Tight contact structures without symplectic fillings are everywhere

TL;DR

This paper demonstrates the widespread existence of tight contact structures that are not symplectically fillable across various dimensions, revealing new insights into contact topology and fillability conditions.

Contribution

It establishes the existence of tight, non-fillable contact structures in all dimensions greater than or equal to three, and constructs examples of Liouville but not Weinstein fillable structures.

Findings

Existence of tight, non-fillable contact structures in all dimensions ≥3.

Construction of Liouville but not Weinstein fillable structures.

Extension of results to cases with vanishing first Chern class.

Abstract

We show that for all $n \ge 3$, any $(2n+1)$-dimensional manifold that admits a tight contact structure, also admits a tight but non-fillable contact structure, in the same almost contact class. For $n=2$, we obtain the same result, provided that the first Chern class vanishes. We further construct Liouville but not Weinstein fillable contact structures on any Weinstein fillable contact manifold of dimension at least $7$ with torsion first Chern class.