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Abstract
Let X be an (n+1)-dimensional manifold, Δ be a one-dimensional foliation on X, and p: X → X / Δ be a quotient map.
We will say that a leaf ω of Δ is special whenever the space of leaves X / Δ is not Hausdorff at ω.
We present necessary and sufficient conditions for the map p: X → X / Δ to be a locally trivial fibration under assumptions that all leaves of Δ are non-compact and the family of all special leaves of Δ is locally finite.
Keywords:
foliation, non-compact surface, fiber bundle, selection
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How to Cite
Максименко, С., & Полулях, Е. (2017). One-dimensional foliations on topological manifolds. Proceedings of the International Geometry Center, 9(2). https://doi.org/10.15673/tmgc.v9i2.277
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References
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14. Andre Haefliger and Georges Reeb. Varietes (non separees) a une dimension et structures feuilletees du plan. Enseignement Math. (2), 3:107-125, 1957.
15 J. Harrison. $C^2$ counterexamples to the Seifert conjecture. Topology, 27(3):249-278, 1988.
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17. James Jenkins and Marston Morse. Contour equivalent pseudoharmonic functions and pseudoconjugates. Amer. J. Math., 74:23-51, 1952.
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24. Sergiy Maksymenko. Stabilizers and orbits of smooth functions. Bull. Sci. Math., 130(4):279-311, 2006.
25. Sergiy Maksymenko and Eugene Polulyakh. Foliations with non-compact leaves on surfaces. Proceedings of Geometric Center, 8(3-4):17-30, 2015.
26. Sergiy Maksymenko and Eugene Polulyakh. Foliations with all non-closed leaves on non-compact surfaces. Methods Funct. Anal. Topology, 22(3):266-282, 2016.
27. Shigenori Matsumoto. Affine flows on 3-manifolds. Mem. Amer. Math. Soc., 162(771):vi+94, 2003.
28. Louis F. McAuley. Completely regular mappings, fiber spaces, the weak bundle properties, and the generalized slicing structure properties. In Topology Seminar (Wisconsin, 1965), pages 219-227. Ann. of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N.J., 1966.
29. Gael Meigniez. Prolongement des homotopies, Q-varietes et cycles tangents. Ann. Inst. Fourier (Grenoble), 47(3):945-965, 1997.
30. Gael Meigniez. Submersions, fibrations and bundles. Trans. Amer. Math. Soc., 354(9):3771-3787 (electronic), 2002.
31. Ernest Michael. Continuous selections. II. Ann. of Math. (2), 64:562-580, 1956.
32. M. Morse. La construction topologique d'un reseau isotherme sur une surface ouverte. J. Math. Pures Appl. (9), 35:67-75, 1956.
33. Marston Morse. The existence of pseudoconjugates on Riemann surfaces. Fund. Math., 39:269-287 (1953), 1952.
34. A. A. Oshemkov. Morse functions on two-dimensional surfaces. Coding of singularities. Trudy Mat. Inst. Steklov., 205(Novye Rezult. v Teor. Topol. Klassif. Integr. Sistem):131-140, 1994.
35. Eugene Polulyakh. Kronrod-reeb graphs of functions on non-compact surfaces. Ukrainian Math. Journal, 67(3):375-396, 2015.
36. Eugene Polulyakh and Iryna Yurchuk. On the pseudo-harmonic functions defined on a disk. Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 80:151, 2009.
37. Georges Reeb. Les espaces localement numeriques non separes et leurs applications a un probleme classique. In Colloque de topologie de {S}trasbourg, 1954-1955, page 12. Institut de Mathematique, Universite de Strasbourg.
38. Ronald H. Rosen. $E^{4}$ is the cartesian product of a totally non-euclidean space and $E^{1}$. Ann. of Math. (2), 73:349-361, 1961.
39. Leonard R. Rubin. A general class of factors of $E^{4}$. Trans. Amer. Math. Soc., 166:215-224, 1972.
40. Stephen B. Seidman. Completely regular mappings with locally compact fiber. Trans. Amer. Math. Soc., 147:461-471, 1970.
