Characterizing isotopic continua in the sphere

Abstract

In this paper we will generalize the following well-known result. Suppose that I is an arc in the complex sphere C * and h : I → C * is an embedding. Then there exists an orientation-preserving homeomorphism H : C * → C * such that H I = h. It follows that h is isotopic to the identity. Suppose X ⊂ C * is an arbitrary, in particular not necessarily locally connected, continuum. In this paper we give necessary and sufficient conditions on an embedding h : X → C * to be extendable to an orientation-preserving homeomorphism of the entire sphere. It follows that in this case h is isotopic to the identity. The proof will make use of partitions of complementary domains U of X, into hyperbolically convex subsets, which have limited distortion under the conformal map ϕ U : D → U on the unit disk.

Fixed Point Theorems for Plane Continua with Applications

“…This extends a well-known result regarding the extension of a holomorphic motion [ST86,Slo91]. In [OV09] this partition is used to give necessary and sufficient conditions to extend a homeomorphism, of an arbitrary planar continuum, over the plane.…”

“…In this section we will apply the results from Section 4.1 to the case that K is a non-separating plane continuum (or, equivalently, that U ∞ = C ∞ \ K is simply connected). The results in this section are essential to [OT07,OV09] but are not used in this paper. The reader who is only interested in the fixed point question can skip this section.…”

“…We will show that the resulting set of hyperbolic geodesics is a closed lamination of U in the sense of Thurston [Thu09]. This version of the Linchpin Theorem, which states that every point in U is either contained in a unique hyperbolic geodesic g in U , or in the interior of an unique hyperbolically convex gap G, both of which are contained in a maximal round ball, is used in [OT07,OV09] to extend a homeomorphism on the boundary of a simply connected domain over the entire domain. To illustrate the usefulness of these partitions we include a simple proof of the Schoenflies Theorem in Section 4.3.…”

Fixed point theorem for non-self maps of regions in the plane

“…Theorem C (Oversteegen and Valkenburg [15]). Let X ⊂Ĉ be compact, simply connected and locally connected, and f : X → Y ⊂Ĉ a homeomorphism.…”

“…More details can be found in [15], for an introduction to Carathéodory's theory of prime ends the reader is referred to Milnor's book [13,Chapter 17].…”