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

On the topology of fillings of contact manifolds and applications