Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. J. M. Aarts and L. G. Oversteegen, The homeomorphism group of the hairy arc, Compositio Math. 96 (1995), no. 3, 283-292. 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, 1776-1785. MR 2519213
  • 3. R. H. Bing and M. Starbird, Linear isotopies in $ E\sp{2}$, Trans. Amer. Math. Soc. 237 (1978), 205-222. 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), 77-111. 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.
  • 7. E. F. Collingwood, A theorem on prime ends, J. London Math. Soc. 31 (1956), 344-349. MR 0080158 (18:201e)
  • 8. R. D. Edwards and R. C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 63-88. MR 0283802 (44:1032)
  • 9. D. B. A. Epstein, Prime ends, Proc. London Math. Soc. (3) 42 (1981), no. 3, 385-414. MR 614728 (83c:30025)
  • 10. -, Curves on $ 2$-manifolds and isotopies, Acta Math. 115 (1966), 83-107. MR 0214087 (35:4938)
  • 11. K. Kawamura, L. G. Oversteegen, and E. D. Tymchatyn, On homogeneous totally disconnected $ 1$-dimensional spaces, Fund. Math. 150 (1996), 97-112. MR 1391294 (97d:54060)
  • 12. R. S. Kulkarni and U. Pinkall, A canonical metric for Möbius structures and its applications, Math. Z. 216 (1994), no. 1, 89-129. MR 1273468 (95b:53017)
  • 13. W. Lewis, Embeddings of the pseudo-arc in $ E\sp{2}$, Pacific J. Math. 93 (1981), no. 1, 115-120. MR 621602 (82h:54056)
  • 14. J. Milnor, Dynamics in one complex variable, third ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309 (2006g:37070)
  • 15. L. G. Oversteegen and E. D. Tymchatyn, Extending isotopies of planar continua, Annals of Mathematics 172 (2010), 2105-2133.
  • 16. P. Seibert, Über die Randstrukturen von Überlagerungsflächen, Math. Nachr. 19 (1958), 339-352. MR 0105497 (21:4238)
  • 17. C. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706 (95b:30008)
  • 18. L. Siebenmann, The Osgood-Schoenflies Theorem revisited, Russian Math. Surveys 60 (2005), 645-672. 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. 3-137. MR 2508255

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 54C20, 57N37, 57N05

Retrieve articles in all journals with MSC (2010): 54C20, 57N37, 57N05


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.

American Mathematical Society