Links of rational singularities, L-spaces and LO fundamental groups

TL;DR

This paper establishes a precise equivalence between rational surface singularities and L-space links, connecting topological, algebraic, and foliation properties of singularity links.

Contribution

It proves that a complex normal surface singularity's link is an L-space if and only if the singularity is rational, linking singularity theory with 3-manifold topology.

Findings

Link of a rational singularity is an L-space.

Non-rational singularity links have left-orderable fundamental groups.

Equivalence between L-space property, left-orderability, and taut foliations.

Abstract

We prove that the link of a complex normal surface singularity is an L--space if and only if the singularity is rational. This via a recent result of Hanselman, J. Rasmussen, S. D. Rasmussen and Watson (proving the conjecture of Boyer, Gordon and Watson), shows that a singularity link is not rational if and only if its fundamental group is left-orderable if and only if it admits a coorientable taut foliation.