Singularity links with exotic Stein fillings

TL;DR

This paper extends the existence of infinitely many exotic Stein fillings to a broader class of contact Seifert fibered 3-manifolds, linking them to complex surface singularities and their canonical contact structures.

Contribution

It generalizes previous results by including singularity links with multiple singular fibers and constructs infinite families of exotic Stein fillings with specific fundamental groups.

Findings

Existence of infinitely many exotic Stein fillings for a larger class of contact Seifert fibered 3-manifolds.

Identification of these 3-manifolds as links of complex surface singularities.

Construction of infinite families of exotic Stein fillings with fixed fundamental groups.

Abstract

In a recent paper of Akhmedov, Etnyre, Mark and Smith, it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifert fibered 3-manifolds with many singular fibers and observe that these 3-manifolds are singularity links. In addition, we prove that the contact structures induced by the Stein fillings are the canonical contact structures on these singularity links. As a consequence, we verify a prediction of Andras Nemethi by providing examples of isolated complex surface singularities whose links with their canonical contact structures admitting infinitely many exotic simply-connected Stein fillings. Moreover, for infinitely many of these contact singularity links and for each positive integer n, we also construct an infinite family of exotic Stein fillings with fixed fundamental group $\mathbb{Z} \oplus \mathbb{Z}_n$.