted by Hol π_{i}(Y, y_{∘}). ◯

Proposition 9.6. The subset hol π_{i}(Y, y_{∘}) and the subgroup Hol π_{i}(Y, y_{∘}) have the following properties:

i) If Y is a homogeneous space of a complex Lie group then hol π_{i}(Y, y_{∘}) = π_{i}(Y, y_{∘}) for every natural i. In particular, the complex orthogonal group O(n + 1, ℂ) acts transitively on ℂS^{m} and hence

hol π_{i}(ℂS^{m}, ζ_{∘}) = π_{i}(ℂS^{m}, ζ_{∘}) ≅ π_{i}(S^{m})

for all natural i and m. ii) If γ ∈ hol π_{i}(Y, y_{∘}) then the whole cyclic subgroup generated by γ is contained

in hol π_{i}(Y, y_{∘}). iii) The isomorphism χ_{i}: π_{i}(Y, y_{1}) → π_{i}(Y, y_{0}) induced by a path χ: [0, 1] → Y

joining points y_{0}, y_{1} ∈ Y carries hol π_{i}(Y, y_{1}) into hol π_{i}(Y, y_{0}).

iv) The homomorphism f_{∗i}: π_{i}(Y, y_{∘}) → π_{i}(Z, z_{∘}) induced by a holomorphic map

f: (Y, y_{∘}) → (Z, z_{∘}) carries hol π_{i}(Y, y_{∘})) into hol π_{i}(Z, z_{∘}).

v) Hol π_{1}(Y, y_{∘}) is a normal subgroup of the fundamental group π_{1}(Y, y_{∘}).

Proof. Property (i) follows from the Cartan–Grauert–Ramspott theorem (see [Car40, Gra58, Rams65]). It follows from (i) that hol π_{i}(ℂS^{i}, ζ_{∘}) = π_{i}(ℂS^{i}, ζ_{∘}) ≅ π_{i}(S^{i}, ζ_{∘}) ≅ ℤ. Hence if h: ℂS^{i} → X is a holomorphic representative of a class γ ∈ hol π_{i}(Y, y_{∘}) and f^{(q)} : ℂS^{i} → ℂS^{i} is a holomorphic map homotopic to a map

S^{i} → S^{i} of degree q ∈ ℤ then the composition ℂS^{i} ⟶^{f^{(q)}} ℂS^{i} ⟶^{h} Y is a holomorphic representative of the class qγ (for i = 1 we use the multiplicative notation γ^{q}); this proves (ii). Properties (iii) and (iv) follow directly from the definitions and (v) is a consequence of (iii). □

9.3. Holomorphic elements of π_{i}(C^{m}(ℂ)). It follows from Proposition 8.3 that the image of any holomorphic map f: ℂS^{i} → C^{m}(ℂ) is contained in the Aut X-orbit of some point Q^{∗} ∈ C^{m}(ℂ).

First, let i ≥ 2. Then hol π_{i}(C^{m}(ℂ)) = π_{i}(C^{m}(ℂ)) = 0 (moreover, it is easily shown that for such i any holomorphic map ℂS^{i} → C^{m}(ℂ) is contractible in the class of holomorphic maps). Thus, on this way no difference occurs in the behavior of higher homotopy group homomorphisms induced by holomorphic maps and continuous ones.

Let us compute holomorphic elements of π_{1}(C^{m}(ℂ)); we use the concept of model holomorphic maps A_{r}, B_{r} introduced in Definition 8.15.

Proposition 9.7. a) For any Q^{∗} ∈ C^{m}(ℂ) = G_{m} there is a special system of generators {a, b} in π_{1}(C^{m}(ℂ)) ≅ B_{m} such that the set hol π_{1}(C^{m}(ℂ), Q^{∗}) of all holomorphic elements in π_{1}(C^{m}(ℂ), Q^{∗}) coincides with H_{m}(a, b).

b) The set hol B_{m} of all elements that for some choice of a basepoint Q^{∗} ∈ C^{m}(ℂ) and an isomorphism B_{m} ≅ π_{1}(C^{m}(ℂ), Q^{∗}) admit holomorphic representatives ℂ^{∗} → C^{m}(ℂ) coincides with the set of all elements of finite order modulo the center CB_{m}.

Proof. For any element h ∈ π_{1}(C^{m}(ℂ), Q^{∗}) let us denote by [h] the free homotopy class of continuous mappings ℂ^{∗} → C^{m}(ℂ) containing some (and hence each one) representative of h. A straightforward geometric consideration shows that a standard system of generators s_{1}, ..., s_{m−1} in the group π_{1}(C^{m}(ℂ), Q^{∗}) = π_{1}(G_{m}, w^{∗}) ≅ B_{m} may be chosen in such a way that the model holomorphic map A = A_{1} : ℂ^{∗} ∋ ζ ↦ (0, ..., 0, 0, −ζ) = t^{m} − ζ belongs to the free homotopy class [a] corresponding to the element a = s_{1} ⋯ s_{m−1} and the map B = B_{1} : ℂ^{∗} ∋ ζ ↦ (0, ..., 0, −ζ, 0) = t^{m} − ζt belongs to the free homotopy class [gbg^{−1}] corresponding to the element gbg^{−1}, where b = as_{1} = s_{1} ⋯ s_{m−1}s_{1} and g is a suitable element of π_{1}(G_{m}^{0}, w_{∗}). Then it is clear that A_{r} ∈ [a^{r}] and ⌊_{r} ∈ g[b^{r}g^{−1}]. On the other hand, by Corollary 8.16, any holomorphic map ℂ^{∗} → C^{m}(ℂ) = G_{m} is homotopic to one of the model maps A_{r}, B_{r} (r ∈ ℤ); this completes the proof of statement (a). In view of the Murasugi theorem quoted above, statement (b) is just a consequence of (a) □

Now we are ready to prove Theorem 9.4.

Proof of Theorem 9.4. Let F: C^{n}(ℂ) → C^{k}(ℂ) be a holomorphic map. By Proposition 9.6(iv), the homomorphism F_{∗}: π_{1}(C^{n}(ℂ), Q^{∘}) → π_{1}(C^{k}(ℂ), F (Q^{∘})) of the fundamental groups induced by F carries hol π_{1}(C^{n}) into hol π_{1}(C^{k}). According to Proposition 9.7(a), this implies that F_{∗}(H_{n}(a, b)) ⊆ H_{k}(a, b) and F_{∗} is special. □

9.4. Special homomorphisms B_{n} → B_{k} and morphisms C^{n}(ℂ) → C^{k}(ℂ). The following result proved in [Lin96b] (Theorem 8.1) clarifies the arithmetical consequences of the assumption that the Artin braid group B_{n} admits a non-cyclic special homomorphism to the Artin braid group B_{k} and, in view of Theorem 9.4, imposes the corresponding restraints on the possible existence of non-cyclic morphisms C^{n}(ℂ) → C^{k}(ℂ).

Theorem 9.8. If n ≠ 4 and there is a non-cyclic special homomorphism B_{n} → B_{k} then n(n − 1) divides k(k − 1). ◯

Theorems 9.4 and 9.8 imply the following

Corollary 9.9. If n ≠ 4 and n(n−1) does not divide k(k −1) then every holomorphic map C^{n}(ℂ) → C^{k}(ℂ) is cyclic. □

Remark 9.10. The group B_{3} possesses non-cyclic special homomorphisms B_{3} → B_{k} for any k that is not forbidden by Theorem 9.8. Moreover, let α, β ∈ B_{3} and a, b ∈ B_{4} be special systems of generators; then the map α ↦ b, β ↦ a^{2} extends uniquely to a surjective (and thereby non-cyclic) homomorphism B_{4} → B_{3}; hence, if 6 divides k(k − 1) then non-cyclic special homomorphisms B_{4} → B_{k} do exist.

Suppose that n > 4 and k is not forbidden by Theorem 9.8, i.e., n(n − 1) divides k(k − 1). Then non-cyclic special homomorphisms B_{n} → B_{k} certainly exist whenever n divides k (see [Lin96b], Section 8.1, Theorem 8.3); however, I do not know whether such homomorphisms can exist when n does not divide k. ◯

10. Dimension of the image

Let n > 4 and let F: C^{n}(X) → C^{n}(X) be a morphism. If F is cyclic then, by Cyclic Map Theorem 8.7, it is orbit-like and hence dim_{ℂ} F (C^{n}(X)) ≤ dim_{ℂ} Aut X = t(X) (see Notation 1.11). If F is non-cyclic then it is tame and hence F ((Aut X)Q) ⊆ (Aut X)Q for all Q ∈ C^{n}(X); thus, each Aut X orbit in C^{n}(X) contains at least one point of its own image and therefore dim_{ℂ} F (C^{n}(X)) ≥ n − dim_{ℂ} Aut X = n − t(X). The next theorem strengthens this estimate.

Theorem 10.1. Let n > 4 and let F: C^{n}(X) → C^{n}(X) be a non-cyclic morphism. Then dim_{ℂ} F (C^{n}(X)) ≥ n − t(X) + 1.

This theorem follows immediately from Tame Map Theorem, the existence of points Q ∈ C^{n}(X) with non-trivial stabilizers St_{Q} and the following local result.

Lemma 10.2. Let T: U → Aut X be a holomorphic map of a non-empty open Aut X invariant subset U ⊆ C^{n}(X) and let F: U → U be the map defined by F (u) = T (u)u for all u ∈ U. Suppose that the stabilizer St_{u^{∗}} of some point u^{∗} ∈ U is non-trivial. There is a non-empty open Aut X invariant subset V ⊆ U such that

(∗) F ((Aut X)v) contains a subset biholomorphic to the open disc 𝔻

for each point v ∈ V Moreover, for such a set V we have dim_{ℂ} F (V) ≥ n − t(X) + 1.

Proof. Notice that dim_{ℂ}(Aut X)u = dim_{ℂ} Aut X = t(X) for each u ∈ U; consider the holomorphic map

f_{u}: Aut X ∋ A ↦ T (Au)Au = F (Au) ∈ (Aut X)u .

Let us show that the differential d_{A}f_{u}: ℂ^{t(X)} → ℂ^{t(X)} of f_{u} at A cannot be zero identically on U × (Aut X). Since Aut X is connected, the assumption that d_{A}f_{u} = 0 on U × (Aut X) implies f_{u} = const for each u ∈ U. I. e., T (Au)Au does not depend on A. Hence T (Au)Au = T (u)u and (T (u))^{−1}T (Au)A ∈ St_{u} on U × (Aut X). Since St_{u} is finite, (T (u))^{−1}T (Au)A cannot depend on A and hence (T (u))^{−1}T (Au)A = (T (u))^{−1}T (u) = 1 for all (u, A) ∈ U × (Aut X). For u = u^{∗} there is a non-trivial A ∈ St_{u^{∗}} and we have 1 = (T (u^{∗}))^{−1}T (Au^{∗})A = (T (u^{∗}))^{−1}T (u^{∗})A = A, which is impossible.

Take (u_{0}, A_{0}) ∈ U × (Aut X) such that d_{A_{0}}f_{u_{0}} ≠ 0; then rank d_{A_{0}}f_{u_{0}} ≥ 1 and, by semi-continuity, rank d_{A}f_{u} ≥ 1 in some open neighborhood U_{0} × A_{0} of (u_{0}, A_{0}). The implicit map theorem implies that for each u ∈ U_{0} the image f_{u}(Aut X) contains a subset biholomorphic to 𝔻.

