Bourgeois contact structures: tightness, fillability and applications

Given a contact structure on a manifold $V$ together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on $V \times \mathbb{T}^2$. We prove that all such structures are universally tight in dimension $5$, independent on whether the original contact manifold is tight or overtwisted. In the planar case, i.e. when the pages of the open book have genus zero, we characterize the $5$-dimensional Bourgeois contact structures admitting strong symplectic fillings: they are precisely those with trivial monodromy. As a consequence, strong fillability is equivalent to Stein fillability in the $5$-dimensional planar case. We also obtain a broad class of new examples of weakly but not strongly fillable contact $5$-manifolds. Lastly, the techniques developed in the $5$-dimensional case also allow to obtain the following result in higher-dimensions: the unit cotangent bundle $S^*\mathbb{T}^n$ of $\mathbb{T}^n$, with its standard contact structure (which is a Bourgeois contact structure), has a unique symplectically aspherical strong filling up to diffeomorphism.