A Criterion for the Topological Equivalence of Two Variable Complex Analytic Function Germs

We show that two analytic function germs (C, 0) → (C, 0) are topologically right equivalent if and only if there is a one-to-one correspondence between the irreducible components of their zero sets, that preserves the multiplicites of these components, their Puiseux pairs, and the intersection numbers of any pairs of distinct components. By Zariski [7] and Burau [1], the topological type of an embedded plane curve singularity (X, 0) ⊂ (C, 0) is determined by the Puiseux pairs of each irreducible component (branch) of this curve and the intersection numbers of any pairs of distinct branches. In this note we show the following Theorem 0.1. Let f , g : (C, 0) → (C, 0) be (not necessarily reduced) analytic function germs. Then f and g are topologically right equivalent if and only if there is a one-to-one correspondence between the irreducible components of their zero sets, that preserves the multiplicites of these components, their Puiseux pairs, and the intersection numbers of any pairs of distinct components. Sketch of the proof. The ”only if” follows from the above cited result of Zariski and Burau. To show ”if” we proceed as follows. We may connect the zero sets (f(0), 0) ⊂ (C, 0) and (g(0), 0) ⊂ (C, 0) by an equisingular (topologically trivial) deformation of plane curve germs (F(0), 0)× P ⊂ (C, 0)× P, where P is a parameter space and F : (C, 0) × P → (C, 0) is analytic. Then, by [9] section 8, the pair (F(0) \ 0 × P, 0 × P ) satisfies Whitney conditions. Consequently, by [2] or [4], the strata (C, 0) × P \ F(0), F(0) \ 0 × P , and 0 × P stratify F as a function with the strong Thom condition wF . This shows, by Thom-Mather theorem, that F is topologically trivial along P . We now give some details. We proceed slightly differently and connect f by an equisingular deformation to a normal family that depends only on the embedded topological type of (f(0), 0) ⊂ (C, 0) and the multiplicities of its branches. 0.1. Deformation of f to a normal family. Fix f : (C, 0) → (C, 0). Choose a system of coordinates so that y = 0 is transverse to the tangent cone to f = 0 at the origin. Let 1991 Mathematics Subject Classification. Primary: 32S15. Secondary: 14B05.