Clearly F ((Aut X)u) = f_{u}(Aut X); hence each point v of the non-empty open Aut X invariant set V = (Aut X)U_{0} satisfies (∗), which implies dim_{ℂ} F (V) ≥ n − t(X) + 1. □

Remark 10.3. The statement of the above lemma may fail if St_{u} = {1} for all u ∈ U. Indeed, let u_{0} ∈ C^{n}(X) and St_{u_{0}} = {1}; then St_{u} = {1} for all u in a small open ball B centered at u_{0}. Let D be a very small open complex disc of dimension n − t(X) contained in B, centered at u_{0} and transversal to the orbit (Aut X)u_{0} at u_{0}. Then [(Aut X)v] ∩ D = {v} for each v ∈ D and the correspondence (A, v) ↦ Av gives rise to the biholomorphic isomorphism f: (Aut X) × D ⟶^{≅} (Aut X)D =^{def} U. The

composition τ = π_{1} ∘ f^{−1} : U ⟶^{f^{−1}} (Aut X) × D ⟶^{π_{1}} Aut X of the inverse isomorphism f^{−1} and the projection π_{1} to the first factor is a holomorphic surjection, the map T: U ∋ u ↦ (τ(u))^{−1} ∈ Aut X is holomorphic, and the map F: U ∋ u ↦ T (u)u carries U to D. Indeed, for each u ∈ U, u = f(A, v) = Av ∈ U, we have τ(u) = (π_{1} ∘ f^{−1})(u) = π_{1}(f^{−1}(f(A, v))) = π_{1}(A, v) = A, T (u) = (τ(u))^{−1} = A^{−1} and, finally, F (u) = T (u)u = A^{−1}Av = v ∈ D so that dim_{ℂ} F (U) = n − t(X). ◯

The linear map L = (L_{1}, ..., L_{n}): C^{n}(ℂ) → C^{n}(ℂ) defined by

L_{i}({q_{1}, ..., q_{n}}) = q_{i} − (q_{1} + ... + q_{n})/n , i = 1, ..., n ,

carries the whole space C^{n}(ℂ) to its (n−1)-dimensional submanifold C_{b}^{n−1}(ℂ) consisting of all balanced configurations (see Definition 8.9). Thus, in the case X = ℂ the lower bound dim_{ℂ} F (C^{n}(ℂ)) ≥ n − t(ℂ) + 1 = n − 1 is the best possible one.

Let L = (L_{1}, ..., L_{n}): C^{n}(ℂ) → C_{b}^{n−1}(ℂ) be the above linear map; the following theorem provides a more precise result about images of non-cyclic endomorphisms, which strengthens Theorem 10.1 in the case X = ℂ.

Theorem 10.4. Let n > 4 and let F: C^{n}(ℂ) → C^{n}(ℂ) be a non-cyclic holomorphic map. Then the linear map L caries the image F (C^{n}(ℂ)) of F on the whole C_{b}^{n−1}(ℂ); more precisely,

L(F (C^{n}(ℂ))) = L(F (C_{b}^{n−1}(ℂ))) = C_{b}^{n−1}(ℂ) .

Proof. By Tame Map Theorem, F (Q) = T (Q)Q, where T: C^{n}(ℂ) → Aff ℂ is a holomorphic map, that is, T (Q): ℂ ∋ z ↦ A(Q)z + B(Q) ∈ ℂ with certain holomorphic functions A: C^{n}(ℂ) → ℂ^{∗} and B: C^{n}(ℂ) → ℂ. The cohomology class d_{n}^{∗} ∈ H^{1}(C^{n}(ℂ), ℤ) ≅ ℤ provided by the discriminant map d_{n}: C^{n}(ℂ) → ℂ^{∗} generates H^{1}(C^{n}(ℂ), ℤ); hence A(Q) = (d_{n}(Q))^{s} exp φ(Q), where s ∈ ℤ and φ is a holomorphic function on C^{n}(ℂ). Take any Q^{∘} = {q_{1}^{∘}, ..., q_{n}^{∘}} ∈ C_{b}^{n−1}(ℂ) and consider the holomorphic function a: ℂ^{∗} ∋ ζ ↦ a(ζ) = ζA(ζQ^{∘}) ∈ ℂ^{∗}. Then a(ζ) = ζA(ζQ^{∘}) = ζ · (d_{n}(ζQ^{∘}))^{s} exp φ(ζQ^{∘}) = ζ^{sn(n−1)+1}(d_{n}(Q^{∘}))^{s} exp φ(ζQ^{∘}), which shows that a ≠ const. Since a does not take the value 0, it follows from the Picard theorem that a(ζ^{∘}) = 1 for some ζ^{∘} ∈ ℂ^{∗}. A straightforward computation shows that G_{i}(ζ^{∘}Q^{∘}) = L_{i}(F (ζ^{∘}Q^{∘}) = ζ^{∘}A(ζ^{∘}Q^{∘})q_{i}^{∘} = a(ζ^{∘})q_{i}^{∘} = q_{i}^{∘} for all i = 1, ..., n, that is, G(ζ^{∘}Q^{∘}) = Q^{∘}, which proves the theorem. □

Remark 10.5. I do not know any example of a non-cyclic holomorphic endomorphism F of C^{n}(ℂℙ^{1}) such that dim_{ℂ} F (C^{n}(ℂℙ^{1})) < n. ◯

11. Some applications to algebraic functions

The n-valued algebraic function [x : y] = U_{n}([z]) defined by the equation

\begin{matrix}ϕ_{n}(x, y; z) = z_{0}x^{n} + z_{1}x^{n−1}y + ... + z_{n}y^{n} = 0 , \\ z = (z_{0}, ..., z_{n}) ∈ ℂ^{n+1} , [z] = [z_{0} : ... : z_{n}] ∈ ℂℙ^{n} , \end{matrix}  (11.1)

and the n-valued entire algebraic function t = u_{n}(w) defined by the equation

p_{n}(t, w) = t^{n} + w_{1}t^{n−1} + ... + w_{n} = 0 , w = (w_{1}, ..., w_{n}) ∈ ℂ^{n} ,  (11.2)

are said to be the universal ones. It is easily seen that any n-valued algebraic function on ℂℙ^{m} (respectively any n-valued entire algebraic function on ℂ^{m}) is a composition of a rational map ℂℙ^{m} → ℂℙ^{n} (respectively a polynomial map ℂ^{m} → ℂ^{n}) and the corresponding universal function. The branch loci of the universal functions are also called universal ones.

In this section we study algebraic functions whose branch loci coincide with a universal one and apply the obtained results to a version of the 13th Hilbert problem for algebraic functions.

11.1. Algebraic functions with the universal branch locus. To avoid the doubling related to our wish to consider both “general” and “entire” algebraic functions, we will use the following notation.

Notation 11.1. We consider the following two cases:

(a) X = ℂℙ^{1}, X^{n}/S(n) = ℂℙ^{n}, U_{n} = U_{n} is the universal n-valued algebraic function on ℂℙ^{n} defined by (11.1) with the branch locus

Σ_{U_{n}} = Σ_{U_{n}} = {[z] ∈ ℂℙ^{n} | D_{ϕ_{n}}([z]) = 0} ,  (a′)

and (X^{n}/S(n)) \ Σ_{U_{n}} = ℂℙ^{n} \ Σ_{U_{n}} ≅ C^{n}(X) = C^{n}(ℂℙ^{1}) ;

(b) X = ℂ, X^{n}/S(n) = ℂ^{n}, U_{n} = u_{n} is the universal n-valued entire algebraic function on ℂ^{n} defined by (11.2) with the branch locus

Σ_{U_{n}} = Σ_{u_{n}} = {w ∈ ℂ^{n} | d_{n}(w) = 0} ,  (b′)

and (X^{n}/S(n)) \ Σ_{U_{n}} = ℂ^{n} \ Σ_{U_{n}} ≅ C^{n}(X) = C^{n}(ℂ^{1}).

In case (a) the term “an m-valued algebraic function F on X^{n}/S(n)” will mean an algebraic function [x : y] = F([z]) on ℂℙ^{n} defined by an equation

φ(x, y; z) = a_{0}(z)x^{m} + a_{1}(z)x^{m−1}y + ... + a_{m}(z)y^{m} = 0 ,

where φ(x, y; z) = a_{0}(z)x^{m} + a_{1}(z)x^{m−1}y + ... + a_{m}(z)y^{m} is a homogeneous binary form whose coefficients a_{0}, ..., a_{m} are coprime homogeneous polynomials in z_{0}, ..., z_{n} of the same positive degree.^{19} The form φ(x, y; z) is called the defining form of the algebraic function F. The zero set Σ_{φ} = {z = (z_{0}, ..., z_{n}) ∈ ℂ^{n+1} | D_{φ}(z) = 0} of the discriminant D_{φ}(z) is said to be degeneration locus of the form φ(x, y; z); in particular, Σ_{φ} contains all common zeros of the coefficients a_{0}, ..., a_{m}. The quotient space Σ_{F} = Σ_{φ}/ℂ^{∗} ⊂ ℂℙ^{n} is the branch locus of the function F (it may happen that D_{φ} = 0 and Σ_{F} = ℂℙ^{n}).

For two algebraic functions [x : y] = G([z]) and [x : y] = F ([z]) with the defining forms ψ(x, y; z) and φ(x, y; z) we say that G is a (multivalued) branch of F and write G ⊆ F if there is a form ω(x, y; z) such that φ(x, y; z) = ψ(x, y; z) · ω(x, y; z). We write F = G if φ(x, y; z) = cψ(x, y; z) for some c ∈ ℂ^{∗}.

Let us notice that if t = f(w) is an algebraic function of w ∈ ℂ^{n} defined by an algebraic equation P (t, w) = A_{0}(w)t^{m} +A_{1}(w)t^{m−1} +...+A_{m}(w) = 0 with coefficients A_{i} ∈ ℂ[w] then one may replace t with x/y, each of w_{i} with z_{i}/z_{0}, and rewrite the function as [x : y] = F ([z]) and the defining equation as φ(x, y; z) = 0, where φ(x, y; z) is a binary form of the type described above.

In case (b) the term “an m-valued algebraic function F on X^{n}/S(n)” will mean an entire algebraic function t = F(w) on ℂ^{n} defined by an equation

P (t; w) = t^{m} + a_{1}(w)t^{m−1} + ... + a_{m}(w) = 0 ,  (b′)

where all a_{i} ∈ ℂ[w]. For two entire algebraic functions t = g(w) and t = f(w) with the defining polynomials Q(t, w) and P (t, w) we say that g is a (multivalued) branch of f and write g ⊆ f if there is a polynomial R(t, w) such that P = Q · R. We write g = f if P = Q.

The following result is in fact equivalent to Tame Map Theorem for regular endomorphisms.

19In this section, all binary forms under consideration are supposed to be of such kind. It is possible that some coefficients a_{i} = 0.

Theorem 11.2. Let n > 4 and let F be an n-valued algebraic function on X^{n}/S(n) with the branch locus Σ_{F} = Σ_{U_{n}} and a non-cyclic monodromy representation

π_{1}((X^{n}/S(n)) \ Σ_{U_{n}}) → S(n) .  (11.3)

Then there exists a rational map T: X^{n}/S(n) → Aut X regular on (X^{n}/S(n)) \ Σ_{U_{n}} such that F(z) = T (z)U_{n}(z) for all z ∈ X^{n}/S(n).

Proof. The algebraic function F is the composition F(z) = U_{n}(A(z)) of the rational map A: X^{n}/S(n) → X^{n}/S(n) given by the coefficients of the equation defining F and the universal function U_{n}. The assumption Σ_{F} = Σ_{U_{n}} implies that A is regular on (X^{n}/S(n))\Σ_{U_{n}} and carries the latter set into itself. Thus, under the identification of the defining equations to the sets of their roots established in Remark 1.9, A may be regarded as a regular endomorphism A′ of the configuration space C^{n}(X). Since the monodromy representation (11.3) is non-cyclic, A′ is non-cyclic as well. By Tame Map Theorem, there is a regular map T′: C^{n}(X) → Aut X such that A′(Q) = T′(Q)Q for all Q ∈ C^{n}(X). Denoting by T the regular morphism (X^{n}/S(n)) \ Σ_{U_{n}} → Aut X corresponding to T′, we obtain that F(z) ≡ U_{n}(A(z)) ≡ T (z)U_{n}(z) meaning that each value of F at z ∈ (X^{n}/S(n)) \ Σ_{U_{n}} is obtained from the corresponding value of U_{n} at z with help of the Möbius transformation T (z). □

The following result is just a simple consequence of Linked Map Theorem (for X = ℂ it has been proved in [Lin96b]).

