
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 2Dimensional Manifold
Bohdana I. Hladysh and Aleksandr O. Prishlyak
Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 4e 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 abovementioned 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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