On Stein fillings of contact torus bundles

TL;DR

This paper studies Stein fillings of contact structures on a broad family of torus bundles over the circle, revealing their topological invariants and classifying their diffeomorphism types, including examples with non-homotopy equivalent fillings.

Contribution

It constructs Stein fillable contact structures on a large family of torus bundles and classifies their Stein fillings based on the geometry of the bundle, including new examples with distinct homotopy types.

Findings

Stein fillings have vanishing first Chern class and Betti number.

All fillings of elliptic bundles are diffeomorphic.

Hyperbolic bundles have finitely many diffeomorphism classes of fillings.

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).