The Reeb Graph of a Map Germ from ℝ³ to ℝ² with Isolated Zeros

Abstract

We consider finitely determined map germs f : (ℝ³, 0) → (ℝ², 0) with f^–1(0) = {0} and we look at the classification of this kind of germ with respect to topological equivalence. By Fukuda's cone structure theorem, the topological type of f can be determined by the topological type of its associated link, which is a stable map from S² to S¹. We define a generalized version of the Reeb graph for stable maps γ : S²→ S¹, which turns out to be a complete topological invariant. If f has corank 1, then f can be seen as a stabilization of a function h₀: (ℝ², 0) → (ℝ, 0), and we show that the Reeb graph is the sum of the partial trees of the positive and negative stabilizations of h₀. Finally, we apply this to give a complete topological description of all map germs with Boardman symbol Σ^{2, 1}.