Theorem 11.3. Let n > t(X) + 1 and let F be an N-valued algebraic function on X^{n}/S(n). If U_{n} ⊆ F then either N = n and F = U_{n} or N > n and F has branch points that are not branch points of U_{n}.

Proof. We consider here only the case X = ℂℙ^{1}. Let φ(x, y; z) be the defining form of the function F. The condition U_{n} ⊆ F means that there is a binary form ω(x, y; z) of degree k = N − n ≥ 0 such that φ(x, y; z) = ϕ_{n}(x, y; z) · ω(x, y; z); in this case D_{ϕ_{n}}(z) = 0 implies D_{φ}(z) = 0, which means that Σ_{F} ⊇ Σ_{U_{n}}.

Assume that k > 0 and Σ_{F} = Σ_{U_{n}}. Then for every z ∈ ℂℙ^{n} \ Σ_{U_{n}} = C^{n}(ℂℙ^{1}) the form φ(x, y; z) is non-degenerate; therefore ω(x, y; z) is non-degenerate as well and has no common roots with ϕ_{n}(x, y; z). This means that the correspondence z ↦ ω(x, y; z) defines a disjoint morphism C^{n}(ℂℙ^{1}) ≅ F^{n} → F^{k} ≅ C^{k}(ℂℙ^{1}), which contradicts Linked Map Theorem 1.12. Thus, Σ_{U_{n}} must be a proper subset of Σ_{F}, which shows that F has branch points that are not branch points of U_{n}. Finally, if k = 0 then N = n, ω = const ≠ 0 and F = U_{n}. □

The problem about existence of an algebraic function with the preassigned branch locus and monodromy group goes back to B. Riemann (in one variable), F. Enriques and O. Zariski (for functions of two variables; see [Enr23, Zar29]). It was discovered that in two variables there are obstacles related to the fundamental group of the complement of the branch curve.

We deal with very distinguished subvarieties Σ_{U_{n}} ⊂ ℂℙ^{n} and Σ_{u_{n}} ⊂ ℂ^{n} and we know in advance that there exist n valued algebraic functions, namely the universal ones U_{n} and u_{n}, with these very branch loci and the standard monodromy representations

\begin{matrix}μ: π_{1}(ℂℙ^{n} \ Σ_{U_{n}}) = π_{1}(C^{n}(ℂℙ^{1})) ≅ B_{n}(S^{2}) → S(n) , \\ μ: π_{1}(ℂ^{n} \ Σ_{u_{n}}) = π_{1}(G_{n}) = π_{1}(C^{n}(ℂ)) ≅ B_{n} → S(n) . \end{matrix}

Theorem 11.2 provides a reasonable description of all non-cyclic n-valued algebraic functions of n variables z ∈ X^{n}/S(n) with the universal branch locus Σ_{U_{n}} in terms of the universal function U_{n}(z) and automorphisms of the target domain X = ℂ or X = ℂℙ^{1} depending regularly on a point z ∈ (X^{n}/S(n)) \ Σ_{U_{n}}. On the other hand, Theorem 11.3 tells that the universal function cannot be a proper branch of an algebraic function with the same branch locus.

11.2. Compositions of multivalued functions. One possible form of the question posed in the 13th Hilbert problem is as follows:

What is the minimal natural number m for which the function u_{n} can be represented as a composition of algebraic functions of m variables?

Since u_{n} is multivalued, the emphasized expression may have quite different meanings. It has been proved in [Arn70c, Lin76, Lin96a] that for certain possible interpretations of the problem the answer is m = n − 1. Here we discuss the corresponding problem for the function [x : y] = U_{n}([z]) and show that in this case m ≥ n − 2.

Following the papers quoted above, we explain first some formalities related to multivalued maps; we do this in a manner convenient for working with non-entire algebraic functions.

Given two topological spaces Y and Z, we use the notation f: Z ⟶^{n} Y for a map defined and continuous on an open dense subset Z_{f} ⊆ Z and carrying it into the space Sym^{m} Y = Y^{m}/S(m); such a map is said to be an m-valued map of Z to Y. The points y_{1}, ..., y_{m} ∈ f(z) are said to be the values of f at z ∈ Z_{f}. The subset Im f ⊆ Y consisting of all values of f at all points z ∈ Z_{f} is called the image of f. The subset Γ_{f} ⊆ Z × Y consisting of all pairs (z, y) such that y is a value of f at z ∈ Z_{f} is called the graph of g.

Let f: Z ⟶^{m} Y and g: Y ⟶^{n} W be multivalued maps; suppose that there are open dense subsets Z′ ⊆ Z_{f} and Y′ ⊆ Y_{g} such that all values of f at any point z ∈ Z′ belong to Y′. The composition

g ∘ f: Z ⟶^{mn} W

is the (mn)-valued map whose values at a point z ∈ Z′ are all mn values of g at the m points of Y′ which are the values of f at z. The complete composition

g ∗ f: Z ⟶^{mn} Y × W

is the (mn)-valued map whose values at a point z ∈ Z′ are all pairs (y, w) such that y is one of the values of f at z and w is one of the values of g at y.

Let k, m, M be natural numbers. Assume that for each i = 1, …, M we are given with an n_{i}-valued algebraic function [x : y] = g_{i}([w]), [w] = [w_{0} : ... : w_{m}] ∈ ℂℙ^{m}, defined by an equation φ_{i}(x, y; [w]) = a_{0}(w)x^{n_{i}} + a_{1}(w)x^{n_{i}−1}y + ... + a_{n_{i}}(w)y^{n_{i}} = 0, where a_{0}, ..., a_{n_{i}} are co-prime homogeneous polynomials of the same positive degree. Assuming that the discriminant D_{φ_{i}}(w) of φ_{i} is not identically zero, we may regard g_{i} as an n_{i}-valued map g_{i}: ℂℙ^{m} ⟶^{n_{i}} ℂℙ^{1} , which is surely defined on the non-empty Zariski open set W_{i} = {w ∈ ℂℙ^{m} | D_{φ_{i}}(w) ≠ 0}.

Furthermore, for each i = 1, …, M let R_{i}: ℂℙ^{k} × (ℂℙ^{1})^{i−1} → ℂℙ^{m} be a rational map such that its regularity domain contains a non-empty Zariski open subset V_{i} with the image R_{i}(V_{i}) ⊆ W_{i}.

Let F_{0} : ℂℙ^{k} → ℂℙ^{k} denote the identity map. Using the compositions ∘ and ∗ of multivalued functions defined above, we construct algebraic functions

G_{i} = g_{i} ∘ R_{i}: ℂℙ^{k} × (ℂℙ^{1})^{i−1} ⟶^{R_{i}} ℂℙ^{m} ⟶^{n_{i}} ℂℙ^{1} , i = 1, ..., M ,

F_{1} = G_{1} ∗ F_{0} : ℂℙ^{k} ^{^{n_{1}}\overline{}}ℂℙ^{k} × ℂℙ^{1} , F_{2} = G_{2} ∗ F_{1} : ℂℙ^{k} ^{\overline{}^{n_{1}n_{2}}} ℂℙ^{k} × (ℂℙ^{1})^{2} , ...................................................................

F_{i} = G_{i} ∗ F_{i}: ℂℙ^{k} ^{\overline{}^{n_{1}n_{2}···n_{i}}} ℂℙ^{k} × (ℂℙ^{1})^{i} , ...................................................................

F_{M−1} = G_{M−1} ∗ F_{M−2} : ℂℙ^{k} ^{\overline{}^{n_{1}n_{2}···n_{M−1}}} ℂℙ^{k} × (ℂℙ^{1})^{M−1} ,

and, finally,

F = F_{M} = G_{M} ∘ F_{M−1} : ℂℙ^{k} ^{\overline{}^{n_{1}n_{2}···n_{M}}} ℂℙ^{1} .

F is a well-defined N = n_{1} ⋯ n_{M}-valued algebraic function on ℂℙ^{k} whenever Im F_{i} ⊆ V_{i+1} for all i = 1, ..., M −1. If this is the case then the collection 𝔄 = {V_{i}, W_{i}, R_{i}, g_{i} | i = 1, ..., M} is said to be a tame algebraic (k, m)-ensemble in ℂℙ^{k}, and the algebraic function F_{𝔄} = F is called the function generated by the ensemble 𝔄. Two tame algebraic (k, m)-ensembles 𝔄 and 𝔄′ in ℂℙ^{k} are said to be equivalent if the functions F_{𝔄} and F_{𝔄′} generated by them coincide.

Some of the algebraic functions g_{i} appearing in a tame ensemble 𝔄 may be singlevalued; the number of those ones which are actually multivalued is called the rank of the ensemble 𝔄.

Example 11.4. For M = 2 we have F (z) = g_{2}{R_{2}[z, g_{1}(R_{1}(z))]}. If 𝔄 is an ensemble of rank 1 then F_{𝔄} = R_{2}(z, g(R_{1}(z)), where [x : y] = g(w) is a multivalued algebraic function and R_{1}, R_{2} are rational. ◯

Definition 11.5. We say that an algebraic function g in ℂℙ^{k} can be represented as a composition of algebraic functions of m variables if there exists a tame algebraic (k, m)-ensemble 𝔄 such that g ⊆ F_{𝔄}, that is, g is a (multivalued) branch of F_{𝔄}. The function F_{𝔄} is called a representing function-composition for g; clearly Σ_{g} ⊆ Σ_{F_{𝔄}}. If the ensemble 𝔄 can be chosen so that g = F_{𝔄} we say that g can be faithfully represented as a composition of algebraic functions of m variables; in this case Σ_{g} = Σ_{F_{𝔄}}. In general, the conditions g ⊆ F_{𝔄} and Σ_{g} = Σ_{F_{𝔄}} do not imply g = F_{𝔄}. ◯ In the following theorem we deal with a representation of the universal algebraic function U_{n} as a composition U_{n} ⊆ F_{𝔄}, which a priori is not supposed to be faithful but satisfies the formally weaker condition Σ_{U_{n}} = Σ_{F_{𝔄}}. We will see however that the conditions U_{n} ⊆ F_{𝔄} and Σ_{U_{n}} = Σ_{F_{𝔄}} imply U_{n} = F_{𝔄}.

Theorem 11.6. For n > max{4, m + 2} the universal n-valued algebraic function [x : y] = U_{n}([z]) cannot be represented as a composition of algebraic functions of m variables in such a way that a representing function-composition F = F_{𝔄} and U_{n} itself have the same branch loci.

