Topological and bilipschitz types of complex surface singularities and their links

TL;DR

The paper addresses how the inner bilipschitz geometry of normal complex surface germs determines their topological invariants and whether topology alone suffices to recover resolution data. It proves that inner bilipschitz homeomorphism implies orientation-preserving homeomorphism of the germs and equality of their minimal plumbing graphs, while the oriented germ type fixes the oriented link type, providing a converse to the Conical Structure Theorem (link as the boundary of the cone over the germ). The authors combine 3-manifold topology (Neumann–Waldhausen–Seifert–Threlfall, Turaev) with Birbrair–Neumann–Pichon inner bilipschitz decompositions (thick/thin zones, inner rates, Milnor-fibration foliations) to show that the inner bilipschitz data determine the dual resolution graph for Hirzebruch–Jung lens spaces $L(p,q)$ and cusp torus bundles over $S^1$, hence the minimal plumbing graph. The results delineate the limits of generalization to non-normal germs or higher dimensions and highlight tautness in the two exceptional link types as a key structural feature.

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.

Figures (4)

• Figure 1: An illustration of the construction used throughout the proof, with the subset $W$ in blue in the cone $C(N)$ on the right-hand side.
• Figure 2: Minimal plumbing graph of the lens space $L(p,q)$.
• Figure 3: Minimal plumbing graphs of the Hirzebruch--Jung singularities $(X,0)$, on the left, and $(Y,0)$, on the right.
• Figure 4: The minimal plumbing graphs of two cusp singularities that are orientation reversing-homeomorphic, while having the same number of thick and thin zones and the same maximal inner rates. Note that these singularities are not hypersurfaces in ${\mathbb C}^3$.

Theorems & Definitions (16)

• Theorem A
• Theorem B
• Theorem C
• Proposition A
• proof
• Lemma B
• proof : Proof of Lemma \ref{['lem:cobordism']}
• proof : End of proof of the implication $\ref{['or_homeo']} \Rightarrow \ref{['or_link']}$ of Theorem \ref{['thm:oriented_topological type for complex surfaces']}
• Remark C
• Remark D