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

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.