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

DOI:
https://doi.org/10.1090/S0002-9939-2010-10830-4

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 is an arc in the complex sphere and is an embedding. Then there exists an orientation-preserving homeomorphism such that . It follows that is isotopic to the identity.

Suppose is an arbitrary, in particular not necessarily locally connected, continuum. In this paper we give necessary and sufficient conditions on an embedding to be extendable to an orientation-preserving homeomorphism of the entire sphere. It follows that in this case is isotopic to the identity. The proof will make use of partitions of complementary domains of , into hyperbolically convex subsets, which have limited distortion under the conformal map 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

Email:
overstee@math.uab.edu

**Kirsten I. S. Valkenburg**

Affiliation:
Faculteit der Exacte Wetenschappen, Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands

Email:
kivalken@few.vu.nl

DOI:
https://doi.org/10.1090/S0002-9939-2010-10830-4

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.