Hyperbolic rational homology spheres not admitting fillable contact structures

TL;DR

This paper constructs an infinite family of hyperbolic rational homology 3-spheres that do not support fillable contact structures, highlighting limitations in the existence of certain contact structures on these manifolds.

Contribution

It introduces an infinite family of hyperbolic rational homology spheres lacking fillable contact structures, expanding understanding of contact topology in hyperbolic 3-manifolds.

Findings

Most of these manifolds admit tight contact structures.

They do not admit any fillable contact structures.

The result provides new examples in contact topology.

Abstract

In this short note, we exhibit an infinite family of hyperbolic rational homology $3$--spheres which do not admit any fillable contact structures. We also note that most of these manifolds do admit tight contact structures.