Twenty years ago Frederic Bourgeois introduced a construction of contact structures on the product of any contact manifold M with a 2-torus given a choice of compatible open book, whose existence was proven by Giroux-Mohsen. In particular, this yielded contact structures on all odd-dimensional tori answering a question of Lutz from the 70’s. A systematic study of these contact manifolds was initiated by Lisi-Marinkovic-Niederkrüger and Gironella, the former asking several questions, which we address in this talk.
In particular, we show that if the initial contact manifold is 3-dimensional the resulting contact structure is tight, independent of the initial contact structure and choice of open book. Furthermore, we show that given ANY contact manifold one can always stabilise the open book so that the resulting contact structure is not strongly symplectically fillable. This then yields (many) examples of weakly but not strongly fillable contact structures in all dimensions. (joint work with F. Gironella and A. Moreno)