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Discriminant of a germ \(\Phi: (\mathbb{C}^2,0)\to (\mathbb{C}^2,0)\) and Seifert fibred manifolds. (English) Zbl 0941.58027
Let \((g,f) : ({\mathbb C}^2,0) \rightarrow ({\mathbb C}^2,0)\) be a map defined by two germs of analytic functions \(f\) and \(g\) with no common branches.
The author defines the set of Jacobian quotients of the map \((g,f)\) which contains all the exponents of leading terms of Puiseux expansions associated with branches of the discriminant curve of the map. Based on the Waldhausen theory of differentiable \(3\)-manifolds, she proves that these rational numbers are topological invariants of the map and can be computed in terms of linking numbers of algebraic knots associated with Seifert fibres of the minimal Waldhausen decomposition of \(S_\epsilon^3\) for the link \(K_{fg},\) where \(S_\epsilon^3\) is the sphere centered at the origin of \({\mathbb C}^2\) and \(K_{fg} = (fg)^{-1}(0) \cap S_\epsilon^3.\)
In the case when \(g\) is a linear form transverse to \(f\), the sets of Jacobian quotients of \((g,f)\) and of the polar quotients of \(f\) coincide. Thus, the author confirms the earlier results from Le Dung Trang, F. Michel and C. Weber [Compos. Math. 72, No. 1, 87-113 (1989; Zbl 0705.32021), Ann. Sci. Éc. Norm. Supér., IV. Sér. 24, No. 2, 141-169 (1991; Zbl 0748.32018)], and some others.
It should be also remarked that a method of computing the Jacobian quotients in terms of contact exponents in the minimal resolution of \((fg)^{-1}(0)\) is described in [H. Maugendre, Ann. Fac. Sci. Toulouse, VI. Sér., Math. 7, No. 3, 497-525 (1998; Zbl 0936.32012)].

MSC:
58K65 Topological invariants on manifolds
57M25 Knots and links in the \(3\)-sphere (MSC2010)
32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
14R15 Jacobian problem
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