Characterizing isotopic continua in the sphere
Authors:
Lex G. Oversteegen and Kirsten I. S. Valkenburg
Journal:
Proc. Amer. Math. Soc. 139 (2011), 14951510
MSC (2010):
Primary 54C20, 57N37; Secondary 57N05
Published electronically:
December 8, 2010
MathSciNet review:
2748444
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Additional Information
Abstract: In this paper we will generalize the following wellknown result. Suppose that is an arc in the complex sphere and is an embedding. Then there exists an orientationpreserving 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 orientationpreserving 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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 J. M. Aarts and L. G. Oversteegen, The homeomorphism group of the hairy arc, Compositio Math. 96 (1995), no. 3, 283292. MR 1327147 (96b:54056)
 2.
 J. Aarts, G. Brouwer and L. Oversteegen, Centerlines of regions in the sphere, Topology and its Appl. 156 (2009), no. 10, 17761785. MR 2519213
 3.
 R. H. Bing and M. Starbird, Linear isotopies in , Trans. Amer. Math. Soc. 237 (1978), 205222. MR 0461510 (57:1495)
 4.
 A. M. Blokh and L. G. Oversteegen, A fixed point theorem for branched covering maps of the plane, Fund. Math. 206 (2009), 77111. MR 2576262
 5.
 A. M. Blokh, R.J. Fokkink, J. C. Mayer, L. G. Oversteegen, and E. D. Tymchatyn, Fixed point theorems for plane continua with applications, arXiv:1004.0214.
 6.
 G. A. Brouwer, Green's functions from a metric point of view, Ph.D. dissertation, University of Alabama at Birmingham, 2005.
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 E. F. Collingwood, A theorem on prime ends, J. London Math. Soc. 31 (1956), 344349. MR 0080158 (18:201e)
 8.
 R. D. Edwards and R. C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 6388. MR 0283802 (44:1032)
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 K. Kawamura, L. G. Oversteegen, and E. D. Tymchatyn, On homogeneous totally disconnected dimensional spaces, Fund. Math. 150 (1996), 97112. MR 1391294 (97d:54060)
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 C. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, SpringerVerlag, Berlin, 1992. MR 1217706 (95b:30008)
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 L. Siebenmann, The OsgoodSchoenflies Theorem revisited, Russian Math. Surveys 60 (2005), 645672. MR 2190924 (2007e:57011)
 19.
 W. P. Thurston, On the geometry and dynamics of iterated rational maps, Complex dynamics: families and friends, A K Peters, Wellesley, MA, 2009, pp. 3137. MR 2508255
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Additional Information
Lex G. Oversteegen
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 352941170
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:
http://dx.doi.org/10.1090/S000299392010108304
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 NSFDMS0906316.
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.
