|
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
-
Bolsinov A.V., Fomenko A.T., Integrable Hamiltonian systems. Geometry, topology, classification, Chapman & Hall/CRC, Boca Raton, FL, 2004.
-
Borodzik M., Némethi A., Ranicki A., Morse theory for manifolds with boundary, Algebr. Geom. Topol. 16 (2016), 971-1023, arXiv:1207.3066.
-
Conner P.E., Floyd E.E., Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 33, Springer-Verlag, Berlin - Göttingen - Heidelberg, 1964.
-
Hladysh B.I., Prishlyak A.O., Functions with nondegenerate critical points on the boundary of a surface, Ukrain. Math. J. 68 (2016), 29-40.
-
Iurchuk I.A., Properties of a pseudo-harmonic function on closed domain, Proc. Internat. Geom. Center 7 (2014), no. 4, 50-59.
-
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.
-
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.
-
Khruzin A., Enumeration of chord diagrams, math.CO/0008209.
-
Kronrod A.S., On functions of two variables, Russian Math. Surveys 5 (1950), 24-134.
-
Lukova-Chuiko N.V., Minimal function on 3-manifolds with boundary, Proc. Internat. Geom. Center 8 (2015), no. 3-4, 46-52.
-
Lukova-Chuiko N.V., Prishlyak A.O., Prishlyak K.O., M-functions on nonoriented surfaces, J. Numer. Appl. Math. (2012), no. 2, 176-185.
-
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.
-
Milnor J.W., Topology from the differentiable viewpoint, The University Press of Virginia, Charlottesville, Va., 1965.
-
Prishlyak A.O., Topological properties of functions on two and three dimensional manifolds, Palmarium. Academic Pablishing.
-
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.
-
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.
-
Sharko V.V., Functions on manifolds. Algebraic and topological aspects, Translations of Mathematical Monographs, Vol. 131, Amer. Math. Soc., Providence, RI, 1993.
-
Stoimenow A., On the number of chord diagrams, Discrete Math. 218 (2000), 209-233.
-
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.
|
|