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

Bohdana I. Hladysh ; Aleksandr O. Prishlyak

Symmetry, integrability and geometry: methods and applications, Tome 13 (2017)

Résumé

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.

Détail

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

Bibliographie

[1] Bolsinov A. V., Fomenko A. T., Integrable {H}amiltonian systems. Geometry, topology, classification, Chapman Hall/CRC, Boca Raton, FL, 2004 | DOI | MR | Zbl

[2] Borodzik M., Némethi A., Ranicki A., “Morse theory for manifolds with boundary”, Algebr. Geom. Topol., 16 (2016), 971–1023, arXiv: 1207.3066 | DOI | MR | Zbl

[3] Conner P. E., Floyd E. E., Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, 33, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1964 | DOI | MR | Zbl

[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 | DOI | MR

[5] Iurchuk I. A., “Properties of a pseudo-harmonic function on closed domain”, Proc. Internat. Geom. Center, 7:4 (2014), 50–59 | DOI

[6] Kadubovskyi A. A., “Enumeration of topologically non-equivalent smooth minimal functions on closed surfaces”, Proceedings of Institute of Mathematics, Kyiv, 12:6 (2015), 105–145 | Zbl

[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, 8:1 (2015), 47–62 | DOI

[8] Khruzin A., Enumeration of chord diagrams, arXiv: math.CO/0008209

[9] Kronrod A. S., “On functions of two variables”, Russian Math. Surveys, 5 (1950), 24–134 | MR | Zbl

[10] Lukova-Chuiko N. V., “Minimal function on 3-manifolds with boundary”, Proc. Internat. Geom. Center, 8:3–4 (2015), 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 | MR | Zbl

[13] Milnor J. W., Topology from the differentiable viewpoint, The University Press of Virginia, Charlottesville, Va., 1965 | MR | Zbl

[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, arXiv: math.GT/9912004 | DOI | MR | Zbl

[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 | MR | Zbl

[17] Sharko V. V., Functions on manifolds. Algebraic and topological aspects, Translations of Mathematical Monographs, 131, Amer. Math. Soc., Providence, RI, 1993 | MR | Zbl

[18] Stoimenow A., “On the number of chord diagrams”, Discrete Math., 218 (2000), 209–233 | DOI | MR | Zbl

[19] Vyatchaninova O. M., “Atoms and molecules of functions with isolated critical points on the boundary of 3-dimensional handlebody”, Proc. Internat. Geom. Center, 5:3–4 (2012), 15–23