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 Ω(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 Ω(M ) and have three critical points has been developed.
We describe all possible (469) topological structures of сodimension one gradient vector fields on the 2-sphere with at most ten singular points. To describe structures, we use a graph whose edges are onedimensional stable manifolds. The saddle-node singularity is specified by selecting a pair of vertices-edge or edge-face, and the saddle connection is specified by a T-vertex.
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