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(σ,ε )what can be said about the symplectically aspherical manifolds .W; bounded by σ ε ?We first extend a theorem of Eliashberg, Floer and McDuff to prove that, under suitable assumptions, the map from Prof.*σ to Prof.*.W / induced by inclusion is surjective. We apply this method in the case of contact manifolds admitting a contact embedding in 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 acontact 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. © Swiss Mathematical Society.