Topological and bilipschitz types of complex surface singularities and their links

TL;DR

This paper establishes that the inner bilipschitz type of complex surface singularities determines their topological type and minimal plumbing graph, linking geometric and topological classifications of singularities and their links.

Contribution

It proves that inner bilipschitz homeomorphisms imply the same topological type and minimal plumbing graph for complex surface germs, and shows the link's topology determines the singularity's topology.

Findings

Inner bilipschitz homeomorphisms preserve topological type.

The link's topology determines the singularity's topological type.

Results apply to Hirzebruch-Jung and cusp singularities.

Abstract

In this paper, we prove that two normal complex surface germs that are inner bilipschitz--but not necessarily orientation-preserving--homeomorphic, have in fact the same oriented topological type and the same minimal plumbing graph. Along the way, we show that the oriented homeomorphism type of an isolated complex surface singularity germ determines the oriented homeomorphism type of its link, providing a converse to the classical Conical Structure Theorem. These results require to study the topology first, and the inner lipschitz geometry later, of Hirzebruch-Jung and cusp singularities, the normal surface singularities whose links are lens spaces and fiber bundles over the circle.