Non-fillability of overtwisted contact manifolds via polyfolds
Wolfgang Schmaltz, Stefan Suhr, Kai Zehmisch
TL;DR
The paper proves that any closed contact manifold admitting a weak symplectic filling is tight, and establishes the strong Weinstein conjecture for concave boundaries of directed symplectic cobordisms when the positive end is weakly fillable and falls into specified classes (e.g., overtwisted, containing a small bordered Legendrian open book with vanishing w2, or arising from a Giroux domain with a disconnected boundary). The authors develop a comprehensive polyfold framework, constructing Deligne–Mumford type moduli spaces of boundary un-noded discs, including skyscraper deformations and desingularisations, and then extend to polyfold perturbations to achieve transversality in the non-semi-positive setting. Central technical innovations include tamed pseudo-convexity via magnetic collars, a holomorphic blocking mechanism at the boundary, and a detailed orientation scheme for CR-operators in the orbifold/polyfold context. The results broaden non-fillability criteria and verify Weinstein-type conclusions under new hypotheses, highlighting the power of polyfold methods in high-dimensional contact and symplectic topology with potential links to Pardon's contact homology framework.
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.