JournalscmhVol. 87, No. 1pp. 41–69

On the topology of fillings of contact manifolds and applications

  • Alexandru Oancea

    Université Pierre et Marie Paris, France
  • Claude Viterbo

    Ecole Normale Superieure, Paris, France
On the topology of fillings of contact manifolds and applications cover
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Abstract

The aim of this paper is to address the following question: given a contact manifold (Σ,ξ)(\Sigma, \xi), what can be said about the symplectically aspherical manifolds (W,ω)(W, \omega) bounded by (Σ,ξ)(\Sigma, \xi)? We first extend a theorem of Eliashberg, Floer and McDuff to prove that, under suitable assumptions, the map from H(Σ)H_{*}(\Sigma) to H(W)H_{*}(W) induced by inclusion is surjective. We apply this method in the case of contact manifolds admitting a contact embedding in R2n\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~\ref{table} for a summary of our results.

Cite this article

Alexandru Oancea, Claude Viterbo, On the topology of fillings of contact manifolds and applications. Comment. Math. Helv. 87 (2012), no. 1, pp. 41–69

DOI 10.4171/CMH/248