On the topology of fillings of contact manifolds and applications

TL;DR

This paper investigates the topological properties of symplectic fillings of contact manifolds, establishing conditions for homology determination and applying these results to obstructions and exotic contact structures.

Contribution

It extends a key theorem to show surjectivity of the homology map and demonstrates homology uniqueness for certain contact manifold fillings.

Findings

Homology of fillings is uniquely determined in many cases.

Surjectivity of the inclusion-induced map from the contact boundary to the filling.

Homology invariance among weakly and subcritical fillings with Stein structures.

Abstract

The aim of this paper is to address the following question: given a contact manifold $(\Sigma, \xi)$, what can be said about the aspherical symplectic manifolds $(W, \omega)$ bounded by $(\Sigma, \xi)$ ? We first extend a theorem of Eliashberg, Floer and McDuff to prove that under suitable assumptions the map from $H_{*}(\Sigma)$ to $H_{*}(W)$ induced by inclusion is surjective. We then apply this method in the case of contact manifolds having a contact embedding in $ {\mathbb R}^{2n}$ or in a subcritical Stein manifold. We prove in many cases that the homology of the fillings is uniquely determined. Finally we use more recent methods of symplectic topology to prove that, if a contact hypersurface has a Stein subcritical filling, then all its weakly subcritical fillings have the same homology. A number of applications are given, from obstructions to the existence of Lagrangian or contact embeddings, to the exotic nature of some contact structures.