On the topology of fillings of contact manifolds and applications

Abstract

The aim of this paper is to address the following question: given a contact manifold $(\Sigma ,\xi )$, what can be said about the symplectically aspherical 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 $H_{\ast }(\Sigma )$ to $H_{\ast }(W)$ induced by inclusion is surjective. We apply this method in the case of contact manifolds admitting 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 subcritical Stein filling, then all its SAWC 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. We refer to the table in Section 7 for a summary of our results.