On the topology of fillings of contact manifolds and applications

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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_{*}(\Sigma) to H_{*}(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.