Contact $(+1)$-surgeries and algebraic overtwistedness

TL;DR

This paper demonstrates that (+1)-surgeries along Legendrian spheres in certain fillable contact manifolds produce algebraically overtwisted manifolds, providing a new proof for the vanishing of contact homology in overtwisted cases.

Contribution

It introduces a method to generate algebraically overtwisted contact manifolds via (+1)-surgeries, extending previous results and offering a new proof of contact homology vanishing.

Findings

(+1)-surgery yields algebraically overtwisted manifolds in flexible fillable contact manifolds.

The construction applies to more general contact manifolds, producing algebraically overtwisted examples.

Provides a symplectic field theory analog of known invariants vanishing results in dimension.

Abstract

We show that a contact $(+1)$-surgery along a Legendrian sphere in a flexibly fillable contact manifold ($c_1=0$ if not subcritical) yields a contact manifold that is algebraically overtwisted if the Legendrian's homology class is not annihilated in the filling. Our construction can also be implemented in more general contact manifolds yielding algebraically overtwisted manifolds through $(+1)$-surgeries. This gives new proof of the vanishing of contact homology for overtwisted contact manifolds. Our result can be viewed as the symplectic field theory analog in any dimension of the vanishing of contact Ozsv\'ath-Szab\'o invariant for $(+1)$-surgeries on two-component Legendrian links proved by Ding, Li, and Wu.