Non-fillability of overtwisted contact manifolds via polyfolds

TL;DR

This paper proves that overtwisted contact manifolds cannot be weakly symplectically filled, confirming tightness, and verifies the strong Weinstein conjecture for certain boundary conditions using polyfold theory.

Contribution

It introduces polyfold techniques to establish non-fillability of overtwisted contact manifolds and verifies the strong Weinstein conjecture in new boundary scenarios.

Findings

Overtwisted contact manifolds are not weakly fillable.

Strong Weinstein conjecture holds for specific boundary conditions.

Polyfolds are effective in contact and symplectic topology proofs.

Abstract

We prove that any weakly symplectically fillable contact manifold is tight. Furthermore we verify the strong Weinstein conjecture for contact manifolds that appear as the concave boundary of a directed symplectic cobordism whose positive boundary satisfies the weak-filling condition and is overtwisted. Similar results are obtained in the presence of bordered Legendrian open books whose binding-complement has vanishing second Stiefel-Whitney class. The results are obtained via polyfolds.