Proof. Suppose to the contrary that there exists a tame algebraic (n, m)-ensemble 𝔄 = {U_{i}, V_{i}, R_{i}, g_{i} | i = 1, ..., M} in ℂℙ^{n} such that U_{n} ⊆ F = F_{𝔄} and the branch locus Σ_{F} of the N-valued algebraic function-composition F = F_{𝔄} defined by 𝔄 coincides with the branch locus Σ_{U_{n}}. Then, by Theorem 11.3, N = n and U_{n} = F. In particular, the monodromy group of F coincides with the monodromy group S(n) of the function U_{n}. This implies that the unbranched n-covering p: Γ_{F} → ℂℙ^{n} \ Σ_{U_{n}} defined by the function F is indecomposable, that is, the projection π cannot be represented as a composition

p = q_{1} ∘ q_{2} : Γ_{F} ⟶^{q_{2}} E ⟶^{q_{1}} ℂℙ^{n} \ Σ_{U_{n}}

of two non-univalent unbranched coverings q_{1},q_{2} (see [Arn70c, Lin76, Lin96a]). This property implies that the ensemble 𝔄 must be of rank 1, that is, it cannot contain more than one multivalued algebraic function and thereby the corresponding representation of the function U_{n} as a superposition must be of the form U_{n}(z) = F (z) = R_{2}(z, g(R_{1}(z)), where [x : y] = g(w) is an n-valued algebraic function on ℂℙ^{m} with a defining binary form

φ(x, y; w) = a_{0}(w)x^{n} + a_{1}(w)x^{n−1}y + ... + a_{n}(w)y^{n} ,  (11.4)

R_{1} : ℂℙ^{n} → ℂℙ^{m} is a rational map regular on ℂℙ^{n} \ Σ_{U_{n}} with the image

R_{1}(ℂℙ^{n} \ Σ_{U_{n}}) ⊆ {w ∈ ℂℙ^{m} | D_{φ}(w) ≠ 0} ,

and R_{2} : ℂℙ^{n} × ℂℙ^{1} → ℂℙ^{1} is a rational function regular at each point (z, [x : y]) ∈ (ℂℙ^{n} \ Σ_{U_{n}}) × ℂℙ^{1} such that φ(x, y; R_{1}(z)) = 0. Let A = (a_{1}, …, a_{n}): ℂℙ^{m} → ℂℙ^{n} be the rational map defined by the coefficients of the binary form (11.4). The identity U_{n}(z) ≡ R_{2}(z, g(R_{1}(z)) shows that A must be regular on {w ∈ ℂℙ^{m} | D_{φ}(w) ≠ 0} and the A-image A(R_{1}(ℂℙ^{n} \ Σ_{U_{n}})) of the set R_{1}(ℂℙ^{n} \ Σ_{U_{n}}) must be contained in ℂℙ^{n} \ Σ_{U_{n}}. Thus, the composition f = A ∘ R_{1},

f: C^{n}(ℂℙ^{1}) = ℂℙ^{n} \ Σ_{U_{n}} ⟶^{R_{1}} {w ∈ ℂℙ^{m} | D_{φ}(w) ≠ 0} ⟶^{A} ℂℙ^{n} \ Σ_{U_{n}} = C^{n}(ℂℙ^{1}) ,

is a regular map. This map is certainly non-cyclic, for otherwise the monodromy group of the algebraic function g((R_{1}(z)) would be cyclic, as well as the monodromy group of the function R_{2}(z, g(R_{1}(z)) = U_{n}(z). Since m < n − 2, we have dim_{ℂ}(f(C^{n}(ℂℙ^{1}))) ≤ m < n − 2, which contradicts Theorem 10.1. □

Remark 11.7. a) The corresponding result for the universal entire algebraic function u_{n} on ℂ^{n} has been proved in [Lin96a]:

Theorem 11.8. For n > max{4, m + 1} the universal n-valued entire algebraic function t = u_{n}(z) cannot be represented as a composition of entire algebraic functions of m variables in such a way that a representing function-composition F = F_{𝔄} and u_{n} itself have the same branch loci.

The proof is also based on Theorems 1.12, 10.1 and 11.3, which in the case X = ℂ have been proved in the paper quoted above.

b) The compositions of entire algebraic functions treated in Theorem 11.8 are defined similarly to what has been done above. Just in the definition of an algebraic (k, m)-ensemble one should consider affine spaces instead of projective ones and work with polynomials instead of rational functions; that is, the operation of division is in fact prohibited. V. I. Arnold many times expressed his conviction that whenever rational functions are admitted u_{n} cannot be faithfully represented as a composition of algebraic functions of less than n − 2 variables. Seemingly, Theorem 11.6 confirms this Arnold’s opinion (and even for a formally wider class of representations without superfluous branch points instead of the faithful ones). ◯

12. Automorphisms

12.1. How to catch automorphisms? By Tame Map Theorem, every noncyclic endomorphisms F of C^{n}(X) is of the form F = F_{T}, F_{T} (Q) = T (Q)Q, where T: C^{n}(X) → Aut X is a morphism uniquely defined by F. It is reasonable to ask the following question: for which morphism T: C^{n}(X) → Aut X the corresponding tame endomorphism F_{T} is an automorphism? It is easy to give the following “answer”:

F_{T} is an automorphism if and only if there exists a morphism S: C^{n}(X) → Aut X such that T (S(Q)Q)S(Q) = S(T (Q)Q)T (Q) = I (the unity of the group Aut X) for all Q ∈ C^{n}(X).

It is unclear how ensure that T satisfies such a condition. It is naturally to look for a relationship between the group multiplication in Aut X and the composition of tame maps. In general, these operations know nothing about each other; however in the special case of Aut X invariant maps C^{n}(X) → Aut X they are in a close connection.

Definition 12.1. As usually, the pointwise product ST: C^{n}(X) → Aut X of two maps S, T: C^{n}(X) → Aut X is defined by (ST)(Q) = S(Q)T (Q), where S(Q)T (Q) denotes the group product of the elements S(Q), T (Q) ∈ Aut X.

A map S: C^{n}(X) → Aut X is said to be Aut X invariant if S(AQ) = S(Q) for all A ∈ Aut X and Q ∈ C^{n}(X). ◯

Lemma 12.2. Let F_{S}, F_{T} : C^{n}(X) → C^{n}(X) be the tame maps induced by two maps S, T: C^{n}(X) → Aut X. If S is Aut X invariant then F_{ST} = F_{S} ∘ F_{T}.

Proof. It suffices to know that S(Q) = S(T (Q)Q) ∀ Q = {q_{1}, ..., q_{n}} ∈ C^{n}(X); indeed, F_{ST} (Q) = {((ST)(Q))q_{1}, ..., ((ST)(Q))q_{n}} = {S(Q)T (Q)q_{1}, ..., S(Q)T (Q)q_{n}} = {S(Q)Aq_{1}, ..., S(Q)Aq_{n}} = {S(AQ)Aq_{1}, ..., S(AQ)Aq_{n}} = F_{S}(AQ) = F_{S}(T (Q)Q)) = F_{S}(F_{T} (Q)), that is, F_{ST} = F_{S} ∘ F_{T}. □

Corollary 12.3. The tame map F_{T} : C^{n}(X) → C^{n}(X) induced by an Aut X invariant morphism T: C^{n}(X) → Aut X is an automorphism of C^{n}(X).

Proof. Set S = T^{−1}, that is, S(Q) = (T (Q))^{−1} for all Q ∈ C^{n}(X). Then S is an Aut X invariant morphism and, by Lemma 12.2, F_{S} ∘ F_{T} = F_{ST} = F_{I} = Id_{C^{n}(X)} = F_{T S} = F_{T} ∘ F_{S}. □

Remark 12.4. If T satisfies T ((T (Q))^{−1}Q) = T (Q) for all Q ∈ C^{n}(X) then F_{T} is an automorphism whose inverse is F_{T^{−1}}. ◯

12.2. Some examples. Corollary 12.3 gives rise to some simple examples.

Example 12.5. Take c ∈ ℂ^{∗} and an Aff ℂ invariant regular function b: C^{n}(ℂ) → ℂ; then the map

F: C^{n}(ℂ) ∋ Q ↦ cQ + b(Q) · 1I ∈ C^{n}(ℂ) , 1I = {1, ..., 1} ∈ ℂ^{n} ,  (12.1)

is a biregular automorphism of C^{n}(ℂ). The inverse map is given by

C^{n}(ℂ) ∋ Q ↦ c′Q + b′(Q) · 1I ∈ C^{n}(ℂ) ,

where c′ = c^{−1} and b′(Q) = −c^{−1}b(Q). For instance, one may take a homogeneous polynomial P of degree rn(n − 1)/2 in n(n − 1)/2 variables ξ = {ξ_{ij} | 1 ≤ i < j ≤ n} and for each Q = {q_{1}, ..., q_{n}} ∈ C^{n}(ℂ) set ξ_{ij}(Q) = q_{i} − q_{j}, b(Q) = P (ξ(Q))/d_{n}^{r}(Q).

A similar example of a biregular automorphism of C^{n}(ℂℙ^{1}) looks as follows: Q = {q_{1}, ..., q_{n}} ↦ {Cq_{1} + B(Q), ..., Cq_{n} + B(Q)}, where C ∈ ℂ^{∗} and B: C^{n}(ℂℙ^{1}) → ℂ is a regular PSL(2, ℂ) invariant function. ◯

Example 12.6. The map

F_{T} : C^{n}(ℂ) ∋ Q ↦ cQ + λa(Q) · 1I ∈ C^{n}(ℂ) , 1I = {1, ..., 1} ∈ ℂ^{n} ,

where a(Q) = ∑_{ζ∈Q} ζ, c ∈ ℂ^{∗} and λ ∈ ℂ, λ ≠ −c/n, provides an example of a tame

biregular automorphism F_{T} of C^{n}(ℂ) corresponding to a non-invariant morphism

T: C^{n}(ℂ) ∋ Q ↦ T (Q) ∈ Aff ℂ , T (Q)w = cw + λa(Q) , w ∈ ℂ .

The inverse map is given by

C^{n}(ℂ) ∋ Q ↦ c′Q + λ′a(Q) · 1I ∈ C^{n}(ℂ) ,

where c′ = 1/c and λ′ = −λ/[c(c + nλ)]. ◯

Example 12.7. Let A(z),B(z) be homogeneous polynomials in z = (z_{0}, ..., z_{n}) ∈ ℂ^{n+1} of degrees 2a(n − 1) and 2b(n − 1) respectively (one or both of A, B may be 0). As in Remark 1.9, let Q ∈ C^{n}(ℂℙ^{1}) be the zero set of a binary projective form ϕ(x, y; z) = z_{0}x^{n} + z_{1}x_{n−1}y + ... + z_{n}y^{n} with the discriminant D_{n}(z). Set h(z) =

\frac{A(z) exp[B(z)/(D_{n}(z))^{b}]}{(D_{n}(z))^{a}} and define the holomorphic map T_{A,B}: C^{n}(ℂℙ^{1}) → SL(2, ℂ)

and the corresponding tame holomorphic endomorphism F_{T_{A,B}}: C^{n}(ℂℙ^{1}) → C^{n}(ℂℙ^{1}) by

T_{A,B}(Q) = ( \begin{matrix}h(z), & h(z) + 1 \\ h(z) − 1, & h(z) \end{matrix} ) , F_{T_{A,B}}(Q) = T_{A,B}(Q)Q .  (12.2)

It is easily seen that F_{T_{A,B}} is homotopic to the automorphism

C^{n}(ℂℙ^{1}) ∋ Q = {q_{1}, ..., q_{n}} ↦ {−q_{1}^{−1}, ..., −q_{n}^{−1}} =^{def} −Q^{−1} ∈ C^{n}(ℂℙ^{1})

in the class of holomorphic endomorphisms of C^{n}(ℂℙ^{1}). ◯

13. Certain coverings of C^{n}(X)

The main objective of this section is describe all unbranched connected k-coverings of the configuration spaces C^{n}(X) for k ≤ 2n. We will use the notation introduced in Remark 1.9.

13.1. Cyclic coverings. The following unbranched holomorphic cyclic coverings will be referred to as the standard ones:

E_{∞}^{n}(ℂ) = {(ζ, w) ∈ ℂ × G_{n} | e^{ζ} = d_{n}(w)}, p_{∞}: E_{∞}^{n}(ℂ) ∋ (ζ, w) ↦ w ∈ G_{n}; (13.1)
E_{k}^{n}(ℂ) = {(ζ, w) ∈ ℂ^{∗} × G_{n} | ζ^{k} = d_{n}(w)}, p_{k}: E_{k}^{n}(ℂ) ∋ (ζ, w) ↦ w ∈ G_{n}; (13.2)
\begin{matrix}E_{2(n−1)}^{n}(ℂℙ^{1}) = \{z = (z_{0}, ... & , z_{n}) ∈ ℂ^{n+1} | D_{n}(z) = 1\}, \\ p_{2(n−1)} : E_{2(n−1)}^{n}(ℂℙ^{1}) ∋ z ↦ ψ([x : y], z) ∈ F^{n}. \end{matrix} (13.3)

For a non-trivial divisor m of 2(n − 1) set k = 2(n − 1)/m; the cyclic subgroup ℤ/mℤ ⊂ ℤ/2(n − 1)ℤ acts freely on E_{2(n−1)}^{n}(ℂℙ^{1}) via

r(z_{0}, ..., z_{n}) =(e^{iπr/(n−1)}z_{0}, ..., e^{iπr/(n−1)}z_{n})

for all r ∈ ℤ/mℤ and all z = (z_{0}, ..., z_{n}) ∈ E_{2(n−1)}^{n}(ℂℙ^{1}) , (13.4)

and the quotient space

E_{k}^{n}(ℂℙ^{1}) = [E_{2(n−1)}^{n}(ℂℙ^{1})]/(ℤ/mℤ)  (13.5)

is an unbranched holomorphic ℤ/kℤ covering of C^{n}(ℂℙ^{1}), which is also referred to as the standard one.

