Automorphisms and endomorphisms of partitions of topological spaces

Authors

DOI:

https://doi.org/10.3842/umzh.v78i1-2.8932

Keywords:

фактор-простір, компактно-відкрита топологія, точка розгалуження, компактно-накриваюче відображення, локальна компактність, гомоморфізм дії

Abstract

UDC 515.1

Let $X$ be a topological space, let $\Delta$ be a partition of $X$, and let $Y =X/\Delta$ be a quotient space with the corresponding quotient topology. Then the automorphism group $\mathcal{H}(\Delta)$ of $\Delta$ (i.e., the homeomorphisms of  $X,$ which permute the elements of partition)  acts in a natural way upon $Y$ by homeomorphisms. We determine the cases in which the corresponding homomorphism of the action $\psi\colon\mathcal{H}(\Delta) \to \mathcal{H}(Y)$ into the group of homeomorphisms of $Y$ is continuous with respect to the compact-open topologies. The obtained results have applications to the foliation theory.

References

1. Mathieu Baillif, Alexandre Gabard, Manifolds: Hausdorffness versus homogeneity, Proc. Amer. Math. Soc., 136, № 3, 1105–1111 (2008); DOI: 10.1090/S0002-9939-07-09100-9.

2. Mathieu Baillif, Alexandre Gabard, David Gauld, Foliations on non-metrisable manifolds: absorption by a Cantor black hole, Proc. Amer. Math. Soc., 142, № 3, 1057–1069 (2014); DOI: 10.1090/S0002-9939-2013-11900-3.

3. Thomas Barthelmé, Andrey Gogolev, A note on self orbit equivalences of Anosov flows and bundles with fiberwise Anosov flows, Math. Res. Lett., 26, № 3, 711–728 (2019); DOI: 10.4310/MRL.2019.v26.n3.a3.

4. C. Bonatti, H. Hattab, E. Salhi, G. Vago, Hasse diagrams and orbit class spaces, Topology Appl., 158, № 6, 729–740 (2011); DOI: 10.1016/j.topol.2010.12.010.

5. Ezzeddine Bouacida, Othman Echi, Ezzeddine Salhi, Feuilletages et topologie spectrale, J. Math. Soc. Japan, 52, № 2, 447–464 (2000); DOI: 10.2969/jmsj/05220447.

6. Michael Brin, Garrett Stuck, Introduction to Dynamical Systems, Cambridge University Press, 1st~ed. (2002).

7. Diego Corro, Fernando Galaz-Garc'a, Myers–Steenrod theorems for metric and singular Riemannian foliations, arXiv:2407.03534 (2024).

8. Lech Drewnowski, True preimages of compact or separable sets for functional analysts, Comment. Math. Univ. Carolin., 61, № 1, 69–82 (2020); DOI: 10.14712/1213-7243.2020.007.

9. Othman Echi, The categories of flows of Set and Top, Topology Appl., 159, № 9, 2357–2366 (2012); DOI: 10.1016/j.topol.2011.11.059.

10. Ryszard Engelking, General topology, Sigma Series in Pure Mathematics, 2nd~ed., Vol. 6, Heldermann Verlag, Berlin (1989).

11. Kazuhiko Fukui, On the homotopy type of some subgroups of $Diff(M^3)$, Japan. J. Math. (N.S.), 2, № 2, 249–267 (1976); DOI: 10.4099/math1924.2.249.

12. Kazuhiko Fukui, Hideki Imanishi, On commutators of foliation preserving homeomorphisms, J. Math. Soc. Japan, 51, № 1, 227–236 (1999); DOI: 10.2969/jmsj/05110227.

13. Kazuhiko Fukui, Hideki Imanishi, On commutators of foliation preserving Lipschitz homeomorphisms, J. Math. Kyoto Univ., 41, № 3, 507–515 (2001); DOI: 10.1215/kjm/1250517615.

14. Paul Gartside, David Gauld, Sina Greenwood, Homogeneous and inhomogeneous manifolds, Proc. Amer. Math. Soc., 136, № 9, 3363–3373 (2008); DOI: 10.1090/S0002-9939-08-09343-X.

15. David Gauld, Non-metrisable manifolds, Springer, Singapore (2014); DOI: 10.1007/978-981-287-257-9.

16. étienne Ghys, Groups acting on the circle, Enseign. Math. (2), 47, № 3-4, 329–407 (2001).

17. C. Godbillon, G. Reeb, Fibrés sur le branchement simple, Enseign. Math. (2), 12, № 12, 277–287 (1966).

18. André Haefliger, Georges Reeb, Variétés (non séparées) à une dimension et structures feuilletées du plan, Enseign. Math. (2), 3, № 3, 107–125 (1957).