41. V. V. Sharko. Smooth and topological equivalence of functions on surfaces. Ukr. Mat. Zh., 55(5):687-700, 2003.
42. V. V. Sharko. Smooth functions on non-compact surfaces. Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 3(3):443-473, arXiv:math/0709.2511, 2006.
43. V. V. Sharko and Yu. Yu. Soroka. Topological equivalence to a projection. Methods Funct. Anal. Topology, 21(1):3-5, 2015.
44. Yuliya Soroka. Homeotopy groups of rooted tree like non-singular foliations on the plane. Methods Funct. Anal. Topology, 22(3):283-294, 2016.
45. I. Tamura. Topologiya sloenii. Mir, Moscow, 1979.
46. Hassler Whitney. Regular families of curves. Ann. of Math. (2), 34(2):244-270, 1933.
47. Hassler Whitney. On regular families of curves. Bull. Amer. Math. Soc., 47:145-147, 1941.
2. R. H. Bing. The cartesian product of a certain non-manifold and a line is $E_{4}$. Bull. Amer. Math. Soc., 64:82-84, 1958.
3. R. H. Bing. The cartesian product of a certain nonmanifold and a line is $E^{4}$. Ann. of Math. (2), 70:399-412, 1959.
4. A. V. Bolsinov and A. T. Fomenko. Vvedenie v topologiyu integriruemykh gamiltonovykh sistem. "Nauka", Moscow, 1997.
5. William M. Boothby. The topology of regular curve families with multiple saddle points. Amer. J. Math., 73:405-438, 1951.
6. William M. Boothby. The topology of the level curves of harmonic functions with critical points. Amer. J. Math., 73:512-538, 1951.
7. L.E.J. Brouwer. Beweis der invarianz des n-dimensionalen gebiets. Mathematische Annalen, 71:305–315, 1912.
8. Alberto Candel and Lawrence Conlon. Foliations. I. Graduate Studies in Mathematics 23. American Mathematical Society, 2000.
9. E. Dyer and M.-E. Hamstrom. Completely regular mappings. Fund. Math., 45:103-118, 1958.
10. C. Godbillon and G. Reeb. Fibres sur le branchement simple. Enseignement Math. (2), 12:277-287, 1966.
11. Claude Godbillon. Feuilletages ayant la propriete du prolongement des homotopies. Ann. Inst. Fourier (Grenoble), 17(fasc. 2):219-260 (1968), 1967.
12. Claude Godbillon. Feuilletages, volume 98 of Progress in Mathematics. Birkhauser Verlag, Basel, 1991. Etudes geometriques. [Geometric studies], With a preface by G. Reeb.
13. Andre Haefliger. Sur les feuilletages des varietes de dimension n par des feuilles fermees de dimension n-1. In Colloque de topologie de {S}trasbourg, 1954-1955, page 8. Institut de Mathematique, Universite de Strasbourg.
14. Andre Haefliger and Georges Reeb. Varietes (non separees) a une dimension et structures feuilletees du plan. Enseignement Math. (2), 3:107-125, 1957.
15 J. Harrison. $C^2$ counterexamples to the Seifert conjecture. Topology, 27(3):249-278, 1988.
16. Witold Hurewicz and Henry Wallman. Dimension Theory. Princeton Mathematical Series, v. 4. Princeton University Press, Princeton, N. J., 1941.
17. James Jenkins and Marston Morse. Contour equivalent pseudoharmonic functions and pseudoconjugates. Amer. J. Math., 74:23-51, 1952.
18. James Jenkins and Marston Morse. Conjugate nets, conformal structure, and interior transformations on open Riemann surfaces. Proc. Nat. Acad. Sci. U. S. A., 39:1261-1268, 1953.
19. James Jenkins and Marston Morse. Curve families $F^*$ locally the level curves of a pseudoharmonic function. Acta Math., 91:1-42, 1954.
20. James Jenkins and Marston Morse. Conjugate nets on an open Riemann surface. In Lectures on functions of a complex variable, pages 123-185. The University of Michigan Press, Ann Arbor, 1955.
