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Characterizing isotopic continua in the sphere

Authors: Lex G. Oversteegen and Kirsten I. S. Valkenburg
Journal: Proc. Amer. Math. Soc. 139 (2011), 1495-1510
MSC (2010): Primary 54C20, 57N37; Secondary 57N05
Published electronically: December 8, 2010
MathSciNet review: 2748444
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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.

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Additional Information

Lex G. Oversteegen
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170

Kirsten I. S. Valkenburg
Affiliation: Faculteit der Exacte Wetenschappen, Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands

Keywords: Isotopy, homeomorphic extension, isotopic continua
Received by editor(s): November 2, 2009
Received by editor(s) in revised form: April 8, 2010
Published electronically: December 8, 2010
Additional Notes: The first author was supported in part by NSF-DMS-0906316.
The second author was supported by the Netherlands Organisation for Scientific Research (NWO), under grant 613.000.551, and thanks the Department of Mathematics at UAB for its hospitality.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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