A pr 2 00 4 Configuration spaces of C and CP 1 : some analytic properties

I mean... You know... We study certain analytic properties of the ordered and unordered configuration spaces Cn o (X) = {(q1, ..., qn) ∈ X | qi 6= qj ∀ i 6= j} and Cn(X) = {Q ⊂ X | #Q = n} of simply connected algebraic curves X = C and X = CP. For n ≥ 3 the braid group Bn(X) = π1(C(X)) of the curve X is non-abelian. A map F : Cn(X) → Ck(X) is called cyclic if the subgroup F∗(π1(C(X))) ⊂ π1(C(X)) is cyclic; otherwise F is called non-cyclic. The diagonal AutX action in X induces AutX actions in Cn o (X) and Cn(X). A morphism (holomorphic or regular) F : Cn(X) → Ck(X) is called orbit-like if F (Cn(X)) ⊂ (AutX)Q for some Q ∈ Ck(X). An endomorphism F of Cn(X) is called tame if there is a morphism T : Cn(X) → AutX such that F (Q) = T (Q)Q for all Q ∈ Cn(X). Tame endomorphisms preserve each AutX orbit in Cn(X). We prove that for n > 4 and k ≥ t(X) := dimC AutX an endomorphism F of Cn(X) is tame if and only if it is non-cyclic, and a morphism F : Cn(X) → Ck(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 Bn(X), including the stability of the pure braid group PBn(X) ⊂ Bn(X) under all non-cyclic endomorphisms of Bn(X). This implies that any non-cyclic endomorphism of Cn(X) lifts to an S(n) equivariant endomorphism of Cn o (X). To study the latter one, we define a new invariant of affine varieties Z. We constract a finite simplicial complex L∆(Z) whose set of vertices L(Z) consists of all non-constant holomorphic functions Z → C \ {0, 1}. In many interesting cases the correspondence Z 7→ 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 (X)) explicitly, compute dimR L∆(C o (X)) and show that any S(n) equivariant endomorphism of Cn o (X) produces the corresponding simplicial self-map of L∆(C o (X)). We determine the S(n) orbits of simplices ∆ ∈ L∆(C o (X)) and find their normal forms with respect to this action. Eventually, this provides a complete description of S(n) equivariant endomorphisms of Cn o (X) and proves Tame Map Theorem. This theorem shows that Cn(X)/AutCn(X) = Cn(X)/AutX and the holomorphic homotopy classes of non-cyclic endomorphisms of Cn(X) are in a natural 1 − 1 correspondence with the homotopy classes of all continuous maps Cn(X) → K(X), where K(X) is a maximal compact subgroup of AutX . It also implies that dimC F (Cn(X)) ≥ n− t(X) + 1 for any n > 4 and any non-cyclic endomorphism F of Cn(X). The study of cyclic maps involves the Liouville property of Cn(X) and the holomorphic direct decomposition Ck o (X) ∼= AutX×Cm o (C\{0, 1}), where m = k− t(X) and Cm o (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 : Cn(X) → Ck(X) there is a point Q ∈ Cn(X) such that Q∩F (Q) 6= ∅ (“Linked Map Theorem”). The proof involves an “explicit” description of the holomorphic universal covering map Π: ÃutX×T(0,m) → Cn(X), where m = n+3− t(X), ÃutX is the universal cover of AutX 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 dimC 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] = Un(z) on CP defined by the ”universal” equation z0x n + z1x y + ... + zny n = 0 cannot be represented as a composition of algebraic functions of m variables in such a way that a representing function-composition F ⊇ Un and Un itself have the same branch loci.