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 Σ2, 1.

Stable bi-maps from closed orientable surfaces to R x R

“…The graph given by M/ ∼ is said to be the Reeb graph (or R-graph) associated to f 1 : M → R (cf. [1], [2]). Definition 2.5.…”

“…Also, we will consider two types of stable maps: f 1 : M → R and f 2 : M → R 2 . Stable maps have been investigated by several authors and have many interesting applications (see [2,3,5,6,7,8,10,11,13], for instance).…”

Stable Bi-Maps on Surfaces and Their Graphs

“…In Singularity theory, many works resort to graph theory to describe local/global invariants and combinatorial models to investigate the recognition and classification problem involving maps or map germs. The authors themselves of this paper have some works in this direction (see, for instance, [3,8]). It is important to highlight that the Singularity theory has many practical applications in Physics, Engineering, Thermodynamic, among others.…”

“…Stable bi-maps means a pair of stable maps, i.e., f 1 : M → R and f 2 : M → R 2 are stable maps. Stable maps have been investigated by several authors and have many interesting applications (see, for instance, [3,4,8,11,13,18]).…”

The dual tree of a fold map germ from to

“…Finally, let us remark that the techniques used in this paper of obtaining the topological classification of real analytic map germs by means of its associated link has been used by both authors and the second author with several collaborators in other cases [1,2,[15][16][17]18].…”