Extensions of normal immersions of 𝑆¹ into 𝑅²

Abstract

Suppose that f : S 1 → R 2 f:{S^1} \to {R^2} is an immersion, i.e., a C 1 {C^1} map such that f ′ f’ is never zero. We call f normal if there are only finitely many self-intersections and these are transverse double points. A normal immersion f can be topologically determined by a finite number of combinatorial invariants. Using these invariants it is possible to give considerable information about extensions of f to D 2 {D^2} . In this paper we give a new set of invariants, inspired by the work of S. Blank, to solve several problems concerning the existence of certain kinds of extensions. The problems solved are as follows: (1) When does f have a light open extension F : D 2 → R 2 F:{D^2} \to {R^2} ? (Recall that light means F − 1 ( y ) {F^{ - 1}}(y) is totally disconnected for all y and open means F maps open sets of the interior of D 2 {D^2} to open sets of R 2 {R^2} .) Because of the work of Stoïlow, the question is equivalent to the following: when does there exist a homeomorphism h : S 1 → S 1 h:{S^1} \to {S^1} , such that fh has an analytic extension to D 2 {D^2} ? (2) Suppose that F : D 2 → R 2 F:{D^2} \to {R^2} is light, open, sense preserving, and, at each point of S 1 {S^1} , F is a local homeomorphism. At each point of the interior of D 2 {D^2} , F is locally topologically equivalent to the power mapping z m {z^m} on D 2 , m ≥ 1 {D^2},m \geq 1 . Points where m > 1 m > 1 are called branch points and m − 1 m - 1 is the multiplicity of the point. There are only a finite number of branch points. The problem is to discover the minimum number of branch points of any properly interior extension of f. Also we can ask what multiplicities can arise for extensions of a given f. (3) Given a normal f, find the maximum number of properly interior extensions of f that are pairwise inequivalent. Since each immersion of the disk is equivalent to a local homeomorphism, the problem of immersion extensions is a special case of this. It is Blank’s solution of the immersion problem that prompted this paper.