Homeotopy groups of rooted tree like non-singular foliations on the plane
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Non-singular foliations, homeotopy groupsAbstract
Let $F$ be a non-singular foliation on the plane with all leaves being closed subsets, $H^{+}(F)$ be the group of homeomorphisms of the plane which maps leaves onto leaves endowed with compact open topology, and $H^{+}_{0}(F)$ be the identity path component of $H^{+}(F)$. The quotient $\pi_0 H^{+}(F) = H^{+}(F)/H^{+}_{0}(F)$ is an analogue of a mapping class group for foliated homeomorphisms. We will describe the algebraic structure of $\pi_0 H^{+}(F)$ under an assumption that the corresponding space of leaves of $F$ has a structure similar to a rooted tree of finite diameter.References
A. V. Bolsinov and A. T. Fomenko, Vvedenie v topologiyu integriruemykh gamiltonovykh sistem (Introduction to the topology of integrable Hamiltonian systems), Nauka, Moscow, 1997 (Russian). MathSciNet
William M. Boothby, The topology of regular curve families with multiple saddle points, Amer. J. Math. 73 (1951), 405-438. MathSciNet
D. B. A. Epstein, Curves on $2$-manifolds and isotopies, Acta Math. 115 (1966), 83-107. MathSciNet
James A. Jenkins and Marston Morse, Contour equivalent pseudoharmonic functions and pseudoconjugates, Amer. J. Math. 74 (1952), 23-51. MathSciNet
Wilfred Kaplan, Regular curve-families filling the plane, I, Duke Math. J. 7 (1940), 154-185. MathSciNet
Wilfred Kaplan, Regular curve-families filling the plane, II, Duke Math J. 8 (1941), 11-46. MathSciNet
K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Panstwowe Wydawnictwo Naukowe, Warsaw, 1966. MathSciNet
Sergiy Maksymenko and Eugene Polulyakh, Foliations with non-compact leaves on surfaces, Proceedings of Geometric Center 8 (2015), no. 3-4, 17-30. arXiv:1512.07809
A. A. Oshemkov, Morse functions on two-dimensional surfaces. Coding of singularities, Proc. Steklov Inst. Math. 205 (1995), no. 4, 119â127. MathSciNet
E. Polulyakh and I. Yurchuk, On the pseudo-harmonic functions defined on a disk, vol. 80, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 2009 (Ukrainian). MathSciNet
E. A. Polulyakh, Kronrod-Reeb graphs of functions on noncompact two-dimensional surfaces. I, Ukrainian Math. J. 67 (2015), no. 3, 431-454. MathSciNet CrossRef
A. O. Prishlyak, Conjugacy of Morse functions on surfaces with values on the line and circle, Ukrainian Math. J. 52 (2000), no. 10, 1623-1627. MathSciNet CrossRef
A. O. Prishlyak, Morse functions with finite number of singularities on a plane, Methods Funct. Anal. Topology 8 (2002), no. 1, 75-78. MathSciNet MFAT Article
V. V. Sharko, Smooth and topological equivalence of functions on surfaces, Ukrainian Math. J. 55 (2003), no. 5, 832-846. MathSciNet CrossRef
V. V. Sharko, Smooth functions on non-compact surfaces, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos. 3 (2006), no. 3, 443-473. arXiv:0709.2511
V. V. Sharko and Yu. Yu. Soroka, Topological equivalence to a projection, Methods Funct. Anal. Topology 21 (2015), no. 1, 3-5. MathSciNet MFAT Article
Hassler Whitney, Regular families of curves, Ann. of Math. (2) 34 (1933), no. 2, 244-270. MathSciNet CrossRef