Since the abelianization B_{n}(X)/B′_{n}(X) is a cyclic group, the equivalence classes of cyclic coverings over C^{n}(X) are in the one-to-one correspondence with the subgroups of B_{n}(X)/B′_{n}(X). This implies the following result.

Theorem 13.1. Every unbranched cyclic covering of C^{n}(X) is equivalent to one of the standard cyclic coverings exhibited above. □

Remark 13.2. a) The ℤ cover E_{∞}^{n}(ℂ) defined by (13.1) is in fact biholomorphic to the algebraic manifold Ẽ_{∞}^{n}(ℂ) = ℂ × {v ∈ ℂ^{n} | d_{n}(v) = 1}. The isomorphism is given by

\begin{aligned}E_{∞}^{n}(ℂ) ∋ (ζ; & w_{1}, w_{2}, ..., w_{n}) \\ & ↦ (ζ; e^{−ζ/n(n−1)}w_{1}, e^{−2ζ/n(n−1)}w_{2}, ..., e^{−nζ/n(n−1)}w_{n}) ∈ Ẽ_{∞}^{n}(ℂ) \end{aligned}

and the covering map Ẽ_{∞}^{n}(ℂ) → G_{n} = C^{n}(ℂ) is defined by

\begin{aligned}Ẽ_{∞}^{n}(ℂ) & ∋ (ζ; v_{1}, v_{2}, ..., v_{n}) \\ & ↦ t^{n} + e^{ζ/n(n−1)}v_{1}t^{n−1} + e^{2ζ/n(n−1)}v_{2}t^{n−2} + ... + e^{nζ/n(n−1)}v_{n} ∈ G_{n} . \end{aligned}

13. CERTAIN COVERINGS OF C^{n}(X) 67

Moreover, π_{1}(E_{∞}^{n}(ℂ)) = π_{1}(Ẽ_{∞}^{n}(ℂ)) ≅ B′_{n} and E_{∞}^{n}(ℂ) is the maximal abelian cover of C^{n}(ℂ).

b) If k ≡ 1 mod n(n − 1) then E_{k}^{n}(ℂ) is isomorphic to C^{n}(ℂ). The biregular isomorphism G_{n} ∋ w = (w_{1}, ..., w_{n}) ↦ (ζ, v) = (ζ; v_{1}, ..., v_{n}) ∈ E_{k}^{n}(ℂ) is given by

ζ = d_{n}(w), v_{1} = (d_{n}(w))^{m}w_{1}, v_{2} = (d_{n}(w))^{2m}w_{2}, ..., v_{n} = (d_{n}(w))^{nm}w_{n} , (13.6)

where m = (k − 1)/n(n − 1). In other words, for m ∈ ℕ and k = mn(n − 1) + 1 the regular endomorphism^{20}

\begin{aligned}p: C^{n}(ℂ) ∋ Q & = \{q_{1}, ..., q_{n}\} \\ & ↦ \{(d_{n}(Q))^{m}q_{1}, ..., (d_{n}(Q))^{m}q_{n}\} = (d_{n}(Q))^{m}Q ∈ C^{n}(ℂ) \end{aligned}  (13.7)

is an unbranched analytic cyclic covering of order k, which is equivalent to the standard cyclic ℤ/kℤ covering defined by (13.2). The induced monomorphism p_{∗}: B_{n} → B_{n} is given by σ_{i} ↦ σ_{i} · c^{m} (i = 1, ..., n − 1), where c = a^{n} = (σ_{1} ⋯ σ_{n−1})^{n} generates the center CB_{n} ≅ ℤ of the group B_{n}. The image of p_{∗} consists of all g ∈ B_{n} such that χ(g) ∈ (mn(n − 1) + 1)ℤ, where χ: B_{n} → ℤ is the abelianization epimorphism.

c) π_{1}(E_{2(n−1)}^{n}(ℂℙ^{1})) ≅ B′_{n}(S^{2}) and E_{2(n−1)}^{n}(ℂℙ^{1}) is the maximal abelian cover of C^{n}(ℂℙ^{1}). ◯

In the rest of this section, assuming that n > 6 and k ≤ 2n, we describe all connected unbranched non-cyclic k covers of C^{n}(X). The monodromy concept provides the oneto-one correspondence between the equivalence classes of such covers and the conjugacy classes of transitive non-cyclic homomorphisms B_{n}(X) → S(k), which are described below.

13.2. Transitive homomorphisms B_{n}(X) → S(k), k ≤ 2n. Artin Theorem quoted in Section 4.3 and Theorem 4.4 give the complete description of transitive homomorphisms B_{n}(X) → S(k) for n ≥ k. Here we present the case n < k ≤ 2n.

Definition 13.3. For i = 1, ..., n − 1 set

φ_{1}(σ_{i}) = \underbrace{(2i − 1, 2i + 2, 2i, 2i + 1)}_{4-cycle};

φ_{2}(σ_{i}) = (1, 2) ⋯ (2i − 3, 2i − 2)\underbrace{(2i − 1, 2i + 1)(2i, 2i + 2)}_{two transpositions}×

× (2i + 3, 2i + 4) ⋯ (2n − 1, 2n);

φ_{3}(σ_{i}) = (1, 2) ⋯ (2i − 3, 2i − 2)\underbrace{(2i − 1, 2i + 2, 2i, 2i + 1)}_{4-cycle}×

× (2i + 3, 2i + 4) ⋯ (2n − 1, 2n).

These formulae determine three non-cyclic transitive homomorphisms φ_{j}: B_{n} → S(2n) (j = 1, 2, 3). Since φ_{2}(σ_{1} ⋯ σ_{n−1}σ_{n−1} ⋯ σ_{1}) = 1, the homomorphism φ_{2} gives also rise to the homomorphism φ_{2} : B_{n}(S^{2}) → S(2n).^{21} All these homomorphisms are referred to as the standard non-cyclic transitive homomorphisms B_{n}(X) → S(2n). ◯

20Here d_{n}(Q) = d_{n}(w(Q)) denotes the discriminant of the polynomial p_{n}(t; w(Q)) = t^{n} + w_{1}(Q)t^{n−1} + ... + w_{n}(Q) = ∏_{ζ∈Q}(t − ζ) corresponding to a point Q ∈ C^{n}(ℂ).

For the braid group B_{n} the next theorem has been proved in [Lin96b] (part (b) for n > 8 only; the cases n = 7 and n = 8 were treated by S. Orevkov [Ore98]). For the sphere braid group B_{n}(S^{2}) both (a) and (b) are easy consequences of the corresponding properties of B_{n} and the presentation of B_{n}(S^{2}) exhibited in Section 4.1.

Theorem 13.4. a) Let 6 < n < k < 2n. Then every transitive homomorphism B_{n}(X) → S(k) is cyclic.

b) For n > 6 every non-cyclic transitive homomorphism B_{n}(X) → S(2n) is conjugate to one of the standard homomorphisms.

Remark 13.5. For any n ≥ 3 and any r ≥ 2 there exist non-cyclic transitive homomorphisms B_{n} → S(rn); let us present some of them.

For any l ∈ ℤ and t ∈ ℕ we denote by |l|_{t} the image of l along the natural epimorphism ℤ → ℤ/tℤ. We identify the set Δ_{n} = {1, ..., n} to the group ℤ/nℤ via {1, ..., n} ∋ m ↔ |m|_{n} − 1 and identify the symmetric groups S(n) and S(ℤ/nℤ). Furthermore, we identify the set Δ_{rn} = {1, ..., rn} to the direct product D(r, n) = (ℤ/rℤ) × (ℤ/nℤ) via

Δ_{rn} ∋ m ↦ (R(m), N(m)) ∈ (ℤ/rℤ) × (ℤ/nℤ) ,
where R(m) = |m − 1|_{r} ∈ ℤ/rℤ and N(m) = |(m − 1 − R(m))/r|_{n} ∈ ℤ/nℤ

and regard S(rn) as the symmetric group S(D(r, n)).

For each x, y ∈ ℤ/rℤ and any i = 1, ..., n − 1 let s_{i} = s_{x,y;i} ∈ S(D(r, n)) denote the permutation that acts on elements (R, N) ∈ D(r, n) = (ℤ/rℤ) × (ℤ/nℤ) as follows:

s_{i}(R, N) = \{ \begin{matrix}(R + y, N)  if N ≠ i − 1, i; \\ (R, N + 1)  if N = i − 1; \\ (R + x, N − 1) if N = i; \end{matrix}  (1 ≤ i ≤ n − 1) .

It is easily shown that the map σ_{i} → s_{x,y;i}, i = 1, ..., n−1, extends to a uniquely defined non-cyclic homomorphism φ_{x,y}: B_{n} → S(D(r, n)) = S(rn). This homomorphism φ_{x,y} is transitive if and only if the elements x, y ∈ ℤ/rℤ generate the whole group ℤ/rℤ, or, which is the same, if and only if x and y are co-prime. However, φ_{x,y} never can be primitive. For more details see [Lin96b]. ◯

13.3. Non-cyclic coverings of C^{n}(X). We start with some simple examples.

Example 13.6. The unbranched connected analytic covering

E_{n}(X) = {(ζ, Q) ∈ X × C^{n}(X) | ζ ∈ Q} , ν_{n}: E_{n}(X) ∋ (ζ, Q) ↦ Q ∈ C^{n}(X) (13.8)

is referred to as the standard non-cyclic n covering of C^{n}(X). In the case X = ℂ it may also be described as

E_{n}(ℂ) = {(ζ, w) ∈ ℂ × G_{n} | p_{n}(ζ, w) = 0} , ν_{n}: E_{n}(ℂ) ∋ (ζ, w) ↦ w ∈ G_{n} , (13.9)

21This is not the case for φ_{1} and φ_{3}.

and in the case X = ℂℙ^{1} as

\begin{matrix}E_{n}(ℂℙ^{1}) = \{([x : y], z) ∈ ℂℙ^{1} × F^{n} | φ([x : y], z) = 0\} , \\ ν_{n}: E_{n}(ℂℙ^{1}) ∋ ([x : y], z) ↦ z = [z_{0} : ... : z_{n}] ∈ F^{n} ⊂ ℂℙ^{n} . \end{matrix}  (13.10)

The monodromy representation corresponding to the covering (13.8) coincides with the standard epimorphism μ: B_{n}(X) → S(n). ◯

Example 13.7. Here is a generalization of the previous example. For m ≤ n set

\begin{matrix}E\binom{n}{m}(X) = \{(O′, Q) ∈ C^{m}(X) × C^{n}(X) | Q′ ⊆ Q\} , \\ ν\binom{n}{m}: E\binom{n}{m}(X) ∋ (Q′, Q) ↦ Q ∈ C^{n}(X) . \end{matrix}  (13.11)

This is an unbranched connected non-cyclic \binom{n}{m}-covering of C^{n}(X). Its monodromy

representation is the composition B_{n}(X) ⟶^{μ} S(n) ⟶^{τ_{m}^{n}} S(\binom{n}{m}), where τ_{m}^{n} stands for the transitive representation corresponding to the natural S(n) action on the set of all m point subsets M ⊂ {1, ..., n}. ◯

Example 13.8. Set p′_{n}(t, w) = nt^{n−1} +(n−1)w_{1}t^{n−2} +...+w_{n−1}. Define the following non-trivial unbranched analytic 2 coverings of the space E_{n}(ℂ):

\begin{matrix}ν^{(1)} : E_{n}^{(1)}(ℂ) = \{(ξ; (ζ, w)) ∈ ℂ^{∗} × E_{n}(ℂ)| ξ^{2} = p′_{n}(ζ, w)\} → E_{n}(ℂ), \\ ν^{(2)} : E_{n}^{(2)}(ℂ) = \{(ξ; (ζ, w)) ∈ ℂ^{∗} × E_{n}(ℂ)| ξ^{2} = d_{n}(w)\} → E_{n}(ℂ), \\ ν^{(3)} : E_{n}^{(3)}(ℂ) = \{(ξ; (ζ, w)) ∈ ℂ^{∗} × E_{n}(ℂ)| ξ^{2} = d_{n}(w)p′_{n}(ζ, w)\} → E_{n}(ℂ), \end{matrix} (13.12)

all with the same natural projection (ξ; (ζ, w)) ↦ (ζ, w). Any non-trivial 2 covering of E_{n}(ℂ) is equivalent to one of the coverings ν^{(j)} : E_{n}^{(j)}(ℂ) → E_{n}(ℂ), j = 1, 2, 3 (see [Lin96b] for the proof; we will not use this fact in what follows).

