On Stein fillings of contact torus bundles

Abstract

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 in F, a Stein fillable contact structure C. We prove that (i) each Stein filling of (Y,C) has vanishing first Chern class and first Betti number, (ii) if Y in F is elliptic then all Stein fillings of (Y,C) are pairwise diffeomorphic and (iii) if Y in F is parabolic or hyperbolic then all Stein fillings of (Y,C) share the same Betti numbers and fall into finitely many diffeomorphism classes. Moreover, for infinitely many hyperbolic torus bundles Y in F we exhibit non-homotopy equivalent Stein fillings of (Y,C).

18 pages, 10 figures. This preprint version differs from the final version which is to appear in the Bulletin of the London Mathematical Society

References in corpus (2)

• Four-dimensional symplectic cobordisms containing three-handles
• Symplectic Divisorial Capping in Dimension 4

Cited by in corpus (4)

• Calabi-Yau Caps, Uniruled Caps and Symplectic Fillings
• On symplectic fillings of virtually overtwisted torus bundles
• Local geometry of symplectic divisors with applications to contact torus bundles
• Circular spherical divisors and their contact topology