On fillings of contact links of quotient singularities

TL;DR

This paper investigates the fillability properties of contact links of quotient singularities using Floer theory, establishing non-existence results and uniqueness for certain classes of singularities.

Contribution

It provides new non-fillability results for contact links of isolated terminal quotient singularities and classifies their exact orbifold fillings.

Findings

Many contact links of quotient singularities are not exactly fillable.

Confirmed a conjecture of Eliashberg regarding non-fillability.

Established uniqueness of orbifold diffeomorphism types of certain fillings.

Abstract

We study several aspects of fillings for links of general quotient singularities using Floer theory, including co-fillings, Weinstein fillings, strong fillings, exact fillings and exact orbifold fillings, focusing on non-existence of exact fillings of contact links of isolated terminal quotient singularities. We provide an extensive list of isolated terminal quotient singularities whose contact links are not exactly fillable, including $\mathbb{C}^n/(\mathbb{Z}/2)$ for $n\ge 3$, which settles a conjecture of Eliashberg, quotient singularities from general cyclic group actions and finite subgroups of $SU(2)$, and all terminal quotient singularities in complex dimension $3$. We also obtain uniqueness of the orbifold diffeomorphism type of exact orbifold fillings of contact links of some isolated terminal quotient singularities.