Branched and Folded Parametrizations of the Sphere

0. This study is addressed to the following genre of topological problems. Let n be a subset of a manifold Wand ç>:2-*II be a parametrization of n by a manifold collection 2 . We seek a factorization 2 -+M-+W9 <p=F o /, where i is an inclusion of S in a manifold M of the same dimension as W and F is a map in a certain class, such that the invariants of (W, II, <p) in some reasonable sense determine (M, F, i) up to topological equivalence. For instance, let II be a closed, but not necessarily simply closed polygon in the complex plane W, 2 the extended real line and F a Schwarz-Christoffel transformation of the Gaussian upper half plane M9 such that the image [cp] of (p=F\H coincides with II. Necessary and sufficient conditions for II to bound a conformai, or more generally, a holomorphic image of a disc were first given by Titus [11]. In view of the Stoïlow-Whyburn [16] theory, it proved more convenient to use light open maps F such that cp is a regular parametrization of a smooth, closed curve II. If the curve lies in general position, the conditions can be expressed in terms of the Whitney [14]-Titus [10] intersection sequence, which is a combinatorial structure on the set of signed self-intersection points AX99) of II. In the last decade considerable progress has been made in the direction of relaxing the specialized aspects of the Picard-Loewner problem solved by Titus. We present here some current work, the precise formulation of some technical definitions and proofs have or will appear elsewhere.