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

## TL;DR

This paper classifies functions with isolated boundary critical points on 2D manifolds, introduces chord diagrams for their neighborhoods, and provides criteria for their global topological equivalence.

## Contribution

It offers a topological classification of boundary-critical functions, constructs chord diagrams, and develops criteria for their global equivalence.

## Key findings

- Classified functions near their critical points.
- Constructed chord diagrams from critical levels.
- Established criteria for global topological equivalence.

## 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.