Kronrod–Reeb Graphs of Functions on Noncompact Two-Dimensional Surfaces. II

Abstract

We consider continuous functions on two-dimensional surfaces satisfying the following conditions: they have a discrete set of local extrema and if a point is not a local extremum, then there exist its neighborhood and a number $n ∈ ℕ$ such that the function restricted to this neighborhood is topologically conjugate to Re $z^n$ in a certain neighborhood of zero. Given $f : M^2 → ℝ$, let $Γ_{K−R} (f)$ be a quotient space of $M^2$ with respect to its partition formed by the components of level sets of the function $f$. It is known that the space $Γ_{K−R} (f)$ is a topological graph if $M^2$ is compact. In the first part of the paper, we introduced the notion of graph with stalks that generalizes the notion of topological graph. For noncompact $M^2$ , we present three conditions sufficient for $Γ_{K−R} (f)$ to be a graph with stalks. In the second part, we prove that these conditions are also necessary in the case $M^2 = ℝ^2$. In the general case, one of our conditions is not necessary. We provide an appropriate example.