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

Bohdana Hladysh, Aleksandr Prishlyak

Published 2017-06-15Version 1

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 the $\Omega(M)$ and have three critical points has been developed.