Bourgeois contact structures: Tightness, fillability and applications

Abstract

AbstractGiven a contact structure on a manifold V together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on $$V \times {\mathbb {T}}^2$$ V × T 2 . We prove that all such structures are universally tight in dimension 5, independent of whether the original contact manifold is itself tight or overtwisted. In arbitrary dimensions, we provide obstructions to the existence of strong symplectic fillings of Bourgeois manifolds. This gives a broad class of new examples of weakly but not strongly fillable contact 5-manifolds, as well as the first examples of weakly but not strongly fillable contact structures in all odd dimensions. These obstructions are particular instances of more general obstructions for $${\mathbb {S}}^1$$ S 1 -invariant contact manifolds. We also obtain a classification result in arbitrary dimensions, namely that the unit cotangent bundle of the n-torus has a unique symplectically aspherical strong filling up to diffeomorphism.