Let us construct a non-trivial unbranched analytic 2 covering of the space E_{n}(ℂℙ^{1}). To this end, take the standard ℤ/2n(n − 1)ℤ covering p_{2(n−1)} : E_{2(n−1)}^{n}(ℂℙ^{1}) → F^{n} described in (13.3). The subgroup ℤ/n(n − 1)ℤ ⊂ ℤ/2n(n − 1)ℤ acts freely on E_{2(n−1)}^{n}(ℂℙ^{1}) by the transformations r(z_{0}, ..., z_{n}) = (e^{iπr/(n−1)}z_{0}, ..., e^{iπr/(n−1)}z_{n}), where r ∈ ℤ/n(n − 1)ℤ and z = (z_{0}, ..., z_{n}) ∈ E_{2(n−1)}^{n}(ℂℙ^{1}). This provides us with the nontrivial unbranched analytic 2 covering

λ: E_{2}^{n}(ℂℙ^{1}) = [E_{2(n−1)}^{n}(ℂℙ^{1})]/(ℤ/n(n − 1)ℤ) → F^{n} = C^{n}(ℂℙ^{1}) .  (13.13)

The desired 2-covering

ν^{(2)} = ν_{n}^{∗}(λ): E_{n}^{(2)}(ℂℙ^{1}) → E_{n}(ℂℙ^{1})  (13.14)

is obtained as the pullback ν^{(2)} = ν_{n}^{∗}(λ) of the covering (13.13) along the mapping ν_{n}: E_{n}(ℂℙ^{1}) → C^{n}(ℂℙ^{1}) defined by (13.10).

The 2-coverings ν^{(j)} : E_{n}^{(j)}(ℂ) → E_{n}(ℂ) (j = 1, 2, 3) and ν^{(2)} : E_{n}^{(2)}(ℂℙ^{1}) → E_{n}(ℂℙ^{1}) defined by (13.8) and (13.14) are referred to as the standard 2-coverings of E_{n}(X). ◯ Example 13.9. We define 2n-coverings μ_{2n}^{(j)}(X): E_{n}^{(j)}(X) → C_{n}(X) as compositions of the 2-coverings ν^{(j)} : E_{n}^{(j)}(X) → E_{n}(X) and the n-covering ν_{n}: E_{n}(X) → C_{n}(X) exhibited in Examples 13.12 and 13.6 respectively. That is,

\begin{matrix}μ_{2n}^{(j)}(ℂ) = ν_{n} ∘ ν^{(j)} : E_{n}^{(j)}(ℂ) ⟶^{ν^{(j)}} E_{n}(ℂ) ⟶^{ν_{n}} C_{n}(ℂ) , j = 1, 2, 3 , \\ μ_{2n}^{(2)}(ℂℙ^{1}) = ν_{n} ∘ ν^{(2)} : E_{n}^{(2)}(ℂℙ^{1}) ⟶^{ν^{(2)}} E_{n}(ℂℙ^{1}) ⟶^{ν_{n}} C_{n}(ℂℙ^{1}) . \end{matrix}  (13.15)

These covering are referred to as the standard non-cyclic 2n-coverings of C_{n}(X).

The monodromy representations φ_{j} = μ_{2n}^{(j)∗}: B_{n}(X) → S(2n) corresponding to the above 2n-coverings of C_{n}(X) are conjugate to the eponymous standard non-cyclic transitive homomorphisms φ_{j}: B_{n}(X) → S(2n) introduced in Definition 13.3. ◯

The description of transitive homomorphisms B_{n}(X) → S(k) provided by Theorem 4.4, Artin Theorem (quoted in Section 4.3) and Theorem 13.4, together with Theorem 13.1 and the latter example, leads the following complete description of unbranched connected k coverings of C^{n}(X) for k ≤ 2n.

Theorem 13.10. Let E = {p: E → C^{n}(X)} be an unbranched connected k-covering.

a) If n > max{k, 4} then E is equivalent to one of the cyclic k-coverings exhibited in Section 13.1.

b) If 6 < n ≤ k ≤ 2n then either E is cyclic and then equivalent to one of the cyclic k-coverings exhibited in Section 13.1 or k = n and E is equivalent to the standard non-cyclic n-covering defined by (13.8) or k = 2n and E is equivalent to one of the standard non-cyclic 2n-coverings exhibited in Example 13.9. In the latter case E splits to the composition of a 2-covering and the standard non-cyclic n-covering. □

14. Configuration spaces of small dimension and some other examples

Configuration spaces of small dimension admit certain exceptional morphisms prohibited by results stated above. In this section we present some of such examples. We will also describe the Eisenstein automorphism mentioned in Section 1.6.

14.1. Ferrari map. The Ferrari map (L. Ferrari, 1545), which carries each polynomial of degree 4 to its cubic resolvent, provides the oldest and undoubtedly the most famous example of a non-cyclic morphism of configuration spaces. This morphism F e: C^{4}(ℂ) → C^{3}(ℂ) arises as follows. For every four distinct q_{1}, q_{2}, q_{3}, q_{4} ∈ ℂ the points z_{1} = (q_{1}−q_{2}−q_{3}+q_{4})^{2}/4, z_{2} = (q_{1}−q_{2}+q_{3}−q_{4})^{2}/4, z_{3} = (q_{1}+q_{2}−q_{3}−q_{4})^{2}/4 are distinct and a permutation of q_{1}, q_{2}, q_{3}, q_{4} leads to a permutation of z_{1}, z_{2}, z_{3}. Thus, we obtain polynomial maps fe: C_{o}^{4}(ℂ) → C_{o}^{3}(ℂ) and F e: C^{4}(ℂ) → C^{3}(ℂ). The morphism F e induces the epimorphism of braid groups F e_{∗}: B_{4} → B_{3}, which carries both σ_{1},σ_{3} to σ_{1} and keeps σ_{2}.

14.2. Disjoint morphisms C^{3}(ℂ) → C^{6}(ℂ) and C^{3}(ℂ) → C^{9}(ℂℙ^{1}). For distinct q_{1}, q_{2}, q_{3} ∈ ℂ define six points μ_{1}^{±}, μ_{2}^{±}, μ_{3}^{±} ∈ ℂ by the quadratic equations

(μ_{1}^{±}−q_{1})^{2} = (q_{1}−q_{2})(q_{1}−q_{3}), (μ_{2}^{±}−q_{2})^{2} = (q_{2}−q_{3})(q_{2}−q_{1}), (μ_{3}^{±}−q_{3})^{2} = (q_{3}−q_{1})(q_{3}−q_{2}).

14. CONFIGURATION SPACES OF SMALL DIMENSION AND SOME OTHER EXAMPLES 71

All nine points q_{i}, μ_{j}^{±}, 1 ≤ i, j ≤ 3, are distinct and a permutation of q’s leads to a permutation of μ’s. Thus, we obtain a disjoint polynomial map

L: C^{3}(ℂ) ∋ {q_{1}, q_{2}, q_{3}} ↦ {μ_{1}^{+}, μ_{1}^{−}, μ_{2}^{+}, μ_{2}^{−}, μ_{3}^{+}, μ_{3}^{−}} ∈ C^{6}(ℂ) ,

which may also be viewed as a polynomial map L: G_{3} → G_{6} that carries any polynomial p_{3}(t, z) = t^{3} + z_{1}t^{2} + z_{2}t + z_{3} ∈ G_{3} to the polynomial p_{6}(t, L(z)) ∈ G_{6}, which is co-prime to p_{3}(t, z) and whose coefficients L_{1}(z), …, L_{6}(z) are as follows:

\begin{aligned}L_{1}(z) & = 2z_{1}; \\ L_{2}(z) & = 5z_{2}; \\ L_{3}(z) & = 20z_{3}; \end{aligned}  \begin{aligned}L_{4}(z) & = 20z_{1}z_{3} − 5z_{2}^{2}; \\ L_{5}(z) & = 8z_{1}^{2}z_{3} − 2z_{1}z_{2}^{2} − 4z_{2}z_{3}; \\ L_{6}(z) & = 4z_{1}z_{2}z_{3} − z_{2}^{3} − 8z_{3}^{2}. \end{aligned}

The first three of these formulae show that L is an embedding; one can check that it is non-cyclic and that the following relation between the discriminants holds:

d_{6}(L(z)) = −4^{9} · [d_{3}(z)]^{5}.

The following simple geometric interpretation of the map L due to E. A. Gorin (personal communication). Let Δ(Q) ⊂ ℂ be the triangle with the vertices at the points q_{1}, q_{2}, q_{3} (it may degenerate to a straightline segment with a distinguished interior point). For each vertex A = q_{i}, let b and c denote the lengths of the sides of Δ(Q) intersecting at A, and let ℓ(A) be the bisector line of the inner angle of Δ(Q) at A. Take two points on ℓ(A) which are \sqrt{bc}-distant from A. Thus, we obtain six points, which coincide with the six points μ_{i}^{±} defined above (see the picture below).

\sqrt{ac}  ^{\sqrt{bc}} c^{\sqrt{ab}}  b a  μ_{1}^{−} q_{3}  q_{1} q_{2} μ_{2}^{+}  μ_{3}^{+} μ_{1}^{+}  μ_{2}^{−} μ_{3}^{−}  ^{•}^{•}^{•} ^{•}  ^{•}^{•}

On the other hand, Yoel Feler has recently noticed that the same morphism L may be defined via the formulae

(\frac{μ_{1}^{±} − q_{2}}{μ_{1}^{±} − q_{3}} )^{2} = \frac{q_{1} − q_{2}}{q_{1} − q_{3}} , (\frac{μ_{2}^{±} − q_{3}}{μ_{2}^{±} − q_{1}} )^{2} = \frac{q_{2} − q_{3}}{q_{2} − q_{1}} , (\frac{μ_{3}^{±} − q_{1}}{μ_{3}^{±} − q_{2}} )^{2} = \frac{q_{3} − q_{1}}{q_{3} − q_{2}} .

This observation let him to construct a disjoint^{22} morphism

F: C^{3}(ℂ) ∋ {q_{1}, q_{2}, q_{3}} ↦ {μ_{1}^{(1)}, μ_{1}^{(2)}, μ_{1}^{(3)}, μ_{2}^{(1)}, μ_{2}^{(2)}, μ_{2}^{(3)}, μ_{3}^{(1)}, μ_{3}^{(2)}, μ_{3}^{(3)}} ∈ C^{9}(ℂℙ^{1}) ,

where μ_{i}^{(j)} (1 ≤ i, j ≤ 3) are defined by the cubic equations

(\frac{μ_{1}^{(j)} − q_{2}}{μ_{1}^{(j)} − q_{3}} )^{3} = (\frac{q_{1} − q_{2}}{q_{1} − q_{3}} )^{2} ,

(\frac{μ_{2}^{(j)} − q_{3}}{μ_{2}^{(j)} − q_{1}} )^{3} = (\frac{q_{2} − q_{3}}{q_{2} − q_{1}} )^{2} ,

(\frac{μ_{3}^{(j)} − q_{1}}{μ_{3}^{(j)} − q_{2}} )^{3} = (\frac{q_{3} − q_{1}}{q_{3} − q_{2}} )^{2} .

