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

Abstract

We consider the topological classification of finitely determined map germs [f] : (R3, 0) → (R2, 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 Sδ1 centered at the origin. The link is a stable map γf: N→ S1, where N is diffeomorphic to a sphere S2 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 Σ 2 , 1.