Commentarii Mathematici Helvetici

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Volume 87, Issue 1, 2012, pp. 41–69
DOI: 10.4171/CMH/248

Published online: 2012-01-16

On the topology of fillings of contact manifolds and applications

Alexandru Oancea[1] and Claude Viterbo[2]

(1) Université Pierre et Marie Paris, France
(2) Ecole Normale Superieure, Paris, France

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~\ref{table} for a summary of our results.

Keywords: Topology of symplectic fillings of contact manifolds, obstructions to contact embeddings

Oancea Alexandru, Viterbo Claude: On the topology of fillings of contact manifolds and applications. Comment. Math. Helv. 87 (2012), 41-69. doi: 10.4171/CMH/248