Non-fillable invariant contact structures on principal circle bundles and left-handed twists

TL;DR

This paper introduces symplectic fractional twists to construct invariant contact structures on principal circle bundles, revealing non-fillable, algebraically overtwisted manifolds that satisfy the Weinstein conjecture.

Contribution

It generalizes Dehn twists to fractional twists and explores their role in creating invariant, non-fillable contact structures on circle bundles.

Findings

Left-handed fractional twists produce non-fillable, algebraically overtwisted contact manifolds.

The constructed manifolds satisfy the Weinstein conjecture.

These manifolds are not related to negative stabilizations or contain bLobs.

Abstract

We define symplectic fractional twists, which generalize Dehn twists, and use these in open books to investigate contact structures. The resulting contact structures are invariant under a circle action, and share several similarities with the invariant contact structures that were studied by Lutz and Giroux. We show that left-handed fractional twists often give rise to non-fillable contact manifolds. These manifolds are in fact "algebraically overtwisted", yet they do not seem to contain bLobs, nor are they directly related to negative stabilizations. We also show that the Weinstein conjecture holds for the non-fillable contact manifolds we construct, and we investigate the symplectic isotopy problem for fractional twists.