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.

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