Lipschitz geometry does not determine embedded topological type

TL;DR

This paper demonstrates that in higher dimensions, Lipschitz outer geometry does not uniquely determine the embedded topological type of hypersurface germs, contrasting with the case of plane curves.

Contribution

It provides the first example of hypersurface germs in $(\,\mathbb C^3)$ with identical Lipschitz outer geometry but different embedded topological types, using superisolated singularities.

Findings

Lipschitz outer geometry does not determine topological type in higher dimensions.

Lipschitz inner geometry of superisolated singularities is topologically invariant.

Constructs examples using Alexander-Zariski pairs with cusp singularities.

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.