The Reeb Graph of a Map Germ from $$\mathbb {R}^3$$ R 3 to $$\mathbb {R}^2$$ R

Abstract

We consider the topological classification of finitely determined map germs [ f ] : (R 3 , 0) → (R 2 , 0) with f −1 (0) = {0}. The case f −1 (0) = {0} was treated in another recent paper by the authors. The main tool used to describe the topological type is the link of [ f ], which is obtained by taking the intersection of its image with a small sphere S 1 δ centered at the origin. The link is a stable map γ f : N → S 1 , where N is diffeomorphic to a sphere S 2 minus 2L disks. We define a complete topological invariant called the generalized Reeb graph. Finally, we apply our results to give a topological description of some map germs with Boardman symbol 2,1 .

The dual tree of a fold map germ from to

“…If rank(α) = 1 we have that j 2 f (0) ∼ (x, y, a 1 z 2 + a 2 xz + a 3 yz, 0), obtaining, following an analogous procedure to the previous case the A 2 -orbits (x, y, z 2 , 0) and (x, y, xz, 0). Finally, if rank(α) = 0 we get that j 2 f (0) ∼ (x, y, 0, 0).…”

“…By induction, χ(Γ ) = 1 and hence χ(Γ) = (v − 1) + 1 − ((v − 2) + 1) = 1, since we are adding a final vertex. We use again induction on the number of vertices to prove (2). For dual graphs with 1 and 2 vertices it is clear from example 4.3.…”

“…If we suppose it is true for v − 1 vertices and we consider a graph Γ with v vertices, by the fact of having a tree [condition (1)] we know that Γ has a final vertex v i connected to another vertex v j . We eliminate v i and obtain a graph Γ with v − 1 vertices that also satisfies conditions ( 1) and (2). By induction hypothesis, Γ is the dual graph of a mosaic (D 2 , K).…”

“…Assume it is true for v − 1 vertices and suppose Γ has v vertices. By condition (2), Γ has at least a final vertex v i of type • connected to another vertex v j . We can eliminate v i and repeat the same procedure used in the previous case.…”

“…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].…”