Extending isotopies of planar continua. (English) Zbl 1221.57035

Oversteegen, Lex G.; Tymchatyn, Edward D.

Ann. Math. (2) 172, No. 3, 2105-2133 (2010).

The paper treats the isotopy extension problem for planar continua, i.e. if \(Z\) is a continuum in the complex plane \(\mathbb{C}\) and \(h^t:Z\to \mathbb{C}, t\in [0,1]\) an isotopy starting at the identity, can \(h^t\) be extended to an isotopy of the plane starting at the identity? In order to solve this question the authors consider the plane as a subset of the complex sphere \(\mathbb{C^*} = \mathbb{C} \cup \{\infty\}\). Using known results related to \(\mathbb{C^*},\) improving and adapting appropriate techniques, in Theorem 7.3 of the paper they obtain:
Suppose that \(h^t:Z\to \mathbb{C}\) is an isotopy of a planar continuum \(Z\), which we consider as a subset of the sphere \(\mathbb{C^*}\) with \(h^0= id|Z\). Then there exists an extension to an isotopy \(H^t:\mathbb{C^*}\to \mathbb{C^*}\) such that \(H^0=id|\mathbb{C^*}.\)
To return to the plane \(\mathbb{C}\) let \(U\) denote the component of \(\mathbb{C^*} \setminus Z\) containing the point at infinity. By composing the isotopy \(H^t\) with an isotopy \(K^t\) of the sphere such that \(K^0=id|\mathbb{C^*}\) and \(K^t|\mathbb{C^*}\setminus U=id|\mathbb{C^*}\setminus U,\) and \(K^t(H^t(\infty))=\infty\) for all \(t\in [0,1]\) one obtains:
Suppose that \(h^t:Z\to \mathbb{C}\) is an isotopy of a planar continuum \(Z\subset \mathbb{C}\) with \(h^0= id|Z\). Then there exists an extension to an isotopy \(H^t:\mathbb{C}\to \mathbb{C}\) such that \(H^0=id|\mathbb{C}.\)

Reviewer: Ivan Ivanšić (Zagreb)

MSC:

57N37	Isotopy and pseudo-isotopy
57N05	Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
54C20	Extension of maps
54F15	Continua and generalizations

Keywords:

extension; isotopy; planar continuum; accessible point; hyperbolic crosscut; metric external ray; holomorphic motion

References:

[1]	J. Aarts, G. Brouwer, and L. G. Oversteegen, ”Centerlines of regions in the sphere,” Topology Appl., vol. 156, iss. 10, pp. 1776-1785, 2009. · Zbl 1173.51007
[2]	K. Astala and G. J. Martin, ”Holomorphic motions,” in Papers on Analysis, Jyväskylä: Univ. Jyväskylä, 2001, vol. 83, pp. 27-40. · Zbl 1001.30020
[3]	R. Baer, ”Kurventypen auf flächen,” J. Reine Angew. Math., vol. 156, pp. 231-246, 1927. · JFM 53.0547.01
[4]	R. Baer, ”Isotopie von kurven auf orientierbaren, geschlossen flächen und ihr zusammenhang mit der topologischen deformation der flächen,” Reine Angew. Math., vol. 159, pp. 101-116, 1928. · JFM 54.0602.05
[5]	A. Beck, Continuous Flows in the Plane, New York: Springer-Verlag, 1974. · Zbl 0295.54001
[6]	H. Bell, ”On fixed point properties of plane continua,” Trans. Amer. Math. Soc., vol. 128, pp. 539-548, 1967. · Zbl 0173.25402
[7]	H. Bell, ”Some topological extensions of plane geometry,” Rev. Colombiana Mat., vol. 9, iss. 3-4, pp. 125-153, 1975. · Zbl 0331.54023
[8]	A. Blokh and L. Oversteegen, ”A fixed point theorem for branched covering maps of the plane,” Fund. Math., vol. 206, pp. 77-111, 2009. · Zbl 1197.54058
[9]	L. Bers and H. L. Royden, ”Holomorphic families of injections,” Acta Math., vol. 157, iss. 3-4, pp. 259-286, 1986. · Zbl 0619.30027
[10]	G. A. Brouwer, ”Green’s functions from a metric point of view,” PhD Thesis , University of Alabama at Birmingham, 2005.
[11]	C. Carathéodory, ”Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten,” Math. Ann., vol. 72, iss. 1, pp. 107-144, 1912. · JFM 43.0760.05
[12]	R. D. Edwards and R. C. Kirby, ”Deformations of spaces of imbeddings,” Ann. Math., vol. 93, pp. 63-88, 1971. · Zbl 0214.50303
[13]	D. B. A. Epstein, ”Curves on \(2\)-manifolds and isotopies,” Acta Math., vol. 115, pp. 83-107, 1966. · Zbl 0136.44605
[14]	P. Fabel, ”Completing Artin’s braid group on infinitely many strands,” J. Knot Theory Ramifications, vol. 14, iss. 8, pp. 979-991, 2005. · Zbl 1116.57002
[15]	R. J. Fokkink, J. C. Mayer, L. G. Oversteegen, and E. D. Tymchatyn, ”The plane fixed problem,” , preprint , 2008.
[16]	S. D. Iliadis, ”Positions of continua on the plane, and fixed points,” Vestnik Moskov. Univ. Ser. I Mat. Meh., vol. 25, iss. 4, pp. 66-70, 1970. · Zbl 0204.31404
[17]	R. S. Kulkarni and U. Pinkall, ”A canonical metric for Möbius structures and its applications,” Math. Z., vol. 216, iss. 1, pp. 89-129, 1994. · Zbl 0813.53022
[18]	Y. M. Lyubich, ”Some typical properties of the dynamics of rational mappings,” Uspekhi Mat. Nauk, vol. 38, iss. 5(233), pp. 197-198, 1983. · Zbl 0598.58028
[19]	J. Milnor, Dynamics in One Complex Variable, Braunschweig: Friedr. Vieweg & Sohn, 1999. · Zbl 0946.30013
[20]	R. Mañé, P. Sad, and D. Sullivan, ”On the dynamics of rational maps,” Ann. Sci. École Norm. Sup., vol. 16, iss. 2, pp. 193-217, 1983. · Zbl 0524.58025
[21]	L. G. Oversteegen and E. D. Tymchatyn, ”Plane strips and the span of continua. I,” Houston J. Math., vol. 8, iss. 1, pp. 129-142, 1982. · Zbl 0506.54022
[22]	C. Pommerenke, Boundary Behaviour of Conformal Maps, New York: Springer-Verlag, 1992, vol. 299. · Zbl 0762.30001
[23]	Z. Slodkowski, ”Holomorphic motions and polynomial hulls,” Proc. Amer. Math. Soc., vol. 111, iss. 2, pp. 347-355, 1991. · Zbl 0741.32009
[24]	D. P. Sullivan and W. P. Thurston, ”Extending holomorphic motions,” Acta Math., vol. 157, iss. 3-4, pp. 243-257, 1986. · Zbl 0619.30026
[25]	W. P. Thurston, ”On the geometry and dynamics of iterated rational maps,” in Complex Dynamics, Schleicher, D. and Selinger, N., Eds., Wellesley, MA: A K Peters, 2009, pp. 3-137. · Zbl 1185.37111
[26]	G. C. Wen, Conformal Mappings and Boundary Value Problems, Providence, RI: Amer. Math. Soc., 1992, vol. 106. · Zbl 0778.30011
[27]	R. L. Wilder, Topology of Manifolds, Providence, R.I.: Amer. Math. Soc., 1963, vol. 27. · Zbl 0117.16204
[28]	G. T. Whyburn, Analytic Topology, New York: Amer. Math. Soc., 1942, vol. 28. · Zbl 0061.39301