On Stein fillings of contact torus bundles

golla, marco; lisca, paolo;

We consider a large family F of torus bundles over the circle, and we use recent work of Li–Mak to construct, on each Y ∈ F, a Stein fillable contact structure ξY . We prove that (i) each Stein filling of (Y, ξY ) has vanishing first Chern class and first Betti number, (ii) if Y ∈ F is elliptic, then all Stein fillings of (Y, ξY ) are pairwise diffeomorphic and (iii) if Y ∈ F is parabolic or hyperbolic, then all Stein fillings of (Y, ξY ) share the same Betti numbers and fall into finitely many diffeomorphism classes. Moreover, for infinitely many hyperbolic torus bundles Y ∈ F we exhibit non-homotopy equivalent Stein fillings of (Y, ξY ).