In terms of polynomials p(t, q) = (t − q_{1})(t − q_{2})(t − q_{3}) ∈ G_{3} and binary projective forms ϕ([x : y], F (q)) ∈ F^{9}, the map F may be defined by the formula

ϕ([x : y], F (q)) =((x − yq_{1})^{3}(q_{2} − q_{3})^{2} − (x − yq_{2})^{3}(q_{3} − q_{1})^{2})
× ((x − yq_{2})^{3}(q_{3} − q_{1})^{2} − (x − yq_{3})^{3}(q_{1} − q_{2})^{2})
× ((x − yq_{3})^{3}(q_{1} − q_{2})^{2} − (x − yq_{1})^{3}(q_{2} − q_{3})^{2}) .

A straightforward computation of the discriminant shows that

D_{ϕ} = 3^{27}[(q_{1} − q_{2})(q_{2} − q_{3})(q_{3} − q_{1})]^{56} .

14.3. Eisenstein automorphism. Here we present three different constructions of the Eisenstein automorphism: the original one [Eis1844], the Cayley construction (1845) and also the new one in the form of a tame morphism C^{3}(ℂℙ^{1}) → C^{3}(ℂℙ^{1}) found by Yoel Feler in 2001.

The discriminant D_{ϕ} of a binary cubic form ϕ(x, y) = z_{0}x^{3} + 3z_{1}x^{2}y + 3z_{2}xy^{2} + z_{3}y^{3} is given by

D_{ϕ} = z_{0}^{2}z_{3}^{2} − 3z_{1}^{2}z_{2}^{2} − 6z_{0}z_{1}z_{2}z_{3} + 4z_{0}z_{2}^{3} + 4z_{1}^{3}z_{3} .

For every binary cubic form ϕ, Eisenstein considers the binary cubic form (Eϕ)(x, y) = w_{0}x^{3}+3w_{1}x^{2}y+3w_{2}xy^{2}+w_{3}y^{3} whose coefficients w_{i} are homogeneous cubic polynomials in z_{0}, ..., z_{3} defined by^{23}

22Meaning that each set Q ∈ C^{3}(ℂ) is disjoint to its image F (Q) ∈ C^{9}(ℂℙ^{1}).

\begin{aligned}w_{0} & = \frac{1}{2} \frac{∂D_{ϕ}}{∂z_{3}} = 2z_{1}^{3} − 3z_{0}z_{1}z_{2} + z_{0}^{2}z_{3} , w_{1} = \frac{1}{6} \frac{∂D_{ϕ}}{∂z_{2}} = 2z_{0}z_{2}^{2} − z_{0}z_{1}z_{3} − z_{1}^{2}z_{2} , \\ w_{2} & = \frac{1}{6} \frac{∂D_{ϕ}}{∂z_{1}} = 2z_{1}^{2}z_{3} − z_{0}z_{2}z_{3} + z_{1}z_{2}^{2} , w_{3} = \frac{1}{2} \frac{∂D_{ϕ}}{∂z_{0}} = 2z_{2}^{3} − 3z_{1}z_{2}z_{3} + z_{0}z_{3}^{2} . \end{aligned}

He proves the identities E(Eϕ) = (D_{ϕ})^{2} · ϕ and

\begin{aligned}D_{Eϕ} & = w_{0}^{2}w_{3}^{2} − 3w_{1}^{2}w_{2}^{2} − 6w_{0}w_{1}w_{2}w_{3} + 4w_{0}w_{2}^{3} + 4w_{1}^{3}w_{3} \\ & = (z_{0}^{2}z_{3}^{2} − 3z_{1}^{2}z_{2}^{2} − 6z_{0}z_{1}z_{2}z_{3} + 4z_{0}z_{2}^{3} + 4z_{1}^{3}z_{3})^{3} = (D_{ϕ})^{3} . \end{aligned}

Thus Eisenstein’s correspondence f ↦ Eϕ determines a birational involution of the space ℂℙ^{3} of all projective cubic binary forms. Moreover, its restriction to the space F_{3} of non-degenerate cubic forms is an involutive biregular automorphism of the latter space:

\begin{matrix}F_{3} \\ _{_{E} _{≅}} & [figure: id] \\ F_{3}  \overline{}^{≅}_{E} F_{3} . \end{matrix}  (14.1)

14.4. Cayley’s form of the Eisenstein automorphism. Take a complex binary cubic form^{24}

ϕ(x, y) = z_{0}x^{3} + 3z_{1}x^{2}y + 3z_{2}xy^{2} + z_{3}y^{3} .

Its Hessian

ψ(x, y) = det (\begin{matrix}∂^{2}ϕ/∂x^{2} , & ∂^{2}ϕ/∂x∂y \\ ∂^{2}ϕ/∂y∂x , & ∂^{2}ϕ/∂y^{2} \end{matrix})

is a binary quadratic form. The Jacobian

J(x, y) = det (\begin{matrix}∂ϕ/∂x , & ∂ϕ/∂y \\ ∂ψ/∂x , & ∂ψ/∂y \end{matrix})

is a binary cubic form.

Eisenstein’s map: E: ϕ ↦ Eϕ = J.

14.5. Feler’s form of the Eisenstein automorphism. For any Q ∈ C^{3}(ℂℙ^{1}) let z(Q) = [z_{0}(Q) : z_{1}(Q) : z_{2}(Q) : z_{3}(Q)] denote the coefficients of the projective binary cubic form ϕ = ϕ([x : y], Q) whose zero set

Z_{ϕ} = {[x : y] ∈ ℂℙ^{1} | ϕ([x : y]) = 0}

coincides with Q, and let D_{ϕ} denote the discriminant of ϕ. The Eisenstein’s automorphism of C^{3}(ℂℙ^{1}) coincides with the tame map

F_{T} : C^{3}(ℂℙ^{1}) ∋ Q ↦ T (Q)Q ∈ C^{3}(ℂℙ^{1})

23Eisenstein has not pointed out that the discriminant D_{ϕ} plays the part of a potential function for the coefficients of the form Eϕ; I have learned this fact from M. Gizatullin.

24There is a minor difference between the automorphism described here and the original Eisenstein form: one has just to interchange x and y in one of the examples.

74 Vladimir Lin that corresponds to the morphism T: C^{3}(ℂℙ^{1}) → PSL(2, ℂ), where

T (Q): ℂℙ^{1} ∋ ζ ↦ \frac{a(Q)ζ + b(Q)}{c(Q)ζ + d(Q)} ∈ ℂℙ^{1}  (14.2)

is the Möbius transformation with the coefficients

\begin{matrix}a(Q) = (z_{1}z_{2} − z_{0}z_{3})/\sqrt{−D_{ϕ}}; \\ b(Q) = 2(z_{2}^{2} − z_{1}z_{3})/\sqrt{−D_{ϕ}}; \\ c(Q) = 2(z_{0}z_{2} − z_{1}^{2})/\sqrt{−D_{ϕ}}; \\ d(Q) = −a(Q) = −(z_{1}z_{2} − z_{0}z_{3})/\sqrt{−D_{ϕ}} . \end{matrix}  (14.3)

It is easily seen that T (Q) ∈ PSL(2, ℂ) is well defined by (14.2) and (14.3).

Bibliography

[Arn70a] V. I. Arnold, Cohomology classes of algebraic functions invariant under Tschirnhausen transformation, Funct. Anal. Appl. 4, n. 1 (1970), 74–75.

[Arn70b] V. I. Arnold, On certain topological invariants of algebraic functions, Trudy Mosk. Mat. Obshch. 21 (1970), 27–46 (in Russian). English translation:

[Arn70c] V. I. Arnold, Topological invariants of algebraic functions, II, Funkt. Analiz i Priloz. 4, no. 2 (1970), 1–9 (in Russian). English translation: Funct. Anal. Appl. 4, no. 2 (1970), 91–98.

[Art25] E. Artin, Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg 4 (1925), 47–72.

[Art47a] E. Artin, Theory of braids, Annals of Math. 48, no. 1 (1947), 101–126.

[Art47b] E. Artin, Braids and permutations, Ann. of Math. 48, no. 3 (1947), 643–649.

[Bir74] J. S. Birman, Braids, links and mapping class groups. Annals of Math. Studies 82. Princeton Univercity Press, Princeton, N. J., 1974.

[BorNar67] A. Borel, R. Narasimhan, Uniqueness conditions for certain holomorphic mappings, Invent. Math. 2 (1967), 247–255.

[Bohn47] F. Bohnenblust, The algebraic braid group, Annals of Math. 48, no. 1 (1947), 127–136.

[Bro76] F. E. Browder, Problems of Present Day Mathematics, Mathematical Developments Arising from Hilbert Problems, Proc. Symp. Pure Math. 28 (F. E. Browder, ed.) Amer. Math. Soc., 1976, 35–80.

[Bur32] W. Burau, Über Zopfinvarianten, Abh. Math. Sem. Univ. Hamburg 9 (1932), 117–124.

[Buh96] J. Buhler, Z. Reichstein On the essential dimension of a finite group, Composito Math. 106, no. 2 (1997), 159–179.

[Car40] H. Cartan, Sur les matrices holomorphes de n variables complexes, J. Math. Pures et Appl. 19 (1940), 1–26.

[Chow48] W.-L. Chow, On the algebraic braid group, Annals of Math. 49, no. 3 (1948), 654–658.

[DanGiz72] V. I. Danilov and M. H. Gizatullin, Automorphisms of affine surfaces II, Math. USSR Izvestija 41, no. 2 (1977),

[Del72] P. Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17, no. 4 (1972),273–302.

[DyeGro81] J. Dyer and E. Grossman, The automorphism groups of the braid groups, Amer. J. Math. 103, no. 6 (1981), 1151–1169.

[Eis1844] G. Eisenstein, Über eine merkwürdige identische Gleichung, Journal für die reine und angewandte Mathematik 27 (1844), 105–106 (see also: Gotthold Eisenstein, Mathematische Werke, Band I, Chelsea, New York, 1975,1989, pp. 26–27).

[EarKra74] C. J. Earle and I. Kra On holomorphic mappings between Teichmüller spaces, in book: Contributions to Analysis. A collection of papers dedicated to L. Bers. L. V. Ahlfors etc., ed., Academic Press, New-York 1974, 107–124.

[EarKra76] C. J. Earle and I. Kra On selections of some families of closed Riemann surfaces, Acta Math. 137 (1976), 49–79

[Enr23] Sulla costrucione delle funcioni algebriche di due variabili possedenti una data curva di diramazione, Annali di matematica pure ed applicata, Ser. 4, 1 (1923), 185–198.

[FadNeu62] E. Fadell and L. Neuwirth, On Configuration spaces, Math. Scand. 10, no. 1 (1962), 111–118.

[FadHus01] E. R. Fadell and S. Y. Husseini, Geometry and topology of configuration spaces. Springer, 2001.

[FeiZie99] E. M. Feichtner, M. Ziegler, The integral cohomology algebras of ordered configuration spaces of spheres, Documenta Math. 5 (2000), 115-139.

[Fin45] P. Finsler, Über die Primzahlen zwischen n und 2n. Speiser-Festschrift, Orell–Füssli, Zürich, 1945.

[Fo51] L. R. Ford, Automorphic Functions, Chelsea, 1951.

[FoxNeu62] R. H. Fox and L. Neuwirth, The braid groups, Math. Scand. 10, no. 1 (1962), 119–126.

[Fuks70] D. B. Fuks, Cohomologies of a braid group mod 2, Funct. Anal. Appl. 4, no. 2 (1970), 91–98.

[Gar] F. Garside, The braid group and other groups, Quart. J. Math. Oxford 20, n. 78 (1969), 235–254.

[Gol74] D. L. Goldsmith, Homotopy of braids in answer to a question of E. Artin, Topology Conference, Lect. Notes Math. 375 (R. F. Dickman, Jr. and P. Fletcher, eds.), Springer, 1974, 91–96.

