Distinguishing graph of function with three critical points on a closed 3-manifold
DOI:
https://doi.org/10.3842/umzh.v77i1.8095Keywords:
Topological equivalence, critical point, 3-manifoldAbstract
UDC 515.1
We investigate a critical-point graph as a topological invariant of an isolated critical point of a smooth function on a 3-manifold. The distinguishing graph, which is a complete topological invariant of functions with three critical points on a closed 3-manifold, is constructed. It specifies the partition of a closed 3-manifold into three three-dimensional disks. We prove the criteria of topological equivalence and the realization theorem. The list of all possible distinguishing graphs whose complexity does not exceed 4 is preseted.
References
V. I. Arnold, Topological classification of Morse functions and generalizations of Hilbert's 16th problem, Math. Phys. and Anal. Geom., 10, 227–236 (2007). DOI: https://doi.org/10.1007/s11040-007-9029-0
A. V. Bolsinov, A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, A CRC Press Company, Boca Raton etc. (2004). DOI: https://doi.org/10.1201/9780203643426
A. T. Fomenko, S. V. Matveeev, Algorithmic and computer methods for three-manifolds, Springer, Netherlands (1997). DOI: https://doi.org/10.1007/978-94-017-0699-5
I. Gelbukh, A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function, Math. Slovaca, 71, № 3, 757–-772 (2021). DOI: https://doi.org/10.1515/ms-2021-0018
B. Hladysh, M. Loseva, A. Pryshlyak, Topological structure of functions with isolated critical points on a 3-manifold, Proc. Int. Geom. Cent., 16, № 3-4, 231–243 (2023); DOI: 10.15673/pigc.v16i3.2512. DOI: https://doi.org/10.15673/pigc.v16i3.2512
B. I. Hladysh, A. O. Pryshlyak, Functions with nondegenerate critical points on the boundary of the surface, Ukr. Math. J., 68, № 1, 29–40 (2016); DOI: 10.1007/s11253-016-1206-5. DOI: https://doi.org/10.1007/s11253-016-1206-5
B. I. Hladysh, A. O. Prishlyak, Topology of functions with isolated critical points on the boundary of a 2-dimensional manifold, SIGMA. Symmetry, Integrability and Geom., Methods and Appl., 13, Article~050 (2017); DOI: 0.3842/ SIGMA.2017.050.
B. I. Hladysh, A. O. Prishlyak, Simple Morse functions on an oriented surface with boundary, J. Math. Phys., Anal., Geom., 15, № 3, 354–368 (2019); DOI: 10.15407/mag15.03.354. DOI: https://doi.org/10.15407/mag15.03.354
A. Kravchenko, S. Maksymenko, Automorphisms of Kronrod–Reeb graphs of Morse functions, Eur. J. Math., 6, № 1, 114–131 (2020). DOI: https://doi.org/10.1007/s40879-019-00379-8
D. P. Lychak, A. O. Prishlyak, Morse functions and flows on nonorientable surfaces, Methods Funct. Anal. and Topology, 15, № 3, 251–258 (2009).
S. Maksymenko, Stabilizers and orbits of smooth functions, Bull. Sci. Math., 130, № 4, 279–311 (2006). DOI: https://doi.org/10.1016/j.bulsci.2005.11.001
S. Maksymenko, Deformations of functions on surfaces by isotopic to the identity diffeomorphisms, Topology and Appl., 282, 107312 (2020). DOI: https://doi.org/10.1016/j.topol.2020.107312
A. Prishlyak, M. Loseva, Topology of optimal flows with collective dynamics on closed orientable surfaces, Proc. Int. Geom. Cent., 13, № 2, 50–67 (2020); DOI: 10.15673/tmgc.v13i2.1731. DOI: https://doi.org/10.15673/tmgc.v13i2.1731
A. O. Prishlyak, On topologically equivalent Morse functions on 3-manifold, Methods Funct. Anal. and Topology, 5, № 3, 49–53 (1999).
A. O. Prishlyak, Conjugacy of Morse functions on surfaces with values on a straight line and circle, Ukr. Math. J., 52, № 10, 1623–1627 (2000); DOI: 10.1023/A, 1010461319703. DOI: https://doi.org/10.1023/A:1010461319703
A. O. Prishlyak, Topological equivalence of Morse–Smale vector fields with beh2 on three-dimensional manifolds, Ukr. Math. J., 54, № 4, 603–612 (2002).
A. O. Prishlyak, Topological equivalence of smooth functions with isolated critical points on a closed surface, Topology and Appl., 119, № 3, 257–267 (2002); DOI: 10.1016/S0166-8641(01)00077-3. DOI: https://doi.org/10.1016/S0166-8641(01)00077-3
G. Reeb, Sur les points singuliers d’une forme de {P}faff complétement intégrable ou d’une fonction numérique, C. R. Math. Acad. Sci. Paris, 222, 847—849 (1946).
A. Savchenko, M. Zarichnyi, Metrization of free groups on ultrametric spaces, Topology and Appl., 157, № 4, 724–729 (2010); DOI: 10.1016/j.topol.2009.08.015. DOI: https://doi.org/10.1016/j.topol.2009.08.015
V. V. Sharko, Functions on manifolds. Algebraic and topological aspects, vol. 131, Transl. Math. Monogr., AMS, Providence, RI (1993). DOI: https://doi.org/10.1090/mmono/131
F. Takens, The minimal number of critical points of a function on a compact manifolds and Lusternic–Schnirelman category, Invent. Math., 6, 197–214 (1968). DOI: https://doi.org/10.1007/BF01404825
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Олександр Пришляк, Володимир Кіосак, Олександр Савченко
This work is licensed under a Creative Commons Attribution 4.0 International License.
