Lipschitz Geometry Does not Determine Embedded Topological Type

Abstract

We investigate the relationships between the Lipschitz outer geometry and the embedded topological type of a hypersurface germ in ($$ {\mathbb{C}}^n $$, 0). It is well known that the Lipschitz outer geometry of a complex plane curve germ determines and is determined by its embedded topological type.We prove that this does not remain true in higher dimensions. Namely, we give two normal hypersurface germs ($$ {{X}}_1 $$, 0) and ($$ {{X}}_2 $$, 0) in ($$ {\mathbb{C}}^3 $$, 0) having the same outer Lipschitz geometry and different embedded topological types. Our pair consist of two superisolated singularities whose tangent cones form an Alexander-Zariski pair having only cusp-singularities. Our result is based on a description of the Lipschitz outer geometry of a superisolated singularity. We also prove that the Lipschitz inner geometry of a superisolated singularity is completely determined by its (non-embedded) topological type, or equivalently by the combinatorial type of its tangent cone.

Dedicated to José Seade for a great occasion. Happy birthday, Pepe!

Mathematics Subject Classification (2000). 14B05, 32S25, 32S05, 57M99.