[GorLin69] E. A. Gorin and V. Ya. Lin, Algebraic equations with continuous coefficients and some problems of the algebraic theory of braids, Matem. Sbornik 78 (120), no. 4 (1969), 579– 610 (in Russian). English translation: Math. USSR Sbornik 7 (1969), 569–596.

[GorLin74] E. A. Gorin and V. Ya. Lin, On separable polynomials over commutative Banach algebras, Dokl. Akad. Nauk SSSR 218, no. 3 (1974), 505–508 (in Russian). English translation: Soviet Math. Dokl. 15, no. 5 (1974), 1357–1361.

[Gra58] H. Grauert, Analytische Faserungen über holomorphvollständigen Räumen, Math. Ann. 135 (1958), 263–273.

[Hall] M. Hall, The theory of groups. McMillan Co., New York, 1959.

[Hub72] J. H. Hubbard, Sur la non-existence de sections analytiques à la courbe universelle de Teichmüller, C.R. Acad. Sci. Paris Sér. A-B 274 (1972), A978–A979.

[Hub76] J. H. Hubbard, Sur les sections analytiques de la courbe universelle de Teichmüller, Mem. Amer. Math. Soc. 4 (1976), no.166, ix+137pp.

[Kal75] S. I. Kaliman, Holomorphic universal covering of the space of polynomials without multiple roots, Funkt. Analiz i Priloz. 9, no. 1 (1975), 71 (in Russian). English translation: Funct. Anal. Appl. 9, no. 1 (1975), 67–68.

[Kal76] S. I. Kaliman, Holomorphic endomorphisms of the manifold of complex polynomials with discriminant 1, Uspehi Mat. Nauk 31, no. 1 (1976), 251–252 (in Russian).

[Kal93] Sh. I. Kaliman The holomorphic universal covers of the spaces of polynomials without multiple roots, Selected translations. Selecta Math. Soviet. 12, no. 4 (1993), 395–405.

[Kle] F. Klein, Vorlesungen über das Ikosaeder und die Aufl” osung der Gleichungen vom f” unften Grade, Birkhäuser Verlag & B. G. Teubner Verlagsgesellschaft, 1993.

[Kurth97] A. Kurth, SL_{2}-equivariant polynomial automorphisms of binary forms, Ann. Inst. Fourier, Grenoble 47, no. 2 (1997), 585–597.

[Lin71] V. Ya. Lin, Algebroidal functions and holomorphic elements of homotopy groups of complex manifolds, Dokl. Akad. Nauk SSSR 201:1 (1971), 28–31 (in Russian). English translation: Soviet Math. Dokl. 12, n. 6 (1971), 1608–1612.

[Lin72a] V. Ya. Lin, On representations of the braid group by permutations, Uspehi Matem. Nauk 27, n. 3 (1972), 192.

[Lin72b] V. Ya. Lin, Algebraic functions with universal discriminant manifolds, Funkt. Analiz i Priloz. 6, no. 1 (1972), 81–82 (in Russian). English translation: Funct. Anal. Appl. 6, n. 1 (1972), 73–75.

[Lin72c] V. Ya. Lin, On superpositions of algebraic functions, Funkt. Analiz i Priloz. 6, no. 3 (1972), 77–78 (in Russian). English translation: Funct. Anal. Appl. 6, n. 3 (1972), 240– 241.

[Lin74] V. Ya. Lin, Representations of braids by permutations, Uspehi Mat. Nauk 29, n. 1 (1974), 173–174 (in Russian).

[Lin76] V. Ya. Lin, Superpositions of algebraic functions, Funkt. Analiz i Priloz. 10, no. 1 (1976), 37–45 (in Russian). English translation: Funct. Anal. Appl. 10, n. 1 (1976), 32–38.

[Lin79] V. Ya. Lin, Artin braids and the groups and spaces connected with them, Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya 17, VINITI, Moscow, 1979, pp. 159–227 (in Russian). English translation: Journal of Soviet Math. 18 (1982), 736–788.

[Lin88] V. Ya. Lin, Liouville coverings of complex spaces and amenable groups, Matem. Sbornik 132 (174), no. 2 (1987), 202–224 (in Russian). English translation: Math. USSR Sbornik 60, n. 1 (1988), 197–216.

[Lin96a] V. Ya. Lin, Around the 13th Hilbert problem for algebraic functions, Israel Math. Conference Proc. 9 (1996), 307–327.

[Lin96b] V. Ya. Lin, Braids, permutations, polynomials–I, Preprint MPI 96-118, Max-Planck-Institut für Mathematik in Bonn, August 1996, 112pp.

[LinZai98] V. Ya. Lin and M. Zaidenberg, Liouville and Carathéodory coverings in Riemannian and complex geometry in book: Voronezh Winter Mathematical School. P. Kuchment & V. Lin eds., Amer. Math. Soc. Transl. (2), 184 (1998), 111–130.

[Mal40] A. I. Malcev, On an isomorphic representation of infinite groups of matrices, Matemat. Sbornik 4 (1940), 405–422 (in Russian).

[Mar45] A. A. Markov, Foundations of the algebraic theory of tresses, Proc. Steklov Math. Institute 16, 1945 (in Russian).

[Mur82] K.Murasugi, Seifert fiber spaces and braid groups, Proc. London Math. Soc. 44 (1982), 71–84.

[MurKur99] K. Murasugi, B. I. Kurpita, A study of braids. Kluwer, 1999.

[HNeum] H. Neumann, Varieties of groups. Springer, 1967.

[Ore98] S. Orevkov, Ttransitive homomorphisms B_{7} → S(14) and B_{8} → S(16), private letter, October 4, 1998.

[Rams65] K. J. Ramspott, Stetige und holomorphe Schnitte in Bündeln mit homogener Faser, Math. Z. 89, no. 3 (1965), 234–246.

[Trost] E. Trost, Primzahlen. Birkhäuser Verlag, Basel–Stuttgart, 1968.

[Vas88] V. A. Vassiliev, Braid group cohomologies and algorithm complexity, Funct. Anal. Appl. 22, no. 3 (1988), 182–189.

[Vas98] V. A. Vassiliev, Complements of Discriminants of Smooth Maps: Topology and Applications. Translations of Mathematical Monographs 98, Amer. Math. Soc., Providence, Rhode Island, 1992.

[Wi64] H. Wielandt, Finite permutation groups. Academic Press, 1964.

[Zar29] O. Zariski, On the problem of existence of algebraic functions of two variables possessing a given branch curve, Amer. J. Math. 51 (1929).

[Zar36] O. Zariski, On the Poincaré group of rational plane curves, Amer. J. Math. 58 (1936), 607–619.

[Zar37a] O. Zariski, The topological discriminant group of a Riemann surface of genus p, Amer. J. Math. 59 (1937), 335–358.

[Zar37b] O. Zariski, A theorem on the Poincaré group of an algebraic hypersurface, Ann. Math. 38 (1937).

[Zin77] V. Zinde, Holomorphic maps of regular orbit spaces of Coxeter groups of series B and D, Sibirsk. Math. J. 18, no. 5 (1977), 1016–1026 (in Russian). English translation:

Index

hol B_{m} 55 t(X) 11 G_{m}^{0} 49 S(n) action in L(q_{n}) 37 S(n) orbits of simplices of cross ratios 38 S(n) orbits of simplices of simple ratios 38 C_{b}^{m−1}(ℂ) 48, 57 C_{b}^{m−1}(ℂ)) 49 D(w) 49

Algebraic functions with the universal branch

locus 60 Artin Lemma 27 Automorphisms of C^{n}(X) 64

Balanced configuration space C_{b}^{m−1}(ℂ) 48, 57 Binary form 10

discriminant of 10

non-degenerate 10

projective 10 Braid groups 7

Artin braid group 7

pure Artin braid group 7 canonical presentations 25 sphere braid group 7

pure sphere braid group 7

Compositions of multivalued functions 61 Configuration spaces

holomorphic universal covering of 20, 23 ordered 7 unordered 7

Cyclic morphisms C^{n}(ℂ) → C^{k}(ℂ) 52

Dimension of L_{△}(C_{o}^{n}(ℂ)) 37 Dimension of L_{△}(C_{o}^{n}(ℂℙ^{1})) 37

Eisenstein 11 Eisenstein’s automorphism

Cayley’s form 73

Feler’s form 73

original form 72 Endomorphism tame 8

Equivariance condition 14

strict 15 Example

biregular automorphisms generated by in-

variant morphisms 65 biregular automorphisms generated by non-

invariant morphisms 65

Finite covers of Liouville spaces 45

Gorin’s relation 26

Holomorphic elements of π_{i}(C^{m}(ℂ)) 55

Holomorphic elements of homotopy groups 54 Holomorphic functions omitting two values

− − − − −, on C_{o}^{n}(X) 18

− − − − −, simplicial complex of 18, 19, 33 Holomorphic maps

−− of ultra-Picard space to C^{m}(X) 46 Solvable −− of Liouville space to C^{m}(X) 46 Holomorphic maps of ultra-Picard space to G_{m}

50 Holomorphic part hol π_{i}(Y) of the homotopy

group π_{i}(Y) 54 Holomorphic part hol B_{m} of B_{m} 55 Homomorphism abelian 8 cyclic 8 solvable 8 transitive 26

Image of non-cyclic endomorphism of C^{n}(ℂ) 58 Invariance property 13 Invariant maps to Aut X 64

Lifting property 12

Liouville and Picard spaces 45

Map abelian 8 cyclic 8 disjoint 11 linked 11 solvable 8

Mixed simplices 37 Model holomorphic maps ℂ^{∗} → C^{m}(ℂ) 51 Morphism

degenerate tame 8

invariant 16

orbit-like 8

Normal form of simplices 38

Parameterization of ℂ^{∗} orbits in G_{m}^{0} 50 Proper divisors 34

Simple and cross ratios 34 Simplices of cross ratios 35 Simplices of simple ratios 35 Simplicial complex of holomorphic functions

omitting two values 33 Simplicial complexes of simple and cross ratios

34 Solvable holomorphic maps of Liouville space

to G_{m} 50 Special homomorphisms B_{m} → B_{k} 53 Special presentations of B_{m} 53 Special system of generators in B_{m} 53 Standard normal series for P B_{n}(X) 23 Standard system of generators in B_{m} 53

Tame algebraic ensembles 61 Tame endomorphism 8

of C_{o}^{n}(X) 16

degenerate 16 Teichmüller space 21 Theorem

Artin Theorem 27

Artin-Markov-Fadell-Fox-Neuwirth 24 Coherence Theorem 17

−−, proof of, 41, 42 Explanation 17

Commutator Theorem 13

−−, proof of, 26 Composition Theorem 13

−−, proof of, 28 Equivariance Theorem 15

−−, proof of, 28 Explanation 15

Equivariant Map Theorem 16 − − −, proof of, 17 Homomorphisms B_{n} → B_{k} induced by holo-

morphic maps C^{n}(ℂ) → C^{k}(ℂ) 54 − − − − −−, proof of, 56 Hubbard-Earl-Kra Theorem 42 Invariance Theorem 13 −−, proof of, 29 − − −− Explanation of 13

invariant morphisms to Aut X and automor-

phisms of C^{n}(X) 65 Kernel Theorem 13 −−, proof of, 28 Lifting Theorem 13 −−, proof of, 29

Linked Map Theorem 11, 42 − − − for C_{o}^{m}(ℂ^{◯ 2}) 43 − − − for C_{o}^{n}(X) 45 proof of − − − for C_{o}^{m}(ℂ^{◯ 2}) 44 proof of − − − for C_{o}^{n}(X) 45

Murasugi Theorem 52

on n-valued algebraic functions with the uni-

versal branch locus 60 on algebraic functions containing the univer-

sal one 60 on compositions without superfluous branch

points 63, 64 on non-cyclic special homomorphisms B_{n} →

B_{k} 56 Surjectivity Theorem 14 −−, proof of, 26 Tame Map Theorem 9 − − −, proof of, 42 strategy of the proof, 12 Torsion of B_{m}/CB_{m} and B_{m}(S^{2}) 52

Ultra-Picard spaces 45

Universal Teichmüller family (curve) 21