Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 050, 17 pages      arXiv:1706.05050      https://doi.org/10.3842/SIGMA.2017.050

Topology of Functions with Isolated Critical Points on the Boundary of a 2-Dimensional Manifold

Bohdana I. Hladysh and Aleksandr O. Prishlyak
Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 4-e Akademika Glushkova Ave., Kyiv, 03127, Ukraine

Received November 18, 2016, in final form June 16, 2017; Published online July 01, 2017

Abstract
This paper focuses on the problem of topological equivalence of functions with isolated critical points on the boundary of a compact surface $M$ which are also isolated critical points of their restrictions to the boundary. This class of functions we denote by $\Omega(M)$. Firstly, we've obtained the topological classification of above-mentioned functions in some neighborhood of their critical points. Secondly, we've constructed a chord diagram from the neighborhood of a critical level. Also the minimum number of critical points of such functions is being considered. And finally, the criterion of global topological equivalence of functions which belong to $\Omega(M)$ and have three critical points has been developed.

Key words: topological classification; isolated boundary critical point; optimal function; chord diagram.

pdf (456 kb)   tex (156 kb)

References

1. Bolsinov A.V., Fomenko A.T., Integrable Hamiltonian systems. Geometry, topology, classification, Chapman & Hall/CRC, Boca Raton, FL, 2004.
2. Borodzik M., Némethi A., Ranicki A., Morse theory for manifolds with boundary, Algebr. Geom. Topol. 16 (2016), 971-1023, arXiv:1207.3066.
3. Conner P.E., Floyd E.E., Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 33, Springer-Verlag, Berlin - Göttingen - Heidelberg, 1964.
4. Hladysh B.I., Prishlyak A.O., Functions with nondegenerate critical points on the boundary of a surface, Ukrain. Math. J. 68 (2016), 29-40.
5. Iurchuk I.A., Properties of a pseudo-harmonic function on closed domain, Proc. Internat. Geom. Center 7 (2014), no. 4, 50-59.
6. Kadubovskyi A.A., Enumeration of topologically non-equivalent smooth minimal functions on closed surfaces, Proceedings of Institute of Mathematics, Kyiv 12 (2015), no. 6, 105-145.
7. Kadubovskyi A.A., On the number of topologically non-equivalent functions with one degenerated saddle critical point on two-dimensional sphere, II, Proc. Internat. Geom. Center 8 (2015), no. 1, 47-62.
8. Khruzin A., Enumeration of chord diagrams, math.CO/0008209.
9. Kronrod A.S., On functions of two variables, Russian Math. Surveys 5 (1950), 24-134.
10. Lukova-Chuiko N.V., Minimal function on 3-manifolds with boundary, Proc. Internat. Geom. Center 8 (2015), no. 3-4, 46-52.
11. Lukova-Chuiko N.V., Prishlyak A.O., Prishlyak K.O., M-functions on nonoriented surfaces, J. Numer. Appl. Math. (2012), no. 2, 176-185.
12. Maksymenko S., Polulyakh E., Foliations with all non-closed leaves on non-compact surfaces, Methods Funct. Anal. Topology 22 (2016), 266-282, arXiv:1606.00045.
13. Milnor J.W., Topology from the differentiable viewpoint, The University Press of Virginia, Charlottesville, Va., 1965.
14. Prishlyak A.O., Topological properties of functions on two and three dimensional manifolds, Palmarium. Academic Pablishing.
15. Prishlyak A.O., Topological equivalence of smooth functions with isolated critical points on a closed surface, Topology Appl. 119 (2002), 257-267, math.GT/9912004.
16. Reeb G., Sur les points singuliers d'une forme de Pfaff complètement intégrable ou d'une fonction numérique, C. R. Acad. Sci. Paris 222 (1946), 847-849.
17. Sharko V.V., Functions on manifolds. Algebraic and topological aspects, Translations of Mathematical Monographs, Vol. 131, Amer. Math. Soc., Providence, RI, 1993.
18. Stoimenow A., On the number of chord diagrams, Discrete Math. 218 (2000), 209-233.
19. Vyatchaninova O.M., Atoms and molecules of functions with isolated critical points on the boundary of 3-dimensional handlebody, Proc. Internat. Geom. Center 5 (2012), no. 3-4, 15-23.