21. Wilfred Kaplan. Regular curve-families filling the plane, I. Duke Math. J., 7:154-185, 1940.
22. Wilfred Kaplan. Regular curve-families filling the plane, II. Duke Math J., 8:11-46, 1941.
23. Jehpill Kim. On (n-1)-dimensional factors of I^{n}. Proc. Amer. Math. Soc., 15:679-680, 1964.
24. Sergiy Maksymenko. Stabilizers and orbits of smooth functions. Bull. Sci. Math., 130(4):279-311, 2006.
25. Sergiy Maksymenko and Eugene Polulyakh. Foliations with non-compact leaves on surfaces. Proceedings of Geometric Center, 8(3-4):17-30, 2015.
26. Sergiy Maksymenko and Eugene Polulyakh. Foliations with all non-closed leaves on non-compact surfaces. Methods Funct. Anal. Topology, 22(3):266-282, 2016.
27. Shigenori Matsumoto. Affine flows on 3-manifolds. Mem. Amer. Math. Soc., 162(771):vi+94, 2003.
28. Louis F. McAuley. Completely regular mappings, fiber spaces, the weak bundle properties, and the generalized slicing structure properties. In Topology Seminar (Wisconsin, 1965), pages 219-227. Ann. of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N.J., 1966.
29. Gael Meigniez. Prolongement des homotopies, Q-varietes et cycles tangents. Ann. Inst. Fourier (Grenoble), 47(3):945-965, 1997.
30. Gael Meigniez. Submersions, fibrations and bundles. Trans. Amer. Math. Soc., 354(9):3771-3787 (electronic), 2002.
31. Ernest Michael. Continuous selections. II. Ann. of Math. (2), 64:562-580, 1956.
32. M. Morse. La construction topologique d'un reseau isotherme sur une surface ouverte. J. Math. Pures Appl. (9), 35:67-75, 1956.
33. Marston Morse. The existence of pseudoconjugates on Riemann surfaces. Fund. Math., 39:269-287 (1953), 1952.
34. A. A. Oshemkov. Morse functions on two-dimensional surfaces. Coding of singularities. Trudy Mat. Inst. Steklov., 205(Novye Rezult. v Teor. Topol. Klassif. Integr. Sistem):131-140, 1994.
35. Eugene Polulyakh. Kronrod-reeb graphs of functions on non-compact surfaces. Ukrainian Math. Journal, 67(3):375-396, 2015.
36. Eugene Polulyakh and Iryna Yurchuk. On the pseudo-harmonic functions defined on a disk. Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 80:151, 2009.
37. Georges Reeb. Les espaces localement numeriques non separes et leurs applications a un probleme classique. In Colloque de topologie de {S}trasbourg, 1954-1955, page 12. Institut de Mathematique, Universite de Strasbourg.
38. Ronald H. Rosen. $E^{4}$ is the cartesian product of a totally non-euclidean space and $E^{1}$. Ann. of Math. (2), 73:349-361, 1961.
39. Leonard R. Rubin. A general class of factors of $E^{4}$. Trans. Amer. Math. Soc., 166:215-224, 1972.
40. Stephen B. Seidman. Completely regular mappings with locally compact fiber. Trans. Amer. Math. Soc., 147:461-471, 1970.
41. V. V. Sharko. Smooth and topological equivalence of functions on surfaces. Ukr. Mat. Zh., 55(5):687-700, 2003.
42. V. V. Sharko. Smooth functions on non-compact surfaces. Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 3(3):443-473, arXiv:math/0709.2511, 2006.
43. V. V. Sharko and Yu. Yu. Soroka. Topological equivalence to a projection. Methods Funct. Anal. Topology, 21(1):3-5, 2015.
44. Yuliya Soroka. Homeotopy groups of rooted tree like non-singular foliations on the plane. Methods Funct. Anal. Topology, 22(3):283-294, 2016.
45. I. Tamura. Topologiya sloenii. Mir, Moscow, 1979.
46. Hassler Whitney. Regular families of curves. Ann. of Math. (2), 34(2):244-270, 1933.
47. Hassler Whitney. On regular families of curves. Bull. Amer. Math. Soc., 47:145-147, 1941.