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 $\mathbb {C}^*$ and $h:I\to \mathbb {C}^*$ is an embedding. Then there exists an orientation-preserving homeomorphism $H:\mathbb {C}^*\to \mathbb {C}^*$ such that $H\restriction I=h$. It follows that $h$ is isotopic to the identity.

Suppose $X\subset \mathbb {C}^*$ is an arbitrary, in particular not necessarily locally connected, continuum. In this paper we give necessary and sufficient conditions on an embedding $h:X\to \mathbb {C}^*$ 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 $\varphi _U:\mathbb {D}\to U$ on the unit disk.