Bourgeois contact structures: tightness, fillability and applications.

Starting from a contact manifold and a supporting open book decomposition, an explicit construction by Bourgeois provides a contact structure in the product of the original manifold with the two-torus. In this talk, we will discuss recent results concerning rigidity and fillability properties of these contact manifolds. For instance, it turns out that Bourgeois contact structures are, in dimension 5, always tight, independent on the rigid/flexible classification of the original contact manifold. Moreover, Bourgeois manifolds associated to suitable monodromies provide new examples of weakly but not strongly fillable contact 5-manifolds. We also present the following application in any dimension: the standard contact structure in the unit cotangent bundle of the n-torus, which is a Bourgeois manifold, admits a unique aspherical filling up to diffeomorphism. This is joint work with Jonathan Bowden and Fabio Gironella.