# Topological structure of functions with isolated critical points on a 3-manifold

## TL;DR

This paper establishes a topological classification of smooth functions with isolated critical points on 3-manifolds by associating each with a unique tree structure, providing a complete invariant for such functions.

## Contribution

It introduces a novel tree-based invariant that completely classifies functions with isolated critical points on closed 3-manifolds.

## Key findings

- Functions are topologically equivalent iff their associated trees are isomorphic
- A complete topological invariant for these functions is constructed
- The invariant simplifies classification of critical points on 3-manifolds

## Abstract

To each isolated critical point of a smooth function on a 3-manifold we put in correspondence a tree (graph without cycles). We will prove that functions are topologically equivalent in the neighborhoods of critical points if and only if the corresponding trees are isomorphic. A complete topological invariant of functions with isolated critical points, on a closed 3-manifold, will be constructed.