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 .

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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.

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