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

Abstract

We consider continuous functions on two-dimensional surfaces satisfying the following conditions: they have a discrete set of local extrema; if a point is not a local extremum, then there exist its neighborhood and a number $n ∈ ℕ$ such that a 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 the level sets of $f$. It is known that, for compact $M^2$, the space $Γ_{K−R} (f)$ is a topological graph. We introduce the notion of graph with stalks, which generalizes the notion of topological graph. For noncompact $M^2$, we establish three conditions sufficient for $Γ_{K−R} (f)$ to be a graph with stalks.