19. Hawete Hattab, Ezzeddine Salhi, Groups of homeomorphisms and spectral topology, Proceedings of the 18th Summer Conference on Topology and its Applications, 28, № 2, 503–526 (2004).

20. Wilfred Kaplan, Regular curve-families filling the plane, I, Duke Math. J., 7, № 8, 154–185 (1940).

21. Wilfred Kaplan, Regular curve-families filling the plane, II, Duke Math. J., 8, 11–46 (1941).

22. Anatole Katok, Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia Math. Appl., 54, Cambridge University Press, Cambridge (1995).

23. O. Khokhliuk, S. Maksymenko, Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 1, J. Homotopy Relat. Struct., 18, № 2-3, 313–356 (2023); DOI: 10.1007/s40062-023-00328-z.

24. Jacek Lech, Ilona Michalik, On the structure of the homeomorphism and diffeomorphism groups fixing a point, Publ. Math. Debrecen, 83, № 3, 435–447 (2013); DOI: 10.5486/PMD.2013.5551.

25. S. Maksymenko, Local inverses of shift maps along orbits of flows, Osaka J. Math., 48, № 2, 415–455 (2011).

26. S. Maksymenko, E. Polulyakh, Yu. Soroka, Homeotopy groups of one-dimensional foliations on surfaces, Proc. Int. Geom. Cent., 10, № 1, 22–46 (2017).

27. Sergiy Maksymenko, Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 2, J.~Homotopy Relat. Struct., 19, № 2, 239–273 (2024); DOI: 10.1007/s40062-024-00346-5.

28. Sergiy Maksymenko, Eugene Polulyakh, Foliations with non-compact leaves on surfaces, Proc. Geom. Cent., 8, № 3-4, 17–30 (2015); DOI: arXiv:1512.07809.

29. Sergiy Maksymenko, Eugene Polulyakh, Foliations with all nonclosed leaves on noncompact surfaces, Methods Funct. Anal. Topology, 22, № 3, 266–282 (2016).

30. Sergiy Maksymenko, Eugene Polulyakh, One-dimensional foliations on topological manifolds, Proc. Geom. Cent., 9, № 2, 1–23 (2016).

31. Sergiy Maksymenko, Eugene Polulyakh, Characterization of striped surfaces, Proc. Int. Geom. Cent., 10, № 2, 24–38 (2017).

32. Sergiy Maksymenko, Eugene Polulyakh, Homeotopy groups of leaf spaces of one-dimensional foliations on non-compact surfaces with non-compact leaves, Proc. Int. Geom. Cent., 14, № 4, 271–290 (2021); DOI: 10.15673/tmgc.v14i4.2204.

33. E. Michael, A theorem on semi-continuous set-valued functions, Duke Math. J., 26, 647–651 (1959); URL: http://projecteuclid.org/euclid.dmj/1077468774.

34. E. Michael, $N_0$-spaces, J. Math. Mech., 15, 983–1002 (1966).

35. E. Michael, K. Nagami, Compact-covering images of metric spaces, Proc. Amer. Math. Soc., 37, 260–266 (1973); DOI: 10.2307/2038744.

36. Ernest Michael, On a theorem of Rudin and Klee, Proc. Amer. Math. Soc., 12, Article~921 (1961); DOI: 10.2307/2034390.

37. Keiô Nagami, Ranges which enable open maps to be compact-covering, Gen. Topol. Appl., 3, 355–367 (1973).

38. Tomasz Rybicki, Homology of the group of leaf preserving homeomorphisms, Demonstr. Math., 29, № 2, 459–464 (1996).

39. Tomasz Rybicki, On the uniform perfectness of groups of bundle homeomorphisms, Arch. Math. (Brno), 55, № 5, 333–339 (2019); DOI: 10.5817/am2019-5-333.

40. Itiro Tamura, Topology of foliations: an introduction, Translations of Mathematical Monographs, Vol. 97, American Mathematical Society, Providence, RI (1992); DOI: 10.1090/mmono/097.

41. Y. Tanaka, Y. Ge, Around quotient compact images of metric spaces, and symmetric spaces, Houston J. Math., 32, № 1, 99–117 (2006).

42. Yoshio Tanaka, Theory of $k$-networks. II, Questions Answers Gen. Topol., 19, № 1, 27–46 (2001).

43. Hassler Whitney, Regular families of curves, Ann. of Math. (2), 34, № 2, 244–270 (1933).

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Published

26.01.2026

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Vol. 78 No. 1-2 (2026)

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Research articles

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Copyright (c) 2026 S. Maksymenko, E. Polulyakh

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How to Cite

Maksymenko, S., and E. Polulyakh. “Automorphisms and Endomorphisms of Partitions of Topological Spaces”. Ukrains’kyi Matematychnyi Zhurnal, vol. 78, no. 1-2, Jan. 2026, pp. 47–70, https://doi.org/10.3842/umzh.v78i1-2.8932.