The Reeb Graph of a Map Germ from ℝ3 to ℝ2 with Isolated Zeros
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Cambridge University Press (CUP)
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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 S 2 to S 1. We define a generalized version of the Reeb graph for stable maps γ : S 2 → S 1, 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 h 0. Finally, we apply this to give a complete topological description of all map germs with Boardman symbol Σ 2, 1.
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Proceedings of the Edinburgh Mathematical Society, v. 60, n. 2, p. 319-348, 2017.
