On Stein fillings of contact torus bundles (Q2788649)

A symplectic/Stein filling of a contact manifold \((Y,\xi)\) is a compact symplectic/Stein manifold such that the boundary is contactomorphic to \((Y,\xi)\) and the contact structure on the boundary is compatible with the symplectic/Stein structure of the manifold. A thorough understanding of the symplectic/Stein fillings of a given contact manifold is currently only accessible in dimension three.NEWLINENEWLINEIn the paper under review, the authors study this question for an important family of \(3\)-manifolds: torus bundles over circle. Even then, a complete answer is out of reach and they focus on the ones that can be realized as the concave boundary of a plumbing of symplectic surfaces and obtain very satisfactory results: they establish the uniqueness/finiteness of diffeomorphism types of symplectic/Stein fillings for these contact manifolds. They also construct infinitely many contact hyperbolic torus bundles with non-homotopy equivalent Stein fillings.